where
Gðψ, AÞ =
1
2
MðψÞ A
j j
2
xy
−
1
2
M ψ xy A
j j
2
,
ð53aÞ
HðAÞ = −
i
4
s
Z 0
− 1
dzA
Z z
− 1
s
*
ðA
*
Þdz
2
4
3
5
y
+ c.c.,
ð53bÞ
operator M = ∂ xx − ∂ yy , ā =
R 0
− 1 adz, and the asterisk denotes the complex
conjugation.
If all fields decay at infinity, then we have
Z
ψGðψ, AÞdxdy = 0,
ð54Þ
therefore, energy E ̄ of the QG component changes in time as
∂ T E ̄ = δq
Z
ψHðAÞdxdy, E ̄ =
1
2
Z
ð∇ h ψÞ
2 dxdy.
ð55a; bÞ
For q ≠ 0 the right-hand side of (55a), is, generally, non-zero whence the
important conclusion follows that without the TA a transfer of energy between the
QG component and inertial oscillations can exist.
Stably-Neutrally Stratified Fluid
Governing Equations
We consider a stably-neutrally stratified fluid of constant depth H, bounded by two
rigid lids and rotating as a whole at a constant angular speed Ω, which, generally, is
not parallel to gravity (directed along the z-axis in Fig. 5). The fluid density ρ, being
continuous, depends on z in the upper layer of depth h 1 − η and is constant in the
lower layer of depth h 2 + η where h 1 , h 2 = H − h 1 are constant mean depths of the
Fig. 5 Schematic
representation of rotating
stably-neutrally stratified fluid
310
G. M. Reznik