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Improved basis set for low frequency plasma waves
P. M. Bellan
1
Received 22 April 2012; revised 16 October 2012; accepted 24 October 2012; published 14 December 2012.
[
1
]
It is shown that the low frequency plasma wave equation can be obtained much more
directly than by the previously used method of solving for the determinant of a matrix
involving the three components of the electric field vector. The more direct method uses a
two-dimensional current density vector space that is precisely equivalent to the previously
used three-dimensional electric field vector space. Unlike the electric field, the current
density is restricted by the quasi-neutrality condition to a two-dimensional vector space.
Comparison with previously obtained dispersion relations is provided and a method is
presented for obtaining exact analytic solutions for the three roots of the cubic dispersion
relation. The commonly used kinetic Alfvén dispersion relation is shown to be valid only
for near-perpendicular propagation in a low beta plasma. It is shown that at a cross-over
point where the perpendicular wave phase velocity equals the ion acoustic velocity, the
coupling between Alfvén and fast modes vanishes and the Alfvén mode reverts to its cold
form even in situations where the Alfvén velocity is smaller than the electron thermal
velocity. A method is prescribed by which measurement of wave electric current density
completely eliminates the space-time ambiguity previously believed to be an unavoidable
shortcoming of single-spacecraft frequency measurements.
Citation:
Bellan, P. M. (2012), Improved basis set for low frequency plasma waves,
J. Geophys. Res.
,
117
, A12219,
doi:10.1029/2012JA017856.
1. Introduction
[
2
] Several apparently different approaches have been
used to describe low frequency waves in a warm magnetized
plasma. While all assume an

exp(
ik
x
x
+
ik
z
z

i
w
t
) wave
dependence and an equilibrium magnetic field
B
¼
B
^
z
,itis
not obvious how these approaches relate to each other. The
algebraic manipulations used have varying complexity and
surprising cancelations sometimes occur after much tedious
algebra. This variation in complexity suggests that the dif-
ferent approaches are effectively using different basis sets in
some multidimensional vector space to characterize the same
physics. A non-optimum basis set would require more
algebra to arrive at the same final result and fortuitous-
appearing cancelations would result. On the other hand,
if an optimum basis set were chosen, algebraic complexity
would be minimized, no surprising cancelations would
occur, and the underlying physics would be more transpar-
ent. The results of various previously used approaches are
first summarized and then a derivation in an optimized basis
set is presented. This is used to identify limitations and
occasional inconsistencies or errors in the previous approa-
ches. These previous approaches are the generalized Ohm
’
s
law method discussed by
Stringer
[1963], the two-fluid
method discussed by
Hollweg
[1999], the 2

2 matrix
method used by
Hasegawa and Uberoi
[1982, section 2.8.3],
Morales and Maggs
[1997], and
Lysak and Lotko
[1996], and
the 3

3 matrix kinetic method used by
Hirose et al.
[2004].
The cold plasma dispersion given by
Stix
[1992] is used as a
reference when considering certain limiting situations. The
previous approaches are all based on deriving a homoge-
neous vector equation involving the vector electric field and
then obtaining a dispersion relation from this vector equation.
The dispersion relations obtained using these various
approaches are listed below:
1.1. Equation (4) of Stringer
[
3
]
Stringer
[1963] places no constraints on the ratio of
wave phase velocity to particle thermal velocity and notes
that the general wave equation has three high frequency
roots where ions play an insignificant role and three low
frequency roots where both ions and electrons are important.
Stringer
[1963] argues that the high frequency roots are
conveniently eliminated by neglecting terms of order
w
/
kc
and so obtains the dispersion relation
cos
2
q

Q
w
2
k
2
v
2
A

cos
2
q

w
2
k
2
c
2
S


Q
w
2
k
2
v
2
A
1

w
2
k
2
c
2
S


¼
1

w
2
k
2
c
2
S

w
2
w
2
ci
cos
2
q
:
ð
1
Þ
Here
Q
¼
1
þ
k
2
c
2
=
w
2
pe
ð
2
Þ
1
Applied Physics, California Institute of Technology, Pasadena,
California, USA.
Corresponding author: P. M. Bellan, Applied Physics, California
Institute of Technology, Pasadena, CA 91125, USA.
(pbellan@its.caltech.edu)
©2012. American Geophysical Union. All Rights Reserved.
0148-0227/12/2012JA017856
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117, A12219,
doi:10.1029/2012JA017856, 2012
A12219
1of14
and cos
q
=
k
z
/
k
. This dispersion relation has been used by
Formisano and Kennel
[1969] and by
Rogers et al.
[2001].
The derivation of equation (1) involves taking the determi-
nant of a fully populated 3

3 matrix involving all three
electric field components and is algebraically quite compli-
cated [see
Stringer
, 1963, Appendix I;
Swanson
, 1989,
section 3.3.1].
1.2. Hollweg Equation (38)
[
4
]
Hollweg
[1999] did not take a determinant, but did
what is mathematically equivalent, namely manipulated a set
of homogenous equations involving different unknowns
until one homogenous equation in one unknown was
obtained. Various approximations were invoked, and in
particular some, but not all, terms of order
w
2
=
w
2
ci
were
dropped. The resulting dispersion relation was Hollweg
’
s
equation (38), namely
w
2
k
2
z
v
2
A

1

w
2
w
2

k
2
v
2
A


b
k
2
v
2
A
w
2

k
2
z
v
2
A


¼
w
2
w
2

k
2
v
2
A

k
2
x
c
2
s
w
2
ci

c
2
w
2
pe
w
2
k
2
z
v
2
A
!
ð
3
Þ
where
b
¼
c
2
s
=
v
2
A
. Equation (3) was claimed to be valid in
the regime
w
≪
w
ci
and for arbitrary
c
s
/
v
A
and for arbitrary
k
z
/
k.
Because the algebraic details leading to equation (3)
were not given but rather only characterized as being based
on
“
three lengthy relations involving the three components
of
d
E
”
, it is not possible to replicate with certainty the
manner in which equation (3) was derived.
1.3. The 2

2 Matrix Approach
[
5
]
Hasegawa and Uberoi
[1982],
Morales and Maggs
[1997], and
Lysak and Lotko
[1996] argued that the 3

3
matrix equation describing warm plasma waves could be
approximated by a 2

2 matrix because the compressional
(i.e., fast) mode could be factored out. This resulted in
w
2
k
2
z
v
2
A

1
k
x
k
z
k
x
k
z
w
2
k
2
z
c
2
s

1

w
2
ci
k
2
z
v
2
A

k
2
x
k
2
z
2
6
6
4
3
7
7
5

~
E
x
~
E
z

¼
0
ð
4
Þ
the determinant of which in the limit
w
2
≫
k
2
z
c
2
s
reduces to
the well-known kinetic Alfvén dispersion relation
w
2
¼
k
2
z
v
2
A
1
þ
k
2
x
c
2
s
w
2
ci

:
ð
5
Þ
Equation (5) was claimed to be valid in the regime
w
≪
w
ci
and
v
Te
≫
w
/
k
z
≫
v
Ti
with no restriction on propa-
gation angle.
1.4. The 3

3 Matrix Method Used by Hirose With
Cold Ion Assumption
[
6
]
Hirose et al.
[2004] evaluated the components of the
dielectric tensor calculated using kinetic theory and obtained
an extremely complicated expression (44 terms) when both
ions and electrons were warm. However, when the ions were
assumed cold, the expression reduced to
w
2
k
2
z
v
2
A
1
1

w
2
!
2
ci

1
i
w
2
k
2
z
v
2
A
w
=
w
ci
1

w
2
!
2
ci
k
x
k
z

i
w
2
k
2
z
v
2
A
w
=
w
ci
1

w
2
!
2
ci
w
2
k
2
z
v
2
A
1
1

w
2
!
2
ci

k
2
k
2
z

i
w
ci
w
w
2
k
2
z
v
2
A
k
x
k
z
k
x
k
z
i
w
ci
w
w
2
k
2
z
v
2
A
k
x
k
z
w
2
k
2
z
c
2
s

1

w
2
ci
k
2
z
v
2
A

k
2
x
k
2
z
2
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
5

~
E
x
~
E
y
~
E
z
2
6
4
3
7
5
¼
0
:
ð
6
Þ
Evaluating the determinant of the matrix in equation (6)
resulted in the amazing result that a quantity 1

w
2
=
w
2
ci

factored out exactly, resulting in the much simpler than
expected dispersion relation
w
2

k
2
z
v
2
A

w
4

w
2
k
2
c
2
s
þ
v
2
A

þ
k
2
v
2
A
k
2
z
c
2
s

¼
k
2
c
2
w
2
pi
w
2
v
2
A
k
2
z
w
2

k
2
c
2
s

:
ð
7
Þ
1.5. Chen and Wu Polynomial Method
[
7
]
Chen and Wu
[2011a] wrote the complete set of two-
fluid equations, assumed
w
≪
w
ci
, and obtained a matrix
equation involving the components of the electric field. By
taking the determinant of this matrix, they obtained a cubic
equation in
W
=
w
/
k
z
v
A
. The coefficients of the cubic equa-
tion were polynomials involving various dimensionless
ratios such as the electron to ion mass ratio,
k
2
?
c
2
s
=
w
2
ci
, and
k
2
?
c
2
=
w
2
pe
.
1.6. Stix Cold Plasma Method
[
8
] For reference, the
Stix
[1992] analysis of cold plasma
waves provides
S

n
2
z

iD
n
x
n
z
iD
S

n
2
0
n
x
n
z
0
P

n
2
x
2
4
3
5

~
E
x
~
E
y
~
E
z
2
4
3
5
¼
0
ð
8
Þ
where
n
=
c
k
/
w
,
S
¼
1

∑
w
2
p
s
=
w
2

w
2
c
s

,
D
¼
∑
w
c
s
=
w
ðÞ
w
2
p
s
=
w
2

w
2
c
s

,
P
¼
1

∑
w
2
p
s
=
w
2
:
Evaluation
of the determinant of the matrix in equation (8) gives
S
sin
2
q
þ
P
cos
2
q

ck
=
w
ðÞ
4

RL
sin
2
q
þ
PS
1
þ
cos
2
q


ck
=
w
ðÞ
2
þ
PRL
¼
0
ð
9
Þ
where
R
=
S
+
D
and
L
=
S

D.
Equation (9) is valid for all
w
and
k
provided
w
/
k
z
≫
v
Te
,
v
Ti
and
k
x
r
L
e
;
i
≪
1. Equation (9)
predicts
k
!
0 when any one of
P
,
R
,or
L
vanish (cutoffs)
and predicts that
k
!
∞
for perpendicular propagation when
S
!
0 (wave resonance).
1.7. Summary of Main Results
[
9
] This paper will demonstrate the following:
BELLAN: LOW FREQUENCY WAVE IMPROVED BASIS
A12219
A12219
2of14
[
10
] 1. By using
~
J
instead of
~
E
as the fundamental quan-
tity, the Stringer result can be derived in a quicker and more
intuitive way than in the original paper.
[
11
] 2. Equation (3) involves an inconsistent retention of
terms of order
w
2
=
w
2
ci
and, contrary to the assertions in
Hollweg
[1999], is valid only if both
c
s
≪
v
A
and propaga-
tion is nearly perpendicular to the magnetic field.
[
12
] 3. Equation (7) is in agreement with the small
w
/
w
ci
limit of equation (1).
[
13
] 4. Equation (4) and hence equation (5) are only cor-
rect for cos
2
q
≪
b
, i.e., only for near-perpendicular propa-
gation in a low
b
plasma.
[
14
] 5. Exact analytic solutions of the form
w
=
w
(
k
) for
the three different roots of the Stringer dispersion relation
are given. These solutions are valid for arbitrary
b
and
arbitrary propagation angle.
[
15
] 6. When
w
2
=
k
2
c
2
s
¼
1, the Alfvén mode decouples
from the fast mode and reverts to its cold plasma character
even if
m
e
/
m
i
<
b
≪
1; this is contrary to the prediction of the
2

2 matrix method and may have some geophysical
implications.
[
16
] 7. Besides describing Alfvén and fast modes, the
Stringer result describes the magnetized ion acoustic dis-
persion relation, the cold ion cyclotron wave dispersion, the
lower hybrid resonance in a plasma where
w
2
pe
≫
w
2
ce
, and
whistler waves.
[
17
] 8. A method is prescribed by which measurement of
~
J
completely eliminates the space-time ambiguity previously
believed to be an unavoidable shortcoming of single-
spacecraft frequency measurements.
[
18
] 9. The effect of finite resistivity (electron-ion colli-
sions) is discussed and it is shown that the modes with
w
2
≪
w
2
ci
are relatively unaffected by resistivity.
[
19
] 10. The relationship to the high-frequency quasi-
longitudinal and quasi-transverse modes predicted by the
Altar-Appleton-Hartree dispersion relation is discussed.
2. Derivation of Stringer Dispersion Using
Optimum Vector Space
[
20
] The derivation of equation (1) by
Stringer
[1963]
involves lengthy algebraic manipulations that eventually
produce three homogeneous equations in the three
unknowns
k

~
E
,
~
E

B
, and
~
E

k

B
. The vanishing of the
determinant of the coefficients of these three equations
provides the dispersion relation given in equation (1). We
show here that equation (1) can be obtained much more
directly by using a different and more natural vector space,
namely
k

~
J
,
k
?

~
J
, and
k

B

~
J
where
~
J
¼
∑
n
s
q
s
~
u
s
is
the electric current density associated with the wave. This
vector space has the immediate advantage that
k

~
J
¼
0for a
quasi-neutral plasma so the system reduces to just two cou-
pled equations in the two unknowns
k
?

~
J
, and
k
?

B

~
J
.
The quasi-neutral set of equations comprise the equations of
motion and continuity for each species
s
,

i
w
m
s
n
~
u
s
¼
nq
s
~
E
þ
~
u
s

B


g
s
k
T
s
i
k
~
n
;
ð
10
Þ

i
w
~
n
þ
i
k

~
u
s
n
¼
0
ð
11
Þ
where
g
s
=1if
w
/
k
z
≪
v
T
s
(isothermal equation of state) and
g
s
=3if
w
/
k
z
≫
v
T
s
(adiabatic equation of state) together
with Faraday
’
s law
i
k

~
E
¼
i
w
~
B
ð
12
Þ
and the pre-Maxwell Ampere
’
s law
i
k

~
B
¼
m
0
~
J
:
ð
13
Þ
The pre-Maxwell Ampere
’
s law provides the quasi-neutrality
condition
k

~
J
¼
0 which is the critical assumption that
enables the method presented here. As shown in Appendix A,
the assumption of quasi-neutrality fails at the cold plasma
L
= 0 cutoff; this failure places the maximum frequency at
which the quasi-neutrality assumption is valid at a value well
above the ion cyclotron frequency.
[
21
] Combination of Faraday
’
s law and the pre-Maxwell
Ampere
’
s law gives
k

k

~
E

¼
i
wm
0
~
J
:
ð
14
Þ
Defining the one-fluid quantities
~
U
¼
∑
m
s
n
u
s
ðÞ
=
∑
m
s
n
and
r
=
∑
nm
s
, and then summing equations (10) over species
gives the one-fluid equation of motion

i
wr
~
U
¼
~
J

B

i
k
X
g
s
k
T
s
~
n
:
ð
15
Þ
It should be noted that this one-fluid model differs in a very
subtle and normally insignificant way from magnetohydro-
dynamics in terms of how temperature is defined. Specifi-
cally, the electron temperature used in (15) is the two-fluid
temperature and so is defined in terms of the random electron
velocities relative to the mean electron velocity. Similarly,
the ion temperature is defined in terms of the random ion
velocities relative to the mean ion velocity. In contrast, in
magnetohydrodynamics the electron and ion temperatures
are each defined with respect to the center of mass velocity of
the entire plasma [see
Bellan
, 2006, section 2.6.2]. If the
electrons and ions have the same mean velocity, there is no
difference between the two-fluid and magnetohydrodynamic
definitions of temperature.
[
22
] Multiplication of equation (11) by
m
s
and summing
over species gives

i
w
~
n
X
m
s
þ
i
k

~
U
r
¼
0
ð
16
Þ
which can be solved to give
~
n
¼
k

~
U
r
w
P
m
s
¼
n
w
k

~
U
:
ð
17
Þ
Dotting equation (15) with
i
k
w
/
r
gives
w
2
k

~
U
¼
i
w
k

~
J

B
r
þ
k
2
k

~
U
X
g
s
n
k
T
s
r
:
ð
18
Þ
Solving for
k

~
U
and using
r
/
n
≃
m
i
gives
k

~
U
¼
i
w
k

~
J

B
rw
2

k
2
c
2
s

ð
19
Þ
BELLAN: LOW FREQUENCY WAVE IMPROVED BASIS
A12219
A12219
3of14
where
c
2
s
¼
P
g
s
k
T
s
=
m
i
defines the sound velocity.
Equation (17) then becomes
~
n
n
¼
i
k

~
J

B
rw
2

k
2
c
2
s

:
ð
20
Þ
Using equation (20), equation (15) reduces to
~
U
¼
i
wr
~
J

B
þ
k
k

~
J

B
c
2
s
w
2

k
2
c
2
s

:
ð
21
Þ
The generalized Ohm
’
s law is obtained as follows: equation (10)
is multiplied by
m
0
q
s
/
m
s
andthensummedoverspecies,terms
of order
m
e
/
m
i
are then discarded, and the approximation
U
?
¼
m
i
~
u
i
þ
m
e
~
u
e
ðÞ
?
m
i
þ
m
e
≃
~
u
i
?
ð
22
Þ
is made so
~
u
e

B
!
~
U

B
þ
~
J

B
=
nq
e
. The generalized
Ohm
’
s law, obtained from these operations and approxima-
tions, is

i
w
c
2
w
2
pe
m
0
~
J
¼
~
E
þ
~
U

B
þ
~
J
nq
e

B

g
e
m
0
c
2
w
2
pe
k
T
e
m
e
q
e
i
k
~
n
:
ð
23
Þ
Substituting for
~
U
in equation (23) and solving for
~
E
gives
~
E
¼
i
w
c
2
w
2
pe
m
0
~
J

i
wr
~
J

B
þ
k
k

~
J

B
c
2
s
w
2

k
2
c
2
s


B

~
J
nq
e

B
þ
g
e
m
0
c
2
w
2
pe
k
T
e
m
e
q
e
i
k
~
n
:
ð
24
Þ
Substituting for
~
E
in equation (14) using equation (24) gives
the sought-after vector equation involving
~
J
only,
k

k

i
w
c
2
w
2
pe
m
0
~
J

i
wr
~
J

B
þ
k
k

~
J

B
c
2
s
w
2

k
2
c
2
s

!
"
(

B

~
J
nq
e

B
#)
¼
i
wm
0
~
J
:
ð
25
Þ
The pressure term in equation (24) was annihilated upon being
crossed with
k
so the only way electron pressure contributes is
as a term in the one-fluid equation of motion. At this point, it is
noted that
B
2
=
m
0
r
¼
v
2
A
,
B
=
m
0
nq
e
¼
w
ci
c
2
=
w
2
pi
, and the
geometric mean frequency is
w
2
gm
¼
w
ce
w
ci
jj¼
w
2
pe
v
2
A
=
c
2
so
equation (25) can be expressed as
k

k

w
2
w
2
gm
~
J
þ
~
J

^
z


^
z
þ
k

^
z
k

~
J

^
zc
2
s
w
2

k
2
c
2
s

"
(
þ
i
w
w
ci
~
J

^
z
#)

w
2
v
2
A
~
J
¼
0
:
ð
26
Þ
Expanding equation (26) gives
k

k
?

~
J
?
þ
i
w
w
ci
k
?

~
J
?

^
z


k
2
w
2
w
2
gm
~
J

~
J
?
þ
k
?

^
z
k
?

~
J
?

^
zc
2
s
w
2

k
2
c
2
s

þ
i
w
w
ci
~
J
?

^
z
!

w
2
v
2
A
~
J
¼
0
:
ð
27
Þ
Dotting equation (27) first with
k
?

^
z
and then with
k
?
gives
two coupled equations involving
k
?

~
J
?
and
k
?

~
J
?

^
z
which can be expressed in matrix form as
w
2
k
2
z
v
2
A
1
þ
k
2
v
2
A
w
2
gm
!

1
i
w
w
ci

i
w
w
ci
w
2
k
2
v
2
A
1
þ
k
2
v
2
A
w
2
gm
!

1

k
2
?
c
2
s
w
2

k
2
c
2
s

2
6
6
6
6
6
4
3
7
7
7
7
7
5

k
?

~
J
?
k
?

~
J
?

^
z
"#
¼
0
:
ð
28
Þ
The determinant of the above matrix gives the dispersion
relation
w
2
k
2
v
2
A
1
þ
k
2
v
2
A
w
2
gm
!

1

k
2
?
c
2
s
w
2

k
2
c
2
s
"#
w
2
k
2
z
v
2
A
1
þ
k
2
v
2
A
w
2
gm
!

1
"#
¼
w
2
w
2
ci
ð
29
Þ
which is identical to equation (1) as can be seen by multiplying
equation (29) through by
w
2
=
k
2
c
2
s

1

cos
2
q
and noting
that
Q
¼
1
þ
k
2
c
2
=
w
2
pe
¼
1
þ
k
2
v
2
A
=
w
2
gm
.Inthelimitwhere
w
/
w
ci
!
0, it is seen that the Alfvén mode has
k
?

~
J
?
finite
while the compressional mode has
k
?

~
J
?

^
z
finite. The
acoustic mode involves
c
2
s
andsoalsoinvolves
k
?

~
J
?

^
z
being finite. The corresponding polarizations of the electric
field are then found using equation (24) with equation (20).
Essentially, we have solved for
~
E
as a function of
~
J
and used
this in equation (14) whereas the traditional approach is to
solve for
~
J
as a function of
~
E
and use this in equation (14); the
method presented here immediately gives a 2

2matrix
whereas the traditional method gives a 3

3 matrix that after
much algebra and many seemingly fortuitous cancelations
reduces to the same dispersion relation as that presented here.
[
23
] For
w
≪
w
ci
equation (29) describes kinetic and
inertial Alfvén waves, the fast mode, and magnetized ion
acoustic waves having
k
2
l
2
D
≪
1. For
w

w
ci
equation (29)
describes the ion cyclotron waves of cold plasma theory and
the electrostatic ion cyclotron waves of warm plasma theory.
For
w
ce
≫
w
>
w
ci
equation (29) describes the lower hybrid
resonance for an over-dense plasma, and whistler waves. It
fails to describe modes where quasi-neutrality is not satis-
fied, namely ion acoustic waves for which
k
2
l
2
D
>
1 and
modes near the
L
= 0 cold plasma wave cutoff. The fact that
the right hand side of equation (29) is
w
2
=
w
2
ci
shows that any
model that purports to describe coupling between the Alfvén
mode (2nd square bracket on left hand side) and the acoustic
mode (embedded in first square bracket on left hand side)
BELLAN: LOW FREQUENCY WAVE IMPROVED BASIS
A12219
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4of14
while dropping terms of order
w
2
=
w
2
ci
as in
Hollweg
[1999]
cannot be correct.
3. Comparison of Equation (29) to Hirose
[
24
]If
w
2
=
w
2
gm
≪
1 equation (29) reduces to
w
2
k
2
z
v
2
A

1

w
2
k
2
v
2
A

1

k
2
?
c
2
s
w
2

k
2
c
2
s

¼
w
2
w
2
ci
ð
30
Þ
which is identical to equation (7) for the situation of cold
ions and warm electrons (i.e., where
c
2
s
¼
k
T
e
=
m
i
); however,
Equations (1) and (29) have the advantage of also being
valid for the situation where both ions and electrons are
warm. Thus, equations (1), (7), and (29) are mutually con-
sistent. Equation (30) clearly shows that the right hand
coupling term is of order
w
2
=
w
2
ci
and so all terms of this order
must be kept in any evaluation of the coupling.
4. Comparison of Hollweg Result to Hirose
and Stringer
[
25
] Rearranging equation (3) and using
b
v
2
A
¼
c
2
s
gives
w
2
k
2
z
v
2
A

1

w
2
k
2
v
2
A

1

k
2
?
c
2
s
w
2

k
2
c
2
s

!
¼
w
2
w
2
ci
k
2
x
c
2
s
k
2
v
2
A
w
2

k
2
v
2
A

w
2

k
2
c
2
s

1

m
e
m
i
w
2
c
2
s
k
2
z

:
ð
31
Þ
which has the same left hand side as equation (30); i.e., the
Q
= 1 limit of equation (1). The respective right hand sides
of equations (30) and (31) agree only if
k
2
x
=
k
2
≃
1,
w
2
≪
k
2
v
2
A
,
and
w
2
≪
k
2
c
2
S
. This requires the propagation angle to be
nearly perpendicular to the magnetic field. Thus, Hollweg
’
s
assertion that equation (3) is valid at all angles and for all
values of
c
s
/
v
A
is not correct. We believe that the reason
equation (30) differs from equation (31) is that some terms
of order
w
2
=
w
2
ci
were retained in
Hollweg
[1999] while others
were discarded.
5. Exact Roots of Equation (29)
[
26
] By defining
x
¼
w
2
k
2
v
2
A
;
b
¼
c
2
s
v
2
A
;
L
¼
k
2
v
2
A
w
2
ci
;
a
¼
cos
2
q
ð
32
Þ
equation (29) becomes
x
Q
a

1

x
Q

x

ba
x

b

¼
x
L
:
ð
33
Þ
This can be expressed as a cubic equation in
x
, namely
x
3

A
x
2
þ
B
x

C
¼
0
ð
34
Þ
where
A
¼
Q
þ
Q
2
b
þ
Q
a
þ
a
L
Q
2
B
¼
a
1
þ
2
Q
b
þ
L
b
ðÞ
Q
2
C
¼
a
2
b
Q
2
:
ð
35
Þ
Equation (34) can be solved exactly for arbitrary
q
by using
a trigonometric substitution method given by
Nickalls
[1993] and previously used in the context of Alfvén waves
by
Chen and Wu
[2011a, 2011b]; however the coefficients
A
,
B
, and
C
used by
Chen and Wu
[2011a, 2011b] differ
from equation (35) here and so appear to be in error (note
that in
Chen and Wu
[2011a, 2011b], the parameter
Q
is the
electron to ion mass ratio which is negligible compared to
unity and so can be dropped from the expressions in
Chen
and Wu
[2011a, 2011b]).
[
27
] On defining
p
¼
3
B

A
2
3
q
¼
9
AB

2
A
3

27
C
27
ð
36
Þ
the exact roots of equation (34) are
x
j
¼
2
ffiffiffiffiffiffiffi

p
3
r
cos
1
3
cos

1
3
q
2
p
ffiffiffiffiffiffiffi

3
p
s
!

2
p
3
j
!
þ
A
3
;
j
¼
0
;
1
;
2
:
ð
37
Þ
These solutions to equation (34) are valid for arbitrary
q
,
w
,
k
,
c
s
, and
v
A
. Choosing
j
= 0, 1, or 2 gives the fast mode,
Alfvén mode, and acoustic mode respectively. A polar plot
of
x
versus angle produces a CMA-like (Clemmow-Mullaly
Allis) plot for given values of
b
and
L
. These plots can be
compared to simpler polynomial expansions such as
equation (5). Figure 1 provides an example of a CMA-like
plot of the three modes for
L
= 0.4 and
b
= 0.4; the solid
lines are plots of
w
2
=
k
2
v
2
A
versus
q
for the three exact roots of
the determinant as given by equation (37). The inner root
(slowest) is the sound wave, the intermediate root is the
Alfvén wave, and the outer root is the fast wave. The non-
solid lines show for comparison various approximations
discussed above and in the following text. In particular, the
MHD Alfvén wave dispersion
w
2
=
k
2
v
2
A
¼
cos
2
q
is indicated
by a line with short dashes (line immediately outside the
solid line labeled
‘
Alfvén
’
) while equation (5) is plotted as a
dotted line slightly to the right of the MHD Alfvén wave
dispersion (corresponding to the prediction of equation (5)
that
w
2
=
k
2
z
v
2
A
always exceeds unity). Figure 2 provides a
zoomed-in view of the lower-left corner of Figure 1 and
shows how equation (5) (dotted line) is always faster than
the MHD mode (dashed line) and only agrees with the exact
solution for
q
!
p
/2. This detailed plot also shows that the
exact Alfvén solution (solid line) is faster than the MHD
solution as
q
!
p
/2 but slower for
q
less than
p
/2. The exact
Alfvén plot also shows that for finite
w
/
w
ci
, propagation in
BELLAN: LOW FREQUENCY WAVE IMPROVED BASIS
A12219
A12219
5of14
the parallel direction is slower than the Alfvén velocity and
in fact, as the ion cyclotron frequency is approached from
below, the parallel propagation velocity slows down to zero.
This is in accordance with the
L
=
∞
cold plasma resonance
whereby for parallel propagation
k
!
∞
as
w
!
w
ci
from
below.
[
28
] Equation (37) can be expressed as
w
w
ci
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
L
ffiffiffiffiffiffiffi

p
3
r
cos
1
3
cos

1
3
q
2
p
ffiffiffiffiffiffiffi

3
p
s
!

2
p
3
j
!
þ
L
A
3
v
u
u
t
;
j
¼
0
;
1
;
2
ð
38
Þ
which is an explicit dispersion relation of the form
w
/
w
ci
=
f
(
k
x
c
/
w
pi
,
k
z
c
/
w
pi
;
b
) where
b
is a parameter, wave numbers
are scaled to the ion skin depth
c
/
w
pi
, and frequency is
scaled to the ion cyclotron frequency.
6. Alfvén Solution
[
29
] In the low frequency limit
w
/
w
ci
!
0 where the right
hand side of equation (29) can be neglected, the left hand
side of equation (29) (and the equivalent, but more concise
expression equation (33)) has two factors, with the root
xa

1
Q

1 = 0 being the Alfvén mode. If
w
/
w
ci
is finite on
the other hand, equation (33) can be re-arranged to be
xa

1
Q

1
¼
x
L
x
Q

x

ba
x

b
:
ð
39
Þ
[
30
] We now consider waves with
w
2
=
k
2
z
v
2
A
being of order
unity in a low
b
plasma. Since
x
¼
w
2
=
k
2
v
2
A
¼
a
this means
that
x
≫
ba
and so a factor
x
cancels from the right hand
side of equation (39) which thus reduces to
xa

1
Q

1
¼
L
Q

1
x

b
:
ð
40
Þ
[
31
] We now restrict consideration to waves where the
perpendicular wavelength is much shorter than the parallel
wavelength, i.e.,
a
is small, in which case
x
≪
1 since
x
/
a
is
of order unity. If
Q
¼
1
þ
k
2
c
2
=
w
2
pe
is of order unity, then
because both
x
and
b
are small, 1/|
x

b
|
≫
Q
and so
equation (40) further simplifies to
xa

1
Q

1
¼
L
b

x
ðÞ
:
ð
41
Þ
Solving for
x
gives
x
a
¼
1
þ
L
b
Q
þ
a
L
¼
1
þ
L
b
1
þ
m
e
=
m
i
þ
a
ðÞ
L
:
ð
42
Þ
Figure 1.
Plot of
w
2
=
k
2
v
2
A
versus angle. Vertical direction
is
q
= 0 and horizontal direction is
q
=
p
/2;
L
= 0.4 and
b
= 0.4 have been used. Exact roots of cubic polynomial
as given by equation (37) are heavy solid lines labeled
‘
Sound
’
,
‘
Alfven
’
, and
‘
Fast
’
respectively. For comparison,
approximate solutions provided by various other models
are shown from left to right as: long dashed line is electro-
static solution given by equation (57), dash-dot line is
equation (63) with the minus sign chosen (cold 2-fluid slow,
i.e., Alfvén mode), medium dashed line is the magnetohy-
drodynamic Alfvén dispersion
w
2
=
k
2
v
2
A
¼
cos
2
q
, dotted line
is traditional Kinetic Alfvén Wave dispersion given by
equation (5), and dash-dot-dot line is cold 2-fluid fast wave
given by equation (63) with plus sign chosen.
Figure 2.
Close-up view of lower-left corner of Figure 1
showing how traditional KAW (dotted line) agrees with
exact Alfvén solution (solid line) for near-perpendicular
angle at which point both are slightly faster than MHD pre-
diction (dashed line). However, at other angles, the classic
KAW disagrees with exact solution which is much slower
than MHD prediction.
BELLAN: LOW FREQUENCY WAVE IMPROVED BASIS
A12219
A12219
6of14
[
32
] The traditional expression for the kinetic Alfvén
wave, equation (5), corresponds to
x
=
a
(1 +
L
b
) and
examination of equation (41) shows this is valid only if
Q
= 1 (i.e.,
k
2
c
2
=
w
2
pe
≪
1) and
x
≪
b
.
Using equation (32) it
is seen that equation (42) corresponds to
w
2
k
2
z
v
2
A
¼
1
þ
k
2
c
2
s
=
w
2
ci
1
þ
m
e
=
m
i
þ
cos
2
q
ðÞ
k
2
v
2
A
w
2
ci
ð
43
Þ
or using
k
2
v
2
A
=
w
2
ci
¼
k
2
c
2
=
w
2
pi
¼
m
i
=
m
e
ðÞ
k
2
c
2
=
w
2
pe
w
2
k
2
z
v
2
A
¼
1
þ
b
m
i
=
m
e
ðÞ
k
2
c
2
=
w
2
pe
1
þ
1
þ
b
m
i
=
m
e
ðÞ
cos
2
q
b
k
2
c
2
=
w
2
pe
:
ð
44
Þ
Equation (43) shows that the traditional kinetic Alfvén dis-
persion fails when cos
2
q
exceeds
m
e
/
m
i
and
k
2
v
2
A
=
w
2
ci
exceeds unity. Furthermore, if cos
2
q
exceeds
b
, then
w
2
=
k
2
z
v
2
A
becomes less than unity in contrast to the pre-
diction of the traditional kinetic Alfvén wave dispersion
that
w
2
=
k
2
z
v
2
A
exceeds unity. Equation (44) is to be con-
trasted to equation (3) of
Lysak and Lotko
[1996] which
omits the term involving (cos
2
q
)/
b
, i.e., is
w
2
=
k
2
z
v
2
A
¼
1
þ
b
m
i
=
m
e
ðÞ
k
2
c
2
=
w
2
pe
=
1
þ
k
2
c
2
=
w
2
pe
. Incidentally, there
appears to be an error in the Figure 1a contour plot in
Lysak
and Lotko
[1996], since for
kc
/
w
pe
= 10 and
b
m
i
/
m
e
= 100,
it is seen that
w
2
=
k
2
z
v
2
A
¼
1
þ
b
m
i
=
m
e
ðÞ
k
2
c
2
=
w
2
pe
=
1
þ
k
2
c
2
=
w
2
pe
¼
1
þ
100

100
ðÞ
=
1
þ
100
ðÞ¼
99
whereas
the contour plot in question has
w
2
=
k
2
z
v
2
A
≈
9. It appears that
what was plotted in this figure was the erroneous quantity
1
þ
b
m
i
=
m
e
ðÞ
1
=
2
k
2
c
2
=
w
2
pe
=
1
þ
k
2
c
2
=
w
2
pe
.
[
33
] The decoupling of the Alfvén wave from the fast
wave when
x
=
b
corresponds to having
w
2
=
k
2
¼
c
2
s
. This
decoupling is evident in equation (18) where it is seen that
k

~
J

B
must vanish if
w
2
=
k
2
¼
c
2
s
. This means that
~
J
y
!
0
and since
m
0
~
J
y
¼
ik
z
~
B
x

ik
x
~
B
z
¼
ik
2
z
=
k
x
þ
k
x

~
B
z
this
means
~
B
z
!
0.
7. Using
~
J to Resolve the Space-Time Ambiguity
of Single-Spacecraft Measurements
[
34
] By invoking
~
J
as the fundamental quantity, the anal-
ysis presented here differs from traditional analyses which
invoked
~
E
as the fundamental quantity. Of course the same
dispersion relation would be obtained no matter which of
ñ
,
~
B
,or
~
U
is declared the fundamental quantity since all these
quantities are proportional to each other. Nevertheless,
choosing
~
J
results in a more transparent analysis because the
determinant of a 2

2 matrix occurs rather than the deter-
minant of a 3

3 matrix. The advantage of using
~
J
results
from imposing quasi-neutrality at the beginning of the
analysis rather than much later.
[
35
] This suggests that an advantage might also be
incurred from using
~
J
in the analysis of measured quantities
and, indeed, this turns out to be the case for the analysis of
spacecraft observations where typically, a spacecraft moves
at some velocity
V
rel
relative to the plasma. The dispersion
relations derived here assume the observer is in the plasma
rest frame but the frequency measured in the spacecraft
frame is Doppler shifted by
k

V
rel
relative to the frequency
observed in the plasma rest frame and also the spacecraft
frame electric field differs from the plasma frame electric
field by
V
rel

B
. In particular, if a prime denotes a quantity
measured in the spacecraft frame and no prime means a
quantity measured in the plasma frame, one has the well-
known relations
w
′
¼
w

k

V
rel
ð
45
Þ
and
E
′
¼
E
þ
V
rel

B
:
ð
46
Þ
It has traditionally been presumed that single spacecraft
measurements cannot resolve how much of the observed
w
′
in equation (45) results from the
w
term and how much from
the

k

V
rel
term. This space-time ambiguity of a single
spacecraft has motivated the use of multispacecraft cluster
missions.
[
36
] Spacecraft measurements have been mainly of
~
E
′
,
~
B
,
ñ
and
w
′
with minimal attention to
~
J
. We propose here a
method whereby a single spacecraft measurement of
~
J
resolves the
w
′
⇔
w
space-time ambiguity. Because
k
,
~
B
and
~
J
are frame-independent, the values of these quantities
measured by a moving spacecraft are the same as what
would be measured by an observer in the plasma frame. The
equilibrium magnetic field unit vector
^
z
is determined from
^
z
¼
B
t
ðÞ
hi
=
B
t
ðÞ
hi
jj
where angle brackets denote time-
average. The wave number
k
associated with any specific
frequency must be orthogonal to both
~
B
and
~
J
associated
with the same frequency since both
k

~
B
¼
0 and
k

~
J
¼
0;
the former condition comes from the solenoidal property of
magnetic fields and the latter condition comes from invo-
cation of quasi-neutrality. This dual orthogonality condition
establishes the unit wave vector to be
^
k
¼
~
B

~
J
~
B
~
J
:
ð
47
Þ
[
37
] The wave vector magnitude
k
and resolution of the

ambiguity are obtained by invoking Faraday
’
s law which is
a frame-independent equation. Faraday
’
s law in the space-
craft frame gives
k
^
k

~
E
′
¼
w
′
~
B
:
ð
48
Þ
Substitution for
^
k
in equation (48) using equation (47) gives

k
~
J

~
B

~
J
~
B

~
E
′
¼
w
′
~
B
:
ð
49
Þ
Either the plus or minus signs is selected in order to make
the left hand side of equation (49) parallel to the right hand
side. The magnitude
k
is then chosen so that the left hand
side of equation (49) equals the right hand side.
k
is thus
fully determined from a single-spacecraft measurement and
can then be used to calculate
k
?

~
J
?
and
k
?

~
J
?

^
z
, the
eigenvector component quantities in equation (28). The
BELLAN: LOW FREQUENCY WAVE IMPROVED BASIS
A12219
A12219
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relative velocity
V
rel
is determined either from knowledge of
the spacecraft orbit or from direct measurement of the mean
vector velocity of the ions impacting the spacecraft. The
plasma frame frequency
w
is then determined unambigu-
ously from equation (45) as
w
=
w
′
+
k

V
rel
; i.e., the space-
time ambiguity has been overcome despite only a single
spacecraft being used. We note that
Korepanov and Dudkin
[1999] proposed an alternate method to determine
k
wherein
Ampere
’
s law rather than Faraday
’
s law is invoked and both
the magnitude and direction of
~
J
are used rather than just the
direction.
[
38
] If it is found that the left and right hand sides of
equation (49) are not in fact parallel, there would have to be
one of the following: a measurement error, an error in the
assumption of quasi-neutrality, or an error in the assumption
that there is a only single
k
associated with
w
′
. Thus, dem-
onstrating parallelism of the right and left hand sides of
equation (49) constitutes an effective validity check for the
procedure.
[
39
] Substitution of equation (20) into equation (24) gives
~
E
¼
i
w
c
2
w
2
pe
m
0
~
J

i
wr
~
J

B
þ
k
k

~
J

B
c
2
s
w
2

k
2
c
2
s


B

~
J
nq
e

B

g
e
m
0
c
2
w
2
pe
k
T
e
m
e
q
e
k
k

~
J

B
rw
2

k
2
c
2
s

:
ð
50
Þ
Equation (50) gives
~
E
as a function of
~
J
,of{
w
,
k
}, and of the
equilibrium quantities
w
2
pe
;
r
;
c
2
s
,
B
, and
T
e
. This predicted
~
E
could be compared with the observed
~
E
¼
~
E
′

V
rel

~
B
as
determined from equation (46). The quasineutrality
assumption could be verified by demonstrating that
ɛ
0
k

~
E
=
q
e
n
jj
≪
1. By using a set of different frequencies
w
′
to determine associated values of
k
and
w
, a dispersion
relation
w
(
k
) would be established from measured quantities
and then compared with the theoretical dispersion relation.
Thus, measurement of
~
J
enables single spacecraft determi-
nation of all relevant wave properties with no space-time
ambiguity.
[
40
] In the limit that electron inertia, Hall current, and
warm plasma effects are negligible, equation (50) reduces to
the MHD relation
~
E
?
!
i
wr
~
J

B


B
¼
iv
2
A
w
m
0
~
J
?
ð
51
Þ
in which case the Alfvén mode has
~
E
?
parallel to
k
?
while
the fast mode has
~
E
?
parallel to
k
?

^
z
. When any or all of
electron inertia, Hall current, and warm plasma effects are
significant, equation (50) shows that the Alfvén mode will
have
~
E
?
deviate from being exactly parallel to
k
?
and the
fast mode will deviate from having
~
E
?
being exactly parallel
to
k
?

^
z
.
8. Some Geophysical Implications
[
41
] Examination of equation (41) shows that the cross-
over from
w
2
=
k
2
z
v
2
A
exceeding unity to being less than unity
occurs when
x
=
b
which, as shown above, corresponds to
having
w
2
=
k
2
¼
c
2
s
. This cross-over is also evident upon
examination of the right-hand side of equation (1) as this
right hand side reverses polarity when
w
2
=
k
2
c
2
s
¼
1 and so
the polarity of any term due to coupling of the fast mode will
reverse polarity. It should be noted that
w
2
=
k
2
¼
c
2
s
is con-
sistent with having
w
2
=
k
2
z
≫
c
2
s
when cos
2
q
is small.
[
42
]
Uritsky et al.
[2009] reported THEMIS measurements
of auroral structures having perpendicular wavelengths
l
x

3

10
5
m and oscillation periods of
t

10
2
s, i.e.,
w
/
k
x

3

10
3
ms

1
. Since ionospheric plasma is pri-
marily oxygen and the electron temperature is 1 eV, the ion
acoustic velocity
c
s

2.5

10
3
ms

1
and so
w
/
k
x

c
s
.
When mapped along a field the perpendicular wavelength
increases as
r
=
r
E
ðÞ
3
=
2
, the temperature increases by about
two orders of magnitude and the species changes from
being primarily oxygen to being primarily hydrogen [
Lysak
and Lotko
, 1996]. In particular, when mapping to a region
where
r

10
r
E
, it is seen that
w
2
=
k
2
c
2
s
¼
f
2
l
2
x
m
i
=
k
T
e
will
remain approximately constant since
l
2
x
increases by

10
3
,
the ion mass drops by 16, and the electron temperature
increases by about 10
2
. Hence, if the condition
x
=
b
holds
in the ionosphere, it will also be approximately true at large
distances. In the ionosphere,
b
m
i
/
m
e
≪
1 so the Alfvén
wave has a cold character (inertial Alfvén wave) whereas as
shown by
Lysak and Lotko
[1996] at large distances from
Earth,
b
m
i
/
m
e
≫
1 and it has been traditionally been pre-
sumed that if this is so the wave is described by equation (5).
However, if
x
=
b
remains approximately true over the length
of the field line, then even though
b
m
i
/
m
e
≫
1, the wave
dispersion will be approximately the same as the cold dis-
persion. As mentioned above, this can be seen from equation
(1) where the right hand side vanishes when
w
2
=
k
2
c
2
s
¼
1and
so the Alfvén mode decouples from the fast mode; it can also
be seen from equation (18). This has implications for the
group velocity
–
as shown by
Morales and Maggs
[1997], the
cold mode group velocity is much more confined to a field
line than the group velocity associated with equation (5).
This will give a tighter mapping of a disturbance at one point
on a field line to an observer at another, distant point on the
same field line.
9. Low Frequency, Electrostatic Limit
[
43
] The validity of equation (33) can be further checked
by showing that it incorporates the quasi-neutral electrostatic
limit, i.e., the electrostatic limit with
k
2
l
2
De
≪
1. This limit
corresponds to the magnetized acoustic mode and is
retrieved by assuming
x
Q
≪
a
in which case
x
Q
≪
1 also;
the assumption corresponds to
w
2
≪
k
2
z
v
2
A
=
1
þ
k
2
c
2
=
w
2
pe
:
Equation (33) reduces to
x

ba
x

b
¼
x
L
ð
52
Þ
which can be recast as
x
b

a

a

1
1
x
L

1
¼
0
ð
53
Þ
or, equivalently,
w
2
k
2
c
2
s

cos
2
q

w
2
sin
2
q
w
2

w
2
ci
¼
0
ð
54
Þ
BELLAN: LOW FREQUENCY WAVE IMPROVED BASIS
A12219
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8of14
which is the magnetized plasma electrostatic ion acoustic
wave dispersion relation with displacement current neglec-
ted. Equation (54) leads to a warm plasma resonance cone
behavior with cone angle
q
c
= cos

1
(
w
/
w
ci
) as observed
experimentally in
Bellan
[1976]. Since the limit
w
2
≪
k
2
z
v
2
A
corresponds to assuming
v
A
!
∞
in Equations (3) and (30)
and since the right hand term of equation (3) is quite dif-
ferent from the right hand side of equation (30) in this limit,
it is clear that equation (3) (i.e., Hollweg
’
s result) fails to
reduce to the electrostatic limit. This distinction between the
electrostatic limits of equations (3) and (30) further supports
the claim that the right hand side of equation (33) is correct
whereas the right hand side of equation (3) is not correct.
[
44
]If
q
=
p
/2, equation (54) becomes the electrostatic ion
cyclotron wave
w
2
¼
w
2
ci
þ
k
2
x
c
2
s
ð
55
Þ
as discussed by
Stix
[1992, section 3
–
6, equation 59].
[
45
] If equation (52) is expressed as a quadratic in
x
x
2
L

1
þ
b
L
ðÞ
x
þ
ba
¼
0
ð
56
Þ
then the ion acoustic mode is the root
x
¼
1
þ
b
L
ðÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
þ
b
L
ðÞ
2

4
ba
L
q
2
L
ð
57
Þ
which is plotted as the long dashed line in the lower left of
Figure 1.
[
46
] The electrostatic ion cyclotron mode is the root
x
¼
1
þ
b
L
ðÞþ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
þ
b
L
ðÞ
2

4
ba
L
q
2
L
:
ð
58
Þ
10. Cold Plasma Ion Cyclotron Wave and Inertial
Alfvén Wave
[
47
] In the limit
x
≫
b
, the finite-
b
terms in equation (33)
and hence the finite temperature terms in equation (29) may
be dropped; this is the cold plasma limit.
[
48
] In the cold plasma limit equation (29) reduces to
w
2
k
2
v
2
A
þ
w
2
w
2
gm
!

1
"#
1
cos
2
q
w
2
k
2
v
2
A
þ
w
2
w
2
gm
!

1
"#
¼
w
2
w
2
ci
:
ð
59
Þ
If
w
2
≪
w
2
gm
Equation (59) can be written as
n
2
x
¼
c
2
v
2
A

n
2
z
1
þ
w
w
ci
c
2
v
2
A

n
2
z
1

w
w
ci
c
2
v
2
A

n
2
z
1

w
2
w
2
ci
ð
60
Þ
where
n
x
=
ck
x
/
w
and
n
z
=
ck
z
/
w
.
Equation (60) is the cold
plasma ion cyclotron wave dispersion given by
Stix
[1992,
section 2
–
5, equation 19].
[
49
]If
w
2
=
w
2
ci
≪
1, then one of the roots of equation (59) is
w
2
¼
cos
2
q
1
k
2
v
2
A
þ
1
w
2
gm
¼
k
2
z
v
2
A
1
þ
k
2
c
2
w
2
pe
ð
61
Þ
which is the inertial Alfvén wave.
[
50
] Alternatively, equation (59) can be expressed as a
quadratic equation. Ignoring finite
b
, equation (33) can be
expressed as the quadratic
Q
2
x
2

Q
þ
a
L
þ
Q
ðÞ
ðÞ
x
þ
a
¼
0
ð
62
Þ
which has the solutions
x
¼
Q
þ
a
L
þ
Q
ðÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Q
þ
a
L
þ
Q
ðÞ
ðÞ
2

4
a
Q
2
q
2
Q
2
:
ð
63
Þ
Choice of the minus sign gives the slow (i.e., Alfvén) mode
while choice of the plus sign gives the fast mode. The slow
mode is plotted as a dash-dot line in Figure1 while the fast
mode is plotted as a dash-dot-dot line.
11. Lower Hybrid Resonance of Over-Dense
Plasma (Hybrid Resonance at
w
gm
)
[
51
]If
x
≫
b
then equation (33) becomes
x
Q
a

1

x
Q

1
ðÞ¼
x
L
ð
64
Þ
If
x
Q
=
x
+ 1, which is equivalent to
w
=
w
gm
, equation (64)
reduces to
x
Q
a
¼
1
þ
L
ð
65
Þ
so if
a
!
0,
x
must also go to zero, corresponding to
k
!
∞
.
Thus, the lower hybrid resonance (i.e.,
S
= 0) in an over-
dense plasma (i.e.,
w
2
pe
≫
w
2
ce
) is retrieved since this reso-
nance corresponds to
k
!
∞
at
w
=
w
gm
for perpendicular
propagation.
12. High Frequency, Whistler Wave Limit
[
52
]If
x
≫
1,
x
≫
b
and
b
is of order unity or smaller, then
(
x

ba
)/(
x

b
)
!
1. This corresponds to frequencies
above the lower hybrid frequency since
x
≫
1 whereas the
lower hybrid frequency has
x
!
0. The first terms in each of
the parentheses in equation (33) dominate the other terms
and so equation (33) reduces to
x
¼
L
a
Q
2
;
ð
66
Þ
i.e.,
w
2
¼
k
4
v
4
A
w
2
ci
1
þ
k
2
c
2
=
w
2
pe
2
cos
2
q
:
ð
67
Þ
BELLAN: LOW FREQUENCY WAVE IMPROVED BASIS
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