Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mso.anu.edu.au/~geoff/AGD/MHD_Eqns.pdf Дата изменения: Fri Feb 12 13:58:32 2016 Дата индексирования: Sun Apr 10 02:34:56 2016 Кодировка: Поисковые слова: п п п п п п п п п п п п п п

Magnetohydrodynamics MHD


References
S. Chandrasekhar: Hydrodynamic and Hydromagnetic Stability T.G. Cowling: Magnetohydrodynamics E. Parker: Magnetic Fields B. Rossi and S. Olbert: Introduction to the Physics of Space T.J.M. Boyd and J.J Sanderson The Physics of Plasmas F. Shu: The Physics of Astrophysics, Vol. 2, Gas Dynamics

2


General physical references
J.D. Jackson: Classical Electrodynamics L.D. Landau & E.M. Lifshitz: The Electrodynamics of Continuous Media E.M. Lifshitz & L.P. Pitaevskii: Physical Kinetics K. Huang: Statistical Mechanics

3


Maxwell's equations (Gaussian units)
Ampere's law

Displacement current Electric current

Gauss's law of electrostatics

No magnetic monopoles

r·B

r · E = 4e =0

1 @B r E + c @t

r B =

4 1 @E Je + c c @t

=0

Faraday's law of induction

Particle equations of motion
dv v m =q E+ B dt c
Lorentz force





mr

Gravitational force 4


Cartesian form of Maxwell's equations
@ Ej @ xj @ Bj @ xj @ Bk ij k @ xj @ Ek 1 @ Bi + @ xj c @t = = = = 4 0 4 1 @ Ei Ji + c c @t 0
e

ij

k

Particle equations of motion dvi vj m = q Ei + ij k Bk dt c

@ m @ xi
5


Energy density, Poynting flux and Maxwell stress tensor
EM S
i

= = = =

EM i B ij

M

M

E ij

=

E2 + B2 = Electromagnetic energy density 8 c ij k Ej Bk = Poynting flux 4 Si = Electromagnetic momentum density 2 c 1 12 Bi Bj B ij = Magnetic component of 4 2 Maxwell stress tensor 12 1 Ei E j E ij = Electric component of 4 2 Maxwell stress tensor
6


Relationships between electromagnetic energy, flux and momentum

@EM @ Si + @t @ xi @ i @ Mij @t @ xj

= =

Ji Ei Jj e Ei + ij k Bk c

7


Momentum equations
Consider the em force acting on a particle vj refers to specific Fi = q Ei + ij k Bk c particle Consider a unit volume of gas and the em force acting on this volume " # # " X X vj em Fi = q Ei + ij k Bk q c where the sum is over all particles within the unit volume. N.B. The velocity here is the particle velocity not the fluid velocity.
8


Momentum (cont'd)
F
em i

=

"

X


q



#

Ei + ij

k

"

X


q

vj

#

Bk

We can identify the following components

so that the electromagnetic force can be written
Jj F = e Ei + ij k Bk c 1 i . e . F = e E + J B c
em i

X


X


q



= e = Electric charge density = Ji = Electric current density

q

vj

9


Momentum (cont'd)
We add the body force to the momentum equations to obtain

dvi = dt

@p @ xi

@ Jj + e Ei + ij k Bk @ xi c

Now use the equation for the conservation of electromagnetic momentum:

1 e E i + c

ij k Jj

Bk =

@ i @ Mij + @t @ xj
10


Momentum (cont'd)
so that the momentum equations become:
Bulk transport of momentum Flux of momentum due to EM field
ij

@ @ (vi + i )+ ( v i v j + p @t @ xj
Mechanical + EM momentum density

Mij ) =

@ @x

i

Flux of momentum due to pressure

Gravitational force

For non-relativistic motions and large conductivity some very useful approximations are possible
11


Limit of infinite conductivity
In the plasma rest frame (denoted by primes), Ohm's law is
0 Ji

=E

0 i

Conductivity

The conductivity of a plasma is very high so that for a finite current 0 Ei 0 This has implications for the lab-frame electric and magnetic field

12


Transformation of electric and magnetic fields
Lorentz transformation of electric and magnetic fields vB 0 E= E+ c = Lorentz factor vE 0 B= B c
If E0 = 0

vB E+ c

=0 vB =O c vB c

)E =

This is the magnetohydrodynamic approximation.

13


Maxwell tensor
Since then
M =O
E ij

E=O



vB c





v2 c2



M

B ij

Hence, we neglect the electric component of the Maxwell tensor. We can also neglect the displacement current.

14


Electromagnetic momentum
We want to compare the electromagnetic momentum density with the matter momentum density, i.e. compare
1 = ij k Ej Bk 4 c 2 vB =O c2



EM i

with

vi vi = O(v )

EM i

EM i =O vi



B2 4c2


15


Electromagnetic momentum
EM i =O vi B2 4c2

As we shall discover later, the quantity
B 2 2 = vA = (Alfven speed) 4
2

where the Alfven speed is a characteristic wave speed within the plasma. We assume that the magnetic field is low enough and/or the density is high enough such that
2 vA c2

and we neglect the electromagnetic momentum density.

1

16


Final form of the momentum equation
Given the above simplifications, the final form of the momentum equations is:
dvi = dt @p @ xi @ @ + @ xi @ xj Bi Bj 4 B2 8

ij

We also have

1 BB curlB B = div 4 4



B I 8

2

where I is the unit tensor
17


Thus the momentum equations can be written:
dv = dt rp 1 r + curlB B 4

This form can often be more useful.

18


Displacement current
Let L be a characteristic length, T a characteristic time and V = L/T a characteristic velocity in the system The equation for the current is:
Ji c = ilm 4 cB O L vV =O c2 @ Bm @ xl O 1 1 @ Ei 4 @ t E vB = T cT

Displacement Current ) Curl B current

19


Displacement current (cont'd)
In the MHD approximation we always put
Ji i.e. J = = c @ Bm ilm 4 @ xl c curl B 4

20


Energy equation
The total electromagnetic energy density is
EM

E since

E

2

E2 + B2 B2 = 8 8 2 v 2 =O B 2 c

In order to derive the total energy equation for a magnetised gas, we add the electromagnetic energy to the total energy and the Poynting flux to the energy flux

21


Energy equation (cont'd)
The final result is 2 @12 B v + + + @t 2 8
+ = h S
i

= = =

where

? vi

=

@ 12 v +h+ vi + Si @ xj 2 ds kT dt +p c ij k Ej Bk 4 2 B2 ? 1 B vi Bj vj Bi = vi 4 4 Component of velocity perpendicular to magnetic field
22


The induction equation
The final equation to consider is the induction equation, which describes the evolution of the magnetic field. We have the following two equations:
Faraday's Law: Infinite conductivity: 1 @B r E + =0 c @t v E= B c

Together these imply the induction equation
@B = curl(v B) @t
23