Nonlinear Zonal Dynamics of Drift and Drift-Alfvén Turbulences in Toama Plasmas Liu Chen, Zhihong Lin, Roscoe B. White and Fulvio Zonca Department of Physics and Astronomy, University of California, Irvine, CA 9717-4575 Princeton Plasma Physics Laboratory, P.O. Box 451, Princeton NJ 8543 ENEA C. R. Frascati, C.P. 65, 44 Frascati, Rome, Italy E-mail contact of main author: liuchen@uci.edu Abstract. The present wor addresses the issue of identifying the major nonlinear physics processes which may regulate drift and drift-alfvén turbulence using a wea turbulence approach. Within this framewor, based upon the nonlinear gyroinetic equation for both electrons and ions, we present an analytic theory for nonlinear zonal dynamics described in terms of two axisymmetric potentials, δφ z and δa z, which spatially depend only on a magnetic flux coordinate. Spontaneous excitation of zonal flows by electrostatic drift microinstabilities is demonstrated both analytically and by direct 3D gyroinetic simulations. Direct comparisons indicate good agreement between analytic expressions of the zonal flow growth rate and numerical simulation results for Ion Temperature Gradient ITG driven modes. Analogously, we show that zonal flows may be spontaneously excited by drift- Alfvén turbulence, in the form of modulational instability of the radial envelope of the mode as well, whereas, in general, excitations of zonal currents are possible but they have little feedbac on the turbulence itself. 1. Introduction In recent years, there has been increasing attention devoted to exploring nonlinear dynamics of zonal flow [1] associated with electrostatic drift-type turbulence [, 3, 4]. On the other hand, despite it being well nown how electrostatic drift modes couple to the electromagnetic shear Alfvén wave as the plasma β or R β increases [5, 6, 7], little effort has been devoted so far to investigating nonlinear zonal dynamics of drift-alfvén turbulence. The present wor addresses the issue of identifying the major nonlinear physics processes which may regulate drift and drift-alfvén turbulence using a wea turbulence approach. Within this framewor, based upon the nonlinear gyroinetic equation [8] for both electrons and ions, we present an analytic theory for nonlinear zonal dynamics described in terms of two axisymmetric potentials, δφ z and δa z, which spatially depend only on a magnetic flux coordinate. Physically, δφ z is associated with zonal flow formation, while δa z corresponds to zonal currents δj z = c/4π δa z. The introduction of a zonal vector potential, δa z, is one of the characteristic differences of the electromagnetic with respect to the electrostatic case. Zonal potentials are characterized by time variations on typical scales which are long compared to the characteristic ones of the drift-alfvén instabilities. This specific ordering of time scales, which formally requires proximity to the marginal stability such that the linear growth rate is smaller than the mode frequency, will be exploited for explicitly manipulating formal expressions in the theoretical analysis. In contrast to other approaches, however, which also assume slow radial variations of the zonal fields z 1 with respect to the typical spatial scale of the bacground turbulence r 1, we generally tae z r, although we still assume r z /z 1for consistency of our eional approach. In this respect our wor is the generalization of Ref. [9], which demonstrated that zonal flows can be spontaneously excited by electrostatic drift turbulence and that these are characterized by z r FIG. 1. In the present
wor, we show that zonal flows in toroidal equilibria can be spontaneously excited via modulations of the radial structure envelope of a single-n coherent drift-wave, with n the toroidal mode number. In this framewor, the turbulent state and the nonlinear couplings among different n s will manifest only via zonal dynamics. Similarly to Ref. [9], the present theory is strictly applicable to toroidal plasma equilibria, where poloidal asymmetry forces each mode to be at least in the linear limit the superposition of many poloidal harmonics m, characterized by the same n. In this respect, the present theoretical analysis is a systematic treatment of the radial mode structure envelope of zonal fields and drift turbulence in the general electromagnetic case, including slow time evolutions and accounting for linear toroidal and nonlinear mode couplings on the same footing. More specifically, we demonstrate that zonal flows δφ z are due to charge separation effects associated with both finite ion Larmor radius and finite ion orbit width effects magnetic curvature, whereas zonal currents δa z are due to parallel electron pressure imbalance cf. also Ref. [1]. Spontaneous excitation of zonal flows by electrostatic drift microinstabilities is demonstrated both analytically and by direct 3D gyroinetic simulations [9]. Direct comparisons indicate good agreement between analytic expressions of the zonal flow growth rate and numerical simulation results for ITG modes. Analogously, we show that zonal flows may be spontaneously excited by drift-alfvén turbulence, in the form of modulational instability of the radial envelope of the mode as well. From the analytic expression for the growth rate of the spontaneously excited zonal flows δφ z we show how no flow generation is expected for a pure shear Alfvén wave, due to the peculiar nature of the Alfvénic state. Meanwhile, we also demonstrate that in general zonal currents are also excited but they have negligible effect on the turbulence itself. The general results obtained within this theoretical model are also applied to Alfvénic oscillations; such as the Kinetic Alfvén Waves and the more recently discussed Alfvén ITG AITG [6] mode.. Theoretical Model Here, we strictly follow Ref. [9] and assume a low-β β =8π/B toroidal equilibrium with major radius R and minor radius a, with typically R /a =1/ɛ 1. For simplicity, we also tae the case of shifted circular magnetic flux surfaces. In this case, we can describe drift wave dynamics in terms of two scalar fields: the scalar potential δφ and the parallel vector potential δa fluctuations. For both fluctuating fields, as stated in the Introduction, we describe the nonlinear dynamic evolution in terms of a four-mode coupling scheme, i.e., each electromagnetic fluctuation is taen to be coherent and composed of a single n drift wave δφ d,δa d and a zonal perturbation δφ z,δa z ; e.g., for scalar potential fluctuations we tae δφ d = δφ + δφ + + δφ δφ =e i nθ dq+inϕ e imϑ φ nq m+c.c., m δφ = ei nθ dq e inϕ+i zdr e imϑ φ e i nθ dq nq m+c.c., m δφ z =e i zdr φ z + c.c., 1 where r, ϕ, ϑ are toroidal coordinates, and an analogue decomposition is assumed for fluctuating parallel vector potentials. Here, θ is the eional describing the radial structure of the drift wave radial envelope and q is the safety factor. Thus, Eq. 1 suggests that zonal fields may be
actually considered as radial modulations of the drift wave envelope, while the modes are simply upper and lower sidebands due to zonal fields modulations of the drift wave [9]. Furthermore, we have adopted the convention that, in the expressions involving the sidebands, the first row in a two component array will refer to the + while the second row will refer to the sideband. The same notation will be used throughout. We first derive nonlinear equations for zonal fields from the quasineutrality condition and parallel Ampère s law. Here, we just report the final results of such derivations in the small ion Larmor radius ρ Li limit: details will be given elsewhere. Contrary to the electrostatic limit, where the electron response to an n perturbation is adiabatic and, thus, only ions contribute to the nonlinear dynamics, electron nonlinearities are important in the general electromagnetic case. Assuming ρ Li 1, the nonlinear coupling coefficients are formally of the Hasegawa- Mima type and the quasineutrality condition reads: t χ iz δφ z = c B ϑ z zρ v A Li α Ψ +α IRe Φ Ψ Ψ +α Φ Ψ ] A A + A A. Here, we have introduced the notations χ iz 1.6q ɛ 1/ z ρ Li [11], α 1+δP i /neδφ, b δψ 1/c t δa, Φ indicates the symmetric Fourier Transform into ballooning space of φ in Eq. 1 and similar notations are used for the other scalar fields. Furthermore, we have omitted for simplicity collisional damping of δφ z [1],... stands for integration over ballooning space, Ψ θ Ψ /qr, θ is the angle-lie coordinate in ballooning space, and A and A indicate the amplitude of radial envelopes of the drift wave and sidebands at the current radial position. Similarly, from the parallel Ampère s law we obtain t δa z = c B ϑ z z δ e 1 α Ψ v A c ] +IRe c Φ Ψ Ψ A A + A A. 3 Here, the presence of δe = c /pe is a consequence of the strong shielding effect of parallel electron current on the electron collisionless sin depth. Furthermore, the forced response in δa z has been neglected since it is of order z / z / /1 α / v A with respect to the spontaneously excited component of Eq. 3. A direct comparison of Eqs. and 3 indicates that both zonal fields may be spontaneously excited except for a pure shear Alfvén wave, for which = v A, α =1and Φ =Ψ. In general, however, zonal flows can be efficiently excited via δφ z, whereas zonal currents or poloidal magnetic fields are strongly reduced because of electron shielding on scale lengths larger that δ e. We also note that, typically, c δa z zδ eδφ z δφ z, v A which will mae it possible to neglect the effect of δa z below. The drift wave nonlinear equations are the quasineutrality condition ne 1+ T i δφ = ej γδh i T i T e eδh e, 4
and the vorticity equation B l l δψ + [ 1 ni 1 Γ B va b i ] b i b iγ b i Γ 1 b i δφ = 4π e c d J δh + b t δa, cb δa e,i + 4π c t e c B b J γj γ J γ δl δh i. 5 In Eqs. 4 and 5, the subscript stands for or depending on whether the drift wave or its sidebands are considered, simple angular bracets... denote velocity space integration, γ v / ci, J is the Bessel function of zero order, l b, b i = ρ Li, Γ,1 b i I,1 b i exp b i, ni and are the ion diamagnetic frequencies associated with - respectively - density and temperature gradients, d is the magnetic drift frequency, = +, δl δφ v /cδa and the fluctuating particle distribution functions have been decomposed in adiabatic and nonadiabatic responses as δf = e m δφ v / F + exp i v b/ c δh. 6 The nonadiabatic response of the particle distribution function, δh, is obtained from the nonlinear gyroinetic equation [8]: t + v l +i d δh =i e m QF J γδl c B b J γ δl δh, QF = F v / + ˆb c F. 7 In Eq. 7, the linear response QF and the generalized E B nonlinearity in the guiding center moving frame δφ δφ v /cδa are readily recognized. Equations 4 and 5 are further simplified when we decompose the linear particle response to the fluctuating fields as [13]: δh LIN = e m J γ QF δψ + δk, 8 where the linearized gyroinetic equation for δk is readily derived from Eq. 7 and may be found in Ref. [13]. It is then readily shown that the quasineutrality condition, Eq. 4, can be cast into the form ne T i { 1+ T i T e [ δφ δψ + e,i i e c B b δφ δh e ej γδk = i 1 ni 1 Γ b i ]} b iγ b i Γ 1 b i e c B b J γj γ J γ δl δh i eδh NL e δψ, 9 where δh NL e indicates the nonlinear nonadiabatic electron response only, which vanishes in the electrostatic limit, as stated above.
Assuming, now, ρ Li 1, consistently with Eqs. and 3, and introducing the notation δk = δk φ δφ δψ+ δk ψ δψ, 1 Eqs. 5 and 9 for the sidebands in the ballooning space can be rewritten as: 1+ T i ej γ δk φ Φ Ψ A T e Φ e,i Ψ + 1 pi b i ej γ δk ψ Ψ A Ψ e,i = i [ c T i ϑ z δφ z 1+ ni T e A Ψ A Φ B T e T i A Ψ Ψ A Φ Ψ ], 11 { θ ϑ θ + A ϑ 4πq R ϑ c 3 4 b i [ 1 pi 3 4 b i 1 pi ] { e d J δk Ψ [ ψ A Ψ + 1 pi e,i A ϑ ] 4πq R e ϑc d J δk Φ Ψ φ A Φ e,i Ψ c ne q R A Φ B T ϑ z δφ z b i i A. 1 Φ 1 pi = 4πi ϑc Equations, 11 and 1, together with Eq. 7 are the basis for our analytic investigations described in the next section. 3. Some Applications In the electrostatic limit [9], Ψ, we obtain from Eq. 11 D S A = i c T i ϑ z δφ z Φ A B T e A, 13 where D S = 1+ T i ej γ δk φ Φ T e Φ e,i Φ Φ 1 14 and [9] D S i D Sr / i Γ z γ d, = z / D Sr / r/ D Sr / is the frequency mismatch, r = nq θ, Γ z = i z and γ d is the sideband damping [9]. Substituting Eq. 13 into Eq., we readily obtain a nonlinear dispersion relation for Γ z, which,in the γ d,γ M limit, reads Γ z = γ d /+ γ M + γ d/4 1/, 15 where γ M =α ɛ 1/ /1.6q T i /T e D Sr / 1 z ρ Li ϑ v thi ea Φ /T i. Including finite zonal flow collisional damping into Eq., ν z 1.5ɛτ ii 1 [1], would have produced
A B γ m /γ γ m /γ eφ /T. θ 1.45.9.58.87 1.16 FIG. 1. : From Ref. [9]. A Analytic prediction of zonal flow growth rate normalized to the linear one γ vs. mode amplitude solid line compared with gyroinetic simulation results. B Zonal flow growth rate vs. θ = z /nq, for fixed mode amplitude. a threshold condition γ M ν zγ d on the modulational instability growth rate Γ z of Eq. 15. In FIG. 1, Γ z as obtained from Eq. 15 is shown to be in good agreement with the results obtained by direct 3D gyroinetic simulations [9] of ITG modes, in which γ d 1.5γ, γ being the linear growth rate of the mode. Nonlinear equations for mode amplitudes have been recently derived [9] and they demonstrate saturation of the linearly unstable modes via coupling to the stable envelope sidebands and oscillatory behaviors in the drift-wave intensity and zonal flows [9]. For electromagnetic modes, and more specifically for Alfvénic-type waves, we typically have Φ Ψ Ψ in Eqs. 11 and 1. In fact, assuming q R 1 [6], we have from Eq. 11 Φ Ψ Φ Ψ A va/ b i Ψ T i /T e + ni / Ψ A i c B ϑ z A Ψ δφ z A Ψ, 16 where we recall that, in the present treatment, stands for an operator in the ballooning space. Substituting bac into Eq. 1, this yields [6] Ψ L M Ψ A =i A { L M = θ θ ϑ ] 4πq R ϑ c c B ϑ z + A e,i ϑ ϑ δφ z 1+ v A [ 1 pi e d J δk ψ A Ψ A Ψ 1 v A / b i T i /T e + ni / v A/ b i T i /T e + ni /, 17 3 4 b i e d J δk φ e,i Equation 17 can be cast into the form D M A =i c A B ϑ z δφ z 1+ K v A A A, Ψ Ψ D M Ψ L Ψ 1 M Ψ ϑ Ψ, 1 pi. 18 K Ψ Ψ Ψ ϑ Ψ ϑ Ψ Ψ 1. 19
From Eq. 19 and Eq., it is possible to derive the nonlinear dispersion relation for Γ z, similar to the electrostatic case. Specifically, using D M = D M+, we obtain: [ Γ z =ϑ ρ z vthi ɛ 1/ ea Ψ ] IIm D M+ 1+K va/ Li A 1.6q T i D M+ [ K α v A K va α IRe / K ] ρ Li, T i /T e + ni / K + K + Ψ + + Ψ Ψ 1. It is straightforward to further specialize Eq. to the case of KAW, for which D M = q R K + /A 1 K ρ Li 3/4+T e/t i. In this case α =1, and defining ˆγ M =ϑρ Liz vthi ɛ 1/ 3 1.6q 4 T e K T ρ eψ Li, i T i 3 ˆ = 4 + T e T zρ Li, 1 i we obtain Γ z,kaw ˆγ M 1 ˆ /ˆγ M. From Eqs. 1 and we see that zonal flows can be spontaneously excited by KAW s and that, as in the electrostatic case, the growth rate Γ z above threshold scales linearly with the wave amplitude. However, the most important feature of KAW s is that they spontaneously generate zonal flows in their propagating region for T e < 3/4T i and in their cut-off region for T e > 3/4T i. Another application of Eq. is to AITG modes [6]. In this case, sufficiently close to the unstable Alfvén continuum accumulation point, D M =Λ + iλδw f, where δw f is the MHD potential energy associated with the mode and Λ is a generalized inertia given by Λ = A 1 pi + q ti A [ 1 ni F / ti T i G/ ti N / ti D/ ti 3 and the functions, F x, Gx, Nx and Dx with x = / ti, ti = v thi /qr, and using the plasma dispersion function Zx, are defined as [6, 14] F x = x x +3/ + x 4 + x +1/ Zx, Gx = x x 4 + x + + x 6 + x 4 /+x +3/4 Zx, Nx = 1 ni [x + 1/ +x Zx ] [ T i x 1/+x + 1/4+x 4 Zx ], 1 Dx = 1+ x T e + 1 ni Zx [ T i x + x 1/ Zx ]. 4 T i With the new definitions γ M =ϑρ z v thi ɛ 1/ Li 1 pi IReΛ eψ, 1.6q T i = z δw f r A A IReΛ / / IReΛ, 5 ],
the zonal flow growth rate induced by AITG is: Γ z,ait G = γ M 1 / γ M. 6 As in the case of KAW, we find a condition for effective excitation of zonal flow by AITG, i.e., > pi, which is the typical case for slightly unstable AITG [14]. Above threshold, also AITG driven zonal flow growth rate scales linearly with the mode amplitude. 4. Conclusions In the present wor, we have demonstrated that zonal flows may be spontaneously generated by a variety of drift and drift-alfvén turbulences and, above their spontaneous excitation threshold, their growth rate typically scales linearly with the mode amplitudes. In the electrostatic limit, good agreement is shown between numerical results from 3D gyroinetic simulations of ITG and the obtained analytic expression [9]. In the same limit, nonlinear equations for mode amplitudes have been recently derived [9] and they demonstrate saturation of the linearly unstable modes via coupling to the stable envelope sidebands and, as a consequence, oscillatory behaviors in the drift-wave intensity and zonal flows [9]. Similar behaviors can also be expected in the general electromagnetic case, which will be analyzed in the near future. References [1] A. Hasegawa, C.G. Maclennan and Y. Kodama, Phys. Fluids, 1, 1979. [] A.M. Dimits et al., Phys. Rev. Lett. 77, 71, 1996. [3] Z. Lin, T.S. Hahm, W.W. Lee, W.M. Tang and R.B. White, Science 81, 1835, 1998. [4] M.A. Beer, Ph.D. dissertation, Princeton University, 1995. [5] J.Y. Kim, W. Horton and J.Q. Dong, Phys. Fluids B 5, 43, 1993. [6] F. Zonca, L. Chen, J.Q. Dong and R.A. Santoro, Phys. Plasmas 6, 1917, 1999. [7] B. Scott, Pl. Phys. Contr. Fus. 39, 1635, 1997. [8] E.A. Frieman and L. Chen, Phys. Fluids 5, 5, 198. [9] L. Chen, Z. Lin and R.B. White, Phys. Plasmas 7, 319,. [1] P.H. Diamond, Private Communication,. [11] M.N. Rosenbluth and F.L. Hinton, Phys. Rev. Lett. 8, 74, 1998. [1] F.L. Hinton and M.N. Rosenbluth, Pl. Phys. Contr. Fus. 41, A653, 1999. [13] L. Chen and A. Hasegawa, J. Geophys. Res. 96, 153, 1991. [14] J.Q. Dong, L. Chen and F. Zonca, Nucl. Fusion 39, 141, 1999.