38
Homogeneous Models of the Ocean Circulation
2. 7 The Linear Problem
Munk's Solution
It is useful to consider the closure of the Sverdrup problem in the linear limit as
a starting point for our general investigation of the relationship between the
dynamics of the boundary current and the interior solution. The linear limit
provides us with an easily accessible solution, and its relation to the interior is
straightforward. The solution was first found by Munk (1950).
In the Munk solution nonlinearity in the boundary layer is completely
ignored. This requires that the boundary-layer Reynolds number (2.3.6) is
small. This requires in turn that the forcing be weak, or the horizontal turbulent
mixing be large. It is not possible to obtain a realistic model of the western
boundary current in such a limit (see, for example, the discussion in Pedlosky
1987), but the solution is of considerable theoretical and historical interest.
Munk also intended his model to describe the vertical average of a baroclinic
ocean with negligible bottom velocities, and the Munk model thus also ignores
the bottom friction term in (2.6.2).
In this case the equation in the boundary layer becomes:
(2.7.1)
For simplicity and without loss of generality we place the western boundary
x = 0. The general solution that satisfies the condition that t/1 is zero on x = 0
and also tends to t/1 1 for x ~ {JM is:
t/J = tfJ1(x,y) [ 1- e-x/ 20 M cos ( '!::)] + C(y)e-xf 20 M sin ( ' !::). (2.7.2)
The function C(y) must be determined by the remaining the boundary
condition, i.e., either (2.4.5), (2.4.6) or (2.4.10).
If the no slip condition is used, so that v = 0 at x = 0, then the solutions
for t/1, v, and ( become:
o/ ~ o/1 [1 - ,-•!"• (cos ;I:+~ ~n ;I:) l
v = ~ t/II e-xf2oM sin (v'3x)
J3 (jM
2{JM
(2.7.3a, b, c)
(= t£e-xf 20 M[cos(~:)- ~sin(~:)]·
Figure 2.7.1 shows the complete circulation pattern in the case when WE
has the form W0sin( ny / L). The Munk layer thickness has been chosen to be
Homogeneous Models of the Ocean Circulation
2. 7 The Linear Problem
Munk's Solution
It is useful to consider the closure of the Sverdrup problem in the linear limit as
a starting point for our general investigation of the relationship between the
dynamics of the boundary current and the interior solution. The linear limit
provides us with an easily accessible solution, and its relation to the interior is
straightforward. The solution was first found by Munk (1950).
In the Munk solution nonlinearity in the boundary layer is completely
ignored. This requires that the boundary-layer Reynolds number (2.3.6) is
small. This requires in turn that the forcing be weak, or the horizontal turbulent
mixing be large. It is not possible to obtain a realistic model of the western
boundary current in such a limit (see, for example, the discussion in Pedlosky
1987), but the solution is of considerable theoretical and historical interest.
Munk also intended his model to describe the vertical average of a baroclinic
ocean with negligible bottom velocities, and the Munk model thus also ignores
the bottom friction term in (2.6.2).
In this case the equation in the boundary layer becomes:
(2.7.1)
For simplicity and without loss of generality we place the western boundary
x = 0. The general solution that satisfies the condition that t/1 is zero on x = 0
and also tends to t/1 1 for x ~ {JM is:
t/J = tfJ1(x,y) [ 1- e-x/ 20 M cos ( '!::)] + C(y)e-xf 20 M sin ( ' !::). (2.7.2)
The function C(y) must be determined by the remaining the boundary
condition, i.e., either (2.4.5), (2.4.6) or (2.4.10).
If the no slip condition is used, so that v = 0 at x = 0, then the solutions
for t/1, v, and ( become:
o/ ~ o/1 [1 - ,-•!"• (cos ;I:+~ ~n ;I:) l
v = ~ t/II e-xf2oM sin (v'3x)
J3 (jM
2{JM
(2.7.3a, b, c)
(= t£e-xf 20 M[cos(~:)- ~sin(~:)]·
Figure 2.7.1 shows the complete circulation pattern in the case when WE
has the form W0sin( ny / L). The Munk layer thickness has been chosen to be