HEAD PREVIOUS

Chapter 4
Fluid Description of Plasma

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 3-d Space)

figures/chap4/volsurface.png
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 nv = J.
Flow Out   =  [Jz ( 0, 0, ∆z/2 ) − Jz ( 0, 0, −∆z/2 ) ]  ∆x ∆y +  x  +  y   .
(4.2)
Expand as Taylor series
Jz (0, 0, η) = Jz (0) +

∂z
Jz  .  η
(4.3)
So,
flow out

∂z
(n vz) ∆z ∆x∆y  +  x  + y
(4.4)
=
∆V  ∇.(n v).
Hence Particle Conservation

∂t
n = ∇ . (n v)
(4.5)
Notice we have essential proved an elementary form of Gauss's theorem



v 
∇ . A d3 r  =  


∂γ 
A . dS .
(4.6)
The expression: `Fluid Description' refers to any simplified plasma treatment which does not keep track of v-dependence of f detail.
  1. Fluid Descriptions are essentially 3-d (r).
  2. Deal with quantities averaged over velocity space (e.g. density, mean velocity, ...).
  3. Omit some important physical processes (but describe others).
  4. Provide tractable approaches to many problems.
  5. Will occupy most of the rest of my lectures.
Fluid Equations can be derived mathematically by taking moments1 of the Boltzmann Equation.
0th moment

d3 v
(4.7)
1st  moment

vd3 v
(4.8)
2nd  moment

vvd3 v
(4.9)
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'

∂t
 n + ∇ . (n v) = 0
(4.10)
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
D

Dt
d

dt

∂t
+ v. ∇
(4.12)
so the continuity equation can be written
D

Dt
n = −n∇.v
(4.13)

4.2.1  Lagrangian & Eulerian Viewpoints

There are essentially 2 views.
  1. Lagrangian. Sit on a fluid element and move with it as fluid moves.
    figures/chap4/lagrange.png
    Figure 4.2: Lagrangean Viewpoint
  2. Eulerian. Sit at a fixed point in space and watch fluid move through your volume element: "identity" of fluid in volume continually changing
    figures/chap4/euler.png
    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

∆V2
  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.)
Now  d

dt
∆V
=
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)

But   d ( ∆x )

dt
=
vx ( ∆x / 2 ) − vx ( − ∆x / 2 )
(4.17)
∆x   ∂vx

∂x
    etc.     ... y     ... z
(4.18)
Hence
d

dt
∆V = ∆V
∂vx

∂x
+ ∂vy

∂y
+ ∂vz

∂z

= ∆V ∇ . v
(4.19)
and so
D

Dt
 n = − n ∇. v
(4.20)
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+ uiB] (ui is individual particle's velocity).
Hence total force on the fluid element due to E-M fields is


i 
 ( q [ E+ uiB ] ) = ∆N  q ( E+ vB)
(4.21)
(Using mean: v = ∑i u / ∆N.)
E-M Force density (per unit volume) is:
FEM = n q (E+ vB).
(4.22)
The total momentum of the element is


i 
m ui = m  ∆N  v = ∆V  mn v
(4.23)
so Momentum Density is mnv.
If no other forces are acting then clearly the equation of motion requires us to set the time derivative of mnv equal to FEM. 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:
figures/chap4/pressure.png
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)
Fp = − ∇ p
(4.26)
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
wi = uiv          `peculiarvelocity
(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, d3w across a surface element ds is


f( w ) m w  d3 w
mommdensity 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) d3 w
(4.29)
The thing in the integral is a tensor. Write
p   =  
m w w f (w) d3 w           (Pressure Tensor)
(4.30)
Then momentum exchange rate is
p  .  ds
(4.31)
Actually, if f(w) is isotropic (e.g. Maxwellian) then
pxy
=

m  wx  wy  f (w) d3w = 0     etc.
(4.32)
and  pxx
=

m wx2  f (w) d3 w ≡ n T ( = pyy = pzz = `p′)
(4.33)
So then the exchange rate is pds. (Scalar Pressure).
Integrate ds over the whole ∆V then x component of momm 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
∇ p        ( = − Fp )
(4.35)
In terms of the momentum equation, either we put ∇p on the momentum derivative side or Fp on force side. The result is the same.
Ignoring Collisions, Momentum Equation is
D

Dt
( m n ∆V v) = [FEM + Fp ] ∆V
(4.36)
Recall that n∆V = ∆N   ;    [D/Dt] (∆N) = 0; so
L.H.S. = m n ∆V   Dv

dt
   .
(4.37)
Thus, substituting for F′s:
           Momentum Equation.
m n Dv

Dt
= m n
v

∂t
+ v. ∇ v
= q n ( E + vB) − ∇p
(4.38)

4.2.4  Momentum Equation: Eulerian Viewpoint

Fixed element in space. Plasma flows through it.
  1. E.M. force on element (per unit vol.)
    FEM = nq ( E+ vB)         as before.
    (4.39)
  2. Momentum flux across boundary (per unit vol)
    =
    ∇.
    m (v+ w) (v+ w)  f(w)  d3w
    (4.40)
    =
    ∇. {
    m (vv+

    vw + wv
    integrates to 0 
    + w w)  f(w)  d3 w }
    (4.41)
    =
    ∇. { m n vv+ p }
    (4.42)
    =
    m n (v. ∇) v + m v[ ∇ . (n v) ] + ∇p
    (4.43)
    (Take isotropic p.)
  3. Rate of change of momentum within element (per unit vol)
    =

    ∂t
    ( m n v)
    (4.44)
Hence, total momentum balance:

∂t
(m n v) + m n (v. ∇) v+ m v[ ∇ . (n v) ] + ∇p = FEM
(4.45)
Use the continuity equation:
∂n

∂t
+ ∇ . (n v) = 0    ,
(4.46)
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+ vB) − ∇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 quasi-neutral 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:
ν12 n1 m1 (v1v2)
(4.49)
Hence we can immediately generalize the momentum equation to
m1 n1
v1

∂t
+ ( v1 . ∇ ) v1
= n1 q1 ( E+ v1B) − ∇p1 − ν12 n1 m1 ( v1v2 )
(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 kth moment equation contains a term which is a (k + 1)th moment.
Continuity,  0th  equation contains v determined by
Momentum, 1st   equation contains p determined by
Energy,       2nd  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:
pn−γ = const.
(4.51)
The value of γ to be taken depends on the heat flux assumption and on the isotropy (or otherwise) of the energy distribution.
Examples
  1. Isothermal: T = const.:      γ = 1.
  2. Adiabatic/Isotropic: 3 degrees of freedom   γ = 5/4.
  3. Adiabatic/1 degree of freedom    γ = 3.
  4. Adiabatic/2 degrees of freedom    γ = 2.
In general, n (l/ 2) δT = − p (δV/V)     (Adiabatic l degrees)
l

2
  δT

T
= − δV

V
=
+ δn

n
(4.52)
So
  δp

p
=
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 space-degrees of freedom. Plasmas are often rather collisionless so compression in 1 dimension often stays confined to 1-degree 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 Two-Fluid Equations

Species j
Plasma Response
  1. Continuity:
    ∂nj

    ∂t
    + ∇ . (njvj) = 0
    (4.55)
  2. Momentum:
    mjnj
    vj

    ∂t
    + ( vj . ∇ ) vj
    = njqj ( E+ vjB) − ∇pj
    -
    ν
     

    jk 
    njmj (vjvk )
    (4.56)
  3. Energy/Equation of State:
    pj nj−γ = const..
    (4.57)
    (j = electrons, ions).
Maxwell's Equations
∇ . B
=
0                      ∇ . E = ρ/ϵo
(4.58)
∇ ∧B
=
μo j + 1

c2
 E

∂t
     ∇ ∧E = −∂B

∂t
(4.59)
With
ρ
=
qe ne + qi ni = e ( −ne + Zni )
(4.60)
j
=
qe ne ve + qi ni vi = e ( −ne ve + Z ni vi )
(4.61)
=
−e ne ( vevi r)      (Quasineutral)
(4.62)
Accounting
Unknowns Equations
ne,ni      2 Continuity e, i     2
ve,vi 6 Momentum e,i 6
pe,pi 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.
∇ . ( ∇ ∧E)
=
0 = −

∂t
( ∇ . B)
(4.63)
and  ∇ . ( ∇ ∧B)
=
0 = μo ∇ . j + 1

c2
 

∂t
 ( ∇ . E) = 1

c2
 

∂t
 
−ρ

ϵo
+ ∇ . 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  Two-Fluid Equilibrium: Diamagnetic Current

     Slab:          [(∂)/(∂x)] ≠ 0      [(∂)/(∂y)], [(∂)/(∂z)] = 0.
Straight B-field:    B = Bz.
Equilibrium:          [(∂)/(∂t)] = 0        (E = − ∇ϕ)
Collisionless:      ν→ 0.
Momentum Equation(s):
mjnj (vj . ∇) vj = nj qj (E+ vjB) − ∇pj
(4.65)
Drop j suffix for now. Then take x,y components:
mn vx d

dx
vx
=
nq (Ex + vy B) − dp

dx
(4.66)
mn vx d

dx
vy
=
nq (0 −vx B)   
(4.67)
Eq 4.67 is satisfied by taking vx = 0. Then 4.66
nq (Ex + vy B) − dp

dx
= 0.
(4.68)
i.e.
vy = − Ex

B
+ 1

nqB
  dp

dx
(4.69)
or, in vector form:
v=

EB

B2

EB drift
 


∇ p

nq
B

B2

Diamagnetic Drift
 
(4.70)
Notice:
Now restore species distinctions and consider electrons plus single ion species i. Quasineutrality says niqi = −neqe. Hence adding solutions
neqeve + niqivi = EB

B2


( niqi + neqe )
=0 
− ∇( pe + pi )∧ B

B2
(4.71)
Hence current density:
j = − ∇( pe + pi ) ∧ B

B2
(4.72)
This is the diamagnetic current. The electric field, E, disappears because of quasineutrality. (General case ∑j qjnjvj = − ∇(∑pj) ∧B/ B2).

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 2-fluid equations:
Continuity  (Cj)
 
∂nj

∂t
+ ∇. (njvj) = 0 .
(4.73)
Momentum  (Mj)
 
mjnj

∂t
+ vj . ∇
vj = njqj ( E+ vjB) − ∇pj + Fjk
(4.74)
(where we just write Fjk = −vjknjmj ( vjvk ) for short.)
Now we rearrange these 4 equations (2 × 2 species) by adding and subtracting appropriately to get new equations governing the new variables:

Mass  Density               ρm
=
neme + nimi
(4.75)
C of M Velocity            V
=
( neme ve + nimivi ) / ρm
(4.76)
Charge density              ρq
=
qene + qini
(4.77)
Electric Current Density  j
=
qeneve +qinivi
(4.78)
=
qe ne ( vevi )  by quasi neutrality
(4.79)
Total  Pressure                p
=
pe + pi
(4.80)
1st equation: take me ×Ce + mi ×Ci

(1)       ∂ρm

∂t
+ ∇ . ( ρm V ) = 0      Mass Conservation
(4.81)
2nd take qe ×Ce + qI ×Ci
(2)       ∂ρq

∂t
+ ∇ . j = 0          Charge Conservation
(4.82)
3rd take Me + Mi. This is a bit more difficult. RHS becomes:

njqj ( E+ vjB) − ∇pj + Fjk = ρq E+ jB− ∇( pe + pi )
(4.83)
(we use the fact that Fei − Fie so no net friction). LHS is


j 
mj nj

∂t
+ vj .∇
vj
(4.84)
The difficulty here is that the convective term is non-linear and so does not easily lend itself to reexpression in terms of the new variables. But note that since me << mi 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 Vvi: the CM velocity is the ion velocity. Thus for the LHS of this momentum equation we take


j 
mini

∂t
+ vj . ∇
vj ≅ ρm

∂t
+ V . ∇
V
(4.85)
so:
(3)      ρm

∂t
+ V . ∇
V = ρq E+ jB− ∇p
(4.86)
Finally we take [(qe)/(me)] Me + [(qi)/(mi)] Mi to get:


j 
nj qj

∂t
+ ( vj . ∇ )
vj =

j 
 { njqj2

mj
( E+ vjB) − qj

mj
∇pj + qj

mj
Fjk }
(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 niqi + neqe = 0) as ρm [(∂)/(∂t)] ( [(j)/(ρm)] ) 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


j 
njqj2

mj
E
=
ne2 qe2
1

neme
+ 1

nimi

E = ne2qe2 mini+mene

nemenimi
E = − qeqi

memi
ρm E,
(4.88)

njqq2

mj
vj
=
neqe2

me
ve + niqi2

mi
vi
=
qeqi

memi
{ neqemi

qi
ve + niqime

qe
vi }
=
qeqi

memi
{ nemeve + nimivi
mi

qi
+ me

qe

( qeneve + qinivi ) }
=
qeqi

memi
{ ρm V
mi

qi
+ me

qe

j }
(4.89)
Also, remembering Fei = −νei nemi (vevi) = − Fie,


j 
qj

mj
Fjk
=

ν
 

ei 

neqe − neqi me

mi

( vevi )
=

ν
 

ei 

1 − qe

qi
me

mi

j
(4.90)
So we get
ρm

∂t

j

ρm

=
qeqi

memi

ρm E+

ρm V
mi

qi
+ me

qe

j

B
qe

me
∇pe qi

mi
∇pi
1 − qe

qi
me

mi


ν
 

ei 
j
(4.91)
Regroup after multiplying by [(memi)/(qeqiρm)]:
E+ VB
=
memi

qeqi
 

∂t

j

ρm

+ 1

ρm

mi

qi
+ me

qe

jB
(4.92)

qe

me
∇pe + qi

mi
∇pi
  memi

ρmqeqi

1 − qe

qi
me

mi

memi

qeqiρm

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.

memi

qeqi

∂t

j

ρm

arises fromelectron inertia.
it will be negligible for low enough frequency.

1

ρm

mi

qi
+ me

qe

jB is called theHall Term.
and arises because current flow in a B-field tends to be diverted across the magnetic field. It is also often dropped but the justification for doing so is less obvious physically.

qi

mi
∇pi term  << qe

me
∇pe for comparable pressures,
and the latter is  ∼  the Hall term; so ignore qi∇pi/mi.
Last term in j has a coefficient, ignoring me/ mi c.f. 1 which is
memi

ν
 

ei 

qeqi ( nimi )
=
me

ν
 

ei 

qe2 ne
= η   the resistivity.
(4.93)
Hence dropping electron inertia, Hall term and pressure, the simplified Ohm's law becomes:
E+ VB= ηj
(4.94)
Final equation needed: state:
pene−γe + pini−γi = constant.
Take quasi-neutrality ⇒ ne ∝ ni ∝ ρm. Take γe = γi, then
p ρm−γ = const.
(4.95)

4.6.1  Summary of Single Fluid Equations: M.H.D.


Mass Conservation:        ∂ρm

∂t
+ ∇( ρm V ) = 0            
(4.96)

Charge  Conservation:       ∂ρq

∂t
+ ∇. j = 0                      
(4.97)

Momentum:       ρm

∂t
+ V . ∇
V = ρq E+ jB− ∇p
(4.98)

Ohms  Law:                  E+ VB = ηj                   
(4.99)

Eq.  of  State:                 p ρm−γ = const.                   
(4.100)

4.6.2  Heuristic Derivation/Explanation

Mass Charge: Obvious.

Momm       

ρm

∂t
+ V . ∇
V

[(rate of change of) || (total momentum density)]
 
=

ρq E
[(Electric) || (body force)] 
+

jB
[(Magnetic Force) || (on current)] 


∇p
Pressure 
(4.101)
Ohm's Law
The electric field `seen' by a moving (conducting) fluid is E+ VB = EV electric field in frame in which fluid is at rest. This is equal to `resistive' electric field ηj:
EV = E+ VB= ηj
(4.102)
The ρqE term is generally dropped because it is much smaller than the jB term. To see this, take orders of magnitude:
∇.E= ρq0     so    ρq  ∼ Eϵ0/L
(4.103)

∇∧B0j
+ 1

c2
E

∂t

    so     σE = j  ∼ B/μ0 L
(4.104)
Therefore
ρqE

jB
 ∼  ϵ0

L

B

μ0σL

2

 
0

B2
 ∼  L2/ c2

0σL2)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 Te=1 eV (σ ≈ 2×103 mho/m), L=1 cm, this ratio is about 10−8.
Conclusion: the ρq E force is much smaller than the jB force for essentially all practical cases. Ignore it.
Normally, also, one uses MHD only for low frequency phenomena, so the Maxwell displacement current, ∂E/c2∂t can be ignored.
Also we shall not need Poisson's equation because that is taken care of by quasi-neutrality.

4.6.3  Maxwell's Equations for MHD Use


∇ . B= 0   ;   ∇ ∧E = − ∂B

∂t
  ;   ∇ ∧B = μoj   .
(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 short-wavelength waves, other techniques: 2-fluid or kinetic are needed. Also, plasma is approx. uniform.
`Macroscopic' Phenomena MHD
`Microscopic' Phenomena 2-Fluid/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
(Momm) → `Force Balance′    0 = jB− ∇ p
(4.107)

Ampere        ∇ ∧B = μo j
(4.108)
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  θ-pinch

figures/chap4/thetapinch.gif
Figure 4.5: θ-pinch configuration.
So called because plasma currents flow in θ-direction.
Use MHD Equations
Take to be ∞ length, uniform in z-dir.
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

Force      ( jB)r
( ∇p )r = 0
(4.109)
Ampere     ( ∇ ∧B)θ
=
( μo j )θ
(4.110)
i.e.      jθ Bz

∂r
p = 0
(4.111)

∂r
Bz
=
μo jθ
(4.112)
Eliminate  j:    − Bz

μo
∂Bz

∂r
∂p

∂r
=
0
(4.113)
i.e.

∂r

Bz2

2 μo
+ p
=
0
(4.114)
Solution     Bz2

2 μo
+ p
=
const.
(4.115)
figures/chap4/magpressure.png
Figure 4.6: Balance of kinetic and magnetic pressure

Bz2

2 μo
+ p = Bz ext2

2 μo
(4.116)
[Recall Single Particle Problem]
Think of these as a pressure equation. Equilibrium says total pressure = const.



Bz2

2 μo

magnetic pressure
 
+

p
kinetic pressure 
= const.
(4.117)
Ratio of kinetic to magnetic pressure is plasma `β'.
β = 2 μo p

Bz2
(4.118)
measures `efficiency' of plasma confinement by B. Want large β for fusion but limited by instabilities, etc.

4.7.2  Z-pinch

figures/chap4/zpinch.png
Figure 4.7: Z-pinch configuration.
so called because j flows in z-direction. Again take to be ∞ length and uniform.
j = jz
^
e
 

z 
      B = Bθ
^
e
 

θ 
(4.119)

Force    ( jB)r − ( ∇p )r = − jz Bθ ∂p

∂r
= 0
(4.120)

Ampere    ( ∇ ∧B)z − ( μo j )z = 1

r

∂r
( r Bθ ) − μojz = 0
(4.121)
Eliminate j:
Bθ

μo r

∂r
( r Bθ ) − ∂p

∂r
= 0
(4.122)
or


Bθ2

μo r

Extra Term
 
+

∂r



Bθ2

2 μo
       +       p

Magnetic + Kinetic pressure
 
= 0
(4.123)
Extra term acts like a magnetic tension force. Arises because B-field lines are curved.
Can integrate equation

b

a 
Bθ2

μo
  dr

r
+
Bθ2

2 μo
+ p(r)
b

a 
= 0
(4.124)
If we choose b to be edge (p(b)=0) and set a=r we get
figures/chap4/concentric.png
Figure 4.8: Radii of integration limits.

p(r) = Bθ2 ( b )

2 μo
Bθ2 ( r )

2 μo
+
b

r 
Bθ2

μo
dr′

r′
(4.125)
Force balance in z-pinch 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 Bθ ) = μo jz ⇒ Bθ = μo jz

2
 r
(4.126)
Hence
p(r)
=
1

2 μo

μo jz

2

2

 
{ b2 − r2 +
b

r 
2 r′dr′}
(4.127)
=
μo jz2

4
{ b2 − r2 }
(4.128)
figures/chap4/parabolic.png
Figure 4.9: Parabolic Pressure Profile.
Also note Bθ (b) = [( μo jzb)/2] so
p = Bθb2

2 μo
2

b2
{ b2 − r2 }
(4.129)

4.7.3  `Stabilized Z-pinch'

Also called `screw pinch', θ−z pinch or sometimes loosely just `z-pinch'.
Z-pinch with some additional Bz as well as Bθ
(Force)r     jθ Bz − jz Bθ

∂r
= 0
(4.130)

Ampere:     

∂r
Bz = μojθ
(4.131)

1

r

∂r
( r bθ ) = μo jz
(4.132)
Eliminate j:

Bz

μo
∂Bz

∂r
Bθ

μo r

∂r
( r Bθ ) − ∂p

∂r
= 0
(4.133)
or


Bθ2

μo r

[(Mag Tension) || (θ only)]
 
+

∂r



B2

2 μo
       +        p

Mag (θ+z) + Kinetic pressure
 
= 0
(4.134)

4.8  Some General Properties of MHD Equilibria

4.8.1  Pressure & Tension


jB− ∇p = 0    :    ∇ ∧B = μo j
(4.135)
We can eliminate j in the general case to get
1

μo
( ∇ ∧B) ∧B= ∇p .
(4.136)
Expand the vector triple product:
∇p = 1

μo
( B. ∇ )B 1

2 μo
∇ B2
(4.137)
put b = [(B)/ | B | ] so that ∇B = ∇B b = B ∇b + b ∇B. Then
∇p
=
1

μo
{ B2 ( b . ∇ ) b + B b ( b . ∇ ) B } − 1

2 μo
∇ B2
(4.138)
=
B2

μo
( b . ∇ )b 1

2 μo
( ∇ − b ( b . ∇ ) ) B2
(4.139)
=
B2

μo
( b . ∇ ) b − ∇
B2

2 μp

(4.140)
Now ∇ ( [(B2)/(2 μo)] ) 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. [ jB− ∇p ] = − B. ∇ p
(4.141)
*Pressure is constant on a field line (in MHD situation).
(Similarly, 0 = j . [ jB− ∇p ] = j . ∇ p.)
figures/chap4/pcontours.png
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 3-d) 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.
figures/chap4/gradpgeom.png
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: Force-Free 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'.
jB= 0
(4.142)
implies j and B are parallel.
i.e.
j = μ(r) B
(4.143)
Current flows along field lines not across. Take divergence:
0
=
∇ . j = ∇ . ( μ( r ) B) = μ( r ) ∇ . B+ ( B. ∇ ) μ
(4.144)
=
( B. ∇ ) μ.
(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: Force-Free Cylindrical Equil.


j ∧B
=
   ⇔    j = μ(r) B
(4.146)
∇ ∧B
=
μo j = μo μ(r) B
(4.147)
This is a somewhat more convenient form because it is linear in B (for specified μ(r)).
Constant−μ:      ∇ ∧B = μo μB
(4.148)
leads to a Bessel function solution

Bz
=
Bo Joo μr)
(4.149)
Bθ
=
Bo J1o μr)
(4.150)
for μo μr > 1st zero of Jo the toroidal field reverses. There are plasma confinement schemes with μ ≅ const. `Reversed Field Pinch'.

4.9  Toroidal Equilibrium

Bend a z-pinch into a torus
figures/chap4/toroidal.png
Figure 4.12: Toroidal z-pinch
Bθ fields due to current are stronger at small R side ⇒ Pressure (Magnetic) Force outwards.
Have to balance this by applying a vertical field Bv to push plasma back by jϕBv.
figures/chap4/vertfield.png
Figure 4.13: The field of a toroidal loop is not an MHD equilibrium. Need to add a vertical field.
Bend a θ-pinch into a torus: Bϕ is stronger at small R side ⇒ outward force.
Cannot be balanced by Bv because no jϕ. No equilibrium for a toroidally symmetric θ-pinch.
Underlying Single Particle reason:
Toroidal θ-pinch has Bϕ only. As we have seen before, curvature drifts are uncompensated in such a configuration and lead to rapid outward motion.
figures/chap4/chargesep.png
Figure 4.14: Charge-separation giving outward drift is equivalent to the lack of MHD toroidal force balance.
We know how to solve this: Rotational Transform: get some Bθ. Easiest way: add jϕ. From MHD viewpoint this allows you to push the plasma back by jϕBv force. Essentially, this is Tokamak.

4.10  Plasma Dynamics (MHD)

When we want to analyze non-equilibrium 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+ VB= 0 .         Ideal  Ohms  Law.
(4.151)
Also
∇ ∧E= − ∂B

∂t
       Faradays  Law.
(4.152)
Together these two equations imply constraints on how the magnetic field can change with time:    Eliminate E:
+ ∇∧(VB) = + B

∂t
(4.153)
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
Φ =


S 
B. ds
(4.154)
Let C and S move with fluid:
figures/chap4/fluxcons.png
Figure 4.15: Motion of contour with fluid gives convective flux derivative term.
Total rate of change of Φ is given by two terms:


Φ
 
=



S 


B

∂t
. ds

Due to changes in B
 
+


(⎜)



C 
B. ( Vd l )

Due to motion of C
 
(4.155)
=



S 
∇ ∧E. ds
(⎜)



C 
( VB) . d l
(4.156)
=

(⎜)



C 
(E+ VB) . d l = 0    by  Ideal  Ohms  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:
figures/chap4/fieldlinesurf.png
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:
  1. Number of field lines ( ≡ flux) through any surface is constant. (Flux Cons.)
  2. 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. z-pinch) does not guarantee a useful confinement scheme because the equil. might be unstable. Ball on hill analogies:
figures/chap4/stabilshape.png
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 [( d2)/(dx2)] { Wpot } < 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 Z-pinch
We know that there is an equilibrium: Is it stable?
Consider a perturbation thus:
figures/chap4/sausage.jpg
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 B2/2 μo pressure and [(B2)/(μor)] tension forces.
figures/chap4/skincurrent.png
Figure 4.19: Skin-current, sharp boundary pinch.
At the place where it pinches in (A)
Bθ and 1/r increase → Mag. pressure & tension increase ⇒ inward force no longer balance by p ⇒ perturbation grows.
At place where it bulges out (B)
Bθ & 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].
figures/chap4/kinkms.jpg
Figure 4.20: Types of kink perturbation.
Generally Helical in form (like a screw thread). Example: m=1    k ≠ 0    z-pinch
figures/chap4/mequal1.jpg
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).
figures/chap4/fieldalign.jpg
Figure 4.22: Alignment of perturbation and field line minimizes bending energy.
Perturbation (Constant surfaces) lies along magnetic field.
Example: θ-pinch type plasma column:
figures/chap4/flutes.jpg
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).
figures/chap4/waterRT.jpg
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 Rayleigh-Taylor instability.
Similar for supporting a plasma by mag field.
(3)  Occur when | B | decreases away from the plasma region.
figures/chap4/RTfieldgrad.jpg
Figure 4.25: Vertical upward field gradient is unstable.

BA2

2 μo
< BB2

2 μo
(4.158)
⇒ Perturbation Grows.
(4)  Occur when field line curvature is towards the plasma (Equivalent to (3) because of ∇ ∧B = 0 in a vacuum).
figures/chap4/curvexamples.jpg
Figure 4.26: Examples of magnetic configurations with good and bad curvature.

4.16  Quick and Simple Analysis of Pinches

θ-pinch      | B | = const. outside pinch
≡ No field line curvature.    Neutral stability
z-pinch     ∇ | 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:
figures/chap4/badgood.png
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
figures/chap4/shearedfield.png
Figure 4.28: Depiction of field shear.
Direction of B changes. A perturbation along B at z3 is not along B at z2 or z1 so it would have to bend field there → Stabilizing effect.
General Principle: Field line bending is stabilizing.
Example: Stabilized z-pinch
Perturbations (e.g. sausage or kink) bend Bz so the tension in Bz 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.
kR = 1:      λ = 2 πR .
(4.159)
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).
figures/chap4/ballooning.png
Figure 4.29: Ballooning modes are localized in the outboard, bad curvature region.

HEAD NEXT