Nonlinear Processes in Two-Fluid Plasmas

Transcription

1 Nonlinear Processes in Two-Fluid Plasmas Ryusuke Numata December, 2003 Graduate School of Frontier Sciences, The University of Tokyo

2 Contents 1 Introduction 1 2 Hall Magnetohydrodynamics and Equilibrium Hall Magnetohydrodynamic Equations Magnetohydrodynamic Waves in Homogeneous Plasmas Double Beltrami Equilibrium Relaxation Process Conservation Laws Variational Principle Force-Free Equilibrium Eigenvalue Problem of the Curl Operator Minimum Energy State Nonlinear Simulation 39 i

3 3.1 Simulation Code Discretization and Time Integration Stability Analysis Numerical Smoothing Simulation Model Nonlinear Simulation Initial Condition and Parameters Relaxed State Relaxation Process Summary Nonlinear Mechanism of Collisionless Resistivity Introduction Model Equations Single Particle Dynamics Chaotic Orbit Lyapunov Exponents Statistical Distribution and Macroscopic Resistivity Velocity Distribution Effective Resistivity ii

4 4.5 Fast Magnetic Reconnection Summary Concluding Remarks 95 iii

5 List of Figures 1.1 Energy spectrum. (a) Kolmogorov spectrum (b) spectrum of dual cascade in two-dimensional turbulence Speedup trend of computers Energy of the force-free solution in a cylinder. The arrows indicate the direction of increasing λa. The solid line shows the (m, n) = (0, 0) mode and the dotted lines show the discrete eigenmodes satisfying the boundary condition (the aspect ratio = 3). The lowest magnetic energy state is the axisymmetric state for Ĥ 8.30, and the helical state of the mode (1, 3) for Ĥ > iv

6 2.2 Typical structures of the force-free state in a cylindrical geometry. Figures show the isosurface and the contour plot in the poloidal cross section of B z. (a) is the minimum energy state, which corresponds to (m, n) = (1, 4), (b) is the higher energy eigenstate of (m, n) = (2, 5) Amplification factors of the FTCS and the Runge-Kutta-Gill scheme. Stability condition demands G(k x) 1 for any k Smoothing functions defined by (3.25). To suppress completely the grid scale errors (k x = π), α must be chosen to 0.5 for the second order smoothing, and for the fourth order smoothing Dispersion relation of the Alfvén whistler wave (2.40). ɛ = 0 corresponds to the shear Alfvén wave, fast/slow mode indicated by +/ shows the electron/ion mode Initial condition of the magnetic field with (n 1, n 2 ) = (3, 3). Columns show the isosurfaces of the magnetic field Isosurfaces of the toroidal magnetic field at time 30τ A, 60τ A, 90τ A,120τ A for ɛ = v

7 3.6 Isosurfaces of the toroidal magnetic field at time 30τ A, 60τ A, 90τ A,120τ A for ɛ = Time evolution of energies. The time scale of relaxation is faster in the Hall-MHD than in the MHD Time evolution of kinetic energy. Left panel shows the absolute value of the total, the parallel/perpendicular component of kinetic energies. Right panel shows the percentages of the parallel/perpendicular components to the total kinetic energy Snapshots of the distribution of the alignment coefficients in the Beltrami condition a, b, which correspond to electron and ion. Electron motion aligns well to its corresponding vorticity, while ion flow deviates from its vorticity outside the magnetic columns Snapshot of pressure and velocity distribution in the poloidal cross section. Dynamic pressure is much less than static pressure because of large initial beta vi

8 3.11 Time evolution of energy (E), helicities (H 1, H 2) and enstrophy (F ). H 2 decrease faster than E. H 1 keeps its conservation better than E and H 2. F initially grows because F is not a constant of motion Spectrum of magnetic energy and magnetic helicity. Energy cascades down to high wavenumber modes, and helicity cascade up to low wavenumber modes A Y-shape magnetic field with l l y /l x = 2 projected onto the x-y plane. (x and y coordinates are normalized to the system size l x ). The chaos region (hatched region) is defined by using the local Lyapunov exponents in Sec Typical particle orbit in a Y-shape magnetic field with l = 1. Dotted line shows the asymptotic line of the magnetic field. Motions are qualitatively the same for both figures, however, the staying times are different for different m A s. In (b), the particle is swept out before it is randomized sufficiently Average staying time in the chaos region (ˆτ 1 ) as a function of m A vii

9 4.4 Local Lyapunov exponents for different subdomains Ω(R) (m A = 0.002). We define the chaos region such that the local Lyapunov exponents have a plateau R Local Lyapunov exponents for different m A s. For larger m A, the local Lyapunov exponents are strongly damped and have no plateau region Velocity distributions in the chaos region (m A = 0.001, l = 0). (a)-(c): distributions of v x, v y, v z, after initial randomization phase with the Gaussian fitting curves. (d)-(f): temporal evolutions of the distributions of v x, v y, v z Standard deviations of the velocity distribution (m A = 0.001, l = 0) Evolution of the average velocity in the direction of the electric field (m A = 0.001, l = 0). The ensemble constants of the particles remaining in the chaos region. The average velocity increases linearly. Dotted line shows a linear fitting curve. Dot-dashed line shows the average velocity weighted by the number of particles (4.14) viii

10 4.9 Evolution of the number of particles in the chaos region. The number decreases exponentially Petschek-type fast reconnection model. The regions (I) and (D) are the ideal MHD and dissipation regions, respectively Hierarchy of the scale in the magnetic reconnection process.. 98 ix

11 Chapter 1 Introduction 1

12 Nonlinear phenomena in two-fluid plasmas are studied in this thesis. Plasmas, which are constituted of electrons and positively charged ions, have a huge hierarchy of structures particle/fluid pictures of electrons/ions. Simultaneous existence, and interaction of structures on disparate scales are the defining characteristic of complexity. The scale hierarchy is likely to prevent the system from being analyzed in terms of noninteracting independent elements. Nonlinearity creates interactions of different scales. Macroscopic structures of plasmas can evolve only with the help of microscopic effects in the scale of particle motion. Ordered structures in the complex system are understood as the self-organization [1] or chaos [2, 3]. Self-organization means the spontaneous generation of large-scale coherent structures. The structure spontaneously arise from homogeneous turbulence reflecting properties of turbulent drive. A well-known example of self-organization in plasma is magnetic field reversal in Reversed-Field Pinch (RFP) device [4], a kind of magnetic containment of plasmas. The field reversal phenomenon is first observed accidentally in the toroidal pinch experiment called Zeta [5]. Plasmas in RFP devices may self-organize the reversed-field configuration, where the magnetic field is reversed in the peripheral region after initial turbulent phase driven by large current. The self-organization of 2

13 plasma is now understood by the variational principle in the framework of the magnetohydrodynamics (MHD) theory. The motion of plasma is characterized by ideal global dynamical invariants; the energy, the magnetic and cross helicity. If the energy decrease faster than two helicities, the minimum energy state will lead the so-called force-free state [6 11]. However, this is not always the case. If the magnetic helicity is small and the cross helicity is large relative to the energy, a self-organized state will appear in the form of an aligned state of the magnetic field and flow field because of the Alfvén wave effect [12 15]. The theory of force-free state is also applied to the magnetic reconnection and energy release in the solar corona [16 18]. Because the coronal magnetic field is anchored in the photosphere, the dynamics in the solar corona is driven by photospheric convection, which is analogous to the RFP driven by external circuit. An important point of the self-organization theory is the selective decay process of dynamical invariants in the turbulent phase. In a fully developed three-dimensional (3D) high-reynolds number hydrodynamic turbulence, the energy spectrum shows power-law property which reflect the similarity. Phenomenological argument by Kolmogorov [19] gives the energy 3

14 spectrum (Fig. 1.1 (a)), E(k) k 5/3. (1.1) However, Kraichnan argued that in 2D hydrodynamic turbulence, the energy spectrum cascade inversely to small wavenumbers, and self-organize large scale structures [20]. The enstrophy directly cascade to small scales and dissipate faster than the energy (Fig. 1.1 (b)). In MHD turbulence the Alfvén effect modifies the basic inertial range scaling [21, 22]. Anisotropy of MHD turbulence has been also discussed [23 28]. The scaling relation of energy spectrum parallel and perpendicular to the local magnetic field differs. Numerical computations have become an indispensable tool in turbulence research. In basic turbulence theory, one usually needs the exact informations down to the dissipative scales (Kolmogorov scales). Direct numerical simulations (DNSs) [29, 30] require computer memory proportional to the ninefourth power of the Reynolds number. For practical applications, one often make approximations at small scales [31 33] to simulate the high-reynolds number turbulence. However, recent development of high-performance computer (Fig. 1.2) [34] enables the DNSs of high Reynolds numbers. The DNS starts from Orszag and Patterson in 1972 (Re 10) [30]. The largest DNS in 2004 is Re 10 4 by Kaneda et al. [35] performed by the earth simula- 4

15 tor [36]. In the MHD turbulence, the problem is more serious because of the nonlinear interaction between the magnetic field and the flow field. Müller and Biskamp [37] have done MHD turbulence simulation of R e The self-organization of plasma including strong flow is one of the topics dealt in this thesis. Flows in plasmas are very common both in laboratory and astrophysical situations. In fusion plasmas, it is indicated recently that there are two types of flows [38 41] streamers and zonal flows. The streamers, which is originated from ejection of flows from the sun, considered to leads large convective loss of plasmas. The zonal flow comes from the notion of air stream of the Jupiter, which is considered to stretch and damp vortices of plasmas and to improve confinements. Magnetic reconnections [42, 43] and jets from accretion disks are another examples of plasma phenomena associated with strong flows. When we study an equilibrium of flowing plasmas, especially flowing perpendicular to the magnetic field, the single-fluid MHD model is not adequate for this purpose. One of the problems is a singularity in an equilibrium equation [44 46]. Following the Grad-Shafranov recipe based on the Clebsch representations of incompressible field, we can generalize the Grad- Shafranov equation to include flows. However, the equation has a serious 5

16 singularity when strong flow exists. Furthermore, if we include compressibility, the equation becomes even more complicated the equation may switch between elliptic and hyperbolic. The hyperbolicity is associated with shocks, and dangerous for plasma confinement. Another problem is related to scale invariance of MHD equation. A small scale structure plays an important role in macroscopic plasma phenomena. Since the Cauchy data of the equilibrium equation is an arbitrary function, it may contain any arbitrary small scale. The well-known Parker s current sheet model [47] can be represented by wrinkles of the Cauchy data. The scope of Cauchy solutions underlying Parker s model may be limited when we consider general non-integrable characteristics in 3D systems. In 3D, the characteristic curves are embedded densely in space, inhomogeneous Cauchy data leads to pathology, and the homogeneous Cauchy data yields only the relatively trivial Taylor relaxed state. In two-fluid plasma, the Hall term, deviation of the ion motion from the electron fluid, leads a nonlinear singular perturbation which may remove above mentioned difficulties. A singular perturbation appears as a term with highest order derivatives multiplied by a small parameter. In standard understanding of physicists, terms with small coefficients may give minor 6

17 contribution to physics, thus, it may be usually neglected. However, the singular perturbation may self-organizes small scale structures of its intrinsic scale. The effects of the small scale structures no longer ignorable. Moreover, it is pointed out by many researchers that the Hall term may play important roles in astrophysical plasmas [48 52]. In Chap. 3, the self-organization processes in Hall-MHD plasmas are investigated by numerical simulations. It is theoretically predicted that the so-called double Beltrami equilibrium [53] may be self-organized in twofluid plasmas. To understand the relaxation process in two-fluid plasma, one may construct the variational principle using the global invariants by analogy with the Taylor theory. The variational principle, however, need more rigorous mathematical treatment [54] concerning the concept of selective decay. In Chap. 4, we studied microscopic particle motion in a inhomogeneous electromagnetic field. In a strongly inhomogeneous magnetic field, conservation of adiabatic invariants is broken. The increase of the degree of freedom can results in chaotic motion. The chaotic motion of particles may be a possible mechanism of producing a collisionless resistivity [55]. The mixing effect of chaos brings about a rapid increase of the kinetic entropy. This process, however, saturates after a short time, so that a mixing process all 7

18 alone cannot lead to a diffusion-type dissipation. In an open system where particles can escape from a domain of the phase space (either through coordinate of momentum axes) after certain staying time, the saturation of the entropy can be avoided, and a continuous dissipation process is achieved. The chaos-induced collisionless resistivity of ions enable fast magnetic reconnection [56]. To apply the chaos-induced resistivity in the fast magnetic reconnection, we have introduced a mesoscopic model and have derived a relation connecting macroscopic scale and the microscopic parameters in a universal form. The kinetic effect produces a strong collisionless dissipation, giving a lower bound for the length scale. Hence, we can remove the unphysical scale length reduction (down to the subgyroradius regime) [57] deduced from the direct application of the single-fluid MHD in the original Petschek model [58], which is incapable of encompassing the huge scale separation. 8

19 E(k) energy input inertial range k -5/3 E(k) inverse cascade k -5/3 forward cascade k -3 dissipation energy input k (a) k (b) dissipation Figure 1.1: Energy spectrum. (a) Kolmogorov spectrum (b) spectrum of dual cascade in two-dimensional turbulence. 9

20 Figure 1.2: Speedup trend of computers. 10

21 Chapter 2 Hall Magnetohydrodynamics and Equilibrium 11

22 2.1 Hall Magnetohydrodynamic Equations We consider two component plasma electron and singly charged ion, and introduce compressible two-fluid magnetohydrodynamic (MHD) equations (see for example [59 61]). Continuity equations and momentum equations for each species are written as t ne + (nev e) = 0, (2.1) t n i + (n i V i ) = 0, (2.2) nme [ t V e + (V e )V e] = en(e + V e B) pe R + Π e, (2.3) [ ] nm i t V i + (V i )V i = en(e + V i B) p i + R + Π i, (2.4) R = ζ(v i V e), (2.5) where n e,i are the number densities, V e,i are the flow velocities, m e,i are the masses, p e,i are the pressures, Π e,i are the viscous stress tensors, e is the elementary charge, and E, B are the electric and the magnetic fields. The subscripts e, i denote an electron and an ion, respectively. The term R represents the momentum transfer between electron and ion, and ζ is a 12

23 positive constant. The viscous stress tensor is given by [ Π = ν V + t ( V ) 2 ] 3 δ ij V + ν δ ij V, (2.6) where ν and ν are the viscosity (ν is often called the second viscosity), t denotes a transpose and δ ij is the Kronecker s delta [62, 63]. In the Navier- Stokes equation for neutral fluids, the second viscosity is often neglected (the Stokes relation) and the divergence of the stress tensor becomes Π = ν V + 1 ν ( V ). (2.7) 3 It is pointed out by Braginskii [64] that the coefficients of the first and second term in (2.7) differ by a factor of the cyclotron frequency normalized by the collision frequency in the presence of strong magnetic field because the motion across the magnetic field is restricted. However, we use (2.7) as the viscous dissipation term for simplicity. The magnetic field B and the electric field E obey the Maxwell s equa- 13

24 tions t B = E, (2.8) t E = c 2 B c 2 µ 0 j, (2.9) B = 0, (2.10) E = 0, (2.11) j = en(v i V e), (2.12) where j is the current density, µ o is the vacuum permeability, and c is the speed of light. We have assumed that plasma is approximately electrically neutral (quasi-neutral), such that ne n i = n. (2.1)-(2.12) and appropriate equations of state for p e,i give the full set of two-fluid MHD equations. Local existence of solutions has been proved [65] for the initial-value problem of the two-fluid equation in the incompressible limit, together with the boundary conditions, n E = 0 on Ω, (2.13) V i,e = 0 on Ω. (2.14) Since the electron mass is very small compared with the ion mass, and can be neglected in many circumstances, we omit the inertia term and the 14

25 viscous term in (2.3). Then, we obtain generalized Ohm s law, E + V e B = 1 en p e 1 R. (2.15) en The correction of the displacement current to the plasma dynamics is also small due to the smallness of the coefficient in a non-relativistic regime. Neglecting the displacement current in (2.9), we obtain B = µ 0 j = µ 0 en(v i V e). (2.16) Inserting (2.15) into (2.4) and (2.8), and eliminate V e using above equation, we obtain nm i [ t V i + (V i )V i ] = 1 ( B) B (p µ i + pe) + Π i, (2.17) 0 [ t B = (V i 1 ] en B) B 1 en 2 n p e (ηj), (2.18) where we have defined the resistivity η ζµ 0 /e 2 n 2. In order to close the set of equation, we must relate the pressure with other variables. If we assume heat exchanges between different parts of the plasma and between the plasma and bodies adjoining it are absent, the motion of 15

26 the ideal plasma must be supposed adiabatic. The adiabatic relation p/ρ γ = const., (2.19) leads the time evolution of pressure t p + (V )p + γp V = 0, (2.20) where γ is the ratio of specific heats. However, the viscous and resistive dissipation heat plasma locally. Thus, we introduce the equation of state by following formula, t p + (V )p + γp V = S, (2.21) where S is the heat source term given by S = (γ 1) [ν V ] ν V 2 + η B 2. (2.22) We summarize the set of Hall-MHD equations in dimensionless form. Variables are normalized by typical scale length L, appropriate measure of the magnetic field B 0 and density n 0 (the ion mass m i is assumed to be unity such that ρ 0 = n 0 ) as x Lx, B B 0 B, n n 0 n, V V A V, t τ A t, p (B 2 0/µ 0 )p, (2.23) 16

27 where V A B 0 / µ 0 n 0 m i is the Alfvén velocity, τ A L/V A. The normalized Hall-MHD equations become t n + (nv ) = 0, (2.24) n[ t V + (V )V ] = j B p + 1 Re t B = [(V ɛ ) n B ( V + 13 ( V ) ), (2.25) ] B + 1 B, (2.26) Rm t p + (V )p + γp V = S, (2.27) [ 1 S = (γ 1) Re V Re V ] B 2, (2.28) Rm B = j = n(v V e), (2.29) where Re V A L/ν is the Reynolds number, Rm V A L/η is the magnetic Reynolds number and ɛ δ i /L (δ i = c/ω pi is the ion collisionless skin depth). We have neglected the electron pressure for simplicity and omit the subscript for ions. 17

28 2.2 Magnetohydrodynamic Waves in Homogeneous Plasmas Let us consider small amplitude wave in the Hall-MHD in the ideal limit Re, Rm. Linearizing the equations (2.24)-(2.27) around a uniform equilibrium where B = B z, V = 0, n and p, we obtain t ñ = n Ṽ, (2.30) t p = γ p Ṽ, (2.31) t Ṽ = 1 n p + 1 n ( B) B, (2.32) t B = ( Ṽ B) ɛ n (( B) B). (2.33) Variables with tilde denote fluctuations. Assume exp(i(k x ωt)) dependence of fluctuations and replace t by iω and by ik, we obtain the following eigenvalue equation, iωξ = Aξ, (2.34) 18

29 where ξ = t (ñ, p, ṽ x, ṽ y, ṽ z, B x, B y, B z ) and 0 0 ik x n ik y n ik z n ik x γ p ik y γ p ik z γ p ik x ik n z B 0 ik x B n n 0 ik y ik z B ik y B n n n A = 0 ik z n 0 0 ik z B ɛk 2 B z ɛk yk z B n n ik z B 0 ɛk 2 B z ɛkzkx B 0 n n 0 0 ik x B iky B 0 ɛk y k z B n ɛk z k x B n 0. (2.35) Since pressure and density are related through the adiabatic relation, the first or the second row can be removed. The remaining 7 7 matrix contains seven eigenvectors and corresponding eigenvalues. From the determinant of the matrix A + iωδ ij, we obtain the dispersion relation [66, 67] ω(ω 2 V 2 Ak 2 z) [ ω 4 (C 2 s + V 2 A)k 2 ω 2 + C 2 s V 2 Ak 2 zk 2] = ɛ 2 V 2 Ak 2 zk 2 ω 3 (ω 2 C 2 s k 2 ), (2.36) where k = kx 2 + ky 2 + kz, 2 V A B/ n, C s γ p/ n are the Alfvén velocity and the sound velocity of the uniform equilibrium in the normalized unit, respectively. If ɛ = 0, we recover the dispersion relation of the MHD 19

30 equations, which gives the entropy wave ω = 0, (2.37) the shear Alfvén wave ω 2 = V 2 A k2 z, (2.38) and the fast and slow magneto-sonic waves ω 2 = 1 2 [ C 2 s + V 2 A )k2 ± ((C s 2 + V A 2 )2 k 4 4CsV 2 2Ak 2zk ) ] (2.39) If we consider waves traveling along the magnetic field, i.e. k = k z, the sound wave (ω = ±Csk) decouples. The dispersion relation of the shear Alfvén wave is modified by the Hall term as ω 2 V 2 A k2 z = ±ɛv A k 2 zω, (2.40) which is called the Alfvén whistler wave. 2.3 Double Beltrami Equilibrium We review the special equilibrium solution to the Hall-MHD equations. In an incompressible, ideal limit, the Hall-MHD equations are described by time evolution of the vector potential A and V. Writing E = φ t A where 20

31 φ is the scalar potential, (2.26), (2.25) translate into t A = (V ɛ B) B (φ ɛpe), (2.41) t (A + ɛv ) = V (B + ɛ V ) (ɛ V ɛp i + φ). (2.42) Taking the curl of (2.41) and (2.42), we can cast the Hall-MHD equations into coupled vortex equations, t ω j (U j ω j ) = 0 (j = 1, 2), (2.43) where j = 1, 2 indicate an electron and an ion. Pairs of generalized vorticities and the corresponding flows are defined by ω 1 = B, ω 2 = B + ɛ V, U 1 = V ɛ B, U 2 = V. (2.44) The simplest equilibrium solution to (2.43) is given under the Beltrami condition [53], that demands alignment of the vorticities along the corresponding flows, U j = µ j ω j (j = 1, 2). (2.45) Writing a = 1/µ 1, b = 1/µ 2, and assuming a,b are constants, the Beltrami condition translates into a system of simultaneous linear equations in B and 21

32 V, B = a(v ɛ B), (2.46) B + ɛ V = bv. (2.47) Combining two equations yields second-order equation for u = B and V, (curl λ + )(curl λ )u = 0, (2.48) where we denote by curl implying an operator, and λ ± = 1 2ɛ [ (b a 1 ) ± ] (b a 1 ) 2 4(1 ba 1 ). (2.49) Because the operators (curl λ ± ) commute, the general solution to (2.48) is given by a linear combination of two Beltrami fields, B = C + G + + C G, (2.50) V = (a 1 + ɛλ + )C + G + + (a 1 + ɛλ )C G, (2.51) where G ± are the Beltrami functions satisfying (curl λ ± )G ± = 0, and C ± are constants. The parameters λ ±, which are the eigenvalues of the curl operators, characterize the spatial scales of the vortices G ±. The Beltrami conditions (2.46), (2.47) give a special class of the general steady state solutions such that U j ω j = 0 = ϕ j (j = 1, 2). (2.52) 22

33 where ϕ j is a certain scalar field corresponding to the energy density given by φ 1 = φ ɛpe (2.53) φ 2 = φ + ɛp i + ɛ V 2 2 (2.54) The latter equality in (2.52), called the generalized Bernoulli conditions, demands that the energy density is uniform in space. Subtracting (2.53) from (2.54), we obtain V 2 + β = const. (2.55) where the beta ratio in the normalized unit is given by β = 2(p i +pe). The relation demands the static pressure is sustained by the dynamic pressure, and suggests an improvement of plasma confinement due to the double Beltrami field. 2.4 Relaxation Process The self-organization process of a plasma into relaxed states may be discussed by a variational principle. Taylor [6] argued that the minimization of the magnetic energy under the constraint of the magnetic helicity conservation 23

34 leads the relaxed state in the MHD model. The variation δ(e m µh) = 0 (2.56) results in the force-free equilibrium B = µb. (2.57) This model is based on the selective decay implying that one of the constants of motion of the ideal limit decays faster than the others. However, the variational principle for the two-fluid MHD requires a more generalized and rigorous arguments [54] Conservation Laws The total energy in the system is given by sum of the magnetic energy Em, the kinetic energy E k, and the thermal energy E t, E = Em + E k + E t = Ω B2 dx + Ω 2 nv 2 p dx + dx. (2.58) Ω γ 1 24

35 The time derivative of the energies are calculated as, d dt E m = E B ds V j Bdx 1 j 2 dx, (2.59) Ω Ω Rm Ω [( d dt E k = 1 Ω 2 nv ) ] 3Re V V + V ( V ) ds + V j Bdx V pdx Ω Ω 1 V 2 dx 4 V 2 dx, (2.60) Re Ω 3Re Ω d dt E pv t = γ 1 ds S p V dx + dx. (2.61) γ 1 Ω Then, the time derivative of the total energy is given by Ω Ω de dt = + Ω Ω Ω Ω E B ds ( 1 2 nv 2 + γ γ 1 p 4 ) 3Re ( V ) V ds V ( V ) ds ( 1 Re V + 4 3Re V + 1 Rm j2 S ) dx.(2.62) γ 1 If the system is surrounded by a rigid perfect conducting wall, (n E = 0, V = 0 at the wall), the total energy conserves through all dynamic process regardless of whether the dissipations exist or not. In the incompressible limit (which corresponds to γ ), the thermal energy is irrelevant. The total energy E = E m + E k, however, monotonically decrease if the dissipations exist. 25

36 The generalized helicities, defined by H j = Ω ω j (curl 1 ω j )dx (j = 1, 2), (2.63) are also global ideal invariants in the incompressible Hall-MHD. Conservation of helicities is proved straightforwardly from the vortex equations (2.43). The time derivative of helicity is given by dh dt = = = Ω Ω Ω The boundary condition, ω t (curl 1 ω)dx + ω (curl 1 ω Ω t )dx (U ω) (curl 1 ω)dx + ω (U ω)dx (U ω) (curl 1 ω) ds. (2.64) Ω n A = 0, n V = 0 on Ω, (2.65) yields the conservation of the helicities Variational Principle There are three global ideal invariants in the incompressible Hall-MHD: E = 1 (B 2 + V 2 )dx, 2 Ω (2.66) H 1 = A Bdx, (2.67) Ω H 2 = (A + ɛv ) (B + ɛ V )dx, (2.68) Ω 26

37 representing the total energy, the electron helicity, and the ion helicity. The minimization of a generalized enstrophy (measure of the complexity or the perturbations), F = 1 (A + ɛv ) 2 dx, (2.69) 2 Ω with keeping E, H 1, and H 2 constant is carried out through the variation δ(f µ 0 E µ 1 H 1 µ 2 H 2 ) = 0, (2.70) where µ 0, µ 1, µ 2 are Lagrange multipliers. The Euler-Lagrange equation becomes (curl Λ 1 )(curl Λ 2 )(curl Λ 3 )u = 0 (u = B or V ), (2.71) and the general solution to (2.71) is linear combination of three Beltrami functions, which is not an equilibrium in general. The DB field is obtained by adjusting E, H 1, H 2 to condense the triple Beltrami field into the double Beltrami field given by two eigenfunctions. The adjustment condition is given by E µ 1H 1 µ 2H 2 = 0. (2.72) The Euler-Lagrange equation of the variation δ(e µ 1H 1 µ 2H 2 ) = 0, which is equivalent to the solution of (2.72), gives the DB equation (2.48). The 27

38 relaxation process is realized by minimizing perturbations, which is scaled by the generalized enstrophy F, with appropriate adjustment of macroscopic variables E, H 1, H Force-Free Equilibrium In this section, we review a structure of the force-free equilibrium, which describes the relaxed state of the MHD model. The force-free equilibrium is given by the eigenfunction of the curl operator. The relaxation process will choose an appropriate eigenvalue for the initial global constants with satisfying given boundary conditions. First, we study the eigenfunction expansion of solenoidal vector fields, and then, discuss the minimum energy state Eigenvalue Problem of the Curl Operator We consider an eigenvalue problem u = λu, (2.73) for a solenoidal field u L 2 σ L 2 σ(ω) {u L 2 (Ω); u = 0 in Ω, n u = 0 on Ω}, (2.74) 28

39 in a bounded 3D domain Ω with smooth boundary Ω. We assume that Ω is multiply connected with cuts Σ j [j = 1,, ν (the first Betti number)], i.e., Ω\ Σ j is simply connected. In multiply connected domain Ω, the curl operator has a point spectrum that covers the entire complex plane [68,69]. This is because of the existence of a nonzero harmonic field h L 2 H, L 2 H(Ω) {h L 2 (Ω); h = 0, h = 0 in Ω, n h = 0 on Ω}, (2.75) which plays the role of an inhomogeneous term in the eigenvalue problem. The solenoidal field u can be decomposed into the harmonic field h and its orthogonal complement u Σ. The latter component is a member of the Hilbert space L 2 Σ(Ω) { w; w H 1 (Ω), w = 0 in Ω, n w = 0 on Ω}. (2.76) The Hilbert space L 2 Σ is identical to the Hilbert space L2 S where L 2 S(Ω) {u L 2 (Ω); u = 0 in Ω, n u = 0 on Ω, Φ j = 0 (j = 1, ν)}, (2.77) and Φ j = u ds (j = 1,, ν) Σ j (2.78) 29

40 is the flux through the cut Σ j. The decomposition L 2 σ = L 2 H L 2 Σ (2.79) is called the Hodge-Kodaira decomposition. The eigenvalue problem now reads as u Σ = λ(u Σ + h). (2.80) If we take h = 0, we find a nontrivial solution only for λ σp, where σp is a countably infinite set of real numbers. The set σp constitutes the point spectrum of the self-adjoint curl operator that is defined in the Hilbert space in L 2 Σ (Ω). For λ / σp, we must invoke h 0 and find a solution u Σ = (curl λ ) 1 h, where the curl denotes the self-adjoint curl operator Minimum Energy State The complete orthogonal set Ψ j ( Ψ j = λ j Ψ j ) spanning L 2 Σ (Ω) allows B = B Σ + B h = j b j Ψ j + B h. (2.81) 30

41 Defining B B h = A Σ and Ag = A A Σ, we may write B h = Ag. The magnetic energy and the magnetic helicity are given by E = 1 2 H = = Ω Ω Ω B 2 dx = 1 2 A Bdx A Σ B Σ dx + 2 Ω B 2 Σdx Ω Ω Ag B Σ dx + B h 2 dx, (2.82) Ω Ag B h dx, (2.83) where the boundary condition n A Σ = 0 and the orthogonality of B Σ and B h are used. The force-free condition demands B Σ = bψ ( Ψ = λψ), and A Σ = b Ψ. Each term in (2.82), (2.83) is calculated as, λ A Σ B Σ dx = 1 B Ω λ Σdx, 2 (2.84) Ω Ag B Σ dx = A Σ B h dx + A Σ Ag ds Ω Ω Ω = 1 B Σ B λ h dx + A Σ Ag ds = 0, (2.85) Ω Ω ( ) 1 Ag B h dx = A g Ω Ω λ B Σ B Σ dx = 1 B λ h B Σ dx + 1 B Σ Ag ds Ag B Σ dx Ω λ Ω Ω = Ag A Σ ds = 0. (2.86) Ω 31

42 Then, we obtain the relations, H = A Σ B Σ dx, (2.87) Ω E = 1 B 2 Σdx B Ω 2 h 2 dx Ω = λ 2 H + 1 Ag B 2 h ds. (2.88) Ω The second term in (2.88) can be expressed by flux variables. The toroidal and the poloidal magnetic flux and the poloidal current are given by Ψ t = Ψp = Ip = B ds = B h ds, (2.89) Sp Sp B ds, (2.90) S t S t B ds. (2.91) where Sp and S t are the poloidal and the toroidal cross section, respectively. The poloidal/toroidal cross section is normal to the toroidal/poloidal direction. By introducing the multi-valued scalar potential φ, which describes the harmonic field as B h = φ, the energy of the harmonic field becomes 1 B 2 h 2 dx = 1 φ B Ω 2 h dx Ω = 1 2 [φ] B h ds = 1 Sp 2 [φ]ψ t, (2.92) where [φ] is the jump of φ across the cut, which does not depend on the coordinate. [φ] is evaluated by integrating φ along the loop in the toroidal 32

43 direction l t, [φ] = φ dl l t = B h dl = = 1 λ = l t S t l t ( 1 λ B Σ B Σ ) dl B Σ ds B Σ ds λ S t S t S t B ds B Σ ds = Ip λψp. (2.93) Thus, the relation between the magnetic energy and the helicity is expressed by E = λ 2 H (I p λψp)ψ t. (2.94) Cylindrical geometry The Chandrasekhar-Kendall (C-K) [70] function is defined by u mn = λ( ψ mn z) + ( ψ mn z), (2.95) where ψ mn = J m (µr)e i(mθ kz) (k = nπ/l; mn {0, 1, 2, }), (2.96) λ = ±(µ 2 + k 2 ) 1/2, (2.97) 33

44 and J m is the Bessel function. The general solution of the force-free equilibrium in a periodic cylinder with the radius a and the length L is given by the C-K function, B = m,n u mn = m,n B mn (r)e i(mθ kz), (2.98) B mn (r) = ib mn Λ mn r (µr) b mn Λ mn θ (µr) b mn J m (µr), (2.99) Λ mn r (µr) = 1 λ + k 2 λ k J m 1(µr) + 1 λ k 2 λ + k J m+1(µr), (2.100) Λ mn θ (µr) = 1 λ + k 2 λ k J m 1(µr) 1 λ k 2 λ + k J m+1(µr), (2.101) The eigenvalue is determined by the boundary condition n B r=a = 0, i.e., Λ mn r (µa) = 1 λ + k 2 λ k J m 1(µa) + 1 λ k 2 λ + k J m+1(µa) = 0. (2.102) For the axisymmetric solution (m, n) = (0, 0), the boundary condition is trivially satisfied. The eigenvalue for the axisymmetric mode is determined from the given toroidal flux. The toroidal flux, the energy, and the helicity for the axisymmetric mode 34

45 are given by Ψ t = 2πa b00 λ J 1(λa), (2.103) ( E 00 = 2πL (b00 a) 2 J1 2 (λa) + J0 2 (λa) 1 ) 2 λa J 0(λa)J 1 (λa), (2.104) ( H 00 = 2πL(b 00 a) 2 λ J1 2 (λa) + J0 2 (λa) 1 ) λa J 0(λa)J 1 (λa), Lb00 λ J 0(λa)Ψ t. (2.105) The normalization of the energy Ê 2 a 2 E and the helicity Ĥ L/2πΨ 2 t a H L/2πΨ 2 t yields Ê 00 = ( (λa)2 J J 1 (λa) 2 1 (λa) 2 + J 0 (λa) 2 1 ) λa J 0(λa)J 1 (λa), (2.106) Ĥ 00 = λê00 1 J 0(λa) J 1 (λa). (2.107) In Fig. 2.1, the solid line is a plot of Ê(λa) vs Ĥ(λa) for the (m, n) = (0, 0) mode. Now suppose we superpose a discrete eigenmode on the (m, n) = (0, 0) mode. Evaluating Ĥ for such a mode, we find that it is of the form ( ) b mn 2 Ĥ(λa) = Ĥ00 (λa) + Ĥmn (λa), (2.108) b 00 and Ĥ mn = 1 λaêmn (2.109) because (m, n) (0, 0) modes contain no toroidal flux. By variating b mn with fixed λa, we draw straight lines originating on the (m, n) = (0, 0) curve 35

46 (the dotted lines in Fig. 2.1). Each dotted line corresponds to the discrete eigenvalue satisfying the boundary condition (2.102) with the aspect ratio L/a = 3. The mixed state with λa = 3.11 is the lowest energy state for Ĥ > 8.30 [6, 71, 72]. Typical spatial structures are shown in Fig

47 E 40 (m,n)=(0,0) (m,n)=(1,3) λa= (m,n)=(1,4) λa=3.11 (m,n)=(1,5) λa=3.18 (m,n)=(2,5) λa=4.71 (m,n)=(3,5) λa= H Figure 2.1: Energy of the force-free solution in a cylinder. The arrows indicate the direction of increasing λa. The solid line shows the (m, n) = (0, 0) mode and the dotted lines show the discrete eigenmodes satisfying the boundary condition (the aspect ratio = 3). The lowest magnetic energy state is the axisymmetric state for Ĥ 8.30, and the helical state of the mode (1, 3) for Ĥ >

48 (a) (m,n)=(1,4) (b) (m,n)=(2,5) Figure 2.2: Typical structures of the force-free state in a cylindrical geometry. Figures show the isosurface and the contour plot in the poloidal cross section of B z. (a) is the minimum energy state, which corresponds to (m, n) = (1, 4), (b) is the higher energy eigenstate of (m, n) = (2, 5). 38

49 Chapter 3 Nonlinear Simulation 39

50 3.1 Simulation Code All numerical simulations in this chapter are carried out on NEC SX-7 system installed in the Theory and Computer Simulation Center of the National Institute for Fusion Science (TCSC, NIFS). The SX-7 is a vectorparallel supercomputer which has 5 nodes. Each node has 256 giga bytes main memory, 32 processor elements (PEs). Processing speed of one PE is about 8.8 giga floating operations per seconds Discretization and Time Integration In order to solve continuous partial differential equations by computers, we must discretize domain into finite grids and assign values of physical quantities on each grid. The most simple, and common method is the finite difference method. We discretize space and time as (x, y, z, t) = (i x, j y, k z, n t) and refer a variable f on the grid i, j, k at time n as fi,j,k n. The Taylor expansion of f with respect to x is given by f(x + x) = f(x) + df dx x + d2 f ( x) 2 x dx 2 x 2 + O(( x) 3 ), (3.1) or in the discrete form, f i+1 = f i + df dx x + d2 f ( x) 2 i dx 2 i 2 + O(( x) 3 ). (3.2) 40

51 We can evaluate the derivatives by choosing appropriate coefficients. Typical methods are summarized in Table. 3.1 [73]. In our simulation code, we use the second order accuracy, central difference methods to evaluate the first and the second order derivatives, df dx = f i+1 f i 1, (3.3) i 2 x d 2 f dx 2 = f i+1 2f i + f i 1. (3.4) i ( x) 2 The time derivative may be evaluated in the similar way. We consider the ordinary differential equation, df dt = L(f). (3.5) The upstream and the central differencing give the following time integration scheme, f n+1 = f n + tl(f n ), (3.6) f n+1 = f n tl(f n ). (3.7) (3.6) is called the Euler method (first order), and (3.7) is called the leap-flog method (second order). However, time goes only positive way. We cannot use variables at future time steps to construct higher order methods. The Runge-Kutta-Gill (RKG) method uses intermediate time steps between n 41

52 Table 3.1: Typical methods to evaluate the derivatives. df dx df dx = f i+1 f i x = f i f i 1 x + O( x) + O( x) df dx df dx df dx df dx df dx df dx = f i+1 f i 1 2 x + O(( x) 2 ) = f i+2+4f i+1 3f i 2 x + O(( x) 2 ) = 3f i 4f i 1 +f i 2 2 x + O(( x) 2 ) = 2f i+1+3f i 6f i 1 +f i 2 6 x + O(( x) 3 ) = f i+2+6f i+1 3f i 2f i 1 6 x + O(( x) 3 ) = f i+2+8f i+1 8f i 1 +f i 2 12 x + O(( x) 4 ) d 2 f = f i+2 2f i+1 +f i + O( x) dx 2 ( x) 2 d 2 f = f i 2f i 1 +f i 2 + O( x) dx 2 ( x) 2 d 2 f dx 2 = f i+1 2f i +f i 1 ( x) 2 + O(( x) 2 ) d 2 f dx 2 = f i+3+4f i+2 5f i+1 +2f i ( x) 2 + O(( x) 2 ) d 2 f dx 2 = 2f i 5f i 1 +4f i 2 f i 3 ( x) 2 + O(( x) 2 ) d 2 f dx 2 = f i+2+16f i+1 30f i +16f i 1 f i 2 12( x) 2 + O(( x) 4 ) 42

53 and n + 1, and achieves the higher order accuracy. [73, 74]. We employ the fourth-order RKG method in the simulation code, given by f n+1 = f n + t 6 [L(f n ) + (2 2)L(f (1) ) + (2 + 2)L(f (2) ) + L(f (3) )], (3.8) where the intermediate steps are evaluated by f (1) = f n + t 2 L(f n ), (3.9) f (2) = f (1) + t 2 L(f (1) ), (3.10) f (3) = f (2) + tl(f (2) ). (3.11) We must mention the reliability of the numerical scheme. It is proved that, for linear scalar equation, the only scheme which converges the numerical solution to the solution of the original equation is stable and consistent scheme (Lax s equivalence theorem [75]). Stability and consistency is necessary and sufficient condition for convergence. The consistency of a finite difference scheme means that the finite difference equation converges to the original equation in the limit that time step and spatial grid spacing goes to zero. The stability of the scheme will be discussed in the next section. 43

54 3.1.2 Stability Analysis We consider, for simplicity, 1D advection-diffusion equation with constant coefficients, f t + c f x = d 2 f x 2, (3.12) where c, d are constants. The FTCS (forward in time central difference in space) scheme, f n+1 i fi n t + c f n i+1 f n i 1 2 x = d f n i+1 2f n i + f n i 1 x 2, (3.13) is used to introduce the stability of the finite difference scheme. According to the Von Neumann, the scheme is stable if each Fourier mode damps in time. Substituting the Fourier decomposition of f n i f n i = k ˆf n (k)e ik xi (3.14) (where ˆf n (k) is the amplitude of the Fourier mode k at time step n) into (3.13) yields [ ˆf n+1 e ik xi = ˆf n e ik xi 1 c t 2 x (eik x e ik x ) + d t x 2 (eik x 2 + e ik x ) ]. (3.15) 44

55 By introducing C c t/ x (the Courant number), D d t/ x 2 (the diffusion number), the amplification factor G ˆf n+1 / ˆf n becomes G = 1 ic sin k x + 2D(cos k x 1). (3.16) The stability condition is given by G 1, which requires, 0 C 2 4D 2 2D 1, (3.17) t x C x2 2D, t, t 2D C, (3.18) 2 x 2D C. (3.19) Since the calculation to derive the amplification factor G of the RKG scheme is rather cumbersome, we give only the results in Fig The stability condition of the RKG method is relaxed compared with the FTCS scheme. We can choose larger t in the RKG method. In a multi-dimensional case, the stability condition may become more restrictive. For example, C/ x must be replaced by C x / x + C y / y + C z / z in 3D. This means that we must choose about three times smaller time step in 3D than in 1D. The stability analysis due to Von Neumann is only applicable to the linear problem. It is almost impossible to derive the exact stability criteria of nonlinear problems. We can evaluate stability of the linearized equations, and can discuss only local stability. 45

56 3.1.3 Numerical Smoothing To suppress numerical error of the grid-size mode, we introduce an artificial smoothing technique [76]. The smoothing procedure is expressed as, f sm i = (1 α)f i + α f i, (3.20) where α is the smoothing parameter which satisfies 0 α 1. The angle bracket denotes average of the nearest grids: 2nd order : f i = f i+1 + f i 1, (3.21) 2 4th order : f i = f i+2 + 4f i+1 + 4f i 1 f i 2. (3.22) 6 The effect of smoothing is evaluated by the Fourier decomposition. Fourier decomposition of (3.20) yields f sm i = (1 α) = (1 α) + α 2 N k=1 N k=1 { N 1 k=0 ˆf(k)e ik xi + α 2 ˆf(k)e ik xi ˆf(k)e ik x(i 1) + If we assume periodicity, we obtain { N N+1 k=2 k=1 ˆf(k)e ik xi + ˆf(k)e ik x(i+1) } N k=1 ˆf(k)e ik xi } (3.23) f sm i = = N k=1 N k=1 ˆf(k)[(1 α)e ik xi + α 2 (eik x(i 1) + e ik x(i+1) )] ˆf(k)e ik xi [(1 α) + α cos(k x)]. (3.24) 46

57 By the smoothing procedure, each Fourier mode ˆf(k) is modified by a factor which depends on k. The factor defines the smoothing function, SM(k) = (1 α) + α cos(k x). (3.25) We plot the smoothing function in Fig The smoothing procedure acts as the low-pass filter and suppress high wavenumber numerical errors. We use the second order smoothing in the simulation code. To suppress unphysical grid-size error, α is set to α = Simulation Model To test the two-fluid self-organization in a general dynamical framework, we have developed a simulation code base to the methods in the previous section, to solve a compressible, dissipative Hall-MHD plasma governed by (2.24)- (2.27). The simulation domain is a rectangular box with size 2a 2a 2πR, surrounded by a rigid perfect conducting wall. The system is periodic along the z axis. The boundary conditions in x, y directions are n B = 0, n ( B) = 0, V = 0 at x, y = ±a. (3.26) To assure tangential components of the electric field vanish, we set ɛ = 0 at the wall. 47

58 In order to confirm a validity of the simulation code, we have solved dispersion relations of purely transversal wave around a uniform equilibrium in a periodic domain. The dispersion relation is given by (2.40). Figure 3.3 shows the dispersion relation for ɛ = 0, 0.1, Results of the numerical code agree well with the analytical curves. Thus, we consider that the code works well. 3.3 Nonlinear Simulation Initial Condition and Parameters The initial condition for nonlinear simulation was a 2D force free equilibrium that is also an equilibrium of the single fluid model [9 11]; 1 k 0 (k 1 B 1 cos k 2 x sin k 1 y + k 2 B 2 cos k 1 x sin k 2 y) B 0 = 1 k 0 (k 2 B 1 sin k 2 x cos k 1 y + k 1 B 2 sin k 1 x cos k 2 y) B 1 cos k 2 x cos k 1 y + B 2 cos k 1 x cos k 2 y,, (3.27) where B 1, B 2 are the amplitudes of two kinds of Fourier modes, k 1 = n 1 π/(2a), k 2 = n 2 π/(2a), and n 1, n 2 are arbitrary integers. Figure 3.4 shows the isosurface of the toroidal magnetic field. Columns in different colors indicate that the directions are opposite. A uniform density n 0 = 1, and uniform 48

59 pressure are assumed. The amplitude of the uniform pressure is given by a parameter β = pdx/ B 2 dx. The initial condition has a flow such that V 0 = (0, 0, M A B z ), where M A is the Alfvén Mach number. We carried out two simulation runs. The simulation domain is implemented on point grids. The parameters are β = 3, M A = 0.5, and (a) ɛ = 0.1, (b) ɛ = Relaxed State Figure 3.5 and 3.6 show snapshots of isosurfaces of the toroidal magnetic field at t = 30τ A, 60τ A, 90τ A, and 120τ A. Because the initial condition is unstable against the kink mode, initially assigned random perturbations exponentially grow and plasma become turbulent. After the turbulent phase the magnetic columns are settled into helically twisted final states (see Sec. 2.5). Figure 3.7 shows the time evolution of the energies. We see that the magnetic energy decreases rapidly in the turbulent phase ((a) t 20 40τ A, (b) t 30 70τ A ), and after that it decreases in constant dissipation rate. The time scale of Hall-MHD is slightly faster compared with the MHD. The global structures (the helical mode number) do not depend on ɛ, and agree with the Taylor s prediction. However, there exists appreciable flow perpen- 49

60 dicular to the magnetic field in the Hall-MHD plasma. Time history of the kinetic energy of the parallel and perpendicular components are shown in Fig A remaining component of the kinetic energy in MHD is almost parallel component, while about 10% of the perpendicular component exists in Hall-MHD. Figure 3.9 shows the snapshots of the distribution of a, b in the Beltrami condition for ɛ = 0.1. The Beltrami conditions are satisfied in the magnetic column. The perpendicular component of the velocity is originated from the Hall current, V ɛ n ( B). (3.28) Figure 3.10 shows the distribution of the pressure and the absolute value of flow in a poloidal cross section at t = 50τ A. The pressure distribution is almost uniform. This is because the kinetic energy is much less than the thermal energy in the relaxed state. The kinetic energy is dissipated by the strong viscosity through the dynamics process Relaxation Process We discuss the relaxation process in terms of the variational principle in the Hall-MHD plasma. Figure 3.11 shows the time evolution of the sum of the magnetic and kinetic energies E, two helicities H 1, H 2 H 2 H 1 and the 50

61 generalized enstrophy F = F Em normalized by the initial values. H 2 is the most fragile quantity, while H 1 is the most conserving one. The simulation result shows the fragility of the conserved quantities is determined by the order of the derivative, and indicates the energy minimization is meaningless in the Hall-MHD. In the relaxation process in Hall-MHD plasma is achieved by minimizing the generalized enstrophy with appropriate adjustment of H 2 to condensate into the DB equilibrium. We also show the energy and helicity spectrum in Fig The low wavenumber mode of energy decrease, while high wavenumber mode of helicity decreases. Because the system is not driving system, forward cascade of the energy makes the energy spectrum lie down and inverse cascade of the helicity makes the helicity spectrum stand up. 3.4 Summary We have developed the Hall-MHD simulation code in a 3D rectangular domain. The Hall term leads the dispersion to the small amplitude wave. The Alfvén whistler wave has a quadratic dispersion and split the Alfvén wave into right-hand polarized electron mode and left-hand polarized ion mode. 51

62 By solving the dispersion relation of the Alfvén whistler wave in a periodic domain, we have validate the simulation code. Nonlinear simulation of the self-organization process in flow-field coupled state have been studied. Comparing the two-fluid relaxed state with that of the single-fluid model, an appreciable flow with a component perpendicular to the magnetic field, which does not exist in the single-fluid relaxed state, was created by the Hall term, and highlights the difference between Hall-MHD and MHD. The relaxation process is realized by minimizing the generalized enstrophy with appropriate adjustment of ion helicity to condensate into the DB field. 52

63 FTCS Runge-Kutta-Gill G(k x) c=1.0, d=0.5 c=0.8, d=0.4 c=0.9, d= k x G(k x) c=1.0, d=0.5 c=1.4, d=0.6 c=1.6, d=0.75 c=1.7, d= k x Figure 3.1: Amplification factors of the FTCS and the Runge-Kutta-Gill scheme. Stability condition demands G(k x) 1 for any k. 53

64 1 0.8 SM(k x) nd: α=0.25 2nd: α=0.5 4th: α= th: α= k x Figure 3.2: Smoothing functions defined by (3.25). To suppress completely the grid scale errors (k x = π), α must be chosen to 0.5 for the second order smoothing, and for the fourth order smoothing. 54

65 ωτα numerical ε=0 ε=0.1 (+,R) ε=0.1 (-,L) ε=0.05 (+,R) ε=0.05 (-,L) analitical ε=0 ε=0.1 (+,R) ε=0.1 (-,L) ε=0.05 (+,R) ε=0.05 (-,L) πRk Figure 3.3: Dispersion relation of the Alfvén whistler wave (2.40). ɛ = 0 corresponds to the shear Alfvén wave, fast/slow mode indicated by +/ shows the electron/ion mode. 55

66 t = 0 [τa] toroidal direction Figure 3.4: Initial condition of the magnetic field with (n 1, n 2 ) = (3, 3). Columns show the isosurfaces of the magnetic field. 56

67 t = 0 [τa] t = 60 [τa] t = 30 [τa] t = 120 [τa] Figure 3.5: Isosurfaces of the toroidal magnetic field at time 30τ A, 60τ A, 90τ A,120τ A for ɛ =

68 t = 0 [τa] t = 60 [τa] t = 30 [τa] t = 120 [τa] Figure 3.6: Isosurfaces of the toroidal magnetic field at time 30τ A, 60τ A, 90τ A,120τ A for ɛ =

69 Energy Magnetic Energy Kinetic Energy Thermal Energy Total Energy time [τa] (a) ε=0.1 Energy Magnetic Energy Kinetic Energy Thermal Energy Total Energy time [τa] (b) ε=0.0 Figure 3.7: Time evolution of energies. The time scale of relaxation is faster in the Hall-MHD than in the MHD. 59

70 Kinetic Energy [VA] ε=0.1 V ε=0.1 V ε=0.1 V ε=0.0 V ε=0.0 V ε=0.0 V time [τa] V / V, V / V [%] ε=0.1 V ε=0.1 V ε=0.0 V ε=0.0 V time [τa] Figure 3.8: Time evolution of kinetic energy. Left panel shows the absolute value of the total, the parallel/perpendicular component of kinetic energies. Right panel shows the percentages of the parallel/perpendicular components to the total kinetic energy. 60

71 (a) distribution of a (b) distribution of b Figure 3.9: Snapshots of the distribution of the alignment coefficients in the Beltrami condition a, b, which correspond to electron and ion. Electron motion aligns well to its corresponding vorticity, while ion flow deviates from its vorticity outside the magnetic columns. 61

72 (a) pressure (b) velocity Figure 3.10: Snapshot of pressure and velocity distribution in the poloidal cross section. Dynamic pressure is much less than static pressure because of large initial beta. 62