Plasma Astrophysics

Chapter 4: Single-Fluid Theory of

Plasma - Magnetohydrodynamics
Yosuke Mizuno
Institute of Astronomy
National Tsing-Hua University
 Exercise 1
• Plasma frequency: ne2
p =
me 0

0 kB Te
• Debye length: D =
e2 n
3/2
1.38 106 Te
• Plasma number: =4 n 3
D =
n1/2
2
36 0 kB T
• Mean free path: mf p
n e2
36 n 4
D D ND
 Exercise 1 (cont.)  
qB
• Gyro frequency: c =
m
mv v
• Larmor radius: rL = =
|q|B c

• Electron volts is energy units = particle’s kinetic energy

• For Larmor radius, we need to get perpendicular components
of velocity.
 Single-Fluid Theory: MHD
• Under certain circumstances, appropriate to consider entire plasma
as a single fluid.
• Do not have any difference between ions and electrons.
• Approach is called magnetohydrodynamics (MHD).
• General method for modeling highly conductive fluids, including
low-density astrophysical plasmas.
• Single-fluid approach appropriate when dealing with slowly
varying conditions.
• MHD is useful when plasma is highly ionized and electrons and
ions are forced to act in unison, either because of frequent
collisions or by the action of a strong external magnetic field.
 Single-fluid equations for fully ionized plasma
• Can combine multiple-fluid equations into a set of equations for a
single fluid.
• Assuming two-specials plasma of electrons and ions (j = e or i):
nj
+ · (nj v j ) = 0 (4.1a)
t
vj
mj n j + (v j · )v j = · P j + qj nj (E + vj B) + Pij (4.1b)
t
• For a fully ionized two-species plasma, total momentum must be
conserved:
Pei = Pie
• As mi >> me the time-scales in continuity and momentum equations
for ions and electrons are very different. The characteristic
frequencies of a plasma, such as plasma frequency or cyclotron
frequency are much larger for electrons.
 Single-fluid equations for fully ionized
plasma (cont.)  
• When plasma phenomena are large-scale (L >> λD) and have
relatively low frequencies (ω << ωplasma and ω << ωcyclotron), on
average plasma is electrically neutral (ni ~ ne). Independent motion
of electrons and ions can then be neglected.
• Can therefore treat plasma as single conducting fluid, whose inertia
is provided by mass of ions.
• Governing equations are obtained by combining eqn (4.1)
• First, define macroscopic parameters of plasma fluid:
m = ne me + ni mi Mass density
e = ne qe + ni qi Charge density
J = ne qe ve + ni qi vi = ne qe (ve vi ) Electric current
v = (ne me v e + ni mi v i )/ m Center of Mass Velocity
P = Pe + Pi Total pressure tensor
 MHD mass and charge conservation
nj
• Using eq (4.1a): +
· (nj v j ) = 0
t
• Multiply by qi and qe and add continuity equations to get:
e
+ · (J ) = 0 Charge conservation
t
• where J is the electric current density: J = ne qe v e + ni qi v i and the
electric charge: e = ne qe + ni qi
• Multiply eq (4.1a) by mi and me,
m Mass conservation /
+ · ( m v) = 0
t continuity equation
• where m = ne me + ni mi is the single-fluid mass density and v is
the fluid mass velocity
v = (ne me v e + ni mi v i )/ m
 MHD equation of motion
• Equation of motion for bulk plasma can be obtained by adding
individual momentum transport equations for ions and electrons.

vj
• LHS of eq(4.1b): mj nj + (v j · )v j
t

• Difficulty is that convective term is non-linear.

• But note that since me << mi contribution of electron momentum is
much less than that from ion. So we ignore it in equation
• Approximation: Center of mass velocity is ion velocity: v v i
• LHS:
vj v
mj n j + (v j · )v j m + (v · )v
t t
 MHD equation of motion (cont.)    
• RHS of eq(4.1b) : · (P e + P i ) + (ne qe + ni qi )E + J B

• In general, second term (Electric body force) is much smaller than J

x B term. So we ignored.
• Therefore, LHS+RHS:

v Equation of motion
m + (v · )v = ·P +J B
t

• For an isotropic plasma, ·P = p where total pressure is p = pe +

pi and
v
m + (v · )v = p+J B Equation of motion
t
 MHD equation of motion (cont.)    
• ρeE term is generally much smaller than J x B term. To see this take
order of magnitudes.
• from Maxwell’s equations:
·E = e/ 0 so e E 0 /L
B = µ0 J so E j B/µ0 L

• Therefore,
2 2
eE 0 B2 Lµ0 L2 /c2 light crossing time
=
jB L µ0 L B2 (µ0 L )2 2 resistive skin time

• This is generally very small number.

• Example: small cold plasma, Te=1eV, L=1cm, this ratio is about 10-8
 Generalized Ohm’s law
• The final single-fluid MHD equation describes the variation of
current density J.
• Consider the momentum equations for electron and ions (eq.4.1b):
vj
mj n j + (v j · )v j = · P j + qj nj (E + vj B) + Pij
t
• Multiple electron equation by qe/me and ion equation by qi/mi and
add: J qe qi
= · Pe · Pi
t me mi
(We ignore second ne qe2 ni qi2
term of LHS as we + + E
me mi
dealing with small
perturbation) ne qe2 ni qi2
+ ve + vi B
me mi
qe qi
+ P ei + P ie
me mi
 Generalized Ohm’s law (cont.)  
In forth term of RHS:
ne qe2 ni qi2
ve + vi
me mi
qe qi n e q e mi ni qi me
= ve + vi
me mi qi qe
qe qi mi me
= ne me v e + ni mi v i + (qe ne v e + qi ni v i )
me mi qi qe
qe qi mi me
= mv + J
me mi qi qe
ne qe2 ni qi2 qe qi
= + v+ + J
me mi me mi
 Generalized Ohm’s law (cont.)  
• For an electrically neutral plasma |qe ne | |qi ni | and using
J = ne qe v e + ni qi v i and v = (ne me v e + ni mi v i )/ m, We can write
J qe qi
= · Pe · Pi
t me mi
ne qe2 ni qi2
+ + (E + v B)
me mi
qe qi
+ + (J B)
me mi
qe qi
+ P ei
me mi
• As me mi qe /me qi /mi and ne qe2 /me ni qi2 /mi . In thermal
equilibrium, kinetic pressures of electrons is similar to ion pressure
(Pe ~ Pi)
J qe ne qe2 qe qe
= · Pe + (E + v B) + (J B) + P ei (4.2)
t me me me me
 Generalized Ohm’s law (cont.)  
• The collisional term can be written: P ei = q 2 n2e (v i v e )
where η is the specific resistivity, q2 relates to fact that collisions
result from Coulomb force between ions (qi) and electrons (qe) and
total momentum transferred to electrons in an elastic collision with
an ion is vi – ve .
• Now qi= - qe and ne = ni and J=neqe(ve-vi), => P ei = ne qe J
• Eq. (4.2) can be written as
J qe ne qe2 qe ne qe2
= · Pe + (E + v B) + (J B) ˆ·J
t me me me me
(4.3)
• Where η is a tensor. This is generalized Ohm’s law
 Generalized Ohm’s law (cont.)  
• For a steady current in a uniform E, J / t = 0, · P = 0 and B = 0
so that
E= J J = 1/ E
• In general form, the electric field E can be found from Eq (4.3):
J B ·P me J
E= v B + + ˆ·J +
ne qe ne qe ne qe t
• Consider right hand side of this equation:
– First term: E associated with plasma motion
– Second term: Hall effect
– Third term: Ambipolar diffusion from E-field generated by pressure gradients
– Fourth term: Ohmic losses/Joule heating by resistivity
– Fifth term: Electron inertia
 One fluid MHD Ohm’s law
• Generalized Ohm’s law
J qe ne qe2 qe ne qe2
= · Pe + (E + v B) + (J B) ˆ·J
t me me me me
• Now assume plasma is isotropic, so that · P = p
Also we neglect Hall effect and Ambipolar diffusion in generalized
Ohm’s law since not important in one-fluid MHD.
For slow variations, J = constant, so can write generalized Ohm’s
law as: n q2 n q2
e e e e
0= (E + v B) J
me me
• Rearranging gives,

J = (E + v B) One-fluid MHD Ohm’s law

• Where σ = 1/η is electrical conductivity

 Simplified MHD equations
• A set of simplified MHD equations can be written:
m
+ ( m v) = 0
t
v
m + (v · )v = p+J B
t
E+v B = J
• Fluid equations must be solved with reduced Maxwell equations
B (displacement current term
B = µ0 J , E=
t is ignored for low
· B = 0, ·E =0 frequency phenomena)

• Here we have assumed that there is no accumulation of charge (i.e.,

ρe = 0)
• Complete set of equations only when equation of state for
relationship between p and n (ρ) is specified.
p m = const
 Plasma β

• The MHD equation of motion contains J x B term, which can given
rise to effects that are similar to those of the pressure term.
1
• Current is given by J = B
µ0
• Taking cross product with the magnetic field,
1 1 B2
J B= ( B) B= (B · )B
µ0 µ0 2
• Inserting into MHD equation of motion
v 1 B2
m + (v · )v = (B · )B p+
t µ0 2µ0
• In second term of RHS, the first term acted on by gradient is plasma
pressure and the second term is magnetic pressure.
2µ0 p
• The dimensionless parameter, plasma β : 2
Plasma beta
B
• β <<1: dominated by magnetization effects
• β>>1: behaves more like a fluid
 The induction equation
• Taking the curl of one-fluid MHD Ohm’s law:
1
E= (v B) + J
• Assuming σ=const. Substituting for J = B/µ0 from Ampere’s law
and using the law of induction equations (Faraday’s law):
B 1
= (v B) + ( B)
t µ0
• The double curl can be expanding from vector identity
B 1 1
= (v B) + ( · B) 2
B
t µ0 µ0
• The second term in R.H.S. is zero by Gauss’s law ( · B = 0 ). So
B 1
= (v B) + 2
B MHD induction equation
t µ0
 The induction equation (cont.)  
• The MHD induction equation, together with fluid mass,
momentum, and energy equations (EoS), a close set of
equations for MHD state variables (ρm, v, p, B)
m
+ ( m v) = 0
t
v 1 B2
m + (v · )v = (B · )B (p + )
t µ0 2µ0
p m = const
B 1
= (v B) + 2
B
t µ0

Here, J = B/µ0
E= v B + J/
 Ideal MHD
• In the case where the conductivity is very high ( ), the
electric field is E= -v x B (motional electric field only). It is
known as ideal Magnetohydrodynamics.
• A set of equations:
m
+ ( m v) = 0
t
v 1 B2
m + (v · )v = (B · )B (p + )
t µ0 2µ0
p m = const
B
= (v B)
t
• This is the most simplest assumption for MHD. But this is
commonly used in Astrophysics.
 The pressure equations
• The above formulation of the ideal MHD equations exploits ρ, v, p, B
as the basic variables
• Equation of states is often replaced by pressure evolution equation.
• It is also work out the evolution equation for the other
thermodynamical variables, such as
– e: internal energy per unit mass (which is equivalent to T)
– s: entropy per unit mass
Cv: specific heat
1 p
e Cv T capacity
1 m
s Cv ln S , S p/ m
• Neglect thermal conduction and heat flow, i.e., considering adiabatic
processes, the entropy convected by the fluid is constant:
Ds DS D p
= 0 , or =0
Dt Dt Dt m
 The pressure equations (cont.)
Apply change rule
D p 1 Dp p D m
= +1 Dt = 0
Dt m m Dt m
Expand D/Dt
1 p 1 p m p
+ (v · )p +1 +1 (v · ) m =0
m t m m t m
p p m
+ (v · )p + (v · ) m =0
t m t
m
But + (v · ) m = m ·v
t
p p
+ (v · )p + ( m · v) = 0
t m

p
+ (v · )p = p ·v Pressure evolution equation
t
 The internal energy equation
• From pressure evolution equations, using equations of state
p=( 1) me

we can write the internal energy equations

e
+ (v · )e = ( 1)e ·v Internal energy equation
t
 Magnetic field behavior in MHD
• MHD induction equation: B 1
= (v B) + 2
B
t µ0
• (v B) Dominant: convection
– Infinite conductivity limit: ideal MHD.
– Flow and field are intimately connected. Field lines convect with the flow.
(flux fleezing)
– The flow response to the field motion via J x B force
• (1/µ0 )
2
B Dominant: Diffusion
– Induction equation takes the form of a diffusion equation.
– Field lines diffuse through the plasma down any field gradient
– No coupling between magnetic field and fluid flow
Here using
– Characteristic Diffusion time: = µ0 L2 = µ0 L2 / = 1/L
• Ratio of the convection term to the diffusion term:
vB/L
Rm = 2
= µ0 vL Magnetic Reynold’s number    
B/µ0 L
 Magnetic field behavior in MHD
Magnetic Reynold’s number     (cont.)  
vB/L Earth’s magetosphere
Rm = 2
= µ0 vL
B/µ0 L
• If Rm is large, convection dominates,
magnetic field frozen into the
plasma.
Else if Rm is small, diffusion
dominates.
• In astrophysics generally, Rm is very
large.
– Solar flare: 108,
– planetary magnetosphere: 1011
• But, not large everywhere
– Thin boundary layers form where Rm~1
and ideal MHD breaks down
 Magnetic field behavior in MHD
(cont.)
• Rewrite continuity equation:
m
= m( · v) (v · ) m
t
– first term describes compression (fluid contracts or expansion)
– Second term describes advection
• The induction equation (ideal MHD) can be written as, using
standard vector identities:
B
= B( · v) (v · )B + (B · )v
t
• Equation is similar to continuity equation.
– First term: compression
– Second term: advection
– Third term: new term describes stretching. It is related magnetic field
amplification
 Flux freezing
• Alfven’s theorem (1947): “field is frozen into the fluid”
• This is extremely important concept in MHD, since it allows us to
study the evolution of the field by finding out about the plasma flow

B
• MHD induction equation: = (v B)
t

• The magnetic flux though a closed loop l : B B · n̂dS

l

Where dS is the area element of any surfaces which has l as a

perimeter. The quantity ΦB is independent of the specific surface
chosen, as can be proven from · B = 0 .
• So the flux freezing law is expressed as: d B = 0
dt

where use total derivative d/dt to indicate that the time derivative is
calculated with respect to fluid elements moving with the flow
 Flux freezing (cont.)  
• The quantity ΦB is not locally defined.
So explicit calculation for its time derivative

• Consider a loop of fluid elements l at two

instants in time, t and t+Δt
• Two surfaces S1 and S2 have l(t) and l(t+Δt)
• “cylinder” S3 generated by the fluid motion between the two instants
of the elements making up l.
• Let ΦB be the flux enclosed by l and ΦB1 be the flux through surface
S1 (similarity for S2 and S3)

• Then d B B2 (t + t) B1 (t)
= lim
dt t 0 t
 Flux freezing (cont.)
• From · B = 0 the net flux through the surfaces at any time is zero
B1 (t + t) + B2 (t + t) + B3 (t + t) = 0

• (Note that negative sign indicated as inward into the volume)

• We can eliminate ΦB2(t+Δt) and use definition of flux in expressing

ΦB1 & ΦB3
d B 1
= lim (B(t + t) B(t)) · n̂dS B · n̂dS
dt t 0 t S1 S3
(4.4)
• The first term in RHS in eq (4.4):
B
· n̂dS
S1 t
 Flux freezing (cont.)  
• The area element for S3 can be written n̂dS = (dl v) t , where dl is
a line element of the loop of fluid elements.
• The second term in RHS of eq (4.4):
B · n̂dS = B · (l v) t = (v B) · dl t
S3 l(t) l(t)

• By using Stokes theorem to convert the line integral to a surface

integral

B · n̂dS = (v B) · n̂dS t
S3 S1
• So finally putting these results into eq(4.4) :
d B B
= (v B) · n̂dS = 0
dt t
 Magnetic pressure and curvature force
• Lorentz force:
1 1 B2
J B= ( B) B= (B · )B
µ0 µ0 2
• First term: magnetic curvature force, which relates to rate of change
of B along the direction of B.
• Second term: magnetic pressure

• To show the role of magnetic curvature force, we consider

B = B b̂ ,where B is the local intensity of B and b̂ is unit vector
• The Lorentz force then becomes
B2 B2 B2
FL = + b̂b̂ · + b̂ · b̂
2µ0 2µ0 µ0
 Magnetic pressure and curvature
force (cont.)  
• Combine first two term:
B2 B2
FL = + b̂ · b̂
2µ0 µ0
• Where is the projection of the gradient operator
on a plane perpendicular to B
• Second term contains the effects of field line
curvature.
B2 B2
• Its magnitude is b̂ · b̂ =
µ0 µ0 Rc

where Rc = 1/|b̂ · b̂| is radius of curvature of path b̂

• ( b̂ · / s is the derivative along a field line )
• The curvature force is directed toward a center of
curvature ( n̂). It is often referred as hoop stress
 Magnetic pressure and curvature force
(cont.)  
• Example of magnetic curvature force
• Consider an pure toroidal (azimuthal) magnetic field, B = B ˆ
in cylindrical coordinates (R, φ, z)
• The strength of B is function of R and z only.
• The unit vector in toroidal (azimuthal) direction ˆ has the property
ˆ · ˆ = R̂/R so that

1 1 B2
(B · )B = R̂
µ0 µ0 R
• The curvature force is directed inward, toward the center of
curvature.
 Magnetic stress tensor
• The most useful alternative form of Lorentz force is in terms of
magnetic stress tensor
• Writing a vector operators in terms of permutation (Levi-Civita)
symbol ε, one has Bm
[( B) B]i = Bk Levi-Civita
ijk jlm
xl symbol is
Bm related to
= ( kl im km il ) Bk Kronecker
xl
1 2 delta
= (Bi Bk B ik )
xk 2
where the summing convention over repeated indices and · B = 0
have been used. Define the magnetic stress tensor M by its
components: 1 2 1
Mij = B ij Bi Bj
2µ0 µ0
• The Lorentz force is written as: 1
( B) B= ·M (4.5)
µ0
 Magnetic stress tensor (cont.)  
• If V is a volume bounded by a closed surface S, eq (4.5) yields by
the divergence theorem
1
( B) BdV = n · M dS
V µ0 S

• Where n is the outward normal to the surface S.

• This shows how the net Lorentz force acting on a volume V of fluid
can be written as an integral of a magnetic stress vector acting on its
surface S
• The force FS exerted by the volume on its surroundings
1 2 1
FS = n·M = B n BBn
2µ0 µ0

• Where Bn = B · n is the component of B along the outward

normal n to the surface of the volume.
 Magnetic stress tensor (cont.)  
• To get the behavior of magnetic stresses,
consider simple case of a uniform magnetic
field, B=Bz
• The force FS in right side of the box is
F right = x̂ · M . The components are
1 2 1 1 2
F right,x = B Bx Bz = B
2µ0 µ0 2µ0 F right,y = F right,z = 0
• The magnetic field exerts a force in the positive x-direction, away
from the volume.
• The force FS in top of the box is
1 2 1 1 2
F top,z = B Bz Bz = B F top,x = F top,y = 0
2µ0 µ0 2µ0
• The magnetic field exerts a force in the negative z-direction, inward
to the volume
 Magnetic stress tensor (cont.)  
• The magnetic pressure makes the volume of magnetic field
expand in the perpendicular directions, x and y. But in the
direction along a magnetic field line the volume would contract.
• Along the field lines the magnetic stress thus acts like a negative
pressure, as in a stretched elastic wire
• This negative stress is referred to as the tension along the
magnetic field lines.
• The stress tensor plays a role analogous like the gas pressure, but
unlike gas pressure is extremely anisotropic.
 Momentum equation
• From equation of motion and continuity equations
v
m + mv · v = ( m v) +v ·( m v) + mv · v
t t
= ( m v) + ·( m vv)
t
• Using definition of magnetic stress tensor, the momentum
equation is ( B B/ µ0 for SI unit)
1 2
( m v) + · m vv + p + B I BB = 0
t 2

M I  is three-dimensional
+ · =0
t identity tensor
Mi = m vi Momentum density
1 2
ij = v v
m i j + p + B ij Bi Bj = 0 Stress tensor
2
Conservation form of ideal MHD equations
m
+ ·( m v) =0 Mass conservation
t
1 2 Momentum
( m v) + · m vv + p + B I BB = 0
t 2 conservation
1 1 2
mv + +
2
t 2
m e
2
B Energy conservation
1
+ · mv
2
+ me + p + B2 v (v · B)B = 0
2
B Magnetic flux conservation
+ · (vB Bv) = 0
t
·B =0
p=( 1) me Ideal equation of state
Neglecting gravity force. ( B B/ µ0 for SI unit)
This form is often used in numerical simulation.
 Poynting flux
• From energy conservation equation, energy flux is
1 1
Y mv
2
+ p v + (B 2 v v · BB)
2 1 µ0
• This compose hydrodynamic part and magnetic part.
• The magnetic part can be transformed:
1
Y em (B 2 v v · BB)
µ0
1
= (v B) B
µ0
= E B
• This is called Poynting flux (Poynting vector), which
represents the flow of electromagnetic energy
 Entropy conservation equation
• The best representation of the conservation form of MHD
equation is in terms of the variables, ρ, v, e and B.
• A peculiar additional variable is the specific entropy s
• For adiabatic process of ideal gas, conservation of entropy is
DS S
+ (v · )S = 0
Dt t
• But this is not in conservation form (but expresses the
conservation of specific entropy co-moving with the fluid)
• A genuine conservation form is obtained by variable ρmS, the
entropy per unit volume

( m S) + ·( m Sv) =0
t
Entropy conservation equation
 Summary
• Single fluid approach is called magnetohydrodynamics (MHD).
• In the case where the conductivity is very high, the electric field is
E= -v x B. It is known as ideal MHD.
• In ideal MHD, magnetic field is frozen into the fluid
• Lorentz force divides two different forces: magnetic pressure &
curvature force
• The induction equation in ideal MHD shows evolution of magnetic
field. It is including compression, advection and stretching
• The induction equation in resistive MHD includes diffusion of
magnetic field.
• From energy conservation equation, energy flux composes
hydrodynamic part and magnetic part. Magnetic part is called
Poynting flux.
  Hydro vs MHD  
MHD equation is shown the coupling of hydrodynamics with
magnetic field
m
+ ( m v) = 0
t
v 1 B2
m + (v · )v = (B · )B (p + )
t µ0 2µ0
p m = const
B
= (v B)
t
MHD equation is recovered hydrodynamic equations when B=0.
m
+ ( m v) = 0
t
v
m + (v · )v = p
t
p m = const
 Hydro vs MHD (cont.)  
• Conservation form of hydrodynamic equations
m
+ ·( m v) =0
t
( m v) + ·[ m vv + pI] = 0
t
1 1
mv + me + + +p v =0
2 2
· m v me
t 2 2
p=( 1) me
 Exercise 2-1
Derivation of conservation form of total energy
v
From equation of motion: + (v · )v + p j B=0
t
v
=> v· + (v · )v + v · p v · (j B) = 0
t
1 2 1 2 1
=> v v + v · v 2 + v · p v · (j B) = 0
t 2 2 t 2

Using continuity equation

1 2 1 2 (1)
=> v + · v v +v· p v · (j B) = 0
t 2 2
 Exercise 2-1 (cont.)
From pressure equation: p
+ (v · )p + p ·v =0
t
Using ideal EoS and continuity equation,
e
+ (v · )e + ( 1)e · v = 0
t
e
=> + (v · )e + ( 1) e · v = 0
t

=> t ( e) e t + (v · )e + p ·v =0
Using continuity equation,

=> ( e) + · ( ev) + p · v = 0 (2)

t
 Exercise 2-1 (cont.)  
From induction equation: B
(v B) = 0
t
B B B
=> · · (v B) = 0
µ0 t µ0

Using D6 =>
B2 1 1
+ · [B (v B)] (v B) · B=0
t 2µ0 µ0 µ0
1
Using D1 & D2=> B ( B) = j B
µ0
B2 1
+ · [(B · B)v (v · B)B] + v · j B=0
t 2µ0 µ0
(3)
 Exercise 2-1 (cont.)  
• (1) + (2) + (3) =0
1 2 B2 1 2 B2 B
v + e+ + · v + e+p+ v (v · B) = 0.
t 2 2µ0 2 µ0 µ0