The Nonlinear Munk Problem
Fig. 2.9.5. Calculation of the winddriven circulation driven by an Ekman
pumping of a single-gyre form for the
case of no-slip for a value of 6/6M of
0(1 ). (From Ierley 1987)
55
the boundary-layer approximation, 8j8x » 8j8y. To examine the connection
between the full vorticity equation, governed by the partial differential
equation (2.3.2), and the boundary-layer problem solved by lerley and
Ruehr, Ierley (1987) examined a "regional" model. That is, he considered a
model in which the Sverdrup theory is solved for first and is supposed to be
valid over the greater part of the basin. The solution in the vicinity of the
western boundary is then determined by solving the vorticity equation in a
domain (Lx, Ly) in which Lx is small compared to the full basin scale but
large compared to the expected scale of the boundary layer. At the eastern
edge of the region the outflow velocity and its vorticity are chosen to exactly match the Sverdrup solution. There is no other forcing for the flow in
that region. At the western, northern, and southern boundaries the conditions of no normal flow and, in this case, slip conditions were employed.
The scale in x was chosen to be 10 {J M, and the scale in y was allowed to
vary so that a= Ly/ Lx = Ly/10[JM becomes another parameter of the calculation in addition to bi/bM. We might expect that as a becomes large, the
results of the regional model would qualitatively approach that of the
boundary-layer problem.
Ierley found that for a of 0(1) boundary-layer type solutions of the regional model exist, for which 8j8x » 8j8y, for values of A (evaluated in terms
of fJI/fJM) that are supercritical, i.e., for A< Ac < 0. Figure 2.9.6a shows a
solution for the regional problem with slip conditions for a value of A= -0.7
whereas the critical value of A for the boundary layer problem is -0.297. In the
outflow region of the flow in the figure a boundary layer solution where x
derivatives exceed y derivatives in size still exists. For a fixed A as a is increased
(i.e., as they length of the region is expanded) they scale in the outflow region
Fig. 2.9.5. Calculation of the winddriven circulation driven by an Ekman
pumping of a single-gyre form for the
case of no-slip for a value of 6/6M of
0(1 ). (From Ierley 1987)
55
the boundary-layer approximation, 8j8x » 8j8y. To examine the connection
between the full vorticity equation, governed by the partial differential
equation (2.3.2), and the boundary-layer problem solved by lerley and
Ruehr, Ierley (1987) examined a "regional" model. That is, he considered a
model in which the Sverdrup theory is solved for first and is supposed to be
valid over the greater part of the basin. The solution in the vicinity of the
western boundary is then determined by solving the vorticity equation in a
domain (Lx, Ly) in which Lx is small compared to the full basin scale but
large compared to the expected scale of the boundary layer. At the eastern
edge of the region the outflow velocity and its vorticity are chosen to exactly match the Sverdrup solution. There is no other forcing for the flow in
that region. At the western, northern, and southern boundaries the conditions of no normal flow and, in this case, slip conditions were employed.
The scale in x was chosen to be 10 {J M, and the scale in y was allowed to
vary so that a= Ly/ Lx = Ly/10[JM becomes another parameter of the calculation in addition to bi/bM. We might expect that as a becomes large, the
results of the regional model would qualitatively approach that of the
boundary-layer problem.
Ierley found that for a of 0(1) boundary-layer type solutions of the regional model exist, for which 8j8x » 8j8y, for values of A (evaluated in terms
of fJI/fJM) that are supercritical, i.e., for A< Ac < 0. Figure 2.9.6a shows a
solution for the regional problem with slip conditions for a value of A= -0.7
whereas the critical value of A for the boundary layer problem is -0.297. In the
outflow region of the flow in the figure a boundary layer solution where x
derivatives exceed y derivatives in size still exists. For a fixed A as a is increased
(i.e., as they length of the region is expanded) they scale in the outflow region