Continuous Models of the Ventilated Thermocline
255
boundary at different depths this boundary condition also avoids the
singularity in the density field found in other treatments of the continuous
problem where all the density surfaces of the moving thermocline outcrop
along the eastern boundary.
The system of equations then is the set ( 4.11.22a,b) subject to the boundary
conditions ( 4.11.24), ( 4.11.25), ( 4.11.27), and the integral condition ( 4.11.39).
This is a very unusual mathematical problem and appears highly overspecified
since the differential equation is second order in density, and there are four
boundary conditions plus an integral condition to satisfy. The correspondence
of the system to the layer model is helpful in understanding that the problem is
in fact sensibly posed. Indeed, the conditions on the boundary are not all
known and must be actually determined by the solution of the problem. As we
have already seen, the base of the thermocline must be determined, i.e., the
density Pn, as a function of latitude and longitude must be found. This is
equivalent to the need to find the pool boundaries of each of the unventilated
layers in the layer formulation of the problem. Those pool boundaries as a
function of layer index are equivalent to the surface z = -D( ¢,e) and must be
found from the solution. Similarly, the potential vorticity in (4.11.22) is not
completely known a priori over the whole water column and must be
determined as part of the solution. This is also true of the surface pressure.
Hence both of the boundary conditions at the surface, i.e., at p = Ps, are
actually used to determine unknown parts of the solution, in particular the
potential vorticity of the newly subducted fluid. The Sverdrup integral
condition is used to determine the vertical extent of the thermocline at each
location, just as we used it in the layer model. It determines the overall depth of
the moving fluid, h, in the layer model after the potential vorticity conservation
relations are used to relate all the thicknesses to h. The continuous model
"works" in exactly the same way as the layer model.
The solution is found numerically by discretizing the governing equations.
The solution is started at the density surface that outcrops furthest north in the
gyre. For illustration purposes we can consider the outcrop lines to be latitude
circles but Huang (1989b) has shown how the argument can be easily
generalized to the case with zonal variation of the surface density.
At each horizontal position (4.11.22a,b) are integrated starting from
p=pn, where the value of Pn is guessed, on a grid of points as shown in
Fig. 4.11.3. The integration proceeds to smaller p, i.e., higher in the water
column. The value of q at each grid point is known from information carried to
that point by streamlines on which q is determined upstream. If the layer is
unventilated, the value of q is determined by the value of the potential vorticity
on the intergyre boundary where the density gradient with depth is known. If
the layer is ventilated, the streamline which intersects the grid point carries the
potential vorticity which has been determined further northward by the
subduction process (we see below how this is done). If the grid point is
informed by a streamline which, although in a ventilated layer, emanates from
the western boundary, the potential vorticity must be specified on that
255
boundary at different depths this boundary condition also avoids the
singularity in the density field found in other treatments of the continuous
problem where all the density surfaces of the moving thermocline outcrop
along the eastern boundary.
The system of equations then is the set ( 4.11.22a,b) subject to the boundary
conditions ( 4.11.24), ( 4.11.25), ( 4.11.27), and the integral condition ( 4.11.39).
This is a very unusual mathematical problem and appears highly overspecified
since the differential equation is second order in density, and there are four
boundary conditions plus an integral condition to satisfy. The correspondence
of the system to the layer model is helpful in understanding that the problem is
in fact sensibly posed. Indeed, the conditions on the boundary are not all
known and must be actually determined by the solution of the problem. As we
have already seen, the base of the thermocline must be determined, i.e., the
density Pn, as a function of latitude and longitude must be found. This is
equivalent to the need to find the pool boundaries of each of the unventilated
layers in the layer formulation of the problem. Those pool boundaries as a
function of layer index are equivalent to the surface z = -D( ¢,e) and must be
found from the solution. Similarly, the potential vorticity in (4.11.22) is not
completely known a priori over the whole water column and must be
determined as part of the solution. This is also true of the surface pressure.
Hence both of the boundary conditions at the surface, i.e., at p = Ps, are
actually used to determine unknown parts of the solution, in particular the
potential vorticity of the newly subducted fluid. The Sverdrup integral
condition is used to determine the vertical extent of the thermocline at each
location, just as we used it in the layer model. It determines the overall depth of
the moving fluid, h, in the layer model after the potential vorticity conservation
relations are used to relate all the thicknesses to h. The continuous model
"works" in exactly the same way as the layer model.
The solution is found numerically by discretizing the governing equations.
The solution is started at the density surface that outcrops furthest north in the
gyre. For illustration purposes we can consider the outcrop lines to be latitude
circles but Huang (1989b) has shown how the argument can be easily
generalized to the case with zonal variation of the surface density.
At each horizontal position (4.11.22a,b) are integrated starting from
p=pn, where the value of Pn is guessed, on a grid of points as shown in
Fig. 4.11.3. The integration proceeds to smaller p, i.e., higher in the water
column. The value of q at each grid point is known from information carried to
that point by streamlines on which q is determined upstream. If the layer is
unventilated, the value of q is determined by the value of the potential vorticity
on the intergyre boundary where the density gradient with depth is known. If
the layer is ventilated, the streamline which intersects the grid point carries the
potential vorticity which has been determined further northward by the
subduction process (we see below how this is done). If the grid point is
informed by a streamline which, although in a ventilated layer, emanates from
the western boundary, the potential vorticity must be specified on that