The long-wave asymptotics (10a, b) are universal in the sense that they also
remain valid in the stratified fluid [7]. We emphasize that the gyroscopic waves are
close to the inertial oscillations if L ≫ H; it is not the case for the surface and
internal gravity waves, which are nearly inertial if L ≫ L R where L, L R are the
horizontal and Rossby scales, respectively. Reznik [20] showed that this property of
the GWs is also valid in the stratified fluid. Usually L R ≫ H, therefore the presence
of the GWs results in the existence of inertial oscillations with shorter horizontal
scales L ≤ L R .
Using the scales L, H, and f
− 1 as the time scale, we write (5a, b) in the
non-dimensional form:
ð∂ tt + 1Þw zz + 2δqw yz + δ
2
ð∇
2
h w tt + q
2 w yy Þ = 0, wj z = 0, − 1 = 0,
ð11a; bÞ
where δ = H ̸ L ≪ 1. The smallness of δ allows a solution to (11a, b) to be sought in
the following asymptotic form:
w = w 0 ðx, y, z, t, T 1 , . . .Þ + δw 1 ðx, y, z, t, T 1 , . . .Þ + . . . ,
ð12Þ
where T n = δ
n t, n = 1, 2, . . . are the slow times.
Substitution of (12) into (11a, b) gives in the lowest order:
ð∂ tt + 1Þw 0zz = 0, wj z = 0, − 1 = 0,
ð13a; bÞ
Whence we have:
w 0 = W 0 ðx, y, z, T 1 , . . .Þe
− it + c.c.;
ð14Þ
c.c. denotes complex-conjugate value. Thus, the lowest order solution is inertial
oscillations modulated by the arbitrary amplitude W 0 ðx, y, z, T 1 , . . .Þ, which depends
on the coordinates and slow time.
The dependence is determined from the first-order equation:
ð∂ tt + 1Þw 1zz = − 2∂ tT 1 w 0zz − 2qw 0yz .
ð15Þ
The correction w 1 is bounded in the “fast” time t if the right-hand side of (15) is
zero whence one obtains:
∂ T 1 W 0zz + iqW 0yz = 0.
ð16Þ
Equation (16) should be solved under the boundary and initial conditions
W 0 j z = 0, − 1 = 0, W 0 j T 1 = 0 = W I =
1
2
w I ðx, y, zÞ.
ð17a; bÞ
Geostrophic Adjustment Beyond the Traditional Approximation
301
remain valid in the stratified fluid [7]. We emphasize that the gyroscopic waves are
close to the inertial oscillations if L ≫ H; it is not the case for the surface and
internal gravity waves, which are nearly inertial if L ≫ L R where L, L R are the
horizontal and Rossby scales, respectively. Reznik [20] showed that this property of
the GWs is also valid in the stratified fluid. Usually L R ≫ H, therefore the presence
of the GWs results in the existence of inertial oscillations with shorter horizontal
scales L ≤ L R .
Using the scales L, H, and f
− 1 as the time scale, we write (5a, b) in the
non-dimensional form:
ð∂ tt + 1Þw zz + 2δqw yz + δ
2
ð∇
2
h w tt + q
2 w yy Þ = 0, wj z = 0, − 1 = 0,
ð11a; bÞ
where δ = H ̸ L ≪ 1. The smallness of δ allows a solution to (11a, b) to be sought in
the following asymptotic form:
w = w 0 ðx, y, z, t, T 1 , . . .Þ + δw 1 ðx, y, z, t, T 1 , . . .Þ + . . . ,
ð12Þ
where T n = δ
n t, n = 1, 2, . . . are the slow times.
Substitution of (12) into (11a, b) gives in the lowest order:
ð∂ tt + 1Þw 0zz = 0, wj z = 0, − 1 = 0,
ð13a; bÞ
Whence we have:
w 0 = W 0 ðx, y, z, T 1 , . . .Þe
− it + c.c.;
ð14Þ
c.c. denotes complex-conjugate value. Thus, the lowest order solution is inertial
oscillations modulated by the arbitrary amplitude W 0 ðx, y, z, T 1 , . . .Þ, which depends
on the coordinates and slow time.
The dependence is determined from the first-order equation:
ð∂ tt + 1Þw 1zz = − 2∂ tT 1 w 0zz − 2qw 0yz .
ð15Þ
The correction w 1 is bounded in the “fast” time t if the right-hand side of (15) is
zero whence one obtains:
∂ T 1 W 0zz + iqW 0yz = 0.
ð16Þ
Equation (16) should be solved under the boundary and initial conditions
W 0 j z = 0, − 1 = 0, W 0 j T 1 = 0 = W I =
1
2
w I ðx, y, zÞ.
ð17a; bÞ
Geostrophic Adjustment Beyond the Traditional Approximation
301