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 (cgs Gaussian units) Displacement current
Electric current Ampere’s law Gauss’s law of electrostatics 4⇡ 1 @E E =4⇡⇢ B = Je + r · e r⇥ c c @t B =0 1 @B r · E + =0 No magnetic monopoles r⇥ c @t Faraday’s law of induction
Particle equations of motion dv v m = q E + B m dt c ⇥ r ⇣ Lorentz force ⌘Gravitational force
4 Cartesian form of Maxwell’s equations
@Ej =4⇡⇢e @xj @B j =0 @xj @Bk 4⇡ 1 @Ei ✏ijk = Ji + @xj c c @t @Ek 1 @Bi ✏ijk + =0 @xj c @t
Equations of motion of a charged particle
dvi vj @ m = q Ei + ✏ijk Bk m dt c @xi ⇣ ⌘ 5 Energy density, Poynting flux and Maxwell stress tensor
E2 + B2 ✏ = = Electromagnetic energy density EM 8⇡ c S = ✏ E B =Poyntingflux i 4⇡ ijk j k S ⇧EM = i = Electromagnetic momentum density i c2 1 1 M B = B B B2 = Magnetic component of ij 4⇡ i j 2 ij ✓ ◆ Maxwell stress tensor 1 1 M E = E E E2 = Electric component of ij 4⇡ i j 2 ij ✓ ◆ Maxwell stress tensor
6 Relationships between electromagnetic energy, flux and momentum
The following relations can be derived from Maxwell’s equations:
@✏EM @Si + = JiEi @t @xi @⇧ @M J i ij = ⇢ E + ✏ j B @t @x e i ijk c k j ✓ ◆
7 Momentum equations
Consider the electromagnetic force acting on a particle v↵ F ↵ = q↵ E + ✏ j B α refers to specific i i ijk c k particle Consider a unit volume of gas and the electromagnetic force acting on this volume
v↵ F em = q↵ E + ✏ q↵ j B i i ijk c k " ↵ # " ↵ # X X 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)
em ↵ ↵ ↵ Fi = q Ei + ✏ijk q vj Bk " ↵ # " ↵ # X X We can identify the following components
↵ q = ⇢e = Electric charge density ↵ X ↵ ↵ q vj = Ji =Electriccurrentdensity ↵ X so that the electromagnetic force can be written J F em = ⇢ E + ✏ j B i e i ijk c k 1 i.e. F = ⇢eE + J B c ⇥ 9 Momentum (cont’d)
We add the body force to the momentum equations to obtain
dvi @p @ Jj ⇢ = ⇢ + ⇢eEi + ✏ijk Bk dt @xi @xi c
Now use the equation for the conservation of electromagnetic momentum:
1 @⇧i @Mij ⇢eEi + ✏ijkJjBk = + c @t @xj
10 Momentum (cont’d) so that the momentum equations become:
Bulk transport of Flux of momentum momentum due to EM field @ @ @ (⇢vi + ⇧i)+ (⇢vivj + p ij Mij)= ⇢ @t @xj @xi Mechanical + Gravitational EM momentum Flux of momentum force density due to pressure
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
Ji0 = Ei0
Conductivity
The conductivity of a plasma is very high so that for a finite current E0 0 i ⇡ 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 v B E0 = E + ⇥ c ✓ ◆ = Lorentz factor v E B0 = B ⇥ c ✓ ◆
If E0 = 0 v B E + ⇥ =0 c v B vB E = ⇥ = ) c O c ✓ ◆ This is the magnetohydrodynamic approximation.
13 Maxwell tensor vB Since E = O c ✓ ◆ then
v2 M E = M B ij O c2 ⇥ ij ✓ ◆ Hence, we neglect the electric component of the Maxwell tensor. We can also neglect the displacement current, as we show later.
14 Electromagnetic momentum
We want to compare the electromagnetic momentum density with the matter momentum density, i.e. compare
1 ⇧EM = ✏ E B with ⇢vi i 4⇡c ijk j k vB2 ⇧EM = ⇢vi = (⇢v) i O c2 O ✓ ◆
⇧EM B2 i = ⇢v O 4⇡⇢c2 i ✓ ◆
15 Electromagnetic momentum
⇧EM B2 i = ⇢v O 4⇡⇢c2 i ✓ ◆ As we shall discover later, the quantity
B2 = v2 = (Alfven speed)2 4⇡⇢ A
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 v2 A 1 c2 ⌧ and we neglect the electromagnetic momentum density. 16 Final form of the momentum equation Given the above simplifications, the final form of the momentum equations is:
dv @p @ @ B B B2 ⇢ i = ⇢ + i j dt @x @x @x 4⇡ 8⇡ ij i i j
We also have 1 BB B2 curlB B =div I 4⇡ ⇥ 4⇡ 8⇡ where I is the unit tensor
17 Thus the momentum equations can be written:
dv 1 ⇢ = p ⇢ + curlB B dt r r 4⇡ ⇥
Either form can be more useful dependent upon circumstances.
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:
c @Bm 1 @Ei Ji = ✏ilm 4⇡ @xl 4⇡ @t cB E vB = O L O T cT ✓ ◆ ✓ ◆ Displacement Current vV = 1 ) Curl B current O c2 ⌧ ✓ ◆
19 Displacement current (cont’d)
In the MHD approximation we always put
c @Bm Ji = ✏ilm 4⇡ @xl c i.e. J = curl B 4⇡
20 Energy equation
The total electromagnetic energy density is
E2 + B2 B2 EEM = 8⇡ ⇡ 8⇡ v2 since E2 = B2 O c2 ✓ ◆
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: @ 1 B2 @ 1 ⇢v2 + ✏ + ⇢ + + ⇢ v2 + h + v + S @t 2 8⇡ @x 2 i i j ✓ ◆ ds = ⇢kT dt ✏ + p h = ⇢ c S = ✏ E B i 4⇡ ijk j k 2 1 2 B = B v B v B = v? 4⇡ i j j i 4⇡ i
where vi? = 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:
1 @B Faraday’s Law: E + =0 r⇥ c @t v Infinite conductivity: E = B c ⇥ Together these imply the induction equation
@B =curl(v B) @t ⇥
23 Summary of MHD equations
Continuity: @⇢ @ + (⇢vi)=0 @t @xi
Momentum:
dv @p @ @ B B B2 ⇢ i = ⇢ + i j dt @x @x @x 4⇡ 8⇡ ij i i j
24 Summary (cont.) Energy:
@ 1 B2 @ 1 ⇢v2 + ✏ + ⇢ + = v2 + h + ⇢v + S @t 2 8⇡ @x 2 i i j ✓ ◆ ds = ⇢kT dt where
✏ + p c B2 h = S = ✏ E B = v? ⇢ i 4⇡ ijk j k 4⇡ i
25 Summary (cont.)
Induction:
@B i =[curl(v B)] @t ⇥ i
@Bi @ = ✏ijk (✏klmvlBm) @t @xj
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