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Homogeneous Models of the Ocean Circulation
2.6 The Western Boundary-Layer Equation
If the Sverdrup theory is to be valid in the interior, it is necessary that a
boundary-layer solution for the western boundary region can be found which
smoothly joins to the Sverdrup interior and at the same time satisfies all the
boundary conditions on the western boundary. In the vicinity of the western
boundary, which for simplicity we place at a constant value of x, i.e., at
x = xw, terms in the vorticity equation neglected in the interior become
important as physical processes associated with the smaller length scales of the
boundary-layer increase in magnitude. On the small scales in x associated with
the boundary layer the estimates of the viscous and nonlinear terms alter as L,
the length scale of the motion, becomes less than the basin scale.
In re-evaluating the importance of various processes and their mathematical representations it is sensible to suppose first of all that derivatives in the x
direction in the boundary layer become large compared with derivatives in the
y direction when operating on the same function. This may not always be true,
but we start with this plausible hypothesis. Then in the vicinity of the western
boundary if:
a a
-»- ox ay
(2.6.1)
the vorticity equation in dimensional form (2.2.9) becomes in the steady state:
(2.6.2)
In the boundary layer tjJ is a rapid function of x so that one or the other of
the terms which are small in the interior become order 1 in the boundary layer.
This means that t/1 is actually a function of:
~=~ (J
where (J is the dimensional boundary layer width so that the derivatives of tjJ
can be estimated as being of the order of tjJ /b.
In the boundary layer the order of magnitude of the streamfunction must
be at least as large as its value in the interior, tjJ b since it joins smoothly to it as
the interior is approached. It could be larger if there is a strong recirculation in
the region of the boundary layer. Thus in the western boundary-current region:
ot/J
( t/11)
(Lfo )
fo
P->0 P- =0 --WE »-WE
OX(J
(J D
D
(2.6.3)
so that the direct action of the forcing, i.e., the vorticity source due to the
Ekman pumping, is negligible in (2.6.2) within the boundary layer. Physically
and mathematically the boundary-layer current is produced only by the
Homogeneous Models of the Ocean Circulation
2.6 The Western Boundary-Layer Equation
If the Sverdrup theory is to be valid in the interior, it is necessary that a
boundary-layer solution for the western boundary region can be found which
smoothly joins to the Sverdrup interior and at the same time satisfies all the
boundary conditions on the western boundary. In the vicinity of the western
boundary, which for simplicity we place at a constant value of x, i.e., at
x = xw, terms in the vorticity equation neglected in the interior become
important as physical processes associated with the smaller length scales of the
boundary-layer increase in magnitude. On the small scales in x associated with
the boundary layer the estimates of the viscous and nonlinear terms alter as L,
the length scale of the motion, becomes less than the basin scale.
In re-evaluating the importance of various processes and their mathematical representations it is sensible to suppose first of all that derivatives in the x
direction in the boundary layer become large compared with derivatives in the
y direction when operating on the same function. This may not always be true,
but we start with this plausible hypothesis. Then in the vicinity of the western
boundary if:
a a
-»- ox ay
(2.6.1)
the vorticity equation in dimensional form (2.2.9) becomes in the steady state:
(2.6.2)
In the boundary layer tjJ is a rapid function of x so that one or the other of
the terms which are small in the interior become order 1 in the boundary layer.
This means that t/1 is actually a function of:
~=~ (J
where (J is the dimensional boundary layer width so that the derivatives of tjJ
can be estimated as being of the order of tjJ /b.
In the boundary layer the order of magnitude of the streamfunction must
be at least as large as its value in the interior, tjJ b since it joins smoothly to it as
the interior is approached. It could be larger if there is a strong recirculation in
the region of the boundary layer. Thus in the western boundary-current region:
ot/J
( t/11)
(Lfo )
fo
P->0 P- =0 --WE »-WE
OX(J
(J D
D
(2.6.3)
so that the direct action of the forcing, i.e., the vorticity source due to the
Ekman pumping, is negligible in (2.6.2) within the boundary layer. Physically
and mathematically the boundary-layer current is produced only by the