Magnetohydrodynamics
Coupling hydrodynamics with Maxwell equations
Jiří Krtička
Masaryk University
Basic equations
Basic assumption
We shall consider plasma as a gas, in which the ionization can not be neglected. In such a gas the hydrodynamical equations are affected by the electromagnetic field. On the other hand, Maxwell equations shall account for the presence of plasma. This leads to a concept of coupled magnetohydrodynamical (MHD) equations.
We shall assume that plasma is electrically neutral, that is, the charge of selected macroscopic volume is zero.
Equation of continuity
We shall assume that plasma consists of: electrons (e), ions (i), and neutrals (n). The equation of continuity holds for all three components,
dne
dt
dn\
~dt
+ V-(neVe) = 0, + V-(AWi) = 0, + V-(nnvn) = 0.
dt
Multiplying these equations by appropriate masses of particles and summing them up, we derive the mean equation of continuity
d
— (neme + A7jA77j + A7nA7n) + V ■ (nemeve + n\m\V\ + nnmnvn) = 0. ot
Introducing the total density p = ^2nama and mean velocity
v =     namava/ (Y nama)i the continuity equation takes the same
form as for one-component fluid,
Equation of continuity for electric charge
Multiplying the three continuity equations by charge of each component, we derive equation of continuity for electric charge in the form of
d
— (neqe + A7j<7j) + v • (neqeve + n\q\V\) = 0.
Introducing the total electric charge density pe = ^2naqa and the current density j = Y1 naQa^a^ the equation of continuity for electric charge has the form of
However, we assume quasineutrality of plasma (pe = 0), therefore the equation of continuity for electric charge simplifies to
div j = 0.
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Equation of motion
The equation of motion holds for each component of the flow separately,
d
(m(Vn(Vv(V) + V (n(Vm(Vv(V ® v(V) =
Pt *-.-<
1
a     Vpn + qn nn E + - q(Y n(Y v(Y x
3 4 5 v-*-
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where we accounted for the Lorentz force (5+6) and for friction between components (7). Summing the equation over individual components:
use the definitions of p and v and the continuity equation, assume va • Vva ^ v • Vv, introduce the total volume force f = J2a introduce the total pressure p = J2a P&>
assume quasineutrality J2a Q^oi — 0 (implying no influence of E), introduce the current density j = Y1 naQa^a^ and assuming Newton's third law = ^> we arr've at
<9v _        _      _    1 . _
p— + pv Vi/ = -Vp + f H—J x B.
at c
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2
3
4
5
6
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Maxwell's equations
We shall write the the Maxwell's equations in cgs units in vacuum (i.e., H = B and E = D) and neglecting the displacement current 1/cdD/dt. From the Ampere's law
,,    1 dD    4tt .
rot H = - —— H--j
c at c
then follows
j = —rotS.
From the equation of motion with frictional term follows the Ohm's law
j =aE
written in the frame where v' = 0. Transforming into a general frame
j = a x B^j .
From this follows for the electric intensity (using the Ampere's law)
i-    j"     1      «      c       «1 «
E = ---v x B =-rotS--v x B.
a     c A-tvct c
Maxwell's equations
From the induction equation
ldB
rot E
c dt
follows using E = 4^rot 8-^x8 the equation for the magnetic field
dB c2
—— =--rot rot B + rot (vxB).
For an ideal plasma (ideal MHD) the conductivity a —>► oo and
— = rot(,xB).
In this case the Ohm's law simplifies to
1
E =   -v x B.
c
A side note: magnetic diffusion
The term inversely proportional to a in the induction equation
dB c2
—— =--rot rot B + rot (v x B)
describes magnetic field diffusion, because
dB c2
dt 47RT
AB
is a diffusion equation. From this equation follows the characteristic diffusion time
r = — L2
which is typically very large for astrophysical plasmas due to large /_. Therefore, in most applications the diffusion term can be safely neglected
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Basic equations of ideal MHD
f+ V>,) = 0
d\/ 1
p— + pv ■ Vi/ = - Vp + f H--(rot 6) x 6
at 47r
- = rot(,xe)
divB = 0 equation for energy
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Some applications
Let us study a closed curve ^ = dS that is co-moving with the plasma and calculate the change of magnetic flux through ^,
This means that the flux remains constant in time for any arbitrary closed curve. In turn, this implies that magnetic field-lines must move with the plasma. Or, in scientific jargon, the field-lines are frozen into the plasma.
Flow along 6 is not affected by the field, while the flow perpendicular to 6 tears down the field. In a tube the product SB is constant, for which the continuity equation pSv = const, implies p ~ B/v.
Two limiting cases
From the momentum equation
d\/ 1
p— + pv Vi/ = -Vp + f H--(rot B)x B
at 47r
two limiting cases follow. For negligible magnetic field
(rot 6) x B/(4tt) ^ 0 the flow is not affected by the magnetic field and
moves freely. But we still have
— = rot(,xB),
and the magnetic field is from the flux freezing condition carried with the flow.
In the oposite case the magnetic field dominates and the flow follows the magnetic field.
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Introducing the magnetic pressure
The magnetic term in the equation of motion can be rewritten as
1 (rotB) x B = — (B- V)B- V fB
4-7T 47T \ 47T
Consequently, the equation of motion takes the form of
From this equation, the term B2/(Att) can be regarded as magnetic pressure.
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Magnetostatics
The magnetic pressure plays a crucial role in magnetostatics, which is an application of MHD on stationary {d/dt = 0) and static (v = 0) systems. For these systems the momentum equation gives
vfp + f-) =f + ^e-ve
V      4tt J 4tt
or
1
Vp = — (rotB) xB + f.
47t
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Magnetostatics: two examples
In absence of external fields f = 0, for vertically constant magnetic field dB/dz = 0 in the z direction the momentum equation simplifies to
B
p=const.
v p +
47t
0
p +
e2
47t
const.
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Magnetostatics: two examples
In absence of external fields f = 0, for vertically constant magnetic field dB/dz = 0 in the z direction the momentum equation simplifies to
B2
0 =4- p H--= const.
H Air
Magnetostatics: two examples
In absence of external fields f = 0, for vertically constant magnetic field dB/dz = 0 in the z direction the momentum equation simplifies to
V  p +
e2
47T
0
P +
e2
47T
const.
For atmosphere with horizontal magnetic field the hydrostatic equilibrium equation reads
dz
P +
e2
47T
For d62/dz < 0 this enables enables to support the matter (p > 0) above the regions of zero density (p = 0). This corresponds to solar prominences.
p>0 p=0
p>0
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Magnetostatics: two examples
In absence of external fields f = 0, for vertically constant magnetic field dB/dz = 0 in the z direction the momentum equation simplifies to
B2
0 ^> p H--= const.
For atmosphere with horizontal magnetic field the hydrostatic equilibrium equation reads
= ~pg>
For d62/dz < 0 this enables enables to support the matter (p > 0) above the regions of zero density (p = 0). This corresponds to solar prominences.
Polarization electric field in the stellar atmosphere
Let us go back to the momentum equations of individual components and assume an isothermal atmosphere in hydrostatic equilibrium without magnetic field (6 = 0) composed of ionized hydrogen only. In such a case the momentum equations simplify to
kTVn\ — m\n\ g — en\ E = 0,
kTVne — mene g + ene E = 0.
Summing these two equation (taking into an account that me <^ m\ and A7j ^ A7e) we derive
-VA7i = -t—g. A7j 2k T
which is ordinary hydrostatic equilibrium equation. Substracting these equations we derive equation for polarization electic field in the form of
Suggested reading
E. Battaner: Astrohysical fluid dynamics L. Mestel: Stellar magnetism
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