J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)271–274
000–000

Spectrum of the Velocity Distribution Function in the Slab Ion
Temperature Gradient Driven Turbulence
SUGAMA Hideo and WATANABE Tomohiko
National Institute for Fusion Science, Graduate University for Advanced Studies, Toki 509-5292, Japan(Received: 10 December 2003 / Accepted: 12 February 2004)

Abstract
A spectral analysis of the velocity distribution function in the slab ion temperature gradient (ITG) driven
turbulence is done by using high-resolution Eulerian kinetic simulation results. It is clarified how the entropy variableassociated with the fine-scale structure of the distribution function is produced by the turbulent heat transport in thepresence of the temperature gradient, transferred from macro to microscales in the velocity space through phasemixing processes, and dissipated by collisions. An interesting analogy between the entropy spectrum in the ITG
turbulence and the spectrum of a passive scalar quantity in a turbulent fluid is pointed out. The entropy spectral
function is analytically derived and confirmed by the simulation result. It is shown that the entropy spectrum obeys apower law in the range that is free from instability sources and collisional dissipation.
Keywords:
slab ITG turbulence, entropy spectrum, gyrokinetic equation, anomalous transport

1. Introduction

2. Basic kinetic equation

Plasma turbulence and resultant anomalous transport
have long been a key issue in the magnetic fusion research[1]. Complexity of the issue partly lies in the fact that, forqualitative study of turbulence in high-temperature plasmas,
we should take account of kinetic characteristics andaccordingly treat not only real-space but also velocity-spacestructures of the particle distribution function. Because oflarge computer memory and time required for calculation of
the distribution function in such turbulent kinetic systems,
detailed investigations of its velocity-space structures inturbulent states have not much been made so far comparedwith those on the real-space structures of fluid variables. In
the present work, combining high-resolution Eulerian kinetic
simulation [2,3] with theoretical analysis, we elucidate thevelocity-space spectral structures in the slab ion temperature
gradient (ITG) driven turbulence [4,5] which causes
anomalous transport in high-temperature plasmas. This typicalexample of plasma turbulence involves a turbulent E × B flow
and anomalous heat transport in the real space as well asphase-mixing and collisional dissipation processes in thevelocity space.
This paper is organized as follows. A basic kinetic
equation for the slab ITG turbulence and geometricalconditions are described in Sec. 2. The spectral analysis of
the distribution function is presented in Sec. 3. There,
analytically derived entropy spectral functions are comparedwith results from numerical simulation of the slab ITG
turbulence. Finally, conclusions are given in Sec. 4.

As in the previous work on the collisionless ITG
turbulence [4], we consider a periodic two-dimensional slab
configuration with translational symmetry in the z-direction,
where the uniform magnetic field is set in the y-z plane suchthat B = B(z̃ + θ ỹ) with θ << 1. Assuming the fluctuation partof the ion distribution function to be give by δfk(v||,v⊥) = f˜k(v||)
FM (v⊥), with the Maxwellian velocity distribution FM ,neglecting the parallel nonlinear term and taking a v⊥-integralof the ion gyrokinetic equation yield the following equationin the wave number space k = (kx,ky),∂ t f˜k + iΘ v|| k y f˜k +[∑ (k k – k k )Ψ f˜' "y xk = k' + k"

' "x yk' k"]= – ik yΨk 1 + ( v||2 – 1 – k 2 )ηi / 2 + Θ v|| FM ( v|| ) + Ci ( f˜k ), (1)where we also have assumed constant density and temperature
gradients of the background ions in the x-direction with muchlarger scale-lengths [Ln ≡ –d(ln n)/dx and LT ≡ –d(ln Ti)/dx]
than the fluctuation wave lengths. The electric potential φk is2related to Ψk by Ψk = e–k /2φk with k2 = k2x + ky2. In addition toeq. (1), the electron Boltzmann relation and the quasineutrality
condition, which are not written here, are used to give aclosed system of governing equations for the slab ITG
turbulence [2,4,5]. In eq. (1), we have used the following
normalization; x = x′/ρi, y = y′/ρi, v|| = v||′/vti, t = t′vti /Ln, f̃ =f̃ ′Lnvti /ρin0, and φ = eφ′Ln/Tiρi, where vti, ρi (= vti /Ωi), Ωi, n0,e, and Ti are the ion thermal velocity, the ion thermal gyro©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: sugama@nifs.ac.jp1271

Sugama H. and Watanabe T., Spectrum of the Velocity Distribution Function in the Slab Ion Temperature Gradient Driven Turbulence

space, respectively, and 〈…〉 denotes the ensemble average.
Using eq. (1), we obtainradius, the ion cyclotron frequency, the background plasma
density, the elementary charge, and the background ion
temperature (Ti = mi vti2 ; mi means the ion mass), respectively.
Prime means a dimensional quantity. Θ is defined as Θ = θ
Ln/ρi. ηi is given by ηi = Ln/LT.
The parallel advection term on the left hand side of eq.
(1) contributes to generation of fine-scale fluctuations of f˜k inthe velocity space, that is, the phase mixing. The instability
drive is contained in the first group of terms on the righthand side of eq. (1). The last term on the right-hand side
denotes the ion-ion collision term for which we employ theLenard-Bernstein model collision operator, Ci ( f˜k) = ν∂v|| [∂v||+ v||] f˜k (v||), with the collision frequency ν normalized by vti /Ln.

dδSn = ηi Qiδn,2 + Jn –1/ 2 − Jn +1/ 2 − 2νnδSn for n ≥ 2, (5)dtwhereJ n –1 / 2 ≡J n +1/ 2 ≡Qi ≡n||M(2)||n=0where Hn(v||) is the Hermite polynomial of order n. In termsof the coefficient fˆk,n in the Hermite-polynomial expansion,eq. (1) is rewritten by∂fˆk ,n∂t+[+ ik yΘ fˆk ,n –1 + (n + 1) fˆk ,n –1]∑ (k k – k k )Ψ fˆ
' "y x' "x yk' k" , nk = k' + k"ηη ⎤⎡= – ik yΨk ⎢δ n,0 ⎛1 – i k 2 ⎞ + δ n,1Θ + δ n,2 i ⎥ ,
⎝⎠
22⎦
⎣

(3)

where δn,m = 1 for n = m and 0 for n ≠ m. The phase mixing
process associated with the parallel streaming of particles are
now represented by the interaction to the adjacent-order[(n – 1) and (n + 1)] terms in the Hermite-polynomial
expansion of the perturbed distribution function with the same
wave number vector k as shown in the second and third termson the left-hand side of eq. (3). On the other hand, the E × B
convection, which is given by the last term on the left-handside, involves the distribution functions of only the same ordern but with different wave number vectors k″. The linear
source terms proportional to Ψ k on the right-hand side
disappear for n ≥ 3, which is the reason why the Hermitepolynomial expansion is employed here. A clear cutoff of thesource like this never occurs if we use the Fourier expansion
in terms of exp(ilv||) (–∞ < l < ∞) as basis functions.
We define the entropy variable δS by [4-8]

δS ≡

∑∫

| f˜k |2dv||k≡∑ ∑ δSk ,n ≡nk≡

2 FM∑ ∑ 2n! | fˆ |1∑ δS ,n∑Θ k (n + 1)! Im fˆ fˆ
y*k , n k , n +1 ,∑ ∫1
k y dv|| (v||2 − 1) Im Ψk f˜– k ,n = 2 ,2 kd(2Θ n | k y | δSn = −2ν n δSn for n ≥ 2.
ndt

(6)
(7)
(8)

(9)

In order to derive the functional form of δSn, we stillneed to specify the n-dependence of 〈|ky |〉n included in eq.
(9). For this purpose, we examine the role of E × B
convection term in eq. (3). Here, we point out the analogy ofour problem to the study by Batchelor on the spectrum of thepassive scalar for wave lengths smaller than the Kolmogorov
scale in the large Prandtl number case [9]. Like the turbulent
passive scalar in small scales, f̂k,n for large n is considered tovary so rapidly that E × B flow acting on f̂k,n is regarded as asteady one which is statistically independent of f̂k,n . Then, thestrain of the steady flow causes the exponential growth of thewave number of the convected variable, k ∝ eγ t [9]. Under
the phase-mixing process described by the second term in theleft-hand side of eq. (1), a factor in the form of exp(–iv||Θky /γ)
is produced in the velocity distribution function. In theHermite-polynomial expansion of this factor, components oforder n ~– (Θky /γ )2 are dominant. Thus, we have the relation,–
Θ|ky| ~– γ √n . Substituting this into eq. (9) and integrating itwith respect to n yield the entropy spectrum δSn,

2k ,nn,and δn,m = 1 (n = m), 0 (n ≠ m). Here, Qi denotes the turbulent
ion heat flux downward in the temperature gradient on theright-hand side of eq. (5), η iQ i represents the entropy
production supplied at n = 2 while –2νnδSn is the collisional
entropy dissipation term. We also find that Jn–1/2 (Jn+1/2)
represents the entropy transfer from the (n – 1)th (nth) to thenth [(n + 1)th] Hermite-polynomial portion.
Now, let us investigate the entropy spectrum δSn in thesteady turbulence, where the left-hand side of eq. (5) vanishes.
Considering small scales in the velocity space (n > 2), we
treat n as a continuous variable and use the approximation
Jn+1/2 – Jn–1/2 ~– dJn /dn to obtain dJn /dn = –2νnδSn. Assumingthat the n-dependence of the phase of f̂k,n is determined
mainly by the phase mixing of the ballistic modes, we can*~put f̂k,n–1/2 f̂k,n–1/2– i(ky /|ky |) | f̂k,n|2. Then, using eqs. (4) and (6),–
we have Jn /δSn ~– 2Θ √n 〈|ky|〉n, where the spectral average〈…〉n ≡ (ΣkδSk,n…)/(ΣkδSk,n) ≡ (Σk〈| f̂k,n|2〉…)/(Σk〈| f̂k,n|2〉) aredefined. Using the relations shown above, we obtain∞k ,nˆ*k , n −1 fk , nk

In order to investigate the velocity-space structure of thedistribution function, we expand f˜k as∑ f˜ H (v )F (v ),yk

3. Spectral analysisf˜k (v|| ) =∑Θ k n! Im fˆ

k(4)n1 n! 〈| fˆ |2〉 and δS ≡ Σ δS represent thewhere δSk,n ≡ —k,nnkk,n2
entropy spectral functions in the (k,n)-space and in the n-2272

Sugama H. and Watanabe T., Spectrum of the Velocity Distribution Function in the Slab Ion Temperature Gradient Driven TurbulenceδSn =⎛ νn ⎞σexp⎜ − ⎟ ,2γ n⎝ γ ⎠102(10)∞where σ ≡ 2ν ∫0 nδSndn ~– 2ν Σ n nδSn represents the collisional
entropy dissipation rate. In the steady state, the entropy
dissipation and production rates balance with other so that σ= ηiQi. We find from eq. (10) that, in the range where neither
entropy production nor collisional dissipation occurs (2 < n<< γ /ν), we expect the power-law of δSn ∝ 1/n with Jn = σ =const. which is analogous to the passive scalar spectrum andits power transfer in the viscous-convective subrange.
In the analytical treatment for derivation of eq. (10),–〈|ky |〉n ∝ √n increases infinitely with n. However, in numericalsimulations, there exists the upper limit of |k|. Even if thepotential amplitude is sufficiently damped at the maximum
wave number in the simulation, still fk,n for large |k| and large
n is continuously produced by the combination of the E × B
convection and the phase mixing process. Therefore,
saturation of 〈|ky |〉n with increasing n is anticipated due to theupper limit of |k|. In this case, taking Θ 〈|k y |〉 n = γ M as
independent of n and using eqs. (4) and (6), we obtain Jn /δSn–= 2 γM √n andδSn =σ3/ 2 ⎞⎛ 2 νnexp ⎜ −⎟,n
⎝ 3 γM ⎠〈|ky|〉n101

0.3√n

100
0.2√n

10−1100Fig. 1

104102100where σ is the same as given after eq. (10).
Now, let us compare the above analytical results withnumerical simulation results. In our simulation, we consider
the case with no zonal flow component (ky = 0) [2] and useparameters (ηi , Θ ) = (10, 2.5). In Fig. 1, the spectrumaveraged wave number 〈|ky |〉n is plotted as a function of nfor (kmax, ν) = (6.4, 2 × 10–3) and for (kmax, ν) = (12.8, 1.25 ×–10–4). We find that 〈|ky |〉n grows nearly in proportional to √nfor smaller values of n (> 2), which agrees with the estimate–of Θ 〈|ky |〉n ~– γ √ n used for derivation of eq. (10). Also,saturation of 〈|ky |〉n for larger n due to the upper limit kmax isseen in Fig. 1. The saturation starts at lower n for smaller
kmax.
From the simulation result with kmax = 6.4 (12.8) in Fig.
1, we obtain γ ~– 0.75 (0.5) and σ ~– 63 (36) for the spectral
function in eq. (10) while, for that in eq. (11), we put γ M =
7.5 (15). The entropy spectra δS n obtained by the same
simulation as in Fig. 1 are compared with those given by thecombination of eqs. (10) and (11) in Fig. 2, where we use
eq. (10) for n < 3 × 102 (103) and eq. (11) for n > 3 × 102(103) in the case of kmax = 6.4 (12.8). In the dissipation range
with large values of n, the spectrum found in the simulation
is well fitted by eq. (11). Since the constant factor in eq. (10)is determined by the constraint 2ν ∫ n δS ndn = σ with σevaluated from the simulation result, the spectrum δSn forlower n evaluated from eq. (10) becomes slightly smaller thanthat in the simulation in order to satisfy the constraint
because, for higher n, the latter spectrum is smaller than theformer due to the effect of finite kmax. Thus, the entropyspectrum observed by the slab ITG turbulence simulation canbe well explained by combining the analytical expressions in10−2

δSn

101

102
n103

104

Spectrum-averaged wave number 〈|ky |〉n for (kmax, ν) =(6.4, 2 × 10–3) and (kmax, ν) = (12.8, 1.25 × 10–4). Upperand lower dotted straight lines represent 〈|ky |〉n = 0.3––√n and 0.2√n , respectively.

(11)

2γ M

ν=1.25×10−4, kmax=12.8
ν=2×10−3, kmax=6.4

ν=1.25×10−4, kmax=12.8(Theory)ν=2×10−3, kmax=6.4(Theory)10−410−610−8100Fig. 2101102103104
Order of Hermitian n105

Entropy spectra δSn (solid and dashed lines) obtainedby the same simulation as in Fig. 1. For comparison,dotted and dot-dashed lines represent the combinationof eqs. (10) and (11). Here, eq. (10) is used for n < 3 ×102 (103) and eq. (11) for n > 3 × 102 (103) in the case ofkmax = 6.4 (12.8).

eqs. (10) and (11). If we can employ an sufficiently high valueof kmax, the simulation will reproduce the spectrum δSn ineq. (10) for the whole range of n > 2.

4. Conclusions
In this paper, the spectral analysis of the velocity
distribution function in the slab ITG turbulence is made byusing the Hermite-polynomial expansion with the Maxwellian
weight function. The entropy variable produced at n = 2 by
the turbulent ion heat flux downward in the temperature
gradient is transfered toward the high-n side by the phasemixing process combined with the turbulent E × B flow andis dissipated by collisions, where n denotes the order of the3273

Sugama H. and Watanabe T., Spectrum of the Velocity Distribution Function in the Slab Ion Temperature Gradient Driven Turbulence

References

Hermite-polynomial expansion. Finding analogy to theturbulent convection of the passive scalar for wave lengths
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and shown to well describe the numerical simulation resultson how δSn (n > 2) depends on n, ν (the collision frequency),and kmax (the maximum wave number used in the simulation).
In the subrange of the n-space where neither entropyproduction nor collisional dissipation occurs, we obtain thepower-law scaling δSn ∝ 1/n, which also resembles the formof the passive-scalar power spectrum ∝ 1/k in the viscousconvective wave-number subrange derived by Batchelor [9].

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Acknowledgments
This work is supported in part by the Japanese Ministryof Education, Culture, Sports, Science, and Technology, Grant
Nos. 12680497 and 14780387.

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