Nonlinear Processes in Two-Fluid Plasmas

Ryusuke Numata

December, 2003

Graduate School of Frontier Sciences,

The University of Tokyo

Contents

1 Introduction 1

2 Hall Magnetohydrodynamics and Equilibrium 11

2.1 Hall Magnetohydrodynamic Equations . . . . . . . . . . . . . 12

2.2 Magnetohydrodynamic Waves in Homogeneous Plasmas . . . . 18

2.3 Double Beltrami Equilibrium . . . . . . . . . . . . . . . . . . 20

2.4 Relaxation Process . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4.1 Conservation Laws . . . . . . . . . . . . . . . . . . . . 24

2.4.2 Variational Principle . . . . . . . . . . . . . . . . . . . 26

2.5 Force-Free Equilibrium . . . . . . . . . . . . . . . . . . . . . . 28

2.5.1 Eigenvalue Problem of the Curl Operator . . . . . . . . 28

2.5.2 Minimum Energy State . . . . . . . . . . . . . . . . . . 30

3 Nonlinear Simulation 39

i

3.1 Simulation Code . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.1.1 Discretization and Time Integration . . . . . . . . . . . 40

3.1.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . 44

3.1.3 Numerical Smoothing . . . . . . . . . . . . . . . . . . . 46

3.2 Simulation Model . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.3 Nonlinear Simulation . . . . . . . . . . . . . . . . . . . . . . . 48

3.3.1 Initial Condition and Parameters . . . . . . . . . . . . 48

3.3.2 Relaxed State . . . . . . . . . . . . . . . . . . . . . . . 49

3.3.3 Relaxation Process . . . . . . . . . . . . . . . . . . . . 50

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4 Nonlinear Mechanism of Collisionless Resistivity 65

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.2 Model Equations . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.3 Single Particle Dynamics . . . . . . . . . . . . . . . . . . . . . 71

4.3.1 Chaotic Orbit . . . . . . . . . . . . . . . . . . . . . . . 71

4.3.2 Lyapunov Exponents . . . . . . . . . . . . . . . . . . . 72

4.4 Statistical Distribution and Macroscopic Resistivity . . . . . . 74

4.4.1 Velocity Distribution . . . . . . . . . . . . . . . . . . . 74

4.4.2 Effective Resistivity . . . . . . . . . . . . . . . . . . . . 75

ii

4.5 Fast Magnetic Reconnection . . . . . . . . . . . . . . . . . . . 78

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5 Concluding Remarks 95

iii

List of Figures

1.1 Energy spectrum. (a) Kolmogorov spectrum (b) spectrum of

dual cascade in two-dimensional turbulence. . . . . . . . . . . 9

1.2 Speedup trend of computers. . . . . . . . . . . . . . . . . . . . 10

2.1 Energy of the force-free solution in a cylinder. The arrows in-

dicate 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 ra-

tio = 3). The lowest magnetic energy state is the axisymmetric

state for H ≤ 8.30, and the helical state of the mode (1, 3) for

H > 8.30. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

iv

2.2 Typical structures of the force-free state in a cylindrical geom-

etry. Figures show the isosurface and the contour plot in the

poloidal cross section of Bz. (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

3.1 Amplification factors of the FTCS and the Runge-Kutta-Gill

scheme. Stability condition demands |G(k∆x)| ≤ 1 for any k. 53

3.2 Smoothing functions defined by (3.25). To suppress com-

pletely the grid scale errors (k∆x = π), α must be chosen to

0.5 for the second order smoothing, and 0.375 for the fourth

order smoothing. . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.3 Dispersion relation of the Alfven whistler wave (2.40). ε = 0

corresponds to the shear Alfven wave, fast/slow mode indi-

cated by +/− shows the electron/ion mode. . . . . . . . . . . 55

3.4 Initial condition of the magnetic field with (n1, n2) = (3, 3).

Columns show the isosurfaces of the magnetic field. . . . . . . 56

3.5 Isosurfaces of the toroidal magnetic field at time 30τA, 60τA,

90τA,120τA for ε = 0.1. . . . . . . . . . . . . . . . . . . . . . 57

v

3.6 Isosurfaces of the toroidal magnetic field at time 30τA, 60τA,

90τA,120τA for ε = 0.0. . . . . . . . . . . . . . . . . . . . . . 58

3.7 Time evolution of energies. The time scale of relaxation is

faster in the Hall-MHD than in the MHD. . . . . . . . . . . . 59

3.8 Time evolution of kinetic energy. Left panel shows the abso-

lute 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

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

3.10 Snapshot of pressure and velocity distribution in the poloidal

cross section. Dynamic pressure is much less than static pres-

sure because of large initial beta. . . . . . . . . . . . . . . . . 62

vi

3.11 Time evolution of energy (E), helicities (H1, H′2) and enstro-

phy (F ′). H ′2 decrease faster than E. H1 keeps its conservation

better than E and H ′2. F ′ initially grows because F ′ is not a

constant of motion. . . . . . . . . . . . . . . . . . . . . . . . . 63

3.12 Spectrum of magnetic energy and magnetic helicity. Energy

cascades down to high wavenumber modes, and helicity cas-

cade up to low wavenumber modes. . . . . . . . . . . . . . . . 64

4.1 A Y-shape magnetic field with ` ≡ `y/`x = 2 projected onto

the x-y plane. (x and y coordinates are normalized to the

system size `x). The chaos region (hatched region) is defined

by using the local Lyapunov exponents in Sec. 4.3.2. . . . . . 85

4.2 Typical particle orbit in a Y-shape magnetic field with ` = 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 mA’s. In (b), the

particle is swept out before it is randomized sufficiently. . . . . 86

4.3 Average staying time in the chaos region (τ1) as a function of

mA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

vii

4.4 Local Lyapunov exponents for different subdomains Ω(R) (mA =

0.002). We define the chaos region such that the local Lya-

punov exponents have a plateau R . 1.0. . . . . . . . . . . . . 88

4.5 Local Lyapunov exponents for different mA’s. For larger mA,

the local Lyapunov exponents are strongly damped and have

no plateau region. . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.6 Velocity distributions in the chaos region (mA = 0.001, ` = 0).

(a)-(c): distributions of vx, vy, vz, after initial randomization

phase with the Gaussian fitting curves. (d)-(f): temporal evo-

lutions of the distributions of vx, vy, vz. . . . . . . . . . . . . . 90

4.7 Standard deviations of the velocity distribution (mA = 0.001, ` =

0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.8 Evolution of the average velocity in the direction of the electric

field (mA = 0.001, ` = 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). . . . . . . . . . . . . . . . . . . . . 92

viii

4.9 Evolution of the number of particles in the chaos region. The

number decreases exponentially. . . . . . . . . . . . . . . . . . 93

4.10 Petschek-type fast reconnection model. The regions (I) and

(D) are the ideal MHD and dissipation regions, respectively. . 94

5.1 Hierarchy of the scale in the magnetic reconnection process. . 98

ix

Chapter 1

Introduction

1

Nonlinear phenomena in two-fluid plasmas are studied in this thesis. Plas-

mas, which are constituted of electrons and positively charged ions, have a

huge hierarchy of structures – particle/fluid pictures of electrons/ions. Si-

multaneous existence, and interaction of structures on disparate scales are

the defining characteristic of complexity. The scale hierarchy is likely to pre-

vent 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 coher-

ent structures. The structure spontaneously arise from homogeneous tur-

bulence 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 rever-

sal 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

plasma is now understood by the variational principle in the framework of

the magnetohydrodynamics (MHD) theory. The motion of plasma is char-

acterized 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 Alfven

wave effect [12–15]. The theory of force-free state is also applied to the mag-

netic 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 de-

cay process of dynamical invariants in the turbulent phase. In a fully de-

veloped three-dimensional (3D) high-Reynolds number hydrodynamic tur-

bulence, the energy spectrum shows power-law property which reflect the

similarity. Phenomenological argument by Kolmogorov [19] gives the energy

3

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 Alfven

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 simu-

lations (DNSs) [29, 30] require computer memory proportional to the nine-

fourth 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 com-

puter (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 ' 104 by Kaneda et al. [35] performed by the earth simula-

4

tor [36]. In the MHD turbulence, the problem is more serious because of the

nonlinear interaction between the magnetic field and the flow field. Muller

and Biskamp [37] have done MHD turbulence simulation of Re ' 103.

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 per-

pendicular to the magnetic field, the single-fluid MHD model is not ade-

quate for this purpose. One of the problems is a singularity in an equilib-

rium 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

singularity when strong flow exists. Furthermore, if we include compressibil-

ity, 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

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 in-

vestigated by numerical simulations. It is theoretically predicted that the

so-called “double Beltrami” equilibrium [53] may be self-organized in two-

fluid 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 rigor-

ous 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, conserva-

tion 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

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 coor-

dinate 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 re-

connection [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 unphys-

ical 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

E(k

)

k

dissipation

energy input

inertial range k-5/3E

(k)

k dissipation

energy input

k-5/3

k-3

inverse cascade

forward cascade

(a) (b)

Figure 1.1: Energy spectrum. (a) Kolmogorov spectrum (b) spectrum of

dual cascade in two-dimensional turbulence.

9

Figure 1.2: Speedup trend of computers.

10

Chapter 2

Hall Magnetohydrodynamics

and Equilibrium

11

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

∂tne +∇ · (neV e) = 0, (2.1)

∂tni +∇ · (niV i) = 0, (2.2)

nme [∂tV e + (V e · ∇)V e]

= −en(E + V e ×B)−∇pe −R +∇ ·←→Π e, (2.3)

nmi[∂tV i + (V i · ∇)V i

]

= en(E + V i ×B)−∇pi + R +∇ ·←→Π i, (2.4)

R = −ζ(V i − V e), (2.5)

where ne,i are the number densities, V e,i are the flow velocities, me,i are the

masses, pe,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

positive constant. The viscous stress tensor is given by

←→Π = ν

[∇V + t(∇V )− 2

3δij∇ · V

]+ ν ′δij∇ · V , (2.6)

where ν and ν ′ are the viscosity (ν ′ is often called the second viscosity), t

denotes a transpose and δij 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

3ν∇(∇ · V ). (2.7)

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

tions

∂tB = −∇×E, (2.8)

∂tE = c2∇×B − c2µ0j, (2.9)

∇ ·B = 0, (2.10)

∇ ·E = 0, (2.11)

j = en(V i − V e), (2.12)

where j is the current density, µo 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 ≈ ni = n.

(2.1)-(2.12) and appropriate equations of state for pe,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

viscous term in (2.3). Then, we obtain generalized Ohm’s law,

E + V e ×B = − 1

en∇pe − 1

enR. (2.15)

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 = µ0j = µ0en(V i − V e). (2.16)

Inserting (2.15) into (2.4) and (2.8), and eliminate V e using above equation,

we obtain

nmi[∂tV i + (V i · ∇)V i

]

=1

µ0

(∇×B)×B −∇(pi + pe) +∇ ·←→Π i, (2.17)

∂tB = ∇×[(V i −

1

en∇×B)×B

]

− 1

en2∇n×∇pe −∇× (ηj) , (2.18)

where we have defined the resistivity η ≡ ζµ0/e2n2.

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

the ideal plasma must be supposed adiabatic. The adiabatic relation

p/ργ = const., (2.19)

leads the time evolution of pressure

∂tp + (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,

∂tp + (V · ∇)p + γp∇ · V = S, (2.21)

where S is the heat source term given by

S = (γ − 1)

[ν|∇ × V |2 +

4

3ν|∇ · 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 B0 and density n0 (the ion mass mi is assumed to be unity

such that ρ0 = n0) as

x → Lx, B → B0B, n → n0n,

V → VAV , t → τAt, p → (B20/µ0)p,

(2.23)

16

where VA ≡ B0/√

µ0n0mi is the Alfven velocity, τA ≡ L/VA. The normalized

Hall-MHD equations become

∂tn +∇ · (nV ) = 0, (2.24)

n[∂tV + (V · ∇)V ]

= j ×B −∇p +1

Re

(∆V +

1

3∇(∇ · V )

), (2.25)

∂tB = ∇×[(

V − ε

n∇×B

)×B

]+

1

Rm∆B, (2.26)

∂tp + (V · ∇)p + γp∇ · V = S, (2.27)

S = (γ − 1)

[1

Re|∇ × V |2 +

4

3Re|∇ · V |2 +

1

Rm|∇ ×B|2

], (2.28)

∇×B = j = n(V − V e), (2.29)

where Re ≡ VAL/ν is the Reynolds number, Rm ≡ VAL/η is the magnetic

Reynolds number and ε ≡ δi/L (δi = c/ωpi is the ion collisionless skin depth).

We have neglected the electron pressure for simplicity and omit the subscript

for ions.

17

2.2 Magnetohydrodynamic Waves in Homo-

geneous 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

∂tn = −n∇ · V , (2.30)

∂tp = −γp∇ · V , (2.31)

∂tV = − 1

n∇p +

1

n(∇× B)× B, (2.32)

∂tB = ∇× (V × B)− ε

n∇× ((∇× B)× B). (2.33)

Variables with tilde “ ” denote fluctuations. Assume exp(i(k · x − ωt)) de-

pendence of fluctuations and replace “∂t” by “−iω” and “∇” by “ik”, we

obtain the following eigenvalue equation,

−iωξ = Aξ, (2.34)

18

where ξ = t(n, p, vx, vy, vz, Bx, By, Bz) and

A =

0 0 −ikxn −ikyn −ikzn 0 0 0

0 0 −ikxγp −ikyγp −ikzγp 0 0 0

0 − ikx

n0 0 0 ikzB

n0 − ikxB

n

0 − iky

n0 0 0 0 ikzB

n− ikyB

n

0 − ikz

n0 0 0 0 0 0

0 0 ikzB 0 0 0 − εk2zBn

εkykzB

n

0 0 0 ikzB 0 εk2zBn

0 − εkzkxBn

0 0 −ikxB −ikyB 0 − εkykzB

nεkzkxB

n0

.

(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+ iωδij, we obtain the dispersion relation [66,67]

ω(ω2 − V 2Ak2

z)[ω4 − (C2

s + V 2A)k2ω2 + C2

s V2Ak2

zk2]

= ε2V 2Ak2

zk2ω3(ω2 − C2

s k2), (2.36)

where k =√

k2x + k2

y + k2z , VA ≡ B/

√n, Cs ≡

√γp/n are the Alfven ve-

locity and the sound velocity of the uniform equilibrium in the normalized

unit, respectively. If ε = 0, we recover the dispersion relation of the MHD

19

equations, which gives the entropy wave

ω = 0, (2.37)

the shear Alfven wave

ω2 = V 2

Ak2z , (2.38)

and the fast and slow magneto-sonic waves

ω2 =1

2

[C2

s + V 2

A)k2 ±((C2

s + V 2

A)2k4 − 4C2sV

2Ak2

zk2) 1

2

]. (2.39)

If we consider waves traveling along the magnetic field, i.e. k = kz, the

sound wave (ω = ±Csk) decouples. The dispersion relation of the shear

Alfven wave is modified by the Hall term as

ω2 − V 2

Ak2z = ±εVAk2

zω, (2.40)

which is called the Alfven 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− ∂tA where

20

φ is the scalar potential, (2.26), (2.25) translate into

∂tA = (V − ε∇×B)×B −∇(φ− εpe), (2.41)

∂t(A + εV ) = V × (B + ε∇× V )−∇(εV 2

2+ εpi + φ). (2.42)

Taking the curl of (2.41) and (2.42), we can cast the Hall-MHD equations

into coupled vortex equations,

∂tωj −∇× (U j × ωj) = 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

ω1 = B, ω2 = B + ε∇× V ,

U 1 = V − ε∇×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 corre-

sponding flows,

U j = µjωj (j = 1, 2). (2.45)

Writing a = 1/µ1, b = 1/µ2, and assuming a,b are constants, the Beltrami

condition translates into a system of simultaneous linear equations in B and

21

V ,

B = a(V − ε∇×B), (2.46)

B + ε∇× 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

2ε

[(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 + ελ+)C+G+ + (a−1 + ελ−)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 × ωj = 0 = ∇ϕj (j = 1, 2). (2.52)

22

where ϕj is a certain scalar field corresponding to the energy density given

by

φ1 = φ− εpe (2.53)

φ2 = φ + εpi + εV 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(pi+pe). The re-

lation 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

leads the relaxed state in the MHD model. The variation

δ(Em − 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 con-

stants 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].

2.4.1 Conservation Laws

The total energy in the system is given by sum of the magnetic energy Em,

the kinetic energy Ek, and the thermal energy Et,

E = Em + Ek + Et =

∫

Ω

1

2B2dx +

∫

Ω

1

2nV 2dx +

∫

Ω

p

γ − 1dx. (2.58)

24

The time derivative of the energies are calculated as,

d

dtEm = −

∫

∂Ω

E ×B · dS −∫

Ω

V · j ×Bdx− 1

Rm

∫

Ω

j2dx, (2.59)

d

dtEk =

∫

∂Ω

[(−1

2nV 2 +

4

3Re∇ · V

)V + V × (∇× V )

]· dS

+

∫

Ω

V · j ×Bdx−∫

Ω

V · ∇pdx

− 1

Re

∫

Ω

|∇ × V |2dx− 4

3Re

∫

Ω

|∇ · V |2dx, (2.60)

d

dtEt = −

∫

∂Ω

pV

γ − 1· dS −

∫

Ω

p∇ · V dx +

∫

Ω

S

γ − 1dx. (2.61)

Then, the time derivative of the total energy is given by

dE

dt= −

∫

∂Ω

E ×B · dS

−∫

∂Ω

(1

2nV 2 +

γ

γ − 1p− 4

3Re(∇ · V )

)V · dS

+

∫

∂Ω

V × (∇× V ) · dS

−∫

Ω

(1

Re|∇ × V |+ 4

3Re|∇ · V |+ 1

Rmj2 − S

γ − 1

)dx.(2.62)

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 = Em + Ek, however, monotonically decrease if the dissipations

exist.

25

The generalized helicities, defined by

Hj =

∫

Ω

ωj · (curl−1ωj)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

∂t· (curl−1ω)dx +

∫

Ω

ω · (curl−1∂ω

∂t)dx

=

∫

Ω

∇× (U × ω) · (curl−1ω)dx +

∫

Ω

ω · (U × ω)dx

=

∫

∂Ω

(U × ω)× (curl−1ω) · dS. (2.64)

The boundary condition,

n×A = 0, n× V = 0 on ∂Ω, (2.65)

yields the conservation of the helicities.

2.4.2 Variational Principle

There are three global ideal invariants in the incompressible Hall-MHD:

E =1

2

∫

Ω

(B2 + V 2)dx, (2.66)

H1 =

∫

Ω

A ·Bdx, (2.67)

H2 =

∫

Ω

(A + εV ) · (B + ε∇× V )dx, (2.68)

26

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

2

∫

Ω

|∇ × (A + εV )|2dx, (2.69)

with keeping E, H1, and H2 constant is carried out through the variation

δ(F − µ0E − µ1H1 − µ2H2) = 0, (2.70)

where µ0, µ1, µ2 are Lagrange multipliers. The Euler-Lagrange equation be-

comes

(curl− Λ1)(curl− Λ2)(curl− Λ3)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, H1, H2 to condense the “triple Beltrami field” into the double

Beltrami field given by two eigenfunctions. The adjustment condition is given

by

E − µ′1H1 − µ′2H2 = 0. (2.72)

The Euler-Lagrange equation of the variation δ(E−µ′1H1−µ′2H2) = 0, which

is equivalent to the solution of (2.72), gives the DB equation (2.48). The

27

relaxation process is realized by minimizing perturbations, which is scaled

by the generalized enstrophy F , with appropriate adjustment of macroscopic

variables E,H1, H2.

2.5 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 satisfy-

ing given boundary conditions. First, we study the eigenfunction expansion

of solenoidal vector fields, and then, discuss the minimum energy state.

2.5.1 Eigenvalue Problem of the Curl Operator

We consider an eigenvalue problem

∇× u = λu, (2.73)

for a solenoidal field u ∈ L2σ

L2σ(Ω) ≡ u ∈ L2(Ω);∇ · u = 0 in Ω,n · u = 0 on ∂Ω, (2.74)

28

in a bounded 3D domain Ω with smooth boundary ∂Ω. We assume that Ω is

multiply connected with cuts Σj [j = 1, · · · , ν (the first Betti number)], i.e.,

Ω\ ∪ Σj 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 ∈ L2H ,

L2H(Ω) ≡ h ∈ L2(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 uΣ. The latter component is a member of the Hilbert

space

L2Σ(Ω) ≡ ∇×w; w ∈ H1(Ω),∇ ·w = 0 in Ω, n×w = 0 on ∂Ω. (2.76)

The Hilbert space L2Σ is identical to the Hilbert space L2

S where

L2S(Ω) ≡ u ∈ L2(Ω);∇ · u = 0 in Ω, n · u = 0 on ∂Ω,

Φj = 0 (j = 1, · · · ν), (2.77)

and

Φj =

∫

Σj

u · dS (j = 1, · · · , ν) (2.78)

29

is the flux through the cut Σj. The decomposition

L2σ = L2

H ⊕ L2Σ (2.79)

is called the Hodge-Kodaira decomposition. The eigenvalue problem now

reads as

∇× uΣ = λ(uΣ + 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 L2Σ(Ω). For λ′ /∈ σp, we must invoke h 6= 0 and find a solution

uΣ = (curl− λ′)−1h, where the curl denotes the self-adjoint curl operator.

2.5.2 Minimum Energy State

The complete orthogonal set Ψj (∇×Ψj = λjΨj) spanning L2Σ(Ω) allows

B = BΣ + Bh =∑

j

bjΨj + Bh. (2.81)

30

Defining B−Bh = ∇×AΣ and Ag = A−AΣ, we may write Bh = ∇×Ag.

The magnetic energy and the magnetic helicity are given by

E =1

2

∫

Ω

B2dx =1

2

∫

Ω

B2Σdx +

1

2

∫

Ω

B2

hdx, (2.82)

H =

∫

Ω

A ·Bdx

=

∫

Ω

AΣ ·BΣdx + 2

∫

Ω

Ag ·BΣdx +

∫

Ω

Ag ·Bhdx, (2.83)

where the boundary condition n×AΣ = 0 and the orthogonality of BΣ and

Bh are used.

The force-free condition demands BΣ = bΨ (∇ ×Ψ = λΨ), and AΣ =

bλΨ. Each term in (2.82), (2.83) is calculated as,

∫

Ω

AΣ ·BΣdx =1

λ

∫

Ω

B2Σdx, (2.84)

∫

Ω

Ag ·BΣdx =

∫

Ω

AΣ ·Bhdx +

∫

∂Ω

AΣ ×Ag · dS

=1

λ

∫

Ω

BΣ ·Bhdx +

∫

∂Ω

AΣ ×Ag · dS = 0, (2.85)

∫

Ω

Ag ·Bhdx =

∫

Ω

Ag ·(

1

λ∇×BΣ −BΣ

)dx

=1

λ

∫

Ω

Bh ·BΣdx +1

λ

∫

∂Ω

BΣ ×Ag · dS −∫

Ω

Ag ·BΣdx

= −∫

∂Ω

Ag ×AΣ · dS = 0. (2.86)

31

Then, we obtain the relations,

H =

∫

Ω

AΣ ·BΣdx, (2.87)

E =1

2

∫

Ω

B2Σdx +

1

2

∫

Ω

B2

hdx

=λ

2H +

1

2

∫

∂Ω

Ag ×Bh · dS. (2.88)

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

Ψt =

∫

SpB · dS =

∫

SpBh · dS, (2.89)

Ψp =

∫

St

B · dS, (2.90)

Ip =

∫

St

∇×B · dS. (2.91)

where Sp and St are the poloidal and the toroidal cross section, respectively.

The poloidal/toroidal cross section is normal to the toroidal/poloidal direc-

tion. By introducing the multi-valued scalar potential φ, which describes the

harmonic field as Bh = ∇φ, the energy of the harmonic field becomes

1

2

∫

Ω

B2

hdx =1

2

∫

Ω

∇φ ·Bhdx

=1

2[φ]

∫

SpBh · dS =

1

2[φ]Ψt, (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

direction `t,

[φ] =

∫

`t

∇φ · d`

=

∫

`t

Bh · d` =

∫

`t

(1

λ∇×BΣ −BΣ) · d`

=1

λ

∫

St

∇×∇×BΣ · dS −∫

St

∇×BΣ · dS

=

∫

St

∇×BΣ · dS − λ

∫

St

B · dS

= Ip − λΨp. (2.93)

Thus, the relation between the magnetic energy and the helicity is expressed

by

E =λ

2H +

1

2(Ip − λΨp)Ψt. (2.94)

Cylindrical geometry

The Chandrasekhar-Kendall (C-K) [70] function is defined by

umn = λ(∇ψmn ×∇z) +∇× (∇ψmn ×∇z), (2.95)

where

ψmn = Jm(µr)ei(mθ−kz) (k = nπ/L; mn ∈ 0, 1, 2, · · · ), (2.96)

λ = ±(µ2 + k2)1/2, (2.97)

33

and Jm is the Bessel function. The general solution of the force-free equilib-

rium in a periodic cylinder with the radius a and the length L is given by

the C-K function,

B =∑m,n

umn =∑m,n

Bmn

(r)ei(mθ−kz), (2.98)

Bmn

(r) =

ibmnΛmnr (µr)

−bmnΛmnθ (µr)

bmnJm(µr)

, (2.99)

Λmnr (µr) =

1

2

√∣∣∣∣λ + k

λ− k

∣∣∣∣Jm−1(µr) +1

2

√∣∣∣∣λ− k

λ + k

∣∣∣∣Jm+1(µr), (2.100)

Λmnθ (µr) =

1

2

√∣∣∣∣λ + k

λ− k

∣∣∣∣Jm−1(µr)− 1

2

√∣∣∣∣λ− k

λ + k

∣∣∣∣Jm+1(µr), (2.101)

The eigenvalue is determined by the boundary condition n ·B|r=a = 0, i.e.,

Λmnr (µa) =

1

2

√∣∣∣∣λ + k

λ− k

∣∣∣∣Jm−1(µa) +1

2

√∣∣∣∣λ− k

λ + k

∣∣∣∣Jm+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

are given by

Ψt = 2πab00

λJ1(λa), (2.103)

E00 = 2πL(b00a)2

2

(J2

1 (λa) + J20 (λa)− 1

λaJ0(λa)J1(λa)

), (2.104)

H00 = 2πL(b00a)2λ

(J2

1 (λa) + J20 (λa)− 1

λaJ0(λa)J1(λa)

),

−Lb00

λJ0(λa)Ψt. (2.105)

The normalization of the energy E ≡ 2 a2

L/2πΨ2tE and the helicity H ≡ a

L/2πΨ2tH

yields

E00 =(λa)2

J1(λa)2

(J1(λa)2 + J0(λa)2 − 1

λaJ0(λa)J1(λa)

), (2.106)

H00 =1

λE00 − J0(λa)

J1(λa). (2.107)

In Fig. 2.1, the solid line is a plot of E(λa) vs H(λa) for the (m,n) = (0, 0)

mode. Now suppose we superpose a discrete eigenmode on the (m,n) = (0, 0)

mode. Evaluating H for such a mode, we find that it is of the form

H(λa) = H00(λa) +

(bmn

b00

)2

Hmn(λa), (2.108)

and

Hmn =1

λaEmn (2.109)

because (m,n) 6= (0, 0) modes contain no toroidal flux. By variating bmn

with fixed λa, we draw straight lines originating on the (m,n) = (0, 0) curve

35

(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

H > 8.30 [6, 71,72]. Typical spatial structures are shown in Fig. 2.2.

36

0

20

40

60

80

100

0 5 10 15 20 25 30

(m,n)=(0,0)(m,n)=(1,3) λa=3.13(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=6.09

H

E

8.30

Figure 2.1: Energy of the force-free solution in a cylinder. The arrows indi-

cate 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 bound-

ary condition (the aspect ratio = 3). The lowest magnetic energy state is the

axisymmetric state for H ≤ 8.30, and the helical state of the mode (1, 3) for

H > 8.30.

37

(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 Bz. (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

Chapter 3

Nonlinear Simulation

39

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 vector-

parallel 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.

3.1.1 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 quan-

tities on each grid. The most simple, and common method is the finite differ-

ence 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 fni,j,k. The Taylor ex-

pansion of f with respect to x is given by

f(x + ∆x) = f(x) +df

dx

∣∣∣∣x

∆x +d2f

dx2

∣∣∣∣x

(∆x)2

2+O((∆x)3), (3.1)

or in the discrete form,

fi+1 = fi +df

dx

∣∣∣∣i

∆x +d2f

dx2

∣∣∣∣i

(∆x)2

2+O((∆x)3). (3.2)

40

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

∣∣∣∣i

=fi+1 − fi−1

2∆x, (3.3)

d2f

dx2

∣∣∣∣i

=fi+1 − 2fi + fi−1

(∆x)2. (3.4)

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,

fn+1 = fn + ∆tL(fn), (3.6)

fn+1 = fn−1 + 2∆tL(fn). (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

Table 3.1: Typical methods to evaluate the derivatives.

df

dx= fi+1−fi

∆x+O(∆x)

df

dx= fi−fi−1

∆x+O(∆x)

df

dx= fi+1−fi−1

2∆x+O((∆x)2)

df

dx= −fi+2+4fi+1−3fi

2∆x+O((∆x)2)

df

dx= 3fi−4fi−1+fi−2

2∆x+O((∆x)2)

df

dx= 2fi+1+3fi−6fi−1+fi−2

6∆x+O((∆x)3)

df

dx= −fi+2+6fi+1−3fi−2fi−1

6∆x+O((∆x)3)

df

dx= −fi+2+8fi+1−8fi−1+fi−2

12∆x+O((∆x)4)

d2f

dx2= fi+2−2fi+1+fi

(∆x)2+O(∆x)

d2f

dx2= fi−2fi−1+fi−2

(∆x)2+O(∆x)

d2f

dx2= fi+1−2fi+fi−1

(∆x)2+O((∆x)2)

d2f

dx2= −fi+3+4fi+2−5fi+1+2fi

(∆x)2+O((∆x)2)

d2f

dx2= 2fi−5fi−1+4fi−2−fi−3

(∆x)2+O((∆x)2)

d2f

dx2= −fi+2+16fi+1−30fi+16fi−1−fi−2

12(∆x)2+O((∆x)4)

42

and n + 1, and achieves the higher order accuracy. [73, 74]. We employ the

fourth-order RKG method in the simulation code, given by

fn+1 = fn

+∆t

6[L(fn) + (2−

√2)L(f (1)) + (2 +

√2)L(f (2)) + L(f (3))], (3.8)

where the intermediate steps are evaluated by

f (1) = fn +∆t

2L(fn), (3.9)

f (2) = f (1) +∆t

2L(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 numer-

ical solution to the solution of the original equation is “stable” and “consis-

tent” 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

3.1.2 Stability Analysis

We consider, for simplicity, 1D advection-diffusion equation with constant

coefficients,

∂f

∂t+ c

∂f

∂x= d

∂2f

∂x2, (3.12)

where c, d are constants. The FTCS (forward in time central difference in

space) scheme,

fn+1i − fn

i

∆t+ c

fni+1 − fn

i−1

2∆x= d

fni+1 − 2fn

i + fni−1

∆x2, (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 fni

fni =

∑

k

fn(k)eik∆xi (3.14)

(where fn(k) is the amplitude of the Fourier mode k at time step n) into

(3.13) yields

fn+1eik∆xi = fneik∆xi

[1− c∆t

2∆x(eik∆x − e−ik∆x)

+d∆t

∆x2(eik∆x − 2 + e−ik∆x)

]. (3.15)

44

By introducing C ≡ c∆t/∆x (the Courant number), D ≡ d∆t/∆x2 (the

diffusion number), the amplification factor G ≡ fn+1/fn 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 ≤ C2 ≤ 4D2 ≤ 2D ≤ 1, (3.17)

∆t ≤ ∆x

C, ∆t ≤ ∆x2

2D, ∆t ≤ 2D

C2, (3.18)

∆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. 3.1. 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 Cx/∆x + Cy/∆y + Cz/∆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

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,

fsmi = (1− α)fi + 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 =fi+1 + fi−1

2, (3.21)

4th order : 〈f〉i =−fi+2 + 4fi+1 + 4fi−1 − fi−2

6. (3.22)

The effect of smoothing is evaluated by the Fourier decomposition. Fourier

decomposition of (3.20) yields

fsmi = (1− α)

N∑

k=1

f(k)eik∆xi +α

2

N∑

k=1

f(k)eik∆xi +N∑

k=1

f(k)eik∆xi

= (1− α)N∑

k=1

f(k)eik∆xi

+α

2

N−1∑

k=0

f(k)eik∆x(i−1) +N+1∑

k=2

f(k)eik∆x(i+1)

(3.23)

If we assume periodicity, we obtain

fsmi =

N∑

k=1

f(k)[(1− α)eik∆xi +α

2(eik∆x(i−1) + eik∆x(i+1))]

=N∑

k=1

f(k)eik∆xi[(1− α) + α cos(k∆x)]. (3.24)

46

By the smoothing procedure, each Fourier mode f(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. 3.2. 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 = 0.5.

3.2 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 ε = 0 at

the wall.

47

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 ε = 0, 0.1, 0.05. Results of the numerical

code agree well with the analytical curves. Thus, we consider that the code

works well.

3.3 Nonlinear Simulation

3.3.1 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];

B0 =

− 1k0

(k1B1 cos k2x sin k1y + k2B2 cos k1x sin k2y)

1k0

(k2B1 sin k2x cos k1y + k1B2 sin k1x cos k2y)

B1 cos k2x cos k1y + B2 cos k1x cos k2y,

, (3.27)

where B1, B2 are the amplitudes of two kinds of Fourier modes, k1 = n1π/(2a),

k2 = n2π/(2a), and n1, n2 are arbitrary integers. Figure 3.4 shows the iso-

surface of the toroidal magnetic field. Columns in different colors indicate

that the directions are opposite. A uniform density n0 = 1, and uniform

48

pressure are assumed. The amplitude of the uniform pressure is given by a

parameter β =∫

pdx/∫

B2dx. The initial condition has a flow such that

V 0 = (0, 0,MABz), where MA is the Alfven Mach number.

We carried out two simulation runs. The simulation domain is imple-

mented on 129×129×256 point grids. The parameters are β = 3, MA = 0.5,

and (a) ε = 0.1, (b) ε = 0.

3.3.2 Relaxed State

Figure 3.5 and 3.6 show snapshots of isosurfaces of the toroidal magnetic field

at t = 30τA, 60τA, 90τA, and 120τA. 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). Fig-

ure 3.7 shows the time evolution of the energies. We see that the magnetic

energy decreases rapidly in the turbulent phase ((a) t ' 20 ∼ 40τA, (b)

t ' 30 ∼ 70τA), 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 ε, and agree

with the Taylor’s prediction. However, there exists appreciable flow perpen-

49

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. 3.8. 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 ε = 0.1. The Beltrami conditions are satisfied in the

magnetic column. The perpendicular component of the velocity is originated

from the Hall current,

V⊥ ' ε

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 = 50τA. 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.

3.3.3 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 H1, H ′2 ≡ H2 −H1 and the

50

generalized enstrophy F ′ = F−Em normalized by the initial values. H ′2 is the

most fragile quantity, while H1 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. 3.12. The low

wavenumber mode of energy decrease, while high wavenumber mode of he-

licity 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 do-

main. The Hall term leads the dispersion to the small amplitude wave. The

Alfven whistler wave has a quadratic dispersion and split the Alfven wave

into right-hand polarized electron mode and left-hand polarized ion mode.

51

By solving the dispersion relation of the Alfven 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

0

0.2

0.4

0.6

0.8

1

1.2

|G

(k∆

x)|

c=1.0, d=0.5c=0.8, d=0.4c=0.9, d=0.3

0 1 2 3 4 5 6

k∆x

0

0.2

0.4

0.6

0.8

1

1.2

|G

(k∆

x)|

0 1 2 3 4 5 6

k∆x

c=1.4, d=0.6c=1.6, d=0.75c=1.7, d=0.8

c=1.0, d=0.5

FTCS Runge-Kutta-Gill

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

SM

(k∆x

)

k∆x

2nd: α=0.25 2nd: α=0.5 4th: α=0.3754th: α=0.1

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

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 0.375 for the fourth order smoothing.

54

ε=0.05 (+,R)

ωτΑ

2πRk02468

1012141618

0 2 4 6 8 10

numerical analitical

ε=0ε=0.1 (+,R)ε=0.1 (-,L)

ε=0.05 (-,L)

ε=0 ε=0.1 (+,R)ε=0.1 (-,L)

ε=0.05 (+,R)ε=0.05 (-,L)

Figure 3.3: Dispersion relation of the Alfven whistler wave (2.40). ε = 0

corresponds to the shear Alfven wave, fast/slow mode indicated by +/−

shows the electron/ion mode.

55

t = 0 [τA]toroidal direction

Figure 3.4: Initial condition of the magnetic field with (n1, n2) = (3, 3).

Columns show the isosurfaces of the magnetic field.

56

t = 0 [τA] t = 60 [τA]

t = 30 [τA] t = 120 [τA]

Figure 3.5: Isosurfaces of the toroidal magnetic field at time 30τA, 60τA,

90τA,120τA for ε = 0.1.

57

t = 0 [τA] t = 60 [τA]

t = 30 [τA] t = 120 [τA]

Figure 3.6: Isosurfaces of the toroidal magnetic field at time 30τA, 60τA,

90τA,120τA for ε = 0.0.

58

0

5

10

15

20

25

30

35

0 50 100 150 200 250

time [τA]

Ene

rgy Magnetic Energy

Kinetic EnergyThermal EnergyTotal Energy

Magnetic EnergyKinetic EnergyThermal EnergyTotal Energy

0

5

10

15

20

25

30

35

0 50 100 150 200 250

time [τA]

Ene

rgy

(a) ε=0.1 (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

0102030405060708090

100

0 20 40 60 80 100 120

time [τA]

ε=0.1 |V|ε=0.1 |V|ε=0.0 |V|ε=0.0 |V|

|V|/

|V|,

|V|/

|V|

[%]

ε=0.1 |V|

ε=0.0 |V|

ε=0.1 |V|ε=0.1 |V|

ε=0.0 |V|ε=0.0 |V|

0 20 40 60 80 100 1200

0.05

0.1

0.15

0.2

0.25

time [τA]

Kin

etic

Ene

rgy

[VA]2

22

22

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

(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

(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

H2’

time [τA]0 20 40 60 80 100 120 140 160 180 200

0

0.5

1

1.5

2

2.5

3

3.5

F’

H1

E

Figure 3.11: Time evolution of energy (E), helicities (H1, H′2) and enstrophy

(F ′). H ′2 decrease faster than E. H1 keeps its conservation better than E

and H ′2. F ′ initially grows because F ′ is not a constant of motion.

63

1 10 100 1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

|ka|

E(k

a)

t=30τΑ

t=40τΑt=100τΑ

|ka|

H(k

a)

1 10 100 1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

t=30τΑt=40τΑ

t=100τΑ

Figure 3.12: Spectrum of magnetic energy and magnetic helicity. Energy

cascades down to high wavenumber modes, and helicity cascade up to low

wavenumber modes.

64

Chapter 4

Nonlinear Mechanism of

Collisionless Resistivity

65

4.1 Introduction

Spatial variations of electromagnetic fields yield nonlinearity in charged par-

ticle dynamics. Chaotic motion of particles is an important mechanism of

producing resistivity in an almost collisionless plasma [55,77,78]. A strongly

inhomogeneous magnetic field including null points breaks the conservation

of adiabatic invariants. The increase of the degree of freedom can result in

chaotic motion of particles. The mixing effect of chaos brings about rapid

increase of the kinetic entropy in a collisionless plasma, which, however, is

not sufficient to yield a diffusion-type dissipation. When a test particle is

confined in a bounded domain of the phase space, the second cumulant of the

velocity distribution saturates after the initial mixing phase, and hence, the

diffusion constant (the time derivative of the second cumulant) diminishes

to zero. However, in an open system where particles can convect into/out a

chaotic region of the phase space (either through coordinate or momentum

axes), particles are heated locally during a certain staying time in the chaos

region, and continuous dissipation process is achieved there [55].

In this chapter, we study the motion of particles in a Y-shape magnetic

field with perpendicular electric field (Fig. 4.1). If particles are far from

the magnetic null point, they are magnetized and describe ordered E ×

66

B drift orbits. However, motion of particles become chaotic in a certain

neighborhood of the null point. By analyzing motion of many particles (we

consider independent particles ignoring collisions), we observe collisionless

heating of particles in the chaos region. This is in marked contrast to the

motion of magnetized particles which cannot gain energy from a stationary

electric field because of the periodicity of motion. Evaluating the average

velocity of particles in the direction parallel to the electric field, we may

estimate the effective collisionless resistivity.

This theory may be applied to various collisionless magnetic diffusion

phenomena, especially to fast magnetic reconnections leading to changes of

magnetic topologies [55, 56, 77–80]. Tearing modes in tokamaks, magnetic

substorms or solar flares are significantly accelerated by “anomalous resis-

tivities”. Wave-particle interactions (through lower hybrid drift instabili-

ties) [79] or stochasticity of magnetic field lines [80] have been studied to

account for enhanced resistivities. The collisionless resistivity studied here

stems in microscopic particle dynamics that cannot be studied by a fluid

model or the Vlasov equation. We consider magnetic null points that un-

magnetize particles and cause highly non-integrable motion. The resultant

rapid entropy production may be applied to overcome the difficulty of the

67

Petschek model of magnetic reconnections [56].

The standard normalization of Newton’s equation of motion shows that

the particle inertial effect (kinetic effect) works in a length scale of the skin

depth (Sec. 4.2). Single particle motion is analyzed in Sec. 4.3. We in-

troduce an ensemble-averaged “local Lyapunov exponent”, and specify the

“chaos region”. The spatial inhomogeneity of the system is essential – the

randomization of orbits and the resultant resistivity are strongly localized in

neighborhoods of null points. Particle convection connects the chaos region

and the outside regular region, and hence, the chaos region (not a priori

determined) must be treated as an open system. In Sec. 4.4, we study the

statistical distribution of chaotic particles. The chaos-induced resistivity is

scaled by the Alfven Mach number, while it does not depend on plasma

temperature. In high temperature plasmas such as solar corona, the present

chaos-induced resistivity is much higher than the classical collisional resis-

tivity.

68

4.2 Model Equations

We consider a 2D Y-shape (and X-shape as the degenerate case) magnetic

field (Fig. 4.1) that can be written as (in Cartesian coordinates)

B =

(b0(y ∓ `y)/`x, b0x/`x, 0) (|y| > `y)

(0, b0x/`x, 0) (|y| ≤ `y)

, (4.1)

where b0, `x and `y are constant numbers. The plane of x = 0, |y| ≤ `y is

called a neutral sheet where the magnetic field vanishes. We apply a constant

electric field in the direction perpendicular to the magnetic field;

E = e0ez = −∇φ, (4.2)

where e0 is a constant number, ez = ∇z and φ is the electro-static potential.

The potentials are given by

Az(x, y) =

b02`x

[(y ∓ `y)2 − x2] (|y| > `y)

− b02`x

x2 (|y| ≤ `y)

,

φ(z) = −e0z.

(4.3)

The Hamiltonian of a single particle includes all three coordinates, so that

the orbit is generally non-integrable (chaotic) [2]. In the neighborhood of the

Y-points, particles describe chaotic orbits receiving almost random sequences

of acceleration and deceleration from the electric field. As we will show later,

69

the resultant randomization of orbits yields a collisionless resistivity. On

the other hands, outside this “chaos region” (to be specified more quanti-

tatively in Sec. 4.3.2), the magnetic field is sufficiently strong to magnetize

particles (the magnetic moment µ ≡ mv2⊥/2B conserves), and the guiding

centers describe E×B drift orbits. The drift motion causes a flow that sup-

plies/extracts particles to/from the chaos region. This convection plays an

essential role to enable a continuous production of heat in the chaos region.

We study collisionless motion of charged particles governed by Newton’s

equation with the given fields (4.1) and (4.2);

mdv

dt= q(E + v ×B), (4.4)

where m and q are the mass and the charge, respectively, and v is the velocity

of a particle. We normalize variables as

x = `xx, B = b0b, t = τAt,

v = vAv, E = (mAvAb0)e,

(4.5)

where b0 is an appropriate measure of the magnetic field, vA is the Alfven

velocity corresponding to b0, τA ≡ `x/vA, and mA is the Alfven Mach num-

ber (mAvA gives the E ×B-drift convection speed). Using the normalized

variables, (4.4) reads

δ

`x

dv

dt= mAe + v × b, (4.6)

70

where δ = vA/ωc is the collisionless skin depth and

b =

(y ∓ `, x, 0) (|y| > `)

(0, x, 0) (|y| ≤ `)

, (4.7)

where ` ≡ `y/`x. The left-hand side of (4.6) gives the “kinetic effect” that

enables deviation from the E ×B-drift motion (principal part of the ideal

MHD flow), resulting in chaotic orbits. In what follows, we take `x = δ to

emphasize the kinetic effect (hence, τ−1A = ωc: cyclotron frequency). If we

define ωc including the sign of the charge (ωc is positive for ions and negative

for electrons), (4.6) can be applied for both ions and electrons.

4.3 Single Particle Dynamics

4.3.1 Chaotic Orbit

Figure 4.2 shows two types of orbits projected onto an x − y plane. In

Fig. 4.2(a), we observe that a particle moves irregularly near the Y-points

while it becomes more regular near the asymptotic line of the magnetic field.

Because of the mirror effect of the magnetic field, a particle goes back to the

chaos region and confined there for a certain time. In Fig. 4.2(b), a larger

electric field (mA = 0.01) is applied. After a relatively short staying time,

71

the particle escapes from the chaos region.

In Fig. 4.3, we plot the average staying time (τ1 ≡ τ1/τA) as a function

of mA. We may approximate τ1 ≈ m−1A . This relation does not depend

significantly on ` as far as ` . 10. In the limit of ` →∞, the magnetic field

becomes 1D and the orbit becomes integrable. Sufficiently large τ1 (relative

to the Lyapunov exponent (see the next section)) is required to yield an

appreciable entropy production in the chaos region. This condition demands

mA . 10−2 (4.8)

4.3.2 Lyapunov Exponents

The maximum Lyapunov exponent of a given orbit characterizes the mean

divergence rate of nearby orbits. In a chaotic system, it provides a quanti-

tative measure of the degree of stochasticity. First, we define a quantity to

measure divergence of two trajectories in a unit time,

χ(t) ≡ 1

∆tlog

|δx(t + ∆t)||δx(t)| , (4.9)

where δx is a distance of initially neighboring two trajectories (a basis vector

in the most divergent direction). Here we set ∆t = 10−2τA, |δx| = 10−2`x.

To detect the principal axis of the Hessian, we reorthonormalize the basis

72

vector using the Gram-Schmidt method to find the most rapidly growing

direction [81]. The conventional maximum Lyapunov exponent is defined by

taking a long-time average of χ(t) over a certain orbit [2, 82]; i.e., choosing

sufficiently small ∆t and |δx|, one may calculate

χ ≡ limN→∞

1

N∆t

N∑n=0

χ(tn)∆t. (4.10)

However, this definition is not suitable when we consider an open system

where the particle staying time is finite. To quantify the degree of stochas-

ticity for a temporally and spatially finite chaotic phase of motion, we take

an ensemble average, instead of the long-time average, to define a temporally

and spatially local maximum Lyapunov exponent (LLE) [83];

χ(t) ≡ 〈χ(t)〉. (4.11)

The ensemble average 〈·〉 is taken for the particles contained in a given region

defined as follows: Particles are initially distributed uniformly in the region

−1.0 < x, y < 1.0 and −0.5 < vx, vy, vz < 0.5 (total number is 5 × 104).

We consider subdomains Ω(R) scaled by the distance from the Y-points;

R ≡√

x2 + (y − `)2. To measure the strength of chaos near the Y-points,

we evaluate LLE for the ensemble of particles remaining in Ω(R). In Fig. 4.4,

we plot LLEs (for different Ω(R)) as functions of t. For larger R (R & 1),

73

the LLE decreases as t increases, simply that Ω(R) contains regions where

particle motion becomes regular. A stationary value of LLE is observed for

R . 1. Hence, we may call Ω(1) the chaos region where the LLE is about

0.25.

In Fig. 4.5, we compare the LLEs for different mA’s (and hence, for dif-

ferent staying times). For mA < 0.01, the LLEs have a plateau at the same

level (LLE ' 0.25). When mA & 0.01, however, the plateau is lost because

all particles are swept out from the chaos region before they are randomized.

4.4 Statistical Distribution and Macroscopic

Resistivity

4.4.1 Velocity Distribution

Macroscopic quantities, such as temperature and resistivity, are calculated

invoking many particles, that obey the same equation of motion with random

initial conditions; initial velocities and positions have a uniform distribution

in the domain of −0.5 < vx, vy, vz < 0.5 and −1.0 < x, y < 1.0. In Figs. 4.6

(a) – (c), we plot the velocity distributions of the particles in the chaos

74

region Ω(1) with the Gaussian fitting curves. In the initial phase (t ∼ χ−1 '

4), particles relax into almost isotropic Gaussian distributions. Temporal

evolutions of the distributions are shown in Figs. 4.6 (d) – (f). The total

number of particles decrease because of the loss of particles from the chaos

region. The peak of distribution of vz gradually shifts in the direction of

the electric field. The velocity distributions have cut off around |v| ∼ 0.8 ,

implying that particles with |v| & 0.8 are lost from the chaos region. Indeed,

if |v| & 0.8, the particle passes the chaos region in a unit time and such a

high energy particle can not be scattered by the Y-points.

Figure 4.7 shows the standard deviations (temperatures) of the velocity

distributions. We observe that the temperatures increase in x and y direc-

tions. In the direction of the electric field (z direction), particles have an

average flow velocity. The distribution is strongly distorted in the region

vz & 1, and the standard deviation decreases.

4.4.2 Effective Resistivity

We may estimate the effective collisionless resistivity from the evolution of the

average velocity in the direction of the electric field (Fig. 4.8). The ensemble

average is taken over the particles in the chaos region Ω(1). Particles in the

75

chaos region are accelerated monotonically; we may write

ˆvz(t) ' αt with α = 3.48× 10−4. (4.12)

However, the number of particles in the chaos region decreases exponentially;

n(t) ' n0 exp(−βt) with β = 2.17× 10−3. (4.13)

We may consider a system where the total number of particles are conserved

with supplying particles with zero average velocity; this model is more ap-

propriate to simulate a convecting system with a driving electric field. Using

(4.12) and (4.13), we obtain the average velocity in such a sustained system;

ˆvw(t) =α

β[1− exp(−βt)]. (4.14)

The macroscopic velocity may be modeled by a dissipative evolution equa-

tion

ρeffdˆvz

dt= mAe− νeff ˆvz, (4.15)

where ˆvz is the normalized average velocity in z direction, ρeff is the effective

mass normalized by the particle mass, and νeff is an effective collision fre-

quency normalized by ωc. To explain the meaning of the effective mass, let

us first assume ρeff = 1. Then, the solution of (4.15) becomes

ˆvz =mA

νeff

[1− exp

(−νeff t)]

. (4.16)

76

Evaluating νeff from the time constant of the numerical result (see (4.14)),

we obtain the saturation level of the velocity;

ˆvsat =mA

νeff

, (4.17)

which translates as E/(vAb0) = νeff(vsat/vA) in the physical units. Comparing

this relation with Ohm’s law (j is the current density, n is the number density,

q is the electric charge)

E = ηeffj = ηeffnqvsat, (4.18)

we obtain

ηeff,1

µ0

= δ2ωcνeff . (4.19)

We can derive another resistivity directly from the saturation level. By sub-

stituting the value of ˆvsat into (4.18), we obtain

ηeff,2

µ0

= δ2ωcmA

ˆvsat

. (4.20)

These two resistivities (ηeff,1 and ηeff,2) do not agree, because (4.15) is a

phenomenological model – the microscopic magnetic force qv×B is absorbed

both in the frictional and inertia terms in the averaged (macroscopic) model.

By adjusting the mass to ρeff in the inertia term of (4.15), the solution of

(4.15) modifies as

ˆvz =mA

νeff

[1− exp

(− νeff

ρeff

t

)]. (4.21)

77

By plugging (4.14) and (4.21), we obtain

νeff = 6.2× 10−3, (4.22)

ρeff = 2.9. (4.23)

The effective resistivity is given by (in physical units),

ηeff

µ0

= δ2ω2cνeff . (4.24)

4.5 Fast Magnetic Reconnection

To apply the chaos-induced resistivity for the theory of flare, we introduce

a “mesoscopic model” of the diffusion region to relate the microscopic and

macroscopic quantities. For the macroscopic configuration, we consider a

Petschek-type shock structure (Fig. 4.10). We assume that the dissipation

region (D) includes multiple chaos regions (Fig. 4.1) that have the length

scale `x = δi. The ion skin depth δi is naturally derived in the two-fluid

model. The “singular perturbation” given by the Hall term can connect

small scales (∼ δi) with large scales. The magnetic and flow velocity fields in

the region (D) may take double Beltrami-type configurations [53] where small

scale vortices couple with large scale fields through the Hall term, producing

78

a jittery structure. At the scale of δi, the kinetic effect of ions produces a

strong collisionless dissipation, while that of electrons is negligible. Simu-

lation studies [84, 85] demonstrate creation of ion-current dominant current

sheets (see also [86]).

Let us summarize the microscopic relations. The staying time τ1 is scaled

by mA (Fig. 4.3), and τ1 must be much longer than the reciprocal of the

Lyapunov exponent, demanding (4.8). In the effective resistivity (4.20), we

estimate νeff ≈ 6 × 10−3. The magnetic field (4.1) must be consistent to

the average current j = η−1eff E through j = µ−1

0 ∇ × B, which reads as

η−1eff mAvAb0 ' b0/(µ0`x) because the ion current is dominant. Using (4.20),

this relation translates as νeff ' mA`x/δi (= mA), which is consistently sat-

isfied with (4.8) and (4.23).

In the macroscopic level, we consider a diffusion region (D) with a thick-

ness 2Lδ and horizontal scale 2L′ (Fig. 4.10). Denoting , the quantities in

the upstreams/downstreams of the flow by subscripts 1 and 2, respectively,

(we use capital letters for the macroscopic quantities), the Rankine-Hugoniot

condition reads |V 2| ≈ V2y ≈ VA, where VA is the Alfven velocity for B1. We

define a macroscopic Alfven Mach number by MA = V1y/VA that scales the

reconnection rate. The divergence-free conditions of the magnetic field and

79

velocity field read

Lδ

L′=

B2x

B1y

=V1x

V2y

= MA. (4.25)

To match the microscopic and macroscopic parameters, we need to in-

troduce a “mesoscopic” model for the dissipation region (D). Assuming a

uniform electric field, we estimate

E = V1xB1y = MAVAB1y ' v0b0 = mAvAb0, (4.26)

which gives MA = mA(b0/B1y)2. The microscopic magnetic field [b0; see

(4.1)] is at most B1y, and hence (4.8) yields a bound

MA . mA . 10−2, (4.27)

which accounts well for the observed time scale of flares.

The energy estimate leads to a relation between the length scales of

macroscopic and microscopic regions. Poynting’s theorem and (4.25) give

∫

D

E · jdx = − 1

µ0

∫

S

E ×B · ndS

=4

µ0

(1−M2A)B2

1yVALδ, (4.28)

where S is the boundary of the dissipation region (D). The left-hand side

of (4.28) represents the total energy dissipation in (D). Using (4.20), we

80

estimate∫

D

E · jdx =N∑

k=1

∫

Ωk

ηeffj2

µ20

dx ' Nδ2i ωciνeff

b20

µ0

, (4.29)

where Ωk (k = 1, · · · , N) are chaos regions included in the diffusion region.

The energy dissipation is dominated by the ion current. Combining (4.28)

and (4.29), we obtain

Lδ ' Nδ2i ωciνeffb2

0

4(1−M2A)B2

1yVA

. Nδi4

νeff . (4.30)

Because Lδ must be larger than δi, the single chaos region (N = 1) cannot

produce a fast (shock-type) reconnection, even though the effective resistivity

is much enhanced.

4.6 Summary

We have shown that Y-points of magnetic fields act as scattering centers

producing a strong effective resistivity for ions. The flow of particles across

the chaos region is essential to yield a diffusion-type collisionless dissipa-

tion. The average staying time of ions in the chaos region, relative to the

temporary local Lyapunov exponent, is the key parameter that restricts the

maximum electric field. To apply this chaos-induced resistivity in a fast

(shock-type) reconnection process, we have introduced a mesoscopic model

81

and have derived a relation (4.30) connecting the macroscopic scale (Lδ) and

the microscopic parameters (δi and νeff) in a “universal form” (Lδ ∝ νeff)

with a mesoscopic coefficient N . At the small scale of the order of δi, the ki-

netic effect (chaos) produces a strong collisionless dissipation, giving a lower

bound for the length scale. Hence, we can remove the unphysical length

scale reduction (down to the sub-gyroradius regime) deduced from the direct

application of the single-fluid MHD in the original Petschek model, which is

incapable of encompassing the huge scale separation.

Chaotic motion of particles in strongly inhomogeneous magnetic fields is a

simple and direct route to produce a collisionless resistivity. In magnetic null

regions, particles are unmagnetized and describe extremely complex orbits,

which invalidates theoretical models based on wave-particle interactions [79]

or stochastic motion of magnetized fluids [80].

In an inhomogeneous magnetic field, charged particles describe chaotic

orbits. The randomization of the phase of cyclotron motion enables particles

to receive net acceleration from the electric field. The phenomenological dis-

sipation equation is introduced to evaluate a current parallel to the electric

field. In a macroscopic model (4.15), the chaotic randomization effect (v×B

term in the particle model (4.4)) is described by both the effective collision

82

frequency and the effective mass. Only the effective collision frequency is

not sufficient to account for the chaos effect. From the effective collision

frequency, we obtain a collisionless effective resistivity. In absence of mutual

interactions among particles, we may construct a stationary state by supply-

ing new particles that are just replicas of particles with shifted initial times.

The velocity distributions of particles broaden, and achieve almost isotropic

Gaussian distribution. The average staying time of particles in the chaos

region, relative to the maximum LLE, is the key parameter that restricts

the maximum electric field (or the Mach number) to yield an appreciable

resistivity – too large electric field sweeps out particles from the chaos region

before they are randomized.

The “chaos-induced resistivity” is proportional to the square of the Alfven

velocity (vA = δωc), which is a function of the magnetic field and the plasma

density. On the other hand, the classical collisional resistivity (Spitzer resis-

tivity [87]) is a function of the electron temperature (∝ T−3/2e ). In a high

temperature plasma, the chaos-induced resistivity (ηeff) can be much larger

than the classical collisional resistivity. For example, if we take typical pa-

rameters of the solar corona (n = 1016m−3, B0 = 10−2T, Te = 102eV), ηeff

is about 104 times larger than the collisional resistivity. The chaos-induced

83

resistivity occurs both in ions and electrons. Ions contribute much larger

dissipation than electrons, because the size of the ion chaos region (scaled

by the skin depth) is much larger than that of electrons. This chaos-induced

resistivity has been applied to explain the fast (shock-type) reconnection

process. Introducing a mesoscopic model in the diffusion region [56], we

may avoid the unphysical scale reduction problem that Petschek’s model [58]

encountered. The bound for the Mach number restricting this collisionless

resistivity explains the time scale of the fast reconnection.

84

2ly

Chaos Region

x

y

2lx

Figure 4.1: A Y-shape magnetic field with ` ≡ `y/`x = 2 projected onto

the x-y plane. (x and y coordinates are normalized to the system size `x).

The chaos region (hatched region) is defined by using the local Lyapunov

exponents in Sec. 4.3.2.

85

(b) mA=0.01(a) mA=0.001

x/lx

-10 -5 0 5 10

y/lx

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -5 0 5 10-10

-8

-6

-4

-2

0

2

4

6

8

10

x/lx

y/lx

Figure 4.2: Typical particle orbit in a Y-shape magnetic field with ` =

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 mA’s. In (b), the particle is swept out before it is

randomized sufficiently.

86

10

100

1000

10000

0.001 0.01

l = 1

mA

= 2= 5=10

τ1/τ

A

Figure 4.3: Average staying time in the chaos region (τ1) as a function of

mA.

87

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 100 200 300 400 500 600

Max

imum

Lya

puno

v E

xpon

ent

R=0.8R=1.0R=1.2R=1.5

time [τA]

Figure 4.4: Local Lyapunov exponents for different subdomains Ω(R) (mA =

0.002). We define the chaos region such that the local Lyapunov exponents

have a plateau R . 1.0.

88

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 100 200 300 400 500 600 700 800 900 1000

Max

imum

Lya

puno

v E

xpon

ent

mA=0.001mA=0.002mA=0.005mA=0.010mA=0.020

time[τA]

Figure 4.5: Local Lyapunov exponents for different mA’s. For larger mA, the

local Lyapunov exponents are strongly damped and have no plateau region.

89

50

100

150

200

250

300

350

400

450

-1 -0.5 0 0.5 1

Num

ber

of P

arti

cles Gaussian

t =10

vx/vA

50

100

150

200

250

300

350

400

450

-1 -0.5 0 0.5 1

Num

ber

of P

arti

cles Gaussian

t =10

vy/vA

50

100

150

200

250

300

350

400

-1 -0.5 0 0.5 1

Num

ber

of P

arti

cles Gaussian

t =10

vz/vA

1

10

100

1000

-1 -0.5 0 0.5 1

Num

ber

of P

arti

cles

vz/vA

t=0

t=406t=200

t=602t=798t=968

1

10

100

1000

-1 -0.5 0 0.5 1

Num

ber

of P

arti

cles

t=0

t=406t=200

t=602t=798t=968

vy/vA

1

10

100

1000

-1 -0.5 0 0.5 1

Num

ber

of P

arti

cles

t=0

t=406t=200

t=602t=798t=968

vx/vA

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4.6: Velocity distributions in the chaos region (mA = 0.001, ` = 0).

(a)-(c): distributions of vx, vy, vz, after initial randomization phase with the

Gaussian fitting curves. (d)-(f): temporal evolutions of the distributions of

vx, vy, vz. 90

0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

0 100 200 300 400 500 600 700 800 900 1000

stan

dard

dev

iati

on

vxvyvz

time[τA]

Figure 4.7: Standard deviations of the velocity distribution (mA = 0.001, ` =

0).

91

Numerical Result

0 100 200 300 400 500 600 700 800 900 10000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

vw(t)=α/β exp(-βt)vz(t)=αt

vz/v

A

time [τA]

Figure 4.8: Evolution of the average velocity in the direction of the electric

field (mA = 0.001, ` = 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).

92

100

1000

10000

100000

0 100 200 300 400 500 600 700 800 900 1000

Num

ber

of P

artic

les

Numerical Resultn(t)=n0exp(-βt)

time [τA]

Figure 4.9: Evolution of the number of particles in the chaos region. The

number decreases exponentially.

93

x

y

shock frontB1yV1x

Region I

2L’Region I

V2yB2x

Region Dv0 b0

2Lδ

Figure 4.10: Petschek-type fast reconnection model. The regions (I) and (D)

are the ideal MHD and dissipation regions, respectively.

94

Chapter 5

Concluding Remarks

95

In this thesis, two nonlinear process in two-fluid plasmas are studied. One

is the self-organization of the Hall-MHD plasmas, and the other is the chaos

of the particle motion in an inhomogeneous electromagnetic field.

In Chap. 2, we have introduced the Hall-MHD equations and theoreti-

cal model of equilibrium. In the Hall-MHD plasmas, the Hall term leads

a singular perturbation which enables the coupling of two different spatial

scale vortices. The resultant double Beltrami equilibrium can span far richer

set of magneto-fluid coupled state than the well-known Taylor’s force-free

state. The relaxation process has been discussed in terms of the variational

principle.

Chapter 3 is devoted to the numerical analysis of the Hall-MHD plasmas.

Based on the theory of the double Beltrami equilibrium, we have developed

the Hall-MHD simulation code and examine the relaxed state. Compared

with the single-fluid MHD, appreciable flow perpendicular to the magnetic

field highlights the difference between the self-organization process in two

models. The relaxation process is achieved by minimizing the generalized

enstrophy with the adjustment of global invariants. In the Hall-MHD plasma,

the energy cannot be the minimizer because more coercive functional, the ion

helicity, dissipate faster than the others as the result of severe adjustment.

96

In Chap. 4, we have analyzed the dissipation mechanism due to the chaos

of particle motion. Inhomogeneity in the ion skin depth scale of the mag-

netic field including null points can acts as the scattering center of particle

orbit and produce collisionless resistivity. We have introduced diagnostic

tools of statistical properties of the chaotic particles such as the local Lya-

punov exponents and specify responsible chaos region. The chaos-induced

resistivity has been applied to the fast magnetic reconnection process. By

introducing the mesoscopic scale structure, we have succeed in construct-

ing self-consistent magnetic reconnection model which explains the fast time

scale magnetic reconnection, and avoids unphysical scale reduction problem

that Petschek’s model encountered. The mesoscopic structure where jittery

structure of the ion skin depth scale may take double Beltrami-type config-

uration. We suggest that the scale hierarchy of the magnetic reconnection

shown in Fig. 5.1 (See Shibata and Tanuma [88], or Kan [89] concerning the

mesoscopic diffusion region.)

97

x

y

shock front

2L’

Macroscopic

Global Scale

Mesoscopic

Diffusion Region

Microscopic

~ Ion skin depth

2δ

Double Beltrami Field ?

2ly

Chaos Region x

y

2lx

Figure 5.1: Hierarchy of the scale in the magnetic reconnection process.

98

Acknowledgments

99

I would like to express my sincerest appreciation to many peoples for their

helpful supports during my PhD work.

First of all, I would like to express my deepest gratitude to my supervi-

sor Professor Zensho Yoshida. His insightful comments based on very wide

and deep knowledge about plasma physics and mathematical physics, have

improved my understandings. Without his guidance, I could not have com-

pleted this work.

I also thank Professor Takaya Hayashi in the National Institute for Fu-

sion Science. During the cooperation with him as the special joint research

fellow, I learned a lot of practical things related to the large scale numerical

simulation using supercomputer.

Professors Yuichi Ogawa, Haruhiko Himura, Junji Morikawa in the Uni-

versity of Tokyo, Ritoku Horiuchi, Sigeo Kida in the National Institute for

Fusion Science kindly gave me many fruitful comments. Professors Vazha

Berezhiani, Nana Shatashvili in Georgian Academy of Science, Hamid Saleem

in Pakistan Institute of Nuclear Science and Technology, Robert Dewar in

the Australian National University were visiting professors in the University

of Tokyo, who kindly teach me a lot of topics in plasma physics through

seminars. Swadesh Mahajan in the University of Texas whom I met in the

100

autumn college on the plasma physics held in the International Center for

Theoretical Physics also kindly gave me suggestions.

I am grateful to Doctors Tomoya Tatsuno, Shigeo Kondoh, Atsushi Ito,

Francesco Volponi, Shuichi Ohsaki and all other members in the laboratory

at Graduate School of Frontier Sciences, the University of Tokyo and the

Theory and Computer Simulation Center in the National Institute for Fusion

Science. I learned a lot of things through the study with them.

Finally, I would express my gratitude to my parents and sister, and all

my friends for their support during my study.

101

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