models, the linear invariants are potential vorticities written in special
non-orthogonal coordinates related to the vector Ω; the invariants are determined by
the initial conditions in a unique way. The steady state related to the invariants is a
geostrophic mode, which is a geostrophically balanced motion parallel to the layer
boundaries. In the barotropic fluid, the motion is columnar, the columns are parallel
to Ω. In the SNS fluid, the same is valid in the homogeneous layer; in the stratified
layer, in the general case, the geostrophic mode is not columnar. The non-traditional
terms are of importance in the QG dynamics if the dominating horizontal scale L of
the initial perturbation is smaller or of the order of the fluid depth, L ≤ H, for the
long-wave perturbation with L ≫ H, the contribution of the terms is small.
In the barotropic model the gyroscopic waves are the only possible wave
motions; in the SNS fluid the wave spectrum consists of internal and gyroscopic
waves (see [20] for details). All the waves are dispersive; therefore, in the process
of linear geostrophic adjustment the motion tends to the geostrophic mode with
increasing time.
Using multiple-time-scale perturbation theory we studied the non-linear adjustment in the long-wave approximation H ≪ L ≤ L R and small Rossby numbers Ro. In
the barotropic case, L R = ∞; in the SNS fluid, L R = HN 0 ̸ f . In this scale range, the
gyroscopic waves are close to weakly dispersive inertial oscillations (confined to the
lower layer in the SNS fluid). The internal waves in the SNS fluid are strongly
dispersive and penetrate into the homogeneous lower layer down to the bottom.
The general scenario of the adjustment is similar to the one with gravity waves (cf.
[24, 29]): an arbitrary perturbation is split in a unique way into slow and fast components evolving with characteristic time scales ðRof Þ
− 1 and f
− 1 , respectively. In
both cases the slow component is close to the geostrophic balance and is not influenced by the fast one on times t ∼ ðfRoÞ
− 1 . In the barotropic model the slow component does not depend on depth and is described by the 2D fluid dynamics equation
for the geostrophic streamfunction. In the SNS fluid, the slow component is governed
by two coupled nonlinear equations of conservation of QG potential vorticity in the
upper and lower layers. The upper and lower layer QG flows are not independent: if at
some moment the QG motion in the lower homogeneous layer vanishes, then at
subsequent times the QG energy is transferred from the upper layer to the lower one.
The fast component consists of the inertial oscillations modulated by the
amplitude depending on coordinates and the slow time, and (in the SNS fluid)
the internal waves. The inertial oscillations are long gyroscopic waves; the
depth-integrated horizontal flow induced by the oscillations is zero. Typical group
speed of the internal waves greatly exceeds the typical horizontal slow velocity U;
at the same time, the group speed of the inertial oscillation is of the order of U. As a
result, in the course of non-linear geostrophic adjustment the internal waves decay
because of dispersion, and the residual flow consists of the QG slow component and
inertial oscillations.
At times t ∼ ðfRoÞ
− 1 the energy of inertial oscillations is conserved but they are
coupled to the slow component: their amplitude obeys an equation with coefficients
depending on the geostrophic streamfunction. Under the TA the inertial oscillations
Geostrophic Adjustment Beyond the Traditional Approximation
329
non-orthogonal coordinates related to the vector Ω; the invariants are determined by
the initial conditions in a unique way. The steady state related to the invariants is a
geostrophic mode, which is a geostrophically balanced motion parallel to the layer
boundaries. In the barotropic fluid, the motion is columnar, the columns are parallel
to Ω. In the SNS fluid, the same is valid in the homogeneous layer; in the stratified
layer, in the general case, the geostrophic mode is not columnar. The non-traditional
terms are of importance in the QG dynamics if the dominating horizontal scale L of
the initial perturbation is smaller or of the order of the fluid depth, L ≤ H, for the
long-wave perturbation with L ≫ H, the contribution of the terms is small.
In the barotropic model the gyroscopic waves are the only possible wave
motions; in the SNS fluid the wave spectrum consists of internal and gyroscopic
waves (see [20] for details). All the waves are dispersive; therefore, in the process
of linear geostrophic adjustment the motion tends to the geostrophic mode with
increasing time.
Using multiple-time-scale perturbation theory we studied the non-linear adjustment in the long-wave approximation H ≪ L ≤ L R and small Rossby numbers Ro. In
the barotropic case, L R = ∞; in the SNS fluid, L R = HN 0 ̸ f . In this scale range, the
gyroscopic waves are close to weakly dispersive inertial oscillations (confined to the
lower layer in the SNS fluid). The internal waves in the SNS fluid are strongly
dispersive and penetrate into the homogeneous lower layer down to the bottom.
The general scenario of the adjustment is similar to the one with gravity waves (cf.
[24, 29]): an arbitrary perturbation is split in a unique way into slow and fast components evolving with characteristic time scales ðRof Þ
− 1 and f
− 1 , respectively. In
both cases the slow component is close to the geostrophic balance and is not influenced by the fast one on times t ∼ ðfRoÞ
− 1 . In the barotropic model the slow component does not depend on depth and is described by the 2D fluid dynamics equation
for the geostrophic streamfunction. In the SNS fluid, the slow component is governed
by two coupled nonlinear equations of conservation of QG potential vorticity in the
upper and lower layers. The upper and lower layer QG flows are not independent: if at
some moment the QG motion in the lower homogeneous layer vanishes, then at
subsequent times the QG energy is transferred from the upper layer to the lower one.
The fast component consists of the inertial oscillations modulated by the
amplitude depending on coordinates and the slow time, and (in the SNS fluid)
the internal waves. The inertial oscillations are long gyroscopic waves; the
depth-integrated horizontal flow induced by the oscillations is zero. Typical group
speed of the internal waves greatly exceeds the typical horizontal slow velocity U;
at the same time, the group speed of the inertial oscillation is of the order of U. As a
result, in the course of non-linear geostrophic adjustment the internal waves decay
because of dispersion, and the residual flow consists of the QG slow component and
inertial oscillations.
At times t ∼ ðfRoÞ
− 1 the energy of inertial oscillations is conserved but they are
coupled to the slow component: their amplitude obeys an equation with coefficients
depending on the geostrophic streamfunction. Under the TA the inertial oscillations
Geostrophic Adjustment Beyond the Traditional Approximation
329