Magnetohydrodynamics

MHD

References

2

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

General physical references

3

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

Maxwell’s equations (cgs Gaussian units)

4

r · E = 4⇡⇢e

r · B = 0

r⇥B =4⇡

cJe +

1c

@E@t

r⇥E +1c

@B@t

= 0

Lorentz force Gravitational force

Displacement current

Faraday’s law of induction

Ampere’s lawGauss’s law of electrostatics

No magnetic monopoles

Particle equations of motion

mdvdt

= q⇣E +

vc⇥B

⌘�mr�

Electric current

Cartesian form of Maxwell’s equations

5

@Ej

@xj= 4⇡⇢e

@Bj

@xj= 0

✏ijk@Bk

@xj=

4⇡

c

Ji +1c

@Ei

@t

✏ijk@Ek

@xj+

1c

@Bi

@t

= 0

m

dvi

dt

= q

⇣Ei + ✏ijk

vj

c

Bk

⌘�m

@�

@xi

Equations of motion of a charged particle

Energy density, Poynting flux and Maxwell stress tensor

6

✏EM =

E2+ B2

8⇡= Electromagnetic energy density

Si =

c

4⇡✏ijkEjBk = Poynting flux

⇧

EMi =

Si

c2= Electromagnetic momentum density

MBij =

1

4⇡

✓BiBj �

1

2

B2�ij

◆= Magnetic component of

Maxwell stress tensor

MEij =

1

4⇡

✓EiEj �

1

2

E2�ij

◆= Electric component of

Maxwell stress tensor

Relationships between electromagnetic energy, flux and momentum

7

@✏EM

@t

+@Si

@xi= �JiEi

@⇧i

@t

� @Mij

@xj= �

✓⇢eEi + ✏ijk

Jj

c

Bk

◆

The following relations can be derived from Maxwell’s equations:

Momentum equations

8

Consider the electromagnetic force acting on a particle

Consider a unit volume of gas and the electromagnetic force acting on this volume

where the sum is over all particles within the unit volume. N.B. The velocity here is the particle velocity not the fluid velocity.

F↵i = q↵

Ei + ✏ijk

v↵j

cBk

�

F emi =

"X

↵

q↵

#Ei + ✏ijk

"X

↵

q↵v↵

j

c

#Bk

α refers to specific particle

Momentum (cont’d)

9

F emi =

"X

↵

q↵

#Ei + ✏ijk

"X

↵

q↵v↵j

#Bk

X

↵

q↵ = ⇢e = Electric charge density

X

↵

q↵v↵j = Ji = Electric current density

We can identify the following components

F emi = ⇢e Ei + ✏ijk

Jj

cBk

so that the electromagnetic force can be written

i.e. F = ⇢eE+1

cJ⇥B

Momentum (cont’d)

10

We add the body force to the momentum equations to obtain

⇢

dvi

dt

= � @p

@xi� ⇢

@�

@xi+ ⇢eEi + ✏ijk

Jj

c

Bk

Now use the equation for the conservation of electromagnetic momentum:

⇢eEi +1

c

✏ijkJjBk = �@⇧i

@t

+@Mij

@xj

Momentum (cont’d)

11

so that the momentum equations become:

@

@t

(⇢vi +⇧i) +@

@xj(⇢vivj + p�ij �Mij) = �⇢

@�

@xi

Bulk transport of momentum

Flux of momentumdue to pressure

Flux of momentumdue to EM field

Gravitationalforce

Mechanical +EM momentumdensity

For non-relativistic motions and large conductivity some veryuseful approximations are possible

Limit of infinite conductivity

12

In the plasma rest frame (denoted by primes), Ohm’s law is

J 0i = �E0

i

Conductivity

The conductivity of a plasma is very high so that for a finite current

E0i ⇡ 0

This has implications for the lab-frame electric and magnetic field

Transformation of electric and magnetic fields

13

Lorentz transformation of electric and magnetic fields

E0 = �✓E +

v ⇥Bc

◆

B0 = �✓B� v ⇥E

c

◆

This is the magnetohydrodynamic approximation.

E +v ⇥B

c= 0

) E = �v ⇥Bc

= O✓

vB

c

◆

If E0 = 0

� = Lorentz factor

Maxwell tensor

14

Since E = O✓

vB

c

◆

then

MEij = O

✓v2

c2

◆⇥MB

ij

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

Electromagnetic momentum

15

We want to compare the electromagnetic momentum density with the matter momentum density, i.e. compare

⇧EMi =

14⇡c

✏ijkEjBk⇢viwith

⇢vi = O(⇢v)

⇧EMi

⇢vi= O

✓B2

4⇡⇢c2

◆

⇧EMi = O

✓vB2

c2

◆

Electromagnetic momentum

16

⇧EMi

⇢vi= O

✓B2

4⇡⇢c2

◆

As we shall discover later, the quantity

B2

4⇡⇢= v2

A = (Alfven speed)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

v2A

c2⌧ 1

and we neglect the electromagnetic momentum density.

Final form of the momentum equation

17

Given the above simplifications, the final form of the momentum equations is:

⇢

dvi

dt

= � @p

@xi� ⇢

@�

@xi+

@

@xj

BiBj

4⇡

� B

2

8⇡

�ij

�

We also have14⇡

curlB⇥B = divBB4⇡

� B2

8⇡I�

where I is the unit tensor

18

Thus the momentum equations can be written:

⇢dvdt

= �rp� ⇢r� +14⇡

curlB⇥B

Either form can be more useful dependent upon circumstances.

Displacement current

19

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

4⇡

✏ilm@Bm

@xl� 1

4⇡

@Ei

@t

O✓

cB

L

◆O

✓E

T

=vB

cT

◆

) Displacement CurrentCurl B current

= O✓

vV

c

2

◆⌧ 1

Displacement current (cont’d)

20

In the MHD approximation we always put

Ji =c

4⇡

✏ilm@Bm

@xl

i.e. J =c

4⇡

curlB

Energy equation

21

The total electromagnetic energy density is

EEM =E2 + B2

8⇡⇡ B2

8⇡

since E2 = O✓

v2

c2

◆B2

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

Energy equation (cont’d)

22

The final result is:

@

@t

1

2

⇢v

2+ ✏ + ⇢� +

B

2

8⇡

�+

@

@xj

⇢

✓1

2

v

2+ h + �

◆vi + Si

�

= ⇢kT

ds

dt

h =

✏ + p

⇢

Si =

c

4⇡

✏ijkEjBk

=

1

4⇡

⇥B

2vi �BjvjBi

⇤=

B

2

4⇡

v

?i

where v

?i = Component of velocity

perpendicular to magnetic field

The induction equation

23

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: r⇥E +

1

c

@B@t

= 0

Infinite conductivity: E = �vc⇥B

Together these imply the induction equation

@B@t

= curl(v ⇥B)

Summary of MHD equations

24

⇢

dvi

dt

= � @p

@xi� ⇢

@�

@xi+

@

@xj

BiBj

4⇡

� B

2

8⇡

�ij

�

@⇢

@t

+@

@xi(⇢vi) = 0

Continuity:

Momentum:

Summary (cont.)

25

Energy:

@

@t

1

2⇢v

2 + ✏+ ⇢�+B

2

8⇡

�=

@

@xj

✓1

2v

2 + h+ �

◆⇢vi + Si

�

= ⇢kT

ds

dt

h =✏+ p

⇢Si =

c

4⇡✏ijkEjBk =

B2

4⇡v?i

where

Summary (cont.)

26

@Bi

@t

= [curl(v ⇥B)]i

@Bi

@t

= ✏ijk@

@xj(✏klmvlBm)

Induction: