84
Homogeneous Models of the Ocean Circulation
the length scale used to calculate the flux is the inertial-scale, which would be
appropriate for streamlines in the outer inertial layer rather than the viscous
sub layer scale, a calculation similar to (2.12.6) yields a flux of vorticity which is
too small by a factor ( (> M j (> 1 ) 3 • This all assumes, of course, that the region of
parameter space where the boundary layer splits its structure between an inertial outer layer and a viscous sublayer is actually realized in calculations
which allow time dependence and eddies. Let us consider this possibility first,
i.e., that the asymptotic separation of scales actually occurs.
Several suggestions have been made for the way in which a possible balance could occur on these streamlines for steady solutions in the nonlinear limit
that would simultaneously retain the Sverdrup interior. Pedlosky (1987) argued
that the damped Rossby waves found in the Moore solution act as a sort of
baffle whose wiggly character increases the length of each streamline sufficiently that the viscous flux of vorticity across the streamlines outside the
sublayer would balance the vorticity input by the wind and pass the vorticity
flux to the sublayer at the boundary where it would be diffused out of the basin.
Pedlosky presented a scaling argument attempting to demonstrate the plausibility of such a balance. Cessi et al. (1990) criticized that scenario and carried
out careful and detailed numerical calculations with a regional model. Cessi et
al. demonstrated that the Moore scenario, in which a Rossby wave field
mediates the transition between an inertial western boundary layer and the
interior, would occur for bJ/bM »1, only if the north-south dimension of the
domain is allowed to increase in the same ratio. Under this frankly unrealistic
condition the necessary dissipation occurs, not in the Rossby wave field but in
a long viscous loop region pressed up against the boundary with a Munk layer
thickness. Note that in the limit where y derivatives tend to zero the nonlinearity vanishes, and the boundary-layer equation reduces to the Munk
problem for arbitrarily large bJ/bM. The dissipation in the Rossby wave field,
in comparison to the loop current, is found to be negligible. The flaw in
Pedlosky's argument lies in attributing to the current in the Rossby wave field a
fluid velocity of the order of the western boundary layer velocity. He assumes
the Rossby wave field is the meandering of a coherent narrow jet with an
inertial layer width. Instead, if the wave field mediates the transition of the
entire northern part of the gyre as in Moore's scenario, the outflow is broader
and is of basin scale in width, and the velocities and vorticity flux are correspondingly smaller by a factor bJ/ L. The flux then fails to balance the input of
vorticity by that amount.
An alternative scenario was suggested by Cessi et al. (1990). They pointed
out that the more realistic configuration of the flow for no-slip boundary
conditions and finite and fixed north-south basin extent is the pattern shown in
Fig. 2.12. 7d. An intense, localized recirculation exists in the northwest corner
of the basin. Streamlines which feed the interior Sverdrup flow navigate around
the edge of the recirculation gyre, dissipating their anomalous vorticity before
rejoining the interior. In the calculation shown in the figure (>I/(> M is of order
unity and a substantial amount of vorticity is actually dissipated in the western
Homogeneous Models of the Ocean Circulation
the length scale used to calculate the flux is the inertial-scale, which would be
appropriate for streamlines in the outer inertial layer rather than the viscous
sub layer scale, a calculation similar to (2.12.6) yields a flux of vorticity which is
too small by a factor ( (> M j (> 1 ) 3 • This all assumes, of course, that the region of
parameter space where the boundary layer splits its structure between an inertial outer layer and a viscous sublayer is actually realized in calculations
which allow time dependence and eddies. Let us consider this possibility first,
i.e., that the asymptotic separation of scales actually occurs.
Several suggestions have been made for the way in which a possible balance could occur on these streamlines for steady solutions in the nonlinear limit
that would simultaneously retain the Sverdrup interior. Pedlosky (1987) argued
that the damped Rossby waves found in the Moore solution act as a sort of
baffle whose wiggly character increases the length of each streamline sufficiently that the viscous flux of vorticity across the streamlines outside the
sublayer would balance the vorticity input by the wind and pass the vorticity
flux to the sublayer at the boundary where it would be diffused out of the basin.
Pedlosky presented a scaling argument attempting to demonstrate the plausibility of such a balance. Cessi et al. (1990) criticized that scenario and carried
out careful and detailed numerical calculations with a regional model. Cessi et
al. demonstrated that the Moore scenario, in which a Rossby wave field
mediates the transition between an inertial western boundary layer and the
interior, would occur for bJ/bM »1, only if the north-south dimension of the
domain is allowed to increase in the same ratio. Under this frankly unrealistic
condition the necessary dissipation occurs, not in the Rossby wave field but in
a long viscous loop region pressed up against the boundary with a Munk layer
thickness. Note that in the limit where y derivatives tend to zero the nonlinearity vanishes, and the boundary-layer equation reduces to the Munk
problem for arbitrarily large bJ/bM. The dissipation in the Rossby wave field,
in comparison to the loop current, is found to be negligible. The flaw in
Pedlosky's argument lies in attributing to the current in the Rossby wave field a
fluid velocity of the order of the western boundary layer velocity. He assumes
the Rossby wave field is the meandering of a coherent narrow jet with an
inertial layer width. Instead, if the wave field mediates the transition of the
entire northern part of the gyre as in Moore's scenario, the outflow is broader
and is of basin scale in width, and the velocities and vorticity flux are correspondingly smaller by a factor bJ/ L. The flux then fails to balance the input of
vorticity by that amount.
An alternative scenario was suggested by Cessi et al. (1990). They pointed
out that the more realistic configuration of the flow for no-slip boundary
conditions and finite and fixed north-south basin extent is the pattern shown in
Fig. 2.12. 7d. An intense, localized recirculation exists in the northwest corner
of the basin. Streamlines which feed the interior Sverdrup flow navigate around
the edge of the recirculation gyre, dissipating their anomalous vorticity before
rejoining the interior. In the calculation shown in the figure (>I/(> M is of order
unity and a substantial amount of vorticity is actually dissipated in the western