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5.1 General Treatment of Linear Waves in Anisotropic Medium

Start with general approach to waves in a linear Medium: Maxwell:

∇ ∧B= μ_o j +

c²

∂E

∂t

; ∇ ∧E = −

∂B

∂t

(5.1)

we keep all the medium's response explicit in j. Plasma is (infinite and) uniform so we Fourier analyze in space and time. That is we seek a solution in which all variables go like

expi ( k . x − ωt ) [real part of]

(5.2)

It is really the linearised equations which we treat this way; if there is some equilibrium field OK but the equations above mean implicitly the perturbations B, E, j, etc.

Fourier analyzed:

i k ∧B= μ_o j+

−iω

c²

E ; i k ∧E = i ωB

(5.3)

Eliminate B by taking k∧ second eq. and ω× 1st

i k ∧(k ∧E) = ωμ_o j−

i ω²

c²

(5.4)

k ∧(k ∧E) +

ω²

c²

E+ i ωμ_o j= 0

(5.5)

Now, in order to get further we must have some relationship between j and E(k,ω). This will have to come from solving the plasma equations but for now we can just write the most general linear relationship j and E as

j=σ . E

(5.6)

σ is the `conductivity tensor'. Think of this equation as a matrix e.g.:

⎛
⎜
⎜
⎜
⎝

j_x

j_y

j_z

⎞
⎟
⎟
⎟
⎠

⎛
⎜
⎜
⎜
⎝

σ_xx

σ_xy

...

σ_zz

⎞
⎟
⎟
⎟
⎠

⎛
⎜
⎜
⎜
⎝

E_x

E_y

E_z

⎞
⎟
⎟
⎟
⎠

(5.7)

This is a general form of Ohm's Law. Of course if the plasma (medium) is isotropic (same in all directions) all off-diagonal σ′s are zero and one gets j = σE.

Thus

k (k . E) − k² E+

ω²

c²

E+ i ωμ_oσ . E= 0

(5.8)

Recall that in elementary E&M, dielectric media are discussed in terms of a dielectric constant ϵ and a "polarization" of the medium, P, caused by modification of atoms. Then

ϵ₀ E=

Displacement

−

Polarization

and ∇ . D =

ρ_ext

external charge

(5.9)

and one writes

P =

susceptibility

ϵ₀ E

(5.10)

Our case is completely analogous, except we have chosen to express the response of the medium in terms of current density, j, rather than "polarization" P For such a dielectric medium, Ampere's law would be written:

μ_o

∇ ∧B= j_ext +

∂D

∂t

∂

∂t

ϵϵ₀ E, if j_ext = 0 ,

(5.11)

where the dielectric constant would be ϵ = 1 + χ.

Thus, the explicit polarization current can be expressed in the form of an equivalent dielectric expression if

j+ ϵ₀

∂E

∂t

= σ . E+ ϵ₀

∂E

∂t

∂

∂t

ϵ₀ ϵ . E

(5.12)

ϵ = 1 +

− i ωϵ₀

(5.13)

Notice the dielectric constant is a tensor because of anisotropy. The last two terms come from the RHS of Ampere's law:

∂

∂t

(ϵ₀ E) .

(5.14)

If we were thinking in terms of a dielectric medium with no explicit currents, only implicit (in ϵ) we would write this [(∂)/(∂t)] (ϵϵ₀ E); ϵ the dielectric constant. Our medium is possibly anisotropic so we need [(∂)/(∂t)] (ϵ₀ϵ . E) dielectric tensor. The obvious thing is therefore to define

ϵ = 1 +

−iωϵ₀

σ = 1 +

i μ_o c²

(5.15)

Then

k ( k . E) − k² E+

ω²

c²

ϵ . E= 0

(5.16)

and we may regard ϵ (k, ω) as the dielectric tensor.

Write the equation as a tensor multiplying E:

K. E= 0

(5.17)

with

K= { k k − k² 1 +

ω²

c²

ϵ }

(5.18)

Again this is a matrix equation i.e. 3 simultaneous homogeneous eqs. for E.

⎛
⎜
⎜
⎜
⎝

K_xx

K_xy

...

K_yx

...

K_zz

⎞
⎟
⎟
⎟
⎠

⎛
⎜
⎜
⎜
⎝

E_x

E_y

E_z

⎞
⎟
⎟
⎟
⎠

= 0

(5.19)

In order to have a non-zero E solution we must have

det |K| = 0 .

(5.20)

This will give us an equation relating k and ω, which tells us about the possible wavelengths and frequencies of waves in our plasma.

5.1.1 Simple Case. Isotropic Medium

σ 1

(5.21)

ϵ 1

(5.22)

Take k in z direction then write out the Dispersion tensor K.

⎛
⎜
⎜
⎜
⎝

⎞
⎟
⎟
⎟
⎠

k k

−

⎛
⎜
⎜
⎜
⎝

k²

⎞
⎟
⎟
⎟
⎠

k²1

⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝

ω²

c²

ω²

c²

ω²

c²

⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠

[(ω²)/(c²)]ϵ

⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣

−k² +

ω²

c²

−k² +

ω²

c²

ω²

c²

⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦

(5.23)

Take determinant:

det | K | =

⎛
⎝

− k² +

ω²

c²

⎞
⎠

ω²

c²

ϵ = 0 .

(5.24)

Two possible types of solution to this dispersion relation:

(A)

−k² +

ω²

c²

ϵ = 0.

(5.25)

⇒

⎛
⎜
⎜
⎜
⎜
⎜
⎝

ω²

c²

⎞
⎟
⎟
⎟
⎟
⎟
⎠

⎛
⎜
⎜
⎜
⎝

E_x

E_y

E_z

⎞
⎟
⎟
⎟
⎠

= 0 ⇒ E_z = 0 .

(5.26)

Electric field is transverse (E. k = 0)

Phase velocity of the wave is

√

(5.27)

This is just like a regular EM wave traveling in a medium with refractive index

N ≡

√

(5.28)

(B)

ω²

c²

ϵ = 0 i.e. ϵ = 0

(5.29)

⇒

⎛
⎜
⎜
⎜
⎝

K_xx

K_yy

⎞
⎟
⎟
⎟
⎠

⎛
⎜
⎜
⎜
⎝

E_x

E_y

E_z

⎞
⎟
⎟
⎟
⎠

= 0 ⇒ E_x = E_y = 0 .

(5.30)

Electric Field is Longitudinal (E∧k = 0 ) E||k.

This has no obvious counterpart in optics etc. because ϵ is not usually zero. In plasmas ϵ = 0 is a relevant solution. Plasmas can support longitudinal waves.

5.1.2 General Case (k in z-direction)

ω²

c²

⎡
⎢
⎢
⎢
⎣

−N² + ϵ_xx

ϵ_xy

ϵ_xz

ϵ_yx

−N² + ϵ_yy

ϵ_yz

ϵ_zx

ϵ_zy

ϵ_zz

⎤
⎥
⎥
⎥
⎦

⎛
⎝

N² =

k² c²

ω²

⎞
⎠

(5.31)

When we take determinant we shall get a quadratic in N² (for given ω) provided ϵ is not explicitly dependent on k. So for any ω there are two values of N². Two `modes'. The polarization E of these modes will be in general partly longitudinal and partly transverse. The point: separation into distinct longitudinal and transverse modes is not possible in anisotropic media (e.g. plasma with B_o).

All we have said applies to general linear medium (crystal, glass, dielectric, plasma). Now we have to get the correct expression for σ and hence ϵ by analysis of the plasma (fluid) equations.

5.2 High Frequency Plasma Conductivity

We want, now, to calculate the current for given (Fourier) electric field E(k,ω), to get the conductivity, σ. It won't be the same as the DC conductivity which we calculated before (for collisions) because the inertia of the species will be important. In fact, provided

ω >>

(5.32)

we can ignore collisions altogether. Do this for simplicity, although this approach can be generalized.

Also, under many circumstances we can ignore the pressure force −∇p. In general will be true if [(ω)/k] >> v_te,i

Remember that we are dealing with uniform steady equilibrium (suffix 0), so the equilibrium has zero electric field E₀=−∇ϕ₀=0, (and pressure gradient ∇p₀=0,) so there is zero equilibrium perpendicular velocity, and we also take v_||0=0. The equilibrium is at rest. Also the wave is a small perturbation to this equilibrium, which formally we could denote with suffix 1, and we keep only first order quantities in the wave equations. This gives a manageable problem with wide applicability. We usually drop the suffixes once we've formulated the problem, recognizing that B and n usually mean the equilibrium quantities and varying things are the wave quantities.

Approximations:

Collisionless

= 0

`Cold Plasma′

∇p = 0

(e.g. T ≅ 0)

Stationary Equil

v₀ = 0

(5.33)

5.2.1 Zero B-field case

To start with take B₀ = 0: Plasma isotropic Momentum equation (for electrons first)

⎡
⎣

∂v

∂t

+ ( v. ∇ )v

⎤
⎦

= n q E

(5.34)

Notice the characteristic of the cold plasma approx. that we can cancel n from this equation and on linearizing get essentially the single particle equation.

∂v₁

∂t

= q E.

(5.35)

The equation can be solved for given ω as

−i ωm

(5.36)

and the current (due to this species, electrons) is

j= n q v=

n q²

−i ωm

(5.37)

So the conductivity is

σ = i

nq²

ωm

(5.38)

and the dielectric constant is

ϵ = 1 +

ωϵ₀

σ = 1 −

⎛
⎝

nq²

m ϵ₀

⎞
⎠

ω²

= 1 + χ

(5.39)

Longitudinal Waves (B₀ = 0)

Dispersion relation we know is

ϵ = 0 = 1 −

⎛
⎝

n q²

m ϵ₀

⎞
⎠

ω²

(5.40)

[Strictly, the ϵ we want here is the total ϵ including both electron and ion contributions to the conductivity. But

σ_e

σ_i

≅

m_i

m_e

(for z = 1)

(5.41)

so to a first approximation, ignore ion motions.]

Solution

ω² =

⎛
⎝

n_e q_e²

m_e ϵ₀

⎞
⎠

(5.42)

In this approximation longitudinal oscillations of the electron fluid have a single unique frequency:

ω_p =

⎛
⎝

n_e e²

m_e ϵ₀

⎞
⎠

¹/₂

(5.43)

This is called the `Plasma Frequency' (more properly ω_pe the `electron' plasma frequency). If we allow for ion motions we get an ion conductivity

σ_i =

i n_i q_i²

ωm_i

(5.44)

and hence

ϵ_tot

1 +

ωϵ₀

( σ_e + σ_i ) = 1 −

⎛
⎝

n_e q_e²

ϵ₀ m_e

n_i q_i²

ϵ₀ m_i

⎞
⎠

ω²

(5.45)

1 − ( ω_pe² + ω_pi² )/ω²

where

ω_pi ≡

⎛
⎝

n_i q_i²

ϵ₀ m_i

⎞
⎠

¹/₂

(5.46)

is the `Ion Plasma Frequency'.

Simple Derivation of Plasma Oscillations

Figure 5.1: Slab derivation of plasma oscillations

Take ions stationary; perturb a slab of plasma by shifting electrons a distance x. Charge built up is n_eq x per unit area. Hence electric field generated

E = −

n_e q_e x

ϵ₀

(5.47)

Equation of motion of electrons

m_e

= −

n_e q_e² x

ϵ₀

;

(5.48)

i.e.

d²x

dt²

⎛
⎝

n_e q_e²

ϵ₀ m_e

⎞
⎠

x = 0

(5.49)

Simple harmonic oscillator with frequency

ω_pe =

⎛
⎝

n_e q_e²

ϵ₀ m_e

⎞
⎠

¹/₂

Plasma Frequency.

(5.50)

The Characteristic Frequency of Longitudinal Oscillations in a plasma. Notice

ω = ω_p for all k in this approx.
Phase velocity [(ω)/k] can have any value.
Group velocity of wave, which is the velocity at which information/energy travel is

v_g = d ω
d k
= 0 !!
(5.51)

In a way, these oscillations can hardly be thought of as a `proper' wave because they do not transport energy or information. (In Cold Plasma Limit). [Nevertheless they do emerge from the wave analysis and with less restrictive approximations do have finite v_g.]

Transverse Waves (B₀ = 0)

Dispersion relation:

− k² +

ω²

c²

ϵ = 0

(5.52)

N²

≡

k² c²

ω²

= ϵ = 1 −( ω_pe² + ω_pi² ) / ω²

≅

1 − ω_pe² / ω²

(5.53)

Figure 5.2: Unmagnetized plasma transverse wave.

Figure 5.3: Alternative dispersion plot.

Alternative expression:

− k² +

ω²

c²

−

ω_p²

c²

= 0

(5.54)

which implies

ω²

ω_p² + k² c²

(5.55)

( ω_p² + k² c² )^¹/₂.

(5.56)

5.2.2 Meaning of Negative N²: Cut Off

When N² < 0 (i.e. for ω < ω_p), N is pure imaginary and hence so is k for real ω. Thus the wave we have found goes like

exp{ ± | k | x − i ωt }

(5.57)

i.e. its space dependence is exponential not oscillatory. Such a wave is said to be `Evanescent' or `Cut Off'. It does not truly propagate through the medium but just damps exponentially.

Example:

Figure 5.4: Wave behaviour at cut-off.

A wave incident on a plasma with ω_p² > ω² is simply reflected, no energy is transmitted through the plasma.

5.3 Cold Plasma Waves (Magnetized Plasma)

Objective: calculate ϵ, K, k(ω), using known plasma equations.

Approximation: Ignore thermal motion of particles.

Applicability: Most situations where (1) plasma pressure and (2) absorption are negligible. Generally requires wave phase velocity >> v_thermal.

5.3.1 Derivation of Dispersion Relation

Can "derive" the cold plasma approx from fluid plasma equations. It is simpler just to say that all particles (of a specific species) just move together obeying Newton's 2nd law:

∂v

∂t

= q (E+ v∧B)

(5.58)

Take the background plasma to have E₀ = 0, B = B₀ and zero velocity. Then all motion is due to the wave and also the wave's magnetic field can be ignored provided the particle speed stays small. (This is a linearization).

∂v

∂t

= q ( E+ v∧B₀) ,

(5.59)

where v, E ∝ expi (k . x − ωt) are wave quantities.

Substitute [(∂)/(∂t)] → − i ω and write out equations. Choose axes such that B₀ = B₀ (0,0,1).

− i ωm v_x

q (E_x + v_y B₀)

− i ωm v_y

q (E_y − v_x B₀)

(5.60)

− i ωm v_z

q E_z

Solve for v in terms of E.

v_x

⎛
⎝

i ωE_x − ΩE_y

ω² − Ω²

⎞
⎠

v_y

⎛
⎝

ΩE_x + i ωE_y

ω² − Ω²

⎞
⎠

(5.61)

v_z

E_z

where Ω = [(q B₀)/m] is the gyrofrequency but its sign is that of the charge on the particle species under consideration.

Since the current is j = q vn = σ . E we can identify the conductivity tensor for the species (j) as:

σ_j =

⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣

q_j² n_j

m_j

iω

ω² − Ω_j²

−

q_j² n_j

m_j

Ω_j

ω² − Ω_j²

q_j² n_j

m_j

Ω_j

ω² − Ω_j²

q_j² n_j

m_j

i ω

ω² − Ω_j²

i q_j²

n_j

⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦

(5.62)

The total conductivity, due to all species, is the sum of the conductivities for each

σ =

∑
j

σ_j

(5.63)

σ_xx

σ_yy =

∑
j

q₁² n_j

m_j

i ω

ω² − Ω_j²

(5.64)

σ_xy

− σ_yx = −

∑
j

q_j² n_j

m_j

Ω_j

ω² − Ω_j²

(5.65)

σ_zz

∑
j

q_j² n_j

m_j

(5.66)

Susceptibility χ = [1/(− i ωϵ₀)] σ. Dielectric tensor ϵ = 1+ χ.

ϵ =

⎡
⎢
⎢
⎢
⎣

ϵ_xx

ϵ_xy

ϵ_yx

ϵ_yy

ϵ_zz

⎤
⎥
⎥
⎥
⎦

⎡
⎢
⎢
⎢
⎣

−i D

i D

⎤
⎥
⎥
⎥
⎦

(5.67)

where

ϵ_xx

ϵ_yy = S = 1 −

∑
j

ω_pj²

ω² − Ω_j²

(5.68)

i ϵ_xy

− i ϵ_yx = D =

∑
j

Ω_j

ω_pj²

ω² − Ω_j²

(5.69)

ϵ_zz

P = 1 −

∑
j

ω_pj²

ω²

(5.70)

and

ω_pj² ≡

q_j² n_j

ϵ₀m_j

(5.71)

is the "plasma frequency" for that species.

S & D stand for "Sum" and "Difference":

S =

(R + L) D =

(R − L)

(5.72)

where R & L stand for "Right-hand" and "Left-hand" and are:

R = 1 −

∑
j

ω_pj²

ω( ω+ Ω_j )

, L = 1 −

∑
j

ω_pj²

ω( ω− Ω_j )

(5.73)

The R & L terms arise in a derivation based on expressing the field in terms of rotating polarizations (right & left) rather than the direct Cartesian approach.

We now have the dielectric tensor from which to obtain the dispersion relation and solve it to get k (ω) and the polarization. Notice, first, that ϵ is indeed independent of k so the dispersion relation (for given ω) is a quadratic in N² (or k²).

Choose convenient axes such that k_y = N_y = 0. Let θ be angle between k and B₀ so that

N_z = N cosθ , N_x = N sinθ .

(5.74)

Then

ω²

c²

⎡
⎢
⎢
⎢
⎣

−N² cos² θ+ S

− i D

N² sinθcosθ

+ i D

− N² + S

N² sinθcosθ

− N² sin² θ+ P

⎤
⎥
⎥
⎥
⎦

(5.75)

and

c²

ω²

|K| = A N⁴ − B N² + C

(5.76)

where

≡

S sin² θ+ P cos² θ

(5.77)

≡

R L sin² θ+ P S (1 + cos² θ)

(5.78)

≡

P R L

(5.79)

Solutions are²

N² =

B ±F

(5.80)

where the discriminant, F, is given by

F² = (RL − PS)² sin⁴ θ+ 4 P² D² cos² θ

(5.81)

after some algebra. The quantity F² is manifestly positive, so N² is real ⇒ "propagating" or "evanescent" no wave absorption for cold plasma.

Solution can also be written (by cunningly making every term in the dispersion relation proportional to cos²θ or sin²θ using cos²+sin²=1)

tan² θ = −

P ( N² − R ) ( N² − L )

( S N² − R L ) ( N² − P )

(5.82)

This compact form makes it easy to identify the dispersion relation at θ = 0 & [(π)/2] i.e. parallel and perpendicular propagation tanθ = 0, ∞.

Parallel: P = 0 , N² = R N² = L

Perp: N² = [R L/S] N² = P .

Wave polarization

Once we have solved for N we can find the electric field E corresponding to this N, which is the wave polarization. Actually we find just the ratios of the electric field components, because the linearized system is independent of the total magnitude of E. From the second row of K we get

E_y

E_x

N²−S

;

(5.83)

and from the third row

E_z

E_x

N²sinθcosθ

N²sin²θ−P

(5.84)

Example: Right-hand wave, parallel propagation θ = 0

N² = R. Polarization: E_z=0, [(E_y)/(E_x)]=i, because R−S=D. Explicitly for a single ion species

N² = 1 −

ω_pe²

ω( ω− | Ω_e | )

−

ω_pi²

ω( ω+ | Ω_i | )

(5.85)

This has a wave resonance N² → ∞ at ω = | Ω_e |, only. Right-hand wave also has a cutoff at R=0, whose solution proves to be

ω = ω_R =

| Ω_e | − | Ω_i |

⎡
⎣

⎛
⎝

| Ω_e | + | Ω_i|

⎞
⎠

+ ω_pe² + ω_pi²

⎤
⎦

1/2

(5.86)

Since m_i >> m_e this can be approximated as:

ω_R ≅

| Ω_e |

⎧
⎨
⎩

1 +

⎛
⎝

1 + 4

ω_pe²

| Ω_e |²

⎞
⎠

¹/₂

⎫
⎬
⎭

(5.87)

This is always above | Ω_e |.

Figure 5.5: The form of the dispersion relation for RH wave.

One can similarly investigate Left-hand wave and perpendicularly propagating waves. The resulting wave resonances and cut-offs depend only upon 2 properties (for specified ion mass) (1) Density ↔ ω_pe² (2) Magnetic Field ↔ | Ω_e |. [Ion values ω_pi, | Ω_i | are got by [(m_i)/(m_e)] factors.]

These resonances and cutoffs are often plotted on a 2-D plane [( | Ω_e |)/(ω)] , [(ω_p²)/(ω²)] ( ∝ B, n) called the

C M A Diagram.

We don't have time for it here.

5.3.2 Hybrid Resonances Perpendicular Propagation

"Extraordinary" wave N² = [R L/S]

Polarization E_z=0, [(E_y)/(E_x)] = [iDS/(RL −S²)]=[(−iS)/D], because RL=S²−D².

N² =

⎡
⎣

( ω+ Ω_e ) ( ω+ Ω_i ) −

ω_pe²

( ω+ Ω_i ) −

ω_pi²

( ω+ Ω_e )

⎤
⎦

⎡
⎣

( ω− Ω_e ) ( ω− Ω_i ) −

ω_pe²

( ω− Ω_i )...

⎤
⎦

( ω² − Ω_e² ) ( ω² − Ω_i²) − ω_pe² ( ω² − Ω_i² ) − ω_pi² ( ω² − Ω_e² )

(5.88)

Resonance is where denominator = 0. Solve the quadratic in ω² and one gets

ω²

⎡
⎣

( ω_pe² + Ω_e² + ω_pi² + Ω_i²)

√

( ω_pe² + Ω_e² + ω_pi² + Ω_i²)² −4(Ω_e²Ω_i²+ω_pe²Ω_i²+ω_pi²Ω_e²)

⎤
⎦

(5.89)

ω_pe² + Ω_e² + ω_pi² + Ω_i²

⎛
√

⎛
⎝

ω_pe² + Ω_e² − ω_pi² − Ω_i²

⎞
⎠

+ ω_pe² ω_pi²

(5.90)

Neglecting terms of order [(m_e)/(m_i)] (e.g. [(ω_pi²)/(ω_pe²)]) one gets solutions

ω_UH²

ω_pe² + Ω_e² Upper Hybrid Resonance.

(5.91)

ω_LH²

Ω_e² ω_pi²

Ω_e² + ω_pe²

Lower Hybrid Resonance..

(5.92)

At very high density, ω_pe² >> Ω_e²

ω_LH² ≅ | Ω_e | | Ω_i |

(5.93)

geometric mean of cyclotron frequencies.

At very low density, ω_pe² << Ω_e²

ω_LH² ≅ ω_pi²

(5.94)

ion plasma frequency

Usually in tokamaks ω_pe² ∼ Ω_e². Intermediate.

Summary Graph (Ω > ω_p)

Figure 5.6: Summary of magnetized dispersion relation for different propagation angles.

Cut-offs are where N² = 0.

Resonances are where N² → ∞.

Intermediate angles of propagation have refractive indices between the θ = 0, [(π)/2] lines, in the shaded areas.

5.3.3 Whistlers

(Ref. R.A. Helliwell, "Whistlers & Related Ionospheric Phenomena," Stanford UP 1965.)

For N² >> 1 the right hand wave can be written

N² ≅

− ω_pe²

ω( ω− | Ω_e | )

, (N = kc/ω)

(5.95)

Group velocity is

v_g =

d ω

⎛
⎝

dω

⎞
⎠

−1

⎡
⎣

dω

⎛
⎝

Nω

⎞
⎠

⎤
⎦

−1

(5.96)

Then since

N =

ω_p

ω^¹/₂ ( | Ω_e | − ω)^¹/₂

(5.97)

we have

d ω

( N ω)

d ω

ω_p ω^¹/₂

( | Ω_e | − ω)^¹/₂

= ω_p

⎧
⎨
⎩

ω^¹/₂ (| Ω_e | − ω)^¹/₂

ω^¹/₂

( | Ω_e | − ω)^³/₂

⎫
⎬
⎭

ω_p/2

( | Ω_e | − ω)^³/₂ ω^¹/₂

{( | Ω_e | − ω) + ω}

ω_p | Ω_e |/2

( | Ω_e | − ω)^³/₂ ω^¹/₂

(5.98)

Thus

v_g =

c 2 ( | Ω_e | − ω)^³/₂ ω^¹/₂

ω_p | Ω_e |

(5.99)

Group Delay is

v_g

∝

ω^¹/₂ ( | Ω_e | − ω)^³/₂

∝

⎛
⎝

| Ω_e |

⎞
⎠

¹/₂

⎛
⎝

1 −

| Ω_e |

⎞
⎠

³/₂

(5.100)

Figure 5.7: Whistler plot of frequency versus delay.

Plot with [ L/(v_g)] as x-axis.

Resulting form explains downward whistle.

Lightning strike ∼ δ-function excites all frequencies.

Lower ones arrive later.

Examples of actual whistler sounds can be obtained from http://www-istp.gsfc.nasa.gov/istp/polar/polar_pwi_sounds.html.

5.4 Thermal Effects on Plasma Waves

The cold plasma approx is only good for high frequency, N² ∼ 1 waves. If ω is low or N² >> 1 one may have to consider thermal effects. From the fluid viewpoint, this means pressure. Write down the momentum equation. (We shall go back to B₀ = 0) linearized

∂v₁

∂t

= n q E₁ − ∇p₁ ;

(5.101)

remember these are the perturbations:

p = p₀ + p₁ .

(5.102)

Fourier Analyse (drop 1's)

m n (−iω) v= nqE− i k p

(5.103)

The key question: how to relate p to v

Answer: Equation of state + Continuity

State

p n^−γ = const. ⇒ ( p₀ + p₁ ) ( n₀ + n₁ )^−γ = p₀ n₀^−γ

(5.104)

Use Taylor Expansion

( p₀ + p₁ ) (n₀ + n₁ )^−γ ≅ p₀ n₀^−γ

⎡
⎣

1 +

p₁

p₀

− γ

n₁

n₀

⎤
⎦

(5.105)

Hence

p₁

p₀

= γ

n₁

n₀

(5.106)

Continuity

∂n

∂t

+ ∇ . ( n v) = 0

(5.107)

Linearise:

∂n₁

∂t

+ ∇ . (n₀ v₁ ) = 0 ⇒

∂n

∂t

+ n₀ ∇ . v = 0

(5.108)

Fourier Transform

− i ωn₁ + n₀ i k . v₁ = 0

(5.109)

i.e.

n₁ = n₀

k . v

(5.110)

Combine State & Continuity

p₁ = p₀ γ

n₁

n₀

= p₀ γ

n₀

k . v

n₀

= p₀ γ

k . v

(5.111)

Hence Momentum becomes

m n₀ ( −i ω) v= n q E−

i k p₀ γ

k . v

(5.112)

Notice Transverse waves have k . v = 0; so they are unaffected by pressure.

Therefore we need only consider the longitudinal wave. However, for consistency let us proceed as before to get the dielectric tensor etc.

Choose axes such that k = k ∧e_z then obviously:

v_x =

i q

ωm

E_x v_y =

i q

ωm

E_y

(5.113)

v_z =

E_z

− i ω+ ( i k² γp₀ / m n₀ ω)

(5.114)

Recall that p₀=n₀T so write p₀/n₀=T. Then

σ =

i n q²

ωm

⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣

1 −

k² γT

ω²m

⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦

(5.115)

ϵ = 1 +

iσ

ϵ₀ ω

⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣

1−

ω_p²

ω²

1 −

ω_p²

ω²

1 −

ω_p²

ω² − k²

γT

⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦

(5.116)

(Taking account only of 1 species, electrons, for now.)

We have confirmed the previous comment that the transverse waves (E_x, E_y) are unaffected. The longitudinal wave is. Notice that ϵ now depends on k as well as ω. This is called `spatial dispersion'.

For completeness, note that the dielectric tensor can be expressed in general tensor notation as

1 −

ω_p²

ω²

⎛
⎜
⎝

1 + k k

⎡
⎢
⎣

1 −

k²

ω²

γT

− 1

⎤
⎥
⎦

⎞
⎟
⎠

1 −

ω_p²

ω²

⎛
⎜
⎝

1 + k k

ω² m

k²γT

− 1

⎞
⎟
⎠

(5.117)

This form shows isotropy with respect to the medium: there is no preferred direction in space for the wave vector k.

But once k is chosen, ϵ is not isotropic. The direction of k becomes a special direction.

Longitudinal Waves: dispersion relation is

ϵ_zz = 0 (as before)

(5.118)

which is

1 −

ω_p²

ω² −

k² γT

= 0

(5.119)

ω² = ω_p² + k²

γT

= ω_p² + k² γv_t² .

(5.120)

[The appropriate value of γ to take is 1 dimensional adiabatic i.e. γ = 3. This seems plausible since the electron motion is 1-d (along k) and may be demonstrated more rigorously by kinetic theory.]

The above dispersion relation is called the Bohm-Gross formula for electron plasma waves. Notice the group velocity:

v_g =

dω

2ω

d ω²

γk v_t²

( ω_p² + γk² v_t² )^¹/₂

≠ 0.

(5.121)

and for kv_t > ω_p this tends to γ^¹/₂ v_t. In this limit energy travels at the electron thermal speed.

5.4.1 Refractive Index Plot

Bohm Gross electron plasma waves:

N² =

c²

γ_e v_te²

⎛
⎝

1 −

ω_p²

ω²

⎞
⎠

(5.122)

Transverse electromagnetic waves:

N² =

⎛
⎝

1 −

ω_p²

ω²

⎞
⎠

(5.123)

These have just the same shape except the electron plasma waves have much larger vertical scale:

Figure 5.8: Refractive Index Plot. Top plot on the scale of the Bohm-Gross Plasma waves. Bottom plot, on the scale of the E-M transverse waves

On the E-M wave scale, the plasma wave curve is nearly vertical. In the cold plasma it was exactly vertical.

We have relaxed the Cold Plasma approximation.

5.4.2 Including the ion response

As an example of the different things which can occur when ions are allowed to move, include longitudinal ion response:

0 = ϵ_zz = 1 −

ω_pe²

ω² −	k² γ_e T_e m_e

−

ω_pi²

ω² −	k² γ_i T_i m_i

(5.124)

This is now a quadratic equation for ω² so there are two ω solutions possible for a given k. One will be in the vicinity of the electron plasma wave solution and the inclusion of ω_pi² which is << ω_pe² will give a small correction.

Second solution will be where the third term is same magnitude as second (both will be >> 1). This will be at low frequency. So we may write the dispersion relation approximately as:

−

ω_pe²

−	k² γ_e T_e m_e

−

ω_pi²

ω² −	k² γ_i T_i m_i

= 0

(5.125)

i.e.

ω²

k²γ_i T_i

m_i

ω_pi²

ω_pe²

k² γ_e T_e

m_e

= k²

⎡
⎣

γ_i T_i + γ_e T_e

m_i

⎤
⎦

(5.126)

[In this case the electrons have time to stream through the wave in 1 oscillation so they tend to be isothermal: i.e. γ_e = 1. What to take for γ_i is less clear, and less important because kinetic theory shows that these waves we have just found are strongly damped unless T_i << T_e.]

These are `ion-acoustic' or `ion-sound' waves

ω²

k²

= c_s²

(5.127)

c_s is the sound speed

c_s² =

γ_i T_i + T_e

m_i

≅

T_e

m_i

(5.128)

Approximately non-dispersive waves with phase velocity c_s.

5.5 Electrostatic Approximation for (Plasma) Waves

The dispersion relation is written generally as

N ∧( N ∧E) + ϵ . E= N ( N . E) − N² E+ ϵ . E= 0

(5.129)

Consider E to be expressible as longitudinal and transverse components E_l, E_t such that N ∧E_l = 0, N . E_t = 0 . Then the dispersion relation can be written

N ( N . E_l ) − N² ( E_l + E_t ) + ϵ . ( E_l + E_t ) = − N² E_t + ϵ . E_t + ϵ .E_l = 0

(5.130)

( N² − ϵ ) . E_t = ϵ . E_l

(5.131)

Now the electric field can always be written as the sum of a curl-free component plus a divergenceless component, e.g. conventionally

− ∇ϕ

[(Curl−free) || Electrostatic]

⋅

[(Divergence−free) || Electromagnetic]

(5.132)

and these may be termed electrostatic and electromagnetic parts of the field.

For a plane wave, these two parts are clearly the same as the longitudinal and transverse parts because

− ∇ϕ = − i k ϕ is longitudinal

(5.133)

and if ∇ . · A = 0 (because ∇ . A = 0 (w.l.o.g.)) then k . · A = 0 so · A is transverse.

`Electrostatic' waves are those that are describable by the electrostatic part of the electric field, which is the longitudinal part: | E_l | >> | E_t |.

If we simply say E_t = 0 then the dispersion relation becomes ϵ . E_l = 0. This is not the most general dispersion relation for electrostatic waves. It is too restrictive. In general, there is a more significant way in which to get solutions where | E_l | >> | E_t |. It is for N² to be very large compared to all the components of ϵ :N² >> ||ϵ||.

If this is the case, then the dispersion relation is approximately

N² E_t = ϵ . E_l ;

(5.134)

E_t is small but not zero.

We can then annihilate the E_t term by taking the N component of this equation; leaving

N . ϵ . E_l = 0 or ( N . ϵ . N ) E_l = 0 : k . ϵ . k = 0 .

(5.135)

When the medium is isotropic there is no relevant difference between the electrostatic dispersion relation:

N . ϵ . N = 0

(5.136)

and the purely longitudinal case ϵ . N = 0. If we choose axes such that N is along ∧z, then the medium's isotropy ensures the off-diagonal components of ϵ are zero so N . ϵ . N = 0 requires ϵ_zz = 0 ⇒ ϵ . N = 0. However if the medium is not isotropic, then even if

N . ϵ . N ( = N² ϵ_zz ) = 0

(5.137)

there may be off-diagonal terms of ϵ that make

ϵ . N ≠ 0

(5.138)

In other words, in an anisotropic medium (for example a magnetized plasma) the electrostatic approximation can give waves that have non-zero transverse electric field (of order ||ϵ|| / N² times E_l) even though the waves are describable in terms of a scalar potential.

To approach this more directly, from Maxwell's equations, applied to a dielectric medium of dielectric tensor ϵ, the electrostatic part of the electric field is derived from the electric displacement

∇ . D = ∇ . ( ϵ₀ ϵ . E) = ρ = 0 (no free charges)

(5.139)

So for plane waves 0 = k . D = k . ϵ . E = i k . ϵ . k ϕ.

The electric displacement, D, is purely transverse (not zero) but the electric field, E then gives rise to an electromagnetic field via ∇ ∧H = ∂D/∂t. If N² >> ||ϵ|| then this magnetic (inductive) component can be considered as a benign passive coupling to the electrostatic wave.

In summary, the electrostatic dispersion relation is k. ϵ . k=0, or in coordinates where k is in the z-direction, ϵ_zz=0.

5.6 Simple Example of MHD Dynamics: Alfven Waves

Ignore Pressure & Resistivity.

D V

= j∧B

(5.140)

E+ V ∧B= 0

(5.141)

Linearize:

V = V₁, B = B₀ + B₁ (B₀ uniform), j = j₁.

(5.142)

∂V

∂t

= j∧B₀

(5.143)

E+ V ∧B₀ = 0

(5.144)

Fourier Transform:

ρ(− i ω) V = j∧B₀

(5.145)

E+ V ∧B₀ = 0

(5.146)

Eliminate V by taking 5.145 ∧B₀ and substituting from 5.146.

E+

− i ωρ

( j∧B₀ ) ∧B₀ = 0

(5.147)

E= −

− i ωρ

{ ( j. B₀ ) B₀ − B₀² j} =

B₀²

− i ωρ

j_⊥

(5.148)

So conductivity tensor can be written (z in B direction).

σ =

− i ωρ

B₀²

⎡
⎢
⎢
⎢
⎣

∞

⎤
⎥
⎥
⎥
⎦

(5.149)

where ∞ implies that E_|| = 0 (because of Ohm's law). Hence Dielectric Tensor

ϵ = 1 +

− i ωϵ₀

⎛
⎝

1 +

ϵ₀ B²

⎞
⎠

⎡
⎢
⎢
⎢
⎣

∞

⎤
⎥
⎥
⎥
⎦

(5.150)

Dispersion tensor in general is:

ω²

c²

[ N N − N² + ϵ ]

(5.151)

Dispersion Relation taking N_⊥ = N_x, N_y=0

|K|=

⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢

−N_||² + 1 +

ϵ₀B²

N_⊥ N_||

−N_||² − N_⊥² + 1 +

ϵ₀B²

N_⊥ N_||

∞

⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢

= 0

(5.152)

Meaning of ∞ is that the cofactor must be zero i.e.

⎛
⎝

− N_||² + 1 +

ϵ₀ B²

⎞
⎠

⎛
⎝

− N² + 1 +

ϵ₀ B²

⎞
⎠

= 0

(5.153)

The 1's here come from Maxwell displacement current and are usually negligible (N_⊥² >> 1). So final waves are

Figure 5.9: Compressional Alfven Wave. Works by magnetic pressure (primarily).

N² = [(ρ)/(ϵ₀ B²)] ⇒ Non-dispersive wave with phase and group velocities

v_p = v_g = c
N
= ⎛
⎝ c² ϵ₀ B²
ρ
⎞
⎠ ¹/₂

= ⎡
⎣ B²
μ₀ ρ
⎤
⎦ ¹/₂

(5.154)
where we call

⎡
⎣ B²
μ₀ ρ
⎤
⎦ ¹/₂

≡ v_A the `Alfven Speed′
(5.155)
Polarization:

E_|| = E_z = 0, E_x = 0. E_y ≠ 0 ⇒ V_y = 0 V_x ≠ 0 (V_z = 0)
(5.156)
Party longitudinal (velocity) wave → Compression "Compressional Alfven Wave".
N_||² = [( ρ)/(ϵ₀ B²)] = [( k_||² c²)/(ω²)]
Any ω has unique k_||. Wave has unique velocity in || direction: v_A.
Polarization

E_z = E_y = 0 E_x ≠ 0 ⇒ V_x = 0 V_y ≠ 0 (V_z = 0)
(5.157)
Transverse velocity: "Shear Alfven Wave".

Figure 5.10: Shear Alfven Wave

Works by field line bending (Tension Force) (no compression).

5.7 Non-Uniform Plasmas and wave propagation

Practical plasmas are not infinite & homogeneous. So how does all this plane wave analysis apply practically?

If the spatial variation of the plasma is slow c.f. the wave length of the wave, then coupling to other waves will be small (negligible).

Figure 5.11: Comparison of sudden and gradualy refractive index change.

For a given ω, slowly varying plasma means N/[dN/dx] >> λ or k N/[dN/dx] >> 1. Locally, the plasma appears uniform.

Even if the coupling is small, so that locally the wave propagates as if in an infinite uniform plasma, we still need a way of calculating how the solution propagates from one place to the other. This is handled by the `WKB(J)' or `eikonal' or `ray optic' or `geometric optics' approximation.

WKBJ solution

Consider the model 1-d wave equation (for field ω)

d² E

d x²

+ k² E = 0

(5.158)

with k now a slowly varying function of x. Seek a solution in the form

E = exp( i ϕ( x ) ) (− i ωt implied)

(5.159)

ϕ is the wave phase ( = kx in uniform plasma).

Differentiate twice

d² E

d x²

= { i

d² ϕ

d x²

−

⎛
⎝

d ϕ

d x

⎞
⎠

} e^iϕ

(5.160)

Substitute into differential equation to obtain

⎛
⎝

d ϕ

d x

⎞
⎠

= k² + i

d² ϕ

d x²

(5.161)

Recognize that in uniform plasma [(d² ϕ)/(d x²)] = 0. So in slightly non-uniform, 1st approx is to ignore this term.

d ϕ

d x

≅ ±k(x)

(5.162)

Then obtain a second approximation by substituting

d² ϕ

d x²

≅ ±

d k

d x

(5.163)

⎛
⎝

d ϕ

d x

⎞
⎠

≅

k² ±i

d k

d x

(5.164)

d ϕ

d x

≅

⎛
⎝

k ±

2 k

d k

d x

⎞
⎠

using Taylor expansion.

(5.165)

Integrate:

ϕ ≅ ±

⌠
⌡

k d x + i ln( k^¹/₂ )

(5.166)

Hence E is

E = e^{i ϕ} =

k^¹/₂

exp

⎛
⎝

±i

⌠
⌡

k d x

⎞
⎠

(5.167)

This is classic WKBJ solution. Originally studied by Green & Liouville (1837), the Green of Green's functions, the Liouville of Sturm Liouville theory.

Basic idea of this approach: (1) solve the local dispersion relation as if in infinite homogeneous plasma, to get k(x), (2) form approximate solution for all space as above.

Phase of wave varies as integral of k d x.

In addition, amplitude varies as [1/(k^¹/₂ )]. This is required to make the total energy flow uniform.

5.8 Two Stream Instability

An example of waves becoming unstable in a non-equilibrium plasma. Analysis is possible using Cold Plasma techniques.

Consider a plasma with two participating cold species but having different average velocities.

These are two "streams".

Species 1

Species 2

. →

Moving.

Stationary .

Speed v

(5.168)

We can look at them in different inertial frames, e.g. species (stream) 2 stationary or 1 stationary (or neither).

We analyse by obtaining the susceptibility for each species and adding together to get total dielectric constant (scalar 1-d if unmagnetized).

In a frame of reference in which it is stationary, a stream j has the (Cold Plasma) susceptibility

χ_j =

− ω_pj²

ω²

(5.169)

If the stream is moving with velocity v_j (zero order) then its susceptibility is

χ_j =

− ω_pj²

( ω− k v_j )²

. (k & v_j in same direction)

(5.170)

Proof from equation of motion:

q_j

m_j

E =

∂

∂t

+ v. ∇

= ( − i ω+ i k . v_j )

= − i ( ω− k v_j )

(5.171)

Current density

j_j = ρ_j v_j + ρ_j .

v_j .

(5.172)

Substitute in

∇ . j+

∂ρ

∂t

i k .

ρ+ i k .

ρ− i ω

= 0

(5.173)

ρ_j

k .

ω− k . v_j

(5.174)

Hence substituting for ~v in terms of E:

− χ_j ϵ₀ ∇ . E=

ρ_j q_j

m_j

k . E

−i ( ω− k .v_j )²

(5.175)

which shows the longitudinal susceptibility is

χ_j = −

ρ_j q_j

m_j ϵ₀

( ω² − k v_j )²

− ω_pj²

( ω− k v_j )²

(5.176)

Proof by transforming frame of reference:

Consider Galilean transformation to a frame moving with the stream at velocity v_j.

x= x′+ v_j t ; t′ = t

(5.177)

expi ( k . x− ωt ) = expi ( k . x′− ( ω− k .v_j ) t′)

(5.178)

So in frame of the stream, ω′ = ω− k . v_j.

Substitute in stationary cold plasma expression:

χ_j = −

ω_pj²

ω^′2

= −

ω_pj²

( ω− k v_j )²

(5.179)

Thus for n streams we have

ϵ = 1 +

∑
j

χ_j = 1 −

∑
j

ω_pj²

( ω− k v_j )²

(5.180)

Longitudinal wave dispersion relation is

ϵ = 0 .

(5.181)

Two streams

0 = ϵ = 1 −

ω_p1²

( ω− k v₁ )²

−

ω_p2²

( ω− k v₂ )²

(5.182)

For given real k this is a quartic in ω. It has the form:

Figure 5.12: Two-stream stability analysis.

If ϵ crosses zero between the wells, then ∃ 4 real solutions for ω. (Case B).

If not, then 2 of the solutions are complex: ω = ω_r ±i ω_i (Case A).

The time dependence of these complex roots is

exp(−i ωt ) = exp( − i ω_r t ±ω_i t ) .

(5.183)

The +ve sign is growing in time: instability.

It is straightforward to show that Case A occurs if

| k (v₂ − v₁) | < [ ω_p1^²/₃ + ω_p2^²/₃ ]^³/₂ .

(5.184)

Small enough k (long enough wavelength) is always unstable.

Simple interpretation (ω_p2² << ω_p1², v₁ = 0) a tenuous beam in a plasma sees a negative ϵ if | k v₂ | < ∼ ω_p1 .

Negative ϵ implies charge perturbation causes E that enhances itself: charge (spontaneous) bunching.

5.9 Kinetic Theory of Plasma Waves

Wave damping is due to wave-particle resonance. To treat this we need to keep track of the particle distribution in velocity space → kinetic theory.

5.9.1 Vlasov Equation

Treat particles as moving in 6-D phase space x position, v velocity. At any instant a particle occupies a unique position in phase space (x,v).

Consider an elemental volume d³ xd³ v of phase space [dx dy dz dv_x dv_y dv_z], at (x, v). Write down an equation that is conservation of particles for this volume

−

∂

∂t

( f d³ xd³ v)

⎡
⎣

v_x f

⎛
⎝

x+ d x

, v

⎞
⎠

− v_x f ( x, v)

⎤
⎦

dy dz d³ v

same for dy, dz

⎡
⎣

a_x f

⎛
⎝

x, v+ d v_x

⎞
⎠

−a_x f ( x, v)

⎤
⎦

d³ xd v_y d v_z

same for d v_y, d v_z

(5.185)

Figure 5.13: Difference in flow across x-surfaces (+y+z).

a is "velocity space motion", i.e. acceleration.

Divide through by d³ xd³ v and take limit

−

∂f

∂t

∂

∂x

( v_x f ) +

∂

∂y

( v_y f ) +

∂

∂z

( v_z f ) +

∂

∂v_x

( a_x f ) +

∂

∂v_y

( a_y f ) +

∂

∂v_z

( a_z f )

∇ . ( vf ) + ∇_v . ( a f )

(5.186)

[Notation: Use [(∂)/(∂x)] ↔ ∇ ; [(∂)/(∂v)] ↔ ∇_v ].

Take this simple continuity equation in phase space and expand:

∂f

∂t

+ ( ∇ . v)f + ( v. ∇ ) f + ( ∇_v .a ) f + ( a . ∇_v ) f = 0 .

(5.187)

Recognize that ∇ means here [(∂)/(∂x)] etc. keeping v constant so that ∇ . v = 0 by definition. So

∂f

∂t

+ v.

∂f

∂x

+ a .

∂f

∂v

= − f ( ∇_v . a )

(5.188)

Now we want to couple this equation with Maxwell's equations for the fields, and the Lorentz force

a =

( E+ v∧B)

(5.189)

Actually we don't want to use the E retaining all the local effects of individual particles. We want a smoothed out field. Ensemble averaged E.

Evaluate

∇_v . a

∇_v .

( E+ v∧B) =

∇_v . ( v∧B)

(5.190)

B. ( ∇_v ∧v) = 0 .

(5.191)

So RHS is zero. However in the use of smoothed out E we have ignored local effect of one particle on another due to the graininess. That is collisions.

Boltzmann Equation:

∂f

∂t

+ v.

∂f

∂x

+ a .

∂f

∂v

⎛
⎝

∂f

∂t

⎞
⎠

collisions

(5.192)

Vlasov Equation ≡ Boltzman Eq without collisions. For electromagnetic forces:

∂f

∂t

+ v.

∂f

∂x

( E+ v∧B)

∂f

∂v

= 0 .

(5.193)

Interpretation:

Distribution function is constant along particle orbit in phase space: [d/dt] f = 0.

f =

∂f

∂t

d x

d t

∂f

∂x

∂v

d t

∂f

∂v

(5.194)

Coupled to Vlasov equation for each particle species we have Maxwell's equations.

Vlasov-Maxwell Equations

∂f_j

∂t

∂f_j

∂x

q_j

m_j

( E+ v∧B) .

∂f_j

∂v_j

= 0

(5.195)

∇ ∧E

− ∂B

∂t

, ∇ ∧B = μ₀ j+

c²

∂E

∂t

(5.196)

∇ . E

ϵ₀

, ∇ . B = 0

(5.197)

Coupling is completed via charge & current densities.

∑
j

q_j n_j =

∑
J

q_j

⌠
⌡

f_j d³v

(5.198)

∑
j

q_j n_j V_j =

∑
j

q_j

⌠
⌡

f_j vd³ v.

(5.199)

Describe phenomena in which collisions are not important, keeping track of the (statistically averaged) particle distribution function.

Plasma waves are the most important phenomena covered by the Vlasov-Maxwell equations.

6-dimensional, nonlinear, time-dependent, integral-differential equations!

5.9.2 Linearized Wave Solution of Vlasov Equation

Unmagnetized Plasma

Linearize the Vlasov Eq by supposing

f₀ (v) + f₁ (v) expi ( k . x− ωt ) , f₁ small.

(5.200)

also E

E₁ expi ( k . x− ωt) B = B₁ expi ( k . x− ωt )

(5.201)

Zeroth order f₀ equation satisfied by [(∂)/(∂t)] , [(∂)/(∂x)] = 0. First order:

− i ωf₁ + v. i k f₁ +

( E₁ +v∧B₁ ) .

∂f₀

∂v

= 0 .

(5.202)

[Note v is not per se of any order, it is an independent variable.]

Solution:

f₁ =

i ( ω− k . v)

( E₁ + v∧B₁ ) .

∂f₀

∂v

(5.203)

For convenience, assume f₀ is isotropic. Then [(∂f₀)/(∂v)] is in direction v so v∧B₁ . [(∂f₀)/(∂v)] = 0

f₁ =

E₁ .

∂f₀

∂v

i ( ω− k . v)

(5.204)

We want to calculate the conductivity σ. Do this by simply integrating:

⌠
⌡

q f₁ vd³ v =

q²

i m

⌠
⌡

∂f₀

∂v

ω− k . v

d³ v . E₁.

(5.205)

Here the electric field has been taken outside the v-integral but its dot product is with ∂f₀/∂v. Hence we have the tensor conductivity,

σ =

q²

i m

⌠
⌡

∂f₀

∂v

ω− k . v

d³ v

(5.206)

Focus on zz component:

1 + χ_zz = ϵ_zz = 1 +

σ_zz

− i ωϵ₀

= 1 +

q²

ωm ϵ₀

⌠
⌡

v_z

∂f₀

∂v_z

ω− k . v

d³ v

(5.207)

Such an expression applies for the conductivity (susceptibility) of each species, if more than one needs to be considered.

It looks as if we are there! Just do the integral!

Now the problem becomes evident. The integrand has a zero in the denominator. At least we can do 2 of 3 integrals by defining the 1-dimensional distribution function

f_z (v_z) ≡

⌠
⌡

f (v) d v_x d v_y (k = k

)

(5.208)

Then

χ =

q²

ωm ϵ₀

⌠
⌡

v_z

∂f_z

∂v_z

ω− k v_z

d v_z

(5.209)

(drop the z suffix from now on. 1-d problem).

How do we integrate through the pole at v = [( ω)/k]? Contribution of resonant particles. Crucial to get right.

Path of velocity integration

First, realize that the solution we have found is not complete. In fact a more general solution can be constructed by adding any solution of

∂f₁

∂t

+ v

∂f₁

∂z

= 0

(5.210)

[We are dealing with 1-d Vlasov equation: [(∂f)/(∂t)] + v [(∂f)/(∂z)] + [q E/m] [(∂f)/(∂v)] = 0.] Solution of this is

f₁ = g (v t − z, v)

(5.211)

where g is an arbitrary function of its arguments. Hence general solution is

f₁ =

∂f₀

∂v

i ( ω− k v )

expi ( k z − ωt ) + g ( v t − z, v )

(5.212)

and g must be determined by initial conditions. In general, if we start up the wave suddenly there will be a transient that makes g non-zero.

So instead we consider a case of complex ω (real k for simplicity) where ω = ω_r + i ω_i and ω_i > 0.

This case corresponds to a growing wave:

exp( − i ωt ) = exp( − i ω_r t + ω_i t )

(5.213)

Then we can take our initial condition to be f₁=0 at t → −∞. This is satisfied by taking g = 0.

For ω_i > 0 the complementary function, g, is zero.

Physically this can be thought of as treating a case where there is a very gradual, smooth start up, so that no transients are generated.

Thus if ω_i > 0, the solution is simply the velocity integral, taken along the real axis, with no additional terms. For

ω_i > 0, χ =

q²

ωm ϵ₀

⌠
⌡

∂f

∂v

ω− k v

d v

(5.214)

where there is now no difficulty about the integration because ω is complex.

Figure 5.14: Contour of integration in complex v-plane.

The pole of the integrand is at v = [(ω)/k] which is above the real axis.

The question then arises as to how to do the calculation if ω_i ≤ 0. The answer is by "analytic continuation", regarding all quantities as complex.

"Analytic Continuation" of χ is accomplished by allowing ω/k to move (e.g. changing the ω_i) but never allowing any poles to cross the integration contour, as things change continuously.

Remember (Fig 5.15)

⌠
(⎜)
⌡

F d z =

∑

residues ×2 πi

(5.215)

(Cauchy's theorem)

Figure 5.15: Cauchy's theorem.

Where residues = lim_{z→ z_k}[F(z)/(z−z_k)] at the poles, z_k, of F(z). We can deform the contour how we like, provided no poles cross it. Hence contour (Fig 5.16)

Figure 5.16: Landau Contour

We conclude that the integration contour for ω_i < 0 is not just along the real v axis. It includes the pole also.

To express our answer in a universal way we use the notation of "Principal Value" of a singular integral defined as the average of paths above and below

℘

⌠
⌡

v − v₀

d v =

⎡
⎣

⌠
⌡

C₁

⌠
⌡

C₂

⎤
⎦

v − v₀

d v

(5.216)

Figure 5.17: Two halves of principal value contour.

Then

χ =

q²

ωm ϵ₀

{ ℘

⌠
⌡

∂f₀

∂v

ω− k v

d v −

2 πi

k²

∂f₀

∂v

⎢
⎢

v = [(ω)/k]

}

(5.217)

Second term is half the normal residue term; so it is half of the integral round the pole.

Figure 5.18: Contour equivalence.

Our expression is only short-hand for the (Landau) prescription:

"Integrate below the pole". (Nautilus).

Contribution from the pole can be considered to arise from the complementary function g(v t − z, v). If g is to be proportional to exp(ikz), then it must be of the form g = exp[i k (z − vt)] h(v) where h(v) is an arbitrary function. To get the result previously calculated, the value of h(v) must be (for real ω)

h(v) = Eπ

∂f₀

∂v

⎢
⎢

^w/_k

⎛
⎝

v −

⎞
⎠

(5.218)

(so that

⌠
⌡

− i ωϵ₀

v g d v =

q²

ωm ϵ₀

⎛
⎝

πi

k²

∂f₀

∂v

⎢
⎢

[(ω)/k]

⎞
⎠

. )

(5.219)

This Dirac delta function says that the complementary function is limited to particles with "exactly" the wave phase speed [(ω)/k]. It is the resonant behaviour of these particles and the imaginary term they contribute to χ that is responsible for wave damping or growth.

We shall see in a moment, that the standard case will be ω_i < 0, so the opposite of the prescription ω_i > 0 that makes g = 0. Therefore there will generally be a complementary function, non-zero, describing resonant effects. We don't have to calculate it explicitly because the Landau prescription takes care of it.

5.9.3 Landau's original approach. (1946)

Corrected Vlasov's assumption that the correct result was just the principal value of the integral. Landau recognized the importance of initial conditions and so used Laplace Transform approach to the problem

(p) =

⌠
⌡

∞

0

e^−pt A(t) dt

(5.220)

The Laplace Transform inversion formula is

A(t) =

2 πi

⌠
⌡

s+i∞

s−i∞

e^pt

(p) dp

(5.221)

where the path of integration must be chosen to the right of any poles of ~A(p) (i.e. s large enough). Such a prescription seems reasonable. If we make ℜ(p) large enough then the ~A(p) integral will presumably exist. The inversion formula can also be proved rigorously so that gives confidence that this is the right approach.

If we identify p → − i ω, then the transform is ~A = ∫e^{i ωt} A(t) dt, which can be identified as the Fourier transform that would give component ~A ∝ e^{−i ωt}, the wave we are discussing. Making ℜ(p) positive enough to be to the right of all poles is then equivalent to making ℑ(ω) positive enough so that the path in ω-space is above all poles, in particular ω_i > ℑ(kv). For real velocity, v, this is precisely the condition ω_i > 0, we adopted before to justify putting the complementary function zero.

Either approach gives the same prescription. It is all bound up with satisfying causality.

5.9.4 Solution of Dispersion Relation

We have the dielectric tensor

ϵ = 1 + χ = 1 +

q²

ωm ϵ₀

⎧
⎨
⎩

℘

⌠
⌡

∂f₀

∂v

ω− k v

d v − πi

k²

∂f₀

∂v

⎢
⎢

[(ω)/k]

⎫
⎬
⎭

(5.222)

for a general isotropic distribution. We also know that the dispersion relation is

⎡
⎢
⎢
⎢
⎣

−N² + ϵ_t

⎤
⎥
⎥
⎥
⎦

= ( −N² + ϵ_t )² ϵ = 0

(5.223)

Giving transverse waves N² = ϵ_t and longitudinal waves ϵ = 0. We need to do the integral and hence get ϵ.

Presumably, if we have done this right, we ought to be able to get back the cold-plasma result as an approximation in the appropriate limits, plus some corrections. We previously argued that cold-plasma is valid if [(ω)/k] >> v_t. So regard [kv/(ω)] as a small quantity and expand:

℘

⌠
⌡

∂f₀

⎛
⎝

1 −

⎞
⎠

⌠
⌡

∂f₀

∂v

⎡
⎣

1 +

⎛
⎝

⎞
⎠

+ ...

⎤
⎦

−1

⌠
⌡

f₀

⎡
⎣

1 +

2 kv

+ 3

⎛
⎝

⎞
⎠

+ ...

⎤
⎦

dv (by parts)

≅

− 1

⎡
⎣

n +

3 n T

k²

ω²

⎤
⎦

+ ...

(5.224)

Here we have assumed we are in the particles' average rest frame (no bulk velocity) so that ∫f₀ vdv = 0 and also we have used the temperature definition

n T =

⌠
⌡

m v² f₀ d v ,

(5.225)

appropriate to one degree of freedom (1-d problem). Ignoring the higher order terms we get:

ϵ = 1 −

ω_p²

ω²

⎧
⎨
⎩

1 + 3

k²

ω²

+ πi

ω²

k²

∂f₀

∂v

⎢
⎢

[(ω)/k]

⎫
⎬
⎭

(5.226)

This is just what we expected. Cold plasma value was ϵ = 1 − [(ω_p²)/(ω²)]. We have two corrections

To real part of ϵ, correction 3 ^T/_m[(k²)/(ω²)] = 3 ( [(v_t)/(v_p)] )² due to finite temperature. We could have got this from a fluid treatment with pressure.
Imaginary part → antihermitian part of ϵ→ dissipation.

Solve the dispersion relation for longitudinal waves ϵ = 0 (again assuming k real ω complex). Assume ω_i << ω_r then

( ω_r + i ω_i )²

≅

ω_r² + 2 ω_r ω_i i = ω_p²

⎧
⎨
⎩

1 + 3

k²

ω²

+ πi

ω²

k²

∂f₀

∂v

⎢
⎢

[(ω)/k]

⎫
⎬
⎭

≅

ω_p²

⎧
⎨
⎩

1 + 3

k²

ω_r²

+ πi

ω_r²

k²

∂f₀

∂v

⎢
⎢

[(ω_r)/k]

⎫
⎬
⎭

(5.227)

Hence

ω_i ≅

2 ω_r i

ω_p² πi

ω_r²

k²

∂f₀

∂v

⎢
⎢

[(ω_r)/k]

=ω_p²

ω_r

k²

∂f₀

∂v

⎢
⎢

[(ω_r)/k]

(5.228)

For a Maxwellian distribution

f₀ =

⎛
⎝

2 πT

⎞
⎠

¹/₂

exp

⎛
⎝

−

m v²

2 T

⎞
⎠

(5.229)

∂f₀

∂v

⎛
⎝

2 πT

⎞
⎠

¹/₂

⎛
⎝

−

m v

⎞
⎠

exp

⎛
⎝

−

m v²

⎞
⎠

n .

(5.230)

So, substituting,

ω_i ≅ − ω_p²

ω_r²

k³

⎛
⎝

2 πT

⎞
⎠

¹/₂

exp

⎛
⎝

−

m ω_r²

2 T k²

⎞
⎠

(5.231)

The difference between ω_r and ω_p may not be important in the outside but ought to be retained inside the exponential since

ω_p²

k²

⎡
⎣

1 + 3

k²

ω_p²

⎤
⎦

m ω_p²

2 T k²

(5.232)

ω_i ≅ − ω_p

⎛
⎝

⎞
⎠

¹/₂

ω_p³

k³

v_t³

exp

⎛
⎝

−

m ω_p²

2 T k²

−

⎞
⎠

(5.233)

Imaginary part of ω is negative ⇒ damping. This is Landau Damping.

Note that we have been treating a single species (electrons by implication) but if we need more than one we simply add to χ. Solution is then more complex.

5.9.5 Direct Calculation of Collisionless Particle Heating

(Landau Damping without complex variables!)

We show by a direct calculation that net energy is transferred to electrons.

Suppose there exists a longitudinal wave

E= E cos(k z − ωt )

(5.234)

Equations of motion of a particle

d v

d t

E cos(k z − ωt )

(5.235)

d z

d t

(5.236)

Solve these assuming E is small by a perturbation expansion v = v₀ + v₁ + ..., z = z₀(t) + z₁(t) + ... .

Zeroth order:

d v₀

d t

= 0 ⇒ v₀ = const , z₀ = z_i + v₀ t

(5.237)

where z_i = const is the initial position.

First Order

d v₁

d t

E cos( k z₀ − ωt ) =

E cos( k ( z_i + v₀ t ) − ωt )

(5.238)

d z₁

d t

v₁

(5.239)

Integrate:

v₁ =

q E

sin( k z_i + [k v₀ − ω] t )

k v₀ − ω

+ const.

(5.240)

take initial conditions to be v₁, v₂ = 0. Then

v₁ =

q E

sin( k z_i + ∆ωt ) − sin( k z_i )

∆ω

(5.241)

where ∆ω ≡ k v₀ − ω, is (-) the frequency at which the particle feels the wave field.

z₁ =

q E

⎡
⎣

cosk z_i − cos( k z_i + ∆ωt )

∆ω²

− t

sink z_i

∆ω

⎤
⎦

(5.242)

(using z₁(0) = 0).

2nd Order (Needed to get energy right)

d v₂

d t

q E

{cos( k z₀ − ωt + k z₁ ) − cos( k z₀t − ωt ) }

q E

{ cos( k z₀ − ωt)cos( k z₁) −sin( k z₀ − ωt)sin( k z₁) − cos( k z₀t − ωt ) }

−

q E

k z₁ sin( k z_i + ∆ωt ) (k z₁ << 1)

(5.243)

Now the gain in kinetic energy of the particle is

m v² −

m v₀²

m { ( v₀ + v₁ + v₂ + ... )² − v₀² }

m { 2 v₀ v₁ + v₁² + 2 v₀ v₂ + higher order }

(5.244)

The total rate of increase of K.E. is the average of the time derivative of this over space, i.e. over z_i.

d t

⎛
⎝

m v²

⎞
⎠

> = m < v₀

d v₁

d t

+ v₁

d v₁

d t

+v₀

d v₂

d t

(5.245)

The z_i average will cancel any component that simply oscillates with z_i, such as sinkz_i or sin(kz_i+∆ωt) cos(kz_i+∆ωt) but not cos² kz_i or sin² kz_i.

< v₀

d v₁

d t

(5.246)

< v₁

d v₁

d t

q² E²

m²

sin( k z_i + ∆ωt ) − sink z_i

∆ω

cos( k z_i + ∆ωt ) >

q² E²

m²

− sin( k z_i + ∆ωt ) cos∆ωt + cos( k z_i + ∆ωt )sin∆ωt

∆ω

cos( k z_i + ∆ωt ) >

q² E²

m²

sin∆ωt

∆ω

cos² ( k z_i + ∆ωt ) > =

q² E²

2m²

sin∆ωt

∆ω

(5.247)

< v₀

d v₂

d t

− q² E²

m²

k v₀ <

⎛
⎝

cosk z_i − cos( k z_i + ∆ωt )

∆ω²

− t

sink z_i

∆ω

⎞
⎠

sin( k z_i + ∆ωt ) >

− q² E²

m²

k v₀ <

sin∆ωt

∆ω²

cos²kz_i − t

cos∆ωt

∆ω

sin² k z_i >

q² E²

m²

k v₀

⎡
⎣

−

sin∆ωt

∆ω²

+ t

cos∆ωt

∆ω

⎤
⎦

(5.248)

Hence

m v² >

q² E²

2 m

⎡
⎣

sin∆ωt

∆ω

− k v₀

sin∆ωt

∆ω²

+ k v₀ t

cos∆ωt

∆ω

⎤
⎦

q² E²

2 m

⎡
⎣

− ωsin∆ωt

∆ω²

ωt

∆ω

cos∆ωt + t cos∆ωt

⎤
⎦

(5.249)

This is the space-averaged power into particles of a specific velocity v₀. We need to integrate over the distribution function. A trick identify helps:

− ω

∆ω²

sin∆ωt

ωt

∆ω

cos∆ωt + t cos∆ωt =

∂

∂∆ω

⎛
⎝

ωsin∆ωt

∆ω

+ sin∆ωt

⎞
⎠

∂

∂v₀

⎛
⎝

ωsin∆ωt

∆ω

+ sin∆ωt

⎞
⎠

(5.250)

Hence power per unit volume is

⌠
⌡

d t

m v² > f ( v₀ ) d v₀

q² E²

2 m k

⌠
⌡

∂

∂v₀

⎛
⎝

ωsin∆ωt

∆ω

+ sin∆ωt

⎞
⎠

f ( v₀ ) d v₀

−

q² E²

2 m k

⌠
⌡

⎛
⎝

ωsin∆ωt

∆ω

+ sin∆ωt

⎞
⎠

∂f

∂v₀

d v₀

(5.251)

As t becomes large, sin∆ωt = sin(k v₀ − ω)t becomes a rapidly oscillating function of v₀. Hence second term of integrand contributes negligibly and the first term,

∝

ωsin∆ωt

∆ω

sin∆ωt

∆ωt

ωt

(5.252)

becomes a highly localized, delta-function-like quantity. That enables the rest of the integrand to be evaluated just where ∆ω = 0 (i.e. k v₀ − ω = 0).

Figure 5.19: Localized integrand function.

So:

P = −

q² E²

2 m k

∂f

∂v

⎢
⎢

[(ω)/k]

⌠
⌡

sinx

d x

(5.253)

Here, x = ∆ωt = ( k v₀ − ω) t; but ∫[ sinx/x] d z = π so

P = − E²

πq² ω

2 m k²

∂f₀

∂v

⎢
⎢

[(ω)/k]

(5.254)

We have shown that there is a net transfer of energy to particles at the resonant velocity [(ω)/k] from the wave. (Positive if [(∂f)/(∂v)]| is negative.)

5.9.6 Physical Picture

∆ω is minus the wave frequency in the particles' (unperturbed) frame of reference, or equivalently it is k v₀′ where v₀′ is particle speed in wave frame of reference. The latter is easier to deal with. ∆ωt = k v₀′t is the phase the particle travels in time t. We found that the energy gain was of the form

⌠
⌡

sin∆ωt

∆ωt

d (∆ωt ) .

(5.255)

Figure 5.20: Phase distance traveled in time t.

This integrand becomes small (and oscillatory) for ∆ωt >> 1. Physically, this means that if particle moves through many wavelengths its energy gain is small. Dominant contribution is from ∆ωt < π. These are particles that move through less than ¹/₂ wavelength during the period under consideration. These are the resonant particles.

Figure 5.21: Dominant contribution

Particles moving slightly faster than wave are slowed down. This is a second-order effect.

Figure 5.22: Particles moving slightly faster than the wave.

Some particles of the same initial velocity v₀ group are being accelerated (A) some slowed (B). A short time after the wave is switched on, particles starting at (A) will be moving slightly faster than those starting at (B). The (A) particles therefore reach the bottom of the well and stop being accelerated sooner than the (B) particles reach the top. So (B) particles lose more momentum than (A) particles gain: on average, faster-than-wave particles are slowed. Similarly particles moving slightly slower than wave (i.e. from right to left in the wave frame) are on average speeded up. Net effect: on average particles move their speed toward that of wave. But this only continues while particles remain within their initial potential well: that is, while they have "caught the wave". When particles have moved half a wavelength their roles are reversed, and the momentum changes are eventually completely undone after they have moved a full period.

Summary: Resonant particles' velocity is drawn toward the wave phase velocity.

Is there net energy when we average both slower and faster particles? Depends which type has most.

Figure 5.23: Damping or growth depends on distribution slope

Our Complex variables wave treatment and our direct particle energy calculation give consistent answers. To show this we need to show energy conservation. Energy density of wave:

< sin² >

[

ϵ₀ | E² |

Electrostatic

m |

Particle Kinetic

]

(5.256)

Magnetic wave energy zero (negligible) for a longitudinal wave. We showed in Cold Plasma treatment that the velocity due to the wave is ~v = [q E/(− i ωm )] Hence

≅

ϵ₀ E²

⎡
⎣

1 +

ω_p²

ω²

⎤
⎦

(again electrons only)

(5.257)

When the wave is damped, it has imaginary part of ω, ω_i and

d t

E²

d E²

d t

= 2 ω_i

(5.258)

Conservation of energy requires that this equal minus the particle energy gain rate, P. Hence

ω_i =

− P

2 W

+ E²

πq² ω

2 m k²

∂f₀

∂v

⎢
⎢

[(ω)/k]

ϵ₀ E²

⎡
⎣

1 +

ω_p²

ω²

⎤
⎦

= ω_p²

k²

∂f₀

∂v

⎢
⎢

[(ω)/k]

1 +

ω_p²

ω²

(5.259)

So for waves such that ω ∼ ω_p, which is the dispersion relation to lowest order, we get

ω_i = ω_p²

ω_r

k²

∂f₀

∂v

⎢
⎢

[(ω_r)/k]

(5.260)

This exactly agrees with the damping calculated from the complex dispersion relation using the Vlasov equation.

This is the Landau damping calculation for longitudinal waves in a (magnetic) field-free plasma. Strictly, just for electron plasma waves.

How does this apply to the general magnetized plasma case with multiple species?

Doing a complete evaluation of the dielectric tensor using kinetic theory is feasible but very heavy algebra. Our direct intuitive calculation gives the correct answer more directly.

5.9.7 Damping Mechanisms

Cold plasma dielectric tensor is Hermitian. [Complex conjugate*, transpose^T = original matrix.] This means no damping (dissipation).

The proof of this fact is simple but instructive. Rate of doing work on plasma per unit volume is P = E. j. However we need to observe notation.

Notation is that E(k, ω) is amplitude of wave which is really ℜ(E(k, ω) expi (k . x−ωt)) and similarly for j. Whenever products are taken: must take real part first. So

ℜ( Eexpi ( k . x− ωt )) . ℜ( jexpi ( k . x− ωt ) )

[ Ee^iϕ + E^* e^−iϕ ] .

[ je^iϕ + j^* e^−iϕ ] ( ϕ = k . x− ωt. )

[ E. je^2iϕ + E. j^* + E^* . j+ E^* . j^* e^−2iϕ ]

(5.261)

The terms e^2iϕ & e^−2iϕ are rapidly varying. We usually average over at least a period. These average to zero. Hence

< P > =

[ E. j^* + E^* . j] =

ℜ( E. j^* )

(5.262)

Now recognize that j =σ . E and substitute

< P > =

[ E.σ^* . E^* + E^* .σ . E]

(5.263)

But for arbitrary matrices and vectors:

A . M . B = B . M^T . A ;

(5.264)

(in our dyadic notation we don't explicitly indicate transposes of vectors). So

E.σ^* . E^* = E^* .σ^{* T} . E

(5.265)

hence

< P > =

E^* . [ σ^*T +σ ] . E

(5.266)

If ϵ = 1 + [ 1/(− i ωϵ₀)]σ is hermitian ϵ^*T = ϵ, then the conductivity tensor is antihermitian σ^*T = −σ (if ω is real). In that case, equation 5.266 shows that 〈P〉 = 0. No dissipation.

Any dissipation of wave energy is associated with a hermitian part of σ and hence an antihermitian part of ϵ. Cold Plasma has none.

Collisions introduce damping. Can be included in equation of motion

d v

d t

= q ( E+ v∧B) − m v ν

(5.267)

where ν is the collision frequency.

Whole calculation can be followed through replacing m(−iω) with m(ν− i ω) everywhere. This introduces complex quantity in S, D, P.

We shall not bother with this because in fusion plasmas collisional damping is usually negligible. See this physically by saying that transit time of a wave is

Size

Speed

∼

1 meter

3 ×10⁺⁸ m/s

≅ 3 ×10⁻⁹ seconds.

(5.268)

(Collision frequency)⁻¹ ∼ 10 μs → 1 ms, depending on T_e, n_e.

When is the conductivity tensor Antihermitian?

Cold Plasma:

ϵ =

⎡
⎢
⎢
⎢
⎣

−iD

⎤
⎥
⎥
⎥
⎦

where

S = 1 −

∑
j

ω_pj²

ω² − Ω_j²

D =

∑
j

Ω_j

ω_pj

ω² − Ω_j²

P = 1 −

∑
j

ω_pj²

ω²

(5.269)

This is manifestly Hermitian if ω is real, and then σ is anti-Hermitian.

This observation is sufficient to show that if the plasma is driven with a steady wave, there is no damping, and k does not acquire a complex part.

Two stream Instability

ϵ_zz = 1 −

∑
j

ω_pj²

(ω− kv_j )²

(5.270)

In this case, the relevant component is Hermitian (i.e. real) if both ω and k are real.

But that just begs the question: If ω and k are real, then there's no damping by definition.

So we can't necessarily detect damping or growth just by inspecting the dieletric tensor form when it depends on both ω and k.

Electrostatic Waves in general have ϵ = 0 which is Hermitian. So really it is not enough to deal with ϵ or χ. We need to deal with σ = − iωϵ₀χ, which indeed has a Hermitian component for the two-stream instability (even though χ is Hermitian) because ω is complex.

5.9.8 Ion Acoustic Waves and Landau Damping

We previously derived ion acoustic waves based on fluid treatment giving

ϵ_zz = 1 −

ω_pe²

ω² −	k²p_eγ_e m_en_e

−

ω_pi²

ω² −	k²p_iγ_i m_in_i

(5.271)

Leading to ω² ≅ k² [ [(γ_iT_i + γ_eT_e)/(m_i )] ].

Kinetic treatment adds the extra ingredient of Landau Damping. Vlasov plasma, unmagnetized:

ϵ_zz = 1 −

ω_pe²

k²

⌠
⌡

v −

∂f_oe

∂v

−

ω_pi²

k²

⌠
⌡

v −

∂f_oi

∂v

(5.272)

Both electron and ion damping need to be considered as possibly important.

Based on our fluid treatment we know these waves will have small phase velocity relative to electron thermal speed. Also c_s is somewhat larger than the ion thermal speed.

Figure 5.24: Distribution functions of ions and electrons near the sound wave speed.

So we adopt approximations

v_te >>

, v_ti < ( < )

(5.273)

and expand in opposite ways.

Ions are in the standard limit, so

χ_i ≅ −

ω_pi²

ω²

⎡
⎣

1 +

3T_i

k²

ω²

+ πi

ω²

k²

n_i

∂f_oi

∂v

⎢
⎢

ω/k

⎤
⎦

(5.274)

Electrons: we regard [(ω)/k] as small and write

℘

⌠
⌡

v −

∂f_oe

∂v

≅

℘

⌠
⌡

∂f_oe

∂v

⌠
⌡

∞

0

∂f_oe

∂v

dv by symmetry

⌠
⌡

∞

0

−

m_e

T_e

f_oe dv forMaxwellian.

−

m_e

T_e

(5.275)

Write F₀ = f₀ / n.

Contribution from the pole is as usual so

χ_e = −

ω_pe²

k²

⎡
⎣

−

m_e

T_e

+ πi

∂F_oe

∂v

⎢
⎢

ω/k

⎤
⎦

(5.276)

Collecting real and imaginary parts (at real ω)

ϵ_r (ω_r)

1 +

ω_pe²

k²

m_e

T_e

−

ω_pi²

ω_r²

⎡
⎣

1 +

3T_i

k²

ω_r²

⎤
⎦

(5.277)

ϵ_i (ω_r)

− π

k²

⎡
⎣

ω_pe²

∂F_oe

∂v

⎢
⎢

ω/k

+ ω_pi²

∂F_oi

∂v

⎢
⎢

ω/k

⎤
⎦

(5.278)

The real part is essentially the same as before. The extra Bohm Gross term in ions appeared previously in the denominator as

ω_pi²

ω² −	k²p_iγ_i m_i

↔

ω_pi²

ω²

⎡
⎣

1 +

3T_i

m_i

k²

ω²

⎤
⎦

(5.279)

Since our kinetic form is based on a rather inaccurate Taylor expansion, it is not clear that it is a better approx. We are probably better off using

ω_pi²

ω²

1 −	3T_ik² m_iω²

(5.280)

Then the solution of ϵ_r(ω_r) = 0 (after some algebra and using λ_De²=(T_e/m_e)/ω_pe²) is

ω_r²

k²

⎡
⎣

3 T_i

m_i

(1 + k² λ_De²)

T_e

m_i

⎤
⎦

(5.281)

as before, but we've proved that the correct choice, keeping the k²λ_De² term (1st term of ϵ_r) is γ_e = 1/(1+k²λ_De) ≅ 1.

The imaginary part of ϵ gives damping.

General way to solve for damping when small

We want to solve ϵ(k,ω) = 0 with ω = ω_r + iω_i , ω_i small.

Taylor expand ϵ about real ω_r:

ϵ(ω)

≅

ϵ(ω_r) + i ω_i

d ϵ

dω

|_{ω_r}

(5.282)

ϵ(ω_r) + i ω_i

∂

∂ω_r

ϵ(ω_r)

(5.283)

Let ω_r be the solution of ϵ_r(ω_r) = 0; then

ϵ_i (ω) = i ϵ_i (ω_r) + i ω_i

∂

∂ω_r

ϵ(ω_r).

(5.284)

This is equal to zero when

ω_i = −

ϵ_i ( ω_r )

∂ϵ( ω_r )

∂ω_r

(5.285)

If, by presumption, ϵ_i << ϵ_r, or more precisely (in the vicinity of ϵ = 0), ∂ϵ_i/∂ω_r << ∂ϵ_r/∂ω_r then this can be written to lowest order:

ω_i = −

ϵ_i ( ω_r )

∂ϵ_r ( ω_r )

∂ω_r

(5.286)

Apply to ion acoustic waves:

∂ϵ_r ( ω_r )

∂ω_r

ω_pi²

ω_r³

⎡
⎣

2 + 4

3T_i

m_i

k²

ω_r²

⎤
⎦

(5.287)

ω_i =

k²

ω_r³

ω_pi²

⎡
⎢
⎣

2 + 4

3T_i

m_i

k²

ω_r²

⎤
⎥
⎦

⎡
⎣

ω_pe²

∂F_oe

∂v

⎢
⎢

ω/k

+ ω_pi²

∂F_oi

∂v

⎢
⎢

ω/k

⎤
⎦

(5.288)

For Maxwellian distributions, using our previous value for ω_r,

∂F_oe

∂v

⎢
⎢

[(ω_r)/k]

⎡
⎣

−

⎛
⎝

m_e

2 πT_e

⎞
⎠

¹/₂

m_ev

T_e

e^{−[(m_ev²)/(2T_e)]}

⎤
⎦

v = [( ω_r)/k]

−

√

2 π

⎛
⎝

m_e

T_e

⎞
⎠

³/₂

⎡
⎣

T_e

m_i(1+k²λ_De²)

3T_i

m_i

⎤
⎦

¹/₂

exp

⎛
⎝

−

m_e

2T_i

⎡
⎣

T_e

m_i(1+k²λ_De²)

3T_i

m_i

⎤
⎦

⎞
⎠

−

√

2 π

⎛
⎝

m_e

T_e

⎞
⎠

³/₂

⎡
⎣

T_e

m_i(1+k²λ_De²)

3T_i

m_i

⎤
⎦

¹/₂

(5.289)

where the exponent is of order m_e/m_i here, and so the exponential is ≅ 1. And for ions

∂F_oi

∂v

⎢
⎢

[(ω_r)/k]

= −

√

2π

m_i

T_i

⎡
⎣

1 + k² λ_D²

3T_i

T_e

⎤
⎦

1/2

⎛
⎝

T_e

T_i

⎞
⎠

¹/₂

exp

⎛
⎝

−

⎡
⎣

T_e

2T_i

1 + k² λ_D²

⎤
⎦

⎞
⎠

(5.290)

Hence

ω_i

ω_r

−

√

2π

ω_r²

k²

⎡
⎢
⎣

2 + 4

3T_i

m_i

k²

ω_r²

⎤
⎥
⎦

⎡
⎣

1 + k² λ_D²

3T_i

T_e

⎤
⎦

1/2

⎡
⎣

m_i

m_e

⎛
⎝

m_e

m_i

⎞
⎠

¹/₂

m_e

T_e

m_i

T_i

⎛
⎝

T_e

T_i

⎞
⎠

¹/₂

exp

⎛
⎝

−

T_e

2T_i

1 + k² λ_D²

−

⎞
⎠

⎤
⎦

−

⎛
√

⎡
⎢
⎣

2 + 4

3T_i

m_i

k²

ω_r²

⎤
⎥
⎦

⎡
⎣

1 + k² λ_D²

3T_i

T_e

⎤
⎦

3/2

⎡
⎣

⎛
⎝

m_e

m_i

⎞
⎠

¹/₂

electron

⎛
⎝

T_e

T_i

⎞
⎠

³/₂

exp

⎛
⎝

−

T_e

2T_i

1 + k² λ_D²

−

⎞
⎠

ion Landau damping

⎤
⎦

(5.291)

[Note: the coefficient on the first line of equation 5.291 for ω_i/ω_r reduces to ≅ −√{π/8} for T_i/T_e << 1 and kλ_De << 1.]

Electron Landau damping of ion acoustic waves is rather small: [(ω_i)/(ω_r)] ∼ √{[(m_e)/(m_i)]} ∼ [1/70].

Ion Landau damping is large, ∼ 1 unless the term in the exponent is large. That is

unless

T_e

T_i

>> 1 .

(5.292)

Physics is that large [(T_e)/(T_i)] pulls the phase velocity of the wave: √{ [(T_e + 3T_i)/(m_i )]} = c_s above the ion thermal velocity v_ti = √{[(T_i)/(m_i)]}. If c_s >> v_ti there are few resonant ions to damp the wave.

[Note. Many texts drop terms of order [(T_i)/(T_e)] early in the treatment, but that is not really accurate. We have kept the [(T_i)/(T_e)] to first order, giving an extra leading coefficient

⎡
⎢
⎣

2 + 4

3T_i

m_i

k²

ω_r²

⎤
⎥
⎦

⎡
⎣

1 + k² λ_D²

3T_i

T_e

⎤
⎦

3/2

≅

⎡
⎣

1 + 3T_i/T_e

1 + 6T_i/T_e

⎤
⎦

⎡
⎣

1 +

3T_i

T_e

⎤
⎦

³/₂

≅ 1 +

T_i

T_e

(5.293)

and an extra term −³/₂ in the exponent. When T_i ∼ T_e we ought really to use full solutions based on the Plasma Dispersion Function.]

5.9.9 Alternative expressions of Dielectric Tensor Elements

This subsection gives some useful algebraic relationships that enable one to transform to different expressions sometimes encountered.

χ_zz

q²

ωmϵ₀

⌠
⌡

∂f₀

∂v

ω− kv

dv =

q²

ω² mϵ₀

⌠
⌡

⎛
⎝

ω− kv

− 1

⎞
⎠

∂f₀

∂v

(5.294)

q²

mϵ₀

k²

⌠
⌡

− v

∂f₀

∂v

(5.295)

ω_p²

k²

⌠
⌡

− v

∂f₀

∂v

(5.296)

ω_p²

k²

⎡
⎢
⎣

℘

⌠
⌡

− v

∂F₀

∂v

dv − πi

∂F₀

∂v

⎢
⎢

[(ω)/k]

⎤
⎥
⎦

(5.297)

where F₀ = [(f₀)/n] is the normalized distribution function. Other elements of χ involve integrals of the form

χ_jl

ωm ϵ₀

q²

⌠
⌡

v_j

∂f₀

∂v_l

ω− k . v

d³ v .

(5.298)

When k is in z-direction, k.v = k_zv_z. (Multi dimensional distribution f₀).

If (e.g., χ_xy) l ≠ z and j ≠ l then the integral over v_l yields ∫[(∂f₀)/(∂v_l)] dv_l = 0. If j = l ≠ z then

⌠
⌡

v_j

∂f₀

∂v_j

dv_j = −

⌠
⌡

f₀dv_j ,

(5.299)

by parts. So, recalling the definition f_z ≡ ∫f dv_x dv_y,

χ_xx = χ_yy

−

q²

ωm ϵ₀

⌠
⌡

f_oz

ω− k.v

dv_z

−

ω_p²

⌠
⌡

F_oz

ω− k . v

dv_z.

(5.300)

The fourth type of element is

χ_xz =

q²

ωm ϵ₀

⌠
⌡

v_x

∂f₀

∂v_z

ω− k_zv_z

d³ v .

(5.301)

This is not zero unless f₀ is isotropic ( = f₀(v)).

If f is isotropic

∂f₀

∂v_z

d f₀

d v

∂v

∂v_z

v_z

d f₀

d v

(5.302)

Then

⌠
⌡

v_x

∂f₀

∂v_z

ω− k_zv_z

d³ v

⌠
⌡

v_x v_z

ω− k_z v_z

d f₀

d³ v

⌠
⌡

v_z

ω− k_z v_z

∂f₀

∂v_x

d³ v = 0

(5.303)

(since the v_x-integral of ∂f₀/∂v_x is zero). Hence for isotropic F₀=f₀/n, with k in the z-direction,

χ =

⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣

−

ω_p²

⌠
⌡

F_oz

ω− k v_z

dv_z

0⁺

−

ω_p²

⌠
⌡

F_oz

ω− k v_z

dv_z

0⁺

ω_p²

⌠
⌡

ω− k v_z

∂F_oz

∂v_z

dv_z

⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦

(5.304)

(and the terms 0⁺ are the ones that need isotropy to make them zero).

ϵ =

⎡
⎢
⎢
⎢
⎣

ϵ_t

ϵ_l

⎤
⎥
⎥
⎥
⎦

(5.305)

where

ϵ_t

1 −

ω_p²

⌠
⌡

F_oz

ω− kv_z

dv_z

(5.306)

ϵ_l

1 −

ω_p²

k²

⌠
⌡

v −

∂F_oz

∂v_z

dv_z

(5.307)

All integrals are along the Landau contour, passing below the pole.

5.9.10 Electromagnetic Waves in unmagnetized Vlasov Plasma

For transverse waves the dispersion relation is

k²c²

ω²

= N² = ϵ_t = 1 −

ω_p²

⌠
⌡

f_oz dv_z

( ω− k_zv_z )

(5.308)

This has, in principle, a contribution from the pole at ω− kv_z = 0. However, for a non-relativistic plasma, thermal velocity is << c and the EM wave has phase velocity ∼ c. Consequently, for all velocities v_z for which f_oz is non-zero kv_z << ω. We have seen with the cold plasma treatment that the wave phase velocity is actually greater than c. Therefore a proper relativistic distribution function will have no particles at all in resonance with the wave.

Therefore:

The imaginary part of ϵ_t from the pole is negligible. And relativistically zero.

ϵ_t

≅

1 −

ω_p²

ω²

⌠
⌡

∞

−∞

f_oz

⎛
⎝

1 +

kv_z

k² v_z²

ω²

+...

⎞
⎠

dv_z

1 −

ω_p²

ω²

⎡
⎣

1 +

k²

ω²

+ ...

⎤
⎦

≅

1 −

ω_p²

ω²

⎡
⎣

1 +

k² v_t²

ω²

⎤
⎦

≅

1 −

ω_p²

ω²

(5.309)

Thermal correction to the refractive index N is small because [(k²v_t²)/(ω²)] << 1.

Electromagnetic waves are hardly affected by Kinetic Theory treatment in unmagnetized plasma. Cold Plasma treatment is generally good enough.

5.10 Experimental Verification of Landau Damping

This last section consists of viewgraphs that may be viewed at expts.pdf.

HEAD

Chapter 5 Electromagnetic Waves in Plasmas

5.1 General Treatment of Linear Waves in Anisotropic Medium

5.1.1 Simple Case. Isotropic Medium

5.1.2 General Case (k in z-direction)

5.2 High Frequency Plasma Conductivity

5.2.1 Zero B-field case

Longitudinal Waves (B0 = 0)

Simple Derivation of Plasma Oscillations

Transverse Waves (B0 = 0)

5.2.2 Meaning of Negative N2: Cut Off

5.3 Cold Plasma Waves (Magnetized Plasma)

5.3.1 Derivation of Dispersion Relation

Wave polarization

Example: Right-hand wave, parallel propagation θ = 0

5.3.2 Hybrid Resonances Perpendicular Propagation

Summary Graph (Ω > ωp)

5.3.3 Whistlers

5.4 Thermal Effects on Plasma Waves

5.4.1 Refractive Index Plot

5.4.2 Including the ion response

5.5 Electrostatic Approximation for (Plasma) Waves

5.6 Simple Example of MHD Dynamics: Alfven Waves

5.7 Non-Uniform Plasmas and wave propagation

5.8 Two Stream Instability

5.9 Kinetic Theory of Plasma Waves

5.9.1 Vlasov Equation

5.9.2 Linearized Wave Solution of Vlasov Equation

Path of velocity integration

5.9.3 Landau's original approach. (1946)

5.9.4 Solution of Dispersion Relation

5.9.5 Direct Calculation of Collisionless Particle Heating

5.9.6 Physical Picture

5.9.7 Damping Mechanisms

When is the conductivity tensor Antihermitian?

5.9.8 Ion Acoustic Waves and Landau Damping

General way to solve for damping when small

5.9.9 Alternative expressions of Dielectric Tensor Elements

5.9.10 Electromagnetic Waves in unmagnetized Vlasov Plasma

5.10 Experimental Verification of Landau Damping

Chapter 5
Electromagnetic Waves in Plasmas

Longitudinal Waves (B₀ = 0)

Transverse Waves (B₀ = 0)

5.2.2 Meaning of Negative N²: Cut Off

Summary Graph (Ω > ω_p)