The Gyrokinetic Description of Microturbulence in Magnetized Plasmas

FL44CH08-Krommes ARI 1 December 2011 7:44

The Gyrokinetic Descriptionof Microturbulence inMagnetized PlasmasJohn A. KrommesPlasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543;email: [email protected]

Annu. Rev. Fluid Mech. 2012. 44:175–201

First published online as a Review in Advance onSeptember 28, 2011

The Annual Review of Fluid Mechanics is online atfluid.annualreviews.org

This article’s doi:10.1146/annurev-fluid-120710-101223

Copyright c© 2012 by Annual Reviews.All rights reserved

0066-4189/12/0115-0175$20.00

Keywords

drift waves, zonal flows, noncanonical Lagrangian methods, gyrokineticsimulations, entropy cascade

Abstract

Nonlinear gyrokinetics is the major formalism used for both the analyticaland numerical descriptions of low-frequency microturbulence in magne-tized plasmas. Its derivation from noncanonical Lagrangian methods andfield-theoretic variational principles is summarized. Basic properties of gy-rokinetic physics are discussed, including polarization and the concept of thegyrokinetic vacuum, equilibrium statistical mechanics, and the two funda-mental constituents of gyrokinetic turbulence, namely drift waves and zonalflows. Numerical techniques are described briefly, and illustrative simulationresults are presented. Advanced topics include the transition to turbulence,the nonlinear saturation of turbulence by coupling to damped gyrokineticeigenmodes, phase-space cascades, subcritical turbulence, and momentumconservation.

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Gyrokinetics: thestudy of fluctuations inmagnetized plasmashaving frequenciesmuch smaller than theion gyrofrequency

GKE: gyrokineticequation

ITER: originally anacronym for theInternationalThermonuclearExperimental Reactor(for discussion, seehttp://www.wikipedia.org/wiki/Iter); it is nowunderstood to mean“the way” in Latin

1. INTRODUCTION

It has been more than two decades since an article focusing on plasma physics appeared in theAnnual Review of Fluid Mechanics (see Similon & Sudan 1990). Since then, enormous progress hasbeen made in all facets of the field, particularly in the theoretical, numerical, and experimental ex-ploration of the consequences of microturbulence for the magnetic confinement of fusion plasmas,but also more recently in astrophysical contexts. These accomplishments speak to a sea changein the method of attack. Whereas early theoretical work on plasma turbulence (beginning in the1960s) employed analytical methods of statistical closure theory (Krommes 2002; see the sidebar)applied to simple models, beginning in the mid-1980s the principal tool used in theoretical andnumerical studies became the nonlinear gyrokinetic formalism. Gyrokinetics, appropriate for thedescription of low-frequency fluctuations in magnetized plasmas, was mentioned only briefly bySimilon & Sudan (1990), whose article appeared 8 years after the seminal derivation by Frieman &Chen (1982) of a nonlinear gyrokinetic equation (GKE). The present review attempts to providea modern perspective. It is specifically intended for an audience of nonplasma physicists, so it doesnot delve deeply into the morass of fusion-related phenomenology. However, it does emphasizethat gyrokinetics is evolving into a quantitatively predictive tool that has already enjoyed signifi-cant successful comparisons with experimental data. After decades of development, the field hasbecome one of the major success stories in plasma-physics research.

The need for gyrokinetics arises from the enormous range of timescales and space scales presentin many plasma configurations (in both the laboratory and space). Let us consider, for example, theITER research device (http://www.iter.org), presently under construction by an internationalconsortium. ITER is a large tokamak, intended for the study of burning plasmas, with a toroidalmagnetic field B of 5.3 Tesla (Green 2003, table 1). The gyrofrequency of deuterium ions in thatfield is ωc i

.= qi B/mi c ≈ 2.5 × 108 s−1 (where .= is used for definitions, qi and mi are the ioncharge and mass, and c is the speed of light), approximately 750 times larger than the characteristicturbulence frequency ω∗ ≈ 3.3 × 105 s−1 (see Section 3.3). An even more dramatic comparison is

BRIEF HISTORY OF MODERN STATISTICAL CLOSURE THEORY FOR PLASMAS

The transition to numerical gyrokinetics was not instantaneous; it substantially overlapped with the development ofanalytical statistical closures for plasmas over a period of several decades. The history and theory of statistical closuresfor magnetized plasmas have been reviewed by Krommes (2002). Modern statistical plasma turbulence theory datesfrom the mid-1970s with the development of Kraichnan’s direct-interaction approximation for plasma physics byDuBois & Espedal (1978) and Krommes (1978). [An overview of Kraichnan’s substantial contributions to statisticalturbulence theory has been given by Eyink & Frisch (2011).] Similon & Sudan (1990) described various early(1980s) plasma applications of the direct-interaction approximation. Some efforts at Markovian statistical closuresfor plasmas were also made during the 1980s (Waltz 1983). But systematic development of Markovian closures, andproper comparison of their predictions with direct numerical simulations, did not occur until the 1990s with thework of Bowman et al. (1993), Bowman & Krommes (1997), and Hu et al. (1995, 1997). Closure theory continuesto be of use in specific contexts such as the theory of zonal flows (Krommes & Kim 2000), provides the naturalframework for qualitative interpretations of numerical and experimental data, inspires diagnostic techniques (Itohet al. 2005), and may serve to motivate useful subgrid-scale models for large eddy simulations (Morel et al. 2011,Smith & Hammett 1997). However, it has always been clear that realistic plasma turbulence [being inhomogeneous,anisotropic, and six dimensional (6D)] will not yield in any quantitative way to analytical statistical closures. By theearly 1980s, the time was ripe for a new tool.

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http://www.wikipedia.org/wiki/Iter



http://www.iter.org


PDF: probabilitydensity function

Magnetic moment(μ): the adiabaticinvariant associatedwith the gyration of acharged particlearound a magneticfield line

Adiabatic invariant:a quantity that isconserved through allorders under slowvariations

with the discharge pulse length τpulse ≈ 400 s: τpulse/(2π/ωc i ) ≈ 1010. Gyrokinetics copes with suchdramatic scale separations by analytically removing the details of the gyromotion and other high-frequency dynamics from consideration. That eliminates many physical processes that are notbelieved to be important for the problem of turbulent transport, and it leads to enormous savingsin computational resources. However, although the underlying idea is simple, both the moderntheoretical formalism and its numerical implementations are sophisticated. I touch on both ofthose facets below, but because of length constraints I can only hint at the enormous volumeof excellent work that has been done. More details can be found in some longer review articles.Brizard & Hahm (2007) focused on the analytical underpinnings, whereas Garbet et al. (2010)were mostly concerned with numerical simulations and their comparison with fusion experiments.A forthcoming review by G.L. Hammett (manuscript in preparation) also addresses those lattertopics. Cary & Brizard (2009) reviewed the modern theory of the closely related problem ofguiding-center (zero-gyroradius) motion. Pedagogical background material on modern issues inplasma turbulence can be found in some articles by Krommes (2006a,b, 2009b, 2010).

2. GYROKINETIC FORMALISM

2.1. Fundamental Particle Kinetic Equation

The generalization of Boltzmann’s equation to include long-ranged electromagnetic interactionsis the plasma kinetic equation [for the probability density function (PDF) f for particles of speciess at position x and velocity v]

∂ fs (x, v, t)∂t

+ v·∇ fs +( q

m

)s

(E + c −1v × B)·∂ fs

∂v= −Cs [ f ], (1)

where C[ f ] denotes a positive-semidefinite collision operator functionally dependent on f (usuallythe Landau form is used). Maxwell’s equations must be adjoined to calculate the self-consistentelectric field E and magnetic field B.

It is assumed that B comprises a large part B0, arising from external coils and (for tokamaks)externally driven plasma currents, and an electromagnetic correction δB. Here I mostly considerthe electrostatic approximation, in which δB is neglected; thus E = −∇ϕ, where ϕ is the electro-static potential obtained from Poisson’s equation −∇2ϕ(x, t) = 4πρ = 4π

∑s (nq )s

∫dv fs (x, v, t)

(with n the mean density). This approximation is inadequate for detailed studies of modern high-pressure devices and some astrophysical situations but still contains an ample amount of importantphysics.

2.2. Basic Gyrokinetic Equations

For strong B0, the rapid gyration arising from the Lorentz v × B0 term, the 6D nature ofEquation 1, and the high-frequency collective oscillations it supports render it intractable for stud-ies of low-frequency nonlinear physics on turbulence timescales. Thus one turns to some sort ofaveraging procedure that removes the fast gyration timescale and reduces the 6D kinetic equationto a 5D one. Together with a particular low-frequency closure of Poisson’s equation that elimi-nates high-frequency collective dynamics, this defines low-frequency gyrokinetics, which I simplycall gyrokinetics in this article. More generally, the particle gyration can merely be segregated(Kolesnikov et al. 2007; Qin et al. 1999, 2000), allowing one to discuss high-frequency gyrokinetics,useful for studies of plasma heating. However, I do not explore that advanced topic here.

For low-frequency motions, a crucial quantity is the magnetic moment or first adiabatic in-variant μ ≈ μ(0) .= 1

2 mv2⊥/ωc (x) associated with the rapid gyration of a charged particle around

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a magnetic field line (Cary & Brizard 2009, Northrop 1963). General results from the theoryof almost-cyclic systems (Kruskal 1962, Lichtenberg & Lieberman 1992) show that μ is asymp-totically conserved even in the presence of weak magnetic inhomogeneities and slowly varyingfields. If weak and slow are indicated by an ordering parameter ε, then μ in principle can bedetermined as an asymptotic expansion through all orders in ε, with μ(0) being the lowest-orderterm. In this regard, a seminal calculation was by Taylor (1967), who found the first-order correc-tion to μ(0) due to the presence of a slowly varying electrostatic wave of arbitrary perpendicularwavelength.

Note that μ is not an exact invariant. Dragt & Finn (1976) discussed μ conservation withina Hamiltonian formulation, relating it to the existence of a Kolmogorov-Arnold-Moser surface.They found stochastic regions in the motion of a charged particle in a dipolar magnetic field.Dubin & Krommes (1982) discussed the interaction of rapid gyration with high harmonics ofperiodic motion on much longer timescales (such as the bounce motion associated with the secondor longitudinal invariant J). They found stochastic layers with widths that scale as exp(−b/ε),where ε is the ratio of the small bounce frequency and large gyrofrequency and b is a constant;because exp(−b/ε) is asymptotic to zero, stochastic wandering in the layers is overlooked by theasymptotic construction of a conserved μ. Lichtenberg & Lieberman (1992) described in moredetail the situation, which is related to Arnold diffusion (Chirikov 1979). One must always keepin mind the possibility that the adiabatic invariance of μ can be broken. Related concerns wererecently expressed by Sugiyama (2008) (for discussion, see Krommes 2009a, Sugiyama 2009).In the material to follow, μ conservation is assumed; that is an excellent approximation for thesituations of interest.

Gyrokinetics amounts to the determination of a change of variables (Catto 1978) from theparticle phase space {x, v} ≡ zi to gyrocenter phase space {X, U , μ, ζ } ≡ zi , together with aclosure approximation to be described. (A mere change of variables cannot alter the physicalcontent of the kinetic equation.) Here X is the gyrocenter position, U is the gyrocenter velocityalong B, and ζ is the gyration phase; the overline (subsequently omitted) signifies developmentas an asymptotic series with its lowest-order form corresponding to that for circular motion.The lowest-order gyrocenter variables are illustrated in Figure 1. In standard gyrokinetics, theordering parameter is ε ∼ ω/ωc i ∼ k‖/k⊥ ∼ VE/c s, where k is a typical fluctuation wave vector,

â

ê2

ê1xX

= ρâρ

ĉ

=v

vĉ

−ζ

x

y

Gyrocenter

Particle

Figure 1Illustration of the lowest-order gyrocenter coordinates. The magnetic field is B = B b; at the position x ofthe particle (blue dot), B is in the z direction. The particle velocity is U b + v⊥ c; if B were constant, theparticle would circle a gyrocenter at position X .= x − ρ ( green square) with angular velocity ζ = ωc , wherethe gyroradius vector is ρ

.= b × v⊥/ωc = ρ a (ρ .= v⊥/ωc ). Instead of resolving vectors onto theorthonormal triad (a, b, c), it is frequently convenient to use a local Cartesian system (e1, e2, b) centered onthe magnetic field line at the position of the particle. The lowest-order magnetic moment isμ(0) .= 1

2 mv2⊥/ωc , and the lowest-order gyrocenter coordinates are {X, U , μ(0), ζ }.

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Pull-back(transformation):operator T thattransforms thegyrocenter PDF Finto the particle PDFf: f = TF

parallel and perpendicular are with respect to B, VE is the E × B velocity, and c s is the soundspeed. [Other orderings are required in some situations (Brizard & Hahm 2007, INI WorkshopGyrokinet. 2010).] The gyrocenter PDF F thus obeys the 6D equation

∂ Fs (X, U , μ, ζ, t)∂t

+ X·∇ F + U∂ F∂U

+ ζ∂ F∂ζ

= −C[F ]. (2)

Here the tilde denotes a ζ-dependent quantity; C[F ] is the transformation of C[ f ]. No derivativewith respect to μ appears because it is (adiabatically) conserved by construction. The gyrocenterdrifts X, U , and ζ follow from the theory; importantly, they are constructed to be independentof ζ (see below). The gyrophase average of Equation 2, denoted by 〈. . .〉ζ , then eliminates thegyration term ∂(. . .)/∂ζ , leaving one with the conventional GKE for the 5D F .= 〈F〉ζ :

∂ F∂t

+ X·∇F + U∂ F∂U

= −〈C[F ]〉ζ . (3)

In the form usually implemented in simulations, the drifts are

X = B−1∗ (B∗U + c q−1 b × ∇H ), U = −(mB∗)−1 B∗·∇H . (4)

Here b .= B/B, B∗.= b ·B∗, B∗

.= B + (mc /q )U ∇ × b, H ≈ H(0) + H

(1), H

(0) .= 12 mU 2 +

μωc (X), H(1) .= q 〈ϕ〉ζ , and

〈ϕ〉ζ (X, μ) .= (2π )−1∫ 2π

0dζ ϕ(X + ρ(ζ )), (5)

where ρ.= ω−1

c b × v⊥ is the gyroradius vector. If periodicity is assumed, Fourier transformationis useful; one finds 〈ϕ〉ζ,k = J0(k⊥v⊥/ωc )ϕk. [Thus gyrokinetics is able to handle wavelengthscomparable to or smaller than the ion gyroradius ρ i; nominally, k⊥ρi = O(1).] Physically, the Besselfunction describes the v⊥- or μ-dependent reduction in effective potential that arises because,during one gyroperiod, a particle samples different phases of a slowly time-varying fluctuation(for more discussion of the resulting nonlinear phase mixing, see Section 5.3). The drifts includestreaming and acceleration along a magnetic field line, the effective E × B velocity 〈V E〉ζ

.=c b × ∇〈ϕ〉ζ /B∗, and the magnetic ∇B and curvature drifts.

If collisions are neglected, Equation 3 appears to be closed in terms of F , but that is illusorybecause the drifts involve self-consistent electromagnetic fields. The theory is not complete untilthe relevant Maxwell equations are solved. Now the charge ρ and current j are given as momentumintegrals over the particle PDF f; however, the GKE evolves the gyrocenter PDF F , and thepull-back transformation T from F to f, f = TF, couples to δ F , where F = F + δ F . A closedsystem is obtained by neglecting the ζ-dependent δ F : f ≈ TF. This can be justified for collisionlessplasmas (Dubin et al. 1983) in terms of a projection procedure and an assumption of ζ-independentinitial conditions. However, neglecting collisions is problematic because they represent the onlytrue dissipative effect in the problem (see further discussion in Section 5.3). When they are retained,Equation 3 is not closed in terms of F because the C in Equation 2 (which describes the physics ofcollisions in the particle, not gyrocenter, phase space) drives a δ F . Fortunately, that correction issmall, so it is generally neglected. The formal gyrokinetic collision operator 〈C〉ζ has been discussedby Brizard (2004). Simplifications such as the model by Abel et al. (2008) are often used in practice.

Although above I focus on the equation for the gyrocenter PDF F , equally important are theform and content of the gyrokinetic Maxwell equations, i.e., the physics of the pull-back T . Thatis discussed in more detail in Section 3.1.

In summary, the asymptotic construction of a conserved μ and the closure f ≈ TF discon-nect gyrokinetics from the Vlasov description, leaving one with a reduced dynamical system that


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PIC: particle in cell

Canonical variables:2N generalizedcoordinates zα whosePoisson brackets are{zα, zβ } = σαβ , whereσ

.= ( 0−I

I0 ), with I the

N × N identity matrix

Noether’s theorem:loosely, the statementthat any symmetry ofthe Lagrangian isassociated with aconservation law

describes the self-consistent collective interactions of gyrocenters and is appropriate for the de-scription of the low-frequency fluctuations that are believed to be important for turbulence inmagnetized plasmas.

2.3. Noncanonical Lagrangian Methods

In principle, the gyrocenter drifts should include corrections through all orders in ε, as should thepull-back T . How to achieve a consistent practical truncation has been a source of confusion, andsome issues remain controversial (see Section 5.5). One therefore requires a systematic methodol-ogy, which has undergone substantial development. Because Brizard & Hahm (2007) and Cary &Brizard (2009) treat the formalism in detail, I shall be brief. Early workers on linear gyrokinetics(e.g., Rutherford & Frieman 1968, Taylor & Hastie 1968, Catto 1978, Antonsen & Lane 1980)analyzed the Vlasov equation perturbatively by treating the magnetic Lorentz force term as large;GKEs then arose as solvability conditions. This method was also followed by Frieman & Chen(1982), who derived a nonlinear GKE (including E × B advection). Their demonstration that aworkable gyro-averaged equation emerges even in the face of nonlinearity was a nontrivial andsignificant result. However, the resulting equations were not in characteristic form, making themunusable for numerical simulation by the particle-in-cell (PIC) method discussed in Section 4.1.1.Lee (1983) derived suitable equations in characteristic form by using a recursive perturbationprocedure, also showing how ion polarization effects appear most naturally in the gyrokineticPoisson equation (see Section 3.1) instead of the kinetic equation; his initial simulations fatheredan enormous effort on PIC gyrokinetics that continues to the present. But years earlier Littlejohn(1979, 1981, 1982) had begun emphasizing the advantages of Hamiltonian techniques and Lieperturbation methods, and his seminal work laid the foundations for many modern developments.In particular, it inspired Dubin et al. (1983) to give a Hamiltonian formulation of self-consistentgyrocenter dynamics in an electrostatic slab (constant B0), clearly spelling out the basic closureprocedure for deriving the low-frequency gyrokinetic Poisson equation.

Further work by Littlejohn (1983) and Cary & Littlejohn (1983) emphasized the utility ofLagrangian methods, in which the equations of motion are derived from an action principleexpressed in terms of noncanonical variables (the use of which leads to considerable technicalsimplifications). Those developments brought to the attention of plasma physicists various notionsof differential geometry (Fecko 2006, Misner et al. 1973), including the use of differential forms.The particle action can be written as S = ∫

γ , where the fundamental one form is γ.= p·dq−H dt,

with H the single-particle Hamiltonian. Because S is indifferent to the particular variables usedin its evaluation, γ can be transformed into any convenient set of variables (e.g., gyrocentercoordinates), which need not be canonical but whose equations of motion nevertheless still followfrom the variational principle δS = 0. One constructs the change of variables perturbatively(possibly after a preparatory transformation to lowest-order gyrocenter variables) in such a waythat μ is conserved order by order. This methodology relies on a form of Noether’s (1918)theorem, which states that if all the coefficients of γ are independent of ζ , then the coefficient ofdζ is conserved (Cary & Brizard 2009, Cary & Littlejohn 1983). That coefficient is precisely μ.Hahm (1988) used the one-form method to derive a nonlinear electrostatic GKE in the presenceof magnetic inhomogeneities, generalizing the slab results of Dubin et al. (1983). Virtually allsubsequent work on modern gyrokinetics, including electromagnetic corrections (Hahm et al.1988), uses some variant of the one-form method. Technically, the perturbation expansions areperformed with the aid of Lie transformations (Brizard & Hahm 2007, Cary 1981, Kaufman 1978).

The noncanonical Lagrangian methods focus on the derivation of the (ζ-independent) gyro-center Hamiltonian H . For example, through the first three orders of expansion (ε−1, ε0, ε1), one

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Zonal flow (ZF):E × B flow (mostlypoloidal) generatedfrom a potential that istoroidally and (mostly)poloidally symmetric

finds electrostatically (with B0 = ∇ × A0) that

γ = [c −1q A0(X)︸︷︷︸O(ε−1)

+ mU b︸︷︷︸O(1)

− μK∗]︸︷︷︸O(ε)

·d X + μ dζ︸︷︷︸O(ε)

− [H(0)

︸︷︷︸O(1)

+ H(1)

]︸︷︷︸O(ε)

dt, (6)

where K∗.= K + 1

2 b(b·∇ × b), K .= (∇e1)·e2. Here K is the so-called gyrogauge vector; e1

and e2 are arbitrary orthogonal unit vectors perpendicular to B that locate the position of thegyrating particle in space (see Figure 1). Littlejohn (1983, 1984, 1988) has clearly interpreted theappearance of K in terms of the invariance of the formalism with respect to a redefinition of ζ .

The second-order H(2)

contains ponderomotive terms such as |∇⊥ϕ|2 [the complete expressionincluding the effects of nonzero pressure (Dubin et al. 1983) is more complicated]. Those termsare related to dielectric polarization (see Section 3.1) and also to Reynolds stresses, which areessentially involved in the generation of zonal flows (ZFs) (see Sections 3.4 and 5.5). Furthercomplications arise when magnetic inhomogeneities are included. Brizard (1989) described thetransformation to gyrocenter variables as two successive steps: First, derive a guiding-center theorythat incorporates the effects of magnetic inhomogeneities (ordered as εB ); second, incorporatesmall fluctuation effects (ordered as εϕ) that destroy the invariance of the guiding-center μ butpreserve a modified invariant μ (see Taylor 1967). When εB and εϕ are ordered with a commonexpansion parameter ε, one must take care to not miss either cross terms of the order of εBεϕ orpurely geometrical terms of the order of ε2

B . Explicit expressions for all terms in a consistent choiceof H

(2)were derived only quite recently in a difficult calculation by Parra & Calvo (2011). However,

A.J. Brizard (private communication, 2011) has stressed that the choice is not unique and that theterms of O(ε2

B ) can be removed from H(2)

in favor of a modification to the symplectic structure.Given (an approximate) H , the gyrocenter drifts follow as z

i = {zi , H }, where {. . .} is anoncanonical Poisson bracket, and the gyrokinetic Maxwell equations are obtained from the pull-back f = TF. In practice, the asymptotic expansions must be truncated in two places: (a) Thedrifts must be truncated in the GKE, and (b) T must be truncated in the Maxwell equations.Doing this consistently is crucial for the preservation of conservation laws, yet how to do so ispossibly unclear. For example, should one truncate at O(εn) for some n in both places, or can onegain accuracy by working out T to higher order than the drifts?

2.4. Gyrokinetic Field Theory

A major advance occurred when it was understood by Sugama (2000) and Brizard (2000) how toderive gyrokinetic-Maxwell systems from field-theoretic variational principles. Brizard’s versionbased on constrained variations is possibly more technically convenient, although it is subtle. Inthese methods, an action functional S is constructed from F , H , and the electromagnetic fields:

S = SEM + SG[F , H (z, Aμ, ∂ν Aμ, . . .)], (7)

where SEM is the electromagnetic Lagrangian, Aμ is the four potential, and the forms of SEM andthe gyrocenter contribution SG are not shown here. Variation with respect to F then leads to the(collisionless) GKE in the form ∂t F + {F , H } = 0, whereas variation with respect to ϕ ≡ A0

leads to the gyrokinetic Poisson equation. (To obtain the usual quasineutrality condition in whichPoisson’s ∇2ϕ is neglected, one ignores the electric-field part of SEM.) The crucial point is thata single, scalar Hamiltonian generates both the GKE and the gyrokinetic Maxwell equations. Hmay be approximate [e.g., correct only through O(εn)], but the variational procedure instills thatapproximation consistently into both the kinetic equation and the Maxwell equations, automat-ically preserving the conservation laws that follow from Noether methods—a point emphasized


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by Scott & Smirnov (2010) and Scott et al. (2010). In particular, functional derivation of H withrespect to ϕ reduces its order by one, equivalent to a truncation of the T in the gyrokinetic Poissonequation to one order lower than that of the drifts retained in the GKE (for more discussion, seeSection 5.5 on momentum conservation).

3. PHYSICAL CONTENT OF GYROKINETICS

The Lagrangian, noncanonical, and field-theoretic derivations of the gyrokinetic-Maxwell systemare elegant and efficient, and they lead to self-consistent equations that capture an enormousamount of the physics of the magnetized plasma, including turbulent fluctuations and transportdriven by gradients in the background profiles of density n, temperature T, and flow u. Most of thatphysics cannot be described in this short article (see Krommes 2006a for some tutorial lectures andSection 4.2 for some illustrations of the contact between simulations and experiments). However,it is instructive to consider some of the basic content of the gyrokinetic system.

3.1. Polarization and the Gyrokinetic Vacuum

Although ζ dependence is rigorously removed from the gyrocenter drifts [a process called “dynam-ical reduction” by Brizard (2008)], gyration-related effects remain in the pull-back transformationthat defines the gyrokinetic Maxwell equations. This redistribution of information is closely relatedto the treatment of dielectric media in classical electromagnetism. As is well known, it is frequentlyconvenient to separate total charge into free charge and polarization charge, the latter describableby a polarization vector P such that Poisson’s equation ∇·E = 4πρ becomes ∇·D = 4πρfree,where D .= E + 4π P and ∇·P = −ρpol. In gyrokinetics, that decomposition arises inevitablyfrom a representation of the dynamics in terms of the gyrocenter F and H , with the gyrocentercharge ρG playing the role of ρfree and the polarization charge arising from the distinction betweenthe particle position and the gyrocenter position. The gyrokinetic quasineutrality condition canbe shown to follow from the variational principle as

∑s

ns

∫p

JFδHδϕ

= 0. (8)

Here∫

p denotes integration over momentum variables, J is the Jacobian of the gyrokinetictransformation, and δ/δϕ denotes the functional derivative with respect to ϕ, which is non-trivial in the presence of spatial gradients of ϕ. Upon writing H = H

(0) + qϕ + �H (ϕ,not 〈ϕ〉ζ , is used here), Equation 8 becomes ρG − ∇·P = 0, where ρG .= ∑

s (nq )s∫

p JFand P .= −∑

s ns∫

JF (∂�H /∂ E + · · ·), with the dots indicating additional terms involvingderivatives with respect to second- and higher-order gradients of ϕ. The most important ped-agogical example involves the zero-gyroradius limit of the H

(2)derived by Dubin et al. (1983):

�H = H(2) = − 1

2 mi V 2E . One finds P = D⊥ E⊥, where D⊥

.= ρ2s /λ2

De = ω2pi/ω

2c i , ρs

.= c s/ωc i isthe so-called sound radius, c s

.= (ZTe/mi )1/2 is the ion sound speed (with Z the atomic numberand Te the electron temperature), λDe

.= (4πne e2/Te )−1/2 is the electron Debye length (with e theelectronic charge), and ωpi

.= (4πni q 2i /mi )1/2 is the ion plasma frequency. Typically D⊥ � 1; this

inequality defines the gyrokinetic regime (Krommes et al. 1986). This substantial polarization is aconsequence of the ion polarization drift Vpol = ω−1

c i ∂t(c E⊥/B), as can be seen by integrating thecontinuity equation for polarization charge ∂tρ

pol +∇·(ni qi Vpol) = 0 in the linearized approxima-tion. It is noteworthy that the (effect of the) polarization drift shows up in the gyrokinetic Poissonequation rather than the GKE. This is illustrated in Figure 2. More generally, polarization also

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VE

Vpol

E

x

y

Particle

Gyrocenter

Figure 2Illustration of the important concept that gyrocenters ( green) move only with the E × B driftV E

.= c E × b/B (and magnetic drifts, for inhomogeneous B). The magnetic field B = B b is in the zdirection. The additional polarization drift Vpol experienced by the ions when E varies in time is taken intoaccount by a polarization charge that appears explicitly in the gyrokinetic Poisson equation.

Gyrokinetic vacuum:the background state,endowed with largedielectric permittivitydue to ionpolarization, in whichgyrocenters move withthe effective E × Band magnetic drifts

includes effects that do not depend on the electric field, such as those driven by the pressuregradient.

Polarization leads to a useful interpretation of the gyrokinetic-Maxwell system as describingthe motion of gyrocenters in a gyrokinetic vacuum. The vacuum state, devoid of gyrocenters,is defined (for the example above) to possess a large dielectric permittivity D⊥ analogous to thepermittivity ε0 of free space. Into that vacuum one places gyrocenters, which move with theE × B and magnetic drifts. This interpretation was first given by Krommes (1993a) and has beendiscussed in various pedagogical papers by Krommes (2006a, 2010).

3.2. Equilibrium Gyrokinetic Statistical Mechanics

Although most interest is in nonequilibrium states (see Section 3.3), it is instructive to considerwhat physics emerges from thermal-equilibrium gyrokinetics. Thermal equilibrium arises fromintrinsically nonlinear interactions, so predictions derived therefrom can be used to test the nonlin-ear routines in simulation codes, a rare opportunity. That was already recognized in pregyrokineticsimulation theory (Birdsall & Langdon 1985), but gyrokinetics is richer and more subtle.

A gas of discrete gyrocenters in thermal equilibrium exhibits fluctuations with properties thatcan be calculated from a gyrokinetic fluctuation-dissipation theorem, which can be formulatedin terms of the wave-number- and frequency-dependent gyrokinetic dielectric function D(k, ω).That was done first by Krommes et al. (1986) for the electrostatic limit and later by Krommes(1993a,b) for weakly electromagnetic fluctuations. One finds that gyrokinetic fluctuations arestrongly suppressed (by the tendency for ion polarization to neutralize charge imbalances) relativeto those of the full many-body plasma.

Even in the absence of discreteness effects, gyrokinetic systems appropriately truncated inwave-number and velocity space possess absolute statistical equilibria, as discussed by Zhu &Hammett (2010). Such equilibria are well known in neutral fluids. For example, in two dimensionsthe conservation of both energy and enstrophy admits two-parameter Gibbsian equilibria withpossible negative temperature states (Kraichnan 1975). [For a review of 2D turbulence with earlierreferences, readers are referred to Kraichnan & Montgomery (1980). Some aspects of the absoluteequilibrium problem were also reviewed by Krommes (2002), and a Monte Carlo method forconstructing states of N gyrocenters with negative temperature was described by Krommes &


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DW: drift wave

HME:Hasegawa-Mimaequation

Polarization-driftnonlinearity: theE × B advection ofthe vorticity of E × Bmotion; thenonlinearity in theHasegawa-Mimaparadigm for driftwaves

Rath (2003).] In a 2D gyrokinetic system truncated in wave number and discretized with N velocitypoints, there are N entropy-related invariants in addition to an energy invariant, and in a nontrivialcalculation Zhu & Hammett were able to find the form of the corresponding Gibbsian equilibriaanalytically. The plethora of invariants leads to modifications of the equilibrium spectrum andimplies nontrivial behavior of forced, dissipative gyrokinetic cascades.

3.3. Drift Waves

Confined plasmas cannot be in thermal equilibrium because they possess profile gradients. Fromthe point of view of basic physics, the most important mode in a nonequilibrium gyrokinetic plasmais the drift wave (DW), supported by a gradient in the background density profile. (Modes involv-ing gradients in the background temperature are considered to be more important in practice.)Although the kinetic effects embodied in the full D(k, ω) are crucial to quantitative calculations oflinear instability, the basic DW can be obtained from a simple fluid description of gyrocenters. Letus consider the cold-ion limit Ti → 0 (which eliminates finite-Larmor-radius effects) and ignoremagnetic inhomogeneities. The density of ion gyrocenters then obeys the continuity equation (ob-tained from the zeroth velocity moment of the collisionless GKE) ∂tni + V E ·∇ni +∇‖(u‖i ni ) = 0.Let us write n = 〈n〉 + δn, where 〈n〉 denotes the background profile. After linearization, thedefinition ∇ ln〈ni 〉 .= −L−1

n x, and the neglect of u‖i because ion inertia is large, this becomes∂t(δni/〈ni 〉) + V∗∂yδ� = 0, where y .= b × x, V∗

.= c Te/e BLn = (ρs/Ln)c s (assumed to be con-stant) is called the diamagnetic velocity, and δ�

.= eδϕ/Te . Electrons are assumed to travel rapidlyalong field lines and to adjust instantaneously to any ambient potential, so they are assigned the adi-abatic or Boltzmann response δne/〈ne 〉 = δ�. Finally, the gyrokinetic Poisson equation relates theion polarization to the net gyrocenter charge: −ρ2

s ∇2⊥δ� = δni/〈ni 〉 − δne/〈ne 〉. These equations

can be combined into a single equation for the potential: (1 − ρ2s ∇2

⊥)∂tδ� + V∗∂yδ� = 0, whichyields the DW dispersion relation ω = �k, where �k

.= ω∗/(1 + k2⊥ρ2

s ) and ω∗(ky ) .= ky V∗.The physics of the wave, illustrated in Figure 3, involves E × B advection of the back-ground density gradient (V∗) in the presence of adiabatic electron response (the 1 in the de-nominator) and ion polarization (the k2

⊥ρ2s term, which leads to wave dispersion; the appear-

ance of ρs rather than ρ i means that polarization is fundamentally not a finite-Larmor-radiuseffect).

When such analysis is repeated without linearization, one is led (Dubin et al. 1983) to theHasegawa-Mima equation (HME), originally derived more tediously by Hasegawa & Mima (1978)from moment equations for the actual particles:

(1 − ρ2s ∇2

⊥)∂t� + V∗∂y� + VE ·∇(−ρ2s ∇2

⊥�) = 0. (9)

This equation is conservative; linear instability and dissipation, which would arise from the con-sideration of nonadiabatic electron response, can be inserted by hand. Note that the vorticity inthe 2D E × B motion is �

.= b·∇ × V E = ωc iρ2s ∇2

⊥� (this point is illustrated in Krommes2006b, figure 2), so the nonlinearity in the HME involves the E × B advection of vorticity. Thateffect is sometimes called the polarization-drift nonlinearity. The HME is more properly calledthe Charney-Hasegawa-Mima equation because it has the same form as the equation for Rossbywaves well known to geophysicists. This observation provides an entree to a large literature on2D geostrophic turbulence (Holloway 1986, Rhines 1979, Vallis 2006), many basic results fromwhich carry over to some of the plasma paradigms.

When the assumption of adiabatic electron response is relaxed, one finds in a slab model withconstant B a universal DW instability, destabilized by inverse electron Landau damping. DWstability is a voluminous and subtle subject. For example, Antonsen (1978) proved that in slab

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E × B

φ(y)

E

x

y

B

+++

<n>(x)

−−−

Direction of modepropagation

δni

Figure 3Illustration of the basic drift-wave mechanism. A constant magnetic field B is in the z direction, and a meandensity gradient is in the −x direction. An electrostatic potential ϕ(y) = − sin(ky y) is depicted. Thatgenerates an electric field E = −∇ϕ. The resulting E × B velocity moves parcels of the background iongyrocenter density (blue hatched square), creating ion density fluctuations δni . By quasineutrality, those areequal to the electron density fluctuations, which track the potential adiabatically because of electron motionalong the field lines. The pluses and minuses indicate an excess and deficiency of electron density δne ,respectively. In the diagram, density fluctuations are being created at the zero of ϕ, where δne = 0; for thedensity fluctuations to remain consistent with ϕ, the mode must propagate upward, in the electrondiamagnetic direction. Figure taken from Krommes (2006a, figure 4.8), Copyright c© 2006, World ScientificPublishing.

ITG: ion-temperature-gradient-driven

geometry a DW is absolutely stable in the presence of arbitrarily small amounts of magneticshear. However, toroidal effects related to the curvature of the magnetic field lines restore insta-bility (Chen & Cheng 1980), which can drive turbulence. [Turbulence can exist even in the face oflinear stability (see Section 5.4).] Modern literature refers to toroidal versions of ion-temperature-gradient-driven (ITG) and electron-temperature-gradient-driven modes, trapped electron modes,etc., and elaborate supercomputer codes have been developed to study their linear physics in re-alistic confinement geometries (Kotschenreuther et al. 1995, Rewoldt et al. 1982). The nonlineargyrokinetic codes described in Section 4 extend those results to the turbulent regime and calcu-late the anomalous transport and spectral characteristics of nonlinearly saturated steady states,quantities that can be directly compared with experiments. A turbulent transport coefficient D istypically compared to the basic gyro-Bohm scaling D ∼ (ρs/L)ρsc s ∝ B−2, where L is a profilescale length. That scaling is a rigorous consequence of dimensional analysis applied to the HME.More generally, it follows heuristically from random-walk considerations that assume that the gy-rokinetic turbulence is local, i.e., possesses a correlation length that scales with ρs and a correlationtime that scales with L/c s = [ω∗(ky = ρ−1

s )]−1.

3.4. Zonal Flows

Another constituent of magnetized plasma turbulence is the nonwavelike ZF. ZFs (frequentlycalled zonal jets in the geophysics literature) are E × B flows that in toroidal geometry stemfrom potential fluctuations that are independent of toroidal angle (and are mostly poloidallysymmetric as well); they are primarily poloidal and have no radial component. Although ZFs donot directly produce transport in the radial direction, they may regulate DW turbulence levels viatheir shearing effect on DW eddies; thus it is generally believed that a larger level of ZFs (frequently


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self-generated by the DWs) is associated with a smaller level of turbulent transport. An extremeexample is afforded by the Dimits-shift regime discussed in Section 5.1, in which turbulence issuppressed completely by the excitation of ZFs. Unfortunately, a thorough discussion of ZFs isimpossible here. For many details, one can refer to the reviews by Diamond et al. (2005), Itohet al. (2006), and Fujisawa (2009). More introductory material can be found in the lectures byKrommes (2006a). The basic ZF and eddy-shearing mechanism are illustrated in Diamond et al.(2005, figures 4 and 5).

Because k‖ = 0 for ZFs, the electron response to a ZF cannot be even approximately adiabatic,as was assumed in the derivation of the HME. [That the proper response strongly enhancesZFs was first emphasized by Hammett et al. (1993).] In the simplest model, the electron densityfluctuation can be taken to entirely vanish for zonal wave numbers. That leads to a generalized ormodified HME (Krommes & Kim 2000) that correctly describes the evolution of zonal vorticityand the coupling of the zonal modes to the DWs via Reynolds stresses.

Frequently ZFs are assumed to be of long wavelength relative to the scales responsible forturbulent transport [see the related work on nonlocal Rossby wave turbulence by Connaughtonet al. (2010)]. When that is done, a detailed connection to prior work in neutral-fluid turbulencetheory can be demonstrated. Krommes & Kim (2000) applied the disparate-scale expansion toa Markovian closure of an equation that can be simply reduced to either the 2D Navier-Stokesequation or the generalized HME by the appropriate choice of a single adiabaticity parameter.Note that the positive growth rate γ nl

q of the mean long-wavelength fluctuation energy due tononlocal interaction with the short scales can be interpreted in terms of a negative eddy viscosity:γ nl

q ≡ −μeddy q 2. For the 2D Navier-Stokes equation, the (generally anisotropic) result for μeddy

reduces in the isotropic limit exactly to the calculation of Kraichnan (1976). In the generalizedHasegawa-Mima limit, the analogous result can be interpreted in terms of (and actually defines)an algorithm based on a wave kinetic equation for the DWs. Seminal work on such an algorithmwas done by Diamond et al. (1998). The more systematic calculation by Krommes & Kim (2000)elucidated the proper form and statistical basis of the wave-kinetic algorithm and corrected someissues related to random Galilean invariance and the role of linear wave dispersion; it incidentallyidentified some technical problems with the important work of Carnevale & Martin (1982) onfield-theoretical methods applied to nonlinear wave dynamics in weakly inhomogeneous media.The difficulties are related to the choice of the plasmon density Nk to be used in the wave kineticequation. Nk(X) is not the formula familiar from linear wave theory; rather, it must be a quantitythat when summed over k and integrated over space is conserved under the DW-ZF interaction.The correct form was found for some special cases by Smolyakov & Diamond (1999). More gen-erally, Krommes & Kolesnikov (2004) proved that Nk is a Casimir invariant in a field-theoreticHamiltonian representation (Morrison 1998) of the advective nonlinearity. These results gener-alize and provide additional perspective on Kraichnan’s deep insights about the nature of negativeeddy viscosity in 2D flow.

Krommes & Kim assumed that the ZFs had zero ensemble mean. An important method thatdoes not require that assumption is the stochastic structural stability theory of Farrell & Ioannou(2003). In that technique, a form of stochastic modeling, one first derives the equation for anindividual realization of a ZF by applying a zonal average to the dynamical equations. Next, theeffects of the short-scale turbulence are modeled by a stochastic forcing. Finally, the zonal averageis replaced by an ensemble average under an ergodic assumption. The resulting closure can predictnontrivial spatial structure as well as temporal fixed points, limit cycles, or chaotic regimes for theZFs. It has been strikingly successful in various comparisons with atmospheric data, such as for theemergence of eddy-driven baroclinic jets (Farrell & Ioannou 2009b), and was applied by Farrell& Ioannou (2009a) to the Hasegawa-Wakatani system of equations (a collisional generalization of

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the HME that is a paradigm for turbulence in a plasma edge). Further remarks on that calculationare made in Section 5.4.

4. GYROKINETIC SIMULATIONS

Numerical solution of the GKE is important both for the exploration of physical processes and forquantitative prediction. I briefly mention various numerical implementations, which have becomehighly developed since their inceptions in the early 1980s, and then provide a few examples of thephysics results and comparisons with experiment that been obtained to date. Extensive additionaldetails and references can be found in the reviews by Garbet et al. (2010) and G.L. Hammett(manuscript in preparation).

4.1. Numerical Methodology

The two basic approaches to the numerical solution of the GKE are (a) the continuum or Vlasovmethod, which treats the GKE as a standard Eulerian partial differential equation (PDE) thatevolves in 5D phase space, and (b) the Lagrangian PIC method. Hybrid Lagrangian-Euleriantechniques have also been developed (Grandgirard et al. 2006). These methods can be applied toeither a full-F simulation, which solves Equation 3, or a δF simulation (Kotschenreuther 1991),which writes F = F0 +δF , analytically inserts a known equilibrium F0, and numerically integratesjust the equation for δF. (This ζ-independent δF differs from the δ F used above.)

Because the integration of a PDE that involves 5D plus time is computationally intensive, analternate gyrofluid approach was developed by Hammett and his coworkers. In that technique,fluid equations are derived from velocity moments of the GKE. The inevitable closure problem(Chapman-Enskog theory is inappropriate for nearly collisionless plasmas) is dealt with by aLandau-fluid closure (Hammett & Perkins 1990), in which unknown moments (e.g., the stresstensor or heat-flow vector) are modeled in such a way that linear response is well reproduced. Themethod has been successful and can be numerically efficient. Key references include Brizard (1992),Dorland & Hammett (1993), Hammett et al. (1993), and Beer & Hammett (1996), and furtherauthoritative discussion is given in the review by G.L. Hammett (manuscript in preparation).However, because the modeling is tricky and can fail to capture certain kinds of nonlinear wave-particle interactions, modern focus has been mostly on simulations of the GKE itself althoughgyrofluid equations are still actively used in areas such as edge turbulence (Scott 2007) and reducedtransport models (Staebler et al. 2007).

4.1.1. The particle-in-cell approach. The PIC approach, pioneered for gyrokinetics by Lee(1983), was the first of several approaches to be implemented. It is based on the fact that the full-FGKE can be written in characteristic form. The characteristic trajectories of many gyrocenters areintegrated forward one time step and then gathered onto a spatial grid for the purpose of computingthe charge and current. [Bessel functions are evaluated by an n-point averaging technique (Lee1987); various important smoothing and interpolation techniques are not described here.] Thismethod avoids the need for a velocity-space grid, is straightforward to program, and is well suitedto massively parallel processing. A noncomprehensive list of significant PIC codes includes GTC(Lin et al. 1998), GEM (Chen & Parker 2003), GTS (Wang et al. 2006), ORB5 ( Jolliet et al.2007), and XGC1 (Chang et al. 2009).

In the δF method, the PIC approach is implemented by assigning a weight wi to the i-th gy-rocenter (called a marker or tracer); w

.= δF/F , where F is the tracer PDF. The equation for w

is derived analytically and then integrated along with the tracer characteristics. The procedure


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is more completely expressed in terms of a two-weight scheme (Hu & Krommes 1994). In col-lisionless theory, it can be shown that the mean-square weight evolves secularly, leading to theso-called entropy paradox in which a statistical observable is changing in time even though byconventional spectral measures the turbulence appears to be saturated. Introduction of collisionsresolves this paradox (Krommes & Hu 1994), which is related to the generation of fine scales invelocity space (see Section 5.3). To control the collisionless limit, Krommes (1999b) suggested analternate approach involving the use of a generalized thermostat (Evans & Morris 1984) or w-stat;that idea was pursued and extended by McMillan et al. (2008).

The PIC method is essentially a Monte Carlo sampling technique (Aydemir 1994, Hu &Krommes 1994), although it was not originally discussed as such. Thus one must contend withsampling noise. Following Krommes’s basic work on gyrokinetic fluctuations (cited in Section 3.2)and calculations of δF noise by Hu & Krommes, further theoretical and numerical analyses ofgyrokinetic noise were made by Nevins et al. (2005), who showed that some PIC simulations maybe noise dominated. Rigorously, the calculation of noise in nonequilibrium situations is nontrivial;the fluctuation-dissipation theorem no longer applies, and discreteness effects may be amplifiedby instabilities and mix nonlinearly with collective effects in complicated ways. Krommes (2007)reviewed the noise issue and discussed the formal problem of calculating nonequilibrium samplingnoise, following a general methodology given by Rose (1979). Although the complete formalismis complicated, the structure of the theory supports the general conclusions drawn by Nevins et al.Noise can be reduced by increasing the number N of markers (but only by

√N ), or by phase-space

smoothing to reduce the weights (Chen & Parker 2007).

4.1.2. The continuum approach. Although the PIC approach is intuitive, it is surprisinglysubtle. An alternate approach is to attack the GKE directly with the aid of advanced numericaltechniques for PDEs. Specific methods are documented in the publications and web pages for themajor continuum codes, a noncomprehensive list of which includes GS2 [http://gs2.sourceforge.net; the linear code of Kotschenreuther et al. (1995) was generalized to nonlinear physicsby Dorland et al. (2000)] and its daughter AstroGK (Numata et al. 2010), which removesgeometry effects from GS2 and is used for studies of astrophysical gyrokinetics; GENE( Jenko et al. 2000; http://gene.rzg.mpg.de); GYRO (Candy & Waltz 2003a,b; https://fusion.gat.com/theory/Gyro); and GT5D (Idomura et al. 2008). Continuum codes have theirown algorithmic challenges. However, they have been effective in attacking practical problemsand have made major progress in the push toward successful comparisons with experiment.

4.2. Illustrative Simulation Results

Figures 4–7 illustrate the kind of results that can be obtained from the modern codes. They includeglobal simulations of existing machines (Figure 4), impressive agreement with experimental data(Figure 5), and studies of turbulence in regions in which magnetic topology changes because ofthe presence of a divertor (Figure 6). Although workstation-sized runs have been useful for manyqualitative physics studies, comparisons with experiment that include realistic geometry and otherpractical effects require massively parallelized processing on the world’s largest supercomputers.Such simulations are not restricted to magnetically confined fusion plasmas. Figure 7 showsan application of a gyrokinetic simulation to the solar wind, and there are other applications tospace physics, e.g., the simulations by Kobayashi et al. (2010) of dipolar systems such as planetarymagnetospheres.

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http://gs2.sourceforge.net/

http://gs2.sourceforge.net/

http://gene.rzg.mpg.de

https://fusion.gat.com/theory/Gyro

https://fusion.gat.com/theory/Gyro


Positive

Negative

Electrostaticpotential

Figure 4Full-torus (global) GENE simulation of a discharge in the TCV tokamak that exhibits an internal transportbarrier. Realistic input data and comprehensive physics are used. The figure displays contours of electrostaticpotential (stream function). In this cross section, the internal transport barrier corresponds to a fairly narrowring near the mid-minor radius of the torus. Figure reprinted with permission from Gorler et al. (2011,figure 8). Copyright c© 2011 by the American Institute of Physics.

Dimits shift:in collisionless ITGturbulence, theamount by which thegradient required forthe onset of turbulenceexceeds the thresholdfor linear instability

5. SOME ADVANCED TOPICS IN GYROKINETICS

All the following topics are still the focus of contemporary research. They demonstrate a niceinterplay between numerical simulation and analytical theory, and they illustrate the richness ofgyrokinetic physics.

5.1. Transition to Turbulence

As simulation codes proliferated, it became important to verify them [here the term verify is usedin a standardized technical sense (e.g., see Greenwald 2010) and should be contrasted with theterm validate (against experiment)], and a standard case (the Cyclone base case) was proposed fordetailed study. Dimits et al. (2000) published the results of comparisons between various codes.Part of that work involved the discovery of what is now called the Dimits shift, which provides awindow into an important area of gyrokinetic physics.

The Dimits shift arises (at least) in studies of the turbulent ion heat flux Q driven by a gradientin the background ion temperature gradient, parameterized here by κ

.= R/LT , with R the majorradius of the torus and LT the temperature-gradient length scale. Linearized gyrokinetics makesa definite prediction for the threshold κlin of linear instability. Conventional arguments suggestedthat turbulence and Q should turn on for κ > κlin; however, the (collisionless) simulations revealedthat Q remained essentially zero for a nonzero range κlin ≤ κ ≤ κ∗ for some κ∗; the differenceof κ∗ from κlin defines the Dimits shift. Dimits et al. gave the correct qualitative interpretation,


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0 0.2 0.4 0.6 0.8

2.5

2.0

1.5

Experimental dataSpline fitTGYRO+GYRO

1.0

0.5

0

3.0

2.0

1.0

0

4

3

2

1

0

Normalized toroidal flux ρ

T i (k

eV)

T e (k

eV)

n e (1

019 m

–3)

Figure 5Predictions (red curves) of the TGYRO code (https://fusion.gat.com/theory/Tgyrooverview) for DIII-Ddischarge 128913 compared with experimental measurements (discrete data points). The electron temperatureis particularly well reproduced. Fits to the experimental data (blue curves) are also shown. For this case, 10simulation radii (10 instances of GYRO) were used. All three profiles (ne, Te, Ti) were evolved by TGYROusing nonlinear GYRO calculations of the turbulent electron particle flux, electron energy flux, and ionenergy flux. Figure courtesy of J. Candy.

which is that in the Dimits-shift regime ITG fluctuations are suppressed by the self-generation ofZFs. Rogers et al. (2000) provided a further compelling analysis (including a discussion of tertiaryinstability) and a simple model that argued in favor of the ZF interpretation. That work wasimportant, as the significance of ZFs (already well known to geophysicists) was just beginning topenetrate the consciousness of the fusion-physics community (Diamond et al. 1998, Rosenbluth& Hinton 1998).

Kolesnikov & Krommes (2005) attempted to study the Dimits shift in a simple model by us-ing the systematic machinery of bifurcation analysis (Guckenheimer & Holmes 1983, Kuznetsov1998), including the use of the center manifold theorem to eliminate rapidly damped modes.ZFs are only very weakly damped by collisions; Kolesnikov & Krommes assumed that they werestrictly undamped. That greatly complicates the analysis. In the absence of ZFs, ITG modes aredestabilized as κ is increased due to a conventional Hopf bifurcation. In their presence, how-ever, additional undamped ZFs remain at the origin in complex λ space [variations like exp(λt)are assumed] as the complex-conjugate pair of ITG eigenvalues crosses the imaginary axis. Thebifurcation is no longer simple Hopf and exhibits peculiar properties that are difficult to analyze.Kolesnikov & Krommes succeeded in analytically calculating a Dimits shift for their model, whichwas a significant proof of principle. However, they employed a Galerkin truncation to reduce theITG PDEs to a small number of coupled amplitudes, and unsurprisingly the predicted shift wassensitive to the truncation. It is probable that a different sort of analysis altogether must be done toproperly obtain the Dimits shift analytically, especially in realistic problems with magnetic shear.Such a calculation remains one of the outstanding problems in plasma theory.

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–1.5 × 102 1.6 × 102–48 57Perturbed electrostatic potential (V)

Figure 6Gyrokinetic simulation of ion-temperature-gradient-driven turbulence by the PIC code XGC1 in an edgepedestal region with the realistic diverted geometry of the DIII-D device. More than 109 marker particleswere used. Such simulations, still in an early stage of development, are difficult because of the change ofmagnetic topology associated with the divertor separatrix. Figure reprinted with permission from Changet al. (2009, figure 6). Copyright c© 2009 by the American Institute of Physics.

Perpendicular magnetic field

Electric fieldPredicted electric energy spectrum

Parallel magnetic fieldPredicted parallel magnetic energy spectrum

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EB (k )

Figure 7One-dimensional energy spectra of the perpendicular magnetic (solid blue), electric (dashed green), and parallelmagnetic (dot-dashed red ) fields from gyrokinetic simulations (using AstroGK) of kinetic Alfven waveturbulence from the ion to electron Larmor radius scales. These spectra demonstrate that kinetic Alfvenwave turbulence can indeed yield energy spectra reaching the electron scales, as found in recent observations(Sahraoui et al. 2009). Thin lines are the perpendicular electric (dashed green) and parallel magnetic energy(dot-dashed red ) spectra predicted from the perpendicular magnetic energy spectrum using the linear kineticAlfven wave eigenfunction, demonstrating that, even in fully developed turbulence, the fluctuations retainthe character of linear wave modes. Figure reprinted with permission from Howes et al. (2011, figure 1).Copyright c© 2011 by the American Physical Society.


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Figure 8Gyrokinetic eigenmode spectrum for ion-temperature-gradient-driven (ITG) fluctuations, demonstratingthe possibility of nonlinear coupling of an unstable ITG mode (red ) to damped eigenmodes. Figure reprintedwith permission from Hatch et al. (2011, figure 2). Copyright c© 2011 by the American Institute of Physics.

5.2. Damped Eigenmodes and the Nature of Plasma Turbulence

In general, sufficiently large drive leads to turbulence. In neutral fluids at high Reynolds num-ber, that is frequently described by the standard paradigm involving long-wavelength energy-containing scales excited by macroscopic instability, an intermediate-scale inertial interval, andvery short dissipation scales (for discussion of a similar situation in plasmas, see Schekochihin et al.2009 and Figure 7). However, plasmas can also behave quite differently. In some cases, especiallyin magnetic fusion, the range of excited scales may be small, possibly no more than a decade; awell-defined inertial range may not exist. The way in which plasma turbulence saturates may beentirely different from the standard Navier-Stokes scenario.

Although these differences have been long appreciated in general terms, only recently have theybeen quantified. Terry and coworkers (Hatch et al. 2011, Terry et al. 2006) have demonstratedthat a new paradigm, involving coupling to damped eigenmodes and not requiring a cascade inwave number, can sometimes be superior. First let us consider an n-field fluid description. (Forthe Hasegawa-Wakatani system, n = 2.) Such a system has n linear eigenmodes, most of whichare typically damped. Nonlinearity can then couple energy in an unstable mode to stable modesat comparable wave numbers, providing a saturation mechanism. Gyrokinetics is richer, as thereare an infinite number of linear eigenmodes. Figure 8 shows a typical eigenmode spectrum forthe gyrokinetic description of ITG turbulence; a single unstable ITG mode can couple to a sea ofdamped eigenmodes (for detailed discussion of Figure 8, see Hatch et al. 2011).

5.3. Entropy, Phase-Space Cascades, and Dissipation

It is well known that the behavior of the dissipative Navier-Stokes equation differs profoundlyfrom that of the conservative Euler equation. The contrast between the collisional and collisionlessGKEs has been discussed analogously by Krommes & Hu (1994), who argued that a numericallyobserved secular growth in the entropy-like quantity

∫d z δF 2/F could be tamed only by the

inclusion of collisional dissipation. That insight was verified by careful simulation measurements byWatanabe & Sugama (2004) and Candy & Waltz (2006). For almost-collisionless physics, effectivedissipation requires the generation of fine scales in velocity space. That can be accomplished by theparallel streaming term in the linearized GKE, as can be seen by an expansion of the v‖ dependence

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of the GKE into Hermite polynomials (Hammett et al. 1993). But nonlinear phase mixing arisingfrom the J0(k⊥v⊥/ωc i ) in the effective potential can be even more efficient. The role of that termin the simultaneous generation of fine perpendicular scales in both position and velocity has beendiscussed by Schekochihin et al. (2009) in conjunction with the gyrokinetic theory of kinetic Alfvencascades, believed to be important in solar-wind physics (Figure 7). That mechanism provides “anonlinear route to dissipation through phase space” (Schekochihin et al. 2008). The analytics leadto concrete predictions for the exponents of self-similar, power-law, phase-space entropy cascades(Plunk et al. 2010). Those have been observed numerically (Navarro et al. 2011, Tatsuno et al.2009) and provide a basis for interpreting some solar-wind spectra (Howes et al. 2008).

5.4. Submarginal and Non-Normal Turbulence

Submarginal or subcritical turbulence exists, by definition, in regimes in which the linear eigen-mode spectrum is entirely stable. This phenomenon, intensively studied in various neutral-fluidsituations such as Poiseuille or planar Couette flow, requires that the linear operator L be non-normal, i.e., [L, L†] �= 0 (Henningson & Reddy 1994). Non-normality can be important even whensome modes are linearly unstable. Submarginal or more generally non-normal turbulence can existin gyrokinetic plasmas for relevant parameters; it may be particularly important to tokamak edges,the physics of which is considered to be of critical importance for the understanding of magneticconfinement. Scott (1992a,b) demonstrated numerically that DW turbulence can be submarginal,and he described a plausible scenario for the nonlinear self-sustainment. Itoh et al. (1996) de-scribed simple closure approximations that suggested that submarginal turbulence can exist quitegenerally. Krommes (1999a) discussed some of the ideas that Waleffe (1995) had developed forthe description of shear flows close to transition, providing an interpretation and generalizationof some earlier plasma work by Drake et al. (1995). Thorough study of the Hasegawa-Wakataniequations, which are non-normal, was done by Farrell & Ioannou (2009a), who used the stochasticstructural stability theory introduced in Section 3.4. Their analytical closure predicted states ofboth high and low transport. [Qualitatively similar regimes have been observed experimentally, asdescribed in the review by Wagner (2007), and are a subject of longstanding interest.] They arguedthat their low-transport states could be accessed through appropriate manipulations of externalparameters. Some details of the results are not definitive because the calculation was not fully en-ergetically self-consistent and the small-scale turbulence was modeled crudely. However, Farrell& Ioannou (1996) have advanced powerful arguments that the predictions should be relatively in-sensitive to the details of the stochastic modeling when the linear operator is non-normal. Furtherapplication of these ideas to models of gyrokinetic turbulence should be a fruitful line of research.

Recently, gyrokinetic simulations were used to obtain a description of ITG heat flux Q pa-rameterized by temperature gradient κT and flow shear κu (Highcock et al. 2010). Nonzero flowshear allows the possibility of submarginal turbulence. Parra et al. (2011) used the unusual shapeof Q(κT , κu) to predict the optimal amount of momentum input that minimizes transport andshowed that it admits the possibility of bifurcations between regimes of high and low transport.Such results provide a fresh look at the mechanisms for the transition between the low and the highmodes as well as the formation of internal transport barriers, understanding of which is importantfor the operation of future devices such as ITER.

5.5. Momentum Conservation and Toroidal Rotation

For numerical studies of microturbulence in toroidal devices, a simulation should be run for atleast a few turbulence autocorrelation times τac so that good statistics can be obtained by time


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averaging, and that is already challenging. Nevertheless, as computational resources continue toimprove, attention is beginning to shift toward simulations on the transport timescale on whichthe macroscopic profiles evolve; this can be orders of magnitude larger than τac. One appealingapproach invokes a multiple-timescale strategy in which a δF turbulence code, run on the τac

scale to obtain local fluxes, is coupled to a coarse-grained transport code that advances meanprofiles on macroscopic timescales [see Figure 5, Barnes et al. (2010), and the work of Sugama &Horton (1997) on the derivation of gyrokinetic transport equations]. But it is at least conceptuallyinteresting to inquire whether a full-F gyrokinetic code can be integrated directly up to transporttimes. In a provocative Ph.D. dissertation, Parra (2009) argued to the contrary in the contextof the calculation of the radial electric field and toroidal rotation. [The latter is of interest formagnetohydrodynamic stability and flow-shear stabilization of microturbulence (Terry 2000).]Two fundamental assumptions were made: (a) the low-flow ordering u/c s = O(ε), and (b) gyro-Bohm transport scaling. Then the basic assertion was that the truncations of the gyrokinetic systemthat are usually implemented in the codes drop higher-order terms that are essential for a correctevaluation of the rotation that develops at long times in axisymmetric toroidal systems. Thisassertion has been highly controversial and has generated a considerable amount of discussion andpublications. Many of those are cited in the overview by Parra & Catto (2010b). (The collectedpapers of those authors serve as an authoritative primer on many facets of gyrokinetics.) The issuereduces to a discussion of the gyrokinetic conservation law for toroidal angular momentum.

In the original derivations of nonlinear gyrokinetics, truncations in principle were made in-dependently in the kinetic equations and in the Poisson equation. That can obviously lead toproblems with conservation laws. As an illustration, Parra & Catto (2010a) revisited the slab cal-culation of Dubin et al. (1983) and showed that if one retains terms through second order in bothplaces, then a spurious nonconservative term emerges in a momentum evolution equation. Theyrecognized that the spurious term would be eliminated if third-order drifts were retained in thekinetic equation. However, those are complicated and possibly impractical to code. Moreover,in the presence of magnetic inhomogeneities, it seemed possible that even higher-order effectswould be required.

The correct way to truncate is given a definitive answer, and in fact is superceded, by thefield-theoretic derivations of gyrokinetics. As described in Section 2.4, truncation should be donedirectly on the action functional S; the use of an n-th-order Hamiltonian in S is equivalent totruncations of O(εn) in the kinetic equation and O(εn−1) in the gyrokinetic Poisson equation.From Noether arguments in the presence of toroidal symmetry, this ensures a conservative formof the equation for toroidal momentum. Left open is the possibility that the momentum fluxesare calculated inaccurately if n is too small [errors of O(1) in the predicted rotation profile maybe possible even if n = 2]. It was thus significant that Scott & Smirnov (2010) derived the exactform of the gyrokinetic conservation law for toroidal angular momentum, expressing the fluxes interms of an arbitrarily accurate Hamiltonian H . [A concise and elegant rederivation of that law byBrizard & Tronko (2011) has been helpful to the author.] Knowledge of that conservation law istechnically useful, as it can serve to verify the fidelity of gyrokinetic simulation codes; related issueshave been discussed authoritatively by Scott et al. (2010). Moreover, it is conceptually important,as it bears on the ordering issue. Scott & Smirnov concluded that first-order polarization effects(related to second-order terms in the Lagrangian) are adequate. However, naive estimates suggestthat, under the assumptions (a) and (b), the Reynolds stresses stemming from H

(3)are of the

same order as those from H(2)

[the latter, nominally O(ε2), must in fact be O(ε3) for gyro-Bohmscaling]. Then second-order polarization terms are necessary, in agreement with a basic concernof Parra & Catto. At the time of writing, the practical importance of H

(3)remained under debate.

The issue would disappear if the actual transport were larger than assumed, and that is the case

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in some important situations such as edge pedestal regions with steep gradients. In any event,although ongoing research should lead to a consensus, it must be emphasized that the full-Fquasineutral method may not be practical for long-time gyrokinetic simulations; alternate δFhybrid approaches may be superior (Parra & Catto 2009, and references therein).

SUMMARY POINTS

1. Nonlinear low-frequency gyrokinetics is the major formalism used to study microturbu-lence in magnetically confined plasmas.

2. The nonlinear GKE (a 5D PDE in gyrocenter phase space) coupled to the gyrokineticMaxwell equations is a unique dynamical system that describes the drift motion of gyro-centers in a gyrokinetic vacuum with large permittivity due to ion polarization.

3. Modern derivations of the gyrokinetic-Maxwell system exploit variational principles,noncanonical Lagrangian methods, and Lie perturbation theory.

4. Gyrokinetics provides a good description of the nonlinear interactions between DWsand self-consistently generated ZFs. The former cause turbulent transport; the latterregulate the turbulence level.

5. Numerical simulation of the nonlinear GKE, although computationally challenging, isbecoming a quantitatively predictive tool.

6. A nonlinear phase-mixing mechanism unique to gyrokinetics is responsible for entropycascade in phase space, which provides a route to collisional dissipation at fine scales.

7. The gyrokinetic conservation law for angular momentum can be used to address difficultissues relating to the development of toroidal rotation on long timescales.

8. Modern nonlinear gyrokinetics is elegant, subtle, and powerful. It should reside in thetoolbox of every plasma physicist.

FUTURE ISSUES

1. Simulations of the edge and scrape-off-layer regions of toroidal devices (a complicatedendeavor because of the change of magnetic topology from closed to open field lines)should continue to be developed.

2. The continued exploration of effective ways of performing gyrokinetic simulations onthe transport timescale is important, with particular focus on momentum transport.

3. An integrated program of theory, simulation, and experiment should be pursued to deeplyunderstand the submarginal and non-normal nature of plasmas.

4. Research on the development of new gyrokinetic simulation algorithms that exploitmodern computer hardware, such as graphical processing units, should be continued.

DISCLOSURE STATEMENT

The author is not aware of any biases that might be perceived as affecting the objectivity of thisreview.


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ACKNOWLEDGMENTS

I am grateful to A. Brizard, S. Brunner, I. Calvo, J. Candy, D. Carati, P. Catto, C.-S. Chang, T.-S.Hahm, G. Hammett, D. Hatch, G. Howes, F. Jenko, J. Johnson, W.W. Lee, J. Parker, F. Parra,G. Plunk, H. Qin, A. Schekochihin, B. Scott, J.B. Taylor, P. Terry, and R. Waltz for educationaldiscussions, illustrative figures, and/or helpful comments on drafts of the manuscript. This workwas supported by U.S. Department of Energy contract DE-AC02-09CH11466. I also enjoyed thehospitality of the Isaac Newton Institute for Mathematical Sciences, Cambridge, which hostedthe 2010 Program on Gyrokinetics in Laboratory and Astrophysical Plasmas. I benefited frominformative interactions with the participants in that program, and I regret that most of theirwork could not be mentioned owing to length constraints. Fortunately, their talks are online (INIWorkshop Gyrokinet. 2010).

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FL44-FrontMatter ARI 1 December 2011 22:45

Annual Review ofFluid Mechanics

Volume 44, 2012Contents

Aeroacoustics of Musical InstrumentsBenoit Fabre, Joel Gilbert, Avraham Hirschberg, and Xavier Pelorson � � � � � � � � � � � � � � � � � � � � 1

Cascades in Wall-Bounded TurbulenceJavier Jimenez � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �27

Large-Eddy-Simulation Tools for Multiphase FlowsRodney O. Fox � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �47

Hydrodynamic Techniques to Enhance Membrane FiltrationMichel Y. Jaffrin � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �77

Wake-Induced Oscillatory Paths of Bodies Freely Risingor Falling in FluidsPatricia Ern, Frederic Risso, David Fabre, and Jacques Magnaudet � � � � � � � � � � � � � � � � � � � � � �97

Flow and Transport in Regions with Aquatic VegetationHeidi M. Nepf � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 123

Electrorheological Fluids: Mechanisms, Dynamics,and Microfluidics ApplicationsPing Sheng and Weijia Wen � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 143

The Gyrokinetic Description of Microturbulence in Magnetized PlasmasJohn A. Krommes � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 175

The Significance of Simple Invariant Solutions in Turbulent FlowsGenta Kawahara, Markus Uhlmann, and Lennaert van Veen � � � � � � � � � � � � � � � � � � � � � � � � � � 203

Modern Challenges Facing Turbomachinery AeroacousticsNigel Peake and Anthony B. Parry � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 227

Liquid Rope CoilingNeil M. Ribe, Mehdi Habibi, and Daniel Bonn � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 249

Dynamics of the Tear FilmRichard J. Braun � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 267

Physics and Computation of Aero-OpticsMeng Wang, Ali Mani, and Stanislav Gordeyev � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 299

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Smoothed Particle Hydrodynamics and Its Diverse ApplicationsJ.J. Monaghan � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 323

Fluid Mechanics of the EyeJennifer H. Siggers and C. Ross Ethier � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 347

Fluid Mechanics of Planktonic MicroorganismsJeffrey S. Guasto, Roberto Rusconi, and Roman Stocker � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 373

Nanoscale Electrokinetics and Microvortices: How MicrohydrodynamicsAffects Nanofluidic Ion FluxHsueh-Chia Chang, Gilad Yossifon, and Evgeny A. Demekhin � � � � � � � � � � � � � � � � � � � � � � � � � 401

Two-Dimensional TurbulenceGuido Boffetta and Robert E. Ecke � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 427

“Vegetable Dynamicks”: The Role of Water in Plant MovementsJacques Dumais and Yoel Forterre � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 453

The Wind in the Willows: Flows in Forest Canopies in Complex TerrainStephen E. Belcher, Ian N. Harman, and John J. Finnigan � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 479

Multidisciplinary Optimization with Applicationsto Sonic-Boom MinimizationJuan J. Alonso and Michael R. Colonno � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 505

Direct Numerical Simulation on the Receptivity, Instability,and Transition of Hypersonic Boundary LayersXiaolin Zhong and Xiaowen Wang � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 527

Air-Entrainment Mechanisms in Plunging Jets and Breaking WavesKenneth T. Kiger and James H. Duncan � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 563

Indexes

Cumulative Index of Contributing Authors, Volumes 1–44 � � � � � � � � � � � � � � � � � � � � � � � � � � � � 597

Cumulative Index of Chapter Titles, Volumes 1–44 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 606

Errata

An online log of corrections to Annual Review of Fluid Mechanics articles may be foundat http://fluid.annualreviews.org/errata.shtml

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Documents

The Gyrokinetic Description of Microturbulence in Magnetized Plasmas