Transcript Ch 4 Hydrodynamics

Fluid Theory: Magnetohydrodynamics (MHD)
MHD: based on moment equations derived from kinetic theory
describing "overall" behavior without gyromotions
Collisionless: collision frequency ~ 1/days
Ideal MHD: Neglecting collision terms in the Boltzmann equation:
(Vlasov equation)
The moment equations are:
Mass conservation (contuity equation):

    u   0
t
Momentum equation
 u

  u    u   p  J  B   g
 t

where u   ms ns u s /  ms ns is the bulk velocity of the fluid

   ns ms is the mass density of the fluid
J =
q n u
s
s
s
is the current density
p is total pressure (definition is more complicated)
g is gravity acceleration, small and neglected.
1
Ideal MHD Field Equations
  E=0
  E= 
B
t
B  0
  B   0 J   0 0
E
t

    u   0
t
 u


  u    u   p  J  B
 t

Variables (neglect gravity):  , p, u, J, E, and B, a total of 14.
Equations: 3 scalar ones, 3 vector ones, a total of 12.
Need 2 more equations.
Adiabatic equation (a simple energy equation)
p    constant
Ohm's law (1st moments of electron and ion charges):
J   (E  u  B )
For collisionless (ideal MHD):  = 
E  u  B
2
Ideal MHD Equation Set for slow processes
(speed much slower that c, and
time scales much slower than ion gyro period)

    u   0
t
1
 u

    u   u   p  (  B)  B
0
 t

  (u  B)=
B
t
B  0
p    constant
B2
(  B)  B  
 (B )B
2
B2
Magnetic pressure: pB =
20
Curvature force:
1
0
(B )B
1
 u

Momentum Equation:  
  u   u   p  pB 
(B  )B
0
 t

3
MHD Waves
Wave perturbation  Q  Q  Q0   Qei ( 't k
r)
/t => i ';  => - ik , b = B 0 /B0
Assume uniform background plasma: Q 0 is not a function of space
keep upto linear terms of the perturbation (small amplitude waves)

     u   0 =>  '   k  u 0   ( ' k  u 0 )  k   u 0
t
  (u  B)  (B  )u  (  B)u  (u  )B  (  u)B
  (u  B)=
B
B
=> ( ' k  u 0 )
 (k   u)b  (b  k ) u
t
B0
(  B)  B  (k   B)  B 0
1
 u

  u    u   p 
(  B)  B
0
 t


 ( ' k  u 0 ) u  Cs2
where C  
2
s
p0
0

B
k  C A2 b  (k 
)
0
B0
,C 
2
A
B02
0  0
C A is called the Alfven speed, Cs is the sound speed.
4
MHD Waves, cont.

 k  u
0
B
( ' k  u 0 )
 (k   u)b  (b  k ) u
( ' k  u 0 )
B0
( ' k  u 0 ) u  Cs2

B
k  C A2b  (k 
)
0
B0
Doppler shift:  : frequency in the plasma frame
 '    k  u0
Remove  and  B, assuming B in zˆ , and k in x-z plane
 2 u  Cs2 (k   u)k  C A2b  {k  [(k   u)b  (b  k ) u]}
 Cs2 (k   u)k  C A2 ( k 2 u x xˆ  k z2 u y y )
Components
 2 u x  Cs2 (k x u x  k z u z )k x  C A2 k 2 u x
 2 u y  C A2 k z2 u y
 2 u z  Cs2 (k x u x  k z u z )k z
5
Rearrange
( 2  Cs2 k x2  C A2 k 2 ) u x  Cs2 k z k x u z  0
( 2  C A2 k z2 ) u y  0
Cs2 k x k z u x  ( 2  Cs2 k z2 ) u z  0
A homogeneous linear algebraic equation set
Condition for a solution is ( u  0, or there is no wave)
(determinant of the coefficient matrix be zero)
 2  Cs2 k x2  C A2 k 2
0
Cs2 k x k z
0
 2  C A2 k z2
0
Cs2 k x k z
0
0
Cs2 k z2   2
or
( 2  C A2 k z2 )[( 2  Cs2 k x2  C A2 k 2 )( 2  Cs2 k z2 )  Cs4 k x2 k z2 ]  0
or
( 2  C A2 k z2 )  0
---- intermediate mode
( 2  Cs2 k x2  C A2 k 2 )( 2  Cs2 k z2 )  Cs4 k x2 k z2  0 ----slow/fast mode
Alfven mode
vi   / k  C A cos 
 =tan -1 (k x / k z )
Fast/slow mode
1
2
v f/s
 ( / k ) 2  [(Cs2  C A2 )  (Cs2  C A2 ) 2  4Cs2C A2 cos 2  ]
2
6
7
Alfvén mode (Intermediate Mode)
• In the Alfven mode, perturbations of B are perpendicular to
both B and k. There is no change in density  or in the
magnitude of B.
• The changes involve only bends of B.
• Alfvén mode waves work to reduce bending of the
magnetic field. They carry field-aligned currents.
8
Fast and Slow Modes
• The fast (+ sign) and slow (- sign) modes that make up the
other two solutions are compressional (i.e. they do change
density and field magnitude).
– Fast waves are produced when the total pressure of the plasma (the
sum of the particle pressure and field pressure) changes locally.
The plasma and magnetic pressures are in phase. This wave
propagates almost isotropically.
9
Fast and Slow Modes – cont.
• For the slow mode the total pressure is
approximately constant across the background
field. Slow mode waves carry energy primarily
along the background field. Field-aligned
gradients in the total pressure drive slow mode
waves.
10
Homework
• 3.1, 3.7, 4.10, 5.2,
• 3.10*, 3.12*, 4.8*
11
1-D discontinuities
Shock Front
Upstream
(low entropy)
v1
Downstream
(high entropy)
v2
12
Rankine-Hugoniot Relations
Steady state MHD equations
 E=q / 0
  E= 0
 B0
  B  0 J
   u   0
  u   u  p  J  B
E  uB  0
1  D : variations occur only in the normal direction nˆ
ˆ.
not in the tangential direction T
  E= 0 =>
E  u  B
ET
 0 => ET  ET2  ET1  0
n
=> un B T  u T Bn  0
Bn
 0 => Bn  Bn2  Bn1  0
n
 un
    u   0 =>
 0 =>  un   2un2  1un1  0
n
 B  0 =>
  u   u  p 
1
0
(  B)  B =>  un u  p 
1
0
(  B)  B  0
Energy conservation equation: slightly more complicated.
13
• The R-H jump conditions are a set of 6 equations. If we want to find
the downstream quantities given the upstream quantities then there are
6 unknowns (  ,vn,,vT,,p,Bn,BT).
• The solutions to these equations are not necessarily shocks. These
conservations laws and a multitude of other discontinuities can also be
described by these equations.
Types of Discontinuities in Ideal MHD
Contact Discontinuity
vn  0 ,Bn  0
Density jumps arbitrary,
all others continuous. No
plasma flow. Both sides
flow together at vT.
Tangential Discontinuity
vn  0 , Bn  0
Complete separation.
Plasma pressure and field
change arbitrarily, but
pressure balance
Rotational Discontinuity
vn  0 , Bn  0
Large amplitude
intermediate wave, field
and flow change direction
but not magnitude.
vn  Bn  0  
1
2
14
Types of Shocks in Ideal MHD
Shock Waves
vn  0
Flow crosses surface
of discontinuity
accompanied by
compression.
Parallel Shock
Bt  0
B unchanged by
shock.
Perpendicular
Shock
Bn  0
P and B increase at
shock
Oblique Shocks
Bt  0, Bn  0
Fast Shock
P, and B increase, B
bends away from
normal
Slow Shock
P increases, B
decreases, B bends
toward normal.
Intermediate
Shock
B rotates 1800 in
shock plane, density
jump in anisotropic
case
15
•Configuration of magnetic field lines for fast and slow shocks. The lines
are closer together for a fast shock, indicating that the field strength
increases. [From Burgess, 1995].
16
• For compressive fast-mode and slow-mode oblique shocks the
upstream and downstream magnetic field directions and the shock
normal all lie in the same plane: coplanarity theorem.
 
nˆ  (Bd  Bu )  0
• The transverse component of the momentum equation can be written
B
as  vn vt  n Bt  0 and Faraday’s Law gives vn Bt  Bn vt  0.
0
• Therefore both Bt and vn Bt are parallel to vt and thus are parallel
to each other.








Expanding vun But  But  vdn Bdt  Bdt  vdn But  Bdt  vun Bdt  But  0
•
Thus Bt  vn Bt  0 .
•
If vn,u  vn,d Bt ,u and Bt ,d must be parallel.



(vn,u  vn,d )(Bt ,u  Bt ,d )  0
• The plane containing one of these vectors and the normal contains both
the upstream and downstream fields.
•
 
 
 
Since (Bu  Bd )  nˆ  0 this means both Bd  Bu and Bu  Bd are
perpendicular to the normal and


 


 
nˆ  ( Bu  Bd )  ( Bu  Bd ) / ( Bu  Bd )  ( Bu  Bd ) 17