Fundamentals of Plasma Physics

5.2 The Landau problem 169

Since the oscillation has been assumed to be weakly damped,ωi<<ωrand so Eq.(5.79)
can be Taylor expanded in the small quantityωi,

Dr(ωr)+iωi

(

dDr

dω

)

ω=ωr

+i

[

Di(ωr)+iωi

(

dDi

dω

)

ω=ωr

]

=0. (5.81)

Sinceωi<<ωr, the real part of Eq.(5.81) is

Dr(ωr)≃ 0. (5.82)

Balancing the two imaginary terms in Eq.(5.81) gives

ωi=−

Di(ωr)

dDr

dω

. (5.83)

Thus, Eqs.(5.82) and (5.80) give the real part of the frequency as

ω^2 r=ω^2 pe

(

1+3

k^2

ω^2 r

κTe

me

)

≃ω^2 pe

(

1+3k^2 λ^2 De

)

(5.84)

while Eqs.(5.83) and (5.80) give the imaginary part of the frequency which is called the
Landau dampingas

ωi = −

√

π

8

ωpe

k^3 λ^3 De

exp

(

−ω^2 /k^2 v^2 Tσ

)

= −

√

π

8

ωpe

k^3 λ^3 De

exp

[

−

(

1+3k^2 λ^2 De

)

/ 2 k^2 λ^2 De

]

.

(5.85)

Since the least damped oscillation goes asexp(pt) = exp(−iωt) = exp(−i(ωr+
iωi)t)= exp(−iωrt+ωit)and Eq.(5.85) gives a negativeωi,this is indeed a damping. It
is interesting to note that while Landau damping was proposed theoretically by Landau in
1949, it took sixteen years before Landau damping was verified experimentally(Malmberg
and Wharton 1964).
What is meant by weak damping v. strong damping? In order to calculateωiit was
assumed thatωiis small compared toωrsuggesting perhaps thatωiis unimportant. How-
ever, even though small,ωican be important, because the factor 2 πaffects the real and
imaginary parts of the wave phase differently. Suppose for example that the imaginary part
of the frequency is 1 / 2 π∼ 1 / 6 the magnitude of the real part. This ratio is surely small
enough to justify the Taylor expansion used in Eq.(5.81) and also to justify theassumption
that the polepjcorresponding to this mode is only slightly to the left of the imaginaryp
axis. Let us calculate how much the wave is attenuated in one periodτ =2π/ωr. This
attenuation will beexp(−|ωi|τ)=exp(− 2 π/6)∼exp(−1)∼ 0. 3 .Thus, the wave ampli-
tude decays to one third its original value in just one period, which is certainly important.

5.2.7 Power relationships

It is premature to calculate the power associated with wave damping, because we do not yet
know how to add up all the energy in the wave. Nevertheless, if we are willing to assume
temporarily that the wave energy is entirely in the wave electric field (it turns out there is