The Nonlinear Munk Problem
51
The boundary-layer equation (2.9.1) is a nonlinear partial differential
equation, and this combination renders the problem very difficult. Progress can
be made if we consider instead the oncoming flow as being constant and either
-1 or + 1. If the actual oncoming flow is as in (2.9 .2), this can be thought of as
a crude attempt at a Taylor series solution in y around either y = 0 or y = 1 in
which we keep only the first term. This is an important point to keep in mind
for it means that our results do not apply all along the boundary layer but are
focused instead on these particular regions near y = ± 1. On the other hand,
we may simply momentarily abandon trying to consider the whole gyre and
instead consider special impinging flows which are constant with latitude in
order to find boundary layer solutions just as we did in the inertial boundary
layer with our choice (2.8.5).
In either case this implies that in the case where the oncoming zonal flow
has velocity -1, we search for solutions of the form:
t/1 = ¢(x)y
(2.9.3)
which when inserted in (2.9.1) yields the ordinary differential equation:
¢"" = A.(¢'¢" - ¢¢1//) + ¢'
(2.9.4)
where a prime denotes a differentiation in x. The problem is now reduced to an
ordinary differential equation in x. It is left for the reader to verify that for the
case where the interior flow is eastward and equal to + 1 the same equation for
the x structure obtains with A. replaced with -A..
The vorticity equation (2.9.4) can be integrated once in x to obtain:
¢ 111 = A.(¢' 2 - ¢¢") +
where we have used the condition that ¢ -> 1 as x goes to oo while the
derivatives of ¢ on the boundary-layer scale go to zero for large x. The
boundary conditions for (2.9.5) are, for no slip:
¢(0) = ¢'(0) = 0
(2.9.6a)
and for slip:
¢(0) = ¢" (0) = 0.
(2.9.6b)
The third boundary condition is set at x = oo, namely that ¢ approaches 1.
Ierley and Ruehr (1986) solved this nonlinear problem numerically by a
"shooting method". That is, if three conditions were set at the origin, the
problem would be an initial value problem which would be very easy to integrate numerically. Instead, however, we have a nonlinear boundary value
problem with two conditions set at the origin and one condition to be satisfied
at oo, and this is much more difficult. The shooting method guesses the third
"missing" initial condition on x = 0 and integrates the equation forward to
x = oo. If, improbably, the guess were correct, and the boundary condition at
51
The boundary-layer equation (2.9.1) is a nonlinear partial differential
equation, and this combination renders the problem very difficult. Progress can
be made if we consider instead the oncoming flow as being constant and either
-1 or + 1. If the actual oncoming flow is as in (2.9 .2), this can be thought of as
a crude attempt at a Taylor series solution in y around either y = 0 or y = 1 in
which we keep only the first term. This is an important point to keep in mind
for it means that our results do not apply all along the boundary layer but are
focused instead on these particular regions near y = ± 1. On the other hand,
we may simply momentarily abandon trying to consider the whole gyre and
instead consider special impinging flows which are constant with latitude in
order to find boundary layer solutions just as we did in the inertial boundary
layer with our choice (2.8.5).
In either case this implies that in the case where the oncoming zonal flow
has velocity -1, we search for solutions of the form:
t/1 = ¢(x)y
(2.9.3)
which when inserted in (2.9.1) yields the ordinary differential equation:
¢"" = A.(¢'¢" - ¢¢1//) + ¢'
(2.9.4)
where a prime denotes a differentiation in x. The problem is now reduced to an
ordinary differential equation in x. It is left for the reader to verify that for the
case where the interior flow is eastward and equal to + 1 the same equation for
the x structure obtains with A. replaced with -A..
The vorticity equation (2.9.4) can be integrated once in x to obtain:
¢ 111 = A.(¢' 2 - ¢¢") +
where we have used the condition that ¢ -> 1 as x goes to oo while the
derivatives of ¢ on the boundary-layer scale go to zero for large x. The
boundary conditions for (2.9.5) are, for no slip:
¢(0) = ¢'(0) = 0
(2.9.6a)
and for slip:
¢(0) = ¢" (0) = 0.
(2.9.6b)
The third boundary condition is set at x = oo, namely that ¢ approaches 1.
Ierley and Ruehr (1986) solved this nonlinear problem numerically by a
"shooting method". That is, if three conditions were set at the origin, the
problem would be an initial value problem which would be very easy to integrate numerically. Instead, however, we have a nonlinear boundary value
problem with two conditions set at the origin and one condition to be satisfied
at oo, and this is much more difficult. The shooting method guesses the third
"missing" initial condition on x = 0 and integrates the equation forward to
x = oo. If, improbably, the guess were correct, and the boundary condition at