J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)11–16
000–000

On the Nature of ETG Turbulence and Cross-Scale Coupling
JENKO Frank
Max-Planck-Institut für Plasmaphysik,
EURATOM-Association, D-85748 Garching, Germany(Received: 12 December 2003 / Accepted: 19 May 2004)

Abstract
Microturbulence in tokamaks and stellarators is studied via gyrokinetic simulation and semi-analytical theory.
The focus is on electron temperature gradient (ETG) turbulence and its interactions with fluctuations at longer
wavelengths. In this context, special attention is paid to the physics of edge transport barriers.
Keywords:
microturbulence, ETG mode, cross-scale coupling, transport barrier

1. Introduction[2]. This is due to the zonal flow mode, characterized byfluctuations of the electrostatic potential with ky = k|| = 0,
which does not participate in the parallel dynamics.
Consequently, the electrons are never adiabatic for this mode
and in the ITG case it must be subtracted out of the adiabatic
term in the equation for the electrostatic potential [3]. By
contrast, in the ETG case, the ions are made adiabatic by
gyromotion, which affects every mode in the spectrum. So
for adiabatic ETG turbulence the zonal flow mode is also
adiabatic, forming the principal distinction between the twobasic models.

Plasma turbulence can be driven by a large number ofmicroinstabilities, including ion temperature gradient (ITG)
modes, trapped electron modes, electron drift waves, andelectron temperature gradient (ETG) modes [1]. While thefirst three are associated with perpendicular spatial scales ofroughly 1–50 ion gyroradii, ETG modes exist on smallerscales of typically 0.1–1 ion gyroradii. So far, almost all effort
in computational and theoretical turbulence studies went into
the investigation of isolated and (sometimes highly) idealized
subsystems. However, with the advent of modern
supercomputers, more comprehensive simulations become
feasible, taking the multi-scale nature of the plasma
turbulence problem seriously. In the present paper, thecharacter of adiabatic ETG turbulence is reviewed, and theinteractions of ETG turbulence with fluctuations at longer
wavelengths are discussed. In this context, special attention
is paid to the physics of edge transport barriers.

2.1 High-amplitude streamers
The manifestation of this effect in nonlinear simulations
is that while adiabatic ITG and ETG computations saturate at
similar normalized levels in the sheared slab case (evenconsistent with mixing length expectations, χ ~ γ max/k⊥2 ~ρ2 vt /L T), toroidal ETG simulations can go to much higher
amplitudes than their ITG counterparts [2]. When this is thecase, they are characterized by a predominance of “streamer”
modes, i.e., radially elongated vortices exhibiting themorphology shown in Fig. 4 of Ref. [2]. At the same time,
the associated electron heat transport (which is always
predominantly electrostatic, both with and without streamers[2,4]) is boosted by up to an order of magnitude or more withrespect to mixing length expectations. Streamers have been
observed both in tokamak [2,5,6] and in stellarator [4,7]geometry if and only if the underlying long-wavelength
instabilities have a clear toroidal (as opposed to slab)
character. For large aspect ratio tokamaks with circular crosssection and small Shafranov shift, this is the case for ŝ ~> 0.4[2]. Here, ŝ and R are, respectively, theand R/LTe >> R/LTcritemagnetic shear and the major radius of the torus. Streamer

2. The character of adiabatic ETG
turbulence
Neglecting Debye shielding effects, ETG and ITG modes
are perfectly isomorphic in the electrostatic and adiabatic
limit. Under these circumstances, it is therefore permissible
to transfer linear results from the one to the other by simply
interchanging the species labels. Consequently, the space andtime scales of ETG modes are set by the electron gyroradius,ρe , and the inverse linear growth rate, L Te /vte . (As will become
clear later in this Section, it is appropriate to use thetemperature scale length L Te as the reference scale, not thedensity scale length L n or the minor/major radius of the torus.)
However, even in the electrostatic and adiabatic limit, thisETG/ITG symmetry is broken in the nonlinear regime due toa subtle difference in the response of the adiabatic species©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: fsj@ipp.mpg.de111

Jenko F., On the Nature of ETG Turbulence and Cross-Scale Coupling

the q-profile.) They can be self-generated by the turbulence
and may in turn act as its dominant nonlinear saturation
mechanism [10]. There is much literature on this for ion scale
turbulence, particularly ITG modes with adiabatic electrons[11,12] as well as edge turbulence [13,14], but much less isknown about their behavior on electron gyroradius scales. For
the stellarator simulation presented in Ref. [7], the time
averaged root-mean-square (RMS) values of the E × B
shearing rate, Ω̃ ≡ ṽ′Ey , and of the magnetic shear fluctuation,s̃ ≡ qRB̃′y /B, are given by Ω̃ rms ≈ 0.12 vte /R ~ 0.3 γ max ands̃ rms ≈ 0.018 where γ max is the maximum linear growth rate.
This is in stark contrast to results from ITG turbulence whereΩ̃ can significantly exceed γ max (e.g., Ω̃ max/γ max ~ 14 in Ref.
[15]). In the ITG case, one obtains the zonal flow saturationmax
only after correcting for the ineffectivecriterion Ω̃ rms <~γness of the high frequency component of Ω̃ . Moreover, thezonal components of ETG turbulence contribute only 1 % or
so to the total φ̃ rms [7]. This is again in contrast to the findingsin the ITG case where zonal modes with kxρi ~ 0.1 tend tocontribute significantly or even dominate the fluctuation freeenergy contained in φ̃ (see, e.g., Ref. [15] and references
therein). It is well known that low-amplitude ITG streamers
are broken up by zonal flows. In the ETG case, however, theself-generated zonal flows are too weak (about 15–20 times
weaker than in the ITG case as we have shown above) to
break up the high-amplitude streamers. This finding is
supported by recent theoretical investigations [16]. Moreover,since magnetic shear variations primarily affect the linear
growth rates of the ETG modes driving the turbulence [17], avalue of s̃ rms ≈ 0.018 is certainly too small for zonal fields toplay a significant role. Similar zonal flow/field saturationlevels as the ones reported here have also been found in ETG
simulations using tokamak geometry. Thus, we may conclude
that, at least for a significant region in parameter space, zonal
modes on ρe scales tend to play a subdominant role in the
turbulent dynamics.

aspect ratios computed via the radial/poloidal anisotropy in
the autocorrelation functions of φ̃ are typically of the orderof 2 [7]. This value seems somewhat low compared with thevisual impression and indicates that refined measurements arecalled for; perhaps those inspired by percolation theory will
prove useful. Nevertheless, we have found the following> 2 thencorollary to hold: if the correlation length ratio λx /λy ~streamers are predominant in the spatial morphology. For
Cyclone Base Case parameters we find the dominant modes
to arise near k y ρe = 0.15 [2,6] and to exibit a phase shift of~ π/3 between φ̃ and T̃, both results commensurate to thecorresponding linear streamer [2]. An analysis of the transfer
rates of E × B energy in the nonlinear saturated state [8]clearly confirms that streamers are driven predominantly by
linear processes and that they may be viewed as residuals of
linear modes. Nonlinear processes like modulational
instabilities need not be invoked to explain streamer
generation.

2.2 Secondary instabilities
Because of their similarity to the underlying linear
modes, the nonlinear saturation of streamers can beinterpreted in the framework of secondary instability theory.
Depending on the importance of slab character of linear
electrostatic ETG/ITG modes (as characterized by the intrinsic
parallel velocity component), one finds that one of two
distinct processes may dominate as nonlinear saturation
mechanism [6]. Perpendicular shear in the parallel flow of
the linear instability drives a (hereafter, “Cowley”) secondary,described in detail in Ref. [9]. Importantly, this secondary is
not sensitive to the form of the adiabatic response and thus
leads to the same (mixing-length type) transport level in bothslab cases. Predominantly curvature driven modes, on theother hand, are broken up by a (hereafter, “Rogers”)
secondary that is driven by the perpendicular shear in the
eigenmode’s perpendicular E × B flow [2,5]. Due to the factthat the adiabatic ion response for the ETG case affects allmodes, the Rogers secondary is significantly weakened on ρe
scales, while on ion scales, it can access the zonal flow modein φ̃ . Because k|| = 0 for this mode, it is not affected by the
adiabatic response and so remains free. This is the essential
difference between the nonlinear dynamics of ETG and ITG
modes. Consequently, for the toroidal case in which theRogers secondary is more relevant, an ETG mode tends tosaturate at a much higher normalized level than either its ITG
counterpart or the corresponding mixing length expectations.
With this enhancement, associated with the more resilient
high-amplitude streamers, ETG-induced transport can, in
principle, be comparable to electron energy transport induced
by nonadiabatic ITG modes and trapped electron modes.

2.4 Semi-analytical transport estimate
Building on these results and insights, a simple
numerical model with a universal fit parameter can be
developed which was shown to agree fairly well with theresults of nonlinear gyrokinetic computations [6]. The basic
idea is to predict the saturation amplitude by balancing
primary (γr) and secondary (γnl) growth rates. The latter arecomputed numerically by using high-amplitude ETG
streamers as pseudo-equilibria. Since γnl is proportional to theamplitude of the primary mode in this regime, the saturation
amplitude can thus be estimated. Furthermore, we takeadvantage of the fact that our computational results are all in
the regime of strong turbulence, where χe ∝ φ̃. This way wearrive at numerical transport estimates which exhibit
reasonable agreement with the results from gyrokinetic
computations [6]. Importantly, neither the linear growth rate
nor the maximal value of γ / 〈k⊥2 〉 predicts the variation foundin the nonlinear computations. Rather, it is the variation of
the secondary growth rate as the linear eigenfunction changes

2.3 Zonal flows and fields
Zonal flows and fields are the E × B velocities andmagnetic fields which result, respectively, from disturbancesin φ̃ and Ã || with a purely radial dependence (kx ≠ 0, ky = k|| =0). (The latter can also be thought of as radial variations in2
12

Jenko F., On the Nature of ETG Turbulence and Cross-Scale Coupling

3.1 Living in the tail of a cascadein response to the equilibrium parameters that correlates withthe difference in the turbulent flux. The secondary growth
rates exhibit a strong dependence on magnetic shear as thebasic character changes from the Rogers (moderate positive
shear) to Cowley (negative shear) type. The good agreement
between the gyrokinetic computation and numerical model
encourages us to persue a semi-analytical treatment of thebalance between long-wavelength primaries and secondaries,
condensing several important pieces of information about thesaturated nonlinear state into simple algebraic formulas. One< R/LTe >> R/LTcrit(algebraic
finds that for ŝ ~> 0.4 and R/Ln ~ecrit
expressions for R/L Te in tokamaks and the stellarator
Wendelstein 7-AS have been derived in Refs. [18] and [4],respectively), transport by adiabatic, electrostatic ETG modes
is approximately given by [8]

qR
0.15
χe =
2
2
3/2
L
τ e (1 + s /2ν )Te

3

ρ e2 vte
L Te

Obviously, the hyperfine-scale turbulence “feels” boththe quiescient background profiles and their modifications dueto the fine-scale fluctuations. In order to better understandthis effect, let us first discuss some basic properties ofcascades in plasma microturbulence. The strong magnetic
field lends the dynamics a quasi-two-dimensional character,such that parallels with two-dimensional Navier-Stokes
turbulence can be drawn [21]. In the latter case, it is known
that there exists a dual cascade: while energy is transferred
to larger scales, enstrophy (i.e., mean-square vorticity, 〈(∇ ×vE)2〉) cascades to smaller scales [22]. Denoting the rate of
the enstrophy cascade by ε, and the eddy size, lifetime, andvelocity by r, τr, and vr, respectively, one can easily establish2
the following relations: ε ~ (vr/r) /τr, τr ~r/vr, kr ~ 1/r.
From these one can derive Kraichnan’s famous energy–3
spectrum of two-dimensional turbulence: E (kr) ∝ ε 2/3kr [22].
Furthermore, for a nearly adiabatic plasma one obtains
nr ∝ φr ∝rvr ∝ ε 1/3kr–2 . This result matches fairly well withnonlinear simulation results (see Ref. [7] and references
therein) which yield exponents of the order of –2.3. Anotherconsequence is that the eddy turnover time τr is independentof the eddy size r. The same is true for the shearing rate
associated with arbitrary (not necessarily zonal) E × B flows:Ω̃ ∝ kr2φ̃r ~ const. Assuming that zonal flows behave thesame way, we expect that this result may be extended topurely radial fluctuations. And indeed, both gyrofluid
simulations of adiabatic ITG turbulence [15] and two-scale
gyrokinetic simulations of co-existing ITG/ETG turbulence[7] show that the zonal flow shearing rate is (almost) scaleindependent. Thus, the comparison of the cascade properties
in plasma microturbulence and in two-dimensional fluid
turbulence seems to be helpful.

ρ e2 vte≈ F (q, s, τ e ) R. (1)L Te L Te

Here, ν = 0.53 q (k y ρe ) + max{0.09, 0.19 ŝ 2 }(q/ τe )(R/LTe) (kyρe)2, ν̂ = ν (ky = kyd ), kyd = 4kyc /3, kycρe = τe1/2 LTe(qR)–1,
and τe = T e /T i. The agreement of this formula with fully
nonlinear simulation results is surprisingly good. Note that
another prediction of this semi-analytical model is that thepoloidal length scale of the dominant modes is given by kyd .
For Cyclone Base Case parameters (including ŝ = 0.8) wefind kyd ~ 0.13, in good agreement to the results of nonlinear
computations, which exhibit k yd = 0.15 ± 0.05 [2,6]. An
analogous treatment of adiabatic ITG turbulence leads to theestimate χi ≈ G (q, ŝ,τe) (ρ i2 vti /LT i ) which is one order downin R/LT compared to the ETG case, Eq. (1). A scaling likethis has indeed been observed in nonlinear computations ofadiabatic ITG turbulence [19,20]. Moreover, the model
finding G (q = 1.4, ŝ = 0.8, τi = 1) ~ 2 is roughly consistent
with the computational results which exhibit a prefactor of
the order of 1.5. It should be kept in mind, however, that incontrast to the ETG case, adiabatic ITG turbulence can becontrolled by zonal modes, especially near the linear
threshold, an effect which is not accounted for by our model.

3.2 E × B shearing effects
To judge the importance of E × B shearing of hyperfinescale eddies in the cascade induced by the fine-scale
turbulence, three questions must be addressed. First, what is
the amplitude/strength of these flows? It sets the value of Ω̃ .
Second, what is the typical time scale of these flows withrespect to that of the modes to be suppressed? It was shown
in Ref. [15] that fast temporal variations of zonal E × B flows
lead to a reduction of their shearing effectiveness. Of course,
these two effects are interdependent, and the latter tends toweaken the effect of the former as the flow amplitude isincreased. For ETG modes, this weakening is expected to besmall due to time-scale separation. Third, what are the linear
growth rates of the hyperfine-scale modes to be suppressed?
As was shown in Sec. 2, in the case of ETG turbulence, therelevant linear growth rates are not the ones around the
maximum of the ky spectrum, but the ones near the longwavelength cut-off. This certainly enhances the role of E × B
shearing effects. In summary, for the shearing of hyperfinescale eddies to be effective, Ω̃ only has to exceed the linear
ETG growth rates at long wavelengths. Whether zonal or nonzonal E × B flows dominate will depend largely on their

3. Interactions between ion and electron
scale turbulence
Having shown that ETG-induced electron heat transport
can be competitive with that from larger scale turbulence –caused, e.g., by nonadiabatic ITG modes, trapped electronmodes, or electron drift waves – it is very interesting andimportant to study the nonlinear interactions between
hyperfine-scale fluctuations at k⊥ ρe <~1<~ k⊥ ρi and fine-scale<
fluctuations at k⊥ ρi ~ 1 << k⊥LT. We certainly expect that theco-existence of these different types of turbulence cannot becharacterized as a simple superposition. In general, there will
be effects of the fine-scale turbulence on the hyperfine scales
and vice versa. In the following, however, we will primarily
assess how the ETG turbulence reacts to the presence of
turbulent fluctuations at longer wavelengths.

313

Jenko F., On the Nature of ETG Turbulence and Cross-Scale Coupling

relative amplitude/strength.

mechanism and may provide a floor (at least) for the electron
thermal transport. To investigate these issues, we have
performed nonlinear gyrokinetic simulations with the genecode [2]. Equilibrium E × B flows have been included asprescribed, sinusoidal variations of vEy, with the amplitudes
and wavelengths chosen in accordance with experimental
findings (typical Mach numbers are in the range of 10 %,and kxρi ~> 0.1). The other physical parameters were R/Ln
= 40, ηi = 2, ηe = 4, q = 2.4, ŝ = 0.8, α = 0.5, Ti /Te = 1, and1/β = mi /me = 1836. Magnetic fluctuations were taken into
account (β and the normalized pressure gradient α werechosen consistently), and a ŝ-α geometry was employed. The
time averaged ky spectra of the electron thermal diffusivity χefor an L-mode case (no equilibrium flows) and an H-mode
case (with equilibrium flows) are shown in Fig. 1. Therein,
one can distinguish contributions from two regions in ky
space: (1) fine-scale fluctuations driven by electron drift
waves (particle trapping is neglected; ITG modes are present
but subdominant for the parameters used here), covering the< 0.5 with a peak around k yρi ~ 0.2; (2)range of ky ρi ~hyperfine-scale turbulence at kyρi > 0.5 due to ETG modes,
peaking around kyρi ~ 1.5. Although it might appear surprising
at first that ETG modes can exist at these relatively longwavelengths, this finding is consistent with the semi-analyticaltheory presented in Sec. 2 when applied to a steep gradientregime. While in the L-mode case, the transport is clearly
dominated by the fine scales (they contribute about 80 % to
the total χe), the situation in the H-mode case is reversed (notethat in this log-log plot, the density of modes increases withk y ). The total transport drops by almost one order ofmagnitude – an effect which can be attributed mostly to the
suppression of the fine-scale turbulence, while the turbulent

3.3 Complex quasi-equilibria
Let us now focus on modifications of the equilibrium
density and temperature profiles by the fine-scale turbulence.
A rough estimate of τr at long wavelengths is given by the
maximum linear growth rate of the fine-scale modes. Since
the latter is smaller than the respective quantity for ETG
modes by the square root of the electron-to-ion mass ratio,
the scale independence of τr implies that the action of thelarger scale fluctuations on the ETG turbulence may be
regarded as slowly varying in time. On the other hand, thespatial structure of the resulting quasi-equilibria may be rather
complex since the local gradients of density and temperature
profiles need not be aligned with the radial direction.
Nevertheless, the most important contributions come from
relatively long wavelengths, such that there is a clear
separation between quasi-equilibrium and fluctuation scales.
This is because the gradients of the disturbances and of the
background are known to be comparable at fine scales while
the fluctuation spectra fall off (at least) like kr–2 [7]. The
contributions of the fine-scale gradients to the total gradients
therefore scale like kr–1. This means that the effective electron
heat diffusivity for ETG turbulence can be approximated bya weighted sum over the respective results for all possible
quasi-equilibria. Since ETG turbulence often exhibits asuperlinear increase of χe with R/LTe (or ηe = Ln /LTe in thecase of steep density profiles), background gradientfluctuations will lead to a (moderate) enhancement of theeffective thermal diffusivity. Note that this effect is opposedto the one induced by E × B shearing.

3.4 Back-reaction on fine scales
So far, we have only discussed how the hyperfine scales
are affected by the fine-scale turbulence. However, thehyperfine-scale turbulence may back-react on the fine scales
via enhanced eddy damping (nonlocal energy transfer in kr
space [23]) or other mechanisms. (Note that indirect backreactions via zonal flows [24] may be judged unlikely, given
the weak zonal flow excitation by ETG turbulence [7,16].)
To address all of the above issues quantitatively requires
comprehensive computations which treat the cross-scaleinteraction self-consistently. One such example will be
discussed next.

3.5 Turbulence in an edge transport barrier
During the formation of an edge transport barrier, theturbulent transport within the barrier region is reduced byabout an order of magnitude. This behavior is generally
attributed to equilibrium E × B shear flows, although theirprecise origin remains unclear. If the corresponding E × B
shearing rate reaches or exceeds the maximum growth rate of
the fine-scale modes, the latter are suppressed. This raises
questions about the role of hyperfine-scale turbulence in the
H-mode edge. Due to their much smaller spatio-temporalscales, ETG modes are hardly affected by the above

Fig. 1

4
14

Gyrokinetic simulations of L-mode and H-mode edge
turbulence: time averaged ky spectra of the electron
thermal diffusivity χe .

Jenko F., On the Nature of ETG Turbulence and Cross-Scale Coupling

Fig. 2

Gyrokinetic simulations of L-mode and H-mode edge turbulence: phase shift between potential and electron temperaturefluctuations as a function of ky .

transport on hyperfine scales remains at L-mode levels.
Nevertheless, one can observe a slight suppression of thehigh-ky part of the spectrum in the presence of strong finescale dynamics. This is also reflected in the ky spectra of thephase shifts between potential and electron temperaturefluctuations shown in Fig. 2. Here, the L-mode case exhibits
more ‘jitter’ at hyperfine scales than its H-mode counterpart,
indicating that the nearly linear ETG modes are disturbed by
the fine-scale turbulence. These results clearly show thatcross-scale coupling mechanisms like the ones describedaboved are active, invalidating a naive superposition approach.
In particular, shearing by E × B flows seems to play animportant role. More details and analyses of thesecomputations will be published elsewhere.

electron modes or nonadiabatic ITG modes.

4. Summary
While for many years, plasma microturbulence researchhas been solely concerned with phenomena whose spatiotemporal scales are set by the ions, there is now a growing
awareness that this limitation has to be overcome in searchof more complete models of turbulent transport in fusion
plasmas. Some recent developments along those lines arepresented in this paper.
First, the physics of adiabatic ETG turbulence isreviewed. Due to a subtle difference in the adiabatic species’
response, the linear symmetry between ETG and ITG modes
is nonlinearly broken. This leads to the emergence of radiallyelongated eddies (or streamers) with large fluctuation
amplitudes in certain parameter regimes. These streamers are
identified as residuals of linear modes, and their saturationlevel can be estimated by a semi-analytical theory based on
the notion of secondary instabilities. Key features of thestreamer-dominated turbulent state (first and foremost theassociated electron heat diffusivity) may thus be predictedeven quantitatively. An important finding is that ETG-induced
transport is boosted by a factor of R/LTe with respect to mixing
length expectations.
Having shown that the hyperfine-scale dynamics cancompete with that on fine scales, we discuss some dynamical
coupling mechanisms between them. As expected, it is foundthat the co-existence of different types of turbulence cannot
be characterized as a simple superposition. The ETG
dynamics mainly perceives the fine-scale turbulence throughE × B shearing and quasistatic modifications of the
background equilibria. The hyperfine scales can in turn backreact on the longer wavelengths via enhanced eddy diffusivity

3.6 Cross-scale coupling and bifurcations
It has been conjectured by S.-I. Itoh et al. [23,25] thatcross-scale coupling mechanisms like the ones describedabove may lead to bifurcation transitions. E.g., the following
scenario may be envisioned. Fine scales can suppress thehyperfine scales via E × B shearing if their amplitude issufficiently large. On the other hand, strongly excited
hyperfine scales can suppress the fine scales through enhanced
eddy damping. This competition triggers bifurcations if thecoupling strength between both ranges of scales exceeds acertain threshold. States in which one set of scales dominates
the other can be shown to result. Moreover, equilibrium shear
flows or other plasma parameters which influence theturbulence level can act as external control parameters. If sucha scenario is indeed applicable to edge transport barriers isstill under investigation. We note in passing that core transportbarriers might also be subject to such transitions. Here, therole of electron drift waves would be played by trapped515

Jenko F., On the Nature of ETG Turbulence and Cross-Scale Coupling

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or other mechanisms. These generic findings und discussions
prove useful for a deeper understanding of the physics oftransport barriers. Here, both hyperfine-scale turbulence andnonlinear cross-scale coupling were observed to play a role.
In particular, gyrokinetic simulations show that ETG
turbulence provides a floor for H-mode edge transport.
Whether the establishment of edge and/or core transportbarriers involves a bifurcation from fine-scale to hyperfinescale turbulence will be addressed in future investigations.

Acknowledgements
I would like to thank S.-I. Itoh and K. Itoh forstimulating discussions.

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