W n = e
ia n z sin b n z, n = 1, 2, . . . .
ð7Þ
Here n is the number of vertical mode equal to the number of zeros of W n ðzÞ, and
a n = −
ff s l
f 2 − σ 2
n
, b n =
σ n κ
f 2 − σ 2
n
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
f 2 − σ 2
n + f ̄
2
s
q
=
nπ
H
, f ̄
s = f s
l
j j
κ
.
ð8a; b; cÞ
The dispersion relation σ n = σ n ðk, lÞ consists of the sub-inertial branch σ
sub
n :
σ
sub
n = f
2
−
1
2ð1 + b ̄
2
n Þ
½ðf
2
− f ̄
2
s Þ +
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
σ 4
0 + 4b ̄
2
n f 2 f ̄
2
s
q
(
) 1 ̸ 2
,
ð9aÞ
and the super-inertial branch σ
sup
n :
σ
sup
n = f
2 +
1
2ð1 + b ̄
2
n Þ
½ − ðf
2
− f ̄
2
s Þ +
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
σ 4
0 + 4b ̄
2
n f 2 f ̄
2
s
q
(
) 1 ̸ 2
,
ð9bÞ
where σ 0 =
ffiffiffiffiffiffiffiffiffiffiffiffiffi
f 2 + f ̄
2
s
q
. The branches are presented in Fig. 2.
In the long-wave approximation κH ≪ 1 both the sub- and super-inertial frequencies (9a, 9b) are close to the inertial frequency f; in this case
σ
sub
n = f −
f s H
2nπ
l
j j + fOðb ̄ − 2
n Þ, σ
sup
n = f +
f s H
2nπ
l
j j + fOðb ̄ − 2
n Þ.
ð10a; bÞ
Fig. 2 Dispersion relation
for the barotropic gyroscopic
waves
300
G. M. Reznik