Here u = ðu, v, wÞ where u, v, w are the velocity components associated with the
x, y, z axes, respectively, the z-axis is directed upward parallel to gravity (origin at
the upper surface); ρ 0 is the fluid density; p the deviation of pressure from the
hydrostatic one; 2Ω = e y f s + e z f where e x , e y , e z are the unit vectors along corresponding axes and f = 2Ω sin φ 0 , f s = 2Ω cos φ 0 .
The velocity field obeys the no-flux conditions at the surface and bottom:
wj z = 0, − H = 0,
ð2Þ
and the initial conditions
ðu, v, wÞ t = 0 = ðu I , v I , w I Þðx, y, zÞ; w I = −
Z z
− H
ð∂ x u I + ∂ y v I Þdz.
ð3a; bÞ
Equations (1a, b) represent the so-called non-traditional f-plane approximation;
in this case φ 0 is the reference latitude around which the west-east, south-north, and
vertical Cartesian coordinates x, y, z are introduced (see Fig. 1). The ratio
q = f s ̸ f = cot φ 0 is assumed to be of the order of unity,
q = Oð1Þ,
ð4Þ
which corresponds to the mid-latitudes.
Linear Gyroscopic Waves
In the absence of the free surface, beta-effect, and stratification the only
wave-generating mechanism is rotation, i.e. only the gyroscopic waves are possible
here. Linearized Eq. (1a, b) can be reduced to one equation for the vertical velocity
Miropol’sky [19]:
ð∂ tt + f
2
Þw zz + ∇
2
h w tt + 2ff s w yz + f
2
s w yy = 0, wj z = 0, − H = 0
ð5a; bÞ
where ∇
2
h = ∂
2
x + ∂
2
y . Any solution to problem (5a, b) can be represented as a
superposition of the wave solutions
w n = W n ðzÞ exp½iðkx + ly − σ n tÞ,
ð6Þ
where k, l are the horizontal wavenumbers κ =
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
k 2 + l 2
p
and amplitude W n has
the form (see [21] for details):
Geostrophic Adjustment Beyond the Traditional Approximation
299
x, y, z axes, respectively, the z-axis is directed upward parallel to gravity (origin at
the upper surface); ρ 0 is the fluid density; p the deviation of pressure from the
hydrostatic one; 2Ω = e y f s + e z f where e x , e y , e z are the unit vectors along corresponding axes and f = 2Ω sin φ 0 , f s = 2Ω cos φ 0 .
The velocity field obeys the no-flux conditions at the surface and bottom:
wj z = 0, − H = 0,
ð2Þ
and the initial conditions
ðu, v, wÞ t = 0 = ðu I , v I , w I Þðx, y, zÞ; w I = −
Z z
− H
ð∂ x u I + ∂ y v I Þdz.
ð3a; bÞ
Equations (1a, b) represent the so-called non-traditional f-plane approximation;
in this case φ 0 is the reference latitude around which the west-east, south-north, and
vertical Cartesian coordinates x, y, z are introduced (see Fig. 1). The ratio
q = f s ̸ f = cot φ 0 is assumed to be of the order of unity,
q = Oð1Þ,
ð4Þ
which corresponds to the mid-latitudes.
Linear Gyroscopic Waves
In the absence of the free surface, beta-effect, and stratification the only
wave-generating mechanism is rotation, i.e. only the gyroscopic waves are possible
here. Linearized Eq. (1a, b) can be reduced to one equation for the vertical velocity
Miropol’sky [19]:
ð∂ tt + f
2
Þw zz + ∇
2
h w tt + 2ff s w yz + f
2
s w yy = 0, wj z = 0, − H = 0
ð5a; bÞ
where ∇
2
h = ∂
2
x + ∂
2
y . Any solution to problem (5a, b) can be represented as a
superposition of the wave solutions
w n = W n ðzÞ exp½iðkx + ly − σ n tÞ,
ð6Þ
where k, l are the horizontal wavenumbers κ =
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
k 2 + l 2
p
and amplitude W n has
the form (see [21] for details):
Geostrophic Adjustment Beyond the Traditional Approximation
299