150
Vertical Structure: Baroclinic Quasi-Geostrophic Models
equation, and (3.10.12) and (3.10.13) impose three boundary conditions in z,
there are more conditions than is normal in a second-order problem. However,
the problem is actually a free boundary problem since D(x, y) is not known a
priori. The boundary of the region in motion is part of the solution and the
additional boundary condition is required to determine D(x, y). A very similar
situation arises in the more general and difficult continuous problem described
in Chapter 4 where the quasi-geostrophic approximation is relaxed, but where
the overall structure of the problem is the same as that confronted in the quasigeostrophic limit.
Our discussion of the layer model allows us to anticipate a useful result. At
each level in z the shape of the bowl of moving fluid is tucked up against the
northwest corner of the basin such that the outermost closed contour at each
depth intersects the northern boundary of the gyre. At that boundary t/J and its
z derivatives are zero so that we can determine y0 as the value of the potential
vorticity, which is then only the planetary vorticity at the northern boundary,
which in nondimensional units is y = 1. Thus we eventually choose:
Yo= 1.
(3.10.16)
We shall see that any other choice of y0 would lead to either a singularity or an
unstable solution. For the moment we hold yo arbitrary.
For simplicity we consider the case of uniform buoyancy frequency for
which N =No. The solution of (3.10.15) subject to (3.10.12) is in that case:
1
2
t/1 = 2(z+D) (Yo- y).
(3.10.17)
The total density field in dimensional variables is
[
2
fo8t/J]
Ptotat =Po 1- N zjg- g 8z (dimensional)
(3.10.18)
which in nondimensional form, using the scales ford and U of (3.10.11), gives
us for the case of constant buoyancy frequency:
Ptotal _ 1 = No
2
d {z + PL8t/J }·
Po
g
fo 8z
(3.10.19)
The smallness of the parameter PL/ fo is required for the validity of the P
plane approximation and the consequent smallness of the density anomaly,
compared to the density distribution in the resting state, is as required by
quasi-geostrophic theory.
The remaining boundary condition is the matching of the vertical velocity
to the given Ekman pumping velocity at z = 0, i.e. (3.10.13). Inserting the
solution (3.10.17) into (3.10.13) yields:
Vertical Structure: Baroclinic Quasi-Geostrophic Models
equation, and (3.10.12) and (3.10.13) impose three boundary conditions in z,
there are more conditions than is normal in a second-order problem. However,
the problem is actually a free boundary problem since D(x, y) is not known a
priori. The boundary of the region in motion is part of the solution and the
additional boundary condition is required to determine D(x, y). A very similar
situation arises in the more general and difficult continuous problem described
in Chapter 4 where the quasi-geostrophic approximation is relaxed, but where
the overall structure of the problem is the same as that confronted in the quasigeostrophic limit.
Our discussion of the layer model allows us to anticipate a useful result. At
each level in z the shape of the bowl of moving fluid is tucked up against the
northwest corner of the basin such that the outermost closed contour at each
depth intersects the northern boundary of the gyre. At that boundary t/J and its
z derivatives are zero so that we can determine y0 as the value of the potential
vorticity, which is then only the planetary vorticity at the northern boundary,
which in nondimensional units is y = 1. Thus we eventually choose:
Yo= 1.
(3.10.16)
We shall see that any other choice of y0 would lead to either a singularity or an
unstable solution. For the moment we hold yo arbitrary.
For simplicity we consider the case of uniform buoyancy frequency for
which N =No. The solution of (3.10.15) subject to (3.10.12) is in that case:
1
2
t/1 = 2(z+D) (Yo- y).
(3.10.17)
The total density field in dimensional variables is
[
2
fo8t/J]
Ptotat =Po 1- N zjg- g 8z (dimensional)
(3.10.18)
which in nondimensional form, using the scales ford and U of (3.10.11), gives
us for the case of constant buoyancy frequency:
Ptotal _ 1 = No
2
d {z + PL8t/J }·
Po
g
fo 8z
(3.10.19)
The smallness of the parameter PL/ fo is required for the validity of the P
plane approximation and the consequent smallness of the density anomaly,
compared to the density distribution in the resting state, is as required by
quasi-geostrophic theory.
The remaining boundary condition is the matching of the vertical velocity
to the given Ekman pumping velocity at z = 0, i.e. (3.10.13). Inserting the
solution (3.10.17) into (3.10.13) yields: