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Nonlinear excitation of coherent structures and associated cross-field transport in a magnetized plasma
Physics Letters A 345 (2005) 191–196 www.elsevier.com/locate/pla
Nonlinear excitation of coherent structures and associated cross-field transport in a magnetized plasma Dastgeer Shaikh a , P.K. Shukla b,∗ a Institute of Geophysics and Planetary Physics, University of California, Riverside, CA 92521, USA b Institut für Theoretische Physik IV, Ruhr-Universität Bochum, D-44780 Bochum, Germany
Received 5 May 2005; accepted 27 June 2005 Available online 5 July 2005 Communicated by V.M. Agranovich
Abstract Nonlinear interactions between zero-frequency electrostatic convective cells (CCs)/zonal flows (ZFs) and magnetostatic (MS) modes in a magnetoplasma are numerically investigated. These nonlinearly coupled flute-like modes can be spontaneously excited, and contribute significantly to the anomalous cross-field transport. While they are degenerate in a linear regime and form purely damped modes, their mutual nonlinear interactions give rise to large-scale coherent structures through a turbulent relaxation that leads to a self-organization via an inverse cascade. Remarkably, we find that nonlinear excitations of large-scale structures give rise to an effective cross-field diffusion coefficient that varies initially as ∼ t 1/2 with time, and enhances the transport level dramatically. 2005 Elsevier B.V. All rights reserved. PACS: 52.35.Mw; 52.35.Fp; 52.35.Ra
Large-scale structures in geophysical [1–4] and strongly magnetized plasmas [5,6] are believed to be of fundamental importance in determining turbulent transport of plasma fields and particles. It has now been a widely held view that the large scale structures, such as ZFs/CCs [7] and jets/streamers, excited [8–11] nonlinearly by the low-frequency Rossby/drift
* Corresponding author.
E-mail addresses: [email protected] (D. Shaikh), [email protected] (P.K. Shukla). 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.06.102
waves can play a crucial role in the regulation of fluid/plasma particle transport. Enormous research has been carried out (e.g., Refs. [3,12,13]) in advancing our understanding of the excitation, dynamical evolution and various nonlinear aspects of these large-scale structures that are based on the Charney–Hasegawa– Mima equation [1,5] for nonlinear Rossby/electron drift waves. Recent laboratory experiments [14–17] have also demonstrated the excitation of these structures in the form of ZFs in magnetically confined toroidal plasmas. Scaling arguments further suggest that transport of energy towards large scales occurs
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due to the inverse cascade insured by the conservation of energy and entrophy in two-dimensional (2D) fluid and plasma turbulence [5,6]. Moreover, understanding the rate of decay of turbulent energy and the formation of large-scale structures is of significant importance in atmospheric systems, in space and astrophysical environments, as well as in the heliosphere and in laboratory experiments. Since zero-frequency 2D CCs [7] and MS modes [18] play a very important role in nonthermal, nonequilibrium magnetoplasmas with regard to the plasma confinement [19], it is our prime goal to investigate their nonlinear dynamics when they are of large amplitudes and interact among themselves. Specifically, here we present computer simulation studies of nonlinearly interacting finite amplitude CCs and MS modes which are governed by a set of nonlinear differential equations. Our computer simulation results reveal that nonlinear interactions between nonthermal CCS and MS modes produce large scale eddies and magnetic filaments, which, in turn, cause anomalous cross-field particle transport in magnetized plasmas. We find that an effective diffusion coefficient Deff associated with the large-scale eddies varies as t 1/2 with time during an early phase of the turbulence. Consequently a rapid particle transport results initially and saturates eventually with time. Notably, the steady-state transport level in our simulations due to the generation of large-scale eddies predicts a reasonably higher value. This is further in agreement with particle-in-cell (PIC) simulations [19] that show that the electron transport due to thermal fluctuations is dominated by the zero-frequency CCs. Furthermore, the experimental demonstration of large scale coherent structures, exhibiting a bursty radial outward transport of the plasma particles across magnetic field lines [15–17], can be attributed to the inverse cascade phenomena described in this Letter. The 2D model describing nonlinear interactions between the CCs and MS modes can be cast in terms of two scalar field variables, viz. φ and Az that form a self-consistent nonlinear coupled set of equations ∂t + µi ∇ 2 + (c/B0 )ˆz × ∇φ · ∇ ∇ 2 φ − VA2 /cB0 zˆ × ∇Az · ∇∇ 2 Az = 0,
(1)
and 1 − λ2e ∇ 2 ∂t + (c/B0 )ˆz × ∇φ · ∇ Az = η∇ 2 Az , (2) when the plasma is cold, or ∂t + (c/B0 )ˆz × ∇ 1 − ρs2 ∇ 2 φ · ∇ Az = η∇ 2 Az , (3) when the plasma is warm [20]. We have denoted µi = 0.3νi ρi2 as the coefficient of the ion gyroviscosity and η = νe λ2e as the plasma resistivity, where νi (νe ) is the ion–ion (electron–ion) collision frequency, ρi is the ion thermal gyroradius, λe = c/ωpe is the electron skin depth, c is the speed of light in vacuum, and ωpe is the electron plasma frequency. The external magnetic field is zˆ B0 , where zˆ is the unit vector along the z axis and B0 is the strength of the magnetic field. For two-dimensional perturbations in the x–y plane, we have ∇ = (∂x , ∂y ). Furthermore, φ is the electrostatic potential, Az is the parallel (to zˆ ) component of the vector potential, ρs = Cs /ωci is the ion-sound gyroradius, Cs is the ion sound speed, ωci is the ion gyrofrequency, and VA is the Alfvén speed. Eq. (1) is the ion vorticity or charge continuity equation (obtained by subtracting the ion continuity equation from the electron continuity equation, using Poisson‘s equation as well as the electron and ion fluid velocities in the drift approximation, viz. |∂t | ωci , and the parallel component of Ampére’s law) in the presence of a nonlinear force arising from the divergence of the product of the parallel electron flow and the sheared magnetic field perturbation. Eq. (2) results from the parallel component of the electron equation of motion in which the electron inertia (the λ2e ∇ 2 -term) dominates the pressure gradient, while Eq. (3) is the parallel electron equation of motion neglecting the electron inertia (viz. λ2e ∇ 2 Az Az ), but retaining the magnetic field-aligned pressure gradient involving the density perturbation n˜ = n0 (c/B0 ωci )∇ 2 φ, which mimics the contribution of the ion polarization drift, where n0 is the unperturbed plasma number density. Furthermore, the ρs2 ∇ 2 term in Eq. (3) arises from the cross coupling of the electron diamagnetic drift and the perturbed magnetic field perturbation. This nonlinear Lorentz force plays a very important role in setting up the inertial spectrum at the ion sound gyroradius scale. The entire set of Eqs. (1)–(3) is valid for ω ωci ωpi , ρi2 ∇ 2 1, Ti < Te , and ρs2 ∇ 2 1, and thus have
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Fig. 1. A dynamical evolution of φ and Az leads to the formation of long-lived stable vortices in a warm plasma that continue to persist in the computational box in the steady state at tωci = 100. A similar, but relatively extended structures are also observed in a cold plasma. The x and y axes are measured in the units of ρs (λe ) for CCs/ZFs (MS) modes in all the simulations.
two distinct scalesizes (namely λe and ρs ) whose importance has been widely accepted in the context of collisionless magnetic reconnection [21], where ωpi is the ion plasma frequency. These equations govern the dynamics of nonthermal low-frequency (in comparison with ωci ) fluctuations in a quasi-neutral dense plasma with ωpi ωci . In the absence of nonlinear interactions, the CCs and MS/magnetic diffusion modes are decoupled. They are purely damped modes, having damping rates 0.3νi k 2 ρi2 , νe k 2 λ2e /(1 + k 2 λ2e ), and νe k 2 λ2e , respectively, where k 2 = kx2 + ky2 . It follows that long scale (k 2 ρi2 1) convective cells/zonal flows and long scale (k 2 λ2e 1) magnetostatic/magnetic diffusion modes would have long lifetimes. On the other hand, short scale (k 2 λ2e > 1) MS modes are short lived and their damping rate is strongly dispersive. Thus, the MS turbulent spectrum is split into two regions, determined by the size of the electron skin depth. However, nonlinear couplings between finite amplitude CCs and MS modes can drive them at nonthermal levels and also introduce interesting effects, as described below. The two-field equations (1) to (3) are integrated numerically, in both the cold and hot regimes, with the help of a fully dealiased pseudospectral scheme. Periodic boundary conditions are imposed along the x- and y-directions. For the cold (warm) plasma case, the space variables are in units of λe (ρs ). The electrostatic potential and the zˆ component of the magnetic vector potential, φ and Az , are discretized in a Fourier space using f (k, t) = k f (r, t) × exp(−ik · r). All fluctuations in our simulations are initialized with a Gaussian random number gener-
ator to ensure that the Fourier components are all spatially uncorrelated and randomly phased. This ensures that the choice of initial state is highly isotropic, i.e., kx ≈ ky at t = 0. Similarly, the boundary conditions (periodic in x, y directions) do not impose any kind of anisotropy. Moreover, the results to be presented here are independent of the size of computational domains, number of Fourier modes, as well as the integration time steps. The initially normalized energy spectrum, peaked at kmin , is chosen to lie within the wavenumber interval kmin < k < kmax /2. During the evolution of our simulations, the turbulence eventually decays through vortex-merging in which like-signed smaller length-scale fluctuations merge to form relatively large-scale fluctuations. The process continues until all merging has occurred to finally form the largest scale coherent vortex dominated by the minimum allowed k in the simulation. The inertial range turbulent cascades in such a manner leads to the formation of large scale structures. The final state, being a large-scale (comparable to our computational box) coherent vortex, is shown in Fig. 1 for a warm plasma case (see φ contours). A similar configuration for the magnetic vector potential (Az ) is also depicted in Fig. 1, when the turbulence has reached its saturated state. For our simulation studies, we choose some typical laboratory parameters: n0 = 3 × 1012 cm−3 , B0 ≈ 1 kG, Te = 10 eV, Ti = 1 eV, and argon plasma, so that the plasma beta ∼ 10−3 , which is larger than the electron to ion mass ratio for an argon plasma. The evolving relaxation of the turbulent fluid is independent of spatial and temporal resolutions, as well as higher turbulent Reynolds numbers. The latter slow
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Fig. 2. The formation of current sheets due to nonlinear interactions between the MS modes and CCs. The parameters are the same as in Fig. 1.
the rate of relaxation, while the qualitative physics remains more or less unaltered. Interestingly, we notice the formation of thin current sheets in the evolution of jz = ∇ 2 Az (see Fig. 2) that is governed typically by the MHD type nonlinearities in the vorticity Eq. (1). The formation of large scale structure through nonlinear interactions can be attributed to an inverse cascade phenomenon. In this process, small scale magnetostatic fluctuations, in an inertial range Fourier space, transfer their spectral energy primarily amongst neighboring Fourier modes. The energy cascades towards smaller scales in our simulations is terminated essentially by viscous damping. The latter is efficient at smaller turbulent scales and have smeared off all the small scale fluctuations in our simulations essentially due to higher order diffusion processes. By contrast, the largest allowed scale is determined typically by the Fourier mode kmin = 2π/L, where L = 10π in each direction. In a cold magnetoplasma, where density fluctuations are insignificant, we find almost similar evolution of large scale eddies (kλe 1) that are generated through energy transfer amongst smaller scale modes (kλe 1). The persistence of large-scale coherent structures, nonetheless, leads to a much steeper spectrum in a spectral space. This is illustrated in Fig. 3 for the warm plasma case, in which the total spectral energy for the random initial perturbation exhibits a nearly Kolmogorov-like flat spectrum (the solid-curve) in an early stage of turbulence. As turbulence relaxes and forms larger scale eddies, the energy migrates towards smaller k-eddies in the spectral space, and consequently leads to a steeper inertial range spectrum in the decaying turbulence (see
Fig. 3. The total energy spectrum grows steeper due to the persistence of large-scale structures (the dashed-curve). The solid curve corresponds to a nearly Kolmogorov-like spectrum during an early phase of the evolution. This demonstrates an inverse cascade of energy, which leads to the formation of coherent structures with kρs 1. The quasistationary spectra is averaged over 75 − 100tωci .
Fig. 4. The total energy associated with the CCs and MS modes decays rapidly during the initial phase of turbulent relaxation. In the process, strong nonlinear interactions lead to a rapid transfer of energy from small-scale to large-scale eddies through an inverse cascade mechanism. The nonlinear transfer of energy is finally terminated by the formation of a largest scale coherent structure in the steady state. The time is measured in a normalized unit as t → tωci , where ωci is ion gyrofrequency.
dashed-curve in Fig. 3). It is further noteworthy from Fig. 3 that the generation of large-scale eddies leads to a sharp cut-off at higher k-modes in the Fourier spectrum. The nonlinear equations (1) to (3), in the absence of dissipation, conserve the total energy and the mean squared magnetic potential, respectively, namely E = (1/2) (∇φ)2 dx dy and A = (1/2) × [Az + (∇Az )2 ] dx dy. In the presence of dissipation, the total energy associated with the magnetostatic modes and convective cells decays eventually with time, viz. ∂(E + A)/∂t = − [(µi /ωci )(∇ 2 φ)2 + η(∇Az )2 ] dx dy. This is shown in Fig. 4 as the tur-
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bulent fluids relax, thereby revealing that the energy decays more rapidly in the initial phase of the evolution. Furthermore, the decay rate (d(E + A)/dt) is very large during an initial phase in which a major part of the total energy in turbulence is dissipated. This is further consistent with the formation of large-scale coherent CCs, in which the energy associated with turbulence must decay in order to relax the turbulence into a well organized coherent structure. It is further instructive to estimate the test-particle transport coefficient associated with a self-consistent evolution of large-scale convective cells and magnetostatic modes. An effective electron diffusion coefficient caused by the latter can be calculated from ∞ Deff = 0 V⊥ (r, t) · V⊥ (r, t + t ) dt , where the angular bracket represents spatial averages. The perpendicular component of the electron fluid velocities in the CCs and MS modes are V⊥ = (c/B0 )ˆz × ∇φ and V⊥ = (v0 /B0 )B⊥ = (v0 /B0 )∇Az × zˆ , respectively, where v0 is some free-streaming electron speed along zˆ (typically 0.04Cs in our simulation). Since the 2D CCs and MS modes are confined in a plane orthogonal to zˆ , the effective cross-field diffusion coefficient, Deff , essentially relates the diffusion processes associated with the transverse motion of electrons in the electromagnetic fields of nonlinearly coupled convective cells and magnetostatic modes. We compute Deff in our simulations, from both the convective and the magnetostatic modes, to measure the turbulent transport that is associated with the coherent nonlinear structures we have reported herein. It is observed that the effective cross-field transport is lower, when the field perturbations are Gaussian. On the other hand, the cross-field diffusion increases rapidly with the eventual formation of longer length-scale structures. This is shown in Fig. 5, which exhibits a t 1/2 dependence of Deff with time. The transport due to the electrostatic CCs cells dominates substantially as depicted by the solid-curve in Fig. 5. Furthermore, in the steady-state, nonlinearly coupled CCs and MS modes form stationary structures, while Deff saturates eventually. Thus, remarkably an enhanced crossfield transport level results primarily due to the emergence of large-scale coherent structures in a magnetized plasma. The enhanced cross-field diffusion coefficient observed in our computer simulations is consistent with the generation of large-scale flows, and they are also in qualitative agreement with the PIC
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Fig. 5. An effective diffusion coefficient evolves initially as Deff ∼ t 1/2 with time (tωci ), thereby indicating that large-scale vortices enhance the transport level significantly. The transport coefficient saturates in the steady-state and predicts a higher value of Deff . The CCs (shown by solid-curve) lead to a larger transport than the MS modes (dashed-curve).
simulations [19] and observations from laboratory observations [15–17]. In summary, we have presented computer simulation studies of nonlinearly interacting 2D CCs and MS modes in a magnetoplasma. Our results show the formation of large-scale coherent structures on account of the energy exchange between CCs and MS modes that accompany enhanced vorticity and electron current filaments. Large scale eddies and filamentary structures, in turn, cause anomalous cross-field electron transport. In particular, large-scale CCs lead to a higher turbulent transport, consistent with the results from PIC simulations [19] and laboratory observations [15–17]. In conclusion, we stress that the results presented here should also be useful for understanding the origin of large-scale coherent structures and associated particle transport in magnetized plasmas, such as those in the Earth’s ionosphere and magnetosphere [22,23] as well as in the solar corona [24], in the solar wind [25], and in forthcoming laboratory experiments [26].
Acknowledgements Dastgeer Shaikh has been supported in part by a NASA grant NAG5-11621 and NAG5-10932 and NSF grant ATM0296113. This work was partially supported by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich 591.
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References [1] J.C. Charney, J. Atmos. Sci. 28 (1971) 1087. [2] G.P. Williams, J. Atmos. Sci. 35 (1978) 1399. [3] E.N. Snezhkin, M.V. Nezlin, Rossby Vortices, Sprial Structures, Solitons: Astrophysics and Plasma Physics in Shallow Water Experiments, Springer-Verlag, Berlin, 1994. [4] F.H. Busse, Chaos 4 (1994) 123; T. Soomre, Phys. Rev. Lett. 75 (1995) 2440. [5] A. Hasegawa, K. Mima, Phys. Rev. Lett. 39 (1977) 205; A. Hasegawa, K. Mima, Phys. Fluids 21 (1978) 87. [6] A. Hasegawa, Y. Kodama, Phys. Rev. Lett. 41 (1978) 1470; A. Hasegawa, et al., Phys. Fluids 22 (1979) 2122. [7] H. Okuda, J.M. Dawson, Phys. Fluids 16 (1973) 480. [8] C.Z. Cheng, H. Okuda, Phys. Rev. Lett. 38 (1977) 708; C.Z. Cheng, H. Okuda, Nucl. Fusion 18 (1978) 587. [9] R.Z. Sagdeev, V.D. Shapiro, V.I. Shevchenko, Sov. J. Plasma Phys. 4 (1978) 306. [10] P.K. Shukla, et al., Phys. Rev. A 23 (1981) 321. [11] D.R. McCarthy, et al., Phys. Rev. Lett. 93 (2004) 065004. [12] W. Horton, Rev. Mod. Phys. 71 (1999) 735. [13] A. Hasegawa, M. Wakatani, Phys. Fluids 25 (1983) 2770.
[14] A. Fujisawa, et al., Phys. Rev. Lett. 93 (2004) 165002. [15] F. Oynes, et al., Phys. Rev. Lett. 75 (1995) 81; Th. Piere, et al., Phys. Rev. Lett. 92 (2004) 065004. [16] A. Fredricksen, et al., Plasma Phys. Controlled Fusion 93 (2003) 215003. [17] M. Spolaore, et al., Phys. Rev. Lett. 93 (2004) 215003. [18] C. Chu, M. Chu, T. Ohkawa, Phys. Rev. Lett. 41 (1978) 653. [19] A.T. Lin, J.M. Dawson, H. Okuda, Phys. Rev. Lett. 41 (1978) 753; A.T. Lin, J.M. Dawson, H. Okuda, Phys. Fluids 23 (1980) 1316; H. Okuda, et al., Phys. Fluids 22 (1979) 1899. [20] P.K. Shukla, et al., Phys. Rep. 105 (1984) 227. [21] D. Grasso, et al., Phys. Rev. Lett. 86 (2001) 5051. [22] R.L. Lysak, J. Geophys. Res. 96 (1991) 1553. [23] K. Stasiewicz, et al., Space Sci. Rev. 92 (2000) 423. [24] L. Golub, J.M. Pasachoff, The Solar Corona, Cambridge Univ. Press, Cambridge, 1997. [25] O.P. Verkhoglyadova, et al., Nonlinear Proc. Geophys. 10 (2003) 335. [26] W. Gekelman, J. Geophys. Res. 104 (1999) 14417.
Nonlinear excitation of coherent structures and associated cross-field transport in a magnetized plasma Dastgeer Shaikh a , P.K. Shukla b,∗ a Institute of Geophysics and Planetary Physics, University of California, Riverside, CA 92521, USA b Institut für Theoretische Physik IV, Ruhr-Universität Bochum, D-44780 Bochum, Germany
Received 5 May 2005; accepted 27 June 2005 Available online 5 July 2005 Communicated by V.M. Agranovich
Abstract Nonlinear interactions between zero-frequency electrostatic convective cells (CCs)/zonal flows (ZFs) and magnetostatic (MS) modes in a magnetoplasma are numerically investigated. These nonlinearly coupled flute-like modes can be spontaneously excited, and contribute significantly to the anomalous cross-field transport. While they are degenerate in a linear regime and form purely damped modes, their mutual nonlinear interactions give rise to large-scale coherent structures through a turbulent relaxation that leads to a self-organization via an inverse cascade. Remarkably, we find that nonlinear excitations of large-scale structures give rise to an effective cross-field diffusion coefficient that varies initially as ∼ t 1/2 with time, and enhances the transport level dramatically. 2005 Elsevier B.V. All rights reserved. PACS: 52.35.Mw; 52.35.Fp; 52.35.Ra
Large-scale structures in geophysical [1–4] and strongly magnetized plasmas [5,6] are believed to be of fundamental importance in determining turbulent transport of plasma fields and particles. It has now been a widely held view that the large scale structures, such as ZFs/CCs [7] and jets/streamers, excited [8–11] nonlinearly by the low-frequency Rossby/drift
* Corresponding author.
E-mail addresses: [email protected] (D. Shaikh), [email protected] (P.K. Shukla). 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.06.102
waves can play a crucial role in the regulation of fluid/plasma particle transport. Enormous research has been carried out (e.g., Refs. [3,12,13]) in advancing our understanding of the excitation, dynamical evolution and various nonlinear aspects of these large-scale structures that are based on the Charney–Hasegawa– Mima equation [1,5] for nonlinear Rossby/electron drift waves. Recent laboratory experiments [14–17] have also demonstrated the excitation of these structures in the form of ZFs in magnetically confined toroidal plasmas. Scaling arguments further suggest that transport of energy towards large scales occurs
192
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due to the inverse cascade insured by the conservation of energy and entrophy in two-dimensional (2D) fluid and plasma turbulence [5,6]. Moreover, understanding the rate of decay of turbulent energy and the formation of large-scale structures is of significant importance in atmospheric systems, in space and astrophysical environments, as well as in the heliosphere and in laboratory experiments. Since zero-frequency 2D CCs [7] and MS modes [18] play a very important role in nonthermal, nonequilibrium magnetoplasmas with regard to the plasma confinement [19], it is our prime goal to investigate their nonlinear dynamics when they are of large amplitudes and interact among themselves. Specifically, here we present computer simulation studies of nonlinearly interacting finite amplitude CCs and MS modes which are governed by a set of nonlinear differential equations. Our computer simulation results reveal that nonlinear interactions between nonthermal CCS and MS modes produce large scale eddies and magnetic filaments, which, in turn, cause anomalous cross-field particle transport in magnetized plasmas. We find that an effective diffusion coefficient Deff associated with the large-scale eddies varies as t 1/2 with time during an early phase of the turbulence. Consequently a rapid particle transport results initially and saturates eventually with time. Notably, the steady-state transport level in our simulations due to the generation of large-scale eddies predicts a reasonably higher value. This is further in agreement with particle-in-cell (PIC) simulations [19] that show that the electron transport due to thermal fluctuations is dominated by the zero-frequency CCs. Furthermore, the experimental demonstration of large scale coherent structures, exhibiting a bursty radial outward transport of the plasma particles across magnetic field lines [15–17], can be attributed to the inverse cascade phenomena described in this Letter. The 2D model describing nonlinear interactions between the CCs and MS modes can be cast in terms of two scalar field variables, viz. φ and Az that form a self-consistent nonlinear coupled set of equations ∂t + µi ∇ 2 + (c/B0 )ˆz × ∇φ · ∇ ∇ 2 φ − VA2 /cB0 zˆ × ∇Az · ∇∇ 2 Az = 0,
(1)
and 1 − λ2e ∇ 2 ∂t + (c/B0 )ˆz × ∇φ · ∇ Az = η∇ 2 Az , (2) when the plasma is cold, or ∂t + (c/B0 )ˆz × ∇ 1 − ρs2 ∇ 2 φ · ∇ Az = η∇ 2 Az , (3) when the plasma is warm [20]. We have denoted µi = 0.3νi ρi2 as the coefficient of the ion gyroviscosity and η = νe λ2e as the plasma resistivity, where νi (νe ) is the ion–ion (electron–ion) collision frequency, ρi is the ion thermal gyroradius, λe = c/ωpe is the electron skin depth, c is the speed of light in vacuum, and ωpe is the electron plasma frequency. The external magnetic field is zˆ B0 , where zˆ is the unit vector along the z axis and B0 is the strength of the magnetic field. For two-dimensional perturbations in the x–y plane, we have ∇ = (∂x , ∂y ). Furthermore, φ is the electrostatic potential, Az is the parallel (to zˆ ) component of the vector potential, ρs = Cs /ωci is the ion-sound gyroradius, Cs is the ion sound speed, ωci is the ion gyrofrequency, and VA is the Alfvén speed. Eq. (1) is the ion vorticity or charge continuity equation (obtained by subtracting the ion continuity equation from the electron continuity equation, using Poisson‘s equation as well as the electron and ion fluid velocities in the drift approximation, viz. |∂t | ωci , and the parallel component of Ampére’s law) in the presence of a nonlinear force arising from the divergence of the product of the parallel electron flow and the sheared magnetic field perturbation. Eq. (2) results from the parallel component of the electron equation of motion in which the electron inertia (the λ2e ∇ 2 -term) dominates the pressure gradient, while Eq. (3) is the parallel electron equation of motion neglecting the electron inertia (viz. λ2e ∇ 2 Az Az ), but retaining the magnetic field-aligned pressure gradient involving the density perturbation n˜ = n0 (c/B0 ωci )∇ 2 φ, which mimics the contribution of the ion polarization drift, where n0 is the unperturbed plasma number density. Furthermore, the ρs2 ∇ 2 term in Eq. (3) arises from the cross coupling of the electron diamagnetic drift and the perturbed magnetic field perturbation. This nonlinear Lorentz force plays a very important role in setting up the inertial spectrum at the ion sound gyroradius scale. The entire set of Eqs. (1)–(3) is valid for ω ωci ωpi , ρi2 ∇ 2 1, Ti < Te , and ρs2 ∇ 2 1, and thus have
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Fig. 1. A dynamical evolution of φ and Az leads to the formation of long-lived stable vortices in a warm plasma that continue to persist in the computational box in the steady state at tωci = 100. A similar, but relatively extended structures are also observed in a cold plasma. The x and y axes are measured in the units of ρs (λe ) for CCs/ZFs (MS) modes in all the simulations.
two distinct scalesizes (namely λe and ρs ) whose importance has been widely accepted in the context of collisionless magnetic reconnection [21], where ωpi is the ion plasma frequency. These equations govern the dynamics of nonthermal low-frequency (in comparison with ωci ) fluctuations in a quasi-neutral dense plasma with ωpi ωci . In the absence of nonlinear interactions, the CCs and MS/magnetic diffusion modes are decoupled. They are purely damped modes, having damping rates 0.3νi k 2 ρi2 , νe k 2 λ2e /(1 + k 2 λ2e ), and νe k 2 λ2e , respectively, where k 2 = kx2 + ky2 . It follows that long scale (k 2 ρi2 1) convective cells/zonal flows and long scale (k 2 λ2e 1) magnetostatic/magnetic diffusion modes would have long lifetimes. On the other hand, short scale (k 2 λ2e > 1) MS modes are short lived and their damping rate is strongly dispersive. Thus, the MS turbulent spectrum is split into two regions, determined by the size of the electron skin depth. However, nonlinear couplings between finite amplitude CCs and MS modes can drive them at nonthermal levels and also introduce interesting effects, as described below. The two-field equations (1) to (3) are integrated numerically, in both the cold and hot regimes, with the help of a fully dealiased pseudospectral scheme. Periodic boundary conditions are imposed along the x- and y-directions. For the cold (warm) plasma case, the space variables are in units of λe (ρs ). The electrostatic potential and the zˆ component of the magnetic vector potential, φ and Az , are discretized in a Fourier space using f (k, t) = k f (r, t) × exp(−ik · r). All fluctuations in our simulations are initialized with a Gaussian random number gener-
ator to ensure that the Fourier components are all spatially uncorrelated and randomly phased. This ensures that the choice of initial state is highly isotropic, i.e., kx ≈ ky at t = 0. Similarly, the boundary conditions (periodic in x, y directions) do not impose any kind of anisotropy. Moreover, the results to be presented here are independent of the size of computational domains, number of Fourier modes, as well as the integration time steps. The initially normalized energy spectrum, peaked at kmin , is chosen to lie within the wavenumber interval kmin < k < kmax /2. During the evolution of our simulations, the turbulence eventually decays through vortex-merging in which like-signed smaller length-scale fluctuations merge to form relatively large-scale fluctuations. The process continues until all merging has occurred to finally form the largest scale coherent vortex dominated by the minimum allowed k in the simulation. The inertial range turbulent cascades in such a manner leads to the formation of large scale structures. The final state, being a large-scale (comparable to our computational box) coherent vortex, is shown in Fig. 1 for a warm plasma case (see φ contours). A similar configuration for the magnetic vector potential (Az ) is also depicted in Fig. 1, when the turbulence has reached its saturated state. For our simulation studies, we choose some typical laboratory parameters: n0 = 3 × 1012 cm−3 , B0 ≈ 1 kG, Te = 10 eV, Ti = 1 eV, and argon plasma, so that the plasma beta ∼ 10−3 , which is larger than the electron to ion mass ratio for an argon plasma. The evolving relaxation of the turbulent fluid is independent of spatial and temporal resolutions, as well as higher turbulent Reynolds numbers. The latter slow
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Fig. 2. The formation of current sheets due to nonlinear interactions between the MS modes and CCs. The parameters are the same as in Fig. 1.
the rate of relaxation, while the qualitative physics remains more or less unaltered. Interestingly, we notice the formation of thin current sheets in the evolution of jz = ∇ 2 Az (see Fig. 2) that is governed typically by the MHD type nonlinearities in the vorticity Eq. (1). The formation of large scale structure through nonlinear interactions can be attributed to an inverse cascade phenomenon. In this process, small scale magnetostatic fluctuations, in an inertial range Fourier space, transfer their spectral energy primarily amongst neighboring Fourier modes. The energy cascades towards smaller scales in our simulations is terminated essentially by viscous damping. The latter is efficient at smaller turbulent scales and have smeared off all the small scale fluctuations in our simulations essentially due to higher order diffusion processes. By contrast, the largest allowed scale is determined typically by the Fourier mode kmin = 2π/L, where L = 10π in each direction. In a cold magnetoplasma, where density fluctuations are insignificant, we find almost similar evolution of large scale eddies (kλe 1) that are generated through energy transfer amongst smaller scale modes (kλe 1). The persistence of large-scale coherent structures, nonetheless, leads to a much steeper spectrum in a spectral space. This is illustrated in Fig. 3 for the warm plasma case, in which the total spectral energy for the random initial perturbation exhibits a nearly Kolmogorov-like flat spectrum (the solid-curve) in an early stage of turbulence. As turbulence relaxes and forms larger scale eddies, the energy migrates towards smaller k-eddies in the spectral space, and consequently leads to a steeper inertial range spectrum in the decaying turbulence (see
Fig. 3. The total energy spectrum grows steeper due to the persistence of large-scale structures (the dashed-curve). The solid curve corresponds to a nearly Kolmogorov-like spectrum during an early phase of the evolution. This demonstrates an inverse cascade of energy, which leads to the formation of coherent structures with kρs 1. The quasistationary spectra is averaged over 75 − 100tωci .
Fig. 4. The total energy associated with the CCs and MS modes decays rapidly during the initial phase of turbulent relaxation. In the process, strong nonlinear interactions lead to a rapid transfer of energy from small-scale to large-scale eddies through an inverse cascade mechanism. The nonlinear transfer of energy is finally terminated by the formation of a largest scale coherent structure in the steady state. The time is measured in a normalized unit as t → tωci , where ωci is ion gyrofrequency.
dashed-curve in Fig. 3). It is further noteworthy from Fig. 3 that the generation of large-scale eddies leads to a sharp cut-off at higher k-modes in the Fourier spectrum. The nonlinear equations (1) to (3), in the absence of dissipation, conserve the total energy and the mean squared magnetic potential, respectively, namely E = (1/2) (∇φ)2 dx dy and A = (1/2) × [Az + (∇Az )2 ] dx dy. In the presence of dissipation, the total energy associated with the magnetostatic modes and convective cells decays eventually with time, viz. ∂(E + A)/∂t = − [(µi /ωci )(∇ 2 φ)2 + η(∇Az )2 ] dx dy. This is shown in Fig. 4 as the tur-
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bulent fluids relax, thereby revealing that the energy decays more rapidly in the initial phase of the evolution. Furthermore, the decay rate (d(E + A)/dt) is very large during an initial phase in which a major part of the total energy in turbulence is dissipated. This is further consistent with the formation of large-scale coherent CCs, in which the energy associated with turbulence must decay in order to relax the turbulence into a well organized coherent structure. It is further instructive to estimate the test-particle transport coefficient associated with a self-consistent evolution of large-scale convective cells and magnetostatic modes. An effective electron diffusion coefficient caused by the latter can be calculated from ∞ Deff = 0 V⊥ (r, t) · V⊥ (r, t + t ) dt , where the angular bracket represents spatial averages. The perpendicular component of the electron fluid velocities in the CCs and MS modes are V⊥ = (c/B0 )ˆz × ∇φ and V⊥ = (v0 /B0 )B⊥ = (v0 /B0 )∇Az × zˆ , respectively, where v0 is some free-streaming electron speed along zˆ (typically 0.04Cs in our simulation). Since the 2D CCs and MS modes are confined in a plane orthogonal to zˆ , the effective cross-field diffusion coefficient, Deff , essentially relates the diffusion processes associated with the transverse motion of electrons in the electromagnetic fields of nonlinearly coupled convective cells and magnetostatic modes. We compute Deff in our simulations, from both the convective and the magnetostatic modes, to measure the turbulent transport that is associated with the coherent nonlinear structures we have reported herein. It is observed that the effective cross-field transport is lower, when the field perturbations are Gaussian. On the other hand, the cross-field diffusion increases rapidly with the eventual formation of longer length-scale structures. This is shown in Fig. 5, which exhibits a t 1/2 dependence of Deff with time. The transport due to the electrostatic CCs cells dominates substantially as depicted by the solid-curve in Fig. 5. Furthermore, in the steady-state, nonlinearly coupled CCs and MS modes form stationary structures, while Deff saturates eventually. Thus, remarkably an enhanced crossfield transport level results primarily due to the emergence of large-scale coherent structures in a magnetized plasma. The enhanced cross-field diffusion coefficient observed in our computer simulations is consistent with the generation of large-scale flows, and they are also in qualitative agreement with the PIC
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Fig. 5. An effective diffusion coefficient evolves initially as Deff ∼ t 1/2 with time (tωci ), thereby indicating that large-scale vortices enhance the transport level significantly. The transport coefficient saturates in the steady-state and predicts a higher value of Deff . The CCs (shown by solid-curve) lead to a larger transport than the MS modes (dashed-curve).
simulations [19] and observations from laboratory observations [15–17]. In summary, we have presented computer simulation studies of nonlinearly interacting 2D CCs and MS modes in a magnetoplasma. Our results show the formation of large-scale coherent structures on account of the energy exchange between CCs and MS modes that accompany enhanced vorticity and electron current filaments. Large scale eddies and filamentary structures, in turn, cause anomalous cross-field electron transport. In particular, large-scale CCs lead to a higher turbulent transport, consistent with the results from PIC simulations [19] and laboratory observations [15–17]. In conclusion, we stress that the results presented here should also be useful for understanding the origin of large-scale coherent structures and associated particle transport in magnetized plasmas, such as those in the Earth’s ionosphere and magnetosphere [22,23] as well as in the solar corona [24], in the solar wind [25], and in forthcoming laboratory experiments [26].
Acknowledgements Dastgeer Shaikh has been supported in part by a NASA grant NAG5-11621 and NAG5-10932 and NSF grant ATM0296113. This work was partially supported by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich 591.
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