nonlinear processes in two-fluid plasmas ( )...(yoshida and mahajan 2002) (17) (18) (19) are...

Graduate School of Frontier Sciences, University of Tokyo

Nonlinear Processes in Two-Fluid Plasmas( )

2004/ 1/22

��

2004/ 1/22 – p.1/41


OutlineWe have studied effects of flows in plasmas by numerical simulations

Two-fluid Effectsfluid description, magnetohydrodynamics

Kinetic Effectsparticle orbit, open (transient) chaos

It is theoretically predicted that structures of the ion collisionless skin depth order will beself-organized in two-fluid plasmas. In such a scale, particles may describe strongchaotic motion.

Self-organization of plasmas with flows

Double Beltrami EquilibriumRelaxation ProcessNonlinear 3D Simulation

Collisionless Magnetic Reconnection

Chaos-Induced Resistivity

Connection with Fluid Model

��

2004/ 1/22 – p.2/41


Self-organization of plasmas with flows

��

2004/ 1/22 – p.3/41


Introduction

Self-Organization of Plasmas with FlowsEffects of flows in Plasmas

Laboratory Plasmas, Fusion Plasmas(Non-neutral Plasmas, Tokamak H-mode boundary layer, Shear-flow Stabilization)

Space Plasmas(Magnetic reconnection, Solar coronae, High beta equilibrium of the Jupiter)

The Hall term in the two-fluid plasma model leads to a nonlinear singular perturbationthat enables the formation of the “double Beltrami” field given by a pair of two differentBeltrami fields.The self-organization of a plasma into relaxed states may be discussed by a variationalprinciple. Taylor assumed the minimization of the magnetic energy under the constant ofthe magnetic helicity, and obtained the force-free state.

DynamicsStatistical properties, Stability, Self-organization of flowing plasmasNonlinearity, Non-Hermitian property, Singular perturbations.

��

2004/ 1/22 – p.4/41


Self-Organization

Self-organization means the spontaneous generation of large-scale coherent structures.Systems of fully developed turbulence exhibit large-scale structure reflecting theproperties of the turbulence drive.

Reversed field configuration in toroidal pinch experiment called Zeta(Bodin and Newton 1980)

Self-organization in MHD turbulence

Force-Free(Woltjer 1958, Taylor 1974, Montgomery 1978)

Alfvénic state(Dobrowolny et. al. 1980)

(Horiuchi and Sato 1985, Stribling and Matthaeus 1991)

Magnetic reconnection and energy release in the solar corona(Heyvaerts and Priest 1984, Browning 1988, Vekstein 1990)

��

2004/ 1/22 – p.5/41


TurbulenceKolmogorov (1941)Hydrodynamic turbulence

� ��

Kraichnan (1967)Inverse cascade in 2D hydrodynamic turbulence

Iroshnikov (1964), Kraichnan (1965)MHD turbulence

� ��

Shebalin et. al. (1983), Montgomery and Matthaes (1995),

Anisotropy of MHD turbulence

Cascade direction

3D 2D

Navier-Stokes�

direct

�

inverse� �

direct

directMHD

�

direct

�

direct��

direct

��

direct��

direct

�

inverse

��

2004/ 1/22 – p.6/41


Computer Simulation

Freedom � � � ��

Amount of calculation � ��

Hydrodynamic TurbulenceOrszag & Patterson (1972)� � ��

Pseudospectral method

Kida & Murakami (1987)� � � � ��

first observation of Kolmogolv’s

� � � � law by DNS

Kaneda et. al. (2003)� � ��

MHD TurbulenceMüller, Biskamp (2000)� � � � � � ��

Speedup trend of computersR. Himeno et. al., J. Plasma Fusion Res. 79, 752 (2003)

��

2004/ 1/22 – p.7/41


Two-Fluid Theory

Single-Fluid Model

Singularity in equilibrium equation

� �

� ��

� � � � ��

d � � ��

(1)

Hameiri (1983), Tasso, et. al. (1998)

Scale invarianceParker’s current sheet model (1996)

Hall effect

Magnetic Reconnection (GEM challenge)MHD Dynamo (Mininni et al 2002)Accretion disks (Wardle 1999, Balbus, Terquem 2001)Generation of magnetic field in early universe (Tajima 1992)

��

2004/ 1/22 – p.8/41


Double Beltrami Equilibrium

Incompressible two-fluid MHD equations can be cast into coupled vortex equations(electron inertia, displacement current are neglected)

��

(2)

where �� are the generalized vorticities and

�� are the corresponding flows,

� � � � � � � � �

� � � � � � � � � (3)

� � ��

(

�� : ion collisionless skin depth,� : speed of light, � � � : ion plasmafrequency,

�

: system size). A simplest equilibrium solution is given by the “Beltramiconditions” ( �� ). Combining two Beltrami conditions yield2nd-order equation for � �

and

,

�� ! � �� (4)

where

#" � %$

& �' � ( � � ) * �' � ( � � � � � � ' � ( � +

.

��

2004/ 1/22 – p.9/41


Double Beltrami Equilibrium

The general solution is described by a linear combination of eigenfunctions of the curloperator

� " (

� � � " � #" � " ) :

� �! ��! � � � � � � � � ( � � � � � �! �! � � ( � � � � � � � � � (5)

The Beltrami conditions give a special class of solution of the general steady statesolution such that ��

(6)

where � � is a certain scalar field corresponding the energy density. The latter equality,called the “generalized Bernoulli condition”, demands the energy density is uniform inspace. From the original electron and ion momentum equations, we find that thegeneralized Bernoulli condition is given by the relation,

��

�� (7)

� � � � ��

2004/ 1/22 – p.10/41


MHD Relaxation Process

Taylor Relaxation

� � ��

(8)

��

� ��

��

� � � (9)

� � � � (10)

Including FlowsCross Helicity

�� (11)

is a global invariant if � � const. or � � � .

� � � � � � � � � ��

(12)

� � � � � � � � � � � �

(13)

� � � � ��

2004/ 1/22 – p.11/41


Relaxation Process

Global Invariants

��

�

� � � � � � � (14)

� � ��

�

� � � (15)

� ��

�

� �� (16)

Variational Principle(Yoshida and Mahajan 2002)

� � � � ��

(17)

� ��

�

� � � � �� (18)

� � � � � � � � � � � � � �

(19)

�� are Lagrange multipliers.

� � � � ��

2004/ 1/22 – p.12/41


Simulation CodeWe consider compressible Hall-MHD equations

� ��

(20)

� � � ��

� � � ��

� � � � �

(21)

� �� & � �

��

� � � � � + � ��

�

(22)

� ��

(23)

� ��

� � � � � � � �

� � � � ��

� � � �

(24)

where �: density, � : number density, �: ion pressure,

� �: Reynolds number,

�� :magnetic Reynolds number, �: ratio of specific heat. We have assumed thequasi-neutrality � � � � � � �, and ��

.

� � � � ��

2004/ 1/22 – p.13/41


Simulation Code

Simulation Domain3 dimensional rectangular domain

� � (� ( � � � (� ( � � ��

Boundary ConditionPeriodic in � directionRigid perfect conducting wall in �� direction

� � �

(25)��

��

(26)

where is a unit normal vector,

� � is a normal derivative, and

�� is the normaland tangential component of the magnetic field, respectively.To assure tangential components of the electric field vanish, we set � � �

at the wall.

Numerical Method2nd order central difference in space, the Runge-Kutta-Gill method for time integration2nd order smoothing to suppress short wave length modeadd a diffusion in the equation of continuity to keep density positive

� � � � ��

2004/ 1/22 – p.14/41


Boundary Conditions (Two-fluid)

Two-fluid MHD equations

� �

e

� ��

e

� � �

e � � �

e

� � � ��

e

� � � � �

e � �� e

�

e� (27)

� �

i

� � � �

i

� � �

i

� � �

i

� � � ��

i

� � � � �

i

� �� i

�

i� (28)

� �

e

� � �

i

� �

(29)

� � � � �

i

� �

e

� � (30)

�� (31)

�� (32)

� � ��

(33)

� � � � �i

� �

e

�

(34)

� � � �

e

� �

i

� � density, �� flow, �� mass, �� pressure, �� viscosity. Subscripts “e, i”represent an electron and ion, respectively. The term

�

represents the momentumtransfer between electron and ion, and

is a positive constant.� magnetic field,

� � electric field,

�� current density, �� are the vacuumpermeability, the vacuum dielectric constant, and the speed of light. The relation

� � � � � ��

holds for �� .

� � � � ��

2004/ 1/22 – p.15/41


Boundary Conditions (Two-fluid)

Existence of solution of the initial value problem of the two-fluid equations with theboundary conditions,

�

e

� �

i

� �

on

� � (35)

� � � �

on

� � (36)

has been proved (local existence)(Giga and Yoshida 1984).The boundary condition � � � �

together with (31) and (32) yields

� � � �� on

� � (37)

� ��

on

� (38)

12 Variables

� �

e� �i�

� � � � 12 Boundary ConditionsHighest order derivative :

� � � � ��

2004/ 1/22 – p.16/41


Boundary Conditions (Hall-MHD)

Neglecting the electron inertia and the displacement current, we obtain the Hall-MHDequations including a hyper resistivity,

Ion Momentum Equation

�

i

� � � �

i

� � �

i

� � �

i

� � � � � ��

i

��

e

�

� � � � � (39)

Induction Equation

��

�

i

��

��

�

� � � � ��

e �

� � �

i

�

�

e

��

(40)

Ampere’s Law

� � � ��

i

� �

e

�

(41)

The highest order term in (40) is

� � � � ��

2004/ 1/22 – p.17/41



Biharmonic OperatorIf the boundary value operator

�

be elliptic and self-adjoint, the boundary value problem,

� � ��

in

� � ��' � �� ' � � �� on

� � (42)

� �� ' � �� ' � � ��

(43)

has a unique solution.The boundary conditions for the biharmonic operator to satisfy the self-adjointness are

��

��

� ��

� ��

� �

� ��

� �

� ��

(44)

� � � � ��

2004/ 1/22 – p.18/41



In a vector form, the following relation holds if

� � � � � � �

.

� � � � ��

��

�

��

� ��

�

� ��

� ��

�

� ��

(45)

Thus, the boundary condition � � � � � � � �, � � � � � � � � �

, or

� �T� � � ��

� �

T� � � � (46)

From � � � � � � � �

,

� � �

(47)

� � � � ��

2004/ 1/22 – p.19/41



The same discussion can be applied for � . From the non-slip boundary condition � � �

,we obtain

� � � � (48)

The condition

� � � � � � �

(49)

is immediately satisfied from � � �

.

� � � � ��

2004/ 1/22 – p.20/41


Numerical Methods

Finite Difference Method

� ��

��

�

�� ! � � � �

� � �

� ��

��

�

�� ! � � � � � � � �

� (50)

Runge-Kutta-Gill ( for

� ��

)

� ! � � � ��

� � � � � � � � � ��

(51)

� � � � � � ��

� � � � � � � � � � � � � ��

� � � � � � � � ��

(52)

Smoothing

�sm� � � � � � � ��

(53)

2nd Order :

� � � �� ! � ��

� (54)

� � � � ��

2004/ 1/22 – p.21/41


Smoothing Function

Smoothing function

� � ��

is expressed as

� � ��

� � � � � � � � ��

� � � � � � �� (55)

SM(k

∆x)

k∆x

0 0.5 1 1.5 2 2.5 3

0

0.2

0.4

0.6

0.8

1

2nd : α = 1/42nd : α = 1/24th : α = 3/84th : α = 0.1

k∆x=πk∆x=π/2k∆x=π/3

x

wave form of mode k

Low Pass Filter

� � � � ��

2004/ 1/22 – p.22/41


Dispersion Relation

In order to check the validity of the code, we have solved the dispersion relation of purelytransversal wave in a periodic domain. In an ideal limit, the dispersion relation for smallperturbations around a uniform equilibrium,

� � � �� areconstants, is given by

� � ) ��

& �� ) * � � � � � +� (56)

which describes the Alfvén whistler wave. The plus and minus signs in the bracketcorrespond to right and left hand polarization, respectively.

ε=0.05 (+,R)

ω

k02468

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)

� � � � ��

2004/ 1/22 – p.23/41


Initial Condition2D force free equilibrium (

� � � � �

)(Horiuchi and Sato 1985)

��

��

��

(57)

��

��

� � � � � � � � � ��

(58)

�� (59)

Flow �� (60)

Density and Pressure are uniform

t/tA=

Bz isosurface

� � � � ��

2004/ 1/22 – p.24/41


Equilibrium State

Aspect Ratio :

� � � � � ( � �

Grid :

� � � � � � � � � � �

� � � � ��

,

� � � � � � � ��

Time :

� � � ��

Isosurface of the toroidal magnetic field. Left: � � � �

(Hall-MHD), Right: � � �

(Single-Fluid MHD)

� � � � ��

2004/ 1/22 – p.25/41


Kinetic Energy

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

� � � �

(Hall-MHD):

�� is more than 10% of

� ��

� � � �

(MHD): No

�� remains

� � � � ��

2004/ 1/22 – p.26/41


Beltrami Conditions

( � � �$

�

� � �

� � � �$�

� � � � ' ��

� � � � � � � � (61)

Time :

� � � ��

(a) distribution of a (b) distribution of b

� � � � ��

2004/ 1/22 – p.27/41


Generalized Ohm’s LawGeneralized Ohm’s Law

� � � � � � � � � � � ��

��

(62)

means that the magnetic field is frozen in to the electron fluid.The effective electric field by the Hall term act on the ion fluid.

� � � � � � ��

��

(63)

� � � � ��

2004/ 1/22 – p.28/41


Variational Principle

Temporal evolution of the conservative quantities and the generalized enstrophynormalized by the initial value for the case of � � � �

Etotal

Em+EkH1H2H2-H1F

time [τA]0 50 100 150 200

00.10.20.30.40.50.60.70.80.9

11.1

� � � � ��

2004/ 1/22 – p.29/41


Energy & Helicity Spectrum

Time :

� � ��

Energy and Helicity Spectrum

� ��

� � � � � � � � � ��

��

� � � � ��

2004/ 1/22 – p.30/41



Time :

� ��


� ��

� � � � � � � � � ��

��

� � � � ��

2004/ 1/22 – p.30/41



Time :

� � ��


� ��

� � � � � � � � � ��

��

� � � � ��

2004/ 1/22 – p.30/41



Time :

� ��


� ��

� � � � � � � � � ��

��

� � � � ��

2004/ 1/22 – p.30/41



Time :

� � ��


� ��

� � � � � � � � � ��

��

� � � � ��

2004/ 1/22 – p.30/41


Spectral Transfer

1 10 100 1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

|k|

E(k

)

t=30τΑ

t=40τΑt=100τΑ

|k|

H(k

)

1 10 100 1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

t=30τΑt=40τΑ

t=100τΑ

Energy forward cascadeHelicity inverse cascade

k

spec

trum

H(k)

E(k)

dissipation region

� � � � ��

2004/ 1/22 – p.31/41


Summary

We have developed a nonlinear 3D Hall-MHD simulation code.

Flow with perpendicular components remains in the two-fluid relaxed state, while itdisappears in the single-fluid MHD model relaxed state.

Beltrami condition for electron is well satisfied. The effective electric field due to the Hall term generates the perpendicular ionflow.

Coercivity is checked for the conserved quantities

� � � � ��

2004/ 1/22 – p.32/41


Collisionless Magnetic Reconnection

� � � � ��

2004/ 1/22 – p.33/41


Background

Collisionless Magnetic ReconnectionMagnetic reconnection phenomena play essential roles in structure formation of plasma

topological evolution of plasma

rapid energy dissipation

Induction Equation � ��

� � �

(64)

Cooperation of the induction and the diffusion term is scaled by a function of

�� (Lundquist number)

Sweet (1958), Parker (1963): thin diffusion region, slow (

� �� )

Petschek (1964): slow shock, fast enough (

�� ),extremely small diffusion region (MHD inapplicable)

Chaotic motion of particles may be a mechanism of collisionless dissipation in diffusionregions where magnetic field is weak. In two-fluid plasmas, an ion skin depth scalestructure including null points may be created.

� � � � ��

2004/ 1/22 – p.34/41


Chaotic Orbitsingle particle orbit in 2D inhomogeneous magnetic field

�

� ��

� � ��

�

� ��

� � (65)

where

�� ,

�� and

�� are constant numbers.

��

� ��

(66)

where

�� is the ion collisionless skin depth, � � is the Alfvén Mach number.

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

x

y

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -5 0 5 10

x

y

-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -5 0 5 10

� � � � ��

2004/ 1/22 – p.35/41


Local Lyapunov Exponents

Divergence rate of two trajectories in a unit time

��

� � � ��

� �� (67)

We define a temporally and spatially local Lyapunov exponent

� � � � � � � �� (68)

where the ensemble average

� is taken for the particles contained in the given region

� � � � � � * � � � � � � � �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

t

2ly

Chaos Region

x

y

2lx

� � � � ��

2004/ 1/22 – p.36/41


Effective Resistivity

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

t

vz

vw(t)=α/β exp(-βt)vz(t)=αt

100

1000

10000

100000

0 100 200 300 400 500 600 700 800 900 1000

t

Num

ber

of P

arti

cles

Numerical Resultn(t)=n0exp(-βt)

We consider a system where the total number of particles is conserved with supplyingparticles with zero average velocity

��

�� (69)

� � � ��

� �� (70)

The effective resistivity is given by

�� .

� � � � ��

2004/ 1/22 – p.37/41


Petschek Model

Microscopic (Chaos) ModelTo stay long time in the chaos region � � � � � �

Spatial scale

��

Macroscopic (Petschek) Model

� � � �

� � � ��

��

� � ��

x

y

shock frontB1yV1x

Region I

2LxRegion I

V2yB2x

Region Dv0 b0

2Ly

� � � � ��

2004/ 1/22 – p.38/41


Connecting Two Models

Electric Field � � � � �� ' � � (71)

�� (72)

which accounts well for the observed time scale of flares.Energy Estimation

��

��

� � � �

�

��

� � � �� (73)

��

��

� �

� � � � � � � � � � � � �'

�� (74)

where

� (

� � ��

) are chaos regions included in the diffusion region.

��

� � � � � � � � � ' �

� � � ��

� � � ��

� � ��

� � � � (75)

� � � � ��

2004/ 1/22 – p.39/41


Hierarchical Structure

x

y

shock front

2L’

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-4 -3 -2 -1 0 1 2 3 4

2lxx

y

2ly

Chaos Region

Macroscopic

Global Scale

Mesoscopic

Diffusion Region

Microscopic

~ Ion skin depth

2δ

Double Beltrami Field ?

Fractal current sheet (Shibata and Tanuma 2001)Patchy reconnection (Kaw 1988)

� � � � ��

2004/ 1/22 – p.40/41


Summary

A collisionless effective resistivity induced by a particle chaotic motion in an opentransient system is calculated.

The spatially and temporally local Lyapunov exponent for an open system toevaluate stochasticity and to specify the chaos region.

The chaos-induced resistivity is applied to the fast magnetic reconnection process.A mesoscopic model is necessary to avoid unphysical scale reduction of thePetschek model.

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2004/ 1/22 – p.41/41

nonlinear processes in two-fluid plasmas ( )...(yoshida and mahajan 2002) (17) (18) (19) are...

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