Numerical Examples
65
dissipation, the stream function and the Ekman pumping must on average be
negatively correlated. If wE < 0, this means ljJ > 0 or an anticyclonic circulation. This in turn means that the velocity circulation integral in (2.11.11) has
the same sign as the Ekman pumping. Therefore in the absence of bottom
friction we must have dQ/ dt/1 < 0. This is the opposite sign than that required
for the Fofonoff mode, and we conclude that resonance with the Fofonoff
mode, is inhibited by the presence of lateral mixing, as long as the flux of
vorticity through the side walls is significant. If the boundary conditions are
superslip conditions, only bottom friction enters to equilibrate the vorticity
balance; the second term in (2.11.11) is absent altogether, and there is no prohibition of resonance with the Fofonoff mode. This is consistent with our
discussion of the geometry of total vorticity isolines.
2.12 Numerical Examples
We now turn our attention to the results of numerical calculations of the wind
driven circulation problem. In particular, we concentrate on the effect of the
magnitude and type of dissipation and the role of the boundary conditions in
shaping the structure of the circulation. On the basis of our discussion up to
now we can expect the role of dissipation to be profound. We are particularly
interested in the limiting case where the boundary-layer Reynolds number is
large, i.e., where {)1 >max ({JM, {)8 ), and where we would expect from naive
scaling arguments that the effects of dissipation would be unimportant.
Because of the numerical difficulty of reaching this limit in the circulation
problem most calculations have tended at most to be in the regime where
inertia and dissipation are of the same importance in determining the
boundary-layer structure. Nevertheless the effects of different dissipation
parameterizations and boundary conditions are still very striking. We later
take up the more difficult question of the limit {)1 » ({JM, bs).
Blandford (1971) was the first to take up the systematic study ofthe role of
boundary conditions and dissipation mechanisms in determining the circulation structure. Previous to his calculations there were two pioneering calculations by Bryan (1963) and Veronis (1966). Bryan used only lateral friction with
a no-slip condition while Veronis employed only bottom friction and could
therefore apply only the no normal flow condition.
A sequence of calculations from the Veronis study are shown in Fig. 2.12.1
for increasing values of bJ/bs. Veronis chose for his forcing an Ekman
pumping (or equivalently a wind stress curl) of the form:
(2.12.1)
At low values of the ratio bJ/bs the solution is indistinguishable from the
linear Stommel solution. Even when the ratio is unity, as in panel a of the
65
dissipation, the stream function and the Ekman pumping must on average be
negatively correlated. If wE < 0, this means ljJ > 0 or an anticyclonic circulation. This in turn means that the velocity circulation integral in (2.11.11) has
the same sign as the Ekman pumping. Therefore in the absence of bottom
friction we must have dQ/ dt/1 < 0. This is the opposite sign than that required
for the Fofonoff mode, and we conclude that resonance with the Fofonoff
mode, is inhibited by the presence of lateral mixing, as long as the flux of
vorticity through the side walls is significant. If the boundary conditions are
superslip conditions, only bottom friction enters to equilibrate the vorticity
balance; the second term in (2.11.11) is absent altogether, and there is no prohibition of resonance with the Fofonoff mode. This is consistent with our
discussion of the geometry of total vorticity isolines.
2.12 Numerical Examples
We now turn our attention to the results of numerical calculations of the wind
driven circulation problem. In particular, we concentrate on the effect of the
magnitude and type of dissipation and the role of the boundary conditions in
shaping the structure of the circulation. On the basis of our discussion up to
now we can expect the role of dissipation to be profound. We are particularly
interested in the limiting case where the boundary-layer Reynolds number is
large, i.e., where {)1 >max ({JM, {)8 ), and where we would expect from naive
scaling arguments that the effects of dissipation would be unimportant.
Because of the numerical difficulty of reaching this limit in the circulation
problem most calculations have tended at most to be in the regime where
inertia and dissipation are of the same importance in determining the
boundary-layer structure. Nevertheless the effects of different dissipation
parameterizations and boundary conditions are still very striking. We later
take up the more difficult question of the limit {)1 » ({JM, bs).
Blandford (1971) was the first to take up the systematic study ofthe role of
boundary conditions and dissipation mechanisms in determining the circulation structure. Previous to his calculations there were two pioneering calculations by Bryan (1963) and Veronis (1966). Bryan used only lateral friction with
a no-slip condition while Veronis employed only bottom friction and could
therefore apply only the no normal flow condition.
A sequence of calculations from the Veronis study are shown in Fig. 2.12.1
for increasing values of bJ/bs. Veronis chose for his forcing an Ekman
pumping (or equivalently a wind stress curl) of the form:
(2.12.1)
At low values of the ratio bJ/bs the solution is indistinguishable from the
linear Stommel solution. Even when the ratio is unity, as in panel a of the