Plasma Waves
S.M.Lea
January 2007
1 General considerations
To consider the different possible normal modes of a plasma, we will usually begin by
assuming that there is an equilibrium in which the plasma parameters such as density and
magnetic ®eld are uniform and constant in time. We will then look at small perturbations
away from this equilibrium, and investigate the time and space dependence of those
perturbations.
The usual notation is to label the equilibrium quantities with a subscript 0; e.g. n0 ;
and the pertrubed quantities with a subscript 1, eg n1 : Then the assumption of small
perturbations is jn1 =n0 j ¿ 1: When the perturbations are small, we can generally ignore
squares and higher powers of these quantities, thus obtaining a set of linear equations for the
unknowns.
These linear equations may be Fourier transformed in both space and time, thus reducing
the differential equations to a set of algebraic equations. Equivalently, we may assume that
each perturbed quantity has the mathematical form
³
´
n1 = n exp i~k ¢ ~x ¡ i!t
(1)
where the real part is implicitly assumed. This form describes a wave. The amplitude n is in
general complex, allowing for a non-zero phase constant Á0 : The vector ~k; called the wave
vector, gives both the direction of propagation of the wave and the wavelength: k = 2¼=¸;
~
! is the angular frequency. There is a relation between
³ ´ ! and k that is determined by the
physical properties of the system. The function ! ~k is called the dispersion relation for
the wave.
A point of constant phase on the wave form moves so that
dÁ ~
= k ¢ ~vÁ ¡ ! = 0
dt
where the wave phase velocity is
!
~v Á = ^
k
(2)
k
The phase speed of a wave may (and often does) exceed the speed of light, since no
information is carried at v Á : Information is carried by a modulation of the wave, in either
amplitude or frequency. For example, a wave pulse may have a Gaussian envelope. For any
1
physical quantity u;
2
2
u (x; t) = Ae¡x =a eikx¡i!t
Thus can be Fourier-transformed to yield a Gaussian in k¡space.
A general disturbance of the system may be written as a superposition of plane waves.
With x¡axis chosen along the direction of propagation for simplicity:
Z +1
1
p
u (x; t) =
A (k) eikx¡i!t dk
2¼ ¡1
But ! is related to k through the dispersion relation:
¯
d! ¯¯
! = ! (k) = ! 0 + (k ¡ k0 )
+¢¢¢
dk ¯
k0
where k0 is the wave number at which A (k) peaks. Then the integral becomes:
"
¯ #
Z +1
1
d! ¯¯
ikx¡i! 0t
u (x; t) = p
A (k) e
exp ¡i (k ¡ k0 )
t dk
dk ¯k 0
2¼ ¡1
" Ã
! #Z
" Ã
¯
¯ !#
+1
1
d! ¯¯
d! ¯¯
= p exp i k0
¡ !0 t
A (k) exp ik x ¡
t
dk
dk ¯k 0
dk ¯ k0
2¼
¡1
" Ã
! # Ã
!
¯
¯
d! ¯¯
d! ¯¯
= exp i k0
¡ !0 t u x ¡
t; 0
dk ¯k 0
dk ¯k 0
Thus, apart from an overall phase factor, the disturbance propagates at the group speed
¯
d! ¯¯
vg =
dk ¯
(3)
k0
Information travels at this group speed.
Additional terms we have neglected in the exponent (involving the higher derivatives
dn !=dkn ) give rise to pulse spreading and other factors generally referred to as dispersion.
(More on this in Phys 704!) Hence the name "dispersion relation".
Our goal will be to identify
³ the
´ different wave modes that occur in the plasma, and to
~
®nd the dispersion relation ! k for each. If the frequency has an imaginary part, that
indicates damping or growth of the wave.
2 Plasma oscillations
In this wave we assume that the initial condition is a cold, uniform, unmagnetized plasma
with n = n0 ; T ´ 0 and ~v0 = 0: Then the equation of motion is:
µ
³
´ ¶
@
~
~
mn0
~v 1 + ~v1 ¢ r ~v 1 = ¡en0 E
@t
We ignore the second term on the left, because it involves a small quantity squared. Then
we use the assumed wave form to obtain
~
¡i!m~v 1 = ¡eE
(4)
2
Next the continuity equation gives
@
~ ¢ (n0 + n1 ) ~v 1 = 0
(n0 + n1 ) + r
@t
@
~ ¢ ~v1 + r
~ ¢ (n1~v1 ) = 0
n1 + n0 r
@t
~ 0 are zero. Again we ignore the last term because it
We used the facts that @n0 =@t and rn
involves the product of two small quantities. Then:
¡i!n1 + n0 i~k ¢ ~v 1 = 0
(5)
Or, solving for n1 ;
~k ¢ ~v 1
(6)
!
Thus the density is not perturbed unless ~v1 has a component along ~k: This means that
transverse waves with ~v1 perpendicular to ~k do not perturb the plasma density.
Finally we use Poissonzs equation, since this is a high-frequency wave:
~ ¢E
~ = ¡ e [n0 ¡ (n0 + n1 )] = ¡ e n1
r
"0
"0
This equation is already linear, and we get:
~ = ¡ e n1
i~k ¢ E
(7)
"0
Now we dot equation (4) with ~k and substitute in from equations (5) and (7).
n1 = n0
¡i!m~k ¢ ~v 1
!n1
¡i!m
n0
~
¡e~k ¢ E
µ
¶
e
¡e ¡
n1
i"0
=
=
!n1
e2
=
n1
n0
"0
Now either n1 = 0 (not the result we want) or
n0 e2
! 2p =
(8)
"0m
The frequency of the disturbance is the plasma frequency !p : Notice that the result is
independent of k: There is no phase speed or group speed: the wave is an oscillation that
does not propagate. As we shall see, introduction of a non-zero temperature causes the
wave to propagate.
We can also see from equation (7) that the electric ®eld is out of phase with the density
by ¼=2; while the velocity is in phase (equation 6).
This oscillation is a fundamental mode of the plasma and has many rami®cations. Our
next step is to begin to investigate its importance.
!m
3
3 Electromagnetic waves in an unmagnetized plasma
To understand the importance of the plasma frequency a bit more, letzs look at the
propagation of electromagnetic waves in the plasma. Maxwellzs equations are already linear.
~ (7) and B,
~
We have Poissonsz equation for E
~1 = 0
i~k ¢ B
(9)
Faradayzs Law
~ = i! B
~1
i~k £ E
and Amperezs law
³
´
~ = ¹ ~j + i!"0 E
~
i~k £ B
0
The current in the plasma is due to the electron and ion motions:
~j = n0 e~v i ¡ (n0 + n1 ) e~ve
(10)
(11)
For high frequency waves we may neglect the ion motion (~v i ' 0) since the massive ions
~ changes. We also ignore the
do not respond rapidly enough as the driving force due to E
product of the two small quantities n1 and ve to get:
~j = ¡n0 e~v e
(12)
so that Amperezs law becomes:
³
´
~ 1 = ¹ ¡n0 e~v e + i!"0 E
~
i~k £ B
0
Finally we need the equation of motion for the electrons:
³
´
~ + ~v e £ B
~1
¡i!m~v1 = ¡e E
(13)
(14)
which reduces to equation (4) when we ignore the non-linear term.
³ Equation (6) shows
´ that the density variations are zero for a transverse wave
~
~ = 0; which is consistent with
~v perpendicular to k , and Poissonzs equation becomes ~k ¢ E
~ (equation 14) . So let us look for
the assumption of a transverse wave with ~v parallel to E
transverse waves.
Cross equation (13) with ~k to get:
³
´
³
´
~1
~
i~k £ ~k £ B
= ¹0 ¡n0 e~k £ ~ve + i!"0 ~k £ E
³
´
~k ~k ¢ B
~ 1 ¡ k2 B
~ 1 = ¡¹ n0 e ~k £ ~ve ¡ ¹ !"0 ! B
~1
0
0
i
where we used Faradayzs law. Further substitutions from equations (9) and (4) give:
Ã
!
~
n0e ~
eE
2~
~1
¡k B1 = ¡¹0
k£
¡ ¹0 "0 ! 2 B
i
i!m
=
=
e2 ~
~1
! B ¡ ¹0 "0 ! 2 B
!m
¡
¢
~1
¹0 "0 ! 2p ¡ ! 2 B
¹0 n0
4
(15)
~ 1 = 0 is uninteresting. The solution with non-zero B
~ has
Again we argue that the solution B
!2
!
= k 2c2 + ! 2p
q
=
k 2 c2 + !2p
(16)
(17)
Thus at high frquencies (! À ! p ) we have the usual vacuum relation ! = ck:
3
2.5
2
omega 1.5
1
0.5
0
0.5
1
However, the wave number
k=
1.5
ck
2
2.5
3
q
! 2 ¡ ! 2p
c
becomes imaginary (k = i°) when ! < ! p : The wave form becomes
exp [i (i°) x ¡ i!t] = e¡ °xe i!t
The wave does not propagate, and the disturbance damps as exp (¡°x). The waves have a
cut-off at ! = ! p :
An electromagnetic wave propagating at ! · !p is able to excite plasma oscillations in
the plasma, thus draining energy out of the wave and into the motion of plasma particles.
Hence the wave damps.
In the ionosphere with n » 10 12 m¡3 ; the plasma frequency is
s
2
1
(10 12 m ¡3 ) (1:6 £ 10 ¡19 C)
fp =
= 9: MHz
2¼ (8:85 £ 10 ¡12 F/m) (9 £ 10 ¡31 kg)
and waves at lower frequencies do not propagate through the ionosphere. This is an intrinsic
limit on our ability to do radio astronomy from the Earthzs surface. If we do build a base on
the moon, a radio observatory will quickly follow!
The refractive index of the plasma is
r
! 2p
c
ck
n=
=
= 1¡ 2
(18)
vÁ
!
!
and is less than one for all ! < 1:
5
1
0.8
n
0.6
0.4
0.2
0
1
2
omega 3
4
5
Snellsz law thus predicts the phenomenon of total external re-ection when an EM wave of
frequency ! < ! p is incident on a plasma. Waves incident on the ionosphere from Earth at
< 9 MHz are re-ected back to earth. X-rays incident on glass from air are also re-ectedw
that is how x-ray mirrors (such as those in the orbiting x-ray observatory AXAF) work.
At the high frequencies of x-rays, the electrons in the glass behave like free electrons in a
plasma.
The group speed is found by differentiating equation (16):
d!
2!
= 2kc2
dk
Thus
r
! 2p
d!
c2
vg =
=
= c 1¡ 2
(19)
dk
vÁ
!
and is less than c at all frequencies ! > ! p :
Equation (15) may also be written
~ 1 = ¹0 "! 2 B
~1
k2 B
and thus we obtain another expression for the plasma dielectric constant, this one valid for
high-frequency transverse oscillations in a ®eld-free plasma:
Ã
!
! 2p
" = "0 1 ¡ 2
(20)
!
Notice that for ! < ! p the dielectric constant is negative. The refractive index is imaginary,
corresponding to absorption of the wave.
4
Langmuir waves
Letzs go back and reconsider the plasma oscillations (longitudinal waves) but now with a
non-zero T : The momentum equation (4) has an extra term due to the pressure gradient:
~ ¡ i~kP 1
¡i!mn0~v1 = ¡en0 E
6
We must add the equation of state
P = K½°
which becomes, to ®rst order,
P 0 + P1
=
K (½0 + ½1 )° ' K ½°0
µ
1+°
½1
½0
¶
µ
¶
½
= P0 1 + ° 1
½0
P0
½
(21)
½0 1
Poissonzs equation and the continuity equation are unchanged. Dotting the momentum
equation with ~k; we get:
~ ¡ ik2 P 1
¡i!mn0~k ¢ ~v1 = ¡en0 ~k ¢ E
P1
=
°
and then using equations (21), (5) and (7), we have
µ
¶
!n1
e
P0
¡i!mn0
= ¡en0 ¡
n1 ¡ik 2° n1 m
n0
i"0
½0
leading to the dispersion relation
kB Te
!2 = ! 2p + k2 °
= ! 2p + k2 c2s
(22)
m
where
r
kB T e
cs = °
m
is the electron sound speed. Equation (22) is the Bohm-Gross dispersion relation for
Langmuir wavesw longitudinal waves in a plasma. Notice its similarity to equation (16) w the
sound speed replaces the speed of light. The phase speed of these waves is
r
!2p
!
vÁ =
=
+ c2s
k
k2
and the group speed is
d!
c2 k
c2
vg =
= s = s
dk
!
vÁ
5 Sound waves
5.1
Fluid sound waves
If we perturb and linearize the momentum equation and the continuity equation for a
®eld-free plasma, we get:
P0
¡i!½0 ~v 1 = ¡i~k° ½1
(23)
½0
and
¡i!½1 + i~k ¢ (½0 ~v 1 ) = 0
(24)
7
Dotting the ®rst equation (23) with ~k and substituting into the second (24), we get:
¡!½1 + ½0
k2 P0
° ½ =0
!½0 ½0 1
and the dispersion relation is
! 2 = k2 °
P0
= k2 c2s
½0
waves with vÁ = vg = cs :
These waves rely on collisions in the -uid to provide the restoring force.
5.2
Ion sound waves
Even if collisions are unimportant, sound waves, being longitudinal waves, generate density
-uctuations which in turn generate electric ®elds that can provide the necessary restoring
force. When ion motion is involved, we know that the waves must be low frequency, so we
can use the plasma approximation, ne ' ni ' n0 : We are still assuming that there is no
magnetic ®eld.
The momentum equation for the ions is:
·
³
´ ¸
@
~ ~v 1 = en0 E
~¡r
~ (P i + P i; 1 )
M n0
~v1 + ~v 1 ¢ r
@t
and for the electrons
·
³
´ ¸
@
~
~ ¡r
~ (Pe + P e;1 )
mn0
~v 1 + ~v1 ¢ r ~v1 = ¡en0 E
@t
The continuity equation is
@
~ ¢ ~v 1 + r
~ ¢ n1~v1 = 0
n1 + n0 r
@t
and linearizing, we get:
@
~ ¢ ~v 1
n1 = ¡n0 r
@t
Thus
@
~ ¢ (~ve1 ¡ ~v i1 )
(ne ¡ ni ) = ¡n0 r
@t
Thus if the ion and electron velocities differ, the densities will become different too. Thus
the plasma approximation also requires ~ve1 = ~vi1 (at least to ®rst order).
Using this result, we add the two linearized momentum equations. The electric ®eld
terms cancel, leaving:
@
~ (Pi;1 + P e;1 )
(m + M ) n0 ~v1 = ¡r
@t
¡i! (m + M ) n0~v1 = ¡i~kn1 (°i kB Ti + ° ekB T e )
8
(Compare this equation with (23)). Dot with ~k; and combine with equation (6) to get:
n1
¡i! (m + M ) n0 !
= ¡ik2 n1 (° ikB T i + °e kB T e )
n0
µ
¶
°i kB Ti + ° ekB T e
! 2 = k2
m+M
There are several things to note about this result.
(25)
1. It is essentially identical to the result for -uid sound waves even though at a microscopic
level there are profound differences. The coupling here is electrostatic not collisional.
2. The electrons move very rapidly, and the distribution may be assumed to be isothermal,
°e = 1:
3. The electron mass is negligible compared with the ion mass in the denominator.
However, Vlasov theory (a detailed study of the effect of the particle velocity
distributions) shows that the wave is strongly damped unless the electron temperature
greatly exceeds the ion temperature. Thus the ion sound speed is determined by the
electron temperature and the ion mass.
r
kB Te
vis =
(26)
M
We may now consider the electric ®eld necessary to effect the coupling. Poissonzs
equation is:
~ = k 2 Á = e (ni ¡ ne)
i~k ¢ E
"0
Now we allow for small differences between the electron and ion densities. The ion density
is given by the continuity equation:
~k ¢ ~v
ni = n0 +
n0
!
while the electrons respond rapidly to the electric ®eld, and so follow the Boltzman relation:
µ
¶
µ
¶
eÁ
eÁ
ne = n0 exp
' n0 1 +
kB T e
kB T e
Thus
Ã
!
en0 ~k ¢ ~v
eÁ
2
k Á=
¡
"0
!
kB T e
Rearranging, we get:
µ
¶
en0 eÁ
en0 ~k ¢ ~v
2
Á k +
=
" 0 kB T e
"0 !
We should recognize the second term in the parentheses as 1=¸ 2D : Now we rewrite the
momentum equation for the ions, substituting this expression for Á in the electric ®eld term
9
~ = ¡ rÁ
~ = ¡i~kÁ)
(E
¡i!M n0~v1
=
¡!M n0~k ¢ ~v1
=
n1
n0
=
!2
=
!M n0 !
¡en0 i~kÁ ¡ i~kn1 ° ikB T i
µ
¶
~ ¢ ~v 1
en0 k
1
¡en0
¡ k 2 n1 °i kB Ti
"0 !
1 + 1=k2 ¸ 2D
µ
¶
en0 n1
1
en0
+ k2 n1 ° i kB T i
"0 n0 1 + 1=k 2 ¸2D
µ
¶
n0
1
° kB T i
e2
+ k2 i
M "0 1 + 1=k2 ¸ 2D
M
Ã
!
2
2
¸D -p
°i kB Ti
k2
+
M
1 + k 2¸ 2D
=
where - p is the ion plasma frequency.
The numerator in the second term is:
"0 kB T e 2 n0
Te
¸ 2D - 2p =
e
= kB
en0 e
M "0
M
Thus the new result is identical to the previous one except for the denominator 1 + k2 ¸ 2D :
Thus the correction is necessary only when k¸ D is not small, that is when the wavelength is
less than or equal to the Debye length. The full wavelength is within the region where we
would expect the plasma approximation to fail. When k¸D À 1 we ®nd
! ' -p
and we have oscillations at the ion plasma frequency. The wave reduces to plasma
oscillations of the ions.
1
0.8
w/wP0.6
0.4
0.2
0
1
2
k x debye length
3
Dispersion relation for ion sound waves
10
4
6 Electrostatic waves in magnetized plasmas
~ 0 in the initial, unperturbed
Now we look at a plasma that has a uniform magnetic ®eld B
state. The motion of the plasma particles is affected by the magnetic ®eld when they try to
~ Thus we expect to ®nd that waves travelling along B
~ and waves travelling
move across B:
~
across B will behave differently.
Electrostatic waves are longitudinal waves, and thus when electrostatic waves propagate
~ and these motions are not
along the magnetic ®eld, the particle motions are also along B;
affected by the magnetic force. Thus the previous dispersion relations for Langmuir waves
(eqn 22) and ion sound waves (eqn 25) are unaffected. We obtain interesting new effects
when the waves propagate perpendicular to the magnetic ®eld.
Note that for electrostatic waves, there is no perturbation to the magnetic ®eld (equation
~ is parallel to ~k:
10) because E
6.1
High frequency electrostatic waves propagating perpendicular to
~0
B
We begin by taking the electron temperature to be zero, as we did when studying Langmuir
waves. With a non-zero magnetic ®eld, we have an additional term in the equation of
motion (4):
~ ¡ e~v £ B
~0
¡i!m~v = ¡eE
(27)
The additional equations we need are the continuity equation in the form (6) and Poissonzs
equation (7). As usual, we dot the equation of motion with ~k :
³
´
~ ¡ e~k ¢ ~v £ B
~0
¡i!m~k ¢ ~v = ¡e~k ¢ E
³
´
~ ¡ eB
~ 0 ¢ ~k £ ~v
= ¡e~k ¢ E
The next step is to get ~k £ ~v from equation (27):
¡i!m~k £ ~v
=
=
³
´
~ ¡ e~k £ ~v £ B
~0
¡e~k £ E
³
´
³
´
~ 0 + eB
~ 0 ~k ¢ ~v
0 ¡ e~v ~k ¢ B
~ = 0 for these waves. Also, for propagation across B
~ 0;
where we used the result that ~k £ E
~k ¢ B
~ 0 = 0: Thus
³
´
~ 0 ~k ¢ ~v
eB
~ ¡ eB
~0 ¢
¡i!m~k ¢ ~v = ¡e~k ¢ E
¡i!m
µ
¶
2
~
~
ek ¢ E
~k ¢ ~v 1 ¡ ! c
=
(28)
!2
i!m
11
Finally we substitute in from the continuity and Poisson equations, to get
µ
¶
µ
¶
n1
!2
e
¡en1
! 1 ¡ c2
=
n0
!
i!m
i"0
! 2 ¡ ! 2c
=
! 2p
Thus we obtain a new oscillation frequency, called the upper hybrid frequency:
! 2H = ! 2c + ! 2p
(29)
It is |hybrid} because it is a mixture of the plasma and cyclotron frequencies. It is a higher
frequency than either because the additional restoring force leads to a higher oscillation
~ the magnetic
frequency. As the electric ®eld in the wave sets the particles moving across B;
force causes them to gyrate. The resulting particle orbits are elliptical.
~
We might ask what happens when the wave propagates at a ®nite angle µ < ¼=2 to B:
We would expect the magnetic force to act, but less strongly, and thus the frequency will
have an intermediate value. We ®nd that as µ ! 0; ! ! the larger of ! p and !c : When
!c > ! p there is a second wave at µ = 0 (the plasma oscillation at ! = ! p ): In fact there
are two oscillations at all intermediate angles, as shown in the ®gure below which shows
!=! c versus cos µ: (See Problem 4-8). (We should be suspicious of the region near ! = 0
because ion motion will become important here.)
2.5
2
1.5
w/wc
1
0.5
0
0.2
0.4 cos theta 0.6
0.8
1
! p = 2!c
The waves do not propagate with T = 0: As for the Langmuir waves, they do propagate
when T > 0; but the analysis is more complicated and we leave it for another day.
6.2
~0
Electrostatic ion waves propagating perpendicular to B
Here we must include the ion motion, but we expect to use the plasma approximation. We
have the two equations of motion, for the electrons
~ ¡ e~v £ B
~0
¡i!m~v = ¡eE
12
and for the ions
~ + e~v £ B
~0
¡i!M ~v = eE
(same ~v because of the plasma approximation) and
~k ¢ ~v
n1 =
n0
!
The method is the same as in the previous section. Dot and cross the equations of motion
with ~k to get both components of ~v: We obtain two versions of equation (28), one for the
ions and one for the electrons:
µ
¶
2
~
e~k ¢ E
~k ¢ ~v 1 ¡ !c
=
!2
i!m
µ
2¶
~
e~k ¢ E
~k ¢ ~v 1 ¡ -c
= ¡
(30)
2
!
i!M
Now divide the two equations, to obtain:
¡ 2
¢
! ¡ ! 2c
M
=¡
(! 2 ¡ - 2c )
m
or
µ
¶
µ
¶
µ
¶
M
M 2
1
1
e2 B20
M
2
2
2 2
! 1+
=
- + ! c = e B0
+ 2 =
1+
m
m c
Mm
m
Mm
m
!2L H
=
!c -c
(31)
This is the lower hybrid frequency. In the previous graph, the lower branch goes to ! LH
not zero.
These waves are almost never observed experimentally because it is necessary that they
~ and experimentally that is impossible to achieve. Suppose that
propagate exactly across B;
~ where µ ¿ 1: Electrons are free to migrate
the waves propagate at an angle ¼=2 ¡ µ to B;
~
~ the
along B; and do so very rapidly. Because the wave is not exactly perpendicular to B;
~
electrostatic potential is not constant along B and the electrons will establish a Boltzmann
distribution by travelling long distances along the ®eld lines. Thus we should replace the
electron density with n1;e = n0 eÁ=kB T ; while retaining equation (6) for the ions.
(Look at the electron density we used before. It is
~k ¢ ~v
~
n0
e~k ¢ E
n0 k2 Á
³
´
n1 =
n0 =
=
!
! i!m 1 ¡ ! c2
im (! 2 ¡ ! 2c )
!2
eÁ k2 (kB Te =m)
kB T e i (! 2 ¡ ! 2c )
µ
¶2
distance travelled per period at v e,th
= nBoltzmann
wavelength
The factor in parentheses must be much greater than one if the argument we have given is
correct, and so nBoltzmann is much less than our previous n1 for the electrons.)
=
n0
13
Now we use the plasma approximation:
~k ¢ ~v
eÁ
ni =
n0 = n0
= ne
!
kB T e
and combine with the equation of motion (30) for the ions:
µ
¶
2
2
2 ~
~ ~
~k ¢ ~v 1 ¡ - c = ¡ ek ¢ E = ek Á = ek k ¢ ~v kB T e
!2
i!M
!M
!M !
e
Thus
kB T e
= k2 vis2
(32)
M
where the speed on the right is the ion sound speed (26). These waves are called
electrostatic ion-cyclotron waves. Unlike lower hybrid oscillations, they are easily observed
experimentally. They propagate with phase speed
! 2 ¡ -2c = k2
vÁ =
and group speed
!
!
= p
v is ! v is as ! ! 1
k
! 2 + - 2c
d!
v2
= is
dk
vÁ
Dispersion relation (32) has the familiar form we have seen before.
~ does the wave vector need to be to get the lower
Exactly how close to perpendicular to B
hybrid oscillations?
From the diagram, we need
dions
v th, ions
m Ti
=
=
delectrons
vth, elec
M Te
p
With equal temperatures, we ®nd µ < 1=1837 = 2 £ 10 ¡2 radians = 1: 1 ± :
sin µ <
14
7 Electromagnetic waves in magnetized plasmas
7.1
~
Electromagnetic waves propagating across B
Electromagnetic waves in an unmagnetized plasma are transverse and thus do not generate
density -uctuations. Thus the plasma particles affect the waves through the currents that
they generate. The same is true for magnetized plasmas, but we will have to be alert for the
possibility that a longitudinal component of the waves may be generated as well.
An additional effect that we did not have to consider with electrostatic waves is the
polarization.
7.1.1 The ordinary wave.
~ 0 ; one polarization has its electric ®eld
When the wave propagates perpendicular to B
~ 0 : This is called the ordinary (or |O} mode) because the plasma particles
vector parallel to B
~ 0 as they are accelerated by the electric ®eld. Thus the magnetic
can move freely along B
®eld has no effect, and we regain the dispersion relation (16). We can see this explicitly
from the electron equation of motion:
³
´
~ + ~v £ B
~0
¡i!m~v = ¡e E
(33)
which gives rise the the current density
~j = ¡n0 e~v
and then Amperezs law gives
³
´
~k £ i~k £ B
~1
³
´
~k ~k ¢ B
~ 1 ¡ k 2B
~1
³
! ~´
= ~k £ ¹0~j ¡ i 2 E
c ¶
µ
¡n
e~
v
!2 ~
0
= ¹0~k £
¡ 2B
1
i
c
~ 1 = 0 (remember that r
~ ¢B
~ 0 must be zero too), we obtain the wave
And since ~k ¢ B
equation:
µ 2
¶
!
2
~ 1 = i¹ n0e~k £ ~v
¡
k
B
(34)
0
c2
15
From the equation of motion:
¡i!m~k £ ~v
=
=
=
h
³
´i
~ + ~k £ ~v £ B
~0
¡e ~k £ E
h ³
´
³
´i
~ 1 ¡ e ~v ~k ¢ B
~0 ¡ B
~ 0 ~k ¢ ~v
¡e!B
~1
¡e!B
~ 0 by assumption, and ~k ¢ ~v = 0 follows immediately from
since ~k is perpendicular to B
equation (33). Substituting into Amperezs law, we ®nd:
µ 2
¶
~
!
2
~ 1 = i¹0 n0 e e! B1
¡k B
2
c
i!m
!2
=
!2p + k2 c2
(35)
7.1.2 The extraordinary wave
~ 0 : Here the electric ®eld drives
Now we look at the case of polarization perpendicular to B
~ 0 ; and thus the magnetic force comes into play. Notice
electron motion perpendicular to B
also that as the electron gyrates, it will have a component of velocity along ~k; and so the
wave cannot be purely transversew it is a mixture.
Letzs begin by solving the equation of motion, because the results will be useful for other
~ 0 : Then the z¡component
waves too. Choose a coordinate system with z¡axis parallel to B
is the easiest:
e
¡i!mvz = ¡eEz ) v z = ¡i
Ez
(36)
!m
The x-and y-component equations are coupled:
³
´
~ 0 = ¡eEx ¡ evy B0
¡i!mvx = ¡eEx ¡ e ~v £ B
³
´x
~
¡i!mv y = ¡eEy ¡ e ~v £ B 0 = ¡eE y + evx B0
y
16
Thus, using the second relation in the ®rst,
¡eE y + evx B0
¡i!m
¡e
B0
B20
2
=
E x + e2
E
¡
e
vx
y
¡i!m
¡!2 m 2
¡! 2 m2
ie
!c
!2
= ¡
E x ¡ e 2 E y + c2 v x
!m
! m
!
e iE x + !!c E y
= ¡
2
!m 1 ¡ ! c2
¡i!mv x
= ¡eE x ¡ eB0
vx
vx
(37)
!
and then using the v y equation
vy
=
!
e
! c e iE x + !c E y
Ey +
2
i!m
i! !m 1 ¡ ! c2
!
=
e
!m
!c
Ex
!
¡ iE y
1¡
(38)
!2
c
!2
From these relation we obtain the components of the current:
jz = ¡nevz = ine
! 2p
e
E z = i" 0 Ez
!m
!
and
jx
=
jy
=
! 2p
ne2 iEx + !!c Ey
¡nev x =
=
"
0
!m 1 ¡ ! 2c2
!
Ã
!!
! 2p iE y ¡ !!c E x
"0
2
!
1 ¡ ! c2
Ã
(39)
iE x +
1¡
!c
Ey
!
!2
c
!2
!
(40)
(41)
!
Note the resonance at ! = ! c :
Now we are ready to ®nd the dispersion relation for the extraordinary (X-) wave. First
~ 0 in the z¡direction, and E
~ in the x ¡ y plane. Then
we choose the x¡axis along ~k , with B
Amperezs law becomes:
µ 2
¶
!
2
~ 1 = ¡i¹ ~k £ ~j
¡
k
B
0
c2
~ 1 is in the z¡ direction. (We
Here the wave number and current lie in the x ¡ y plane, so B
~ 1 has no x¡component, because its divergence is zero.) Thus
already know B
Ã
!
µ 2
¶
!2p iE y ¡ !!c E x
!
2
¡ k Bz = ¡i¹0 kj y = ¡i¹0 k"0
(42)
2
c2
!
1 ¡ !! c2
~ ; we use Faradayzs law:
To ®nd the components of E
~k £ E
~
kE y
~1
= !B
= !Bz
17
(43)
We are short one equation, and that must be Poissonzs equation, which determines the
~:
longitudinal component of E
ikE x
=
=
n1 e
e ~k ¢ ~v
e
k
= ¡ n0
= ¡ n0 vx
"0
"0
!
"0 !
!
e
k e iEx + !c E y
n0
2
"0 ! !m 1 ¡ ! c2
¡
!
iE x
=
! 2p iE x + !!c E y
! 2 1 ¡ ! 2c2
!
Thus, rearranging and using Faradayzs law (43):
Ã
!
!2p
!2c
! c !2p
Ex 1 ¡ 2 ¡ 2
=
Ey
!
!
i! !2
µ
¶
! 2H
! c ! 2p
! c !2p
Ex 1 ¡ 2
= ¡i
E
=
¡i
Bz
y
!
! !2
k !2
Substituting into Amperezs law (42), we have:
0
1
µ 2
¶µ
¶
!2
2
p
2
2
!
!
!
i
!
2
p @
¡ k2
1 ¡ c2 Bz = ¡ 2
i! + i c ³ ! 2 ´ A Bz
c2
!
c !
! 1 ¡ !H
!2
2
2
2
2
¡ 2
¢
!
! !
!
¡ 2
¢
1 ¡ ! p2 ¡ ! c2 + ! p2 ! c2
! ¡ !2p
2 2
2 ³
2
´³
´ = !p 2
! ¡c k
= !p
!2
!2
(! ¡ ! 2H )
1¡ c 1¡ H
!2
So the dispersion relation for the X-wave is
2
2 2
! =c k +
! 2p
!2
¡
¢
! 2 ¡ ! 2p
(! 2 ¡ ! 2H )
(44)
OK, letzs investigate this dispersion relation. First note that for ! À ! p; !H we retrieve
the zero-®eld result (16), and as ! ! 1; we get back the vacuum relation ! = ck: Now
letzs see whether there are any stop bands w bands of frequencies at which the wave cannot
propagate.
Cut-offs occur where k = 0: The frequencies are then given by:
¡ 2
¢
!2
! ¡ !2p
1 ¡ ! p2
2
2
2
! = !p 2
= !p
(45)
!2
!2
(! ¡ ! 2H )
1¡ p ¡ c
!2
Rearranging and dividing, we ®nd:
! 2p
!2
= 1¡
18
! 2c =!2
1¡
!2
p
!2
!2
or
Ã
! 2p
1¡ 2
!
!2
=
!2c
!2
and taking the square root:
! 2p
!c
=§
!2
!
There are two roots, so we get two quadratics for ! :
1¡
! 2 § !! c ¡ !2p = 0
with the four solutions:
!=
¨! c §
(46)
q
! 2c + 4!2p
2
Negative frequencies do not have physical meaning, so the two meaningful solutions are
´
1 ³q 2
!R =
! c + 4! 2p + !c
(47)
2
and
´
1 ³q 2
!L =
!c + 4! 2p ¡ ! c
(48)
2
The subscripts refer to R for right and L forpleft. Wezll see why later. ¡p
¢
Let y = !=! p ; ! c =! p = 2; !H =! p = 5: Then we have !!Rp = 12
8 + 2 = 2: 4142
¡p
¢
and !!Lp = 12
8 ¡ 2 = : 41421: Then the plot of ! versus k looks like:
4
3
y
2
1
0
1
2
x
3
4
! versus k for the X-mode
There is a stop band between ! R and ! H and again below ! L :
We can gain more insight by looking at the refractive index:
¡
¢
! 2p ! 2 ¡ ! 2p
c 2 k2
2
n =
= 1¡ 2 2
!2
! (! ¡ ! 2H )
¡
¢
¡
¢
2
2
2
! ! ¡ ! H ¡ !2p !2 ¡ !2p
=
! 2 (! 2 ¡ ! 2H )
19
Comparing with equation (45), we see that we can factor the numerator:
¡ 2
¢¡
¢
! ¡ !2R ! 2 ¡ ! 2L
2
n =
! 2 (! 2 ¡ ! 2H )
This con®rms our previous result that there are cutoffs at !R and ! L ; and shows that the
stop bands (n2 < 0) occur for ! R > ! > ! H and ! < ! L : (To obtain these results, note
that ! R > ! H > !L )
There is also a resonance (n ! 1) at ! = ! H . (Also n2 ! ¡1 at ! = 0; but this
one arises because we have neglected ion motion.) Note also that n ! 1 as ! ! 1:
The plot looks like:
2
1.8
1.6
1.4
1.2
n^2 1
0.8
0.6
0.4
0.2
0
1
2
3
4
w/wp
5
6
7
8
n2 versus ! for the X-wave
7.2
~ 0:
Electromagnetic waves propagating along B
~ 0 have the electric ®eld vector in the plane
Transverse waves propagating along B
~ 0; and thus the electron motions, while affected by the magnetic force,
perpendicular to B
remain transverse. Thus this wave is purely electromagnetic. We again choose our z¡axis
along the magnetic ®eld direction. We can use our previous results (37) and (38) (or
equivalently 40 and 41) for the particle motions. Now Faradayzs law gives both components
~ ; but there are also two components of B
~1 :
of E
kEy = ¡!Bx and kEx = !B y
Amperezs law (34) has two components. Using relation (38) for the electron velocity, the
x¡component is
µ 2
¶
!
2
¡ k Bx = i¹0 kjy
(49)
c2
Ã
!
µ 2
¶
! 2p iE y ¡ !!c E x
!
k
2
¡
¡
k
E
=
i¹
k"
(50)
y
0
0
!2
c2
!
!
1 ¡ ! c2
20
and similarly for the y¡component
µ 2
¶
!
2
¡
k
By
c2
µ 2
¶
!
k
2
¡
k
Ex
2
c
!
=
¡i¹0 kjx
=
!2p
¡i¹0 k"0
!
(51)
Ã
iE x +
1¡
!c
Ey
!
!2
c
!2
!
(52)
Rearrange (50) to get:
µ
¶
¡ 2
¢
!c ´
! 2c
2 2
iE y ¡
Ex = ¡ ! ¡ k c
1 ¡ 2 Ey
!
!
µ
µ
¶¶
¡ 2
¢
!2c
2 !c
2
2 2
i! p E x + E y ! p ¡ ! ¡ k c
1¡ 2
=0
!
!
i! 2p
and (52) becomes
³
(53)
µ
¶
³
¢
!c ´ ¡ 2
!2
¡i! 2p iE x +
E y = ! ¡ k2 c2 1 ¡ c2 E x
!
!
µ
µ
¶¶
¡
¢
!2
!c
E x ! 2p ¡ ! 2 ¡ k2 c2 1 ¡ c2
¡ i! 2p E y = 0
(54)
!
!
For a non-zero solution for E x and E y ; the determinant of the coef®cients in equations (53)
and (54) must be zero, so:
µ
¶¸2
³ ! ´2 ·
¡ 2
¢
! 2c
2 c
2
2 2
!p
¡ !p ¡ ! ¡ k c
1¡ 2
=0
!
!
or, taking the square root,
·
µ
¶¸
¡
¢
!c
!2
!2p
= § ! 2p ¡ !2 ¡ k2 c2 1 ¡ c2
!
!
µ
¶
³
´
2
¡
¢
!c
!
! 2p 1 §
=
! 2 ¡ k2 c2 1 ¡ c2
!
!
³
´
¡
¢
!
c
! 2p =
! 2 ¡ k2 c 2 1 §
!
and so we obtain the dispersion relation
!2 = k2 c2 + ¡
! 2p
1§
!c
!
¢
(55)
There is a resonance at ! = ! c for one of the two possible frequencies. Note again that
for ! À ! c we get back the result for an unmagnetized plasma, and we retrieve ! = ck as
! ! 1: The cutoffs (k = 0) are at
!2p
¢
!2 = ¡
1 § !!c
³
!c ´
!2 1 §
= ! 2p
!
which is equation (46), with the same two solutions ! R (bottom sign) and ! L (top sign).
21
Now letzs solve for the electric ®eld components, using equation (53) with our solution for !
inserted:
"
µ
¶#
! 2p
! 2c
2 !c
2
¢ 1¡ 2
i! p E x + E y ! p ¡ ¡
= 0
!
1 § !!c
!
h
³
!c
! c ´i
i Ex + E y 1 ¡ 1 ¨
= 0
!
³ !! ´
!c
c
i Ex + E y §
= 0
!
!
E y = ¨iE x
This solution corresponds to circularly polarized waves. If
¡
¢
E x = E0 cos !t = Re E 0 e¡i!t
then
Ey
¡
¢
Re ¨iE0 e ¡i!t = ¨E 0 Re (i cos !t + sin !t)
=
=
¨E 0 sin !t
~ vector rotates counter-clockwise, and with the minus
Thus with the plus (lower) sign, the E
(upper) sign the vector rotates clockwise. The waves have circular polarization. Putting
your thumb along the direction of propagation (the z¡direction), your right hand gives the
direction of rotation with the plus sign- these are right-hand circular waves. Conversely,
with the minus sign you need your left handw these are left hand circular waves. Thus the
RH circular wave has a cutoff at ! R and the left hand circular wave has a cutoff at !L :
For the plot, let !p =! c = 2:
4
3
w/wp
2
1
0
1
2 ck/wp 3
4
5
! versus k for R and L circular waves
The refractive index is:
! 2p =!2
c 2 k2
=
1
¡
¡
¢
(56)
!2
1 § !!c
and there is a resonance at ! = !c for the right circular wave. In this wave the electric ®eld
n2 =
22
circulates in the same direction as the electron, and when ! = ! c the ®eld is able to transfer
energy to the electron continuously, producing the resonance. Below we shall show that
when ion motion is included the left circular wave exhibits a resonance at - c :
14
12
10
8
n^2
6
4
2
0
1
-2
2
w/wc 3
4
5
!p =! c =
n2 versus !=!c for R (dashed) and L circular waves
5
4
3
n^2
2
1
0
1
2 w/wc 3
4
5
-1
p
-2
2
! p =! c = 1=2
n2 versus !=! c
7.3
Observational effects
The dispersion relations that we have derived for EM waves propagating in plasmas have
23
interesting observational consequences.
7.3.1 Plasma dispersion
When a signal has a well-de®ned time-structure, the dispersion relation can affect how we
observe it. For example, radio pulsars produce a well-de®ned pulse over a range of radio
frequencies ranging from MHz to GHz. The different frequencies travel at different group
speeds, so the pulse does not arrive at Earth at the same time at each frequency.
In an unmagnetized plasma (or for ! À !c ) we have the dispersion relation
! 2 = k 2 c2 + ! 2p
and thus
2!
d!
dk
d!
dk
= 2kc2
r
!2p
cq 2
= vg =
! ¡ !2p = c 1 ¡ 2
!
à !
!
2
1 !p
' c 1¡
for ! À ! p
2 !2
Thus the pulses at higher frequencies travel faster, and arrive sooner, than those at lower
frequencies.
In the interstellar medium, the electron density is about 0:05 electrons/cm3 , so the
plasma frequency is
s
s
1
ne2
1
(5 £ 104 m ¡3 ) (1:6 £ 10 ¡19 C)2
fp =
=
2¼ "0 m
2¼ (8:85 £ 10¡ 12 F/m) (9 £ 10¡ 31 kg)
= 2: £ 10 3 Hz,
much less than the observed frequencies in the radio band.
6£10 ¡19 C
¡6
The magnetic ®eld is about 10 ¡6 Gauss, so !c ' 1:9£10
G = 1: 8 £ 10 5 Hz is
¡31 kg 10
also much less than the observed frequencies.
The time taken for a pulse to reach earth is
L
T =
vg
where L is the distance to the pulsar. The difference in arrival times between two pulses at
24
frequencies f 1 and f2 is
¢T
=
=
=
2
3
L
L
L
1
1
5
¡
= 4
¡
2
vg 1
v g2
c 1 ¡ 1 f p2
1 fp
1
¡
2 f 12
2 f22
Ã
!
2
2
fp
L fp
¡ 2
2
2c f 1
f2
µ
¶
L 2 1
1
f
¡ 2
2c p f 12
f2
(57)
For a typical pulsar at a distance of approximately 1 kpc= 3 £ 10 19 m, the time delay
between two pulses at 1 and 5 Ghz is
µ
¶2 µ
¶
¢2
3 £ 10 19 m ¡
1
1
3
¢T =
2:
£
10
Hz
1
¡
6 £ 10 8 m/s
10 9 Hz
25
= 0: 192 s
This time delay is easily detectable.
Since fp2 / n0; measurement of the time delay provides a measurement of the plasma
density along the line of sight to the pulsar. See. eg, The Astrophysical Journal, Volume
645, Issue 1, pp. 303-313, Astrophysical Journal, Part 1 (ISSN 0004-637X), vol. 385, Jan.
20, 1992, p. 273-281
7.3.2 Faraday Rotation
When a wave travels along the magnetic ®eld, we have shown that the normal modes are the
left-and right circularly polarized waves. A wave that is emitted as a linearly polarized wave
will travel as the sum of two circularly polarized waves. We write the electric ®eld vector
for a wave travelling in the z¡direction as:
~ = E0 ^
E
x exp (ikz ¡ i!t)
where as usual the real part is assumed, and we have chosen the x¡axis along the electric
®eld direction. We decompose the wave at the source into two circularly polarized waves:
µ
¶
x
^+i^
y
x¡i^
^
y
~
E (0) = E 0
+
2
2
~ R + iE
~L
= E
The right and left circularly polarized waves have different phase speeds, so at some distance
z from the source, we have:
µ
¶
x+i^
^
y
x¡i^
^
y
~
E (z) = E 0
exp (ikR z) +
exp (ikL z) e ¡i!t
2
2
µ
¶
x
^
y
^
= E0
(exp (ikR z) + exp (ikL z)) + i (exp (ikRz) ¡ exp (ikL z)) e¡i!t
2
2
25
The angle between the electric ®eld vector and the x¡axis is given by:
Ey
exp (ikRz) ¡ exp (ikL z)
tan µ =
=i
Ex
exp (ikRz) + exp (ikL z)
which is not zero for z > 0 if kR 6= kL : Thus the wave vector rotates as the wave
propagates. This phenomenon is called Faraday rotation.
We use the dispersion relation (55) (or equivalently (56)) to get
c2 k2R;L
! 2p =! 2
¡
¢
=
1
¡
!2
1 § !!c
First letzs assume ! À !p ; !c which is always true in astronomical applications (but may
not be in lab situations- so beware!). Then we can expand to get
s
! 2p =!2
ckR;L
1 ! 2p =! 2
¢
¡
¢
=
1¡ ¡
=
1
¡
!
2 1 § !!c
1 § !!c
2
1 !p ³
!c ´
= 1¡
1¨
(58)
2
2!
!
where the top sign refers to the left circular wave and the bottom to the right circular wave.
The two results differ only in the small term in ! 2p ! c =!3 :
Letzs rewrite our result for the angle to exhibit its dependence on the difference of the
two wave numbers:
£
¤
Ey
exp (ikR z) ¡ exp (ikL z) exp ¡i z2 (kL + kR )
£
¤
tan µ =
=i
Ex
exp (ikR z) + exp (ikL z) exp ¡i z2 (kL + kR )
£
¤
£
¤
1 exp i (kR ¡ kL ) z2 ¡ exp ¡i (kR ¡ kL ) z2
£
¤
£
¤
= ¡
i exp i (kR ¡ kL ) z2 + exp ¡i (kR ¡ kL ) z2
£¡
¢¤
h
sin kL ¡ kR z2
zi
£
¤ = tan (kL ¡ kR )
=
z
2
cos (kL ¡ kR ) 2
Thus
Using our result (58), we ®nd
h
zi
µ = (kL ¡ kR )
§ 2n¼
2
"
#
!
1 ! 2p ³
! c ´ 1 !2p ³
!c ´
kL ¡ kR =
¡
1¡
+
1+
c
2 !2
!
2 !2
!
2 ³
´
2
! p! c
! !p !c
=
= 2
2
c!
!
! c
n0 e2 eB0 1
e3 n0 B0
=
=
2
"0 m m ! c
"0 m 2 c ! 2
and thus the rotation angle determines the product n0 B0 of the plasma density and magnetic
®eld strength.
e3 n0 B0 z
µ=
§ 2n¼
" 0 m2 c ! 2 2
26
where z is the distance between source and observer. Measurements at several frequencies
are necessary to remove the ambiguity due to the unknown factor 2n¼:
For a pulsar at a distance of 1 kpc with n0 = 0.05 cm ¡3 and B0 = 3 ¹G, the expected
rotation is:
2
µ
=
=
=
e3
(2¼)
n 0 B0 2 z
2"0 m2 c
f
2¼ 2 e3
z
n 0 B0 2
"0 m 2 c
f
¡
¢3
2¼ 2 1:6 £ 10 ¡19 C
(8:85 £
10¡ 12
= 3: 76 £ 10 7
F/m) (9 £
10¡ 31
3
C
n 0 B0 z
F ¢ kg 2 f 2
2
kg) (3 £
10 8
n 0 B0 z
m/s) f 2
Now a Farad is a C/V and a V/m is a N/C, so a F = C2 /J, thus the units are
C¢J¢s/kg2 =C¢kg¢m2 /kg 2 .s= C¢m 2 =kg¢s: Now put in the numbers for the physical parameters
n0 ; B0 and z :
¡
¢¡
¢¡
¢
2 5 £ 10 4 m ¡3
3 £ 10 ¡10 T 3 £ 10 19 m
7C ¢m
µ = 3: 76 £ 10
kg ¢ s
f2
22
1: 692 £ 10 C ¢ T
=
f2
kg ¢ s
Now a N equals a C¢T¢m/s (from the force law), so the units now are
C ¢T
N
1
=
= 2
kg ¢ s
kg ¢ m
s
which is exactly what we need!
At f = 1 GHz we get
1: 7 £ 10 22
= 1: 7 £ 104 radians
1018
which shows that we do indeed expect the wave to rotate around more than once under some
circumstances.
See, eg, The Astrophysical Journal, Volume 642, Issue 2, pp. 868-881. (2006)
µ=
7.3.3 Whistlers
Letzs look at waves on the lower branch of the R-wave dispersion relation, i.e. at frequencies
below !c : (This is the dashed "bucket" in the upper left corner of the n2 versus ! plots.)
Here we have the dispersion relation
!2 =! 2
!2 =! 2
c2 k2R
¡ p ! ¢ = 1 + ¡! p
¢
=
1
¡
c
!2
1 ¡ !c
¡1
!
At frequencies well below the cyclotron frequency, we can approximate the denominator to
27
get:
¶
!
!c
and if the frequency ! is also well below ! p ; we will have the approximate relation
r
!
ckR ' ! p
!c
or equivalently
(ck)2 ! c
!=
! 2p
These waves have a phase speed
p
!
!! c
vÁ =
=c
k
!p
and a group speed
r
p
d!
c2 k! c
c!c
!
!! c
vg =
=2
=
2
!
=
2c
= 2vÁ
p
2
2
dk
!p
!p
!c
!p
c2 k2R ' ! 2 + ! 2p
µ
In the ionosphere at 5 RE the plasma frequency is about 10 4 Hz and the cyclotron frequency
is about 2.9 MHz. Thus waves in the kHz band satisfy the constraints we have imposed,
and would have this group speed. The group speed decreases with frequency, thus a pulsed
signal would arrive at the observer high frequency ®rst, and lower frequencies later. Put
through an audio ampli®er, the signal would sound like a whistle, hence the name: whistler.
During WWI radio operators used frequencies around 10 kHz and heard whistling
signals that they interpreted as enemy shells. They turned out to be lighning pulses from the
southern hemisphere that had travelled along the ®eld lines to the north.
See also Journal of Geophysical Research, vol. 86, June 1, 1981, p. 4471-4492. for
example.
Or The Astrophysical Journal, Volume 610, Issue 1, pp. 550-571 (2004)
7.4
Low frequency waves: propagation along the magnetic ®eld.
For low-frequency waves we need to include the ion motion in the current.
~j = n0 e (~vi ¡ ~ve )
~ (purely transverse waves) there is no density perturbation,
For waves propagating along B
and no electrostatic ®eld. We may use all the results from our previous derivation, provided
that we adjust the current accordingly. We may also use equations (37), and (38) for the ion
28
motion, if we make the appropriate changes to the charge and mass of the particle. Thus:
"
Ã
!#
e iE x ¡ -!c E y
e iE x + !!c E y
jx = n0 e
¡ ¡
-2
!M
!m 1 ¡ ! 2c2
1 ¡ ! c2
!
Ã
!
2 iE ¡ - c E
2 iE + ! c E
n0 e
n0 e
x
y
x
y
!
!
=
+
-2
!2
c
c
!M
!m
1¡ 2
1¡ 2
- 2p iE x ¡
=
!
!
-c
E
y
!
-2
c
!2
1¡
Ã
-2p
iE x
-2
!
1 ¡ c2
=
!
!
! 2p iEx + !!c E y
+
!2
!
1 ¡ ! c2
!
Ã
!
! 2p
! 2p ! c
-2p - c
Ey
+
+ 2
¡
!2
!2
-2
!
1 ¡ ! c2
1 ¡ ! c2
1 ¡ ! 2c
and similarly for the y¡component:
"
#
¡e ¡ -!c E x ¡ iEy
e !!c E x ¡ iE y
jy = n0 e
¡
2
c
!M
!m 1 ¡ ! 2c2
1¡ 2
!
"
#!
-c
!c
1
E
+
iE
E
¡
iE
x
y
x
y
=
- 2p !
¡ !2p !
-2
!2
!
1 ¡ ! c2
1 ¡ ! c2
"
Ã
!
Ã
!#
- 2p - c
! 2p ! c
-2p
! 2p
1 Ex
=
¡
+ iE y
+
-2
!2
-2
!2
! !
1 ¡ c2
1 ¡ c2
1 ¡ c2
1 ¡ c2
!
!
!
!
and hence Amperezs law takes the form (cf 52):
Ã
!
Ã
!
¡
¢
-2p
! 2p
! 2p ! c
- 2p - c
Ey
Ex
+
¡i
¡
= ! 2 ¡ k2 c 2 E x
-2
!2
!2
-2
c
c
c
c
!
1 ¡ !2
1 ¡ !2
1 ¡ !2
1 ¡ !2
and (50)
Ey
Ã
-2p
1¡
-2
c
!2
+
! 2p
1¡
!2
c
!2
!
Ex
+i
!
Ã
! 2p ! c
1¡
giving the dispersion relation
Ã
!2
- 2p
!2p
2
2 2
+
¡! +k c
-2
!2
1 ¡ ! c2
1 ¡ ! c2
-2p
1¡
-2
c
!2
+
! 2p
1¡
!2
c
!2
¡ !2 + k2 c2
29
!2
c
!2
=
¡
- 2p - c
1¡
1
!2
-2
c
!2
Ã
!
¡
¢
= ! 2 ¡ k2 c 2 E y
! 2p ! c
- 2p - c
!2
¡
!2
-2
1 ¡ ! c2
1 ¡ ! c2
Ã
!
!2p !c
-2p -c
1
= §
¡
-2
! 1 ¡ ! 2c2
1 ¡ ! c2
!
Rearranging:
!
2
=
=
2 2
k c +
k2 c 2 +
- 2p
1¡
-2
c
!2
- 2p
1 ¨ -!c
µ
-c
1§
!
+
¶
+
! 2p
1 § !!c
! 2p
1¡
!2
c
!2
³
1¨
!c ´
!
(59)
We obtain a second resonance at the ion cyclotron frequency, but this time it is the L wave
that has the resonance. The refractive index is:
-2p
! 2p
c 2 k2
n2 =
=
1
¡
¡
(60)
!2
! (! § -c ) ! (! ¨ !c )
Taking the mass ratio to be 3 rather than the real value of about 2000, and !p =! c = 2; the
plot now looks like this:
20
18
16
14
12
n^2 10
8
6
4
2
0
1
2
w/wc
3
n2 versus !=!c : solid line- L waveu dashed, R wave
Below: same plot with ! p =! c = 1/2 rather than 2:
30
4
5
5
4
3
n^2
2
1
0
0.5
1
1.5
w/wc
2
2.5
n2 versus !=!c : solid line- L waveu dashed, R wave
Notice that the resonance at ! = 0 has disappeared, and both R and L circular waves
have the same n as ! ! 0:
7.5
~0:
Low frequency propagation across B
~ along B
~ 0 (O-wave), the ions as well as the electrons can
When the wave is polarized with E
move freely along the ®eld lines, so we expect to get only a minor change to the previous
dispersion relation. The current is
µ
¶
~j = n0 e (~v i ¡ ~v e ) = n0 e e ¡ ¡e E
~
M
m
µ
¶
1
1 ~
2
= n0 e
+
E
M
m
Combining with Amperezs law, we get the dispersion relation:
! 2 = c2 k2 + ! 2p + -2p
q
Since there is no propagation at frequencies less than ! p (now strictly ! 2p + - 2p ) we
never have to worry about low frequency waves.
~ £B
~ drift is parallel to ~k;
The other polarization (X-wave) is more interesting. The E
suggesting that we will have a longitudinal component to this wave. We already saw this for
31
3
the high frequency waves. So letzs include the pressure term. The equation of motion is:
³
´
~ + ~v £ B
~ 0 ¡ i°kB T i~k n1
¡i!M ~v i = e E
n0
and the continuity equation gives the by-now familiar expression for n1 :
~k ¢ ~v
n1
=
n0
!
Notice that the density perturbation is more important at low frequencies. Then
³
´
~
~ + ~v £ B
~ 0 ¡ i°kB T i~k k ¢ ~v
¡i!M ~v i = e E
!
Since these are low frequency waves, wezll also use the plasma approximation: ni ' ne
and consequently ~k ¢ ~vi = ~k ¢ ~v e : At low frequencies, then, the electrostatic ®eld is shielded
~ = 0: Taking the dot product of the equation of motion with ~k; we get:
and ~k ¢ E
´
2
e ~ ³
~k ¢ ~v i =
~ 0 ¡ °kB T ik ~k ¢ ~v
k ¢ ~v £ B
2
¡i!M
¡! M
µ
¶
³
´
2
°k
T
k
e
~k ¢ ~v i 1 ¡ B i
~k £ ~v ¢ B
~0
=
! 2M
¡i!M
~ along z: :
or, in components, with x along ~k and B
µ
¶
°kB T ik2
eB0
-c
vix 1 ¡
=
viy =
viy
!2M
¡i!M
¡i!
The y¡component is the same as before:
eB0
´ c viy
¡i!M viy = eEy ¡ ev ix B0 = eE y ¡ ³
° kB Ti k2 ¡i!
1 ¡ !2 M
The solution for vy is modi®ed:
0
viy @1 ¡ ³
eB0
!2
¡
°k B Tik 2
M
The parenthesis in the denominator is:
´
1
-c
e
A=i
Ey
M
!M
! 2 ¡ k2 v 2i;th
and thus
Ã
!
2
!2 ¡ k2 vi;th
e
v iy = i
Ey
2
!M
!2 ¡ -2c ¡ k 2 vi;th
The result for electrons is the same with the usual switch of e ! ¡e and m ! M:
Then substituting into Amperezs law, we have:
µ 2
¶
!
2
~ 1 = ¡i¹0~k £ ~j = ¡i¹0 ~k £ n0 e (~vi ¡ ~ve )
¡k B
c2
32
The interesting component is the z¡component:
µ 2
¶
!
2
¡
k
Bz = ¡i¹0 n0 ek (viy ¡ vey )
c2
Ã
Ã
!
Ã
!!
! 2 ¡ k 2 v2i;th
! 2 ¡ k2 v 2e;th
e
e
= ¡i¹0 n0 ek i
Ey
+i
Ey
2
!M
! 2 ¡ - 2c ¡ k2 v 2i;th
!m
!2 ¡ !2c ¡ k2 ve;
th
and ®nally Faradayzs law gives us
So the dispersion relation is
2
2 2
! ¡c k =
-2p
Ã
Ey =
!
Bz
k
2
!2 ¡ k 2 vi;th
!
2
!2 ¡ -2c ¡ k 2 vi;th
+
! 2p
Ã
! 2 ¡ k2 v 2e;th
2
! 2 ¡ ! 2c ¡ k 2 ve;th
!
(61)
Check that we have the right limit as T ! 0:
Now we recall that these are low frequency waves (! ¿ ! p ): We are going to make the
additional assumptions:
! c À kv th;e
(the electrons travel a distance much less than a wavelength in one gyro-period) and also
! ¿ -c ¿ !c
and
kvth; i ¿ -c
With these approximations, the dispersion relation simpli®es:
Ã
!
Ã
!
2
2 2
2
2 2
!
¡
k
v
!
¡
k
v
i;th
e;th
!2 ¡ c2 k 2 = - 2p
+ ! 2p
¡- 2c
¡! 2c
Ã
!
Ã
!
- 2p
!2p
- 2p
! 2p 2
!2 1 + 2 + 2
= k 2 c2 + 2 v 2i;th + 2 ve;
-c
!c
-c
!c th
The ratio
- 2p
n0 e2 M 2
n0 M
=
"0 M e 2 B20
"0 B02
2
and similarly for (!p =! c ) : Thus the dispersion relation is:
µ
¶
µ
¶
n0 (M + m)
n0 M kB Ti
n0 m kB T e
2
2
2
! 1+
= k c +
+
"0
B02
"0 B02 M
"0 B 20 m
µ
¶
µ
¶
½
n0
2
2
2
! 1+
= k c +
kB (Ti + T e)
"0 B02
"0 B02
- 2c
=
(62)
De®ne a new speed called the Alfven speed:
2
vA
=
B02
¹0 ½
33
(63)
Then
¢
2 + v2
vA
s
! =k c
(v 2A + c2 )
It is often the case that vA ¿ c; in which case the relation simpli®es again, to give:
¡ 2
¢
! 2 = k 2 vA
+ v 2s
2
2 2
¡
(64)
(65)
The dispersion relation (64) or (65) decribes magnetosonic waves.
~0:
Very Low frequency propagation along B
7.6
We start with our previous result (59)
!2 = k2 c2 +
! 2p
-2p
+
1 ¨ !!c
1 § -!c
Now we approximate for ! ¿ - c ; (and hence also ! ¿ !c ): Then
!2
=
=
=
!! 2
!- 2
³ p ´ §
³ p ´
! c 1 ¨ !!c
-c 1 § -!c
µ
¶
µ
¶
!2p
-2p
!
!2
!
!2
k2 c 2 ¨ !
1§
+ 2 §!
1¨
+ 2
!c
!c
!c
-c
-c
-c
2
2 2
2 3
2
2 2
2 3
!
!
!
!
!
!
p
p
p
p
p
p!
k2 c 2 ¨ !
¡
¨
§
!
¡
+
2
3
2
!c
!c
!c
-c
-c
- 3c
k2 c 2 ¨
Now
!2p
-2p
n0 e 2 m
n0 e
=
=
=
!c
"0 m eB0
"0 B0
-c
and thus the ®rst order terms in ! cancel. Dropping the third order terms, we get:
! 2 = k 2 c2 ¡
Now
(compare 62). Thus
or
! 2p ! 2
- 2p ! 2
¡
!2c
-2c
µ
¶2
! 2p
n0 e2
m
n0 m
=
=
2
!c
"0 m eB 0
"0 B02
µ
¶
n0 (m + M )
2
! 1+
= k2 c 2
"0
B02
! = kv A
where
vA = p
c
1 + ½="0 B 2
34
(66)
is the Alfven speed. Letzs compare with equation (63). From (66),
1
1
1
2
vA;2
=
=
½ =
2
2
¹0 "o (1 + ½="0 B )
¹0 "o + ¹0 B2
1=c + 1=v 2A;1
B2
½¹0 (1 + B 2 =½¹0 c2 )
Thus the two expressions are the same when
=
B2
¿1
½¹0 c2
This is almost always the case. However, there are some extreme astrophysical situations
(near pulsars, for example) where B is very large and ½ is very small and this ratio
approaches 1. In these cases we have to use the more exact result (66).
Now letzs take a closer look at what is going on in this wave. The wave vector is
~ 0 ; and both the electric and magnetic ®elds of the wave lie in the x ¡ y plane. In
along B
the low-frequency limit, both the L and R circular waves travel at the same speed. This
~ in the x¡direction,
means that linearly polarized waves are also normal modes. With E
~ 1 = 1 ~k £ E
~ = k E y^ is in the y¡direction. The plasma particles undergo an E
~ £B
~ drift
B
!
!
which is (to ®rst order)
~ £B
~0
E
E
~v E =
=
(¡^
y)
2
B0
B0
~ 1 causes the magnetic ®eld to change direction, and since B
~ 1 has a wave
The perturbation B
form, the ®eld line has a ripple:
~ 1 oscillates, the ®eld line ripple moves back and forth. The ®eld line behaves like a
As B
vibrating string. The speed of the sideways motion is
B1
!B1
E
v = vÁ
=
=
B0
kB0
B0
which is the same as the particle drift speed. Thus the particles and the ®eld lines move
35
together. This phenomenon is sometimes called -ux freezing as the magnetic -ux is frozen
into the plasma.
The Alfven speed
s
B2
½¹0
may be viewed as the speed of a wave on a string with tension
v=
B2
¹0
and density ½: Wezll return to these ideas later when we study magnetohydrodynamics
(MHD).
The Alfven speed plays an important role in plasmas because -ows that are superAlfvenic often behave like supersonic -ows in ordinary -uids. (Notice that as T ! 0 the
magnetosonic wave also travels at the Alfven speed. ) For example, shock waves may form.
The solar wind parameters are: speed about 400 km/s near the Earth, density n » 5
cm¡3 ; proton cyclotron frequency (from ACE news release) of about 0.17 Hz.
T mag =
eB
:17 £ 2¼ £ 1:7 £ 10 ¡27 kg
kg
= 0:17 £ 2¼ rad/s ) B =
= :11349 £ 10 ¡7
¡19
m
s (1:6 £ 10
C)
s¢C
Now the units: 1 C¢T¢m/s=1 N=1 kg¢m/s2 and so 1 T = 1 kg/s¢C. Thus we have B of about
10 ¡8 T. The Alfven speed is then
vA
»
=
q¡
10 ¡8 T
5 £ 10 6 £ 1:7 £ 10 ¡27 kg/m
9: 676 £ 10 4
kg/s ¢ C
kg
m¢ C
¢
3
4¼ £ 10¡7 N/A 2
= 9: 7 £ 10 4 m/s
= 9: 6759 £ 10 4 r³
T
kg kg¢m
m3 A 2s 2
´
Thus the solar wind speed of about 4 £ 10 5 m/s exceeds the Alfven speed.
7.7
Low frequency waves at intermediate angles.
When waves propagate along the magnetic ®eld we get electrostatic waves (ion-acoustic
waves) propagating at phase speed vs ; and electromagnetic Alfven waves, travelling at phase
speed vA : Propagating across B we get the hybrid magnetosonic waves: (the X-mode is a
mixture of electromagnetic and electrostatic components). (The
p O-wave does not propagate
2 : . At intermediate
at low-frequency.) The speed of the magnetosonic waves is vs2 + vA
angles we expect the waves to transition from one mode to another. The Alfven wave
p
does propagate at all angles to B; and the phase speed is Bk = ½¹0 = vA cos µ where Bk
is the component of B parallel to the wave vector ~k: When v A > vs ; the magnetosonic
wave transitions to the Alfven wave as µ ! 0; but when v s > vA ; the magnetosonic wave
transitions to the acoustic wave. Wezll be able to investigate these low frequency waves
more easily when we study MHD (magnetohydrodynamics).
36
There are still 3 waves here. The lowest branch is called the modi®ed Alfven wave.
37