Advanced Plasma

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Advanced Plasma Physics Notes 2008 Spike April 6, 2008 Contents 1 Lecture 1: Kinetic Theory 3 1.1 Liouville’s Theorem ....................................... 3 1.2 μ-space .............................................. 3 1.3 μ-space vs. Γ-space ........................................ 3 1.4 The BBGKY Hierarchy ..................................... 4 2 Lecture 2: Derivation of BBGKY Hierarchy 4 2.1 BBGKY Hierarchy ........................................ 5 3 Lecture 3 6 3.1 Dilute Systems .......................................... 6 3.2 Boltzmann Collision Integral from the BBGKY Viewpoint .................. 7 3.3 Time’s Arrow ........................................... 8 4 Lecture 4 8 4.1 Boltzmann Collision Operator .................................. 8 4.1.1 Boltzmann Collision Operator .............................. 9 4.2 Plasma ............................................... 9 5 Lecture 5 10 5.1 Electron Scattering by Ions ................................... 10 5.2 Fokker-Planck Equation ..................................... 11 6 Lecture 6 12 6.1 More on the collision operator: BGK (Bahatnagar-Gross-Krook) Equation ......... 12 6.2 Vlasov Equation ......................................... 12 6.3 Debye Length ........................................... 13 7 Lecture 7 15 7.1 Weakly and Strongly Coupled Plasmas ............................. 15 7.2 Dielectric Properties of a Plasma ................................ 15 7.3 Charge and Current Density ................................... 16 8 Lecture 8 17 8.1 Landau Prescription ....................................... 17 8.2 Charge Density .......................................... 17 8.3 Plasma Dispersion function ................................... 18 9 Lecture 9 20 9.1 Plasma Dielectric Constant ................................... 20 9.2 Plasma Dispersion Relation ................................... 20 9.3 Dispersion Relation for Longitudinal Waves .......................... 21 9.4 Dispersion Relation for Transverse waves ............................ 21 9.5 Transverse Waves ......................................... 22 1

Transcript of Advanced Plasma

Page 1: Advanced Plasma

Advanced Plasma Physics Notes 2008

Spike

April 6, 2008

Contents

1 Lecture 1: Kinetic Theory 3

1.1 Liouville’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 µ-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 µ-space vs. Γ-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 The BBGKY Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Lecture 2: Derivation of BBGKY Hierarchy 4

2.1 BBGKY Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Lecture 3 6

3.1 Dilute Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Boltzmann Collision Integral from the BBGKY Viewpoint . . . . . . . . . . . . . . . . . . 73.3 Time’s Arrow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Lecture 4 8

4.1 Boltzmann Collision Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.1.1 Boltzmann Collision Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.2 Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5 Lecture 5 10

5.1 Electron Scattering by Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.2 Fokker-Planck Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

6 Lecture 6 12

6.1 More on the collision operator: BGK (Bahatnagar-Gross-Krook) Equation . . . . . . . . . 126.2 Vlasov Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.3 Debye Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

7 Lecture 7 15

7.1 Weakly and Strongly Coupled Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.2 Dielectric Properties of a Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.3 Charge and Current Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

8 Lecture 8 17

8.1 Landau Prescription . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.2 Charge Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.3 Plasma Dispersion function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

9 Lecture 9 20

9.1 Plasma Dielectric Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.2 Plasma Dispersion Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.3 Dispersion Relation for Longitudinal Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 219.4 Dispersion Relation for Transverse waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.5 Transverse Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

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9.6 Longitudinal Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.7 Physical Origin of Landau Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

10 Lecture 10: MHD 23

10.1 Hydrodynamic limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2310.2 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.3 Momentum Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.4 Multi-Element Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2610.5 Bulk Fluid Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

11 Lecture 11 27

11.1 Ohm’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2711.2 Energy Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2811.3 Transport Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2811.4 Hydro-Magnetic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

12 Lecture 12: Ideal MHD Approximation 30

12.1 Pressure Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3012.2 Incompressible Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3112.3 Bernoulli’s Equation - Steady Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3112.4 Kelvin’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3212.5 Alfven Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

13 Lecture 13: Magnetosonic Waves 33

14 Lecture 14: Magneto-Hydrodynamic Waves 35

14.1 M.H.D. Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

15 Lecture 15: M.H.D. Shocks 38

15.0.1 Alfven Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3915.1 Boltzmann’s Equation as a Fokker-Planck (Landau’s Equation) . . . . . . . . . . . . . . . 39

16 Lecture 16 41

16.1 Rosenbluth, Macdonald and Judd Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4216.1.1 Relationship with Landau’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 45

17 Lecture 17: Calculation of the Transport Coefficients 46

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1 Lecture 1: Kinetic Theory

The motion of individual particles can be considered as an ensemble. We take as our starting pointHamilton’s equation for the dynamics of particles. Consider a system of N particles, where N is verylarge. Each particle has qi, pi(≡ r, v). We can now consider a space, Γ, which consists of the co-ordinateaxis of qi, pi of all of the particles. (A 6N dimensional space). The particle state (as determined by qi, pi)appears as a point, and the particle trajectory appears as a line in the Γ space. The probability functiongives the probability that a particle will be found in the box formed by dq1dq2 . . . dqN at q1p1 . . . qNpNis given by Prob = F (q1p1q2p2 . . . qNpN )dq1dp1 . . . dqNdpN .

If the particle position is initially at some known point, we can say that

F (q1p1q2p2 . . . qNpN , t) = ΠNt=1δ(qi1 −Qi1(t))δ(pi1 − Pi1(t)) . . . (1)

where Qi1(t), Pi1(t) are the integrals of the Hamiltonian equation in time, Qi1(t), Pi1(t) is the classicaltrajectory of particle i.

In fact, although we cannot describe the initial conditions exactly, we do know a probability distri-bution for the particles. We can use this to form an ensemble of paths in phase space (6N, qi, pi). Wefollow the motion of the ensemble in phase space.

1.1 Liouville’s Theorem

As we follow the motion of the probability cloud, Liouville’s theorem states that:

DF

DT= 0 =

∂F

∂t+ ΣNi=1qi

(∂F

∂qi+ pi

∂F

∂pi

)

(2)

This follows directly from Hamilton’s equation for a conservative system:

pi = −∂H∂qi

; qi =∂H

∂pi(3)

This only works if the system is conservative. If we add a magnetic field, so that the system isnon-conservative, Hamilton’s equations are still applicable, and Liouville’s theorem is valid.

N.B. So for, the motion of the system is completely thermodynamically reversible. However, a realplasma is not reversible. How is this?

1.2 µ-space

This is a six-dimensional phase space: (r, v, t).The probability distribution function, f (r, v, t) gives the probability of finding any particle in the

volume element (dr, dv) at (r, v) as f (r, v, t) dr dv.

1.3 µ-space vs. Γ-space

Consider the probability in Γ-space of finding particle 1 in r1, v1, dr1, dv1 to be Adr1, dv1 and the otherparticles anywhere in phase space.

A =w

2· · ·

w

NF (v1, r1, v2, r2, . . . , vN , rN , t)dr2, dv2, . . . , drN , dvN (4)

N.B. There is no dr1, dv1 in this equation.But: the particles are indistinguishable; all particles have the same probability of being in the same

space element.

f (1)(r1, v1, t) = f (r1, v1, t) = NA = Nw

2· · ·

w

NF (v1, r1, v2, r2, . . . , vN , rN , t)dr2, dv2, . . . , drN , dvN (5)

But: the integration from r2 → rN removes all of the correlation information, for example f (r1, v1, t)contains no enhanced probability for finding particle 1 in the neighbourhood of 2. In this case, theprobability distributions of the particles are totally independent of each other.

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Consider the probability of finding a particle in r1, dr1, v1, dv1 and a second in r2, dr2, v2, dv2 si-multaneously. This is the two-particle distribution function: f (2)(r1, v1, r2, v2, t)dr1, dv1, dr2, dv2. Thisdistribution includes correlation, i.e the probability of finding one particle is influenced by that of asecond, the distributions of the particles are not independent, one depends on the other.

Correlation is very important. There are two types of correlation in plasmas. Collisional correlationis short-range, and there is also a long-range correlation. In dense plasmas there may also be a third,short-range correlation.

In terms of Γ-space distribution:

f (2)(r1, r2, v1, v2) =N !

(N − 2)!2!

w

3· · ·

w

NF (v1, r1, v2, r2, . . . , vN , rN , t)dr3, dv3, . . . , drN , dvN (6)

where

N !

(N − 2)!2!=N(N − 1)

2(7)

This can be generalised to any number of particles.The general particle distribution function: r1, v1, dr1, dv1, . . . , rx, vx, drx, dvx gives:

f (q)(r1, v1, . . . , rq, vq) =N !

(N − q)!q!w

q+1· · ·

w

NF (r1, v1, . . . , rq, vq, rq+1, vq+1, . . . , rN , vN )drq+1, dvq+1, . . . , drN , dvN

(8)gives a set of N-fold distribution functions in µ-space. The complete set contains all the information

in the Γ-space distribution. Calculating this is very difficult.If the particles are independent (i.e. there is no interaction between the particles) then

f (2)(r1, v1, r2, v2) = f (1)(r1, v1)f(1)(r2, v2) (9)

If the particle-particle interactions are weak, the motions are nearly independent, and the one-particledistribution particle is nearly adequate. We can therefor work with a truncated set of distributions.

1.4 The BBGKY Hierarchy

A set of equations which enable the calculation of f (q) from f (q−1), f (q+1). Eventually this set must betruncated with some f (q+1) assumed to be small. For dilute gas plasmas we can truncate at f (2), whichremoves correlations. However, the full set of µ-space distributions are still reversible. The truncationmust also introduce irreversibility.

2 Lecture 2: Derivation of BBGKY Hierarchy

Forces on a particle:Externally applied force: Fi. e.g. gravity. (N.B. F = distribution function. Note lack of indices)Internal forces: forces from mutual interactions: F ij , in general a function of the separation of the

particles: F ij(|ri − rj |).

∂F

∂t+ Σvi ·

∂F

∂ri+ Σvi ·

∂F

∂vi= −ΣNi=1Σ

Nj=1,j 6=i

F ijm· ∂F∂vi

(10)

Remember: F is a symmetric function w.r.t. the interchange of particles.Multiply (10) by N !

(N−n)!

Integrate over drn+1dvn+1 . . . drNdvN to yield f (n)(r1, v1, . . . , rn, vn)

N !

(N − n)!

w ∂F

∂tdrn+1dvn+1 . . . drNdvN =

∂f (n)

∂t(r1, v1, . . . , rN , vN ) (11)

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Integrals for i > n which contain derivatives with respect to one of the variables of integration, maybe integrated by parts to give a surface integral at the boundary and a different volume integral, whosevalues are zero, for example

wvi ·

∂F

∂ridri =

zdSi · viF −

wdriF

∂vi∂rr

= 0

since limr→∞

F = 0, and vi and ri are independent variables.

N !

(N − n)!

wvi ·

∂F

∂ridrn+1dvn+1 . . . drNdvN =

vi ·∂f (n)

∂ri(r1, v1, . . . , rN , vN ) if i ≤ n

0 if i > n

Noting that the only external force which depends on the velocity is the Lorentz force due to anexternal magnetic field v ∧B, there is no component parallel to the velocity and therefore ∂F i/∂vi = 0

N !

(N − n)!

1

m

wF i ·

∂F

∂vidrn+1dvn+1 . . . drNdvN =

1

mF i ·

∂f (n)

∂vi(r1, v1, . . . , rN , vN ) if i ≤ n

0 if i > n

The mutual interaction between particles is treated by breaking the sum into three parts

ΣNi=1ΣNj=1,j 6=i = Σni=1Σ

nj=1,j 6=i + Σni=1Σ

Nj=n+1,j 6=i + ΣNi=n+1Σ

Nj=1,j 6=i (12)

N !

(N − n)!

w F ij(ri − rj)m

· ∂F∂vi

drn+1, dvn+1, . . . , drN , dvN =F ij(ri − rj)

m· ∂f

(n)

∂vi(13)

where i ≤ n and j ≤ n.

N !

(N − n)!

w F ijm· ∂F∂vi

drn+1 dvn+1, . . . , drN , dvN =1

(N − n)

w F ij(ri − rj)m

· ∂f(n+1)

∂vidrj dvj (14)

where i ≤ n and j > n.Total contribution to the sum: multiply by (N-n) to take into account identical particles.

N !

(N − n)!

w F ij(ri − rj)m

=∂F

∂vidrn+1 dvn+1 . . . , drN dvN (15)

Now, substitute these into the Liouville equation:

∂f (n)

∂t+Σni=1vi·

∂f (n)

∂ri+Σni=1

F im·∂f

(n)

∂vi+Σni=1Σj=i,j 6=i

F ijm· f

(n)

∂vi= −Σni=1Σ

Nj=n+1

w∂rj∂vj

F ij(ri − rj)m

·∂f(n+1)

∂vi(16)

2.1 BBGKY Hierarchy

To obtain f (n) you need f (n+1). The complete set f 1) . . . f (N) are equivalent to the Liouville equation.We need to introduce closure: Limit the number of n, usually to n = 1 (or n = 2 for dilute systems).Using n = 1 we get:

∂f (1)

∂t+ v · ∂f

(1)

∂r1+F

m· ∂f

(1)

∂v= − 1

m

wF12(r1 − r2) ·

∂r1

f (2)(r1, v1, r2, v2)

dr2dv2

︸ ︷︷ ︸

(17)

where the third term is the collision term, giving the interaction force. If it is assumed that there is noexternal force the Boltzmann equation is recovered.

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∂f (2)

∂t+ v1 ·

∂f (2)

∂r1+ v2 ·

∂f (2)

∂r2+

1

m

F12(r1 − r2) ·∂f (1)

∂v1

+ F21(r2 − r1) ·∂f (2)

∂v2

=

−w [

F13(r1 − r3) ·∂f (n+1)

∂r1+ F23(r2 − r3) ·

∂f (n+1)

∂r2

]

dv3dr3 (18)

Find an approximation based on physical considerations for the collision term. What are the con-straints on the physics to get a sensible collision term?

3 Lecture 3

Consider the streaming derivative for the one particle distribution

Df (1)

Dt≡ ∂f (1)

∂t+ v1 ·

∂f (1)

∂r1+F 1

m· ∂f

(1)

∂v1︸ ︷︷ ︸

(19)

In the fluid dynamic analogy, this is the Lagrangian time derivative for the ’fluid’ flowing in phasespace.’Driving’ term for the velocity, namely the acceleration, is determined by the external forces alone.In addition the time variation of the one particle distribution depends on an additional term determinedby the interaction forces, which itself depends of the two particle derivative.

n = 1 can also be written as:

Df (1)

Dt=∂f (1)

∂t

∣∣∣∣∣coll

(20)

3.1 Dilute Systems

If the system is dilute there are three well separated time scales:

1. Collision time: τ0 ∼ av where a = range of intermolecular force.

2. Collision frequency (time between collisions): t0 ∼ λv where λ = the mean free path.

3. Macro. time: θ0 ∼ Lvs

where L = Lab scale, and Vs = speed of sound.

Since

• The range of the interparticle force a is very small

• The mean free path is relatively large λ≫ a is large

• The scale of the laboratory apparatus is extremely large L≫ λ

Consequently τ0 ≪ t0 ≪ θ0We prepare the system in some state at t = 0, not in equilibrium. Rapid changes occur over times τ0

due to the interparticle force. The multiple particle distribution functions f (s) s ≥ 2 change rapidly overthese time scales, and establish an equilibrium amongst themselves. On the other hand the one particledistribution f (1) on changes more slowly due to the streaming derivative and changes only over as timet0 as interactions take place.

We are usually not interested in changes over τ0, as it is too small and gives a level of detail, whichis not required. However changes over t0 determining the one particle distribution directly relate toexperimentally measurable quantities such as temperature/density etc. We can therefore average thedetailed fluctuations over times ≫ τ0 without losing any important information.

When this averaging is carried out, the behaviour of the multiple particle distribution functionsrapidly achieve the equilibrium forms associated with the instantaneous one particle distribution. Since

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this equilibration time is very short, we imagine that multiple particle distributions, on averaging, havea time depending, whose time dependence is determined by that of the one particle distribution

f (s)(r1, v1, . . . , rs, vs, t) ≡ f (s)(

r1, v1, . . . , rs, vs)∣∣f (1)(t)

)

(21)

where s ≥ 2.As a result the equation for n = 1 becomes

Df (1)

Dt= A

(

r, v,∣∣f (1)

)

=∂f (1)

∂t

∣∣∣∣∣coll

(22)

We still have reversibility problems. There is no arrow of time. A loss of reversibility can be introducedby introducing a direction of time by which ’before’ and ’after’ are clearly distinguished. This assumes”molecular chaos”:

1. Before a collision (2-body) the 2 bodies are statistically independent.

2. After the collision they are correlated.

3. In time the correlation is lost between collisions. .

As a result, the particles involved in the collisions are always uncorrelated before the collision.

3.2 Boltzmann Collision Integral from the BBGKY Viewpoint

For dilute gases collisions are binary and completed over very short timescales.Assumptions:

i 3-body (and higher) terms are negligible.

ii Time variation during collision is negligible.

iii Spatial variation over the collision range (a) is negligible.

iv Change in velocity due to external forces during collisions is negligible.

Take n = 2:

(ii)

∂f (2)

∂t+ v · ∂f

(2)

∂r+ v′ · ∂f

(2)

∂r′+F

m· ∂f

(2)

∂v+F ′

m· ∂f

(2)

∂v′+

: (iv)F (r − r′

m· ∂f

(2)

∂v+

: (iv)F (r − r′)

m· ∂f

(2)

∂v1

= − 1

m

w

: (i)

F (r − r′′) · ∂f(3)

∂r+

: (i)

F (r′ − r′′′) · ∂f(3)

∂r′

dr

′′dr′′′ (23)

Since there is no spatial inhomogeneity (iii), a small displacement ρ leaves the two particle distributionunchanged.

f (2)(r, v, r′, v′, t) = f (2)((r + ρ), v, (r′ + ρ), v′, t) (24)

We can now differentiate with respect to ρ:

∂f (2)

∂r+∂f (2)

∂r′= 0 (25)

So (23) becomes:

−(v − v′)∂f(2)

∂r′+F (r − r′)

m

∂f (2)

∂v+F (r′ − r)

m

∂f (2)

∂v′= 0 (26)

For n = 1:

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∂f (1)′

∂t+ v · ∂f

(1)′

∂r+F

m

∂f (1)′

∂v= −

w w︷ ︸︸ ︷(

F (r′ − r)m

∂f (2)

∂v

)

dr′dv′

︸ ︷︷ ︸

(27)

where the left hand side is J , and the integrand of the right hand side can be substituted from (26) togive:

J = −w w [

(v − v′) · ∂f(2)

∂r′− F (r′ − r)

m· ∂f

(2)

∂v′

]

dr′dv′ (28)

integrate the second term over v′, using Gauss’s theorem.

w F (r′ − r)m

· ∂f(2)

∂v′dv′ = 0 (29)

Integrate the first term over r′, can use Gauss to give:

J = −wdv′

zdS′ · (v − v′)f (2) −

wdr′

:0f (2)

∂r′(v − v′) (30)

It is possible to cancel the second term on the left hand side as v & v′ are not functions of r′. Thesurface integral is over the collision volume.

Consider the collision volume as a sphere of radius a, centred on the centre of mass. V is the relativevelocity, where V = v−v′,which is unchanged in magnitude in an elastic collision. We consider (different)

pairs of particles with the same velocities after the collision as those before.∩r ′ is the radius vector of

the outgoing particle at the surface on top (exit),∪r ′ is the radius vector of the incoming particle at

the surface on the bottom (entry). The element dS′ of the surface corresponding to the entry and exitparticles is projected into dω on the equatorial plane. Thus dω is the element of the collision crosssection.

3.3 Time’s Arrow

Assumption (v) Molecular chaos. Incoming particles are statistically independent.

f (2)(r, v, r′, v′) = f (1)(r, v) f (1)(r′, v′) (31)

This applies to the entry, but not the exit, particles.For an elastic collision: Conservation of total momentum and energy, therefore centre-of-mass velocity

is constant and relative velocity is rotated through a scattering angle χ. χ can be specified from thecross-section as σ = σ(v, χ)dΩ. Hence the velocities after the collision v and v′ can be calculated.

From Liouville’s theorem the two particle distribution function after the collision must be equal toits value before the collision for the same two particles. Thus if r, v and r′, v′ are the initial particlepositions and velocities which give rise to velocities r, v and r′, v′ respectively after collision:

f (2)(r, v, r′, v′) = f (1)(r, v) f (1)(r′, v′) (32)

4 Lecture 4

4.1 Boltzmann Collision Operator

From the closing of the BBGKY Hierarchy:

J =wdv′V χ = f (2)(r, v,

∩r ′, v′, t)− f (2)(r, v, ∪r ′, v′, t)dw (33)

where v = relative velocity, dw = differential cross-section for collision through an angle χ.We make the additional assumption of molecular chaos, where two particles have such a long history

before the collision that they are essentially uncorrelated before the collision.

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Before Collision:

f (2)(r, v,∪r ′, v′) = f (1)(r, v) f (1)(r ′, v′) (34)

Liouville’s Theorem: Distribution function f (2) is unchanged through the collision.After collision assuming Molecular Chaos:

f (2)(r, v,∩r ′, v′, t) = f (1)(r, v, t)f (1)(r′, v′, t) (35)

r, v and r′, v′ after collision result from r, v, r′, v′ before collision.

Since the collision volume is very small, r = r′ = r = r′ =∪r ′ =

∩r ′

J =wdv′dw

f (1)(r, v, t) f (1)(r, v′, t)− f (1)(r, v, t) f (1)(r, v′, t)

(36)

where v and v′ lead to v and v′ after the collision.

J =wdv′dΩσ(V, χ)

f (1)(r, v) f (1)(r, v′)− f (1)(r, v) f (1)(r, v′)

(37)

where dw = σ(v, χ)dΩ is the differential cross section for scattering through solid angle dΩ for a relativevelocity V and angle of scatter χ.

4.1.1 Boltzmann Collision Operator

Boltzmann Collision Operator (for a 1 particle distribution function):

∂f

∂t+ v

∂f

∂r+F

m

∂f

∂v=∂f

∂t

∣∣∣∣coll

(38)

where f ≡ f (1) and∂f∂t

∣∣∣coll

= J . [Note that henceforward we shall omit distribution function superscripts

unless necessary].The rate of change of f due to collisions = the rate of collisions bringing particles into the phase

space volume drdv - rate of collisions exiting drdv.The number of collisions exiting drdv per unit time = f (r, v)drdv

rf (r′, v′)σV dv′dr′

Number of collisions entering the volume element r v r′ v′ ⇒ r v r′ v′

wf (r, v)f (r′, v′)σ V dr dv dr′ dv′

The volume element is given by the Jacobian, for which drdvdr′dv′ = drdvdr′dv′ follows from thedynamics.

Hence we obtain the same result as before

∂f

∂t

∣∣∣∣coll

=w [f (r, v)f (r, v′)− f (r, v)f (r, v′)

]σV dv′dΩ (39)

All occur at the same place in space (due to the collision volume being small) so r is the same for all.

4.2 Plasma

Long range forces between particles (Coulomb Force):Coulomb Force: φ(r, r′) ∝ 1

|r−r′| violates the short-range assumption.

Impose closure on the BBGKY: Only at short range is the Coulomb force strong. Assume a dilute

plasma - can separate a strong field region and consider two body collisions only in this region. Outsidethis region we consider the field as weak, we can therefore treat the field as a perturbation, followingmany body collisions.

Strong collision region: collisions are essentially binary. Impact parameter less than of order of 90o

scattering by a Coulomb field, b ∼ bmin, Landau length.Weak collision region: Multi-particle, cooperative region Impact parameter bmin > b > λD (Debye

Length).(e.g. plasma waves).

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Collective region: Impact parameter b ? λD. Interaction characterised by collective behaviour re-sulting in plasma waves due to non-local self-generated fields.

For small collisions: The particle wanders through the plasma with numerous small deflections - manybody collisions: (interaction through the coulomb field.

Electron Ion scattering: Momentum change of the particle is the sum of many terms. ∆p = ∆p1

+∆p

2+ · · ·+ ∆p

N. Interacting particles are randomly distributed (molecular chaos) giving ∆p

x≈ 0 but

∆p2x 6= 0. The statistical distribution of the particle deflections. These are randomly distributed in angle,

Gaussian with variance proportional to length of path and proportional to time. The R.M.S. Deflectionis proportional to the square root of time.

In the remainder of this course we shall consider dilute plasma in which the particle density issufficiently small that correlations are almost unimportant. However we cannot neglect them altogetheras they form the basis for both collisions and collective behaviour. For dilate plasma we may consideronly the one particle distribution function (f)(1) greatly simplifying calculations.

5 Lecture 5

5.1 Electron Scattering by Ions

We consider the scattering of electrons by randomly distributed stationary ions.The impulse of each ion on the electron: ∆p

iTotal impulse due to many ions i = Σi∆piIf the electron is reasonably well separated from the ions (i.e. the ions are far apart) the impulse from

each ion is small, and can be treated as a perturbation on the motion of the electron. The motion istherefore not much disturbed by each impulse. Since each perturbation is small it acts independently ofthe others despite being simultaneous. (This condition is only true for distant collisions). Since the ionsare randomly distributed, the net impulse on the electron will be zero, as long as the path is sufficientlylong. However Σi|∆pi|

2 6= 0. (In a particular direction j, Σi|∆pij |2 yields the mean square impulse in

the direction j.)

∆p2j =

(Σi∆pij)2

N(40)

where N is the number of ions encounteredHence the mean squared impulse in time δt:

∆p2x =

Σi(∆pix)2 +

: 0

2Σi6=jΣ(∆pix∆pjx)

N(41)

=w w

[∆px(b)]2Nvbdbdθdt (42)

where theta is the azimuthal angle and b the impact parameter.The mean squared deflection angle φ is given by:

∆φ2x =

∆p2x

(mv)2(43)

where (mv)2 is the momentum of the electron.Since the electron-ion interaction is weak, the electron path is approximately a straight line z = vt.

The transverse impulse at impact parameter b is:

∆pxi =qQ

4πε0

w ∞

−∞

bdz

(b2 + z2)32

=2qQ

4πε0bv(44)

∆pi =2qQ

4πε0bmv2(45)

where Q is the ion charge and q the electron charge.

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∆φ2x =

wbdb

wdθ

[2qQ cos θ

4πε0bmv2

]2

Nvdt (46)

=4πq2Q2N

(4πε0)2m2v3

w db

bdt (47)

whererdbb is a logarithmic singularity, giving problems for b both large and small.

We now look at the case where b is large. Here we start to get some screening of ion potential byother electrons, up to b > λD (see later). (If b > λD there is no impulse, this gives an upper limit on theintegral of λD.)

Turning our attention to the case where b is small, the deflection will be very large, and the assumptionof small angle deflection is violated. We can treat the problem as a series of two particle collisions usingthe Rutherford scattering formula (equivalent to a Boltzmann collision term). It is found that this resultis equivalent to a cut-off at bmin (the Landau parameter/distance). This is the distance that correspondsto 90o scattering.

bmin =qQ

4πε0mv2(48)

∆φ2x =

4πq2Q2N

(4πε0)2m2v3

w λD

bmin

db

bdt (49)

=4πq2Q2N

(4πε0)2m2v3ln

[λDbmin

]

(50)

Let ln Λ = ln(λD

bmin

)

known as the Spitzer parameter/Coulomb logarithm. (∼ 5− 10). The physical

significance of Λ is that it is the ratio of the number of small-angle large-impact collisions to the numberof large-angle small-impact collisions. Post-hoc justification of the use of perturbation is provided by Λbeing large.

∆p2x,∆φ

2x lead directly to the Fokker-Planck equation, a method for calculating the distribution

function.

5.2 Fokker-Planck Equation

Multiple small interactions that change the distribution. We are interested in the probability functionthat an electron with velocity v undergoes a collision that changes its velocity by (∆v, d∆v) namelyψ(∆v)d∆v in a time dt.

f (r, v, t) =w [f (r, (v −∆v), (t− dt)

]ψ(v,∆v)d∆v (51)

If ∆v is small we can expect, via the Taylor series:

f (r, v, t) =rd(∆v)f (r, v, (t− dt))ψ(v,∆v)−

−∆v[∂f

∂vψ + f ∂ψ∂v

]

+ 12∆vi∆vj

[

ψ∂2f

∂vi∂vj+ 2

∂f

∂vi

∂ψ∂vj

+ f ∂2ψ∂vi∂vj

]

wψ(v,∆v)d(∆v) = 1 (52)

∂f

∂t

∣∣∣∣coll

= − ∂

∂vi

[f∆vi

]+

1

2

∂2

∂vi∂vj

[f∆vi∆vj

](53)

where the first term in square brackets is the coefficient of dynamical friction and the second is thecoefficient of diffusion.

∆vi

∆vi∆vj

=1

∆t

wψ(v,∆v)

∆vi

∆vi∆vj

d(∆v)

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Referring to the mean change in the velocity on the top line, and the mean change in the product of twovelocities on the bottom.

For electron scattering from stationary ions, we may use the expressions obtained earlier:

∆vx = ∆y ≈ 0 (54)

∆v2x = ∆v2

y 6≈ 0 (55)

∆vx∆vy ≈ 0 (56)

where x and y are perpendicular to the electron motion. Writing ⊥ for the total perpendicular velocitycomponent and ‖ for the parallel we note that because the collisions are elastic: 1

2m(v2‖ + 2v2

⊥) = isunchanged. Hence

∆v⊥2 =8πq2Q2N ln Λ

(4πǫ0)2m2v

∆v‖ = − 12

1v∆v

2⊥ =

(4πq2Q2N ln Λ

4πǫ0)2m2v2

∆v2‖ ≈ 0

(57)

If we add these back into (53) we get the Fokker-Planck equation.We will discuss electron-electron collisions later.If the plasma is non-isotropic ∆vx may not be zero, and ∆vx∆vy may also not be zero.In equilibrium (i.e. thermodynamic balance) (53):

∂f

∂t

∣∣∣∣coll

= 0 (58)

which fixes the numerical constants.

6 Lecture 6

6.1 More on the collision operator: BGK (Bahatnagar-Gross-Krook) Equa-tion

Defining the relaxation time that the distribution takes to go into a Maxwellian:

∂f

∂t

∣∣∣∣coll

= −1

τ(f − f0) (59)

Where f0 is Maxwellian, and τ is the relaxation time.

6.2 Vlasov Equation

Thus far we have considered collisional effects, where the electron is scattered by short range effects. (i.e.by randomly distributed scattering fields, l < λD).

Now, we look at co-operative effects: scattering by self-generated fields. If the interaction is overlonger distances (l > ΛD) the system acts more like a fluid, and we lose graininess. This gives rise tothe Vlasov equation describing the distribution function (f)(1).

Imagine that the plasma is subdivided. em is constant, where e is the electronic charge and m the

electron mass. There are no changes on the macroscopic scale, and we hold the total charge and massconstant. Now let the number of particles, N , tend to infinity, but with total charge and mass constant.e,m→ 0. Consider the static fields, which will produce an acceleration:

a(0)i = − e

m

[

E(0) + vi ×B(0)]

(60)

where em is a constant.

Now consider the interactions between two particles, i and j:

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a(j)i = − e2

4πε0m

(ri − rj)|ri − rj |3

∼ e2

m→ 0 (61)

In this limit two body effects are small, so we assume that the particles are weakly correlated, i.e.independent of one another.

f (2)(r1, v1, r2, v2) = f (1)(r1, v1)f(1)(r2, v2) (62)

Put this into the 1st term of the BBGKY:

∂f (1)

∂t+ v1 ·

∂f (1)

∂r1+[

a(0) +wa(2)1 · f (1)(r2, v2, t)dr2dv2

]

· ∂f(1)

∂v1

= 0 (63)

where a(0) is the static field term, and the integral is over the self-consistent field term.

Substituting for a(2)1 we introduce the self-consistent electric field:

Eself(r,t) = − e

4πε0

w r − r′|r − r′|3

︸ ︷︷ ︸

f (1)(r′, v′, t)dr′dv′ (64)

where the terms over the brace are equal to a(2).Similarly we may include a self-generated magnetic field arising from the current in the plasma Bself ,

which will add an additional acceleration v ×Bself to a(2).

∂f

∂t+ v · ∂f

∂r+ a · ∂f

∂v= 0 (65)

where we have omitted the (unnecessary) superscript on the distribution function, and

a = − e

m

[

E0 + Eself + v × (B0 +Bself )]

(66)

and Eself and Bself are found by integrating over the local charge density and current density throughoutthe plasma.

This is the Vlasov Equation.The Vlasov equation is collision free, fluid-like long range interactions. The interaction term is

collective, but non-collisional.Empirically the collisions can be added by adding an extra term to the Vlasov equation:

∂f

∂t+ v · ∂f

∂r+ a · ∂f

∂v=∂f

∂t

∣∣∣∣coll

(67)

The collision free Vlasov equation gives wave solutions. The collision term adds damping.

6.3 Debye Length

Length over which individual particle ”graininess” exists. Consider a plasma of temperature Ti for theions and Te for the electrons. The ion charge is Ze and the electron density is n0

e. The plasma will bequasi-neutral: ni ≈ ne

Z . The electrons moving around the ions experience an attractive electro-staticfield. We would therefore expect the density of electrons close to the ions to be greater than the ambientdensity:

ne = n0e exp

[eφ

kTe

]

; ni = n0i exp

[−ZeφkTi

]

(68)

Consider the value around a single ion: charge Ze at r = 0. Assume that the interparticle ion distanceis small compared to the penetration distance of the field, i.e. that electrons are not fixed to particularions. Over long times and large distances we will lose the graininess of the individual particles. Thestatistical distribution of the electrons will give a slowly varying potential, φ, described by Poisson’sequation:

∇2φ = − 1

ε0[Znie− nee] (69)

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Spherically symmetric near an ion:

1

r2d

dr

(

r2dφ

dr

)

= − 1

ε0n0e

[

exp

[−ZeφkTi

]

− exp

[eφ

kTe

]]

(70)

Boundary condition 1: Near the ion: φ = Ze4πε0r

as r → 0Boundary condition 2: Far from the ion field must vanish: φ→ 0 as r →∞.

1

r2d

dr

(

r2dφ

dr

)

=n0ee

2

ε0

1

kTe+

Z

kTi

︸ ︷︷ ︸

φ =φ

λ2D

(71)

Substitute:

φ =1

xf (72)

x =r

λD(73)

to obtaind2f

dx2= f (74)

subject to

f (0) = 1 (75)

f (x)→ 0 as x→∞ (76)

(77)

whose solution isf (x) ≈ exp−x (78)

and thus

φ =Ze

4πε0r2exp

(

− r

λD

)

(79)

where λ2D =

n0ee

2

ε0

[1kTe

+ ZkTi

]

is the Debye length.

We may write this expression in alternative convenient forms:

λ2D =

ε0kTenee2

1 +ZkTekTi

(80)

Divide this into Electron and Ion parts:

λ2e = ε0kTe

nee2

λ2i = ε0kTi

nie2

1

λ2D

=1

λ2e

+1

λ2i

The field falls off rapidly as r & λD, screening the Coulomb field, and reducing the range of the fieldfrom ∞ to ∼ λD.

The electron screening of fields is more general over λD. Imagine a wall at the edge of the plasma.The electron rate to the wall is 1

4nv, where the electrons are lost. This produces an electron-deficientarea adjacent to the wall, which will create a field. This field will retard electrons and accelerate ions,and the area over which this violation of charge neutrality occurs is known as the sheath. It extends adistance λD from the wall.

The Debye length measures the penetration of electric fields into the plasma. However, the Debyelength only well formed if the number of particles in the Debye Sphere is large, or alternatively that theDebye length is much larger than the interparticle separation.

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7 Lecture 7

7.1 Weakly and Strongly Coupled Plasmas

The analysis leading to the Debye shielding concept is only applicable to a weakly coupled plasma wherethe thermal energies are large compared to electro-static energies

The conditions for Debye Length analysis:

1. Dilute Plasma

2. Over a Debye Length the particles should form a statistically averaged continuum. (i.e. n− 1

3

i <<λD).

If the plasma is dense: λD > n− 1

3

i ⇒ we get a strongly correlated plasma. These plasmas are highlystructured. (We can also get ”dusty” plasmas). The strength of correlation is defined via the correlationcoefficient:

n− 1

3

i

λD=

n− 1

3

i√(ε0kTeZ2nie2

)

2

(81)

Γ =Z2n

13

i e2

ε0kTe(82)

If Γ > 1 then the plasma is strongly correlated.

Γ ≈

Z2e2/

ε0n−1/3i

kTe→ Electrostatic potential energy between particles

Thermal Energy per particle(83)

Therefore the plasma is strongly correlated when the electrostatic energy is greater than the thermalenergy. Strongly coupled plasmas are unusual, and very hard to make plasmas where Γ ≈ 1↔ 100 ish.In this case the distributing can no longer be written in terms of a single particle distribution function asthe higher order terms are not small. Calculations of these systems are very difficult, and Monte-Carlomethods must often be used. (Note that an ionic crystal is essentially a strongly coupled plasma.)

7.2 Dielectric Properties of a Plasma

• Unmagnetised, Warm, Dilute

• Long Wavelengths (>> λD)

• Coherent/Cooperative motion → Vlasov Equation

Consider an imposed E.M. wave field with wave vector k and frequency ω.

ℜE0 exp [i(ωt− k · r)] (84)

The response of the plasma to this field is given by:

∂f

∂t+ v · ∂f

∂r+

q

mE · ∂f

∂v= 0

(∂f

∂t

∣∣∣∣coll

(85)

Assume that the applied field is weak: f = f0 + f1 (perturbation) where f0 is the ambient distribution(Maxwellian) and f1 is the perturbation. This gives equations for the electrons and ions. As the ion massis large, the effect on ions is much weaker than that on electrons. We therefore neglect ion motion:

∂f1∂t

+ v · ∂f1∂r− e

mE0 exp i(ωt− k · r) · ∂f0

∂v= 0 (86)

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Writing f1(v, r, t) as a Fourier transform f1(v, k, ω)

f1(v, r, t) = f1(v, k, ω) exp i(ωt− k · r) (87)

i(ω − r · v)f1 =e

mE0 ·

∂f0∂v

(88)

f1 =

emE0 ·

∂f0∂v

i(ω − k · v) (89)

Denominator singular when v‖ (v component ‖ to k) satisfies:

kv‖ = ω (90)

We can treat this in one of three ways to maintain physical reality:

1. An initial value problem: Switch field on at t = 0⇒ Laplace Transforms.

2. Switch on: slowly increase field from t = −∞. Introduce a term exp (γt) into the amplitude.

3. Add in collisions via BGK term with a collision time τ (which can be very large).

All three cases change the denominator of (89) to [i(ω − k · v) + ǫ] where ǫ = γ for case 2, or 1τ for

case 3, but is a very small quantity (→ 0). This explicitly introduces causality, and removes reversibility.

7.3 Charge and Current Density

We calculate the Fourier components of the charge and current density. These are wave terms with thesame frequency and wave number as the applied field.

ρ(r, t) = ρ(k, ω)

j(r, t) = j(k, ω)

exp [i(ωt− k · r)] (91)

ρ− = −ewf1dv ← over all velocity space (92)

= −e2

mlimǫ→0

w E0 ·∂f0∂v dv

[i(ω − k · v) + ǫ](93)

ρ− = −e2

mlimǫ→0

w E0 ·∂f0∂v (−i(ω − k · v) + ǫ)

(ω − k · v)2 + ǫ2dv (94)

= −e2

mlimǫ→0

w E0 ·∂f0∂v [−i(ω − k · v)]

(ω − k · v)2 + ǫ2dv

−e2

mlimǫ→0

w ǫE0 ·∂f0∂v

(ω − k · v)2 + ǫ2dv (95)

= −e2

mP

w E0 ·∂f0∂v

[i(ω − k · v)]dv −e

m2

π

k

wE0

∂f0∂v

dv⊥

∣∣∣∣v‖= ω

k

(96)

For the first integral we introduce the Principal Value of the integral as ǫ→ 0

Pw ∞

−∞

f (x)

xdx = lim

ǫ→0

∞w

−∞

xf (x)

(x2 + ǫ2)dx = lim

δ→0

w −δ

−∞

f (x)

xdx+

w ∞

δ

f (x)

xdx

(97)

provided the function f (x) is continuous at x = 0. For the second integral we note that for (ω − kv‖)not small, the integrand is small, therefore the dominant contribution occurs when kv‖ = ω. Hence

limǫ→0

w E0 ·∂f0∂v

(ω − k · v‖)2 + ǫ2dv‖ = lim

ǫ→0E0 ·

∂f0∂v

∣∣∣∣v‖= ω

k

ǫw ∞

−∞

dv‖(ω − k · v‖)2 + ǫ2

= E0 ·∂f0∂V

∣∣∣∣v‖= ω

k1ǫ

ǫ

π

k(98)

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using the standard integral∞w

−∞

1

a2 + x2dx =

π

a

This yields the Landau Prescription (standard):

w 1

x= P

w 1

x+ iπδ(x) (99)

where δ(x) is the Dirac delta function.

8 Lecture 8

8.1 Landau Prescriptionw

L

1

x= P

w 1

x+ iπδ(x) (100)

1. Oscillating part of the waves: exp [i(ωt− k · r)]. If this is exp [i(k · r − ωt)] the plus in the LandauPrescription becomes a minus.

2. δ(x) term with +i is a direct consequence of causality: expression of time’s arrow.

3. Singularity is when v‖ = ωk , when particle velocity matches the phase velocity - resonance.

8.2 Charge Density

ρ− = −ewf dv = −e

2

mlimΣ→0

w E0 ·∂f0∂v dv

i(ω − k · v) = −e2

m

w

L

E0 ·∂f0∂V dV

i(ω − k · v) (101)

f0 is symmetric in v →. This is an even function of v.∂f0∂v‖

,∂f0∂v1

,∂f0∂v2

are all odd in v‖, v1 and v2 respectively. (v1, v2) = v⊥.

v‖∂f0∂v‖

is even.

Integrating:

w E0 ·∂f0∂V

∣∣∣1dv1

(i(ω − k · v)) = 0 =w E0 ·

∂f0∂V

∣∣∣1dv1

(i(ω − k · v‖))(102)

ρ− = −e2

mE0‖

w

L

∂f0∂v‖

dv‖

(i(ω − k · v‖))(103)

j is the charge × velocity.

j = −e2

m

w v E0 ·∂f0∂v dv

i(ω − k · v‖)(104)

j‖ = −e2

mE0‖

w v‖∂f0∂v‖

dv

i(ω − k · v‖)(105)

˜j1,2 = −e2

mE01,2

w v1,2∂f0∂v1,2

dv

i(ω − kv‖)(106)

The charge density and current densities are not independent, but must satisfy an equation of continuity

∂ρ

∂t+∇ · j = 0 (← Current continuity equation) (107)

i(ωρ− k · j) = −e2

mE0‖

w

*1i(ω − k · v)i(ω − k · v)

70

∂f0∂v‖

dv (108)

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We have two decoupled terms:

• ‖ longitudinal parallel to the field propagation direction, k

• ⊥ transverse, perpendicular parallel to the field propagation direction, k, two components (1,2)

Introduce the dielectric constant: ε(k, ω).From electrostatics:

D = εE = ε0E + P (109)

where P = dipole moment per unit volume.

j =∂P

∂t(Rate of change of polarisation),P is an oscillating term. (110)

= iωP 0 (111)

∂D

∂t= ε0

∂E

∂t+ j (112)

= ε∂E

∂t(113)

iωε = iωε0 +j

E0

(114)

where ω2p = nee

2

ε0m, and f0 =

f0ne

. f0 is the number of electrons per unit volume of phase space, ne is the

electron density, and f0 is the probability of finding the electron per unit volume.

ε‖ε0

= 1 +ω2p

ω2

w v‖∂ f0∂v‖

dv

(1− kv‖ω )

(115)

= 1 +ω2p

ω2

(ω2

k2

) w v‖∂ f0∂v‖

dv

(ωk − v‖)

(116)

ε⊥ε0

= 1 +ω2p

ω2

w v⊥∂ f0∂v⊥

dv

(1− kv‖ω )

(117)

= 1−ω2p

ω2

ω

k

w f0dv

(ωk − v‖)(118)

ε‖ε0

= 1 +ω2p

ωφ1

k

)

(119)

ε⊥ε0

= 1−ω2p

ωφ2

k

)

(120)

φ1(x) = x2w

L

∂ f0∂v‖

dv

(x− v‖)(121)

φ2(x) = xw

L

f0

(x− v‖)dv (122)

φ1, φ2 are components of the plasma dispersion function:

8.3 Plasma Dispersion function

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2 4 6 8 z

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

G

ℜ Gℑ Gℜ G′ℑ G′

0

Figure 1: Plot of the real and imaginary components of the plasma dispersion function G(z)and its derivative G′(z) for real values of the argument z.

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G(z) = π− 12

w

L

e−ζ2

(z − ζ)dζ (123)

I(z) = P

(

π− 12

w e−ζ2

(z − ζ)dζ)

(124)

= −I(−z) (125)

= 2e−z2w z

0et

2

dt (Dawson’s Integral) (126)

G(z) = 2e−z2w z

0et

2

dt+ iπt12 e−z

2

(127)

G′(z) = −2π− 12

w ξe−ξ2

i(z − ξ) = 2(1− zG(z)) (128)

ε‖ε0

= 1 +ω2

p

ω2 z2G′(z)

ε⊥ε0

= 1− ω2p

ω2 zG(z)

z =

ω

k

( m

2kT

) 12

T is the electron temperatureFor small z:

G(z) ≈ iπ 12 e−z

2

+ 2z(1− 2z2

3+

4z4

15− . . . ) (129)

For large z:

G(z) =: very small

iπ12 ηe−z

2

+ z−1(1 +1

2z2+

3

4z4+ . . . ) (130)

where η = 0, 1, or 2 if ℑz < 0, = 0, or > 0

9 Lecture 9

9.1 Plasma Dielectric Constant

There are two. One parallel to the wave:

ε‖ε0

= 1 +ω2p

ω2z2G(z) (131)

and one perpendicular to the wave:

ε⊥ε0

= 1−ω2p

ω2zG(z) (132)

where:

z =ω

k

( m

2kT

) 12

(133)

G′(z) = 2(1− zG(z)) (134)

9.2 Plasma Dispersion Relation

This is the relationship between frequency and wavelength. Written as D(ω, k).Start from Maxwell’s equations:

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1

µ0∇×B = j + ε0

∂E

∂t(135)

∇× E = −∂B∂t

(136)

∇ ·B = 0 (137)

∇ ·E =ρ

ε0(138)

−∇× (∇× E) = µ0

∂j

∂t+ µ0ε0

∂E

∂t(139)

∇2E − ε0µ0∂2E

∂t2= µ0

∂j

∂t+

1

ε0∇p (140)

Add the plane waves:E

jtildeρ

exp [i(ωt− k · r]

(

k2 − ω2

c2

)

· E = − 1

ε0

(1

c2ωj − kρ

)

(141)

9.3 Dispersion Relation for Longitudinal Waves

Take the scalar product k · E:

(

k2 − ω2

c2

)

k · E = − i

ε0

(1

c2ωk · j − k2ρ

)

(142)

Apply equation of continuity of current:

∂ρ

∂t+∇ · j = 0 (143)

i(ωρ− k · j) = 0 (144)

ω

ck · j − k2ρ =

1

ω

[ω2

c2− k2

]

k · j (145)

(

k2 − ω2

c2

)

k · E = − 1

iωε0

(

k2 − ω2

c2

)

k · j (146)

Now introduce the dielectric function:

ε‖E‖ = ε0E‖ +1

iωj‖ (147)

[

k2 − ω2

c2

]

ε‖k · E = 0 (148)

If k2 = ω2

c2 ← this is an electro-magnetic wave in free space. Not the right solution.

If k · E = 0 ← there is no applied field: Also a wrong solution.Finally, if ε‖(k, ω) = 0 ← this is the dispersion relationship for longitudinal waves.

9.4 Dispersion Relation for Transverse waves

Take the vector product with k

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(

k2 − ω2

c2

)

k × E = − i

ε0c2ωk × j =

ω2

c21

iωε0k × j (149)

ε⊥E⊥ = ε0E⊥ +1

iωj⊥ ← Use this to eliminate j term above (150)

[

k2 − ω2

c2ε⊥ε0

]

k × E = 0 (151)

If k × E = 0 there is no applied wave.The dispersion relation for transverse waves is therefore given by:

k2 =ω2

c2ε⊥ε0

(152)

9.5 Transverse Wavesω

k=

c√

ε⊥ε0

(153)

If the phase velocity ωk >> Thermal Velocity

(√2kTm

)

, then:

z =ωk

√2kTm

≫ 1 (154)

and we can use the large z approximation for the plasma dispersion relation.The large z approximation:

G(z) ≈ iπ 12 e−z

2

+ z−1

(

1 +1

2z2+

3

4z4+ . . .

)

(155)

The imaginary part of this approximation is negligible, and there is no damping.

ε⊥ε0

= 1−ω2p

ω2zG(z) (156)

= 1−ω2p

ω2

(

1 +1

2z2+

3

4z4+ . . .

)

(157)

≈ 1−ω2p

ω2(Due to z being very large) (158)

ω

k=

c√(

1− ω2p

ω2

) (159)

> c (160)

The phase velocity greater than the speed of light (ω > ωp). The phase velocity is real, otherwisethe light is strongly attenuated. Therefore light will only propagate if the density is less than the criticaldensity. (ωp = ω).

The group velocity is given by:

dk= c

√(

1−ω2p

ω2

)

(< c) (161)

The information is therefore transferred slower than the speed of light.Transverse waves ≡ electro-magnetic waves in plasmas.Collisional damping does occur, resulting in inverse bremsstrahling or collisional absorption. (In this

analysis the damping term is zero).

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9.6 Longitudinal Waves

ε‖ = 0 (162)

1

2G′(z) + k2λ2

D = 0 (163)

zG(z) = 1 + k2λ2D (164)

Here z is complex, and ω is complex due to the damping:

ω = ωr + iγ︸︷︷︸

(165)

where the indicated term is the damping term.Assume a small kλD (i.e. the wavelength is long compared to the Debye length, λ≫ λD). This gives

that z is large.

G(z) = iπ− 12 e−z

2

+ z−1

(

1 +1

2z2+

3

4z4+ . . .

)

(166)

The imaginary term is small. The asymptotic form of G(z) gives

ω2 = ω2p +

3

2v2θk

2 = ω2p(1 + 3k2λ2

D) (167)

where vθ =√

2kTe

m . This result is the Bohm-Gross frequency. In a cold plasma ω2 = ω2p → cold plasma

wave. 3k2λ2D is the warm plasma correction. Warm plasma changes the resonant frequency, introducing

a wavelength dependence.Landau damping is given by:

γ = ℑ(ω) =(π

8

) 12 ωp

(k3λ3D)

exp

[

− 1

2k2λ2D

− 3

2

]

(168)

This is weak for long wavelengths. We get strong damping when kλD ? 1. No wave is formed for veryshort wavelengths (if kλD ? 1) as it is damped before oscillation.

9.7 Physical Origin of Landau Damping

The imaginary part is due to the resonance where v‖ = ωk (phase velocity). This depends on the gradient

of the density distribution at v‖ = ωk

(

∂f0∂v‖

∣∣∣v‖= ω

k

)

.

The wave tends to accelerate (pull along...) particles with v ‖< ωk , whilst it slows particles with

v‖ >ωk . The slow particles are accelerated, the fast decelerate. However, there are more slow particles

than fast, which leeches energy from the wave, giving wave damping.In the frame of the wave, the field has a well, in which particles can become trapped. The wave

changes phase slightly due to the trapped electrons, which introduces phase incoherence. This alsodamps the wave.

10 Lecture 10: MHD

10.1 Hydrodynamic limit

Consider the system in an averaged sense - averaged over the distribution:

n =wf (v)dv (169)

where n is the number density. This gives the particle numbers per unit density.

nmu =wmvf (v)dv (170)

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where nmu is the momentum per unit volume.

nE =w 1

2mv2f (v)dv (171)

where nE is the energy of a particle. The velocity of a particle has 2 parts. The averaged flow velocity(u), and the random (thermal) velocity (w).

v = u+ w (172)

The thermal energy is given by:

nΣ =w 1

2mw2f (v)dv (173)

Since the mean of w is zero:

nmw =wwf (v)dv = 0 (174)

Therefore energy is given by:

E = ǫ+1

2mu2 (175)

where

ǫ =1

2mw2 =

3

2nkT

if the distribution is isotropic.Use the Boltzmann equation as a starting point to form averages. First define a quantity Q(v):

Q(r, t) =w Q(v)f (r, v, t)dv

n(r, t)(176)

The force on the particles:F = ma = q(E + v ×B)−m∇φ (177)

Where φ is the gravitational potential. Consider the moments of v × B about terms in the Boltzmannequation:

wQ∂f

∂tdv = ∂

∂t

rQf dv =

∂(Qn)

∂t(178)

wQv · ∂f

∂r∂v = ∂

∂r

rQvf (v)dv =

∂r(nQv) (179)

wQa · ∂f

∂vdv =

rQai

∂f∂vidv = −

wf∂

∂vi(Qai)dv (180)

after interaction by parts, noting that boundary terms at v → ∞ : f → 0. Since the only forces onthe particle depending on velocity are those due to the magnetic field, which have no component in thedirection xi depending on vi. Therefore

∂ai∂xi

= 0 (181)

wQa · ∂f

∂vdv = −na · ∂Q

∂v(182)

wQ∂f

∂t

∣∣∣∣coll

dv =∂

∂t(nQ)

∣∣∣∣coll

(183)

since the particle number, momentum and energy are conserved in the collision. Inserting the momentuminto the Boltzmann equation gives:

∂t(nQ) +

∂r(nQv)− na · ∂Q

∂v=

∂t(nQ)

∣∣∣∣coll

(184)

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10.2 Continuity Equation

Q(v) = 1

∂n

∂t+

∂r(nv) = 0 (185)

This excludes ionisation events, and is the equation for both electrons and ions (different number density).

10.3 Momentum Equation

Q(v) = mv (186)

For a single element fluid (e.g. a gas):

∂t(nmv) +

∂rj(nmvivj)−

nFim

∂vj(mvj) = 0 (187)

in Cartesian tensor notation. F is the total force on each particle, m its mass.

vivj = uiuj + wiwj +*0uiwj +*0

wiuj (188)

where nmuiuj is the momentum flux, nmwiwj is the momentum transfer (stress tensor), and the otherterms cancel to zero because the average of w is zero.

nmwiwj = −ψij (189)

where ψi,j is the total stress tensor.If the fluid is assumed to be isotropic: wiwj = 0 where i 6= j. and the only stress is the hydrodynamicpressure, acting inwards:

ψij = −pδij = −1

3nmw2δij (190)

where

δi,j =

1 if (i = j)0 otherwise

is the Kroneker deltaThere is another stress term, associated with viscosity in a velocity gradient, when the fluid is non-

isotropic:

σij = 2µ(εij −1

3εiiδij) + ξεijδij (191)

εij =1

2

(∂ui∂rj

+∂uj∂ri

)

(192)

and hence the total stressψi,j = σi,j − pδi,j (193)

Making use of the equation of continuity we obtain the familiar equation from fluid mechanics

nm∂u

∂t+ nm(u · ∇)u − ∂

∂xiψi,j − nF = 0 (194)

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10.4 Multi-Element Fluid

Momentum is exchanged between particles in a collision:

∂t

[nQ]

coll≡ ∂

∂t[nmv]coll 6= 0 (195)

Introduce an exchange term: P ij = average momentum exchange from particle i to particle j.(sec/unit volume). From particle i:

nimi

[∂ui∂t

+

(

ui ·∂

∂r

)

ui

]

>+ ∂∂rpi

∂rl[ψil]− niF i = P ij (196)

In plasma viscosity is generally negligible:

∂rl⇒ ∂

∂r(pi) = ∇pi (197)

Replacing the external force by that due to the electric and magnetic fields

nimi

[∂ui∂t

+ (ui · ∇)ui

]

+∇pi = zeni[E + ui ×B] + Pil (198)

neme

[∂ue∂t

+ (ue · ∇)ud

]

+∇pd = −ene[E + vi ×B] + Pil (199)

where the first equation is for ions, and the second is for electrons. The viscous terms and gravity areneglected.

10.5 Bulk Fluid Equation

ne ≈ zni → quasi-neutrality → Debye length is short.Equation of continuity

∂t(nimi) +

∂t(neme) ≈ ∂ρ

∂t +∇ · (ρu) = 0 (200)

u =(miui+meue)

mi+me← Mean flow velocity (201)

ρ = nimi + neme ← Density (202)

Momentum Equation

ρ

[∂u

∂t+ (u · ∇)u

]

= −∇p+︷ ︸︸ ︷

qE + j ×B (203)

is Euler’s Equation, with added E.M. field force.

q = zeni − ene ≈ 0← Charge Density (204)

j = zeniui − eneui ← Current Density (205)

me ≪ mi (206)

u ≈ ui (207)

If the plasma is quasi-neutral:

∇ · ui ≈ ∇ · ue (208)

and ui and ue differ only by a rotational vector, i.e. the difference in velocity vector forms closed loops inthe plasma. Since the bulk velocity is approximately equal to the ion velocity is small, these are currentloops of electrons independent of bulk motion, governed by Ohms Law:

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me ≪ mi (209)

me

nee2

[

zeni∂ui∂t− ene

∂ue∂t

]

=me

nee2∂j

∂t(210)

= E + u×B −j ×Bnee

+∇penee

+Piene≈ 0 (211)

ue = u−j

nee(212)

ue ≫ ui (213)

The∂j

∂t term is usually negligible. The generalised field

E′ = E + u×B −j ×Bnee

+∇penee

(214)

In an isotropic (non-magnetised plasma), the momentum exchange

P ienee

∝ (ui − ue) (215)

∝ j (216)

= −j

σ(where σ is the conductivity) (217)

= E′ (eqn.213) (218)

11 Lecture 11

11.1 Ohm’s Law

This derives from the current equation. The current is carried by electrons, as they are lighter than theions, and is caused by the difference between the electron velocity and mean speed:

j = ene(u− ue) (219)

me

nee2

[

zeeni∂ni∂t− eue

∂ue∂t

]

=me

nee2 ≈ 0(almost always)

∂j

∂t(220)

=

effective electrical field (with electron pressure)︷ ︸︸ ︷

E︸︷︷︸

E-Field

+ u×B︸ ︷︷ ︸

Lorentz force (induced)

−j ×Bnee︸ ︷︷ ︸

Hall term

+P ienee

+∇penee︸︷︷︸

electron pressure

(221)

Ohm’s law:

E =j

σ(222)

where σ is the conductivity (scalar in an unmagnetised plasma, second order tensor when magnetised).We expect P ie ∝ j (current density), leading to, assuming that the medium is isotropic, Ohm’s law:

1

σ= −Pie

nee(223)

This is negative because we are dealing with negative charges.

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Thermo-electricity, the Thompson effect, is current driven by a temperature gradient.

P ienee

= −j

σ+α

σ∇T (224)

where α is the thermo-electric coefficient. In a magnetic field σ, α are both tensors with three independentcomponents. (Again, assume a non-isotropic medium in the presence of a magnetic field):

σ =

σ‖ 0 00 σ⊥ σ∧0 −σ∧ σ⊥

whereσ‖ = electrical field along magnetic fieldσ⊥ = electrical field perpendicular to the magnetic fieldσ∧ = cross-product component arising when E and B are normal to each other

11.2 Energy Equations

Set Q = 12mv

2, and ions, electrons and E.M. field, we get the force balance equation:

∂U

∂t+∇ ·W = 0 (225)

Where U is the energy density, and w is the energy flux:

U =3

2nikTi +

3

2nekTe +

1

2ρu2 +

ε0E2

2+B2

2µ0(226)

W =

[3

2nikTi + pi

]

︸ ︷︷ ︸

Enthalpy of the ions

ui +

[3

2nekTe + pe

]

︸ ︷︷ ︸

Enthalpy of the electrons

ue +

[1

2miniu

2i

]

︸ ︷︷ ︸

ion K.E. density

ui +

[1

2meneu

2e

]

︸ ︷︷ ︸

electron K.E. density

ue (227)

+1

µ0E ×B

︸ ︷︷ ︸

Poynting vector

− σijvj︸ ︷︷ ︸

viscous work done

+ q︸︷︷︸

thermal flux

(228)

Where q is the energy flux due to thermal transport, and is the heat transfer term in the above equation.

q = −κ∇T − βE′ (229)

Where E′ is the net effective electric field. In a magnetised, non-isotropic plasma κ and β are tensors,where

β = αTe +5

2

kTeeσ (230)

where α is the same as that in equation(224) as required by the Onsager relations (reciprocal kineticprocess).

11.3 Transport Coefficients

For a small perturbation, the flux is proportional to the appropriate driving force: Conductivity, ThermalConductivity, Thermo-electric Coefficient, Viscosity1. These are all collisional terms, and, if there aremagnetic fields as well, these are also all tensors.

If you apply a force to a particle distribution, the distribution will be distorted away from theMaxwellian. Provided the force is weak, the distortion will be small (perturbation), and the flux will belinearly proportional to the force. If the force is strong, this linearity will break down.

Distribution function can be written with a perturbation f1 on a steady state Maxwellian f0

1Only associated with ions, usually neglected

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f = f0 + f1 (231)

Applied to electrons for σ, κ, α, and to ions for η. If we assume that f1 is small, the conditions on fare given by:

0 =∂f0∂t

∣∣∣∣c

(232)

This gives a Maxwellian for f0.

∂f0∂t

+ v · ∂f0∂r

+ a · ∂f0∂v

︸ ︷︷ ︸

Force

= C(f0|f1))︸ ︷︷ ︸

Collision term

(233)

where C(f0|f1) is the first order perturbation term introduced by f1. Hence knowing that (f)0 isMaxwellian yields the l.h.s. from which the perturbed distribution function f1 can be calculated. Thesolution of f1 gives the perturbed distribution, and f1 gives you the flux.

11.4 Hydro-Magnetic Approximation

We assume that the plasma has infinite conductivity, σ → ∞ (or, more correctly, that the magneticReynolds number Rm(= σLu) → ∞, where L is the characteristic length, and u is the characteristicvelocity). Since j is finite, this gives E′ = 0. We assume that there is no Hall term, and no electronpressure term ⇒ E + u×B = 0. This leads to Faradays Law (by taking the curl) for the flux Φ :

∂B

∂t= ∇× (u×B) (234)

We now consider through an arbitrary loop S which is fixed in the fluid, i.e. made up of the samefluid particles for all time:

Φ =z

AB · dA (235)

where A is the area vector, which changes in time as the fluid moves. The area changes due to themotion:

dA

dt=

z

Au× dl (236)

where the integral is around the loopHence

dt=

z

A

∂B

∂t︸︷︷︸

©1·dA+

zB · u× dl︸ ︷︷ ︸

©2(237)

The first term©1 is due to time variation in the magnetic field, the second©2 the result of flux sweptout by the motion.

Thus changing the order of terms in the scalar triple product and using Stokes Theorem:

dt= 0 (238)

and the flux remains constant through the loop.Consider a tube of force with walls parallel to B. Put a loop into a wall of the tube. Φ is therefore 0

since the normal component of B is zero . This remains constant (= 0) as the plasma moves, as there isno B through the loop. The loop is arbitrary, moves with the fluid along a stream line, and

uB ·dA = 0

for all such loops, so the fluid, which ties the loop, also follows a flux line in addition to the stream lines.It may also be said that the flux lines are tied to the motion, the ”frozen in” condition.

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∇× (u×B) = (B ×∇)u− (u · ∇)B +:0u∇ · B −B∇ · u (239)

∂ρ

∂t+ (u · ∇)ρ = −ρ∇ · u (240)

d

dt

[B

ρ

]

=d

dt

[B

ρ

]

+ u · ∇[B

ρ

]

(241)

=

(B

ρ· ∇)

u (242)

Consider the movement of an element ∆l fixed in the field:

d

dt[∆l] =

∂t[∆l] + u · ∇[∆l] = (∆l · ∇)u (243)

If ∆l initially lies on a field line, then it will continue to do so.

B ∝ ρ∆l (244)

12 Lecture 12: Ideal MHD Approximation

12.1 Pressure Balance

If the plasma is at rest, the acceleration term can be rewritten as:

ρdu

dt= 0 (245)

= −∇p+ j ×B (246)

⇒ ∇p = j ×B (247)

If we use Maxwell’s laws:

j =1

µ0(∇×B) (248)

⇒ ∇p = j ×B (249)

=1

µ0(∇×B)×B (250)

= − 1

µ0

1

2∇(B2)︸ ︷︷ ︸

Mag. Pressure Term

− (B · ∇)B︸ ︷︷ ︸

Longitudinal Stress

(251)

where the longitudinal stress acts along the field lines.For many applications (B · ∇)B = 0, as we are only interested in the ability of the magnetic field to

constrain the plasma. The pressure balance equation is therefore:

p+1

2µ0B2 = Constant (252)

The plasma β, which, for magnetic confinement devices, relates to the stability etc, is defined as:

β =Pmax(B2max2µ0

) (253)

For all confined plasmas, β < 1, and should be significantly less than one.

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In the the adiabatic limit when: σ →∞, and q ≈ 0, i.e no dissipation or entropy increase the energyequation reduces to the adiabatic gas law:

3

2

dp

dt+

5

2

p

ρ

dt= 0 (254)

whered

dt=

∂t+ ucdot∇ (255)

is the time derivative moving with the plasma.

Hence(pρ

) 53

= constant

12.2 Incompressible Approximation

The equation of motion:

ρ

∂u

∂t+ (∇ · u)× u+

1

2∇(u2)

︸ ︷︷ ︸

(u·∇)u

= −∇(

p+1

2µ0B2

)

+1

µ0(B · ∇)B (256)

and the vorticity ω = ∇× v. If the last term is zero, i.e. j × B can be written as a potential, the flowmay be described in terms of the usual fluid mechanics relations.

12.3 Bernoulli’s Equation - Steady Flow

For a fluid:

1

2u2 +

p

ρ= Constant along a streamline (257)

If we take the scalar product of u with the equation of motion:

ρu · ∇[p

ρ+

B2

2µ0ρ+

1

2u2

]

=1

µ0u · (B · ∇)B︸ ︷︷ ︸

Mag. Stress Term

(258)

where the magnetic stress term must be much less than 1 for Bernoulli to apply. If we integrate along astreamline, we find that:

p

ρ+B2

2µ0+

1

2u2 = Constant (259)

We now re-write the stress term in terms of a component along the field lines, and another normalto the field lines. Let b be the unit vector in the field direction, and u be the unit vector normal to thefield line:

1

µ0u · (B · ∇)B =

1

µ0

u · uB2

R︸ ︷︷ ︸

Normal to the Field line

+ (u · b)(b · ∇(1

2B2)

︸ ︷︷ ︸

Along the field line

(260)

Where R = radius of curvature of the field line.

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12.4 Kelvin’s Theorem

The circulation around a loop fixed in the field is constant, and given by:

Γ =zu · dl =

z∇× u︸ ︷︷ ︸

Vorticity

·ds (261)

We now take the curl of out equation of motion (∇×∇φ = 0):

ρ∇×[∂u

∂t+ ω × u

]

=1

µ0∇× [(B · ∇)B] (262)

If we integrate the left hand side over the loop:

d

dt

zω · ds = 0→ recover Kelvin’s theorem (263)

But the right hand side gives a non-zero contribution (Kelvin’s theorem fails in the plasma). Theplasma generates vorticity. In contrast to a fluid, a magnetised plasma can support shear, i.e. the fieldimposes a degree of ‘stiffness’ to the plasma. In consequence although fluids only support longitudinalwaves (only sound waves, no transverse waves), a plasma allows transverse waves (via vorticity) whichhave vorticity associated with them, in addition to the longitudinal waves.

Transverse:∂Bx∂z6= 0 → Non-zero curl = Vorticity (264)

Longitudinal:∂ux∂z

= 0 : ux = 0 (265)

Solids also support both longitudinal and transverse waves (e.g. earthquakes).

12.5 Alfven Waves

The simplest possible case of transverse waves: uniform incompressible infinite conductivity plasma,moving at a constant velocity, u, in a uniform field,B. We move to the rest frame of the plasma. Theplasma is perturbed by small changes in velocity u′ and field B′. The linearised equations are:

ρ∂u′

∂t= −∇p′ + 1

µ0(∇×B′)×B (266)

Adding Maxwell gives:

∂B′

∂t= −∇× E = (B′ · ∇)u′ (267)

where E = u×B′.We now differentiate (266) with respect to time, and substitute for (267):

ρ∂2u′

∂t2= −∇

(∂p′

∂t

)

+1

µ0∇× [(B · ∇)u′] ×B (268)

Now take the curl:

ρ∂2ω′

∂t2=

1

µ0(B · ∇)[(B · ∇)ω′] (269)

N.B. B is a constant.Introduce a plane wave:

ω′ = ω0 exp [i(Ωt− k · r)] (270)

where ω is the vorticity, and Ω is the angular frequency.The phase velocity is given by:

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V =Ω

k= CA cos θ (271)

Where CA is the Alfven velocity, and θ is the angle between k and B, the angle of propagation withrespect to the field lines.

CA =B√µ0ρ

(272)

String tension: B2

µ0, String mass: ρ.

∂2ω′

∂t2=

1

µ0ρ(B · ∇)[(B · ∇)ω′] = Vorticity (273)

The only direction in which ω′ varies is along B. Since ω = ∇× u, this implies that particle velocity u′

is perpendicular to the magnetic field superimposed on the steady background motion.For transverse waves: ∇ · u′ = 0, as these are incompressible waves anyway, and there is no need to

make the incompressible approximation.

13 Lecture 13: Magnetosonic Waves

An incompressible plasma can only support transverse waves (Alfven waves). Plasmas are, however,compressible. This adds longitudinal (sonic) waves. We expect that the new waves will be mixedlongitudinal and transverse. Consider a perturbation on an ambient condition assuming that the plasmais initially at rest:

• Density: ρ0 + ρ (ρ is the perturbation, ρ0 is the ambient condition

• Pressure: p0 + p

• Magnetic field: B0 +B

• Velocity: v (v0 = 0 after changing the frame to the plasma rest frame)

• Displacement: ξ, where v =dξ

dt .

Linearising the M.H.D. equations, and assuming adiabatic flow:

∂ρ

∂t+ ρ0∇ · v = 0 (274)

ρ0 ·∂v

∂t= −∇p+

1

µ0(∇×B)×B0 (275)

∂p

∂t= γ

p0

ρ0

∂ρ0

∂t(276)

E + v ×B = 0 (277)

∂B

∂t= −∇× E (278)

v =dξ

dt=∂ξ

∂tThe other term is neglected here) (279)

Here, (274) becomes:

ρ+ ρ0∇ · ξ = 0 (280)

If we combine (276) and (280) we get:

p = γp0

ρ0ρ = −γp0∇ · ξ (281)

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Combining (277), (278) and (279) gives:

∂B

∂t= ρ0

(∂ξ

∂t×B0

)

(282)

⇒ B = ∇× (ξ ×B0) (283)

∂2ξ

∂t2− γ p0

ρ0∇(∇ · ξ) − 1

µ0ρ0(∇×B)×B0 = 0 (284)

Let:

(∇×B)×B0 =∇[B2

0∇ · ξ − (B0 · ∇)(B0 · ξ)]+ (B0 · ∇2)ξ −B0

[(B0 · ∇)(∇ · ξ)

](285)

Therefore:

ρ0

∂2ξ

∂t2+ γp0∇(∇ · ξ)− 1

µ0∇[B2

0∇ · ξ − (B0 · ∇)(B0 · ξ]− 1

µ0(B0 · ∇)2ξ +

1

µ0B0(B0 · ∇)(∇ · ξ) = 0(286)

We now introduce the Alfven speed:

C2A =

B20

µ0ρ0(287)

The sound speed:

C2s =

γp0

ρ0(288)

And the ratio:

α =C2s

C2A

=(γ

2

)

β (289)

where β = p0“

B2

2µ0

This gives:

1

C2A

∂2ξ

∂t2−∇

[(1 + α)∇ · ξ − (b · ∇)(b · ξ)

]− (b · ∇)2ξ + (b · ∇)(∇ · ξ)b = 0 (290)

where b =B0

|B0|, and is the unit vector in the B-direction.

N.B.∂2ξ

∂t2 is not parallel to ξ, so the wave is not longitudinal. Indeed, it must include an Alfven wave,which is transverse.

We now introduce a plane wave: ξ = ξ0exp [i(ωt− k · r)]. The equation will contain 3 vector com-

ponents. We could go to the three Cartesian coordinates are obtain equations of the form:

V 2

C2A

ξi +∑

j

Aijξj = 0 (291)

where V is the phase velocity, and Aij(b, k, α) is what we need to solve for the wave.Forming the components of the wave in three directions: k, (⊥ to k in plane (k, b)), and (⊥ to k and

b).

• Form k · ξ: gives components parallel to k.

• Form b · ξ: gives components parallel to b.

• Form (b · k) · ξ: gives components perpendicular to both b and k.

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1st:

k · ξ ⇒[V 2

C2A

− (1 + α)

]

k · ξ + (b · k)(b · ξ) = 0 (292)

2nd:

b · ξ ⇒ V 2

C2A

b · ξ − α(b · k)(k · ξ) = 0 (293)

3rd:

(b · k) · ξ ⇒[V 2

C2A

− (b · k)2]

b · k × ξ = 0 (294)

These three equations must be satisfied by any general wave. If we have a wave with ξ parallel tob× k, i.e. perpendicular to both b and k, we get:

[V

CA

]

= ±b · k (295)

The phase velocity V = CA cos θ where (θ) is the angle between b and k. These are the Alfven waves.They are characterised by having displacement, ξ perpendicular to b and k. Two coupled equationswhere ξ is in the plane of k and b:

(V

CA

)4

− (1 + α)

(V

CA

)2

+ α cos2 θ = 0 (296)

Obtaining consistence for the equations for (k · ξ) and (b · ξ). This is a quadratic equation for V 2. Thereare two distinct waves, each of which can move forwards or backwards (fast or slow waves).

V 2± =

1

2

(C2A + C2

s )±√

(C2A + C2

s )− 4C2AC

2s cos2 θ

︸ ︷︷ ︸

Two waves, neither all longitudinal nor all transverse

(297)

If ± = + this is a fast wave, if ± = − this is a slow wave, with the condition that k, b, ξ all couple.

14 Lecture 14: Magneto-Hydrodynamic Waves

1

C2A

∂2ξ

∂t2−∇[(1 + α)∇ · ξ − (b · ∇)(b · ξ)]− (b · ∇)2ξ + (b · ∇)(∇ · ξ)b = 0 (298)

Introduce a plasma wave:ξ = ξ0exp

[i 2πλ (V t− k · r)

]where v is the phase velocity, and k is the unit

vector in the wave direction.Form the terms k · ξ, b · ξ, (b · k) · ξ).

[V 2

C2A

− (1 + α)

]

k · ξ + (b · k)(b · ξ) = 0 (299)

V 2

C2A

b · ξ − α(b · k)(k · ξ) = 0 (300)

[V 62

C2A

− (b · k)2]

b · k × ξ = 0 (301)

Case 1: ξ perpendicular to b and k:

k · ξ = b · ξ = 0 (302)

b · k × ξ 6= 0 (303)

V 2 = (b · k)2C2A (304)

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Let θ be the angle between b and k.

V 2 = C2A cos2 θ ← Alfven waves (305)

Case 2: ξ lies in the plane of b and k.

b · k × ξ = 0 (306)

The remaining two equations are coupled.There are two waves, since there are two solutions. We require consistency between the two equations.

[V 2

C2A

− (1 + α)

]V 2

C2A

+ α(b · k)2 = 0 (307)

V 4

C4A

− (1 + α)V 2

C2A

+ α cos2 θ = 0 (308)

(309)

Solutions:

V 2± =

1

2

(C2A + C2

s )±√

[(C2A + C2

s )− 4C2AC

2s cos2 θ]

(310)

Two waves, fast (+) and slow (−). These have both longitudinal and transverse components.If θ = 0→ k is parallel to b.

V+ =1

2

C2A + C2

s + |C2A − C2

s | = Larger of CA or Cs (311)

V− =1

2

C2A + C2

s − |C2A − C2

s | = Smaller of CA or Cs (312)

If θ = π2 → b is perpendicular to k.

V+ =√

(C2A + C2

s ) (313)

V− = 0 (314)

This is a purely longitudinal wave, with no second wave. The magnetic pressure adds to the kineticpressure, like a sound wave.

The longitudinal and transverse component (to the magnetic field) are given by: k‖ξ‖ for longitudinal,and k⊥ξ⊥ for transverse.

k⊥ · ξ⊥ = −

[V 2

C2A

− α]

[V 2

C2A

− (1 + α)]k‖ · ξ‖ (315)

k⊥ · ξ⊥k‖ · ξ‖

=(1 + α)±

[(1 + α)2 − 4α cos2 θ]

(1 + α)±√

[(1 + α)2 − 4α cos2 θ](316)

This will be greater than zero for a fast wave, and less than zero for a slow wave.The major difference between the waves is that in fast waves the longitudinal and transverse parts

are in phase, but in slow waves they are out of phase. Given an arbitrary initial condition, displacementand velocity, the relative phases fix the initial fast and slow components.

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14.1 M.H.D. Shocks

In fluid dynamics a large amplitude sound wave goes into a shock:

c2 =γp

ρ(317)

p ∝ ρ(1γ ) (318)

⇒ c2 ≈ ρ(1−γ)

γ (319)

Thus as the density increases, the sound speed increases and waves from the back build up into asharply rising front - shock wave.

Large amplitude magneto-sonic waves can develop into shock waves. We consider shocks with normalin the x direction. The complexity of the shock wave is much increased due to the magnetic field. i.e.the wave is not only longitudinal, which means that the tangential symmetry of the the gas dynamicsis lost. We get flows along the surface of the shock (y,z directions). Let us look at the jump conditionsin the rest frame of the shock, remembering that on either side of the shock front we have p1, ρ1, B1, u1

and p2, ρ2, B2, u2, where u is the flow velocity, u = (u, v, w).Conservation of mass

ρ1u1 = ρ2u2 (A)

Conservation of momentum normal to the shock front

p1 − ρ1u21 +

B2y1 +B2

z1

2µ0= p2 − ρ2u

22 +

B2y2

2µ0+B2z2

2µ0(B)

Conservation of momentum transverse to the shock

ρ1u1v1 −Bx1By1µ0

= ρ2u2v2 −Bx2By2µ0

y-direction

ρ1u1w1 −Bx1Bz1µ0

= ρ2u2w2 −Bx2Bz2µ0

z-direction

(C)

Energy:

ρ1u1

[

h1 +1

2(u2

1 + v21 + w2

1)

]

+E1 ×B1

µ0

∣∣∣∣x

= ρ2u2

[

h2 +1

2(u2

2 + v22 + w2

2)

]

+E2 ×B2

µ0

∣∣∣∣x

(D)

where h = ǫ+ pρ = enthalpy per unit mass, or specific enthalpy.

The transverse electric and normal magnetic fields are both continuous across the shock:

Et1 = Et2 and Bx1 = Bx2 (E)

We can simplify these equations. The velocity components may be set by an appropriate transforma-tion to lie in the (x, y) plane only. A further transformation in y allows either the transverse momentumflux or the transverse electric field to be zeroed (normally the transverse electric field). We can assumeinfinite conductivity - Ideal M.H.D.- so that E+ v×B → 0. The transverse field equations now become:

u1By1 − v1Bx1 = u2By2 − v2Bx2u1Bz1 − w1Bx1 = u2Bz2 − w2Bx2

(F)

As a result the energy equation (E) becomes :

ρ1u1

[

h1 +1

2(u2

1 + v21 +w

21)

]

= ρ2u2

[

h2 +1

2(u2

2 + v22 +w

22)

]

+u1

µ0

(

B2y1 +

B2z1

)

+u2

µ0

(

B2y2 +

B2z2

)

− v1Bx1By1µ0

−w1

µ0Bx1Bz1 −v2

Bx2By2µ0

−w2

µ0Bx2Bz2

(D’)

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following the frame transformations.Example 1: Flow normal to shock on Entry:

v1 = 0 (320)

By1 = 0 (321)

Bx1 = Bx2 = Bx (322)

⇒ By2 = µ0 ρ2 u2v2Bx

via momentum (323)

=v2Bxu2

via transverse electric field (324)

Either v2 = By2 = 0, a normal gas dynamic shock, or u22 =

B2x2

µ0ρ2⇒ By2 6= v2 6= 0, switch on shock.

15 Lecture 15: M.H.D. Shocks

ρ1u1 = ρ2u2 = j (325)

p1ρ1u21 +

B2y1

2µ0+B2z1

2µ0= p2 + ρ2u

22 +

B2y2

2µ0+B2z2

2µ0(326)

ρ1u1v1 −Bx1By1µ0

= ρ2u2v2 −Bx2By2µ0

(327)

ρ1u1w1 −Bx1Bz1µ0

= ρ2u2w2 −Bx2Bz2µ0

(328)

ρ1u1

(

h1 +1

2(u2

1 + v21 + w2

1)

)

+E1 ×B1

µ0

∣∣∣∣x

= ρ2u2

(

h2 +1

2(u2

2 + v22 + w2

2)

)

+E2 ×B2

µ0

∣∣∣∣x

(329)

Due to the assumption that the plasma has infinite conductivity:

Et1 = Et2 (330)

Bx1 = Bx2 (331)

E + v ×B = 0 (332)

⇒ u1By1 − v1Bx1 = u2By2 − v2Bx2 (333)

⇒ u1Bz1 − w1Bx1 = u2Bz2 − w2Bx2 (334)

If v1 = 0, By1 = 0 There is no upstream variation in either flow or field.

By2 =µ0ρ2u2v2

Bx(335)

=v2Bxu2

(336)

⇒ u22 =

B2x

µoρ2(337)

or v2 = By2 = 0 (338)

Where (337) and (338) represent two possible solutions:v2 = By2 = 0→ No transverse field, same as an ordinary gas dynamic shock.u2 = Bx√

µ0ρ2→ Transverse fields, switched-on shock.

In practice, the solution is determined by the stability and evolution of the shock. 1st get the lesslikely shock, and then get the second shock. The shock involving the largest entropy change dominatesthe interaction (by experiment). For weak shocks, this is the ordinary gas dynamics shock. Past a criticalpoint, increased strength = switched-on shock.

The three types of sonic waves, Alfven, Fast and Slow, can all produce shocks, and all have their owntype of shock. This produces very complex situations.

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15.0.1 Alfven Wave

Consider v, w ∼ By, Bz: therefore v/By/v = w/Bz = α. Assume that

B2y1 +B2

z1 = B2y2 +B2

z2

and thereforev2y1 + v2

z1 = v2y2 + v2

z2

Defining the change across the shock by ∆, thus

∆By = By2 −By1

Hence we obtain from the jump condition (C)

By ∆By = −Bz ∆Bz

∆v =1

µojBx∆By

∆w =1

µ0jBx ∆Bz

Making use of the initial assumptions and conditions (F) we obtain

u1By1 − v1Bx = κ1By1 = u2By2 − v2Bx = κ2By2

u1Bz1 − w1Bx = κ1Bz1 = u2Bz2 − w2Bx = κ2Bz2

where

j = ρ1 u1 = ρ2 u2 ; κ1,2 = u1,2 − α andv

w=ByBz

Thus the field terms in (D’) may be equated to

∆[u(B2

x +B2y)− v(BxBy)− w(BxBz)

]= 0

It is easy to show that there exist solutions where the flow variables normal to the shock remainunchanged, i.e. the disturbance is transverse along the shock,

κ1 = κ2

u1 = u2

ρ1 = ρ2

p1 = p2

Hence the wave speed is given by

u∆By = Bx ∆v =B2x

µoρu∆By (339)

The wave speed,√

B2x

µ0ρ, is the Alfven speed. This results in a discontinuous jump wave moving at

the Alfven speed, a finite amplitude Alfven wave. Thus far, we have not included collisions, and areconsidering only collision free shocks. Where does the entropy change originate? Microscopic turbulence.

15.1 Boltzmann’s Equation as a Fokker-Planck (Landau’s Equation)

Boltzmann’s equation is a binary collision equation. The Fokker-Planck equation is a many body equa-tion. Our route here will be to start with weak collisions and small impulses, and add velocity slowly andlinearly. A large number of small binaries is a many body collision. We use Boltzmann, but with weakcollisions only, as there are many more weak collisions than strong collisions. We assume two identical

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particles, here electron-electron scattering, with velocity v1, v2, relative velocity u = v2 − v1, and centre

of mass velocity V =v1+v2

2 . The cross-section for coulomb scattering through θ is:

σ =

(e2

2mr

)2 [

u sin

2

)]−4

(340)

where mr = 12m - electron reduced mass.

The Boltzmann collision integral:

I =

(e2

m

)2 wdv2

wsin θdθdφ

u[u sin

(θ2

)]4

f

(

v1 −1

2∆u

)

f

(

v2 +1

2∆u

)

− f (v1)f (v2)

(341)

uf = OuO is a rotation (342)

∆u = uf − u (343)

|uf | = |u| (344)

v1 ⇒(

v1 −1

2∆u

)

(345)

v2 ⇒(

v2 −1

2∆u

)

(346)

Unit vector l is parallel to u. Normal to l are orthogonal unit vectors l are m.n.

∆u = 2u sin

2

)

−l sin(θ

2

)

+m cos

2

)

cos (φ) + n sin

2

)

sin (φ)

(347)

since magnitude of u is unchanged magnitude on scatter (elastic collision).If the scatter is weak, ∆u is small, we can expand terms via the Taylor series:

f

(

v1 −1

2∆u

)

f

(

v2 +1

2∆u

)

− f (v1)f (v2)

= f (v1)f (v2)−

f (v1)f (v2) +

[

f (v1)∂f

∂v

∣∣∣∣v2

+ f (v2)∂f

∂v

∣∣∣∣v1

]

1

2∆u

+1

2

[

f (v1)∂2f

∂vi∂vj

∣∣∣∣v2

− 2∂f

∂vi

∣∣∣∣v1

∂f

∂vj

∣∣∣∣v2

+ f (v2)∂2f

∂vi∂vj

∣∣∣∣v2

]

1

2∆ui

1

2∆uj (348)

Integration over the azimuthal angle φ eliminates terms in cosφ, sinφ, cosφ sinφ since

w 2π

0dφ cosφ =

w 2π

0dφ sinφ =

w 2π

0dφ sinφ cosφ = 0

Hence substituting

I = 8π

(e2

m

)2 wdv2

w π

0d

2

)

cos

2

) −uiu3 sin

(θ2

)

[

f (v1)∂f

∂vi

∣∣∣∣v2

− f (v2)∂f

∂vi

∣∣∣∣v1

+

+1

2

[

f (v1)∂2f

∂vi∂vj

∣∣∣∣v2

− 2∂f

∂vi

∣∣∣∣v1

∂f

∂vj

∣∣∣∣v2

+ f (v2)∂2f

∂vi∂vj

∣∣∣∣v2

]

×

× 1

u

(

u2u1

u2sin

2

)

+1

2(mimj + ninj)

(

1

sin(θ2

) − sin

2

)))]

(349)

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Page 41: Advanced Plasma

Butr π0

cos ( θ2 )d(

θ2 )

sin ( θ2 )

→ diverges at small and large θ. Physically this is due to the neglect of correlations

at small θ ≈ 0 (Debye shielding). As before we cut-off at the integral for small impact parameters at theDebye length corresponding to

θmin ≈λDbmin

(350)

Hence

ln

sin(θmax

2

)

sin(θmin

2

)

= ln Λ (351)

ln Λ is large.

16 Lecture 16

I = 8π

(e2

m

)2 wdv2

w π

0d

2

)

cos

2

) −uiu3 sin

(θ2

)

[

f (v1)∂f

∂vi

∣∣∣∣v2

− f (v2)∂f

∂vi

∣∣∣∣v1

+

+1

2

[

f (v1)∂2f

∂vi∂vj

∣∣∣∣v2

− 2∂f

∂vi

∣∣∣∣v1

∂f

∂vj

∣∣∣∣v2

+ f (v2)∂2f

∂vi∂vj

∣∣∣∣v2

]

×

× 1

u

(

u2u1

u2sin

2

)

+1

2(mimj + ninj)

(

1

sin(θ2

) − sin

2

)))]

(352)

w π

0

cos(θ2

)d(θ2

)

sin(θ2

) → ln

sin(θmax

2

)

sin(θmin

2

)

= ln Λ Spitzer/Coulomb Log. (353)

w π

0d

2

)

cos

2

)

sin

2

)

=1

2(354)

Assume ln Λ is large.mimj + ninj is the projection operator. It projects a vector, a, on to the plane perpendicular to g.

[mimj + ninj ]a = mi(m · a) + ni(n · a)⇒ Component of a lying in m, n plane. (355)

= a− l(l · a) (356)

In tensor notation (357)

mimj + ninj = δij − lilj N.B. l is parallel to u. (358)

mimj + ninj = δij −uiuju2

(359)

New Tensor: (360)

ωij =1

u

(

δij −uiuju2

)

(361)

∂ωij∂ui

=−2uiu3

N.B. Repeated index = summation (362)

I = 2π

(e2

m

)2

ln Λwdv2

2

[

f (v1)∂f

∂vi

∣∣∣∣v2

− f (v2)∂f

∂vi

∣∣∣∣v1

]

∂ωij∂ui

+∂

∂vj

∣∣∣∣v1

[

f (v2)∂f

∂vi

∣∣∣∣v1

− f (v1)∂f

∂vi

∣∣∣∣v2

]ωij

+

[

f (v1)∂2f

∂vi∂vj

∣∣∣∣v2

− ∂f

∂vi

∣∣∣∣v1

∂f

∂vj

∣∣∣∣v2

]

ωij

(363)

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Noting∂ωij∂ui

=−∂ωij∂vij

=∂ωij∂v2j

and integrating by parts as necessary, we obtain Landau’s equation:

I = 2π

(e2

m

)2

ln Λ∂

∂vij

wdv2

[

f (v2)∂f

∂vi

∣∣∣∣v1

− f (v1)∂f

∂vi

∣∣∣∣v2

]

ωij (364)

Landau’s Equation

∂vj

wdv2

[

f (v2)∂f

∂vi

∣∣∣∣v1

− f (v1)∂f

∂vi

∣∣∣∣v2

]

ωij

= − ∂

∂vj

f (v1)

[wdv2

∂f

∂vi

∣∣∣∣v2

]

ωij

+∂2

∂vj∂vi

f (v1)[w

dv2f (v2)ωij

]

− ∂

∂vij

f (v1)wdv2f (v2)

∂ωij

∂vij

= −2∂

∂vij

f (vi)

wdv2

∂f

∂vi

∣∣∣∣v2

ωij

︸ ︷︷ ︸

<∆vi,change in vi>

+∂2

∂vi∂vj

f (v1)[w

dv2f (v2)ωij

]

︸ ︷︷ ︸

<∆vi∆vj>

(365)

(366)

Since∂ωij∂v1i

=∂ωij∂v2j

16.1 Rosenbluth, Macdonald and Judd Form

The detailed calculation of the collision terms foll wing Coulomb scattering is straightforward, butcomplicated (Rosenbluth, Macdonald and Judd,Phys Rev 107, 1 1957). It leads to an equivalent resultas that of Landau for electron-electron collisions, but is more general as the particles may be dissimilar.

We consider the case of two particles of masses m and m′ moving with velocities v and v′. As is wellknown the collision occurs with scattering about the centre of mass with rotation of the relative velocityvector V The velocity of each particle in terms of the centre of mass velocity U and the relative velocityV are

v = U +m′

m+m′V

v′ = U +m

m+m′V

Hence the change in the velocity v as a result of the collision

∆v =m′

m+m′ ∆V

The scattering gives rise to velocity changes in the plane perpendicular,⊥, and parallel, ‖, to theinitial relative velocity V as above. The cross section for scatter through an angle φ is modified by theinclusion of the reduced mass m

m =mm′

m+m′

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but is otherwise unchanged. Therefore we may use the averages we have already calculated but with themass replaced by the reduced mass.

However the relative velocity V changes in direction and the centre of mass velocity U must beremoved before we can relate the particle velocities to their values in the laboratory frame. To do thiswe identify a set of directions with respect to V , namely λ parallel to V and µ and ν perpendicular toV so that

∆Vλ = −V (1− cos θ) = 2V sin2 (φ/2)

∆Vµ = V sinφ cosχ = 2V sin(φ/2) cos(φ/2) cosχ

∆Vν = V sinφ sinχ = 2V sin(φ/2) cos(φ/2) sinχ (367)

χ being the rotation angle in the plane µ, ν perpendicular to V .Using either the small angle approach as in section 5.1 or directly from the full Rutherford cross

section as in section 15.1

σ(Ω) dΩ =Z2 e4

4m2 V 4csc4 (φ/2) dΩ (368)

for particles of charge e and Ze. The resultant averages are carried out as described in section 5.1, butin the centre of mass frame

〈∆Vλ〉 = 〈∆V‖〉 = −4πZ2e4

m2V 2ln Λ

〈∆Vµ〉 = 〈∆Vν〉 = 〈12∆V⊥〉 = 0

〈∆Vλ2〉 = 〈∆V‖2〉 = 0

〈∆Vµ2〉 = 〈∆Vν2〉 = 〈12∆V⊥

2〉 =4πZ2e4

m2Vln Λ (369)

The unit vectors in the directions λ, µ and ν are

λ =V

Vµ =

k ∧ VVx

2 + Vy2 and ν = λ ∧ µ

where (ı, j, k) are the unit vectors in the (x, y, z) directionsThus ∆V in the laboratory set of Cartesian co-ordinates (x, y, z) may written

∆Vx = (ı · λ)∆Vλ + (ı · µ)∆Vµ + (ı · ν)∆Vν

with similar expressions for ∆Vy and ∆Vz . Thus taking the average

〈∆Vx〉 = (ı · λ)〈∆Vλ〉+ (ı · µ):0〈∆Vµ〉+ (ı · ν):0〈∆Vν〉

=VxV〈∆V‖〉 (370)

with similar expressions for 〈Vy〉 and 〈Vz〉.The product terms 〈∆Vx ∆Vy〉 etc. are obtained in a similar manner, noting that several terms cancel

to zero

〈∆Vx ∆Vy〉 =⟨ [

(ı · λ)∆Vλ + (ı · µ)∆Vµ + (ı · ν)∆Vν] [

(j · λ)∆Vλ + (j · µ)∆Vµ + (j · ν)∆Vν] ⟩

=⟨ [(

ı · λ) (

j · λ)]

∆Vλ

2 +[(

ı · λ)(

j · µ)

+(ı · µ

) (

j · λ)]

∆Vλ ∆Vµ

+[(ı · µ

) (

j · µ)]

∆Vµ2 +

[

(ı · ν)(

j · λ)

+(

ı · λ)(

j · ν)]

∆Vλ ∆Vν

+[

(ı · ν)(

j · ν)]

∆Vν2 +

[(ı · µ

) (

j · ν)

+ (ı · ν)(

j · µ)]

∆Vµ ∆Vν

=[(ı · µ

) (

j · µ)

+ (ı · ν)(

j · ν)] 1

2

⟨∆V⊥

2⟩

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Page 44: Advanced Plasma

Hence generalising

⟨∆Vi ∆Vj

⟩= [µi µj + νi νj ]

1

2

⟨∆V⊥

2⟩

= [δij − λi λj ]1

2

⟨∆V⊥

2⟩

(371)

since the components of the unit vectors λ, µ, ν on the orthogonal set of unit vectors ı, j, k namely(λi, λj , λk), (µi, µj , µk) and (νi, νj , νk) respectively satisfy the condition

λiλj + µiµj + νiνj = δij

Returning to the laboratory frame, the change in the particle averages are given by

⟨∆ vi

⟩=

m′

m+m′⟨∆Vi

= −4 π Z2 e4 ln Λ

mm

ViV 3

=4 π Z2 e4 ln Λ

mm

∂Vi

(1

V

)

(372)

since V =√

(vi − v′i) (vi − v′i) and

⟨∆vi ∆vj

⟩=

4 π Z2 e4 ln Λ

m2

[

δij −Vi VjV 2

]

=4 π Z2 e4 ln Λ

m2

∂2V

∂Vi ∂Vj(373)

These terms are written in terms of the functions

g(v) =∑

wdv′f(v′)|v − v′| (374)

h(v) =∑

m+m′

m′

wdv′f(v′)|v − v′|−1 (375)

the sums being taken over all the perturbing species. The derivatives of g(v) and h(v) yield

∂g

∂vi= −

wdv′ f(v′)

ViV

∂2g

∂vi ∂vj=

wdv′ f(v′)

[

δij −Vi VjV 2

]

(376)

∂h

∂vi= −

m+m′

m

wdv′ f(v′)

ViV 3

(377)

Substituting we obtain the MacDonald, Rosenbluth and Judd form of the Fokker-Planck collisionoperator.

⟨∆vi

⟩=

4 π Z2 e4 ln Λ

m2

∂h

∂vi(378)

⟨∆vi∆vj

⟩=

4 π Z2 e4 ln Λ

m2

∂2g

∂vi ∂vj(379)

In polar co-ordinates, without symmetry, a similar analysis gives basically a similar, but more com-plicated, result.

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16.1.1 Relationship with Landau’s Formula

This result is easily related to the Landau formula for identical particles m = m′, section 15.1. Usingthe same symbols as before u ≡ V , and v1 ≡ v and v2 ≡ v′ we obtain I, the integral involved the finalsolution.

ωij =1

u

[

δij −uiuju2

]

(380)

∂ωij∂uij

=−∂ωij∂vij

(381)

=∂ωij∂v2j

(382)

= −2uiu3

(383)

Define:

g(v1) =wdv2 f (v2)u (384)

h(v1) = 2wdv2

f

(v2)u (385)

∂g

∂v1i= −

wdv2 f (v2)

uiu

] (386)

∂2g

∂v1i∂v1j=

wdv2 f (v2)

[

δij −uiuju2

]

(387)

=wdv2 f (v2)ωij (388)

∂h

∂v1i= −2

wdv2 f (v2)

uiu

(389)

=wdv2 f (v2)

∂ωij∂v1j

(390)

= −wdv2 f (v2)

∂ωij∂v2j

(391)

∂3g

∂v1i∂v1j∂v1j=

wdv2f (v2)

∂ωij∂v1j

(392)

=∂h

∂v1i(393)

wdv2

[

f (v0)∂f

∂v1j− f (v1)

∂f

∂v2j

]

ωij =wdv2

[

f (v2)∂f

∂v1j+ f (v1)

∂f

∂v2j

]

ωij

+2wdv2 f (v1) f (v2)

∂ωij∂v2j

(394)

=∂f

∂v1j

∂2g

∂v1i∂v1j+ f (v1)

∂3g

∂v1j∂v1j∂v1j− 2f (v1)

∂h

∂v1j(395)

Hence

I = 4πe4

m2ln Λ

− ∂

∂v1i

[

f (v1)∂h

∂v1i

]

︸ ︷︷ ︸

<∆vi>

+1

2

∂2

∂v1i∂v1j

[

f (v1)∂2g

∂v1i∂v1j

]

︸ ︷︷ ︸

<∆vi∆vj>

(396)

gives the dynamical friction and the diffusionSpherically symmetric distribution for f (v):

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Page 46: Advanced Plasma

I = (4π)2e4

m2ln Λ

1

v2

d

dv

αf + βdf

dv

(397)

where:

α =w v

0f (v′)v′2dv′ (398)

β =1

3

[1

v

w v

0f (v′)v′4dv′ + v2

w ∞

vf (v′)v′dv′

]

(399)

For dissimilar particles with mass m and m′, use the reduced mass: mm′

m+m′ . The constant in h(v) is

changed to by m+m′

m . i.e. replace the 2 by m+m′

m in the front of the definition.)

17 Lecture 17: Calculation of the Transport Coefficients

The transport coefficients are: Electrical Conductivity, Thermal Conductivity, and Viscosity. Let usimagine that the plasma is weakly perturbed by an external agency/force, such as a temperature gradient.We now have the ambient thermal distribution, and the perturbation to the distribution induced by theforce. We assume that there is no net flux associated with the background (isotropic distribution). Wewish to find the flux from the perturbation, this gives the properties of the plasma. We would like tobe able to calculate the perturbation due to a force on the distribution function, and calculate the fluxfrom this perturbation. We assume steady state before the perturbation:

f (v)︸︷︷︸

From Distribution

= f0(v)︸ ︷︷ ︸

Initial Maxwellian

+ f1(v)︸ ︷︷ ︸

Perturbation

(400)

Boltzmann equation:

0due to steady state

∂f

∂t+ v · ∂f

∂r− a0 ·

∂f

∂v=∂f

∂t

∣∣∣∣c

→ I (401)

Apply (400), neglect terms of second order and higher → To first order:

f0

[mv2

2kt− 5

2

]1

Tvi∂T

∂xi− f0

eZ

kTEivi = I(f1, f

′0) + I(f0, f

′1) (402)

Solve for f1. In general, the system has cylindrical symmetry if there is no magnetic field present.This means that f1 ∝ cos θ, where θ is the poloidal angle with respect to the field direction.

f1 = D(v) cos θ (403)

where D is some function of v. For more details see Spitzer and Harm (Phys Rev 89, 977,1952), andCohen, Spitzer and Routly (Phys Rev 80, 230, 1950).

46