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)

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 = [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,

flow out

≅

∂

∂z

(n v_z) ∆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

⌠
⌡

∇ . 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.

Fluid Descriptions are essentially 3-d (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 moments¹ of the Boltzmann Equation.

0^th 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

≡

∂

∂t

+ v. ∇

(4.12)

so the continuity equation can be written

n = −n∇.v

(4.13)

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

n =

∆N

∆V

= −

∆N

∆V²

∆V = − n

∆V

d ∆V

(4.14)

since total number of particles in volume element (∆N) is constant (we are moving with them). (∆V = ∆x ∆y ∆z.)

Now

∆V

d∆x

∆y ∆z +

d ∆y

∆z ∆x +

d ∆z

∆y ∆x

(4.15)

∆V

⎡
⎣

∆x

d ∆x

∆y

d ∆y

∆x

d ∆z

⎤
⎦

(4.16)

But

d ( ∆x )

v_x ( ∆x / 2 ) − v_x ( − ∆x / 2 )

(4.17)

≅

∆x

∂v_x

∂x

etc. ... y ... z

(4.18)

Hence

∆V = ∆V

⎡
⎣

∂v_x

∂x

∂v_y

∂y

∂v_z

∂z

⎤
⎦

= ∆V ∇ . v

(4.19)

and so

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+ u_i ∧B] (u_i is individual particle's velocity).

Hence total force on the fluid element due to E-M fields is

∑
i

( q [ E+ u_i ∧B ] ) = ∆N q ( E+ v∧B)

(4.21)

(Using mean: v = ∑_i u / ∆N.)

E-M Force density (per unit volume) is:

F_EM = n q (E+ v∧B).

(4.22)

The total momentum of the element is

∑
i

m u_i = 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 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)

F_p = − ∇ 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

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, d³w across a surface element ds is

f( w ) m w d³ w

mom^mdensity 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

p_xy

⌠
⌡

m w_x w_y f (w) d³w = 0 etc.

(4.32)

and p_xx

⌠
⌡

m w_x² f (w) d³ w ≡ n T ( = p_yy = p_zz = `p′)

(4.33)

So then the exchange rate is pds. (Scalar Pressure).

Integrate ds over the whole ∆V then x component of mom^m exchange rate is

⎛
⎝

∆x

⎞
⎠

∆y ∆z − − p

⎛
⎝

− ∆x

⎞
⎠

∆y ∆z = ∆V ( ∇ p )_x

(4.34)

and so

Total momentum loss rate due to exchange across the boundary per unit volume is

∇ p ( = − F_p )

(4.35)

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

( m n ∆V v) = [F_EM + F_p ] ∆V

(4.36)

Recall that n∆V = ∆N ; [D/Dt] (∆N) = 0; so

L.H.S. = m n ∆V

(4.37)

Thus, substituting for F′s:

Momentum Equation.

m n

= 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) 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.)

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 = F_EM

(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+ 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 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 (v₁ − v₂)

(4.49)

Hence we can immediately generalize the momentum equation to

m₁ n₁

⎡
⎣

∂v₁

∂t

+ ( v₁ . ∇ ) v₁

⎤
⎦

= n₁ q₁ ( E+ v₁ ∧B) − ∇p₁ − ν₁₂ n₁ m₁ ( v₁ − v₂ )

(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:

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

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)

δT

− δV

δn

(4.52)

δp

δn

δT

⎛
⎝

1 +

⎞
⎠

δn

(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

Continuity:

∂n_j
∂t
+ ∇ . (n_jv_j) = 0
(4.55)
Momentum:

m_jn_j ⎡
⎣ ∂v_j
∂t
+ ( v_j . ∇ ) v_j ⎤
⎦ = n_jq_j ( E+ v_j ∧B) − ∇p_j −
-

ν

jk
n_jm_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

∇ . B

0 ∇ . E = ρ/ϵ_o

(4.58)

∇ ∧B

μ_o j +

c²

∂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.

∇ . ( ∇ ∧E)

0 = −

∂

∂t

( ∇ . B)

(4.63)

and ∇ . ( ∇ ∧B)

0 = μ_o ∇ . j +

c²

∂

∂t

( ∇ . E) =

c²

∂

∂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 Two-Fluid Equilibrium: Diamagnetic Current

Slab: [(∂)/(∂x)] ≠ 0 [(∂)/(∂y)], [(∂)/(∂z)] = 0.

Straight B-field: B = B∧z.

Equilibrium: [(∂)/(∂t)] = 0 (E = − ∇ϕ)

Collisionless: ν→ 0.

Momentum Equation(s):

m_jn_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:

mn v_x

v_x

nq (E_x + v_y B) −

(4.66)

mn v_x

v_y

nq (0 −v_x B)

(4.67)

Eq 4.67 is satisfied by taking v_x = 0. Then 4.66 →

nq (E_x + v_y B) −

= 0.

(4.68)

i.e.

v_y =

− E_x

nqB

(4.69)

or, in vector form:

E∧B

B²

E∧B drift

−

∇ p

∧

B²

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_iq_i = −n_eq_e. Hence adding solutions

n_eq_ev_e + n_iq_iv_i =

E∧B

B²

( n_iq_i + n_eq_e )

− ∇( p_e + p_i )∧

B²

(4.71)

Hence current density:

j = − ∇( p_e + p_i ) ∧

B²

(4.72)

This is the diamagnetic current. The electric field, E, disappears because of quasineutrality. (General case ∑_j q_jn_jv_j = − ∇(∑p_j) ∧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 (C_j)

∂n_j

∂t

+ ∇. (n_jv_j) = 0 .

(4.73)

Momentum (M_j)

m_jn_j

⎛
⎝

∂

∂t

+ v_j . ∇

⎞
⎠

v_j = n_jq_j ( E+ v_j ∧B) − ∇p_j + F_jk

(4.74)

(where we just write F_jk = −―v_jkn_jm_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:

Mass Density ρ_m

n_em_e + n_im_i

(4.75)

C of M Velocity V

( n_em_e v_e + n_im_iv_i ) / ρ_m

(4.76)

Charge density ρ_q

q_en_e + q_in_i

(4.77)

Electric Current Density j

q_en_ev_e +q_in_iv_i

(4.78)

q_e n_e ( v_e−v_i ) by quasi neutrality

(4.79)

Total Pressure p

p_e + p_i

(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_jq_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 non-linear 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_in_i

⎛
⎝

∂

∂t

+ v_j . ∇

⎞
⎠

v_j ≅ ρ_m

⎛
⎝

∂

∂t

+ 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_jq_j²

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_iq_i + n_eq_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

∑
j

n_jq_j²

m_j

n_e² q_e²

⎛
⎝

n_em_e

n_im_i

⎞
⎠

E = n_e²q_e²

m_in_i+m_en_e

n_em_en_im_i

E = −

q_eq_i

m_em_i

ρ_m E,

(4.88)

∑

n_jq_q²

m_j

v_j

n_eq_e²

m_e

v_e +

n_iq_i²

m_i

v_i

q_eq_i

m_em_i

{

n_eq_em_i

q_i

v_e +

n_iq_im_e

q_e

v_i }

−

q_eq_i

m_em_i

{ n_em_ev_e + n_im_iv_i −

⎛
⎝

m_i

q_i

m_e

q_e

⎞
⎠

( q_en_ev_e + q_in_iv_i ) }

−

q_eq_i

m_em_i

{ ρ_m V −

⎛
⎝

m_i

q_i

m_e

q_e

⎞
⎠

j }

(4.89)

Also, remembering F_ei = −―ν_ei n_em_i (v_e−v_i) = − F_ie,

∑
j

q_j

m_j

F_jk

−

⎛
⎝

n_eq_e − n_eq_i

m_e

m_i

⎞
⎠

( v_e − v_i )

−

⎛
⎝

1 −

q_e

q_i

m_e

m_i

⎞
⎠

(4.90)

So we get

ρ_m

∂

∂t

⎛
⎝

ρ_m

⎞
⎠

−

q_eq_i

m_em_i

⎡
⎣

ρ_m E+

⎧
⎨
⎩

ρ_m V −

⎛
⎝

m_i

q_i

m_e

q_e

⎞
⎠

⎫
⎬
⎭

∧B

⎤
⎦

−

q_e

m_e

∇p_e −

q_i

m_i

∇p_i −

⎛
⎝

1 −

q_e

q_i

m_e

m_i

⎞
⎠

(4.91)

Regroup after multiplying by [(m_em_i)/(q_eq_iρ_m)]:

E+ V ∧B

−

m_em_i

q_eq_i

∂

∂t

⎛
⎝

ρ_m

⎞
⎠

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_em_i

ρ_mq_eq_i

−

⎛
⎝

1 −

q_e

q_i

m_e

m_i

⎞
⎠

m_em_i

q_eq_iρ_m

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_em_i

q_eq_i

∂

∂t

⎛
⎝

ρ_m

⎞
⎠

arises fromelectron inertia.

it will be negligible for low enough frequency.

ρ_m

⎛
⎝

m_i

q_i

m_e

q_e

⎞
⎠

j ∧B 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.

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

m_em_i

q_eq_i ( n_im_i )

m_e

q_e² n_e

= η the resistivity.

(4.93)

Hence dropping electron inertia, Hall term and pressure, the simplified Ohm's law becomes:

E+ V ∧B= ηj

(4.94)

Final equation needed: state:

p_en_e^−γ_e + p_in_i^−γ_i = constant.

Take quasi-neutrality ⇒ n_e ∝ n_i ∝ ρ_m. Take γ_e = γ_i, then

p ρ_m^−γ = const.

(4.95)

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)

Ohm′s Law: E+ V ∧B = ηj

(4.99)

Eq. of State: p ρ_m^−γ = const.

(4.100)

4.6.2 Heuristic Derivation/Explanation

Mass Charge: Obvious.

Mom^m

ρ_m

⎛
⎝

∂

∂t

+ V . ∇

⎞
⎠

[(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 ηj:

E_V = E+ V ∧B= ηj

(4.102)

The ρ_qE term is generally dropped because it is much smaller than the j∧B term. To see this, take orders of magnitude:

∇.E= ρ_q/ϵ₀ so ρ_q ∼ Eϵ₀/L

(4.103)

∇∧B=μ₀j

⎛
⎝

c²

∂E

∂t

⎞
⎠

so σE = j ∼ B/μ₀ L

(4.104)

Therefore

ρ_qE

∼

ϵ₀

⎛
⎝

μ₀σL

⎞
⎠

Lμ₀

B²

∼

L²/ c²

(μ₀σL²)²

⎛
⎝

light transit time

resistive skin time

⎞
⎠

(4.105)

This is generally a very small number. For example, even for a small cold plasma, say T_e=1 eV (σ ≈ 2×10³ 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/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 = μ_oj .

(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

(Mom^m) → `Force Balance′ 0 = j ∧B− ∇ p

(4.107)

Ampere ∇ ∧B = μ_o 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

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 ( j ∧B)_r

−

( ∇p )_r = 0

(4.109)

Ampere ( ∇ ∧B)_θ

( μ_o j )_θ

(4.110)

i.e. j_θ B_z

−

∂

∂r

p = 0

(4.111)

−

∂

∂r

B_z

μ_o j_θ

(4.112)

Eliminate j: −

B_z

μ_o

∂B_z

∂r

−

∂p

∂r

(4.113)

i.e.

∂

∂r

⎛
⎝

B_z²

2 μ_o

+ p

⎞
⎠

(4.114)

Solution

B_z²

2 μ_o

+ p

const.

(4.115)

Figure 4.6: Balance of kinetic and magnetic pressure

B_z²

2 μ_o

+ p =

B_z ext²

2 μ_o

(4.116)

[Recall Single Particle Problem]

Think of these as a pressure equation. Equilibrium says total pressure = const.

B_z²

2 μ_o

magnetic pressure

kinetic pressure

= const.

(4.117)

Ratio of kinetic to magnetic pressure is plasma `β'.

β =

2 μ_o p

B_z²

(4.118)

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

4.7.2 Z-pinch

Figure 4.7: Z-pinch configuration.

so called because j flows in z-direction. Again take to be ∞ length and uniform.

j = j_z

B = B_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 =

∂

∂r

( r B_θ ) − μ_oj_z = 0

(4.121)

Eliminate j:

B_θ

μ_o r

∂

∂r

( r B_θ ) −

∂p

∂r

= 0

(4.122)

B_θ²

μ_o r

Extra Term

∂

∂r

⎛
⎝

B_θ²

2 μ_o

+ 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

B_θ²

μ_o

⎡
⎣

B_θ²

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_θ² ( b )

2 μ_o

−

B_θ² ( r )

2 μ_o

⌠
⌡

b

r

B_θ²

μ_o

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.

∂

∂r

( r B_θ ) = μ_o j_z ⇒ B_θ =

μ_o j_z

(4.126)

Hence

p(r)

2 μ_o

⎛
⎝

μ_o j_z

⎞
⎠

{ b² − r² +

⌠
⌡

b

r

2 r′dr′}

(4.127)

μ_o j_z²

{ b² − r² }

(4.128)

Figure 4.9: Parabolic Pressure Profile.

Also note B_θ (b) = [( μ_o j_zb)/2] so

p =

B_θb²

2 μ_o

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 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_oj_θ

(4.131)

∂

∂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)

B_θ²

μ_o r

[(Mag Tension) || (θ only)]

∂

∂r

⎛
⎝

B²

2 μ_o

+ p

⎞
⎠

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

μ_o

( ∇ ∧B) ∧B= ∇p .

(4.136)

Expand the vector triple product:

∇p =

μ_o

( B. ∇ )B−

2 μ_o

∇ B²

(4.137)

put b = [(B)/ | B | ] so that ∇B = ∇B b = B ∇b + b ∇B. Then

∇p

μ_o

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

2 μ_o

∇ B²

(4.138)

B²

μ_o

( b . ∇ )b −

2 μ_o

( ∇ − b ( b . ∇ ) ) B²

(4.139)

B²

μ_o

( b . ∇ ) b − ∇_⊥

⎛
⎝

B²

2 μ_p

⎞
⎠

(4.140)

Now ∇_⊥ ( [(B²)/(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 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.

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'.

j ∧B= 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:

∇ . 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

μ_o j = μ_o 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 = μ_o μB

(4.148)

leads to a Bessel function solution

B_z

B_o J_o (μ_o μr)

(4.149)

B_θ

B_o J₁ (μ_o μr)

(4.150)

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 z-pinch into a torus

Figure 4.12: Toroidal z-pinch

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 θ-pinch 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 θ-pinch.

Underlying Single Particle reason:

Toroidal θ-pinch 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: 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 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 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+ 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:

+ ∇∧(V ∧B) = +

∂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 =

⌠
⌡

B. ds

(4.154)

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:

⋅

⌠
⌡

∂B

∂t

. ds

Due to changes in B

⌠
(⎜)
⌡

B. ( V ∧d l )

Due to motion of C

(4.155)

−

⌠
⌡

∇ ∧E. ds −

⌠
(⎜)
⌡

( V ∧B) . d l

(4.156)

−

⌠
(⎜)
⌡

(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. z-pinch) 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 [( d²)/(dx²)] { 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 Z-pinch

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 B²/2 μ_o pressure and [(B²)/(μ_or)] tension forces.

Figure 4.19: Skin-current, 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 z-pinch

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: θ-pinch 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 Rayleigh-Taylor 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_A²

2 μ_o

B_B²

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

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:

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 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 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.

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).

Figure 4.29: Ballooning modes are localized in the outboard, bad curvature region.

HEAD

Chapter 4 Fluid Description of Plasma

4.1 Particle Conservation (In 3-d Space)

4.2 Fluid Motion

4.2.1 Lagrangian & Eulerian Viewpoints

4.2.2 Momentum (Conservation) Equation

4.2.3 Pressure Force

4.2.4 Momentum Equation: Eulerian Viewpoint

4.2.5 Effect of Collisions

4.3 The Key Question for Momentum Equation:

4.4 Summary of Two-Fluid Equations

4.5 Two-Fluid Equilibrium: Diamagnetic Current

4.6 Reduction of Fluid Approach to the Single Fluid Equations

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

4.6.2 Heuristic Derivation/Explanation

4.6.3 Maxwell's Equations for MHD Use

4.7 MHD Equilibria

4.7.1 θ-pinch

4.7.2 Z-pinch

4.7.3 `Stabilized Z-pinch'

4.8 Some General Properties of MHD Equilibria

4.8.1 Pressure & Tension

4.8.2 Magnetic Surfaces

4.8.3 `Current Surfaces'

4.8.4 Low β equilibria: Force-Free Plasmas

Example: Force-Free Cylindrical Equil.

4.9 Toroidal Equilibrium

4.10 Plasma Dynamics (MHD)

4.11 Flux Conservation

4.12 Field Line Motion

4.13 MHD Stability

4.14 General Perturbations of Cylindrical Equil.

4.15 General Principles Governing Instabilities

4.16 Quick and Simple Analysis of Pinches

Chapter 4
Fluid Description of Plasma