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Introduction Plasma is a complex fluid that supports many plasma wave modes Restoring forces include kinetic pressure and electric and magnetic forces Wave phenomena are important for heating plasmas instabilities and diagnostics etc

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Chapter 7 MHD PLASMA WAVES 7.1 Introduction Plasma is a complex fluid that supports many plasma wave modes. Restoring forces include kinetic pressure and electric and magnetic forces. Wave phenomena are important for heating plasmas, instabilities and diagnostics etc. In vacuum, there is only one wave mode - the electromagnetic wave with ?/k = c and having oscillating E and B components perpendicular to k. In air, both sound waves and electromagnetic waves propagate. In plasma, both elec- trostatic waves and electromagnetic waves will propagate. In the former case, the electric field perturbation associated with the wave is parallel to the wave propoa- gation direction E ? k so that there are no magnetic perturbations associated with the wave: ??E = ik?E = i?B = 0 While in air, sound waves propagate through collisions, in a highy ionized plasma, these collisions occur through the wave electric fields. There are a great variety of possible plasma waves modes, since the wave phase velocity depends on both the wave frequency and its angle of propagation with respect to the background magnetic field. Important characteristic frequencies are ? pe , ? ce and ? ci . 7.2 Waves Though conventionally we write n = n˜ exp (ik.r ? i?t) 3?D n = n˜ exp (ikx? i?t) 1?D

  • phase velocity

  • field lines

  • wave

  • plasma

  • magnetic field

  • waves

  • restoring force

  • alfven wave

  • perturbations associated

  • feels magnetic


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Chapter 7
MHD PLASMA WAVES
7.1 Introduction
Plasma is a complex fluid that supports many plasma wave modes. Restoring
forces include kinetic pressure and electric and magnetic forces. Wave phenomena
are important for heating plasmas, instabilities and diagnostics etc.
In vacuum, there is only one wave mode - the electromagnetic wave with
ω/k = c and having oscillatingE andB components perpendicular tok.Inair,
both sound waves and electromagnetic waves propagate. In plasma, both elec-
trostatic waves and electromagnetic waves will propagate. In the former case, the
electric field perturbation associated with the wave is parallel to the wave propoa-
gation direction E k so that there are no magnetic perturbations associated
with the wave:
∇×E =ik×E =iωB =0
While in air, sound waves propagate through collisions, in a highy ionized plasma,
these collisions occur through the wave electric fields.
There are a great variety of possible plasma waves modes, since the wave phase
velocity depends on both the wave frequency and its angle of propagation with
respect to the background magnetic field. Important characteristic frequencies
are ω , ω and ω .pe ce ci
7.2 Waves
Though conventionally we write
n =˜ n exp (ik.r− iωt)3− D
n =˜ n exp (ikx− iωt)1− D148
the observable quantity is n =˜ ncos(kx− ωt). The exponential notation is
useful for analysis of linear systems where Fourier synthesis and superposition
are valid.
7.2.1 Phase velocity
The phase velocity is the velocity on the wave of a point of constant phase. Thus
kx− ωt = constant

ω constant
⇒ x = t + (7.1)
k k
and
w
v = (7.2)φ
k
is the phase velocity.
The wave complex amplitude carries the phase information
˜E = E cos (kx− ωt + δ)
˜⇒ E exp [i(kx− ωt + δ)]
˜= E exp (ikx− iωt) (7.3)c
where the wave amplitude is the complex quantity:
˜ ˜E =E exp (iδ). (7.4)c
By the principle of superposition an arbitrary composite time varying waveform
is constructed from its Fourier components:

∞ dω˜E(t)= E (ω)exp(ik.r− iωt) (7.5)c
−∞ 2π
where the complex amplitude is now frequency dependent.
Henceforth we shall omit the tilde notation that distinguishes the complex
Fourier amplitude. The meaning should be evident from the context.
In order to take advantage of superposition and linear theory, we shall usually
“linearize” the fluid equations by performing a Taylor series expansion about
equilibrium values u , E , B etc. and keep only first order perturbations u ,0 0 0 1
E ,B etc. The perturbations are the wave-related quantities. Thus1 1
n = n + n exp (ikx− iωt)0 1
or simply n = n + n with the phasor understood. Non-linear combinations of0 1
small terms are always dropped e.g. n u , as these are of second order in small1 1
quantities.7.3 Overview of plasma ion waves 149
7.2.2 Group velocity
Information is usually encoded on a carrier wave as either a modulation of its
phase or amplitude (or polarization). A simple amplitude modulated wave can
be constructed by combining two carriers of slightly different frequency ω and
ω+dω. The resulting beat pattern (the information) travels at the group velocity

v = <c (7.6)g
dk
The group velocity is closely related to the concept of Poynting flux which we
will encounter later. The distinction between phase and group velocities is shown
schematically in Fig. 7.1
v =dω/dkg
ω
v = ω/kφ
Dispersion curve
k
Figure 7.1: The phase and group velocities of a wave can be determined from its
dispersion relation.
7.3 Overview of plasma ion waves
The treatment given here largely follows the account given in [7]. We commence
by briefly reviewing the propagation of sound waves in air. The extension to
a compressible magnetized plasma is intuitively clear. The formal derivations
follow.150
7.3.1 Sound waves in air
Variations in air pressure and density obey the adiabatic law
∇p ∇ρ
= γ (7.7)
p ρ
or

γp
∇p = ∇ρ
ρ
2≡ v ∇ρ (7.8)S
where v is the adiabatic sound speed:S
1/2
γp
v =S
ρ
1/2
γk TB
= . (7.9)
m
We can draw an analogy from this for MHD waves — i.e. waves in a compressible
conducting fluid in a magnetic field.
7.3.2 Alfv´en waves
2We have established that the MHD fluid feels magnetic tension B /µ along0
2the field lines and an isotropic pressure B /2µ . The field lines thus behave as0
mass-loaded strings under tension – plasma particles are tied to the field lines.
We therefore expect the magnetic field to execute transverse perturbations when
perturbed which propagate with velocity
1/2 /22tension B
V = = . (7.10)A
density µ ρ0
This is the Alfv´en velocity. It was encountered earlier in relation to the polariza-
tion drift (see Sec. 4.3.1). Figure 7.2 illustrates the transverse nature of the fluid
motion and the frozen magnetic lines of force. There are no density or pressure
fluctuations associated with this wave. The wave is often called the torsional or
shear Alfv´en wave.
7.3.3 Ion acoustic and Magnetoacoustic waves
By further analogy, we should also expect longitudinal oscillations due to pressure
fluctuations. For motion of the particles and propagation of the wave along the7.3 Overview of plasma ion waves 151
B
k
Figure 7.2: Torisonal Alfv´en waves in a compressible conducting MHD fluid prop-
agating along the lines of force. The fluid motion and magnetic perturbations
are normal to the field lines.
field there will be no field perturbation since the particles are free to move in
this direction (B =0 ⇒ electrostatic wave). These waves will therefore be1
compresional waves, called ion acoustic waves propagating at velocity
1/2
γ k T + γ k Te B e i B i
V = (7.11)S
mi
along the field lines as shown in Fig. 7.3.
In the direction perpendicular to B a new type of longitudinal oscillation
is made possible by the magnetic restoring force (magnetic pressure). This is
the magnetoacoustic, magnetosonic or simply compressional wave that involves
compression and rarefaction of the magnetic lines of force as well as the plasma.
This wave propagates at velocity V that satisfiesM

2B 2∇ p + = V ∇ρ (7.12)M2µ0

2d B2⇒ V = p +M dρ 2µ0 ρ0

2d B2= V + (7.13)S dρ 2µ0 ρ0
where ρ is the background density of the unperturbed fluid. Observe that we0
have included the magnetic pressure in the restoring force. Since particles are
tied to field lines, B/ρ = B /ρ and we have0 0

2 2d B ρ2 2 0V = V +M S 2dρ 2µ ρ0 0 ρ0152
B
k
Rarefaction
Compression
λ
Figure 7.3: Longitudinal sound waves propagate along the magnetic field lines in
a compressible conducting magnetofluid.
2 2= V + V (7.14)S A
where as previously, V is the Alfv´en velocity. The nature of the magnetoacousticA
wave is illustrated in Fig. 7.4
7.4 Mathematical treatment of MHD waves
Our starting point is the set of ideal MHD fluid equations which, for convenience,
we reproduce here:
∂ρ
+∇.(ρu) = 0 continuity (7.15)
∂t
∂u
ρ + ρ(u.∇)u = j×B−∇p force (7.16)
∂t
2∇p = V ∇ρ equation of state (7.17)S
∇×B = µ j Ampere s law (7.18)0
∂B ∇×E = − Faraday s law (7.19)
∂t
E +u×B =0 Ohmslaw. (7.20)
Combining equations (7.16)-(7.18) yields
∂u 12ρ + ρ(u.∇)u =−V ∇ρ + (∇×B)×B (7.21)S
∂t µ07.4 Mathematical treatment of MHD waves 153
B
k
Figure 7.4: The magnetoacoustic wave propagates perpendicularly to B com-
pressing and releasing both the lines of force and the conducting fluid which is
tied to the field.
while (7.19) and (7.20) give
∂B
=∇×(u×B). (7.22)
∂t
We now make a perturbation expansion about the equilibrum values
ρ → ρ + ρ0 1
B → B +B0 1
u → u1
where u = 0 (fluid is at rest). With these substitutions, we linearize equations0
(7.15), (7.21) and (7.22):
∂ρ1
+ ρ ∇.u = 0 (7.23)0 1
∂t
∂u 11 2ρ + V ∇ρ + B ×∇×B = 0 (7.24)0 1 0 1S∂t µ0
∂B1
−∇×(u×B)=0. (7.25)1 0
∂t
The linearized equations can now be used to develop a wave equation foru .1
First we differentiate Eq. (7.24) with respect to time:
2∂ u ∂ρ 1 ∂B1 1 12ρ + V ∇ + B ×∇× =0. (7.26)0 0S2∂t ∂t µ ∂t0154
We then substitute from Eq. (7.23) and Eq. (7.25) to obtain
2∂ u1 2− V ∇(∇.u )+V ×{∇×[∇×(u×V )]} = 0 (7.27)1 A 1 AS2∂t
wherewehavedefined
B0
V = . (7.28)A 1/2(µ ρ )0 0
We now assume a plane wave solution
u (r,t)=u exp (ik.r− iωt) (7.29)1 1
and replace the derivatives

∇→ ik →−iω
∂t
to obtain the dispersion relation
2 2−ω u + V (k.u )k−V ×{k×[k×(u×V )]}=0. (7.30)1 1 A 1 AS
Expanding the vector triple cross product using
A×(B×C)=(A.C)B− (A.B)C (7.31)
finally gives
2 2 2−ω u +(V + V )(k.u )k+(k.V )[(k.V )u − (V .u )k− (k.u )V ]=0.1 1 A A 1 A 1 1 AS A
(7.32)
This complex expression has simple solutions!
7.4.1 Propagation perpendicular to B0
This is the magnetoacoustic wave. For K ⊥ B we have k.V =0 and the0 A
dispersion relation reduces to
2 2 2−ω u +(V + V )(k.u )k =0. (7.33)1 1S A
The vector nature of the equation requires thatu – the perturbed fluid velocity1
– be parallel to the propagation direction k and hence k.u = ku .Thewaveis1 1
longitudinal in nature and its dispersion relation becomes
ω 2 2 1/2v = =(V + V ) (7.34)φ S A
k
which is the same as Eq. (7.14) derived above.7.4 Mathematical treatment of MHD waves 155
The magnetic field perturbation associated with the wave is obtained using
Eq. (7.25):
−ωB −k×(u×B )=0. (7.35)1 1 0
Expanding the triple product and using k.B = 0 (i.e. k.V =0) gives0 A
u1
B = B (7.36)1 0

and the perturbation is parallel to the background field. Ohm’s law can be used
to obtain the electric field perturbation
E =−u×B . (7.37)1 1 0
Summarizing the wave properties we find
B ⊥ku k (compressional)1 1
E ⊥B E ⊥u (like transverse electric too!) (7.38)1 1 1 1
The wave produces compressions and rarefactions in theB field without magnetic
line bending. Since the fluid is ideal (infinite conductivity), the magnetic lines and
fluid move together in the direction ofk. The restoring force involves pressure and
magnetic stresses working together. The various perturbed wave components are
shown in Fig. 7.5. When displacement current is included in Maxwell’s equation
1 ∂E
∇×B = µ j + (7.39)0 2c ∂t
(asisrequired for frequencies approacing ω where the polarization drift is im-ci
portant), the phase velocity becomes
1/22 2V + VA Sv = . (7.40)φ 2 21+ V /cA
This reduces to v = V for V c.φ M A
7.4.2 Propagation parallel to B0
ForkB we have k.V = kV and Eq. (7.32) reduces to0 A A

2VS2 2 2 2V − ω )u + − 1 k (V .u )V = 0 (7.41)(k 1 A 1 AA 2VA
This relation supports two possible wave motions. Let us take u B.The1 0
dispersion relation gives
ω
v = = V (7.42)φ S
k156
B0
B1
k
u uu1 1 1
E1
Figure 7.5: The perturbed components associated with the compressional mag-
netoacoustic wave propagating perpendicular toB0
which is the longitudinal ion acoustic, or sound wave encountered earlier. There
are no perturbed components E , j andB associated with this wave.1 11
A transverse wave, in which the fluid velocity perturbation is perpendicular
toB is the other possibility. We set u .B =0 andu .k = 0 to obtain0 1 0 1
ω
v = = V (7.43)φ A
k
which is the torsional Alfv´en wave dispersion relation. All of the Alfv´en waves
exhibit no dispersion i.e. v is independent of ω and the group velocity is theφ
same as the phase velocity.
The magnetic perturbation accompanying this wave is found using Eq. (7.35):
u1
B =− B . (7.44)1 0

Thus the perturbation isantiparallel to the fluid perturbation velocity (perpendic-
ular toB ). The electric perturbation is again given by Eq. (7.37). Summarizing0
the wave properties we find
B ⊥ku ⊥k (transverse magnetic)1 1
E ⊥B E ⊥k (like transverse electric too!). (7.45)1 1 1
There are no fluctuations in ρ although the fluid and magnetic field lines oscillate
together. The relationship of the perturbed quantities to the wave are shown in
Fig. 7.6.
For the torsional wave, the fluid and magnetic perturbations are in antiphase,
indicating an exchange of energy between the fluid kinetic energy and the mag-
2netic energy (i.e. p + B /2µ is constant):0
2 2 2 2B B u B u 111 0 0 1 2= = = ρu12 22µ 2µ v 2µ V 20 0 0φ A