42
Homogeneous Models of the Ocean Circulation
the basin. Indeed, since (2.7.4) is equally valid for this boundary condition, the
flux of vorticity entering is exactly the same as in the case of the no-slip boundary
condition. Although the fluid slips along the boundary, the vorticity gradient at
x = 0 is the same in each case, namely -1p1 (0,y)/c5~. It is important to note that
since b~ = AH/ {3, the flux of vorticity into the basin by diffusion is independent
of AH, which is the physical content of (2.7.4b).
On the other hand, if we tried to apply the boundary condition (2.4.10),
there would be no boundary layer solution possible, as (2.7.4) shows us directly,
for if(2.4.10) is used in (2.7.4), a contradiction is reached immediately. This is
the first indication that the circulation is sensitive to the boundary conditions
applied.
It might be thought that this sensitivity is an artifact of the linear solution
that imposes the local balance (2.7.4). This is not the case, as we see below.
Stommel's Solution
We include here for the sake of completeness the solution to the linear problem
when lateral friction is ignored, and only bottom friction is kept. This is
Stommel's (1948) original model for the western intensification. Note that in
this case the order of the problem is lowered, and the boundary layer equation
becomes:
(2.7.6)
which can satisfy only the no normal flow condition on x = 0 and the condition
that the solution merges smoothly with t/1 I when x » bs = r / {3.
The solution in this case is:
t/1 = t/II(x,y) ( 1 - e-xf!Js)
V =~I e-xf!Js.
s
(2.7.7a,b)
In the absence of horizontal diffusion of vorticity the input of vorticity by
the wind via the Ekman pumping must be balanced by a draining away of
vorticity in the boundary layer through bottom friction. Analysis similar to
that which leads to (2.7.4) requires, in the case of bottom friction, that:
v(O,y) = fo re W£(x',y)dx'.
rD} 0
(2.7.8)
If (2.5.3) and (2.7.7b) are used, we can see that (2.7.8) is satisfied automatically. Thus, in linear theory there is no difficulty for sufficient dissipation
to occur locally in the boundary layer. A vorticity balance is always achieved
locally between the interior source and the boundary layer dissipation which
Homogeneous Models of the Ocean Circulation
the basin. Indeed, since (2.7.4) is equally valid for this boundary condition, the
flux of vorticity entering is exactly the same as in the case of the no-slip boundary
condition. Although the fluid slips along the boundary, the vorticity gradient at
x = 0 is the same in each case, namely -1p1 (0,y)/c5~. It is important to note that
since b~ = AH/ {3, the flux of vorticity into the basin by diffusion is independent
of AH, which is the physical content of (2.7.4b).
On the other hand, if we tried to apply the boundary condition (2.4.10),
there would be no boundary layer solution possible, as (2.7.4) shows us directly,
for if(2.4.10) is used in (2.7.4), a contradiction is reached immediately. This is
the first indication that the circulation is sensitive to the boundary conditions
applied.
It might be thought that this sensitivity is an artifact of the linear solution
that imposes the local balance (2.7.4). This is not the case, as we see below.
Stommel's Solution
We include here for the sake of completeness the solution to the linear problem
when lateral friction is ignored, and only bottom friction is kept. This is
Stommel's (1948) original model for the western intensification. Note that in
this case the order of the problem is lowered, and the boundary layer equation
becomes:
(2.7.6)
which can satisfy only the no normal flow condition on x = 0 and the condition
that the solution merges smoothly with t/1 I when x » bs = r / {3.
The solution in this case is:
t/1 = t/II(x,y) ( 1 - e-xf!Js)
V =~I e-xf!Js.
s
(2.7.7a,b)
In the absence of horizontal diffusion of vorticity the input of vorticity by
the wind via the Ekman pumping must be balanced by a draining away of
vorticity in the boundary layer through bottom friction. Analysis similar to
that which leads to (2.7.4) requires, in the case of bottom friction, that:
v(O,y) = fo re W£(x',y)dx'.
rD} 0
(2.7.8)
If (2.5.3) and (2.7.7b) are used, we can see that (2.7.8) is satisfied automatically. Thus, in linear theory there is no difficulty for sufficient dissipation
to occur locally in the boundary layer. A vorticity balance is always achieved
locally between the interior source and the boundary layer dissipation which