Large scale instabilities in the electromagnetic drift wave turbulence and transport

Large scale structures play an important role in self-organization of drift wave turbulence. Large scale perturbations of plasma flow and magnetic field are spontaneously generated in generic electromagnetic drift wave turbulence via the action of Reynolds stress and electromotive force. Initial lar...

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Date:	2002
Main Authors:	Smolyakov, A.I., Diamond, P.H.
Format:	Article
Language:	English
Published:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2002
Series:	Вопросы атомной науки и техники
Subjects:	Basic plasma physics
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/80313
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Cite this:	Large scale instabilities in the electromagnetic drift wave turbulence and transport / A.I. Smolyakov, P.H. Diamond // Вопросы атомной науки и техники. — 2002. — № 4. — С. 96-99. — Бібліогр.: 23 назв. — англ

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spelling	nasplib_isofts_kiev_ua-123456789-803132025-02-09T14:02:01Z Large scale instabilities in the electromagnetic drift wave turbulence and transport Smolyakov, A.I. Diamond, P.H. Basic plasma physics Large scale structures play an important role in self-organization of drift wave turbulence. Large scale perturbations of plasma flow and magnetic field are spontaneously generated in generic electromagnetic drift wave turbulence via the action of Reynolds stress and electromotive force. Initial large scale perturbations are amplifed by positive feedback due to the modulations of wave packets by the shearing effect of the large scale flow and/or by the perturbed large scale magnetic field. As a result, the propagation of small scale wave packets is accompanied by the instability of a low frequency, long wavelength components. Anomalous transport due to drift wave turbulence may also be unstable with respect to the large scale perturbations of plasma profile. In this case the instability occurs as a result of a positive feedback response of the anomalous flux to the large scale variations of plasma temperature. 2002 Article Large scale instabilities in the electromagnetic drift wave turbulence and transport / A.I. Smolyakov, P.H. Diamond // Вопросы атомной науки и техники. — 2002. — № 4. — С. 96-99. — Бібліогр.: 23 назв. — англ 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/80313 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Basic plasma physics Basic plasma physics
spellingShingle	Basic plasma physics Basic plasma physics Smolyakov, A.I. Diamond, P.H. Large scale instabilities in the electromagnetic drift wave turbulence and transport Вопросы атомной науки и техники
description	Large scale structures play an important role in self-organization of drift wave turbulence. Large scale perturbations of plasma flow and magnetic field are spontaneously generated in generic electromagnetic drift wave turbulence via the action of Reynolds stress and electromotive force. Initial large scale perturbations are amplifed by positive feedback due to the modulations of wave packets by the shearing effect of the large scale flow and/or by the perturbed large scale magnetic field. As a result, the propagation of small scale wave packets is accompanied by the instability of a low frequency, long wavelength components. Anomalous transport due to drift wave turbulence may also be unstable with respect to the large scale perturbations of plasma profile. In this case the instability occurs as a result of a positive feedback response of the anomalous flux to the large scale variations of plasma temperature.
format	Article
author	Smolyakov, A.I. Diamond, P.H.
author_facet	Smolyakov, A.I. Diamond, P.H.
author_sort	Smolyakov, A.I.
title	Large scale instabilities in the electromagnetic drift wave turbulence and transport
title_short	Large scale instabilities in the electromagnetic drift wave turbulence and transport
title_full	Large scale instabilities in the electromagnetic drift wave turbulence and transport
title_fullStr	Large scale instabilities in the electromagnetic drift wave turbulence and transport
title_full_unstemmed	Large scale instabilities in the electromagnetic drift wave turbulence and transport
title_sort	large scale instabilities in the electromagnetic drift wave turbulence and transport
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2002
topic_facet	Basic plasma physics
url	https://nasplib.isofts.kiev.ua/handle/123456789/80313
citation_txt	Large scale instabilities in the electromagnetic drift wave turbulence and transport / A.I. Smolyakov, P.H. Diamond // Вопросы атомной науки и техники. — 2002. — № 4. — С. 96-99. — Бібліогр.: 23 назв. — англ
series	Вопросы атомной науки и техники
work_keys_str_mv	AT smolyakovai largescaleinstabilitiesintheelectromagneticdriftwaveturbulenceandtransport AT diamondph largescaleinstabilitiesintheelectromagneticdriftwaveturbulenceandtransport
first_indexed	2025-11-26T15:09:32Z
last_indexed	2025-11-26T15:09:32Z
_version_	1849866088380104704
fulltext	Large Scale Instabilities in the Electromagnetic Drift Wave Turbulence and Transport A.I. Smolyakov1,2, P.H. Diamond3 1Department of Physics and Engineering Physics, University of Saskatchewan, Saskatoon, Canada 2 Institute of Nuclear Fusion, Russian Research Center “Kurchatov Institute”, Moscow, Russia 3Depatment of Physics, University of California at San Diego, La Jolla, CA USA Large scale structures play an important role in self-organization of drift wave turbulence. Large scale perturbations of plasma flow and magnetic field are spontaneously generated in generic electromagnetic drift wave turbulence via the action of Reynolds stress and electromotive force. Initial large scale perturbations are amplifed by positive feedback due to the modulations of wave packets by the shearing effect of the large scale flow and/or by the perturbed large scale magnetic field. As a result, the propagation of small scale wave packets is accompanied by the instability of a low frequency, long wavelength components. Anomalous transport due to drift wave turbulence may also be unstable with respect to the large scale perturbations of plasma profile. In this case the instability occurs as a result of a positive feedback response of the anomalous flux to the large scale variations of plasma temperature. INTRODUCTION It has recently been realized that generation of the large scale shear flow (zonal flow) plays an important role in self-regulation of the drift wave turbulence [1- 3]. Spontaneously excited large scale flows occur as a result of the intrinsic non-ambipolarity of the ra- dial plasma flow, or, in other words, due to the ra- dial momentum flux [4,5]. The transfer of wave en- ergy towards the long wavelength region and the for- mation of large scale structures (zonal flows and con- vective cells) may be viewed as a result of the inverse cascade in two-dimensional and quasi two-dimensional geostrophic fluids [6]. The strongly sheared flow as- sociated with localized structures leads to turbulence suppression and enhancement of confinement in a toka- mak. Zonal flows are defined here as poloidal and toroidally symmetric (qz = qθ = 0) perturbations with a finite radial scale q−1r larger than the scale of the un- derlying small scale turbulence, qr kr, q is the wave vector for large scale motions, k is the wave-vector of small scale turbulence, and r, θ, and z are axis of a straight cylindrical tokamak. Opposite to zonal flows is another class of coher- ent large scale structures, streamers which are radially elongated formation with short poloidal wavelength, qθ >> qr. The latter structures may be an underlying mechanism for nonlocal, radially extended transport events or avalanches that have been recently identified in turbulent transport [7-10]. Both types of structures are nonlinearly generated in drift wave turbulence. In the electromagnetic case, generation of the large scale magnetic field is also possible [11-13] and can be considered as a fast dynamo process [14]. Magnetic structures (magnetic islands and magnetic streamers) have been studied as a possible mechanism of regula- tion and enhancement of the electron transport [15,16]. We consider the dynamics of small scale drift wave turbulence coupled with slow, large scale perturba- tions of the electrostatic potential, magnetic field, and plasma temperature. We demonstrate how the large scale coherent structures in plasma flow, magnetic field and plasma temperature may be spontaneously gen- erated. For simplicity we consider them separately, though in general case, the large scale magnetic field, shear flows and transport events may be coupled [17]. SHEAR FLOW INSTABILITY Instabilities of the shear flow can be considered by using a simple two-dimensional model for electron drift waves ∂ ∂t +V0 ·∇ eφ Te +V∗ ·∇eφ Te −ρ2s ∂ ∂t +V0 ·∇+VE ·∇ ∇2⊥ eφ Te = 0. (1) Here, ρ2s is the ion-sound Larmor radius and V∗ = θV∗ is the electron diamagnetic drift velocity. Nonlinear equation (1) is similar to the Hasegawa-Mima model except the term V0 · ∇(eφ/Te) which is retained be- cause the plasma density do not follow Boltzman dis- tribution for large scale modes (similar situation oc- curs in the sheared magnetic field for modes with k → 0 at the rational surface). Coupled dynamics of large scale flow and small scale turbulence are considered, so that the electrostatic potential φ is a sum of fluc- tuating φ and mean φ quantities, V0 = cb×∇φ/B0, VE = cb ×∇φ/B0. The mean potential is the aver- age of the total potential over fast, small scale vari- ables and depends only on slow variables X and T , φ = φ(X, T ). The evolution equation for the mean flow is ∂ ∂T ∇2⊥φ = − c B0 φ,∇2⊥φ , (2) which shows that the large scale flow is driven by small scale fluctuations via Reynolds stress forces. Here, {a, b} = ∂ra∂θb− ∂θa∂rb is the Poisson bracket. Coupling of small scale fluctuations to the mean flow is described by the kinetic equation for wave pack- ets ∂Nk ∂T + ∂ωk ∂k · ∂Nk ∂X − ∂ωk ∂X · ∂Nk ∂k = S, (3) where Nk = Nk(X, T ) is the adiabatic action invari- ant, and the exact form of Nk is model dependent. For the model given by Eq. (1), the wave frequency is ωk = kθV0 + ωlk, where ω l k = kθV∗/(1 + k2⊥ρ 2) is the local wave frequency, and the mean flow V0 enters the total frequency ωk as a simple Doppler shift. The source term in (3) describes the wave growth and damping due to linear and nonlinear mechanisms. We assume that small scale turbulence is close to a stationary state, so that S → 0. Coupled equations (2,3) can be solved to show that the modulations of the wave packets and zonal flow V0 are unstable [2]. We consider equations (2),(3) lin- earized for small perturbations (Nk,φ) ∼ exp(−iΩT + iqr),where q ≡ qr = −i∂/∂r is the radial wave vector of the large scale perturbation. Then, Eq. (2) takes the form −iΩφ = c B0 krkθ \|φk\|2 d2k. (4) The modulation of Nk is calculated from (3) Nk = − c B0 q2φkθ ∂N0 k ∂kr i Ω− qVg , (5) where Vg = ∂ω/∂kr. Using (5) in (4), we obtain the following equation [2] −iΩ = −q2c2s d2k k2θρ 2 s (1 + k2⊥ρ2s)2 kr ∂N0 k ∂kr i Ω− qVgr . (6) The resonant type instability is obtained from (5) by using the resonant function R = i/(Ω − qVg) → πδ(Ω−qVg) or its broadened counterpart i/(Ω−qVg+ ∆ωk) (for a white noise source, this can be taken as 1/∆ωk, where ∆ωk is nonlinear broadening due to the wave-wave interaction). For the case of the narrow res- onant function approximated by a delta-function, the growth rate of the resonant instability is γq = −q2c2s d2k k2θρ 2 s (1 + k2⊥ρ2s)2 kr ∂N0 k ∂kr πδ(Ω− qVg). (7) The condition ∂N0 k/∂kr < 0 is required for instability. When the growth rate of the instability becomes large compared to the characteristic frequency spread for the background fluctuations, individual Nk com- ponents contribute to the instability coherently. In- sight into this mechanism can be provided by a sim- ple case of a monochromatic wave packet with N0 k = N0δ(k − k0), with k0 = (kr0, kθ0). In this case after some transformations we obtain [18] (Ω− qVgr)2 = q2c2s k2θ N0 k 2kθV∗ ∂Vg ∂kr , (8) Note that the criterion for the instability is thus N0 k (2kθV∗) −1∂Vg/∂kr < 0. It can readily be seen that the coherent (“hydrodynamic”) instability has a larger growth rate compared to that of the resonant instabil- ity (7). Resonant growth of large scale perturbations can be described as negative viscosity instability [19]. In the “hydrodynamic” regime the instability if of the reactive type due to the interaction of negative and positive energy modes. ROBUST FAST DYNAMO IN ALFVEN TURBULENCE Modulational instability of small scale electromag- netic fluctuations may also lead to the generation of large scale magnetic structures in a turbulent magne- tized plasma [11-17]. The large scale magnetic field is driven by the mean electromotive force term in Ohm’s law, v×B. As an example we consider a collisionless Alfven wave turbulence in the presence of an ambient mag- netic field B0 = B0z. A spontaneous excitation of large scale magnetic fields B = ∇ψ × z, where ψ = ψ(X), is a result of coupling of small scale tur- bulence and the initial perturbation of the mean field. Large scale random magnetic field refracts wave pack- ets of the Alfven waves and, thus, modulates spec- trum of the turbulence. Modulated spectrum reacts back on the generated field via correlation between the perturbed small scale components of electrostatic and magnetic potentials. The latter provides electromotive force in the mean Ohm’s law. To consider the evolution of a large scale magnetic field we consider a simple case of the mean Ohm’s law with the main contribution v×B and neglecting other terms which can be important, such as contrubutions due to the electron pressure ∂ ∂t ψ = − c B0 z ·∇φ×∇ψ = c B0 ∂ ∂X Ry − c B0 ∂ ∂Y Rx , (9) where Ry = φkikyψ−k , and Rx = φkikxψ−k . It is obvious that a finite phase be- tween φ and ψ is required for the generation of the large scale magnetic field. At a linear stage such a phase shift occurs due to mode grow/damping associ- ated with dissipation effects. Our initial equations do not contain such dissipation/wave particle interaction effects. In two-fluid model dissipative effects will ap- pear in the momentum balance equation as electron viscosity and in the energy balance equation as a heat flux. However, quasineutrality equation (or density conservation equation) in general form remains correct even in the presence of the dissipation. For illustration purposes we consider Alfven waves neglecting all drift effects. Therefore from the quasinetrality equations we have φk = k c (ωk − iηk) ωk2 + η2k v2A c2 ψk, (10) where ωk is a real part of the wave frequency, and ηk is the imaginary part of the mode frequency. In the linear (transient) regime ηk is due to the mode growth rate/damping while in the nonlinear steady- state regime ηk is due to the nonlinear mode inter- action and negative. As an estimate, eddy turn-over time can be used for this parameter. More detailed estimates can be done using various strong turbulence closure schemes [20]. Then we have Ry = φkikyψ−k = k ηkky ωk2 + η2k \|ψk\|2 , (11) and similarly Rx can be calculated. The response of \|ψk\|2 to the variations of the mean vector potential ψ can be found by using the wave kinetic equation in the form ∂ ∂t Nk + ∂ ∂k ωk + δk δω δk · ∂ ∂x Nk − ∂ ∂x ωk + δk δω δk · ∂ ∂k Nk = 0. (12) General solution of this equation can be wrrit- ten as an integral along the characteristics Nk = Nk(x,k, t) = N0 k (k0,x0) where x0= x0(k,x, t) and k0 = k0(k,x, t) are inverse of the characteristics x = x(x0,k0, t) and k = k(x0,k0, t) defined by the equations dx dt = ∂ ∂k ωk + δk δω δk , (13) dk dt = − ∂ ∂x ωk + δk δω δk . (14) In the linear approximation,k = k0 + δk, dδk dt = − ∂ ∂x δk δω δk , (15) where δk = −B−10 z ·∇ψ × k, and δω/δk = k v2A/ω. Then in the linear approximation ∂2 ∂t2 ψ = c B0 ∂ ∂X Ay − ∂ ∂Y Ax , (16) where Ay = k kyηk ωk2 + η2k ∂ ∂t δNk, Ax = k kxηk ωk2 + η2k ∂ ∂t δNk. Perturbations of the wave action density can be found δNk = N 0 k (k−δk)−N0 k (k) −∂N 0 k ∂k · δk Then for ∂/∂X = 0 we have ∂2 ∂t2 ψ = − ∂3 ∂Y 3 ψ ηk ω2k + η2k k2xk 2 z ω ∂N0 k ∂ky . (17) Remarkable is that the instability occurs for arbi- trary signs of ∂N0 k/∂ky and q that can be interpreted as a robust alpha-effect. Further investigation is war- ranted here to explore whether this type of dynamo may by a subject of quenching due to the backreac- tion of large scale magnetic field [21]. In the above analysis we have implicitly assumed that the wave action invariant can be written Nk ∼ \|ψk\|2. In general, more exact expression is required that will change the final expression for the growth rate, though it does not affect the basic mechanism of the instability. LARGE SCALE TRANSPORT EVENT INSTABILITY Recently, detailed studies of the distrubution of transport events in turbulent plasmas have shown that relatively infrequent but large transport events may provide a dominant contrubution to the overall plasma transport [7,8]. Dynamical model of anomalous trans- port due to weakly excited long wavelength modes, “avalanches” or streamers, has been suggested in Ref. 22. Nonlinear instability leading to the formation of strongly anisotropic, poloidally localized streamers has been investigated in Refs. 7 and 9. Here we study the large scale instability of plasma temperature profile as a result of a positive feedback from background small scale fluctuations. As an example of the generation of large scale instabilities in the anomalous transport we consider a model transport equation (temperature) with a local diffusive operator ∂ ∂t T + ∂ ∂x D ∂ ∂x T = P. (18) We assume that a local diffusion coefficient can be written in the form D = D0 Nk, where Nk is suitably normalized dimensionless action invari- ant Nk ∼ e \|φk\|2 /T. We consider a stability of the diffusion equation (18) with respect to global per- turbations on a scale length comparable to the de- vice scale and much larger than the turbulence cor- relation length. Under this assumption the evolu- tion of the background turbulent field can be mod- eled by the wave kinetic equation (3). For sim- plicity we consider a simple one dimensional case. Then general solution of (3) can be written in the form of the integral over trajectories Nk(k, x, t) = N0 k (k0, x0) = N0 k (k0(k, x, t), x0(k, x, t), t) , where k0(k, x, t) and x0(k, x, t) are inverse functions to k = k(k0, x0, t) and x = x(k0, x0, t) which are solutions of the ray equations (subscript x is omitted below) To illustrate the instability we consider perturba- tions δT and δNk. Then we have ∂ ∂t δT + ∂ ∂x D0 δNk ∂T0 ∂x = 0. (19) One can show that the effect of the variation of the temperature under constant D can be neglected for typical parameters. The perturbation of the wave action can be found as δNk = N0 k (k− δk)−N0 k (k) = −δk∂N0 k/δk where is found from equations d dt δk = − ∂ ∂x δω = − ∂ ∂x ∂ω ∂T δT, (20) d dt = ∂ ∂t + ∂ω ∂k ∂ ∂x . Then in the Fourier form we have δk = q Ω− qVg ∂ω ∂T δT. (21) We obtain for the contribution of the individual wave vector to the growth rate Ω(Ω− qVg) = −q2D0 ∂ω ∂T T0 ∂N0 k ∂kx . (22) Instability occurs for ∂ω/∂T T0∂N 0 k/∂kx > 0. One can show that further evolution of the insta- bility within Eqs (18) and (3) leads to the formation of singularities [22]. SUMMARY It is shown here that large scale structures such as strongly sheared flow, magnetic islands and streamers, as well as localized transport events may develop as a result of secondary instabilities of the saturated drift wave turbulence. The instabilities develop due to a positive feedback on large scale formations from mod- ulations of wave packets. We examined instability for the basic drift waves as well as for the electromagnetic fluctuations where generation of large scale magnetic field is possible. We have also shown that anomalous transport may be unstable leading to development of global plasma profile perturbations that may be rele- vant to large scale transport events, or avalanches. Long wavelength structures, such as decsribed above, appear to be a crucial element of the self- organized drift wave turbulence due to a dominant role of nonlocal interactions [23]. The large scale modes are driven by the energy cascade from small scales, while the spectrum of small scale drift wave fluctuations is modified by the back reaction of large scales ( such as random shearing [2]). It results in the coupled nonlin- ear system for large and small scale components. Fixed points of this system define saturated states of the drift wave turbulence and respective transport coeffi- cients. Transitions between different fixed states may be related to sudden changes in transport (bursting phenomena) frequently observed in gyrokinetic simu- lations of tokamak transport [7]. REFERENCES [1] M.N. Rosenbluth and F.L. Hinton, Phys Rev. Lett. 80, 724 (1998). [2] P. H. Diamond, M. N. Rosenbluth, F. L. Hinton et al., 17th IAEA Fusion Energy Conference, Yokohama, Japan, 1998, IAEA-CN-69/TH3/1. [3] Z. Lin, T.S. Hahm, W.W. Lee, W.M. Tang and R.B. White, Science 281, 1835 (1998). T.S. Hahm, M.A. Beer, Z. Lin, et al., Phys. Plasmas 6, 922 (1999); L. Chen, Z. Lin, and R. White, Phys Plasmas 7, 3129 (2000). [4] P. H. Diamond and Y.-B. Kim, Phys. Fluids B 3, 1626 (1991) [5] W.M. Solomon and M.G. Shats, Phys.Rev. Lett. 87, 195003-1 (2001); M.G. Shats and W.M. Solomon, New J. Phys. 4, 30.1 (2002); V. Naulin, New J. Phys. 4, 28.1 (2002). [6] R.Z. Sagdeev, V.D. Shapiro, V.I. Shevchenko, Sov. J. Plasma Physics 4, 551 (1978); A. Hasegawa, M. Wakatani, Phys Rev. Lett. 59, 1581 (1987). [7] P.H. Diamond, S. Champeaux, M. Malkov, et al. Nuclear Fusion 41, 1067 (2001). [8] P. A. Politzer, Phys. Rev. Lett. 84, 1192 (2000); P. Beyer, S. Benkadda, X. Garbet, and P. H. Diamond, ibid. 85, 4892 (2000); J. F. Drake, P. N. Guzdar, and A. B. Hassam, ibid. 61, 2205 (1988). [9] S. Champeaux and P. H. Diamond, Phys. Letters A 288, 214 (2001). [10] R. Singh, P.K. Kaw, and J. Weiland, Nuclear Fu- sion 41, 1219 (2001). [11] A. Smolyakov, P.H. Diamond, I. Gruzinov, et al., in Theory of Fusion Plasmas, ed. by J. W. Connor, So- cieta Italiana di Fisica, Editrice Compositori Bologna, 199-211, 2000. [12] L. Chen, Z. Lin, R.B. White, and F. Zonca, Nu- clear Fusion 41, 747 (2001). [13] P.N. Guzdar, R.G. Kleva, A.Das, and P.K. Kaw, Phys. Rev. Lett. 87, 015001 (2001). [14] I. Gruzinov, A. Das, P.H. Diamond, and A.I. Smolyakov, Fast zonal field dynamo in collisionless ki- netic Alfven wave turbulence, to appear in Phys Lett. A (2002) [15] C. Holland and P. H. Diamond, Phys. Plasmas 9, 3857 (2002). [16] A. Smolyakov, P. Diamond, and Y. Kishimoto Phys. Plasmas 9, 3826 (2002). [17] V.P. Lakhin, Generation of zonal flows and large scale magnetic field in drift-Alfven turbulence, to ap- pear in Plasma Phys. Reports (2001). [18] A.I. Smolyakov, P.H. Diamond, and V. I. Shevchenko, Phys. Plasmas 7, 1349 (2000). [19] A.V. Chechkin, M.I. Kopp, V.V. Yanovsky, A.V. Tur. JETP 86, 357 (1998); A.V. Tur, A.V. Chechkin, V.V. Yanovsky Phys Fluids B 4, 3513 (1992). [20] V.P. Lakhin, Phys Lett. A 268, 149(2000); V.P. Lakhin, Plasma Phys. Reports 27, 777 (2001). [21] A. Gruzinov and P.H. Diamond, Phys. Rev. Lett. 72. 1651 (1994). [22] P. H. Diamond and T. S. Hahm, Phys. Plasmas 2, 3640 (1995); D. E. Newman, B. A. Carreras, P. H. Diamond, and T. S. Hahm, Phys. Plasmas 3, 1858 (1996); B. A. Carreras, D. Newman, V. E. Lynch, and P. H. Diamond, Phys. Plasmas 3, 2903 (1996). [23] S.V. Nazarenko, A.C. Newell, S. Galtier, Physica D 152-153, 646 (2201); A. M. Balk, S. V. Nazarenko, V.E. Zakharov, Sov. Phys. JETP 71, 249 (1990); D.Yu. Manin, S.V. Nazarenko, Phys Fluid 6, 1158 (1994).

Large scale instabilities in the electromagnetic drift wave turbulence and transport

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