Fundamentals of Plasma Physics

130 Chapter 4. Elementary plasma waves

For self-consistency, it is necessary to havec^2 s>> κTi 0 /mi;if this were not true,

the ion acoustic wave would becomeω^2 =3k^2 κTi 0 /miwhich would violate the

assumption thatω/k >>

√

κTi 0 /mi.The conditionc^2 s>> κTi 0 /mi is the same

asTe>> Tiso ion acoustic waves can only propagate when the electrons are much

hotter than the ions. This issue will be further explored when ion acousticwaves are

re-examined from the Vlasov point of view.

4.2.2 Electromagnetic (incompressible) waves

The compressional waves discussed in the previous section were obtained bytaking the
divergence of Eq. (4.7). An arbitrary vector fieldVcan always be decomposed into a
gradient of a potential and a solenoidal part, i.e., it can always be written asV=∇ψ
+∇×QwhereψandQcan be determined fromV.The potential gradient∇ψhas zero
curl and so describes a conservative field whereas the solenoidal term∇×Qhas zero
divergence and describes a non-conservative field. Because Coulomb gauge is being used,
the−∇φterm on the right hand side of Eq.(4.8) is the only conservative field;the−∂A/∂t
term is the solenoidal or non-conservative field.
Waves involving finiteAhave coupled electric and magnetic fields and are a generaliza-
tion of vacuum electromagnetic waves such as light or radio waves. ThesefiniteAwaves
are variously called electromagnetic, transverse, or incompressible waves. Since no elec-
trostatic potential is involved,∇·E=0and the plasma remains neutral. BecauseA=0,
these waves involve electric currents.
Since the electromagnetic waves are solenoidal, the−∇φterm in Eq. (4.7) is superflu-
ous and can be eliminated by taking the curl of Eq. (4.7) giving

∂

∂t

∇×(mσnσuσ 1 )=−qσnσ

∂B 1

∂t

. (4.38)

To obtain an equation involving currents, Eq.(4.38) is integrated with respect to time, mul-
tiplied byqσ/mσ,and then summed over species to give

∇×J 1 =−ε 0 ω^2 pB 1 (4.39)

where
ω^2 p=

∑

σ

ω^2 pσ. (4.40)

However, Ampere’s law can be written in the form

J 1 =

1

μ 0

∇×B 1 −ε 0

∂E 1

∂t

(4.41)

which, after substitution into Eq. (4.39), gives

∇×

(

∇×B 1 −

1

c^2

∂E 1

∂t

)

=−

ω^2 p

c^2

B 1. (4.42)

Using the vector identity∇×(∇×Q)=∇(∇·Q)−∇^2 Qand Faraday’s law this be-
comes

∇^2 B 1 =

1

c^2

∂^2 B 1

∂t^2

+

ω^2 p

c^2

B 1. (4.43)