Similarly to the barotropic case the slow QG component does not depend on the
fast ageostrophic waves on times ∼ 1 ̸ δ, whereas evolution of the inertial oscillations depends on the geostrophic streamfunction p
−
g . At the same time, it follows
from (140) and (122) that
∂
∂T 1
Z
dxdy
Z − h 1
− 1
dz A
j j
2 = 0,
ð144Þ
i.e., the total energy of inertial oscillations is conserved along with that of the QG
component. We note, however, that the conservation takes place on times ∼ 1 ̸ δ,
on longer times the “non-traditional” terms in the equations of motion can, in
principle, give rise to energy exchange between the components as it takes place in
the barotropic fluid (see section “Barotropic Model”). If this is the case, the energy
of QG motion in the lower layer can be transferred to the inertial oscillations. Of
course, the mechanism of dissipation of QG energy in a homogeneous layer is
highly speculative and it would be useful to verify it numerically using a
non-hydrostatic model without the TA.
It was shown in subsection “Non-dimensional Equations and the Lowest-Order
Solution” that condition (122) forbids the inertial signal in the vertical velocity field
to penetrate into the stratified layer. Equation (140) “supports” limitation (122): it is
readily to show integrating (140) over z from –1 to −h 1 that if (122) is satisfied at
some moment T 0 then it is valid for all times T > T 0 . If the initial conditions do not
satisfy (108) and, therefore, (123), then the analogous screening in horizontal
velocity is provided by a non-stationary boundary layer developing in the upper
layer near the interface. Generally, condition (123) is not supported by (140)
because of the “non-traditional” term in (140), i.e., the quantity Aj z = − h 1 ceases to
be zero even if (123) is satisfied at some moment. This means that without the TA
the near interface boundary layer develops at any initial conditions.
Summary and Discussion
We have examined geostrophic adjustment in a rotating fluid of constant depth
confined between two rigid lids. The angular speed of rotation Ω does not coincide
in direction with the gravity; the traditional and hydrostatic approximations are not
used. Two models were considered: the barotropic one and the SNS fluid consisting
of a stratified upper layer with N ≫ f and a homogeneous lower layer, the density
and other fields being continuous at the interface between the layers. In both the
cases, the wave spectra contain the gyroscopic waves which exist due to rotation
and are susceptible to the non-traditional terms in equations of motion.
Geostrophic adjustment is a particular case of more general wave adjustment
[23], which takes place in a physical system possessing in the linear approximation
linear invariants and linear wave solutions harmonically depending on time. In our
328
G. M. Reznik
fast ageostrophic waves on times ∼ 1 ̸ δ, whereas evolution of the inertial oscillations depends on the geostrophic streamfunction p
−
g . At the same time, it follows
from (140) and (122) that
∂
∂T 1
Z
dxdy
Z − h 1
− 1
dz A
j j
2 = 0,
ð144Þ
i.e., the total energy of inertial oscillations is conserved along with that of the QG
component. We note, however, that the conservation takes place on times ∼ 1 ̸ δ,
on longer times the “non-traditional” terms in the equations of motion can, in
principle, give rise to energy exchange between the components as it takes place in
the barotropic fluid (see section “Barotropic Model”). If this is the case, the energy
of QG motion in the lower layer can be transferred to the inertial oscillations. Of
course, the mechanism of dissipation of QG energy in a homogeneous layer is
highly speculative and it would be useful to verify it numerically using a
non-hydrostatic model without the TA.
It was shown in subsection “Non-dimensional Equations and the Lowest-Order
Solution” that condition (122) forbids the inertial signal in the vertical velocity field
to penetrate into the stratified layer. Equation (140) “supports” limitation (122): it is
readily to show integrating (140) over z from –1 to −h 1 that if (122) is satisfied at
some moment T 0 then it is valid for all times T > T 0 . If the initial conditions do not
satisfy (108) and, therefore, (123), then the analogous screening in horizontal
velocity is provided by a non-stationary boundary layer developing in the upper
layer near the interface. Generally, condition (123) is not supported by (140)
because of the “non-traditional” term in (140), i.e., the quantity Aj z = − h 1 ceases to
be zero even if (123) is satisfied at some moment. This means that without the TA
the near interface boundary layer develops at any initial conditions.
Summary and Discussion
We have examined geostrophic adjustment in a rotating fluid of constant depth
confined between two rigid lids. The angular speed of rotation Ω does not coincide
in direction with the gravity; the traditional and hydrostatic approximations are not
used. Two models were considered: the barotropic one and the SNS fluid consisting
of a stratified upper layer with N ≫ f and a homogeneous lower layer, the density
and other fields being continuous at the interface between the layers. In both the
cases, the wave spectra contain the gyroscopic waves which exist due to rotation
and are susceptible to the non-traditional terms in equations of motion.
Geostrophic adjustment is a particular case of more general wave adjustment
[23], which takes place in a physical system possessing in the linear approximation
linear invariants and linear wave solutions harmonically depending on time. In our
328
G. M. Reznik