2. MHD Equations
2.1 Introduction
Many processes caused
by magnetic field (B)
Sun is NOT a normal gas
Sun is in 4th state of matter
("PLASMA")
behaves differently from
normal gas:
B and plasma -- coupled (intimate, subtle)
B exerts force on plasma -- stores energy
MOST of UNIVERSE is PLASMA:
Ionosphere --> Sun (8 light mins)
Learn basic behaviour of plasma from Sun
Magnetic Field Effects
E.g., A Sunspot
B exerts a force:
-- creates intricate structure
* ____________________*
E.g., A Prominence
Magnetic tube
w. cool plasma
B --> Thermal
Blanket +
Stability
What is global equilibrium? / fine structure ?
*_______*
E.g., a
Coronal Mass
Ejection
QuickTime™ and a
decompressor
are needed to see this picture.
*_________
_____
*
E.g., A Solar Flare
(from TRACE)
B stores energy converted to other
forms
QuickTime™ and a
decompressor
are needed to see this picture.
•_ _ _ _ _ _ _
_______
_ _ _ _ _ _ _*
2.2 Flux Tubes & Field Lines
Magnetic Field Line -- Curve with tangent in
direction of B.
Equation:
dx dy dz
In 2D: * _ _ _ _ _ _ * or in 3D: B  B  B
x
y
z
Magnetic Flux Tube
-- Surface generated by set of field lines
intersecting simple closed curve.
Strength (F) -- magnetic flux crossing a
section
i.e., * _ _ _ _ _ _ _ *
But
.B  0
--> No flux is created/destroyed inside flux tube
So F   B.dS is constant along tube
Ex 2.1
Prove the above result that,
if .B  0 , then
F   B.dS
is constant
along a flux tube.
F   B.dS
If cross-section is small,
* _ _ _ _ _*
B lines closer --> A smaller + B increases
Thus, when sketching field lines,
ensure they are closer when B is stronger
To sketch magnetic field lines:
(i)
Solve
dy By

dx Bx
(ii) Sketch one field line
(iii)  Sketch other field lines, remembering
that B increases as the field lines
become closer
(iv) Put arrows on the field lines
EXAMPLE
Sketch the field lines for Bx  y,
(i) Eqn. of field lines:
*__________*
(ii) Sketch a
few field lines:
? arrows, spacing

dy By

dx Bx
By  x

(iii) Directions of arrows:
(Bx  y, By  x)
(iv) Spacing (Bx  y, By  x)
At origin B = 0.* _ _ _ _ _ _ _ _ _ _ _ _ _ _ *
Magnetic reconnection & energy conversion
**Examples
Ex 2.2 Sketch the field lines
for
(a) By=x
(b) Bx=1, By=x
Ex 2.3 Sketch the field lines for
(a) Bx=y, By=a2x
(b) Bx=y, By=-a2x
2.3 Plasma Theory

-- the study of the interaction between a magnetic field
and a plasma, treated as a continuous medium/set of p’cles

But there are different ways of modelling a plasma:
(i) MHD -- fluid eqns + Maxwell
(ii) 2-fluid-- electron/ion fluid eqns + Maxwell
(iii) Kinetic -- distribution function for
each species of particle
Eqns of Magnetohydrodynamics
Magnetohydrodynamics (MHD)
Unification of Eqns of:
(a) Maxwell
B/ 
=
j   D /  t,
.B = 0,
  E =   B /  t,
.D =  c ,
where
B =  H, D =  E,
E= j/.
(b) Fluid Mechanics
dv
Motion

  p,
dt
d
Continuity
 .v = 0,
dt
Perfect gas
p  R  T,
Energy eqn.
.............
where d / dt =  / t + v.
or (D / Dt)
In MHD
 1. Assume v << c
--> Neglect * _ _ _ *
  B/ = j

2. Extra E on plasma moving
E +

(1)
* _ _ _ _* =
j/
 (2)
3. Add magnetic force
dv

dt
  p  * _ _ _ _ *
 Eliminate E and j: take curl (2), use (1) for j
2.4 Induction Equation
B
    E =   (v  B  j /  )
t
   (v  B)    (  B)
   (v  B)   B,
2
where _ _ _ _ _ _ is magnetic diffusivity
Describes:
how B moves with plasma / diffuses through it
N.B.
• In MHD, v and B are * _ _ _ _ _ _ _ _ _ _*:
• Induction eqn
+ eqn of motion
B
2
   (v  B)    B
t
dv

dt
•
are
  p 
j B
--> basic processes
j =   B /  and E =  v  B + j / 
secondary variables
INDUCTION EQUATION
B
2
   (v  B)    B
t
I
II
•
•
B changes due to transport + diffusion
I
II

L0 v0

 Rm
-- * _ _ _ _ _ _ _
______*
2

=
1
m
/s, L0 = 105 m, v0 = 103 m/s --> Rm = 108
eg,
• I >> II in most of Solar System -->
B frozen to plasma -- keeps its energy
Except Reconnection -- j & B large
(a) If Rm << 1
 The induction equation reduces to
B
2
  B
t

B is governed by a diffusion equation
--> field variations on a scale L0
diffuse away on time * _ _ _ _ _*

with speed vd  L0 /td  L
0
 E.g.: sunspot (  = 1 m2/s, L0 = 106 m), td = 1012
sec;

(b) If Rm >> 1
The induction equation reduces to
B
   (v  B)
t
and Ohm's law -->
E + vB = 0
Magnetic field is “* _ _ _ _ _ _ _ _ _ _ _ _ _*”
Magnetic Flux Conservation:
Magnetic Field Line Conservation:
2.5 EQUATION of MOTION
dv

  p 
dt
(1)
(2)
(2)
(i)
 
(3)

When  <<1,
j B +  g
(3)
(4)
p
2
B / (2  ) * _ _ _ _ _ _ _ *
j  B dominates
(ii) (1)  (3)  v  vA 
B

* _ _ _ _ _ _*
Typical Values on Sun
Photosphere
Corona
N (m-3)
1023
1015
T (K)
6000
106
B (G)
5 - 103
10

106 - 1
10-3
vA (km/s)
0.05 - 10
103
[N (m-3) = 106 N (cm-3),
B (G) = 104 B (tesla)
 = 3.5 x 10 -21 N T/B2, vA = 2 x 109 B/N1/2]
Magnetic force:
j  B = (  B) 
= (B.)
B

B

B 
  
2 
2
Magnetic field lines have a
Tension B2/  ----> * _ _ _ _ _ _ _ _ _ _*
Pressure B2/(2 )----> * _ _ _ _ _ _ _ _ _ _ *
*EXAMPLE
B 
j  B = (B.)
  

2 
2

 Bx
B 
(j  B)x = (Bx
+ By )



x
y 
x  2  
B
2
**Examples
Find Magnetic Pressure force, Magnetic Tension
force and j x B force for
Ex 2.4
ˆ
(a) B = x y
(b)
B = y xˆ + x yˆ
Hydrostatic Equilibrium
dv

dt
(1)
  p 
(2)
j  B + g
(3)
(4)
 In most of corona, (3) dominates
 Along B, (3) = 0, so (2) + (4) important

(2)
p0 / L0

 1 for
(4)
0 g
L0
 H

p0
 0 g * _ _ _ _ _ _ _ _ _*
Example
Suppose g = - g zˆ
MHS Eqm. along B:
dp
   g,
dz
where  = p / (R T ).
dp
p
RT
So
 , H 
.
dz
H
g
T  const 
p  p0 e
z / H
p  p0 e
z / H
On Earth H = 9 km,
so on munro (1 km) p = 0.9 p0

or on Everest (9 km) p = 0.37 p0
T = 5000 K, H = * _ _ _ _ _*;
T = 2 x 106 K, H = * _ _ _ _ _ *
When is MHD valid ?
 = constant,
We assumed in deriving MHD eqns -- v<<c,
and plasma continuous
 Can treat plasma as a continuous medium when
1
2
L  mpf
n 
 T  
 300  6   17 3  km
 10 K   10 m 
Chromosphere
Corona
 When L  mpf
“collide” with 
B
(T  10 , n  10 ) L  3 cm
4
20
T  10 , n  10 
6
16
L  30km
MHD can still be valid when particles
L  ri =v j m / (eB) (ion gyroradius)
ri = 1 m(corona)
 MHD equations can be derived by taking integrals of a
kinetic equation for particles (but tricky)