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PHYSICS OF PLASMAS

VOLUME 7, NUMBER 8

AUGUST 2000

LETTERS The purpose of this Letters section is to provide rapid dissemination of important new results in the fields regularly covered by Physics of Plasmas. Results of extended research should not be presented as a series of letters in place of comprehensive articles. Letters cannot exceed four printed pages in length, including space allowed for title, figures, tables, references and an abstract limited to about 100 words. There is a three-month time limit, from date of receipt to acceptance, for processing Letter manuscripts. Authors must also submit a brief statement justifying rapid publication in the Letters section.

Excitation of zonal flow by drift waves in toroidal plasmas Liu Chen Department of Physics and Astronomy, University of California, Irvine, California 92697

Zhihong Lin and Roscoe White Plasma Physics Laboratory, Princeton University, P.O. Box 451, Princeton, New Jersey 08543

共Received 16 March 2000; accepted 25 April 2000兲 An analytical dispersion relation is derived which shows that, in toroidal plasmas, zonal flows can be spontaneously excited via modulations in the radial envelope of a single-n coherent drift wave, with n the toroidal mode number. Predicted instability features are verified by three-dimensional global gyrokinetic simulations of the ion-temperature-gradient mode. Nonlinear equations for mode amplitudes demonstrate saturation of the linearly unstable pump wave and nonlinear oscillations of the drift-wave intensity and zonal flows, with a parameter-dependent period doubling route to chaos. © 2000 American Institute of Physics. 关S1070-664X共00兲01208-8兴

plasma with the usual radial (r), poloidal 共␪兲, and toroidal 共␾兲 coordinates. Here R and a are, respectively, the major and minor radii. Electrostatic fluctuations are taken to be coherent and composed of a single n (n⫽0) drift wave, ␦ ␾ d and a zonal flow mode ␦ ␾ z ; that is, ␦ ␾ d ⫽ ␾ 0 ⫹ ␦ ␾ ⫹ ⫹ ␦ ␾ ⫺ ⫹c.c.,

Recent three-dimensional 共3-D兲 gyrokinetic1,2 and gyrofluid3 simulations in toroidal plasmas have demonstrated that zonal flows4 play a crucial role in regulating the nonlinear evolution of electrostatic drift-wave instabilities such as the ion temperature gradient 共ITG兲 modes and, as a consequence, the level of the anomalous ion thermal transport. Zonal flows correspond to potentials which spatially depend only on the radius r and contain temporal variations with time scales longer than that of the drift waves. Recent gyrokinetic simulations5 have shown that zonal flows could be spontaneously excited by ITG turbulence held at constant level, suggesting parametric instability processes as the generation mechanism. Diamond et al.6 have proposed the modulational instability of drift-wave turbulence 共‘‘plasmons’’兲 in a slab-geometry treatment. Those authors also noted that unstable zonal flows can couple back to the drift waves and proposed a predator-prey model for the nonlinear self-regulation of the drift wave turbulence. In the present letter, we show, both analytically and by direct 3-D gyrokinetic simulations, that zonal flows can be readily excited via the modulational instability of a single-n coherent drift wave in toroidal plasmas, with n the toroidal mode number. We note that our theory is strictly applicable to toroidal geometry. Specifically, the drift wave, while having only a single n value, contains many poloidal harmonics 共m’s兲 which are toroidally coupled. Thus the modulation corresponds to that of the radial envelope describing the magnitude of each poloidal harmonic. In this respect the zonal flow can be regarded as the radial envelope mode. Consider a large aspect ratio ( ⑀ ⫽a/RⰆ1) tokamak 1070-664X/2000/7(8)/3129/4/$17.00

␾ 0 共 r,t 兲 ⫽e ⫺i(n ␾ ⫹ ␻ 0 t) 兺 ⌽ 0 共 m⫺nq 兲 e im ␪ ,

共1兲

␦ ␾ ⫾ ⫽e i(⫿n ␾ ⫺( ␻ z ⫾ ␻ 0 )t⫹K z r) 兺 ⌽ ⫾ 共 m⫺nq 兲 e im ␪ ,

共2兲

m

m

and ␦ ␾ z ⫽⌽ z e i(K z r⫺ ␻ z t) ⫹c.c.. Thus ␾ 0 is the pump drift wave and ␻ 0 its eigenmode frequency; ␦ ␾ ⫹ and ␦ ␾ ⫺ are, respectively, the upper and lower sidebands produced by the modulation in the radial envelope due to ␦ ␾ z at frequency ␻ z and radial wave number K z . We have assumed nⰇ1 and adopted the ballooning mode representation7 in which K z ⫽nq ⬘ ␪ 0 , q⫽rB ␾ /RB ␪ is the safety factor, and 0⭐ ␪ 0 ⭐ ␲ is the Bloch phase shift. The pump mode ␾ 0 has ␪ 0 ⫽0 共i.e., a flat radial envelope兲, which is, for a given n, usually the linearly most unstable mode. On the other hand, ␦ ␾ ⫹ and ␦ ␾ ⫺ have ␪ 0 ⫽0, giving radial envelope modulations. Typically they are linearly stable for moderate values of ␪ 0 . 3 We are thus dealing with a four-wave coupling process among ␾ 0 , ␦ ␾ ⫹ , ␦ ␾ ⫺ , and ␦ ␾ z . Three wave parametric excitation of zonal flows can be shown to be rather ineffective due to the frequency and wave number matching constraints. Since electrons are adiabatic for the n⫽0 drift waves, only ions contribute to the nonlinear physics. ␦ ⌽ z is then 3129

© 2000 American Institute of Physics

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coupled to ⌽ 0 and ␦ ␾ ⫾ , and the nonlinear coupling coefficient is formally of the Hasegawa–Mima type,8–10 i.e., 共 ⫺i ␻ z ⫹ ␯ z 兲 ␹ iz ⌽ z ⫽g

冓

兺m 关 a ⫹ ⌽ *0 ⌽ ⫹ ⫺a ⫺ ⌽ 0 ⌽ ⫺ 兴

冔

, 共3兲

2 2 2 where g⫽ (c/2B) ␣ i ␳ 2i k ␪ K z , a ⫹ ⫽k 0⬜ ⫺k ⫹⬜ , a ⫺ ⫽k 0⬜ 2 2 11 3/2 2 2 2 ⫺1 12 ⫺k ⫺⬜ , ␹ iz ⯝1.6⑀ K z ␳ i B ␾ /B ␪ , ␯ z ⫽(1.5⑀␶ ii ) , k0⬜ ⫽ ␪ˆ k ␪ ⫹inq ⬘ rˆ ⳵ ␨ , k ␪ ⫽nq/r 0 , r 0 refers to one reference mode surface, ␨ ⫽m⫺nq corresponds to the fast radial vari1/2 Ad ␨ is an averaging able, k⫾⬜ ⫽rˆ K z ⫾k0⬜ , and 具 A 典 ⫽ 兰 ⫺1/2 with respect to r 0 , ␣ i ⯝ ␦ P 0⬜ /(Ne⌽ 0 )⫹1 and ␦ P 0⬜ is the perturbed perpendicular pressure due to ⌽ 0 . The detailed expression for ␣ i depends on the specific drift wave mode and plasma parameters, e.g., ␣ i ⯝1⫹ ␶ ⫹ ␩ i ␶ , ␩ i ⫽d ln Ti /d ln N for the electron drift wave, and ␣ i ⯝ ␶ (1 ⫹ ␩ i )/ 关 (3 ␶ ⫺1)L n /R⫹1/2兴 ⫹1 for ITG in the fluid ion local approximation.3 Here L ⫺1 n ⫽d ln N/dr and ␶ ⫽T i /T e . In deriving Eq. 共3兲 we have assumed 兩 k⬜ ␳ i 兩 ⬍1 with ␳ i the ion gyroradius, 兩 ␻ z 兩 ⬍ ␻ GAM , and averaged over the geodesic acoustic mode.13 The nonlinear coupling of ␦ ␾ ⫾ to ⌽ 0 and ␦ ⌽ z can be straightforwardly calculated using the nonlinear gyrokinetic equation9,10 and the quasineutrality condition. We have, denoting ␻ ⫹ ⫽ ␻ z ⫹ ␻ 0 ,

冋

L⫹ ⌽ ⫹ ⫽ ␻ ⫹ 共 1⫹ ␶ 兲 ⌽ ⫹ ⫺ 共 T i /e 兲

冓冕

l d 3 vJ 0 ␦ G i⫹

冔册

⫽⫺i 共 c/B 兲 k ␪ K z ␶ ⌽ 0 共 ␨ 兲 ⌽ z ,

共4兲

l satisfies the linear gyrokinetic equation14,15 where ␦ G i⫹ l ⫽⫺h⌽ ⫹ 共 v 储 b"“⫺i ␻ ⫹ ⫹ik⫹⬜ "vd 兲 ␦ G i⫹

共5兲

with h⫽( ␻ ⫹ ⫺ ␻ i )F Mi J 0 e/T i and b⫽B0 /B 0 , vd is the * magnetic “B 0 and curvature drift, J 0 ⫽J 0 (k ⫹⬜ ␳ i ), ␻ * 2 ⫽ ␻ in 关 1⫹ ␩ i ( v /(2 v 2it )⫺3/2) 兴 , ␻ in ⫽k ␪ ␳ it v it /L n , and * * F Mi is the Maxwellian ion distribution with v it the ion thermal velocity. L⫹ ⫽L( ␻ ⫹ ,k⫹⬜ , ␨ ) and L is just the linear operator for the drift wave eigenmode. In particular, L0 ⌽ 0 ⫽L( ␻ 0 ,k0⬜ , ␨ )⌽ 0 ⫽0 with ␻ 0 the eigenmode frequency. In deriving the nonlinear response in Eq. 共4兲 we have assumed fluid ions. Since Eq. 共4兲 depends only on ␨ we can solve it by first Fourier transforming to the along-field-line ballooning ˆ 0 ( ␩ ) equal the coordinate ␩. Letting ⌽ ⫹ ⫽A ⫹ ⌽ 0 ( ␨ ) and ⌽ Fourier transform of ⌽ 0 ( ␨ ) we readily find A ⫹ ⫽⫺ick ␪ K z ␶ ⌽ z /BD ⫹ ,

共6兲

ˆ* ˆ ˆ ˆ 2 with where D ⫹ ( ␻ ⫹ ,k ␪ ,K z )⬅ 具具 ⌽ 0 L⫹ ⌽ 0 典典 / 具具 兩 ⌽ 0 兩 典典 , ⬁ ˆ ˆ 具具 A 典典 ⫽ 兰 ⫺⬁ d ␩ A, and L⫹ ⫽L( ␻ ⫹ , ␪ k ␪ ⫹rˆ (k ␪ sˆ ␩ ⫹K z ), ⫺i ⳵ ␩ ) is the corresponding linear drift wave operator in the ␩ coordinate, sˆ ⫽q ⬘ r/q the local shear. Letting ⍀ ⫹ ⫽ ␻ (k ␪ ,K z ) be the eigenfrequency for the upper side band and noting that 兩 ␻ z 兩 , 兩 ⍀ ⫹ ⫺ ␻ 0 兩 Ⰶ ␻ 0 , D ⫹ can then be approximated as D ⫹ ⯝( ⳵ D 0r / ⳵␻ 0 )( ␻ z ⫹⌬ ⫹i ␥ d ), where D 0r is the Hermitian part of D 0 , ⳵ D 0r / ⳵␻ 0 ⯝ ␶ , ⌬⫽ ␻ 0 ⫺⍀ ⫹r is the frequency mismatch, and ␥ d ⫽⫺⍀ ⫹i is the sideband damping rate.

Similar analysis can also be carrried out for the lower sideband. Thus we find ⌽ ⫺ ⫽A ⫺ ⌽ 0 ( ␨ ), A ⫺ ⫽ick ␪ K z ␶ ⌽ z /BD ⫺

共7兲

and D ⫺ ⫽( ⳵ D 0r / ⳵␻ 0 )( ␻ z ⫺⌬⫹i ␥ d ). Here ⍀ ⫹ ⫽⍀ ⫺ ⫽ ␻ (k ␪ ,⫺K z ) due to the up–down symmetry. Substituting ⌽ ⫾ ⫽A ⫾ ⌽ 0 into Eq. 共3兲 and noting that 具 兺 m 兩 ⌽ 0 兩 2 典 ⬁ ˆ 0 兩 2 典典 we finally obtain the desired lin⫽ 兰 ⫺⬁ 兩 ⌽ 0 兩 2 d ␨ ⫽ 具具 兩 ⌽ ear dispersion relation for the modulational instability 2 ⌫ z⫹ ␯ z⫽ ␥ M 共 ⌫ z⫹ ␥ d 兲/ 关 ⌬ 2⫹共 ⌫ z⫹ ␥ d 兲2兴 ,

共8兲

2 ␥M ⫽( ␣ i /1.6 ⑀ 3/2)

where we have let ⫺i ␻ z ⫽⌫ z and ˆ 0 /T e 兩 2 典典 . With appropriate ␣ i ⫻(B ␪ k ␪ c s K z ␳ s /B ␾ ) 2 具具 兩 e⌽ Eq. 共8兲 is valid for various branches of drift waves such as the electron drift wave or ITG. It can be solved for ⌫ z in two limits. In the 兩 ⌬ 兩 ⬍ ␥ d , ␥ M limit, we have 2 ⌫ z ⫽⫺ 共 ␥ d ⫹ ␯ z 兲 /2⫹ 关 ␥ M ⫹ 共 ␥ d ⫺ ␯ z 兲 2 /4兴 1/2.

共9兲

2 ⯝ ␯ z␥ d , Thus while the instability has a threshold at ␥ M strong growth with ⌫ z ⯝ ␥ M only sets in when ␥ Mⲏ ␥ d /2. On the other hand, for ␥ M , 兩 ⌬ 兩 ⬎ ␥ d , ␯ z we have 2 1/2 ⌫ z ⯝ ␥ M共 1⫺⌬ 2 / ␥ M 兲 .

共10兲

Again, strong growth with ⌫ z ⯝ ␥ M sets in when ␥ Mⲏ 兩 ⌬ 兩 . ˆ 0 /T e 兩 , the coherent moduNote that for ⌫ z ⯝ ␥ M⬀ 兩 K z ␳ s 兩兩 e⌽ lational instability is generally much stronger than that of the ˆ 0 /T e 兩 2 . Furdrift wave ‘‘plasmons’’ with growth rate ⬀ 兩 e⌽ 2 2 thermore, 兩 ⌬ 兩 ⯝K z ( ⳵ D 0r / ⳵ k 0⬜ )/2( ⳵ D 0r / ⳵␻ 0 )⯝ ␻ 0 K z2 ␳ s2 (1 ⫹ ␶ )/2, and ␥ d generally increases with K z ␳ s . ⌫ z , for a ˆ 0 兩 , can then be expected to first increase with K z ␳ s given 兩 ⌽ but eventually decrease at large K z ␳ s . While the exact (K z ␳ s ) m at which ⌫ z maximizes cannot be predicted, it in ˆ 0兩 . general will increase with the pump wave amplitude 兩 ⌽ Thus far away from linear marginal stability one expects ˆ 0 兩 , and modulational instastrong linear instability, larger 兩 ⌽ bilities peaked around Bloch phase shifts ␪ 0m ⬃O(1), i.e., (K z ␳ s ) m ⬃k ␪ ␳ s sˆ . That is, with strong linear drive the radial scale lengths of zonal flows and drift wave envelopes should be on the order of a typical distance between adjacant mode rational surfaces. The predicted modulational instability features have been observed in 3-D global gyrokinetic simulations of ITG modes using the gyrokinetic toroidal code.2 These nonlinear simulations keep only a single toroidal mode n⫽0 initially. The starting fluctuation level is very low to allow linear ITG eigenmode structure to be formed before nonlinear saturation. When the ITG mode grows to a desired amplitude, an external damping is applied so that the mode amplitude stays constant. Zonal flow with a single radial mode number is now self-consistently included. We observe exponential growth of zonal flow until it reaches a high level where the ITG mode is suppressed. The radial envelope modulation of the ITG mode correlates with the zonal flow radial structure. As shown in Fig. 1共A兲, the growth rate of zonal flow with a fixed radial mode number linearly depends on the ITG mode amplitude. These results are for amplitudes which are small compared to saturation levels. At larger amplitude ITG nonlinear effects appear. Analytical prediction of zonal flow

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Phys. Plasmas, Vol. 7, No. 8, August 2000

Excitation of zonal flow by drift waves in toroidal . . .

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FIG. 2. Values of Z k , ␦ ⫽2, ⌫ d ⫽2.

FIG. 1. Gyrokinetic simulation results of zonal flow growth rate 共A兲 vs ITG mode amplitude for fixed ␪ 0 , and 共B兲 vs ␪ 0 for fixed ITG amplitude, normalized to ITG growth, ␥ 0 . The line in 共A兲 is the solution of Eq. 共9兲.

growth rate from the solution of Eq. 共9兲 is shown by the solid line in Fig. 1共A兲. In the analytical calculation, the sideband damping rate is estimated from simulations to be ␥ d ⬃1.5␥ 0 including both intrinsic damping and externally applied damping, and ␥ 0 is the pump ITG intrinsic linear growth rate. For a fixed ITG mode amplitude, measured zonal flow growth rate increases linearly with radial mode number 共or ␪ 0 兲 for small ␪ 0 and decreases for large ␪ 0 , as shown in Fig. 1共B兲, consistent with theory. We now consider the nonlinear evolution of this modulation instability. As ␦ ␾ z and ␦ ␾ ⫾ exponentiate in amplitude, they will nonlinearly couple and induce damping in the pump wave amplitude. Replacing ␻ 0 by ␻ 0 ⫹i ⳵ t , letting ˆ 0 /T e 兩 2 典典 ⫽A 20 , and including the linear growth rate ␥ 0 具具 兩 e⌽ the equation for A 0 (t) becomes

冉

冊

d cT e ␶ k ␪ K z ⫺ ␥ 0 A 0 ⫽⫺ 共 A ⌽ ⫹A ⫹ ⌽ z* 兲 . dt eB ⳵ D 0r / ⳵␻ 0 ⫺ z

dS ⫽⫺⌫ d S⫹Z P cos共 ⌿ 兲 , d␶

共13兲

dZ ⫽⫺ ␥ z Z⫹2 PS cos共 ⌿ 兲 , d␶

共14兲

PZ d⌿ sin共 ⌿ 兲 , ⫽␦⫺ d␶ S

共15兲

with ⌫ d ⫽ ␥ d / ␥ 0 , ␥ z ⫽ ␯ z / ␥ 0 , and ␦ ⫽⌬/ ␥ 0 . These equations are similar to those for three-wave coupling,16 hence we anticipate similar behavior, such as the existence of a stable attractor and a period doubling route to chaos. Introduce an associated one-dimensional map by defining times ␶ k at successive zeros of dZ/d ␶ . A numerical plot of the values of Z k in steady state 共after transients have died away兲 is shown in Fig. 2 for ␦ ⫽2, ⌫ d ⫽2. Equations 共12兲–

共11兲

* , and ⌽ z are given by Equations governing A ⫹ , A ⫺ ⫽A ⫹ Eqs. 共3兲, 共4兲, and 共6兲 and noting that D ⫾ ⯝i ⳵ D 0r / ⳵␻ 0 (d/dt⫿i⌬⫹ ␥ d ). Using dimensionless time ␶ ⫽ ␥ 0 t and performing straightforward normalizations such that A 0 ⬀ P, A ⫹ ⬀Se i⌿(t) , and ⌽ z ⬀Z, we find dP ⫽ P⫺2ZS cos共 ⌿ 兲 , d␶

共12兲

FIG. 3. Numerical frequencies, linear values ␻ l , ␥ l , and damping ␯ near the fixed point, ␦ ⫽2, ⌫ d ⫽2.

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Present turbulence simulations have ⌫ d ⬃1, with values of ␥ z and ␦ placing them in the stable fixed point domain. The oscillations observed are thus probably nonlinear transient decay to the fixed point, with the decay time much longer than the simulation time. The drift wave intensity is I d ⫽ P 20 ⫹2S 20 . Assuming weak turbulence scaling of ␹ i ⬀I d , where ␹ i is the anomalous ion thermal transport coefficient, we find that in the stable domain ␹ i ⬀ ␥ z ⬀ ␯ ii consistent with the trend observed in simulations.5 In the future we will explore the route to chaos with nonlinear simulations and examine its implications for ␹ i . Finally, we note that, assuming nonlinear interactions among n⫽0 toroidal drift modes are ignorable, the present results, obtained for a single-n mode, can be readily generalized to a spectrum of multiple-n toroidal modes. FIG. 4. Destabilization of the fixed point.

共15兲 have a fixed point attractor for ␥ z ⱗ0.3. For 0.3ⱗ ␥ z ⱗ0.58 the attractor is a stable limit cycle with the bounding values of Z given by the two branches in Fig. 2. The initial bifurcation of the stable fixed point into the limit cycle corresponds also to period doubling, as can be seen in the plot of associated frequencies, Fig. 3. The two frequencies correspond to a rapid increase in Z to the upper value followed by a slow decay to the lower. Frequency, damping, and growth about the fixed point ␻ l , ␥ l , obtained in the following, are also shown. Apparent chaos sets in for ␥ z ⲏ0.75. The fixed point of Eqs. 共12兲–共15兲 is readily found to be Z 0 ⫽ 冑( ␦ 2 ⫹⌫ 2d )/(2⌫ d ), P 0 ⫽ 冑␥ z Z 0 , S 0 ⫽ P 0 / 冑2⌫ d , sin ⌿0 ⫽␦/冑␦ 2 ⫹⌫ 2d . Linearizing Eqs. 共12兲–共15兲 about the fixed point we find for the complex frequency

␻ 4 ⫺ic 3 ␻ 3 ⫹c 2 ␻ 2 ⫺ic 1 ␻ ⫹c 0 ⫽0,

共16兲

with c 0 ⫽4 ␥ z (⌫ 2d ⫹ ␦ 2 ), c 1 ⫽c 0 /⌫ d , c 2 ⫽ ␥ z ␦ 2 /⌫ d ⫺ ␥ z ⌫ d ⫺ ␦ 2 /⌫ d ⫹⌫ d ⫺⌫ 2d ⫺ ␦ 2 , c 3 ⫽1⫺2⌫ d ⫺ ␥ z . Destabilization of the fixed point is obtained by requiring that the frequency be real, giving c 1 c 2 c 3 ⫽c 21 ⫹c 0 c 23 with the frequency given by ␻ 2 ⫽⫺c 1 /c 3 . In Fig. 4 is shown the domain in which the stable fixed point exists. The boundaries of the stable domain at ␥ z ⫽0 are given by ⌫ d ⫽1/2 and ⌫ 3d ⫽ ␦ 2 (⌫ d ⫹1). For small ␥ z and ␻ the real frequency and the damping behave as ␻ ⯝A 冑␥ z , ␯ ⯝B ␥ z , with A ⫽2 冑(⌫ 2d ⫹ ␦ 2 )/(⌫ 2d ⫹ ␦ 2 ⫺⌫ d ⫹ ␦ 2 /⌫ d ) and B⫽2( ␦ 2 ⫺⌫ 3d ⫹ ␦ 4 /⌫ d ⫹ ␦ 4 /⌫ 2d )/(⌫ 2d ⫹ ␦ 2 ⫺⌫ d ⫹ ␦ 2 /⌫ d ) 2 . The ratio of damping to frequency is shown in Fig. 3.

ACKNOWLEDGMENTS

This work was supported by U.S. Department of Energy Grant No. DE-FG03-94ER54271 and under Contract No. DE-AC02-76-CHO3073. The authors acknowledge useful discussions with M. A. Beer, P. H. Diamond, T. S. Hahm, F. L. Hinton, and F. Zonca. 1

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