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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: