Substituting (128) into (125a) and neglecting small terms we have:
ξ∂ ξξξ ŵ b + 3∂ ξξ ŵ b − i∇
2
h ŵ b = 0.
ð129Þ
In terms of the Fourier amplitude ŵ ̃ b (see (112)) (129) is rewritten as:
ξ∂ ξξξ ŵ ̃ b + 3∂ ξξ ŵ ̃ b + iκ
2 ŵ ̃ b = 0.
ð130Þ
For the boundary layer to exist the solution ŵ ̃ 0 to (130) satisfying the conditions
ŵ ̃ 0 → 0 as ξ → ∞; ∂ ξ ŵ ̃ 0
ξ = 0 = 1,
ð131a; bÞ
must exist. Analysis in Reznik [22] confirms possibility of such a solution. The
corresponding Fourier amplitude w̄ ̃ is given by:
w̄ ̃ b = Cðk, lÞŵ ̃ 0 ðk, l, ξÞ ̸
ffi ffi
t
p
, C = 0.5m(ẇ
+
I + iw
+
I Þ z = − h 1 .
ð132a; bÞ
The boundary layer near the surface z = 0 is similar to that near the interface
[22].
Slow Evolution of the QG Component and Inertial
Oscillations
The lowest-order solution constructed in subsection “Non-dimensional Equations
and the Lowest-Order Solution” is the sum of a time-independent geostrophic
component, lower-layer inertial oscillations, and dispersive internal waves; the
non-stationary boundary layers are the result of the joint impact of the internal
waves with very short vertical lengths. To derive the solution of the lowest-order
system (90a, b, c, 90d, e), (91a, 91b, c, 91d) depending on slow times one should
“allow” parameters related to the initial fields to depend on the slow times. The
geostrophic part of the solution is determined by the PV Π
± in the layers (see (93a,
93b)) directly related to the initial fields u I , v I , ρ I . In what follows we assume that:
Π
± = Π
±
ðx, y, z, T 1 , T 2 , . . .Þ, Π
±
ðx, y, z, 0, 0, . . .Þ = Π
±
I .
ð133a; bÞ
Similarly, for the internal waves the initial fields ðw
+
I , ẇ
+
I Þðx, y, zÞ are replaced
by the functions ðw
+
s , ẇ
+
s Þðx, y, z, T 1 , T 2 , . . .Þ with
ðw
+
s , ẇ
+
s Þðx, y, z, 0, 0, . . .Þ = ðw
+
I , ẇ
+
I Þðx, y, zÞ.
ð134Þ
Geostrophic Adjustment Beyond the Traditional Approximation
325
ξ∂ ξξξ ŵ b + 3∂ ξξ ŵ b − i∇
2
h ŵ b = 0.
ð129Þ
In terms of the Fourier amplitude ŵ ̃ b (see (112)) (129) is rewritten as:
ξ∂ ξξξ ŵ ̃ b + 3∂ ξξ ŵ ̃ b + iκ
2 ŵ ̃ b = 0.
ð130Þ
For the boundary layer to exist the solution ŵ ̃ 0 to (130) satisfying the conditions
ŵ ̃ 0 → 0 as ξ → ∞; ∂ ξ ŵ ̃ 0
ξ = 0 = 1,
ð131a; bÞ
must exist. Analysis in Reznik [22] confirms possibility of such a solution. The
corresponding Fourier amplitude w̄ ̃ is given by:
w̄ ̃ b = Cðk, lÞŵ ̃ 0 ðk, l, ξÞ ̸
ffi ffi
t
p
, C = 0.5m(ẇ
+
I + iw
+
I Þ z = − h 1 .
ð132a; bÞ
The boundary layer near the surface z = 0 is similar to that near the interface
[22].
Slow Evolution of the QG Component and Inertial
Oscillations
The lowest-order solution constructed in subsection “Non-dimensional Equations
and the Lowest-Order Solution” is the sum of a time-independent geostrophic
component, lower-layer inertial oscillations, and dispersive internal waves; the
non-stationary boundary layers are the result of the joint impact of the internal
waves with very short vertical lengths. To derive the solution of the lowest-order
system (90a, b, c, 90d, e), (91a, 91b, c, 91d) depending on slow times one should
“allow” parameters related to the initial fields to depend on the slow times. The
geostrophic part of the solution is determined by the PV Π
± in the layers (see (93a,
93b)) directly related to the initial fields u I , v I , ρ I . In what follows we assume that:
Π
± = Π
±
ðx, y, z, T 1 , T 2 , . . .Þ, Π
±
ðx, y, z, 0, 0, . . .Þ = Π
±
I .
ð133a; bÞ
Similarly, for the internal waves the initial fields ðw
+
I , ẇ
+
I Þðx, y, zÞ are replaced
by the functions ðw
+
s , ẇ
+
s Þðx, y, z, T 1 , T 2 , . . .Þ with
ðw
+
s , ẇ
+
s Þðx, y, z, 0, 0, . . .Þ = ðw
+
I , ẇ
+
I Þðx, y, zÞ.
ð134Þ
Geostrophic Adjustment Beyond the Traditional Approximation
325