Integral Conditions
57
a tight recirculating gyre in the northwest corner of the gyre which mediates the
return of the boundary-layer flow to the interior. The problem of understanding the relation between the interior Sverdrup flow and the western
boundary region can therefore be expected to involve regions of flow much
more complex than standard boundary-layer regions matched to a Sverdrup
interior. The need for the fluid to sufficiently dissipate vorticity along its path
after entering the western region of nonlinear flow evidently requires the path
of the fluid to become substantially distorted into a permanent recirculating
eddy before rejoining the interior. This makes the problem of understanding
the relationship between the interior and the boundary layer unexpectedly
complex. It further emphasizes the important role of dissipation in determining
the structure of the circulation pattern although for the parameter settings
studied here the effects are limited to the region near the western boundary.
Given the sensitivity of the circulation to the mechanism of dissipation, it
must also be anticipated that the pattern of the circulation depends on the type
of dissipation (lateral mixing or bottom friction) and the magnitude of the
dissipation compared to the inertial effects and to the boundary conditions (as
in the problem studied by Ierley and Ruehr where the no slip boundary layer
solutions persist to higher Reynolds numbers than do the slip solutions in the
outflow region). Determining exactly what the sensitivity to these conditions is
requires us to examine a range of numerical experiments that have been carried
out by several investigators over the past few decades. Before describing these
experiments in detail we continue with our preparation by discussing several
general integral constraints on the circulation and the implications that they
have for different types of dissipation and different boundary conditions.
2.10 Integral Conditions
Integral Statements of Energy and Vorticity
Consider the vorticity equation (2.2.9) and imagine first that the solution has
reached a steady state. Each streamline must be closed, and of course there is
no flow across streamlines. We integrate over the area S enclosed by a
streamline (see Fig. 2.10.1). Note that the boundary of the basin is also a
streamline to which the following analysis must also apply.
When the advective term is integrated over the area S we obtain:
11 J(ljJ, \l 2 l/J + {3y)dx dy = 11 u· \l(( + f3y)dx dy
= 11 \l· (il{( + {3y} )dx dy = i~ u· n{( + [3y}ds
=0.
(2.10.1)
57
a tight recirculating gyre in the northwest corner of the gyre which mediates the
return of the boundary-layer flow to the interior. The problem of understanding the relation between the interior Sverdrup flow and the western
boundary region can therefore be expected to involve regions of flow much
more complex than standard boundary-layer regions matched to a Sverdrup
interior. The need for the fluid to sufficiently dissipate vorticity along its path
after entering the western region of nonlinear flow evidently requires the path
of the fluid to become substantially distorted into a permanent recirculating
eddy before rejoining the interior. This makes the problem of understanding
the relationship between the interior and the boundary layer unexpectedly
complex. It further emphasizes the important role of dissipation in determining
the structure of the circulation pattern although for the parameter settings
studied here the effects are limited to the region near the western boundary.
Given the sensitivity of the circulation to the mechanism of dissipation, it
must also be anticipated that the pattern of the circulation depends on the type
of dissipation (lateral mixing or bottom friction) and the magnitude of the
dissipation compared to the inertial effects and to the boundary conditions (as
in the problem studied by Ierley and Ruehr where the no slip boundary layer
solutions persist to higher Reynolds numbers than do the slip solutions in the
outflow region). Determining exactly what the sensitivity to these conditions is
requires us to examine a range of numerical experiments that have been carried
out by several investigators over the past few decades. Before describing these
experiments in detail we continue with our preparation by discussing several
general integral constraints on the circulation and the implications that they
have for different types of dissipation and different boundary conditions.
2.10 Integral Conditions
Integral Statements of Energy and Vorticity
Consider the vorticity equation (2.2.9) and imagine first that the solution has
reached a steady state. Each streamline must be closed, and of course there is
no flow across streamlines. We integrate over the area S enclosed by a
streamline (see Fig. 2.10.1). Note that the boundary of the basin is also a
streamline to which the following analysis must also apply.
When the advective term is integrated over the area S we obtain:
11 J(ljJ, \l 2 l/J + {3y)dx dy = 11 u· \l(( + f3y)dx dy
= 11 \l· (il{( + {3y} )dx dy = i~ u· n{( + [3y}ds
=0.
(2.10.1)