is sought as an expansion in the vertical modes, which here are the eigenfunctions
of the problem:
W zz + b
2 N
2 W = 0, Wj z = 0 = 0, m(WÞj z = − h 1 = 0.
ð113Þ
Eigenfunctions W n , eigenvalues b n , and the corresponding wave frequencies σ n
are readily found in the case N = const (cf. (68)–(70)):
W n = sin b n z, b n = s n ̸ h 1 , σ n =
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 + κ 2 ̸ b 2
n
q
, n = 1, 2, . . .
ð114Þ
here s n is the n-th root of (69).
Amplitude w̃
+
a can be written as (see Reznik [22] for details):
w̃
+
a = ∑
∞
n = 1
½w̃
+
In cos σ n t + ðẇ ̃
+
In ̸ σ n Þ sin σ n tŠW n ðzÞ.
ð115Þ
We see that the upper layer ageostrophic component is the superposition of long
super-inertial internal waves ∼ exp½iðkx + ly − σ n tފ considered in subsection
“Linear Wave Modes”. The function w
+
a does not contain the inertial oscillations
(cf. subsection “Some Properties of the Ageostrophic Solution”) and at the interface
any finite partial sum of the series (115) satisfies boundary conditions (111) with
zero r.h.s. This means that in the very close vicinity of the interface z = − h 1 the
wave modes with very large numbers n play an important role. Since
b n → ∞, σ n → 1 for n → ∞, vertical scales and frequencies of these modes are close
to zero and to the inertial frequency, respectively. Joint effect of these modes forms
the near interface boundary layer considered below in subsection “Boundary
Layer”.
Motion in the Homogeneous Lower Layer
It readily follows from (95a, b, c, 95f) that in the lower layer:
∂ z ð∂ t u
−
a − v
−
a Þ = 0, ∂ z ð∂ t v
−
a + u
−
a Þ = 0,
ð116a; bÞ
whence
U
−
a = U
−
aI ðx, y, zÞe
− it + Uðx, y, tÞ.
ð117Þ
Here, U = ū + iv̄ is still unknown depth-independent complex velocity and
U
−
a = u
−
a + iv
−
a , U
−
aI = u
−
aI + iv
−
aI .
ð118a; bÞ
Integrating (117) over z from –1 to − h 1 and using (99), (105) one finds:
322
G. M. Reznik