The single particle approach gets to be horribly complicated, as we have seen.
Basically we need a more statistical approach because we can't follow each particle separately. If the details of the distribution function in velocity space are important we have to stay with the Boltzmann equation. It is a kind of particle conservation equation.
4.1 Particle Conservation (In 3d Space)
Figure 4.1: Elementary volume for particle conservation
Number of particles in box ∆x ∆y ∆z is the volume, ∆V = ∆x ∆y ∆z, times the density n. Rate of change of number is is equal to the number flowing across the boundary per unit time, the flux. (In absence of sources.)
−  ∂
∂t  [∆x ∆y ∆z n] = Flow Out across boundary. 
 (4.1) 
Take particle velocity to be
v(
r) [no random velocity, only flow] and origin at the center of the box refer to flux density as n
v =
J.
Flow Out = [J_{z} ( 0, 0, ∆z/2 ) − J_{z} ( 0, 0, −∆z/2 ) ] ∆x ∆y + x + y . 
 (4.2) 
Expand as Taylor series
J_{z} (0, 0, η) = J_{z} (0) +  ∂
∂z  J_{z} . η 
 (4.3) 
So,



 ∂
∂z  (n v_{z}) ∆z ∆x∆y + x + y 
  (4.4) 



 

Hence
Particle Conservation
Notice we have essential proved an elementary form of Gauss's theorem
 ⌠
⌡ 
v  ∇ . A d^{3} r =  ⌠
⌡ 
∂γ  A . dS . 
 (4.6) 
The expression:
`Fluid Description' refers to any simplified plasma treatment which does
not keep track of vdependence of f detail.
 Fluid Descriptions are essentially 3d (r).
 Deal with quantities averaged over velocity space (e.g. density, mean velocity, ...).
 Omit some important physical processes (but describe others).
 Provide tractable approaches to many problems.
 Will occupy most of the rest of my lectures.
Fluid Equations can be derived mathematically by taking moments
^{1} of the Boltzmann Equation.
These lead, respectively, to (0) Particle (1) Momentum (2) Energy conservation equations.
We shall adopt a more direct `physical' approach.
4.2 Fluid Motion
The motion of a fluid is described by a vector velocity field
v(
r) (which is the mean velocity of all the individual particles which make up the fluid at
r). Also the particle density n(r) is required. We are here discussing the motion of fluid of a
single type of particle of mass/charge, m/q so the charge and mass density are qn and mn respectively.
The particle conservation equation we already know. It is also sometimes called the `Continuity Equation'
It is also possible to expand the ∇. to get:
 ∂
∂t  n + (v. ∇) n + n ∇ . v= 0 
 (4.11) 
The significance, here, is that the first two terms are the `convective derivative" of n
so the continuity equation can be written
4.2.1 Lagrangian & Eulerian Viewpoints
There are essentially 2 views.
 Lagrangian. Sit on a fluid element and move with it as fluid moves.
Figure 4.2: Lagrangean Viewpoint
 Eulerian. Sit at a fixed point in space and watch fluid move through your volume element: "identity" of fluid in volume continually changing
Figure 4.3: Eulerian Viewpoint
[(∂)/(∂t)] means rate of change at fixed point (Euler).
[ D/Dt] ≡ [d/dt] ≡ [( ∂)/(∂t )] + v. ∇ means rate of change at moving point (Lagrange).
v.∇ = [dx/(∂t)] [(∂)/(∂x)]+[dy/(∂t)] [(∂)/(∂y)]+[dz/(∂t)] [(∂)/(∂z)] : change due to motion.
Our derivation of continuity was Eulerian. From the Lagrangian view
 D
Dt  n =  d
dt   ∆N
∆V  = −  ∆N
∆V^{2}   d
dt  ∆V = − n  1
∆V   d ∆V
dt  
 (4.14) 
since total number of particles in volume element (∆N) is constant (we are moving with them). (∆V = ∆x ∆y ∆z.)



 d∆x
dt  ∆y ∆z +  d ∆y
dt  ∆z ∆x +  d ∆z
dt  ∆y ∆x 
  (4.15) 



∆V  ⎡
⎣  1
∆x   d ∆x
dt  +  1
∆y   d ∆y
dt  +  1
∆x   d ∆z
dt  ⎤
⎦  
  (4.16) 




v_{x} ( ∆x / 2 ) − v_{x} ( − ∆x / 2 ) 
  (4.17) 



∆x  ∂v_{x}
∂x  etc. ... y ... z 
  (4.18) 

Hence
 d
dt  ∆V = ∆V  ⎡
⎣  ∂v_{x}
∂x  +  ∂v_{y}
∂y  +  ∂v_{z}
∂z  ⎤
⎦  = ∆V ∇ . v 
 (4.19) 
and so
Lagrangian Continuity. Naturally, this is the same equation as Eulerian when one puts [D/Dt] = [(∂)/(∂t)] +
v. ∇.
The quantity − ∇ .
v is the rate of (Volume) compression of element.
4.2.2 Momentum (Conservation) Equation
Each of the particles is acted on by the Lorentz force q [
E+
u_{i} ∧
B] (
u_{i} is individual particle's velocity).
Hence total force on the fluid element due to EM fields is

∑i  ( q [ E+ u_{i} ∧B ] ) = ∆N q ( E+ v∧B) 
 (4.21) 
(Using mean:
v = ∑
_{i} u / ∆N.)
EM Force density (per unit volume) is:
The total momentum of the element is

∑i  m u_{i} = m ∆N v = ∆V mn v 
 (4.23) 
so
Momentum Density is mn
v.
If no other forces are acting then clearly the equation of motion requires us to set the time derivative of mn
v equal to
F_{EM}. Because we want to retain the identity of the particles under consideration we want D/Dt i.e. the convective derivative (Lagrangian picture).
In general there are additional forces acting.
(1) Pressure (2) Collisional Friction.
4.2.3 Pressure Force
In a gas p( = nT) is the force per unit area arising from thermal motions. The surrounding fluid exerts this force on the element:
Figure 4.4: Pressure forces on opposite faces of element.
Net force in x direction is



p ( ∆x / 2 ) ∆y ∆z + p ( − ∆x / 2 ) ∆y ∆z 
  (4.24) 



− ∆x ∆y ∆z  ∂p
∂x  = − ∆V  ∂p
∂x  = − ∆V ( ∇p )_{x} 
  (4.25) 

So (isotropic) pressure
force density (/unit vol)
How does this arise in our picture above?
Answer: Exchange of momentum by particle thermal motion across the element boundary.
Although in Lagrangian picture we move with the element (as defined by mean velocity
v) individual particles also have thermal velocity so that the additional velocity they have is
w_{i} = u_{i} − v `peculiar′velocity 
 (4.27) 
Because of this, some cross the element boundary and exchange momentum with outside. (Even though there is no net change of number of particles in element.) Rate of exchange of momentum due to particles with peculiar velocity
w, d
^{3}w across a surface element
ds is

f( w ) m w d^{3} w
mom^{m}density at w  × 
w . ds
flow rate across ds  
 (4.28) 
Integrate over distrib function to obtain the total momentum exchange rate:
ds .  ⌠
⌡  m w w f (w) d^{3} w 
 (4.29) 
The thing in the integral is a tensor. Write
p =  ⌠
⌡  m w w f (w) d^{3} w (Pressure Tensor) 
 (4.30) 
Then momentum exchange rate is
Actually, if f(
w) is isotropic (e.g. Maxwellian) then



 ⌠
⌡  m w_{x} w_{y} f (w) d^{3}w = 0 etc. 
  (4.32) 



 ⌠
⌡  m w_{x}^{2} f (w) d^{3} w ≡ n T ( = p_{yy} = p_{zz} = `p′) 
  (4.33) 

So then the exchange rate is p
ds. (Scalar Pressure).
Integrate
ds over the whole ∆V then x component of mom
^{m} exchange rate is
p  ⎛
⎝  ∆x
2  ⎞
⎠  ∆y ∆z − − p  ⎛
⎝  − ∆x
2  ⎞
⎠  ∆y ∆z = ∆V ( ∇ p )_{x} 
 (4.34) 
and so
Total momentum loss rate due to exchange across the boundary per unit volume is
In terms of the momentum equation, either we put ∇p on the momentum derivative side or
F_{p} on force side. The result is the same.
Ignoring Collisions, Momentum Equation is
 D
Dt  ( m n ∆V v) = [F_{EM} + F_{p} ] ∆V 
 (4.36) 
Recall that n∆V = ∆N ; [D/Dt] (∆N) = 0; so
Thus, substituting for
F′s:
Momentum Equation.
m n  Dv
Dt  = m n  ⎛
⎝  ∂v
∂t  + v. ∇ v  ⎞
⎠  = q n ( E + v∧B) − ∇p 
 (4.38) 
4.2.4 Momentum Equation: Eulerian Viewpoint
Fixed element in space. Plasma flows through it.
 E.M. force on element (per unit vol.)
F_{EM} = nq ( E+ v∧B) as before. 
 (4.39) 
 Momentum flux across boundary (per unit vol)



∇.  ⌠
⌡  m (v+ w) (v+ w) f(w) d^{3}w 
  (4.40) 



∇. {  ⌠
⌡  m (vv+ 
vw + wv
integrates to 0  + w w) f(w) d^{3} w } 
  (4.41) 



  (4.42) 



m n (v. ∇) v + m v[ ∇ . (n v) ] + ∇p 
  (4.43) 

(Take isotropic p.)
 Rate of change of momentum within element (per unit vol)
Hence, total momentum balance:
 ∂
∂t  (m n v) + m n (v. ∇) v+ m v[ ∇ . (n v) ] + ∇p = F_{EM} 
 (4.45) 
Use the continuity equation:
to cancel the third term and part of the 1st:
 ∂
∂t  (m n v) + m v( ∇ . ( n v) ) = m v{  ∂n
∂t  + ∇ . ( n v) } + m n  ∂v
∂t  = m n  ∂v
∂t  
 (4.47) 
Then take ∇p to RHS to get final form:
Momentum Equation:
m n  ⎡
⎣  ∂v
∂t  + ( v. ∇ ) v  ⎤
⎦  = nq ( E+ v∧B) − ∇p . 
 (4.48) 
As before, via Lagrangian formulation. (Collisions have been ignored.)
4.2.5 Effect of Collisions
First notice that
like particle collisions
do not change the total momentum (which is averaged over all particles of that species).
Collisions between
unlike particles
do exchange momentum between the species. Therefore once we realize that any quasineutral plasma consists of at least two different species (electrons and ions) and hence two different interpenetrating fluids we may need to account for another momentum loss (gain) term.
The rate of momentum density loss by species 1 colliding with species 2 is:
ν_{12} n_{1} m_{1} (v_{1} − v_{2}) 
 (4.49) 
Hence we can immediately generalize the
momentum equation to
m_{1} n_{1}  ⎡
⎣  ∂v_{1}
∂t  + ( v_{1} . ∇ ) v_{1}  ⎤
⎦  = n_{1} q_{1} ( E+ v_{1} ∧B) − ∇p_{1} − ν_{12} n_{1} m_{1} ( v_{1} − v_{2} ) 
 (4.50) 
With similar equation for species 2.
4.3 The Key Question for Momentum Equation:
What do we take for p?
Basically p = nT is determined by energy balance, which will tell how T varies. We could write an energy equation in the same way as momentum. However, this would then contain a term for heat flux, which would be unknown. In general, the k
^{th} moment equation contains a term which is a (k + 1)
^{th} moment.
Continuity, 0
^{th} equation contains
v determined by
Momentum, 1
^{st} equation contains p determined by
Energy, 2
^{nd} equation contains Q determined by ...
In order to get a sensible result we have to
truncate this hierarchy. Do this by some sort of assumption about the heat flux. This will lead to an
Equation of State:
The value of γ to be taken depends on the heat flux assumption and on the isotropy (or otherwise) of the energy distribution.
Examples
 Isothermal: T = const.: γ = 1.
 Adiabatic/Isotropic: 3 degrees of freedom γ = ^{5}/_{4}.
 Adiabatic/1 degree of freedom γ = 3.
 Adiabatic/2 degrees of freedom γ = 2.
In general, n (
l/ 2) δT = − p (δV/V) (Adiabatic
l degrees)
So



 δn
n  +  δT
T  =  ⎛
⎝  1 +  2
l  ⎞
⎠   δn
n  , 
  (4.53) 

i.e.
p n ^{−( 1 + [2/(l)])} = const. 
 (4.54) 
In a normal gas, which `holds together' by collisions, energy is rapidly shared between 3 spacedegrees of freedom. Plasmas are often rather collisionless so compression in 1 dimension often stays confined to 1degree of freedom. Sometimes heat transport is so rapid that the isothermal approach is valid. It depends on the exact situation; so let's leave γ undefined for now.
4.4 Summary of TwoFluid Equations
Species j
Plasma Response
 Continuity:
 ∂n_{j}
∂t  + ∇ . (n_{j}v_{j}) = 0 
 (4.55) 
 Momentum:
m_{j}n_{j}  ⎡
⎣  ∂v_{j}
∂t  + ( v_{j} . ∇ ) v_{j}  ⎤
⎦  = n_{j}q_{j} ( E+ v_{j} ∧B) − ∇p_{j} − 

ν

jk  n_{j}m_{j} (v_{j} − v_{k} ) 
 (4.56) 
 Energy/Equation of State:
p_{j} n_{j}^{−γ} = const.. 
 (4.57) 
(j = electrons, ions).
Maxwell's Equations



  (4.58) 



μ_{o} j +  1
c^{2}   ∂E
∂t  ∇ ∧E =  −∂B
∂t  
  (4.59) 

With



q_{e} n_{e} + q_{i} n_{i} = e ( −n_{e} + Zn_{i} ) 
  (4.60) 



q_{e} n_{e} v_{e} + q_{i} n_{i} v_{i} = e ( −n_{e} v_{e} + Z n_{i} v_{i} ) 
  (4.61) 



−e n_{e} ( v_{e} − v_{i} r) (Quasineutral) 
  (4.62) 

Accounting
Unknowns  Equations 
n_{e},n_{i}  2  Continuity e, i  2 
v_{e},v_{i}  6  Momentum e,i  6 
p_{e},p_{i}  2  State e,i  2 
E, B  6  Maxwell  8 
 16   18 
but 2 of Maxwell (∇. equs) are redundant because can be deduced from others: e.g.



  (4.63) 



0 = μ_{o} ∇ . j +  1
c^{2}   ∂
∂t  ( ∇ . E) =  1
c^{2}   ∂
∂t   ⎛
⎝  −ρ
ϵ_{o}  + ∇ . E  ⎞
⎠  
  (4.64) 

So 16 equs for 16 unknowns.
Equations still very difficult and complicated mostly because it is
Nonlinear
In some cases can get a tractable problem by `linearizing'. That means, take some known equilibrium solution and suppose the deviation (perturbation) from it is small so we can retain only the 1st linear terms and not the others.
4.5 TwoFluid Equilibrium: Diamagnetic Current
Slab: [(∂)/(∂x)] ≠ 0 [(∂)/(∂y)], [(∂)/(∂z)] = 0.
Straight Bfield:
B =
B∧z.
Equilibrium: [(∂)/(∂t)] = 0 (E = − ∇ϕ)
Collisionless: ν→ 0.
Momentum Equation(s):
m_{j}n_{j} (v_{j} . ∇) v_{j} = n_{j} q_{j} (E+ v_{j} ∧B) − ∇p_{j} 
 (4.65) 
Drop j suffix for now. Then take x,y components:



nq (E_{x} + v_{y} B) −  dp
dx  
  (4.66) 



  (4.67) 

Eq 4.67 is satisfied by taking v
_{x} = 0. Then 4.66 →
nq (E_{x} + v_{y} B) −  dp
dx  = 0. 
 (4.68) 
i.e.
v_{y} =  − E_{x}
B  +  1
nqB   dp
dx  
 (4.69) 
or, in vector form:
v= 
E∧B drift
 − 
Diamagnetic Drift
 
 (4.70) 
Notice:
 In magnetic field (⊥) fluid velocity is determined by component of momentum equation orthogonal to it (and to B).
 Additional drift (diamagnetic) arises in standard F ∧B form from pressure force.
 Diagmagnetic drift is opposite for opposite signs of charge (electrons vs. ions).
Now restore species distinctions and consider electrons plus single ion species i. Quasineutrality says n
_{i}q
_{i} = −n
_{e}q
_{e}. Hence adding solutions
n_{e}q_{e}v_{e} + n_{i}q_{i}v_{i} =  E∧B
B^{2}  
( n_{i}q_{i} + n_{e}q_{e} )
=0  − ∇( p_{e} + p_{i} )∧  B
B^{2}  
 (4.71) 
Hence current density:
j = − ∇( p_{e} + p_{i} ) ∧  B
B^{2}  
 (4.72) 
This is the diamagnetic current. The electric field,
E, disappears because of quasineutrality. (General case ∑
_{j} q
_{j}n
_{j}v
_{j} = − ∇(∑p
_{j}) ∧
B/ B
^{2}).
4.6 Reduction of Fluid Approach to the Single Fluid Equations
So far we have been using fluid equations which apply to electrons and ions
separately. These are called
`Two Fluid' equations because we always have to keep track of both fluids separately.
A further simplification is possible and useful sometimes by combining the electron and ion equations together to obtain equations governing the plasma viewed as a
`Single Fluid'.
Recall 2fluid equations:



 ∂n_{j}
∂t  + ∇. (n_{j}v_{j}) = 0 . 
  (4.73) 



m_{j}n_{j}  ⎛
⎝  ∂
∂t  + v_{j} . ∇  ⎞
⎠  v_{j} = n_{j}q_{j} ( E+ v_{j} ∧B) − ∇p_{j} + F_{jk} 
  (4.74) 

(where we just write
F_{jk} = −
―v_{jk}n
_{j}m
_{j} (
v_{j} −
v_{k} ) for short.)
Now we rearrange these 4 equations (2 × 2 species) by adding and subtracting appropriately to get new equations governing the new variables:



  (4.75) 



( n_{e}m_{e} v_{e} + n_{i}m_{i}v_{i} ) / ρ_{m} 
  (4.76) 



  (4.77) 

Electric Current Density j 


q_{e}n_{e}v_{e} +q_{i}n_{i}v_{i} 
  (4.78) 



q_{e} n_{e} ( v_{e}−v_{i} ) by quasi neutrality 
  (4.79) 



  (4.80) 

1st equation: take m
_{e} ×C
_{e} + m
_{i} ×C
_{i} →
(1)  ∂ρ_{m}
∂t  + ∇ . ( ρ_{m} V ) = 0 Mass Conservation 
 (4.81) 
2nd take q
_{e} ×C
_{e} + q
_{I} ×C
_{i} →
(2)  ∂ρ_{q}
∂t  + ∇ . j = 0 Charge Conservation 
 (4.82) 
3rd take M
_{e} + M
_{i}. This is a bit more difficult. RHS becomes:
 ∑  n_{j}q_{j} ( E+ v_{j} ∧B) − ∇_{pj} + F_{jk} = ρ_{q} E+ j ∧B− ∇( p_{e} + p_{i} ) 
 (4.83) 
(we use the fact that F
_{ei} − F
_{ie} so no
net friction). LHS is

∑j  m_{j} n_{j}  ⎛
⎝  ∂
∂t  + v_{j} .∇  ⎞
⎠  v_{j} 
 (4.84) 
The difficulty here is that the convective term is nonlinear and so does not easily lend itself to reexpression in terms of the new variables. But note that since m
_{e} << m
_{i} the contribution from electron momentum is usually much less than that from ions. So we ignore it in this equation. To the same degree of approximation
V ≅
v_{i}: the CM velocity is the ion velocity. Thus for the LHS of this momentum equation we take

∑j  m_{i}n_{i}  ⎛
⎝  ∂
∂t  + v_{j} . ∇  ⎞
⎠  v_{j} ≅ ρ_{m}  ⎛
⎝  ∂
∂t  + V . ∇  ⎞
⎠  V 
 (4.85) 
so:
(3) ρ_{m}  ⎛
⎝  ∂
∂t  + V . ∇  ⎞
⎠  V = ρ_{q} E+ j ∧B− ∇p 
 (4.86) 
Finally we take [(q
_{e})/(m
_{e})] M
_{e} + [(q
_{i})/(m
_{i})] M
_{i} to get:

∑j  n_{j} q_{j}  ⎡
⎣  ∂
∂t  + ( v_{j} . ∇ )  ⎤
⎦  v_{j} = 
∑j  {  n_{j}q_{j}^{2}
m_{j}  ( E+ v_{j} ∧B) −  q_{j}
m_{j}  ∇p_{j} +  q_{j}
m_{j}  F_{jk} } 
 (4.87) 
Again several difficulties arise which it is not very profitable to deal with rigorously. Observe that the LHS can be written (using quasineutrality n
_{i}q
_{i} + n
_{e}q
_{e} = 0) as ρ
_{m} [(∂)/(∂t)] ( [(
j)/(ρ
_{m})] ) provided we discard the term in (
v. ∇ )
v. (Think of this as a linearization of this question.) [The (
v. ∇)
v convective term is a term which is not satisfactorily dealt with in this approach to the single fluid equations.]
In the R.H.S. we use quasineutrality again to write



n_{e}^{2} q_{e}^{2}  ⎛
⎝  1
n_{e}m_{e}  +  1
n_{i}m_{i}  ⎞
⎠  E = n_{e}^{2}q_{e}^{2}  m_{i}n_{i}+m_{e}n_{e}
n_{e}m_{e}n_{i}m_{i}  E = −  q_{e}q_{i}
m_{e}m_{i}  ρ_{m} E, 
  (4.88) 

 ∑   n_{j}q_{q}^{2}
m_{j}  v_{j} 


 n_{e}q_{e}^{2}
m_{e}  v_{e} +  n_{i}q_{i}^{2}
m_{i}  v_{i} 
 



 q_{e}q_{i}
m_{e}m_{i}  {  n_{e}q_{e}m_{i}
q_{i}  v_{e} +  n_{i}q_{i}m_{e}
q_{e}  v_{i} } 
 



−  q_{e}q_{i}
m_{e}m_{i}  { n_{e}m_{e}v_{e} + n_{i}m_{i}v_{i} −  ⎛
⎝  m_{i}
q_{i}  +  m_{e}
q_{e}  ⎞
⎠  ( q_{e}n_{e}v_{e} + q_{i}n_{i}v_{i} ) } 
 



−  q_{e}q_{i}
m_{e}m_{i}  { ρ_{m} V −  ⎛
⎝  m_{i}
q_{i}  +  m_{e}
q_{e}  ⎞
⎠  j } 
  (4.89) 

Also, remembering
F_{ei} = −
―ν
_{ei} n
_{e}m
_{i} (
v_{e}−
v_{i}) = −
F_{ie},



− 
ν

ei   ⎛
⎝  n_{e}q_{e} − n_{e}q_{i}  m_{e}
m_{i}  ⎞
⎠  ( v_{e} − v_{i} ) 
 



− 
ν

ei   ⎛
⎝  1 −  q_{e}
q_{i}   m_{e}
m_{i}  ⎞
⎠  j 
  (4.90) 

So we get



−  q_{e}q_{i}
m_{e}m_{i}   ⎡
⎣  ρ_{m} E+  ⎧
⎨
⎩  ρ_{m} V −  ⎛
⎝  m_{i}
q_{i}  +  m_{e}
q_{e}  ⎞
⎠  j  ⎫
⎬
⎭  ∧B  ⎤
⎦  
 



 q_{e}
m_{e}  ∇p_{e} −  q_{i}
m_{i}  ∇p_{i} −  ⎛
⎝  1 −  q_{e}
q_{i}   m_{e}
m_{i}  ⎞
⎠  
ν

ei  j 
  (4.91) 

Regroup after multiplying by [(m
_{e}m
_{i})/(q
_{e}q
_{i}ρ
_{m})]:



−  m_{e}m_{i}
q_{e}q_{i}   ∂
∂t   ⎛
⎝  j
ρ_{m}  ⎞
⎠  +  1
ρ_{m}   ⎛
⎝  m_{i}
q_{i}  +  m_{e}
q_{e}  ⎞
⎠  j ∧B 
  (4.92) 



−  ⎛
⎝  q_{e}
m_{e}  ∇p_{e} +  q_{i}
m_{i}  ∇p_{i}  ⎞
⎠   m_{e}m_{i}
ρ_{m}q_{e}q_{i}  −  ⎛
⎝  1 −  q_{e}
q_{i}   m_{e}
m_{i}  ⎞
⎠   m_{e}m_{i}
q_{e}q_{i}ρ_{m}  
ν

ei  j 
 

Notice that this is an equation relating the Electric field in the frame moving with the fluid (L.H.S.) to things depending on current
j i.e. this is a generalized form of
Ohm's Law.
One essentially never deals with this full generalized Ohm's law. Make some approximations recognizing the physical significance of the various R.H.S. terms.
 m_{e}m_{i}
q_{e}q_{i}   ∂
∂t   ⎛
⎝  j
ρ_{m}  ⎞
⎠  arises fromelectron inertia. 

it will be negligible for low enough frequency.
 1
ρ_{m}   ⎛
⎝  m_{i}
q_{i}  +  m_{e}
q_{e}  ⎞
⎠  j ∧B is called theHall Term. 

and arises because current flow in a Bfield tends to be diverted across the magnetic field. It is also often dropped but the justification for doing so is less obvious physically.
 q_{i}
m_{i}  ∇p_{i} term <<  q_{e}
m_{e}  ∇p_{e} for comparable pressures, 

and the latter is ∼ the Hall term; so ignore q
_{i}∇p
_{i}/m
_{i}.
Last term in
j has a coefficient, ignoring m
_{e}/ m
_{i} c.f. 1 which is

q_{e}q_{i} ( n_{i}m_{i} )  = 
q_{e}^{2} n_{e}  = η the resistivity. 
 (4.93) 
Hence dropping electron inertia, Hall term and pressure, the simplified Ohm's law becomes:
Final equation needed: state:
p_{e}n_{e}^{−γe} + p_{i}n_{i}^{−γi} = constant. 

Take quasineutrality ⇒ n
_{e} ∝ n
_{i} ∝ ρ
_{m}. Take γ
_{e} = γ
_{i}, then
4.6.1 Summary of Single Fluid Equations: M.H.D.
Mass Conservation:  ∂ρ_{m}
∂t  + ∇( ρ_{m} V ) = 0 
 (4.96) 
Charge Conservation:  ∂ρ_{q}
∂t  + ∇. j = 0 
 (4.97) 
Momentum: ρ_{m}  ⎛
⎝  ∂
∂t  + V . ∇  ⎞
⎠  V = ρ_{q} E+ j ∧B− ∇p 
 (4.98) 
Eq. of State: p ρ_{m}^{−γ} = const. 
 (4.100) 
4.6.2 Heuristic Derivation/Explanation
Mass Charge: Obvious.
Mom^{m} 
[(rate of change of)  (total momentum density)]
 = 
ρ_{q} E
[(Electric)  (body force)]  + 
j ∧B
[(Magnetic Force)  (on current)]  − 
∇p
Pressure  
 (4.101) 
Ohm's Law
The electric field `seen' by a moving (conducting) fluid is
E+
V ∧
B =
E_{V} electric field in frame in which fluid is at rest. This is equal to `resistive' electric field η
j:
The ρ
_{q}E term is generally dropped because it is much smaller than the
j∧
B term. To see this, take orders of magnitude:
∇.E= ρ_{q}/ϵ_{0} so ρ_{q} ∼ Eϵ_{0}/L 
 (4.103) 
∇∧B=μ_{0}j  ⎛
⎝  +  1
c^{2}   ∂E
∂t  ⎞
⎠  so σE = j ∼ B/μ_{0} L 
 (4.104) 
Therefore
 ρ_{q}E
jB  ∼  ϵ_{0}
L   ⎛
⎝  B
μ_{0}σL  ⎞
⎠  2
  Lμ_{0}
B^{2}  ∼  L^{2}/ c^{2}
(μ_{0}σL^{2})^{2}  =  ⎛
⎝  light transit time
resistive skin time  ⎞
⎠  2
 . 
 (4.105) 
This is generally a very small number. For example, even for a small cold plasma, say T
_{e}=1 eV (σ ≈ 2×10
^{3} mho/m), L=1 cm, this ratio is about 10
^{−8}.
Conclusion: the ρ
_{q} E force is
much smaller than the
j∧
B force for essentially all practical cases. Ignore it.
Normally, also, one uses MHD only for low frequency phenomena, so the Maxwell displacement current, ∂
E/c
^{2}∂t can be ignored.
Also we shall not need Poisson's equation because that is taken care of by quasineutrality.
4.6.3 Maxwell's Equations for MHD Use
∇ . B= 0 ; ∇ ∧E =  − ∂B
∂t  ; ∇ ∧B = μ_{o}j . 
 (4.106) 
The MHD equations find their major use in studying macroscopic magnetic confinement problems. In Fusion we want somehow to confine the plasma pressure away from the walls of the chamber, using the magnetic field. In studying such problems MHD is the major tool.
On the other hand if we focus on a small section of the plasma as we do when studying shortwavelength waves, other techniques: 2fluid or kinetic are needed. Also, plasma is approx. uniform.
`Macroscopic' Phenomena MHD
`Microscopic' Phenomena 2Fluid/Kinetic
4.7 MHD Equilibria
Study of how plasma can be `held' by magnetic field. Equilibrium ⇒
V = [(∂)/(∂t)] = 0. So equations reduce. Mass and Faraday's law are ∼ automatic. We are left with
(Mom^{m}) → `Force Balance′ 0 = j ∧B− ∇ p 
 (4.107) 
Plus ∇ .
B = 0, ∇ .
j = 0.
Notice that provided we don't ask questions about Ohm's law.
E doesn't come into MHD equilibrium.
These deceptively simple looking equations are the subject of much of Fusion research. The hard part is taking into account complicated geometries.
We can do some useful calculations on simple geometries.
4.7.1 θpinch
Figure 4.5: θpinch configuration.
So called because plasma currents flow in θdirection.
Use MHD Equations
Take to be ∞ length, uniform in zdir.
By symmetry
B has only z component.
By symmetry
j has only θ comp.
By symmetry ∇p has only r comp.
So we only need



  (4.109) 



  (4.110) 



  (4.111) 



  (4.112) 

Eliminate j: −  B_{z}
μ_{o}   ∂B_{z}
∂r  −  ∂p
∂r  


  (4.113) 

i.e.

 ∂
∂r   ⎛
⎝  B_{z}^{2}
2 μ_{o}  + p  ⎞
⎠  


  (4.114) 

Solution  B_{z}^{2}
2 μ_{o}  + p 


  (4.115) 

Figure 4.6: Balance of kinetic and magnetic pressure
 B_{z}^{2}
2 μ_{o}  + p =  B_{z ext}^{2}
2 μ_{o}  
 (4.116) 
[Recall Single Particle Problem]
Think of these as a pressure equation. Equilibrium says
total pressure = const.

magnetic pressure
 + 
p
kinetic pressure  = const. 
 (4.117) 
Ratio of kinetic to magnetic pressure is plasma `β'.
measures `efficiency' of plasma confinement by B. Want large β for fusion but limited by instabilities, etc.
4.7.2 Zpinch
Figure 4.7: Zpinch configuration.
so called because
j flows in zdirection. Again take to be ∞ length and uniform.
j = j_{z} 
^
e

z  B = B_{θ} 
^
e

θ  
 (4.119) 
Force ( j ∧B)_{r} − ( ∇p )_{r} = − j_{z} B_{θ} −  ∂p
∂r  = 0 
 (4.120) 
Ampere ( ∇ ∧B)_{z} − ( μ_{o} j )_{z} =  1
r   ∂
∂r  ( r B_{θ} ) − μ_{o}j_{z} = 0 
 (4.121) 
Eliminate j:
 B_{θ}
μ_{o} r   ∂
∂r  ( r B_{θ} ) −  ∂p
∂r  = 0 
 (4.122) 
or

Extra Term
 +  ∂
∂r  
 ⎛
⎝  B_{θ}^{2}
2 μ_{o}  + p  ⎞
⎠ 
Magnetic + Kinetic pressure
 = 0 
 (4.123) 
Extra term acts like a magnetic
tension force. Arises because Bfield lines are
curved.
Can integrate equation
 ⌠
⌡  b
a   B_{θ}^{2}
μ_{o}   dr
r  +  ⎡
⎣  B_{θ}^{2}
2 μ_{o}  + p(r)  ⎤
⎦  b
a  = 0 
 (4.124) 
If we choose b to be edge (p(b)=0) and set a=r we get
Figure 4.8: Radii of integration limits.
p(r) =  B_{θ}^{2} ( b )
2 μ_{o}  −  B_{θ}^{2} ( r )
2 μ_{o}  +  ⌠
⌡  b
r   B_{θ}^{2}
μ_{o}   dr′
r′  
 (4.125) 
Force balance in zpinch is somewhat more complicated because of the tension force. We can't choose p(r) and j(r) independently; they have to be self consistent.
Example j = const.
 1
r   ∂
∂r  ( r B_{θ} ) = μ_{o} j_{z} ⇒ B_{θ} =  μ_{o} j_{z}
2  r 
 (4.126) 
Hence



 1
2 μ_{o}   ⎛
⎝  μ_{o} j_{z}
2  ⎞
⎠  2
 { b^{2} − r^{2} +  ⌠
⌡  b
r  2 r′dr′} 
  (4.127) 



 μ_{o} j_{z}^{2}
4  { b^{2} − r^{2} } 
  (4.128) 

Figure 4.9: Parabolic Pressure Profile.
Also note B
_{θ} (b) = [( μ
_{o} j
_{z}b)/2] so
p =  B_{θb}^{2}
2 μ_{o}   2
b^{2}  { b^{2} − r^{2} } 
 (4.129) 
4.7.3 `Stabilized Zpinch'
Also called `screw pinch', θ−z pinch or sometimes loosely just `zpinch'.
Zpinch with some additional B
_{z} as well as B
_{θ}
(Force)_{r} j_{θ} B_{z} − j_{z} B_{θ} −  ∂
∂r  = 0 
 (4.130) 
Ampere:  ∂
∂r  B_{z} = μ_{o}j_{θ} 
 (4.131) 
 1
r   ∂
∂r  ( r b_{θ} ) = μ_{o} j_{z} 
 (4.132) 
Eliminate j:
−  B_{z}
μ_{o}   ∂B_{z}
∂r  −  B_{θ}
μ_{o} r   ∂
∂r  ( r B_{θ} ) −  ∂p
∂r  = 0 
 (4.133) 
or

[(Mag Tension)  (θ only)]
 +  ∂
∂r  
Mag (θ+z) + Kinetic pressure
 = 0 
 (4.134) 
4.8 Some General Properties of MHD Equilibria
4.8.1 Pressure & Tension
j ∧B− ∇p = 0 : ∇ ∧B = μ_{o} j 
 (4.135) 
We can eliminate
j in the
general case to get
Expand the vector triple product:
∇p =  1
μ_{o}  ( B. ∇ )B−  1
2 μ_{o}  ∇ B^{2} 
 (4.137) 
put
b = [(
B)/  B  ] so that ∇
B = ∇B
b = B ∇
b +
b ∇B. Then



 1
μ_{o}  { B^{2} ( b . ∇ ) b + B b ( b . ∇ ) B } −  1
2 μ_{o}  ∇ B^{2} 
  (4.138) 



 B^{2}
μ_{o}  ( b . ∇ )b −  1
2 μ_{o}  ( ∇ − b ( b . ∇ ) ) B^{2} 
  (4.139) 



 B^{2}
μ_{o}  ( b . ∇ ) b − ∇_{⊥}  ⎛
⎝  B^{2}
2 μ_{p}  ⎞
⎠  
  (4.140) 

Now ∇
_{⊥} ( [(B
^{2})/(2 μ
_{o})] ) is the perpendicular (to
B) derivative of
magnetic pressure and (
b . ∇)
b is the
curvature of the magnetic field line giving
tension.
 (
b .∇ )
b  has value
^{1}/
_{R}. R: radius of curvature.
4.8.2 Magnetic Surfaces
0 = B. [ j ∧B− ∇p ] = − B. ∇ p 
 (4.141) 
*Pressure is constant on a field line (in MHD situation).
(Similarly, 0 =
j . [
j ∧
B− ∇p ] =
j . ∇ p.)
Figure 4.10: Contours of pressure.
Consider some arbitrary volume in which ∇p ≠ 0. That is, some plasma of whatever shape. Draw contours (surfaces in 3d) on which p = const. At any point on such an isoberic surface ∇p is perp to the surface. But
B. ∇ p = 0 implies that
B is also perp to ∇p.
Figure 4.11: B is perpendicular to ∇p and so lies in the isobaric surface.
Hence
B lies in the surface p = const.
In equilibrium
isobaric surfaces are
`magnetic surfaces'.
[This argument does not work if p = const. i.e. ∇ p = 0. Then there need be no magnetic surfaces.]
4.8.3 `Current Surfaces'
Since
j . ∇ p = 0 in equilibrium the same argument applies to current density. That is
j lies in the surface p = const.
Isobaric Surfaces are `Current Surfaces'.
Moreover it is clear that
`Magnetic Surfaces' are `Current Surfaces'.
(since both coincide with isobaric surfaces.)
[It is important to note that the existence of magnetic surfaces is guaranteed only in the MHD approximation when ∇p ≠ 0 > Taking account of corrections to MHD we may not have magnetic surfaces even if ∇p ≠ 0.]
4.8.4 Low β equilibria: ForceFree Plasmas
In many cases the ratio of kinetic to magnetic pressure is small, β << 1 and we can approximately
ignore ∇p. Such an equilibrium is called `force free'.
implies
j and
B are parallel.
i.e.
Current flows
along field lines
not across. Take divergence:



∇ . j = ∇ . ( μ( r ) B) = μ( r ) ∇ . B+ ( B. ∇ ) μ 
  (4.144) 



  (4.145) 

The ratio j/B = μ is
constant along field lines.
μ is constant on a magnetic surface. If there are no surfaces, μ is constant
everywhere.
Example: ForceFree Cylindrical Equil.
This is a somewhat more convenient form because it is linear in
B (for specified μ(r)).
Constant−μ: ∇ ∧B = μ_{o} μB 
 (4.148) 
leads to a Bessel function solution
for μ
_{o} μr > 1st zero of J
_{o} the toroidal field reverses. There are plasma confinement schemes with μ ≅ const. `Reversed Field Pinch'.
4.9 Toroidal Equilibrium
Bend a zpinch into a torus
Figure 4.12: Toroidal zpinch
B
_{θ} fields due to current are stronger at small R side ⇒ Pressure (Magnetic) Force
outwards.
Have to balance this by applying a
vertical field B_{v} to push plasma back by
j_{ϕ} ∧
B_{v}.
Figure 4.13: The field of a toroidal loop is not an MHD equilibrium. Need to add a vertical field.
Bend a θpinch into a torus: Bϕ is stronger at small R side ⇒ outward force.
Cannot be balanced by B_{v} because no j
_{ϕ}. No equilibrium for a toroidally symmetric θpinch.
Underlying Single Particle reason:
Toroidal θpinch has B
_{ϕ} only. As we have seen before, curvature drifts are uncompensated in such a configuration and lead to rapid
outward motion.
Figure 4.14: Chargeseparation giving outward drift is equivalent to the lack of MHD toroidal force balance.
We know how to solve this: Rotational Transform: get some B
_{θ}. Easiest way: add
j_{ϕ}. From MHD viewpoint this allows you to push the plasma back by
j_{ϕ} ∧
B_{v} force. Essentially, this is Tokamak.
4.10 Plasma Dynamics (MHD)
When we want to analyze
nonequilibrium situations we must retain the momentum terms. This will give a dynamic problem. Before doing this, though, let us analyse some purely Kinematic Effects.
`Ideal MHD' ⇔ Set eta = 0 in Ohm's Law.
A good approximation for high frequencies, i.e. times shorter than resistive decay time.
E+ V ∧B= 0 . Ideal Ohm′s Law. 
 (4.151) 
Also
∇ ∧E=  − ∂B
∂t  Faraday′s Law. 
 (4.152) 
Together these two equations imply constraints on how the magnetic field can change with time: Eliminate
E:
This shows that the changes in
B are completely determined by the flow,
V.
4.11 Flux Conservation
Consider an arbitrary closed contour C and spawning surface S in the fluid.
Flux linked by C is
Let C and S move with fluid:
Figure 4.15: Motion of contour with fluid gives convective flux derivative term.
Total rate of change of Φ is given by two terms:



 ⌠
⌡ 
S  
Due to changes in B
 + 
Due to motion of C
 
  (4.155) 



−  ⌠
⌡ 
S  ∇ ∧E. ds −  ⌠
(⎜)
⌡ 
C  ( V ∧B) . d l 
  (4.156) 



−  ⌠
(⎜)
⌡ 
C  (E+ V ∧B) . d l = 0 by Ideal Ohm′s Law. 
  (4.157) 

Flux through any surface moving with fluid is conserved.
4.12 Field Line Motion
Think of a field line as the intersection of two surfaces both tangential to the field everywhere:
Figure 4.16: Field line defined by intersection of two flux surfaces tangential to field.
Let surfaces move with fluid.
Since all parts of surfaces had zero flux crossing at start, they also have zero after, (by flux conserv.).
Surfaces are tangent after motion
⇒ Their intersection defines a field line after.
We think of the new field line as the same line as the old one (only moved).
Thus:
 Number of field lines ( ≡ flux) through any surface is constant. (Flux Cons.)
 A line of fluid that starts as a field line remains one.
4.13 MHD Stability
The fact that one can find an MHD
equilibrium (e.g. zpinch) does not guarantee a useful confinement scheme because the equil. might be unstable. Ball on hill analogies:
Figure 4.17: Potential energy curves
An equilibrium is
unstable if the curvature of the `Potential energy surface' is downward away from equil. That is if [( d
^{2})/(dx
^{2})] { W
_{pot} } < 0.
In MHD the potential energy is Magnetic + Kinetic Pressure (usually mostly magnetic).
If we can find
any type of perturbation which
lowers the potential energy then the equil is
unstable. It will not remain but will rapidly be lost.
Example Zpinch
We know that there is an equilibrium: Is it stable?
Consider a perturbation thus:
Figure 4.18: 'Sausage' instability
Simplify the picture by taking the current all to flow in a skin. We know that the pressure is supported by the combination of B
^{2}/2 μ
_{o} pressure and [(B
^{2})/(μ
_{o}r)] tension forces.
Figure 4.19: Skincurrent, sharp boundary pinch.
At the place where it pinches
in (A)
B
_{θ} and
^{1}/
_{r} increase → Mag. pressure & tension increase ⇒ inward force no longer balance by p ⇒ perturbation grows.
At place where it bulges
out (B)
B
_{θ} &
^{1}/
_{r} decrease → Pressure & tension ⇒ perturbation grows.
Conclusion a small perturbation induces a force tending to increase itself.
Unstable ( ≡ δW < 0).
4.14 General Perturbations of Cylindrical Equil.
Look for things which go like exp[i(kz + mθ)]. [Fourier (Normal Mode) Analysis].
Figure 4.20: Types of kink perturbation.
Generally Helical in form (like a screw thread). Example: m=1 k ≠ 0 zpinch
Figure 4.21: Driving force of a kink. Net force tends to increase perturbation. Unstable.
4.15 General Principles Governing Instabilities
(1) They try
not to bend field lines. (Because bending takes energy).
Figure 4.22: Alignment of perturbation and field line minimizes bending energy.
Perturbation (Constant surfaces) lies
along magnetic field.
Example: θpinch type plasma column:
Figure 4.23: `Flute' or `Interchange' modes.
Preferred Perturbations are `Flutes' as per Greek columns → `Flute Instability.' [Better name: `Interchange Instability', arises from idea that plasma and vacuum change places.]
(2) Occur when a `heavier' fluid is supported by a `lighter' (Gravitational analogy).
Figure 4.24: Inverted water glass analogy. Rayleigh Taylor instability.
Why does water fall out of an inverted glass? Air pressure could sustain it but does not because of RayleighTaylor instability.
Similar for supporting a plasma by mag field.
(3) Occur when  B  decreases
away from the plasma region.
Figure 4.25: Vertical upward field gradient is unstable.
 B_{A}^{2}
2 μ_{o}  <  B_{B}^{2}
2 μ_{o}  
 (4.158) 
⇒ Perturbation Grows.
(4) Occur when field line curvature is
towards the plasma (Equivalent to (3) because of ∇ ∧
B = 0 in a vacuum).
Figure 4.26: Examples of magnetic configurations with good and bad curvature.
4.16 Quick and Simple Analysis of Pinches
θpinch  B  = const. outside pinch
≡ No field line curvature.
Neutral stability
zpinch ∇  B  away from plasma outside
≡ Bad Curvature (Towards plasma) ⇒
Instability.
Generally it is difficult to get the curvature to be good everywhere. Often it is sufficient to make it good
on average on a field line. This is referred to as `Average Minimum B'. Tokamak has this.
General idea is that if field line is only in bad curvature over part of its length then to perturb in that region and not in the good region requires field line bending:
Figure 4.27: Parallel localization of perturbation requires bending.
But bending is
not preferred. So this may stabilize.
Possible way to stabilize configuration with bad curvature:
Shear
Shear of Field Lines
Figure 4.28: Depiction of field shear.
Direction of B changes. A perturbation along B at z
_{3} is
not along
B at z
_{2} or z
_{1} so it would have to
bend field there → Stabilizing effect.
General Principle: Field line bending is stabilizing.
Example: Stabilized zpinch
Perturbations (e.g. sausage or kink)
bend B_{z} so the tension in B
_{z} acts as a restoring force to prevent instability. If wave length very
long bending is less. ⇒ Least stable tends to be longest wave length.
Example: `Cylindrical Tokamak'
Tokamak is in some ways like a periodic cylindrical stabilized pinch. Longest allowable wave length = 1 turn round torus the long way, i.e.
Express this in terms of a toroidal mode number, n (s.t. perturbation ∝ expi (nϕ+ mθ): ϕ =
^{z}/
_{R} n = kR.
Most unstable mode
tends to be n=1.
[Careful! Tokamak has important toroidal effects and some modes can be localized in the bad curvature region (n ≠ 1).
Figure 4.29: Ballooning modes are localized in the outboard, bad curvature region.