J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)84–88
000–000

A Roadmap to Recent Developments in the Theory of
Turbulent Generation of Long-Wavelength FlowsKROMMES John A.
Plasma Physics Laboratory, Princeton University, P.O. Box 451, Princeton, NJ 08543 USA(Received: 9 December 2003 / Accepted: 1 April 2004)

Abstract
A brief overview is given of recent results in the theory of turbulent generation of long-wavelength fluctuations;references are provided to more detailed calculations in the literature. Emphasis is placed on conceptual foundations
and technical tools, which can be simply illustrated by studies of the scalar (one-field) generalized Hasegawa-Mima
equation. Then a functional Hamiltonian formalism is discussed as a way of efficiently handling systems of coupledfields. Casimir invariants are shown to be the appropriate “plasmon densities”; a tensor spectral-balance equationbuilds in the conservation constraints automatically. A concise expression can be obtained for the nonlinear growthrate in terms of the spectral density, a tensor generalization of the triad interaction time, and the metric tensor andstructure constants of the Hamiltonian structure.
Keywords:
convective cell, zonal flow, drift wave, Hasegawa-Mima equation, EDQNM, Reynolds stress, eddy viscosity,
Hamiltonian, Casimir invariant

1. Introduction

ZF’s) as well as streamers (qx = 0). Convective cells havehighly nonadiabatic electron response [4]. One is concernedwith the nonlinear generation rate g q of such fluctuations,which is one ingredient in the ultimate determination of theself-consistent turbulent steady state.
In understanding the role of ZF’s and CC’s in the saturation of DW turbulence, one general approach could be thedevelopment of nonlinear variational principles [5] that yield
rigorous bounds on DW transport. Although preparatory
groundwork has been performed [6-8] and an attempt has
been made on the realistic problem of ion-temperaturegradient-driven turbulence [9], the theory is not yet insatisfactory shape. Here I will not discuss such bounding
theories explicitly, but will rather focus on a different classof nonlinear energy principles for the CC growth rate. Those
have intriguing implications for the more general problem.
A more technical motivation concerns the distinction
between (i) the growth rate of modulational instability, and(ii) the nonlinear growth rate g q in a statistically steady state.
Many authors have focused on modulational instability, butthat assumes that DW’s have already grown to a quasi-steadystate while the CC’s are not yet excited. This is valid only in
an initial transient stage of evolution; in a statistically steadystate, the techniques described in the subsequent sections must
be employed.

The theory of turbulent generation of long-wavelength
flows is a topic of considerable current interest because of its
connections to the general theory of drift-wave (DW) saturation and turbulent transport. In this paper I briefly discuss
some of the recent developments in this area, including (i)the detailed study of the generalized Hasegawa-Mima
equation (GHME) by Krommes and Kim (KK) [1]; (ii) the
derivation of nonlinear energy theorems using field-theoretic
techniques [1,2]; and (iii) the Hamiltonian description of thegeneration process [2], which is especially well suited fortreating systems of coupled partial differential equations but
also yields further insights even for one-field theories. The
emphasis is on conceptual foundations and technical tools;
details can be found in the cited references.
The specific calculations reported here have been conducted in slab geometry. They must not be applied blindly totoroidal situations [3], as there are important differences ofboth interpretation and detail (specifically, the wave numberk || parallel to the magnetic field depends on the radial
coordinate). With that caviat, I shall continue to use slab
language for simplicity. I believe the technical tools arepotentially useful quite generally, and various paradoxes andconceptual difficulties also have toroidal counterparts.
In slab geometry, by convective cell (CC) I shall mean
any fluctuation (wavevector q) with q|| = 0. That includes as
special cases the particularly important zonal flows (qy = 0;

©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: krommes@princeton.edu
184Krommes J.A., A Roadmap to Recent Developments in the Theory of Turbulent Generation of Long-Wavelength Flows

2. Statistical closure theory and theconvective-cell growth rate

including polarization-drift corrections, namely w klin = w */(1 + k2) (w * is the diamagnetic frequency). However, use ofa linear frequency in this context is troubling because the CC
generation process is nonlinear. Although linear physics iscertainly relevant for determining the ultimate spectral
properties of the saturated state, the work of KK showed
clearly that the form of g q does not depend explicitly on theproperties of linear waves; those enter only implicitly throughthe DW–CC triad interaction time and the ultimate form of
the spectrum. In an attempt to better understand the foundations of the algorithm proposed in Ref. [11], KK used fieldtheory techniques to discuss the systematic derivation of anonlinear energy principle for g q. It is useful here to sketch
the intuitive content of the formalism. Refer to Fig. 1 andassume that the system is statistically homogeneous and in
stedy state. (These restrictions are not fundamental.) Let ydenote the field variable(s). One expects that the CC growthrate g q is related to some sort of variation of the DW energy
due to the presence of the CC’s. Now the CC’s randomly
modulate the DW’s; however, there is no mean field associated with the CC’s because of the homogeneity. Furthermore,in a steady state the mean rate-of-change of the DW energy
must vanish. Thus it is not the case that, for example, g q µ·–·eDWÒ. Clearly one must study the behavior of the DW
fluctuations in the presence of the CC’s, but one cannot look“too microscopically,” e.g., consider ·d ẽDW/d ỹCCÒ, since there
will be cancellations due to the random phases of the
turbulence. Rather, one needs to calculate the mean response
function (Green’s function) [10] for the turbulence. Thisobservation [15] is a nonequilibrium generalization of thewell-known fluctuation-dissipation theorem of equilibrium
statistical mechanics, which relates the two-time correlation
function C of the random background fluctuations to the mean
response function for infinitesimal perturbations away fromthat background. The technical solution that is now a standard
tool of analytical statistical dynamics [15] is to break thesymmetry by inserting an external source ĥ. Now all statisticalobservables depend on ĥ, ·y Ò ĥ  0, and two-point correlations(including energy spectra) can be considered to be functionally dependent on the induced mean field: C = C [·yÒ ].
This is illustrated in Fig. 2. It can be shown that g q is related

Although robust Markovian closure theory has been wellunderstood for some time (many references can be found in
the recent review by Krommes [10]), a more heuristic analytical theory that focused specifically on ZF generation was
initiated relatively recently by Diamond et al. [11]. That work
was important and inspirational, but raised numerous
questions concerning technical details (including the sign ofthe result), physical interpretations, and the relationship tomore systematic closure theory. The subsequent work by
Smolyakov and Diamond [12] on conserved quantities was
also significant. Motivated by those works, KK [1] performed
a detailed analysis of the GHME. They first discussed a direct
asymptotic expansion of the eddy-damped quasinormal
Markovian (EDQNM) formula [10] for nonlinear coherent
damping (the turbulent generalization of collisional drag inclassical Langevin theory), using the small parameter e =· q/k<< 1, where k refers to the DW’s. That calculation (surprisingly nontrivial in the general anisotropic case) has important
links to the theory of negative eddy viscosity in 2D NavierStokes turbulence [13]; KK were able to make a quantitative
connection. Because nonlinearity leads to interacting
wavevector triads k + p + q = 0, it is easily seen for fixed k
and |q| that the basic interaction involves a DW (k), two DW
sidebands (p1 and p2), and a CC (q and –q). However, thecalculation is not of a modulational instability with fixed DW
pump. Such analyses do not conserve energy between the
DW’s and CC’s, which can lead to unforeseen and unphysicalconclusions, including a spurious change of sign of the eddy
viscosity (see the appendix of Ref. [1]). The derivation fromthe EDQNM closure also shows that the appropriate“propagator” is the triad interaction time qk,–k,q between themodes k, –k, and q; this observation resolves a difficulty
regarding overall sign and aids in the construction of an
appropriate physical algorithm. Part of that algorithm involves
consideration of weakly inhomogeneous spectral balance
equations; KK also provided a reformulation and generalization of the results of Ref. [12] in terms of the standard
Poisson-bracket formalism conventionally used in weakly
inhomogeneous statistical dynamics, thereby resolving aparadox in the original theory [14]. One resulting insight isthat it is unnecessary to formulate spectral balance equationsin terms of an “action” or “plasmon density” invariant; thespectral density of any dependent variable can be used. Thatis particularly significant when n-field systems are considered,
as discussed in Sec. 4.

3. Field theory and the modulation
algorithm
In Ref. [11] it was noted that g q was determined in termsof q ·  kZk, where Zk was a certain quantity conserved by the
DW’s under the CC modulation. In the specific case of the
GHME [1], Zk = (1 + k2)2 ·dj k2Ò. It has been noted [12] thatthe expression for Zk is reproduced by the formula Zk µ kyEk/
wklin, where Ek =· (1 + k2) ·dj k2Ò and wklin is the DW frequency

Fig. 1

2
85

Modulation and energy transfer in homogeneous,steady-state turbulence. All mean fields vanish andthere is no net energy transfer between the DW’s andCC’s.

Krommes J.A., A Roadmap to Recent Developments in the Theory of Turbulent Generation of Long-Wavelength Flows

Fig. 2the nonlinearities as much as possible. Seminal work by
Morrison and Greene [18] showed that various Eulerian
PDE’s have a Hamiltonian structure; several decades ofsubsequent work was reviewed by Morrison [19]. I will showthat the Hamiltonian formalism can be usefully applied to thecalculation of g q and leads to an important new insight about
the nature of the plasmon density. The methodology isparticularly useful for systems of coupled PDE’s, where one
is led to very compact and elegant results.
In ordinary (noncanonical) Hamiltonian mechanics, thegeneralized coordinates evolve according to z· i = {z i, H},
where H (z) is the Hamiltonian and {A, B} =· ( zi A) Jij ( z j B)
(the antisymmetric matrix J being known as the Poisson
tensor or the cosymplectic form). The use of noncanonicalvariables in plasma physics was pioneered by Littlejohn, exploited by Dubin et al. [20] in their noncanonical Hamiltonian
formulation of the nonlinear gyrokinetic–Poisson system, andlater generalized elegantly by Hahm [21] and others (forfurther references, see App. C of Ref. [10]). When applied toan Eulerian PDE for a field y (x, t), an infinite-dimensional
generalization is required (i Æ x). One may further generalize
to a vector y of variables such as density or temperature. [Iwill also label those variables by an i (y  y i), although thisi should not be confused with the label of the original
generalized coordinates.] One now requires a Hamiltonian
functional H[y ] and an appropriate bracket {·, ·}, which mustsatisfy antisymmetry ({A, B} = –{B, A}) and the Jacobi
identity ({{A, B}, C} + {{B, C}, A} + {{C, A}, B} = 0). Thenty = {y, H}. Motivated by the characteristic structure of theE ¥ B advective nonlinearity, I define {A, B} =· ẑ · —A ¥ —B.
This “ordinary” bracket has the property A[ B, C ] = [ A, B]C ,where the overline denotes the integral over all space. I thenchooseAs ĥ is varied, the symmetry of the physical state ĥ = 0
is broken, ·yĥÒ  0, and C = C [·yÒ].

to the first-order variation of the mean nonlinearity withrespect to the induced mean field, calculated in the physicallimit ĥ = 0. Technically, this result is related to the eliminationof disconnected graphs that leads to the famous Dyson
equation [16]. For much more information about such fieldtheoretic techniques, see Ref. [10].
Physically, the mean nonlinearity is just the Reynolds
stress S, and proper understanding of the method of sources
just described is relevant to the interpretation of g q in termsof S. Historically, S was defined to be the mean nonlinearity
for a time-evolving flow in inhomogeneous geometry and is
still used in that context in engineering-type modeling. In
homogeneous turbulence S vanishes, but as discussed above
the relevant quantity is not S itself but rather the variation of
S due to an induced mean field. As an example, consider the
Reynolds stress on CC’s described by the GHME. Then S is
the average of the nonlinear term describing advection ofvorticity w =· —^2 j . That average can be transformed [17] into
an expression involving the Cartesian components of the E ¥B velocity (primes denote the DW’s):

H = 1 y i gijy j = 1 y iy i ,(2)

È dA dB {A, B} = S rs [y ] Í r ,s .
Î dy dy 

(3)

2

(1)

2

The symmetric matrix gij is called the metric tensor and canbe used to raise and lower indices, creating an appealing
covariant formalism; note the covariant expression of H in
Eq. (2). The symmetric matrix S ij is called the structure matrix
and is taken to be linear in the fields: S ij[y ] = S ijky k. The S ijk
are called the structure constants; they are constrained [2] inorder that the Jacobi identity be satisfied. Using these formulas, one can demonstrate Hamiltonian structure for a numberof the popular systems of PDE’s of current interest [2],including, for example, the system of finite-b collisional DW
equations studied in Ref. [22].
It is easy to show that the bracket structure just definedconserves H. Other quantities may be conserved as well. A
Casimir invariant C is defined as any functional that is conserved without regard to the form of the Hamiltonian. That

Now for homogeneous, isotropic turbulence each of theunderlined terms vanishes by symmetry; there is no net
Reynolds stress. But the situation is quite different when the
DW turbulence is modulated by a CC with wavevector q.
Then neither of the underlined terms vanishes because theperturbed fluctuations are anisotropic (in a way dependent onq). For arbitrary q, both terms are of the same order and must
both be retained in order to ensure a proper, rotationallycovariant calculation of g q. This argument is explained in
more detail in Ref. [17].

4. Hamiltonian description
The CC growth rate g q is determined by nonlinear
interactions. It is therefore useful to exploit the properties of386Krommes J.A., A Roadmap to Recent Developments in the Theory of Turbulent Generation of Long-Wavelength Flows

is, a Casimir is conserved purely because of the structure of
the bracket. An example of a prospective Casimir is C =1 i i . Then conservation of H and C can be proven very
2y ysimilarly by manipulating their time derivatives into contractions of symmetric and antisymmetric forms:

 tH = - S ij k [y k ,y j ]y i = - S ij k y k [y j ,y i ] = 0;(4)

 t C = - S ij k [y k ,y j ]y i = - S ij k [y i ,y k ]y j = 0,

(5)

The scalar CC growth rate itself follows as a contraction of
H ij
SH with the spectral tensor C according to g q = –[(S
–q) ¥
k
C
– ji (q)]/ C– k (q).
The remaining manipulations involve only a modest,
though detailed, amount of algebra: (i) write the tensor
spectral-balance equation for Ckij; (ii) obtain formulas for the
nonlinear modulation frequencies in terms of the structure
constants Sijk (one must be careful about co- and contravariant
indices, which cannot be passed indiscriminately throughPoisson brackets; the action of those k-dependent brackets on
the metric tensor is related to the energy loss of the DW’s to
the CC’s due to wave-number refraction); (iii) write the–·equation for e DW = –12 t(C ii)DW; (iv) carry out the required
functional derivatives; (v) insert an appropriate triadq̂ ; (vi) manipulate to obtain the final forinteraction tensor q
mula [2], which except for contractions over tensor indices is
identical in form with the result of the scalar calculations [1].
I stress that in this derivation one does not need to guess oreven be inspired about the form of a possibly preferred
plasmon density; conservation of the proper Casimir(s) is built
in. The elegant and concise way in which the formalism deals
with those invariants provides a powerful argument in favorof further applications of the Hamiltonian methodology.
In the n-field formalism, the triad interaction timeq̂  q ijkl, which must not
generalizes to a fourth-rank tensor q
be treated cavalierly. An important preliminary exercise is todemonstrate that a nontrivial n-field model (such as the electrostatic Hasegawa-Wakatani equations [23]) reduces correctlyin the adiabatic DW limit to the GHME result; this is not
obvious because the tensor theory couches g q in terms of q ·kCkij, whereas Zk is a k-dependent weighted contraction of
Ckij. The resolution is that qq̂ cannot be assumed to be diagonal.
In the adiabatic limit it can be shown to approach a singularmatrix times the scalar qk,–k,q; this nontrivial matrix structureis one way in which the physics of linear theory enters theformula for g q.

the last result holding if S ijk = S kji. Notice in particular how
the proof of C conservation does not involve the covariant
component yj. Also note that this particular definition of C is
not covariant in form, so a linear transformation of variables
would change both its value and the specific symmetry
property of the S ijk that ensures that C remains conserved. In
general, it is a nontrivial task to identify all Casimir invariantsof a given Hamiltonian structure.
Casimir conservation furnishes an important constraint
on the dynamics, which are no longer allowed to explore theentire energy “surface”. Consider in particular the interactionof DW’s (denoted by primes) and CC’s (denoted by underlines). Projecting the dynamics onto the DW subspace leads
to(6)
Actually, such a decomposition holds for any projection into
two orthogonal subspaces, but it is particularly useful when ascale separation holds between the two classes of fluctuations.
Term (a) describes the advection of DW’s by CC’s, term (b)describes the inverse process of CC’s advected by DW’s, andterm (c) describes DW self-interactions. When the DW’s are
assumed to be of relatively short wavelength, term (b) is
negligible, so term (a) describes the principal DW–CC interaction. Because y j = d H /d y j and the proof of Casimir––conservation does not involve this covariant component of
the field, this interaction conserves the DW Casimir(s). In
particular, for the case of the GHME (which can easily beshown to be Hamiltonian), Z is a Casimir. This argument links
the rather abstract concept of Casimir invariance with thequite physical process of nonlinear advection involving two
disparate scales.
The same projection process can be exploited to derivea generalized Poynting theorem and nonlinear energy
principle; for the details, see Ref. [2]. The fundamental result
is that if Q =· ·y iÒ and P =· ·y iÒ , then second variations of the–– – –· –mean DW energy e DW with respect to Q or P produce co- or––contravariant versions of the Hermitian part of the coherent
damping tensor S, e.g.,

Explicit statistical calculations on the generalized
Hasegawa-Mima equation, including systematic asymptotic
reduction of Markovian closure theory, application of themethod of sources to the derivation of nonlinear energy
principles, and proper derivation of weakly inhomogeneous
spectral balance equations, have led to deeper understandingof the asymptotic problem defined by the nonlinear interactions of long-wavelength convective cells with shortwavelength drift waves. Further application of an infinitedimensional Hamiltonian formalism shows that the DW’sconserve Casimir invariants under the CC modulation. For
multiple fields, the nonlinear CC generation process isconstrained by the conservation of possibly multiple Casimirs,
but a succinct formula can be obtained [2] in which those
constraints are neatly built in. Applications of that formula
are in progress [24]. It is hoped that the conceptual ideas andtechnical tools discussed here will stimulate further research
into both toroidal generalizations and robust variational

2(S ) = 12 d Pd*ėd P .
H ijq

5. Summary and conclusions

DW

q ,i

q, j

(7)

4
87

Krommes J.A., A Roadmap to Recent Developments in the Theory of Turbulent Generation of Long-Wavelength Flows

descriptions of steady-state microturbulence.

[9] C.-B. Kim and K.-O. Choi, Phys. Plasmas 3, 4463
(1996).
[10] J.A. Krommes, Phys. Rep. 360, 1 (2002).
[11] P.H. Diamond et al., 17th IAEA Fusion Energy Conference (International Atomic Energy Agency, Vienna,1998), pp. 1421, AEA-CN-69/TH3/1.
[12] A.I. Smolyakov and P.H. Diamond, Phys. Plasmas 6,4410 (1999).
[13] R.H. Kraichnan, J. Atmos. Sci. 33, 1521 (1976).
[14] G.F. Carnevale and P.C. Martin, Geophys. Astrophys.
Fluid Dynamics 20, 131 (1982).
[15] P.C. Martin, E.D. Siggia and H.A. Rose, Phys. Rev. A 8,
423 (1973).
[16] F.J. Dyson, Phys. Rev. 75, 486 (1949).
[17] J.A. Krommes, Phys. Plasmas 11, 1744 (2004).
[18] P.J. Morrison and J.M. Greene, Phys. Rev. Lett. 45, 790(1980).
[19] P.J. Morrison, Rev. Mod. Phys. 70, 467 (1998).
[20] D.H.E. Dubin et al., Phys. Fluids 26, 3524 (1983).
[21] T.S. Hahm, Phys. Fluids 31, 2670 (1988).
[22] S.J. Camargo, B.D. Scott and D. Biskamp, Phys. Plasmas
3, 3912 (1996).
[23] A. Hasegawa and M. Wakatani, Phys. Rev. Lett. 50, 682(1983).
[24] R.A. Kolesnikov and J.A. Krommes, Bull. Am. Phys.
Soc. 48(9), 73 (2003).

Acknowledgements
In various phases of this work I enjoyed collaborations
with C.-B. Kim and R.A. Kolesnikov. I am grateful for
informative discussions with B. Scott, T.-S. Hahm, and G.
Hammett. This research was supported by U.S. Departmentof Energy Contract No. DE-AC02-76-CHO-3073.

References[1] J.A. Krommes and C.-B. Kim, Phys. Rev. E 62, 8508
(2000).
[2] J.A. Krommes and R.A. Kolesnikov, Hamiltonian description of zonal-flow generation, Phys. Plasmas 11, L29(2004).
[3] L. Chen, Z. Lin and R. White, Phys. Plasmas 7, 3129
(2000).
[4] G.W. Hammett et al., Plasma Phys. Control. Fusion 35,
973 (1993).
[5] F.H. Busse, Adv. Appl. Mech. 18, 77 (1978).
[6] J.A. Krommes and R.A. Smith, Ann. Phys. (N.Y.) 177,
246 (1987).
[7] C.-B. Kim and J.A. Krommes, J. Stat. Phys. 53, 1103
(1988).
[8] C.-B. Kim and J.A. Krommes, Phys. Rev. A 42, 7487
(1990).

588