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

Diamond, P.H.

Smolyakov, A.I.

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

Large Scale Instabilities in the Electromagnetic Drift WaveTurbulence and TransportA.I. Smolyakov1,2, P.H. Diamond31Department of Physics and Engineering Physics, University of Saskatchewan, Saskatoon, Canada2 Institute of Nuclear Fusion, Russian Research Center “Kurchatov Institute”, Moscow, Russia3Depatment of Physics, University of California at San Diego, La Jolla, CA USALarge scale structures play an important role in self-organization of drift wave turbulence. Large scaleperturbations of plasma flow and magnetic field are spontaneously generated in generic electromagnetic driftwave turbulence via the action of Reynolds stress and electromotive force. Initial large scale perturbations areamplifed by positive feedback due to the modulations of wave packets by the shearing eﬀect of the large scaleflow and/or by the perturbed large scale magnetic field. As a result, the propagation of small scale wave packetsis accompanied by the instability of a low frequency, long wavelength components. Anomalous transport dueto 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 largescale variations of plasma temperature.INTRODUCTIONIt has recently been realized that generation of thelarge scale shear flow (zonal flow) plays an importantrole in self-regulation of the drift wave turbulence [1-3]. Spontaneously excited large scale flows occur asa 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 inversecascade in two-dimensional and quasi two-dimensionalgeostrophic fluids [6]. The strongly sheared flow as-sociated with localized structures leads to turbulencesuppression and enhancement of confinement in a toka-mak. Zonal flows are defined here as poloidal andtoroidally symmetric (qz = qθ = 0) perturbations witha finite radial scale q−1r larger than the scale of the un-derlying small scale turbulence, qr  kr, q is the wavevector for large scale motions, k is the wave-vector ofsmall scale turbulence, and r, θ, and z are axis of astraight cylindrical tokamak.Opposite to zonal flows is another class of coher-ent large scale structures, streamers which are radiallyelongated formation with short poloidal wavelength,qθ >> qr. The latter structures may be an underlyingmechanism for nonlocal, radially extended transportevents or avalanches that have been recently identifiedin turbulent transport [7-10]. Both types of structuresare nonlinearly generated in drift wave turbulence.In the electromagnetic case, generation of the largescale magnetic field is also possible [11-13] and can beconsidered as a fast dynamo process [14]. Magneticstructures (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 waveturbulence coupled with slow, large scale perturba-tions of the electrostatic potential, magnetic field, andplasma temperature. We demonstrate how the largescale coherent structures in plasma flow, magnetic fieldand 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 INSTABILITYInstabilities of the shear flow can be considered byusing a simple two-dimensional model for electron driftwaves∂∂t+V0 ·∇ehφTe+V∗ ·∇ehφTe−ρ2s∂∂t+V0 ·∇+ hVE ·∇∇2⊥eφTe= 0. (1)Here, ρ2s is the ion-sound Larmor radius and V∗ = eθV∗is the electron diamagnetic drift velocity. Nonlinearequation (1) is similar to the Hasegawa-Mima modelexcept the term V0 · ∇(ehφ/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 kn→0 at the rational surface). Coupled dynamics of largescale flow and small scale turbulence are considered,so that the electrostatic potential φ is a sum of fluc-tuating hφ and mean φ quantities, V0 = cb×∇φ/B0,hVE = cb ×∇hφ/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 flowis∂∂T∇2⊥φ = −cB0qhφ,∇2⊥hφr, (2)which shows that the large scale flow is driven by smallscale fluctuations via Reynolds stress forces. Here,{a, b} = ∂ra∂θb− ∂θa∂rb is the Poisson bracket.Coupling of small scale fluctuations to the meanflow 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 isωk = kθV0 + ωlk, where ωlk = kθV∗/(1 + k2⊥ρ2) is thelocal wave frequency, and the mean flow V0 enters thetotal frequency ωk as a simple Doppler shift.The source term in (3) describes the wave growthand damping due to linear and nonlinear mechanisms.We assume that small scale turbulence is close to astationary state, so that S → 0.Coupled equations (2,3) can be solved to show thatthe modulations of the wave packets and zonal flowV0 are unstable [2]. We consider equations (2),(3) lin-earized for small perturbations ( hNk,φ) ∼ exp(−iΩT +iqr),where q ≡ qr = −i∂/∂r is the radial wave vectorof the large scale perturbation. Then, Eq. (2) takesthe form−iΩφ = cB0]krkθ |φk|2 d2k. (4)The modulation of hNk is calculated from (3)hNk = −cB0q2φkθ∂N0k∂kriΩ− qVg, (5)where Vg = ∂ω/∂kr. Using (5) in (4), we obtain thefollowing equation [2]−iΩ = −q2c2s]d2kk2θρ2s(1 + k2⊥ρ2s)2kr∂N0k∂kriΩ− 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 as1/∆ωk, where ∆ωk is nonlinear broadening due to thewave-wave interaction). For the case of the narrow res-onant function approximated by a delta-function, thegrowth rate of the resonant instability isγq = −q2c2s]d2kk2θρ2s(1 + k2⊥ρ2s)2kr∂N0k∂krπδ(Ω− qVg).(7)The condition ∂N0k/∂kr < 0 is required for instability.When the growth rate of the instability becomeslarge compared to the characteristic frequency spreadfor 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 N0k =N0δ(k − k0), with k0 = (kr0, kθ0). In this case aftersome transformations we obtain [18](Ω− qVgr)2 = q2c2s k2θN0k2kθV∗∂Vg∂kr, (8)Note that the criterion for the instability is thusN0k (2kθV∗)−1∂Vg/∂kr < 0. It can readily be seen thatthe coherent (“hydrodynamic”) instability has a largergrowth rate compared to that of the resonant instabil-ity (7). Resonant growth of large scale perturbationscan be described as negative viscosity instability [19].In the “hydrodynamic” regime the instability if of thereactive type due to the interaction of negative andpositive energy modes.ROBUST FAST DYNAMO IN ALFVENTURBULENCEModulational instability of small scale electromag-netic fluctuations may also lead to the generation oflarge scale magnetic structures in a turbulent magne-tized plasma [11-17]. The large scale magnetic field isdriven by the mean electromotive force term in Ohm’slaw, hv× hB.As an example we consider a collisionless Alfvenwave turbulence in the presence of an ambient mag-netic field B0 = B0ez. A spontaneous excitationof large scale magnetic fields B = ∇ψ × ez, whereψ = 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 reactsback on the generated field via correlation between theperturbed small scale components of electrostatic andmagnetic potentials. The latter provides electromotiveforce in the mean Ohm’s law.To consider the evolution of a large scale magneticfield we consider a simple case of the mean Ohm’s lawwith the main contribution hv× hB and neglecting otherterms which can be important, such as contrubutionsdue to the electron pressure∂∂tψ = − cB0ez ·∇eφ×∇eψ=cB0∂∂XRy −cB0∂∂YRx , (9)where Ry =Seφkikyeψ−k, and Rx =Seφkikxeψ−k. It is obvious that a finite phase be-tween φ and ψ is required for the generation of thelarge scale magnetic field. At a linear stage such aphase shift occurs due to mode grow/damping associ-ated with dissipation eﬀects. Our initial equations donot contain such dissipation/wave particle interactioneﬀects. In two-fluid model dissipative eﬀects will ap-pear in the momentum balance equation as electronviscosity and in the energy balance equation as a heatflux. However, quasineutrality equation (or densityconservation equation) in general form remains correcteven in the presence of the dissipation. For illustrationpurposes we consider Alfven waves neglecting all drifteﬀects. Therefore from the quasinetrality equationswe haveφk =knc (ωk − iηk)ωk2 + η2kv2Ac2ψk, (10)where ωk is a real part of the wave frequency, andηk is the imaginary part of the mode frequency. Inthe linear (transient) regime ηk is due to the modegrowth 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-overtime can be used for this parameter. More detailedestimates can be done using various strong turbulenceclosure schemes [20].Then we haveRy =[eφkikyeψ−k =[ knηkkyωk2 + η2k|ψk|2 , (11)and similarly Rx can be calculated. The response of|ψk|2 to the variations of the mean vector potential ψcan be found by using the wave kinetic equation in theform∂∂tNk +∂∂kωk + δknδωδkn· ∂∂xNk− ∂∂xωk + δknδωδkn· ∂∂kNk = 0. (12)General solution of this equation can be wrrit-ten as an integral along the characteristics Nk =Nk(x,k, t) = N0k (k0,x0) where x0= x0(k,x, t) andk0 = k0(k,x, t) are inverse of the characteristicsx = x(x0,k0, t) and k = k(x0,k0, t) defined by theequationsdxdt=∂∂kωk + δknδωδkn, (13)dkdt= − ∂∂xωk + δknδωδkn. (14)In the linear approximation,k = k0 + δk,dδkdt= − ∂∂xδknδωδkn, (15)where δkn = −B−10 ez ·∇ψ × k, and δω/δkn = knv2A/ω.Then in the linear approximation∂2∂t2ψ =cB0∂∂XAy −∂∂Y[Ax, (16)whereAy =[ knkyηkωk2 + η2k∂∂tδNk,Ax =knkxηkωk2 + η2k∂∂tδNk.Perturbations of the wave action density can be foundδNk = N0k (k−δk)−N0k (k) * −∂N0k∂k· δkThen for ∂/∂X = 0 we have∂2∂t2ψ = − ∂3∂Y 3ψ[ ηkω2k + η2kk2xk2zω∂N0k∂ky. (17)Remarkable is that the instability occurs for arbi-trary signs of ∂N0k/∂ky and q that can be interpretedas a robust alpha-eﬀect. Further investigation is war-ranted here to explore whether this type of dynamomay by a subject of quenching due to the backreac-tion of large scale magnetic field [21].In the above analysis we have implicitly assumedthat the wave action invariant can be written Nk ∼|ψk|2. In general, more exact expression is requiredthat will change the final expression for the growthrate, though it does not aﬀect the basic mechanism ofthe instability.LARGE SCALE TRANSPORT EVENTINSTABILITYRecently, detailed studies of the distrubution oftransport events in turbulent plasmas have shown thatrelatively infrequent but large transport events mayprovide a dominant contrubution to the overall plasmatransport [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 ofstrongly anisotropic, poloidally localized streamers hasbeen investigated in Refs. 7 and 9. Here we study thelarge scale instability of plasma temperature profile asa result of a positive feedback from background smallscale fluctuations. As an example of the generationof large scale instabilities in the anomalous transportwe consider a model transport equation (temperature)with a local diﬀusive operator∂∂tT +∂∂xD∂∂xT = P. (18)We assume that a local diﬀusion coeﬃcient canbe written in the form D = D0SNk, where Nkis suitably normalized dimensionless action invari-ant Nk ∼ e |φk|2 /T. We consider a stability of thediﬀusion 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 theform of the integral over trajectories Nk(k, x, t) =N0k (k0, x0) = N0k (k0(k, x, t), x0(k, x, t), t) , wherek0(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 ofthe ray equations (subscript x is omitted below)To illustrate the instability we consider perturba-tions δT and δNk. Then we have∂∂tδT +∂∂xD0[δNk∂T0∂x= 0. (19)One can show that the eﬀect of the variation of thetemperature under constant D can be neglected fortypical parameters.The perturbation of the wave action can be foundas δNk = N0k (k− δk)−N0k (k) = −δk∂N0k/δk where isfound from equationsddtδk = − ∂∂xδω = − ∂∂x∂ω∂TδT, (20)ddt=∂∂t+∂ω∂k∂∂x.Then in the Fourier form we haveδk =qΩ− qVg∂ω∂TδT. (21)We obtain for the contribution of the individual wavevector to the growth rateΩ(Ω− qVg) = −q2D0∂ω∂TT30∂N0k∂kx. (22)Instability occurs for ∂ω/∂T T30∂N0k/∂kx > 0.One can show that further evolution of the insta-bility within Eqs (18) and (3) leads to the formationof singularities [22].SUMMARYIt is shown here that large scale structures such asstrongly sheared flow, magnetic islands and streamers,as well as localized transport events may develop as aresult of secondary instabilities of the saturated driftwave turbulence. The instabilities develop due to apositive feedback on large scale formations from mod-ulations of wave packets. We examined instability forthe basic drift waves as well as for the electromagneticfluctuations where generation of large scale magneticfield is possible. We have also shown that anomaloustransport may be unstable leading to development ofglobal plasma profile perturbations that may be rele-vant to large scale transport events, or avalanches.Long wavelength structures, such as decsribedabove, appear to be a crucial element of the self-organized drift wave turbulence due to a dominant roleof nonlocal interactions [23]. The large scale modes aredriven by the energy cascade from small scales, whilethe spectrum of small scale drift wave fluctuations ismodified by the back reaction of large scales ( such asrandom shearing [2]). It results in the coupled nonlin-ear system for large and small scale components. Fixedpoints of this system define saturated states of thedrift wave turbulence and respective transport coeﬃ-cients. Transitions between diﬀerent fixed states maybe related to sudden changes in transport (burstingphenomena) 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 etal., 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 andR.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, andA. B. 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Large scale instabilities in the electromagnetic drift wave turbulence and transport

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Large scale instabilities in the electromagnetic drift wave turbulence and transport

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)