THE ASTROPHYSICAL JOURNAL SUPPLEMENT SERIES, 182:310 (2009) [e-print arXiv:0704.0044]
Preprint typeset using LATEX style emulateapj v. 08/22/09
ASTROPHYSICAL GYROKINETICS: KINETIC AND FLUID TURBULENT CASCADES IN MAGNETIZED WEAKLY
COLLISIONAL PLASMAS
A. A. SCHEKOCHIHIN,1,2 S. C. COWLEY,2,3 W. DORLAND,4 G. W. HAMMETT,5 G. G. HOWES,6 E. QUATAERT,7 AND T. TATSUNO4
Submitted April 1, 2007; accepted February 21, 2009; published May 6, 2009
ABSTRACT
This paper presents a theoretical framework for understanding plasma turbulence in astrophysical plasmas. It
is motivated by observations of electromagnetic and density fluctuations in the solar wind, interstellar medium
and galaxy clusters, as well as by models of particle heating in accretion disks. All of these plasmas and many
others have turbulent motions at weakly collisional and collisionless scales. The paper focuses on turbulence in
a strong mean magnetic field. The key assumptions are that the turbulent fluctuations are small compared to the
mean field, spatially anisotropic with respect to it and that their frequency is low compared to the ion cyclotron
frequency. The turbulence is assumed to be forced at some system-specific outer scale. The energy injected at
this scale has to be dissipated into heat, which ultimately cannot be accomplished without collisions. A kinetic
cascade develops that brings the energy to collisional scales both in space and velocity. The nature of the
kinetic cascade in various scale ranges depends on the physics of plasma fluctuations that exist there. There are
four special scales that separate physically distinct regimes: the electron and ion gyroscales, the mean free path
and the electron diffusion scale. In each of the scale ranges separated by these scales, the fully kinetic problem
is systematically reduced to a more physically transparent and computationally tractable system of equations,
which are derived in a rigorous way. In the “inertial range” above the ion gyroscale, the kinetic cascade
separates into two parts: a cascade of Alfvénic fluctuations and a passive cascade of density and magnetic-field-
strength fluctuations. The former are governed by the Reduced Magnetohydrodynamic (RMHD) equations at
both the collisional and collisionless scales; the latter obey a linear kinetic equation along the (moving) field
lines associated with the Alfvénic component (in the collisional limit, these compressive fluctuations become
the slow and entropy modes of the conventional MHD). In the “dissipation range” below ion gyroscale, there
are again two cascades: the kinetic-Alfvén-wave (KAW) cascade governed by two fluid-like Electron Reduced
Magnetohydrodynamic (ERMHD) equations and a passive cascade of ion entropy fluctuations both in space
and velocity. The latter cascade brings the energy of the inertial-range fluctuations that was Landau-damped at
the ion gyroscale to collisional scales in the phase space and leads to ion heating. The KAW energy is similarly
damped at the electron gyroscale and converted into electron heat. Kolmogorov-style scaling relations are
derived for all of these cascades. The relationship between the theoretical models proposed in this paper and
astrophysical applications and observations is discussed in detail.
Subject headings: magnetic fields—methods: analytical—MHD—plasmas—turbulence
1. INTRODUCTION
As observations of velocity, density and magnetic fields in
astrophysical plasmas probe ever smaller scales, turbulence—
i.e., broadband disordered fluctuations usually characterized
by power-law energy spectra—emerges as a fundamental and
ubiquitous feature.
One of the earliest examples of observed turbulence in
space was the detection of a Kolmogorov k−5/3 spectrum
of magnetic fluctuations in the solar wind over a fre-
quency range of about three decades (first reported by
Matthaeus & Goldstein 1982; Bavassano et al. 1982 and con-
firmed to a high degree of accuracy by a multitude of subse-
Electronic address: a.schekochihin1@physics.ox.ac.uk
1 Rudolf Peierls Centre for Theoretical Physics, University of Oxford,
Oxford OX1 3NP, UK.
2 Plasma Physics, Blackett Laboratory, Imperial College, Lon-
don SW7 2AZ, UK.
3 Euratom/UKAEA Fusion Association, Culham Science Centre, Abing-
ton OX14 3DB, UK.
4 Department of Physics, University of Maryland, College Park,
MD 20742-3511.
5 Princeton Plasma Physics Laboratory, Princeton, NJ 08543-0451.
6 Department of Physics and Astronomy, University of Iowa, Iowa City,
IA 52242-1479.
7 Department of Astronomy, University of California, Berkeley, CA
94720-3411.
quent observations, e.g., Marsch & Tu 1990a; Horbury et al.
1996; Leamon et al. 1998; Bale et al. 2005; see Fig. 1). An-
other famous example in which the Kolmogorov power law
appears to hold is the electron density spectrum in the inter-
stellar medium (ISM)—in this case it emerges from observa-
tions by various methods in several scale intervals and, when
these are pieced together, the power law famously extends
over as many as 12 decades of scales (Armstrong et al. 1981,
1995; Lazio et al. 2004), a record that has earned it the name
of “the Great Power Law in the Sky.” Numerous other mea-
surements in space and astrophysical plasmas, from the mag-
netosphere to galaxy clusters, result in Kolmogorov (or con-
sistent with Kolmogorov) spectra but also show steeper power
laws at very small (microphysical) scales (these observations
are discussed in more detail in § 8).
Power-law spectra spanning broad bands of scales are
symptomatic of the fundamental role of turbulence as a mech-
anism of transferring energy from the outer scale(s) (hence-
forth denoted L), where the energy is injected to the inner
scale(s), where it is dissipated. As these scales tend to be
widely separated in astrophysical systems, one way for the
system to bridge this scale gap is to fill it with fluctuations; the
power-law spectra then arise due to scale invariance at the in-
termediate scales. Besides being one of the more easily mea-
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2 SCHEKOCHIHIN ET AL.
surable characteristics of the multi-scale nature of turbulence,
power-law (and, particularly, Kolmogorov) spectra evoke a
number of fundamental physical ideas that lie at the heart of
the turbulence theory: universality of small-scale physics, en-
ergy cascade, locality of interactions, etc. In this paper, we
shall revisit and generalize these ideas for the problem of ki-
netic plasma turbulence,8 so it is perhaps useful to remind the
reader how they emerge in a standard argument that leads to
the k−5/3 spectrum (Kolmogorov 1941; Obukhov 1941).
1.1. Kolmogorov Turbulence
Suppose the average energy per unit time per unit volume
that the system dissipates is ε. This energy has to be trans-
ferred from some (large) outer scale L at which it is injected
to some (small) inner scale(s) at which the dissipation occurs
(see § 1.5). It is assumed that in the range of scales interme-
diate between the outer and the inner (the inertial range), the
statistical properties of the turbulence are universal (indepen-
dent of the macrophysics of injection or of the microphysics
of dissipation), spatially homogeneous and isotropic and the
energy transfer is local in scale space. The flux of kinetic en-
ergy through any inertial-range scale λ is independent of λ:
∼ ε = const, (1)
where the (constant) density of the medium is absorbed into
ε, uλ is the typical velocity fluctuation associated with the
scale λ, and τλ is the cascade time.
9 Since interactions are
assumed local, τλ must be expressed in terms of quantities
associated with scale λ. It is then dimensionally inevitable
that τλ ∼ λ/uλ (the nonlinear interaction time, or turnover
time), so we get
uλ ∼ (ελ)1/3. (2)
This corresponds to a k−5/3 spectrum of kinetic energy.
1.2. MHD Turbulence and Critical Balance
That astronomical data appear to point to a ubiquitous na-
ture of what, in its origin, is a dimensional result for the turbu-
lence in a neutral fluid, might appear surprising. Indeed, the
astrophysical plasmas in question are highly conducting and
support magnetic fields whose energy is at least comparable
to the kinetic energy of the motions. Let us consider a situa-
tion where the plasma is threaded by a uniform dynamically
strong magnetic field B0 (the mean, or guide, field; see § 1.3
for a brief discussion of the validity of this assumption). In
the presence of such a field, there is no dimensionally unique
way of determining the cascade time τλ because besides the
nonlinear interaction time λ/uλ, there is a second character-
istic time associated with the fluctuation of size λ, namely the
Alfvén time l‖λ/vA, where vA is the Alfvén speed and l‖λ is
the typical scale of the fluctuation along the magnetic field.
The first theories of magnetohydrodynamic (MHD) turbu-
lence (Iroshnikov 1963; Kraichnan 1965; Dobrowolny et al.
1980) calculated τλ by assuming an isotropic cascade (l‖λ ∼
λ) of weakly interacting Alfvén-wave packets (τλ ≫ l‖λ/vA)
8 An outline of a Kolmogorov-style approach to kinetic turbulence was
given in a recent paper by Schekochihin et al. (2008b). It can be read as a
conceptual introduction to the present paper, which is much more detailed
and covers a much broader set of topics.
9 This is the version of Kolmogorov’s theory due to Obukhov 1941.
FIG. 1.— Spectra of electric and magnetic fluctuations in the solar wind
at 1 AU (see Table 1 for the solar-wind parameters corresponding to this
plot). This figure is adapted with permission from Fig. 3 of Bale et al. (2005)
(copyright 2005 by the American Physical Society). We have added the
reference slopes for Alfvén-wave and kinetic-Alfvén-wave turbulence in
bold dashed (red) lines and labeled “KRMHD,” “GK ions,” and “ERMHD”
the wavenumber intervals where these analytical descriptions are valid (see
§ 3, § 5 and § 7).
and obtained a k−3/2 spectrum. The failure of the ob-
served spectra to conform to this law (see references above)
and especially the observational (see references at the end
of this subsection) and experimental (Robinson & Rusbridge
1971; Zweben et al. 1979) evidence of anisotropy of MHD
fluctuations led to the isotropy assumption being discarded
(Montgomery & Turner 1981).
The modern form of MHD turbulence theory is commonly
associated with the names of Goldreich & Sridhar (1995,
1997, henceforth, GS). It can be summarized as follows. As-
sume that (a) all electromagnetic perturbations are strongly
anisotropic, so that their characteristic scales along the mean
field are much larger than those across it, l‖λ ≫ λ, or, in terms
of wavenumbers, k‖ ≪ k⊥; (b) the interactions between the
Alfvén-wave packets are strong and the turbulence at suffi-
ciently small scales always arranges itself in such a way that
the Alfvén timescale and the perpendicular nonlinear interac-
tion timescale are comparable to each other, i.e.,
ω ∼ k‖vA ∼ k⊥u⊥, (3)
where ω is the typical frequency of the fluctuations and u⊥
is the velocity fluctuation perpendicular to the mean field.
Taken scale by scale, this assumption, known as the critical
balance, removes the dimensional ambiguity of the MHD tur-
bulence theory. Thus, the cascade time is τλ ∼ l‖λ/vA ∼λ/uλ,
whence
uλ∼ (εl‖λ/vA)1/2 ∼ (ελ)1/3 , (4)
l‖λ∼ l
2/3, (5)
where l0 = v
A/ε. The scaling relation (4) is equivalent to a
⊥ spectrum of kinetic energy, while Eq. (5) quantifies the
anisotropy by establishing the relationship between the per-
pendicular and parallel scales. Note that Eq. (4) implies that
in terms of the parallel wavenumbers, the kinetic-energy spec-
trum is ∼ k−2‖ .
The above considerations apply to Alfvénic fluctuations,
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 3
i.e., perpendicular velocities and magnetic-field perturbations
from the mean given (at each scale) by δB⊥ ∼ u⊥
4πρ0,
where ρ0 is the mean mass density of the plasma (see
Fig. 1 and discussion in § 8.1.1). Other low-frequency
MHD modes—slow waves and the entropy mode—turn
out to be passively advected by the Alfvénic component
of the turbulence (this follows from the anisotropy; see
Lithwick & Goldreich 2001, and §§ 2.4-2.6, § 5.5, and § 6.3
for further discussion of the compressive fluctuations).
As we have mentioned above, the anisotropy was, in
fact, incorporated into MHD turbulence theory already by
Montgomery & Turner (1981). However, these authors’ view
differed from the GS theory in that they thought of MHD
turbulence as essentially two dimensional, described by a
Kolmogorov-like cascade (Fyfe et al. 1977), with an admix-
ture of Alfvén waves having some spectrum in k‖ unrelated
to the perpendicular structure of the turbulence (note that
Higdon 1984, while adopting a similar view, anticipated the
scaling relation (5), but did not seem to consider it to be any-
thing more than the confirmation of an essentially 2D nature
of the turbulence). In what we are referring to here as GS
turbulence, the 2D and Alfvénic fluctuations are not separate
components of the turbulence. The turbulence is three dimen-
sional, with correlations parallel and perpendicular to the (lo-
cal) mean field related at each scale by the critical balance
assumption.
Indeed, intuitively, we cannot have k‖vA ≪ k⊥u⊥: the tur-
bulence cannot be any more 2D than allowed by the critical
balance because fluctuations in any two planes perpendicu-
lar to the mean field can only remain correlated if an Alfvén
wave can propagate between them in less than their perpen-
dicular decorrelation time. In the opposite limit, weakly inter-
acting Alfén waves with fixed k‖ and ω = k‖vA ≫ k⊥u⊥ can be
shown to give rise to an energy cascade towards smaller per-
pendicular scales where the turbulence becomes strong and
Eq. (3) is satisfied (Goldreich & Sridhar 1997; Galtier et al.
2000; Yousef et al. 2009). Thus, there is a natural tendency
towards critical balance in a system containing nonlinearly in-
teracting Alfvén waves. We will see in what follows that crit-
ical balance may, in fact, be taken as a general physical prin-
ciple relating parallel scales (associated with linear propaga-
tion) and perpendicular scales (associated with nonlinear in-
teraction) in anisotropic plasma turbulence (see § 7.5, § 7.9.4,
§ 7.10.3).
We emphasize that, the anisotropy of astrophysical plasma
turbulence is an observed phenomenon. It is seen most clearly
in the spacecraft measurements of the turbulent fluctuations
in the solar wind (Belcher & Davis 1971; Matthaeus et al.
1990; Bieber et al. 1996; Dasso et al. 2005; Bigazzi et al.
2006; Sorriso-Valvo et al. 2006; Horbury et al. 2005, 2008;
Osman & Horbury 2007; Hamilton et al. 2008) and in the
magnetosheath Sahraoui et al. (2006); Alexandrova et al.
(2008b). In a recent key development, solar-wind data anal-
ysis by Horbury et al. (2008) approaches quantitative cor-
roboration of the critical balance conjecture by confirm-
ing the scaling of the spectrum with the parallel wavenum-
ber ∼ k−2
that follows from the first scaling relation in
Eq. (4). Anisotropy is also observed indirectly in the
ISM (Wilkinson et al. 1994; Trotter et al. 1998; Rickett et al.
2002; Dennett-Thorpe & de Bruyn 2003), including recently
in molecular clouds (Heyer et al. 2008), and, with unambigu-
ous consistency, in numerical simulations of MHD turbulence
(Shebalin et al. 1983; Oughton et al. 1994; Cho & Vishniac
2000; Maron & Goldreich 2001; Cho et al. 2002; Müller et al.
2003).10
1.3. MHD Turbulence with and without a Mean Field
In the discussion above, treating MHD turbulence as tur-
bulence of Alfvénic fluctuations depended on assuming the
presence of a mean (guide) field B0 that is strong compared to
the magnetic fluctuations, δB/B0 ∼ u/vA ≪ 1. We will also
need this assumption in the formal developments to follow
(see § 2.1, § 3.1). Is it legitimate to expect that such a spatially
regular field will be generically present? Kraichnan (1965) ar-
gued that in a generic situation in which all magnetic fields are
produced by the turbulence itself via the dynamo effect, one
could assume that the strongest field will be at the outer scale
and that this field will play the role of an (approximately) uni-
form guide field for the Alfvén waves in the inertial range.
Formally, this amounts to assuming that in the inertial range,
≪ 1, k‖L ≪ 1. (6)
It is, however, by no means obvious that this should be true.
When a strong mean field is imposed by some external mech-
anism, the turbulent motions cannot bend it significantly, so
only small perturbations are possible and δB ≪ B0. In con-
trast, without a strong imposed field, the energy density of the
magnetic fluctuations is at most comparable to the kinetic-
energy density of the plasma motions, which are then suffi-
ciently energetic to randomly tangle the field, so δB ≫ B0.
In the weak-mean-field case, the dynamically strong
stochastic magnetic field is a result of saturation of the
small-scale, or fluctuation, dynamo—amplification of mag-
netic field due to random stretching by the turbulent mo-
tions (see review by Schekochihin & Cowley 2007). The
definitive theory of this saturated state remains to be dis-
covered. Both physical arguments and numerical evidence
(Schekochihin et al. 2004; Yousef et al. 2007) suggest that
the magnetic field in this case is organized in folded flux
sheets (or ribbons). The length of these folds is compara-
ble to the outer scale, while the scale of the field-direction
reversals transverse to the fold is determined by the dissipa-
tion physics: in MHD with isotropic viscosity and resistiv-
ity, it is the resistive scale.11 Although Alfvén waves prop-
10 The numerical evidence is much less clear on the scaling of the
spectrum. The fact that the spectrum is closer to k
than to k
in numerical simulations (Maron & Goldreich 2001; Müller et al. 2003;
Mason et al. 2007; Perez & Boldyrev 2008, 2009; Beresnyak & Lazarian
2008b) prompted Boldyrev (2006) to propose a scaling argument that allows
an anisotropic Alfvénic turbulence with a k
spectrum. His argument is
based on the conjecture that the fluctuating velocity and magnetic fields tend
to partially align at small scales, an idea that has had considerable numeri-
cal support (Maron & Goldreich 2001; Beresnyak & Lazarian 2006, 2008b;
Mason et al. 2006; Matthaeus et al. 2008a). The alignment weakens nonlin-
ear interactions and alters the scalings. Another modification of the GS the-
ory leading to an anisotropic k
spectrum was proposed by Gogoberidze
(2007), who assumed that MHD turbulence with a strong mean field is dom-
inated by non-local interactions with the outer scale. However, in both argu-
ments, the basic assumption that the turbulence is strong is retained. This is
the main assumption that we make in this paper: the critical balance conjec-
ture (3) is used below not as a scaling prescription but in a weaker sense of
an ordering assumption, i.e., we simply take the wave propagation terms in
the equations to be comparable to the nonlinear terms. It is not hard to show
that the results derived in what follows remain valid whether or not the align-
ment is present. We note that observationally, only in the solar wind does one
measure the spectra with sufficient accuracy to state that they are consistent
with k
but not with k
(see § 8.1.1).
11 In weakly collisional astrophysical plasmas, such a description is not
4 SCHEKOCHIHIN ET AL.
agating along the folds may exist (Schekochihin et al. 2004;
Schekochihin & Cowley 2007), the presence of the small-
scale direction reversals means that there is no scale-by-scale
equipartition between the velocity and magnetic fields: while
the magnetic energy is small-scale dominated due to the di-
rection reversals,12 the kinetic energy should be contained pri-
marily at the outer scale, with some scaling law in the inertial
range.
Thus, at the current level of understanding we have to as-
sume that there are two asymptotic regimes of MHD turbu-
lence: anisotropic Alfvénic turbulence with δB ≪ B0 and
isotropic MHD turbulence with small-scale field reversals and
δB ≫ B0. In this paper, we shall only discuss the first regime.
The origin of the mean field may be external (as, e.g., in the
solar wind, where it is the field of the Sun) or due to some
form of mean-field dynamo (rather than small-scale dynamo),
as usually expected for galaxies (see, e.g., Shukurov 2007).
Note finally that the condition δB ≪ B0 need not be satis-
fied at the outer scale and in fact is not satisfied in most space
or astrophysical plasmas, where more commonly δB ∼ B0 at
the outer scale. This, however, is sufficient for the Kraich-
nan hypothesis to hold and for an Alfénic cascade to be set
up, so at small scales (in the inertial range and beyond), the
assumptions (6) are satisfied.
1.4. Kinetic Turbulence
The GS theory of MHD turbulence (§ 1.2) allows us to
make sense of the magnetized turbulence observed in cosmic
plasmas exhibiting the same statistical scaling as turbulence
in a neutral fluid (although the underlying dynamics are very
different in these two cases!). However, there is an aspect of
the observed astrophysical turbulence that undermines the ap-
plicability of any type of fluid description: in most cases, the
inertial range where the Kolmogorov scaling holds extends to
scales far below the mean free path deep into the collisionless
regime. For example, in the case of the solar wind, the mean
free path is close to 1 AU, so all scales are collisionless—
an extreme case, which also happens to be the best studied,
thanks to the possibility of in situ measurements (see § 8).
The proper way of treating such plasmas is using kinetic
theory, not fluid equations. The basis for the application of the
MHD fluid description to them has been the following well
known result from the linear theory of plasma waves: while
the fast, slow and entropy modes are damped at the mean-
free-path scale both by collisional viscosity (Braginskii 1965,
see § 6.1.2) and by collisionless wave–particle interactions
(Barnes 1966, see § 6.2.2), the Alfvén waves are only damped
at the ion gyroscale. It has, therefore, been assumed that the
MHD description, inasmuch as it concerns the Alfvén-wave
cascade, can be extended to the ion gyroscale, with the un-
derstanding that this cascade is decoupled from the damped
cascades of the rest of the MHD modes. This approach and
its application to the turbulence in the ISM are best explained
by Lithwick & Goldreich (2001).
While the fluid description may be sufficient to under-
stand the Alfvénic fluctuations in the inertial range, it is cer-
applicable: the field reversal scale is most probably determined by more
complicated and as yet poorly understood kinetic plasma effects; below this
scale, an Alfvénic turbulence of the kind discussed in this paper may exist
(Schekochihin & Cowley 2006).
12 See Haugen et al. (2004) for an alternative view. Note also that the
numerical evidence cited above pertains to forced simulations. In decaying
MHD turbulence simulations, the magnetic energy does indeed appear to be
at the outer scale (Biskamp & Müller 2000), so one might expect an Alfvénic
cascade deep in the inertial range.
tainly inadequate for everything else: the compressive fluctu-
ations in the inertial range and turbulence in the dissipation
range (below the ion gyroscale), where power-law spectra are
also detected (e.g., Denskat et al. 1983; Leamon et al. 1998;
Czaykowska et al. 2001; Smith et al. 2006; Sahraoui et al.
2006; Alexandrova et al. 2008a,b, see also Fig. 1). The fun-
damental challenge that a comprehensive theory of astrophys-
ical plasma turbulence must meet is to give the full account of
how the turbulent fluctuation energy injected at the outer scale
is cascaded to small scales and deposited into particle heat.
We shall see (§ 3.4 and § 3.5) that the familiar concept of an
energy cascade can be generalized in the kinetic framework
as the kinetic cascade of a single quantity that we call the
generalized energy (see also Schekochihin et al. 2008b, and
references therein). The small scales developed in the pro-
cess are small scales both in the position and velocity space.
The fundamental reason for this is the low collisionality of
the plasma: since heating cannot ultimately be accomplished
without collisions, large gradients in phase space are neces-
sary for the collisions to be effective.
The idea of a generalized energy cascade in phase space
as the engine of kinetic plasma turbulence is the central con-
cept of this paper. In order to understand the physics of the
kinetic cascade in various scale ranges, we derive in what fol-
lows a hierarchy of simplified, yet rigorous, reduced kinetic,
fluid and hybrid descriptions. While the full kinetic theory
of turbulence is very difficult to handle either analytically or
numerically, the models we derive are much more tractable.
For all, the regimes of applicability (scale/parameter ranges,
underlying assumptions) are clearly stated. In each of these
regimes, the kinetic cascade splits into several channels of en-
ergy transfer, some of them familiar (e.g., the Alfvénic cas-
cade, § 5.3 and § 5.4), others conceptually new (e.g., the ki-
netic cascade of collisionless compressive fluctuations, § 6.2,
or the entropy cascade, §§ 7.9-7.12).
So as to introduce this theoretical framework in a way that
is both analytically systematic and physically intelligible, let
us first consider the characteristic scales that are relevant to
the problem of astrophysical turbulence (§ 1.5). The models
we derive are previewed in § 1.6, at the end of which the plan
of further developments is given.
1.5. Scales in the Problem
1.5.1. Outer Scale
It is a generic feature of turbulent systems that energy is
injected via some large-scale mechanism: “large scale” here
means some scale (or a range of scales) comparable to the
size of the system, depending on its global properties, and
much larger than the microphysical scales at which energy
is dissipated and converted into heat (§ 1.5.2). Examples of
large-scale stirring of turbulent fluctuations include the solar
activity in the corona (launching Alfvén waves to produce
turbulence in the solar wind); supernova explosions in the
ISM (e.g., Norman & Ferrara 1996; Ferrière 2001); the mag-
netorotational instability in accretion disks (Balbus & Hawley
1998); merger events, galaxy wakes and active galactic
nuclei in galaxy clusters (e.g., Subramanian et al. 2006;
Enßlin & Vogt 2006; Chandran 2005a). Since in this paper
we are concerned with the local properties of astrophysical
plasmas, let us simply assume that energy injection occurs at
some characteristic outer scale L. All further considerations
will apply to scales that are much smaller than L and we will
assume that the particular character of the energy injection
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 5
FIG. 2.— Partition of the wavenumber space by characteristic scales. The wavenumbers are normalized by l0 ∼ v
A/ε, where ε is the total power input (see
§ 1.2). Dotted line shows the path an Alfvén-wave cascade starting at the outer scale L ∼ l0 takes through the wavenumber space. We also show the regions of
validity of the three tertiary approximations. They all require k‖ ≪ k⊥ (anisotropic fluctuations) and k‖ρi ≪ 1 (i.e., k‖vthi ≪ Ωi, low-frequency limit). Reduced
MHD (RMHD, § 2) is valid when k⊥ρi ≪ k‖λmfpi ≪ (me/mi)
1/2 (strongly magnetized collisional limit, adiabatic electrons). The regions of validity of Kinetic
Reduced MHD (KRMHD, § 5) and Electron Reduced MHD (ERMHD, § 7) lie within that of the isothermal electron/gyrokinetic ion approximation (Fig. 4) with
the additional requirement that k⊥ρi ≪ min(1,k‖λmfpi) (strongly magnetized ions) for KRMHD or k⊥ρi ≫ 1 (unmagnetized ions) for ERMHD. The collisional
limit of KRMHD (§ 6.1 and Appendix D), (me/mi)1/2 ≪ k‖λmfpi ≪ 1, is similar to RMHD, except electrons are isothermal. The dotted line is the scaling of k‖
vs k⊥ from critical balance in both the Alfvén-wave [§ 1.2, Eq. (5)] and kinetic-Alfvén-wave [§ 7.5, Eq. (241)] regimes.
does not matter at these small scales.
In most astrophysical situations, one cannot assume that
equilibrium quantities such as density, temperature, mean ve-
locity and mean magnetic field are uniform at the outer scale.
However, at scales much smaller than L, the gradients of the
small-scale fluctuating fields are much larger than the outer-
scale gradients (although the fluctuation amplitudes are much
smaller; for the mean magnetic field, this assumption is dis-
cussed in some detail in § 1.3), so we may neglect the equi-
librium gradients and consider the turbulence to be homoge-
neous. Specifically, this is a good assumption if k‖L ≫ 1
[Eq. (6)], i.e., not only the perpendicular scales but also the
much larger parallel ones are still shorter than the outer scale.
Note that we cannot generally assume that the outer-scale en-
ergy injection is anisotropic, so the anisotropy is also the prop-
erty of small scales only.
1.5.2. Microscales
There are four microphysical scales that mark the transi-
tions between distinct physical regimes:
Electron diffusion scale. — At k‖λmfpi(mi/me)
1/2 ≫ 1, the
electron response is isothermal (§ 4.4, Appendix A.4). At
k‖λmfpi(mi/me)
1/2 ≪ 1, it is adiabatic (§ 4.8.4, Appendix A.3).
Mean free path. — At k‖λmfpi ≫ 1, the plasma is collisionless.
In this regime, wave–particle interactions can damp compres-
sive fluctuations via Barnes damping (§ 6.2.2), so kinetic de-
scription becomes essential. At k‖λmfpi ≪ 1, the plasma is
collisional and fluid-like (§ 6.1, Appendices A and D).
Ion gyroscale. — At k⊥ρi ≪ 1, ions (as well as the electrons)
are magnetized and the magnetic field is frozen into the ion
flow (the E × B velocity field). At k⊥ρi ∼ 1, ions can ex-
change energy with electromagnetic fluctuations via wave–
particle interactions (and ion heating eventually occurs via a
kinetic ion-entropy cascade, see §§ 7.9-7.10). At k⊥ρi ≫ 1,
the ions are unmagnetized and have a Boltzmann response
(§ 7.2). Note that the ion inertial scale di = ρi/
βi is compara-
ble to the ion gyroscale unless the plasma beta βi = 8πniTi/B
is very different from unity. In the theories developed below,
di does not play a special role except in the limit of Ti ≪ Te,
which is not common in astrophysical plasmas (see further
discussion in § 7.1 and Appendix E).
Electron gyroscale. — At k⊥ρe ≪ 1, electrons are magnetized
and the magnetic field is frozen into the electron flow (§ 4,
§ 7, Appendix C). At k⊥ρe ∼ 1, the electrons absorb the
energy of the electromagnetic fluctuations via wave–particle
interactions (leading to electron heating via a kinetic electron-
entropy cascade, see § 7.12).
Typical values of these scales and of several other key pa-
rameters are given in Table 1. In Fig. 2, we show how the
wavenumber space, (k⊥,k‖), is divided by these scales into
several domains, where the physics is different. Further parti-
tioning of the wavenumber space results from comparing k⊥ρi
and k‖λmfpi (k⊥ρi ≪ k‖λmfpi is the limit of strong magnetiza-
6 SCHEKOCHIHIN ET AL.
TABLE 1
REPRESENTATIVE PARAMETERS FOR ASTROPHYSICAL PLASMAS.
Parameter
Solar
1 AU(a)
ionized
ISM(b)
Accretion
flow near
Sgr A∗(c)
Galaxy
clusters
(cores)(d)
ne = ni, cm−3 30 0.5 106 6× 10−2
Te, K ∼ Ti(e) 8000 1011 3× 107
Ti, K 5× 105 8000 ∼ 1012(f) ?(e)
B, G 10−4 10−6 30 7× 10−6
βi 5 14 4 130
vthi, km/s 90 10 10
5 700
vA, km/s 40 3 7× 10
U , km/s(f) ∼ 10 ∼ 10 ∼ 104 ∼ 102
L, km(f) ∼ 105 ∼ 1015 ∼ 108 ∼ 1017
(mi/me)1/2λmfpi, km 10
10 2× 108 4× 1010 4× 1016
λmfpi, km
(g) 3× 108 6× 106 109 1015
ρi, km 90 1000 0.4 104
ρe, km 2 30 0.003 200
a Values for slow wind (mean flow speed Vsw = 350 km/s in this case)
measured by Cluster spacecraft and taken from Bale et al. (2005), ex-
cept the value of Te, which they do not report, but which is expected
to be of the same order as Ti (Newbury et al. 1998). Note that the
data interval studied by Bale et al. (2005) is slightly atypical, with βi
higher than usual in the solar wind (the full range of βi variation in
the solar wind is roughly between 0.1 and 10; see Howes et al. 2008a
for another, perhaps more typical, fiducial set of slow-wind parameters
and Appendix A of the review by Bruno & Carbone 2005 for slow- and
fast-wind parameters measured by Helios 2). However, we use their pa-
rameter values as our representative example because the spectra they
report show with particular clarity both the electric and magnetic fluc-
tuations in both the inertial and dissipation ranges (see Fig. 1). See
further discussion in § 8.1 and § 8.2.
b Typical values (see, e.g., Norman & Ferrara 1996; Ferrière 2001). See
discussion in § 8.4.
c Values based on observational constraints for the radio-emitting
plasma around the Galactic Center (Sgr A∗) as interpreted by
Loeb & Waxman (2007) (see also Quataert 2003). See discussion in
§ 8.5.
d Values for the core region of the Hydra A cluster taken from
Enßlin & Vogt (2006); see Schekochihin & Cowley 2006 for a consis-
tent set of numbers for the hot plasmas outside the cores. See discussion
in § 8.6.
e We assume Ti ∼ Te for these estimates.
f Rough order-of-magnitude estimate.
g Defined λmfpi = vthi/νii, where νii is given by Eq. (52).
tion, see Appendix A.2) and, most importantly, from com-
paring parallel and perpendicular wavenumbers. As we ex-
plained above, observational and numerical evidence tells us
that Alfvénic turbulence is anisotropic, k‖ ≪ k⊥. In Fig. 2,
we sketch the path the turbulent cascade is expected to take
in the wavenumber space (we use the scalings of k‖ with k⊥
that follow from the GS argument for the Alfvén waves and an
analogous argument for the kinetic Alfvén waves, reviewed in
§ 1.2 and § 7.5, respectively).
1.6. Kinetic and Fluid Models
What is the correct analytical description of the turbulent
plasma fluctuations along the (presumed) path of the cascade?
As we promised above, it is going to be possible to simplify
the full kinetic theory substantially. These simplifications can
be obtained in the form of a hierarchy of approximations and
as these emerge, specific physical mechanisms that control the
turbulent cascade in various physical regimes become more
transparent.
Gyrokinetics (§ 3). — The starting point for these develop-
ments and the primary approximation in the hierarchy is gy-
rokinetics, a low-frequency kinetic theory resulting from av-
eraging over the cyclotron motion of the particles. Gyroki-
netics is appropriate for the study of subsonic plasma turbu-
lence in virtually all astrophysically relevant parameter ranges
(Howes et al. 2006). For fluctuations at frequencies lower
than the ion cyclotron frequency, ω ≪ Ωi, gyrokinetics can
be systematically derived by making use of the following two
assumptions, which also underpin the GS theory (§ 1.2): (a)
anisotropy of the turbulence, so ǫ∼ k‖/k⊥ is used as the small
parameter, and (b) strong interactions, i.e., the fluctuation am-
plitudes are assumed to be such that wave propagation and
nonlinear interaction occur on comparable timescales: from
Eq. (3), u⊥/vA ∼ ǫ. The first of these assumptions implies that
fluctuations at Alfvénic frequencies satisfy ω ∼ k‖vA ≪ Ωi
even when their perpendicular scale is such k⊥ρi ∼ 1. This
makes gyrokinetics an ideal tool both for analytical theory and
for numerical studies of astrophysical plasma turbulence; the
numerical approaches are also made attractive by the long ex-
perience of gyrokinetic simulations accumulated in the fusion
research and by the existence of publicly available gyroki-
netic codes (Kotschenreuther et al. 1995; Jenko et al. 2000;
Candy & Waltz 2003; Chen & Parker 2003). A concise re-
view of gyrokinetics is provided in § 3 (see Howes et al. 2006
for a detailed derivation). The reader is urged to pay partic-
ular attention to § 3.4 and § 3.5, where the concept of the ki-
netic cascade of generalized energy is introduced and the par-
ticle heating in gyrokinetics is discussed (Appendix F intro-
duces additional conservation laws that arise in 2D and some-
times also in 3D). This establishes the conceptual framework
in which most of the subsequent physical arguments are pre-
sented. The region of validity of gyrokinetics is illustrated
in Fig. 3: it covers virtually the entire path of the turbulent
cascade, except the largest (outer) scales, where one cannot
assume anisotropy. Note that the two-fluid theory, which is
the starting point for the MHD theory (see Appendix A), is
not a good description at collisionless scales. It is important
to mention, however, that the formulation of gyrokinetics that
we adopt, while appropriate for treating fluctuations at col-
lisionless scales, does nevertheless require a certain (weak)
degree of collisionality (see discussion in § 3.1.3 and an ex-
tended treatment of collisions in gyrokinetics in Appendix B).
Isothermal Electron Fluid (§ 4). — While gyrokinetics con-
stitutes a significant simplification, it is still a fully kinetic
description. Further progress towards simpler models is
achieved by showing that, for parallel scales smaller than the
electron diffusion scale, k‖λmfpi ≫ (me/mi)1/2, and perpen-
dicular scales larger than the electron gyroscale, k⊥ρe ≪ 1,
the electrons are a magnetized isothermal fluid while ions
must be treated (gyro)kinetically. This is the secondary
approximation in our hierarchy, derived in § 4 via an asymp-
totic expansion in (me/mi)
1/2 (see also Appendix C.1). The
plasma is described by the ion gyrokinetic equation and two
fluid-like equations that contain electron dynamics—these
are summarized in § 4.9. The region of validity of this
approximation is illustrated in Fig. 4: it does not capture the
dissipative effects around the electron diffusion scale or the
electron heating, but it remains uniformly valid as the cascade
passes from collisional to collisionless scales and also as it
crosses the ion gyroscale.
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 7
In order to elucidate the nature of the turbulence above and
below the ion gyroscale, we derive two tertiary approxima-
tions, one of which is valid for k⊥ρi ≪ 1 (§ 5 and § 6) and the
other for k⊥ρi ≫ 1 (§ 7; see also Appendix C, which gives
a non-rigorous, non-gyrokinetic, but perhaps more intuitive,
derivation of the results of § 4 and § 7.2).
Kinetic Reduced MHD (§ 5 and § 6). — On scales above the ion
gyroscale, known as the “inertial range” we demonstrate that
the decoupling of the Alfvén-wave cascade and its indiffer-
ence to both collisional and collisionless damping are explicit
and analytically provable properties. We show rigorously that
the Alfvén-wave cascade is governed by a closed set of two
fluid-like equations for the stream and flux functions—the Re-
duced Magnetohydrodynamics (RMHD)—independently of
the collisionality (§ 5.3 and § 5.4; the derivation of RMHD
from MHD and its properties are presented in § 2). The cas-
cade proceeds via interaction of oppositely propagating wave
packets and is decoupled from the density and magnetic-field-
strength fluctuations (the “compressive” modes; in the colli-
sional limit, these are the entropy and slow modes; see § 6.1
and Appendix D). The latter are passively mixed by the
Alfvén waves, but, unlike in the fluid (collisional) limit, this
passive cascade is governed by a (simplified) kinetic equa-
tion for the ions (§ 5.5). Together with RMHD, it forms
a hybrid fluid-kinetic description of magnetized turbulence
in a weakly collisional plasma, which we call Kinetic Re-
duced MHD (KRMHD). The KRMHD equations are sum-
marized in § 5.7. Their collisional and collisionless limits
are explored in § 6.1 and § 6.2, respectively. Whereas the
Alfvén waves are undamped in this approximation, the com-
pressive fluctuations are subject to damping both in the col-
lisional (Braginskii 1965 viscous damping, § 6.1.2) and colli-
sionless (Barnes 1966 damping, § 6.2.2) limits. In the colli-
sionless limit, the compressive component of the turbulence is
a simple example of an essentially kinetic turbulence, includ-
ing such features as conservation of generalized energy de-
spite collisionless damping and (parallel) phase mixing, pos-
sibly leading to ion heating (§§ 6.2.3-6.2.5). How strongly
the compressive fluctuations are damped depends on the par-
allel scale of these fluctuations. Since the ion kinetic equation
turns out to be linear along the moving field lines associated
with the Alfvén waves, the compressive fluctuations do not, in
the absence of finite-gyroradius effects, develop small parallel
scales and their cascade may be only weakly damped above
the ion gyroscale—this is discussed in § 6.3.
Electron Reduced MHD (§ 7). — At the ion gyroscale, the
Alfvénic and the compressive cascades are no longer decou-
pled and their energy is partially damped via collisionless
wave–particle interactions (§ 7.1). This part of the energy
is channeled into ion heat. The rest of it is converted into
a cascade of kinetic Alfvén waves (KAW). This cascade ex-
tends through what is known as the “dissipation range” to
the electron gyroscale, where its turn comes to be damped via
wave–particle interaction and transferred into electron heat.
The KAW turbulence is again anisotropic with k‖ ≪ k⊥. It
is governed by a pair of fluid-like equations, also derived
from gyrokinetics. We call them Electron Reduced MHD
(ERMHD). In the high-beta limit, they coincide with the re-
duced (anisotropic) form of the previously known Electron
MHD (Kingsep et al. 1990). The ERMHD equations are de-
rived in § 7.2 (see also Appendix C.2) and the KAW cas-
cade is considered in §§ 7.3-7.5. The fate of the inertial-
range energy collisionlessly damped at the ion gyroscale is
investigated in §§ 7.9-7.11; an analogous consideration for
the KAW energy damped at the electron gyroscale is pre-
sented in § 7.12. In these sections, we introduce the no-
tion of the entropy cascade—a nonlinear phase-mixing pro-
cess whereby the collisionless damping occurring at the ion
and electron gyroscales is made irreversible and particles are
heated. This part of the cascade is purely kinetic and its salient
feature is the particle distribution functions developing small
scales in the gyrokinetic phase space. Note that besides deriv-
ing rigorous sets of equations for the dissipation-range turbu-
lence, § 7 also presents a number of Kolmogorov-style scaling
predictions—both for the KAW cascade (§ 7.5) and for the en-
tropy cascade (§ 7.9.2, § 7.10.2, § 7.10.4, § 7.12).
Hall Reduced MHD (Appendix E). — The reduced (anisotropic)
form of the popular Hall MHD system can be derived as a
special limit of gyrokinetics (k⊥ρi ≪ 1, Ti ≪ Te, βi ≪ 1).
The resulting Hall Reduced MHD (HRMHD) equations
are a convenient model for some purposes because they
simultaneously capture the cold-ion, low-beta limits of both
the KRMHD and ERMHD systems. However, they are
usually not strictly applicable in space and astrophysical
plasmas of interest, where ions are rarely cold and βi is not
particularly low. The HRMHD equations are derived in § E.1,
the kinetic cascade of generalized energy in the Hall limit is
discussed in § E.2, and the circumstances under which the ion
inertial and ion sound scales become important in theories of
plasma turbulence are summarized in § E.4. Theories of the
dissipation-range turbulence based on Hall MHD are briefly
discussed in § 8.2.6.
The regions of validity of the tertiary approximations—
KRMHD and ERMHD—are illustrated in Fig. 2. In this fig-
ure, we also show the region of validity of the RMHD sys-
tem derived from the standard compressible MHD equations
by assuming anisotropy of the turbulence and strong inter-
actions. This derivation is the fluid analog of the derivation
of gyrokinetics. We present it in § 2, before embarking on
the gyrokinetics-based path outlined above, in order to make
a connection with the conventional MHD treatment and to
demonstrate with particular simplicity how the assumption of
anisotropy leads to a reduced fluid system in which the decou-
pling of the cascades of the Alfvén waves and of the compres-
sive modes is manifest (Appendix A extends this derivation
to Braginskii 1965 two-fluid equations in the limit of strong
magnetization; it also works out rigorously the transition from
the fluid limit to the KRMHD equations).
The main formal developments of this paper are contained
in §§ 3-7. The outline given above is meant to help the reader
navigate these sections. In § 8, we discuss at some length
how our results apply to various astrophysical plasmas with
weak collisionality: the solar wind and the magnetosheath,
the ISM, accretion disks, and galaxy clusters (§ 8.1 and § 8.2
can also be read as an overall summary of the paper in light
of the evidence available from space-plasma measurements).
Finally, in § 9, we provide a brief epilogue and make a few
remarks about future directions of inquiry.
2. REDUCED MHD AND THE DECOUPLING OF TURBULENT
CASCADES
8 SCHEKOCHIHIN ET AL.
Consider the equations of compressible MHD
= −ρ∇·u, (7)
B ·∇B
, (8)
= 0, s =
, γ =
, (9)
= B ·∇u − B∇·u, (10)
where ρ is the mass density, u velocity, p pressure, B magnetic
field, s the entropy density, and d/dt = ∂/∂t + u ·∇ (the con-
ditions under which these equations are valid are discussed in
Appendix A). Consider a uniform static equilibrium with a
straight mean field in the z direction, so
ρ = ρ0 + δρ, p = p0 + δp, B = B0ẑ + δB, (11)
where ρ0, p0, and B0 are constants. In what follows, the sub-
scripts ‖ and ⊥ will be used to denote the projections of fields,
variables and gradients on the mean-field direction ẑ and onto
the plane (x,y) perpendicular to this direction, respectively.
2.1. RMHD Ordering
As we explained in the Introduction, observational and nu-
merical evidence makes it safe to assume that the turbulence
in such a system will be anisotropic with k‖ ≪ k⊥ (at scales
smaller than the outer scale, k‖L ≫ 1; see § 1.3 and § 1.5.1).
Let us, therefore, introduce a small parameter ǫ ∼ k‖/k⊥ and
carry out a systematic expansion of Eqs. (7-10) in ǫ. In this
expansion, the fluctuations are treated as small, but not arbi-
trarily so: in order to estimate their size, we shall adopt the
critical-balance conjecture (3), which is now treated not as a
detailed scaling prescription but as an ordering assumption.
This allows us to introduce the following ordering:
∼ δB⊥
∼ ǫ, (12)
where vA = B0/
4πρ0 is the Alfvén speed. Note that this
means that we order the Mach number
, (13)
where cs = (γp0/ρ0)
1/2 is the speed of sound and
is the plasma beta, which is ordered to be order unity in the ǫ
expansion (subsidiary limits of high and low β can be taken
after the ǫ expansion is done; see § 2.4).
In Eq. (12), we made two auxiliary ordering assump-
tions: that the velocity and magnetic-field fluctuations
have the character of Alfvén and slow waves (δB⊥/B0 ∼
u⊥/vA, δB‖/B0 ∼ u‖/vA) and that the relative amplitudes
of the Alfvén-wave-polarized fluctuations (δB⊥/B0, u⊥/vA),
slow-wave-polarized fluctuations (δB‖/B0, u‖/vA) and den-
sity/pressure/entropy fluctuations (δρ/ρ0, δp/p0) are all the
same order. Strictly speaking, whether this is the case depends
on the energy sources that drive the turbulence: as we shall
see, if no slow waves (or entropy fluctuations) are launched,
none will be present. However, in astrophysical contexts, the
outer-scale energy input may be assumed random and, there-
fore, comparable power is injected into all types of fluctua-
tions.
We further assume that the characteristic frequency of the
fluctuations is ω∼ k‖vA [Eq. (3)], meaning that the fast waves,
for which ω ≃ k⊥(v2A + c2s )1/2, are ordered out. This restric-
tion must be justified empirically. Observations of the solar-
wind turbulence confirm that it is primarily Alfvénic (see,
e.g., Bale et al. 2005) and that its compressive component is
substantially pressure-balanced (Roberts 1990; Burlaga et al.
1990; Marsch & Tu 1993; Bavassano et al. 2004, see Eq. (22)
below). A weak-turbulence calculation of compressible MHD
turbulence in low-beta plasmas (Chandran 2005b) suggests
that only a small amount of energy is transferred from the fast
waves to Alfvén waves with large k‖. A similar conclusion
emerges from numerical simulations (Cho & Lazarian 2002,
2003). As the fast waves are also expected to be subject to
strong collisionless damping and/or to strong dissipation after
they steepen into shocks, we eliminate them from our con-
sideration of the problem and concentrate on low-frequency
turbulence.
2.2. Alfvén Waves
We start by observing that the Alfvén-wave-polarized
fluctuations are two-dimensionally solenoidal: since, from
Eq. (7),
∇·u = − d
= O(ǫ2) (15)
and ∇·δB = 0 exactly, separating the O(ǫ) part of these diver-
gences gives ∇⊥ ·u⊥ = 0 and ∇⊥ · δB⊥ = 0. To lowest order
in the ǫ expansion, we may, therefore, express u⊥ and δB⊥ in
terms of scalar stream (flux) functions:
u⊥ = ẑ×∇⊥Φ,
= ẑ×∇⊥Ψ. (16)
Evolution equations for Φ and Ψ are obtained by substituting
the expressions (16) into the perpendicular parts of the induc-
tion equation (10) and the momentum equation (8)—of the
latter the curl is taken to annihilate the pressure term. Keep-
ing only the terms of the lowest order, O(ǫ2), we get
+{Φ,Ψ}= vA
, (17)
∇2⊥Φ+
Φ,∇2⊥Φ
∇2⊥Ψ+
Ψ,∇2⊥Ψ
, (18)
where {Φ,Ψ} = ẑ · (∇⊥Φ×∇⊥Ψ) and we have taken into
account that, to lowest order,
+ u⊥ ·∇⊥ =
+{Φ, · · ·} , (19)
b̂ ·∇= ∂
·∇⊥ =
{Ψ, · · ·} . (20)
Here b̂ = B/B0 is the unit vector along the perturbed field line.
Equations (17-18) are known as the Reduced Magne-
tohydrodynamics (RMHD). The first derivations of these
equations (in the context of fusion plasmas) are due to
Kadomtsev & Pogutse (1974) and to Strauss (1976). These
were followed by many systematic derivations and gener-
alizations employing various versions and refinements of
the basic expansion, taking into account the non-Alfvénic
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 9
modes (which we will do in § 2.4), and including the ef-
fects of spatial gradients of equilibrium fields (e.g., Strauss
1977; Montgomery 1982; Hazeltine 1983; Zank & Matthaeus
1992; Kinney & McWilliams 1997; Bhattacharjee et al. 1998;
Kruger et al. 1998). A comparative review of these expansion
schemes and their (often close) relationship to ours is outside
the scope of this paper. One important point we wish to em-
phasize is that we do not assume the plasma beta [defined in
Eq. (14)] to be either large or small.
Equations (17) and (18) form a closed set, meaning that the
Alfvén-wave cascade decouples from the slow waves and den-
sity fluctuations. It is to the turbulence described by Eqs. (17-
18) that the GS theory outlined in § 1.2 applies.13 In § 5.3, we
will show that Eqs. (17) and (18) correctly describe inertial-
range Alfvénic fluctuations even in a collisionless plasma,
where the full MHD description [Eqs. (7-10)] is not valid.
2.3. Elsasser Fields
The MHD equations (7-10) in the incompressible limit
(ρ = const) acquire a symmetric form if written in terms of
the Elsasser fields z± = u± δB/
4πρ (Elsasser 1950). Let
us demonstrate how this symmetry manifests itself in the re-
duced equations derived above.
We introduce Elsasser potentials ζ± = Φ±Ψ, so that z±⊥ =
ẑ×∇⊥ζ±. For these potentials, Eqs. (17-18) become
∇2⊥ζ±∓ vA
∇2⊥ζ± = −
ζ+,∇2⊥ζ−
ζ−,∇2⊥ζ+
∓∇2⊥ {ζ+, ζ−}
. (21)
These equations show that the RMHD has a simple set of ex-
act solutions: if ζ− = 0 or ζ+ = 0, the nonlinear term vanishes
and the other, non-zero, Elsasser potential is simply a fluc-
tuation of arbitrary shape and magnitude propagating along
the mean field at the Alfvén speed vA: ζ
± = f±(x,y,z∓ vAt).
These solutions are finite-amplitude Alfvén-wave packets of
arbitrary shape. Only counterpropagating such solutions can
interact and thereby give rise to the Alfvén-wave cascade
(Kraichnan 1965). Note that these interactions are conserva-
tive in the sense that the “+” and “−” waves scatter off each
other without exchanging energy.
Note that the individual conservation of the “+” and “−”
waves’ energies means that the energy fluxes associated
with these waves need not be equal, so instead of a sin-
gle Kolmogorov flux ε assumed in the scaling arguments
13 The Alfvén-wave turbulence in the RMHD system has been stud-
ied by many authors. Some of the relevant numerical investigations are
due to Kinney & McWilliams (1998), Dmitruk et al. (2003), Oughton et al.
(2004), Rappazzo et al. (2007, 2008), Perez & Boldyrev (2008, 2009). An-
alytical theory has mostly been confined to the weak-turbulence paradigm
(Ng & Bhattacharjee 1996, 1997; Bhattacharjee & Ng 2001; Galtier et al.
2002; Lithwick & Goldreich 2003; Galtier & Chandran 2006; Nazarenko
2008). We note that adopting the critical balance [Eq. (3)] as an ordering
assumption for the expansion in k‖/k⊥ does not preclude one from subse-
quently attempting a weak-turbulence approach: the latter should simply be
treated as a subsidiary expansion. Indeed, implementing the anisotropy as-
sumption on the level of MHD equations rather than simultaneously with
the weak-turbulence closure (Galtier et al. 2000) significantly reduces the
amount of algebra. One should, however, bear in mind that the weak-
turbulence approximation always breaks down at some sufficiently small
scale—namely, when k⊥ ∼ (vA/U)
L, where L is the outer scale of the
turbulence, U velocity at the outer scale, and k‖ the parallel wavenum-
ber of the Alfvén waves (see Goldreich & Sridhar 1997 or the review by
Schekochihin & Cowley 2007). Below this scale, interactions cannot be as-
sumed weak.
reviewed in § 1.2, we could have ε+ 6= ε−. The GS the-
ory can be generalized to this case of imbalanced Alfvénic
cascades (Lithwick et al. 2007; Beresnyak & Lazarian 2008a;
Chandran 2008), but here we will focus on the balanced tur-
bulence, ε+ ∼ ε−. If one considers the turbulence forced
in a physical way (i.e., without forcing the magnetic field,
which would break the flux conservation), the resulting cas-
cade would always be balanced. In the real world, imbal-
anced Alfvénic fluxes are measured in the fast solar wind,
where the influence of initial conditions in the solar atmo-
sphere is more pronounced, while the slow-wind turbulence
is approximately balanced (Marsch & Tu 1990a; see also re-
views by Tu & Marsch 1995; Bruno & Carbone 2005 and ref-
erences therein).
2.4. Slow Waves and the Entropy Mode
In order to derive evolution equations for the remaining
MHD modes, let us first revisit the perpendicular part of the
momentum equation and use Eq. (12) to order terms in it. In
the lowest order, O(ǫ), we get the pressure balance
B0δB‖
= 0 ⇒ δp
. (22)
Using Eq. (22) and the entropy equation (9), we get
, (23)
where s0 = p0/ρ
0 . Now, substituting Eq. (15) for ∇·u in the
parallel component of the induction equation (10), we get
− b̂ ·∇u‖ = 0. (24)
Combining Eqs. (23) and (24), we obtain
1 + c2s/v
b̂ ·∇u‖, (25)
1 + v2A/c2s
b̂ ·∇u‖. (26)
Finally, we take the parallel component of the momentum
equation (8) and notice that, due to the pressure balance (22)
and to the smallness of the parallel gradients, the pressure
term is O(ǫ3), while the inertial and tension terms are O(ǫ2).
Therefore,
= v2Ab̂ ·∇
. (27)
Equations (26-27) describe the slow-wave-polarized fluctu-
ations, while Eq. (23) describes the zero-frequency entropy
mode, which is decoupled from the slow waves.14 The non-
linearity in Eqs. (26-27) enters via the derivatives defined in
14 For other expansion schemes leading to reduced sets of equations for
these “compressive” fluctuations see references in § 2.2. Note that the na-
ture of the density fluctuations described above is distinct from the so called
“pseudosound” density fluctuations that arise in the “nearly incompress-
ible” MHD theories (Montgomery et al. 1987; Matthaeus & Brown 1988;
Matthaeus et al. 1991; Zank & Matthaeus 1993). The “pseudosound” is es-
sentially the density response caused by the nonlinear pressure fluctuations
calculated from the incompressibility constraint. The resulting density fluc-
tuations are second order in Mach number and, therefore, order ǫ2 in our
expansion [see Eq. (13)]. The passive density fluctuations derived in this sec-
tion are order ǫ and, therefore, supersede the “pseudosound” (see review by
Tu & Marsch 1995 for a discussion of the relevant solar-wind evidence).
10 SCHEKOCHIHIN ET AL.
Eqs. (19-20) and is due solely to interactions with Alfvén
waves. Thus, both the slow-wave and the entropy-mode cas-
cades occur via passive scattering/mixing by Alfvén waves, in
the course of which there is no energy exchange between the
cascades.
Note that in the high-beta limit, cs ≫ vA [see Eq. (14)], the
entropy mode is dominated by density fluctuations [Eq. (23),
cs ≫ vA], which also decouple from the slow-wave cascade
[Eq. (25), cs ≫ vA]. and are passively mixed by the Alfvén-
wave turbulence:
= 0. (28)
The high-beta limit is equivalent to the incompressible ap-
proximation for the slow waves.
In § 5.5, we will derive a kinetic description for the inertial-
range compressive fluctuations (density and magnetic-field
strength), which is more generally valid in weakly collisional
plasmas and which reduces to Eqs. (26-27) in the collisional
limit (see Appendix D). While these fluctuations will in gen-
eral satisfy a kinetic equation, they will remain passive with
respect to the Alfvén waves.
2.5. Elsasser Fields for the Slow Waves
The original Elsasser (1950) symmetry was derived for in-
compressible MHD equations. However, for the “compres-
sive” slow-wave fluctuations, we may introduce generalized
Elsasser fields:
= u‖±
. (29)
Straightforwardly, the evolution equation for these fields is
∓ vA√
1 + v2A/c2s
1∓ 1√
1 + v2A/c2s
ζ+,z±
1± 1√
1 + v2A/c2s
ζ−,z±
. (30)
In the high-beta limit (vA ≪ cs), the generalized Elsasser
fields (29) become the parallel components of the conven-
tional incompressible Elsasser fields. We see that only in this
limit do the slow waves interact exclusively with the counter-
propagating Alfvén waves, and so only in this limit does set-
ting ζ− = 0 or ζ+ = 0 gives rise to finite-amplitude slow-wave-
packet solutions z±
= f±(x,y,z∓ vAt) analogous to the finite-
amplitude Alfvén-wave packets discussed in § 2.3.15 For gen-
eral β, the phase speed of the slow waves is smaller than that
of the Alfvén waves and, therefore, Alfvén waves can “catch
up” and interact with the slow waves that travel in the same
direction. All of these interactions are of scattering type and
involve no exchange of energy.
2.6. Scalings for Passive Fluctuations
15 Obviously, setting both ζ± = 0 does always enable these finite-
amplitude slow-wave solutions. More non-trivially, such finite-amplitude so-
lutions exist in the Lagrangian frame associated with the Alfvén waves—this
is discussed in detail in § 6.3.
The scaling of the passively mixed scalar fields introduced
above is slaved to the scaling of the Alfvénic fluctuations.
Consider for example the entropy mode [Eq. (23)]. As
in Kolmogorov–Obukhov theory (see § 1.1), one assumes a
local-in-scale-space cascade of scalar variance and a constant
flux εs of this variance. Then, analogously to Eq. (1),
v2thi
∼ εs. (31)
Since the cascade time is τ−1λ ∼ u⊥ ·∇⊥ ∼ vA/l‖λ ∼ ε/u2⊥λ,
)1/2 u⊥λ
, (32)
so the scalar fluctuations have the same scaling as the turbu-
lence that mixes them (Obukhov 1949; Corrsin 1951). In GS
turbulence, the scalar-variance spectrum should, therefore, be
⊥ (Lithwick & Goldreich 2001). The same argument ap-
plies to all passive fields.
It is the (presumably) passive electron-density spectrum
that provides the main evidence of the k−5/3 scaling in the in-
terstellar turbulence (Armstrong et al. 1981, 1995; Lazio et al.
2004, see further discussion in § 8.4.1). The explanation of
this spectrum in terms of passive mixing of the entropy mode,
originally proposed by Higdon (1984), was developed on the
basis of the GS theory by Lithwick & Goldreich (2001). The
turbulent cascade of the compressive fluctuations and the rel-
evant solar-wind data is discussed further in § 6.3. In partic-
ular, it will emerge that the anisotropy of these fluctuations
remains a non-trivial issue: is there an analog of the scaling
relation (5)? The scaling argument outlined above does not
invoke any assumptions about the relationship between the
parallel and perpendicular scales of the compressive fluctu-
ations (other than the assumption that they are anisotropic).
Lithwick & Goldreich (2001) argue that the parallel scales of
the Alfvénic fluctuations will imprint themselves on the pas-
sively advected compressive ones, so Eq. (5) holds for the
latter as well. In § 6.3, we examine this conclusion in view
of the solar-wind evidence and of the fact that the equations
for the compressive modes become linear in the Lagrangian
frame associated with the Alfvénic turbulence.
2.7. Five RMHD Cascades
Thus, the anisotropy and critical balance (3) taken as
ordering assumptions lead to a neat decomposition of the
MHD turbulent cascade into a decoupled Alfvén-wave cas-
cade and cascades of slow waves and entropy fluctuations pas-
sively scattered/mixed by the Alfvén waves. More precisely,
Eqs. (23), (21) and (30) imply that, for arbitrary β, there are
five conserved quantities:16
W±AW =
d3rρ0|∇⊥ζ±|2 (Alfven waves), (33)
W±sw =
d3rρ0|z±‖ |
2 (slow waves), (34)
(entropy fluctuations).(35)
16 Note that magnetic helicity of the perturbed field is not an invariant of
RMHD, except in two dimensions (see Appendix F.4). In 2D, there is also
conservation of the mean square flux,
d3r |Ψ|2 (see Appendix F.2).
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 11
W +AW and W
AW are always cascaded by interaction with each
other, Ws is passively mixed by W
AW and W
AW, W
sw are pas-
sively scattered by W∓AW and, unless β ≫ 1, also by W
This is an example of splitting of the overall energy cascade
into several channels (recovered as a particular case of the
more general kinetic cascade in Appendix D.2)—a concept
that will repeatedly arise in the kinetic treatment to follow.
The decoupling of the slow- and Alfvén-wave cascades in
MHD turbulence was studied in some detail and confirmed
in direct numerical simulations by Maron & Goldreich (2001,
for β ≫ 1) and by Cho & Lazarian (2002, 2003, for a range
of values of β). The derivation given in § 2.2 and § 2.4 (cf.
Lithwick & Goldreich 2001) provides a straightforward theo-
retical basis for these results, assuming anisotropy of the tur-
bulence (which was also confirmed in these numerical stud-
ies).
It turns out that the decoupling of the Alfvén-wave cascade
that we demonstrated above for the anisotropic MHD turbu-
lence is a uniformly valid property of plasma turbulence at
both collisional and collisionless scales and that this cascade
is correctly described by the RMHD equations (17-18) all the
way down to the ion gyroscale, while the fluctuations of den-
sity and magnetic-field strength do not satisfy simple fluid
evolution equations anymore and require solving the kinetic
equation. In order to prove this, we adopt a kinetic descrip-
tion and apply to it the same ordering (§ 2.1) as we used to
reduce the MHD equations. The kinetic theory that emerges
as a result is called gyrokinetics.
3. GYROKINETICS
The gyrokinetic formalism was first worked out
for linear waves by Rutherford & Frieman (1968)
and by Taylor & Hastie (1968) (see also Catto 1978;
Antonsen & Lane 1980; Catto et al. 1981) and subsequently
extended to the nonlinear regime by Frieman & Chen (1982).
Rigorous derivations of the gyrokinetic equation based on
the Hamiltonian formalism were developed by Dubin et al.
(1983, electrostatic) and Hahm et al. (1988, electromagnetic).
This approach is reviewed in Brizard & Hahm (2007). A
more pedestrian, but perhaps also more transparent exposition
of the gyrokinetics in a straight mean field can be found in
Howes et al. (2006), who also provide a detailed explanation
of the gyrokinetic ordering in the context of astrophysical
plasma turbulence and a treatment of the linear waves and
damping rates. Here we review only the main points so as
to allow the reader to understand the present paper without
referring elsewhere.
In general, a plasma is completely described by the distribu-
tion function fs(t,r,v)—the probability density for a particle
of species s (= i,e) to be found at the spatial position r mov-
ing with velocity v. This function obeys the kinetic Vlasov–
Landau (or Boltzmann) equation
+ v ·∇ fs +
· ∂ fs
, (36)
where qs and ms are the particle’s charge and mass, c is the
speed of light, and the right-hand side is the collision term
(quadratic in f ). The electric and magnetic fields are
E = −∇ϕ−
, B = ∇×A. (37)
The first equality is Faraday’s law uncurled, the second
the magnetic-field solenoidality condition; we shall use the
Coulomb gauge, ∇·A = 0. The fields satisfy the Poisson and
the Ampère–Maxwell equations with the charge and current
densities determined by fs(t,r,v):
∇·E = 4π
qsns = 4π
d3v fs, (38)
∇×B − 1
d3vv fs. (39)
3.1. Gyrokinetic Ordering and Dimensionless Parameters
As in § 2 we set up a static equilibrium with a uniform mean
field, B0 = B0ẑ, E0 = 0, assume that the perturbations will be
anisotropic with k‖ ≪ k⊥ (at scales smaller than the outer
scale, k‖L ≫ 1; see § 1.3 and § 1.5.1), and construct an expan-
sion of the kinetic theory around this equilibrium with respect
to the small parameter ǫ ∼ k‖/k⊥. We adopt the ordering ex-
pressed by Eqs. (3) and (12), i.e., we assume the perturbations
to be strongly interacting Alfvén waves plus electron density
and magnetic-field-strength fluctuations.
Besides ǫ, several other dimensionless parameters are
present, all of which are formally considered to be of order
unity in the gyrokinetic expansion: the electron–ion mass ra-
tio me/mi, the charge ratio
Z = qi/|qe| = qi/e (40)
(for hydrogen, this is 1, which applies to most astrophysical
plasmas of interest to us), the temperature ratio17
τ = Ti/Te, (41)
and the plasma (ion) beta
v2thi
8πniTi
, (42)
where vthi = (2Ti/mi)
1/2 is the ion thermal speed and the total
β was defined in Eq. (14) based on the total pressure p = niTi +
neTe. We shall occasionally also use the electron beta
8πneTe
βi. (43)
The total beta is β = βi +βe.
3.1.1. Wavenumbers and Frequencies
As we want our theory to be uniformly valid at all (perpen-
dicular) scales above, at or below the ion gyroscale, we order
k⊥ρi ∼ 1, (44)
where ρi = vthi/Ωi is the ion gyroradius, Ωi = qiB0/cmi the ion
cyclotron frequency. Note that
ρi. (45)
17 It can be shown that equilibrium temperatures change on the timescale
∼ (ǫ2ω)−1 (Howes et al. 2006). On the other hand, from standard theory
of collisional transport (e.g., Helander & Sigmar 2002), the ion and elec-
tron temperatures equalize on the timescale ∼ ν−1ie ∼ (mi/me)
1/2ν−1ii [see
Eq. (51)]. Therefore, τ can depart from unity by an amount of order
ǫ2(ω/νii)(mi/me)1/2. In our ordering scheme [Eq. (49)], this is O(ǫ2) and,
therefore, we should simply set τ = 1 + O(ǫ2). However, we shall carry the
parameter τ because other ordering schemes are possible that permit arbitrary
values of τ . These are appropriate to plasmas with very weak collisions. For
example, in the solar wind, τ appears to be order unity but not exactly 1
(Newbury et al. 1998), while in accretion flows near the black hole, some
models predict τ ≫ 1 (see § 8.5).
12 SCHEKOCHIHIN ET AL.
FIG. 3.— Regions of validity in the wavenumber space of two primary approximations—the two-fluid (Appendix A.1) and gyrokinetic (§ 3). The gyrokinetic
theory holds when k‖ ≪ k⊥ and ω ≪ Ωi [when k‖ ≪ k⊥ < ρ
i , the second requirement is automatically satisfied for Alfvén, slow and entropy modes; see
Eq. (46)]. The two-fluid equations hold when k‖λmfpi ≪ 1 (collisional limit) and k⊥ρi ≪ 1 (magnetized plasma). Note that the gyrokinetic theory holds for all
but the very largest (outer) scales, where anisotropy cannot be assumed.
Assuming Alfvénic frequencies implies
∼ k⊥ρi√
ǫ. (46)
Thus, gyrokinetics is a low-frequency limit that averages over
the timescales associated with the particle gyration. Because
we have assumed that the fluctuations are anisotropic and have
(by order of magnitude) Alfvénic frequencies, we see from
Eq. (46) that their frequency remains far below Ωi at all scales,
including the ion and even electron gyroscale—the gyroki-
netics remains valid at all of these scales and the cyclotron-
frequency effects are negligible (cf. Quataert & Gruzinov
1999).
3.1.2. Fluctuations
Equation (3) allows us to order the fluctuations of the scalar
potential: on the one hand, we have from Eq. (3) u⊥ ∼ ǫvA; on
the other hand, the plasma mass flow velocity is (to the lowest
order) the E×B drift velocity of the ions, u⊥ ∼ cE⊥/B0 ∼
ck⊥ϕ/B0, so
ǫ. (47)
All other fluctuations (magnetic, density, parallel velocity) are
ordered according to Eq. (12).
Note that the ordering of the flow velocity dictated by
Eq. (3) means that we are considering the limit of small Mach
numbers:
M ∼ u
. (48)
This means that the gyrokinetic description in the form used
below does not extend to large sonic flows that can be
present in many astrophysical systems. It is, in principle,
possible to extend the gyrokinetics to systems with sonic
flows (e.g., in the toroidal geometry; see Artun & Tang 1994;
Sugama & Horton 1997). However, we do not follow this
route because such flows belong to the same class of non-
universal outer-scale features as background density and tem-
perature gradients, system-specific geometry etc.—these can
all be ignored at small scales, where the turbulence should be
approximately homogeneous and subsonic (as long as k‖L ≫
1, see discussion in § 1.5.1).
3.1.3. Collisions
Finally, we want our theory to be valid both in the colli-
sional and the collisionless regimes, so we do not assume
ω to be either smaller or larger than the (ion) collision fre-
quency νii:
k‖λmfpi√
∼ 1, (49)
where λmfpi = vthi/νii is the ion mean free path (this order-
ing can actually be inferred from equating the gyrokinetic en-
tropy production terms to the collisional entropy production;
see extended discussion in Howes et al. 2006). Note that the
ordering (49) holds on the understanding that we have ordered
k⊥ρi ∼ 1 [Eq. (44)] because the fluctuation frequency can de-
pend on k⊥ρi in the dissipation range (see § 7.3).
Other collision rates are related to νii via a set of standard
formulae (see, e.g., Helander & Sigmar 2002), which will be
useful in what follows:
νei = Zνee =
τ 3/2
νii, (50)
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 13
νie =
τ 3/2
νii, (51)
νii =
2πZ4e4ni lnΛ
, (52)
where lnΛ is the Coulomb logarithm and the numerical factor
in the definition of νie has been inserted for future notational
convenience (see Appendix A). We always define
λmfpi =
, λmfpe =
λmfpi. (53)
The ordering of the collision frequency expressed by
Eq. (49) means that collisions, while not dominant as in
the fluid description (Appendix A), are still retained in
the version of the gyrokinetic theory adopted by us. Their
presence is required in order for us to be able to assume
that the equilibrium distribution is Maxwellian [Eq. (54)
below] and for the heating and entropy production to be
treated correctly (§ 3.4 and § 3.5). However, our ordering of
collisions and of the fluctuation amplitudes (§ 3.1.2) imposes
certain limitations: thus, we cannot treat the class of nonlinear
phenomena involving particle trapping by parallel-varying
fluctuations, non-Maxwellian tails of particle distributions,
plasma instabilities arising from the equilibrium pressure
anisotropies (mirror, firehose) and their possible nonlinear
evolution to large amplitudes (see discussion in § 8.3).
The region of validity of the gyrokinetic approximation in
the wavenumber space is illustrated in Fig. 3—it embraces
all of the scales that are expected to be traversed by the
anisotropic energy cascade (except the scales close to the
outer scale).
As we explained above, me/mi, βi, k⊥ρi and k‖λmfpi (or
ω/νii) are assigned order unity in the gyrokinetic expansion.
Subsidiary expansions in small me/mi (§ 4) and in small or
large values of the other three parameters (§§ 5-7) can be car-
ried out at a later stage as long as their values are not so large
or small as to interfere with the primary expansion in ǫ. These
expansions will yield simpler models of turbulence with more
restricted domains of validity than gyrokinetics.
3.2. Gyrokinetic Equation
Given the gyrokinetic ordering introduced above, the ex-
pansion of the distribution function up to first order in ǫ can
be written as
fs(t,r,v) = F0s(v) −
qsϕ(t,r)
F0s(v) + hs(t,Rs,v⊥,v‖). (54)
To zeroth order, it is a Maxwellian:18
F0s(v) =
(πv2ths)
v2ths
, vths =
, (55)
with uniform density n0s and temperature T0s and no mean
flow. As will be explained in more detail in § 3.5, F0s has a
slow time dependence via the equilibrium temperature, T0s =
T0s(ǫ
2t). This reflects the slow heating of the plasma as the tur-
bulent energy is dissipated. However, T0s can be treated as a
constant with respect to the time dependence of the first-order
18 The use of isotropic equilibrium is a significant idealization—this is
discussed in more detail in § 8.3.
distribution function (the timescale of the turbulent fluctua-
tions). The first-order part of the distribution function is com-
posed of the Boltzmann response [second term in Eq. (54), or-
dered in Eq. (47)] and the gyrocenter distribution function hs.
The spatial dependence of the latter is expressed not by the
particle position r but by the position Rs of the particle gy-
rocenter (or guiding center)—the center of the ring orbit that
the particle follows in a strong guide field:
Rs = r +
v⊥× ẑ
. (56)
Thus, some of the velocity dependence of the distribution
function is subsumed in the Rs dependence of hs. Explicitly,
hs depends only on two velocity-space variables: it is cus-
tomary in the gyrokinetic literature for these to be chosen as
the particle energy εs = msv
2/2 and its first adiabatic invari-
ant µs = msv
⊥/2B0 (both conserved quantities to two lowest
orders in the gyrokinetic expansion). However, in a straight
uniform guide field B0ẑ, the pair (v⊥,v‖) is a simpler choice,
which will mostly be used in what follows (we shall some-
times find an alternative pair, v and ξ = v‖/v, useful, especially
where collisions are concerned). It must be constantly kept in
mind that derivatives of hs with respect to the velocity-space
variables are taken at constant Rs, not at constant r.
The function hs satisfies the gyrokinetic equation:
{〈χ〉Rs ,hs} =
qsF0s
∂〈χ〉Rs
where
χ(t,r,v) = ϕ−
v⊥ ·A⊥
, (58)
the Poisson brackets are defined in the usual way:
{〈χ〉Rs ,hs} = ẑ ·
∂〈χ〉Rs
× ∂hs
, (59)
and the ring average notation is introduced:
〈χ(t,r,v)〉Rs =
t,Rs −
v⊥× ẑ
, (60)
where ϑ is the angle in the velocity space taken in the plane
perpendicular to the guide field B0ẑ. Note that, while χ is
a function of r, its ring average is a function of Rs. Note
also that the ring averages depend on the species index, as
does the gyrocenter variable Rs. Equation (57) is derived by
transforming the first-order kinetic equation to the gyrocenter
variable (56) and ring averaging the result (see Howes et al.
2006, or the references given at the beginning of § 3). The
ring-averaged collision integral (∂hs/∂t)c is discussed in Ap-
pendix B.
3.3. Field Equations
To Eq. (57), we must append the equations that determine
the electromagnetic field, namely, the potentials ϕ(t,r) and
A(t,r) that enter the expression for χ [Eq. (58)]. In the
non-relativistic limit (vthi ≪ c), these are the plasma quasi-
neutrality constraint [which follows from the Poisson equa-
tion (38) to lowest order in vthi/c]:
qsδns =
n0s +
d3v〈hs〉r
14 SCHEKOCHIHIN ET AL.
and the parallel and perpendicular parts of Ampère’s law
[Eq. (39) to lowest order in ǫ and in vthi/c]:
∇2⊥A‖ = −
j‖ = −
d3vv‖〈hs〉r, (62)
∇2⊥δB‖ = −
∇⊥× j⊥
d3v〈v⊥hs〉r
, (63)
where we have used δB‖ = ẑ · (∇⊥×A⊥) and dropped the dis-
placement current. Since field variables ϕ, A‖ and δB‖ are
functions of the spatial variable r, not of the gyrocenter vari-
able Rs, we had to determine the contribution from the gy-
rocenter distribution function hs to the charge distribution at
fixed r by performing a gyroaveraging operation dual to the
ring average defined in Eq. (60):
〈hs(t,Rs,v⊥,v‖)〉r =
t,r +
v⊥× ẑ
,v⊥,v‖
In other words, the velocity-space integrals in Eqs. (61-63)
are performed over hs at constant r, rather than constant Rs.
If we Fourier transform hs in Rs, the gyroaveraging operation
takes a simple mathematical form:
〈hs〉r =
〈eik·Rs〉rhsk(t,v⊥,v‖)
eik·r
ik · v⊥× ẑ
hsk(t,v⊥,v‖)
eik·rJ0(as)hsk(t,v⊥,v‖), (65)
where as = k⊥v⊥/Ωs and J0 is a Bessel function that arose
from the angle integral in the velocity space. In Eq. (63), an
analogous calculation taking into account the angular depen-
dence of v⊥ leads to
δB‖ = −
eik·r
d3vmsv
J1(as)
hsk(t,v⊥,v‖).
Note that Eq. (63) [and, therefore, Eq. (66)] is the gyroki-
netic equivalent of the perpendicular pressure balance that ap-
peared in § 2 [Eq. (22)]:
B0δB‖
= ∇⊥ ·
d3v〈ẑ× v⊥hs〉r
= ∇⊥ ·
t,r +
v⊥× ẑ
,v⊥,v‖
= −∇⊥∇⊥ :
d3vms〈v⊥v⊥ hs〉r = −∇⊥∇⊥ : δP⊥,(67)
where we have integrated by parts with respect to the gyroan-
gle ϑ and used ∂v⊥/∂ϑ = ẑ× v⊥, ∂2v⊥/∂ϑ2 = −v⊥ (cf. the
Appendix of Roach et al. 2005).
Once the fields are determined, they have to be substi-
tuted into χ [Eq. (58)] and the result ring averaged [Eq. (60)].
Again, we emphasize that ϕ, A‖ and δB‖ are functions of r,
while 〈χ〉Rs is a function of Rs. The transformation is ac-
complished via a calculation analogous to the one that led to
Eqs. (65) and (66):
〈χ〉Rs =
eik·Rs〈χ〉Rs ,k, (68)
〈χ〉Rs ,k = J0(as)
v‖A‖k
v2ths
J1(as)
. (69)
The last equation establishes a correspondence between the
Fourier transforms of the fields with respect to r and the
Fourier transform of 〈χ〉Rs with respect to Rs.
3.4. Generalized Energy and the Kinetic Cascade
As promised in § 1.4, the central unifying concept of this
paper is now introduced.
If we multiply the gyrokinetic equation (57) by T0shs/F0s
and integrate over the velocities and gyrocenters, we find that
the nonlinear term conserves the variance of hs and
d3Rs qs
∂〈χ〉Rs
T0shs
. (70)
Let us now sum this equation over all species. The first term
on the right-hand side is
∂〈χ〉Rs
d3v〈hs〉r −
d3v〈vhs〉r
d3rE · j, (71)
where we have used Eq. (61) and Ampère’s law [Eqs. (62-
63)] to express the integrals of hs. The second term on the
right-hand side is the total work done on plasma per unit time.
Using Faraday’s law [Eq. (37)] and Ampère’s law [Eq. (39)],
it can be written as
d3rE · j = − d
|δB|2
+ Pext, (72)
where Pext ≡ −
d3rE · jext is the total power injected into the
system by the external energy sources (outer-scale stirring; in
terms of the Kolmogorov energy flux ε used in the scaling
arguments in § 1.2, Pext = Vmin0iε, where V is the system vol-
ume). Combining Eqs. (70-72), we find (Howes et al. 2006)
T0s〈h2s 〉r
|δB|2
= Pext +
T0shs
. (73)
W is a positive definite quantity—this becomes explicit if we
use Eq. (61) to express it in terms of the total perturbed distri-
bution function δ fs = −qsϕF0s/T0s + hs [see Eq. (54)]:
T0sδ f
|δB|2
. (74)
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 15
We will refer to W as the generalized energy. We use this
term to emphasize the role of W as the cascaded quantity in
gyrokinetic turbulence (see below). This quantity is, in fact,
the gyrokinetic version of a collisionless kinetic invariant var-
iously referred to as the generalized grand canonical poten-
tial (see Hallatschek 2004, who points out the fundamental
role of this quantity in plasma turbulence simulations) or free
energy (e.g., Fowler 1968; Scott 2007). The non-magnetic
part of W is related to the perturbed entropy of the sys-
tem (Krommes & Hu 1994; Sugama et al. 1996; Howes et al.
2006; Schekochihin et al. 2008b, see discussion in § 3.5).19
Equation (73) is a conservation law of the generalized en-
ergy: Pext is the source and the second term on the right-hand
side, which is negative definite, represents collisional dissi-
pation. This suggests that we might think of kinetic plasma
turbulence in terms of the generalized energy W injected by
the outer-scale stirring and dissipated by collisions. In or-
der for the dissipation to be important, the collisional term in
Eq. (73) has to become comparable to Pext. This can happen
in two ways:
1. At collisional scales (k‖λmfpi ∼ 1) due to deviations of
the perturbed distribution function from a local per-
turbed Maxwellian (see § 6.1 and Appendix D);
2. At collisionless scales (k‖λmfpi ≫ 1) due the develop-
ment of small scales in the velocity space—large gra-
dients in v‖ (see § 6.2.4) or v⊥ (which is accompanied
by the development of small perpendicular scales in the
position space; see § 7.9.1).
Thus, the dissipation is only important at particular (small)
scales, which are generally well separated from the outer
scale. The generalized energy is transferred from the outer
scale to the dissipation scales via a nonlinear cascade. We
shall call it the kinetic cascade. It is analogous to the energy
cascade in fluid or MHD turbulence, but a conceptually new
feature is present: the small scales at which dissipation hap-
pens are small scales both in the velocity and position space.
Whereas the large gradients in v‖ are produced by the lin-
ear parallel phase mixing, whose role in the kinetic dissipa-
tion processes has been appreciated for some time (Landau
1946; Hammett et al. 1991; Krommes & Hu 1994; Krommes
1999; Watanabe & Sugama 2004, see § 6.2.4), the emergence
of large gradients in v⊥ is due to an essentially nonlinear
phase mixing mechanism (§ 7.9.1). At spatial scales smaller
than the ion gyroradius, this nonlinear perpendicular phase
mixing turns out to be a faster and, therefore, presumably the
dominant way of generating small-scale structure in the veloc-
ity space. It was anticipated in the development of gyrofluid
moment hierarchies by Dorland & Hammett (1993). Here we
treat it for the first time as a phase-space turbulent cascade:
this is done in § 7.9 and § 7.10 (see also Schekochihin et al.
2008b).
In the sections that follow, we shall derive particular forms
of W for various limiting cases of the gyrokinetic theory
(§ 4.7, § 5.6, § 6.2.5, § 7.8, Appendices D.2 and E.2). We
shall see that the kinetic cascade of W is, indeed, a direct
generalization of the more familiar fluid cascades (such as
19 Note also that a quadratic form involving both the perturbed distribution
function and the electromagnetic field appears, in a more general form than
Eq. (74), in the formulation of the energy principle for the Kinetic MHD
approximation (Kruskal & Oberman 1958; Kulsrud 1962, 1964). Regarding
the relationship between Kinetic MHD and gyrokinetics, see footnote 23.
the RMHD cascades discussed in § 2) and that W contains
the energy invariants of the fluid models in the appropriate
limits. In these limits, the cascade of the generalized en-
ergy will split into several decoupled cascades, as it did in
the case of RMHD (§ 2.7). Whenever one of the physically
important scales (§ 1.5.2) is crossed and a change of physical
regime occurs, these cascades are mixed back together into
the overall kinetic cascade of W , which can then be split in
a different way as it emerges on the “opposite side” of the
transition region in the scale space. The conversion of the
Alfvénic cascade into the KAW cascade and the entropy cas-
cade at k⊥ρi ∼ 1 is the most interesting example of such a
transition, discussed in § 7.
The generalized energy appears to be the only quadratic
invariant of gyrokinetics in three dimensions; in two dimen-
sions, many other invariants appear (see Appendix F).
3.5. Heating and Entropy
In a stationary state, all of the the turbulent power injected
by the external stirring is dissipated and thus transferred into
heat. Mathematically, this is expressed as a slow increase in
the temperature of the Maxwellian equilibrium. In gyrokinet-
ics, the heating timescale is ordered as ∼ (ǫ2ω)−1.
Even though the dissipation of turbulent fluctuations may
be occurring “collisionlessly” at scales such that k‖λmfpi ≫ 1
(e.g., via wave–particle interaction at the ion gyroscale; § 7.1),
the resulting heating must ultimately be effected with the help
of collisions. This is because heating is an irreversible process
and it is a small amount of collisions that make “collisionless”
damping irreversible. In other words, slow heating of the
Maxwellian equilibrium is equivalent to entropy production
and Boltzmann’s H-theorem rigorously requires collisions to
make this possible. Indeed, the total entropy of species s is
Ss = −
d3v fs ln fs
F0s lnF0s +
δ f 2s
+ O(ǫ3), (75)
where we took
d3rδ fs = 0. It is then not hard to show that
T0shs
where the overlines mean averaging over times longer than
the characteristic time of the turbulent fluctuations ∼ ω−1
but shorter than the typical heating time ∼ (ǫ2ω)−1 (see
Howes et al. 2006; Schekochihin et al. 2008b for a detailed
derivation of this and related results on heating in gyroki-
netics; see also earlier discussions of the entropy production
in gyrokinetics by Krommes & Hu 1994; Krommes 1999;
Sugama et al. 1996). We have omitted the term describing the
interspecies collisional temperature equalization. Note that
both sides of Eq. (76) are order ǫ2ω.
If we now time average Eq. (73) in a similar fashion, the
left-hand side vanishes because it is a time derivative of a
quantity fluctuating on the timescale ∼ ω−1 and we confirm
that the right-hand side of Eq. (76) is simply equal to the av-
erage power Pext injected by external stirring. The import of
Eq. (76) is that it tells us that heating can only be effected
by collisions, while Eq. (73) implies that the injected power
gets to the collisional scales in velocity and position space by
means of a kinetic cascade of generalized energy.
16 SCHEKOCHIHIN ET AL.
The first term in the expression for the generalized energy
(74) is −
s T0sδSs, where δSs is the perturbed entropy [see
Eq. (75)]. The second term in Eq. (74) is magnetic energy.
Collisionless damping of electromagnetic fluctuations can be
thought of as a redistribution of the generalized energy, trans-
ferring the electromagnetic energy into entropy fluctuations,
while the total W is conserved (a simple example of how that
happens for collisionless compressive fluctuations in the iner-
tial range is worked out in § 6.2.3).
The contribution to the perturbed entropy from the gy-
rocenter distribution is the integral of −h2s/2F0s, whose
evolution equation (70) can be viewed as the gyrokinetic
version of the H-theorem. The first term on the right-hand
side of this equation represents the wave–particle interaction
(collisionless damping). Under time average, it is related to
the work done on plasma [Eq. (71)] and hence to the average
externally injected power Pext via time-averaged Eq. (72).
In a stationary state, this is balanced by the second term in the
right-hand side of Eq. (70), which is the collisional-heating,
or entropy-production, term that also appears in Eq. (76).
Thus, the generalized energy channeled by collisionless
damping into entropy fluctuations is eventually converted
into heat by collisions. The sub-gyroscale entropy cascade,
which brings the perturbed distribution function hs to col-
lisional scales, will be discussed further in § 7.9 and § 7.10
(see also Schekochihin et al. 2008b).
This concludes a short primer on gyrokinetics necessary
(and sufficient) for adequate understanding of what is to fol-
low. Formally, all further analytical derivations in this paper
are simply subsidiary expansions of the gyrokinetics in the pa-
rameters we listed in § 3.1: in § 4, we expand in (me/mi)
in § 5 in k⊥ρi (followed by further subsidiary expansions in
large and small k‖λmfpi in § 6), and in § 7 in 1/k⊥ρi.
4. ISOTHERMAL ELECTRON FLUID
In this section, we carry out an expansion of the electron gy-
rokinetic equation in powers of (me/mi)
1/2 ≃ 0.02 (for hydro-
gen plasma). In virtually all cases of interest, this expansion
can be done while still considering
βi, k⊥ρi, and k‖λmfpi to
be order unity.21 Note that the assumption k⊥ρi ∼ 1 together
with Eq. (45) mean that
k⊥ρe ∼ k⊥ρi(me/mi)1/2 ≪ 1, (77)
i.e., the expansion in (me/mi)
1/2 means also that we are
considering scales larger than the electron gyroradius. The
idea of such an expansion of the electron kinetic equation
has been utilized many times in plasma physics literature.
The mass-ratio expansion of the gyrokinetic equation in a
form very similar to what is presented below is found in
Snyder & Hammett (2001).
20 Note that Eq. (72) is valid not only in the integral form but also indi-
vidually for each wavenumber: indeed, using the Fourier-transformed Fara-
day and Ampère’s laws, we have Ek · j
k + E
k · jk = Ek · j
ext,k + E
k · jext,k −
(1/4π)∂|δBk|2/∂t. In a stationary state, time averaging eliminates the time
derivative of the magnetic-fluctuation energy, so Ek · j∗k + E
k · jk = 0 at all k
except those corresponding to the outer scale, where the external energy in-
jection occurs. This means that below the outer scale, the work done on one
species balances the work done on the other. The wave–particle interaction
term in the gyrokinetic equation is responsible for this energy exchange.
21 One notable exception is the LAPD device at UCLA, where β ∼ 10−4 −
10−3 (due mostly to the electron pressure because the ions are cold, τ ∼
0.1, so βi ∼ βe/10; see, e.g., Morales et al. 1999; Carter et al. 2006). This
interferes with the mass-ratio expansion.
The primary import of this section will be technical: we
shall dispense with the electron gyrokinetic equation and thus
prepare the necessary ground for further approximations. The
main results are summarized in § 4.9. A reader who is only
interested in following qualitatively the major steps in the
derivation may skip to this summary.
4.1. Ordering the Terms in the Kinetic Equation
In view of Eq. (77), ae ≪ 1, so we can expand the Bessel
functions arising from averaging over the electron ring mo-
tion:
J0(ae) = 1 −
a2e + · · · ,
J1(ae)
a2e + · · ·
. (78)
Keeping only the lowest-order terms of the above expansions
in Eq. (69) for 〈χ〉Re , then substituting this 〈χ〉Re and qe =
−e in the electron gyrokinetic equation, we get the following
kinetic equation for the electrons, accurate up to and including
the first order in (me/mi)
1/2 (or in k⊥ρe):
︸ ︷︷ ︸
︸ ︷︷ ︸
v2the
︸ ︷︷ ︸
︸ ︷︷ ︸
v2the
︸ ︷︷ ︸
︸ ︷︷ ︸
. (79)
Note that ϕ, A‖, δB‖ in Eq. (79) are taken at r = Re. We
have indicated the lowest order to which each of the terms
enters if compared with v‖∂he/∂z. In order to obtain these
estimates, we have assumed that the physical ordering intro-
duced in § 3.1 holds with respect to the subsidiary expansion
in (me/mi)
1/2 as well as for the primary gyrokinetic expansion
in ǫ, so we can use Eqs. (3) and (12) to order terms with re-
spect to (me/mi)
1/2. We have also made use of Eqs. (45), (47),
and of the following three relations:
∼ vthe
, (80)
(v‖/c)A‖
∼ vtheδB⊥
, (81)
v2the
βi. (82)
The collision term is estimated to be zeroth order because [see
Eqs. (49), (50)]
k‖λmfpi
. (83)
The consequences of other possible orderings of the collision
terms are discussed in § 4.8. We remind the reader that all
dimensionless parameters except k‖/k⊥ ∼ ǫ and (me/mi)1/2
are held to be order unity.
We now let he = h
e + h
e + . . . and carry out the expansion
to two lowest orders in (me/mi)
4.2. Zeroth Order
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 17
To zeroth order, the electron kinetic equation is
v‖b̂ ·∇h(0)e = v‖
∂h(0)e
, (84)
where we have assembled the terms in the left-hand side to
take the form of the derivative of the distribution function
along the perturbed magnetic field:
b̂ ·∇ = ∂
·∇ = ∂
A‖, · · ·
. (85)
We now multiply Eq. (84) by h(0)e /F0e and integrate over v and
r (since we are only retaining lowest-order terms, the distinc-
tion between r and Re does not matter here). Since ∇·B = 0,
the left-hand side vanishes (assuming that all perturbations are
either periodic or vanish at the boundaries) and we get
h(0)e
∂h(0)e
‖e = 0.
The right-hand side of this equation is zero because the
electron flow velocity is zero in the zeroth order, u(0)
(1/n0e)
d3vv‖h
e = 0. This is a consequence of the paral-
lel Ampére’s law [Eq. (62)], which can be written as follows
u‖e =
4πen0e
∇2⊥A‖ + u‖i, (87)
where
u‖i =
eik·r
d3vv‖J0(ai)hik. (88)
The three terms in Eq. (87) can be estimated as follows
∼ ǫvthe
ǫ, (89)
∼ ǫ, (90)
c∇2⊥A‖
4πen0evA
∼ k⊥ρi
ǫ, (91)
where we have used the fundamental ordering (12) of the slow
waves (u‖i ∼ ǫvA) and Alfvén waves (δB⊥ ∼ ǫB0). Thus, the
two terms in the right-hand side of Eq. (87) are one order of
(me/mi)
1/2 smaller than u(0)
‖e , which means that to zeroth order,
the parallel Ampère’s law is u(0)
‖e = 0.
The collision operator in Eq. (86) contains electron–
electron and electron–ion collisions. To lowest order in
(me/mi)
1/2, the electron–ion collision operator is simply the
pitch-angle scattering operator [see Eq. (B20) in Appendix B
and recall that u‖i is first order]. Therefore, we may then
rewrite Eq. (86) as follows
h(0)e
Cee[h
νeiD (v)
1 − ξ2
∂h(0)e
= 0. (92)
Both terms in this expression are negative definite and must,
therefore, vanish individually. This implies that h(0)e must be
a perturbed Maxwellian distribution with zero mean veloc-
ity (this follows from the proof of Boltzmann’s H theorem;
see, e.g., Longmire 1963), i.e., the full electron distribution
function to zeroth order in the mass-ratio expansion is [see
Eq. (54)]:
fe = F0e +
+ h(0)e =
2πTe/me
, (93)
where ne = n0e + δne, Te = T0e + δTe. Expanding around the
unperturbed Maxwellian F0e, we get
h(0)e =
v2the
F0e, (94)
where the fields are taken at r = Re. Now substitute this so-
lution back into Eq. (84). The collision term vanishes and the
remaining equation must be satisfied at all values of v. This
gives
+ b̂ ·∇ϕ= b̂ ·∇T0e
, (95)
b̂ ·∇δTe
= 0. (96)
The collision term is neglected in Eq. (95) because, for h(0)e
given by Eq. (94), it vanishes to zeroth order.
4.3. Flux Conservation
Equation (95) implies that the magnetic flux is conserved
and magnetic-field lines cannot be broken to lowest order in
the mass-ratio expansion. Indeed, we may follow Cowley
(1985) and argue that the left-hand side of Eq. (95) is minus
the projection of the electric field on the total magnetic field
[see Eq. (37)], so we have
E · b̂ = −b̂ ·∇
; (97)
hence the total electric field is
Î − b̂b̂
and Faraday’s law becomes
= −c∇×E = ∇× (ueff ×B) , (99)
ueff =
E +∇T0e
×B, (100)
i.e., the magnetic field lines are frozen into the velocity field
ueff. In Appendix C.1, we show that this effective velocity is
the part of the electron flow velocity ue perpendicular to the
total magnetic field B [see Eq. (C6)].
The flux conservation is broken in the higher orders of the
mass-ratio expansion. In the first order, Ohmic resistivity for-
mally enters in Eq. (95) (unless collisions are even weaker
than assumed so far; if they are downgraded one order as is
done in § 4.8.3, resistivity enters in the second order). In the
second order, the electron inertia and the finiteness of the elec-
tron gyroradius also lead to unfreezing of the flux. This can be
seen formally by keeping second-order terms in Eq. (79), mul-
tiplying it by v‖ and integrating over velocities. The relative
importance of these flux unfreezing mechanisms is evaluated
in § 7.7.
18 SCHEKOCHIHIN ET AL.
4.4. Isothermal Electrons
Equation (96) mandates that the perturbed electron temper-
ature must remain constant along the perturbed field lines.
Strictly speaking, this does not preclude δTe varying across
the field lines. However, we shall now assume δTe = const (has
no spatial variation), which is justified, e.g., if the field lines
are stochastic. Assuming that no spatially uniform perturba-
tions exist, we may set δTe = 0. Equation (94) then reduces
h(0)e =
F0e(v), (101)
or, using Eq. (54),
δ fe =
F0e(v). (102)
Hence follows the equation of state for isothermal electrons:
δpe = T0eδne. (103)
4.5. First Order
We now integrate Eq. (79) over the velocity space and retain
the lowest (first) order terms only. Using Eq. (101), we get
A‖,u‖e
= 0, (104)
where the parallel electron velocity is first order:
u‖e = u
d3vv‖h
e . (105)
The velocity-space integral of the collision term does not enter
because it is subdominant by at least one factor of (me/mi)
indeed, as shown in Appendix B.1, the velocity integration
leads to an extra factor of k2⊥ρ
e , so that
∼ νeik2⊥ρ2e
i νii
, (106)
where we have used Eqs. (45) and (50). The collision term
is subdominant because of the ordering of the ion collision
frequency given by Eq. (49).
4.6. Field Equations
Using Eq. (101) and qi = Ze, n0e = Zn0i, T0e = T0i/τ ,
we derive from the quasi-neutrality equation (61) [see also
Eq. (65)]
eik·r
d3vJ0(ai)hik, (107)
and, from the perpendicular part of Ampère’s law [Eq. (66),
using also Eq. (107)],
eik·r
J0(ai) +
v2thi
J1(ai)
. (108)
The parallel electron velocity, u‖e, is determined from the par-
allel part of Ampère’s law, Eq. (87).
The ion distribution function hi that enters these equations
has to be determined by solving the ion gyrokinetic equation:
Eq. (57) with s = i.
4.7. Generalized Energy
The generalized energy (§ 3.4) for the case of isothermal
electrons is calculated by substituting Eq. (102) into Eq. (74):
T0iδ f
n0eT0e
|δB|2
, (109)
where δ fi = hi −
Zeϕ/T0i
F0i [see Eq. (54)].
4.8. Validity of the Mass-Ratio Expansion
Let us examine the range of spatial scales in which the
equations derived above are valid. In carrying out the ex-
pansion in (me/mi)
1/2, we ordered k⊥ρi ∼ 1 [Eq. (77)] and
k‖λmfpi ∼ 1 [Eq. (83)]. Formally, this means that the perpen-
dicular and parallel wavelengths of the perturbations must not
be so small or so large as to interfere with the mass ratio ex-
pansion. We now discuss the four conditions that this require-
ment leads to and whether any of them can be violated without
destroying the validity of the equations derived above.
4.8.1. k⊥ρi ≪ (mi/me)
This is equivalent to demanding that k⊥ρe ≪ 1, a condition
that was, indeed, essential for the expansion to hold [Eq. (78)].
This is not a serious limitation because electrons can be con-
sidered well magnetized at virtually all scales of interest for
astrophysical applications. However, we do forfeit the de-
tailed information about some important electron physics at
k⊥ρe ∼ 1: for example such effects as wave damping at the
electron gyroscale and the electron heating (although the total
amount of the electron heating can be deduced by subtracting
the ion heating from the total energy input). The breaking of
the flux conservation (resistivity) is also an effect that requires
incorporation of the finite electron gyroscale physics.
4.8.2. k⊥ρi ≫ (me/mi)
If this condition is broken, the small-k⊥ρi expansion, car-
ried out in § 5, must, formally speaking, precede the mass-
ratio expansion. However, it turns out that the small-
k⊥ρi expansion commutes with the mass-ratio expansion
(Schekochihin et al. 2007, see also footnote 23), so we
may use the equations derived in §§ 4.2-4.6 when k⊥ρi .
(me/mi)
4.8.3. k‖λmfpi ≪ (mi/me)
Let us consider what happens if this condition is broken
and k‖λmfpi & (mi/me)
1/2. In this case, the collisions be-
come even weaker and the expansion procedure must be mod-
ified. Namely, the collision term picks up one extra order of
(me/mi)
1/2, so it is first order in Eq. (79). To zeroth order,
the electron kinetic equation no longer contains collisions: in-
stead of Eq. (84), we have
v‖b̂ ·∇h(0)e = v‖
. (110)
We may seek the solution of this equation in the form h(0)e =
H(t,Re)F0e + h
e,hom, where H(t,Re) is an unknown function to
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 19
FIG. 4.— Region of validity in the wavenumber space of the secondary approximation—isothermal electrons and gyrokinetic ions (§ 4). It is the region
of validity of the gyrokinetic approximation (Fig. 3) further circumscribed by two conditions: k‖λmfpi ≫ (me/mi)
1/2 (isothermal electrons) and k⊥ρe ≪ 1
(magnetized electrons). The region of validity of the strongly magnetized two-fluid theory (Appendix A.2) is also shown. It is the same as for the full two-fluid
theory plus the additional constraint k⊥ρi ≪ k‖λmfpi. The region of validity of MHD (or one-fluid theory) is the subset of this with k‖λmfpi ≪ (me/mi)
(adiabatic electrons).
be determined and h(0)e,hom is the homogeneous solution satis-
fying
b̂ ·∇h(0)e,hom = 0, (111)
i.e., h(0)e,hom must be constant along the perturbed magnetic
field. This is a generalization of Eq. (96). Again assuming
stochastic field lines, we conclude that h(0)e,hom is independent
of space. If we rule out spatially uniform perturbations, we
may set h(0)e,hom = 0. The unknown function H(t,Re) is readily
expressed in terms of δne and ϕ:
d3vh(0)e ⇒ H =
, (112)
so h(0)e is again given by Eq. (101), so the equations derived
in §§ 4.2-4.6 are unaltered. Thus, the mass-ratio expansion
remains valid at k‖λmfpi & (mi/me)
4.8.4. k‖λmfpi ≫ (me/mi)
If the parallel wavelength of the fluctuations is so long that
this is violated, k‖λmfpi . (me/mi)
1/2, the collision term in
Eq. (79) is minus first order. This is the lowest-order term in
the equation. Setting it to zero obliges h(0)e to be a perturbed
Maxwellian again given by Eq. (94). Instead of Eq. (84), the
zeroth-order kinetic equation is
v‖b̂ ·∇h(0)e = v‖
∂h(1)e
. (113)
Now the collision term in this order contains h(1)e , which
can be determined from Eq. (113) by inverting the colli-
sion operator. This sets up a perturbation theory that in due
course leads to the Reduced MHD version of the general
MHD equations—this is what was considered in § 2. Equa-
tion (96) no longer needs to hold, so the electrons are not
isothermal. In this true one-fluid limit, both electrons and
ions are adiabatic with equal temperatures [see Eq. (115) be-
low]. The collisional transport terms in this limit (parallel
and perpendicular resistivity, viscosity, heat fluxes, etc.) were
calculated [starting not from gyrokinetics but from the gen-
eral Vlasov–Landau equation (36)] in exhaustive detail by
Braginskii (1965). His results and the way RMHD emerges
from them are reviewed in Appendix A.
In physical terms, the electrons can no longer be isothermal
if the parallel electron diffusion time becomes longer than the
characteristic time of the fluctuations (the Alfvén time):
vtheλmfpik2‖
⇔ k‖λmfpi .
. (114)
Furthermore, under a similar condition, electron and ion tem-
peratures must equalize: this happens if the ion–electron col-
lision time is shorter than the Alfvén time,
⇔ k‖λmfpi .
(115)
(see Lithwick & Goldreich 2001 for a discussion of these con-
ditions in application to the ISM).
4.9. Summary
The original gyrokinetic description introduced in § 3 was
a system of two kinetic equations [Eq. (57)] that evolved the
electron and ion distribution functions he, hi and three field
20 SCHEKOCHIHIN ET AL.
equations [Eqs. (61-63)] that related ϕ, A‖ and δB‖ to he and
hi. In this section, we have taken advantage of the smallness
of the electron mass to treat the electrons as an isothermal
magnetized fluid, while ions remained fully gyrokinetic.
In mathematical terms, we solved the electron kinetic equa-
tion and replaced the gyrokinetics with a simpler closed sys-
tem of equations that evolve 6 unknown functions: ϕ, A‖, δB‖,
δne, u‖e and hi. These satisfy two fluid-like evolution equa-
tions (95) and (104), three integral relations (107), (108), and
(87) which involve hi, and the kinetic equation (57) for hi.
The system is simpler because the full electron distribution
function has been replaced by two scalar fields δne and u‖e.
We now summarize this new system of equations: denoting
ai = k⊥v⊥/Ωi, we have
+ b̂ ·∇ϕ= b̂ ·∇T0e
, (116)
+ b̂ ·∇u‖e = −
,(117)
eik·r
d3vJ0(ai)hik, (118)
u‖e =
4πen0e
∇2⊥A‖ +
eik·r
d3vv‖J0(ai)hik, (119)
eik·r
J0(ai) +
v2thi
J1(ai)
, (120)
and Eq. (57) for s = i and ion–ion collisions only:
{〈χ〉Ri ,hi} =
∂〈χ〉Ri
F0i + 〈Cii[hi]〉Ri ,
(121)
where 〈Cii[. . .]〉Ri is the gyrokinetic ion–ion collision oper-
ator (see Appendix B) and the ion–electron collisions have
been neglected to lowest order in (me/mi)
1/2 [see Eq. (51)].
Note that Eqs. (116-117) have been written in a compact form,
where
+ uE ·∇ =
{ϕ, · · ·} (122)
is the convective derivative with respect to the E×B drift ve-
locity, uE = −c∇⊥ϕ× ẑ/B0, and
b̂ ·∇ = ∂
·∇ = ∂
A‖, · · ·
(123)
is the gradient along the total magnetic field (mean field plus
perturbation).
The generalized energy conserved by Eqs. (116-121) is
given by Eq. (109).
It is worth observing that the left-hand side of Eq. (116) is
simply minus the component of the electric field along the to-
tal magnetic field [see Eq. (37)]. This was used in § 4.3 to
prove that the magnetic flux described by Eq. (116) is exactly
conserved (see § 7.7 for a discussion of scales at which this
conservation is broken). Equation (116) is the projection of
the generalized Ohm’s law onto the total magnetic field—the
right-hand side of this equation is the so-called thermoelec-
tric term. This is discussed in more detail in Appendix C.1,
where we also show that Eq. (117) is the parallel part of Fara-
day’s law and give a qualitative non-gyrokinetic derivation of
Eqs. (116-117).
We will refer to Eqs. (116-121) as the equations of isother-
mal electron fluid. They are valid in a broad range of scales:
the only constraints are that k‖ ≪ k⊥ (gyrokinetic order-
ing, § 3.1), k⊥ρe ≪ 1 (electrons are magnetized, § 4.8.1) and
k‖λmfpi ≫ (me/mi)1/2 (electrons are isothermal, § 4.8.4). The
region of validity of Eqs. (116-121) in the wavenumber space
is illustrated in Fig. 4. A particular advantage of this hybrid
fluid-kinetic system is that it is uniformly valid across the
transition from magnetized to unmagnetized ions (i.e., from
k⊥ρi ≪ 1 to k⊥ρi ≫ 1).
5. TURBULENCE IN THE INERTIAL RANGE: KINETIC RMHD
Our goal in this section is to derive a reduced set of equa-
tions that describe the magnetized plasma in the limit of small
k⊥ρi. Before we proceed with an expansion in k⊥ρi, we need
to make a formal technical step, the usefulness of which will
become clear shortly. A reader with no patience for this or
any of the subsequent technical developments may skip to the
summary at the end of this section (§ 5.7).
5.1. A Technical Step
Let us formally split the ion gyrocenter distribution function
into two parts:
v⊥ ·A⊥
F0i + g
eik·Ri
J0(ai)
v2thi
J1(ai)
F0i + g. (124)
Then g satisfies the following equation, obtained by substitut-
ing Eq. (124) and the expression for ∂A‖/∂t that follows from
Eq. (116) into the ion gyrokinetic equation (121):
{〈χ〉Ri ,g}− 〈Cii[g]〉Ri
︸ ︷︷ ︸
A‖,ϕ− 〈ϕ〉Ri
︸ ︷︷ ︸
+ b̂ ·∇
v⊥ ·A⊥
︸ ︷︷ ︸
v⊥ ·A⊥
︸ ︷︷ ︸
. (125)
In the above equation, we have used compact notation in
writing out the nonlinear terms: e.g.,
A‖,ϕ− 〈ϕ〉Ri
A‖(r),ϕ(r)
〈A‖〉Ri ,〈ϕ〉Ri
, where the first Poisson
bracket involves derivatives with respect to r and the second
with respect to Ri.
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 21
The field equations (118-120) rewritten in terms of g are
−Γ1(αi)
︸ ︷︷ ︸
1 −Γ0(αi)
]Zeϕk
︸ ︷︷ ︸
d3vJ0(ai)gk
︸ ︷︷ ︸
, (126)
4πen0e
k2⊥A‖k
︸ ︷︷ ︸
d3vv‖J0(ai)gk
︸ ︷︷ ︸
= u‖ki, (127)
︸ ︷︷ ︸
Γ2(αi) +
︸ ︷︷ ︸
1 −Γ1(αi)
]Zeϕk
︸ ︷︷ ︸
v2thi
J1(ai)
︸ ︷︷ ︸
, (128)
where ai = k⊥v⊥/Ωi, αi = k
i /2 and we have defined
Γ0(αi) =
d3v [J0(ai)]
2 F0i
= I0(αi)e
−αi = 1 −αi + · · · , (129)
Γ1(αi) =
v2thi
J0(ai)
J1(ai)
F0i = −Γ′0(αi)
= [I0(αi) − I1(αi)] e−αi = 1 −
αi + · · · , (130)
Γ2(αi) =
v2thi
J1(ai)
F0i = 2Γ1(αi). (131)
Underneath each term in Eqs. (125-128), we have indicated
the lowest order in k⊥ρi to which this term enters.
5.2. Subsidiary Ordering in k⊥ρi
In order to carry out a subsidiary expansion in small k⊥ρi,
we must order all terms in Eqs. (95-104) and (125-128) with
respect to k⊥ρi. Let us again assume, like we did when ex-
panding the electron equation (§ 4), that the ordering intro-
duced for the gyrokinetics in § 3.1 holds also for the sub-
sidiary expansion in k⊥ρi. First note that, in view of Eq. (47),
we must regard Zeϕ/T0i to be minus first order:
. (132)
Also, as δB⊥/B0 ∼ ǫ [Eq. (12)],
(v‖/c)A‖
∼ vthiδB⊥
βi, (133)
so ϕ and (v‖/c)A‖ are same order.
Since u‖ = u‖i (electrons do not contribute to the mass flow),
assuming that slow waves and Alfvén waves have comparable
energies implies u‖i ∼ u⊥. As u‖i is determined by the second
equality in Eq. (127), we can order g [using Eq. (12)]:
, (134)
so g is zeroth order in k⊥ρi. Similarly, δne/n0e ∼ δB‖/B0 ∼ ǫ
are zeroth order in k⊥ρi—this follows directly from Eq. (12).
Together with Eq. (3), the above considerations allow us to
order all terms in our equations. The ordering of the collision
term involving ϕ is explained in Appendix B.2.
5.3. Alfvén Waves: Kinetic Derivation of RMHD
We shall now show that the RMHD equations (17-18) hold
in this approximation. There is a simple correspondence be-
tween the stream and flux functions defined in Eq. (16) and
the electromagnetic potentials ϕ and A‖:
ϕ, Ψ = −
4πmin0i
. (135)
The first of these definitions says that the perpendicular flow
velocity u⊥ is the E×B drift velocity; the second definition
is the standard MHD relation between the magnetic flux func-
tion and the parallel component of the vector potential.
5.3.1. Derivation of Eq. (17)
Deriving Eq. (17) is straightforward: in Eq. (95), we retain
only the lowest—minus first—order terms (those that contain
ϕ and A‖). The result is
= 0. (136)
Using Eq. (135) and the definition of the Alfvén speed, vA =
4πmin0i, we get Eq. (17). By the argument of § 4.3,
Eq. (136) expresses the fact that that magnetic-field lines are
frozen into the E×B velocity field, which is the mean flow
velocity associated with the Alfvén waves (see § 5.4).
5.3.2. Derivation of Eq. (18)
As we are about to see, in order to derive Eq. (18), we have
to separate the first-order part of the k⊥ρi expansion. The
easiest way to achieve this, is to integrate Eq. (125) over the
velocity space (keeping r constant) and expand the resulting
equation in small k⊥ρi. Using Eqs. (126) and (127) to express
the velocity-space integrals of g, we get
1 −Γ0(αi)
]Zeϕk
︸ ︷︷ ︸
−Γ1(αi)
︸ ︷︷ ︸
4πen0e
k2⊥A‖k
︸ ︷︷ ︸
d3vJ0(ai){〈χ〉Ri ,g}k
︸ ︷︷ ︸
d3vJ0(ai)
v⊥ ·A⊥
︸ ︷︷ ︸
22 SCHEKOCHIHIN ET AL.
. (137)
Underneath each term, the lowest order in k⊥ρi to which it
enters is shown. We see that terms containing ϕ are all first
order, so it is up to this order that we shall retain terms. The
collision term integrated over the velocity space picks up two
extra orders of k⊥ρi (see Appendix B.1), so it is second order
and can, therefore, be dropped. As a consequence of quasi-
neutrality, the zeroth-order part of the above equation exactly
coincides with Eq. (104), i.e, δni/n0i = δne/n0e satisfy the
same equation. Indeed, neglecting second-order terms (but
not first-order ones!), the nonlinear term in Eq. (137) (the last
term on the left-hand side) is
d3vv‖g
v2thi
, (138)
and, using Eqs. (126-128) to express velocity-space integrals
of g in the above expression, we find that the zeroth-order part
of the nonlinearity is the same as the nonlinearity in Eq. (104),
while the first-order part is
ρ2i ∇2⊥
4πen0e
∇2⊥A‖
, (139)
where we have used the expansion (129) of Γ0(αi) and con-
verted it back into x space.
Thus, if we subtract Eq. (104) from Eq. (137), the remain-
der is first order and reads
ρ2i ∇2⊥
ρ2i ∇2⊥
4πen0e
∇2⊥A‖ −
4πen0e
∇2⊥A‖
= 0. (140)
Multiplying Eq. (140) by 2T0i/Zeρ
i and using Eq. (135), we
get the second RMHD equation (18).
We have established that the Alfvén-wave component of the
turbulence is decoupled and fully described by the RMHD
equations (17) and (18). This result is the same as that in
§ 2.2 but now we have proven that collisions do not affect the
Alfvén waves and that a fluid-like description only requires
k⊥ρi ≪ 1 to be valid.
5.4. Why Alfvén Waves Ignore Collisions
Let us write explicitly the distribution function of the ion
gyrocenters [Eq. (124)] to two lowest orders in k⊥ρi:
〈ϕ〉Ri F0i +
v2thi
F0i + g + · · · , (141)
where, up to corrections of order k2⊥ρ
i , the ring-averaged
scalar potential is 〈ϕ〉Ri = ϕ(Ri), the scalar potential taken at
the position of the ion gyrocenter. Note that in Eq. (141), the
first term is minus first order in k⊥ρi [see Eq. (132)], the sec-
ond and third terms are zeroth order [Eq. (134)], and all terms
of first and higher orders are omitted. In order to compute the
full ion distribution function given by Eq. (54), we have to
convert hi to the r space. Keeping terms up to zeroth order,
we get
〈ϕ〉Ri ≃
ϕ(Ri) =
ϕ(r) +
v⊥× ẑ
·∇ϕ(r) + · · ·
ϕ(r) +
2v⊥ ·uE
v2thi
+ . . . , (142)
where uE = −c∇ϕ(r)× ẑ/B0, the E×B drift velocity. Sub-
stituting Eq. (142) into Eq. (141) and then Eq. (141) into
Eq. (54), we find
fi = F0i +
2v⊥ ·uE
v2thi
F0i +
v2thi
F0i + g + · · · . (143)
The first two terms can be combined into a Maxwellian
with mean perpendicular flow velocity u⊥ = uE . These are
the terms responsible for the Alfvén waves. The remaining
terms, which we shall denote δ f̃i, are the perturbation of the
Maxwellian in the moving frame of the Alfvén waves—they
describe the passive (compressive) component of the turbu-
lence (see § 5.5). Thus, the ion distribution function is
(πv2thi)
(v⊥ − uE)2 + v2‖
+ δ f̃i. (144)
This sheds some light on the indifference of Alfvén waves
to collisions: Alfvénic perturbations do not change the
Maxwellian character of the ion distribution. Unlike in a neu-
tral fluid or gas, where viscosity arises when particles trans-
port the local mean momentum a distance ∼ λmfpi, the parti-
cles in a magnetized plasma instantaneously take on the lo-
cal E×B velocity (they take a cyclotron period to adjust, so,
roughly speaking, ρi plays the role of the mean free path).
Thus, there is no memory of the mean perpendicular motion
and, therefore, no perpendicular momentum transport.
Some readers may find it illuminating to notice that
Eq. (140) can be interpreted as stating simply ∇·j = 0: the first
two terms represent the divergence of the polarization current,
which is perpendicular to the magnetic field;22 the last two
terms are b̂ ·∇ j‖. No contribution to the current arises from
the collisional term in Eq. (137) as ion–ion collisions cause
no particle transport to lowest order in k⊥ρi.
5.5. Compressive Fluctuations
The equations that describe the density (δne) and magnetic-
field-strength (δB‖) fluctuations follow immediately from
Eqs. (125-128) if only zeroth-order terms are kept. In these
equations, terms that involve ϕ and A‖ also contain factors
∼ k2⊥ρ2i and are, therefore, first-order [with the exception of
the nonlinearity on the left-hand side of Eq. (125)]. The fact
that 〈Cii[〈ϕ〉Ri F0i]〉Ri in Eq. (125) is first order is proved in Ap-
pendix B.2. Dropping these terms along with all other contri-
butions of order higher than zeroth and making use of Eq. (69)
22 The polarization-drift velocity is formally higher order than uE in the
gyrokinetic expansion. However, since uE does not produce any current,
the lowest-order contribution to the perpendicular current comes from the
polarization drift. The higher-order contributions to the gyrocenter distri-
bution function did not need to be calculated explicitly because the informa-
tion about the polarization charge is effectively carried by the quasi-neutrality
condition (61). We do not belabor this point because, in our approach, the no-
tion of polarization charge is only ever brought in for interpretative purposes,
but is not needed to carry out calculations. For further qualitative discussion
of the role of the polarization charge and polarization drift in gyrokinetics,
we refer the reader to Krommes 2006 and references therein.
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 23
to write out 〈χ〉Ri , we find that Eq. (125) takes the form
+ v‖ b̂ ·∇
v2thi
v2thi
, (145)
where we have used definitions (122-123) of the convective
time derivative d/dt and the total gradient along the magnetic
field b̂ · ∇ to write our equation in a compact form. Note
that, in view of the correspondence between Φ, Ψ and ϕ, A‖
[Eq. (135)], these nonlinear derivatives are the same as those
defined in Eqs. (19-20). The collision term in the right-hand
side of the above equation is the zeroth-order limit of the gy-
rokinetic ion–ion collision operator: a useful model form of it
is given in Appendix B.3 [Eq. (B18)].
To zeroth order, Eqs. (126-128) are
d3vg, (146)
d3vv‖g, (147)
v2thi
g. (148)
Note that u‖ is not an independent quantity—it can be com-
puted from the ion distribution but is not needed for the deter-
mination of the latter.
Equations (145-148) evolve the ion distribution function
g, the “slow-wave quantities” u‖, δB‖, and the density fluc-
tuations δne. The nonlinearities in Eq. (145), contained in
d/dt and b̂ ·∇, involve the Alfvén-wave quantities Φ and Ψ
(or, equivalently, ϕ and A‖) determined separately and inde-
pendently by the RMHD equations (17-18). The situation
is qualitatively similar to that in MHD (§ 2.4), except now
a kinetic description is necessary—Eqs. (145-148) replace
Eqs. (25-27)—and the nonlinear scattering/mixing of the slow
waves and the entropy mode by the Alfvén waves takes the
form of passive advection of the distribution function g. The
density and magnetic-field-strength fluctuations are velocity-
space moments of g.
Another way to understand the passive nature of the com-
pressive component of the turbulence discussed above is to
think of it as the perturbation of a local Maxwellian equilib-
rium associated with the Alfvén waves. Indeed, in § 5.4, we
split the full ion distribution function [Eq. (144)] into such a
local Maxwellian and its perturbation
δ f̃i = g +
v2thi
F0i. (149)
It is this perturbation that contains all the information about
the compressive component; the second term in the above ex-
pression enforces to lowest order the conservation of the first
adiabatic invariant µi = miv
⊥/2B. In terms of the function
(149), Eqs. (145-148) take a somewhat more compact form
(cf. Schekochihin et al. 2007):
δ f̃i −
v2thi
+ v‖b̂ ·∇
δ f̃i +
δ f̃i
, (150)
FIG. 5.— Channels of the kinetic cascade of generalized energy (§ 3.4)
from large to small scales: see § 2.7 and Appendix D.2 (inertial range,
collisional regime), § 5.6 and § 6.2.5 (inertial range, collisionless regime),
§ 7.8 and § 7.12 (dissipation range). Note that some ion heating probably
also results from the collisional and collisionless damping of the compressive
fluctuations in the inertial range (see § 6.1.2 and § 6.2.4).
d3vδ f̃i, (151)
v2thi
δ f̃i. (152)
5.6. Generalized Energy: Three KRMHD Cascades
The generalized energy (§ 3.4) in the limit k⊥ρi ≪ 1 is cal-
culated by substituting into Eq. (109) the perturbed ion dis-
tribution function δ fi = 2v⊥ · uEF0i/v2thi + δ f̃i [see Eqs. (143)
and (149)]. After performing velocity integration, we get
min0iu
n0iT0i
δ f̃ 2i
=WAW +Wcompr. (153)
We see that the kinetic energy of the Alfvénic fluctuations
has emerged from the ion-entropy part of the generalized en-
ergy. The first two terms in Eq. (153) are the total (kinetic
plus magnetic) energy of the Alfvén waves, denoted WAW. As
we learned from § 5.3, it cascades independently of the rest of
the generalized energy, Wcompr, which contains the compres-
sive component of the turbulence (§ 5.5) and is the invariant
conserved by Eqs. (150-152).
In terms of the potentials used in our discussion of RMHD
in § 2, we have
WAW =
min0i
|∇⊥Φ|2 + |∇⊥Ψ|2
min0i
|∇⊥ζ+|2 + |∇⊥ζ−|2
=W +AW +W
AW (154)
whereW +AW and W
AW are the energies of the “+” and “−” waves
[Eq. (33)], which, as we know from § 2.3, cascade by scatter-
ing off each other but without exchanging energy.
Thus, the kinetic cascade in the limit k⊥ρi ≪ 1 is split, in-
dependently of the collisionality, into three cascades: of W +AW,
24 SCHEKOCHIHIN ET AL.
W −AW and Wcompr. The compressive cascade is, in fact, split
into three independent cascades—the splitting is different in
the collisional limit (Appendix D.2) and in the collisionless
one (§ 6.2.5). Figure 5 schematically summarizes both the
splitting of the kinetic cascade that we have worked out so far
and the upcoming developments.
5.7. Summary
In § 4, gyrokinetics was reduced to a hybrid fluid-kinetic
system by means of an expansion in the electron mass, which
was valid for k⊥ρe ≪ 1. In this section, we have further re-
stricted the scale range by taking k⊥ρi ≪ 1 and as a result have
been able to achieve a further reduction in the complexity of
the kinetic theory describing the turbulent cascades. The re-
duced theory derived here evolves 5 unknown functions: Φ,
Ψ, δB‖, δne and g. The stream and flux functions, Φ and Ψ
are related to the fluid quantities (perpendicular velocity and
magnetic field perturbations) via Eq. (16) and to the electro-
magnetic potentials ϕ, A‖ via Eq. (135). They satisfy a closed
system of equations, Eqs. (17-18), which describe the decou-
pled cascade of Alfvén waves. These are the same equations
that arise from the MHD approximations, but we have now
proven that their validity does not depend on the assumption
of high collisionality (the fluid limit) and extends to scales
well below the mean free path, but above the ion gyroscale.
The physical reasons for this are explained in § 5.4. The den-
sity and magnetic-field-strength fluctuations (the “compres-
sive” fluctuations, or the slow waves and the entropy mode in
the MHD limit) now require a kinetic description in terms of
the ion distribution function g [or δ f̃i, Eq. (149)], evolved by
the kinetic equation (145) [or Eq. (150)]. The kinetic equation
contains δne and δB‖, which are, in turn calculated in terms
of the velocity-space integrals of g via Eqs. (146) and (148)
[or Eqs. (151) and (152)]. The nonlinear evolution (turbulent
cascade) of g, δB‖ and δne is due solely to passive advection
of g by the Alfvén-wave turbulence.
Let us summarize the new set of equations:
= vAb̂ ·∇Φ, (155)
∇2⊥Φ= vAb̂ ·∇∇2⊥Ψ, (156)
+ v‖ b̂ ·∇
v2thi
v2thi
, (157)
v2thi
(158)
v2thi
g, (159)
where
+{Φ, · · ·} , b̂ ·∇ = ∂
{Ψ, · · ·} . (160)
An explicit form of the collision term in the right-hand side of
Eq. (157) is provided in Appendix B.3 [Eq. (B18)].
The generalized energy conserved by Eqs. (155-159) is
given by Eq. (153). The kinetic cascade is split, the Alfvénic
cascade proceeding independently of the compressive one
(see Fig. 5).
The decoupling of the Alfvénic cascade is manifested by
Eqs. (155-156) forming a closed subset. As already noted in
§ 4.9, Eq. (155) is the component of Ohm’s law along the total
magnetic field, B ·E = 0. Equation (156) can be interpreted as
the evolution equation for the vorticity of the perpendicular
plasma flow velocity, which is the E×B drift velocity.
We shall refer to the system of equations (155-159) as Ki-
netic Reduced Magnetohydrodynamics (KRMHD).23 It is a
hybrid fluid-kinetic description of low-frequency turbulence
in strongly magnetized weakly collisional plasma that is uni-
formly valid at all scales satisfying k⊥ρi ≪ min(1,k‖λmfpi)
(ions are strongly magnetized)24 and k‖λmfpi ≫ (me/mi)1/2
(electrons are isothermal), as illustrated in Fig. 2. Therefore,
it smoothly connects the collisional and collisionless regimes
and is the appropriate theory for the study of the turbulent cas-
cades in the inertial range. The KRMHD equations generalize
rather straightforwardly to plasmas that are so collisionless
that one cannot assume a Maxwellian equilibrium distribu-
tion function (Chen et al. 2009)—a situation that is relevant
in some of the solar-wind measurements (see further discus-
sion in § 8.3).
KRMHD describe what happens to the turbulent cascade at
or below the ion gyroscale—we shall move on to these scales
in § 7, but first we would like to discuss the turbulent cascades
of density and magnetic-field-strength fluctuations and their
damping by collisional and collisionless mechanisms.
6. COMPRESSIVE FLUCTUATIONS IN THE INERTIAL RANGE
Here we first derive the nonlinear equations that govern
the evolution of the compressive (density and magnetic-field-
strength) fluctuations in the collisional (k‖λmfpi ≪ 1, § 6.1 and
Appendix D) and collisionless (k‖λmfpi ≫ 1, § 6.2) limits, dis-
cuss the linear damping that these fluctuations undergo in the
two limits and work out the form the generalized energy takes
for compressive fluctuations (which is particularly interesting
in the collisionless limit, §§ 6.2.3-6.2.5). As in previous sec-
tions, an impatient reader may skip to § 6.3 where the results
of the previous two subsections are summarized and the im-
plications for the structure of the turbulent cascades of the
density and field-strength fluctuations are discussed.
6.1. Collisional Regime
6.1.1. Equations
In the collisional regime, k‖λmfpi ≪ 1, the fluid limit is re-
covered by expanding Eqs. (155-159) in small k‖λmfpi. The
calculation that is necessary to achieve this is done in Ap-
pendix D (see also Appendix A.4). The result is a closed set
23 The term is introduced by analogy with a popular fluid-kinetic system
known as Kinetic MHD, or KMHD (see Kulsrud 1964, 1983). KMHD is de-
rived for magnetized plasmas (ρi ≪ λmfpi) under the assumption that kρs ≪ 1
and ω ≪ Ωs but without assuming either strong anisotropy (k‖ ≪ k⊥) or
small fluctuations (|δB| ≪ B0). The KRMHD equations (155-159) can be
recovered from KMHD by applying to it the GK-RMHD ordering [Eq. (12)
and § 3.1] and an expansion in (me/mi)1/2 (Schekochihin et al. 2007). This
means that the k⊥ρi expansion (§ 5), which for KMHD is the primary ex-
pansion, commutes with the gyrokinetic expansion (§ 3) and the (me/mi)1/2
expansion (§ 4), both of which preceded it in this paper.
24 The condition k⊥ρi ≪ k‖λmfpi must be satisfied because in our esti-
mates of the collision terms (Appendix B.2) we took k⊥ρi ≪ 1 while assum-
ing that k‖λmfpi ∼ 1.
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 25
of three fluid equations that evolve δB‖, δne and u‖:
= b̂ ·∇u‖ +
, (161)
= v2Ab̂ ·∇
+ ν‖ib̂ ·∇
b̂ ·∇u‖
, (162)
+κ‖ib̂ ·∇
b̂ ·∇δTi
, (163)
where
ν‖ib̂ ·∇u‖
, (164)
and ν‖i and κ‖i are the coefficients of parallel ion viscosity
and thermal diffusivity, respectively. The viscous and ther-
mal diffusion are anisotropic because plasma is magnetized,
λmfpi ≫ ρi (Braginskii 1965). The method of calculation of
ν‖i and κ‖i is explained in Appendix D.3. Here we shall ig-
nore numerical prefactors of order unity and give order-of-
magnitude values for these coefficients:
ν‖i ∼ κ‖i ∼
v2thi
∼ vthiλmfpi. (165)
If we set ν‖i = κ‖i = 0, Eqs. (161-164) are the same as the
RMHD equations of § 2 with the sound speed defined as
cs = vA
. (166)
This is the natural definition of cs for the case of adiabatic
ions, whose specific heat ratio is γi = 5/3, and isothermal elec-
trons, whose specific heat ratio is γe = 1 [because δpe = T0eδne;
see Eq. (103)]. Note that Eq. (164) is equivalent to the
pressure balance [Eq. (22) of § 2] with p = niTi + neTe and
δpe = T0eδne.
As in § 2, the fluctuations described by Eqs. (161-164) sep-
arate into the zero-frequency entropy mode and the left- and
right-propagating slow waves with
ω = ±
1 + v2A/c2s
(167)
[see Eq. (30)]. All three are cascaded independently of each
other via nonlinear interaction with the Alfvén waves. In Ap-
pendix D.2, we show that the generalized energy Wcompr for
this system, given in § 5.6, splits into the three familiar invari-
ants W +sw, W
sw, and Ws, defined by Eqs. (34-35) (see Fig. 5).
6.1.2. Dissipation
The diffusion terms add dissipation to the equations. Be-
cause diffusion occurs along the field lines of the total mag-
netic field (mean field plus perturbation), the diffusive terms
are nonlinear and the dissipation process also involves interac-
tion with the Alfvén waves. We can estimate the characteristic
parallel scale at which the diffusion terms become important
by balancing the nonlinear cascade time and the typical diffu-
sion time:
k‖vA ∼ vthiλmfpik2‖ ⇔ k‖λmfpi ∼ 1/
βi, (168)
where we have used Eq. (165).
Technically speaking, the cutoff given by Eq. (168) always
lies in the range of k‖ that is outside the region of validity
of the small-k‖λmfpi expansion adopted in the derivation of
Eqs. (161-163). In fact, in the low-beta limit, the collisional
cutoff falls manifestly in the collisionless scale range, i.e.,
the collisional (fluid) approximation breaks down before the
slow-wave and entropy cascades are damped and one must use
the collisionless (kinetic) limit to calculate the damping (see
§ 6.2.2). The situation is different in the high-beta limit: in
this case, the expansion in small k‖λmfpi can be reformulated
as an expansion in small 1/
βi and the cutoff falls within the
range of validity of the fluid approximation. Equations (161-
163) in this limit are
= b̂ ·∇u‖, (169)
= v2Ab̂ ·∇
+ ν‖ib̂ ·∇
b̂ ·∇u‖
, (170)
1 + Z/τ
5/3 + Z/τ
κ‖ib̂ ·∇
b̂ ·∇δne
. (171)
As in § 2 [Eq. (28)], the density fluctuations [Eq. (171)] have
decoupled from the slow waves [Eqs. (169-170)]. The former
are damped by thermal diffusion, the latter by viscosity. The
corresponding linear dispersion relations are
ω = −i
1 + Z/τ
5/3 + Z/τ
‖, (172)
ω =±k‖vA
ν‖ik‖
. (173)
Equation (172) describes strong diffusive damping of the den-
sity fluctuations. The slow-wave dispersion relation (173) has
two distinct regimes:
1. When k‖ < 2vA/ν‖i, it describes viscously damped slow
waves. In particular, in the limit k‖λmfpi ≪ 1/
βi, we
ω ≃±k‖vA − i
. (174)
2. For k‖ > 2vA/ν‖i, both solutions become purely imag-
inary, so the slow waves are converted into aperiodic
decaying fluctuations. The stronger-damped (diffusive)
branch has ω ≃ −iν‖ik2‖, the weaker-damped one has
ω ≃ −i v
∼ − i
λmfpi
∼ − i√
λmfpi
. (175)
This damping effect is called viscous relaxation. It is
valid until k‖λmfpi ∼ 1, where it is replaced by the col-
lisionless damping discussed in § 6.2.2 [see Eq. (190)].
The viscous and thermal-diffusive dissipation mechanisms
described above lead, in the limits where they are efficient, to
ion heating via the standard fluid (collisional) route, involving
the development of small parallel scales in the position space,
but not in velocity space (see § 3.4 and § 3.5).
6.2. Collisionless Regime
6.2.1. Equations
In the collisionless regime, k‖λmfpi ≫ 1, the collision inte-
gral in the right-hand side of the kinetic equation (157) can be
26 SCHEKOCHIHIN ET AL.
neglected. The v⊥ dependence can then be integrated out of
Eq. (157). Indeed, let us introduce the following two auxiliary
functions:
Gn(v‖) = −
dv⊥ v⊥
v2thi
g, (176)
GB(v‖) = −
dv⊥ v⊥
v2thi
g. (177)
In terms of these functions,
dv‖Gn,
dv‖GB (178)
and Eq. (157) reduces to the following two coupled one-
dimensional kinetic equations
+ v‖b̂ ·∇Gn = −
v‖FM(v‖)
×b̂ ·∇
, (179)
+ v‖b̂ ·∇GB =
v‖FM(v‖)
×b̂ ·∇
, (180)
where FM(v‖) = (1/
πvthi)exp(−v2‖/v
thi) is a one-dimensional
Maxwellian. This system can be diagonalized, so it splits into
two decoupled equations
+v‖b̂ ·∇G± =
v‖FM(v‖)
b̂ ·∇
dv′‖ G
±(v′‖), (181)
where
± = −
(182)
and we have introduced a new pair of functions
G+ = GB +
Gn, G
− = Gn +
GB, (183)
where
σ = 1 +
. (184)
Equation (181) describes two decoupled kinetic cascades,
which we will discuss in greater detail in §§ 6.2.3-6.2.5.
6.2.2. Collisionless Damping
Fluctuations described by Eq. (181) are subject to collision-
less damping. Indeed, let us linearize Eq. (181), Fourier trans-
form in time and space, divide through by −i(ω − k‖v‖), and
integrate over v‖. This gives the following dispersion relation
(the “−” branch is for G−, the “+” branch for G+)
ζiZ (ζi) = Λ
± − 1, (185)
FIG. 6.— Schematic log-log plot (artist’s impression) of the ratio of the
damping rate of magnetic-field-strength fluctuations to the Alfvén frequency
k‖vA in the high-beta limit [see Eqs. (173) and (190)]. In Barnes et al. (2009),
this plot is reproduced via a direct numerical solution of the linearized ion
gyrokinetic equation with collisions.
where ζi = ω/|k‖|vthi = ω/|k‖|vA
βi and we have used the
plasma dispersion function (Fried & Conte 1961)
Z (ζi) =
x − ζi
(186)
(the integration is along the Landau contour). This function is
not to be confused with the ion charge parameter Z = qi/e.
Formally, Eq. (185) has an infinite number of solutions.
When βi ∼ 1, they are all strongly damped with damping rates
Im(ω) ∼ |k‖|vthi ∼ |k‖|vA, so the damping time is compara-
ble to the characteristic timescale on which the Alfvén waves
cause these fluctuations to cascade to smaller scales.
It is interesting to consider the high- and low-beta limits.
High-Beta Limit. — When βi ≫ 1, we have in Eq. (185)
− − 1≃−2
, G− ≃ Gn, (187)
+ − 1≃
, G+ ≃ GB +
Gn. (188)
The “−” branch corresponds to the density fluctuations. The
solution of Eq. (185) has Im(ζi) ∼ 1, so these fluctuations are
strongly damped:
ω ∼ −i|k‖|vA
βi. (189)
The damping rate is much greater than the Alfvénic rate k‖vA
of the nonlinear cascade. In contrast, for the “+” branch, the
damping rate is small: it can be obtained by expanding Z(ζi) =
π + O (ζi), which gives25
ω = −i
|k‖|vthi√
|k‖|vA√
. (190)
Since Gn is strongly damped, Eq. (188) implies G
+ ≃ GB, i.e.,
the fluctuations that are damped at the rate (190) are predom-
inantly of the magnetic-field strength. The damping rate is a
25 This is the gyrokinetic limit (k‖/k⊥ ≪ 1) of the more general damping
effect known in astrophysics as the Barnes (1966) damping and in plasma
physics as transit-time damping. We remind the reader that our approach was
to carry out the gyrokinetic expansion (in small k‖/k⊥) first, and then take
the high-beta limit as a subsidiary expansion. A more standard approach in
the linear theory of plasma waves is to take the limit of high βi while treating
k‖/k⊥ as an arbitrary quantity. A detailed calculation of the damping rates
done in this way can be found in Foote & Kulsrud (1979).
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 27
constant (independent of k‖) small fraction ∼ 1/
βi of the
Alfvénic cascade rate.
In Fig. 6, we give a schematic plot of the damping rate of the
magnetic-field-strength fluctuations (slow waves) connecting
the fluid and kinetic limits for βi ≫ 1.
Low-Beta Limit. — When βi ≪ 1, we have
− − 1≃−
, G− ≃ Gn +
GB, (191)
+ − 1≃ 2
, G+ ≃ GB. (192)
For the “−” branch, we again have Im(ζi) ∼ 1, so
ω ∼ −i|k‖|vA
βi, (193)
which now is much smaller than the Alfvénic cascade rate
k‖vA. For the “+” branch (predominantly the field-strength
fluctuations), we seek a solution with ζ = −iζ̃i and ζ̃i ≫ 1.
Then Eq. (185) becomes ζiZ(ζi) ≃ 2
π ζ̃i exp(ζ̃i) = 2/βi. Up
to logarithmically small corrections, this gives ζ̃i ≃
| lnβi|,
whence
ω ∼ −i|k‖|vA
βi| lnβi|. (194)
While this damping rate is slightly greater than that of the “−”
branch, it is still much smaller than the Alfvénic cascade rate.
6.2.3. Collisionless Invariants
Equation (181) obeys a conservation law, which is very easy
to derive. Multiplying Eq. (181) by G±/FM and integrating
over space and velocities and performing integration by parts
in the right-hand side, we get
(G±)2
b̂ ·∇
dv‖v‖G
±. (195)
On the other hand, integrating Eq. (181) over v‖ gives
± = −b̂ ·∇
dv‖v‖G
±. (196)
Using this to express the right-hand side of Eq. (195) as a full
time derivative, we find
dW±compr
= 0, (197)
where the two invariants are
W±compr =
n0iT0i
(G±)2
(198)
It is useful (and always possible) to split
G± = FM
± + G̃±, (199)
where
dv‖G̃
± = 0 by construction. Then
W±compr =
n0iT0i
(G̃±)2
. (200)
Written in this form, the two invariants W±compr are mani-
festly positive definite quantities because Λ+ > 1 and Λ− < 0.
The invariants regulate the two decoupled kinetic cascades of
compressive fluctuations in the collisionless regime. The col-
lisionless damping derived in § 6.2.2 leads to exponential de-
cay of the density and field-strength fluctuations, or, equiva-
lently, of
±, while conserving W±compr. This means that
the damping is merely a redistribution of the conserved quan-
tity W±compr: the first term in Eq. (200) grows to compensate
for the decay of the second.
6.2.4. Linear Parallel Phase Mixing
In dynamical terms, how does the kinetic system Eq. (181)
arrange for the integral of the distribution function G±(v‖) to
decay while allowing its norm to grow? This is a very well
known phenomenon of (linear) phase mixing (Landau 1946;
Hammett et al. 1991; Krommes & Hu 1994; Krommes 1999;
Watanabe & Sugama 2004). To put it in simple terms, the
solution of the linearized Eq. (181) consists of the inhomoge-
neous part, which contains the collisionless damping and the
homogeneous part (solution of the left-hand side = 0) given by
G± ∝ e−ik‖v‖t , the so-called ballistic response (this is also the
nonlinear solution if t and k‖ are interpreted as Lagrangian
variables in the frame of the Alfvén waves; see § 6.3). As
time goes on, this part of the solution becomes increasingly
oscillatory in v‖, so its velocity integral tends to zero, while
its amplitude does not decay. It is such ballistic contributions
that make up the G̃± term in Eq. (200).
As the velocity gradient of G̃± increases with time,
∂G̃±/∂v‖ ∼ k‖tG±, at some point it can become sufficiently
large to activate the collision integral [the right-hand side of
Eq. (157)], which has so far been neglected. This way the col-
lisionless damping of compressive fluctuations can be turned
into ion heating—a simple example of a more general prin-
ciple of how electromagnetic fluctuation energy is transferred
into heat via the entropy part of the generalized energy (§ 3.5).
Indeed, we will prove in § 6.2.5 that the invariants W±compr are
constituent parts of the overall generalized energy functional
for the compressive fluctuations, so their cascade to small
scales in phase space is part of the overall kinetic cascade in-
troduced in § 3.4.
It is not entirely clear how efficient is the parallel-phase-
mixing route to ion heating and, therefore, whether the colli-
sionlessly damped energy of compressive fluctuations ends up
in the ion heat or rather reaches the ion gyroscale and couples
back to the Alfvénic component of the turbulence (§ 7.1). The
answer to this question will depend on whether compressive
fluctuations can develop large k‖—a non-trivial issue further
discussed in § 6.3.
6.2.5. Generalized Energy: Three Collisionless Cascades
We will now show how the generalized energy for com-
pressive fluctuations in the collisionless regime incorporates
the two invariants derived in § 6.2.3.
Rewriting the compressive part of the KRMHD generalized
energy [Eq. (153)] in terms of the function g [see Eq. (149)],
we get
Wcompr =
n0iT0i
. (201)
28 SCHEKOCHIHIN ET AL.
Using Eqs. (178) and (183), we can express δne and δB‖ in
terms of
± as follows
, (202)
, (203)
where σ was defined in Eq. (184) and
. (204)
In order to express g in terms of G±, we have to reconstruct
the v⊥ dependence of g, which we integrated out at the begin-
ning of § 6.2.1.
Let us represent the distribution function as follows
πv2thi
e−xĝ(x,v‖), ĝ(x,v‖) =
Ll(x)Gl(v‖), (205)
where x = v2⊥/v
thi and we have expanded ĝ in Laguerre poly-
nomials Ll(x) = (e
x/l!)(dl/dxl)xle−x. Since Laguerre polyno-
mials are orthogonal, the first term in Eq. (201) splits into a
sum of “energies” associated with the expansion coefficients:
. (206)
The expansion coefficients are determined via the Laguerre
transform:
Gl(v‖) =
dxe−xLl(x)ĝ(x,v‖). (207)
As L0 = 1 and L1 = 1 − x, it is easy to see that δne and δB‖ can
be expressed as linear combinations of
dv‖G0 and
dv‖G1
[see Eqs. (176-178)]. Using Eqs. (176), (177), and (183), we
can show that
G0 = −
+G+ +
σ − 1 −
, (208)
σΛ+G+ −
, (209)
where G± satisfy Eq. (181). As follows from Eq. (157) (ne-
glecting the collision integral), all higher-order expansion co-
efficients satisfy a simple homogeneous equation:
+ v‖b̂ ·∇Gl = 0, l > 1. (210)
Thus, the distribution function can be explicitly written in
terms of G±:
G0(v‖) +
v2thi
G1(v‖)
πv2thi
thi + g̃, (211)
where G0 and G1 are given by Eqs. (208-209) and g̃ com-
prises the rest of the Laguerre expansion (all Gl with l > 1),
i.e., it is the homogeneous solution of Eq. (157) that does not
contribute to either density or magnetic-field strength:
+ v‖b̂ ·∇g̃ = 0,
d3v g̃ = 0,
v2thi
g̃ = 0. (212)
Now substituting Eqs. (208) and (209) into Eq. (206) and
then substituting the result and Eqs. (202-203) into Eq. (201),
we find after some straightforward manipulations
Wcompr =
T0ig̃
(Λ+)2W +compr
(Λ−)2W −compr, (213)
where κ is defined by Eq. (204) and W±compr are the two inde-
pendent invariants that we derived in § 6.2.3. Thus, the gener-
alized energy for compressive fluctuations splits into three in-
dependently cascading parts: W±compr associated with the den-
sity and magnetic-field-strength fluctuations and a purely ki-
netic part given by the first term in Eq. (213) (see Fig. 5).
The dynamical evolution of this purely kinetic component is
described by Eq. (212)—it is a passively mixed, undamped
ballistic-type mode.
All three cascade channels lead to small perpendicular spa-
tial scales via passive mixing by the Alfvénic turbulence and
also to small scales in v‖ via the parallel phase mixing pro-
cess discussed in § 6.2.4 (note that g̃ is subject to this process
as well).
6.3. Parallel and Perpendicular Cascades
Let us return to the kinetic equation (157) and transform
it to the Lagrangian frame associated with the velocity field
u⊥ = ẑ×∇⊥Φ of the Alfvén waves: (t,r) → (t,r0), where
r(t,r0) = r0 +
dt ′u⊥(t
′,r(t ′,r0)). (214)
In this frame, the convective derivative d/dt defined in
Eq. (160) turns into ∂/∂t, while the parallel spatial gradient
b̂ ·∇ can be calculated by employing the Cauchy solution for
the perturbed magnetic field δB⊥ = ẑ×∇⊥Ψ:
b̂(t,r) = ẑ +
δB⊥(t,r)
= b̂(0,r0) ·∇0r, (215)
where r is given by Eq. (214) and ∇0 = ∂/∂r0. Then
b̂ ·∇ = b̂(0,r0) ·
·∇ = b̂(0,r0) ·∇0 =
, (216)
where s0 is the arc length along the perturbed magnetic field
taken at t = 0 [if δB⊥(0,r0) = 0, s0 = z0]. Thus, in the La-
grangian frame associated with the Alfvénic component of
the turbulence, Eq. (157) is linear. This means that, if the
effect of finite ion gyroradius is neglected, the KRMHD sys-
tem does not give rise to a cascade of density and magnetic-
field-strength fluctuations to smaller scales along the moving
(perturbed) field lines, i.e., b̂ · ∇δne and b̂ · ∇δB‖ do not in-
crease. In contrast, there is a perpendicular cascade (cascade
in k⊥): the perpendicular wandering of field lines due to the
Alfvénic turbulence causes passive mixing of δne and δB‖ in
the direction transverse to the magnetic field (see § 2.6 for a
quick recapitulation of the standard scaling argument on the
passive cascade that leads to a k
⊥ in the perpendicular di-
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 29
FIG. 7.— Lagrangian mixing of passive fields: fluctuations develop small
scales across, but not along the exact field lines.
rection). Figure 7 illustrates this situation.26
We emphasize that this lack of nonlinear refinement of the
scale of δne and δB‖ along the moving field lines is a particu-
lar property of the compressive component of the turbulence,
not shared by the Alfvén waves. Indeed, unlike Eq. (157), the
RMHD equations (155-156), do not reduce to a linear form
under the Lagrangian transformation (214), so the Alfvén
waves should develop small scales both across and along the
perturbed magnetic field.
Whether the density and magnetic-field-strength fluctua-
tions develop small scales along the magnetic field has direct
physical and observational consequences. Damping of these
fluctuations, both in the collisional and collisionless regimes,
discussed in § 6.1.2 and § 6.2.2, respectively, depends pre-
cisely on their scale along the perturbed field: indeed, the
linear results derived there are exact in the Lagrangian frame
(214). To summarize these results, the damping rate of δne
and δB‖ at βi ∼ 1 is
γ∼ vthiλmfpik2‖0, k‖0λmfpi ≪ 1, (217)
γ∼ vthik‖0, k‖0λmfpi ≫ 1, (218)
where k‖0 ∼ b̂ ·∇ is the wavenumber along the perturbed field
(i.e., if there is no parallel cascade, the wavenumber of the
large-scale stirring).
Whether this damping cuts off the cascades of δne and δB‖
depends on the relative magnitudes of the damping rate γ for
a given k⊥ and the characteristic rate at which the Alfvén
waves cause δne and δB‖ to cascade to higher k⊥. This rate
is ωA ∼ k‖AvA, where k‖A is the parallel wave number of the
Alfvén waves that have the same k⊥. Since the Alfvén waves
do have a parallel cascade, assuming scale-by-scale critical
balance (3) leads to [Eq. (5)]
k‖A ∼ k
0 . (219)
If, in contrast to the Alfvén waves, δne and δB‖ have no par-
allel cascade, k‖0 does not grow with k⊥, so, for large enough
k⊥, k‖0 ≪ k‖A and γ≪ωA. This means that, despite the damp-
ing, the density and field-strength fluctuations should have
perpendicular cascades extending to the ion gyroscale.
The validity of the argument at the beginning of this sec-
tion that ruled out the parallel cascade of δne and δB‖ is not
quite as obvious as it might appear. Lithwick & Goldreich
(2001) argued that the dissipation of δne and δB‖ at the ion
gyroscale would cause these fluctuations to become uncorre-
lated at the same parallel scales as the Alfvénic fluctuations by
which they are mixed, i.e., k‖0 ∼ k‖A. The damping rate then
becomes comparable to the cascade rate, cutting off the cas-
cades of density and field-strength fluctuations at k‖λmfpi ∼ 1.
The corresponding perpendicular cutoff wavenumber is [see
26 Note that effectively, there is also a cascade in k‖ if the latter is mea-
sured along the unperturbed field—more precisely, a cascade in kz. This is
due to the perpendicular deformation of the perturbed magnetic field by the
Alfvén-wave turbulence: since ∇⊥ grows while b̂ ·∇ remains the same, we
have from Eq. (123) ∂/∂z ≃ −(δB⊥/B0) ·∇⊥.
Eq. (219)]
k⊥ ∼ l1/20 λ
mfpi . (220)
Asymptotically speaking, in a weakly collisional plasma,
this cutoff is far above the ion gyroscale, k⊥ρi ≪ 1. How-
ever, the relatively small value of λmfpi in the warm ISM,
which was the main focus of Lithwick & Goldreich 2001,
meant that the numerical value of the perpendicular cutoff
scale given by Eq. (220) was, in fact, quite close both to
the ion gyroscale (see Table 1) and to the observational es-
timates for the inner scale of the electron-density fluctuations
in the ISM (Spangler & Gwinn 1990; Armstrong et al. 1995).
Thus, it was not possible to tell whether Eq. (220), rather than
k⊥ ∼ ρ−1i , represented the correct prediction.
The situation is rather different in the nearly collision-
less case of the solar wind, where the cutoff given by
Eq. (220) would mean that very little density or field-
strength fluctuations should be detected above the ion gy-
roscale. Observations do not support such a conclu-
sion: the density fluctuations appear to follow a k−5/3 law
at all scales larger than a few times ρi (Lovelace et al.
1970; Woo & Armstrong 1979; Celnikier et al. 1983, 1987;
Coles & Harmon 1989; Marsch & Tu 1990b; Coles et al.
1991), consistently with the expected behavior of an un-
damped passive scalar field (see § 2.6). An extended range
of k−5/3 scaling above the ion gyroscale is also observed for
the fluctuations of the magnetic-field strength (Marsch & Tu
1990b; Bershadskii & Sreenivasan 2004; Hnat et al. 2005;
Alexandrova et al. 2008a).
These observational facts suggest that the cutoff formula
(220) does not apply. This does not, however, conclusively
vitiate the Lithwick & Goldreich (2001) theory. Heuristically,
their argument is plausible, although it is, perhaps, useful
to note that in order for the effect of the perpendicular dis-
sipation terms, not present in the KRMHD equations (157-
159), to be felt, the density and field-strength fluctuations
should reach the ion gyroscale in the first place. Quanti-
tatively, the failure of the compressive fluctuations in the
solar wind to be damped could still be consistent with the
Lithwick & Goldreich (2001) theory because of the relative
weakness of the collisionless damping, especially at low beta
(§ 6.2.2)—the explanation they themselves favor. The way to
check observationally whether this explanation suffices would
be to make a comparative study of the compressive fluctua-
tions for solar-wind data with different values of βi. If the
strength of the damping is the decisive factor, one should al-
ways see cascades of both δne and δB‖ at low βi, no cascades
at βi ∼ 1, and a cascade of δB‖ but not δne at high βi (in
this limit, the damping of the density fluctuations is strong,
of the field-strength weak; see § 6.2.2). If, on the other hand,
the parallel cascade of the compressive fluctuations is intrin-
sically inefficient, very little βi dependence is expected and a
perpendicular cascade should be seen in all cases.
Obviously, an even more direct observational (or numer-
ical) test would be the detection or non-detection of near-
perfect alignment of the density and field-strength structures
with the moving field lines (not with the mean magnetic
field—see footnote 26), but it is not clear how to measure
this reliably. It is interesting, in this context, that in near-
the-Sun measurements, the density fluctuations are reported
to have the form of highly anisotropic filaments aligned with
the magnetic field (Armstrong et al. 1990; Grall et al. 1997;
Woo & Habbal 1997). Another intriguing piece of observa-
30 SCHEKOCHIHIN ET AL.
tional evidence is the discovery that the local structure of the
magnetic-field-strength and density fluctuations at 1 AU is, in
a certain sense, correlated with the solar cycle (Kiyani et al.
2007; Hnat et al. 2007; Wicks et al. 2009)—this suggests a
dependence on initial conditions that is absent in the Alfvénic
fluctuations and that presumably should also disappear in the
compressive fluctuations if the latter are fully mixed both in
the perpendicular and parallel directions.
7. TURBULENCE IN THE DISSIPATION RANGE: ELECTRON RMHD
AND THE ENTROPY CASCADE
7.1. Transition at the Ion Gyroscale
The validity of the theory discussed in § 5 and § 6 breaks
down when k⊥ρi ∼ 1. As the ion gyroscale is approached,
the Alfvén waves are no longer decoupled from the rest of
the plasma dynamics. All modes now contain perturbations
of density and magnetic-field strength and can be collision-
lessly damped. Because of the low-frequency nature of the
Alfvén-wave cascade, ω ≪ Ωi even at k⊥ρi ∼ 1 [Eq. (46)],
so the ion cyclotron resonance (ω − k‖v‖ = ±Ωi) is not im-
portant, while the Landau one (ω = k‖v‖) is. The linear the-
ory of this collisionless damping in the gyrokinetic approx-
imation is worked out in detail in Howes et al. (2006) (see
also Gary & Borovsky 2008). Figure 8 shows the solutions of
their dispersion relation that illustrate how the Alfvén wave
becomes a dispersive kinetic Alfvén wave (KAW) (see § 7.3)
and collisionless damping becomes important as the ion gy-
roscale is reached.
We stress that this transition occurs at the ion gyroscale, not
at the ion inertial scale di = ρi/
βi (except in the limit of cold
ions, τ = T0i/T0e ≪ 1; see Appendix E). This statement is true
even when βi is not order unity, as illustrated in Fig. 8: for the
three cases plotted there, k⊥di = 1 corresponds to k⊥ρi = 0.1,
1 and 10 for βi = 0.01, 1 and 100, respectively, but there is
no trace of the ion inertial scale in the solutions of the linear
dispersion relation. Nonlinearly, in the limit βi ≪ 1, we may
consider the scales k⊥di ∼ 1 and expand the gyrokinetics in
k⊥ρi = k⊥di
βi ≪ 1 in a way similar to how it was done in § 5
and obtain precisely the same results: Alfvénic fluctuations
described by the RMHD equations and compressive fluctua-
tions passively advected by them and satisfying the reduced
kinetic equation derived in § 5.5. Thus, even though di ≫ ρi
at low beta, there is no change in the nature of the turbulent
cascade until k⊥ρi ∼ 1 is reached.
The nonlinear theory of what happens at k⊥ρi ∼ 1 is very
poorly understood. It is, however, possible to make progress
by examining what kind of fluctuations emerge on the other
side of the transition, at k⊥ρi ≫ 1. As we will demonstrate
below, it turns out that another turbulent cascade—this time
of KAW—is possible in this so-called dissipation range. It
can transfer the energy of KAW-like fluctuations down to the
electron gyroscale, where electron Landau damping becomes
important (see Howes et al. 2006). Some observational evi-
dence of KAW is, indeed, available in the solar wind and the
magnetosphere (Bale et al. 2005; Grison et al. 2005, see fur-
ther discussion in § 8.2.4). Below we derive the equations that
describe KAW-like fluctuations in the scale range k⊥ρi ≫ 1,
k⊥ρe ≪ 1 (§ 7.2) and work out a Kolmogorov-style scaling
theory for this cascade (§ 7.5).
Because of the presence of the collisionless damping at the
ion gyroscale, only a certain fraction of the turbulent power
arriving there from the inertial range is converted into the
KAW cascade, while the rest is Landau-damped. The damp-
ing leads to the heating of the ions, but the process of deposit-
ing the collisionlessly damped fluctuation energy into the ion
heat is non-trivial because, as we explained in § 3.5, collisions
do need to play a role in order for true heating to occur. As
we explained in § 3.5 and will see specifically for the dissi-
pation range in § 7.8, the electromagnetic-fluctuation energy
does not disappear as a result of the Landau damping but is
converted into ion entropy fluctuations, while the generalized
energy is conserved. Collisions are then accessed and ion
heating achieved via a purely kinetic phenomenon: the ion
entropy cascade in phase space (nonlinear phase mixing), for
which a theory is developed in § 7.9 and § 7.10. A similar pro-
cess of conversion of the KAW energy into electron entropy
fluctuations and then electron heat is treated in § 7.12.
Figure 5 illustrates the routes energy takes from the ion gy-
roscale towards heating. Crucially, it is at k⊥ρi ∼ 1 that it
is decided how much energy would eventually go into the
ions and how much into electrons.27 How this distribution
of energy depends on plasma parameters (βi and T0i/T0e)
is an open theoretical question28 of considerable astrophys-
ical interest: e.g., the efficiency of ion heating is a key un-
known in the theory of advection-dominated accretion flows
(Quataert & Gruzinov 1999, see discussion in § 8.5) and of
the solar corona (e.g., Cranmer & van Ballegooijen 2003); we
will also see in § 7.11 that it may determine the form of the
observed dissipation-range spectra in space plasmas.
A short summary of this section is given in § 7.14.
7.2. Equations of Electron Reduced MHD
The derivation is straightforward: when ai ∼ k⊥ρi ≫ 1, all
Bessel functions in Eqs. (118-120) are small, so the integrals
of the ion distribution function vanish and Eqs. (118-120) be-
, (221)
u‖e =
4πen0e
∇2⊥A‖ = −
ρi∇2⊥Ψ√
, u‖i = 0, (222)
, (223)
where we used the definitions (135) of the stream and flux
functions Φ and Ψ.
These equations are a reflection of the fact that, for k⊥ρi ≫
1, the ion response is effectively purely Boltzmann, with the
gyrokinetic part hi contributing nothing to the fields or flows
[see Eq. (54) with hi omitted; hi does, however, play an impor-
tant role in the energy balance and ion heating, as explained
in §§ 7.8-7.10 below]. The Boltzmann response for ion den-
sity is expressed by Eq. (221). Equation (222) states that the
parallel ion flow velocity can be neglected. Finally, Eq. (223)
expresses the pressure balance for Boltzmann (and, therefore,
isothermal) electrons [Eq. (103)] and ions: if we write
B0δB‖
= −δpi − δpe = −T0iδni − T0eδne, (224)
27 Some of the energy of compressive fluctuations may go into ion heat via
collisional (§ 6.1.2) or collisionless (§ 6.2.2) damping of these fluctuations
in the inertial range. Whether this is a significant ion heating mechanism
depends on the efficiency of the parallel cascade (see § 6.2.4 and § 6.3).
28 How much energy is converted into ion entropy fluctuations in the pro-
cess of a nonlinear turbulent cascade is not necessarily directly related to the
strength of the linear collisionless damping.
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 31
FIG. 8.— Numerical solutions of the linear gyrokinetic dispersion relation (for a detailed treatment of the linear theory, see Howes et al. 2006) showing the
transition from the Alfvén wave to KAW between the inertial range (k⊥ρi ≪ 1) and the dissipation range (k⊥ρi & 1). We show three cases: low beta (βi = 0.01),
βi = 1, and high beta (βi = 100). In all three cases, τ = 1 and Z = 1. Bold solid lines show the real frequency ω, bold dashed lines the damping rate γ, both
normalized by k‖vA (in gyrokinetics, ω/k‖vA and γ/k‖vA are functions of k⊥ only). Dotted lines show the asymptotic KAW solution (230). Horizontal solid line
shows the Alfvén wave ω = k‖vA. Vertical solid lines show k⊥ρi = 1 and k⊥ρe = 1. Note that the damping can be considered strong if the characteristic decay
time is comparable or shorter than the wave period, i.e., γ/ω & 1/2π. Thus, in these plots, the damping at k⊥ρi ∼ 1 is relatively weak for βi = 1, relatively
strong for low beta and very strong for high beta.
it follows that
, (225)
which, combined with Eq. (221), gives Eq. (223). We remind
the reader that the perpendicular Ampère’s law, from which
Eq. (223) was derived [Eq. (66) via Eq. (120)] is, in gyrokinet-
ics, indeed equivalent to the statement of perpendicular pres-
sure balance (see § 3.3).
Substituting Eqs. (221-223) into Eqs. (116-117), we obtain
the following closed system of equations
b̂ ·∇Φ, (226)
2 +βi
1 + Z/τ
) b̂ ·∇
ρ2i ∇2⊥Ψ
. (227)
Note that, using Eq. (223), Eqs. (226) and (227) can be recast
as two coupled evolution equations for the perpendicular and
parallel components of the perturbed magnetic field, respec-
tively [Eqs. (C10) in Appendix C.2].
We shall refer to Eqs. (226-227) as Electron Reduced MHD
(ERMHD). They are related to the Electron Magnetohydrody-
namics (EMHD)—a fluid-like approximation that evolves the
magnetic field only and arises if one assumes that the mag-
netic field is frozen into the electron flow velocity ue, while
the ions are immobile, ui = 0 (Kingsep et al. 1990):
4πen0e
∇× [(∇×B)×B] . (228)
As explained in Appendix C.2, the result of applying the
RMHD/gyrokinetic ordering (§ 2.1 and § 3.1) to Eq. (228),
where B = B0ẑ + δB and
ẑ×∇⊥Ψ+ ẑ
, (229)
coincides with our Eqs. (226-227) in the effectively incom-
pressible limits of βi ≫ 1 or βe = βiZ/τ ≫ 1. When betas are
arbitrary, density fluctuations cannot be neglected compared
to the magnetic-field-strength fluctuations [Eq. (225)] and
give rise to perpendicular ion flows with ∇·ui 6= 0. Thus, our
ERMHD system constitutes the appropriate generalization of
EMHD for low-frequency anisotropic fluctuations without the
assumption of incompressibility.
A (more tenuous) relationship also exists between our
ERMHD system and the so-called Hall MHD, which, like
EMHD, is based on the magnetic field being frozen into
the electron flow, but includes the ion motion via the stan-
dard MHD momentum equation [Eq. (8)]. Strictly speak-
ing, Hall MHD can only be used in the limit of cold ions,
τ = T0i/T0e ≪ 1 (see, e.g., Ito et al. 2004; Hirose et al. 2004,
and Appendix E), in which case it can be shown to reduce
to Eqs. (226-227) in the appropriate small-scale limit (Ap-
pendix E). Although τ ≪ 1 is not a natural assumption for
most space and astrophysical plasmas, Hall MHD has, due to
its simplicity, been a popular theoretical paradigm in the stud-
ies of space and astrophysical plasma turbulence (see § 8.2.6).
We have therefore devoted Appendix E to showing how this
approximation fits into the theoretical framework proposed
here: namely, we derive the anisotropic low-frequency ver-
sion of the Hall MHD approximation from gyrokinetics under
the assumption τ ≪ 1 and discuss the role of the ion inertial
and ion sound scales, which acquire physical significance in
this limit. However, outside this Appendix, we assume τ ∼ 1
everywhere and shall not use Hall MHD.
The validity of the ERMHD equations as a model for
plasma dynamics in the dissipation range is further discussed
in § 7.6.
7.3. Kinetic Alfvén Waves
The linear modes supported by ERMHD are kinetic Alfvén
waves (KAW) with frequencies
ωk = ±
1 + Z/τ
2 +βi
1 + Z/τ
) k⊥ρik‖vA. (230)
This dispersion relation is illustrated in Fig. 8: note that the
transition from Alfvén waves to dispersive KAW always oc-
curs at k⊥ρi ∼ 1, even when βi ≪ 1 or βi ≫ 1. In the latter
case, there is a sharp frequency jump at the transition (accom-
panied by very strong ion Landau damping).
The eigenfunctions corresponding to the two waves with
32 SCHEKOCHIHIN ET AL.
FIG. 9.— Polarization of the kinetic Alfvén wave, see Eqs. (232) and (233).
frequencies (230) are
2 +βi
∓ k⊥Ψk. (231)
Using Eqs. (229) and (223), the perturbed magnetic-field vec-
tor can be expressed as follows
= −iẑ× k⊥
1 + Z/τ
2 +βi
1 + Z/τ
(232)
so, for a single “+” or “−” wave (corresponding to Θ−k = 0 or
k = 0, respectively), δBk rotates in the plane perpendicular
to the wave vector k⊥ clockwise with respect to the latter,
while the wave propagates parallel or antiparallel to the guide
field (Fig. 9).
The waves are elliptically right-hand polarized. Indeed, us-
ing Eq. (223), the perpendicular electric field is:
E⊥k = −ik⊥ϕ+
−ik⊥ + ẑ×k⊥
ϕ (233)
(cf. Gary 1986; Hollweg 1999). The second term is small in
the gyrokinetic expansion, so this is a very elongated ellipse
(Fig. 9).
7.4. Finite-Amplitude Kinetic Alfvén Waves
As we are about to argue for a critically balanced KAW
turbulence in a fashion analogous to the GS theory for the
Alfvén waves (§ 1.2), it is a natural question to ask how simi-
lar the nonlinear properties of a putative KAW cascade will be
to an Alfvén-wave cascade. As in the case of Alfvén waves,
there are two counterpropagating linear modes [Eqs. (230)
and (231)], and it turns out that certain superpositions of these
modes (KAW packets) are also exact nonlinear solutions of
Eqs. (226-227). Let us show that this is the case.
We might look for the nonlinear solutions of Eqs. (226-227)
by requiring that the nonlinear terms vanish. Since b̂ · ∇ =
∂/∂z + (1/vA){Ψ, · · ·}, this gives
{Ψ,Φ} = 0 ⇒ Ψ = c1Φ, (234)
{Ψ,ρ2i ∇2⊥Ψ} = 0 ⇒ ρ2i ∇2⊥Ψ = c2Ψ, (235)
where c1 and c2 are constants. Whether such solutions are
possible is determined by substituting Eqs. (234) and (235)
into Eqs. (226) and (227) and demanding that the two result-
ing linear equations be consistent with each other (both equa-
tions now just evolve Ψ). This is achieved if29
c21 = −
2 +βi
, (236)
so real solutions exist if c2 < 0. In particular, wave pack-
ets consisting of KAW given by one of the linear eigen-
modes (231) with an arbitrary shape in z but confined to a
single shell |k⊥| = k⊥ = const, satisfy Eqs. (234-236) with
c2 = −k2⊥ρ
i . This outcome is, in fact, only mildly non-trivial:
in gyrokinetics, the Poisson bracket nonlinearity [Eq. (59)]
vanishes for any monochromatic (in k⊥) mode because the
Poisson bracket of two modes with wavenumbers k⊥ and k′⊥
is ∝ ẑ · (k⊥ × k′⊥). Therefore, any monochromatic solution
of the linearized equations is also an exact nonlinear solution.
As we have shown above, a superposition of monochromatic
KAW that have a fixed k⊥, or, somewhat more generally, sat-
isfy Eq. (235) with a fixed c2, is still an exact solution.
Note that a similar procedure applied to the RMHD equa-
tions (17-18) returns the Elsasser solutions: perturbations of
arbitrary shape that satisfy Φ = ±Ψ. The physical difference
between these finite-amplitude Alfven-wave packets and the
finite-amplitude KAW packets discussed above is that non-
linear interactions can occur not just between counterpropa-
gating KAW but also between copropagating ones—a natural
conclusion because KAW are dispersive (their group velocity
along the guide field is ∝ vAk⊥ρi), so copropagating waves
with different k⊥ can “catch up” with each other and inter-
act.30
7.5. Scalings for KAW Turbulence
A scaling theory for the turbulence described by Eqs. (221-
227) can be constructed along the same lines as the GS theory
for the Alfvén-wave turbulence (§ 1.2). Namely, we shall as-
sume that the turbulence below the ion gyroscale consists of
KAW-like fluctuations with k‖ ≪ k⊥ (Quataert & Gruzinov
1999) and that the interactions between them are critically
balanced (Cho & Lazarian 2004), i.e., that the propagation
time and nonlinear interaction time are comparable at every
scale. We stress that none of these assumptions are, strictly
speaking, inevitable31 (and, in fact, neither were they in-
evitable in the case of Alfvén waves). Since we have de-
rived Eqs. (226-227) from gyrokinetics, the anisotropy of
the fluctuations described by these equations is hard-wired,
but it is not guaranteed that the actual physical cascade be-
low the ion gyroscale is indeed anisotropic, although anal-
ysis of solar-wind measurements does seem to indicate that
29 Formally speaking, c1 and c2 can depend on t and z. If this is allowed,
we still recover Eq. (236), but in addition to it, we get the evolution equation
c1∂c1/∂t = vA(1 + Z/τ )∂c1/∂z. This allows c1 = const, but there are, of
course, other solutions. We shall not consider them here.
30 The calculation above is analogous to the calculation by
Mahajan & Krishan (2005) for incompressible Hall MHD (i.e., essen-
tially, the high-βe limit of the equations discussed in Appendix E), but
the result is more general in the sense that it holds at arbitrary ion and
electron betas. The Mahajan–Krishan solution in the EMHD limit amounts
to noticing that Eq. (228) becomes linear for force-free (Beltrami) magnetic
perturbations, ∇× δB = λδB. Substituting Eq. (229) into this equation
and using Eq. (223), we see that the force-free equation is equivalent
to Eqs. (234-236) if c2 = −λ2 and the incompressible limit (βi ≫ 1 or
βe = βiZ/τ ≫ 1) is taken.
31 In fact, the EMHD turbulence was thought to be weak by several au-
thors, who predicted a k−2 spectrum of magnetic energy assuming isotropy
(Goldreich & Reisenegger 1992) or k
for the anisotropic case (Voitenko
1998; Galtier & Bhattacharjee 2003; Galtier 2006).
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 33
at least a significant fraction of it is (see Leamon et al.
1998; Hamilton et al. 2008). Numerical simulations based
on Eq. (228) (Biskamp et al. 1996, 1999; Ghosh et al. 1996;
Ng et al. 2003; Cho & Lazarian 2004; Shaikh & Zank 2005)
have revealed that the spectrum of magnetic fluctuations
scales as k
⊥ , the outcome consistent with the assumptions
stated above. Let us outline the argument that leads to this
scaling.
First assume that the fluctuations are KAW-like and that Θ+
and Θ− [Eq. (231)] have similar scaling. This implies
1 +βi
Φλ (237)
(for the purposes of scaling arguments and order-of-
magnitude estimates, we set Z/τ = 1, but keep the βi de-
pendence so low- and high-beta limits could be recovered if
necessary). The fact that fixed-k⊥ KAW packets, which sat-
isfy Eq. (237) with λ = 1/k⊥, are exact nonlinear solutions
of the ERMHD equations (§ 7.4) lends some credence to this
assumption.
Assuming scale-space locality of interactions implies a
constant-flux KAW cascade: analogously to Eq. (1),
(Ψλ/λ)
τKAWλ
∼ (1 +βi)(Φλ/ρi)
τKAWλ
∼ εKAW = const, (238)
where τKAWλ is the cascade time and εKAW is the KAW energy
flux proportional to the fraction of the total flux ε (or the total
turbulent power Pext; see § 3.4) that was converted into the
KAW cascade at the ion gyroscale.
Using Eqs. (226-227) and Eq. (237), it is not hard to see
that the characteristic nonlinear decorrelation time is λ2/Φλ.
If the turbulence is strong, then this time is comparable to
the inverse KAW frequency [Eq. (230)] scale by scale and we
may assume the cascade time is comparable to either:
τKAWλ ∼
1 +βi
. (239)
In other words, this says that ∂/∂z ∼ (δB⊥/B0) ·∇⊥ and so
δB⊥λ/B0 ∼ λ/l‖λ (note that the last relation confirms that
our scaling arguments do not violate the gyrokinetic ordering;
see § 2.1 and § 3.1). Equation (239) is the critical-balance as-
sumption for KAW. As in the case of the Alfvén waves (§ 1.2),
we might argue physically that the critical balance is set up be-
cause the parallel correlation length l‖λ is determined by the
condition that a wave can propagate the distance l‖λ in one
nonlinear decorrelation time corresponding to the perpendic-
ular correlation length λ.
Combining Eqs. (238) and (239), we get the desired scaling
relations for the KAW turbulence:
(εKAW
)1/3 vA
(1 +βi)1/3
2/3, (240)
(1 +βi)1/6
, (241)
where l0 = v
A/ε, as in § 1.2. The first of these scaling relations
is equivalent to a k
⊥ spectrum of magnetic energy, the sec-
ond quantifies the anisotropy (which is stronger than for the
GS turbulence). Both scalings were confirmed in the numer-
ical simulations of Cho & Lazarian (2004)—it is their detec-
tion of the scaling (241) that makes a particularly strong case
that KAW turbulence is not weak and that the critical balance
hypothesis applies.
For KAW-like fluctuations, the density [Eq. (221)] and
magnetic field [Eqs. (223) and (231)] have the same spec-
trum as the scalar potential, i.e., k
⊥ , while the electric field
E ∼ k⊥ϕ has a k−1/3⊥ spectrum. The solar-wind fluctuation
spectra reported by Bale et al. (2005) indeed are consistent
with a transition to KAW turbulence around the ion gyroscale:
k−5/3 magnetic and electric-field power spectra at kρi ≪ 1 are
replaced, for kρi & 1, with what appears to be consistent with
a k−7/3 scaling for the magnetic-field spectrum and a k−1/3 for
the electric one (see Fig. 1). A similar result is recovered in
fully gyrokinetic simulations with βi = 1, τ = 1 (Howes et al.
2008b). However, not all solar-wind observations are quite as
straightforwardly supportive of the notion of the KAW cas-
cade and much steeper magnetic-fluctuation spectra have also
been reported (e.g., Denskat et al. 1983; Leamon et al. 1998;
Smith et al. 2006). Possible reasons for this will emerge in
§ 7.6 and § 7.11 and the solar-wind data are further discussed
in § 8.2.4 and § 8.2.5.
7.6. Validity of the Electron RMHD and the Effect of Electron
Landau Damping
The ERMHD equations derived in § 7 are valid provided
k⊥ρi ≫ 1 and also provided it is sufficient to use the leading
order in the mass-ratio expansion (isothermal electrons; see
§ 4). In particular, this means that the electron Landau damp-
ing is neglected. Asymptotically speaking, this is a rigorous
limit, but one must be cautious in applying it to real plas-
mas. Since the width of the scale range where k⊥ρi ≫ 1 and
k⊥ρe ≪ 1 is only ∼ (mi/me)1/2 ≃ 43, for some values of the
plasma parameters (T0i/T0e and βi) there may not be a very
broad interval of scales where the electron Landau damping
is truly negligible. Consider, for example, the low-beta limit,
βi ≪ 1. In this limit, the KAW frequency is ω ∼ k⊥ρik‖vA
[Eq. (230)]. The electron Landau damping becomes impor-
tant when ω ∼ k‖vthe, or k⊥ρe ∼
βi ≪ 1, so the ERMHD
approximation breaks down and, consequently, the KAW cas-
cade, if any, should be interrupted well before the electron
gyroscale is reached. Figure 8 shows the solution of the
full gyrokinetic dispersion relation (Howes et al. 2006) for
small, unity and large βi. One can judge for which scales and
how well (or how badly) the ERMHD approximation holds
from the precision with which the exact frequency follows the
asymptotic solution Eq. (230) and from the relative strength
of the damping compared to the real frequency of the waves.
Non-negligible electron Landau damping may affect turbu-
lence spectra because one can no longer assume a constant
flux of KAW energy as we did in § 7.5. To evaluate the conse-
quences of this effect, Howes et al. (2008a) constructed a sim-
ple model of spectral energy transfer and concluded that Lan-
dau damping leads to steepening of the KAW spectra—one
of several possible reasons for steep dissipation-range spectra
observed in space plasmas (see also § 7.11).
7.7. Unfreezing of Flux
As ERMHD is a limit of the isothermal-electron-fluid sys-
tem (§ 4), the magnetic-field lines remain unbroken (see
§ 4.3). Within the orderings employed above (small mass ra-
tio, νii ∼ ω, βi ∼ 1, τ ∼ 1), the flux unfreezes only in the
vicinity of the electron gyroscale. It is interesting to evaluate
somewhat more precisely the scale at which this happens as a
function of plasma parameters.
34 SCHEKOCHIHIN ET AL.
Physically, there are three kinds of mechanisms by which
the flux conservation is broken: electron inertia, the effects of
finite electron gyroradius, and Ohmic resistivity. Let us take
the v‖ moment of the electron gyrokinetic equation [Eq. (57),
s = e, integration at constant r] and use Eq. (222) to evaluate
the inertial term in the resulting parallel electron momentum
equation:
d2e∇2⊥A‖, (242)
where de = ρe/
βe is the electron inertial scale and βe =
Zβi/τ . Comparing this with the ∂A‖/∂t term in the right-
hand side of the electron momentum equation, we see that the
electron inertia becomes important when k⊥ρe ∼
βe. The
finite-gyroradius effects enter when k⊥ρe ∼ 1. Thus, at low
βe, the electron inertia becomes important above the electron
gyroscale, whereas at high βe, the finite-gyroradius effects en-
ter first. Finally, the Ohmic resistivity comes from the colli-
sion term (see Appendix B.4):
d3vv‖
νeiu‖e ∼ νeik2⊥d2e A‖. (243)
Thus, resistivity starts to act when k⊥de ∼ (ω/νei)1/2. Using
the KAW frequency [Eq. (230)] to estimate ω and assuming
that τ is not small, we get
k⊥ρe ∼ k‖λmfpi
1 +βi
. (244)
Thus, the resistive scale can only be larger the electron gy-
roscale if the plasma is collisional (k‖λmfpi ≪ 1) and/or elec-
trons are much colder than ions (τ ≫ 1) and/or βi ≪ 1. Note
if only the last of these conditions is satisfied, the electron
inertia still becomes important at larger scales than resistivity.
7.8. Generalized Energy: KAW and Entropy Cascades
The generalized energy (§ 3.4) in the limit k⊥ρi ≫ 1 is cal-
culated by substituting Eqs. (221) and (223) into Eq. (109):
T0i〈h2i 〉r
n0iT0i
=Whi +WKAW. (245)
Here the first term, Whi , is the total variance of hi, which is
proportional to minus the entropy of the ion gyrocenter distri-
bution (see § 3.5) and whose cascade to collisional scales will
be discussed in § 7.9 and § 7.10. The remaining two terms are
the independently cascaded KAW energy:
WKAW =
min0i
|∇⊥Ψ|2
min0i
|Θ+|2 + |Θ−|2
. (246)
Although we can write WKAW as the sum of the energies of
the “+” and “−” linear KAW eigenmodes [Eq. (231)], which
are also exact nonlinear solutions (§ 7.4), the two do not cas-
cade independently and can exchange energy. Note that the
ERMHD equations also conserve
d3rΨΦ, which is readily
interpreted as the helicity of the perturbed magnetic field (see
Appendix F.3). However, it does not affect the KAW cascade
discussed in § 7.5 because it can be argued to have a tendency
to cascade inversely (Appendix F.6).
Comparing the way the generalized energy is split above
and below the ion gyroscale (see § 5.6 for the k⊥ρi ≪ 1 limit),
we interpret what happens at the k⊥ρi ∼ 1 transition as a redis-
tribution of the power that arrived from large scales between
a cascade of KAW and a cascade of the (minus) gyrocenter
entropy in the phase space (see Fig. 5). The latter cascade
is the way in which the energy diverted from the electromag-
netic fluctuations by the collisionless damping (wave–particle
interaction) can be transferred to the collisional scales and de-
posited into heat (§ 7.1). The concept of entropy cascade as
the key agent in the heating of the plasma was introduced in
§ 3.5, where we promised a more detailed discussion later on.
We now proceed to this discussion.
7.9. Entropy Cascade
The ion-gyrocenter distribution function hi satisfies the ion
gyrokinetic equation (121), where ion–electron collisions are
neglected under the mass-ratio expansion. At k⊥ρi ≫ 1, the
dominant contribution to 〈χ〉Ri comes from the electromag-
netic fluctuations associated with KAW turbulence. Since
the KAW cascade is decoupled from the entropy cascade, hi
is a passive tracer of the ring-averaged KAW turbulence in
phase space. Expanding the Bessel functions in the expres-
sion for 〈χ〉Ri ,k [ai ≫ 1 in Eq. (69) with s = i] and making
use of Eqs. (222-223) and of the KAW scaling Ψ ∼ Φ/k⊥ρi
[Eq. (231)], it is not hard to show that
〈χ〉Ri ,k ≃
〈ϕ〉Ri ,k =
J0(ai)Φk
, (247)
where
J0(ai) ≃
, ai = k⊥ρi
, (248)
so hi satisfies [Eq. (121)]
+{〈Φ〉Ri ,hi} =
βiρivA
∂〈Φ〉Ri
F0i + 〈Cii[hi]〉Ri
(249)
with the conservation law [Eq. (70), s = i]
βi ρivA
∂〈Φ〉Ri
hi 〈Cii[hi]〉Ri
. (250)
7.9.1. Nonlinear Perpendicular Phase Mixing
The wave–particle interaction term (the first term on the
right hand sides of these two equations) will shortly be seen
to be subdominant at k⊥ρi ≫ 1. It represents the source of
the invariant Whi due to the collisionless damping at the ion
gyroscale of some fraction of the energy arriving from the in-
ertial range. In a stationary turbulent state, we should have
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 35
FIG. 10.— Nonlinear perpendicular phase-mixing mechanism: the
gyrocenter distribution function at Ri of particles with velocities v⊥ and v
is mixed by turbulent fluctuations of the potential Φ (E×B flows) averaged
over particle orbits separated by a distance greater than the correlation length
of Φ.
dWhi/dt = 0 and this source should be balanced on average by
the (negative definite) collisional dissipation term ( = heating;
see § 3.5). This balance can only be achieved if hi develops
small scales in the velocity space and carries the generalized
energy, or, in this case, entropy, to scales in the phase space at
which collisions are important. A quick way to see this is by
recalling that the collision operator has two velocity deriva-
tives and can only balance the terms on the left-hand side of
Eq. (249) if
∼ ω ⇒ δv
, (251)
where ω is the characteristic frequency of the fluctuations
of hi. If νii ≪ ω, δv/vthi ≪ 1. This is certainly true for
k⊥ρi ∼ 1: taking ω ∼ k‖vA and using k‖λmfpi ≫ 1 (which
is the appropriate limit at and below the ion gyroscale for
most of the plasmas of interest; cf. footnote 24), we have
νii/ω ∼
βi/k‖λmfpi ≪ 1.
The condition (251) means that the collision rate can be ar-
bitrarily small—this will always be compensated by the suf-
ficiently fine velocity-space structure of the distribution func-
tion to produce a finite amount of entropy production (heat-
ing) independent of νii in the limit νii → +0. The situa-
tion bears some resemblance to the emergence of small spa-
tial scales in neutral-fluid turbulence with arbitrarily small
but non-zero viscosity (Kolmogorov 1941). The analogy
is not perfect, however, because the ion gyrokinetic equa-
tion (249) does not contain a nonlinear interaction term that
would explicitly cause a cascade in the velocity space. In-
stead, the (ring-averaged) KAW turbulence mixes hi in the
gyrocenter space via the nonlinear term in Eq. (249), so hi
will have small-scale structure in Ri on characteristic scales
much smaller than ρi. Let us assume that the dominant non-
linear effect is a local interaction of the small-scale fluctua-
tions of hi with the similarly small-scale component of 〈Φ〉Ri .
Since ring averaging is involved and k⊥ρi is large, the val-
ues of 〈Φ〉Ri corresponding to two velocities v and v′ will
come from spatially decorrelated electromagnetic fluctuations
if k⊥v⊥/Ωi and k⊥v
⊥/Ωi [the argument of the Bessel function
in Eq. (247)] differ by order unity, i.e., for
|v⊥ − v′⊥|
(252)
(see Fig. 10). This relation gives a correspondence between
the decorrelation scales of hi in the position and velocity
space. Combining Eqs. (252) and (251), we see that there is
a collisional cutoff scale determined by k⊥ρi ∼ (ω/νii)1/2 ≫
1.32 The cutoff scale is much smaller than the ion gyroscale.
In the range between these scales, collisional dissipation is
small. The ion entropy fluctuations are transferred across this
scale range by means of a cascade, for which we will con-
struct a scaling theory in § 7.9.2 (and, for the case without the
background KAW turbulence, in § 7.10).
It is important to emphasize that no matter how small the
collisional cutoff scale is, all of the generalized energy chan-
neled into the entropy cascade at the ion gyroscale eventually
reaches it and is converted into heat. Note that the rate at
which this happens is in general amplitude-dependent because
the process is nonlinear, although we will argue in § 7.9.4 (see
also § 7.10.3) that the nonlinear cascade time and the parallel
linear propagation (particle streaming) time are related by a
critical-balance-like condition (we will also argue there that
the linear parallel phase mixing, which can generate small
scales in v‖, is a less efficient process than the nonlinear per-
pendicular one discussed above).
It is interesting to note the connection between the entropy
cascade and certain aspects of the gyrofluid closure formal-
ism developed by Dorland & Hammett (1993). In their the-
ory, the emergence of small scales in v⊥ manifested itself as
the growth of high-order v⊥ moments of the gyrocenter distri-
bution function. They correctly identified this effect as a con-
sequence of the nonlinear perpendicular phase mixing of the
gyrocenter distribution function caused by a perpendicular-
velocity-space spread in the ring-averaged E ×B velocities
(given by 〈uE〉Ri = ẑ×∇〈Φ〉Ri in our notation) arising at and
below the ion gyroscale.
7.9.2. Scalings
Since entropy is a conserved quantity, we will follow the
well trodden Kolmogorov path, assume locality of interac-
tions in scale space and constant entropy flux, and conclude,
analogously to Eq. (1),
v8thi
h2iλ ∼ εh = const, (253)
where εh is the entropy flux proportional to the fraction of the
total turbulent power ε (or Pext; see § 3.4) that was diverted
into the entropy cascade at the ion gyroscale, and is the cas-
cade time that we now need to find.
By the critical-balance assumption, the decorrelation time
of the electromagnetic fluctuations in KAW turbulence is
comparable at each scale to the KAW period at that scale and
to the nonlinear interaction time [Eq. (239)]:
τKAWλ ∼
(1 +βi)1/3
. (254)
The characteristic time associated with the nonlinear term in
Eq. (249) is longer than τKAWλ by a factor of (ρi/λ)
1/2 due to
the ring averaging, which reduces the strength of the nonlinear
interaction. This weakness of the nonlinearity makes it pos-
sible to develop a systematic analytical theory of the entropy
32 Another source of small-scale spatial smoothing comes from the per-
pendicular gyrocenter-diffusion terms ∼ −νii(v/vthi)2k2⊥ρ
i hik that arise in
the ring-averaged collision operators, e.g., the second term in the model
operator (B13). These terms again enforce a cutoff wavenumber such that
k⊥ρi ∼ (ω/νii)
1/2 ≫ 1.
36 SCHEKOCHIHIN ET AL.
cascade (Schekochihin & Cowley 2009). It is also possible
to estimate the cascade time via a more qualitative argument
analogous to that first devised by Kraichnan (1965) for the
weak turbulence of Alfvén waves: during each KAW correla-
tion time τKAWλ, the nonlinearity changes the amplitude of hi
by only a small amount:
∆hiλ ∼ (λ/ρi)1/2hiλ ≪ hiλ; (255)
these changes accumulate with time as a random walk,
so after time t, the cumulative change in amplitude is
∆hiλ(t/τKAWλ)
1/2; finally, the cascade time t = is the time
after which the cumulative change in amplitude is compara-
ble to the amplitude itself, which gives, using Eq. (254),
τKAWλ ∼
(1 +βi)1/3
. (256)
Substituting this into Eq. (253), we get
hiλ ∼
v3thi
)1/2(
(1 +βi)1/6√
(257)
which corresponds to a k
⊥ spectrum of entropy.
In the argument presented above, we assumed that the scal-
ing of hi was determined by the nonlinear mixing of hi by
the ring-averaged KAW fluctuations rather than by the wave–
particle interaction term on the right-hand side of Eq. (249).
We can now confirm the validity of this assumption. The
change in amplitude of hi in one KAW correlation time τKAWλ
due to the wave–particle interaction term is
∆hiλ∼
v3thi
βiρivA
∼ n0i
v3thi
(εKAW
)1/3 1√
βi (1 +βi)1/3
7/6, (258)
where we have used Eq. (240). Comparing this with Eq. (255)
and using Eq. (257), we see that ∆hiλ in Eq. (258) is a factor
of (λ/ρi)
1/2 smaller than ∆hiλ due to the nonlinear mixing.
7.9.3. Phase-Space Cutoff
To work out the cutoff scales both in the position and veloc-
ity space, we use Eqs. (251) and (252): in Eq. (251), ω ∼ 1/,
where is the characteristic decorrelation time of hi given by
Eq. (256); using Eq. (252), we find the cutoffs:
∼ (νiiτρi )3/5 = Do−3/5, (259)
where τρi is the cascade time [Eq. (256)] taken at λ = ρi.
By a recently established convention, the dimensionless num-
ber Do = 1/νiiτρi is called the Dorland number. It plays
the role of Reynolds number for kinetic turbulence, mea-
suring the scale separation between the ion gyroscale and
the collisional dissipation scale (Schekochihin et al. 2008b;
Tatsuno et al. 2009a,b).
7.9.4. Parallel Phase Mixing
Another assumption, which was made implicitly, was that
the parallel phase mixing due to the second term on the left-
hand side of Eq. (249) could be ignored. This requires jus-
tification, especially because it is with this “ballistic” term
that one traditionally associates the emergence of small-scale
structure in the velocity space (e.g., Krommes & Hu 1994;
Krommes 1999; Watanabe & Sugama 2004). The effect of
the parallel phase mixing is to produce small scales in veloc-
ity space δv‖ ∼ 1/k‖t. Let us assume that the KAW turbu-
lence imparts its parallel decorrelation scale to hi and use the
scaling relation (241) to estimate k‖ ∼ l−1‖λ. Then, after one
cascade time [Eq. (256)], hi is decorrelated on the parallel
velocity scales
βi(1 +βi)
∼ 1. (260)
We conclude that the nonlinear perpendicular phase mixing
[Eq. (259)] is more efficient than the linear parallel one. Note
that up to a βi-dependent factor Eq. (260) is equivalent to a
critical-balance-like assumption for hi in the sense that the
propagation time is comparable to the cascade time, or k‖v‖ ∼
−1 [see Eq. (249)].
7.10. Entropy Cascade in the Absence of KAW Turbulence
It is not currently known how one might determine ana-
lytically what fraction of the turbulent power arriving from
the inertial range to the ion gyroscale is channeled into the
KAW cascade and what fraction is dissipated via the kinetic
ion-entropy cascade introduced in § 7.9 (perhaps it can only
be determined by direct numerical simulations). It is cer-
tainly a fact that in many solar-wind measurements, the rel-
atively shallow magnetic-energy spectra associated with the
KAW cascade (§ 7.5) fail to appear and much steeper spectra
are detected (close to k−4; see Leamon et al. 1998; Smith et al.
2006). In view of this evidence, it is interesting to ask what
would be the nature of electromagnetic fluctuations below the
ion gyroscale if the KAW cascade failed to be launched, i.e.,
if all (or most) of the turbulent power were directed into the
entropy cascade (i.e., if W ≃Whi in § 7.8).
7.10.1. Equations
It is again possible to derive a closed set of equations for all
fluctuating quantities.
Let us assume (and verify a posteriori; § 7.10.4) that the
characteristic frequency of such fluctuations is much lower
than the KAW frequency [Eq. (230)] so that the first term in
Eq. (116) is small and the equation reduces to the balance of
the other two terms. This gives
, (261)
meaning that the electrons are purely Boltzmann [he = 0 to
lowest order; see Eq. (101)]. Then, from Eq. (118),
ρivthi
eik·r
d3vJ0(ai)hik (262)
Using Eq. (262), we find from Eq. (120) that the field-
strength fluctuations are
eik·r
v2thi
J1(ai)
hik, (263)
which is smaller than Zeϕ/T0i by a factor of βi/k⊥ρi.
Therefore, we can neglect δB‖/B0 compared to δne/n0e in
Eq. (117). Using Eq. (261), we get what is physically the
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 37
electron continuity equation:
+ b̂ ·∇
4πen0e
∇2⊥A‖ + u‖i
= 0, (264)
u‖i =
eik·r
d3vv‖J0(ai)hik. (265)
Note that in terms of the stream and flux functions, Eq. (264)
takes the form
ρ2i ∇2⊥Ψ =
, (266)
where we have approximated b̂ · ∇ ≃ ∂/∂z, which will, in-
deed, be shown to be correct in § 7.10.4.
Together with the ion gyrokinetic equation, which deter-
mines hi, Eqs. (261-264) form a closed set. They describe
low-frequency fluctuations of the density and electromagnetic
field due solely to the presence of fluctuations of hi below the
ion gyroscale.
It follows from Eq. (263) that δB‖/B0 contributes subdom-
inantly to 〈χ〉Ri [Eq. (69) with s = i and ai ≫ 1]. It will be
verified a posteriori (§ 7.10.4) that the same is true for A‖.
Therefore, Eqs. (247) and (249) continue to hold, as in the
case with KAW. This means that Eqs. (249) and (262) form
a closed subset. Thus the kinetic ion-entropy cascade is self-
regulating in the sense that hi is no longer passive (as it was
in the presence of KAW turbulence; § 7.9) but is mixed by
the ring-averaged “electrostatic” fluctuations of the scalar po-
tential, which themselves are produced by hi according to
Eq. (262).
The magnetic fluctuations are passive and determined by
the electrostatic and entropy fluctuations via Eqs. (263)
and (264).
7.10.2. Scalings
From Eq. (262), we can establish a correspondence between
Φλ and hiλ (the electrostatic fluctuations and the fluctuations
of the ion-gyrocenter distribution function):
Φλ ∼ ρivthi
hiλλ, (267)
where the factor of (λ/ρi)
1/2 comes from the Bessel function
[Eq. (248)] and the factor of (δv⊥/vthi)
1/2 results from the
v⊥ integration of the oscillatory factor in the Bessel function
times hi, which decorrelates on small scales in the velocity
space and, therefore, its integral accumulates in a random-
walk-like fashion. The velocity-space scales are related to the
spatial scales via Eq. (252), which was arrived at by an ar-
gument not specific to KAW-like fluctuations and, therefore,
continues to hold.
Using Eq. (267), we find that the wave–particle interaction
term in the right-hand side of Eq. (249) is subdominant: com-
paring it with ∂hi/∂t shows that it is smaller by a factor of
(λ/ρi)
3/2 ≪ 1. Therefore, it is the nonlinear term in Eq. (249)
that controls the scalings of hiλ and Φλ.
We now assume again the scale-space locality and con-
stancy of the entropy flux, so Eq. (253) holds. The cascade
(decorrelation) time is equal to the characteristic time associ-
ated with the nonlinear term in Eq. (249): ∼ (ρi/λ)1/2λ2/Φλ.
Substituting this into Eq. (253) and using Eq. (267), we ar-
rive at the desired scaling relations for the entropy cascade
(Schekochihin et al. 2008b):
v3thi
)1/3 1√
1/6, (268)
)1/3 vthi√
7/6, (269)
)1/3 √
1/3, (270)
where l0 = v
A/ε, as in § 1.2. Note that since the existence
of this cascade depends on it not being overwhelmed by the
KAW fluctuations, we should have εKAW ≪ ε and εh = ε −
εKAW ≈ ε.
The scaling for the ion-gyrocenter distribution function,
Eq. (268), implies a k
⊥ spectrum—the same as for the KAW
turbulence [Eq. (257)]. The scaling for the the cascade time,
Eq. (270), is also similar to that for the KAW turbulence
[Eq. (256)]. Therefore the velocity- and gyrocenter-space cut-
offs are still given by Eq. (259), where τρi is now given by
Eq. (270) taken at λ = ρi.
A new feature is the scaling of the scalar potential, given by
Eq. (269), which corresponds to a k
−10/3
⊥ spectrum (unlike the
KAW spectrum, § 7.5). This is a measurable prediction for the
electrostatic fluctuations: the implied electric-field spectrum
⊥ . From Eq. (261), we also conclude that the density
fluctuations should have the same spectrum as the scalar po-
tential, k
−10/3
⊥ —another measurable prediction.
The scalings derived above for the spectra of the ion
distribution function and of the scalar potential have been
confirmed in the numerical simulations by Tatsuno et al.
(2009a,b), who studied decaying electrostatic gyrokinetic tur-
bulence in two spatial dimensions. They also found velocity-
space scalings in accord with Eq. (252) (using a spectral
representation of the correlation functions in the v⊥ space
based on the Hankel transform of the distribution function;
see Plunk et al. 2009).
7.10.3. Parallel Cascade and Parallel Phase Mixing
We have again ignored the ballistic term (the second on
the left-hand side) in Eq. (249). We will estimate the effi-
ciency of the parallel spatial cascade of the ion entropy and of
the associated parallel phase mixing by making a conjecture
analogous to the critical balance: assuming that any two per-
pendicular planes only remain correlated provided particles
can stream between them in one nonlinear decorrelation time
(cf. § 1.2 and § 7.9.4), we conclude that the parallel particle-
streaming frequency k‖v‖ should be comparable at each scale
to the inverse nonlinear time −1, so
k‖vthi ∼ 1. (271)
As we explained in § 7.9.4, the parallel scales in the velocity
space generated via the ballistic term are related to the parallel
wavenumbers by δv‖ ∼ 1/k‖t. From Eq. (271), we find that
after one cascade time , the typical parallel velocity scale is
δv‖/vthi ∼ 1, so the parallel phase mixing is again much less
efficient than the perpendicular one.
Note that Eq. (271) combined with Eq. (270) means that the
anisotropy is again characterized by the scaling relation k‖ ∼
⊥ , similarly to the case of KAW turbulence [see Eq. (241)
and § 7.9.4].
38 SCHEKOCHIHIN ET AL.
7.10.4. Scalings for the Magnetic Fluctuations
The scaling law for the fluctuations of the magnetic-field
strength follows immediately from Eqs. (263) and (269):
ρivthi
−11/6
13/6, (272)
whence the spectrum of these fluctuations is k
−16/3
The scaling of A‖ (the perpendicular magnetic fluctuations)
depends on the relation between k‖ and k⊥. Indeed, the ratio
between the first and the third terms on the left-hand side of
Eq. (264) [or, equivalently, between the first and second terms
on the right-hand side of Eq. (266)] is ∼
k‖vthi
. For a crit-
ically balanced cascade, this makes the two terms comparable
[Eq. (271)]. Using the first term to work out the scaling for the
perpendicular magnetic fluctuations, we get, using Eq. (269),
ρivthi
−11/6
13/6, (273)
which is the same scaling as for δB‖/B0 [Eq. (272)].
Using Eq. (273) together with Eqs. (269) and (270), it is
now straightforward to confirm the three assumptions made
in § 7.10.1 that we promised to verify a posteriori:
1. In Eq. (116), ∂A‖/∂t ≪ cb̂ ·∇ϕ, so Eq. (261) holds (the
electrons remain Boltzmann). This means that no KAW
can be excited by the cascade.
2. δB⊥/B0 ≪ k‖/k⊥, so b̂ ·∇ ≃ ∂/∂z in Eq. (264). This
means that field lines are not significantly perturbed.
3. In the expression for 〈χ〉Ri [Eq. (69)], v‖A‖/c ≪ ϕ, so
Eq. (249) holds. This means that the electrostatic fluc-
tuations dominate the cascade.
7.11. Cascades Superposed?
The spectra of magnetic fluctuations obtained in § 7.10.4
are very steep—steeper, in fact, than those normally observed
in the dissipation range of the solar wind (§ 8.2.5). One might
speculate that the observed spectra may be due to a superposi-
tion of the two cascades realizable below the ion gyroscale: a
high-frequency cascade of KAW (§ 7.5) and a low-frequency
cascade of electrostatic fluctuations due to the ion entropy
fluctuations (§ 7.10). Such a superposition could happen if
the power going into the KAW cascade is relatively small,
εKAW ≪ ε. One then expects an electrostatic cascade to be
set up just below the ion gyroscale with the KAW cascade
superseding it deeper into the dissipation range. Comparing
Eqs. (240) and (269), we can estimate the position of the spec-
tral break:
k⊥ρi ∼
ε/εKAW
. (274)
Since ρi/ρe ∼ (τmi/me)1/2/Z is not a very large number, the
dissipation range is not very wide. It is then conceivable that
the observed spectra are not true power laws but simply non-
asymptotic superpositions of the electrostatic and KAW spec-
tra with the observed range of “effective” spectral exponents
due to varying values of the spectral break (274) between the
two cascades.33
33 Several alternative theories that aim to explain the dissipation-range
spectra exist: see § 8.2.6.
The value of εKAW/ε specific to any particular set of param-
eters (βi, τ , etc.) is set by what happens at k⊥ρi ∼ 1 (§ 7.1;
see § 8.2.2, § 8.2.5, and § 8.5 for further discussion).
7.12. Below the Electron Gyroscale: The Last Cascade
Finally, let us consider what happens when k⊥ρe ≫ 1. At
these scales, we have to return to the full gyrokinetic sys-
tem of equations. The quasi-neutrality [Eq. (61)], parallel
[Eq. (62)] and perpendicular [Eq. (66)] Ampère’s law become
eik·r
d3vJ0(ae)hek, (275)
4πen0e
∇2⊥A‖ =
eik·r
d3vv‖J0(ae)hek, (276)
eik·r
v2the
J1(ae)
hek, (277)
where βe = βiZ/τ . We have discarded the velocity integrals
of hi both because the gyroaveraging makes them subdom-
inant in powers of (me/mi)
1/2 and because the fluctuations
of hi are damped by collisions [assuming the collisional cut-
off given by Eq. (259) lies above the electron gyroscale]. To
Eqs. (275-277), we must append the gyrokinetic equation for
he [Eq. (57) with s = e], thus closing the system.
The type of turbulence described by these equations is very
similar to that discussed in § 7.10. It is easy to show from
Eqs. (275-277) that
. (278)
Hence the magnetic fluctuations are subdominant in the ex-
pression for 〈χ〉Re [Eq. (69) with s = e and ae ≫ 1], so
〈χ〉Re ≃ 〈ϕ〉Re . The electron gyrokinetic equation then is
{〈ϕ〉Re ,he} =
, (279)
where the wave–particle interaction term in the right-hand
side has been dropped because it can be shown to be small
via the same argument as in § 7.10.2.
Together with Eq. (275), Eq. (279) describes the kinetic cas-
cade of electron entropy from the electron gyroscale down to
the scale at which electron collisions can dissipate it into heat.
This cascade the result of collisionless damping of KAW at
k⊥ρe ∼ 1, whereby the power in the KAW cascade is con-
verted into the electron-entropy fluctuations: indeed, in the
limit k⊥ρe ≫ 1, the generalized energy is simply
= Whe (280)
(see Fig. 5).
The same scaling arguments as in § 7.10.2 apply and scaling
relations analogous to Eqs. (268-270), and (272) duly follow:
v3the
(εKAW
1/6, (281)
(εKAW
vthe l
7/6, (282)
)1/3(
, (283)
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 39
(εKAW
−11/6
13/6, (284)
where l0 = v
A/ε, as in § 1.2. The formula for the collisional
cutoffs in the wavenumber and velocity space is analogous to
Eq. (259):
∼ (νeiτρe )3/5, (285)
where τρe is the cascade time (283) taken at λ = ρe.
7.13. Validity of Gyrokinetics in the Dissipation Range
As the kinetic cascade takes the (generalized) energy to ever
smaller scales, the frequency ω of the fluctuations increases.
In applying the gyrokinetic theory, one must be mindful of
the need for this frequency to stay smaller than Ωi. Using
the scaling formulae for the characteristic times of the fluc-
tuations derived above [Eqs. (254), (270) and (283)], we can
determine the conditions for ω ≪ Ωi. Thus, for the gyroki-
netic theory to be valid everywhere in the inertial range, we
must have
k⊥ρi ≪ β3/4i
(286)
at all scales down to k⊥ρi ∼ 1, i.e., ρi/l0 ≪ β3/2i , not a very
stringent condition.
Below the ion gyroscale, the KAW cascade (§ 7.5) remains
in the gyrokinetic regime as long as
k⊥ρi ≪
i (1 +βi)
(287)
(we are assuming Ti/Te ∼ 1 everywhere). The condition for
this still to be true at the electron gyroscale is
i (1 +βi)
. (288)
The ion entropy fluctuations passively mixed by the KAW tur-
bulence (§ 7.9) satisfy Eq. (287) at all scales down to the ion
collisional cutoff [Eq. (259)] if
λmfpi
i (1 +βi)
. (289)
Note that the condition for the ion collisional cutoff to lie
above the electron gyroscale is
λmfpi
βi(1 +βi)1/3
)5/6(
(290)
In the absence of KAW turbulence, the pure ion-entropy cas-
cade (§ 7.10) remains gyrokinetic for
k⊥ρi ≪ β3/2i
. (291)
This is valid at all scales down to the ion collisional cutoff
provided λmfpi/l0 ≪ β3i (l0/ρi), an extremely weak condition,
which is always satisfied. This is because the ion-entropy
fluctuations in this case have much lower frequencies than in
the KAW regime. The ion collisional cutoff lies above the
electron gyroscale if, similarly to Eq. (290),
λmfpi
)5/6(
. (292)
If the condition (290) is satisfied, all fluctuations of the
ion distribution function are damped out above the electron
gyroscale. This means that below this scale, we only need
the electron gyrokinetic equation to be valid, i.e., ω ≪ Ωe.
The electron-entropy cascade (§ 7.12), whose characteristic
timescale is given by Eq. (283), satisfies this condition for
k⊥ρe ≪
β3/2e
. (293)
This is valid at all scales down to the electron
collisional cutoff [Eq. (285)] provided λmfpe/l0 ≪
(ε/εKAW)
2β3e (mi/me)
3(l0/ρe), which is always satisfied.
Within the formal expansion we have adopted (k⊥ρi ∼ 1
and k‖λmfpi ∼
βi), it is not hard to see that λmfpi/l0 ∼ ǫ2
and ρi/l0 ∼ ǫ3. Since all other parameters (me/mi, βi, βe
etc.) are order unity with respect to ǫ, all of the above con-
ditions for the validity of the gyrokinetics are asymptotically
correct by construction. However, in application to real as-
trophysical plasmas, one should always check whether this
construction holds. For example, substituting the relevant pa-
rameters for the solar wind shows that the gyrokinetic ap-
proximation is, in fact, likely to start breaking down some-
where between the ion and electron gyroscales (Howes et al.
2008a).34 This releases a variety of high-frequency wave
modes, which may be participating in the turbulent cascade
around and below the electron gyroscale (see, e.g., the recent
detailed observations of these scales in the magnetosheath by
Mangeney et al. 2006; Lacombe et al. 2006 or the early mea-
surements of high-frequency fluctuations in the solar wind by
Denskat et al. 1983; Coroniti et al. 1982).
7.14. Summary
In this section, we have analyzed the turbulence in the dissi-
pation range, which turned out to have many more essentially
kinetic features than the inertial range.
At the ion gyroscale, k⊥ρi ∼ 1, the kinetic cascade rear-
ranged itself into two distinct components: part of the (gener-
alized) energy arriving from the inertial range was collision-
lessly damped, giving rise to a purely kinetic cascade of ion-
entropy fluctuations, the rest was converted into a cascade of
Kinetic Alfvén Waves (KAW) (Fig. 5; see § 7.1 and § 7.8).
The KAW cascade is described by two fluid-like equa-
tions for two scalar functions, the magnetic flux function
Ψ = −A‖/
4πmin0i and the scalar potential, expressed, for
continuity with the results of § 5, in terms of the function
Φ = (c/B0)ϕ. The equations are (see § 7.2)
b̂ ·∇Φ, (294)
2 +βi
1 + Z/τ
) b̂ ·∇
ρ2i ∇2⊥Ψ
, (295)
where b̂ · ∇ = ∂/∂z + (1/vA){Ψ, · · ·}. The density and
34 See this paper also for a set of numerical tests of the validity of gy-
rokinetics in the dissipation range, a linear theory of the conversion of KAW
into ion-cyclotron-damped Bernstein waves, and a discussion of the potential
(un)importance of ion cyclotron damping for the dissipation of turbulence.
40 SCHEKOCHIHIN ET AL.
magnetic-field-strength fluctuations are directly related to the
scalar potential:
. (296)
We call Eqs. (294-296) the Electron Reduced Magnetohydro-
dynamics (ERMHD).
The ion-entropy cascade is described by the ion gyrokinetic
equation:
+{〈Φ〉Ri ,hi} = 〈Cii[hi]〉Ri . (297)
The ion distribution function is mixed by the ring-averaged
scalar potential and undergoes a cascade both in the velocity
and gyrocenter space—this phase-space cascade is essential
for the conversion of the turbulent energy into the ion heat,
which can ultimately only be done by collisions (see § 7.9).
If the KAW cascade is strong (its power εKAW is an order-
unity fraction of the total injected turbulent power ε), it de-
termines Φ in Eq. (297), so the ion-entropy cascade is passive
with respect to the KAW turbulence. Equations (294-295) and
(297) form a closed system that determines the three func-
tions Φ, Ψ, hi, of which the latter is slaved to the first two.
One can also compute δne and δB‖, which are proportional
to Φ [Eq. (296)]. The generalized energy conserved by these
equations is given by Eq. (245).
If the KAW cascade is weak (εKAW ≪ ε), the ion-entropy
cascade dominates the turbulence in the dissipation range and
drives low-frequency mostly electrostatic fluctuations, with a
subdominant magnetic component. These are given by the
following relations (see § 7.10)
ρivthi
2(1 + τ/Z)
eik·r
d3vJ0(ai)hik, (298)
ρivthi
, (299)
eik·r ×
1 + Z/τ
J0(ai)
hik, (300)
eik·r
v2thi
J1(ai)
hik, (301)
where ai = k⊥v⊥/Ωi, Equations (297) and (298) form a closed
system for Φ and hi. The rest of the fields, namely δne, Ψ and
δB‖, are slaved to hi via Eqs. (299-301).
The fluid and kinetic models summarized above are valid
between the ion and electron gyroscales. Below the electron
gyroscale, the collisionless damping of the KAW cascade con-
verts it into a cascade of electron entropy, similar in nature to
the ion-entropy cascade (§ 7.12).
The KAW cascade and the low-frequency turbulence asso-
ciated with the ion-entropy cascade have distinct scaling be-
haviors. For the KAW cascade, the spectra of the electric,
density and magnetic fluctuations are (§ 7.5)
EE (k⊥) ∝ k−1/3⊥ , En(k⊥) ∝ k
⊥ , EB(k⊥) ∝ k
⊥ . (302)
For the ion- and electron-entropy cascades (§ 7.9 and § 7.12),
EE (k⊥) ∝ k−4/3⊥ , En(k⊥) ∝ k
−10/3
⊥ , EB(k⊥) ∝ k
−16/3
(303)
We argued in § 7.11 that the observed spectra in the dissipa-
tion range of the solar wind could be the result of a superpo-
sition of these two cascades, although a number of alternative
theories exist (§ 8.2.6).
8. DISCUSSION OF ASTROPHYSICAL APPLICATIONS
We have so far only occasionally referred to some relevant
observational evidence for space and astrophysical plasmas.
We now discuss in more detail how the theoretical framework
laid out above applies to real plasma turbulence in space.
Although we will discuss the interstellar medium, accre-
tion disks and galaxy clusters towards the end of this sec-
tion, the most rewarding source of observational information
about plasma turbulence in astrophysical conditions is the so-
lar wind and the magnetosheath because only there direct in
situ measurements of all the interesting quantities are possi-
ble. Measurements of the fluctuating magnetic and velocity
fields in the solar wind have been available since the 1960s
(Coleman 1968) and a vast literature now exists on their spec-
tra, anisotropy, Alfvénic character and many other aspects (a
short recent review is Horbury et al. 2005; two long ones are
Tu & Marsch 1995; Bruno & Carbone 2005). It is not our
aim here to provide a comprehensive survey of what is known
about plasma turbulence in the solar wind. Instead, we shall
limit our discussion to a few points that we consider impor-
tant in light of the theoretical framework proposed in this pa-
per.35 As we do this, we shall provide copious references to
the main body of the paper, so this section can be read as a
data-oriented guide to it, aimed both at a thorough reader who
has arrived here after going through the preceding sections
and an impatient one who has skipped to this one hoping to
find out whether there is anything of “practical” use in the
theoretical developments above.
8.1. Inertial-Range Turbulence in the Solar Wind
In the inertial range, i.e., for k⊥ρi ≪ 1, the solar-wind turbu-
lence should be described by the reduced hybrid fluid-kinetic
theory derived in § 5 (KRMHD). Its applicability hinges on
three key assumptions: (i) the turbulence is Alfvénic, i.e., con-
sists of small (δB/B0 ≪ 1) low-frequency (ω ∼ k‖vA ≪ Ωi)
perturbations of an ambient mean magnetic field and corre-
sponding velocity fluctuations; (ii) it is strongly anisotropic,
k⊥ ≫ k‖; (iii) the equilibrium distribution can be approxi-
mated or, at least, reasonably modeled by a Maxwellian with-
out loss of essential physics (this will be discussed in § 8.3).
If these assumptions are satisfied, KRMHD (summarized in
§ 5.7) is a rigorous set of dynamical equations for the inertial
range, a set of Kolmogorov-style scaling predictions for the
Alfvénic component of the turbulence can be produced (the
GS theory, reviewed in § 1.2), while to the compressive fluc-
tuations, the considerations of § 6 apply. So let us examine
the observational evidence.
8.1.1. Alfvénic Nature of the Turbulence
The presence of Alfvén waves in the solar wind was re-
ported already the early works of Unti & Neugebauer (1968)
and Belcher & Davis (1971). Alfvén waves are detected al-
ready at very low frequencies (large scales)—and, at these
35 An extended quantitative discussion of the applicability of the gyroki-
netic theory to the turbulence in the slow solar wind was given by Howes et al.
(2008a).
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 41
low frequencies, have a k−1 spectrum.36 This spectrum cor-
responds to a uniform distribution of scales/frequencies of
waves launched by the coronal activity of the Sun. Nonlin-
ear interaction of these waves gives rise to an Alfvénic tur-
bulent cascade of the type that was discussed above. The ef-
fective outer scale of this cascade can be detected as a spec-
tral break where the k−1 scaling steepens to the Kolmogorov
slope k−5/3 (see Bavassano et al. 1982; Marsch & Tu 1990a;
Horbury et al. 1996 for fast-wind results on the spectral break;
for a discussion of the effective outer scale in the slow wind
at 1 AU, see Howes et al. 2008a). The particular scale at
which this happens increases with the distance from the Sun
(Bavassano et al. 1982), reflecting the more developed state
of the turbulence at later stages of evolution. At 1 AU, the
outer scale is roughly in the range of 105 − 106 km; the k−5/3
range extends down to scales/frequencies that correspond to a
few times the ion gyroradius (102 − 103 km; see Table 1).
The range between the outer scale (the spectral break) and
the ion gyroscale is the inertial range. In this range, δB/B0 de-
creases with scale because of the steep negative spectral slope.
Therefore, the assumption of small fluctuations, δB/B0 ≪ 1,
while not necessarily true at the outer scale, is increasingly
better satisfied further into the inertial range (cf. § 1.3).
Are these fluctuations Alfvénic? In a plasma such as the
solar wind, they ought to be because, as showed in § 5.3, for
k⊥ρi ≪ 1, these fluctuations are rigorously described by the
RMHD equations. The magnetic flux is frozen into the ion
motions, so displacing a parcel of plasma should produce a
matching (Alfvénic) perturbation of the magnetic field line
and vice versa: in an Alfvén wave, u⊥ = ±δB⊥/
4πmin0i.
The strongest confirmation that this is indeed true for the
inertial-range fluctuations in the solar wind was achieved by
Bale et al. (2005), who compared the spectra of electric and
magnetic fluctuations and found that they both scale as k−5/3
and follow each other with remarkable precision (see Fig. 1).
The electric field is a very good measure of the perpendicular
velocity field because, for k⊥ρi ≪ 1, the plasma velocity is
the E×B drift velocity, u⊥ = cE× ẑ/B0 (see § 5.4).
This picture of agreement between basic theory and ob-
servations is upset in a disturbing fashion by an extraordi-
nary recent result by Chapman & Hnat (2007); Podesta et al.
(2006) and J. E. Borovsky (2008, private communication),
who claim different spectral indices for velocity and mag-
netic fluctuations—k−3/2 and k−5/3, respectively. This result
is puzzling because if it is asymptotically correct in the iner-
tial range, it implies either u⊥ ≫ δB⊥ or u⊥ ≪ δB⊥ and it is
not clear how perpendicular velocity fluctuations in a near-
ideal plasma could fail to produce Alfvénic displacements
and, therefore, perpendicular magnetic field fluctuations with
matching energies. Plausible explanations may be either that
the velocity field in these measurements is polluted by a non-
Alfvénic component parallel to the magnetic field (although
data analysis by Chapman & Hnat 2007 does not support this)
or that the flattening of the velocity spectrum is due to some
form of a finite-gyroradius effect or even an energy injection
into the velocity fluctuations at scales approaching the ion
gyroscale (e.g., from the pressure-anisotropy-driven instabili-
36 Inferred from the frequency spectrum f −1 via the Taylor (1938) hypoth-
esis, f ∼ k ·Vsw , where Vsw is the mean velocity at which the wind blows
past the spacecraft. The Taylor hypothesis is a good assumption for the so-
lar wind because Vsw (∼ 800 km/s in the fast wind, ∼ 300 km/s in the slow
wind) is highly supersonic, super-Alfvénic and far exceeds the fluctuating
velocities.
ties, § 8.3).
8.1.2. Energy Spectrum
How solid is the statement that the observed spectrum
has a k−5/3 scaling? In individual measurements of the
magnetic-energy spectra, very high accuracy is claimed
for this scaling: the measured spectral exponent is be-
tween 1.6 and 1.7; agreement with Kolmogorov value 1.67
is often reported to be within a few percent (see, e.g.,
Horbury et al. 1996; Leamon et al. 1998; Bale et al. 2005;
Narita et al. 2006; Alexandrova et al. 2008a; Horbury et al.
2008)). There is a somewhat wider scatter of spectral in-
dices if one considers large sets of measurement intervals
(Smith et al. 2006), but overall, the observational evidence
does not appear to be consistent with a k
⊥ spectrum consis-
tently found in the MHD simulations with a strong mean field
(Maron & Goldreich 2001; Müller et al. 2003; Mason et al.
2007; Perez & Boldyrev 2008, 2009; Beresnyak & Lazarian
2008b) and defended on theoretical grounds in the recent
modifications of the GS theory by Boldyrev (2006) and by
Gogoberidze (2007) (see footnote 10). This discrepancy be-
tween observations and simulations remains an unresolved
theoretical issue. It is probably best addressed by numeri-
cal modeling of the RMHD equations (§ 2.2) and by a de-
tailed comparison of the structure of the Alfvénic fluctuations
in such simulations and in the solar wind.
8.1.3. Anisotropy
Building up evidence for anisotropy of turbulent fluctua-
tions has progressed from merely detecting their elongation
along the magnetic field (Belcher & Davis 1971)—to fitting
data to an ad hoc model mixing a 2D perpendicular and a
1D parallel (“slab”) turbulent components in some propor-
tion37 (Matthaeus et al. 1990; Bieber et al. 1996; Dasso et al.
2005; Hamilton et al. 2008)—to formal systematic unbiased
analyses showing the persistent presence of anisotropy at all
scales (Bigazzi et al. 2006; Sorriso-Valvo et al. 2006)— to di-
rect measurements of three-dimensional correlation functions
(Osman & Horbury 2007)—and finally to computing spectral
exponents at fixed angles between k and B0 (Horbury et al.
2008). The latter authors appear to have achieved the first
direct quantitative confirmation of the GS theory by demon-
strating that the magnetic-energy spectrum scales as k
wavenumbers perpendicular to the mean field and as k−2
wavenumbers parallel to it [consistent with the first scaling
relation in Eq. (4)]. This is the closest that observations have
got to confirming the GS relation k‖ ∼ k
⊥ [see Eq. (5)] in a
real astrophysical turbulent plasma.
8.1.4. Compressive Fluctuations
According to the theory developed in § 5, the density and
magnetic-field-strength fluctuations are passive, energetically
decoupled from and mixed by the Alfvénic cascade (§ 5.5;
these are slow and entropy modes in the collisional MHD
limit—see § 2.4 and § 6.1). These fluctuations are expected to
be pressure-balanced, as expressed by Eq. (22) or, more gen-
erally in gyrokinetics, by Eq. (67). There is, indeed, strong
37 These techniques originate from the view of MHD turbulence as a su-
perposition of a 2D turbulence and an admixture of Alfvén waves (Fyfe et al.
1977; Montgomery & Turner 1981). As we discussed in § 1.2, we consider
the Goldreich & Sridhar (1995, 1997) view of a critically balanced Alfvénic
cascade to be better physically justified.
42 SCHEKOCHIHIN ET AL.
evidence that magnetic and thermal pressures in the solar
wind are anticorrelated, although there are some indications
of the presence of compressive, fast-wave-like fluctuations as
well (Roberts 1990; Burlaga et al. 1990; Marsch & Tu 1993;
Bavassano et al. 2004).
Measurements of density and field-strength fluctua-
tions done by a variety of different methods both at
1 AU (Celnikier et al. 1983, 1987; Marsch & Tu 1990b;
Bershadskii & Sreenivasan 2004; Hnat et al. 2005;
Kellogg & Horbury 2005; Alexandrova et al. 2008a) and
near the Sun (Lovelace et al. 1970; Woo & Armstrong 1979;
Coles & Harmon 1989; Coles et al. 1991) show fluctuation
levels of order 10% and spectra that appear to have a k−5/3
scaling above scales of order 102 − 103 km, which approxi-
mately corresponds to the ion gyroscale. The Kolmogorov
value of the spectral exponent is, as in the case of Alfvénic
fluctuations, measured quite accurately in individual cases
(1.67 ± 0.03 in Celnikier et al. 1987). Interestingly, the
higher-order structure function exponents measured for the
magnetic-field strength show that it is a more intermittent
quantity than the velocity or the vector magnetic field (i.e.,
than the Alfvénic fluctuations) and that the scaling expo-
nents are quantitatively very close to the values found for
passive scalars in neutral fluids (Bershadskii & Sreenivasan
2004; Bruno et al. 2007). One might argue that this lends
some support to the theoretical expectation of passive
magnetic-field-strength fluctuations.
Considering that in the collisionless regime these fluctua-
tions are supposed to be subject to strong kinetic damping
(§ 6.2.2), the presence of well-developed Kolmogorov-like
and apparently undamped turbulent spectra is more surprising
than has perhaps been publicly acknowledged. An extended
discussion of this issue was given in § 6.3. Without the in-
clusion of the dissipation effects associated with the finite ion
gyroscale, the passive cascade of the density and field strength
is purely perpendicular to the (exact) local magnetic field and
does not lead to any scale refinement along the field. This im-
plies highly anisotropic field-aligned structures, whose length
is determined by the initial conditions (i.e., conditions in the
corona). The kinetic damping is inefficient for such fluctua-
tions. While this would seem to explain the presence of fully
fledged power-law spectra, it is not entirely obvious that the
parallel cascade is really absent once dissipation is taken into
account (Lithwick & Goldreich 2001), so the issue is not yet
settled. This said, we note that there is plenty of evidence of
a high degree of anisotropy and field alignment of the den-
sity microstructure in the inner solar wind and outer corona
(e.g., Armstrong et al. 1990; Grall et al. 1997; Woo & Habbal
1997). There is also evidence that the local structure of the
compressive fluctuations at 1 AU is correlated with the coro-
nal activity, implying some form of memory of initial condi-
tions (Kiyani et al. 2007; Hnat et al. 2007; Wicks et al. 2009).
We note, finally, that whether compressive fluctuations in
the inertial range can develop short parallel scales should also
tell us how much ion heating can result from their damping
(see § 6.2.4).
8.2. Dissipation-Range Turbulence in the Solar Wind and the
Magnetosheath
At scales approaching the ion gyroscale, k⊥ρi ∼ 1, effects
associated with the finite extent of ion gyroorbits start to
matter. Observationally, this transition manifests itself as a
clear break in the spectrum of magnetic fluctuations, with the
inertial-range k−5/3 scaling replaced by a steeper slope (see
Fig. 1). While the electrons at these scales can be treated as
an isothermal fluid (as long as we are considering fluctuations
above the electron gyroscale, k⊥ρe ≪ 1; see § 4), the fully
gyrokinetic description (§ 3) has to be adopted for the ions.
It is, indeed, to understand plasma dynamics at and around
k⊥ρi ∼ 1 that gyrokinetics was first designed in fusion plasma
theory (Frieman & Chen 1982; Brizard & Hahm 2007). In or-
der for gyrokinetics and further dissipation-range approxima-
tions that follow from it (§ 7) to be a credible approach in
the solar wind and other space plasmas, it has to be estab-
lished that fluctuations at and below the ion gyroscale are still
strongly anisotropic, k‖ ≪ k⊥. If that is the case, then their
frequencies (ω∼ k‖vAk⊥ρi, see § 7.3) will still be smaller than
the cyclotron frequency in at least a part of the “dissipation
range”38—the range of scales k⊥ρi & 1 (see § 7.13).
Note that additional information about the dissipation-
range turbulence can be extracted from the measurements in
the magnetosheath—while scales above the ion gyroscale are
probably non-universal there, the dissipation range appears to
display universal behavior, mostly similar to the solar wind
(see, e.g., Alexandrova 2008). This complements the obser-
vational picture emerging from the solar-wind data and al-
lows us to learn more as fluctuation amplitudes in the mag-
netosheath are larger and much smaller scales can be probed
than in the solar wind (Mangeney et al. 2006; Lacombe et al.
2006; Alexandrova et al. 2008b).
8.2.1. Anisotropy
We know with a fair degree of certainty that the fluctu-
ations that cascade down to the ion gyroscale from the in-
ertial range are strongly anisotropic (§ 8.1.3). While it ap-
pears likely that the anisotropy persists at k⊥ρi ∼ 1, it is ex-
tremely important to have a clear verdict on this assumption
from solar wind measurements. While Leamon et al. (1998)
and, more recently, Hamilton et al. (2008) did present some
evidence that magnetic fluctuations in the solar wind have a
degree of anisotropy below the ion gyroscale, no definitive
study similar to Horbury et al. (2008) or Bigazzi et al. (2006);
Sorriso-Valvo et al. (2006) exists as yet. In the magne-
tosheath, where the dissipation-range scales are easier to mea-
sure than in the solar wind, recent analysis by Sahraoui et al.
(2006); Alexandrova et al. (2008b) does show evidence of
strong anisotropy.
Besides confirming the presence of the anisotropy, it would
be interesting to study its scaling characteristics: e.g., check
the scaling prediction k‖ ∼ k
⊥ [Eq. (241); see also § 7.9.4
and § 7.10.3] in a similar fashion as the GS relation k‖ ∼ k
[Eq. (5)] was corroborated by Horbury et al. (2008).
In this paper, we have proceeded on the assumption that
the anisotropy, and, therefore, low frequencies (ω ≪ Ωi) do
characterize fluctuations in the dissipation range—or, at least,
that the low-frequency anisotropic fluctuations are a signifi-
cant energy cascade channel and can be considered decoupled
from any possible high-frequency dynamics.
8.2.2. Transition at the Ion Gyroscale: Collisionless Damping and
Heating
38 This term, customary in the space-physics literature, is somewhat of
a misnomer because, as we have seen in § 7, rich dissipationless turbulent
dynamics are present in this range alongside what is normally thought of as
dissipation.
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 43
If the fluctuations at the ion gyroscale have k‖ ≪ k⊥ and
ω ≪ Ωi (§ 8.2.1), they are not subject to the cyclotron res-
onance (ω − k‖v‖ = ±Ωi), but are subject to the Landau one
(ω = k‖v‖). Alfvénic fluctuations at the ion gyroscale are
no longer decoupled from the compressive fluctuations and
can be Landau-damped (§ 7.1). It seems plausible that it
is the inflow of energy from the Alfvénic cascade that ac-
counts for a pronounced local flattening of the spectrum of
density fluctuations in the solar wind observed just above
the ion gyroscale (Woo & Armstrong 1979; Celnikier et al.
1983, 1987; Coles & Harmon 1989; Marsch & Tu 1990b;
Coles et al. 1991; Kellogg & Horbury 2005).39
In energetic terms, Landau damping amounts to a redis-
tribution of generalized energy from electromagnetic fluctu-
ations to entropy fluctuations (§ 3.4, § 7.8). This gives rise
to the entropy cascade, ultimately transferring the Landau-
damped energy into ion heat (§ 3.5, § 7.9 and § 7.10). How-
ever, only part of the inertial-range cascade is so damped be-
cause an alternative, electron, cascade channel exists: the ki-
netic Alfvén waves (§§ 7.2-7.8). The energy transferred into
the KAW-like fluctuations can cascade to the electron gy-
roscale, where it is Landau damped on electrons, converting
first into the electron entropy cascade and then electron heat
(§ 7.12).
Thus, the transition at the ion gyroscale ultimately de-
cides in what proportion the turbulent energy arriving from
the inertial range is distributed between the ion and electron
heat. How the fraction of power going into either depends on
parameters—βi, Ti/Te, amplitudes, . . . —is a key unanswered
question both in space and astrophysical (see, e.g., § 8.5) plas-
mas. Gyrokinetics appears to be an ideal tool for addressing
this question both analytically and numerically (Howes et al.
2008b). Within the framework outlined in this paper, the min-
imal model appropriate for studying the transition at the ion
gyroscale is the system of equations for isothermal electrons
and gyrokinetic ions derived in § 4 (it is summarized in § 4.9).
8.2.3. Ion Gyroscale vs. Ion Inertial Scale
It is often assumed in the space physics literature that it is
at the ion inertial scale, di = ρi/
βi, rather than at the ion gy-
roscale ρi that the spectral break between the inertial and dis-
sipation range occurs. The distinction between di and ρi be-
comes noticeable when βi is significantly different from unity,
a relatively rare occurrence in the solar wind. While some at-
tempts to determine at which of these two scales a spectral
break between the inertial and dissipation ranges occurs have
produced claims that di is a more likely candidate (Smith et al.
2001), more comprehensive studies of the available data sets
conclude basically that it is hard to tell (Leamon et al. 2000;
Markovskii et al. 2008).
In the gyrokinetic approach advocated in this paper, the ion
inertial scale does not play a special role (see § 7.1). The only
parameter regime in which di does appear as a special scale
is Ti ≪ Te (“cold ions”), when the Hall MHD approximation
can be derived in a systematic way (see Appendix E). This,
however, is not the right limit for the solar wind or most other
astrophysical plasmas of interest because ions are rarely cold.
Hall MHD is discussed further in § 8.2.6 and Appendix E.
8.2.4. KAW Turbulence
39 Celnikier et al. (1987) proposed that the flattening might be a k−1 spec-
trum analogous to Batchelor’s spectrum of passive scalar variance in the
viscous-convective range. We think this analogy cannot apply because den-
sity is not passive at or below the ion gyroscale.
If gyrokinetics is valid at scales k⊥ρi & 1 (i.e., if k‖ ≪ k⊥,
ω ≪ Ωi and it is acceptable to at least model the equilibrium
distribution as a Maxwellian; see § 8.3), the electromagnetic
fluctuations below the ion gyroscale will be described by the
fluid approximation that we derived in § 7.2 and referred to
ERMHD. The wave solutions of this system of equations are
the kinetic Alfvén waves (§§ 7.3-7.4) and it is possible to ar-
gue for a GS-style critically balanced cascade of KAW-like
electromagnetic fluctuations (§ 7.5) between the ion and elec-
tron gyroscales (Landau damped on electrons at k⊥ρe ∼ 1; the
expression for the KAW damping rate in the gyrokinetic limit
is given in Howes et al. 2006; see also Fig. 8).
Individual KAW have, indeed, been detected in space plas-
mas (e.g., Grison et al. 2005). What about KAW turbulence?
How does one tell whether any particular spectral slope one is
measuring corresponds to the KAW cascade or fits some alter-
native scheme for the dissipation-range turbulence (§ 8.2.6)?
It appears to be a sensible program to look for specific rela-
tionships between different fields predicted by theory (§ 7.2)
and for the corresponding spectral slopes and scaling relations
for the anisotropy (§ 7.5). This means that simultaneous mea-
surements of magnetic, electric, density and magnetic-field-
strength fluctuations are needed.
For the solar wind, the spectra of electric and magnetic
fluctuations below the ion gyroscale reported by Bale et al.
(2005) are consistent with the k−1/3 and k−7/3 scalings pre-
dicted for an anisotropic critically balanced KAW cascade
(§ 7.5; see Fig. 1 for theoretical scaling fits superimposed
on a plot taken from Bale et al. 2005; note, however, that
Bale et al. 2005 themselves interpreted their data in a some-
what different way and that their resolution was in any case
not sufficient to be sure of the scalings). They were also able
to check that their fluctuations satisfied the KAW dispersion
relation—for critically balanced fluctuations, this is, indeed,
plausible. Magnetic-fluctuation spectra recently reported by
Alexandrova et al. (2008a) are only slightly steeper than the
theoretical k−7/3 KAW spectrum. These authors also find a
significant amount of magnetic-field-strength fluctuations in
the dissipation range, with a spectrum that follows the same
scaling—this is again consistent with the theoretical picture
of KAW turbulence [see Eq. (223)]. Measurements reported
by Czaykowska et al. (2001); Alexandrova et al. (2008b) for
the magnetosheath appear to present a similar picture.
The density spectra measured by Celnikier et al. (1983,
1987) steepen below the ion gyroscale following the flattened
segment around k⊥ρi ∼ 1 (discussed in § 8.2.2). For a KAW
cascade, the density spectrum should be k−7/3 (§ 7.5); with-
out KAW, k−10/3 (§ 7.10.2). The slope observed in the papers
cited above appears to be somewhat shallower even than k−2
(cf. a similar result by Spangler & Gwinn 1990 for the ISM;
see § 8.4.1), but, given imperfect resolution, neither seriously
in contradiction with the prediction based on the KAW cas-
cade, nor sufficient to corroborate it. Unfortunately, we have
not found published simultaneous measurements of density-
and magnetic- or electric-fluctuation spectra.
8.2.5. Variability of the Spectral Slope
While many measurements consistent with the KAW pic-
ture can be found, there are also many in which the spectra
are much steeper (Denskat et al. 1983; Leamon et al. 1998).
Analysis of a large set of measurements of the magnetic-
fluctuation spectra in the dissipation range of the solar wind
reveals a wide spread in the spectral indices: roughly between
44 SCHEKOCHIHIN ET AL.
−1 and −4 (Smith et al. 2006). There is evidence of a weak
positive correlation between steeper dissipation-range spectra
and higher ion temperatures (Leamon et al. 1998) or higher
cascade rates calculated from the inertial range (Smith et al.
2006). This suggests that a larger amount of ion heating may
correspond to a fully or partially suppressed KAW cascade,
which is in line with our view of the ion heating and the KAW
cascade as the two competing channels of the overall kinetic
cascade (§ 7.8). With a weakened KAW cascade, all or part of
the dissipation range would be dominated by the ion entropy
cascade—a purely kinetic phenomenon manifested by pre-
dominantly electrostatic fluctuations and very steep magnetic-
energy spectra (§ 7.10). This might account both for the steep-
ness of the observed spectra and for the spread in their indices
(§ 7.11), although many other theories exist (see § 8.2.6).
While we may thus have a plausible argument, this is not
yet a satisfactory quantitative theory that would allow us to
predict when the KAW cascade is present and when it is not or
what dissipation-range spectrum should be expected for given
values of the solar-wind parameters (βi, Ti/Te, etc.). Resolu-
tion of this issue again appears to hinge on the question of how
much turbulent power is diverted into the ion entropy cascade
(equivalently, into ion heat) at the ion gyroscale (see § 8.2.2).
8.2.6. Alternative Theories of the Dissipation Range
A number of alternative theories and models have been put
forward to explain the observed spectral slopes (and their vari-
ability) in the dissipation range. It is not our aim to review or
critique them all in detail, but perhaps it is useful to provide a
few brief comments about some of them in light of the theo-
retical framework constructed in this paper.
This entire theoretical framework hinges on adopting gy-
rokinetics as a valid description or, at least, a sensible model
that does not miss any significant channels of energy cascade
and dissipation. While we obviously believe this to be the
right approach, it is worth spelling out what effects are left
out “by construction.”
Parallel Alfvén-wave cascade and ion cyclotron damping. — The
use of gyrokinetics assumes that fluctuations stay anisotropic
at all scales, k‖ ≪ k⊥, and, therefore, ω ≪ Ωi, so the
cyclotron resonances are ordered out. However, if one
insists on routing the Alfvén-wave energy into a paral-
lel cascade, e.g., by forcibly setting k⊥ = 0, it is pos-
sible to construct a weak turbulence theory in which it
is dissipated by the ion cyclotron damping (Yoon & Fang
2008). Numerical simulations of 3D MHD turbulence do
not support the possibility of a parallel Alfvén-wave cascade
(Shebalin et al. 1983; Oughton et al. 1994; Cho & Vishniac
2000; Maron & Goldreich 2001; Cho et al. 2002; Müller et al.
2003). Solar-wind evidence that the perpendicular cascade
dominates is quite strong for the inertial range (§ 8.1.3) and
less so for the dissipation range (§ 8.2.1). While, as stated
in § 8.2.1, one cannot yet definitely claim that observations
tell us that ω ≪ Ωi at k⊥ρi ∼ 1, it has been argued that
observations do not appear to be consistent with cyclotron
damping being the main mechanism for the dissipation of
the inertial-range Alfvénic turbulence at the ion gyroscale
(Leamon et al. 1998, 2000; Smith et al. 2001). Ion-cyclotron
resonance could conceivably be reached somewhere in the
dissipation range (see § 7.13). At this point gyrokinetics will
formally break down, although, as argued by Howes et al.
(2008a, see their § 3.6), this does not necessarily mean that
ion cyclotron damping will become the dominant dissipation
channel for the turbulence.
Parallel whistler cascade. — A parallel magnetosonic/whistler
cascade eventually damped by the electron cyclotron
resonance (Stawicki et al. 2001) is also excluded in the
construction of gyrokinetics. The whistler cascade has
been given some consideration in the Hall MHD approxi-
mation (further discussed at the end of this section). Both
weak-turbulence theory (Galtier 2006) and 3D numerical
simulations (Cho & Lazarian 2004) concluded that, like
in MHD, the turbulent cascade is highly anisotropic, with
perpendicular energy transfer dominating over the parallel
one.40 The same conclusion appears to have been reached
in recent 2D kinetic PIC simulations by Gary et al. (2008);
Saito et al. (2008). Thus, the turbulence again seems to be
driven into the gyrokinetically accessible regime.
While theory and numerical simulations appear to make
arguing in favor of a parallel cascade and cyclotron heat-
ing difficult, there exists some observational evidence in sup-
port of them, especially for the near-Sun solar wind (e.g.,
Harmon & Coles 2005). Thus, the presence or relative im-
portance of the cyclotron heating in the solar wind and, more
generally, the mechanism(s) responsible for the observed per-
pendicular ion heating (Marsch et al. 1983) remain a largely
open problem. Besides the theories mentioned above, many
other ideas have been proposed, some of which attempted
to reconcile the dominance of the low-frequency perpendic-
ular cascade with the possibility of cyclotron heating (e.g.,
Chandran 2005b; Markovskii et al. 2006; see Hollweg 2008
for a concise recent review of the problem).
Mirror cascade. — Sahraoui et al. (2006) analyzed a set of
Cluster multi-spacecraft measurements in the magnetosheath
and reported a broad power-law (∼ k−8/3) spectrum of mirror
structures at and below the ion gyroscale. They claim that
these are not KAW-like fluctuations because their frequency
is zero in the plasma frame. Although these structures are
highly anisotropic with k‖ ≪ k⊥, they cannot be described by
the gyrokinetic theory in its present form because δB‖/B0 is
very large (∼ 40%, occasionally reaching unity) and because
the particle trapping by fluctuations, which is likely to be
important in the nonlinear physics of the mirror instabil-
ity (Kivelson & Southwood 1996; Pokhotelov et al. 2008;
Rincon et al. 2009), is ordered out in gyrokinetics. Thus, if a
“mirror cascade” exists, it is not captured in our description.
More generally, the effect of the pressure-anisotropy-driven
instabilities on the turbulence in the dissipation range is a
wide open area, requiring further analytical effort (see § 8.3).
If k‖ ≪ k⊥, ω ≪ Ωi, and δB/B0 ≪ 1 are accepted for the
dissipation range and plasma instabilities at the ion gyroscale
(§ 8.3) are ignored, the formal gyrokinetic theory and its
asymptotic consequences derived above should hold. There
are two essential features of the linear physics at and below
the ion gyroscale that must play some role: the collisionless
(Landau) damping and the dispersive nature of the wave so-
lutions (see Fig. 8 and § 7.3; cf., e.g., Leamon et al. 1999;
Stawicki et al. 2001). Both of these features have been em-
ployed to explain the spectral break at the ion gyroscale and
the spectral slopes below it.
40 It is possible to produce a parallel cascade artificially by running 1D
simulations (Matthaeus et al. 2008b).
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 45
Landau damping and instrumental effects. — In most of our dis-
cussion, (§ 7, §§ 8.2.4-8.2.5), we effectively assumed that the
Landau damping is only important at k⊥ρi ∼ 1 and k⊥ρe ∼ 1,
but not in between, so we could talk about asymptotic scal-
ings and dissipationless cascades. However, as was noted
in § 7.6, a properly asymptotic scaling behavior in the dis-
sipation range is probably impossible in nature because the
scale separation between the ion and electron gyroscales is
only about (mi/me)
1/2 ≃ 43. In particular, there is not always
a wide scale interval where the kinetic damping is negligi-
bly small (especially at low βi; see Fig. 8; cf. Leamon et al.
1999). Howes et al. (2008a) proposed a model of how the
presence of damping combined with instrumental effects (a
resolution floor) could lead to measured spectra that look like
power laws steeper than k−7/3, with the effective spectral ex-
ponent depending on plasma parameters (we refer the reader
to that paper for a discussion of how this compares with pre-
vious models of a similar kind, e.g., Li et al. 2001). A key
physical assumption of theirs and similar models is that the
amount of power drained from the Alfvén-wave and KAW
cascades into the ion heat is set by the strength of the linear
damping. Whether this is justified is not yet clear.
Hall and Electron MHD. — If Landau damping is deemed
unimportant in some part of the dissipation range (which
can be true in some regimes; see Fig. 8 and Howes et al.
2006, 2008a,b) and the wave dispersion is considered to
be the salient feature, it might appear that a fluid, rather
than kinetic, description should be sufficient. Hall MHD
(Mahajan & Yoshida 1998) or its kdi ≫ 1 limit the Electron
MHD (Kingsep et al. 1990) have been embraced by many au-
thors as such a description, suitable both for analytical argu-
ments (Goldreich & Reisenegger 1992; Krishan & Mahajan
2004; Gogoberidze 2005; Galtier & Bhattacharjee 2003;
Galtier 2006; Alexandrova et al. 2008a) and numerical sim-
ulations (Biskamp et al. 1996, 1999; Ghosh et al. 1996;
Ng et al. 2003; Cho & Lazarian 2004; Shaikh & Zank 2005;
Galtier & Buchlin 2007; Matthaeus et al. 2008b).
To what extent does this constitute an approach alterna-
tive to (and better than?) gyrokinetics (as suggested, e.g., by
Matthaeus et al. 2008b)? For fluctuations with k‖ ≪ k⊥, Hall
MHD is merely a particular limit of gyrokinetics: βi ≪ 1 and
Ti/Te ≪ 1 (cold-ion limit; see Appendix E). If k‖ is not small
compared to k⊥, then the gyrokinetics is not valid, while Hall
MHD continues to describe the cold-ion limit correctly (e.g.,
Ito et al. 2004; Hirose et al. 2004), capturing in particular the
whistler branch of the dispersion relation. However, as we
have already mentioned above, the dominance of the perpen-
dicular energy transfer (k‖ ≪ k⊥) is supported both by weak-
turbulence theory for Hall MHD (Galtier 2006) and by 3D
numerical simulations of the Electron MHD (Cho & Lazarian
2004).
Thus, the gyrokinetic theory and its rigorous limits, such
as ERMHD (§ 7.2), supersede Hall MHD for anisotropic tur-
bulence. Since ions are generally not cold in the solar wind
(or any other plasma discussed here), Hall MHD is not for-
mally a relevant approximation. It also entirely misses the
kinetic damping and the associated entropy cascade channel
leading to particle heating (§ 7.1, § 7.9 and § 7.10). However,
Hall MHD does capture the Alfvén waves becoming disper-
sive and numerical simulations of it do show a spectral break,
although, technically speaking, at the wrong scale (di instead
of ρi; see § 7.1). Although Hall MHD cannot be rigorously
used as quantitative theory of the spectral break and the asso-
ciated change in the nature of the turbulent cascade, the Hall
MHD equations in the limit kdi ≫ 1 are mathematically sim-
ilar to our ERMHD equations (see § 7.2 and Appendix E) to
within constant coefficients probably not essential for quali-
tative models of turbulence. Therefore, results of numerical
simulations of Hall and Electron MHD cited above are di-
rectly useful for understanding the KAW cascade—and, in-
deed, in the limit kdi ≫ 1, kde ≪ 1, they are mostly consistent
with the scaling arguments of § 7.5.
Alfvén vortices. — Finally we mention an argument pertaining
to the dissipation-range spectra that is not based on energy
cascades at all. Based on the evidence of Alfvén vortices in
the magnetosheath, Alexandrova (2008) speculated that steep
power-law spectra observed in the dissipation range at least
in some cases could reflect the geometry of the ion-gyroscale
structures rather than a local energy cascade. If Alfvén vor-
tices are a common feature, this possibility cannot be ex-
cluded. However, the resulting geometrical spectra are quite
steep (k−4 and steeper), so they can become important only
if the KAW cascade is weak or suppressed—somewhat simi-
larly to the steep spectra associated with the entropy cascade
(§ 7.11).
8.3. Is Equilibrium Distribution Isotropic and Maxwellian?
In rigorous theoretical terms, the weakest point of this pa-
per is the use of a Maxwellian equilibrium. Formally, this is
only justified when the collisions are weak but not too weak:
we ordered the collision frequency as similar to the fluctu-
ation frequency [Eq. (49)]. This degree of collisionality is
sufficient to prove that a Maxwellian equilibrium distribution
F0s(v) does indeed emerge in the lowest order of the gyroki-
netic expansion (Howes et al. 2006). This argument works
well for plasmas such as the ISM (§ 8.4), where collisions are
weak (λmfpi ≫ ρi) but non-negligible (λmfpi ≪ L). In space
plasmas, the mean free path is of the order of 1 AU—the dis-
tance between the Sun and the Earth (see Table 1). Strictly
speaking, in so highly collisionless a plasma, the equilib-
rium distribution does not have to be either Maxwellian or
isotropic.
The conservation of the first adiabatic invariant, µ = v2⊥/2B,
suggests that temperature anisotropy with respect to the
magnetic-field direction (T0⊥ 6= T0‖) may exist. When the
relative anisotropy is larger than (roughly) 1/βi, it triggers
several very fast growing plasma instabilities: most promi-
nently the firehose (T0⊥ < T0‖) and mirror (T0⊥ > T0‖) modes
(e.g., Gary et al. 1976). Their growth rates peak around the
ion gyroscale, thus giving rise to additional energy injection
at k⊥ρi ∼ 1.
No definitive analytical theory of how these fluctuations sat-
urate, cascade and affect the equilibrium distribution has been
proposed. It appears to be a reasonable expectation that the
fluctuations resulting from temperature anisotropy will satu-
rate by limiting this anisotropy. This idea has some support
in solar-wind observations: while the degree of anisotropy
of the core particle distribution functions varies consider-
ably between data sets, the observed anisotropies do seem
to populate the part of the parameter plane (T0⊥/T0‖,βi) cir-
cumscribed in a rather precise way by the marginal stabil-
ity boundaries for the mirror and firehose (Gary et al. 2001;
Kasper et al. 2002; Marsch et al. 2004; Hellinger et al. 2006;
Matteini et al. 2007).41
41 Note that Kellogg et al. (2006) measure the electric-field fluctuations
46 SCHEKOCHIHIN ET AL.
If we want to study turbulence in data sets that do not lie
too close to these stability boundaries, assuming an isotropic
Maxwellian equilibrium distribution [Eq. (54)] is probably
an acceptable simplification, although not an entirely rigor-
ous one. Further theoretical work is clearly possible on this
subject: thus, it is not a problem to formulate gyrokinetics
with an arbitrary equilibrium distribution (Frieman & Chen
1982) and starting from that, once can generalize the results
of this paper (for the KRMHD system, § 5, this has been done
by Chen et al. 2009). Treating the instabilities themselves
might prove more difficult, requiring the gyrokinetic order-
ing to be modified and the expansion carried to higher orders
to incorporate features that are not captured by gyrokinetics,
e.g., short parallel scales (Rosin et al. 2009), particle trap-
ping (Pokhotelov et al. 2008; Rincon et al. 2009), or nonlin-
ear finite-gyroradius effects (Califano et al. 2008). Note that
the theory of the dissipation-range turbulence will probably
need to be modified to account for the additional energy in-
jection from the instabilities and for the (yet unclear) way in
which this energy makes its way to dissipation and into heat.
Besides the anisotropies, the particle distribution functions
in the solar wind (especially the electron one) exhibit non-
Maxwellian suprathermal tails (see Maksimovic et al. 2005;
Marsch 2006, and references therein). These contain small
(∼ 5% of the total density) populations of energetic particles.
Both the origin of these particles and their effect on turbulence
have to be modeled kinetically. Again, it is possible to formu-
late gyrokinetics for general equilibrium distributions of this
kind and examine the interaction between them and the turbu-
lent fluctuations, but we leave such a theory outside the scope
of this paper.
Thus, much remains to be done to incorporate realistic equi-
librium distribution functions into the gyrokinetic description
of the solar wind plasma. In the meanwhile, we believe that
the gyrokinetic theory based on a Maxwellian equilibrium dis-
tribution as presented in this paper, while idealized and imper-
fect, is nevertheless a step forward in the analytical treatment
of the space-plasma turbulence compared to the fluid descrip-
tions that have prevailed thus far.
8.4. Interstellar Medium
While the solar wind is unmatched by other astrophysical
plasmas in the level of detail with which turbulence in it can
be measured, the interstellar medium (ISM) also offers an ob-
server a number of ways of diagnosing plasma turbulence,
which, in the case of the ISM, is thought to be primarily ex-
cited by supernova explosions (Norman & Ferrara 1996). The
accuracy and resolution of this analysis are due to improve
rapidly thanks to many new observatories, e.g., LOFAR,42
Planck (Enßlin et al. 2006), and, in more distant future, the
SKA (Lazio et al. 2004).
The ISM is a spatially inhomogeneous environment consist-
ing of several phases that have different temperatures, densi-
ties and degrees of ionization (Ferrière 2001).43 We will use
the Warm ISM phase (see Table 1) as our fiducial interstel-
lar plasma and discuss briefly what is known about the two
main observationally accessible quantities—the electron den-
sity and magnetic fields—and how this information fits into
in the ion-cyclotron frequency range, estimate the resulting velocity-space
diffusion and argue that it is sufficient to isotropize the ion distribution
42 http://www.lofar.org
43 And, therefore, different degrees of importance of the neutral particles
and the associated ambipolar damping effects—these will not be discussed
here; see Lithwick & Goldreich 2001.
the theoretical framework proposed here.
8.4.1. Electron Density Fluctuations
The electron-density fluctuations inferred from the inter-
stellar scintillation measurements appear to have a spectrum
with an exponent ≃ −1.7, consistent with the Kolmogorov
scaling (Armstrong et al. 1981, 1995; Lazio et al. 2004; see,
however, dissenting evidence by Smirnova et al. 2006, who
claim a spectral exponent closer to −1.5). This holds over
about 5 decades of scales: λ ∈ (105,1010) km. Other observa-
tional evidence at larger and smaller scales supports the case
for this presumed inertial range to be extended over as many
as 12 decades: λ ∈ (102,1015) km, a fine example of scale
separation that prompted an impressed astrophysicist to dub
the density scaling “The Great Power Law in the Sky.” The
upper cutoff here is consistent with the estimates of the su-
pernova scale of order 100 pc—presumably the outer scale of
the turbulence (Norman & Ferrara 1996) and also roughly the
scale height of the galactic disk (obviously the upper bound
on the validity of any homogeneous model of the ISM tur-
bulence). The lower cutoff is an estimate for the inner scale
below which the logarithmic slope of the density spectrum
steepens to about −2 (Spangler & Gwinn 1990).
Higdon (1984) was the first to realize that the electron-
density fluctuations in the ISM could be attributed to a cas-
cade of a passive tracer mixed by the ambient turbulence (the
MHD entropy mode; see § 2.6). This idea was brought to ma-
turity by Lithwick & Goldreich (2001), who studied the pas-
sive cascades of the slow and entropy modes in the frame-
work of the GS theory (see also Maron & Goldreich 2001).
If the turbulence is assumed anisotropic, as in the GS theory,
the passive nature of the density fluctuations with respect to
the decoupled Alfvén-wave cascade becomes a rigorous re-
sult both in MHD (§ 2.4) and, as we showed above, in the
more general gyrokinetic description appropriate for weakly
collisional plasmas (§ 5.5). Anisotropy of the electron-density
fluctuations in the ISM is, indeed, observationally supported
(Wilkinson et al. 1994; Trotter et al. 1998; Rickett et al. 2002;
Dennett-Thorpe & de Bruyn 2003; Heyer et al. 2008, see also
Lazio et al. 2004 for a concise discussion), although detailed
scale-by-scale measurements are not currently possible.
If the underlying Alfvén-wave turbulence in the ISM has
⊥ spectrum, as predicted by GS, so should the elec-
tron density (see § 2.6). As we discussed in § 6.3, the phys-
ical nature of the inner scale for the density fluctuations de-
pends on whether they have a cascade in k‖ and are effi-
ciently damped when k‖λmfpi ∼ 1 or fail to develop small
parallel scales and can, therefore, reach k⊥ρi ∼ 1. The ob-
servationally estimated inner scale is consistent with the ion
gyroscale, ρi ∼ 103 km (see Table 1; note that the ion iner-
tial scale di = ρi/
βi is similar to ρi at the moderate values
of βi characteristic of the ISM—see further discussion of the
(ir)relevance of di in § 7.1, § 8.2.3 and Appendix E). How-
ever, since the mean free path in the ISM is not huge (Ta-
ble 1), it is not possible to distinguish this from the perpen-
dicular cutoff k−1⊥ ∼ λ
mfpiL
−1/2 ∼ 500 km implied by the par-
allel cutoff at k‖λmfpi ∼ 1 [see Eq. (220)], as advocated by
Lithwick & Goldreich (2001). Note that the relatively short
mean free path means that much of the scale range spanned by
the Great Power Law in the Sky is, in fact, well described by
the MHD approximation either with adiabatic (§ 2) or isother-
mal (§ 6.1 and Appendix D) electrons.
Below the ion gyroscale, the −2 spectral exponent reported
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 47
by Spangler & Gwinn (1990) is measured sufficiently impre-
cisely to be consistent with the −7/3 expected for the density
fluctuations in the KAW cascade (§ 7.5). However, given the
high degree of uncertainty about what happens in this “dis-
sipation range” even in the much better resolved case of the
solar wind (§ 8.2), it would probably be wise to reserve judg-
ment until better data are available.
8.4.2. Magnetic Fluctuations
The second main observable type of turbulent fluctuations
in the ISM are the magnetic fluctuations, accessible indirectly
via the measurements of the Faraday rotation of the polar-
ization angle of the pulsar light travelling through the ISM.
The structure function of the rotation measure (RM) should
have the Kolmogorov slope of 2/3 if the magnetic fluctua-
tions are due to Alfvénic turbulence described by the GS the-
ory. There is a considerable uncertainty in interpreting the
available data, primarily due to insufficient spatial resolution
(rarely better than a few parsec). Structure function slopes
consistent with 2/3 have been reported (Minter & Spangler
1996), but, depending on where one looks, shallower struc-
ture functions that seem to steepen at scales of a few parsec
are also observed (Haverkorn et al. 2004).
A recent study by Haverkorn et al. (2005) detected an in-
teresting trend: the RM structure functions computed for re-
gions that lie in the galactic spiral arms are nearly perfectly
flat down to the resolution limit, while in the interarm regions,
they have detectable slopes (although these are mostly shal-
lower that 2/3). Observations of magnetic fields in external
galaxies also reveal a marked difference in the magnetic-field
structure between arms and interarms: the spatially regular
(mean) fields are stronger in the interarms, while in the arms,
the stochastic fields dominate (Beck 2007). This qualitative
difference between the magnetic-field structure in the arms
and interarms has been attributed to smaller effective outer
scale in the arms (∼ 1 pc, compared to ∼ 102 pc in the in-
terarms; see Haverkorn et al. 2008) or to the turbulence in the
arms and interarms belonging to the two distinct asymptotic
regimes described in § 1.3: closer to the anisotropic Alfvénic
turbulence with a strong mean field in the interarms and to the
isotropic saturated state of small-scale dynamo in the arms
(Schekochihin et al. 2007).
8.5. Accretion Disks
Accretion of plasma onto a central black hole or neutron
star is responsible for many of the most energetic phenomena
observed in astrophysics (see, e.g., Narayan & Quataert 2005
for a review). It is now believed that a linear instability of dif-
ferentially rotating plasmas—the magnetorotational instabil-
ity (MRI)—amplifies magnetic fields and gives rise to MHD
turbulence in astrophysical disks (Balbus & Hawley 1998).
Magnetic stresses due to this turbulence transport angular mo-
mentum, allowing plasma to accrete. The MRI converts the
gravitational potential energy of the inflowing plasma into
turbulence at the outer scale that is comparable to the scale
height of the disk. This energy is then cascaded to small
scales and dissipated into heat—powering the radiation that
we see from accretion flows. Fluid MHD simulations show
that the MRI-generated turbulence in disks is subsonic and
has β ∼ 10 − 100. Thus, on scales much smaller than the scale
height of the disk, homogeneous turbulence in the parameter
regimes considered in this paper is a valid idealization and
the kinetic models developed above should represent a step
forward compared to the purely fluid approach.
Turbulence is not yet directly observable in disks, so mod-
els of turbulence are mostly used to produce testable predic-
tions of observable properties of disks such as their X-ray and
radio emission. One of the best observed cases is the (pre-
sumed) accretion flow onto the black hole coincident with the
radio source Sgr A∗ in the center of our Galaxy (see review
by Quataert 2003).
Depending on the rate of heating and cooling in the inflow-
ing plasma (which in turn depend on accretion rate and other
properties of the system under consideration), there are differ-
ent models that describe the physical properties of accretion
flows onto a central object. In one class of models, a geometri-
cally thin optically thick accretion disk (Shakura & Sunyaev
1973), the inflowing plasma is cold and dense and well de-
scribed as an MHD fluid. When applied to Sgr A∗, these
models produce a prediction for its total luminosity that is
several orders of magnitude larger than observed. Another
class of models, which appears to be more consistent with the
observed properties of Sgr A∗, is called radiatively inefficient
accretion flows (RIAFs; see Rees et al. 1982; Narayan & Yi
1995 and review by Quataert 2003 of the applications and ob-
servational constraints in Sgr A∗). In these models, the in-
flowing plasma near the black hole is believed to adopt a two-
temperature configuration, with the ions (Ti ∼ 1011 − 1012 K)
hotter than the electrons (Te ∼ 109 − 1011 K).44 The electron
and ion thermodynamics decouple because the densities are
so low that the temperature equalization time ∼ ν−1ie is longer
than the time for the plasma to flow into the black hole. Thus,
like the solar wind, RIAFs are macroscopically collisionless
plasmas (see Table 1 for plasma parameters in the Galactic
center; note that these parameters are so extreme that the gy-
rokinetic description, while probably better than the fluid one,
cannot be expected to be rigorously valid; at the very least, it
needs to be reformulated in a relativistic form). At the high
temperatures appropriate to RIAFs, electrons radiate energy
much more efficiently than the ions (by virtue of their much
smaller mass) and are, therefore, expected to contribute dom-
inantly to the observed emission, while the thermal energy of
the ions is swallowed by the black hole. Since the plasma
is collisionless, the electron heating by turbulence largely de-
termines the thermodynamics of the electrons and thus the
observable properties of RIAFs. The question of which frac-
tion of the turbulent energy goes into ion and which into elec-
tron heating is, therefore, crucial for understanding accretion
flows—and the answer to this question depends on the de-
tailed properties of the small-scale kinetic turbulence (e.g.,
Quataert & Gruzinov 1999; Sharma et al. 2007), as well as on
the linear properties of the collisionless MRI (Quataert et al.
2002; Sharma et al. 2003).
Since all of the turbulent power coming down the cascade
must be dissipated into either ion or electron heat, it is re-
ally the amount of generalized energy diverted at the ion gy-
roscale into the ion entropy cascade (§§ 7.8-7.9) that decides
how much energy is left to heat the electrons via the KAW
cascade (§§ 7.2-7.5, § 7.12). Again, as in the case of the solar
wind (§ 8.2.2 and § 8.2.5), the transition around the ion gy-
roscale from the Alfvénic turbulence at k⊥ρi ≪ 1 to the KAW
turbulence at k⊥ρi ≫ 1 emerges as a key unsolved problem.
8.6. Galaxy Clusters
44 It is partly with this application in mind that we carried the general
temperature ratio in our calculations; see footnote 17.
48 SCHEKOCHIHIN ET AL.
Galaxy clusters are the largest plasma objects in the Uni-
verse. Like the other examples discussed above, the intraclus-
ter plasma is in the weakly collisional regime (see Table 1).
Fluctuations of electron density, temperature and of magnetic
fields are measured in clusters by X-ray and radio observa-
tories, but the resolution is only just enough to claim that a
fairly broad scale range of fluctuations exists (Schuecker et al.
2004; Vogt & Enßlin 2005). No power-law scalings have yet
been established beyond reasonable doubt.
What fundamentally hampers quantitative modeling of tur-
bulence and related effects in clusters is that we do not have
a definite theory of the basic properties of the intracluster
medium: its (effective) viscosity, magnetic diffusivity or ther-
mal conductivity. In a weakly collisional and strongly mag-
netized plasma, all of these depend on the structure of the
magnetic field (Braginskii 1965), which is shaped by the tur-
bulence. If (or at scales where) a reasonable a priori assump-
tion can be made about the field structure, further analytical
progress is possible: thus, the theoretical models presented in
this paper assume that the magnetic field is a sum of a slowly
varying in space “mean field” and small low-frequency per-
turbations (δB ≪ B0).
In fact, since clusters do not have mean fields of any mag-
nitude that could be considered dynamically significant, but
do have stochastic fields, the outer-scale MHD turbulence in
clusters falls into the weak-mean-field category (see § 1.3).
The magnetic field should be highly filamentary, organized
in long folded direction-reversing structures. It is not cur-
rently known what determines the reversal scale.45 Obser-
vations, while tentatively confirming the existence of very
long filaments (Clarke & Enßlin 2006), suggest that the re-
versal scale is much larger than the ion gyroscale: thus, the
magnetic-energy spectrum for the Hydra A cluster core re-
ported by Vogt & Enßlin (2005) peaks at around 1 kpc, com-
pared to ρi ∼ 105 km. Below this scale, an Alfvén-wave cas-
cade should exist (as is, indeed, suggested by Vogt & Enßlin’s
spectrum being roughly consistent with k−5/3 at scales below
the peak). As these scales are collisionless (λmfpi ∼ 100 pc in
the cores and ∼ 10 kpc in the bulk of the clusters), it is to this
turbulence that the theory developed in this paper should be
applicable.
Another complication exists, similar to that discussed in
§ 8.3: pressure anisotropies could give rise to fast plasma
instabilities whose growth rate peaks just above the ion gy-
roscale. As was pointed out by Schekochihin et al. (2005),
these are, in fact, an inevitable consequence of any large-scale
fluid motions that change the strength of the magnetic field.
Although a number of interesting and plausible arguments
can be made about the way the instabilities might determine
the magnetic-field structure (Schekochihin & Cowley 2006;
Schekochihin et al. 2008a; Rosin et al. 2009; Rincon et al.
2009), it is not currently understood how the small-scale
fluctuations resulting from these instabilities coexist with the
Alfvénic cascade.
The uncertainties that result from this imperfect under-
standing of the nature of the intracluster medium are exempli-
fied by the problem of its thermal conductivity. The magnetic-
field reversal scale in clusters is certainly not larger than the
electron diffusion scale, (mi/me)
1/2λmfpi, which varies from a
45 See Schekochihin & Cowley (2006) for a detailed presentation of our
views on the interplay between turbulence, magnetic field and plasma ef-
fects in cluster; for further discussions and disagreements, see Enßlin & Vogt
(2006); Subramanian et al. (2006); Brunetti & Lazarian (2007).
few kpc in the cores to a few hundred kpc in the bulk. There-
fore, one would expect that the approximation of isothermal
electron fluid (§ 4) should certainly apply at all scales below
the reversal scale, where δB ≪ B0 presumably holds. Even
this, however, is not absolutely clear. One could imagine
the electrons being effectively adiabatic if (or in the regions
where) the plasma instabilities give rise to large fluctuations
of the magnetic field (δB/B0 ∼ 1) at the ion gyroscale re-
ducing the mean free path to λmfpi ∼ ρi (Schekochihin et al.
2008a; Rosin et al. 2009; Rincon et al. 2009). Such fluctu-
ations cannot be described by the gyrokinetics in its cur-
rent form. The current state of the observational evidence
does not allow one to exclude either of these possibilities.
Both isothermal (Fabian et al. 2006; Sanders & Fabian 2006)
and non-isothermal (Markevitch & Vikhlinin 2007) coherent
structures that appear to be shocks are observed. Disordered
fluctuations of temperature can also be detected, which allows
one to infer an upper limit for the scale at which the isothermal
approximation can start being valid: thus, Markevitch et al.
(2003) find temperature variations at all scales down to ∼
100 kpc, which is the statistical limit that defines the spa-
tial resolution of their temperature map. In none of these or
similar measurements is the magnetic field data available that
would make possible a pointwise comparison of the magnetic
and thermal structure.
Because of this lack of information about the state of the
magnetized plasma in clusters, theories of the intracluster
medium are not sufficiently constrained by observations, so
no one theory is in a position to prevail. This uncertain state
of affairs might be improved by analyzing the observationally
much better resolved case of the solar wind, which should be
quite similar to the intracluster medium at very small scales
(except for somewhat lower values of βi in the solar wind).
9. CONCLUSION
In this paper, we have considered magnetized plasma tur-
bulence in the astrophysically prevalent regime of weak col-
lisionality. We have shown how the energy injected at the
outer scale cascades in phase space, eventually to increase the
entropy of the system and heat the particles. In the process,
we have explained how one combines plasma physics tools—
in particular, the gyrokinetic theory—with the ideas of a tur-
bulent cascade of energy to arrive at a hierarchy of tractable
models of turbulence in various physically distinct scale in-
tervals. These models represent the branching pathways of a
generalized energy cascade in phase space (the “kinetic cas-
cade”; see Fig. 5) and make explicit the “fluid” and “kinetic”
aspects of plasma turbulence.
A detailed outline of these developments was given in the
Introduction. Intermediate technical summaries were pro-
vided in § 4.9, § 5.7, and § 7.14. An astrophysical summary
and discussion of the observational evidence was given in § 8,
with a particular emphasis on space plasmas (§§ 8.1-8.3). Our
view of how the transformation of the large-scale turbulent
energy into heat occurs was encapsulated in the concept of
a kinetic cascade of generalized energy. It was previewed in
§ 1.4 and developed quantitatively in §§ 3.4-3.5, § 4.7, § 5.6,
§§ 6.2.3-6.2.5, §§ 7.8-7.12, Appendices D.2 and E.2.
Following a series of analytical contributions that set
up a theoretical framework for astrophysical gyrokinetics
(Howes et al. 2006, 2008a; Schekochihin et al. 2007, 2008b,
and this paper), an extensive program of fluid, hybrid fluid-
kinetic, and fully gyrokinetic46 numerical simulations of mag-
netized plasma turbulence is now underway (for the first re-
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 49
sults of this program, see Howes et al. 2008b; Tatsuno et al.
2009a,b). Careful comparisons of the fully gyrokinetic
simulations with simulations based on the more readily
computable models derived in this paper (RMHD—§ 2,
isothermal electron fluid—§ 4, KRMHD—§ 5, ERMHD—
§ 7, HRMHD—Appendix E) as well as with the numerical
studies based on various Landau fluid (Snyder et al. 1997;
Goswami et al. 2005; Ramos 2005; Sharma et al. 2006, 2007;
Passot & Sulem 2007) and gyrofluid (Hammett et al. 1991;
Dorland & Hammett 1993; Snyder & Hammett 2001; Scott
2007) closures appear to be the way forward in developing a
comprehensive numerical model of the kinetic turbulent cas-
cade from the outer scale to the electron gyroscale. Of the
many astrophysical plasmas to which these results apply, the
solar wind and, perhaps, the magnetosheath, due to the high
quality of turbulence measurements possible in them, appear
to be the most suitable test beds for direct and detailed quan-
titative comparisons of the theory and simulation results with
observational evidence. The objective of all this work remains
a quantitative characterization of the scaling-range properties
(spectra, anisotropy, nature of fluctuations and their interac-
tions), the ion and electron heating, and the transport proper-
ties of the magnetized plasma turbulence.
We thank O. Alexandrova, S. Bale, J. Borovsky, T. Carter,
S. Chapman, C. Chen, E. Churazov, T. Enßlin, A. Fabian,
A. Finoguenov, A. Fletcher, M. Haverkorn, B. Hnat, T. Hor-
bury, K. Issautier, C. Lacombe, M. Markevitch, K. Osman,
T. Passot, F. Sahraoui, A. Shukurov, and A. Vikhlinin for
helpful discussions of experimental and observational data;
I. Abel, M. Barnes, D. Ernst, J. Hastie, P. Ricci, C. Roach,
and B. Rogers for discussions of collisions in gyrokinetics;
and G. Plunk for discussions of the theory of gyrokinetic tur-
bulence in two spatial dimensions. The authors’ travel was
supported by the US DOE Center for Multiscale Plasma Dy-
namics and by the Leverhulme Trust (UK) International Aca-
demic Network for Magnetized Plasma Turbulence. A.A.S.
was supported in part by a PPARC/STFC Advanced Fellow-
ship and by the STFC Grant ST/F002505/1. He also thanks
the UCLA Plasma Group for its hospitality on several occa-
sions. S.C.C. and W.D. thank the Kavli Institute for The-
oretical Physics and the Aspen Center for Physics for their
hospitality. G.W.H. was supported by the US DOE contract
DE-AC02-76CH03073. G.G.H. and T.T. were supported by
the US DOE Center for Multiscale Plasma Dynamics. E.Q.
and G.G.H. were supported in part by the David and Lucille
Packard Foundation.
46 Using the publicly available GS2 code (developed originally for fusion
applications; see http://gs2.sourceforge.net) and the purpose-built AstroGK
code (see http://www.physics.uiowa.edu/~ ghowes/astrogk/).
APPENDIX
A. BRAGINSKII’S TWO-FLUID EQUATIONS AND REDUCED MHD
Here we explain how the standard one-fluid MHD equations used in § 2 and the collisional limit of the KRMHD system
(§ 6.1, derived in Appendix D) both emerge as limiting cases of the two-fluid theory. For the case of anisotropic fluctuations,
k‖/k⊥ ≪ 1, all of this can, of course, be derived from gyrokinetics, but it is useful to provide a connection to the more well
known fluid description of collisional plasmas.
A.1. Two-Fluid Equations
The rigorous derivation of the fluid equations for a collisional plasma was done in the classic paper of Braginskii (1965). His
equations, valid for ω/νii ≪ 1, k‖λmfpi ≪ 1, k⊥ρi ≪ 1 (see Fig. 3), evolve the densities ns, mean velocities us and temperatures
Ts of each plasma species (s = i,e):
+ us ·∇
ns = −ns∇·us, (A1)
+ us ·∇
us = −∇ps −∇· Π̂s + qsns
us ×B
+ Fs, (A2)
+ us ·∇
Ts = −ps∇·us −∇·Γs − Π̂s : ∇us + Qs, (A3)
where ps = nsTs and the expressions for the viscous stress tensor Π̂s, the friction force Fs, the heat flux Γs and the interspecies heat
exchange Qs are given in Braginskii (1965). Equations (A1-A3) are complemented with the quasi-neutrality condition, ne = Zni,
and the Faraday and Ampère laws, which are (in the non-relativistic limit)
= −c∇×E, j = ene(ui − ue) =
∇×B. (A4)
Because of quasi-neutrality, we only need one of the continuity equations, say the ion one. We can also use the electron momen-
tum equation [Eq. (A2), s = e] to express E, which we then substitute into the ion momentum equation and the Faraday law. The
resulting system is
= −ρ∇·u, (A5)
−∇· Π̂+ B ·∇B
+ ue ·∇
ue, (A6)
u×B − j×B
c∇· Π̂e
+ ue ·∇
, (A7)
http://gs2.sourceforge.net
http://www.physics.uiowa.edu/~
50 SCHEKOCHIHIN ET AL.
where ρ = mini, u = ui, p = pi + pe, Π̂ = Π̂i + Π̂e, ue = u − j/ene, ne = Zni, d/dt = ∂/∂t + u ·∇. The ion and electron temperatures
continue to satisfy Eq. (A3).
A.2. Strongly Magnetized Limit
In this form, the two-fluid theory starts resembling the standard one-fluid MHD, which was our starting point in § 2: Eqs. (A5-
A7) already look similar to the continuity, momentum and induction equations. The additional terms that appear in these equations
and the temperature equations (A3) are brought under control by considering how they depend on a number of dimensionless
parameters: ω/νii, k‖λmfpi, k⊥ρi, (me/mi)
1/2. While all these are small in Braginskii’s calculation, no assumption is made as to
how they compare to each other. We now specify that
k‖λmfpi√
, k⊥ρi ≪ k‖λmfpi ∼
≪ 1 (A8)
(see Fig. 4). Note that the first of these relations is equivalent to assuming that the fluctuation frequencies are Alfvénic—the same
assumption as in gyrokinetics [Eq. (49)]. The second relation in Eq. (A8) will be referred to by us as the strongly magnetized
limit. Under the assumptions (A8), the two-fluid equations reduce to the following closed set:47
= −ρ∇·u, (A10)
b̂b̂ : ∇u − 1
b̂b̂ρν‖i
b̂b̂ : ∇u − 1
B ·∇B
, (A11)
= B ·∇u − B∇·u, (A12)
Ti∇·u +
b̂ρκ‖ib̂ ·∇Ti
− νie (Ti − Te) +
miν‖i
b̂b̂ : ∇u − 1
, (A13)
Te∇·u +
b̂ρκ‖eb̂ ·∇Te
νie (Te − Ti) , (A14)
where ν‖i = 0.90vthiλmfpi is the parallel ion viscosity, κ‖i = 2.45vthiλmfpi parallel ion thermal diffusivity, κ‖e = 1.40vtheλmfpe ∼
Z2/τ 5/2
(mi/me)
1/2κ‖i parallel electron thermal diffusivity [here λmfpi = vthi/νii with νii defined in Eq. (52)], and νie ion–electron
collision rate [defined in Eq. (51)]. Note that the last term in Eq. (A13) represents the viscous heating of the ions.
A.3. One-Fluid Equations (MHD)
If we now restrict ourselves to the low-frequency regime where ion–electron collisions dominate over all other terms in the
ion-temperature equation (A13),
k‖λmfpi√
≪ 1 (A15)
[see Eqs. (A8) and (51)], we have, to lowest order in this new subsidiary expansion, Ti = Te = T . We can now write p = (ni +ne)T =
(1 + Z)ρT/mi and, adding Eqs. (A13) and (A14), find the equation for pressure:
p∇·u = ∇·
b̂neκ‖eb̂ ·∇T
miν‖i
b̂b̂ : ∇u − 1
, (A16)
where we have neglected the ion thermal diffusivity compared to the electron one, but kept the ion heating term to maintain
energy conservation. Equation (A16) together with Eqs. (A10-A12) constitutes the conventional one-fluid MHD system. With
the dissipative terms [which are small because of Eq. (A15)] neglected, this was the starting point for our fluid derivation of
RMHD in § 2.
Note that the electrons in this regime are adiabatic because the electron thermal diffusion is small
∼ k‖λmfpi
≪ 1, (A17)
47 The structure of the momentum equation (A11) is best understood by realizing that ρν‖i
b̂b̂ : ∇u −∇·u/3
= p⊥ − p‖ , the difference between the perpen-
dicular and parallel (ion) pressures. Since the total pressure is p = (2/3)p⊥ + (1/3)p‖ , Eq. (A11) can be written
p⊥ − p‖
B ·∇B
. (A9)
This is the general form of the momentum equation that is also valid for collisionless plasmas, when k⊥ρi ≪ 1 but k‖λmfpi is order unity or even large.
Equation (A9) together with the continuity equation (A11), the induction equation (A12) and a kinetic equation for the particle distribution function (from the
solution of which p⊥ and p‖ are determined) form the system known as Kinetic MHD (KMHD, see Kulsrud 1964, 1983). The collisional limit, k‖λmfpi ≪ 1, of
KMHD is again Eqs. (A10-A14).
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 51
provided Eq. (A15) holds and βi is order unity. If we take βi ≫ 1 instead, we can still satisfy Eq. (A15), so Ti = Te follows from
the ion temperature equation (A13) and the one-fluid equations emerge as an expansion in high βi. However, these equations now
describe two physical regimes: the adiabatic long-wavelength regime that satisfies Eq. (A17) and the shorter-wavelength regime
in which (me/mi)
βi ≪ k‖λmfpi ≪ (me/mi)1/2
βi, so the fluid is isothermal, T = T0 = const, p = [(1+Z)T0/mi]ρ = c2sρ [Eq. (9)
holds with γ = 1].
A.4. Two-Fluid Equations with Isothermal Electrons
Let us now consider the regime in which the coupling between the ion and electron temperatures is small and the electron
diffusion is large [the limit opposite to Eqs. (A15) and (A17)]:
k‖λmfpi√
∼ k‖λmfpi
≫ 1, (A18)
Then the electrons are isothermal, Te = T0e = const (with the usual assumption of stochastic field lines, so b̂ · ∇Te = 0 implies
∇Te = 0, as in § 4.4), while the ion temperature satisfies
Ti∇·u +
b̂ρκ‖ib̂ ·∇Ti
miν‖i
b̂b̂ : ∇u − 1
. (A19)
Equation (A19) together with Eqs. (A10-A12) and p = ρ(Ti + ZT0e)/mi are a closed system that describes an MHD-like fluid
of adiabatic ions and isothermal electrons. Applying the ordering of § 2.1 to these equations and carrying out an expansion in
k‖/k⊥ ≪ 1 entirely analogously to the way it was done in § 2, we arrive at the RMHD equations (17-18) for the Alfvén waves
and the following system for the compressive fluctuations (slow and entropy modes):
+ b̂ ·∇u‖ = 0, (A20)
− v2Ab̂ ·∇
= ν‖i b̂ ·∇
b̂ ·∇u‖ +
, (A21)
= κ‖ib̂ ·∇
b̂ ·∇
, (A22)
and the pressure balance
b̂ ·∇u‖ +
. (A23)
Recall that these equations, being the consequence of Braginskii’s two-fluid equations (§ A.1), are an expansion in k‖λmfpi ≪ 1
correct up to first order in this small parameter. Since the dissipative terms are small, we can replace (d/dt)δρ/ρ0 in the viscous
terms of Eqs. (A21) and (A23) by its value computed from Eqs. (A20), (A22) and (A23) in neglect of dissipation: (d/dt)δρ/ρ0 =
−b̂ · ∇u‖/(1 + c2s/v2A) [cf. Eq. (25)], where the speed of sound cs is defined by Eq. (166). Substituting this into Eqs. (A21)
and (A23), we recover the collisional limit of KRMHD derived in Appendix D, see Eqs. (D18-D20) and (D22).
B. COLLISIONS IN GYROKINETICS
The general collision operator that appears in Eq. (36) is (Landau 1936)
= 2π lnΛ
q2s q
fs′ (v
∂ fs(v)
fs(v)
∂ fs′ (v′)
, (B1)
where w = v − v′ and lnΛ is the Coulomb logarithm. We now take into account the expansion of the distribution function (54),
use the fact that the collision operator vanishes when it acts on a Maxwellian, and retain only first-order terms in the gyrokinetic
expansion. This gives us the general form of the collision term in Eq. (57): it is the ring-averaged linearized form of the Landau
collision operator (B1), (∂hs/∂t)c = 〈Cs[h]〉Rs , where
Cs[h] = 2π lnΛ
q2s q
F0s′(v
hs(v) − F0s(v)
hs′(v
. (B2)
Note that the velocity derivatives are taken at constant r, i.e., the gyrocenter distribution functions that appear in the integrand
should be understood as hs(v)≡ hs(t,r+v⊥× ẑ/Ωs,v⊥,v‖). The explicit form of the gyrokinetic collision operator can be derived
in k space as follows:
eik·Rhk
eik·rCs
e−ik·ρhk
eik·Rs
eik·ρs(v)Cs
e−ik·ρhk
, (B3)
52 SCHEKOCHIHIN ET AL.
where ρs(v) = −v⊥× ẑ/Ωs and Rs = r−ρs(v). Angle brackets with no subscript refer to averages over the gyroangle ϑ of quantities
that do not depend on spatial coordinates. Note that inside the operator Cs[. . .], h occurs both with index s and velocity v and
with index s′ and velocity v′ (over which summation/integration is done). In the latter case, ρ = ρs′(v′) = −v′⊥× ẑ/Ωs′ in the
exponential factor inside the operator.
Most of the properties of the collision operator that are used in the main body of this paper to order the collision terms
can be established in general, already on the basis of Eq. (B3) (§§ B.1-B.2). If the explicit form of the collision operator is
required, we could, in principle, perform the ring average on the linearized operator C [Eq. (B2)] and derive an explicit form of
(∂hs/∂t)c. In practice, in gyrokinetics, as in the rest of plasma physics, the full collision operator is only used when it is absolutely
unavoidable. In most problems of interest, further simplifications are possible: the same-species collisions are often modeled by
simpler operators that share the full collision operator’s conservation properties (§ B.3), while the interspecies collision operators
are expanded in the electron–ion mass ratio (§ B.4).
B.1. Velocity-Space Integral of the Gyrokinetic Collision Operator
Many of our calculations involve integrating the gyrokinetic equation (57) over the velocity space while keeping r constant.
Here we estimate the size of the integral of the collision term when k⊥ρs ≪ 1. Using Eq. (B3),
d3veik·r−ik·ρs(v)
eik·ρs(v)Cs
e−ik·ρhk
eik·r2π
dv⊥ v⊥
e−ik·ρs(v)
eik·ρs(v)Cs
e−ik·ρhk
eik·r
e−ik·ρs(v)
eik·ρs(v)Cs
e−ik·ρhk
eik·r
d3vJ0(as)e
ik·ρs(v)Cs
e−ik·ρhk
eik·r
1 − ik · v⊥× ẑ
k · v⊥× ẑ
+ . . .
e−ik·ρhk
. (B4)
Since the (linearized) collision operator Cs conserves particle number, the first term in the expansion vanishes. The operator
Cs = Css +Css′ is a sum of the same-species collision operator [the s′ = s part of the sum in Eq. (B2)] and the interspecies collision
operator (the s′ 6= s part). The former conserves total momentum of the particles of species s, so it gives no contribution to the
second term in the expansion in Eq. (B4). Therefore,
d3v〈〈Css[hs]〉Rs〉r ∼ νssk
s δns. (B5)
The interspecies collisions do contribute to the second term in Eq. (B4) due to momentum exchange with the species s′. This
contribution is readily inferred from the standard formula for the linearized friction force (see, e.g., Helander & Sigmar 2002):
d3vvCss′
e−ik·ρhk
S (v)e
−ik·ρs(v)hsk + ms′νs
S (v)e
−ik·ρs′ (v)hs′k
, (B6)
S (v) =
2πn0s′q
s′ lnΛ
(vths
vths′
vths′
erf ′
vths′
, (B7)
where erf(x) = (2/
dy exp(−y2) is the error function. From this, via a calculation of ring averages analogous to Eq. (B17),
we get
−ik ·
v⊥× ẑ
e−ik·ρhk
S (v)
ik ·ρs(v)e−ik·ρs(v)
hsk +
S (v)
ik ·ρs′(v)e−ik·ρs′ (v)
S (v)asJ1(as)hsk +
S (v)as′J1(as′)hs′k
∼ νss′k2⊥ρ2sδns + νs′sk2⊥ρ2s′δns′ . (B8)
For the ion–electron collisions (s = i, s′ = e), using Eqs. (45) and (51), we find that both terms are ∼ (me/mi)1/2νiik2⊥ρ2i δni.
Thus, besides an extra factor of k2⊥ρ
i , the ion–electron collisions are also subdominant by one order in the mass-ratio expansion
compared to the ion–ion collisions. The same estimate holds for the interspecies contributions to the third and fourth terms in
Eq. (B4). In a similar fashion, the integral of the electron–ion collision operator (s = e, s′ = i), is ∼ νeik2⊥ρ2eδne, which is the same
order as the integral of the electron–electron collisions.
The conclusion of this section is that, both for ion and for electron collisions, the velocity-space integral (at constant r) of the
gyrokinetic collision operator is higher order than the collision operator itself by two orders of k⊥ρs. This is the property that we
relied on in neglecting collision terms in Eqs. (104) and (137).
B.2. Ordering of Collision Terms in Eqs. (125) and (137)
In § 5, we claimed that the contribution to the ion–ion collision term due to the (Ze〈ϕ〉Ri/T0i)F0i part of the ion distribution
function [Eq. (124)] was one order of k⊥ρi smaller than the contributions from the rest of hi. This was used to order collision
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 53
terms in Eqs. (125) and (137). Indeed, from Eq. (B3),
Ze〈ϕ〉Ri
eik·Ri
eik·ρiCii
e−ik·ρi J0(ai)F0i
]〉 Zeϕk
eik·Ri
eik·ρiCii
1 − ik ·ρi −
(k ·ρi)2 −
+ · · ·
∼ νiik2⊥ρ2i
F0i. (B9)
This estimate holds because, as it is easy to ascertain using Eq. (B2), the operator Cii annihilates the first two terms in the
expansion and only acts non-trivially on an expression that is second order in k⊥ρi. With the aid of Eq. (47), the desired ordering
of the term (B9) in Eq. (125) follows. When Eq. (B9) is integrated over velocity space, the result picks up two extra orders in
k⊥ρi [a general effect of integrating the gyroaveraged collision operator over the velocity space; see Eq. (B4)]:
Ze〈ϕ〉Ri
∼ νiik4⊥ρ4i
, (B10)
so the resulting term in Eq. (137) is third order, as stated in § 5.3.
B.3. Model Pitch-Angle-Scattering Operator for Same-Species Collisions
A popular model operator for same-species collisions that conserves particle number, momentum, and energy is constructed
by taking the test-particle pitch-angle-scattering operator and correcting it with an additional term that ensures momentum con-
servation (Rosenbluth et al. 1972; see also Helander & Sigmar 2002):
CM[hs] = ν
D (v)
1 − ξ2
) ∂hs
1 − ξ2
2v ·U[hs]
v2ths
, U[hs] =
d3vvνssD (v)hs
d3v (v/vths)2νssD (v)F0s(v)
, (B11)
νssD (v) = νss
(vths
v2ths
erf ′
, νss =
2πn0sq
s lnΛ
, (B12)
where the velocity derivatives are at constant r. The gyrokinetic version of this operator is (cf. Catto & Tsang 1977;
Dimits & Cohen 1994)
〈CM[hs]〉Rs =
eik·RsνssD (v)
1 − ξ2
) ∂hsk
v2(1 + ξ2)
4v2ths
s hsk + 2
v⊥J1(as)U⊥[hsk] + v‖J0(as)U‖[hsk]
v2ths
, (B13)
U⊥[hsk] =
d3vv⊥J1(as)νssD (v)hsk(v⊥,v‖)
d3v (v/vths)2νssD (v)F0s(v)
, U‖[hsk] =
d3vv‖J0(as)ν
D (v)hsk(v⊥,v‖)
d3v (v/vths)2νssD (v)F0s(v)
where as = k⊥v⊥/Ωs. The velocity derivatives are now at constant Rs. The spatial diffusion term appearing in the ring-averaged
collision operator is physically due to the fact that a change in a particle’s velocity resulting from a collision can lead to a change
in the spatial position of its gyrocenter.
In order to derive Eq. (B13), we use Eq. (B3). Since, ρs(v) =
1 − ξ2 sinϑ+ ŷv
1 − ξ2 cosϑ
/Ωs, it is not hard to see
e−ik·ρs(v)hsk = e
−ik·ρs(v)
1 − ξ2
ik⊥ ·
v⊥× ẑ
e−ik·ρs(v)hsk = e
−ik·ρ(v)
ik⊥ ·v⊥
hsk. (B14)
Therefore,
eik·ρs(v)
1 − ξ2
e−ik·ρs(v)hsk
1 − ξ2
) ∂hsk
k2⊥hsk,
eik·ρs(v)
e−ik·ρs(v)hsk
1 − ξ2
k2⊥hsk.
(B15)
Combining these formulae, we obtain the first two terms in Eq. (B13). Now let us work out the U term:
eik·ρs(v)v ·
d3v′ v′νssD (v
′)e−ik·ρs(v
′)hsk
v′⊥,v
veik·ρs(v)
dv′⊥ v
dv′‖ν
v′e−ik·ρs(v
v′⊥,v
(B16)
Since
ve±ik·ρs(v)
= ẑv‖
e±ik·ρs(v)
v⊥e±ik·ρs(v)
, where
e±ik·ρs(v)
= J0(as) and
±ik·ρs(v)
= ẑ×
v⊥× ẑ
∓ik⊥ ·
v⊥× ẑ
= ±iΩsẑ×
∓ik⊥ ·
v⊥× ẑ
= ±i ẑ×k⊥
v⊥J1(as), (B17)
we obtain the third term in Eq. (B13).
54 SCHEKOCHIHIN ET AL.
It is useful to give the lowest-order form of the operator (B13) in the limit k⊥ρs ≪ 1:
〈CM[hs]〉Rs = ν
D (v)
1 − ξ2
) ∂hs
d3v′v′
νssD (v
′)hs(v
d3v′v′2νssD (v
′)F0s(v′)
+ O(k2⊥ρ
s ). (B18)
This is the operator that can be used in the right-hand side of Eq. (145) (as, e.g., is done in the calculation of collisional transport
terms in Appendix D.3).
In practical numerical computations of gyrokinetic turbulence, the pitch-angle scattering operator is not sufficient because the
distribution function develops small scales not only in ξ but also in v (M. Barnes, W. Dorland and T. Tatsuno 2006, unpublished).
This is, indeed, expected because the phase-space entropy cascade produces small scales in v⊥, rather than just in ξ (see § 7.9.1).
In order to provide a cut off in v, an energy-diffusion operator must be added to the pitch-angle-scattering operator derived above.
A numerically tractable model gyrokinetic energy-diffusion operator was proposed by Abel et al. (2008); Barnes et al. (2009).48
B.4. Electron–Ion Collision Operator
This operator can be expanded in me/mi and to the lowest order is (see, e.g., Helander & Sigmar 2002)
Cei[h] = ν
D (v)
1 − ξ2
) ∂he
1 − ξ2
2v ·ui
v2the
, νeiD (v) = νei
(vthe
. (B19)
The corrections to this form are O(me/mi). This is second order in the expansion of § 4 and, therefore, we need not keep these
corrections. The operator (B19) is mathematically similar to the model operator for the same-species collisions [Eq. (B13)]. The
gyrokinetic version of this operator is derived in the way analogous to the calculation in Appendix B.3. The result is
〈Cei[h]〉Re =
eik·ReνeiD (v)
1 − ξ2
) ∂hek
v2(1 + ξ2)
4v2the
v2the
J1(ae)
2v′2⊥
v2thi
hik +
2v‖J0(ae)u‖ki
v2the
. (B20)
At scales not too close to the electron gyroscale, namely, such that k⊥ρe ∼ (me/mi)1/2, the second and third terms are manifestly
second order in (me/mi)
1/2, so have to be neglected along with other O(me/mi) contributions to the electron–ion collisions.
49 The
remaining two terms are first order in the mass-ratio expansion: the first term vanishes for he = h
e [Eq. (101)], so its contribution
is first order; in the fourth term, we can use Eq. (87) to express u‖i in terms of quantities that are also first order. Keeping only
the first-order terms, the gyrokinetic electron–ion collision operator is
〈Cei[h]〉Re = ν
D (v)
1 − ξ2
) ∂h(1)e
2v‖u‖i
v2the
. (B21)
Note that the ion drag term is essential to represent the ion–electron friction correctly and, therefore, to capture the Ohmic
resistivity (which, however, is rarely more important for unfreezing flux than the electron inertia and the finiteness of the electron
gyroradius; see § 7.7).
C. A HEURISTIC DERIVATION OF THE ELECTRON EQUATIONS
Here we show how the equations (116-117) of § 4 and the ERMHD equations (226-227) of § 7 can be derived heuristically
from electron fluid dynamics and a number of physical assumptions, without the use of gyrokinetics (§ C.1). This derivation is
not rigorous. Its role is to provide an intuitive route to the isothermal electron fluid and ERMHD approximations.
C.1. Derivation of Eqs. (116-117)
We start with the following three equations:
= −c∇×E, ∂ne
+∇· (neue) = 0, E +
ue ×B
. (C1)
These are Faraday’s law, the electron continuity equation, and the generalized Ohm’s law, which is the electron momentum
equation with all electron inertia terms neglected (i.e., effectively, the lowest order in the expansion in the electron mass me). The
electron pressure is assumed to be scalar by fiat (this can be justified in certain limits: for example in the collisional limit, as in
Appendix A, or for the isothermal electron fluid approximation derived in § 4). The electron-pressure term in the right-hand side
of Ohm’s law is sometimes called the thermoelectric term. We now assume the same static uniform equilibrium, E0 = 0, B0 = B0ẑ,
that we have used throughout this paper and apply to Eqs. (C1) the fundamental ordering discussed in § 3.1.
48 The collision operator now used the GS2 and AstroGK codes (see footnote 46) is their energy-diffusion operator plus the pitch-angle-scattering opera-
tor (B13).
49 The third term in Eq. (B20) is, in fact, never important: at the electron scales, k⊥ρe ∼ 1, it is negligible because of the Bessel function in the velocity
integral (Abel et al. 2008).
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 55
First consider the projection of Ohm’s law onto the total magnetic field B, use the definition of E [Eq. (37)], and keep the
leading-order terms in the ǫ expansion:
E · b̂ = − 1
b̂ ·∇pe ⇒
+ b̂ ·∇ϕ = b̂ ·∇ δpe
. (C2)
This turns into Eq. (116) if we also assume isothermal electrons, δpe = T0eδne [see Eq. (103)].
With the aid of Ohm’s law, Faraday’s law turns into
= ∇× (ue ×B) = −ue ·∇B + B ·∇ue − B∇·ue. (C3)
Keeping the leading-order terms, we find, for the components of Eq. (C3) perpendicular and parallel to the mean field,
+ u⊥e ·∇⊥
= b̂ ·∇u⊥e,
+ u⊥e ·∇⊥
= b̂ ·∇u‖e. (C4)
In the last equation, we have used the electron continuity equation to write
∇·ue = −
+ u⊥e ·∇⊥
. (C5)
From Ohm’s law, we have, to lowest order,
u⊥e = −ẑ×
E⊥ +∇⊥
= ẑ×∇⊥
. (C6)
Using this expression in the second of the equations (C4) gives
− b̂ ·∇u‖e =
, (C7)
where d/dt is defined in the usual way [Eq. (122)]. Assuming isothermal electrons (δpe = T0eδne) annihilates the second term on
the right-hand side and turns the above equation into Eq. (117). As for the first of the equations (C4), the use of Eq. (C6) and
substitution of δB⊥ = −ẑ×∇⊥A‖ turns it into the previously derived Eq. (C2), whence follows Eq. (116).
Thus, we have shown that Eqs. (116-117) can be derived as a direct consequence of Faraday’s law, electron fluid dynamics
(electron continuity equation and the electron force balance, a. k. a. the generalized Ohm’s law), and the assumption of isothermal
electrons—all taken to the leading order in the gyrokinetic ordering given in § 3.1 (i.e., assuming strongly interacting anisotropic
fluctuations with k‖ ≪ k⊥).
We have just proved that Eqs. (116) and (117) are simply the perpendicular and parallel part, respectively, of Eq. (C3). The
latter equation means that the magnetic-field lines are frozen into the electron flow velocity ue, i.e., the flux is conserved, the
result formally proven in § 4.3 [see Eq. (99)].
C.2. Electron MHD and the Derivation of Eqs. (226-227)
One route to Eqs. (226-227), already explained in § 7.2, is to start with Eqs. (C2) and (C7) and assume Boltzmann electrons
and ions and the total pressure balance. Another approach, more standard in the literature on the Hall and Electron MHD, is to
start with Eq. (C3), which states that the magnetic field is frozen into the electron flow. The electron velocity can be written in
terms of the ion velocity and the current density, and the latter then related to the magnetic field via Ampère’s law:
ue = ui −
= ui −
4πene
∇×B. (C8)
To the leading order in ǫ, the perpendicular and parallel parts of Eq. (C3) are Eqs. (C4), respectively, where the perpendicular and
parallel electron velocities are [from Eq. (C8)]
u⊥e = u⊥i +
4πen0e
ẑ×∇⊥δB‖, u‖e = u‖i +
4πen0e
∇2⊥A‖. (C9)
The relative size of the two terms in each of these expressions is controlled by the size of k⊥di, where di = ρi/
βi is the ion
inertial scale. When k⊥di ≫ 1, we may set ui = 0. Note, however, that the ion motion is not totally neglected: indeed, in the
second of the equations (C4), the δne/ne terms comes, via Eq. (C5), from the divergence of the ion velocity [from Eq. (C8),
∇·ui = ∇·ue]. To complete the derivation, we relate δne to δB‖ via the assumption of total pressure balance, as explained in
§ 7.2, giving us Eq. (225). Substituting this equation and Eqs. (C9) into Eqs. (C4), we obtain
= v2Adi b̂ ·∇
1 + 2/βi(1 + Z/τ )
b̂ ·∇∇2⊥Ψ, (C10)
where Ψ = −A‖/
4πmin0i. Equations (C10) evolve the perturbed magnetic field. These equations become the ERMHD equations
(226-227) if δB‖/B0 is expressed in terms of the scalar potential via Eq. (223).
56 SCHEKOCHIHIN ET AL.
Note that there are two special limits in which the assumption of immobile ions suffices to derive Eqs. (C10) from Eq. (C3)
without the need for the pressure balance: βi ≫ 1 (incompressible ions) or τ = T0i/T0e ≪ 1 (cold ions) but βe = βiZ/τ ≫ 1. In both
cases, Eq. (225) shows that δne/n0e ≪ δB‖/B0, so the density perturbation can be ignored and the coefficient of the right-hand
side of the second of the equations (C10) is equal to 1. The limit of cold ions is discussed further in Appendix E.
D. FLUID LIMIT OF THE KINETIC RMHD
Taking the fluid (collisional) limit of the KRMHD system (summarized in § 5.7) means carrying out another subsidiary
expansion—this time in k‖λmfpi ≪ 1. The expansion only affects the equations for the density and magnetic-field-strength
fluctuations (§ 5.5) because the Alfvén waves are indifferent to collisional effects.
The calculation presented below follows a standard perturbation algorithm used in the kinetic theory of gases and in plasma
physics to derive fluid equations with collisional transport coefficients (Chapman & Cowling 1970). For magnetized plasma,
this calculation was carried out in full generality by Braginskii (1965), whose starting point was the full plasma kinetic theory
[Eqs. (36-39)]. While what we do below is, strictly speaking, merely a particular case of his calculation (see Appendix A), it has
the advantage of relative simplicity and also serves to show how the fluid limit is recovered from the gyrokinetic formalism—a
demonstration that we believe to be of value.
It will be convenient to use the KRMHD system written in terms of the function δ f̃i = g + (v2⊥/v
thi)(δB‖/B0)F0i, which is the
perturbation of the local Maxwellian in the frame of the Alfvén waves [Eqs. (150-152)]. We want to expand Eq. (150) in powers
of k‖λmfpi, so we let δ f̃i = δ f̃
i + δ f̃
i + . . ., δB‖ = δB
+ δB(1)
+ . . ., etc.
D.1. Zeroth Order: Ideal Fluid Equations
Since [see Eq. (49)]
k‖λmfpi√
k‖vthi
∼ k‖λmfpi, (D1)
to zeroth order Eq. (150) becomes
δ f̃ (0)i
= 0. The zero mode of the collision operator is a Maxwellian. Therefore, we
may write the full ion distribution function up to zeroth order in k‖λmfpi as follows [see Eq. (144)]
2πTi/mi
mi[(v⊥ − uE)2 + (v‖ − u‖)2]
, (D2)
where ni = n0i + δni and Ti = T0i + δTi include both the unperturbed quantities and their perturbations. The E×B drift velocity uE
comes from the Alfvén waves (see § 5.4) and does not concern us here. Since the perturbations δni, u‖ and δTi are small in the
original gyrokinetic expansion, Eq. (D2) is equivalent to
δ f̃ (0)i =
δn(0)e
v2thi
δT (0)i
v2thi
F0i, (D3)
where we have used quasi-neutrality to replace δni/n0i = δne/n0e. This automatically satisfies Eq. (151), while Eq. (152) gives us
an expression for the ion-temperature perturbation:
δT (0)i
δn(0)e
δB(0)
. (D4)
Note that this is consistent with the interpretation of the perpendicular Ampère’s law [Eq. (63), which is the progenitor of
Eq. (152)] as the pressure balance [see Eq. (67)]: indeed, recalling that the electron pressure perturbation is δpe = T0eδne
[Eq. (103)], we have
= −δpe − δpi = −δneT0e − δniT0i − n0iδTi, (D5)
whence follows Eq. (D4) by way of quasi-neutrality (Zni = ne) and the definitions of Z, τ , βi [Eqs. (40-42)].
Since the collision operator conserves particle number, momentum and energy, we can obtain evolution equations for δn(0)e /n0e,
and δB(0)
/B0 by multiplying Eq. (150) by 1, v‖, v
2/v2thi, respectively, and integrating over the velocity space. The three
moments that emerge this way are
d3vδ f̃ (0)i =
δn(0)e
d3vv‖δ f̃
i = u
v2thi
δ f̃ (0)i =
δn(0)e
δT (0)i
. (D6)
The three evolution equations for these moments are
δn(0)e
δB(0)
+ b̂ ·∇u(0)
= 0, (D7)
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 57
du(0)
− v2A b̂ ·∇
δB(0)
= 0, (D8)
δn(0)e
δT (0)i
δB(0)
b̂ ·∇u(0)
= 0. (D9)
These allow us to recover the fluid equations we derived in § 2.4: Eq. (D8) is the parallel component of the MHD momentum
equation (27); combining Eqs. (D7), (D9) and (D4), we obtain the continuity equation and the parallel component of the induction
equation—these are the same as Eqs. (25) and (26):
δn(0)e
1 + c2s/v
b̂ ·∇u(0)
δB(0)
1 + v2A/c2s
b̂ ·∇u(0)
, (D10)
where the sound speed cs is defined by Eq. (166). From Eqs. (D7) and (D9), we also find the analog of the entropy equation (23):
δT (0)i
δn(0)e
δs(0)
δs(0)
δT (0)i
δn(0)e
δn(0)e
δB(0)
. (D11)
This implies that the temperature changes due to compressional heating only.
D.2. Generalized Energy: Five RMHD Cascades Recovered
We now calculate the generalized energy by substituting δ f̃i from Eq. (D3) into Eq. (153) and using Eqs. (D4) and (D11):
min0iu
min0iu
n0iT0i
1 + Z/τ
5/3 + Z/τ
=W +AW +W
AW +W
sw +W
n0iT0i
1 + Z/τ
5/3 + Z/τ
Ws. (D12)
The first two terms are the Alfvén-wave energy [Eq. (154)]. The following two terms are the slow-wave energy, which splits into
the independently cascaded energies of “+” and “−” waves (see § 2.5):
WSW = W
sw +W
min0i
|z+‖|
2 + |z−‖|
. (D13)
The last term is the total variance of the entropy mode. Thus, we have recovered the five cascades of the RMHD system (§ 2.7;
Fig. 5 maps out the fate of these cascades at kinetic scales).
D.3. First Order: Collisional Transport
Now let us compute the collisional transport terms for the equations derived above. In order to do this, we have to determine
the first-order perturbed distribution function δ f̃ (1)i , which satisfies [see Eq. (150)]
δ f̃ (1)i
δ f̃ (0)i −
v2thi
δB(0)
+ v‖ b̂ ·∇
δ f̃ (0)i +
δn(0)e
. (D14)
We now use Eq. (D3) to substitute for δ f̃ (0)i and Eqs. (D10-D11) and (D8) to compute the time derivatives. Equation (D14)
becomes
δ f̃ (1)i
1 − 3ξ2
v2thi
2/3 + c2s/v
1 + c2s/v
b̂ ·∇u(0)
v2thi
b̂ ·∇δT
F0i(v), (D15)
where ξ = v‖/v. Note that the right-hand side gives zero when multiplied by 1, v‖ or v
2 and integrated over the velocity space, as
it must do because the collision operator in the left-hand side conserves particle number, momentum and energy.
Solving Eq. (D15) requires inverting the collision operator. While this can be done for the general Landau collision operator
(see Braginskii 1965), for our purposes, it is sufficient to use the model operator given in Appendix B.3, Eq. (B18). This simplifies
calculations at the expense of an order-one inaccuracy in the numerical values of the transport coefficients. As the exact value of
these coefficients will never be crucial for us, this is an acceptable loss of precision. Inverting the collision operator in Eq. (D15)
then gives
δ f̃ (1)i =
ν iiD(v)
1 − 3ξ2
v2thi
2/3 + c2s/v
1 + c2s/v
b̂ ·∇u(0)
v2thi
b̂ ·∇δT
F0i(v), (D16)
58 SCHEKOCHIHIN ET AL.
where ν iiD(v) is a collision frequency defined in Eq. (B12) and we have chosen the constants of integration in such a way that the
three conservation laws are respected:
d3vδ f̃ (1)i = 0,
d3vv‖δ f̃
i = 0,
d3vv2δ f̃ (1)i = 0. These relations mean that δn
e = 0,
= 0, δT (1)i = 0 and that, in view of Eq. (152), we have
δB(1)
2/3 + c2s/v
1 + c2s/v
ν‖ib̂ ·∇u‖, (D17)
where ν‖i is defined below [Eq. (D21)]. Equations (D16-D17) are now used to calculate the first-order corrections to the moment
equations (D7-D9). They become
+ b̂ ·∇u‖ = 0, (D18)
− v2Ab̂ ·∇
2/3 + c2s/v
1 + c2s/v
ν‖i b̂ ·∇
b̂ ·∇u‖
, (D19)
= κ‖ib̂ ·∇
b̂ ·∇δTi
, (D20)
where we have introduced the coefficients of parallel viscosity and parallel thermal diffusivity:
ν‖i =
ν iiD(v)v
F0i(v), κ‖i =
ν iiD(v)v
v2thi
F0i(v). (D21)
All perturbed quantities are now accurate up to first order in k‖λmfpi. Note that in Eq. (D19), we used Eq. (D17) to express
δB(0)
= δB‖ − δB
. We do the same in Eq. (D4) and obtain
2/3 + c2s/v
1 + c2s/v
ν‖ib̂ ·∇u‖
. (D22)
This equation completes the system (D18-D20), which allows us to determine δne, u‖, δTi and δB‖. In § 6.1, we use the equations
derived above, but absorb the prefactor (2/3 + c2s/v
A)/(1 + c
A) into the definition of ν‖i. The same system of equations can also
be derived from Braginskii’s two-fluid theory (Appendix A.4), from which we can borrow the quantitatively correct values of the
viscosity and ion thermal diffusivity: ν‖i = 0.90v
thi/νii, κ‖i = 2.45v
thi/νii, where νii is defined in Eq. (52).
E. HALL REDUCED MHD
The popular Hall MHD approximation consists in assuming that the magnetic field is frozen into the electron flow velocity
[Eq. (C3)]. The latter is calculated from the ion flow velocity and the current determined by Ampère’s law [Eq. (C8)]:
4πen0e
, (E1)
where the ion flow velocity ui satisfies the conventional MHD momentum equation (8). The Hall MHD is an appealing theoretical
model that appears to capture both the MHD behavior at long wavelengths (when ue ≃ ui) and some of the kinetic effects that
become important at small scales due to decoupling between the electron and ion flows (the appearance of dispersive waves)
without bringing in the full complexity of the kinetic theory. However, unlike the kinetic theory, it completely ignores the
collisionless damping effects and suggests that the key small-scale physical change is associated with the ion inertial scale
di = ρi/
βi (or, when βe ≪ 1, the ion sound scale ρs = ρi
Z/2τ ; see § E.3), rather than the ion gyroscale ρi. Is this an acceptable
model for plasma turbulence? Figure 8 illustrates the fact that at τ ∼ 1, the ion inertial scale does not play a special role linearly,
the MHD Alfvén wave becomes dispersive at the ion gyroscale, not at di, and that the collisionless damping cannot in general
be neglected. A detailed comparison of the Hall MHD linear dispersion relation with full hot plasma dispersion relation leads to
the conclusion that Hall MHD is only a valid approximation in the limit of cold ions, namely, τ = T0i/T0e ≪ 1 (Ito et al. 2004;
Hirose et al. 2004). In this Appendix, we show that a reduced (low-frequency, anisotropic) version of Hall MHD can, indeed, be
derived from gyrokinetics in the limit τ ≪ 1.50 This demonstrates that the Hall MHD model fits into the theoretical framework
proposed in this paper as a special limit. However, the parameter regime that gives rise to this special limit is not common in
space and astrophysical plasmas of interest.
E.1. Gyrokinetic Derivation of Hall Reduced MHD
Let us start with the equations of isothermal electron fluid, Eqs. (116-121), i.e., work within the assumptions that allowed us
to carry out the mass-ratio expansion (§ 4.8). In Eq. (120) (perpendicular Ampère’s law, or gyrokinetic pressure balance), taking
50 Note that, strictly speaking, our ordering of the collision frequency does not allow us to take this limit (see footnote 17), but this is a minor betrayal of rigor,
which does not, in fact, invalidate the results.
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 59
the limit τ ≪ 1 gives
eik·r
d3vJ0(ai)hik
, (E2)
where we have used Eq. (118) to express the hi integral and the expression for the electron beta βe = βiZ/τ . Note that the above
equation is simply the statement of a balance between the magnetic and electron thermal pressure (the ions are relatively cold,
so they have fallen out of the pressure balance). Using Eq. (E2) to express δne in terms of δB‖ in Eqs. (116) and (117) and also
substituting for u‖e from Eq. (119) [or, equivalently, Eq. (87)], we get
= vAb̂ ·∇
Φ+ vAdi
1 + 2/βe
b̂ ·∇
u‖i − di∇2⊥Ψ
, (E3)
where we have used our usual definitions of the stream and flux functions [Eq. (135)] and of the full derivatives [Eq. (160)].
These equations determine the evolution of the magnetic field, but we still need the ion gyrokinetic equation (121) to calculate
the ion motion (Φ = cϕ/B0 and u‖i) via Eqs. (118) and (88). There are two limits in which the ion kinetics can be reduced to
simple fluid models.
E.1.1. High-Ion-Beta Limit, βi ≫ 1
In this limit, k⊥ρi = k⊥di
βi ≫ 1 as long as k⊥di is not small. Then the ion motion can be neglected because it is averaged out
by the Bessel functions in Eqs. (118) and (88)—in the same way as in § 7.2. So we get Φ = (τ/Z)vAdiδB‖/B0 [using Eq. (E2);
this is the τ ≪ 1 limit of Eq. (223)] and u‖i = 0. Noting that βe = βiZ/τ ≫ 1 in this limit, we find that Eqs. (E3) reduce to
= v2Adi b̂ ·∇
= −di b̂ ·∇∇2⊥Ψ, (E4)
which is the τ ≪ 1 limit of our ERMHD equations (226-227) [or, equivalently, Eqs. (C10)].
E.1.2. Low-Ion-Beta Limit, βi ∼ τ ≪ 1 (the Hall Limit)
This limit is similar to the RMHD limit worked out in § 5: we take, for now, k⊥di ∼ 1 and βe ∼ 1 (in which subsidiary
expansions can be carried out later), and expand the ion gyrokinetics in k⊥ρi = k⊥di
βi ≪ 1. Note that ordering βe ∼ 1 means
that we have ordered βi ∼ τ ≪ 1. We now proceed analogously to the way we did in § 5: express the ion distribution in terms of
the g function defined by Eq. (124) and, using the relation (E2) between δB‖/B0 and δne/n0e, write Eqs. (125-127) as follows:
︸ ︷︷ ︸
︸ ︷︷ ︸
v⊥ ·A⊥
︸ ︷︷ ︸
− 〈Cii[g]〉Ri
︸ ︷︷ ︸
A‖,ϕ− 〈ϕ〉Ri
︸ ︷︷ ︸
+b̂ ·∇
︸ ︷︷ ︸
v⊥ ·A⊥
︸ ︷︷ ︸
F0i +
v⊥ ·A⊥
︸ ︷︷ ︸
,(E5)
Γ1(αi) +
︸ ︷︷ ︸
1 −Γ0(αi)
]Zeϕk
︸ ︷︷ ︸
d3vJ0(ai)gk
︸ ︷︷ ︸
, u‖ki
d3vv‖J0(ai)gk
︸ ︷︷ ︸
. (E6)
All terms in these equations can be ordered with respect to the small parameter
βi (an expansion subsidiary to the gyrokinetic
expansion in ǫ and the Hall expansion in τ ≪ 1). The lowest order to which they enter is indicated underneath each term. The
ordering we use is the same as in § 5.2, but now we count the powers of
βi and order formally k⊥di ∼ 1 and βe ∼ 1. It is easy
to check that this ordering can be summarized as follows
and that the ion and electron terms in Eqs. (E3) are comparable under this ordering, so their competition is retained (in fact, this
could be used as the underlying assumption behind the ordering). The fluctuation frequency continues to be ordered as the Alfvén
frequency, ω ∼ k‖vA. The collision terms are ordered via ω/νii ∼ k‖λmfpi/
βi and k‖λmfpi ∼ 1, although the latter assumption is
not essential for what follows, because collisions turn out to be negligible and it is fine to take k‖λmfpi ≫ 1 from the outset and
neglect them completely.
In Eqs. (E6), we use Eqs. (129) and (130) to write 1 −Γ0(αi) ≃ αi = k2⊥ρ2i /2 and Γ1(αi) ≃ 1. These equations imply that if we
expand g = g(−1) + g(0) + . . ., we must have
d3vg(−1) = 0, so the contribution to the right-hand side of the first of the equations
60 SCHEKOCHIHIN ET AL.
(E6) (the quasi-neutrality equation) comes from g(0), while the parallel ion flow is determined by g(−1). Retaining only the lowest
(minus first) order terms in Eq. (E5), we find the equation for g(−1), the v‖ moment of which gives an equation for u‖i:
∂g(−1)
{ϕ,g(−1)} = 2
v‖b̂ ·∇
F0i ⇒
= v2Ab̂ ·∇
. (E8)
Now integrating Eq. (E5) over the velocity space (at constant r), using the first of the equations (E6) to express the integral of
g(0), and retaining only the lowest (zeroth) order terms, we find
ρ2i ∇2⊥
+ b̂ ·∇u‖i = 0 ⇒
∇2⊥Φ = vAb̂ ·∇∇2⊥Ψ, (E9)
where we have used the second of the equations (E3) to express the time derivative of δB‖/B0.
Together with Eqs. (E3), Eqs. (E8) and (E9) form a closed system, which it is natural to call Hall Reduced MHD (HRMHD)
because these equations can be straightforwardly derived by applying the RMHD ordering (§ 2.1) to the MHD equations (8-10)
with the induction equation (10) replaced by Eq. (E1). Indeed, Eqs. (E8) and (E9) exactly coincide with Eqs. (27) and (18), which
are the parallel and perpendicular components of the MHD momentum equation (8) under the RMHD ordering; Eqs. (E3) should
be compared Eqs. (17) and (26) while noticing that, in the limit τ ≪ 1, the sound speed is cs = vA
βe/2 [see Eq. (166)]. The
incompressible case (Mahajan & Yoshida 1998) is recovered in the subsidiary limit βe ≫ 1 (i.e., 1 ≫ βi ≫ τ ).
E.2. Generalized Energy for Hall RMHD and the Passive Entropy Mode
To work out the generalized energy (§ 3.4) for the HRMHD regime, we start with the generalized energy for the isothermal
electron fluid [Eq. (109)] and use Eq. (E2) to express the density perturbation:
T0iδ f
, (E10)
where δB⊥ = ẑ×∇⊥Ψ. The perturbed ion distribution function can be written in the same form as it was done in § 5.4 [Eq. (143)]:
to lowest order in the
βi expansion (§ E.1.2),
δ f (−1)i =
2v⊥ ·u⊥
v2thi
F0i + g(−1) =
2v⊥ ·u⊥
v2thi
F0i +
2v‖u‖i
v2thi
F0i + g̃, (E11)
where u⊥ = ẑ×∇⊥Φ. The last equality above is achieved by noticing that, since g(−1) satisfies Eq. (E8), we may split it into a
perturbed Maxwellian with parallel velocity u‖i and the remainder: g
(−1) = 2v‖u‖iF0i/v
thi + g̃. Then g̃ is the homogeneous solution
of the leading-order kinetic equation [see Eq. (E8)]:
+{Φ, g̃} = 0,
d3v g̃ = 0. (E12)
Substituting Eq. (E11) into Eq. (E10) and keeping only the leading-order terms in the
βi expansion, we get
min0iu
min0iu
T0ig̃
. (E13)
The first four terms are the energy of the Alfvénic and slow-wave-polarized fluctuations [cf. Eq. (D12)]. Unlike in RMHD, these
are not decoupled in HRMHD, unless a further subsidiary long-wavelength limit is taken (see § E.4). It is easy to verify that the
sum of these four terms is indeed conserved by Eqs. (E3), (E8) and (E9). The last term in Eq. (E13) is an individually conserved
kinetic quantity. Its conservation reflects the fact that g̃ is decoupled from the wave dynamics and passively advected by the
Alfvénic velocities via Eq. (E12).51
The passive kinetic mode g̃ can be thought of as a kinetic version of the MHD entropy mode and, indeed, reduces to it if the
collision operator in Eq. (E5) is upgraded to the leading order by orderingω/νii ∼ 1 (i.e., by considering long parallel wavelengths,
k‖λmfpi ∼
βi). In such a collisional limit, g̃ has to be a perturbed Maxwellian with no density or velocity perturbation [because
d3vg̃ = 0, while the velocity perturbation is explicitly separated from g̃ in Eq. (E11)]. Therefore,
v2thi
F0i ⇒
T0ig̃
n0iT0i
δT 2i
T 20i
. (E14)
This is to be compared with the βi ∼ τ ≪ 1 limit of Eqs. (D11) and (D12). As we have established, in the
βi expansion,
δTi = δT
i , δni = δn
i , δB‖ = δB
, so to lowest order δs/s0 = δTi/T0i and Eq. (E14) describes the entropy mode in the Hall limit.
51 A similar splitting of the generalized energy cascade into a fluid-like cascade plus a passive cascade of a zero-density part of the distribution function occurs
in the Hasegawa–Mima regime, which is the electrostatic version of the Hall limit (Plunk et al. 2009).
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 61
E.3. Hall RMHD Dispersion Relation
Linearizing the Hall RMHD equations (E3), (E8) and (E9) (derived in § E.1.2 assuming the ordering βi ∼ τ ≪ 1), we obtain
the following dispersion relation:52
ω2 − k2‖v
1 + 2/βe
= ω2k2‖v
1 + 2/βe
. (E15)
When the coupling term on the right-hand side is negligible, k⊥di/
1 + 2/βe ≪ 1, we recover the MHD Alfvén wave, ω2 = k2‖v
and the MHD slow wave, ω2 = k2
v2A/(1 + v
s ) [Eq. (167)], where cs = vA
βe/2 in the limit τ ≪ 1 [Eq. (166)]. In the opposite
limit, we get the kinetic Alfvén wave, ω2 = k2
i /(1 + 2/βe) [same as Eq. (230) with τ ≪ 1].
The solution of the dispersion relation (E15) is
1 + 2/βe
 . (E16)
The corresponding eigenfunctions then satisfy53
Ψ = −
Φ+ vAdi
, u‖i = −
, Φ = −
Ψ. (E17)
Equation (E16) takes a particularly simple form in the subsidiary limits of high and low electron beta βe = βiZ/τ :
βe ≫ 1 : ω2 = k2‖v
 , βe ≪ 1 : ω2 = k2‖v
1 + k2⊥ρ
and ω2 =
1 + k2⊥ρ
, (E18)
where ρs = di
βe/2 = ρi
Z/2τ = cs/Ωi is called the ion sound scale. The Alfvén wave and the slow wave (known as the ion
acoustic wave in the limit of τ ≪ 1, βe ≪ 1) become dispersive at the ion inertial scale (k⊥di ∼ 1) when βe ≫ 1 and at the ion
sound scale (k⊥ρs ∼ 1) when βe ≪ 1.
E.4. Summary of Hall RMHD and the Role of the Ion Inertial and Ion Sound Scales
We have shown that in the limit of cold ions and low ion beta (βi ∼ τ ≪ 1, “the Hall limit”), gyrokinetic turbulence can be
described by five scalar functions: the stream and flux functions Φ and Ψ for the Alfvénic fluctuations, the parallel velocity and
magnetic-field perturbations u‖i and δB‖ for the slow-wave-polarized fluctuations, and g̃, the zero-density, zero-velocity part of
the ion distribution function, which is the kinetic version of the MHD entropy mode. The first four of these functions satisfy a
closed set of four fluid-like equations, derived in § E.1 and collected here:
= vAb̂ ·∇
Φ+ vAdi
1 + 2/βe
b̂ ·∇
u‖i − di∇2⊥Ψ
, (E19)
∇2⊥Φ = vAb̂ ·∇∇2⊥Ψ,
= v2Ab̂ ·∇
. (E20)
We call these equations the Hall Reduced Magnetohydrodynamics (HRMHD). To fully account for the generalized energy cas-
cade, one must append to the four HRMHD equations the fifth, kinetic equation (E12) for g̃, which is energetically decoupled
from HRMHD and slaved to the Alfvénic velocity fluctuations (§ E.2).
The equations given above are valid above the ion gyroscale, k⊥ρi ≪ 1. They contain a special scale, di/
1 + 2/βe, which
is the ion inertial scale di for βe ≫ 1 and the ion sound scale ρs = cs/Ωi for βe ≪ 1. As becomes clear from the linear theory
(§ E.3), the Alfvén and slow waves become dispersive at this scale. Nonlinearly, this scale marks the transition from the regime
in which the Alfvénic and slow-wave-polarized fluctuations are decoupled to the regime in which they are mixed. Namely, when
k⊥di/
1 + 2/βe ≪ 1, HRMHD turns into RMHD: Eqs. (E19) become Eqs. (17) and (26), while Eqs. (E20) remain unchanged
and identical to Eqs. (18) and (27); in the opposite limit, k⊥di/
1 + 2/βe ≫ 1, the ion motion decouples from the magnetic-field
evolution and Eqs. (E19) turn into the ERMHD equations (226-227).
Since we are considering the case βi ≪ 1, both di and ρs are much larger than the ion gyroscale ρi. In the opposite limit of
βi ≫ 1 (§ E.1.1), while di is the only scale that appears explicitly in Eqs. (E4), we have di ≪ ρi and the equations themselves
represent the dynamics at scales much smaller than the ion gyroscale, so the transition between the RMHD and ERMHD regimes
occurs at k⊥ρi ∼ 1. The same is true for βi ∼ 1, when di ∼ ρi. The ion sound scale ρs ≫ ρi does not play a special role when
52 The full gyrokinetic dispersion relation in a similar limit was worked out in Howes et al. (2006), Appendix D.2.1.
53 Note that wave packets with |k⊥| = k⊥ and satisfying Eq. (E17) with k‖vA/ω as a function of k⊥ given by Eq. (E16) are exact nonlinear solutions of
the HRMHD equations (E3) and (E8-E9). This can be shown via a calculation analogous to that in § 7.3 (for the incompressible Hall MHD, this was done by
Mahajan & Krishan 2005).
62 SCHEKOCHIHIN ET AL.
βi is not small: it is not hard to see that for k⊥ρs ∼ 1, the ion motion terms in Eqs. (E19) dominate and we simply recover the
inertial-range KRMHD model (§ 5) by expanding in k⊥ρi = k⊥ρs
2τ/Z ≪ 1.
Various theories of the dissipation-range turbulence based on Hall and Electron MHD are further discussed in § 8.2.6.
F. TWO-DIMENSIONAL INVARIANTS IN GYROKINETICS
Since gyrokinetics is in a sense a “quasi-two-dimensional” approximation, it is natural to inquire if this gives rise to additional
conservation properties (besides the conservation of the generalized energy discussed in § 3.4) and how they are broken by the
presence of parallel propagation terms. It is important to emphasize that, except in a few special cases, these invariants are only
invariants in 2D, so gyrokinetic turbulence in 2D and 3D has fundamentally different properties, despite its seemingly “quasi-2D”
nature. It is, therefore, generally not correct to think of the gyrokinetic turbulence (or its special case the MHD turbulence) as
essentially a 2D turbulence with an admixture of parallel-propagating waves (Fyfe et al. 1977; Montgomery & Turner 1981).
In this Appendix, we work out the 2D invariants. Without attempting to present a complete analysis of the 2D conservation
properties of gyrokinetics, we limit our discussion to showing how some more familiar fluid invariants (most notably, magnetic
helicity) emerge from the general 2D invariants in the appropriate asymptotic limits.
F.1. General 2D Invariants
In deriving the generalized energy invariant, we used the fact that
d3Rs hs{〈χ〉Rs ,hs} = 0, so Eq. (57) after multiplication
by T0shs/F0s and integration over space contains no contribution from the Poisson-bracket nonlinearity. Since we also have∫
d3Rs 〈χ〉Rs{〈χ〉Rs ,hs} = 0, multiplying Eq. (57) by qs〈χ〉Rs and integrating over space has a similar outcome. Subtracting the
latter integrated equation from the former and rearranging terms gives
qs〈χ〉Rs
= qsv‖
d3Rs 〈χ〉Rs
qs〈χ〉Rs
. (F1)
We see that in a purely 2D situation, when ∂/∂z = 0, we have an infinite family of invariants Is = Is(v⊥,v‖) whose conservation (for
each species and for every value of v⊥ and v‖!) is broken only by collisions. In 3D, the parallel particle streaming (propagation)
term in the gyrokinetic equation generally breaks these invariants, although special cases may arise in which the first term on the
right-hand side of Eq. (F1) vanishes and a genuine 3D invariant appears.
F.2. “A2
-Stuff”
Let apply the mass-ratio expansion (§ 4.1) to Eq. (F1) for electrons. Using the solution (101) for the electron distribution
function, we find
T0eF0e
v2the
d3rA‖
v2the
+ · · ·
= −ev‖
v2the
F0e −
∂h(1)e
d3rA‖
, (F2)
where we have kept terms to two leading orders in the expansion. To lowest order, the above equation reduces to
d3rA‖
. (F3)
This equation can also be obtained directly from Eq. (116) (multiply by A‖ and integrate). In 2D, it expresses a well known
conservation law of the “A2‖-stuff.” As this 2D invariant exists already on the level of the mass-ratio expansion of the electron
kinetics, with no assumptions about the ions, it is inherited both by the RMHD equations in the limit of k⊥ρi ≪ 1 (§ 5.3) and
by the ERMHD equations in the limit of k⊥ρi ≫ 1 (§ 7.2). In the former limit, δne/n0e on the right-hand side of Eq. (F3) is
negligible (under the ordering explained in § 5.2); in the latter limit, it is expressed in terms of ϕ via Eq. (221). The conservation
of “A2‖-stuff” is a uniquely 2D feature, broken by the parallel propagation term in 3D.
F.3. Magnetic Helicity in the Electron Fluid
If we now divide Eq. (F2) through by ev‖/c and integrate over velocities, we get, after some integrations by parts, another
relation that becomes a conservation law in 2D and that can also easily be derived directly from the equations of the isothermal
electron fluid (116-117):
d3rA‖
. (F4)
In the ERMHD limit k⊥ρi ≫ 1 (§ 7.2), we use Eqs. (221-223) to simplify the above equation and find that the integral on the
right-hand side vanishes and we get a genuine 3D conservation law:
d3rA‖δB‖ = 0. (F5)
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 63
This can also be derived directly from the ERMHD equations (226-227) [using Eq. (223)]. The conserved quantity is readily seen
to be the helicity of the perturbed magnetic field:
d3rA · δB =
∇⊥×A‖ẑ
+ A‖δB‖
A‖ẑ · (∇⊥×A⊥) + A‖δB‖
d3rA‖δB‖. (F6)
F.4. Magnetic Helicity in the RMHD Limit
Unlike in the case of ERMHD, the helicity of the perturbed magnetic field in RMHD is conserved only in 2D. This is because
the induction equation for the perturbed field has an inhomogeneous term associated with the mean field [Eq. (10) with B =
B0ẑ + δB] (this issue has been extensively discussed in the literature; see Matthaeus & Goldstein 1982; Stribling et al. 1994;
Berger 1997; Montgomery & Bates 1999; Brandenburg & Matthaeus 2004). Directly from the induction equation or from its
RMHD descendants Eqs. (17) and (26), we obtain [note the definitions (135)]
d3rA‖δB‖ =
1 + v2A/c2s
, (F7)
so helicity is conserved only if ∂/∂z = 0.
For completeness, let us now show that this 2D conservation law is a particular case of Eq. (F1) for ions. Let us consider the
inertial range (k⊥ρi ≪ 1). We substitute Eq. (124) into Eq. (F1) for ions and expand to two leading orders in k⊥ρi using the
ordering explained in § 5.2:
v‖〈A‖〉Ri
Z2e2v2
d3rA‖g + · · ·
Z2e2v2
d3rA‖
v2thi
+ Zev‖
d3rA‖
. (F8)
The lowest-order terms in the above equations (all proportional to v2‖F0i) simply reproduce the 2D conservation of “A
‖-stuff,”
given by Eq. (F3). We now subtract Eq. (F3) multiplied by (Zev‖/c)
2F0i/T0i from Eq. (F8). This leaves us with
d3rA‖g = c
+ v‖F0i
v2thi
d3rA‖
. (F9)
This equation is a general 2D conservation law of the KRMHD equations (see § 5.7) and can also be derived directly from them.
If we integrate it over velocities and use Eqs. (146) and (147), we simply recover Eq. (F4). However, since Eq. (F9) holds for
every value of v‖ and v⊥, it carries much more information than Eq. (F4).
To make connection to MHD, let us consider the fluid (collisional) limit of KRMHD worked out in Appendix D. The distribu-
tion function to lowest order in the k‖λmfpi ≪ 1 expansion is g = −(v2⊥/v2thi)δB‖/B0 +δ f̃
i , where δ f̃
i is the perturbed Maxwellian
given by Eq. (D3). We can substitute this expression into Eq. (F9). Since in this expansion the collision integral is applied to δ f̃ (1)i
and is the same order as the rest of the terms (see § D.3), conservation laws are best derived by taking 1, v‖, and v
2/v2thi moments
of Eq. (F9) so as to make the collision term vanish. In particular, multiplying Eq. (F9) by 1 + (2τ/3Z)v2/v2thi, integrating over
velocities and using Eqs. (D4) and (D6), we obtain the evolution equation for
d3rA‖δB‖, which coincides with Eq. (F7). Note
that, either proceeding in an analogous way, one can derive similar equations for
d3rA‖δne and
d3rA‖u‖—these are also 2D
invariants of the RMHD system, broken in 3D by the presence of the propagation terms. The same result can be derived directly
from the evolution equations (D8) and (D10).
F.5. Electrostatic Invariant
Interestingly, the existence of the general 2D invariants introduced in § F.1 alongside the generalized energy invariant given by
Eq. (73) means that one can construct a 2D invariant of gyrokinetics that does not involve any velocity-space quantities. In order
to do that, one must integrate Eq. (F1) over velocities, sum over species, and subtract Eq. (73) from the resulting equation (thus
removing the h2s integrals). The result is not particularly edifying in the general case, but it takes a simple form if one considers
electrostatic perturbations (δB = 0). In this case, χ = ϕ, and the manipulations described above lead to the following equation
d3v Is −W
q2s n0s
1−Γ0(αs)
|ϕk|2 =
d3rE‖ j‖ −
d3Rs 〈ϕ〉Rs
, (F10)
where E‖ = −∂ϕ/∂z, αs = k2⊥ρ
s/2 and Γ0 is defined by Eq. (129). In 2D, E‖ = 0 and the above equation expresses a conservation
law broken only by collisions. The complete derivation and analysis of 2D conservation properties of gyrokinetics in the electro-
static limit, including the invariant (F10), the electrostatic version of Eq. (F1), and their consequences for scalings and cascades,
was given by Plunk et al. (2009). Here we briefly consider a few relevant limits.
For k⊥ρi ≪ 1, we have Γ0(α) = 1 −αs + . . ., so the invariant given by Eq. (F10) is simply the kinetic energy of the E×B flows:
s(msn0s/2)
d3r |∇⊥Φ|2, where Φ = cϕ/B0. In the limit k⊥ρi ≫ 1, k⊥ρe ≪ 1, we have Y = −n0i
d3rZ2e2ϕ2/2T0i. In
64 SCHEKOCHIHIN ET AL.
the limit k⊥ρe ≫ 1, we have Y = −(1 + Z/τ )n0e
d3re2ϕ2/2T0e. Whereas we are not interested in electrostatic fluctuations in the
inertial range, electrostatic turbulence in the dissipation range was discussed in § 7.10 and § 7.12. The electrostatic 2D invariant
in the limits k⊥ρi ≫ 1, k⊥ρe ≪ 1 and k⊥ρe ≫ 1 can also be derived directly from the equations given there [in the former limit,
use Eq. (264) to express u‖i in terms of j‖ in order to get Eq. (F10)].
Note that, taken separately and integrated over velocities, Eq. (F1) for ions (when k⊥ρi ≫ 1, k⊥ρe ≪ 1) and for electrons
(when k⊥ρe ≫ 1), reduces to lowest order to the statement of 3D conservation of
d3Ri T0ih2i /2F0i [Whi in Eq. (245)] and∫
d3Re T0eh2e/2F0e [Eq. (280)], respectively.
F.6. Implications for Turbulent Cascades and Scalings
Since invariants other than the generalized energy or its constituent parts are present in 2D and, in some limits, also in 3D, one
might ask how their presence affects the turbulent cascades and scalings. As an example, let us consider the magnetic helicity in
KAW turbulence, which is a 3D invariant of the ERMHD equations (§ F.3).
A Kolmogorov-style analysis of a local KAW cascade based on a constant flux of helicity gives (proceeding as in § 7.5):
τKAWλ
1 +βi
τKAWλ
1 +βi
∼ εH = const ⇒ Φλ ∼
(1 +βi)1/6
1/3, (F11)
where εH is the helicity flux (omitting constant dimensional factors, the helicity is now defined as
d3rΨΦ and assumed to be
non-zero). This corresponds to a k
⊥ spectrum of magnetic energy.
In order to decide whether we expect the scalings to be determined by the constant-helicity flux or by the constant-energy
flux (as assumed in § 7.5), we adapt a standard argument originally due to Fjørtoft (1953). If the helicity flux of the KAW
turbulence originating at the ion gyroscale (via partial conversion from the inertial-range turbulence; see § 7) is εH , its energy
flux is εKAW ∼ εH [set λ = ρi in Eq. (F11) and compare with Eq. (238)]. If the cascade between the ion and electron gyroscales
is controlled by maintaining a constant flux of helicity, then the helicity flux arriving to the electron gyroscale is still εH , while
the associated energy flux is εHρi/ρe ≫ εKAW, i.e., more energy arrives to ρe than there was at ρi! This is clearly impossible in
a stationary state. The way to resolve this contradiction is to conclude that the helicity cascade is, in fact, inverse (i.e., directed
towards larger scales), while the energy cascade is direct (to smaller scales). A similar argument based on the constancy of the
energy flux εKAW then leads to the conclusion that the helicity flux arriving to the electron gyroscale is εKAWρe/ρi ≪ εH ∼ εKAW,
i.e., the helicity indeed does not cascade to smaller scales. It does not, in fact, cascade to large scales either because the ERMHD
equations are not valid above the ion gyroscale and the helicity of the perturbed magnetic field in the inertial range is not a 3D
invariant (§ F.4). The situation would be different if an energy source existed either at the electron gyroscale or somewhere in
between ρe and ρi. In such a case, one would expect an inverse helicity cascade and the consequent shallower scaling [Eq. (F11)]
between the energy-injection scale and the ion gyroscale.
Other invariants introduced above can in a similar fashion be argued to give rise to inverse cascades in the hypothetical 2D
situations where they are valid and provided there is energy injection at small scales (for the electrostatic case, see Plunk et al.
2009 and numerical simulations by Tatsuno et al. 2009b). The view of turbulence advanced in this paper does not generally
allow for this to happen. First, the fundamentally 3D nature of the turbulence is imposed via the critical balance conjecture and
supported by the argument that “two dimensionality” can only be maintained across parallel distances that do not exceed the
distance a parallel-propagating wave (or parallel-streaming particles) travels over one nonlinear decorrelation time (see § 1.2,
§ 7.5 and § 7.10.3). Secondly, the lack of small-scale energy injection was assumed at the outset. This can, however, be violated
in real astrophysical plasmas by various small-scale plasma instabilities (e.g., triggered by pressure anisotropies; see discussion
in § 8.3). Treatment of such effects falls outside the scope of this paper and remains a matter for future work.
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  We present a theoretical framework for plasma turbulence in astrophysical
plasmas (solar wind, interstellar medium, galaxy clusters, accretion disks).
The key assumptions are that the turbulence is anisotropic with respect to the
mean magnetic field and frequencies are low compared to the ion cyclotron
frequency. The energy injected at the outer scale scale has to be converted
into heat, which ultimately cannot be done without collisions. A KINETIC
CASCADE develops that brings the energy to collisional scales both in space and
velocity. Its nature depends on the physics of plasma fluctuations. In each of
the physically distinct scale ranges, the kinetic problem is systematically
reduced to a more tractable set of equations. In the "inertial range" above the
ion gyroscale, the kinetic cascade splits into a cascade of Alfvenic
fluctuations, which are governed by the RMHD equations at both the collisional
and collisionless scales, and a passive cascade of compressive fluctuations,
which obey a linear kinetic equation along the moving field lines associated
with the Alfvenic component. In the "dissipation range" between the ion and
electron gyroscales, there are again two cascades: the kinetic-Alfven-wave
(KAW) cascade governed by two fluid-like Electron RMHD equations and a passive
phase-space cascade of ion entropy fluctuations. The latter cascade brings the
energy of the inertial-range fluctuations that was damped by collisionless
wave-particle interaction at the ion gyroscale to collisional scales in the
phase space and leads to ion heating. The KAW energy is similarly damped at the
electron gyroscale and converted into electron heat. Kolmogorov-style scaling
relations are derived for these cascades. Astrophysical and space-physical
applications are discussed in detail.

Introduction, observational and nu-
merical evidence makes it safe to assume that the turbulence
in such a system will be anisotropic with k‖ ≪ k⊥ (at scales
smaller than the outer scale, k‖L ≫ 1; see § 1.3 and § 1.5.1).
Let us, therefore, introduce a small parameter ǫ ∼ k‖/k⊥ and
carry out a systematic expansion of Eqs. (7-10) in ǫ. In this
expansion, the fluctuations are treated as small, but not arbi-
trarily so: in order to estimate their size, we shall adopt the
critical-balance conjecture (3), which is now treated not as a
detailed scaling prescription but as an ordering assumption.
This allows us to introduce the following ordering:
∼ δB⊥
∼ ǫ, (12)
where vA = B0/
4πρ0 is the Alfvén speed. Note that this
means that we order the Mach number
, (13)
where cs = (γp0/ρ0)
1/2 is the speed of sound and
is the plasma beta, which is ordered to be order unity in the ǫ
expansion (subsidiary limits of high and low β can be taken
after the ǫ expansion is done; see § 2.4).
In Eq. (12), we made two auxiliary ordering assump-
tions: that the velocity and magnetic-field fluctuations
have the character of Alfvén and slow waves (δB⊥/B0 ∼
u⊥/vA, δB‖/B0 ∼ u‖/vA) and that the relative amplitudes
of the Alfvén-wave-polarized fluctuations (δB⊥/B0, u⊥/vA),
slow-wave-polarized fluctuations (δB‖/B0, u‖/vA) and den-
sity/pressure/entropy fluctuations (δρ/ρ0, δp/p0) are all the
same order. Strictly speaking, whether this is the case depends
on the energy sources that drive the turbulence: as we shall
see, if no slow waves (or entropy fluctuations) are launched,
none will be present. However, in astrophysical contexts, the
outer-scale energy input may be assumed random and, there-
fore, comparable power is injected into all types of fluctua-
tions.
We further assume that the characteristic frequency of the
fluctuations is ω∼ k‖vA [Eq. (3)], meaning that the fast waves,
for which ω ≃ k⊥(v2A + c2s )1/2, are ordered out. This restric-
tion must be justified empirically. Observations of the solar-
wind turbulence confirm that it is primarily Alfvénic (see,
e.g., Bale et al. 2005) and that its compressive component is
substantially pressure-balanced (Roberts 1990; Burlaga et al.
1990; Marsch & Tu 1993; Bavassano et al. 2004, see Eq. (22)
below). A weak-turbulence calculation of compressible MHD
turbulence in low-beta plasmas (Chandran 2005b) suggests
that only a small amount of energy is transferred from the fast
waves to Alfvén waves with large k‖. A similar conclusion
emerges from numerical simulations (Cho & Lazarian 2002,
2003). As the fast waves are also expected to be subject to
strong collisionless damping and/or to strong dissipation after
they steepen into shocks, we eliminate them from our con-
sideration of the problem and concentrate on low-frequency
turbulence.
2.2. Alfvén Waves
We start by observing that the Alfvén-wave-polarized
fluctuations are two-dimensionally solenoidal: since, from
Eq. (7),
∇·u = − d
= O(ǫ2) (15)
and ∇·δB = 0 exactly, separating the O(ǫ) part of these diver-
gences gives ∇⊥ ·u⊥ = 0 and ∇⊥ · δB⊥ = 0. To lowest order
in the ǫ expansion, we may, therefore, express u⊥ and δB⊥ in
terms of scalar stream (flux) functions:
u⊥ = ẑ×∇⊥Φ,
= ẑ×∇⊥Ψ. (16)
Evolution equations for Φ and Ψ are obtained by substituting
the expressions (16) into the perpendicular parts of the induc-
tion equation (10) and the momentum equation (8)—of the
latter the curl is taken to annihilate the pressure term. Keep-
ing only the terms of the lowest order, O(ǫ2), we get
+{Φ,Ψ}= vA
, (17)
∇2⊥Φ+
Φ,∇2⊥Φ
∇2⊥Ψ+
Ψ,∇2⊥Ψ
, (18)
where {Φ,Ψ} = ẑ · (∇⊥Φ×∇⊥Ψ) and we have taken into
account that, to lowest order,
+ u⊥ ·∇⊥ =
+{Φ, · · ·} , (19)
b̂ ·∇= ∂
·∇⊥ =
{Ψ, · · ·} . (20)
Here b̂ = B/B0 is the unit vector along the perturbed field line.
Equations (17-18) are known as the Reduced Magne-
tohydrodynamics (RMHD). The first derivations of these
equations (in the context of fusion plasmas) are due to
Kadomtsev & Pogutse (1974) and to Strauss (1976). These
were followed by many systematic derivations and gener-
alizations employing various versions and refinements of
the basic expansion, taking into account the non-Alfvénic
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 9
modes (which we will do in § 2.4), and including the ef-
fects of spatial gradients of equilibrium fields (e.g., Strauss
1977; Montgomery 1982; Hazeltine 1983; Zank & Matthaeus
1992; Kinney & McWilliams 1997; Bhattacharjee et al. 1998;
Kruger et al. 1998). A comparative review of these expansion
schemes and their (often close) relationship to ours is outside
the scope of this paper. One important point we wish to em-
phasize is that we do not assume the plasma beta [defined in
Eq. (14)] to be either large or small.
Equations (17) and (18) form a closed set, meaning that the
Alfvén-wave cascade decouples from the slow waves and den-
sity fluctuations. It is to the turbulence described by Eqs. (17-
18) that the GS theory outlined in § 1.2 applies.13 In § 5.3, we
will show that Eqs. (17) and (18) correctly describe inertial-
range Alfvénic fluctuations even in a collisionless plasma,
where the full MHD description [Eqs. (7-10)] is not valid.
2.3. Elsasser Fields
The MHD equations (7-10) in the incompressible limit
(ρ = const) acquire a symmetric form if written in terms of
the Elsasser fields z± = u± δB/
4πρ (Elsasser 1950). Let
us demonstrate how this symmetry manifests itself in the re-
duced equations derived above.
We introduce Elsasser potentials ζ± = Φ±Ψ, so that z±⊥ =
ẑ×∇⊥ζ±. For these potentials, Eqs. (17-18) become
∇2⊥ζ±∓ vA
∇2⊥ζ± = −
ζ+,∇2⊥ζ−
ζ−,∇2⊥ζ+
∓∇2⊥ {ζ+, ζ−}
. (21)
These equations show that the RMHD has a simple set of ex-
act solutions: if ζ− = 0 or ζ+ = 0, the nonlinear term vanishes
and the other, non-zero, Elsasser potential is simply a fluc-
tuation of arbitrary shape and magnitude propagating along
the mean field at the Alfvén speed vA: ζ
± = f±(x,y,z∓ vAt).
These solutions are finite-amplitude Alfvén-wave packets of
arbitrary shape. Only counterpropagating such solutions can
interact and thereby give rise to the Alfvén-wave cascade
(Kraichnan 1965). Note that these interactions are conserva-
tive in the sense that the “+” and “−” waves scatter off each
other without exchanging energy.
Note that the individual conservation of the “+” and “−”
waves’ energies means that the energy fluxes associated
with these waves need not be equal, so instead of a sin-
gle Kolmogorov flux ε assumed in the scaling arguments
13 The Alfvén-wave turbulence in the RMHD system has been stud-
ied by many authors. Some of the relevant numerical investigations are
due to Kinney & McWilliams (1998), Dmitruk et al. (2003), Oughton et al.
(2004), Rappazzo et al. (2007, 2008), Perez & Boldyrev (2008, 2009). An-
alytical theory has mostly been confined to the weak-turbulence paradigm
(Ng & Bhattacharjee 1996, 1997; Bhattacharjee & Ng 2001; Galtier et al.
2002; Lithwick & Goldreich 2003; Galtier & Chandran 2006; Nazarenko
2008). We note that adopting the critical balance [Eq. (3)] as an ordering
assumption for the expansion in k‖/k⊥ does not preclude one from subse-
quently attempting a weak-turbulence approach: the latter should simply be
treated as a subsidiary expansion. Indeed, implementing the anisotropy as-
sumption on the level of MHD equations rather than simultaneously with
the weak-turbulence closure (Galtier et al. 2000) significantly reduces the
amount of algebra. One should, however, bear in mind that the weak-
turbulence approximation always breaks down at some sufficiently small
scale—namely, when k⊥ ∼ (vA/U)
L, where L is the outer scale of the
turbulence, U velocity at the outer scale, and k‖ the parallel wavenum-
ber of the Alfvén waves (see Goldreich & Sridhar 1997 or the review by
Schekochihin & Cowley 2007). Below this scale, interactions cannot be as-
sumed weak.
reviewed in § 1.2, we could have ε+ 6= ε−. The GS the-
ory can be generalized to this case of imbalanced Alfvénic
cascades (Lithwick et al. 2007; Beresnyak & Lazarian 2008a;
Chandran 2008), but here we will focus on the balanced tur-
bulence, ε+ ∼ ε−. If one considers the turbulence forced
in a physical way (i.e., without forcing the magnetic field,
which would break the flux conservation), the resulting cas-
cade would always be balanced. In the real world, imbal-
anced Alfvénic fluxes are measured in the fast solar wind,
where the influence of initial conditions in the solar atmo-
sphere is more pronounced, while the slow-wind turbulence
is approximately balanced (Marsch & Tu 1990a; see also re-
views by Tu & Marsch 1995; Bruno & Carbone 2005 and ref-
erences therein).
2.4. Slow Waves and the Entropy Mode
In order to derive evolution equations for the remaining
MHD modes, let us first revisit the perpendicular part of the
momentum equation and use Eq. (12) to order terms in it. In
the lowest order, O(ǫ), we get the pressure balance
B0δB‖
= 0 ⇒ δp
. (22)
Using Eq. (22) and the entropy equation (9), we get
, (23)
where s0 = p0/ρ
0 . Now, substituting Eq. (15) for ∇·u in the
parallel component of the induction equation (10), we get
− b̂ ·∇u‖ = 0. (24)
Combining Eqs. (23) and (24), we obtain
1 + c2s/v
b̂ ·∇u‖, (25)
1 + v2A/c2s
b̂ ·∇u‖. (26)
Finally, we take the parallel component of the momentum
equation (8) and notice that, due to the pressure balance (22)
and to the smallness of the parallel gradients, the pressure
term is O(ǫ3), while the inertial and tension terms are O(ǫ2).
Therefore,
= v2Ab̂ ·∇
. (27)
Equations (26-27) describe the slow-wave-polarized fluctu-
ations, while Eq. (23) describes the zero-frequency entropy
mode, which is decoupled from the slow waves.14 The non-
linearity in Eqs. (26-27) enters via the derivatives defined in
14 For other expansion schemes leading to reduced sets of equations for
these “compressive” fluctuations see references in § 2.2. Note that the na-
ture of the density fluctuations described above is distinct from the so called
“pseudosound” density fluctuations that arise in the “nearly incompress-
ible” MHD theories (Montgomery et al. 1987; Matthaeus & Brown 1988;
Matthaeus et al. 1991; Zank & Matthaeus 1993). The “pseudosound” is es-
sentially the density response caused by the nonlinear pressure fluctuations
calculated from the incompressibility constraint. The resulting density fluc-
tuations are second order in Mach number and, therefore, order ǫ2 in our
expansion [see Eq. (13)]. The passive density fluctuations derived in this sec-
tion are order ǫ and, therefore, supersede the “pseudosound” (see review by
Tu & Marsch 1995 for a discussion of the relevant solar-wind evidence).
10 SCHEKOCHIHIN ET AL.
Eqs. (19-20) and is due solely to interactions with Alfvén
waves. Thus, both the slow-wave and the entropy-mode cas-
cades occur via passive scattering/mixing by Alfvén waves, in
the course of which there is no energy exchange between the
cascades.
Note that in the high-beta limit, cs ≫ vA [see Eq. (14)], the
entropy mode is dominated by density fluctuations [Eq. (23),
cs ≫ vA], which also decouple from the slow-wave cascade
[Eq. (25), cs ≫ vA]. and are passively mixed by the Alfvén-
wave turbulence:
= 0. (28)
The high-beta limit is equivalent to the incompressible ap-
proximation for the slow waves.
In § 5.5, we will derive a kinetic description for the inertial-
range compressive fluctuations (density and magnetic-field
strength), which is more generally valid in weakly collisional
plasmas and which reduces to Eqs. (26-27) in the collisional
limit (see Appendix D). While these fluctuations will in gen-
eral satisfy a kinetic equation, they will remain passive with
respect to the Alfvén waves.
2.5. Elsasser Fields for the Slow Waves
The original Elsasser (1950) symmetry was derived for in-
compressible MHD equations. However, for the “compres-
sive” slow-wave fluctuations, we may introduce generalized
Elsasser fields:
= u‖±
. (29)
Straightforwardly, the evolution equation for these fields is
∓ vA√
1 + v2A/c2s
1∓ 1√
1 + v2A/c2s
ζ+,z±
1± 1√
1 + v2A/c2s
ζ−,z±
. (30)
In the high-beta limit (vA ≪ cs), the generalized Elsasser
fields (29) become the parallel components of the conven-
tional incompressible Elsasser fields. We see that only in this
limit do the slow waves interact exclusively with the counter-
propagating Alfvén waves, and so only in this limit does set-
ting ζ− = 0 or ζ+ = 0 gives rise to finite-amplitude slow-wave-
packet solutions z±
= f±(x,y,z∓ vAt) analogous to the finite-
amplitude Alfvén-wave packets discussed in § 2.3.15 For gen-
eral β, the phase speed of the slow waves is smaller than that
of the Alfvén waves and, therefore, Alfvén waves can “catch
up” and interact with the slow waves that travel in the same
direction. All of these interactions are of scattering type and
involve no exchange of energy.
2.6. Scalings for Passive Fluctuations
15 Obviously, setting both ζ± = 0 does always enable these finite-
amplitude slow-wave solutions. More non-trivially, such finite-amplitude so-
lutions exist in the Lagrangian frame associated with the Alfvén waves—this
is discussed in detail in § 6.3.
The scaling of the passively mixed scalar fields introduced
above is slaved to the scaling of the Alfvénic fluctuations.
Consider for example the entropy mode [Eq. (23)]. As
in Kolmogorov–Obukhov theory (see § 1.1), one assumes a
local-in-scale-space cascade of scalar variance and a constant
flux εs of this variance. Then, analogously to Eq. (1),
v2thi
∼ εs. (31)
Since the cascade time is τ−1λ ∼ u⊥ ·∇⊥ ∼ vA/l‖λ ∼ ε/u2⊥λ,
)1/2 u⊥λ
, (32)
so the scalar fluctuations have the same scaling as the turbu-
lence that mixes them (Obukhov 1949; Corrsin 1951). In GS
turbulence, the scalar-variance spectrum should, therefore, be
⊥ (Lithwick & Goldreich 2001). The same argument ap-
plies to all passive fields.
It is the (presumably) passive electron-density spectrum
that provides the main evidence of the k−5/3 scaling in the in-
terstellar turbulence (Armstrong et al. 1981, 1995; Lazio et al.
2004, see further discussion in § 8.4.1). The explanation of
this spectrum in terms of passive mixing of the entropy mode,
originally proposed by Higdon (1984), was developed on the
basis of the GS theory by Lithwick & Goldreich (2001). The
turbulent cascade of the compressive fluctuations and the rel-
evant solar-wind data is discussed further in § 6.3. In partic-
ular, it will emerge that the anisotropy of these fluctuations
remains a non-trivial issue: is there an analog of the scaling
relation (5)? The scaling argument outlined above does not
invoke any assumptions about the relationship between the
parallel and perpendicular scales of the compressive fluctu-
ations (other than the assumption that they are anisotropic).
Lithwick & Goldreich (2001) argue that the parallel scales of
the Alfvénic fluctuations will imprint themselves on the pas-
sively advected compressive ones, so Eq. (5) holds for the
latter as well. In § 6.3, we examine this conclusion in view
of the solar-wind evidence and of the fact that the equations
for the compressive modes become linear in the Lagrangian
frame associated with the Alfvénic turbulence.
2.7. Five RMHD Cascades
Thus, the anisotropy and critical balance (3) taken as
ordering assumptions lead to a neat decomposition of the
MHD turbulent cascade into a decoupled Alfvén-wave cas-
cade and cascades of slow waves and entropy fluctuations pas-
sively scattered/mixed by the Alfvén waves. More precisely,
Eqs. (23), (21) and (30) imply that, for arbitrary β, there are
five conserved quantities:16
W±AW =
d3rρ0|∇⊥ζ±|2 (Alfven waves), (33)
W±sw =
d3rρ0|z±‖ |
2 (slow waves), (34)
(entropy fluctuations).(35)
16 Note that magnetic helicity of the perturbed field is not an invariant of
RMHD, except in two dimensions (see Appendix F.4). In 2D, there is also
conservation of the mean square flux,
d3r |Ψ|2 (see Appendix F.2).
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 11
W +AW and W
AW are always cascaded by interaction with each
other, Ws is passively mixed by W
AW and W
AW, W
sw are pas-
sively scattered by W∓AW and, unless β ≫ 1, also by W
This is an example of splitting of the overall energy cascade
into several channels (recovered as a particular case of the
more general kinetic cascade in Appendix D.2)—a concept
that will repeatedly arise in the kinetic treatment to follow.
The decoupling of the slow- and Alfvén-wave cascades in
MHD turbulence was studied in some detail and confirmed
in direct numerical simulations by Maron & Goldreich (2001,
for β ≫ 1) and by Cho & Lazarian (2002, 2003, for a range
of values of β). The derivation given in § 2.2 and § 2.4 (cf.
Lithwick & Goldreich 2001) provides a straightforward theo-
retical basis for these results, assuming anisotropy of the tur-
bulence (which was also confirmed in these numerical stud-
ies).
It turns out that the decoupling of the Alfvén-wave cascade
that we demonstrated above for the anisotropic MHD turbu-
lence is a uniformly valid property of plasma turbulence at
both collisional and collisionless scales and that this cascade
is correctly described by the RMHD equations (17-18) all the
way down to the ion gyroscale, while the fluctuations of den-
sity and magnetic-field strength do not satisfy simple fluid
evolution equations anymore and require solving the kinetic
equation. In order to prove this, we adopt a kinetic descrip-
tion and apply to it the same ordering (§ 2.1) as we used to
reduce the MHD equations. The kinetic theory that emerges
as a result is called gyrokinetics.
3. GYROKINETICS
The gyrokinetic formalism was first worked out
for linear waves by Rutherford & Frieman (1968)
and by Taylor & Hastie (1968) (see also Catto 1978;
Antonsen & Lane 1980; Catto et al. 1981) and subsequently
extended to the nonlinear regime by Frieman & Chen (1982).
Rigorous derivations of the gyrokinetic equation based on
the Hamiltonian formalism were developed by Dubin et al.
(1983, electrostatic) and Hahm et al. (1988, electromagnetic).
This approach is reviewed in Brizard & Hahm (2007). A
more pedestrian, but perhaps also more transparent exposition
of the gyrokinetics in a straight mean field can be found in
Howes et al. (2006), who also provide a detailed explanation
of the gyrokinetic ordering in the context of astrophysical
plasma turbulence and a treatment of the linear waves and
damping rates. Here we review only the main points so as
to allow the reader to understand the present paper without
referring elsewhere.
In general, a plasma is completely described by the distribu-
tion function fs(t,r,v)—the probability density for a particle
of species s (= i,e) to be found at the spatial position r mov-
ing with velocity v. This function obeys the kinetic Vlasov–
Landau (or Boltzmann) equation
+ v ·∇ fs +
· ∂ fs
, (36)
where qs and ms are the particle’s charge and mass, c is the
speed of light, and the right-hand side is the collision term
(quadratic in f ). The electric and magnetic fields are
E = −∇ϕ−
, B = ∇×A. (37)
The first equality is Faraday’s law uncurled, the second
the magnetic-field solenoidality condition; we shall use the
Coulomb gauge, ∇·A = 0. The fields satisfy the Poisson and
the Ampère–Maxwell equations with the charge and current
densities determined by fs(t,r,v):
∇·E = 4π
qsns = 4π
d3v fs, (38)
∇×B − 1
d3vv fs. (39)
3.1. Gyrokinetic Ordering and Dimensionless Parameters
As in § 2 we set up a static equilibrium with a uniform mean
field, B0 = B0ẑ, E0 = 0, assume that the perturbations will be
anisotropic with k‖ ≪ k⊥ (at scales smaller than the outer
scale, k‖L ≫ 1; see § 1.3 and § 1.5.1), and construct an expan-
sion of the kinetic theory around this equilibrium with respect
to the small parameter ǫ ∼ k‖/k⊥. We adopt the ordering ex-
pressed by Eqs. (3) and (12), i.e., we assume the perturbations
to be strongly interacting Alfvén waves plus electron density
and magnetic-field-strength fluctuations.
Besides ǫ, several other dimensionless parameters are
present, all of which are formally considered to be of order
unity in the gyrokinetic expansion: the electron–ion mass ra-
tio me/mi, the charge ratio
Z = qi/|qe| = qi/e (40)
(for hydrogen, this is 1, which applies to most astrophysical
plasmas of interest to us), the temperature ratio17
τ = Ti/Te, (41)
and the plasma (ion) beta
v2thi
8πniTi
, (42)
where vthi = (2Ti/mi)
1/2 is the ion thermal speed and the total
β was defined in Eq. (14) based on the total pressure p = niTi +
neTe. We shall occasionally also use the electron beta
8πneTe
βi. (43)
The total beta is β = βi +βe.
3.1.1. Wavenumbers and Frequencies
As we want our theory to be uniformly valid at all (perpen-
dicular) scales above, at or below the ion gyroscale, we order
k⊥ρi ∼ 1, (44)
where ρi = vthi/Ωi is the ion gyroradius, Ωi = qiB0/cmi the ion
cyclotron frequency. Note that
ρi. (45)
17 It can be shown that equilibrium temperatures change on the timescale
∼ (ǫ2ω)−1 (Howes et al. 2006). On the other hand, from standard theory
of collisional transport (e.g., Helander & Sigmar 2002), the ion and elec-
tron temperatures equalize on the timescale ∼ ν−1ie ∼ (mi/me)
1/2ν−1ii [see
Eq. (51)]. Therefore, τ can depart from unity by an amount of order
ǫ2(ω/νii)(mi/me)1/2. In our ordering scheme [Eq. (49)], this is O(ǫ2) and,
therefore, we should simply set τ = 1 + O(ǫ2). However, we shall carry the
parameter τ because other ordering schemes are possible that permit arbitrary
values of τ . These are appropriate to plasmas with very weak collisions. For
example, in the solar wind, τ appears to be order unity but not exactly 1
(Newbury et al. 1998), while in accretion flows near the black hole, some
models predict τ ≫ 1 (see § 8.5).
12 SCHEKOCHIHIN ET AL.
FIG. 3.— Regions of validity in the wavenumber space of two primary approximations—the two-fluid (Appendix A.1) and gyrokinetic (§ 3). The gyrokinetic
theory holds when k‖ ≪ k⊥ and ω ≪ Ωi [when k‖ ≪ k⊥ < ρ
i , the second requirement is automatically satisfied for Alfvén, slow and entropy modes; see
Eq. (46)]. The two-fluid equations hold when k‖λmfpi ≪ 1 (collisional limit) and k⊥ρi ≪ 1 (magnetized plasma). Note that the gyrokinetic theory holds for all
but the very largest (outer) scales, where anisotropy cannot be assumed.
Assuming Alfvénic frequencies implies
∼ k⊥ρi√
ǫ. (46)
Thus, gyrokinetics is a low-frequency limit that averages over
the timescales associated with the particle gyration. Because
we have assumed that the fluctuations are anisotropic and have
(by order of magnitude) Alfvénic frequencies, we see from
Eq. (46) that their frequency remains far below Ωi at all scales,
including the ion and even electron gyroscale—the gyroki-
netics remains valid at all of these scales and the cyclotron-
frequency effects are negligible (cf. Quataert & Gruzinov
1999).
3.1.2. Fluctuations
Equation (3) allows us to order the fluctuations of the scalar
potential: on the one hand, we have from Eq. (3) u⊥ ∼ ǫvA; on
the other hand, the plasma mass flow velocity is (to the lowest
order) the E×B drift velocity of the ions, u⊥ ∼ cE⊥/B0 ∼
ck⊥ϕ/B0, so
ǫ. (47)
All other fluctuations (magnetic, density, parallel velocity) are
ordered according to Eq. (12).
Note that the ordering of the flow velocity dictated by
Eq. (3) means that we are considering the limit of small Mach
numbers:
M ∼ u
. (48)
This means that the gyrokinetic description in the form used
below does not extend to large sonic flows that can be
present in many astrophysical systems. It is, in principle,
possible to extend the gyrokinetics to systems with sonic
flows (e.g., in the toroidal geometry; see Artun & Tang 1994;
Sugama & Horton 1997). However, we do not follow this
route because such flows belong to the same class of non-
universal outer-scale features as background density and tem-
perature gradients, system-specific geometry etc.—these can
all be ignored at small scales, where the turbulence should be
approximately homogeneous and subsonic (as long as k‖L ≫
1, see discussion in § 1.5.1).
3.1.3. Collisions
Finally, we want our theory to be valid both in the colli-
sional and the collisionless regimes, so we do not assume
ω to be either smaller or larger than the (ion) collision fre-
quency νii:
k‖λmfpi√
∼ 1, (49)
where λmfpi = vthi/νii is the ion mean free path (this order-
ing can actually be inferred from equating the gyrokinetic en-
tropy production terms to the collisional entropy production;
see extended discussion in Howes et al. 2006). Note that the
ordering (49) holds on the understanding that we have ordered
k⊥ρi ∼ 1 [Eq. (44)] because the fluctuation frequency can de-
pend on k⊥ρi in the dissipation range (see § 7.3).
Other collision rates are related to νii via a set of standard
formulae (see, e.g., Helander & Sigmar 2002), which will be
useful in what follows:
νei = Zνee =
τ 3/2
νii, (50)
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 13
νie =
τ 3/2
νii, (51)
νii =
2πZ4e4ni lnΛ
, (52)
where lnΛ is the Coulomb logarithm and the numerical factor
in the definition of νie has been inserted for future notational
convenience (see Appendix A). We always define
λmfpi =
, λmfpe =
λmfpi. (53)
The ordering of the collision frequency expressed by
Eq. (49) means that collisions, while not dominant as in
the fluid description (Appendix A), are still retained in
the version of the gyrokinetic theory adopted by us. Their
presence is required in order for us to be able to assume
that the equilibrium distribution is Maxwellian [Eq. (54)
below] and for the heating and entropy production to be
treated correctly (§ 3.4 and § 3.5). However, our ordering of
collisions and of the fluctuation amplitudes (§ 3.1.2) imposes
certain limitations: thus, we cannot treat the class of nonlinear
phenomena involving particle trapping by parallel-varying
fluctuations, non-Maxwellian tails of particle distributions,
plasma instabilities arising from the equilibrium pressure
anisotropies (mirror, firehose) and their possible nonlinear
evolution to large amplitudes (see discussion in § 8.3).
The region of validity of the gyrokinetic approximation in
the wavenumber space is illustrated in Fig. 3—it embraces
all of the scales that are expected to be traversed by the
anisotropic energy cascade (except the scales close to the
outer scale).
As we explained above, me/mi, βi, k⊥ρi and k‖λmfpi (or
ω/νii) are assigned order unity in the gyrokinetic expansion.
Subsidiary expansions in small me/mi (§ 4) and in small or
large values of the other three parameters (§§ 5-7) can be car-
ried out at a later stage as long as their values are not so large
or small as to interfere with the primary expansion in ǫ. These
expansions will yield simpler models of turbulence with more
restricted domains of validity than gyrokinetics.
3.2. Gyrokinetic Equation
Given the gyrokinetic ordering introduced above, the ex-
pansion of the distribution function up to first order in ǫ can
be written as
fs(t,r,v) = F0s(v) −
qsϕ(t,r)
F0s(v) + hs(t,Rs,v⊥,v‖). (54)
To zeroth order, it is a Maxwellian:18
F0s(v) =
(πv2ths)
v2ths
, vths =
, (55)
with uniform density n0s and temperature T0s and no mean
flow. As will be explained in more detail in § 3.5, F0s has a
slow time dependence via the equilibrium temperature, T0s =
T0s(ǫ
2t). This reflects the slow heating of the plasma as the tur-
bulent energy is dissipated. However, T0s can be treated as a
constant with respect to the time dependence of the first-order
18 The use of isotropic equilibrium is a significant idealization—this is
discussed in more detail in § 8.3.
distribution function (the timescale of the turbulent fluctua-
tions). The first-order part of the distribution function is com-
posed of the Boltzmann response [second term in Eq. (54), or-
dered in Eq. (47)] and the gyrocenter distribution function hs.
The spatial dependence of the latter is expressed not by the
particle position r but by the position Rs of the particle gy-
rocenter (or guiding center)—the center of the ring orbit that
the particle follows in a strong guide field:
Rs = r +
v⊥× ẑ
. (56)
Thus, some of the velocity dependence of the distribution
function is subsumed in the Rs dependence of hs. Explicitly,
hs depends only on two velocity-space variables: it is cus-
tomary in the gyrokinetic literature for these to be chosen as
the particle energy εs = msv
2/2 and its first adiabatic invari-
ant µs = msv
⊥/2B0 (both conserved quantities to two lowest
orders in the gyrokinetic expansion). However, in a straight
uniform guide field B0ẑ, the pair (v⊥,v‖) is a simpler choice,
which will mostly be used in what follows (we shall some-
times find an alternative pair, v and ξ = v‖/v, useful, especially
where collisions are concerned). It must be constantly kept in
mind that derivatives of hs with respect to the velocity-space
variables are taken at constant Rs, not at constant r.
The function hs satisfies the gyrokinetic equation:
{〈χ〉Rs ,hs} =
qsF0s
∂〈χ〉Rs
where
χ(t,r,v) = ϕ−
v⊥ ·A⊥
, (58)
the Poisson brackets are defined in the usual way:
{〈χ〉Rs ,hs} = ẑ ·
∂〈χ〉Rs
× ∂hs
, (59)
and the ring average notation is introduced:
〈χ(t,r,v)〉Rs =
t,Rs −
v⊥× ẑ
, (60)
where ϑ is the angle in the velocity space taken in the plane
perpendicular to the guide field B0ẑ. Note that, while χ is
a function of r, its ring average is a function of Rs. Note
also that the ring averages depend on the species index, as
does the gyrocenter variable Rs. Equation (57) is derived by
transforming the first-order kinetic equation to the gyrocenter
variable (56) and ring averaging the result (see Howes et al.
2006, or the references given at the beginning of § 3). The
ring-averaged collision integral (∂hs/∂t)c is discussed in Ap-
pendix B.
3.3. Field Equations
To Eq. (57), we must append the equations that determine
the electromagnetic field, namely, the potentials ϕ(t,r) and
A(t,r) that enter the expression for χ [Eq. (58)]. In the
non-relativistic limit (vthi ≪ c), these are the plasma quasi-
neutrality constraint [which follows from the Poisson equa-
tion (38) to lowest order in vthi/c]:
qsδns =
n0s +
d3v〈hs〉r
14 SCHEKOCHIHIN ET AL.
and the parallel and perpendicular parts of Ampère’s law
[Eq. (39) to lowest order in ǫ and in vthi/c]:
∇2⊥A‖ = −
j‖ = −
d3vv‖〈hs〉r, (62)
∇2⊥δB‖ = −
∇⊥× j⊥
d3v〈v⊥hs〉r
, (63)
where we have used δB‖ = ẑ · (∇⊥×A⊥) and dropped the dis-
placement current. Since field variables ϕ, A‖ and δB‖ are
functions of the spatial variable r, not of the gyrocenter vari-
able Rs, we had to determine the contribution from the gy-
rocenter distribution function hs to the charge distribution at
fixed r by performing a gyroaveraging operation dual to the
ring average defined in Eq. (60):
〈hs(t,Rs,v⊥,v‖)〉r =
t,r +
v⊥× ẑ
,v⊥,v‖
In other words, the velocity-space integrals in Eqs. (61-63)
are performed over hs at constant r, rather than constant Rs.
If we Fourier transform hs in Rs, the gyroaveraging operation
takes a simple mathematical form:
〈hs〉r =
〈eik·Rs〉rhsk(t,v⊥,v‖)
eik·r
ik · v⊥× ẑ
hsk(t,v⊥,v‖)
eik·rJ0(as)hsk(t,v⊥,v‖), (65)
where as = k⊥v⊥/Ωs and J0 is a Bessel function that arose
from the angle integral in the velocity space. In Eq. (63), an
analogous calculation taking into account the angular depen-
dence of v⊥ leads to
δB‖ = −
eik·r
d3vmsv
J1(as)
hsk(t,v⊥,v‖).
Note that Eq. (63) [and, therefore, Eq. (66)] is the gyroki-
netic equivalent of the perpendicular pressure balance that ap-
peared in § 2 [Eq. (22)]:
B0δB‖
= ∇⊥ ·
d3v〈ẑ× v⊥hs〉r
= ∇⊥ ·
t,r +
v⊥× ẑ
,v⊥,v‖
= −∇⊥∇⊥ :
d3vms〈v⊥v⊥ hs〉r = −∇⊥∇⊥ : δP⊥,(67)
where we have integrated by parts with respect to the gyroan-
gle ϑ and used ∂v⊥/∂ϑ = ẑ× v⊥, ∂2v⊥/∂ϑ2 = −v⊥ (cf. the
Appendix of Roach et al. 2005).
Once the fields are determined, they have to be substi-
tuted into χ [Eq. (58)] and the result ring averaged [Eq. (60)].
Again, we emphasize that ϕ, A‖ and δB‖ are functions of r,
while 〈χ〉Rs is a function of Rs. The transformation is ac-
complished via a calculation analogous to the one that led to
Eqs. (65) and (66):
〈χ〉Rs =
eik·Rs〈χ〉Rs ,k, (68)
〈χ〉Rs ,k = J0(as)
v‖A‖k
v2ths
J1(as)
. (69)
The last equation establishes a correspondence between the
Fourier transforms of the fields with respect to r and the
Fourier transform of 〈χ〉Rs with respect to Rs.
3.4. Generalized Energy and the Kinetic Cascade
As promised in § 1.4, the central unifying concept of this
paper is now introduced.
If we multiply the gyrokinetic equation (57) by T0shs/F0s
and integrate over the velocities and gyrocenters, we find that
the nonlinear term conserves the variance of hs and
d3Rs qs
∂〈χ〉Rs
T0shs
. (70)
Let us now sum this equation over all species. The first term
on the right-hand side is
∂〈χ〉Rs
d3v〈hs〉r −
d3v〈vhs〉r
d3rE · j, (71)
where we have used Eq. (61) and Ampère’s law [Eqs. (62-
63)] to express the integrals of hs. The second term on the
right-hand side is the total work done on plasma per unit time.
Using Faraday’s law [Eq. (37)] and Ampère’s law [Eq. (39)],
it can be written as
d3rE · j = − d
|δB|2
+ Pext, (72)
where Pext ≡ −
d3rE · jext is the total power injected into the
system by the external energy sources (outer-scale stirring; in
terms of the Kolmogorov energy flux ε used in the scaling
arguments in § 1.2, Pext = Vmin0iε, where V is the system vol-
ume). Combining Eqs. (70-72), we find (Howes et al. 2006)
T0s〈h2s 〉r
|δB|2
= Pext +
T0shs
. (73)
W is a positive definite quantity—this becomes explicit if we
use Eq. (61) to express it in terms of the total perturbed distri-
bution function δ fs = −qsϕF0s/T0s + hs [see Eq. (54)]:
T0sδ f
|δB|2
. (74)
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 15
We will refer to W as the generalized energy. We use this
term to emphasize the role of W as the cascaded quantity in
gyrokinetic turbulence (see below). This quantity is, in fact,
the gyrokinetic version of a collisionless kinetic invariant var-
iously referred to as the generalized grand canonical poten-
tial (see Hallatschek 2004, who points out the fundamental
role of this quantity in plasma turbulence simulations) or free
energy (e.g., Fowler 1968; Scott 2007). The non-magnetic
part of W is related to the perturbed entropy of the sys-
tem (Krommes & Hu 1994; Sugama et al. 1996; Howes et al.
2006; Schekochihin et al. 2008b, see discussion in § 3.5).19
Equation (73) is a conservation law of the generalized en-
ergy: Pext is the source and the second term on the right-hand
side, which is negative definite, represents collisional dissi-
pation. This suggests that we might think of kinetic plasma
turbulence in terms of the generalized energy W injected by
the outer-scale stirring and dissipated by collisions. In or-
der for the dissipation to be important, the collisional term in
Eq. (73) has to become comparable to Pext. This can happen
in two ways:
1. At collisional scales (k‖λmfpi ∼ 1) due to deviations of
the perturbed distribution function from a local per-
turbed Maxwellian (see § 6.1 and Appendix D);
2. At collisionless scales (k‖λmfpi ≫ 1) due the develop-
ment of small scales in the velocity space—large gra-
dients in v‖ (see § 6.2.4) or v⊥ (which is accompanied
by the development of small perpendicular scales in the
position space; see § 7.9.1).
Thus, the dissipation is only important at particular (small)
scales, which are generally well separated from the outer
scale. The generalized energy is transferred from the outer
scale to the dissipation scales via a nonlinear cascade. We
shall call it the kinetic cascade. It is analogous to the energy
cascade in fluid or MHD turbulence, but a conceptually new
feature is present: the small scales at which dissipation hap-
pens are small scales both in the velocity and position space.
Whereas the large gradients in v‖ are produced by the lin-
ear parallel phase mixing, whose role in the kinetic dissipa-
tion processes has been appreciated for some time (Landau
1946; Hammett et al. 1991; Krommes & Hu 1994; Krommes
1999; Watanabe & Sugama 2004, see § 6.2.4), the emergence
of large gradients in v⊥ is due to an essentially nonlinear
phase mixing mechanism (§ 7.9.1). At spatial scales smaller
than the ion gyroradius, this nonlinear perpendicular phase
mixing turns out to be a faster and, therefore, presumably the
dominant way of generating small-scale structure in the veloc-
ity space. It was anticipated in the development of gyrofluid
moment hierarchies by Dorland & Hammett (1993). Here we
treat it for the first time as a phase-space turbulent cascade:
this is done in § 7.9 and § 7.10 (see also Schekochihin et al.
2008b).
In the sections that follow, we shall derive particular forms
of W for various limiting cases of the gyrokinetic theory
(§ 4.7, § 5.6, § 6.2.5, § 7.8, Appendices D.2 and E.2). We
shall see that the kinetic cascade of W is, indeed, a direct
generalization of the more familiar fluid cascades (such as
19 Note also that a quadratic form involving both the perturbed distribution
function and the electromagnetic field appears, in a more general form than
Eq. (74), in the formulation of the energy principle for the Kinetic MHD
approximation (Kruskal & Oberman 1958; Kulsrud 1962, 1964). Regarding
the relationship between Kinetic MHD and gyrokinetics, see footnote 23.
the RMHD cascades discussed in § 2) and that W contains
the energy invariants of the fluid models in the appropriate
limits. In these limits, the cascade of the generalized en-
ergy will split into several decoupled cascades, as it did in
the case of RMHD (§ 2.7). Whenever one of the physically
important scales (§ 1.5.2) is crossed and a change of physical
regime occurs, these cascades are mixed back together into
the overall kinetic cascade of W , which can then be split in
a different way as it emerges on the “opposite side” of the
transition region in the scale space. The conversion of the
Alfvénic cascade into the KAW cascade and the entropy cas-
cade at k⊥ρi ∼ 1 is the most interesting example of such a
transition, discussed in § 7.
The generalized energy appears to be the only quadratic
invariant of gyrokinetics in three dimensions; in two dimen-
sions, many other invariants appear (see Appendix F).
3.5. Heating and Entropy
In a stationary state, all of the the turbulent power injected
by the external stirring is dissipated and thus transferred into
heat. Mathematically, this is expressed as a slow increase in
the temperature of the Maxwellian equilibrium. In gyrokinet-
ics, the heating timescale is ordered as ∼ (ǫ2ω)−1.
Even though the dissipation of turbulent fluctuations may
be occurring “collisionlessly” at scales such that k‖λmfpi ≫ 1
(e.g., via wave–particle interaction at the ion gyroscale; § 7.1),
the resulting heating must ultimately be effected with the help
of collisions. This is because heating is an irreversible process
and it is a small amount of collisions that make “collisionless”
damping irreversible. In other words, slow heating of the
Maxwellian equilibrium is equivalent to entropy production
and Boltzmann’s H-theorem rigorously requires collisions to
make this possible. Indeed, the total entropy of species s is
Ss = −
d3v fs ln fs
F0s lnF0s +
δ f 2s
+ O(ǫ3), (75)
where we took
d3rδ fs = 0. It is then not hard to show that
T0shs
where the overlines mean averaging over times longer than
the characteristic time of the turbulent fluctuations ∼ ω−1
but shorter than the typical heating time ∼ (ǫ2ω)−1 (see
Howes et al. 2006; Schekochihin et al. 2008b for a detailed
derivation of this and related results on heating in gyroki-
netics; see also earlier discussions of the entropy production
in gyrokinetics by Krommes & Hu 1994; Krommes 1999;
Sugama et al. 1996). We have omitted the term describing the
interspecies collisional temperature equalization. Note that
both sides of Eq. (76) are order ǫ2ω.
If we now time average Eq. (73) in a similar fashion, the
left-hand side vanishes because it is a time derivative of a
quantity fluctuating on the timescale ∼ ω−1 and we confirm
that the right-hand side of Eq. (76) is simply equal to the av-
erage power Pext injected by external stirring. The import of
Eq. (76) is that it tells us that heating can only be effected
by collisions, while Eq. (73) implies that the injected power
gets to the collisional scales in velocity and position space by
means of a kinetic cascade of generalized energy.
16 SCHEKOCHIHIN ET AL.
The first term in the expression for the generalized energy
(74) is −
s T0sδSs, where δSs is the perturbed entropy [see
Eq. (75)]. The second term in Eq. (74) is magnetic energy.
Collisionless damping of electromagnetic fluctuations can be
thought of as a redistribution of the generalized energy, trans-
ferring the electromagnetic energy into entropy fluctuations,
while the total W is conserved (a simple example of how that
happens for collisionless compressive fluctuations in the iner-
tial range is worked out in § 6.2.3).
The contribution to the perturbed entropy from the gy-
rocenter distribution is the integral of −h2s/2F0s, whose
evolution equation (70) can be viewed as the gyrokinetic
version of the H-theorem. The first term on the right-hand
side of this equation represents the wave–particle interaction
(collisionless damping). Under time average, it is related to
the work done on plasma [Eq. (71)] and hence to the average
externally injected power Pext via time-averaged Eq. (72).
In a stationary state, this is balanced by the second term in the
right-hand side of Eq. (70), which is the collisional-heating,
or entropy-production, term that also appears in Eq. (76).
Thus, the generalized energy channeled by collisionless
damping into entropy fluctuations is eventually converted
into heat by collisions. The sub-gyroscale entropy cascade,
which brings the perturbed distribution function hs to col-
lisional scales, will be discussed further in § 7.9 and § 7.10
(see also Schekochihin et al. 2008b).
This concludes a short primer on gyrokinetics necessary
(and sufficient) for adequate understanding of what is to fol-
low. Formally, all further analytical derivations in this paper
are simply subsidiary expansions of the gyrokinetics in the pa-
rameters we listed in § 3.1: in § 4, we expand in (me/mi)
in § 5 in k⊥ρi (followed by further subsidiary expansions in
large and small k‖λmfpi in § 6), and in § 7 in 1/k⊥ρi.
4. ISOTHERMAL ELECTRON FLUID
In this section, we carry out an expansion of the electron gy-
rokinetic equation in powers of (me/mi)
1/2 ≃ 0.02 (for hydro-
gen plasma). In virtually all cases of interest, this expansion
can be done while still considering
βi, k⊥ρi, and k‖λmfpi to
be order unity.21 Note that the assumption k⊥ρi ∼ 1 together
with Eq. (45) mean that
k⊥ρe ∼ k⊥ρi(me/mi)1/2 ≪ 1, (77)
i.e., the expansion in (me/mi)
1/2 means also that we are
considering scales larger than the electron gyroradius. The
idea of such an expansion of the electron kinetic equation
has been utilized many times in plasma physics literature.
The mass-ratio expansion of the gyrokinetic equation in a
form very similar to what is presented below is found in
Snyder & Hammett (2001).
20 Note that Eq. (72) is valid not only in the integral form but also indi-
vidually for each wavenumber: indeed, using the Fourier-transformed Fara-
day and Ampère’s laws, we have Ek · j
k + E
k · jk = Ek · j
ext,k + E
k · jext,k −
(1/4π)∂|δBk|2/∂t. In a stationary state, time averaging eliminates the time
derivative of the magnetic-fluctuation energy, so Ek · j∗k + E
k · jk = 0 at all k
except those corresponding to the outer scale, where the external energy in-
jection occurs. This means that below the outer scale, the work done on one
species balances the work done on the other. The wave–particle interaction
term in the gyrokinetic equation is responsible for this energy exchange.
21 One notable exception is the LAPD device at UCLA, where β ∼ 10−4 −
10−3 (due mostly to the electron pressure because the ions are cold, τ ∼
0.1, so βi ∼ βe/10; see, e.g., Morales et al. 1999; Carter et al. 2006). This
interferes with the mass-ratio expansion.
The primary import of this section will be technical: we
shall dispense with the electron gyrokinetic equation and thus
prepare the necessary ground for further approximations. The
main results are summarized in § 4.9. A reader who is only
interested in following qualitatively the major steps in the
derivation may skip to this summary.
4.1. Ordering the Terms in the Kinetic Equation
In view of Eq. (77), ae ≪ 1, so we can expand the Bessel
functions arising from averaging over the electron ring mo-
tion:
J0(ae) = 1 −
a2e + · · · ,
J1(ae)
a2e + · · ·
. (78)
Keeping only the lowest-order terms of the above expansions
in Eq. (69) for 〈χ〉Re , then substituting this 〈χ〉Re and qe =
−e in the electron gyrokinetic equation, we get the following
kinetic equation for the electrons, accurate up to and including
the first order in (me/mi)
1/2 (or in k⊥ρe):
︸ ︷︷ ︸
︸ ︷︷ ︸
v2the
︸ ︷︷ ︸
︸ ︷︷ ︸
v2the
︸ ︷︷ ︸
︸ ︷︷ ︸
. (79)
Note that ϕ, A‖, δB‖ in Eq. (79) are taken at r = Re. We
have indicated the lowest order to which each of the terms
enters if compared with v‖∂he/∂z. In order to obtain these
estimates, we have assumed that the physical ordering intro-
duced in § 3.1 holds with respect to the subsidiary expansion
in (me/mi)
1/2 as well as for the primary gyrokinetic expansion
in ǫ, so we can use Eqs. (3) and (12) to order terms with re-
spect to (me/mi)
1/2. We have also made use of Eqs. (45), (47),
and of the following three relations:
∼ vthe
, (80)
(v‖/c)A‖
∼ vtheδB⊥
, (81)
v2the
βi. (82)
The collision term is estimated to be zeroth order because [see
Eqs. (49), (50)]
k‖λmfpi
. (83)
The consequences of other possible orderings of the collision
terms are discussed in § 4.8. We remind the reader that all
dimensionless parameters except k‖/k⊥ ∼ ǫ and (me/mi)1/2
are held to be order unity.
We now let he = h
e + h
e + . . . and carry out the expansion
to two lowest orders in (me/mi)
4.2. Zeroth Order
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 17
To zeroth order, the electron kinetic equation is
v‖b̂ ·∇h(0)e = v‖
∂h(0)e
, (84)
where we have assembled the terms in the left-hand side to
take the form of the derivative of the distribution function
along the perturbed magnetic field:
b̂ ·∇ = ∂
·∇ = ∂
A‖, · · ·
. (85)
We now multiply Eq. (84) by h(0)e /F0e and integrate over v and
r (since we are only retaining lowest-order terms, the distinc-
tion between r and Re does not matter here). Since ∇·B = 0,
the left-hand side vanishes (assuming that all perturbations are
either periodic or vanish at the boundaries) and we get
h(0)e
∂h(0)e
‖e = 0.
The right-hand side of this equation is zero because the
electron flow velocity is zero in the zeroth order, u(0)
(1/n0e)
d3vv‖h
e = 0. This is a consequence of the paral-
lel Ampére’s law [Eq. (62)], which can be written as follows
u‖e =
4πen0e
∇2⊥A‖ + u‖i, (87)
where
u‖i =
eik·r
d3vv‖J0(ai)hik. (88)
The three terms in Eq. (87) can be estimated as follows
∼ ǫvthe
ǫ, (89)
∼ ǫ, (90)
c∇2⊥A‖
4πen0evA
∼ k⊥ρi
ǫ, (91)
where we have used the fundamental ordering (12) of the slow
waves (u‖i ∼ ǫvA) and Alfvén waves (δB⊥ ∼ ǫB0). Thus, the
two terms in the right-hand side of Eq. (87) are one order of
(me/mi)
1/2 smaller than u(0)
‖e , which means that to zeroth order,
the parallel Ampère’s law is u(0)
‖e = 0.
The collision operator in Eq. (86) contains electron–
electron and electron–ion collisions. To lowest order in
(me/mi)
1/2, the electron–ion collision operator is simply the
pitch-angle scattering operator [see Eq. (B20) in Appendix B
and recall that u‖i is first order]. Therefore, we may then
rewrite Eq. (86) as follows
h(0)e
Cee[h
νeiD (v)
1 − ξ2
∂h(0)e
= 0. (92)
Both terms in this expression are negative definite and must,
therefore, vanish individually. This implies that h(0)e must be
a perturbed Maxwellian distribution with zero mean veloc-
ity (this follows from the proof of Boltzmann’s H theorem;
see, e.g., Longmire 1963), i.e., the full electron distribution
function to zeroth order in the mass-ratio expansion is [see
Eq. (54)]:
fe = F0e +
+ h(0)e =
2πTe/me
, (93)
where ne = n0e + δne, Te = T0e + δTe. Expanding around the
unperturbed Maxwellian F0e, we get
h(0)e =
v2the
F0e, (94)
where the fields are taken at r = Re. Now substitute this so-
lution back into Eq. (84). The collision term vanishes and the
remaining equation must be satisfied at all values of v. This
gives
+ b̂ ·∇ϕ= b̂ ·∇T0e
, (95)
b̂ ·∇δTe
= 0. (96)
The collision term is neglected in Eq. (95) because, for h(0)e
given by Eq. (94), it vanishes to zeroth order.
4.3. Flux Conservation
Equation (95) implies that the magnetic flux is conserved
and magnetic-field lines cannot be broken to lowest order in
the mass-ratio expansion. Indeed, we may follow Cowley
(1985) and argue that the left-hand side of Eq. (95) is minus
the projection of the electric field on the total magnetic field
[see Eq. (37)], so we have
E · b̂ = −b̂ ·∇
; (97)
hence the total electric field is
Î − b̂b̂
and Faraday’s law becomes
= −c∇×E = ∇× (ueff ×B) , (99)
ueff =
E +∇T0e
×B, (100)
i.e., the magnetic field lines are frozen into the velocity field
ueff. In Appendix C.1, we show that this effective velocity is
the part of the electron flow velocity ue perpendicular to the
total magnetic field B [see Eq. (C6)].
The flux conservation is broken in the higher orders of the
mass-ratio expansion. In the first order, Ohmic resistivity for-
mally enters in Eq. (95) (unless collisions are even weaker
than assumed so far; if they are downgraded one order as is
done in § 4.8.3, resistivity enters in the second order). In the
second order, the electron inertia and the finiteness of the elec-
tron gyroradius also lead to unfreezing of the flux. This can be
seen formally by keeping second-order terms in Eq. (79), mul-
tiplying it by v‖ and integrating over velocities. The relative
importance of these flux unfreezing mechanisms is evaluated
in § 7.7.
18 SCHEKOCHIHIN ET AL.
4.4. Isothermal Electrons
Equation (96) mandates that the perturbed electron temper-
ature must remain constant along the perturbed field lines.
Strictly speaking, this does not preclude δTe varying across
the field lines. However, we shall now assume δTe = const (has
no spatial variation), which is justified, e.g., if the field lines
are stochastic. Assuming that no spatially uniform perturba-
tions exist, we may set δTe = 0. Equation (94) then reduces
h(0)e =
F0e(v), (101)
or, using Eq. (54),
δ fe =
F0e(v). (102)
Hence follows the equation of state for isothermal electrons:
δpe = T0eδne. (103)
4.5. First Order
We now integrate Eq. (79) over the velocity space and retain
the lowest (first) order terms only. Using Eq. (101), we get
A‖,u‖e
= 0, (104)
where the parallel electron velocity is first order:
u‖e = u
d3vv‖h
e . (105)
The velocity-space integral of the collision term does not enter
because it is subdominant by at least one factor of (me/mi)
indeed, as shown in Appendix B.1, the velocity integration
leads to an extra factor of k2⊥ρ
e , so that
∼ νeik2⊥ρ2e
i νii
, (106)
where we have used Eqs. (45) and (50). The collision term
is subdominant because of the ordering of the ion collision
frequency given by Eq. (49).
4.6. Field Equations
Using Eq. (101) and qi = Ze, n0e = Zn0i, T0e = T0i/τ ,
we derive from the quasi-neutrality equation (61) [see also
Eq. (65)]
eik·r
d3vJ0(ai)hik, (107)
and, from the perpendicular part of Ampère’s law [Eq. (66),
using also Eq. (107)],
eik·r
J0(ai) +
v2thi
J1(ai)
. (108)
The parallel electron velocity, u‖e, is determined from the par-
allel part of Ampère’s law, Eq. (87).
The ion distribution function hi that enters these equations
has to be determined by solving the ion gyrokinetic equation:
Eq. (57) with s = i.
4.7. Generalized Energy
The generalized energy (§ 3.4) for the case of isothermal
electrons is calculated by substituting Eq. (102) into Eq. (74):
T0iδ f
n0eT0e
|δB|2
, (109)
where δ fi = hi −
Zeϕ/T0i
F0i [see Eq. (54)].
4.8. Validity of the Mass-Ratio Expansion
Let us examine the range of spatial scales in which the
equations derived above are valid. In carrying out the ex-
pansion in (me/mi)
1/2, we ordered k⊥ρi ∼ 1 [Eq. (77)] and
k‖λmfpi ∼ 1 [Eq. (83)]. Formally, this means that the perpen-
dicular and parallel wavelengths of the perturbations must not
be so small or so large as to interfere with the mass ratio ex-
pansion. We now discuss the four conditions that this require-
ment leads to and whether any of them can be violated without
destroying the validity of the equations derived above.
4.8.1. k⊥ρi ≪ (mi/me)
This is equivalent to demanding that k⊥ρe ≪ 1, a condition
that was, indeed, essential for the expansion to hold [Eq. (78)].
This is not a serious limitation because electrons can be con-
sidered well magnetized at virtually all scales of interest for
astrophysical applications. However, we do forfeit the de-
tailed information about some important electron physics at
k⊥ρe ∼ 1: for example such effects as wave damping at the
electron gyroscale and the electron heating (although the total
amount of the electron heating can be deduced by subtracting
the ion heating from the total energy input). The breaking of
the flux conservation (resistivity) is also an effect that requires
incorporation of the finite electron gyroscale physics.
4.8.2. k⊥ρi ≫ (me/mi)
If this condition is broken, the small-k⊥ρi expansion, car-
ried out in § 5, must, formally speaking, precede the mass-
ratio expansion. However, it turns out that the small-
k⊥ρi expansion commutes with the mass-ratio expansion
(Schekochihin et al. 2007, see also footnote 23), so we
may use the equations derived in §§ 4.2-4.6 when k⊥ρi .
(me/mi)
4.8.3. k‖λmfpi ≪ (mi/me)
Let us consider what happens if this condition is broken
and k‖λmfpi & (mi/me)
1/2. In this case, the collisions be-
come even weaker and the expansion procedure must be mod-
ified. Namely, the collision term picks up one extra order of
(me/mi)
1/2, so it is first order in Eq. (79). To zeroth order,
the electron kinetic equation no longer contains collisions: in-
stead of Eq. (84), we have
v‖b̂ ·∇h(0)e = v‖
. (110)
We may seek the solution of this equation in the form h(0)e =
H(t,Re)F0e + h
e,hom, where H(t,Re) is an unknown function to
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 19
FIG. 4.— Region of validity in the wavenumber space of the secondary approximation—isothermal electrons and gyrokinetic ions (§ 4). It is the region
of validity of the gyrokinetic approximation (Fig. 3) further circumscribed by two conditions: k‖λmfpi ≫ (me/mi)
1/2 (isothermal electrons) and k⊥ρe ≪ 1
(magnetized electrons). The region of validity of the strongly magnetized two-fluid theory (Appendix A.2) is also shown. It is the same as for the full two-fluid
theory plus the additional constraint k⊥ρi ≪ k‖λmfpi. The region of validity of MHD (or one-fluid theory) is the subset of this with k‖λmfpi ≪ (me/mi)
(adiabatic electrons).
be determined and h(0)e,hom is the homogeneous solution satis-
fying
b̂ ·∇h(0)e,hom = 0, (111)
i.e., h(0)e,hom must be constant along the perturbed magnetic
field. This is a generalization of Eq. (96). Again assuming
stochastic field lines, we conclude that h(0)e,hom is independent
of space. If we rule out spatially uniform perturbations, we
may set h(0)e,hom = 0. The unknown function H(t,Re) is readily
expressed in terms of δne and ϕ:
d3vh(0)e ⇒ H =
, (112)
so h(0)e is again given by Eq. (101), so the equations derived
in §§ 4.2-4.6 are unaltered. Thus, the mass-ratio expansion
remains valid at k‖λmfpi & (mi/me)
4.8.4. k‖λmfpi ≫ (me/mi)
If the parallel wavelength of the fluctuations is so long that
this is violated, k‖λmfpi . (me/mi)
1/2, the collision term in
Eq. (79) is minus first order. This is the lowest-order term in
the equation. Setting it to zero obliges h(0)e to be a perturbed
Maxwellian again given by Eq. (94). Instead of Eq. (84), the
zeroth-order kinetic equation is
v‖b̂ ·∇h(0)e = v‖
∂h(1)e
. (113)
Now the collision term in this order contains h(1)e , which
can be determined from Eq. (113) by inverting the colli-
sion operator. This sets up a perturbation theory that in due
course leads to the Reduced MHD version of the general
MHD equations—this is what was considered in § 2. Equa-
tion (96) no longer needs to hold, so the electrons are not
isothermal. In this true one-fluid limit, both electrons and
ions are adiabatic with equal temperatures [see Eq. (115) be-
low]. The collisional transport terms in this limit (parallel
and perpendicular resistivity, viscosity, heat fluxes, etc.) were
calculated [starting not from gyrokinetics but from the gen-
eral Vlasov–Landau equation (36)] in exhaustive detail by
Braginskii (1965). His results and the way RMHD emerges
from them are reviewed in Appendix A.
In physical terms, the electrons can no longer be isothermal
if the parallel electron diffusion time becomes longer than the
characteristic time of the fluctuations (the Alfvén time):
vtheλmfpik2‖
⇔ k‖λmfpi .
. (114)
Furthermore, under a similar condition, electron and ion tem-
peratures must equalize: this happens if the ion–electron col-
lision time is shorter than the Alfvén time,
⇔ k‖λmfpi .
(115)
(see Lithwick & Goldreich 2001 for a discussion of these con-
ditions in application to the ISM).
4.9. Summary
The original gyrokinetic description introduced in § 3 was
a system of two kinetic equations [Eq. (57)] that evolved the
electron and ion distribution functions he, hi and three field
20 SCHEKOCHIHIN ET AL.
equations [Eqs. (61-63)] that related ϕ, A‖ and δB‖ to he and
hi. In this section, we have taken advantage of the smallness
of the electron mass to treat the electrons as an isothermal
magnetized fluid, while ions remained fully gyrokinetic.
In mathematical terms, we solved the electron kinetic equa-
tion and replaced the gyrokinetics with a simpler closed sys-
tem of equations that evolve 6 unknown functions: ϕ, A‖, δB‖,
δne, u‖e and hi. These satisfy two fluid-like evolution equa-
tions (95) and (104), three integral relations (107), (108), and
(87) which involve hi, and the kinetic equation (57) for hi.
The system is simpler because the full electron distribution
function has been replaced by two scalar fields δne and u‖e.
We now summarize this new system of equations: denoting
ai = k⊥v⊥/Ωi, we have
+ b̂ ·∇ϕ= b̂ ·∇T0e
, (116)
+ b̂ ·∇u‖e = −
,(117)
eik·r
d3vJ0(ai)hik, (118)
u‖e =
4πen0e
∇2⊥A‖ +
eik·r
d3vv‖J0(ai)hik, (119)
eik·r
J0(ai) +
v2thi
J1(ai)
, (120)
and Eq. (57) for s = i and ion–ion collisions only:
{〈χ〉Ri ,hi} =
∂〈χ〉Ri
F0i + 〈Cii[hi]〉Ri ,
(121)
where 〈Cii[. . .]〉Ri is the gyrokinetic ion–ion collision oper-
ator (see Appendix B) and the ion–electron collisions have
been neglected to lowest order in (me/mi)
1/2 [see Eq. (51)].
Note that Eqs. (116-117) have been written in a compact form,
where
+ uE ·∇ =
{ϕ, · · ·} (122)
is the convective derivative with respect to the E×B drift ve-
locity, uE = −c∇⊥ϕ× ẑ/B0, and
b̂ ·∇ = ∂
·∇ = ∂
A‖, · · ·
(123)
is the gradient along the total magnetic field (mean field plus
perturbation).
The generalized energy conserved by Eqs. (116-121) is
given by Eq. (109).
It is worth observing that the left-hand side of Eq. (116) is
simply minus the component of the electric field along the to-
tal magnetic field [see Eq. (37)]. This was used in § 4.3 to
prove that the magnetic flux described by Eq. (116) is exactly
conserved (see § 7.7 for a discussion of scales at which this
conservation is broken). Equation (116) is the projection of
the generalized Ohm’s law onto the total magnetic field—the
right-hand side of this equation is the so-called thermoelec-
tric term. This is discussed in more detail in Appendix C.1,
where we also show that Eq. (117) is the parallel part of Fara-
day’s law and give a qualitative non-gyrokinetic derivation of
Eqs. (116-117).
We will refer to Eqs. (116-121) as the equations of isother-
mal electron fluid. They are valid in a broad range of scales:
the only constraints are that k‖ ≪ k⊥ (gyrokinetic order-
ing, § 3.1), k⊥ρe ≪ 1 (electrons are magnetized, § 4.8.1) and
k‖λmfpi ≫ (me/mi)1/2 (electrons are isothermal, § 4.8.4). The
region of validity of Eqs. (116-121) in the wavenumber space
is illustrated in Fig. 4. A particular advantage of this hybrid
fluid-kinetic system is that it is uniformly valid across the
transition from magnetized to unmagnetized ions (i.e., from
k⊥ρi ≪ 1 to k⊥ρi ≫ 1).
5. TURBULENCE IN THE INERTIAL RANGE: KINETIC RMHD
Our goal in this section is to derive a reduced set of equa-
tions that describe the magnetized plasma in the limit of small
k⊥ρi. Before we proceed with an expansion in k⊥ρi, we need
to make a formal technical step, the usefulness of which will
become clear shortly. A reader with no patience for this or
any of the subsequent technical developments may skip to the
summary at the end of this section (§ 5.7).
5.1. A Technical Step
Let us formally split the ion gyrocenter distribution function
into two parts:
v⊥ ·A⊥
F0i + g
eik·Ri
J0(ai)
v2thi
J1(ai)
F0i + g. (124)
Then g satisfies the following equation, obtained by substitut-
ing Eq. (124) and the expression for ∂A‖/∂t that follows from
Eq. (116) into the ion gyrokinetic equation (121):
{〈χ〉Ri ,g}− 〈Cii[g]〉Ri
︸ ︷︷ ︸
A‖,ϕ− 〈ϕ〉Ri
︸ ︷︷ ︸
+ b̂ ·∇
v⊥ ·A⊥
︸ ︷︷ ︸
v⊥ ·A⊥
︸ ︷︷ ︸
. (125)
In the above equation, we have used compact notation in
writing out the nonlinear terms: e.g.,
A‖,ϕ− 〈ϕ〉Ri
A‖(r),ϕ(r)
〈A‖〉Ri ,〈ϕ〉Ri
, where the first Poisson
bracket involves derivatives with respect to r and the second
with respect to Ri.
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 21
The field equations (118-120) rewritten in terms of g are
−Γ1(αi)
︸ ︷︷ ︸
1 −Γ0(αi)
]Zeϕk
︸ ︷︷ ︸
d3vJ0(ai)gk
︸ ︷︷ ︸
, (126)
4πen0e
k2⊥A‖k
︸ ︷︷ ︸
d3vv‖J0(ai)gk
︸ ︷︷ ︸
= u‖ki, (127)
︸ ︷︷ ︸
Γ2(αi) +
︸ ︷︷ ︸
1 −Γ1(αi)
]Zeϕk
︸ ︷︷ ︸
v2thi
J1(ai)
︸ ︷︷ ︸
, (128)
where ai = k⊥v⊥/Ωi, αi = k
i /2 and we have defined
Γ0(αi) =
d3v [J0(ai)]
2 F0i
= I0(αi)e
−αi = 1 −αi + · · · , (129)
Γ1(αi) =
v2thi
J0(ai)
J1(ai)
F0i = −Γ′0(αi)
= [I0(αi) − I1(αi)] e−αi = 1 −
αi + · · · , (130)
Γ2(αi) =
v2thi
J1(ai)
F0i = 2Γ1(αi). (131)
Underneath each term in Eqs. (125-128), we have indicated
the lowest order in k⊥ρi to which this term enters.
5.2. Subsidiary Ordering in k⊥ρi
In order to carry out a subsidiary expansion in small k⊥ρi,
we must order all terms in Eqs. (95-104) and (125-128) with
respect to k⊥ρi. Let us again assume, like we did when ex-
panding the electron equation (§ 4), that the ordering intro-
duced for the gyrokinetics in § 3.1 holds also for the sub-
sidiary expansion in k⊥ρi. First note that, in view of Eq. (47),
we must regard Zeϕ/T0i to be minus first order:
. (132)
Also, as δB⊥/B0 ∼ ǫ [Eq. (12)],
(v‖/c)A‖
∼ vthiδB⊥
βi, (133)
so ϕ and (v‖/c)A‖ are same order.
Since u‖ = u‖i (electrons do not contribute to the mass flow),
assuming that slow waves and Alfvén waves have comparable
energies implies u‖i ∼ u⊥. As u‖i is determined by the second
equality in Eq. (127), we can order g [using Eq. (12)]:
, (134)
so g is zeroth order in k⊥ρi. Similarly, δne/n0e ∼ δB‖/B0 ∼ ǫ
are zeroth order in k⊥ρi—this follows directly from Eq. (12).
Together with Eq. (3), the above considerations allow us to
order all terms in our equations. The ordering of the collision
term involving ϕ is explained in Appendix B.2.
5.3. Alfvén Waves: Kinetic Derivation of RMHD
We shall now show that the RMHD equations (17-18) hold
in this approximation. There is a simple correspondence be-
tween the stream and flux functions defined in Eq. (16) and
the electromagnetic potentials ϕ and A‖:
ϕ, Ψ = −
4πmin0i
. (135)
The first of these definitions says that the perpendicular flow
velocity u⊥ is the E×B drift velocity; the second definition
is the standard MHD relation between the magnetic flux func-
tion and the parallel component of the vector potential.
5.3.1. Derivation of Eq. (17)
Deriving Eq. (17) is straightforward: in Eq. (95), we retain
only the lowest—minus first—order terms (those that contain
ϕ and A‖). The result is
= 0. (136)
Using Eq. (135) and the definition of the Alfvén speed, vA =
4πmin0i, we get Eq. (17). By the argument of § 4.3,
Eq. (136) expresses the fact that that magnetic-field lines are
frozen into the E×B velocity field, which is the mean flow
velocity associated with the Alfvén waves (see § 5.4).
5.3.2. Derivation of Eq. (18)
As we are about to see, in order to derive Eq. (18), we have
to separate the first-order part of the k⊥ρi expansion. The
easiest way to achieve this, is to integrate Eq. (125) over the
velocity space (keeping r constant) and expand the resulting
equation in small k⊥ρi. Using Eqs. (126) and (127) to express
the velocity-space integrals of g, we get
1 −Γ0(αi)
]Zeϕk
︸ ︷︷ ︸
−Γ1(αi)
︸ ︷︷ ︸
4πen0e
k2⊥A‖k
︸ ︷︷ ︸
d3vJ0(ai){〈χ〉Ri ,g}k
︸ ︷︷ ︸
d3vJ0(ai)
v⊥ ·A⊥
︸ ︷︷ ︸
22 SCHEKOCHIHIN ET AL.
. (137)
Underneath each term, the lowest order in k⊥ρi to which it
enters is shown. We see that terms containing ϕ are all first
order, so it is up to this order that we shall retain terms. The
collision term integrated over the velocity space picks up two
extra orders of k⊥ρi (see Appendix B.1), so it is second order
and can, therefore, be dropped. As a consequence of quasi-
neutrality, the zeroth-order part of the above equation exactly
coincides with Eq. (104), i.e, δni/n0i = δne/n0e satisfy the
same equation. Indeed, neglecting second-order terms (but
not first-order ones!), the nonlinear term in Eq. (137) (the last
term on the left-hand side) is
d3vv‖g
v2thi
, (138)
and, using Eqs. (126-128) to express velocity-space integrals
of g in the above expression, we find that the zeroth-order part
of the nonlinearity is the same as the nonlinearity in Eq. (104),
while the first-order part is
ρ2i ∇2⊥
4πen0e
∇2⊥A‖
, (139)
where we have used the expansion (129) of Γ0(αi) and con-
verted it back into x space.
Thus, if we subtract Eq. (104) from Eq. (137), the remain-
der is first order and reads
ρ2i ∇2⊥
ρ2i ∇2⊥
4πen0e
∇2⊥A‖ −
4πen0e
∇2⊥A‖
= 0. (140)
Multiplying Eq. (140) by 2T0i/Zeρ
i and using Eq. (135), we
get the second RMHD equation (18).
We have established that the Alfvén-wave component of the
turbulence is decoupled and fully described by the RMHD
equations (17) and (18). This result is the same as that in
§ 2.2 but now we have proven that collisions do not affect the
Alfvén waves and that a fluid-like description only requires
k⊥ρi ≪ 1 to be valid.
5.4. Why Alfvén Waves Ignore Collisions
Let us write explicitly the distribution function of the ion
gyrocenters [Eq. (124)] to two lowest orders in k⊥ρi:
〈ϕ〉Ri F0i +
v2thi
F0i + g + · · · , (141)
where, up to corrections of order k2⊥ρ
i , the ring-averaged
scalar potential is 〈ϕ〉Ri = ϕ(Ri), the scalar potential taken at
the position of the ion gyrocenter. Note that in Eq. (141), the
first term is minus first order in k⊥ρi [see Eq. (132)], the sec-
ond and third terms are zeroth order [Eq. (134)], and all terms
of first and higher orders are omitted. In order to compute the
full ion distribution function given by Eq. (54), we have to
convert hi to the r space. Keeping terms up to zeroth order,
we get
〈ϕ〉Ri ≃
ϕ(Ri) =
ϕ(r) +
v⊥× ẑ
·∇ϕ(r) + · · ·
ϕ(r) +
2v⊥ ·uE
v2thi
+ . . . , (142)
where uE = −c∇ϕ(r)× ẑ/B0, the E×B drift velocity. Sub-
stituting Eq. (142) into Eq. (141) and then Eq. (141) into
Eq. (54), we find
fi = F0i +
2v⊥ ·uE
v2thi
F0i +
v2thi
F0i + g + · · · . (143)
The first two terms can be combined into a Maxwellian
with mean perpendicular flow velocity u⊥ = uE . These are
the terms responsible for the Alfvén waves. The remaining
terms, which we shall denote δ f̃i, are the perturbation of the
Maxwellian in the moving frame of the Alfvén waves—they
describe the passive (compressive) component of the turbu-
lence (see § 5.5). Thus, the ion distribution function is
(πv2thi)
(v⊥ − uE)2 + v2‖
+ δ f̃i. (144)
This sheds some light on the indifference of Alfvén waves
to collisions: Alfvénic perturbations do not change the
Maxwellian character of the ion distribution. Unlike in a neu-
tral fluid or gas, where viscosity arises when particles trans-
port the local mean momentum a distance ∼ λmfpi, the parti-
cles in a magnetized plasma instantaneously take on the lo-
cal E×B velocity (they take a cyclotron period to adjust, so,
roughly speaking, ρi plays the role of the mean free path).
Thus, there is no memory of the mean perpendicular motion
and, therefore, no perpendicular momentum transport.
Some readers may find it illuminating to notice that
Eq. (140) can be interpreted as stating simply ∇·j = 0: the first
two terms represent the divergence of the polarization current,
which is perpendicular to the magnetic field;22 the last two
terms are b̂ ·∇ j‖. No contribution to the current arises from
the collisional term in Eq. (137) as ion–ion collisions cause
no particle transport to lowest order in k⊥ρi.
5.5. Compressive Fluctuations
The equations that describe the density (δne) and magnetic-
field-strength (δB‖) fluctuations follow immediately from
Eqs. (125-128) if only zeroth-order terms are kept. In these
equations, terms that involve ϕ and A‖ also contain factors
∼ k2⊥ρ2i and are, therefore, first-order [with the exception of
the nonlinearity on the left-hand side of Eq. (125)]. The fact
that 〈Cii[〈ϕ〉Ri F0i]〉Ri in Eq. (125) is first order is proved in Ap-
pendix B.2. Dropping these terms along with all other contri-
butions of order higher than zeroth and making use of Eq. (69)
22 The polarization-drift velocity is formally higher order than uE in the
gyrokinetic expansion. However, since uE does not produce any current,
the lowest-order contribution to the perpendicular current comes from the
polarization drift. The higher-order contributions to the gyrocenter distri-
bution function did not need to be calculated explicitly because the informa-
tion about the polarization charge is effectively carried by the quasi-neutrality
condition (61). We do not belabor this point because, in our approach, the no-
tion of polarization charge is only ever brought in for interpretative purposes,
but is not needed to carry out calculations. For further qualitative discussion
of the role of the polarization charge and polarization drift in gyrokinetics,
we refer the reader to Krommes 2006 and references therein.
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 23
to write out 〈χ〉Ri , we find that Eq. (125) takes the form
+ v‖ b̂ ·∇
v2thi
v2thi
, (145)
where we have used definitions (122-123) of the convective
time derivative d/dt and the total gradient along the magnetic
field b̂ · ∇ to write our equation in a compact form. Note
that, in view of the correspondence between Φ, Ψ and ϕ, A‖
[Eq. (135)], these nonlinear derivatives are the same as those
defined in Eqs. (19-20). The collision term in the right-hand
side of the above equation is the zeroth-order limit of the gy-
rokinetic ion–ion collision operator: a useful model form of it
is given in Appendix B.3 [Eq. (B18)].
To zeroth order, Eqs. (126-128) are
d3vg, (146)
d3vv‖g, (147)
v2thi
g. (148)
Note that u‖ is not an independent quantity—it can be com-
puted from the ion distribution but is not needed for the deter-
mination of the latter.
Equations (145-148) evolve the ion distribution function
g, the “slow-wave quantities” u‖, δB‖, and the density fluc-
tuations δne. The nonlinearities in Eq. (145), contained in
d/dt and b̂ ·∇, involve the Alfvén-wave quantities Φ and Ψ
(or, equivalently, ϕ and A‖) determined separately and inde-
pendently by the RMHD equations (17-18). The situation
is qualitatively similar to that in MHD (§ 2.4), except now
a kinetic description is necessary—Eqs. (145-148) replace
Eqs. (25-27)—and the nonlinear scattering/mixing of the slow
waves and the entropy mode by the Alfvén waves takes the
form of passive advection of the distribution function g. The
density and magnetic-field-strength fluctuations are velocity-
space moments of g.
Another way to understand the passive nature of the com-
pressive component of the turbulence discussed above is to
think of it as the perturbation of a local Maxwellian equilib-
rium associated with the Alfvén waves. Indeed, in § 5.4, we
split the full ion distribution function [Eq. (144)] into such a
local Maxwellian and its perturbation
δ f̃i = g +
v2thi
F0i. (149)
It is this perturbation that contains all the information about
the compressive component; the second term in the above ex-
pression enforces to lowest order the conservation of the first
adiabatic invariant µi = miv
⊥/2B. In terms of the function
(149), Eqs. (145-148) take a somewhat more compact form
(cf. Schekochihin et al. 2007):
δ f̃i −
v2thi
+ v‖b̂ ·∇
δ f̃i +
δ f̃i
, (150)
FIG. 5.— Channels of the kinetic cascade of generalized energy (§ 3.4)
from large to small scales: see § 2.7 and Appendix D.2 (inertial range,
collisional regime), § 5.6 and § 6.2.5 (inertial range, collisionless regime),
§ 7.8 and § 7.12 (dissipation range). Note that some ion heating probably
also results from the collisional and collisionless damping of the compressive
fluctuations in the inertial range (see § 6.1.2 and § 6.2.4).
d3vδ f̃i, (151)
v2thi
δ f̃i. (152)
5.6. Generalized Energy: Three KRMHD Cascades
The generalized energy (§ 3.4) in the limit k⊥ρi ≪ 1 is cal-
culated by substituting into Eq. (109) the perturbed ion dis-
tribution function δ fi = 2v⊥ · uEF0i/v2thi + δ f̃i [see Eqs. (143)
and (149)]. After performing velocity integration, we get
min0iu
n0iT0i
δ f̃ 2i
=WAW +Wcompr. (153)
We see that the kinetic energy of the Alfvénic fluctuations
has emerged from the ion-entropy part of the generalized en-
ergy. The first two terms in Eq. (153) are the total (kinetic
plus magnetic) energy of the Alfvén waves, denoted WAW. As
we learned from § 5.3, it cascades independently of the rest of
the generalized energy, Wcompr, which contains the compres-
sive component of the turbulence (§ 5.5) and is the invariant
conserved by Eqs. (150-152).
In terms of the potentials used in our discussion of RMHD
in § 2, we have
WAW =
min0i
|∇⊥Φ|2 + |∇⊥Ψ|2
min0i
|∇⊥ζ+|2 + |∇⊥ζ−|2
=W +AW +W
AW (154)
whereW +AW and W
AW are the energies of the “+” and “−” waves
[Eq. (33)], which, as we know from § 2.3, cascade by scatter-
ing off each other but without exchanging energy.
Thus, the kinetic cascade in the limit k⊥ρi ≪ 1 is split, in-
dependently of the collisionality, into three cascades: of W +AW,
24 SCHEKOCHIHIN ET AL.
W −AW and Wcompr. The compressive cascade is, in fact, split
into three independent cascades—the splitting is different in
the collisional limit (Appendix D.2) and in the collisionless
one (§ 6.2.5). Figure 5 schematically summarizes both the
splitting of the kinetic cascade that we have worked out so far
and the upcoming developments.
5.7. Summary
In § 4, gyrokinetics was reduced to a hybrid fluid-kinetic
system by means of an expansion in the electron mass, which
was valid for k⊥ρe ≪ 1. In this section, we have further re-
stricted the scale range by taking k⊥ρi ≪ 1 and as a result have
been able to achieve a further reduction in the complexity of
the kinetic theory describing the turbulent cascades. The re-
duced theory derived here evolves 5 unknown functions: Φ,
Ψ, δB‖, δne and g. The stream and flux functions, Φ and Ψ
are related to the fluid quantities (perpendicular velocity and
magnetic field perturbations) via Eq. (16) and to the electro-
magnetic potentials ϕ, A‖ via Eq. (135). They satisfy a closed
system of equations, Eqs. (17-18), which describe the decou-
pled cascade of Alfvén waves. These are the same equations
that arise from the MHD approximations, but we have now
proven that their validity does not depend on the assumption
of high collisionality (the fluid limit) and extends to scales
well below the mean free path, but above the ion gyroscale.
The physical reasons for this are explained in § 5.4. The den-
sity and magnetic-field-strength fluctuations (the “compres-
sive” fluctuations, or the slow waves and the entropy mode in
the MHD limit) now require a kinetic description in terms of
the ion distribution function g [or δ f̃i, Eq. (149)], evolved by
the kinetic equation (145) [or Eq. (150)]. The kinetic equation
contains δne and δB‖, which are, in turn calculated in terms
of the velocity-space integrals of g via Eqs. (146) and (148)
[or Eqs. (151) and (152)]. The nonlinear evolution (turbulent
cascade) of g, δB‖ and δne is due solely to passive advection
of g by the Alfvén-wave turbulence.
Let us summarize the new set of equations:
= vAb̂ ·∇Φ, (155)
∇2⊥Φ= vAb̂ ·∇∇2⊥Ψ, (156)
+ v‖ b̂ ·∇
v2thi
v2thi
, (157)
v2thi
(158)
v2thi
g, (159)
where
+{Φ, · · ·} , b̂ ·∇ = ∂
{Ψ, · · ·} . (160)
An explicit form of the collision term in the right-hand side of
Eq. (157) is provided in Appendix B.3 [Eq. (B18)].
The generalized energy conserved by Eqs. (155-159) is
given by Eq. (153). The kinetic cascade is split, the Alfvénic
cascade proceeding independently of the compressive one
(see Fig. 5).
The decoupling of the Alfvénic cascade is manifested by
Eqs. (155-156) forming a closed subset. As already noted in
§ 4.9, Eq. (155) is the component of Ohm’s law along the total
magnetic field, B ·E = 0. Equation (156) can be interpreted as
the evolution equation for the vorticity of the perpendicular
plasma flow velocity, which is the E×B drift velocity.
We shall refer to the system of equations (155-159) as Ki-
netic Reduced Magnetohydrodynamics (KRMHD).23 It is a
hybrid fluid-kinetic description of low-frequency turbulence
in strongly magnetized weakly collisional plasma that is uni-
formly valid at all scales satisfying k⊥ρi ≪ min(1,k‖λmfpi)
(ions are strongly magnetized)24 and k‖λmfpi ≫ (me/mi)1/2
(electrons are isothermal), as illustrated in Fig. 2. Therefore,
it smoothly connects the collisional and collisionless regimes
and is the appropriate theory for the study of the turbulent cas-
cades in the inertial range. The KRMHD equations generalize
rather straightforwardly to plasmas that are so collisionless
that one cannot assume a Maxwellian equilibrium distribu-
tion function (Chen et al. 2009)—a situation that is relevant
in some of the solar-wind measurements (see further discus-
sion in § 8.3).
KRMHD describe what happens to the turbulent cascade at
or below the ion gyroscale—we shall move on to these scales
in § 7, but first we would like to discuss the turbulent cascades
of density and magnetic-field-strength fluctuations and their
damping by collisional and collisionless mechanisms.
6. COMPRESSIVE FLUCTUATIONS IN THE INERTIAL RANGE
Here we first derive the nonlinear equations that govern
the evolution of the compressive (density and magnetic-field-
strength) fluctuations in the collisional (k‖λmfpi ≪ 1, § 6.1 and
Appendix D) and collisionless (k‖λmfpi ≫ 1, § 6.2) limits, dis-
cuss the linear damping that these fluctuations undergo in the
two limits and work out the form the generalized energy takes
for compressive fluctuations (which is particularly interesting
in the collisionless limit, §§ 6.2.3-6.2.5). As in previous sec-
tions, an impatient reader may skip to § 6.3 where the results
of the previous two subsections are summarized and the im-
plications for the structure of the turbulent cascades of the
density and field-strength fluctuations are discussed.
6.1. Collisional Regime
6.1.1. Equations
In the collisional regime, k‖λmfpi ≪ 1, the fluid limit is re-
covered by expanding Eqs. (155-159) in small k‖λmfpi. The
calculation that is necessary to achieve this is done in Ap-
pendix D (see also Appendix A.4). The result is a closed set
23 The term is introduced by analogy with a popular fluid-kinetic system
known as Kinetic MHD, or KMHD (see Kulsrud 1964, 1983). KMHD is de-
rived for magnetized plasmas (ρi ≪ λmfpi) under the assumption that kρs ≪ 1
and ω ≪ Ωs but without assuming either strong anisotropy (k‖ ≪ k⊥) or
small fluctuations (|δB| ≪ B0). The KRMHD equations (155-159) can be
recovered from KMHD by applying to it the GK-RMHD ordering [Eq. (12)
and § 3.1] and an expansion in (me/mi)1/2 (Schekochihin et al. 2007). This
means that the k⊥ρi expansion (§ 5), which for KMHD is the primary ex-
pansion, commutes with the gyrokinetic expansion (§ 3) and the (me/mi)1/2
expansion (§ 4), both of which preceded it in this paper.
24 The condition k⊥ρi ≪ k‖λmfpi must be satisfied because in our esti-
mates of the collision terms (Appendix B.2) we took k⊥ρi ≪ 1 while assum-
ing that k‖λmfpi ∼ 1.
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 25
of three fluid equations that evolve δB‖, δne and u‖:
= b̂ ·∇u‖ +
, (161)
= v2Ab̂ ·∇
+ ν‖ib̂ ·∇
b̂ ·∇u‖
, (162)
+κ‖ib̂ ·∇
b̂ ·∇δTi
, (163)
where
ν‖ib̂ ·∇u‖
, (164)
and ν‖i and κ‖i are the coefficients of parallel ion viscosity
and thermal diffusivity, respectively. The viscous and ther-
mal diffusion are anisotropic because plasma is magnetized,
λmfpi ≫ ρi (Braginskii 1965). The method of calculation of
ν‖i and κ‖i is explained in Appendix D.3. Here we shall ig-
nore numerical prefactors of order unity and give order-of-
magnitude values for these coefficients:
ν‖i ∼ κ‖i ∼
v2thi
∼ vthiλmfpi. (165)
If we set ν‖i = κ‖i = 0, Eqs. (161-164) are the same as the
RMHD equations of § 2 with the sound speed defined as
cs = vA
. (166)
This is the natural definition of cs for the case of adiabatic
ions, whose specific heat ratio is γi = 5/3, and isothermal elec-
trons, whose specific heat ratio is γe = 1 [because δpe = T0eδne;
see Eq. (103)]. Note that Eq. (164) is equivalent to the
pressure balance [Eq. (22) of § 2] with p = niTi + neTe and
δpe = T0eδne.
As in § 2, the fluctuations described by Eqs. (161-164) sep-
arate into the zero-frequency entropy mode and the left- and
right-propagating slow waves with
ω = ±
1 + v2A/c2s
(167)
[see Eq. (30)]. All three are cascaded independently of each
other via nonlinear interaction with the Alfvén waves. In Ap-
pendix D.2, we show that the generalized energy Wcompr for
this system, given in § 5.6, splits into the three familiar invari-
ants W +sw, W
sw, and Ws, defined by Eqs. (34-35) (see Fig. 5).
6.1.2. Dissipation
The diffusion terms add dissipation to the equations. Be-
cause diffusion occurs along the field lines of the total mag-
netic field (mean field plus perturbation), the diffusive terms
are nonlinear and the dissipation process also involves interac-
tion with the Alfvén waves. We can estimate the characteristic
parallel scale at which the diffusion terms become important
by balancing the nonlinear cascade time and the typical diffu-
sion time:
k‖vA ∼ vthiλmfpik2‖ ⇔ k‖λmfpi ∼ 1/
βi, (168)
where we have used Eq. (165).
Technically speaking, the cutoff given by Eq. (168) always
lies in the range of k‖ that is outside the region of validity
of the small-k‖λmfpi expansion adopted in the derivation of
Eqs. (161-163). In fact, in the low-beta limit, the collisional
cutoff falls manifestly in the collisionless scale range, i.e.,
the collisional (fluid) approximation breaks down before the
slow-wave and entropy cascades are damped and one must use
the collisionless (kinetic) limit to calculate the damping (see
§ 6.2.2). The situation is different in the high-beta limit: in
this case, the expansion in small k‖λmfpi can be reformulated
as an expansion in small 1/
βi and the cutoff falls within the
range of validity of the fluid approximation. Equations (161-
163) in this limit are
= b̂ ·∇u‖, (169)
= v2Ab̂ ·∇
+ ν‖ib̂ ·∇
b̂ ·∇u‖
, (170)
1 + Z/τ
5/3 + Z/τ
κ‖ib̂ ·∇
b̂ ·∇δne
. (171)
As in § 2 [Eq. (28)], the density fluctuations [Eq. (171)] have
decoupled from the slow waves [Eqs. (169-170)]. The former
are damped by thermal diffusion, the latter by viscosity. The
corresponding linear dispersion relations are
ω = −i
1 + Z/τ
5/3 + Z/τ
‖, (172)
ω =±k‖vA
ν‖ik‖
. (173)
Equation (172) describes strong diffusive damping of the den-
sity fluctuations. The slow-wave dispersion relation (173) has
two distinct regimes:
1. When k‖ < 2vA/ν‖i, it describes viscously damped slow
waves. In particular, in the limit k‖λmfpi ≪ 1/
βi, we
ω ≃±k‖vA − i
. (174)
2. For k‖ > 2vA/ν‖i, both solutions become purely imag-
inary, so the slow waves are converted into aperiodic
decaying fluctuations. The stronger-damped (diffusive)
branch has ω ≃ −iν‖ik2‖, the weaker-damped one has
ω ≃ −i v
∼ − i
λmfpi
∼ − i√
λmfpi
. (175)
This damping effect is called viscous relaxation. It is
valid until k‖λmfpi ∼ 1, where it is replaced by the col-
lisionless damping discussed in § 6.2.2 [see Eq. (190)].
The viscous and thermal-diffusive dissipation mechanisms
described above lead, in the limits where they are efficient, to
ion heating via the standard fluid (collisional) route, involving
the development of small parallel scales in the position space,
but not in velocity space (see § 3.4 and § 3.5).
6.2. Collisionless Regime
6.2.1. Equations
In the collisionless regime, k‖λmfpi ≫ 1, the collision inte-
gral in the right-hand side of the kinetic equation (157) can be
26 SCHEKOCHIHIN ET AL.
neglected. The v⊥ dependence can then be integrated out of
Eq. (157). Indeed, let us introduce the following two auxiliary
functions:
Gn(v‖) = −
dv⊥ v⊥
v2thi
g, (176)
GB(v‖) = −
dv⊥ v⊥
v2thi
g. (177)
In terms of these functions,
dv‖Gn,
dv‖GB (178)
and Eq. (157) reduces to the following two coupled one-
dimensional kinetic equations
+ v‖b̂ ·∇Gn = −
v‖FM(v‖)
×b̂ ·∇
, (179)
+ v‖b̂ ·∇GB =
v‖FM(v‖)
×b̂ ·∇
, (180)
where FM(v‖) = (1/
πvthi)exp(−v2‖/v
thi) is a one-dimensional
Maxwellian. This system can be diagonalized, so it splits into
two decoupled equations
+v‖b̂ ·∇G± =
v‖FM(v‖)
b̂ ·∇
dv′‖ G
±(v′‖), (181)
where
± = −
(182)
and we have introduced a new pair of functions
G+ = GB +
Gn, G
− = Gn +
GB, (183)
where
σ = 1 +
. (184)
Equation (181) describes two decoupled kinetic cascades,
which we will discuss in greater detail in §§ 6.2.3-6.2.5.
6.2.2. Collisionless Damping
Fluctuations described by Eq. (181) are subject to collision-
less damping. Indeed, let us linearize Eq. (181), Fourier trans-
form in time and space, divide through by −i(ω − k‖v‖), and
integrate over v‖. This gives the following dispersion relation
(the “−” branch is for G−, the “+” branch for G+)
ζiZ (ζi) = Λ
± − 1, (185)
FIG. 6.— Schematic log-log plot (artist’s impression) of the ratio of the
damping rate of magnetic-field-strength fluctuations to the Alfvén frequency
k‖vA in the high-beta limit [see Eqs. (173) and (190)]. In Barnes et al. (2009),
this plot is reproduced via a direct numerical solution of the linearized ion
gyrokinetic equation with collisions.
where ζi = ω/|k‖|vthi = ω/|k‖|vA
βi and we have used the
plasma dispersion function (Fried & Conte 1961)
Z (ζi) =
x − ζi
(186)
(the integration is along the Landau contour). This function is
not to be confused with the ion charge parameter Z = qi/e.
Formally, Eq. (185) has an infinite number of solutions.
When βi ∼ 1, they are all strongly damped with damping rates
Im(ω) ∼ |k‖|vthi ∼ |k‖|vA, so the damping time is compara-
ble to the characteristic timescale on which the Alfvén waves
cause these fluctuations to cascade to smaller scales.
It is interesting to consider the high- and low-beta limits.
High-Beta Limit. — When βi ≫ 1, we have in Eq. (185)
− − 1≃−2
, G− ≃ Gn, (187)
+ − 1≃
, G+ ≃ GB +
Gn. (188)
The “−” branch corresponds to the density fluctuations. The
solution of Eq. (185) has Im(ζi) ∼ 1, so these fluctuations are
strongly damped:
ω ∼ −i|k‖|vA
βi. (189)
The damping rate is much greater than the Alfvénic rate k‖vA
of the nonlinear cascade. In contrast, for the “+” branch, the
damping rate is small: it can be obtained by expanding Z(ζi) =
π + O (ζi), which gives25
ω = −i
|k‖|vthi√
|k‖|vA√
. (190)
Since Gn is strongly damped, Eq. (188) implies G
+ ≃ GB, i.e.,
the fluctuations that are damped at the rate (190) are predom-
inantly of the magnetic-field strength. The damping rate is a
25 This is the gyrokinetic limit (k‖/k⊥ ≪ 1) of the more general damping
effect known in astrophysics as the Barnes (1966) damping and in plasma
physics as transit-time damping. We remind the reader that our approach was
to carry out the gyrokinetic expansion (in small k‖/k⊥) first, and then take
the high-beta limit as a subsidiary expansion. A more standard approach in
the linear theory of plasma waves is to take the limit of high βi while treating
k‖/k⊥ as an arbitrary quantity. A detailed calculation of the damping rates
done in this way can be found in Foote & Kulsrud (1979).
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 27
constant (independent of k‖) small fraction ∼ 1/
βi of the
Alfvénic cascade rate.
In Fig. 6, we give a schematic plot of the damping rate of the
magnetic-field-strength fluctuations (slow waves) connecting
the fluid and kinetic limits for βi ≫ 1.
Low-Beta Limit. — When βi ≪ 1, we have
− − 1≃−
, G− ≃ Gn +
GB, (191)
+ − 1≃ 2
, G+ ≃ GB. (192)
For the “−” branch, we again have Im(ζi) ∼ 1, so
ω ∼ −i|k‖|vA
βi, (193)
which now is much smaller than the Alfvénic cascade rate
k‖vA. For the “+” branch (predominantly the field-strength
fluctuations), we seek a solution with ζ = −iζ̃i and ζ̃i ≫ 1.
Then Eq. (185) becomes ζiZ(ζi) ≃ 2
π ζ̃i exp(ζ̃i) = 2/βi. Up
to logarithmically small corrections, this gives ζ̃i ≃
| lnβi|,
whence
ω ∼ −i|k‖|vA
βi| lnβi|. (194)
While this damping rate is slightly greater than that of the “−”
branch, it is still much smaller than the Alfvénic cascade rate.
6.2.3. Collisionless Invariants
Equation (181) obeys a conservation law, which is very easy
to derive. Multiplying Eq. (181) by G±/FM and integrating
over space and velocities and performing integration by parts
in the right-hand side, we get
(G±)2
b̂ ·∇
dv‖v‖G
±. (195)
On the other hand, integrating Eq. (181) over v‖ gives
± = −b̂ ·∇
dv‖v‖G
±. (196)
Using this to express the right-hand side of Eq. (195) as a full
time derivative, we find
dW±compr
= 0, (197)
where the two invariants are
W±compr =
n0iT0i
(G±)2
(198)
It is useful (and always possible) to split
G± = FM
± + G̃±, (199)
where
dv‖G̃
± = 0 by construction. Then
W±compr =
n0iT0i
(G̃±)2
. (200)
Written in this form, the two invariants W±compr are mani-
festly positive definite quantities because Λ+ > 1 and Λ− < 0.
The invariants regulate the two decoupled kinetic cascades of
compressive fluctuations in the collisionless regime. The col-
lisionless damping derived in § 6.2.2 leads to exponential de-
cay of the density and field-strength fluctuations, or, equiva-
lently, of
±, while conserving W±compr. This means that
the damping is merely a redistribution of the conserved quan-
tity W±compr: the first term in Eq. (200) grows to compensate
for the decay of the second.
6.2.4. Linear Parallel Phase Mixing
In dynamical terms, how does the kinetic system Eq. (181)
arrange for the integral of the distribution function G±(v‖) to
decay while allowing its norm to grow? This is a very well
known phenomenon of (linear) phase mixing (Landau 1946;
Hammett et al. 1991; Krommes & Hu 1994; Krommes 1999;
Watanabe & Sugama 2004). To put it in simple terms, the
solution of the linearized Eq. (181) consists of the inhomoge-
neous part, which contains the collisionless damping and the
homogeneous part (solution of the left-hand side = 0) given by
G± ∝ e−ik‖v‖t , the so-called ballistic response (this is also the
nonlinear solution if t and k‖ are interpreted as Lagrangian
variables in the frame of the Alfvén waves; see § 6.3). As
time goes on, this part of the solution becomes increasingly
oscillatory in v‖, so its velocity integral tends to zero, while
its amplitude does not decay. It is such ballistic contributions
that make up the G̃± term in Eq. (200).
As the velocity gradient of G̃± increases with time,
∂G̃±/∂v‖ ∼ k‖tG±, at some point it can become sufficiently
large to activate the collision integral [the right-hand side of
Eq. (157)], which has so far been neglected. This way the col-
lisionless damping of compressive fluctuations can be turned
into ion heating—a simple example of a more general prin-
ciple of how electromagnetic fluctuation energy is transferred
into heat via the entropy part of the generalized energy (§ 3.5).
Indeed, we will prove in § 6.2.5 that the invariants W±compr are
constituent parts of the overall generalized energy functional
for the compressive fluctuations, so their cascade to small
scales in phase space is part of the overall kinetic cascade in-
troduced in § 3.4.
It is not entirely clear how efficient is the parallel-phase-
mixing route to ion heating and, therefore, whether the colli-
sionlessly damped energy of compressive fluctuations ends up
in the ion heat or rather reaches the ion gyroscale and couples
back to the Alfvénic component of the turbulence (§ 7.1). The
answer to this question will depend on whether compressive
fluctuations can develop large k‖—a non-trivial issue further
discussed in § 6.3.
6.2.5. Generalized Energy: Three Collisionless Cascades
We will now show how the generalized energy for com-
pressive fluctuations in the collisionless regime incorporates
the two invariants derived in § 6.2.3.
Rewriting the compressive part of the KRMHD generalized
energy [Eq. (153)] in terms of the function g [see Eq. (149)],
we get
Wcompr =
n0iT0i
. (201)
28 SCHEKOCHIHIN ET AL.
Using Eqs. (178) and (183), we can express δne and δB‖ in
terms of
± as follows
, (202)
, (203)
where σ was defined in Eq. (184) and
. (204)
In order to express g in terms of G±, we have to reconstruct
the v⊥ dependence of g, which we integrated out at the begin-
ning of § 6.2.1.
Let us represent the distribution function as follows
πv2thi
e−xĝ(x,v‖), ĝ(x,v‖) =
Ll(x)Gl(v‖), (205)
where x = v2⊥/v
thi and we have expanded ĝ in Laguerre poly-
nomials Ll(x) = (e
x/l!)(dl/dxl)xle−x. Since Laguerre polyno-
mials are orthogonal, the first term in Eq. (201) splits into a
sum of “energies” associated with the expansion coefficients:
. (206)
The expansion coefficients are determined via the Laguerre
transform:
Gl(v‖) =
dxe−xLl(x)ĝ(x,v‖). (207)
As L0 = 1 and L1 = 1 − x, it is easy to see that δne and δB‖ can
be expressed as linear combinations of
dv‖G0 and
dv‖G1
[see Eqs. (176-178)]. Using Eqs. (176), (177), and (183), we
can show that
G0 = −
+G+ +
σ − 1 −
, (208)
σΛ+G+ −
, (209)
where G± satisfy Eq. (181). As follows from Eq. (157) (ne-
glecting the collision integral), all higher-order expansion co-
efficients satisfy a simple homogeneous equation:
+ v‖b̂ ·∇Gl = 0, l > 1. (210)
Thus, the distribution function can be explicitly written in
terms of G±:
G0(v‖) +
v2thi
G1(v‖)
πv2thi
thi + g̃, (211)
where G0 and G1 are given by Eqs. (208-209) and g̃ com-
prises the rest of the Laguerre expansion (all Gl with l > 1),
i.e., it is the homogeneous solution of Eq. (157) that does not
contribute to either density or magnetic-field strength:
+ v‖b̂ ·∇g̃ = 0,
d3v g̃ = 0,
v2thi
g̃ = 0. (212)
Now substituting Eqs. (208) and (209) into Eq. (206) and
then substituting the result and Eqs. (202-203) into Eq. (201),
we find after some straightforward manipulations
Wcompr =
T0ig̃
(Λ+)2W +compr
(Λ−)2W −compr, (213)
where κ is defined by Eq. (204) and W±compr are the two inde-
pendent invariants that we derived in § 6.2.3. Thus, the gener-
alized energy for compressive fluctuations splits into three in-
dependently cascading parts: W±compr associated with the den-
sity and magnetic-field-strength fluctuations and a purely ki-
netic part given by the first term in Eq. (213) (see Fig. 5).
The dynamical evolution of this purely kinetic component is
described by Eq. (212)—it is a passively mixed, undamped
ballistic-type mode.
All three cascade channels lead to small perpendicular spa-
tial scales via passive mixing by the Alfvénic turbulence and
also to small scales in v‖ via the parallel phase mixing pro-
cess discussed in § 6.2.4 (note that g̃ is subject to this process
as well).
6.3. Parallel and Perpendicular Cascades
Let us return to the kinetic equation (157) and transform
it to the Lagrangian frame associated with the velocity field
u⊥ = ẑ×∇⊥Φ of the Alfvén waves: (t,r) → (t,r0), where
r(t,r0) = r0 +
dt ′u⊥(t
′,r(t ′,r0)). (214)
In this frame, the convective derivative d/dt defined in
Eq. (160) turns into ∂/∂t, while the parallel spatial gradient
b̂ ·∇ can be calculated by employing the Cauchy solution for
the perturbed magnetic field δB⊥ = ẑ×∇⊥Ψ:
b̂(t,r) = ẑ +
δB⊥(t,r)
= b̂(0,r0) ·∇0r, (215)
where r is given by Eq. (214) and ∇0 = ∂/∂r0. Then
b̂ ·∇ = b̂(0,r0) ·
·∇ = b̂(0,r0) ·∇0 =
, (216)
where s0 is the arc length along the perturbed magnetic field
taken at t = 0 [if δB⊥(0,r0) = 0, s0 = z0]. Thus, in the La-
grangian frame associated with the Alfvénic component of
the turbulence, Eq. (157) is linear. This means that, if the
effect of finite ion gyroradius is neglected, the KRMHD sys-
tem does not give rise to a cascade of density and magnetic-
field-strength fluctuations to smaller scales along the moving
(perturbed) field lines, i.e., b̂ · ∇δne and b̂ · ∇δB‖ do not in-
crease. In contrast, there is a perpendicular cascade (cascade
in k⊥): the perpendicular wandering of field lines due to the
Alfvénic turbulence causes passive mixing of δne and δB‖ in
the direction transverse to the magnetic field (see § 2.6 for a
quick recapitulation of the standard scaling argument on the
passive cascade that leads to a k
⊥ in the perpendicular di-
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 29
FIG. 7.— Lagrangian mixing of passive fields: fluctuations develop small
scales across, but not along the exact field lines.
rection). Figure 7 illustrates this situation.26
We emphasize that this lack of nonlinear refinement of the
scale of δne and δB‖ along the moving field lines is a particu-
lar property of the compressive component of the turbulence,
not shared by the Alfvén waves. Indeed, unlike Eq. (157), the
RMHD equations (155-156), do not reduce to a linear form
under the Lagrangian transformation (214), so the Alfvén
waves should develop small scales both across and along the
perturbed magnetic field.
Whether the density and magnetic-field-strength fluctua-
tions develop small scales along the magnetic field has direct
physical and observational consequences. Damping of these
fluctuations, both in the collisional and collisionless regimes,
discussed in § 6.1.2 and § 6.2.2, respectively, depends pre-
cisely on their scale along the perturbed field: indeed, the
linear results derived there are exact in the Lagrangian frame
(214). To summarize these results, the damping rate of δne
and δB‖ at βi ∼ 1 is
γ∼ vthiλmfpik2‖0, k‖0λmfpi ≪ 1, (217)
γ∼ vthik‖0, k‖0λmfpi ≫ 1, (218)
where k‖0 ∼ b̂ ·∇ is the wavenumber along the perturbed field
(i.e., if there is no parallel cascade, the wavenumber of the
large-scale stirring).
Whether this damping cuts off the cascades of δne and δB‖
depends on the relative magnitudes of the damping rate γ for
a given k⊥ and the characteristic rate at which the Alfvén
waves cause δne and δB‖ to cascade to higher k⊥. This rate
is ωA ∼ k‖AvA, where k‖A is the parallel wave number of the
Alfvén waves that have the same k⊥. Since the Alfvén waves
do have a parallel cascade, assuming scale-by-scale critical
balance (3) leads to [Eq. (5)]
k‖A ∼ k
0 . (219)
If, in contrast to the Alfvén waves, δne and δB‖ have no par-
allel cascade, k‖0 does not grow with k⊥, so, for large enough
k⊥, k‖0 ≪ k‖A and γ≪ωA. This means that, despite the damp-
ing, the density and field-strength fluctuations should have
perpendicular cascades extending to the ion gyroscale.
The validity of the argument at the beginning of this sec-
tion that ruled out the parallel cascade of δne and δB‖ is not
quite as obvious as it might appear. Lithwick & Goldreich
(2001) argued that the dissipation of δne and δB‖ at the ion
gyroscale would cause these fluctuations to become uncorre-
lated at the same parallel scales as the Alfvénic fluctuations by
which they are mixed, i.e., k‖0 ∼ k‖A. The damping rate then
becomes comparable to the cascade rate, cutting off the cas-
cades of density and field-strength fluctuations at k‖λmfpi ∼ 1.
The corresponding perpendicular cutoff wavenumber is [see
26 Note that effectively, there is also a cascade in k‖ if the latter is mea-
sured along the unperturbed field—more precisely, a cascade in kz. This is
due to the perpendicular deformation of the perturbed magnetic field by the
Alfvén-wave turbulence: since ∇⊥ grows while b̂ ·∇ remains the same, we
have from Eq. (123) ∂/∂z ≃ −(δB⊥/B0) ·∇⊥.
Eq. (219)]
k⊥ ∼ l1/20 λ
mfpi . (220)
Asymptotically speaking, in a weakly collisional plasma,
this cutoff is far above the ion gyroscale, k⊥ρi ≪ 1. How-
ever, the relatively small value of λmfpi in the warm ISM,
which was the main focus of Lithwick & Goldreich 2001,
meant that the numerical value of the perpendicular cutoff
scale given by Eq. (220) was, in fact, quite close both to
the ion gyroscale (see Table 1) and to the observational es-
timates for the inner scale of the electron-density fluctuations
in the ISM (Spangler & Gwinn 1990; Armstrong et al. 1995).
Thus, it was not possible to tell whether Eq. (220), rather than
k⊥ ∼ ρ−1i , represented the correct prediction.
The situation is rather different in the nearly collision-
less case of the solar wind, where the cutoff given by
Eq. (220) would mean that very little density or field-
strength fluctuations should be detected above the ion gy-
roscale. Observations do not support such a conclu-
sion: the density fluctuations appear to follow a k−5/3 law
at all scales larger than a few times ρi (Lovelace et al.
1970; Woo & Armstrong 1979; Celnikier et al. 1983, 1987;
Coles & Harmon 1989; Marsch & Tu 1990b; Coles et al.
1991), consistently with the expected behavior of an un-
damped passive scalar field (see § 2.6). An extended range
of k−5/3 scaling above the ion gyroscale is also observed for
the fluctuations of the magnetic-field strength (Marsch & Tu
1990b; Bershadskii & Sreenivasan 2004; Hnat et al. 2005;
Alexandrova et al. 2008a).
These observational facts suggest that the cutoff formula
(220) does not apply. This does not, however, conclusively
vitiate the Lithwick & Goldreich (2001) theory. Heuristically,
their argument is plausible, although it is, perhaps, useful
to note that in order for the effect of the perpendicular dis-
sipation terms, not present in the KRMHD equations (157-
159), to be felt, the density and field-strength fluctuations
should reach the ion gyroscale in the first place. Quanti-
tatively, the failure of the compressive fluctuations in the
solar wind to be damped could still be consistent with the
Lithwick & Goldreich (2001) theory because of the relative
weakness of the collisionless damping, especially at low beta
(§ 6.2.2)—the explanation they themselves favor. The way to
check observationally whether this explanation suffices would
be to make a comparative study of the compressive fluctua-
tions for solar-wind data with different values of βi. If the
strength of the damping is the decisive factor, one should al-
ways see cascades of both δne and δB‖ at low βi, no cascades
at βi ∼ 1, and a cascade of δB‖ but not δne at high βi (in
this limit, the damping of the density fluctuations is strong,
of the field-strength weak; see § 6.2.2). If, on the other hand,
the parallel cascade of the compressive fluctuations is intrin-
sically inefficient, very little βi dependence is expected and a
perpendicular cascade should be seen in all cases.
Obviously, an even more direct observational (or numer-
ical) test would be the detection or non-detection of near-
perfect alignment of the density and field-strength structures
with the moving field lines (not with the mean magnetic
field—see footnote 26), but it is not clear how to measure
this reliably. It is interesting, in this context, that in near-
the-Sun measurements, the density fluctuations are reported
to have the form of highly anisotropic filaments aligned with
the magnetic field (Armstrong et al. 1990; Grall et al. 1997;
Woo & Habbal 1997). Another intriguing piece of observa-
30 SCHEKOCHIHIN ET AL.
tional evidence is the discovery that the local structure of the
magnetic-field-strength and density fluctuations at 1 AU is, in
a certain sense, correlated with the solar cycle (Kiyani et al.
2007; Hnat et al. 2007; Wicks et al. 2009)—this suggests a
dependence on initial conditions that is absent in the Alfvénic
fluctuations and that presumably should also disappear in the
compressive fluctuations if the latter are fully mixed both in
the perpendicular and parallel directions.
7. TURBULENCE IN THE DISSIPATION RANGE: ELECTRON RMHD
AND THE ENTROPY CASCADE
7.1. Transition at the Ion Gyroscale
The validity of the theory discussed in § 5 and § 6 breaks
down when k⊥ρi ∼ 1. As the ion gyroscale is approached,
the Alfvén waves are no longer decoupled from the rest of
the plasma dynamics. All modes now contain perturbations
of density and magnetic-field strength and can be collision-
lessly damped. Because of the low-frequency nature of the
Alfvén-wave cascade, ω ≪ Ωi even at k⊥ρi ∼ 1 [Eq. (46)],
so the ion cyclotron resonance (ω − k‖v‖ = ±Ωi) is not im-
portant, while the Landau one (ω = k‖v‖) is. The linear the-
ory of this collisionless damping in the gyrokinetic approx-
imation is worked out in detail in Howes et al. (2006) (see
also Gary & Borovsky 2008). Figure 8 shows the solutions of
their dispersion relation that illustrate how the Alfvén wave
becomes a dispersive kinetic Alfvén wave (KAW) (see § 7.3)
and collisionless damping becomes important as the ion gy-
roscale is reached.
We stress that this transition occurs at the ion gyroscale, not
at the ion inertial scale di = ρi/
βi (except in the limit of cold
ions, τ = T0i/T0e ≪ 1; see Appendix E). This statement is true
even when βi is not order unity, as illustrated in Fig. 8: for the
three cases plotted there, k⊥di = 1 corresponds to k⊥ρi = 0.1,
1 and 10 for βi = 0.01, 1 and 100, respectively, but there is
no trace of the ion inertial scale in the solutions of the linear
dispersion relation. Nonlinearly, in the limit βi ≪ 1, we may
consider the scales k⊥di ∼ 1 and expand the gyrokinetics in
k⊥ρi = k⊥di
βi ≪ 1 in a way similar to how it was done in § 5
and obtain precisely the same results: Alfvénic fluctuations
described by the RMHD equations and compressive fluctua-
tions passively advected by them and satisfying the reduced
kinetic equation derived in § 5.5. Thus, even though di ≫ ρi
at low beta, there is no change in the nature of the turbulent
cascade until k⊥ρi ∼ 1 is reached.
The nonlinear theory of what happens at k⊥ρi ∼ 1 is very
poorly understood. It is, however, possible to make progress
by examining what kind of fluctuations emerge on the other
side of the transition, at k⊥ρi ≫ 1. As we will demonstrate
below, it turns out that another turbulent cascade—this time
of KAW—is possible in this so-called dissipation range. It
can transfer the energy of KAW-like fluctuations down to the
electron gyroscale, where electron Landau damping becomes
important (see Howes et al. 2006). Some observational evi-
dence of KAW is, indeed, available in the solar wind and the
magnetosphere (Bale et al. 2005; Grison et al. 2005, see fur-
ther discussion in § 8.2.4). Below we derive the equations that
describe KAW-like fluctuations in the scale range k⊥ρi ≫ 1,
k⊥ρe ≪ 1 (§ 7.2) and work out a Kolmogorov-style scaling
theory for this cascade (§ 7.5).
Because of the presence of the collisionless damping at the
ion gyroscale, only a certain fraction of the turbulent power
arriving there from the inertial range is converted into the
KAW cascade, while the rest is Landau-damped. The damp-
ing leads to the heating of the ions, but the process of deposit-
ing the collisionlessly damped fluctuation energy into the ion
heat is non-trivial because, as we explained in § 3.5, collisions
do need to play a role in order for true heating to occur. As
we explained in § 3.5 and will see specifically for the dissi-
pation range in § 7.8, the electromagnetic-fluctuation energy
does not disappear as a result of the Landau damping but is
converted into ion entropy fluctuations, while the generalized
energy is conserved. Collisions are then accessed and ion
heating achieved via a purely kinetic phenomenon: the ion
entropy cascade in phase space (nonlinear phase mixing), for
which a theory is developed in § 7.9 and § 7.10. A similar pro-
cess of conversion of the KAW energy into electron entropy
fluctuations and then electron heat is treated in § 7.12.
Figure 5 illustrates the routes energy takes from the ion gy-
roscale towards heating. Crucially, it is at k⊥ρi ∼ 1 that it
is decided how much energy would eventually go into the
ions and how much into electrons.27 How this distribution
of energy depends on plasma parameters (βi and T0i/T0e)
is an open theoretical question28 of considerable astrophys-
ical interest: e.g., the efficiency of ion heating is a key un-
known in the theory of advection-dominated accretion flows
(Quataert & Gruzinov 1999, see discussion in § 8.5) and of
the solar corona (e.g., Cranmer & van Ballegooijen 2003); we
will also see in § 7.11 that it may determine the form of the
observed dissipation-range spectra in space plasmas.
A short summary of this section is given in § 7.14.
7.2. Equations of Electron Reduced MHD
The derivation is straightforward: when ai ∼ k⊥ρi ≫ 1, all
Bessel functions in Eqs. (118-120) are small, so the integrals
of the ion distribution function vanish and Eqs. (118-120) be-
, (221)
u‖e =
4πen0e
∇2⊥A‖ = −
ρi∇2⊥Ψ√
, u‖i = 0, (222)
, (223)
where we used the definitions (135) of the stream and flux
functions Φ and Ψ.
These equations are a reflection of the fact that, for k⊥ρi ≫
1, the ion response is effectively purely Boltzmann, with the
gyrokinetic part hi contributing nothing to the fields or flows
[see Eq. (54) with hi omitted; hi does, however, play an impor-
tant role in the energy balance and ion heating, as explained
in §§ 7.8-7.10 below]. The Boltzmann response for ion den-
sity is expressed by Eq. (221). Equation (222) states that the
parallel ion flow velocity can be neglected. Finally, Eq. (223)
expresses the pressure balance for Boltzmann (and, therefore,
isothermal) electrons [Eq. (103)] and ions: if we write
B0δB‖
= −δpi − δpe = −T0iδni − T0eδne, (224)
27 Some of the energy of compressive fluctuations may go into ion heat via
collisional (§ 6.1.2) or collisionless (§ 6.2.2) damping of these fluctuations
in the inertial range. Whether this is a significant ion heating mechanism
depends on the efficiency of the parallel cascade (see § 6.2.4 and § 6.3).
28 How much energy is converted into ion entropy fluctuations in the pro-
cess of a nonlinear turbulent cascade is not necessarily directly related to the
strength of the linear collisionless damping.
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 31
FIG. 8.— Numerical solutions of the linear gyrokinetic dispersion relation (for a detailed treatment of the linear theory, see Howes et al. 2006) showing the
transition from the Alfvén wave to KAW between the inertial range (k⊥ρi ≪ 1) and the dissipation range (k⊥ρi & 1). We show three cases: low beta (βi = 0.01),
βi = 1, and high beta (βi = 100). In all three cases, τ = 1 and Z = 1. Bold solid lines show the real frequency ω, bold dashed lines the damping rate γ, both
normalized by k‖vA (in gyrokinetics, ω/k‖vA and γ/k‖vA are functions of k⊥ only). Dotted lines show the asymptotic KAW solution (230). Horizontal solid line
shows the Alfvén wave ω = k‖vA. Vertical solid lines show k⊥ρi = 1 and k⊥ρe = 1. Note that the damping can be considered strong if the characteristic decay
time is comparable or shorter than the wave period, i.e., γ/ω & 1/2π. Thus, in these plots, the damping at k⊥ρi ∼ 1 is relatively weak for βi = 1, relatively
strong for low beta and very strong for high beta.
it follows that
, (225)
which, combined with Eq. (221), gives Eq. (223). We remind
the reader that the perpendicular Ampère’s law, from which
Eq. (223) was derived [Eq. (66) via Eq. (120)] is, in gyrokinet-
ics, indeed equivalent to the statement of perpendicular pres-
sure balance (see § 3.3).
Substituting Eqs. (221-223) into Eqs. (116-117), we obtain
the following closed system of equations
b̂ ·∇Φ, (226)
2 +βi
1 + Z/τ
) b̂ ·∇
ρ2i ∇2⊥Ψ
. (227)
Note that, using Eq. (223), Eqs. (226) and (227) can be recast
as two coupled evolution equations for the perpendicular and
parallel components of the perturbed magnetic field, respec-
tively [Eqs. (C10) in Appendix C.2].
We shall refer to Eqs. (226-227) as Electron Reduced MHD
(ERMHD). They are related to the Electron Magnetohydrody-
namics (EMHD)—a fluid-like approximation that evolves the
magnetic field only and arises if one assumes that the mag-
netic field is frozen into the electron flow velocity ue, while
the ions are immobile, ui = 0 (Kingsep et al. 1990):
4πen0e
∇× [(∇×B)×B] . (228)
As explained in Appendix C.2, the result of applying the
RMHD/gyrokinetic ordering (§ 2.1 and § 3.1) to Eq. (228),
where B = B0ẑ + δB and
ẑ×∇⊥Ψ+ ẑ
, (229)
coincides with our Eqs. (226-227) in the effectively incom-
pressible limits of βi ≫ 1 or βe = βiZ/τ ≫ 1. When betas are
arbitrary, density fluctuations cannot be neglected compared
to the magnetic-field-strength fluctuations [Eq. (225)] and
give rise to perpendicular ion flows with ∇·ui 6= 0. Thus, our
ERMHD system constitutes the appropriate generalization of
EMHD for low-frequency anisotropic fluctuations without the
assumption of incompressibility.
A (more tenuous) relationship also exists between our
ERMHD system and the so-called Hall MHD, which, like
EMHD, is based on the magnetic field being frozen into
the electron flow, but includes the ion motion via the stan-
dard MHD momentum equation [Eq. (8)]. Strictly speak-
ing, Hall MHD can only be used in the limit of cold ions,
τ = T0i/T0e ≪ 1 (see, e.g., Ito et al. 2004; Hirose et al. 2004,
and Appendix E), in which case it can be shown to reduce
to Eqs. (226-227) in the appropriate small-scale limit (Ap-
pendix E). Although τ ≪ 1 is not a natural assumption for
most space and astrophysical plasmas, Hall MHD has, due to
its simplicity, been a popular theoretical paradigm in the stud-
ies of space and astrophysical plasma turbulence (see § 8.2.6).
We have therefore devoted Appendix E to showing how this
approximation fits into the theoretical framework proposed
here: namely, we derive the anisotropic low-frequency ver-
sion of the Hall MHD approximation from gyrokinetics under
the assumption τ ≪ 1 and discuss the role of the ion inertial
and ion sound scales, which acquire physical significance in
this limit. However, outside this Appendix, we assume τ ∼ 1
everywhere and shall not use Hall MHD.
The validity of the ERMHD equations as a model for
plasma dynamics in the dissipation range is further discussed
in § 7.6.
7.3. Kinetic Alfvén Waves
The linear modes supported by ERMHD are kinetic Alfvén
waves (KAW) with frequencies
ωk = ±
1 + Z/τ
2 +βi
1 + Z/τ
) k⊥ρik‖vA. (230)
This dispersion relation is illustrated in Fig. 8: note that the
transition from Alfvén waves to dispersive KAW always oc-
curs at k⊥ρi ∼ 1, even when βi ≪ 1 or βi ≫ 1. In the latter
case, there is a sharp frequency jump at the transition (accom-
panied by very strong ion Landau damping).
The eigenfunctions corresponding to the two waves with
32 SCHEKOCHIHIN ET AL.
FIG. 9.— Polarization of the kinetic Alfvén wave, see Eqs. (232) and (233).
frequencies (230) are
2 +βi
∓ k⊥Ψk. (231)
Using Eqs. (229) and (223), the perturbed magnetic-field vec-
tor can be expressed as follows
= −iẑ× k⊥
1 + Z/τ
2 +βi
1 + Z/τ
(232)
so, for a single “+” or “−” wave (corresponding to Θ−k = 0 or
k = 0, respectively), δBk rotates in the plane perpendicular
to the wave vector k⊥ clockwise with respect to the latter,
while the wave propagates parallel or antiparallel to the guide
field (Fig. 9).
The waves are elliptically right-hand polarized. Indeed, us-
ing Eq. (223), the perpendicular electric field is:
E⊥k = −ik⊥ϕ+
−ik⊥ + ẑ×k⊥
ϕ (233)
(cf. Gary 1986; Hollweg 1999). The second term is small in
the gyrokinetic expansion, so this is a very elongated ellipse
(Fig. 9).
7.4. Finite-Amplitude Kinetic Alfvén Waves
As we are about to argue for a critically balanced KAW
turbulence in a fashion analogous to the GS theory for the
Alfvén waves (§ 1.2), it is a natural question to ask how simi-
lar the nonlinear properties of a putative KAW cascade will be
to an Alfvén-wave cascade. As in the case of Alfvén waves,
there are two counterpropagating linear modes [Eqs. (230)
and (231)], and it turns out that certain superpositions of these
modes (KAW packets) are also exact nonlinear solutions of
Eqs. (226-227). Let us show that this is the case.
We might look for the nonlinear solutions of Eqs. (226-227)
by requiring that the nonlinear terms vanish. Since b̂ · ∇ =
∂/∂z + (1/vA){Ψ, · · ·}, this gives
{Ψ,Φ} = 0 ⇒ Ψ = c1Φ, (234)
{Ψ,ρ2i ∇2⊥Ψ} = 0 ⇒ ρ2i ∇2⊥Ψ = c2Ψ, (235)
where c1 and c2 are constants. Whether such solutions are
possible is determined by substituting Eqs. (234) and (235)
into Eqs. (226) and (227) and demanding that the two result-
ing linear equations be consistent with each other (both equa-
tions now just evolve Ψ). This is achieved if29
c21 = −
2 +βi
, (236)
so real solutions exist if c2 < 0. In particular, wave pack-
ets consisting of KAW given by one of the linear eigen-
modes (231) with an arbitrary shape in z but confined to a
single shell |k⊥| = k⊥ = const, satisfy Eqs. (234-236) with
c2 = −k2⊥ρ
i . This outcome is, in fact, only mildly non-trivial:
in gyrokinetics, the Poisson bracket nonlinearity [Eq. (59)]
vanishes for any monochromatic (in k⊥) mode because the
Poisson bracket of two modes with wavenumbers k⊥ and k′⊥
is ∝ ẑ · (k⊥ × k′⊥). Therefore, any monochromatic solution
of the linearized equations is also an exact nonlinear solution.
As we have shown above, a superposition of monochromatic
KAW that have a fixed k⊥, or, somewhat more generally, sat-
isfy Eq. (235) with a fixed c2, is still an exact solution.
Note that a similar procedure applied to the RMHD equa-
tions (17-18) returns the Elsasser solutions: perturbations of
arbitrary shape that satisfy Φ = ±Ψ. The physical difference
between these finite-amplitude Alfven-wave packets and the
finite-amplitude KAW packets discussed above is that non-
linear interactions can occur not just between counterpropa-
gating KAW but also between copropagating ones—a natural
conclusion because KAW are dispersive (their group velocity
along the guide field is ∝ vAk⊥ρi), so copropagating waves
with different k⊥ can “catch up” with each other and inter-
act.30
7.5. Scalings for KAW Turbulence
A scaling theory for the turbulence described by Eqs. (221-
227) can be constructed along the same lines as the GS theory
for the Alfvén-wave turbulence (§ 1.2). Namely, we shall as-
sume that the turbulence below the ion gyroscale consists of
KAW-like fluctuations with k‖ ≪ k⊥ (Quataert & Gruzinov
1999) and that the interactions between them are critically
balanced (Cho & Lazarian 2004), i.e., that the propagation
time and nonlinear interaction time are comparable at every
scale. We stress that none of these assumptions are, strictly
speaking, inevitable31 (and, in fact, neither were they in-
evitable in the case of Alfvén waves). Since we have de-
rived Eqs. (226-227) from gyrokinetics, the anisotropy of
the fluctuations described by these equations is hard-wired,
but it is not guaranteed that the actual physical cascade be-
low the ion gyroscale is indeed anisotropic, although anal-
ysis of solar-wind measurements does seem to indicate that
29 Formally speaking, c1 and c2 can depend on t and z. If this is allowed,
we still recover Eq. (236), but in addition to it, we get the evolution equation
c1∂c1/∂t = vA(1 + Z/τ )∂c1/∂z. This allows c1 = const, but there are, of
course, other solutions. We shall not consider them here.
30 The calculation above is analogous to the calculation by
Mahajan & Krishan (2005) for incompressible Hall MHD (i.e., essen-
tially, the high-βe limit of the equations discussed in Appendix E), but
the result is more general in the sense that it holds at arbitrary ion and
electron betas. The Mahajan–Krishan solution in the EMHD limit amounts
to noticing that Eq. (228) becomes linear for force-free (Beltrami) magnetic
perturbations, ∇× δB = λδB. Substituting Eq. (229) into this equation
and using Eq. (223), we see that the force-free equation is equivalent
to Eqs. (234-236) if c2 = −λ2 and the incompressible limit (βi ≫ 1 or
βe = βiZ/τ ≫ 1) is taken.
31 In fact, the EMHD turbulence was thought to be weak by several au-
thors, who predicted a k−2 spectrum of magnetic energy assuming isotropy
(Goldreich & Reisenegger 1992) or k
for the anisotropic case (Voitenko
1998; Galtier & Bhattacharjee 2003; Galtier 2006).
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 33
at least a significant fraction of it is (see Leamon et al.
1998; Hamilton et al. 2008). Numerical simulations based
on Eq. (228) (Biskamp et al. 1996, 1999; Ghosh et al. 1996;
Ng et al. 2003; Cho & Lazarian 2004; Shaikh & Zank 2005)
have revealed that the spectrum of magnetic fluctuations
scales as k
⊥ , the outcome consistent with the assumptions
stated above. Let us outline the argument that leads to this
scaling.
First assume that the fluctuations are KAW-like and that Θ+
and Θ− [Eq. (231)] have similar scaling. This implies
1 +βi
Φλ (237)
(for the purposes of scaling arguments and order-of-
magnitude estimates, we set Z/τ = 1, but keep the βi de-
pendence so low- and high-beta limits could be recovered if
necessary). The fact that fixed-k⊥ KAW packets, which sat-
isfy Eq. (237) with λ = 1/k⊥, are exact nonlinear solutions
of the ERMHD equations (§ 7.4) lends some credence to this
assumption.
Assuming scale-space locality of interactions implies a
constant-flux KAW cascade: analogously to Eq. (1),
(Ψλ/λ)
τKAWλ
∼ (1 +βi)(Φλ/ρi)
τKAWλ
∼ εKAW = const, (238)
where τKAWλ is the cascade time and εKAW is the KAW energy
flux proportional to the fraction of the total flux ε (or the total
turbulent power Pext; see § 3.4) that was converted into the
KAW cascade at the ion gyroscale.
Using Eqs. (226-227) and Eq. (237), it is not hard to see
that the characteristic nonlinear decorrelation time is λ2/Φλ.
If the turbulence is strong, then this time is comparable to
the inverse KAW frequency [Eq. (230)] scale by scale and we
may assume the cascade time is comparable to either:
τKAWλ ∼
1 +βi
. (239)
In other words, this says that ∂/∂z ∼ (δB⊥/B0) ·∇⊥ and so
δB⊥λ/B0 ∼ λ/l‖λ (note that the last relation confirms that
our scaling arguments do not violate the gyrokinetic ordering;
see § 2.1 and § 3.1). Equation (239) is the critical-balance as-
sumption for KAW. As in the case of the Alfvén waves (§ 1.2),
we might argue physically that the critical balance is set up be-
cause the parallel correlation length l‖λ is determined by the
condition that a wave can propagate the distance l‖λ in one
nonlinear decorrelation time corresponding to the perpendic-
ular correlation length λ.
Combining Eqs. (238) and (239), we get the desired scaling
relations for the KAW turbulence:
(εKAW
)1/3 vA
(1 +βi)1/3
2/3, (240)
(1 +βi)1/6
, (241)
where l0 = v
A/ε, as in § 1.2. The first of these scaling relations
is equivalent to a k
⊥ spectrum of magnetic energy, the sec-
ond quantifies the anisotropy (which is stronger than for the
GS turbulence). Both scalings were confirmed in the numer-
ical simulations of Cho & Lazarian (2004)—it is their detec-
tion of the scaling (241) that makes a particularly strong case
that KAW turbulence is not weak and that the critical balance
hypothesis applies.
For KAW-like fluctuations, the density [Eq. (221)] and
magnetic field [Eqs. (223) and (231)] have the same spec-
trum as the scalar potential, i.e., k
⊥ , while the electric field
E ∼ k⊥ϕ has a k−1/3⊥ spectrum. The solar-wind fluctuation
spectra reported by Bale et al. (2005) indeed are consistent
with a transition to KAW turbulence around the ion gyroscale:
k−5/3 magnetic and electric-field power spectra at kρi ≪ 1 are
replaced, for kρi & 1, with what appears to be consistent with
a k−7/3 scaling for the magnetic-field spectrum and a k−1/3 for
the electric one (see Fig. 1). A similar result is recovered in
fully gyrokinetic simulations with βi = 1, τ = 1 (Howes et al.
2008b). However, not all solar-wind observations are quite as
straightforwardly supportive of the notion of the KAW cas-
cade and much steeper magnetic-fluctuation spectra have also
been reported (e.g., Denskat et al. 1983; Leamon et al. 1998;
Smith et al. 2006). Possible reasons for this will emerge in
§ 7.6 and § 7.11 and the solar-wind data are further discussed
in § 8.2.4 and § 8.2.5.
7.6. Validity of the Electron RMHD and the Effect of Electron
Landau Damping
The ERMHD equations derived in § 7 are valid provided
k⊥ρi ≫ 1 and also provided it is sufficient to use the leading
order in the mass-ratio expansion (isothermal electrons; see
§ 4). In particular, this means that the electron Landau damp-
ing is neglected. Asymptotically speaking, this is a rigorous
limit, but one must be cautious in applying it to real plas-
mas. Since the width of the scale range where k⊥ρi ≫ 1 and
k⊥ρe ≪ 1 is only ∼ (mi/me)1/2 ≃ 43, for some values of the
plasma parameters (T0i/T0e and βi) there may not be a very
broad interval of scales where the electron Landau damping
is truly negligible. Consider, for example, the low-beta limit,
βi ≪ 1. In this limit, the KAW frequency is ω ∼ k⊥ρik‖vA
[Eq. (230)]. The electron Landau damping becomes impor-
tant when ω ∼ k‖vthe, or k⊥ρe ∼
βi ≪ 1, so the ERMHD
approximation breaks down and, consequently, the KAW cas-
cade, if any, should be interrupted well before the electron
gyroscale is reached. Figure 8 shows the solution of the
full gyrokinetic dispersion relation (Howes et al. 2006) for
small, unity and large βi. One can judge for which scales and
how well (or how badly) the ERMHD approximation holds
from the precision with which the exact frequency follows the
asymptotic solution Eq. (230) and from the relative strength
of the damping compared to the real frequency of the waves.
Non-negligible electron Landau damping may affect turbu-
lence spectra because one can no longer assume a constant
flux of KAW energy as we did in § 7.5. To evaluate the conse-
quences of this effect, Howes et al. (2008a) constructed a sim-
ple model of spectral energy transfer and concluded that Lan-
dau damping leads to steepening of the KAW spectra—one
of several possible reasons for steep dissipation-range spectra
observed in space plasmas (see also § 7.11).
7.7. Unfreezing of Flux
As ERMHD is a limit of the isothermal-electron-fluid sys-
tem (§ 4), the magnetic-field lines remain unbroken (see
§ 4.3). Within the orderings employed above (small mass ra-
tio, νii ∼ ω, βi ∼ 1, τ ∼ 1), the flux unfreezes only in the
vicinity of the electron gyroscale. It is interesting to evaluate
somewhat more precisely the scale at which this happens as a
function of plasma parameters.
34 SCHEKOCHIHIN ET AL.
Physically, there are three kinds of mechanisms by which
the flux conservation is broken: electron inertia, the effects of
finite electron gyroradius, and Ohmic resistivity. Let us take
the v‖ moment of the electron gyrokinetic equation [Eq. (57),
s = e, integration at constant r] and use Eq. (222) to evaluate
the inertial term in the resulting parallel electron momentum
equation:
d2e∇2⊥A‖, (242)
where de = ρe/
βe is the electron inertial scale and βe =
Zβi/τ . Comparing this with the ∂A‖/∂t term in the right-
hand side of the electron momentum equation, we see that the
electron inertia becomes important when k⊥ρe ∼
βe. The
finite-gyroradius effects enter when k⊥ρe ∼ 1. Thus, at low
βe, the electron inertia becomes important above the electron
gyroscale, whereas at high βe, the finite-gyroradius effects en-
ter first. Finally, the Ohmic resistivity comes from the colli-
sion term (see Appendix B.4):
d3vv‖
νeiu‖e ∼ νeik2⊥d2e A‖. (243)
Thus, resistivity starts to act when k⊥de ∼ (ω/νei)1/2. Using
the KAW frequency [Eq. (230)] to estimate ω and assuming
that τ is not small, we get
k⊥ρe ∼ k‖λmfpi
1 +βi
. (244)
Thus, the resistive scale can only be larger the electron gy-
roscale if the plasma is collisional (k‖λmfpi ≪ 1) and/or elec-
trons are much colder than ions (τ ≫ 1) and/or βi ≪ 1. Note
if only the last of these conditions is satisfied, the electron
inertia still becomes important at larger scales than resistivity.
7.8. Generalized Energy: KAW and Entropy Cascades
The generalized energy (§ 3.4) in the limit k⊥ρi ≫ 1 is cal-
culated by substituting Eqs. (221) and (223) into Eq. (109):
T0i〈h2i 〉r
n0iT0i
=Whi +WKAW. (245)
Here the first term, Whi , is the total variance of hi, which is
proportional to minus the entropy of the ion gyrocenter distri-
bution (see § 3.5) and whose cascade to collisional scales will
be discussed in § 7.9 and § 7.10. The remaining two terms are
the independently cascaded KAW energy:
WKAW =
min0i
|∇⊥Ψ|2
min0i
|Θ+|2 + |Θ−|2
. (246)
Although we can write WKAW as the sum of the energies of
the “+” and “−” linear KAW eigenmodes [Eq. (231)], which
are also exact nonlinear solutions (§ 7.4), the two do not cas-
cade independently and can exchange energy. Note that the
ERMHD equations also conserve
d3rΨΦ, which is readily
interpreted as the helicity of the perturbed magnetic field (see
Appendix F.3). However, it does not affect the KAW cascade
discussed in § 7.5 because it can be argued to have a tendency
to cascade inversely (Appendix F.6).
Comparing the way the generalized energy is split above
and below the ion gyroscale (see § 5.6 for the k⊥ρi ≪ 1 limit),
we interpret what happens at the k⊥ρi ∼ 1 transition as a redis-
tribution of the power that arrived from large scales between
a cascade of KAW and a cascade of the (minus) gyrocenter
entropy in the phase space (see Fig. 5). The latter cascade
is the way in which the energy diverted from the electromag-
netic fluctuations by the collisionless damping (wave–particle
interaction) can be transferred to the collisional scales and de-
posited into heat (§ 7.1). The concept of entropy cascade as
the key agent in the heating of the plasma was introduced in
§ 3.5, where we promised a more detailed discussion later on.
We now proceed to this discussion.
7.9. Entropy Cascade
The ion-gyrocenter distribution function hi satisfies the ion
gyrokinetic equation (121), where ion–electron collisions are
neglected under the mass-ratio expansion. At k⊥ρi ≫ 1, the
dominant contribution to 〈χ〉Ri comes from the electromag-
netic fluctuations associated with KAW turbulence. Since
the KAW cascade is decoupled from the entropy cascade, hi
is a passive tracer of the ring-averaged KAW turbulence in
phase space. Expanding the Bessel functions in the expres-
sion for 〈χ〉Ri ,k [ai ≫ 1 in Eq. (69) with s = i] and making
use of Eqs. (222-223) and of the KAW scaling Ψ ∼ Φ/k⊥ρi
[Eq. (231)], it is not hard to show that
〈χ〉Ri ,k ≃
〈ϕ〉Ri ,k =
J0(ai)Φk
, (247)
where
J0(ai) ≃
, ai = k⊥ρi
, (248)
so hi satisfies [Eq. (121)]
+{〈Φ〉Ri ,hi} =
βiρivA
∂〈Φ〉Ri
F0i + 〈Cii[hi]〉Ri
(249)
with the conservation law [Eq. (70), s = i]
βi ρivA
∂〈Φ〉Ri
hi 〈Cii[hi]〉Ri
. (250)
7.9.1. Nonlinear Perpendicular Phase Mixing
The wave–particle interaction term (the first term on the
right hand sides of these two equations) will shortly be seen
to be subdominant at k⊥ρi ≫ 1. It represents the source of
the invariant Whi due to the collisionless damping at the ion
gyroscale of some fraction of the energy arriving from the in-
ertial range. In a stationary turbulent state, we should have
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 35
FIG. 10.— Nonlinear perpendicular phase-mixing mechanism: the
gyrocenter distribution function at Ri of particles with velocities v⊥ and v
is mixed by turbulent fluctuations of the potential Φ (E×B flows) averaged
over particle orbits separated by a distance greater than the correlation length
of Φ.
dWhi/dt = 0 and this source should be balanced on average by
the (negative definite) collisional dissipation term ( = heating;
see § 3.5). This balance can only be achieved if hi develops
small scales in the velocity space and carries the generalized
energy, or, in this case, entropy, to scales in the phase space at
which collisions are important. A quick way to see this is by
recalling that the collision operator has two velocity deriva-
tives and can only balance the terms on the left-hand side of
Eq. (249) if
∼ ω ⇒ δv
, (251)
where ω is the characteristic frequency of the fluctuations
of hi. If νii ≪ ω, δv/vthi ≪ 1. This is certainly true for
k⊥ρi ∼ 1: taking ω ∼ k‖vA and using k‖λmfpi ≫ 1 (which
is the appropriate limit at and below the ion gyroscale for
most of the plasmas of interest; cf. footnote 24), we have
νii/ω ∼
βi/k‖λmfpi ≪ 1.
The condition (251) means that the collision rate can be ar-
bitrarily small—this will always be compensated by the suf-
ficiently fine velocity-space structure of the distribution func-
tion to produce a finite amount of entropy production (heat-
ing) independent of νii in the limit νii → +0. The situa-
tion bears some resemblance to the emergence of small spa-
tial scales in neutral-fluid turbulence with arbitrarily small
but non-zero viscosity (Kolmogorov 1941). The analogy
is not perfect, however, because the ion gyrokinetic equa-
tion (249) does not contain a nonlinear interaction term that
would explicitly cause a cascade in the velocity space. In-
stead, the (ring-averaged) KAW turbulence mixes hi in the
gyrocenter space via the nonlinear term in Eq. (249), so hi
will have small-scale structure in Ri on characteristic scales
much smaller than ρi. Let us assume that the dominant non-
linear effect is a local interaction of the small-scale fluctua-
tions of hi with the similarly small-scale component of 〈Φ〉Ri .
Since ring averaging is involved and k⊥ρi is large, the val-
ues of 〈Φ〉Ri corresponding to two velocities v and v′ will
come from spatially decorrelated electromagnetic fluctuations
if k⊥v⊥/Ωi and k⊥v
⊥/Ωi [the argument of the Bessel function
in Eq. (247)] differ by order unity, i.e., for
|v⊥ − v′⊥|
(252)
(see Fig. 10). This relation gives a correspondence between
the decorrelation scales of hi in the position and velocity
space. Combining Eqs. (252) and (251), we see that there is
a collisional cutoff scale determined by k⊥ρi ∼ (ω/νii)1/2 ≫
1.32 The cutoff scale is much smaller than the ion gyroscale.
In the range between these scales, collisional dissipation is
small. The ion entropy fluctuations are transferred across this
scale range by means of a cascade, for which we will con-
struct a scaling theory in § 7.9.2 (and, for the case without the
background KAW turbulence, in § 7.10).
It is important to emphasize that no matter how small the
collisional cutoff scale is, all of the generalized energy chan-
neled into the entropy cascade at the ion gyroscale eventually
reaches it and is converted into heat. Note that the rate at
which this happens is in general amplitude-dependent because
the process is nonlinear, although we will argue in § 7.9.4 (see
also § 7.10.3) that the nonlinear cascade time and the parallel
linear propagation (particle streaming) time are related by a
critical-balance-like condition (we will also argue there that
the linear parallel phase mixing, which can generate small
scales in v‖, is a less efficient process than the nonlinear per-
pendicular one discussed above).
It is interesting to note the connection between the entropy
cascade and certain aspects of the gyrofluid closure formal-
ism developed by Dorland & Hammett (1993). In their the-
ory, the emergence of small scales in v⊥ manifested itself as
the growth of high-order v⊥ moments of the gyrocenter distri-
bution function. They correctly identified this effect as a con-
sequence of the nonlinear perpendicular phase mixing of the
gyrocenter distribution function caused by a perpendicular-
velocity-space spread in the ring-averaged E ×B velocities
(given by 〈uE〉Ri = ẑ×∇〈Φ〉Ri in our notation) arising at and
below the ion gyroscale.
7.9.2. Scalings
Since entropy is a conserved quantity, we will follow the
well trodden Kolmogorov path, assume locality of interac-
tions in scale space and constant entropy flux, and conclude,
analogously to Eq. (1),
v8thi
h2iλ ∼ εh = const, (253)
where εh is the entropy flux proportional to the fraction of the
total turbulent power ε (or Pext; see § 3.4) that was diverted
into the entropy cascade at the ion gyroscale, and is the cas-
cade time that we now need to find.
By the critical-balance assumption, the decorrelation time
of the electromagnetic fluctuations in KAW turbulence is
comparable at each scale to the KAW period at that scale and
to the nonlinear interaction time [Eq. (239)]:
τKAWλ ∼
(1 +βi)1/3
. (254)
The characteristic time associated with the nonlinear term in
Eq. (249) is longer than τKAWλ by a factor of (ρi/λ)
1/2 due to
the ring averaging, which reduces the strength of the nonlinear
interaction. This weakness of the nonlinearity makes it pos-
sible to develop a systematic analytical theory of the entropy
32 Another source of small-scale spatial smoothing comes from the per-
pendicular gyrocenter-diffusion terms ∼ −νii(v/vthi)2k2⊥ρ
i hik that arise in
the ring-averaged collision operators, e.g., the second term in the model
operator (B13). These terms again enforce a cutoff wavenumber such that
k⊥ρi ∼ (ω/νii)
1/2 ≫ 1.
36 SCHEKOCHIHIN ET AL.
cascade (Schekochihin & Cowley 2009). It is also possible
to estimate the cascade time via a more qualitative argument
analogous to that first devised by Kraichnan (1965) for the
weak turbulence of Alfvén waves: during each KAW correla-
tion time τKAWλ, the nonlinearity changes the amplitude of hi
by only a small amount:
∆hiλ ∼ (λ/ρi)1/2hiλ ≪ hiλ; (255)
these changes accumulate with time as a random walk,
so after time t, the cumulative change in amplitude is
∆hiλ(t/τKAWλ)
1/2; finally, the cascade time t = is the time
after which the cumulative change in amplitude is compara-
ble to the amplitude itself, which gives, using Eq. (254),
τKAWλ ∼
(1 +βi)1/3
. (256)
Substituting this into Eq. (253), we get
hiλ ∼
v3thi
)1/2(
(1 +βi)1/6√
(257)
which corresponds to a k
⊥ spectrum of entropy.
In the argument presented above, we assumed that the scal-
ing of hi was determined by the nonlinear mixing of hi by
the ring-averaged KAW fluctuations rather than by the wave–
particle interaction term on the right-hand side of Eq. (249).
We can now confirm the validity of this assumption. The
change in amplitude of hi in one KAW correlation time τKAWλ
due to the wave–particle interaction term is
∆hiλ∼
v3thi
βiρivA
∼ n0i
v3thi
(εKAW
)1/3 1√
βi (1 +βi)1/3
7/6, (258)
where we have used Eq. (240). Comparing this with Eq. (255)
and using Eq. (257), we see that ∆hiλ in Eq. (258) is a factor
of (λ/ρi)
1/2 smaller than ∆hiλ due to the nonlinear mixing.
7.9.3. Phase-Space Cutoff
To work out the cutoff scales both in the position and veloc-
ity space, we use Eqs. (251) and (252): in Eq. (251), ω ∼ 1/,
where is the characteristic decorrelation time of hi given by
Eq. (256); using Eq. (252), we find the cutoffs:
∼ (νiiτρi )3/5 = Do−3/5, (259)
where τρi is the cascade time [Eq. (256)] taken at λ = ρi.
By a recently established convention, the dimensionless num-
ber Do = 1/νiiτρi is called the Dorland number. It plays
the role of Reynolds number for kinetic turbulence, mea-
suring the scale separation between the ion gyroscale and
the collisional dissipation scale (Schekochihin et al. 2008b;
Tatsuno et al. 2009a,b).
7.9.4. Parallel Phase Mixing
Another assumption, which was made implicitly, was that
the parallel phase mixing due to the second term on the left-
hand side of Eq. (249) could be ignored. This requires jus-
tification, especially because it is with this “ballistic” term
that one traditionally associates the emergence of small-scale
structure in the velocity space (e.g., Krommes & Hu 1994;
Krommes 1999; Watanabe & Sugama 2004). The effect of
the parallel phase mixing is to produce small scales in veloc-
ity space δv‖ ∼ 1/k‖t. Let us assume that the KAW turbu-
lence imparts its parallel decorrelation scale to hi and use the
scaling relation (241) to estimate k‖ ∼ l−1‖λ. Then, after one
cascade time [Eq. (256)], hi is decorrelated on the parallel
velocity scales
βi(1 +βi)
∼ 1. (260)
We conclude that the nonlinear perpendicular phase mixing
[Eq. (259)] is more efficient than the linear parallel one. Note
that up to a βi-dependent factor Eq. (260) is equivalent to a
critical-balance-like assumption for hi in the sense that the
propagation time is comparable to the cascade time, or k‖v‖ ∼
−1 [see Eq. (249)].
7.10. Entropy Cascade in the Absence of KAW Turbulence
It is not currently known how one might determine ana-
lytically what fraction of the turbulent power arriving from
the inertial range to the ion gyroscale is channeled into the
KAW cascade and what fraction is dissipated via the kinetic
ion-entropy cascade introduced in § 7.9 (perhaps it can only
be determined by direct numerical simulations). It is cer-
tainly a fact that in many solar-wind measurements, the rel-
atively shallow magnetic-energy spectra associated with the
KAW cascade (§ 7.5) fail to appear and much steeper spectra
are detected (close to k−4; see Leamon et al. 1998; Smith et al.
2006). In view of this evidence, it is interesting to ask what
would be the nature of electromagnetic fluctuations below the
ion gyroscale if the KAW cascade failed to be launched, i.e.,
if all (or most) of the turbulent power were directed into the
entropy cascade (i.e., if W ≃Whi in § 7.8).
7.10.1. Equations
It is again possible to derive a closed set of equations for all
fluctuating quantities.
Let us assume (and verify a posteriori; § 7.10.4) that the
characteristic frequency of such fluctuations is much lower
than the KAW frequency [Eq. (230)] so that the first term in
Eq. (116) is small and the equation reduces to the balance of
the other two terms. This gives
, (261)
meaning that the electrons are purely Boltzmann [he = 0 to
lowest order; see Eq. (101)]. Then, from Eq. (118),
ρivthi
eik·r
d3vJ0(ai)hik (262)
Using Eq. (262), we find from Eq. (120) that the field-
strength fluctuations are
eik·r
v2thi
J1(ai)
hik, (263)
which is smaller than Zeϕ/T0i by a factor of βi/k⊥ρi.
Therefore, we can neglect δB‖/B0 compared to δne/n0e in
Eq. (117). Using Eq. (261), we get what is physically the
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 37
electron continuity equation:
+ b̂ ·∇
4πen0e
∇2⊥A‖ + u‖i
= 0, (264)
u‖i =
eik·r
d3vv‖J0(ai)hik. (265)
Note that in terms of the stream and flux functions, Eq. (264)
takes the form
ρ2i ∇2⊥Ψ =
, (266)
where we have approximated b̂ · ∇ ≃ ∂/∂z, which will, in-
deed, be shown to be correct in § 7.10.4.
Together with the ion gyrokinetic equation, which deter-
mines hi, Eqs. (261-264) form a closed set. They describe
low-frequency fluctuations of the density and electromagnetic
field due solely to the presence of fluctuations of hi below the
ion gyroscale.
It follows from Eq. (263) that δB‖/B0 contributes subdom-
inantly to 〈χ〉Ri [Eq. (69) with s = i and ai ≫ 1]. It will be
verified a posteriori (§ 7.10.4) that the same is true for A‖.
Therefore, Eqs. (247) and (249) continue to hold, as in the
case with KAW. This means that Eqs. (249) and (262) form
a closed subset. Thus the kinetic ion-entropy cascade is self-
regulating in the sense that hi is no longer passive (as it was
in the presence of KAW turbulence; § 7.9) but is mixed by
the ring-averaged “electrostatic” fluctuations of the scalar po-
tential, which themselves are produced by hi according to
Eq. (262).
The magnetic fluctuations are passive and determined by
the electrostatic and entropy fluctuations via Eqs. (263)
and (264).
7.10.2. Scalings
From Eq. (262), we can establish a correspondence between
Φλ and hiλ (the electrostatic fluctuations and the fluctuations
of the ion-gyrocenter distribution function):
Φλ ∼ ρivthi
hiλλ, (267)
where the factor of (λ/ρi)
1/2 comes from the Bessel function
[Eq. (248)] and the factor of (δv⊥/vthi)
1/2 results from the
v⊥ integration of the oscillatory factor in the Bessel function
times hi, which decorrelates on small scales in the velocity
space and, therefore, its integral accumulates in a random-
walk-like fashion. The velocity-space scales are related to the
spatial scales via Eq. (252), which was arrived at by an ar-
gument not specific to KAW-like fluctuations and, therefore,
continues to hold.
Using Eq. (267), we find that the wave–particle interaction
term in the right-hand side of Eq. (249) is subdominant: com-
paring it with ∂hi/∂t shows that it is smaller by a factor of
(λ/ρi)
3/2 ≪ 1. Therefore, it is the nonlinear term in Eq. (249)
that controls the scalings of hiλ and Φλ.
We now assume again the scale-space locality and con-
stancy of the entropy flux, so Eq. (253) holds. The cascade
(decorrelation) time is equal to the characteristic time associ-
ated with the nonlinear term in Eq. (249): ∼ (ρi/λ)1/2λ2/Φλ.
Substituting this into Eq. (253) and using Eq. (267), we ar-
rive at the desired scaling relations for the entropy cascade
(Schekochihin et al. 2008b):
v3thi
)1/3 1√
1/6, (268)
)1/3 vthi√
7/6, (269)
)1/3 √
1/3, (270)
where l0 = v
A/ε, as in § 1.2. Note that since the existence
of this cascade depends on it not being overwhelmed by the
KAW fluctuations, we should have εKAW ≪ ε and εh = ε −
εKAW ≈ ε.
The scaling for the ion-gyrocenter distribution function,
Eq. (268), implies a k
⊥ spectrum—the same as for the KAW
turbulence [Eq. (257)]. The scaling for the the cascade time,
Eq. (270), is also similar to that for the KAW turbulence
[Eq. (256)]. Therefore the velocity- and gyrocenter-space cut-
offs are still given by Eq. (259), where τρi is now given by
Eq. (270) taken at λ = ρi.
A new feature is the scaling of the scalar potential, given by
Eq. (269), which corresponds to a k
−10/3
⊥ spectrum (unlike the
KAW spectrum, § 7.5). This is a measurable prediction for the
electrostatic fluctuations: the implied electric-field spectrum
⊥ . From Eq. (261), we also conclude that the density
fluctuations should have the same spectrum as the scalar po-
tential, k
−10/3
⊥ —another measurable prediction.
The scalings derived above for the spectra of the ion
distribution function and of the scalar potential have been
confirmed in the numerical simulations by Tatsuno et al.
(2009a,b), who studied decaying electrostatic gyrokinetic tur-
bulence in two spatial dimensions. They also found velocity-
space scalings in accord with Eq. (252) (using a spectral
representation of the correlation functions in the v⊥ space
based on the Hankel transform of the distribution function;
see Plunk et al. 2009).
7.10.3. Parallel Cascade and Parallel Phase Mixing
We have again ignored the ballistic term (the second on
the left-hand side) in Eq. (249). We will estimate the effi-
ciency of the parallel spatial cascade of the ion entropy and of
the associated parallel phase mixing by making a conjecture
analogous to the critical balance: assuming that any two per-
pendicular planes only remain correlated provided particles
can stream between them in one nonlinear decorrelation time
(cf. § 1.2 and § 7.9.4), we conclude that the parallel particle-
streaming frequency k‖v‖ should be comparable at each scale
to the inverse nonlinear time −1, so
k‖vthi ∼ 1. (271)
As we explained in § 7.9.4, the parallel scales in the velocity
space generated via the ballistic term are related to the parallel
wavenumbers by δv‖ ∼ 1/k‖t. From Eq. (271), we find that
after one cascade time , the typical parallel velocity scale is
δv‖/vthi ∼ 1, so the parallel phase mixing is again much less
efficient than the perpendicular one.
Note that Eq. (271) combined with Eq. (270) means that the
anisotropy is again characterized by the scaling relation k‖ ∼
⊥ , similarly to the case of KAW turbulence [see Eq. (241)
and § 7.9.4].
38 SCHEKOCHIHIN ET AL.
7.10.4. Scalings for the Magnetic Fluctuations
The scaling law for the fluctuations of the magnetic-field
strength follows immediately from Eqs. (263) and (269):
ρivthi
−11/6
13/6, (272)
whence the spectrum of these fluctuations is k
−16/3
The scaling of A‖ (the perpendicular magnetic fluctuations)
depends on the relation between k‖ and k⊥. Indeed, the ratio
between the first and the third terms on the left-hand side of
Eq. (264) [or, equivalently, between the first and second terms
on the right-hand side of Eq. (266)] is ∼
k‖vthi
. For a crit-
ically balanced cascade, this makes the two terms comparable
[Eq. (271)]. Using the first term to work out the scaling for the
perpendicular magnetic fluctuations, we get, using Eq. (269),
ρivthi
−11/6
13/6, (273)
which is the same scaling as for δB‖/B0 [Eq. (272)].
Using Eq. (273) together with Eqs. (269) and (270), it is
now straightforward to confirm the three assumptions made
in § 7.10.1 that we promised to verify a posteriori:
1. In Eq. (116), ∂A‖/∂t ≪ cb̂ ·∇ϕ, so Eq. (261) holds (the
electrons remain Boltzmann). This means that no KAW
can be excited by the cascade.
2. δB⊥/B0 ≪ k‖/k⊥, so b̂ ·∇ ≃ ∂/∂z in Eq. (264). This
means that field lines are not significantly perturbed.
3. In the expression for 〈χ〉Ri [Eq. (69)], v‖A‖/c ≪ ϕ, so
Eq. (249) holds. This means that the electrostatic fluc-
tuations dominate the cascade.
7.11. Cascades Superposed?
The spectra of magnetic fluctuations obtained in § 7.10.4
are very steep—steeper, in fact, than those normally observed
in the dissipation range of the solar wind (§ 8.2.5). One might
speculate that the observed spectra may be due to a superposi-
tion of the two cascades realizable below the ion gyroscale: a
high-frequency cascade of KAW (§ 7.5) and a low-frequency
cascade of electrostatic fluctuations due to the ion entropy
fluctuations (§ 7.10). Such a superposition could happen if
the power going into the KAW cascade is relatively small,
εKAW ≪ ε. One then expects an electrostatic cascade to be
set up just below the ion gyroscale with the KAW cascade
superseding it deeper into the dissipation range. Comparing
Eqs. (240) and (269), we can estimate the position of the spec-
tral break:
k⊥ρi ∼
ε/εKAW
. (274)
Since ρi/ρe ∼ (τmi/me)1/2/Z is not a very large number, the
dissipation range is not very wide. It is then conceivable that
the observed spectra are not true power laws but simply non-
asymptotic superpositions of the electrostatic and KAW spec-
tra with the observed range of “effective” spectral exponents
due to varying values of the spectral break (274) between the
two cascades.33
33 Several alternative theories that aim to explain the dissipation-range
spectra exist: see § 8.2.6.
The value of εKAW/ε specific to any particular set of param-
eters (βi, τ , etc.) is set by what happens at k⊥ρi ∼ 1 (§ 7.1;
see § 8.2.2, § 8.2.5, and § 8.5 for further discussion).
7.12. Below the Electron Gyroscale: The Last Cascade
Finally, let us consider what happens when k⊥ρe ≫ 1. At
these scales, we have to return to the full gyrokinetic sys-
tem of equations. The quasi-neutrality [Eq. (61)], parallel
[Eq. (62)] and perpendicular [Eq. (66)] Ampère’s law become
eik·r
d3vJ0(ae)hek, (275)
4πen0e
∇2⊥A‖ =
eik·r
d3vv‖J0(ae)hek, (276)
eik·r
v2the
J1(ae)
hek, (277)
where βe = βiZ/τ . We have discarded the velocity integrals
of hi both because the gyroaveraging makes them subdom-
inant in powers of (me/mi)
1/2 and because the fluctuations
of hi are damped by collisions [assuming the collisional cut-
off given by Eq. (259) lies above the electron gyroscale]. To
Eqs. (275-277), we must append the gyrokinetic equation for
he [Eq. (57) with s = e], thus closing the system.
The type of turbulence described by these equations is very
similar to that discussed in § 7.10. It is easy to show from
Eqs. (275-277) that
. (278)
Hence the magnetic fluctuations are subdominant in the ex-
pression for 〈χ〉Re [Eq. (69) with s = e and ae ≫ 1], so
〈χ〉Re ≃ 〈ϕ〉Re . The electron gyrokinetic equation then is
{〈ϕ〉Re ,he} =
, (279)
where the wave–particle interaction term in the right-hand
side has been dropped because it can be shown to be small
via the same argument as in § 7.10.2.
Together with Eq. (275), Eq. (279) describes the kinetic cas-
cade of electron entropy from the electron gyroscale down to
the scale at which electron collisions can dissipate it into heat.
This cascade the result of collisionless damping of KAW at
k⊥ρe ∼ 1, whereby the power in the KAW cascade is con-
verted into the electron-entropy fluctuations: indeed, in the
limit k⊥ρe ≫ 1, the generalized energy is simply
= Whe (280)
(see Fig. 5).
The same scaling arguments as in § 7.10.2 apply and scaling
relations analogous to Eqs. (268-270), and (272) duly follow:
v3the
(εKAW
1/6, (281)
(εKAW
vthe l
7/6, (282)
)1/3(
, (283)
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 39
(εKAW
−11/6
13/6, (284)
where l0 = v
A/ε, as in § 1.2. The formula for the collisional
cutoffs in the wavenumber and velocity space is analogous to
Eq. (259):
∼ (νeiτρe )3/5, (285)
where τρe is the cascade time (283) taken at λ = ρe.
7.13. Validity of Gyrokinetics in the Dissipation Range
As the kinetic cascade takes the (generalized) energy to ever
smaller scales, the frequency ω of the fluctuations increases.
In applying the gyrokinetic theory, one must be mindful of
the need for this frequency to stay smaller than Ωi. Using
the scaling formulae for the characteristic times of the fluc-
tuations derived above [Eqs. (254), (270) and (283)], we can
determine the conditions for ω ≪ Ωi. Thus, for the gyroki-
netic theory to be valid everywhere in the inertial range, we
must have
k⊥ρi ≪ β3/4i
(286)
at all scales down to k⊥ρi ∼ 1, i.e., ρi/l0 ≪ β3/2i , not a very
stringent condition.
Below the ion gyroscale, the KAW cascade (§ 7.5) remains
in the gyrokinetic regime as long as
k⊥ρi ≪
i (1 +βi)
(287)
(we are assuming Ti/Te ∼ 1 everywhere). The condition for
this still to be true at the electron gyroscale is
i (1 +βi)
. (288)
The ion entropy fluctuations passively mixed by the KAW tur-
bulence (§ 7.9) satisfy Eq. (287) at all scales down to the ion
collisional cutoff [Eq. (259)] if
λmfpi
i (1 +βi)
. (289)
Note that the condition for the ion collisional cutoff to lie
above the electron gyroscale is
λmfpi
βi(1 +βi)1/3
)5/6(
(290)
In the absence of KAW turbulence, the pure ion-entropy cas-
cade (§ 7.10) remains gyrokinetic for
k⊥ρi ≪ β3/2i
. (291)
This is valid at all scales down to the ion collisional cutoff
provided λmfpi/l0 ≪ β3i (l0/ρi), an extremely weak condition,
which is always satisfied. This is because the ion-entropy
fluctuations in this case have much lower frequencies than in
the KAW regime. The ion collisional cutoff lies above the
electron gyroscale if, similarly to Eq. (290),
λmfpi
)5/6(
. (292)
If the condition (290) is satisfied, all fluctuations of the
ion distribution function are damped out above the electron
gyroscale. This means that below this scale, we only need
the electron gyrokinetic equation to be valid, i.e., ω ≪ Ωe.
The electron-entropy cascade (§ 7.12), whose characteristic
timescale is given by Eq. (283), satisfies this condition for
k⊥ρe ≪
β3/2e
. (293)
This is valid at all scales down to the electron
collisional cutoff [Eq. (285)] provided λmfpe/l0 ≪
(ε/εKAW)
2β3e (mi/me)
3(l0/ρe), which is always satisfied.
Within the formal expansion we have adopted (k⊥ρi ∼ 1
and k‖λmfpi ∼
βi), it is not hard to see that λmfpi/l0 ∼ ǫ2
and ρi/l0 ∼ ǫ3. Since all other parameters (me/mi, βi, βe
etc.) are order unity with respect to ǫ, all of the above con-
ditions for the validity of the gyrokinetics are asymptotically
correct by construction. However, in application to real as-
trophysical plasmas, one should always check whether this
construction holds. For example, substituting the relevant pa-
rameters for the solar wind shows that the gyrokinetic ap-
proximation is, in fact, likely to start breaking down some-
where between the ion and electron gyroscales (Howes et al.
2008a).34 This releases a variety of high-frequency wave
modes, which may be participating in the turbulent cascade
around and below the electron gyroscale (see, e.g., the recent
detailed observations of these scales in the magnetosheath by
Mangeney et al. 2006; Lacombe et al. 2006 or the early mea-
surements of high-frequency fluctuations in the solar wind by
Denskat et al. 1983; Coroniti et al. 1982).
7.14. Summary
In this section, we have analyzed the turbulence in the dissi-
pation range, which turned out to have many more essentially
kinetic features than the inertial range.
At the ion gyroscale, k⊥ρi ∼ 1, the kinetic cascade rear-
ranged itself into two distinct components: part of the (gener-
alized) energy arriving from the inertial range was collision-
lessly damped, giving rise to a purely kinetic cascade of ion-
entropy fluctuations, the rest was converted into a cascade of
Kinetic Alfvén Waves (KAW) (Fig. 5; see § 7.1 and § 7.8).
The KAW cascade is described by two fluid-like equa-
tions for two scalar functions, the magnetic flux function
Ψ = −A‖/
4πmin0i and the scalar potential, expressed, for
continuity with the results of § 5, in terms of the function
Φ = (c/B0)ϕ. The equations are (see § 7.2)
b̂ ·∇Φ, (294)
2 +βi
1 + Z/τ
) b̂ ·∇
ρ2i ∇2⊥Ψ
, (295)
where b̂ · ∇ = ∂/∂z + (1/vA){Ψ, · · ·}. The density and
34 See this paper also for a set of numerical tests of the validity of gy-
rokinetics in the dissipation range, a linear theory of the conversion of KAW
into ion-cyclotron-damped Bernstein waves, and a discussion of the potential
(un)importance of ion cyclotron damping for the dissipation of turbulence.
40 SCHEKOCHIHIN ET AL.
magnetic-field-strength fluctuations are directly related to the
scalar potential:
. (296)
We call Eqs. (294-296) the Electron Reduced Magnetohydro-
dynamics (ERMHD).
The ion-entropy cascade is described by the ion gyrokinetic
equation:
+{〈Φ〉Ri ,hi} = 〈Cii[hi]〉Ri . (297)
The ion distribution function is mixed by the ring-averaged
scalar potential and undergoes a cascade both in the velocity
and gyrocenter space—this phase-space cascade is essential
for the conversion of the turbulent energy into the ion heat,
which can ultimately only be done by collisions (see § 7.9).
If the KAW cascade is strong (its power εKAW is an order-
unity fraction of the total injected turbulent power ε), it de-
termines Φ in Eq. (297), so the ion-entropy cascade is passive
with respect to the KAW turbulence. Equations (294-295) and
(297) form a closed system that determines the three func-
tions Φ, Ψ, hi, of which the latter is slaved to the first two.
One can also compute δne and δB‖, which are proportional
to Φ [Eq. (296)]. The generalized energy conserved by these
equations is given by Eq. (245).
If the KAW cascade is weak (εKAW ≪ ε), the ion-entropy
cascade dominates the turbulence in the dissipation range and
drives low-frequency mostly electrostatic fluctuations, with a
subdominant magnetic component. These are given by the
following relations (see § 7.10)
ρivthi
2(1 + τ/Z)
eik·r
d3vJ0(ai)hik, (298)
ρivthi
, (299)
eik·r ×
1 + Z/τ
J0(ai)
hik, (300)
eik·r
v2thi
J1(ai)
hik, (301)
where ai = k⊥v⊥/Ωi, Equations (297) and (298) form a closed
system for Φ and hi. The rest of the fields, namely δne, Ψ and
δB‖, are slaved to hi via Eqs. (299-301).
The fluid and kinetic models summarized above are valid
between the ion and electron gyroscales. Below the electron
gyroscale, the collisionless damping of the KAW cascade con-
verts it into a cascade of electron entropy, similar in nature to
the ion-entropy cascade (§ 7.12).
The KAW cascade and the low-frequency turbulence asso-
ciated with the ion-entropy cascade have distinct scaling be-
haviors. For the KAW cascade, the spectra of the electric,
density and magnetic fluctuations are (§ 7.5)
EE (k⊥) ∝ k−1/3⊥ , En(k⊥) ∝ k
⊥ , EB(k⊥) ∝ k
⊥ . (302)
For the ion- and electron-entropy cascades (§ 7.9 and § 7.12),
EE (k⊥) ∝ k−4/3⊥ , En(k⊥) ∝ k
−10/3
⊥ , EB(k⊥) ∝ k
−16/3
(303)
We argued in § 7.11 that the observed spectra in the dissipa-
tion range of the solar wind could be the result of a superpo-
sition of these two cascades, although a number of alternative
theories exist (§ 8.2.6).
8. DISCUSSION OF ASTROPHYSICAL APPLICATIONS
We have so far only occasionally referred to some relevant
observational evidence for space and astrophysical plasmas.
We now discuss in more detail how the theoretical framework
laid out above applies to real plasma turbulence in space.
Although we will discuss the interstellar medium, accre-
tion disks and galaxy clusters towards the end of this sec-
tion, the most rewarding source of observational information
about plasma turbulence in astrophysical conditions is the so-
lar wind and the magnetosheath because only there direct in
situ measurements of all the interesting quantities are possi-
ble. Measurements of the fluctuating magnetic and velocity
fields in the solar wind have been available since the 1960s
(Coleman 1968) and a vast literature now exists on their spec-
tra, anisotropy, Alfvénic character and many other aspects (a
short recent review is Horbury et al. 2005; two long ones are
Tu & Marsch 1995; Bruno & Carbone 2005). It is not our
aim here to provide a comprehensive survey of what is known
about plasma turbulence in the solar wind. Instead, we shall
limit our discussion to a few points that we consider impor-
tant in light of the theoretical framework proposed in this pa-
per.35 As we do this, we shall provide copious references to
the main body of the paper, so this section can be read as a
data-oriented guide to it, aimed both at a thorough reader who
has arrived here after going through the preceding sections
and an impatient one who has skipped to this one hoping to
find out whether there is anything of “practical” use in the
theoretical developments above.
8.1. Inertial-Range Turbulence in the Solar Wind
In the inertial range, i.e., for k⊥ρi ≪ 1, the solar-wind turbu-
lence should be described by the reduced hybrid fluid-kinetic
theory derived in § 5 (KRMHD). Its applicability hinges on
three key assumptions: (i) the turbulence is Alfvénic, i.e., con-
sists of small (δB/B0 ≪ 1) low-frequency (ω ∼ k‖vA ≪ Ωi)
perturbations of an ambient mean magnetic field and corre-
sponding velocity fluctuations; (ii) it is strongly anisotropic,
k⊥ ≫ k‖; (iii) the equilibrium distribution can be approxi-
mated or, at least, reasonably modeled by a Maxwellian with-
out loss of essential physics (this will be discussed in § 8.3).
If these assumptions are satisfied, KRMHD (summarized in
§ 5.7) is a rigorous set of dynamical equations for the inertial
range, a set of Kolmogorov-style scaling predictions for the
Alfvénic component of the turbulence can be produced (the
GS theory, reviewed in § 1.2), while to the compressive fluc-
tuations, the considerations of § 6 apply. So let us examine
the observational evidence.
8.1.1. Alfvénic Nature of the Turbulence
The presence of Alfvén waves in the solar wind was re-
ported already the early works of Unti & Neugebauer (1968)
and Belcher & Davis (1971). Alfvén waves are detected al-
ready at very low frequencies (large scales)—and, at these
35 An extended quantitative discussion of the applicability of the gyroki-
netic theory to the turbulence in the slow solar wind was given by Howes et al.
(2008a).
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 41
low frequencies, have a k−1 spectrum.36 This spectrum cor-
responds to a uniform distribution of scales/frequencies of
waves launched by the coronal activity of the Sun. Nonlin-
ear interaction of these waves gives rise to an Alfvénic tur-
bulent cascade of the type that was discussed above. The ef-
fective outer scale of this cascade can be detected as a spec-
tral break where the k−1 scaling steepens to the Kolmogorov
slope k−5/3 (see Bavassano et al. 1982; Marsch & Tu 1990a;
Horbury et al. 1996 for fast-wind results on the spectral break;
for a discussion of the effective outer scale in the slow wind
at 1 AU, see Howes et al. 2008a). The particular scale at
which this happens increases with the distance from the Sun
(Bavassano et al. 1982), reflecting the more developed state
of the turbulence at later stages of evolution. At 1 AU, the
outer scale is roughly in the range of 105 − 106 km; the k−5/3
range extends down to scales/frequencies that correspond to a
few times the ion gyroradius (102 − 103 km; see Table 1).
The range between the outer scale (the spectral break) and
the ion gyroscale is the inertial range. In this range, δB/B0 de-
creases with scale because of the steep negative spectral slope.
Therefore, the assumption of small fluctuations, δB/B0 ≪ 1,
while not necessarily true at the outer scale, is increasingly
better satisfied further into the inertial range (cf. § 1.3).
Are these fluctuations Alfvénic? In a plasma such as the
solar wind, they ought to be because, as showed in § 5.3, for
k⊥ρi ≪ 1, these fluctuations are rigorously described by the
RMHD equations. The magnetic flux is frozen into the ion
motions, so displacing a parcel of plasma should produce a
matching (Alfvénic) perturbation of the magnetic field line
and vice versa: in an Alfvén wave, u⊥ = ±δB⊥/
4πmin0i.
The strongest confirmation that this is indeed true for the
inertial-range fluctuations in the solar wind was achieved by
Bale et al. (2005), who compared the spectra of electric and
magnetic fluctuations and found that they both scale as k−5/3
and follow each other with remarkable precision (see Fig. 1).
The electric field is a very good measure of the perpendicular
velocity field because, for k⊥ρi ≪ 1, the plasma velocity is
the E×B drift velocity, u⊥ = cE× ẑ/B0 (see § 5.4).
This picture of agreement between basic theory and ob-
servations is upset in a disturbing fashion by an extraordi-
nary recent result by Chapman & Hnat (2007); Podesta et al.
(2006) and J. E. Borovsky (2008, private communication),
who claim different spectral indices for velocity and mag-
netic fluctuations—k−3/2 and k−5/3, respectively. This result
is puzzling because if it is asymptotically correct in the iner-
tial range, it implies either u⊥ ≫ δB⊥ or u⊥ ≪ δB⊥ and it is
not clear how perpendicular velocity fluctuations in a near-
ideal plasma could fail to produce Alfvénic displacements
and, therefore, perpendicular magnetic field fluctuations with
matching energies. Plausible explanations may be either that
the velocity field in these measurements is polluted by a non-
Alfvénic component parallel to the magnetic field (although
data analysis by Chapman & Hnat 2007 does not support this)
or that the flattening of the velocity spectrum is due to some
form of a finite-gyroradius effect or even an energy injection
into the velocity fluctuations at scales approaching the ion
gyroscale (e.g., from the pressure-anisotropy-driven instabili-
36 Inferred from the frequency spectrum f −1 via the Taylor (1938) hypoth-
esis, f ∼ k ·Vsw , where Vsw is the mean velocity at which the wind blows
past the spacecraft. The Taylor hypothesis is a good assumption for the so-
lar wind because Vsw (∼ 800 km/s in the fast wind, ∼ 300 km/s in the slow
wind) is highly supersonic, super-Alfvénic and far exceeds the fluctuating
velocities.
ties, § 8.3).
8.1.2. Energy Spectrum
How solid is the statement that the observed spectrum
has a k−5/3 scaling? In individual measurements of the
magnetic-energy spectra, very high accuracy is claimed
for this scaling: the measured spectral exponent is be-
tween 1.6 and 1.7; agreement with Kolmogorov value 1.67
is often reported to be within a few percent (see, e.g.,
Horbury et al. 1996; Leamon et al. 1998; Bale et al. 2005;
Narita et al. 2006; Alexandrova et al. 2008a; Horbury et al.
2008)). There is a somewhat wider scatter of spectral in-
dices if one considers large sets of measurement intervals
(Smith et al. 2006), but overall, the observational evidence
does not appear to be consistent with a k
⊥ spectrum consis-
tently found in the MHD simulations with a strong mean field
(Maron & Goldreich 2001; Müller et al. 2003; Mason et al.
2007; Perez & Boldyrev 2008, 2009; Beresnyak & Lazarian
2008b) and defended on theoretical grounds in the recent
modifications of the GS theory by Boldyrev (2006) and by
Gogoberidze (2007) (see footnote 10). This discrepancy be-
tween observations and simulations remains an unresolved
theoretical issue. It is probably best addressed by numeri-
cal modeling of the RMHD equations (§ 2.2) and by a de-
tailed comparison of the structure of the Alfvénic fluctuations
in such simulations and in the solar wind.
8.1.3. Anisotropy
Building up evidence for anisotropy of turbulent fluctua-
tions has progressed from merely detecting their elongation
along the magnetic field (Belcher & Davis 1971)—to fitting
data to an ad hoc model mixing a 2D perpendicular and a
1D parallel (“slab”) turbulent components in some propor-
tion37 (Matthaeus et al. 1990; Bieber et al. 1996; Dasso et al.
2005; Hamilton et al. 2008)—to formal systematic unbiased
analyses showing the persistent presence of anisotropy at all
scales (Bigazzi et al. 2006; Sorriso-Valvo et al. 2006)— to di-
rect measurements of three-dimensional correlation functions
(Osman & Horbury 2007)—and finally to computing spectral
exponents at fixed angles between k and B0 (Horbury et al.
2008). The latter authors appear to have achieved the first
direct quantitative confirmation of the GS theory by demon-
strating that the magnetic-energy spectrum scales as k
wavenumbers perpendicular to the mean field and as k−2
wavenumbers parallel to it [consistent with the first scaling
relation in Eq. (4)]. This is the closest that observations have
got to confirming the GS relation k‖ ∼ k
⊥ [see Eq. (5)] in a
real astrophysical turbulent plasma.
8.1.4. Compressive Fluctuations
According to the theory developed in § 5, the density and
magnetic-field-strength fluctuations are passive, energetically
decoupled from and mixed by the Alfvénic cascade (§ 5.5;
these are slow and entropy modes in the collisional MHD
limit—see § 2.4 and § 6.1). These fluctuations are expected to
be pressure-balanced, as expressed by Eq. (22) or, more gen-
erally in gyrokinetics, by Eq. (67). There is, indeed, strong
37 These techniques originate from the view of MHD turbulence as a su-
perposition of a 2D turbulence and an admixture of Alfvén waves (Fyfe et al.
1977; Montgomery & Turner 1981). As we discussed in § 1.2, we consider
the Goldreich & Sridhar (1995, 1997) view of a critically balanced Alfvénic
cascade to be better physically justified.
42 SCHEKOCHIHIN ET AL.
evidence that magnetic and thermal pressures in the solar
wind are anticorrelated, although there are some indications
of the presence of compressive, fast-wave-like fluctuations as
well (Roberts 1990; Burlaga et al. 1990; Marsch & Tu 1993;
Bavassano et al. 2004).
Measurements of density and field-strength fluctua-
tions done by a variety of different methods both at
1 AU (Celnikier et al. 1983, 1987; Marsch & Tu 1990b;
Bershadskii & Sreenivasan 2004; Hnat et al. 2005;
Kellogg & Horbury 2005; Alexandrova et al. 2008a) and
near the Sun (Lovelace et al. 1970; Woo & Armstrong 1979;
Coles & Harmon 1989; Coles et al. 1991) show fluctuation
levels of order 10% and spectra that appear to have a k−5/3
scaling above scales of order 102 − 103 km, which approxi-
mately corresponds to the ion gyroscale. The Kolmogorov
value of the spectral exponent is, as in the case of Alfvénic
fluctuations, measured quite accurately in individual cases
(1.67 ± 0.03 in Celnikier et al. 1987). Interestingly, the
higher-order structure function exponents measured for the
magnetic-field strength show that it is a more intermittent
quantity than the velocity or the vector magnetic field (i.e.,
than the Alfvénic fluctuations) and that the scaling expo-
nents are quantitatively very close to the values found for
passive scalars in neutral fluids (Bershadskii & Sreenivasan
2004; Bruno et al. 2007). One might argue that this lends
some support to the theoretical expectation of passive
magnetic-field-strength fluctuations.
Considering that in the collisionless regime these fluctua-
tions are supposed to be subject to strong kinetic damping
(§ 6.2.2), the presence of well-developed Kolmogorov-like
and apparently undamped turbulent spectra is more surprising
than has perhaps been publicly acknowledged. An extended
discussion of this issue was given in § 6.3. Without the in-
clusion of the dissipation effects associated with the finite ion
gyroscale, the passive cascade of the density and field strength
is purely perpendicular to the (exact) local magnetic field and
does not lead to any scale refinement along the field. This im-
plies highly anisotropic field-aligned structures, whose length
is determined by the initial conditions (i.e., conditions in the
corona). The kinetic damping is inefficient for such fluctua-
tions. While this would seem to explain the presence of fully
fledged power-law spectra, it is not entirely obvious that the
parallel cascade is really absent once dissipation is taken into
account (Lithwick & Goldreich 2001), so the issue is not yet
settled. This said, we note that there is plenty of evidence of
a high degree of anisotropy and field alignment of the den-
sity microstructure in the inner solar wind and outer corona
(e.g., Armstrong et al. 1990; Grall et al. 1997; Woo & Habbal
1997). There is also evidence that the local structure of the
compressive fluctuations at 1 AU is correlated with the coro-
nal activity, implying some form of memory of initial condi-
tions (Kiyani et al. 2007; Hnat et al. 2007; Wicks et al. 2009).
We note, finally, that whether compressive fluctuations in
the inertial range can develop short parallel scales should also
tell us how much ion heating can result from their damping
(see § 6.2.4).
8.2. Dissipation-Range Turbulence in the Solar Wind and the
Magnetosheath
At scales approaching the ion gyroscale, k⊥ρi ∼ 1, effects
associated with the finite extent of ion gyroorbits start to
matter. Observationally, this transition manifests itself as a
clear break in the spectrum of magnetic fluctuations, with the
inertial-range k−5/3 scaling replaced by a steeper slope (see
Fig. 1). While the electrons at these scales can be treated as
an isothermal fluid (as long as we are considering fluctuations
above the electron gyroscale, k⊥ρe ≪ 1; see § 4), the fully
gyrokinetic description (§ 3) has to be adopted for the ions.
It is, indeed, to understand plasma dynamics at and around
k⊥ρi ∼ 1 that gyrokinetics was first designed in fusion plasma
theory (Frieman & Chen 1982; Brizard & Hahm 2007). In or-
der for gyrokinetics and further dissipation-range approxima-
tions that follow from it (§ 7) to be a credible approach in
the solar wind and other space plasmas, it has to be estab-
lished that fluctuations at and below the ion gyroscale are still
strongly anisotropic, k‖ ≪ k⊥. If that is the case, then their
frequencies (ω∼ k‖vAk⊥ρi, see § 7.3) will still be smaller than
the cyclotron frequency in at least a part of the “dissipation
range”38—the range of scales k⊥ρi & 1 (see § 7.13).
Note that additional information about the dissipation-
range turbulence can be extracted from the measurements in
the magnetosheath—while scales above the ion gyroscale are
probably non-universal there, the dissipation range appears to
display universal behavior, mostly similar to the solar wind
(see, e.g., Alexandrova 2008). This complements the obser-
vational picture emerging from the solar-wind data and al-
lows us to learn more as fluctuation amplitudes in the mag-
netosheath are larger and much smaller scales can be probed
than in the solar wind (Mangeney et al. 2006; Lacombe et al.
2006; Alexandrova et al. 2008b).
8.2.1. Anisotropy
We know with a fair degree of certainty that the fluctu-
ations that cascade down to the ion gyroscale from the in-
ertial range are strongly anisotropic (§ 8.1.3). While it ap-
pears likely that the anisotropy persists at k⊥ρi ∼ 1, it is ex-
tremely important to have a clear verdict on this assumption
from solar wind measurements. While Leamon et al. (1998)
and, more recently, Hamilton et al. (2008) did present some
evidence that magnetic fluctuations in the solar wind have a
degree of anisotropy below the ion gyroscale, no definitive
study similar to Horbury et al. (2008) or Bigazzi et al. (2006);
Sorriso-Valvo et al. (2006) exists as yet. In the magne-
tosheath, where the dissipation-range scales are easier to mea-
sure than in the solar wind, recent analysis by Sahraoui et al.
(2006); Alexandrova et al. (2008b) does show evidence of
strong anisotropy.
Besides confirming the presence of the anisotropy, it would
be interesting to study its scaling characteristics: e.g., check
the scaling prediction k‖ ∼ k
⊥ [Eq. (241); see also § 7.9.4
and § 7.10.3] in a similar fashion as the GS relation k‖ ∼ k
[Eq. (5)] was corroborated by Horbury et al. (2008).
In this paper, we have proceeded on the assumption that
the anisotropy, and, therefore, low frequencies (ω ≪ Ωi) do
characterize fluctuations in the dissipation range—or, at least,
that the low-frequency anisotropic fluctuations are a signifi-
cant energy cascade channel and can be considered decoupled
from any possible high-frequency dynamics.
8.2.2. Transition at the Ion Gyroscale: Collisionless Damping and
Heating
38 This term, customary in the space-physics literature, is somewhat of
a misnomer because, as we have seen in § 7, rich dissipationless turbulent
dynamics are present in this range alongside what is normally thought of as
dissipation.
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 43
If the fluctuations at the ion gyroscale have k‖ ≪ k⊥ and
ω ≪ Ωi (§ 8.2.1), they are not subject to the cyclotron res-
onance (ω − k‖v‖ = ±Ωi), but are subject to the Landau one
(ω = k‖v‖). Alfvénic fluctuations at the ion gyroscale are
no longer decoupled from the compressive fluctuations and
can be Landau-damped (§ 7.1). It seems plausible that it
is the inflow of energy from the Alfvénic cascade that ac-
counts for a pronounced local flattening of the spectrum of
density fluctuations in the solar wind observed just above
the ion gyroscale (Woo & Armstrong 1979; Celnikier et al.
1983, 1987; Coles & Harmon 1989; Marsch & Tu 1990b;
Coles et al. 1991; Kellogg & Horbury 2005).39
In energetic terms, Landau damping amounts to a redis-
tribution of generalized energy from electromagnetic fluctu-
ations to entropy fluctuations (§ 3.4, § 7.8). This gives rise
to the entropy cascade, ultimately transferring the Landau-
damped energy into ion heat (§ 3.5, § 7.9 and § 7.10). How-
ever, only part of the inertial-range cascade is so damped be-
cause an alternative, electron, cascade channel exists: the ki-
netic Alfvén waves (§§ 7.2-7.8). The energy transferred into
the KAW-like fluctuations can cascade to the electron gy-
roscale, where it is Landau damped on electrons, converting
first into the electron entropy cascade and then electron heat
(§ 7.12).
Thus, the transition at the ion gyroscale ultimately de-
cides in what proportion the turbulent energy arriving from
the inertial range is distributed between the ion and electron
heat. How the fraction of power going into either depends on
parameters—βi, Ti/Te, amplitudes, . . . —is a key unanswered
question both in space and astrophysical (see, e.g., § 8.5) plas-
mas. Gyrokinetics appears to be an ideal tool for addressing
this question both analytically and numerically (Howes et al.
2008b). Within the framework outlined in this paper, the min-
imal model appropriate for studying the transition at the ion
gyroscale is the system of equations for isothermal electrons
and gyrokinetic ions derived in § 4 (it is summarized in § 4.9).
8.2.3. Ion Gyroscale vs. Ion Inertial Scale
It is often assumed in the space physics literature that it is
at the ion inertial scale, di = ρi/
βi, rather than at the ion gy-
roscale ρi that the spectral break between the inertial and dis-
sipation range occurs. The distinction between di and ρi be-
comes noticeable when βi is significantly different from unity,
a relatively rare occurrence in the solar wind. While some at-
tempts to determine at which of these two scales a spectral
break between the inertial and dissipation ranges occurs have
produced claims that di is a more likely candidate (Smith et al.
2001), more comprehensive studies of the available data sets
conclude basically that it is hard to tell (Leamon et al. 2000;
Markovskii et al. 2008).
In the gyrokinetic approach advocated in this paper, the ion
inertial scale does not play a special role (see § 7.1). The only
parameter regime in which di does appear as a special scale
is Ti ≪ Te (“cold ions”), when the Hall MHD approximation
can be derived in a systematic way (see Appendix E). This,
however, is not the right limit for the solar wind or most other
astrophysical plasmas of interest because ions are rarely cold.
Hall MHD is discussed further in § 8.2.6 and Appendix E.
8.2.4. KAW Turbulence
39 Celnikier et al. (1987) proposed that the flattening might be a k−1 spec-
trum analogous to Batchelor’s spectrum of passive scalar variance in the
viscous-convective range. We think this analogy cannot apply because den-
sity is not passive at or below the ion gyroscale.
If gyrokinetics is valid at scales k⊥ρi & 1 (i.e., if k‖ ≪ k⊥,
ω ≪ Ωi and it is acceptable to at least model the equilibrium
distribution as a Maxwellian; see § 8.3), the electromagnetic
fluctuations below the ion gyroscale will be described by the
fluid approximation that we derived in § 7.2 and referred to
ERMHD. The wave solutions of this system of equations are
the kinetic Alfvén waves (§§ 7.3-7.4) and it is possible to ar-
gue for a GS-style critically balanced cascade of KAW-like
electromagnetic fluctuations (§ 7.5) between the ion and elec-
tron gyroscales (Landau damped on electrons at k⊥ρe ∼ 1; the
expression for the KAW damping rate in the gyrokinetic limit
is given in Howes et al. 2006; see also Fig. 8).
Individual KAW have, indeed, been detected in space plas-
mas (e.g., Grison et al. 2005). What about KAW turbulence?
How does one tell whether any particular spectral slope one is
measuring corresponds to the KAW cascade or fits some alter-
native scheme for the dissipation-range turbulence (§ 8.2.6)?
It appears to be a sensible program to look for specific rela-
tionships between different fields predicted by theory (§ 7.2)
and for the corresponding spectral slopes and scaling relations
for the anisotropy (§ 7.5). This means that simultaneous mea-
surements of magnetic, electric, density and magnetic-field-
strength fluctuations are needed.
For the solar wind, the spectra of electric and magnetic
fluctuations below the ion gyroscale reported by Bale et al.
(2005) are consistent with the k−1/3 and k−7/3 scalings pre-
dicted for an anisotropic critically balanced KAW cascade
(§ 7.5; see Fig. 1 for theoretical scaling fits superimposed
on a plot taken from Bale et al. 2005; note, however, that
Bale et al. 2005 themselves interpreted their data in a some-
what different way and that their resolution was in any case
not sufficient to be sure of the scalings). They were also able
to check that their fluctuations satisfied the KAW dispersion
relation—for critically balanced fluctuations, this is, indeed,
plausible. Magnetic-fluctuation spectra recently reported by
Alexandrova et al. (2008a) are only slightly steeper than the
theoretical k−7/3 KAW spectrum. These authors also find a
significant amount of magnetic-field-strength fluctuations in
the dissipation range, with a spectrum that follows the same
scaling—this is again consistent with the theoretical picture
of KAW turbulence [see Eq. (223)]. Measurements reported
by Czaykowska et al. (2001); Alexandrova et al. (2008b) for
the magnetosheath appear to present a similar picture.
The density spectra measured by Celnikier et al. (1983,
1987) steepen below the ion gyroscale following the flattened
segment around k⊥ρi ∼ 1 (discussed in § 8.2.2). For a KAW
cascade, the density spectrum should be k−7/3 (§ 7.5); with-
out KAW, k−10/3 (§ 7.10.2). The slope observed in the papers
cited above appears to be somewhat shallower even than k−2
(cf. a similar result by Spangler & Gwinn 1990 for the ISM;
see § 8.4.1), but, given imperfect resolution, neither seriously
in contradiction with the prediction based on the KAW cas-
cade, nor sufficient to corroborate it. Unfortunately, we have
not found published simultaneous measurements of density-
and magnetic- or electric-fluctuation spectra.
8.2.5. Variability of the Spectral Slope
While many measurements consistent with the KAW pic-
ture can be found, there are also many in which the spectra
are much steeper (Denskat et al. 1983; Leamon et al. 1998).
Analysis of a large set of measurements of the magnetic-
fluctuation spectra in the dissipation range of the solar wind
reveals a wide spread in the spectral indices: roughly between
44 SCHEKOCHIHIN ET AL.
−1 and −4 (Smith et al. 2006). There is evidence of a weak
positive correlation between steeper dissipation-range spectra
and higher ion temperatures (Leamon et al. 1998) or higher
cascade rates calculated from the inertial range (Smith et al.
2006). This suggests that a larger amount of ion heating may
correspond to a fully or partially suppressed KAW cascade,
which is in line with our view of the ion heating and the KAW
cascade as the two competing channels of the overall kinetic
cascade (§ 7.8). With a weakened KAW cascade, all or part of
the dissipation range would be dominated by the ion entropy
cascade—a purely kinetic phenomenon manifested by pre-
dominantly electrostatic fluctuations and very steep magnetic-
energy spectra (§ 7.10). This might account both for the steep-
ness of the observed spectra and for the spread in their indices
(§ 7.11), although many other theories exist (see § 8.2.6).
While we may thus have a plausible argument, this is not
yet a satisfactory quantitative theory that would allow us to
predict when the KAW cascade is present and when it is not or
what dissipation-range spectrum should be expected for given
values of the solar-wind parameters (βi, Ti/Te, etc.). Resolu-
tion of this issue again appears to hinge on the question of how
much turbulent power is diverted into the ion entropy cascade
(equivalently, into ion heat) at the ion gyroscale (see § 8.2.2).
8.2.6. Alternative Theories of the Dissipation Range
A number of alternative theories and models have been put
forward to explain the observed spectral slopes (and their vari-
ability) in the dissipation range. It is not our aim to review or
critique them all in detail, but perhaps it is useful to provide a
few brief comments about some of them in light of the theo-
retical framework constructed in this paper.
This entire theoretical framework hinges on adopting gy-
rokinetics as a valid description or, at least, a sensible model
that does not miss any significant channels of energy cascade
and dissipation. While we obviously believe this to be the
right approach, it is worth spelling out what effects are left
out “by construction.”
Parallel Alfvén-wave cascade and ion cyclotron damping. — The
use of gyrokinetics assumes that fluctuations stay anisotropic
at all scales, k‖ ≪ k⊥, and, therefore, ω ≪ Ωi, so the
cyclotron resonances are ordered out. However, if one
insists on routing the Alfvén-wave energy into a paral-
lel cascade, e.g., by forcibly setting k⊥ = 0, it is pos-
sible to construct a weak turbulence theory in which it
is dissipated by the ion cyclotron damping (Yoon & Fang
2008). Numerical simulations of 3D MHD turbulence do
not support the possibility of a parallel Alfvén-wave cascade
(Shebalin et al. 1983; Oughton et al. 1994; Cho & Vishniac
2000; Maron & Goldreich 2001; Cho et al. 2002; Müller et al.
2003). Solar-wind evidence that the perpendicular cascade
dominates is quite strong for the inertial range (§ 8.1.3) and
less so for the dissipation range (§ 8.2.1). While, as stated
in § 8.2.1, one cannot yet definitely claim that observations
tell us that ω ≪ Ωi at k⊥ρi ∼ 1, it has been argued that
observations do not appear to be consistent with cyclotron
damping being the main mechanism for the dissipation of
the inertial-range Alfvénic turbulence at the ion gyroscale
(Leamon et al. 1998, 2000; Smith et al. 2001). Ion-cyclotron
resonance could conceivably be reached somewhere in the
dissipation range (see § 7.13). At this point gyrokinetics will
formally break down, although, as argued by Howes et al.
(2008a, see their § 3.6), this does not necessarily mean that
ion cyclotron damping will become the dominant dissipation
channel for the turbulence.
Parallel whistler cascade. — A parallel magnetosonic/whistler
cascade eventually damped by the electron cyclotron
resonance (Stawicki et al. 2001) is also excluded in the
construction of gyrokinetics. The whistler cascade has
been given some consideration in the Hall MHD approxi-
mation (further discussed at the end of this section). Both
weak-turbulence theory (Galtier 2006) and 3D numerical
simulations (Cho & Lazarian 2004) concluded that, like
in MHD, the turbulent cascade is highly anisotropic, with
perpendicular energy transfer dominating over the parallel
one.40 The same conclusion appears to have been reached
in recent 2D kinetic PIC simulations by Gary et al. (2008);
Saito et al. (2008). Thus, the turbulence again seems to be
driven into the gyrokinetically accessible regime.
While theory and numerical simulations appear to make
arguing in favor of a parallel cascade and cyclotron heat-
ing difficult, there exists some observational evidence in sup-
port of them, especially for the near-Sun solar wind (e.g.,
Harmon & Coles 2005). Thus, the presence or relative im-
portance of the cyclotron heating in the solar wind and, more
generally, the mechanism(s) responsible for the observed per-
pendicular ion heating (Marsch et al. 1983) remain a largely
open problem. Besides the theories mentioned above, many
other ideas have been proposed, some of which attempted
to reconcile the dominance of the low-frequency perpendic-
ular cascade with the possibility of cyclotron heating (e.g.,
Chandran 2005b; Markovskii et al. 2006; see Hollweg 2008
for a concise recent review of the problem).
Mirror cascade. — Sahraoui et al. (2006) analyzed a set of
Cluster multi-spacecraft measurements in the magnetosheath
and reported a broad power-law (∼ k−8/3) spectrum of mirror
structures at and below the ion gyroscale. They claim that
these are not KAW-like fluctuations because their frequency
is zero in the plasma frame. Although these structures are
highly anisotropic with k‖ ≪ k⊥, they cannot be described by
the gyrokinetic theory in its present form because δB‖/B0 is
very large (∼ 40%, occasionally reaching unity) and because
the particle trapping by fluctuations, which is likely to be
important in the nonlinear physics of the mirror instabil-
ity (Kivelson & Southwood 1996; Pokhotelov et al. 2008;
Rincon et al. 2009), is ordered out in gyrokinetics. Thus, if a
“mirror cascade” exists, it is not captured in our description.
More generally, the effect of the pressure-anisotropy-driven
instabilities on the turbulence in the dissipation range is a
wide open area, requiring further analytical effort (see § 8.3).
If k‖ ≪ k⊥, ω ≪ Ωi, and δB/B0 ≪ 1 are accepted for the
dissipation range and plasma instabilities at the ion gyroscale
(§ 8.3) are ignored, the formal gyrokinetic theory and its
asymptotic consequences derived above should hold. There
are two essential features of the linear physics at and below
the ion gyroscale that must play some role: the collisionless
(Landau) damping and the dispersive nature of the wave so-
lutions (see Fig. 8 and § 7.3; cf., e.g., Leamon et al. 1999;
Stawicki et al. 2001). Both of these features have been em-
ployed to explain the spectral break at the ion gyroscale and
the spectral slopes below it.
40 It is possible to produce a parallel cascade artificially by running 1D
simulations (Matthaeus et al. 2008b).
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 45
Landau damping and instrumental effects. — In most of our dis-
cussion, (§ 7, §§ 8.2.4-8.2.5), we effectively assumed that the
Landau damping is only important at k⊥ρi ∼ 1 and k⊥ρe ∼ 1,
but not in between, so we could talk about asymptotic scal-
ings and dissipationless cascades. However, as was noted
in § 7.6, a properly asymptotic scaling behavior in the dis-
sipation range is probably impossible in nature because the
scale separation between the ion and electron gyroscales is
only about (mi/me)
1/2 ≃ 43. In particular, there is not always
a wide scale interval where the kinetic damping is negligi-
bly small (especially at low βi; see Fig. 8; cf. Leamon et al.
1999). Howes et al. (2008a) proposed a model of how the
presence of damping combined with instrumental effects (a
resolution floor) could lead to measured spectra that look like
power laws steeper than k−7/3, with the effective spectral ex-
ponent depending on plasma parameters (we refer the reader
to that paper for a discussion of how this compares with pre-
vious models of a similar kind, e.g., Li et al. 2001). A key
physical assumption of theirs and similar models is that the
amount of power drained from the Alfvén-wave and KAW
cascades into the ion heat is set by the strength of the linear
damping. Whether this is justified is not yet clear.
Hall and Electron MHD. — If Landau damping is deemed
unimportant in some part of the dissipation range (which
can be true in some regimes; see Fig. 8 and Howes et al.
2006, 2008a,b) and the wave dispersion is considered to
be the salient feature, it might appear that a fluid, rather
than kinetic, description should be sufficient. Hall MHD
(Mahajan & Yoshida 1998) or its kdi ≫ 1 limit the Electron
MHD (Kingsep et al. 1990) have been embraced by many au-
thors as such a description, suitable both for analytical argu-
ments (Goldreich & Reisenegger 1992; Krishan & Mahajan
2004; Gogoberidze 2005; Galtier & Bhattacharjee 2003;
Galtier 2006; Alexandrova et al. 2008a) and numerical sim-
ulations (Biskamp et al. 1996, 1999; Ghosh et al. 1996;
Ng et al. 2003; Cho & Lazarian 2004; Shaikh & Zank 2005;
Galtier & Buchlin 2007; Matthaeus et al. 2008b).
To what extent does this constitute an approach alterna-
tive to (and better than?) gyrokinetics (as suggested, e.g., by
Matthaeus et al. 2008b)? For fluctuations with k‖ ≪ k⊥, Hall
MHD is merely a particular limit of gyrokinetics: βi ≪ 1 and
Ti/Te ≪ 1 (cold-ion limit; see Appendix E). If k‖ is not small
compared to k⊥, then the gyrokinetics is not valid, while Hall
MHD continues to describe the cold-ion limit correctly (e.g.,
Ito et al. 2004; Hirose et al. 2004), capturing in particular the
whistler branch of the dispersion relation. However, as we
have already mentioned above, the dominance of the perpen-
dicular energy transfer (k‖ ≪ k⊥) is supported both by weak-
turbulence theory for Hall MHD (Galtier 2006) and by 3D
numerical simulations of the Electron MHD (Cho & Lazarian
2004).
Thus, the gyrokinetic theory and its rigorous limits, such
as ERMHD (§ 7.2), supersede Hall MHD for anisotropic tur-
bulence. Since ions are generally not cold in the solar wind
(or any other plasma discussed here), Hall MHD is not for-
mally a relevant approximation. It also entirely misses the
kinetic damping and the associated entropy cascade channel
leading to particle heating (§ 7.1, § 7.9 and § 7.10). However,
Hall MHD does capture the Alfvén waves becoming disper-
sive and numerical simulations of it do show a spectral break,
although, technically speaking, at the wrong scale (di instead
of ρi; see § 7.1). Although Hall MHD cannot be rigorously
used as quantitative theory of the spectral break and the asso-
ciated change in the nature of the turbulent cascade, the Hall
MHD equations in the limit kdi ≫ 1 are mathematically sim-
ilar to our ERMHD equations (see § 7.2 and Appendix E) to
within constant coefficients probably not essential for quali-
tative models of turbulence. Therefore, results of numerical
simulations of Hall and Electron MHD cited above are di-
rectly useful for understanding the KAW cascade—and, in-
deed, in the limit kdi ≫ 1, kde ≪ 1, they are mostly consistent
with the scaling arguments of § 7.5.
Alfvén vortices. — Finally we mention an argument pertaining
to the dissipation-range spectra that is not based on energy
cascades at all. Based on the evidence of Alfvén vortices in
the magnetosheath, Alexandrova (2008) speculated that steep
power-law spectra observed in the dissipation range at least
in some cases could reflect the geometry of the ion-gyroscale
structures rather than a local energy cascade. If Alfvén vor-
tices are a common feature, this possibility cannot be ex-
cluded. However, the resulting geometrical spectra are quite
steep (k−4 and steeper), so they can become important only
if the KAW cascade is weak or suppressed—somewhat simi-
larly to the steep spectra associated with the entropy cascade
(§ 7.11).
8.3. Is Equilibrium Distribution Isotropic and Maxwellian?
In rigorous theoretical terms, the weakest point of this pa-
per is the use of a Maxwellian equilibrium. Formally, this is
only justified when the collisions are weak but not too weak:
we ordered the collision frequency as similar to the fluctu-
ation frequency [Eq. (49)]. This degree of collisionality is
sufficient to prove that a Maxwellian equilibrium distribution
F0s(v) does indeed emerge in the lowest order of the gyroki-
netic expansion (Howes et al. 2006). This argument works
well for plasmas such as the ISM (§ 8.4), where collisions are
weak (λmfpi ≫ ρi) but non-negligible (λmfpi ≪ L). In space
plasmas, the mean free path is of the order of 1 AU—the dis-
tance between the Sun and the Earth (see Table 1). Strictly
speaking, in so highly collisionless a plasma, the equilib-
rium distribution does not have to be either Maxwellian or
isotropic.
The conservation of the first adiabatic invariant, µ = v2⊥/2B,
suggests that temperature anisotropy with respect to the
magnetic-field direction (T0⊥ 6= T0‖) may exist. When the
relative anisotropy is larger than (roughly) 1/βi, it triggers
several very fast growing plasma instabilities: most promi-
nently the firehose (T0⊥ < T0‖) and mirror (T0⊥ > T0‖) modes
(e.g., Gary et al. 1976). Their growth rates peak around the
ion gyroscale, thus giving rise to additional energy injection
at k⊥ρi ∼ 1.
No definitive analytical theory of how these fluctuations sat-
urate, cascade and affect the equilibrium distribution has been
proposed. It appears to be a reasonable expectation that the
fluctuations resulting from temperature anisotropy will satu-
rate by limiting this anisotropy. This idea has some support
in solar-wind observations: while the degree of anisotropy
of the core particle distribution functions varies consider-
ably between data sets, the observed anisotropies do seem
to populate the part of the parameter plane (T0⊥/T0‖,βi) cir-
cumscribed in a rather precise way by the marginal stabil-
ity boundaries for the mirror and firehose (Gary et al. 2001;
Kasper et al. 2002; Marsch et al. 2004; Hellinger et al. 2006;
Matteini et al. 2007).41
41 Note that Kellogg et al. (2006) measure the electric-field fluctuations
46 SCHEKOCHIHIN ET AL.
If we want to study turbulence in data sets that do not lie
too close to these stability boundaries, assuming an isotropic
Maxwellian equilibrium distribution [Eq. (54)] is probably
an acceptable simplification, although not an entirely rigor-
ous one. Further theoretical work is clearly possible on this
subject: thus, it is not a problem to formulate gyrokinetics
with an arbitrary equilibrium distribution (Frieman & Chen
1982) and starting from that, once can generalize the results
of this paper (for the KRMHD system, § 5, this has been done
by Chen et al. 2009). Treating the instabilities themselves
might prove more difficult, requiring the gyrokinetic order-
ing to be modified and the expansion carried to higher orders
to incorporate features that are not captured by gyrokinetics,
e.g., short parallel scales (Rosin et al. 2009), particle trap-
ping (Pokhotelov et al. 2008; Rincon et al. 2009), or nonlin-
ear finite-gyroradius effects (Califano et al. 2008). Note that
the theory of the dissipation-range turbulence will probably
need to be modified to account for the additional energy in-
jection from the instabilities and for the (yet unclear) way in
which this energy makes its way to dissipation and into heat.
Besides the anisotropies, the particle distribution functions
in the solar wind (especially the electron one) exhibit non-
Maxwellian suprathermal tails (see Maksimovic et al. 2005;
Marsch 2006, and references therein). These contain small
(∼ 5% of the total density) populations of energetic particles.
Both the origin of these particles and their effect on turbulence
have to be modeled kinetically. Again, it is possible to formu-
late gyrokinetics for general equilibrium distributions of this
kind and examine the interaction between them and the turbu-
lent fluctuations, but we leave such a theory outside the scope
of this paper.
Thus, much remains to be done to incorporate realistic equi-
librium distribution functions into the gyrokinetic description
of the solar wind plasma. In the meanwhile, we believe that
the gyrokinetic theory based on a Maxwellian equilibrium dis-
tribution as presented in this paper, while idealized and imper-
fect, is nevertheless a step forward in the analytical treatment
of the space-plasma turbulence compared to the fluid descrip-
tions that have prevailed thus far.
8.4. Interstellar Medium
While the solar wind is unmatched by other astrophysical
plasmas in the level of detail with which turbulence in it can
be measured, the interstellar medium (ISM) also offers an ob-
server a number of ways of diagnosing plasma turbulence,
which, in the case of the ISM, is thought to be primarily ex-
cited by supernova explosions (Norman & Ferrara 1996). The
accuracy and resolution of this analysis are due to improve
rapidly thanks to many new observatories, e.g., LOFAR,42
Planck (Enßlin et al. 2006), and, in more distant future, the
SKA (Lazio et al. 2004).
The ISM is a spatially inhomogeneous environment consist-
ing of several phases that have different temperatures, densi-
ties and degrees of ionization (Ferrière 2001).43 We will use
the Warm ISM phase (see Table 1) as our fiducial interstel-
lar plasma and discuss briefly what is known about the two
main observationally accessible quantities—the electron den-
sity and magnetic fields—and how this information fits into
in the ion-cyclotron frequency range, estimate the resulting velocity-space
diffusion and argue that it is sufficient to isotropize the ion distribution
42 http://www.lofar.org
43 And, therefore, different degrees of importance of the neutral particles
and the associated ambipolar damping effects—these will not be discussed
here; see Lithwick & Goldreich 2001.
the theoretical framework proposed here.
8.4.1. Electron Density Fluctuations
The electron-density fluctuations inferred from the inter-
stellar scintillation measurements appear to have a spectrum
with an exponent ≃ −1.7, consistent with the Kolmogorov
scaling (Armstrong et al. 1981, 1995; Lazio et al. 2004; see,
however, dissenting evidence by Smirnova et al. 2006, who
claim a spectral exponent closer to −1.5). This holds over
about 5 decades of scales: λ ∈ (105,1010) km. Other observa-
tional evidence at larger and smaller scales supports the case
for this presumed inertial range to be extended over as many
as 12 decades: λ ∈ (102,1015) km, a fine example of scale
separation that prompted an impressed astrophysicist to dub
the density scaling “The Great Power Law in the Sky.” The
upper cutoff here is consistent with the estimates of the su-
pernova scale of order 100 pc—presumably the outer scale of
the turbulence (Norman & Ferrara 1996) and also roughly the
scale height of the galactic disk (obviously the upper bound
on the validity of any homogeneous model of the ISM tur-
bulence). The lower cutoff is an estimate for the inner scale
below which the logarithmic slope of the density spectrum
steepens to about −2 (Spangler & Gwinn 1990).
Higdon (1984) was the first to realize that the electron-
density fluctuations in the ISM could be attributed to a cas-
cade of a passive tracer mixed by the ambient turbulence (the
MHD entropy mode; see § 2.6). This idea was brought to ma-
turity by Lithwick & Goldreich (2001), who studied the pas-
sive cascades of the slow and entropy modes in the frame-
work of the GS theory (see also Maron & Goldreich 2001).
If the turbulence is assumed anisotropic, as in the GS theory,
the passive nature of the density fluctuations with respect to
the decoupled Alfvén-wave cascade becomes a rigorous re-
sult both in MHD (§ 2.4) and, as we showed above, in the
more general gyrokinetic description appropriate for weakly
collisional plasmas (§ 5.5). Anisotropy of the electron-density
fluctuations in the ISM is, indeed, observationally supported
(Wilkinson et al. 1994; Trotter et al. 1998; Rickett et al. 2002;
Dennett-Thorpe & de Bruyn 2003; Heyer et al. 2008, see also
Lazio et al. 2004 for a concise discussion), although detailed
scale-by-scale measurements are not currently possible.
If the underlying Alfvén-wave turbulence in the ISM has
⊥ spectrum, as predicted by GS, so should the elec-
tron density (see § 2.6). As we discussed in § 6.3, the phys-
ical nature of the inner scale for the density fluctuations de-
pends on whether they have a cascade in k‖ and are effi-
ciently damped when k‖λmfpi ∼ 1 or fail to develop small
parallel scales and can, therefore, reach k⊥ρi ∼ 1. The ob-
servationally estimated inner scale is consistent with the ion
gyroscale, ρi ∼ 103 km (see Table 1; note that the ion iner-
tial scale di = ρi/
βi is similar to ρi at the moderate values
of βi characteristic of the ISM—see further discussion of the
(ir)relevance of di in § 7.1, § 8.2.3 and Appendix E). How-
ever, since the mean free path in the ISM is not huge (Ta-
ble 1), it is not possible to distinguish this from the perpen-
dicular cutoff k−1⊥ ∼ λ
mfpiL
−1/2 ∼ 500 km implied by the par-
allel cutoff at k‖λmfpi ∼ 1 [see Eq. (220)], as advocated by
Lithwick & Goldreich (2001). Note that the relatively short
mean free path means that much of the scale range spanned by
the Great Power Law in the Sky is, in fact, well described by
the MHD approximation either with adiabatic (§ 2) or isother-
mal (§ 6.1 and Appendix D) electrons.
Below the ion gyroscale, the −2 spectral exponent reported
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 47
by Spangler & Gwinn (1990) is measured sufficiently impre-
cisely to be consistent with the −7/3 expected for the density
fluctuations in the KAW cascade (§ 7.5). However, given the
high degree of uncertainty about what happens in this “dis-
sipation range” even in the much better resolved case of the
solar wind (§ 8.2), it would probably be wise to reserve judg-
ment until better data are available.
8.4.2. Magnetic Fluctuations
The second main observable type of turbulent fluctuations
in the ISM are the magnetic fluctuations, accessible indirectly
via the measurements of the Faraday rotation of the polar-
ization angle of the pulsar light travelling through the ISM.
The structure function of the rotation measure (RM) should
have the Kolmogorov slope of 2/3 if the magnetic fluctua-
tions are due to Alfvénic turbulence described by the GS the-
ory. There is a considerable uncertainty in interpreting the
available data, primarily due to insufficient spatial resolution
(rarely better than a few parsec). Structure function slopes
consistent with 2/3 have been reported (Minter & Spangler
1996), but, depending on where one looks, shallower struc-
ture functions that seem to steepen at scales of a few parsec
are also observed (Haverkorn et al. 2004).
A recent study by Haverkorn et al. (2005) detected an in-
teresting trend: the RM structure functions computed for re-
gions that lie in the galactic spiral arms are nearly perfectly
flat down to the resolution limit, while in the interarm regions,
they have detectable slopes (although these are mostly shal-
lower that 2/3). Observations of magnetic fields in external
galaxies also reveal a marked difference in the magnetic-field
structure between arms and interarms: the spatially regular
(mean) fields are stronger in the interarms, while in the arms,
the stochastic fields dominate (Beck 2007). This qualitative
difference between the magnetic-field structure in the arms
and interarms has been attributed to smaller effective outer
scale in the arms (∼ 1 pc, compared to ∼ 102 pc in the in-
terarms; see Haverkorn et al. 2008) or to the turbulence in the
arms and interarms belonging to the two distinct asymptotic
regimes described in § 1.3: closer to the anisotropic Alfvénic
turbulence with a strong mean field in the interarms and to the
isotropic saturated state of small-scale dynamo in the arms
(Schekochihin et al. 2007).
8.5. Accretion Disks
Accretion of plasma onto a central black hole or neutron
star is responsible for many of the most energetic phenomena
observed in astrophysics (see, e.g., Narayan & Quataert 2005
for a review). It is now believed that a linear instability of dif-
ferentially rotating plasmas—the magnetorotational instabil-
ity (MRI)—amplifies magnetic fields and gives rise to MHD
turbulence in astrophysical disks (Balbus & Hawley 1998).
Magnetic stresses due to this turbulence transport angular mo-
mentum, allowing plasma to accrete. The MRI converts the
gravitational potential energy of the inflowing plasma into
turbulence at the outer scale that is comparable to the scale
height of the disk. This energy is then cascaded to small
scales and dissipated into heat—powering the radiation that
we see from accretion flows. Fluid MHD simulations show
that the MRI-generated turbulence in disks is subsonic and
has β ∼ 10 − 100. Thus, on scales much smaller than the scale
height of the disk, homogeneous turbulence in the parameter
regimes considered in this paper is a valid idealization and
the kinetic models developed above should represent a step
forward compared to the purely fluid approach.
Turbulence is not yet directly observable in disks, so mod-
els of turbulence are mostly used to produce testable predic-
tions of observable properties of disks such as their X-ray and
radio emission. One of the best observed cases is the (pre-
sumed) accretion flow onto the black hole coincident with the
radio source Sgr A∗ in the center of our Galaxy (see review
by Quataert 2003).
Depending on the rate of heating and cooling in the inflow-
ing plasma (which in turn depend on accretion rate and other
properties of the system under consideration), there are differ-
ent models that describe the physical properties of accretion
flows onto a central object. In one class of models, a geometri-
cally thin optically thick accretion disk (Shakura & Sunyaev
1973), the inflowing plasma is cold and dense and well de-
scribed as an MHD fluid. When applied to Sgr A∗, these
models produce a prediction for its total luminosity that is
several orders of magnitude larger than observed. Another
class of models, which appears to be more consistent with the
observed properties of Sgr A∗, is called radiatively inefficient
accretion flows (RIAFs; see Rees et al. 1982; Narayan & Yi
1995 and review by Quataert 2003 of the applications and ob-
servational constraints in Sgr A∗). In these models, the in-
flowing plasma near the black hole is believed to adopt a two-
temperature configuration, with the ions (Ti ∼ 1011 − 1012 K)
hotter than the electrons (Te ∼ 109 − 1011 K).44 The electron
and ion thermodynamics decouple because the densities are
so low that the temperature equalization time ∼ ν−1ie is longer
than the time for the plasma to flow into the black hole. Thus,
like the solar wind, RIAFs are macroscopically collisionless
plasmas (see Table 1 for plasma parameters in the Galactic
center; note that these parameters are so extreme that the gy-
rokinetic description, while probably better than the fluid one,
cannot be expected to be rigorously valid; at the very least, it
needs to be reformulated in a relativistic form). At the high
temperatures appropriate to RIAFs, electrons radiate energy
much more efficiently than the ions (by virtue of their much
smaller mass) and are, therefore, expected to contribute dom-
inantly to the observed emission, while the thermal energy of
the ions is swallowed by the black hole. Since the plasma
is collisionless, the electron heating by turbulence largely de-
termines the thermodynamics of the electrons and thus the
observable properties of RIAFs. The question of which frac-
tion of the turbulent energy goes into ion and which into elec-
tron heating is, therefore, crucial for understanding accretion
flows—and the answer to this question depends on the de-
tailed properties of the small-scale kinetic turbulence (e.g.,
Quataert & Gruzinov 1999; Sharma et al. 2007), as well as on
the linear properties of the collisionless MRI (Quataert et al.
2002; Sharma et al. 2003).
Since all of the turbulent power coming down the cascade
must be dissipated into either ion or electron heat, it is re-
ally the amount of generalized energy diverted at the ion gy-
roscale into the ion entropy cascade (§§ 7.8-7.9) that decides
how much energy is left to heat the electrons via the KAW
cascade (§§ 7.2-7.5, § 7.12). Again, as in the case of the solar
wind (§ 8.2.2 and § 8.2.5), the transition around the ion gy-
roscale from the Alfvénic turbulence at k⊥ρi ≪ 1 to the KAW
turbulence at k⊥ρi ≫ 1 emerges as a key unsolved problem.
8.6. Galaxy Clusters
44 It is partly with this application in mind that we carried the general
temperature ratio in our calculations; see footnote 17.
48 SCHEKOCHIHIN ET AL.
Galaxy clusters are the largest plasma objects in the Uni-
verse. Like the other examples discussed above, the intraclus-
ter plasma is in the weakly collisional regime (see Table 1).
Fluctuations of electron density, temperature and of magnetic
fields are measured in clusters by X-ray and radio observa-
tories, but the resolution is only just enough to claim that a
fairly broad scale range of fluctuations exists (Schuecker et al.
2004; Vogt & Enßlin 2005). No power-law scalings have yet
been established beyond reasonable doubt.
What fundamentally hampers quantitative modeling of tur-
bulence and related effects in clusters is that we do not have
a definite theory of the basic properties of the intracluster
medium: its (effective) viscosity, magnetic diffusivity or ther-
mal conductivity. In a weakly collisional and strongly mag-
netized plasma, all of these depend on the structure of the
magnetic field (Braginskii 1965), which is shaped by the tur-
bulence. If (or at scales where) a reasonable a priori assump-
tion can be made about the field structure, further analytical
progress is possible: thus, the theoretical models presented in
this paper assume that the magnetic field is a sum of a slowly
varying in space “mean field” and small low-frequency per-
turbations (δB ≪ B0).
In fact, since clusters do not have mean fields of any mag-
nitude that could be considered dynamically significant, but
do have stochastic fields, the outer-scale MHD turbulence in
clusters falls into the weak-mean-field category (see § 1.3).
The magnetic field should be highly filamentary, organized
in long folded direction-reversing structures. It is not cur-
rently known what determines the reversal scale.45 Obser-
vations, while tentatively confirming the existence of very
long filaments (Clarke & Enßlin 2006), suggest that the re-
versal scale is much larger than the ion gyroscale: thus, the
magnetic-energy spectrum for the Hydra A cluster core re-
ported by Vogt & Enßlin (2005) peaks at around 1 kpc, com-
pared to ρi ∼ 105 km. Below this scale, an Alfvén-wave cas-
cade should exist (as is, indeed, suggested by Vogt & Enßlin’s
spectrum being roughly consistent with k−5/3 at scales below
the peak). As these scales are collisionless (λmfpi ∼ 100 pc in
the cores and ∼ 10 kpc in the bulk of the clusters), it is to this
turbulence that the theory developed in this paper should be
applicable.
Another complication exists, similar to that discussed in
§ 8.3: pressure anisotropies could give rise to fast plasma
instabilities whose growth rate peaks just above the ion gy-
roscale. As was pointed out by Schekochihin et al. (2005),
these are, in fact, an inevitable consequence of any large-scale
fluid motions that change the strength of the magnetic field.
Although a number of interesting and plausible arguments
can be made about the way the instabilities might determine
the magnetic-field structure (Schekochihin & Cowley 2006;
Schekochihin et al. 2008a; Rosin et al. 2009; Rincon et al.
2009), it is not currently understood how the small-scale
fluctuations resulting from these instabilities coexist with the
Alfvénic cascade.
The uncertainties that result from this imperfect under-
standing of the nature of the intracluster medium are exempli-
fied by the problem of its thermal conductivity. The magnetic-
field reversal scale in clusters is certainly not larger than the
electron diffusion scale, (mi/me)
1/2λmfpi, which varies from a
45 See Schekochihin & Cowley (2006) for a detailed presentation of our
views on the interplay between turbulence, magnetic field and plasma ef-
fects in cluster; for further discussions and disagreements, see Enßlin & Vogt
(2006); Subramanian et al. (2006); Brunetti & Lazarian (2007).
few kpc in the cores to a few hundred kpc in the bulk. There-
fore, one would expect that the approximation of isothermal
electron fluid (§ 4) should certainly apply at all scales below
the reversal scale, where δB ≪ B0 presumably holds. Even
this, however, is not absolutely clear. One could imagine
the electrons being effectively adiabatic if (or in the regions
where) the plasma instabilities give rise to large fluctuations
of the magnetic field (δB/B0 ∼ 1) at the ion gyroscale re-
ducing the mean free path to λmfpi ∼ ρi (Schekochihin et al.
2008a; Rosin et al. 2009; Rincon et al. 2009). Such fluctu-
ations cannot be described by the gyrokinetics in its cur-
rent form. The current state of the observational evidence
does not allow one to exclude either of these possibilities.
Both isothermal (Fabian et al. 2006; Sanders & Fabian 2006)
and non-isothermal (Markevitch & Vikhlinin 2007) coherent
structures that appear to be shocks are observed. Disordered
fluctuations of temperature can also be detected, which allows
one to infer an upper limit for the scale at which the isothermal
approximation can start being valid: thus, Markevitch et al.
(2003) find temperature variations at all scales down to ∼
100 kpc, which is the statistical limit that defines the spa-
tial resolution of their temperature map. In none of these or
similar measurements is the magnetic field data available that
would make possible a pointwise comparison of the magnetic
and thermal structure.
Because of this lack of information about the state of the
magnetized plasma in clusters, theories of the intracluster
medium are not sufficiently constrained by observations, so
no one theory is in a position to prevail. This uncertain state
of affairs might be improved by analyzing the observationally
much better resolved case of the solar wind, which should be
quite similar to the intracluster medium at very small scales
(except for somewhat lower values of βi in the solar wind).
9. CONCLUSION
In this paper, we have considered magnetized plasma tur-
bulence in the astrophysically prevalent regime of weak col-
lisionality. We have shown how the energy injected at the
outer scale cascades in phase space, eventually to increase the
entropy of the system and heat the particles. In the process,
we have explained how one combines plasma physics tools—
in particular, the gyrokinetic theory—with the ideas of a tur-
bulent cascade of energy to arrive at a hierarchy of tractable
models of turbulence in various physically distinct scale in-
tervals. These models represent the branching pathways of a
generalized energy cascade in phase space (the “kinetic cas-
cade”; see Fig. 5) and make explicit the “fluid” and “kinetic”
aspects of plasma turbulence.
A detailed outline of these developments was given in the
Introduction. Intermediate technical summaries were pro-
vided in § 4.9, § 5.7, and § 7.14. An astrophysical summary
and discussion of the observational evidence was given in § 8,
with a particular emphasis on space plasmas (§§ 8.1-8.3). Our
view of how the transformation of the large-scale turbulent
energy into heat occurs was encapsulated in the concept of
a kinetic cascade of generalized energy. It was previewed in
§ 1.4 and developed quantitatively in §§ 3.4-3.5, § 4.7, § 5.6,
§§ 6.2.3-6.2.5, §§ 7.8-7.12, Appendices D.2 and E.2.
Following a series of analytical contributions that set
up a theoretical framework for astrophysical gyrokinetics
(Howes et al. 2006, 2008a; Schekochihin et al. 2007, 2008b,
and this paper), an extensive program of fluid, hybrid fluid-
kinetic, and fully gyrokinetic46 numerical simulations of mag-
netized plasma turbulence is now underway (for the first re-
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 49
sults of this program, see Howes et al. 2008b; Tatsuno et al.
2009a,b). Careful comparisons of the fully gyrokinetic
simulations with simulations based on the more readily
computable models derived in this paper (RMHD—§ 2,
isothermal electron fluid—§ 4, KRMHD—§ 5, ERMHD—
§ 7, HRMHD—Appendix E) as well as with the numerical
studies based on various Landau fluid (Snyder et al. 1997;
Goswami et al. 2005; Ramos 2005; Sharma et al. 2006, 2007;
Passot & Sulem 2007) and gyrofluid (Hammett et al. 1991;
Dorland & Hammett 1993; Snyder & Hammett 2001; Scott
2007) closures appear to be the way forward in developing a
comprehensive numerical model of the kinetic turbulent cas-
cade from the outer scale to the electron gyroscale. Of the
many astrophysical plasmas to which these results apply, the
solar wind and, perhaps, the magnetosheath, due to the high
quality of turbulence measurements possible in them, appear
to be the most suitable test beds for direct and detailed quan-
titative comparisons of the theory and simulation results with
observational evidence. The objective of all this work remains
a quantitative characterization of the scaling-range properties
(spectra, anisotropy, nature of fluctuations and their interac-
tions), the ion and electron heating, and the transport proper-
ties of the magnetized plasma turbulence.
We thank O. Alexandrova, S. Bale, J. Borovsky, T. Carter,
S. Chapman, C. Chen, E. Churazov, T. Enßlin, A. Fabian,
A. Finoguenov, A. Fletcher, M. Haverkorn, B. Hnat, T. Hor-
bury, K. Issautier, C. Lacombe, M. Markevitch, K. Osman,
T. Passot, F. Sahraoui, A. Shukurov, and A. Vikhlinin for
helpful discussions of experimental and observational data;
I. Abel, M. Barnes, D. Ernst, J. Hastie, P. Ricci, C. Roach,
and B. Rogers for discussions of collisions in gyrokinetics;
and G. Plunk for discussions of the theory of gyrokinetic tur-
bulence in two spatial dimensions. The authors’ travel was
supported by the US DOE Center for Multiscale Plasma Dy-
namics and by the Leverhulme Trust (UK) International Aca-
demic Network for Magnetized Plasma Turbulence. A.A.S.
was supported in part by a PPARC/STFC Advanced Fellow-
ship and by the STFC Grant ST/F002505/1. He also thanks
the UCLA Plasma Group for its hospitality on several occa-
sions. S.C.C. and W.D. thank the Kavli Institute for The-
oretical Physics and the Aspen Center for Physics for their
hospitality. G.W.H. was supported by the US DOE contract
DE-AC02-76CH03073. G.G.H. and T.T. were supported by
the US DOE Center for Multiscale Plasma Dynamics. E.Q.
and G.G.H. were supported in part by the David and Lucille
Packard Foundation.
46 Using the publicly available GS2 code (developed originally for fusion
applications; see http://gs2.sourceforge.net) and the purpose-built AstroGK
code (see http://www.physics.uiowa.edu/~ ghowes/astrogk/).
APPENDIX
A. BRAGINSKII’S TWO-FLUID EQUATIONS AND REDUCED MHD
Here we explain how the standard one-fluid MHD equations used in § 2 and the collisional limit of the KRMHD system
(§ 6.1, derived in Appendix D) both emerge as limiting cases of the two-fluid theory. For the case of anisotropic fluctuations,
k‖/k⊥ ≪ 1, all of this can, of course, be derived from gyrokinetics, but it is useful to provide a connection to the more well
known fluid description of collisional plasmas.
A.1. Two-Fluid Equations
The rigorous derivation of the fluid equations for a collisional plasma was done in the classic paper of Braginskii (1965). His
equations, valid for ω/νii ≪ 1, k‖λmfpi ≪ 1, k⊥ρi ≪ 1 (see Fig. 3), evolve the densities ns, mean velocities us and temperatures
Ts of each plasma species (s = i,e):
+ us ·∇
ns = −ns∇·us, (A1)
+ us ·∇
us = −∇ps −∇· Π̂s + qsns
us ×B
+ Fs, (A2)
+ us ·∇
Ts = −ps∇·us −∇·Γs − Π̂s : ∇us + Qs, (A3)
where ps = nsTs and the expressions for the viscous stress tensor Π̂s, the friction force Fs, the heat flux Γs and the interspecies heat
exchange Qs are given in Braginskii (1965). Equations (A1-A3) are complemented with the quasi-neutrality condition, ne = Zni,
and the Faraday and Ampère laws, which are (in the non-relativistic limit)
= −c∇×E, j = ene(ui − ue) =
∇×B. (A4)
Because of quasi-neutrality, we only need one of the continuity equations, say the ion one. We can also use the electron momen-
tum equation [Eq. (A2), s = e] to express E, which we then substitute into the ion momentum equation and the Faraday law. The
resulting system is
= −ρ∇·u, (A5)
−∇· Π̂+ B ·∇B
+ ue ·∇
ue, (A6)
u×B − j×B
c∇· Π̂e
+ ue ·∇
, (A7)
http://gs2.sourceforge.net
http://www.physics.uiowa.edu/~
50 SCHEKOCHIHIN ET AL.
where ρ = mini, u = ui, p = pi + pe, Π̂ = Π̂i + Π̂e, ue = u − j/ene, ne = Zni, d/dt = ∂/∂t + u ·∇. The ion and electron temperatures
continue to satisfy Eq. (A3).
A.2. Strongly Magnetized Limit
In this form, the two-fluid theory starts resembling the standard one-fluid MHD, which was our starting point in § 2: Eqs. (A5-
A7) already look similar to the continuity, momentum and induction equations. The additional terms that appear in these equations
and the temperature equations (A3) are brought under control by considering how they depend on a number of dimensionless
parameters: ω/νii, k‖λmfpi, k⊥ρi, (me/mi)
1/2. While all these are small in Braginskii’s calculation, no assumption is made as to
how they compare to each other. We now specify that
k‖λmfpi√
, k⊥ρi ≪ k‖λmfpi ∼
≪ 1 (A8)
(see Fig. 4). Note that the first of these relations is equivalent to assuming that the fluctuation frequencies are Alfvénic—the same
assumption as in gyrokinetics [Eq. (49)]. The second relation in Eq. (A8) will be referred to by us as the strongly magnetized
limit. Under the assumptions (A8), the two-fluid equations reduce to the following closed set:47
= −ρ∇·u, (A10)
b̂b̂ : ∇u − 1
b̂b̂ρν‖i
b̂b̂ : ∇u − 1
B ·∇B
, (A11)
= B ·∇u − B∇·u, (A12)
Ti∇·u +
b̂ρκ‖ib̂ ·∇Ti
− νie (Ti − Te) +
miν‖i
b̂b̂ : ∇u − 1
, (A13)
Te∇·u +
b̂ρκ‖eb̂ ·∇Te
νie (Te − Ti) , (A14)
where ν‖i = 0.90vthiλmfpi is the parallel ion viscosity, κ‖i = 2.45vthiλmfpi parallel ion thermal diffusivity, κ‖e = 1.40vtheλmfpe ∼
Z2/τ 5/2
(mi/me)
1/2κ‖i parallel electron thermal diffusivity [here λmfpi = vthi/νii with νii defined in Eq. (52)], and νie ion–electron
collision rate [defined in Eq. (51)]. Note that the last term in Eq. (A13) represents the viscous heating of the ions.
A.3. One-Fluid Equations (MHD)
If we now restrict ourselves to the low-frequency regime where ion–electron collisions dominate over all other terms in the
ion-temperature equation (A13),
k‖λmfpi√
≪ 1 (A15)
[see Eqs. (A8) and (51)], we have, to lowest order in this new subsidiary expansion, Ti = Te = T . We can now write p = (ni +ne)T =
(1 + Z)ρT/mi and, adding Eqs. (A13) and (A14), find the equation for pressure:
p∇·u = ∇·
b̂neκ‖eb̂ ·∇T
miν‖i
b̂b̂ : ∇u − 1
, (A16)
where we have neglected the ion thermal diffusivity compared to the electron one, but kept the ion heating term to maintain
energy conservation. Equation (A16) together with Eqs. (A10-A12) constitutes the conventional one-fluid MHD system. With
the dissipative terms [which are small because of Eq. (A15)] neglected, this was the starting point for our fluid derivation of
RMHD in § 2.
Note that the electrons in this regime are adiabatic because the electron thermal diffusion is small
∼ k‖λmfpi
≪ 1, (A17)
47 The structure of the momentum equation (A11) is best understood by realizing that ρν‖i
b̂b̂ : ∇u −∇·u/3
= p⊥ − p‖ , the difference between the perpen-
dicular and parallel (ion) pressures. Since the total pressure is p = (2/3)p⊥ + (1/3)p‖ , Eq. (A11) can be written
p⊥ − p‖
B ·∇B
. (A9)
This is the general form of the momentum equation that is also valid for collisionless plasmas, when k⊥ρi ≪ 1 but k‖λmfpi is order unity or even large.
Equation (A9) together with the continuity equation (A11), the induction equation (A12) and a kinetic equation for the particle distribution function (from the
solution of which p⊥ and p‖ are determined) form the system known as Kinetic MHD (KMHD, see Kulsrud 1964, 1983). The collisional limit, k‖λmfpi ≪ 1, of
KMHD is again Eqs. (A10-A14).
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 51
provided Eq. (A15) holds and βi is order unity. If we take βi ≫ 1 instead, we can still satisfy Eq. (A15), so Ti = Te follows from
the ion temperature equation (A13) and the one-fluid equations emerge as an expansion in high βi. However, these equations now
describe two physical regimes: the adiabatic long-wavelength regime that satisfies Eq. (A17) and the shorter-wavelength regime
in which (me/mi)
βi ≪ k‖λmfpi ≪ (me/mi)1/2
βi, so the fluid is isothermal, T = T0 = const, p = [(1+Z)T0/mi]ρ = c2sρ [Eq. (9)
holds with γ = 1].
A.4. Two-Fluid Equations with Isothermal Electrons
Let us now consider the regime in which the coupling between the ion and electron temperatures is small and the electron
diffusion is large [the limit opposite to Eqs. (A15) and (A17)]:
k‖λmfpi√
∼ k‖λmfpi
≫ 1, (A18)
Then the electrons are isothermal, Te = T0e = const (with the usual assumption of stochastic field lines, so b̂ · ∇Te = 0 implies
∇Te = 0, as in § 4.4), while the ion temperature satisfies
Ti∇·u +
b̂ρκ‖ib̂ ·∇Ti
miν‖i
b̂b̂ : ∇u − 1
. (A19)
Equation (A19) together with Eqs. (A10-A12) and p = ρ(Ti + ZT0e)/mi are a closed system that describes an MHD-like fluid
of adiabatic ions and isothermal electrons. Applying the ordering of § 2.1 to these equations and carrying out an expansion in
k‖/k⊥ ≪ 1 entirely analogously to the way it was done in § 2, we arrive at the RMHD equations (17-18) for the Alfvén waves
and the following system for the compressive fluctuations (slow and entropy modes):
+ b̂ ·∇u‖ = 0, (A20)
− v2Ab̂ ·∇
= ν‖i b̂ ·∇
b̂ ·∇u‖ +
, (A21)
= κ‖ib̂ ·∇
b̂ ·∇
, (A22)
and the pressure balance
b̂ ·∇u‖ +
. (A23)
Recall that these equations, being the consequence of Braginskii’s two-fluid equations (§ A.1), are an expansion in k‖λmfpi ≪ 1
correct up to first order in this small parameter. Since the dissipative terms are small, we can replace (d/dt)δρ/ρ0 in the viscous
terms of Eqs. (A21) and (A23) by its value computed from Eqs. (A20), (A22) and (A23) in neglect of dissipation: (d/dt)δρ/ρ0 =
−b̂ · ∇u‖/(1 + c2s/v2A) [cf. Eq. (25)], where the speed of sound cs is defined by Eq. (166). Substituting this into Eqs. (A21)
and (A23), we recover the collisional limit of KRMHD derived in Appendix D, see Eqs. (D18-D20) and (D22).
B. COLLISIONS IN GYROKINETICS
The general collision operator that appears in Eq. (36) is (Landau 1936)
= 2π lnΛ
q2s q
fs′ (v
∂ fs(v)
fs(v)
∂ fs′ (v′)
, (B1)
where w = v − v′ and lnΛ is the Coulomb logarithm. We now take into account the expansion of the distribution function (54),
use the fact that the collision operator vanishes when it acts on a Maxwellian, and retain only first-order terms in the gyrokinetic
expansion. This gives us the general form of the collision term in Eq. (57): it is the ring-averaged linearized form of the Landau
collision operator (B1), (∂hs/∂t)c = 〈Cs[h]〉Rs , where
Cs[h] = 2π lnΛ
q2s q
F0s′(v
hs(v) − F0s(v)
hs′(v
. (B2)
Note that the velocity derivatives are taken at constant r, i.e., the gyrocenter distribution functions that appear in the integrand
should be understood as hs(v)≡ hs(t,r+v⊥× ẑ/Ωs,v⊥,v‖). The explicit form of the gyrokinetic collision operator can be derived
in k space as follows:
eik·Rhk
eik·rCs
e−ik·ρhk
eik·Rs
eik·ρs(v)Cs
e−ik·ρhk
, (B3)
52 SCHEKOCHIHIN ET AL.
where ρs(v) = −v⊥× ẑ/Ωs and Rs = r−ρs(v). Angle brackets with no subscript refer to averages over the gyroangle ϑ of quantities
that do not depend on spatial coordinates. Note that inside the operator Cs[. . .], h occurs both with index s and velocity v and
with index s′ and velocity v′ (over which summation/integration is done). In the latter case, ρ = ρs′(v′) = −v′⊥× ẑ/Ωs′ in the
exponential factor inside the operator.
Most of the properties of the collision operator that are used in the main body of this paper to order the collision terms
can be established in general, already on the basis of Eq. (B3) (§§ B.1-B.2). If the explicit form of the collision operator is
required, we could, in principle, perform the ring average on the linearized operator C [Eq. (B2)] and derive an explicit form of
(∂hs/∂t)c. In practice, in gyrokinetics, as in the rest of plasma physics, the full collision operator is only used when it is absolutely
unavoidable. In most problems of interest, further simplifications are possible: the same-species collisions are often modeled by
simpler operators that share the full collision operator’s conservation properties (§ B.3), while the interspecies collision operators
are expanded in the electron–ion mass ratio (§ B.4).
B.1. Velocity-Space Integral of the Gyrokinetic Collision Operator
Many of our calculations involve integrating the gyrokinetic equation (57) over the velocity space while keeping r constant.
Here we estimate the size of the integral of the collision term when k⊥ρs ≪ 1. Using Eq. (B3),
d3veik·r−ik·ρs(v)
eik·ρs(v)Cs
e−ik·ρhk
eik·r2π
dv⊥ v⊥
e−ik·ρs(v)
eik·ρs(v)Cs
e−ik·ρhk
eik·r
e−ik·ρs(v)
eik·ρs(v)Cs
e−ik·ρhk
eik·r
d3vJ0(as)e
ik·ρs(v)Cs
e−ik·ρhk
eik·r
1 − ik · v⊥× ẑ
k · v⊥× ẑ
+ . . .
e−ik·ρhk
. (B4)
Since the (linearized) collision operator Cs conserves particle number, the first term in the expansion vanishes. The operator
Cs = Css +Css′ is a sum of the same-species collision operator [the s′ = s part of the sum in Eq. (B2)] and the interspecies collision
operator (the s′ 6= s part). The former conserves total momentum of the particles of species s, so it gives no contribution to the
second term in the expansion in Eq. (B4). Therefore,
d3v〈〈Css[hs]〉Rs〉r ∼ νssk
s δns. (B5)
The interspecies collisions do contribute to the second term in Eq. (B4) due to momentum exchange with the species s′. This
contribution is readily inferred from the standard formula for the linearized friction force (see, e.g., Helander & Sigmar 2002):
d3vvCss′
e−ik·ρhk
S (v)e
−ik·ρs(v)hsk + ms′νs
S (v)e
−ik·ρs′ (v)hs′k
, (B6)
S (v) =
2πn0s′q
s′ lnΛ
(vths
vths′
vths′
erf ′
vths′
, (B7)
where erf(x) = (2/
dy exp(−y2) is the error function. From this, via a calculation of ring averages analogous to Eq. (B17),
we get
−ik ·
v⊥× ẑ
e−ik·ρhk
S (v)
ik ·ρs(v)e−ik·ρs(v)
hsk +
S (v)
ik ·ρs′(v)e−ik·ρs′ (v)
S (v)asJ1(as)hsk +
S (v)as′J1(as′)hs′k
∼ νss′k2⊥ρ2sδns + νs′sk2⊥ρ2s′δns′ . (B8)
For the ion–electron collisions (s = i, s′ = e), using Eqs. (45) and (51), we find that both terms are ∼ (me/mi)1/2νiik2⊥ρ2i δni.
Thus, besides an extra factor of k2⊥ρ
i , the ion–electron collisions are also subdominant by one order in the mass-ratio expansion
compared to the ion–ion collisions. The same estimate holds for the interspecies contributions to the third and fourth terms in
Eq. (B4). In a similar fashion, the integral of the electron–ion collision operator (s = e, s′ = i), is ∼ νeik2⊥ρ2eδne, which is the same
order as the integral of the electron–electron collisions.
The conclusion of this section is that, both for ion and for electron collisions, the velocity-space integral (at constant r) of the
gyrokinetic collision operator is higher order than the collision operator itself by two orders of k⊥ρs. This is the property that we
relied on in neglecting collision terms in Eqs. (104) and (137).
B.2. Ordering of Collision Terms in Eqs. (125) and (137)
In § 5, we claimed that the contribution to the ion–ion collision term due to the (Ze〈ϕ〉Ri/T0i)F0i part of the ion distribution
function [Eq. (124)] was one order of k⊥ρi smaller than the contributions from the rest of hi. This was used to order collision
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 53
terms in Eqs. (125) and (137). Indeed, from Eq. (B3),
Ze〈ϕ〉Ri
eik·Ri
eik·ρiCii
e−ik·ρi J0(ai)F0i
]〉 Zeϕk
eik·Ri
eik·ρiCii
1 − ik ·ρi −
(k ·ρi)2 −
+ · · ·
∼ νiik2⊥ρ2i
F0i. (B9)
This estimate holds because, as it is easy to ascertain using Eq. (B2), the operator Cii annihilates the first two terms in the
expansion and only acts non-trivially on an expression that is second order in k⊥ρi. With the aid of Eq. (47), the desired ordering
of the term (B9) in Eq. (125) follows. When Eq. (B9) is integrated over velocity space, the result picks up two extra orders in
k⊥ρi [a general effect of integrating the gyroaveraged collision operator over the velocity space; see Eq. (B4)]:
Ze〈ϕ〉Ri
∼ νiik4⊥ρ4i
, (B10)
so the resulting term in Eq. (137) is third order, as stated in § 5.3.
B.3. Model Pitch-Angle-Scattering Operator for Same-Species Collisions
A popular model operator for same-species collisions that conserves particle number, momentum, and energy is constructed
by taking the test-particle pitch-angle-scattering operator and correcting it with an additional term that ensures momentum con-
servation (Rosenbluth et al. 1972; see also Helander & Sigmar 2002):
CM[hs] = ν
D (v)
1 − ξ2
) ∂hs
1 − ξ2
2v ·U[hs]
v2ths
, U[hs] =
d3vvνssD (v)hs
d3v (v/vths)2νssD (v)F0s(v)
, (B11)
νssD (v) = νss
(vths
v2ths
erf ′
, νss =
2πn0sq
s lnΛ
, (B12)
where the velocity derivatives are at constant r. The gyrokinetic version of this operator is (cf. Catto & Tsang 1977;
Dimits & Cohen 1994)
〈CM[hs]〉Rs =
eik·RsνssD (v)
1 − ξ2
) ∂hsk
v2(1 + ξ2)
4v2ths
s hsk + 2
v⊥J1(as)U⊥[hsk] + v‖J0(as)U‖[hsk]
v2ths
, (B13)
U⊥[hsk] =
d3vv⊥J1(as)νssD (v)hsk(v⊥,v‖)
d3v (v/vths)2νssD (v)F0s(v)
, U‖[hsk] =
d3vv‖J0(as)ν
D (v)hsk(v⊥,v‖)
d3v (v/vths)2νssD (v)F0s(v)
where as = k⊥v⊥/Ωs. The velocity derivatives are now at constant Rs. The spatial diffusion term appearing in the ring-averaged
collision operator is physically due to the fact that a change in a particle’s velocity resulting from a collision can lead to a change
in the spatial position of its gyrocenter.
In order to derive Eq. (B13), we use Eq. (B3). Since, ρs(v) =
1 − ξ2 sinϑ+ ŷv
1 − ξ2 cosϑ
/Ωs, it is not hard to see
e−ik·ρs(v)hsk = e
−ik·ρs(v)
1 − ξ2
ik⊥ ·
v⊥× ẑ
e−ik·ρs(v)hsk = e
−ik·ρ(v)
ik⊥ ·v⊥
hsk. (B14)
Therefore,
eik·ρs(v)
1 − ξ2
e−ik·ρs(v)hsk
1 − ξ2
) ∂hsk
k2⊥hsk,
eik·ρs(v)
e−ik·ρs(v)hsk
1 − ξ2
k2⊥hsk.
(B15)
Combining these formulae, we obtain the first two terms in Eq. (B13). Now let us work out the U term:
eik·ρs(v)v ·
d3v′ v′νssD (v
′)e−ik·ρs(v
′)hsk
v′⊥,v
veik·ρs(v)
dv′⊥ v
dv′‖ν
v′e−ik·ρs(v
v′⊥,v
(B16)
Since
ve±ik·ρs(v)
= ẑv‖
e±ik·ρs(v)
v⊥e±ik·ρs(v)
, where
e±ik·ρs(v)
= J0(as) and
±ik·ρs(v)
= ẑ×
v⊥× ẑ
∓ik⊥ ·
v⊥× ẑ
= ±iΩsẑ×
∓ik⊥ ·
v⊥× ẑ
= ±i ẑ×k⊥
v⊥J1(as), (B17)
we obtain the third term in Eq. (B13).
54 SCHEKOCHIHIN ET AL.
It is useful to give the lowest-order form of the operator (B13) in the limit k⊥ρs ≪ 1:
〈CM[hs]〉Rs = ν
D (v)
1 − ξ2
) ∂hs
d3v′v′
νssD (v
′)hs(v
d3v′v′2νssD (v
′)F0s(v′)
+ O(k2⊥ρ
s ). (B18)
This is the operator that can be used in the right-hand side of Eq. (145) (as, e.g., is done in the calculation of collisional transport
terms in Appendix D.3).
In practical numerical computations of gyrokinetic turbulence, the pitch-angle scattering operator is not sufficient because the
distribution function develops small scales not only in ξ but also in v (M. Barnes, W. Dorland and T. Tatsuno 2006, unpublished).
This is, indeed, expected because the phase-space entropy cascade produces small scales in v⊥, rather than just in ξ (see § 7.9.1).
In order to provide a cut off in v, an energy-diffusion operator must be added to the pitch-angle-scattering operator derived above.
A numerically tractable model gyrokinetic energy-diffusion operator was proposed by Abel et al. (2008); Barnes et al. (2009).48
B.4. Electron–Ion Collision Operator
This operator can be expanded in me/mi and to the lowest order is (see, e.g., Helander & Sigmar 2002)
Cei[h] = ν
D (v)
1 − ξ2
) ∂he
1 − ξ2
2v ·ui
v2the
, νeiD (v) = νei
(vthe
. (B19)
The corrections to this form are O(me/mi). This is second order in the expansion of § 4 and, therefore, we need not keep these
corrections. The operator (B19) is mathematically similar to the model operator for the same-species collisions [Eq. (B13)]. The
gyrokinetic version of this operator is derived in the way analogous to the calculation in Appendix B.3. The result is
〈Cei[h]〉Re =
eik·ReνeiD (v)
1 − ξ2
) ∂hek
v2(1 + ξ2)
4v2the
v2the
J1(ae)
2v′2⊥
v2thi
hik +
2v‖J0(ae)u‖ki
v2the
. (B20)
At scales not too close to the electron gyroscale, namely, such that k⊥ρe ∼ (me/mi)1/2, the second and third terms are manifestly
second order in (me/mi)
1/2, so have to be neglected along with other O(me/mi) contributions to the electron–ion collisions.
49 The
remaining two terms are first order in the mass-ratio expansion: the first term vanishes for he = h
e [Eq. (101)], so its contribution
is first order; in the fourth term, we can use Eq. (87) to express u‖i in terms of quantities that are also first order. Keeping only
the first-order terms, the gyrokinetic electron–ion collision operator is
〈Cei[h]〉Re = ν
D (v)
1 − ξ2
) ∂h(1)e
2v‖u‖i
v2the
. (B21)
Note that the ion drag term is essential to represent the ion–electron friction correctly and, therefore, to capture the Ohmic
resistivity (which, however, is rarely more important for unfreezing flux than the electron inertia and the finiteness of the electron
gyroradius; see § 7.7).
C. A HEURISTIC DERIVATION OF THE ELECTRON EQUATIONS
Here we show how the equations (116-117) of § 4 and the ERMHD equations (226-227) of § 7 can be derived heuristically
from electron fluid dynamics and a number of physical assumptions, without the use of gyrokinetics (§ C.1). This derivation is
not rigorous. Its role is to provide an intuitive route to the isothermal electron fluid and ERMHD approximations.
C.1. Derivation of Eqs. (116-117)
We start with the following three equations:
= −c∇×E, ∂ne
+∇· (neue) = 0, E +
ue ×B
. (C1)
These are Faraday’s law, the electron continuity equation, and the generalized Ohm’s law, which is the electron momentum
equation with all electron inertia terms neglected (i.e., effectively, the lowest order in the expansion in the electron mass me). The
electron pressure is assumed to be scalar by fiat (this can be justified in certain limits: for example in the collisional limit, as in
Appendix A, or for the isothermal electron fluid approximation derived in § 4). The electron-pressure term in the right-hand side
of Ohm’s law is sometimes called the thermoelectric term. We now assume the same static uniform equilibrium, E0 = 0, B0 = B0ẑ,
that we have used throughout this paper and apply to Eqs. (C1) the fundamental ordering discussed in § 3.1.
48 The collision operator now used the GS2 and AstroGK codes (see footnote 46) is their energy-diffusion operator plus the pitch-angle-scattering opera-
tor (B13).
49 The third term in Eq. (B20) is, in fact, never important: at the electron scales, k⊥ρe ∼ 1, it is negligible because of the Bessel function in the velocity
integral (Abel et al. 2008).
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 55
First consider the projection of Ohm’s law onto the total magnetic field B, use the definition of E [Eq. (37)], and keep the
leading-order terms in the ǫ expansion:
E · b̂ = − 1
b̂ ·∇pe ⇒
+ b̂ ·∇ϕ = b̂ ·∇ δpe
. (C2)
This turns into Eq. (116) if we also assume isothermal electrons, δpe = T0eδne [see Eq. (103)].
With the aid of Ohm’s law, Faraday’s law turns into
= ∇× (ue ×B) = −ue ·∇B + B ·∇ue − B∇·ue. (C3)
Keeping the leading-order terms, we find, for the components of Eq. (C3) perpendicular and parallel to the mean field,
+ u⊥e ·∇⊥
= b̂ ·∇u⊥e,
+ u⊥e ·∇⊥
= b̂ ·∇u‖e. (C4)
In the last equation, we have used the electron continuity equation to write
∇·ue = −
+ u⊥e ·∇⊥
. (C5)
From Ohm’s law, we have, to lowest order,
u⊥e = −ẑ×
E⊥ +∇⊥
= ẑ×∇⊥
. (C6)
Using this expression in the second of the equations (C4) gives
− b̂ ·∇u‖e =
, (C7)
where d/dt is defined in the usual way [Eq. (122)]. Assuming isothermal electrons (δpe = T0eδne) annihilates the second term on
the right-hand side and turns the above equation into Eq. (117). As for the first of the equations (C4), the use of Eq. (C6) and
substitution of δB⊥ = −ẑ×∇⊥A‖ turns it into the previously derived Eq. (C2), whence follows Eq. (116).
Thus, we have shown that Eqs. (116-117) can be derived as a direct consequence of Faraday’s law, electron fluid dynamics
(electron continuity equation and the electron force balance, a. k. a. the generalized Ohm’s law), and the assumption of isothermal
electrons—all taken to the leading order in the gyrokinetic ordering given in § 3.1 (i.e., assuming strongly interacting anisotropic
fluctuations with k‖ ≪ k⊥).
We have just proved that Eqs. (116) and (117) are simply the perpendicular and parallel part, respectively, of Eq. (C3). The
latter equation means that the magnetic-field lines are frozen into the electron flow velocity ue, i.e., the flux is conserved, the
result formally proven in § 4.3 [see Eq. (99)].
C.2. Electron MHD and the Derivation of Eqs. (226-227)
One route to Eqs. (226-227), already explained in § 7.2, is to start with Eqs. (C2) and (C7) and assume Boltzmann electrons
and ions and the total pressure balance. Another approach, more standard in the literature on the Hall and Electron MHD, is to
start with Eq. (C3), which states that the magnetic field is frozen into the electron flow. The electron velocity can be written in
terms of the ion velocity and the current density, and the latter then related to the magnetic field via Ampère’s law:
ue = ui −
= ui −
4πene
∇×B. (C8)
To the leading order in ǫ, the perpendicular and parallel parts of Eq. (C3) are Eqs. (C4), respectively, where the perpendicular and
parallel electron velocities are [from Eq. (C8)]
u⊥e = u⊥i +
4πen0e
ẑ×∇⊥δB‖, u‖e = u‖i +
4πen0e
∇2⊥A‖. (C9)
The relative size of the two terms in each of these expressions is controlled by the size of k⊥di, where di = ρi/
βi is the ion
inertial scale. When k⊥di ≫ 1, we may set ui = 0. Note, however, that the ion motion is not totally neglected: indeed, in the
second of the equations (C4), the δne/ne terms comes, via Eq. (C5), from the divergence of the ion velocity [from Eq. (C8),
∇·ui = ∇·ue]. To complete the derivation, we relate δne to δB‖ via the assumption of total pressure balance, as explained in
§ 7.2, giving us Eq. (225). Substituting this equation and Eqs. (C9) into Eqs. (C4), we obtain
= v2Adi b̂ ·∇
1 + 2/βi(1 + Z/τ )
b̂ ·∇∇2⊥Ψ, (C10)
where Ψ = −A‖/
4πmin0i. Equations (C10) evolve the perturbed magnetic field. These equations become the ERMHD equations
(226-227) if δB‖/B0 is expressed in terms of the scalar potential via Eq. (223).
56 SCHEKOCHIHIN ET AL.
Note that there are two special limits in which the assumption of immobile ions suffices to derive Eqs. (C10) from Eq. (C3)
without the need for the pressure balance: βi ≫ 1 (incompressible ions) or τ = T0i/T0e ≪ 1 (cold ions) but βe = βiZ/τ ≫ 1. In both
cases, Eq. (225) shows that δne/n0e ≪ δB‖/B0, so the density perturbation can be ignored and the coefficient of the right-hand
side of the second of the equations (C10) is equal to 1. The limit of cold ions is discussed further in Appendix E.
D. FLUID LIMIT OF THE KINETIC RMHD
Taking the fluid (collisional) limit of the KRMHD system (summarized in § 5.7) means carrying out another subsidiary
expansion—this time in k‖λmfpi ≪ 1. The expansion only affects the equations for the density and magnetic-field-strength
fluctuations (§ 5.5) because the Alfvén waves are indifferent to collisional effects.
The calculation presented below follows a standard perturbation algorithm used in the kinetic theory of gases and in plasma
physics to derive fluid equations with collisional transport coefficients (Chapman & Cowling 1970). For magnetized plasma,
this calculation was carried out in full generality by Braginskii (1965), whose starting point was the full plasma kinetic theory
[Eqs. (36-39)]. While what we do below is, strictly speaking, merely a particular case of his calculation (see Appendix A), it has
the advantage of relative simplicity and also serves to show how the fluid limit is recovered from the gyrokinetic formalism—a
demonstration that we believe to be of value.
It will be convenient to use the KRMHD system written in terms of the function δ f̃i = g + (v2⊥/v
thi)(δB‖/B0)F0i, which is the
perturbation of the local Maxwellian in the frame of the Alfvén waves [Eqs. (150-152)]. We want to expand Eq. (150) in powers
of k‖λmfpi, so we let δ f̃i = δ f̃
i + δ f̃
i + . . ., δB‖ = δB
+ δB(1)
+ . . ., etc.
D.1. Zeroth Order: Ideal Fluid Equations
Since [see Eq. (49)]
k‖λmfpi√
k‖vthi
∼ k‖λmfpi, (D1)
to zeroth order Eq. (150) becomes
δ f̃ (0)i
= 0. The zero mode of the collision operator is a Maxwellian. Therefore, we
may write the full ion distribution function up to zeroth order in k‖λmfpi as follows [see Eq. (144)]
2πTi/mi
mi[(v⊥ − uE)2 + (v‖ − u‖)2]
, (D2)
where ni = n0i + δni and Ti = T0i + δTi include both the unperturbed quantities and their perturbations. The E×B drift velocity uE
comes from the Alfvén waves (see § 5.4) and does not concern us here. Since the perturbations δni, u‖ and δTi are small in the
original gyrokinetic expansion, Eq. (D2) is equivalent to
δ f̃ (0)i =
δn(0)e
v2thi
δT (0)i
v2thi
F0i, (D3)
where we have used quasi-neutrality to replace δni/n0i = δne/n0e. This automatically satisfies Eq. (151), while Eq. (152) gives us
an expression for the ion-temperature perturbation:
δT (0)i
δn(0)e
δB(0)
. (D4)
Note that this is consistent with the interpretation of the perpendicular Ampère’s law [Eq. (63), which is the progenitor of
Eq. (152)] as the pressure balance [see Eq. (67)]: indeed, recalling that the electron pressure perturbation is δpe = T0eδne
[Eq. (103)], we have
= −δpe − δpi = −δneT0e − δniT0i − n0iδTi, (D5)
whence follows Eq. (D4) by way of quasi-neutrality (Zni = ne) and the definitions of Z, τ , βi [Eqs. (40-42)].
Since the collision operator conserves particle number, momentum and energy, we can obtain evolution equations for δn(0)e /n0e,
and δB(0)
/B0 by multiplying Eq. (150) by 1, v‖, v
2/v2thi, respectively, and integrating over the velocity space. The three
moments that emerge this way are
d3vδ f̃ (0)i =
δn(0)e
d3vv‖δ f̃
i = u
v2thi
δ f̃ (0)i =
δn(0)e
δT (0)i
. (D6)
The three evolution equations for these moments are
δn(0)e
δB(0)
+ b̂ ·∇u(0)
= 0, (D7)
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 57
du(0)
− v2A b̂ ·∇
δB(0)
= 0, (D8)
δn(0)e
δT (0)i
δB(0)
b̂ ·∇u(0)
= 0. (D9)
These allow us to recover the fluid equations we derived in § 2.4: Eq. (D8) is the parallel component of the MHD momentum
equation (27); combining Eqs. (D7), (D9) and (D4), we obtain the continuity equation and the parallel component of the induction
equation—these are the same as Eqs. (25) and (26):
δn(0)e
1 + c2s/v
b̂ ·∇u(0)
δB(0)
1 + v2A/c2s
b̂ ·∇u(0)
, (D10)
where the sound speed cs is defined by Eq. (166). From Eqs. (D7) and (D9), we also find the analog of the entropy equation (23):
δT (0)i
δn(0)e
δs(0)
δs(0)
δT (0)i
δn(0)e
δn(0)e
δB(0)
. (D11)
This implies that the temperature changes due to compressional heating only.
D.2. Generalized Energy: Five RMHD Cascades Recovered
We now calculate the generalized energy by substituting δ f̃i from Eq. (D3) into Eq. (153) and using Eqs. (D4) and (D11):
min0iu
min0iu
n0iT0i
1 + Z/τ
5/3 + Z/τ
=W +AW +W
AW +W
sw +W
n0iT0i
1 + Z/τ
5/3 + Z/τ
Ws. (D12)
The first two terms are the Alfvén-wave energy [Eq. (154)]. The following two terms are the slow-wave energy, which splits into
the independently cascaded energies of “+” and “−” waves (see § 2.5):
WSW = W
sw +W
min0i
|z+‖|
2 + |z−‖|
. (D13)
The last term is the total variance of the entropy mode. Thus, we have recovered the five cascades of the RMHD system (§ 2.7;
Fig. 5 maps out the fate of these cascades at kinetic scales).
D.3. First Order: Collisional Transport
Now let us compute the collisional transport terms for the equations derived above. In order to do this, we have to determine
the first-order perturbed distribution function δ f̃ (1)i , which satisfies [see Eq. (150)]
δ f̃ (1)i
δ f̃ (0)i −
v2thi
δB(0)
+ v‖ b̂ ·∇
δ f̃ (0)i +
δn(0)e
. (D14)
We now use Eq. (D3) to substitute for δ f̃ (0)i and Eqs. (D10-D11) and (D8) to compute the time derivatives. Equation (D14)
becomes
δ f̃ (1)i
1 − 3ξ2
v2thi
2/3 + c2s/v
1 + c2s/v
b̂ ·∇u(0)
v2thi
b̂ ·∇δT
F0i(v), (D15)
where ξ = v‖/v. Note that the right-hand side gives zero when multiplied by 1, v‖ or v
2 and integrated over the velocity space, as
it must do because the collision operator in the left-hand side conserves particle number, momentum and energy.
Solving Eq. (D15) requires inverting the collision operator. While this can be done for the general Landau collision operator
(see Braginskii 1965), for our purposes, it is sufficient to use the model operator given in Appendix B.3, Eq. (B18). This simplifies
calculations at the expense of an order-one inaccuracy in the numerical values of the transport coefficients. As the exact value of
these coefficients will never be crucial for us, this is an acceptable loss of precision. Inverting the collision operator in Eq. (D15)
then gives
δ f̃ (1)i =
ν iiD(v)
1 − 3ξ2
v2thi
2/3 + c2s/v
1 + c2s/v
b̂ ·∇u(0)
v2thi
b̂ ·∇δT
F0i(v), (D16)
58 SCHEKOCHIHIN ET AL.
where ν iiD(v) is a collision frequency defined in Eq. (B12) and we have chosen the constants of integration in such a way that the
three conservation laws are respected:
d3vδ f̃ (1)i = 0,
d3vv‖δ f̃
i = 0,
d3vv2δ f̃ (1)i = 0. These relations mean that δn
e = 0,
= 0, δT (1)i = 0 and that, in view of Eq. (152), we have
δB(1)
2/3 + c2s/v
1 + c2s/v
ν‖ib̂ ·∇u‖, (D17)
where ν‖i is defined below [Eq. (D21)]. Equations (D16-D17) are now used to calculate the first-order corrections to the moment
equations (D7-D9). They become
+ b̂ ·∇u‖ = 0, (D18)
− v2Ab̂ ·∇
2/3 + c2s/v
1 + c2s/v
ν‖i b̂ ·∇
b̂ ·∇u‖
, (D19)
= κ‖ib̂ ·∇
b̂ ·∇δTi
, (D20)
where we have introduced the coefficients of parallel viscosity and parallel thermal diffusivity:
ν‖i =
ν iiD(v)v
F0i(v), κ‖i =
ν iiD(v)v
v2thi
F0i(v). (D21)
All perturbed quantities are now accurate up to first order in k‖λmfpi. Note that in Eq. (D19), we used Eq. (D17) to express
δB(0)
= δB‖ − δB
. We do the same in Eq. (D4) and obtain
2/3 + c2s/v
1 + c2s/v
ν‖ib̂ ·∇u‖
. (D22)
This equation completes the system (D18-D20), which allows us to determine δne, u‖, δTi and δB‖. In § 6.1, we use the equations
derived above, but absorb the prefactor (2/3 + c2s/v
A)/(1 + c
A) into the definition of ν‖i. The same system of equations can also
be derived from Braginskii’s two-fluid theory (Appendix A.4), from which we can borrow the quantitatively correct values of the
viscosity and ion thermal diffusivity: ν‖i = 0.90v
thi/νii, κ‖i = 2.45v
thi/νii, where νii is defined in Eq. (52).
E. HALL REDUCED MHD
The popular Hall MHD approximation consists in assuming that the magnetic field is frozen into the electron flow velocity
[Eq. (C3)]. The latter is calculated from the ion flow velocity and the current determined by Ampère’s law [Eq. (C8)]:
4πen0e
, (E1)
where the ion flow velocity ui satisfies the conventional MHD momentum equation (8). The Hall MHD is an appealing theoretical
model that appears to capture both the MHD behavior at long wavelengths (when ue ≃ ui) and some of the kinetic effects that
become important at small scales due to decoupling between the electron and ion flows (the appearance of dispersive waves)
without bringing in the full complexity of the kinetic theory. However, unlike the kinetic theory, it completely ignores the
collisionless damping effects and suggests that the key small-scale physical change is associated with the ion inertial scale
di = ρi/
βi (or, when βe ≪ 1, the ion sound scale ρs = ρi
Z/2τ ; see § E.3), rather than the ion gyroscale ρi. Is this an acceptable
model for plasma turbulence? Figure 8 illustrates the fact that at τ ∼ 1, the ion inertial scale does not play a special role linearly,
the MHD Alfvén wave becomes dispersive at the ion gyroscale, not at di, and that the collisionless damping cannot in general
be neglected. A detailed comparison of the Hall MHD linear dispersion relation with full hot plasma dispersion relation leads to
the conclusion that Hall MHD is only a valid approximation in the limit of cold ions, namely, τ = T0i/T0e ≪ 1 (Ito et al. 2004;
Hirose et al. 2004). In this Appendix, we show that a reduced (low-frequency, anisotropic) version of Hall MHD can, indeed, be
derived from gyrokinetics in the limit τ ≪ 1.50 This demonstrates that the Hall MHD model fits into the theoretical framework
proposed in this paper as a special limit. However, the parameter regime that gives rise to this special limit is not common in
space and astrophysical plasmas of interest.
E.1. Gyrokinetic Derivation of Hall Reduced MHD
Let us start with the equations of isothermal electron fluid, Eqs. (116-121), i.e., work within the assumptions that allowed us
to carry out the mass-ratio expansion (§ 4.8). In Eq. (120) (perpendicular Ampère’s law, or gyrokinetic pressure balance), taking
50 Note that, strictly speaking, our ordering of the collision frequency does not allow us to take this limit (see footnote 17), but this is a minor betrayal of rigor,
which does not, in fact, invalidate the results.
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 59
the limit τ ≪ 1 gives
eik·r
d3vJ0(ai)hik
, (E2)
where we have used Eq. (118) to express the hi integral and the expression for the electron beta βe = βiZ/τ . Note that the above
equation is simply the statement of a balance between the magnetic and electron thermal pressure (the ions are relatively cold,
so they have fallen out of the pressure balance). Using Eq. (E2) to express δne in terms of δB‖ in Eqs. (116) and (117) and also
substituting for u‖e from Eq. (119) [or, equivalently, Eq. (87)], we get
= vAb̂ ·∇
Φ+ vAdi
1 + 2/βe
b̂ ·∇
u‖i − di∇2⊥Ψ
, (E3)
where we have used our usual definitions of the stream and flux functions [Eq. (135)] and of the full derivatives [Eq. (160)].
These equations determine the evolution of the magnetic field, but we still need the ion gyrokinetic equation (121) to calculate
the ion motion (Φ = cϕ/B0 and u‖i) via Eqs. (118) and (88). There are two limits in which the ion kinetics can be reduced to
simple fluid models.
E.1.1. High-Ion-Beta Limit, βi ≫ 1
In this limit, k⊥ρi = k⊥di
βi ≫ 1 as long as k⊥di is not small. Then the ion motion can be neglected because it is averaged out
by the Bessel functions in Eqs. (118) and (88)—in the same way as in § 7.2. So we get Φ = (τ/Z)vAdiδB‖/B0 [using Eq. (E2);
this is the τ ≪ 1 limit of Eq. (223)] and u‖i = 0. Noting that βe = βiZ/τ ≫ 1 in this limit, we find that Eqs. (E3) reduce to
= v2Adi b̂ ·∇
= −di b̂ ·∇∇2⊥Ψ, (E4)
which is the τ ≪ 1 limit of our ERMHD equations (226-227) [or, equivalently, Eqs. (C10)].
E.1.2. Low-Ion-Beta Limit, βi ∼ τ ≪ 1 (the Hall Limit)
This limit is similar to the RMHD limit worked out in § 5: we take, for now, k⊥di ∼ 1 and βe ∼ 1 (in which subsidiary
expansions can be carried out later), and expand the ion gyrokinetics in k⊥ρi = k⊥di
βi ≪ 1. Note that ordering βe ∼ 1 means
that we have ordered βi ∼ τ ≪ 1. We now proceed analogously to the way we did in § 5: express the ion distribution in terms of
the g function defined by Eq. (124) and, using the relation (E2) between δB‖/B0 and δne/n0e, write Eqs. (125-127) as follows:
︸ ︷︷ ︸
︸ ︷︷ ︸
v⊥ ·A⊥
︸ ︷︷ ︸
− 〈Cii[g]〉Ri
︸ ︷︷ ︸
A‖,ϕ− 〈ϕ〉Ri
︸ ︷︷ ︸
+b̂ ·∇
︸ ︷︷ ︸
v⊥ ·A⊥
︸ ︷︷ ︸
F0i +
v⊥ ·A⊥
︸ ︷︷ ︸
,(E5)
Γ1(αi) +
︸ ︷︷ ︸
1 −Γ0(αi)
]Zeϕk
︸ ︷︷ ︸
d3vJ0(ai)gk
︸ ︷︷ ︸
, u‖ki
d3vv‖J0(ai)gk
︸ ︷︷ ︸
. (E6)
All terms in these equations can be ordered with respect to the small parameter
βi (an expansion subsidiary to the gyrokinetic
expansion in ǫ and the Hall expansion in τ ≪ 1). The lowest order to which they enter is indicated underneath each term. The
ordering we use is the same as in § 5.2, but now we count the powers of
βi and order formally k⊥di ∼ 1 and βe ∼ 1. It is easy
to check that this ordering can be summarized as follows
and that the ion and electron terms in Eqs. (E3) are comparable under this ordering, so their competition is retained (in fact, this
could be used as the underlying assumption behind the ordering). The fluctuation frequency continues to be ordered as the Alfvén
frequency, ω ∼ k‖vA. The collision terms are ordered via ω/νii ∼ k‖λmfpi/
βi and k‖λmfpi ∼ 1, although the latter assumption is
not essential for what follows, because collisions turn out to be negligible and it is fine to take k‖λmfpi ≫ 1 from the outset and
neglect them completely.
In Eqs. (E6), we use Eqs. (129) and (130) to write 1 −Γ0(αi) ≃ αi = k2⊥ρ2i /2 and Γ1(αi) ≃ 1. These equations imply that if we
expand g = g(−1) + g(0) + . . ., we must have
d3vg(−1) = 0, so the contribution to the right-hand side of the first of the equations
60 SCHEKOCHIHIN ET AL.
(E6) (the quasi-neutrality equation) comes from g(0), while the parallel ion flow is determined by g(−1). Retaining only the lowest
(minus first) order terms in Eq. (E5), we find the equation for g(−1), the v‖ moment of which gives an equation for u‖i:
∂g(−1)
{ϕ,g(−1)} = 2
v‖b̂ ·∇
F0i ⇒
= v2Ab̂ ·∇
. (E8)
Now integrating Eq. (E5) over the velocity space (at constant r), using the first of the equations (E6) to express the integral of
g(0), and retaining only the lowest (zeroth) order terms, we find
ρ2i ∇2⊥
+ b̂ ·∇u‖i = 0 ⇒
∇2⊥Φ = vAb̂ ·∇∇2⊥Ψ, (E9)
where we have used the second of the equations (E3) to express the time derivative of δB‖/B0.
Together with Eqs. (E3), Eqs. (E8) and (E9) form a closed system, which it is natural to call Hall Reduced MHD (HRMHD)
because these equations can be straightforwardly derived by applying the RMHD ordering (§ 2.1) to the MHD equations (8-10)
with the induction equation (10) replaced by Eq. (E1). Indeed, Eqs. (E8) and (E9) exactly coincide with Eqs. (27) and (18), which
are the parallel and perpendicular components of the MHD momentum equation (8) under the RMHD ordering; Eqs. (E3) should
be compared Eqs. (17) and (26) while noticing that, in the limit τ ≪ 1, the sound speed is cs = vA
βe/2 [see Eq. (166)]. The
incompressible case (Mahajan & Yoshida 1998) is recovered in the subsidiary limit βe ≫ 1 (i.e., 1 ≫ βi ≫ τ ).
E.2. Generalized Energy for Hall RMHD and the Passive Entropy Mode
To work out the generalized energy (§ 3.4) for the HRMHD regime, we start with the generalized energy for the isothermal
electron fluid [Eq. (109)] and use Eq. (E2) to express the density perturbation:
T0iδ f
, (E10)
where δB⊥ = ẑ×∇⊥Ψ. The perturbed ion distribution function can be written in the same form as it was done in § 5.4 [Eq. (143)]:
to lowest order in the
βi expansion (§ E.1.2),
δ f (−1)i =
2v⊥ ·u⊥
v2thi
F0i + g(−1) =
2v⊥ ·u⊥
v2thi
F0i +
2v‖u‖i
v2thi
F0i + g̃, (E11)
where u⊥ = ẑ×∇⊥Φ. The last equality above is achieved by noticing that, since g(−1) satisfies Eq. (E8), we may split it into a
perturbed Maxwellian with parallel velocity u‖i and the remainder: g
(−1) = 2v‖u‖iF0i/v
thi + g̃. Then g̃ is the homogeneous solution
of the leading-order kinetic equation [see Eq. (E8)]:
+{Φ, g̃} = 0,
d3v g̃ = 0. (E12)
Substituting Eq. (E11) into Eq. (E10) and keeping only the leading-order terms in the
βi expansion, we get
min0iu
min0iu
T0ig̃
. (E13)
The first four terms are the energy of the Alfvénic and slow-wave-polarized fluctuations [cf. Eq. (D12)]. Unlike in RMHD, these
are not decoupled in HRMHD, unless a further subsidiary long-wavelength limit is taken (see § E.4). It is easy to verify that the
sum of these four terms is indeed conserved by Eqs. (E3), (E8) and (E9). The last term in Eq. (E13) is an individually conserved
kinetic quantity. Its conservation reflects the fact that g̃ is decoupled from the wave dynamics and passively advected by the
Alfvénic velocities via Eq. (E12).51
The passive kinetic mode g̃ can be thought of as a kinetic version of the MHD entropy mode and, indeed, reduces to it if the
collision operator in Eq. (E5) is upgraded to the leading order by orderingω/νii ∼ 1 (i.e., by considering long parallel wavelengths,
k‖λmfpi ∼
βi). In such a collisional limit, g̃ has to be a perturbed Maxwellian with no density or velocity perturbation [because
d3vg̃ = 0, while the velocity perturbation is explicitly separated from g̃ in Eq. (E11)]. Therefore,
v2thi
F0i ⇒
T0ig̃
n0iT0i
δT 2i
T 20i
. (E14)
This is to be compared with the βi ∼ τ ≪ 1 limit of Eqs. (D11) and (D12). As we have established, in the
βi expansion,
δTi = δT
i , δni = δn
i , δB‖ = δB
, so to lowest order δs/s0 = δTi/T0i and Eq. (E14) describes the entropy mode in the Hall limit.
51 A similar splitting of the generalized energy cascade into a fluid-like cascade plus a passive cascade of a zero-density part of the distribution function occurs
in the Hasegawa–Mima regime, which is the electrostatic version of the Hall limit (Plunk et al. 2009).
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 61
E.3. Hall RMHD Dispersion Relation
Linearizing the Hall RMHD equations (E3), (E8) and (E9) (derived in § E.1.2 assuming the ordering βi ∼ τ ≪ 1), we obtain
the following dispersion relation:52
ω2 − k2‖v
1 + 2/βe
= ω2k2‖v
1 + 2/βe
. (E15)
When the coupling term on the right-hand side is negligible, k⊥di/
1 + 2/βe ≪ 1, we recover the MHD Alfvén wave, ω2 = k2‖v
and the MHD slow wave, ω2 = k2
v2A/(1 + v
s ) [Eq. (167)], where cs = vA
βe/2 in the limit τ ≪ 1 [Eq. (166)]. In the opposite
limit, we get the kinetic Alfvén wave, ω2 = k2
i /(1 + 2/βe) [same as Eq. (230) with τ ≪ 1].
The solution of the dispersion relation (E15) is
1 + 2/βe
 . (E16)
The corresponding eigenfunctions then satisfy53
Ψ = −
Φ+ vAdi
, u‖i = −
, Φ = −
Ψ. (E17)
Equation (E16) takes a particularly simple form in the subsidiary limits of high and low electron beta βe = βiZ/τ :
βe ≫ 1 : ω2 = k2‖v
 , βe ≪ 1 : ω2 = k2‖v
1 + k2⊥ρ
and ω2 =
1 + k2⊥ρ
, (E18)
where ρs = di
βe/2 = ρi
Z/2τ = cs/Ωi is called the ion sound scale. The Alfvén wave and the slow wave (known as the ion
acoustic wave in the limit of τ ≪ 1, βe ≪ 1) become dispersive at the ion inertial scale (k⊥di ∼ 1) when βe ≫ 1 and at the ion
sound scale (k⊥ρs ∼ 1) when βe ≪ 1.
E.4. Summary of Hall RMHD and the Role of the Ion Inertial and Ion Sound Scales
We have shown that in the limit of cold ions and low ion beta (βi ∼ τ ≪ 1, “the Hall limit”), gyrokinetic turbulence can be
described by five scalar functions: the stream and flux functions Φ and Ψ for the Alfvénic fluctuations, the parallel velocity and
magnetic-field perturbations u‖i and δB‖ for the slow-wave-polarized fluctuations, and g̃, the zero-density, zero-velocity part of
the ion distribution function, which is the kinetic version of the MHD entropy mode. The first four of these functions satisfy a
closed set of four fluid-like equations, derived in § E.1 and collected here:
= vAb̂ ·∇
Φ+ vAdi
1 + 2/βe
b̂ ·∇
u‖i − di∇2⊥Ψ
, (E19)
∇2⊥Φ = vAb̂ ·∇∇2⊥Ψ,
= v2Ab̂ ·∇
. (E20)
We call these equations the Hall Reduced Magnetohydrodynamics (HRMHD). To fully account for the generalized energy cas-
cade, one must append to the four HRMHD equations the fifth, kinetic equation (E12) for g̃, which is energetically decoupled
from HRMHD and slaved to the Alfvénic velocity fluctuations (§ E.2).
The equations given above are valid above the ion gyroscale, k⊥ρi ≪ 1. They contain a special scale, di/
1 + 2/βe, which
is the ion inertial scale di for βe ≫ 1 and the ion sound scale ρs = cs/Ωi for βe ≪ 1. As becomes clear from the linear theory
(§ E.3), the Alfvén and slow waves become dispersive at this scale. Nonlinearly, this scale marks the transition from the regime
in which the Alfvénic and slow-wave-polarized fluctuations are decoupled to the regime in which they are mixed. Namely, when
k⊥di/
1 + 2/βe ≪ 1, HRMHD turns into RMHD: Eqs. (E19) become Eqs. (17) and (26), while Eqs. (E20) remain unchanged
and identical to Eqs. (18) and (27); in the opposite limit, k⊥di/
1 + 2/βe ≫ 1, the ion motion decouples from the magnetic-field
evolution and Eqs. (E19) turn into the ERMHD equations (226-227).
Since we are considering the case βi ≪ 1, both di and ρs are much larger than the ion gyroscale ρi. In the opposite limit of
βi ≫ 1 (§ E.1.1), while di is the only scale that appears explicitly in Eqs. (E4), we have di ≪ ρi and the equations themselves
represent the dynamics at scales much smaller than the ion gyroscale, so the transition between the RMHD and ERMHD regimes
occurs at k⊥ρi ∼ 1. The same is true for βi ∼ 1, when di ∼ ρi. The ion sound scale ρs ≫ ρi does not play a special role when
52 The full gyrokinetic dispersion relation in a similar limit was worked out in Howes et al. (2006), Appendix D.2.1.
53 Note that wave packets with |k⊥| = k⊥ and satisfying Eq. (E17) with k‖vA/ω as a function of k⊥ given by Eq. (E16) are exact nonlinear solutions of
the HRMHD equations (E3) and (E8-E9). This can be shown via a calculation analogous to that in § 7.3 (for the incompressible Hall MHD, this was done by
Mahajan & Krishan 2005).
62 SCHEKOCHIHIN ET AL.
βi is not small: it is not hard to see that for k⊥ρs ∼ 1, the ion motion terms in Eqs. (E19) dominate and we simply recover the
inertial-range KRMHD model (§ 5) by expanding in k⊥ρi = k⊥ρs
2τ/Z ≪ 1.
Various theories of the dissipation-range turbulence based on Hall and Electron MHD are further discussed in § 8.2.6.
F. TWO-DIMENSIONAL INVARIANTS IN GYROKINETICS
Since gyrokinetics is in a sense a “quasi-two-dimensional” approximation, it is natural to inquire if this gives rise to additional
conservation properties (besides the conservation of the generalized energy discussed in § 3.4) and how they are broken by the
presence of parallel propagation terms. It is important to emphasize that, except in a few special cases, these invariants are only
invariants in 2D, so gyrokinetic turbulence in 2D and 3D has fundamentally different properties, despite its seemingly “quasi-2D”
nature. It is, therefore, generally not correct to think of the gyrokinetic turbulence (or its special case the MHD turbulence) as
essentially a 2D turbulence with an admixture of parallel-propagating waves (Fyfe et al. 1977; Montgomery & Turner 1981).
In this Appendix, we work out the 2D invariants. Without attempting to present a complete analysis of the 2D conservation
properties of gyrokinetics, we limit our discussion to showing how some more familiar fluid invariants (most notably, magnetic
helicity) emerge from the general 2D invariants in the appropriate asymptotic limits.
F.1. General 2D Invariants
In deriving the generalized energy invariant, we used the fact that
d3Rs hs{〈χ〉Rs ,hs} = 0, so Eq. (57) after multiplication
by T0shs/F0s and integration over space contains no contribution from the Poisson-bracket nonlinearity. Since we also have∫
d3Rs 〈χ〉Rs{〈χ〉Rs ,hs} = 0, multiplying Eq. (57) by qs〈χ〉Rs and integrating over space has a similar outcome. Subtracting the
latter integrated equation from the former and rearranging terms gives
qs〈χ〉Rs
= qsv‖
d3Rs 〈χ〉Rs
qs〈χ〉Rs
. (F1)
We see that in a purely 2D situation, when ∂/∂z = 0, we have an infinite family of invariants Is = Is(v⊥,v‖) whose conservation (for
each species and for every value of v⊥ and v‖!) is broken only by collisions. In 3D, the parallel particle streaming (propagation)
term in the gyrokinetic equation generally breaks these invariants, although special cases may arise in which the first term on the
right-hand side of Eq. (F1) vanishes and a genuine 3D invariant appears.
F.2. “A2
-Stuff”
Let apply the mass-ratio expansion (§ 4.1) to Eq. (F1) for electrons. Using the solution (101) for the electron distribution
function, we find
T0eF0e
v2the
d3rA‖
v2the
+ · · ·
= −ev‖
v2the
F0e −
∂h(1)e
d3rA‖
, (F2)
where we have kept terms to two leading orders in the expansion. To lowest order, the above equation reduces to
d3rA‖
. (F3)
This equation can also be obtained directly from Eq. (116) (multiply by A‖ and integrate). In 2D, it expresses a well known
conservation law of the “A2‖-stuff.” As this 2D invariant exists already on the level of the mass-ratio expansion of the electron
kinetics, with no assumptions about the ions, it is inherited both by the RMHD equations in the limit of k⊥ρi ≪ 1 (§ 5.3) and
by the ERMHD equations in the limit of k⊥ρi ≫ 1 (§ 7.2). In the former limit, δne/n0e on the right-hand side of Eq. (F3) is
negligible (under the ordering explained in § 5.2); in the latter limit, it is expressed in terms of ϕ via Eq. (221). The conservation
of “A2‖-stuff” is a uniquely 2D feature, broken by the parallel propagation term in 3D.
F.3. Magnetic Helicity in the Electron Fluid
If we now divide Eq. (F2) through by ev‖/c and integrate over velocities, we get, after some integrations by parts, another
relation that becomes a conservation law in 2D and that can also easily be derived directly from the equations of the isothermal
electron fluid (116-117):
d3rA‖
. (F4)
In the ERMHD limit k⊥ρi ≫ 1 (§ 7.2), we use Eqs. (221-223) to simplify the above equation and find that the integral on the
right-hand side vanishes and we get a genuine 3D conservation law:
d3rA‖δB‖ = 0. (F5)
KINETIC TURBULENCE IN MAGNETIZED PLASMAS 63
This can also be derived directly from the ERMHD equations (226-227) [using Eq. (223)]. The conserved quantity is readily seen
to be the helicity of the perturbed magnetic field:
d3rA · δB =
∇⊥×A‖ẑ
+ A‖δB‖
A‖ẑ · (∇⊥×A⊥) + A‖δB‖
d3rA‖δB‖. (F6)
F.4. Magnetic Helicity in the RMHD Limit
Unlike in the case of ERMHD, the helicity of the perturbed magnetic field in RMHD is conserved only in 2D. This is because
the induction equation for the perturbed field has an inhomogeneous term associated with the mean field [Eq. (10) with B =
B0ẑ + δB] (this issue has been extensively discussed in the literature; see Matthaeus & Goldstein 1982; Stribling et al. 1994;
Berger 1997; Montgomery & Bates 1999; Brandenburg & Matthaeus 2004). Directly from the induction equation or from its
RMHD descendants Eqs. (17) and (26), we obtain [note the definitions (135)]
d3rA‖δB‖ =
1 + v2A/c2s
, (F7)
so helicity is conserved only if ∂/∂z = 0.
For completeness, let us now show that this 2D conservation law is a particular case of Eq. (F1) for ions. Let us consider the
inertial range (k⊥ρi ≪ 1). We substitute Eq. (124) into Eq. (F1) for ions and expand to two leading orders in k⊥ρi using the
ordering explained in § 5.2:
v‖〈A‖〉Ri
Z2e2v2
d3rA‖g + · · ·
Z2e2v2
d3rA‖
v2thi
+ Zev‖
d3rA‖
. (F8)
The lowest-order terms in the above equations (all proportional to v2‖F0i) simply reproduce the 2D conservation of “A
‖-stuff,”
given by Eq. (F3). We now subtract Eq. (F3) multiplied by (Zev‖/c)
2F0i/T0i from Eq. (F8). This leaves us with
d3rA‖g = c
+ v‖F0i
v2thi
d3rA‖
. (F9)
This equation is a general 2D conservation law of the KRMHD equations (see § 5.7) and can also be derived directly from them.
If we integrate it over velocities and use Eqs. (146) and (147), we simply recover Eq. (F4). However, since Eq. (F9) holds for
every value of v‖ and v⊥, it carries much more information than Eq. (F4).
To make connection to MHD, let us consider the fluid (collisional) limit of KRMHD worked out in Appendix D. The distribu-
tion function to lowest order in the k‖λmfpi ≪ 1 expansion is g = −(v2⊥/v2thi)δB‖/B0 +δ f̃
i , where δ f̃
i is the perturbed Maxwellian
given by Eq. (D3). We can substitute this expression into Eq. (F9). Since in this expansion the collision integral is applied to δ f̃ (1)i
and is the same order as the rest of the terms (see § D.3), conservation laws are best derived by taking 1, v‖, and v
2/v2thi moments
of Eq. (F9) so as to make the collision term vanish. In particular, multiplying Eq. (F9) by 1 + (2τ/3Z)v2/v2thi, integrating over
velocities and using Eqs. (D4) and (D6), we obtain the evolution equation for
d3rA‖δB‖, which coincides with Eq. (F7). Note
that, either proceeding in an analogous way, one can derive similar equations for
d3rA‖δne and
d3rA‖u‖—these are also 2D
invariants of the RMHD system, broken in 3D by the presence of the propagation terms. The same result can be derived directly
from the evolution equations (D8) and (D10).
F.5. Electrostatic Invariant
Interestingly, the existence of the general 2D invariants introduced in § F.1 alongside the generalized energy invariant given by
Eq. (73) means that one can construct a 2D invariant of gyrokinetics that does not involve any velocity-space quantities. In order
to do that, one must integrate Eq. (F1) over velocities, sum over species, and subtract Eq. (73) from the resulting equation (thus
removing the h2s integrals). The result is not particularly edifying in the general case, but it takes a simple form if one considers
electrostatic perturbations (δB = 0). In this case, χ = ϕ, and the manipulations described above lead to the following equation
d3v Is −W
q2s n0s
1−Γ0(αs)
|ϕk|2 =
d3rE‖ j‖ −
d3Rs 〈ϕ〉Rs
, (F10)
where E‖ = −∂ϕ/∂z, αs = k2⊥ρ
s/2 and Γ0 is defined by Eq. (129). In 2D, E‖ = 0 and the above equation expresses a conservation
law broken only by collisions. The complete derivation and analysis of 2D conservation properties of gyrokinetics in the electro-
static limit, including the invariant (F10), the electrostatic version of Eq. (F1), and their consequences for scalings and cascades,
was given by Plunk et al. (2009). Here we briefly consider a few relevant limits.
For k⊥ρi ≪ 1, we have Γ0(α) = 1 −αs + . . ., so the invariant given by Eq. (F10) is simply the kinetic energy of the E×B flows:
s(msn0s/2)
d3r |∇⊥Φ|2, where Φ = cϕ/B0. In the limit k⊥ρi ≫ 1, k⊥ρe ≪ 1, we have Y = −n0i
d3rZ2e2ϕ2/2T0i. In
64 SCHEKOCHIHIN ET AL.
the limit k⊥ρe ≫ 1, we have Y = −(1 + Z/τ )n0e
d3re2ϕ2/2T0e. Whereas we are not interested in electrostatic fluctuations in the
inertial range, electrostatic turbulence in the dissipation range was discussed in § 7.10 and § 7.12. The electrostatic 2D invariant
in the limits k⊥ρi ≫ 1, k⊥ρe ≪ 1 and k⊥ρe ≫ 1 can also be derived directly from the equations given there [in the former limit,
use Eq. (264) to express u‖i in terms of j‖ in order to get Eq. (F10)].
Note that, taken separately and integrated over velocities, Eq. (F1) for ions (when k⊥ρi ≫ 1, k⊥ρe ≪ 1) and for electrons
(when k⊥ρe ≫ 1), reduces to lowest order to the statement of 3D conservation of
d3Ri T0ih2i /2F0i [Whi in Eq. (245)] and∫
d3Re T0eh2e/2F0e [Eq. (280)], respectively.
F.6. Implications for Turbulent Cascades and Scalings
Since invariants other than the generalized energy or its constituent parts are present in 2D and, in some limits, also in 3D, one
might ask how their presence affects the turbulent cascades and scalings. As an example, let us consider the magnetic helicity in
KAW turbulence, which is a 3D invariant of the ERMHD equations (§ F.3).
A Kolmogorov-style analysis of a local KAW cascade based on a constant flux of helicity gives (proceeding as in § 7.5):
τKAWλ
1 +βi
τKAWλ
1 +βi
∼ εH = const ⇒ Φλ ∼
(1 +βi)1/6
1/3, (F11)
where εH is the helicity flux (omitting constant dimensional factors, the helicity is now defined as
d3rΨΦ and assumed to be
non-zero). This corresponds to a k
⊥ spectrum of magnetic energy.
In order to decide whether we expect the scalings to be determined by the constant-helicity flux or by the constant-energy
flux (as assumed in § 7.5), we adapt a standard argument originally due to Fjørtoft (1953). If the helicity flux of the KAW
turbulence originating at the ion gyroscale (via partial conversion from the inertial-range turbulence; see § 7) is εH , its energy
flux is εKAW ∼ εH [set λ = ρi in Eq. (F11) and compare with Eq. (238)]. If the cascade between the ion and electron gyroscales
is controlled by maintaining a constant flux of helicity, then the helicity flux arriving to the electron gyroscale is still εH , while
the associated energy flux is εHρi/ρe ≫ εKAW, i.e., more energy arrives to ρe than there was at ρi! This is clearly impossible in
a stationary state. The way to resolve this contradiction is to conclude that the helicity cascade is, in fact, inverse (i.e., directed
towards larger scales), while the energy cascade is direct (to smaller scales). A similar argument based on the constancy of the
energy flux εKAW then leads to the conclusion that the helicity flux arriving to the electron gyroscale is εKAWρe/ρi ≪ εH ∼ εKAW,
i.e., the helicity indeed does not cascade to smaller scales. It does not, in fact, cascade to large scales either because the ERMHD
equations are not valid above the ion gyroscale and the helicity of the perturbed magnetic field in the inertial range is not a 3D
invariant (§ F.4). The situation would be different if an energy source existed either at the electron gyroscale or somewhere in
between ρe and ρi. In such a case, one would expect an inverse helicity cascade and the consequent shallower scaling [Eq. (F11)]
between the energy-injection scale and the ion gyroscale.
Other invariants introduced above can in a similar fashion be argued to give rise to inverse cascades in the hypothetical 2D
situations where they are valid and provided there is energy injection at small scales (for the electrostatic case, see Plunk et al.
2009 and numerical simulations by Tatsuno et al. 2009b). The view of turbulence advanced in this paper does not generally
allow for this to happen. First, the fundamentally 3D nature of the turbulence is imposed via the critical balance conjecture and
supported by the argument that “two dimensionality” can only be maintained across parallel distances that do not exceed the
distance a parallel-propagating wave (or parallel-streaming particles) travels over one nonlinear decorrelation time (see § 1.2,
§ 7.5 and § 7.10.3). Secondly, the lack of small-scale energy injection was assumed at the outset. This can, however, be violated
in real astrophysical plasmas by various small-scale plasma instabilities (e.g., triggered by pressure anisotropies; see discussion
in § 8.3). Treatment of such effects falls outside the scope of this paper and remains a matter for future work.
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