Dynamics of Fully Nonlinear Drift Wave-Zonal Flow Turbulence System in Plasmas
aa r X i v : . [ phy s i c s . p l a s m - ph ] O c t Dynamics of Fully Nonlinear Drift Wave-Zonal Flow TurbulenceSystem in Plasmas
P. K. Shukla ∗ Institut f¨ur Theoretische Physik IV,Fakult¨at f¨ur Physik und Astronomie,Ruhr-Universit¨at Bochum, D-44780 Bochum, Germany
Dastgeer Shaikh † Department of Physics and Center for Space Plasma and Aeronomic Research,The University of Alabama in Huntsville, Huntsville. Alabama, 35899 (Received 3 October 2009; Accepted 8 October 2009)
Abstract
We present numerical simulations of fully nonlinear drift wave-zonal flow (DW-ZF) turbulencesystems in a nonuniform magnetoplasma. In our model, the drift wave (DW) dynamics is pseudo-three-dimensional (pseudo-3D) and accounts for self-interactions among finite amplitude DWs andtheir coupling to the two-dimensional (2D) large amplitude zonal flows (ZFs). The dynamics ofthe 2D ZFs in the presence of the Reynolds stress of the pseudo-3D DWs is governed by thedriven Euler equation. Numerical simulations of the fully nonlinear coupled DW-ZF equationsreveal that shortscale DW turbulence leads to nonlinear saturated dipolar vortices, whereas theZF sets in spontaneously and is dominated by a monopolar vortex structure. The ZFs are found tosuppress the cross-field turbulent particle transport. The present results provide a better model forunderstanding the coexistence of short- and large-scale coherent structures, as well as associatedsubdued cross-field particle transport in magnetically confined fusion plasmas.
PACS numbers: 52.35.Kt,52.35.Mw,52.25.Fi,52.35.Ra ∗ Electronic address: [email protected] † Electronic address: [email protected]
1t is widely recognized that the presence of large scale sheared flows [1, 2, 3] [also referredto as convective cells (CCs) or zonal flows (ZFs)] is detrimental to regulating the cross-field turbulent transport in magnetically confined fusion plasmas.The ZF is characterized bypoloidally and toroidally symmetric structure with radial variation, and the relative zonalflow potential fluctuation (in comparison with T e /e , where T e is the electron temperatureand e is the magnitude of the electron charge) is much smaller than the relative zonal flowdensity perturbation (in comparison with the equilibrium plasma number density n ). Thelarge scale Zonal jets also occur in various planetary atmosphere, where they are nonlinearlygenerated by the Rossby waves [4, 5], and influence the atmospheric wind circulation [6, 7].In magnetically confined fusion plasmas, there exist free energy sources in the form ofdensity, temperature,and magnetic field inhomogeneities, which are responsible for excitingthe low-frequency (in comparison with the ion gyrofrequency), short scale (of the orderof the ion gyroradius or the ion sound gyroradius) DW-like fluctuations [8, 9, 10]. Thelinearly growing drift modes interact among themselves and attain large amplitudes in duecourse of time. The Reynolds stress of finite amplitude DWs, in turn, nonlinearly generateconvective cells (CCs) and sheared flows/ZFs [11, 12, 13, 14, 15, 16, 17], via three-wave decayand modulational instabilities [12], respectively. There are recent review articles presentingthe status of theoretical and simulation works [17], as well as experimental observations[18] concerning the dynamics of DW-ZF turbulence system. Specifically, some numericalsimulations [17] lend support to the experimental observation that the DW turbulence andtransport levels are reduced in the presence of the sheared flows/ZFs.Recently, Guo et al. [19] used the governing equations of Ref. [12] for the DW-CCturbulence system to investigate the radial spreading of the DW-ZF turbulence via solitonformation. However, the authors of Ref. [19] completely neglected self-interactions amongdrift waves and zonal flows, which are very important in the study of nonlinearly coupledfinite amplitude drift and zonal flow disturbances in nonuniform magnetoplasmas.In this Letter, we present simulation results of fully nonlinear DW-ZF turbulence systems,which exhibit the coexistence of drift dipolar vortices and a radially symmetric monopolarzonal flow vortex. The effect of the latter on the cross-field turbulent transport is examined.Our investigation is based on the governing equations for the DW-ZF turbulence systemsthat incorporate the Hasegawa-Mima (HM) self-interaction nonlinearity [20] in the nonlineardynamics of the DW which are nonlinearly exciting CCs/ZFs. Furthermore, we also account2or nonlinear interactions among the CCs/ZFs and obtain the driven Euler equation forthe dynamics of finite amplitude CCs/ZFs. The generalization of the governing equationsfor fully nonlinear DW-ZF turbulence systems is rather essential for the investigation ofthe formation of coherent nonlinear structures that control the transport properties andconfinement of tokamak plasmas.We consider a nonuniform magnetoplasma in an external magnetic field ˆ z B , where ˆ z isthe unit vector along the z − axis in a Cartesian coordinate system and B is the strength ofthe homogeneous magnetic field. The density gradient ∂n /∂x is along the x − axis. In thepresence of the finite amplitude low-frequency (in comparison with the ion gyrofrequency ω ci = eB /m i c , where m i is the ion mass and c is the speed of light in vacuum) electrostaticDWs and ZFs, the perpendicular (to ˆ z component of the electron and ion fluid velocities[13] are, respectively, u de ⊥ ≈ cB ˆ z × ∇ φ − cB n e ˆ z × ∇ ( n e T e ) ≡ u dEB + u dDe , (1) u ze ⊥ ≈ ( c/B )ˆ z × ∇ ψ ≡ u zEB , (2) u di ⊥ ≈ u dEB + u dDi − cB ω ci ∂∂t + ν in − . ν ii ρ i ∇ ⊥ + u dEB · ∇ + u dDi · ∇ ! ∇ ⊥ φ (3) − cB ω ci h ( u dEB · ∇ ) ∇ ⊥ ψ + ( u zEB · ∇ ) ∇ ⊥ φ i , and u zi ⊥ ≈ u zEB − cB ω ci " ∂∂t + ν in − . ν ii ρ i ∇ ⊥ ! ∇ ⊥ ψ + D ( u dEB · ∇ ) ∇ ⊥ φ E , (4)where the superscripts d and z represents quantities associated with the DWs and ZFs,respectively, φ and ψ are the electrostatic potentials of the DWs and ZFs, respectively, n e and n i are the electron and ion number densities, respectively, u dDi = ( c/eB n i )ˆ z × ∇ ( n i T i )is the ion diamagnetic drift velocity, T i is the ion temperature, ν in ( ν ii is the ion-neutral(ion-ion) collision frequency, ρ i = V T i /ω ci is the ion gyro-thermal radius, and V T i is theion thermal speed, We stress that the self-interaction nonlinearities of the DWs and ZFsare retained in the fluid velocities (3) and (4), respectively. The angular brackets denoteaveraging over one period of the DWs.Assuming that (cid:12)(cid:12)(cid:12) ( ∂/∂t ) + u dEB · ∇ (cid:12)(cid:12)(cid:12) ≪ ν en , where ν en is the electron-neutral collision fre-quency, we obtain from the parallel (to ˆ z component) of the electron momentum equation the3agnetic field-aligned electron fluid velocity u dez ≈ (1 /m e ν en ) ∂ (cid:16) eφ − T e n de /n (cid:17) /∂z , where n e = ( n e − n ) ≪ n . We can now insert u dez into the electron continuity equation to obtain " ∂∂t + V T e ν en ∂ ∂z + (cid:16) u dEB + u zEB (cid:17) · ∇ n de + u dEB · ∇ n + n em e ν en ∂ φ∂z = 0 , (5)where V T e = ( T e /m e ) / is the electron thermal speed and m e is the electron mass. Fur-thermore, substituting for the ion fluid velocity from (3) into the ion continuity equation wehave " ∂∂t + (cid:16) u dEB + u zEB (cid:17) · ∇ n di + u dEB · ∇ ( n + n zi ) (6) − n cB ω ci " ∂∂t + ν in − . ν ii ρ i ∇ ⊥ + u dEB · ∇ ! ∇ ⊥ φ + ∇ · ( u Did · ∇ ) ∇ ⊥ φ − n cB ω ci h ( u dEB · ∇ ) ∇ ⊥ ψ + ( u zEB · ∇ ) ∇ ⊥ φ i = 0 , where the magnetic field-aligned ion dynamics has been ignored, thereby isolating the ionsound waves from our system. The ion density perturbation associated with the ZFs is n zi = ( n c/B ω ci ) ∇ ⊥ ψ .Equations (5) and (6), which govern the dynamics of collisional drift waves [21] in thepresence of zonal flows, are closed by assuming n de ≈ n di ≡ n , which is a valid approxima-tion in plasmas with ω pi ≫ ω ci , where ω pi is the ion plasma frequency. In the linear limit,without the ZFs, Eqs. (5) and (6) yield the DW frequency ω k = − k y c s ρ s /L n (1 + k ⊥ ρ s ) andthe growth rate γ k ( ≪ ω k ), which are much larger than the damping rate ν in + 0 . ν ii k ⊥ ρ i .The growth rate is γ k = ν en ω k k ⊥ /ω LH k z (1 + k ⊥ ρ s ), where c s = ( T e /m i ) / is the ion soundspeed, ρ s = c s /ω ci is the sound gyroradius, ω LH = ( ω ce ω ci ) / is the lower- hybrid reso-nance frequency, ω ce = eB /m e c is the electron gyrofrequency, L n = ( ∂ ln n /∂x ) − is thescale-length of the density gradient, and k = k ⊥ + ˆ z k z is the wave vector.The equation for the ZFs is obtained by inserting (2) and (4) into the electron andion continuity equations, and inserting the resultant equations into the Poisson equation,obtaining the driven [by the DW Reynolds stress; the last term in the left-hand side of Eq.(7)] damped (by the ion-neutral collision and ion-gyroviscosity effects) ZF equation ∂∂t + ν in − . ν ii ρ i ∇ ⊥ + u zEB · ∇ ! ∇ ⊥ ψ + D ( u dEB · ∇ ) ∇ ⊥ φ E = 0 . (7)For the collisionless DWs, we assume that | ( ∂φ/∂t ) + ( u dEB + u zEB ) · ∇ φ | ≪ ( V T e /ν en ) ∇ ⊥ φ and ˆ z × ∇ n · ∇ φ ≪ ( ω ce /ν en ) n | ∂ φ/∂z | , and obtain from (5) the Boltzmann law for the4lectron number density perturbation n e = n eφ/T e . The latter can be inserted into (6)by assuming that n di = n de , so that we have fully nonlinear equation for the DWs in thepresence of ZFs ∂φ∂t − c s ρ s L n ∂φ∂y − ρ s " ∂∂t + ν in − . ν ii ρ i ∇ ⊥ + cB (1 + σ ) (ˆ z × ∇ φ ) · ∇ ∇ ⊥ φ (8)+ cB (ˆ z × ∇ ψ ) · ∇ (cid:16) φ − ρ s ∇ ⊥ φ (cid:17) = 0 , where σ = T i /T e .We normalize the time and space variables by ω ci − and ρ s , as well as φ and ψ by T e ,and the collision frequencies by ω ci . In the normalized units, we can rewrite (7) and (8) as,respectively, " ∂∂t + ν in ω ci − . ν ii ω ci σ ∇ ⊥ + (ˆ z × ∇ ψ ) · ∇ ∇ ⊥ ψ + D (ˆ z × ∇ φ · ∇ ) ∇ ⊥ φ E = 0 , (9)and ∂φ∂t − ρ s L n ∂φ∂y − " ∂∂t + ν in ω ci − . ν ii ω ci σ ∇ ⊥ + (1 + σ )(ˆ z × ∇ φ ) · ∇ ∇ ⊥ φ (10)+(ˆ z × ∇ ψ ) · ∇ (cid:16) φ − ∇ ⊥ φ (cid:17) = 0 . We have developed a 2D code to numerically integrate the system of equations (9) and(10), which describe the self-consistent evolution of the DW-ZF turbulence systems. We havechosen ν in /ω ci = 0 . ν ii /ω ci = 0 . σ = 0 .
1, and ρ s /L n = 0 .
01. Numerical descritizationemploys the spatial derivative in Fourier spectral space, while time is descritized using time-split integration algorithm, as prescribed in Ref. [22]. Periodic boundary conditions areused along the x and y directions. A fixed time integration step is used. The conservation ofenergy [23] is used to check the numerical accuracy and validity of our numerical code duringthe nonlinear evolution of the small scale drift wave fluctuations and zonal flows. We alsomake sure that the initial fluctuations are isotropic and do not influence any anisotropicflow during the evolution. Anisotropic flows in the evolution can, however, be generatedfrom a k y = 0 mode that is excited as a result of the nonlinear interactions between theZFs and small scale DW turbulence. The ZF and DW fields are initialized with a smallamplitude and uniform isotropic random spectral distribution of Fourier modes in a 2Dcomputational domain. These fields further evolve through Eqs. (9) and (10) under theinfluence of nonlinear interactions. Intrinsically, the set of Eqs. (9) and (10) possesses5 IG. 1: Evolution of mode structures in our coupled DW-ZF turbulence model from an initialrandom distribution. In the presence of self-interaction terms, zonal flows are enhanced and quenchthe DW turbulence more efficiently. Numerical resolution is 256 , box size is 2 π × π . parametrically unstable modes involving short scale drift waves and zonal flows. In theearly phase of simulations, we obtain the growth of small scale DWs. We have carried outtwo characteristically distinct sets of simulations by switching on and off the self-interactionterms. This enables us to gain considerable insight into the physics of generation of zonalflows and associated transport level in the coupled DW-ZF turbulence systems.To gain insight into the characteristics nonlinear interactions in our coupled DW-ZFturbulence model, we examine the Hasegawa-Mima-Wakatani (HMW) model [20, 21] thatdescribes the electrostatic drift waves in an inhomogeneous magnetoplasma. First, the ionpolarization drift nonlinearity in the HMW model, viz. ˆ z × ∇ φ · ∇∇ φ , signifies the self-interaction Reynolds stress that plays a critical role in the formation of the ZFs [12]. Thisnonlinearity is basically responsible for the generations of the ZFs. Secondly, since in thecollisionless DW dynamics, the electron density perturbation follows a Boltzmann law dueto the rapid thermalization of electrons along ˆ z , the nonlinearity ˆ z × ∇ ψ · ∇ φ comes from6
10 20 30 40 50−4−20246810 t ( Ω i ) l og Σ k y = | φ ( k x , k y ) | self−interactionNo self−interaction Evolution of Zonal Flow
FIG. 2: The self-consistent generation of zonal flows is shown. In the presence of the self-interactionnonlinearity, zonal flows are generated rapidly and their saturated level is also enhanced whencompared with the evolution without the self-interaction nonlinearity. the cross-coupling of the ZF’s E z ⊥ × ˆ z particle motion with the drift wave density fluctuationin our model. The role of this nonlinearity has traditionally been identified as a source ofsuppressing the intensity of the nonlinear flows in the DW turbulence [23]. Nevertheless, thepresence of the linear inhomogenous background can modify the nonlinear mode couplings ina subtle manner. Our objective here is to understand the latter in the context of the coupledDW-ZF turbulence system. The initially isotropic and homogeneous spectral distributionassociated with potential fluctuations, as described above, evolve dynamically following theset of Eqs. (9) and (10).The small amplitude initial drift wave fluctuations are subject to the modulational in-stability on account of their nonlinear coupling with ZFs. The parametrically unstablefluctuations grow rapidly during the early phase of the evolution. The instability eventuallysaturates via the nonlinear mode couplings in which the DW Reynolds stress, in concertwith other nonlinearities in Eqs. (9) and (10), play a critical role. The mode couplingsduring the nonlinear phase of the evolution leads to the formation of non-symmetric zonal7 Ω i ) D i ff u s i v i t y Self−interactionNo Self−interaction
FIG. 3: Evolution of the cross-field diffusivity in the presence (blue curve), as well as in the absence(red) of the self-interaction term. The cross-field diffusivity with and without zonal flows is shownby the dashed and solid curves, respectively. Clearly, the presence of the self-interaction termenhances zonal flows, which dramatically reduce the cross-field diffusivity. flow structures. This is shown in Fig. 1. Our simulations exhibit that the self-interactionterms not only suppress the modulational instability on a rapid timescales, but they alsoregulate the generation of the ZFs (see, Figs 1 b, c, e, f). The extent and amplitude ofthe ZFs in Figs. 1(b) and (e) [with the self-interaction] are larger than that of (c) and(f) [without the self-interaction]. The final (i.e. steady-state) structures, nonetheless, showthe formation of a predominantly dipolar vortex in the DW fluctuations, while the ZFs aredominated by a large-scale monopolar-vortex motions. It is noteworthy that the absence ofthe self-interaction contaminates the flows with more small scale structures (See Figs. 1 cand f).We next investigate the quantitative evolution of the ZFs, which is depicted in Fig.2. The spectral transfer of energy in the ZF is estimated from P k | φ ( k x , k y = 0) | . Thelatter describes a pile up of energy in the k y = 0 mode that is summed up over the entireturbulent spectrum. We find from our simulations that the presence of the self-interactionnonlinearity rapidly suppresses the linear phase of the modulational instability of the DWs.Hence, the linear phase during the evolution terminates rapidly when compared with theno-self-interaction case. Alternatively, the modulational growth rate is enhanced, and it8aturates on a rapid timescales. Consequently, the presence of the self-interactions gives riseto an enhanced level of ZFs, as shown in Fig. 2.A direct consequence of the enhanced ZFs is to markedly suppress the cross-field turbulenttransport, because the sheared flows (in the poloidal direction) associated with the ZFs tearapart the DW fluctuations/eddies, and thereby keeping their amplitudes low. We havecomputed the cross-field diffusivity in our simulations by using the ion fluxes involvingthe Boltzmann electron density perturbation and the linear ion polarization drift velocityassociated with the nonthermal DWs. The cross-field ion diffusivity reads D = D B X k k y ρ s | φ k | (1 + k ⊥ ρ s ) , (11)where D B = cT e /eB is the Bohm diffusion coefficient and φ k is the spectral potentialdistribution of the DWs. Note that the ion thermal diffusivity, as defined above, is subduedin the presence of the ZFs, since the latter eventually suppress the DW turbulence so thatthe steady state cross-field transport level is reduced. Consistent with this scenario, wefind, from our simulations, that the onset of the ZFs quenches the cross-field turbulenttransport, as shown by the dashed-curve in Fig. 3. Furthermore, due to the vanishingpoloidal wavenumber of the ZFs, the sheared flows do not cause any cross-field turbulenttransport in magnetized plasmas.In summary, the most notable point that emerges from our simulations of the coupledDW-ZF turbulence system is the importance and significance of the self-interaction nonlin-earity in modeling the low-frequency DW turbulence that is believed to be a critical sourcefor heat and energy losses. A most realistic and accurate understanding of the latter is,therefore, essential for the building and the performance of the next generation controlledthermonuclear fusion reactors, such as the ITER. In the work, we have, for the first time,brought about the importance of self-interaction processes and their role with regard to thecross-field turbulent transport in high-temperature plasmas of thermonuclear fusion devices(e.g. tokamaks). For our purposes, we used a new set of nonlinear equations for the coupledDW-ZF turbulence system, which is a generalization of Ref. 7, by including self-interactionsamong DWs which drives finite amplitude ZFs. Numerical simulations of the newly obtainednonlinear equations reveal that the coupled DW-ZF turbulence system evolves in the formof short-scale drift dipolar vortices and a large scale monopolar zonal flow structure. Thesimultaneous presence of the dipolar and monopolar vortices is responsible for a subdued9ross-field turbulent transport in a magnetically confined fusion plasma.This research was partially supported by the Deutsche Forschungsgemeinschaft throughthe project SH21/3-1 of the Research Unit 1048. [1] Z. Lin, T. S. Hahm, W. W. Lee, W. M. Tang, and R. B. White, Science , 1835 (1998).[2] A. Hasegawa, Adv. Phys. , 1 (1985).[3] W. Horton and A. Hasegawa, Chaos , 227 (1994).[4] R. Z. Sagdeev, V. D. Shapiro, and V. I. Shevchenko, Sov. Astron. Lett. , 279 (1981); A.Hasegawa, J. Phys. Soc. Jpn. , 1930 (1983); M. V. Nezlin, Sov. Phys. Usp. , 807 (1986).[5] P. K. Shukla and L. Stenflo, Phys. Lett. A , 84 (2003); A. M. Friedman, A. B. Mikhailovskii,and R. Z. Sagdeev, ibid. , 84 (2007); A. B. Mikhailovskii, et al. , ibid.
218 (2007).[6] J. N. Koshyk and K. Hamilton, J. Atmos. Sci. , 329 (2001).[7] C. A. Jones, J. Rotvig, and A. Abdulrahman, Geophys. Res. Lett. , 1731 (2003);doi: 10.1029/2003GL016980; P. L. Read et al. , Geophys. Res.Lett. , 122701 (2004);doi:10.1029/2004GL020106.[8] B. B. Kadomtsev, Plasma Turbulence (Academic, New York, 1965).[9] W. Horton, Rev. Mod. Phys. , 735 (1999).[10] J. Weiland, Collective Modes in Inhomogeneous Plasma: Kinetic and Advanced Fluid Theory (IOP Publishing, Bristol, 2000).[11] R. Z. Sagdeev, V. D. Shapiro, and V. I. Shevchenko, Zh. Eksp. Teor. Fiz. Pisma Red. , 361(1978) [JETP Lett. , 390 (1978); Fiz. Plazmy , 551 (1978) [Sov. J. Plasma Phys. , 306(1978)].[12] P. K. Shukla, M. Y. Yu, H. U. Rahman, and K. H. Spatschek, Phys. Rev. A , 321 (1981).[13] P. K. Shukla, M. Y. Yu, H. U. Rahman, and K. H. Spatschek, Phys. Rep. , 227 (1984).[14] A. I. Smolyakov, P. H. Diamond, and M. Malkov, Phys. Rev. Lett. , 491 (2000).[15] L. Chen, Z. Lin, and R. White, Phys. Plasmas , 3129 (2000).[16] P. K. Shukla, and L. Stenflo, Eur. Phys. J. D , 103, 2002.[17] P. H. Diamond et al. , Plasma Phys. Controlled Fusion , R35 (2005).[18] A. Fujisawa, Nucl. Fusion , 013001 (2009).[19] Z. Guo, L. Chen, and F. Zonca, Phys. Rev. Lett. , 055002 (2009).
20] A. Hasegawa and K. Mima, Phys. Rev. Lett. , 205 (1977); K. Mima and A. Hasegawa,Phys. Fluids , 81 (1978).[21] A. Hasegawa and M Wakatani, Phys. Rev. Lett. , 682 (1983); ibid. , 1581 (1987).[22] T. J. Chung, Computational Fluid Dynamics , Cambridge University Press, 2002.[23] D. Shaikh, R. Singh, H. Nordman, J. Weiland, A. Rogister, Phys. Rev. E , 036408 (2002)., 036408 (2002).