Continuous Models of the Ventilated Thermocline
251
As a differential equation in p the boundary conditions for the system
(4.11.22) are rather unusual mathematically but rather straightforward
physically. At the base of the thermocline, i.e., at z = -D, the density and
the pressure must be continuous with that of the resting fluid. Thus for each ¢
and e in the gyre, at z = -D, where the density is PD (and which is a function of
horizontal position):
n=nD(p); z=ZD(P) at P=PD·
( 4.11.24)
Of course we do not know ahead of time where the base of the thermocline
is, and therefore where the boundary is in either z or p. That is, we are dealing
in density coordinates with a free boundary problem, and an important part of
the solution is the determination at each latitude and longitude of the density,
PD, which sets the base of the thermocline. Since the density is continuous, the
velocity (but not the vertical shear) must be continuous across the surface
z = -D, and thus the velocity vanishes there as it does in the abyss. Therefore it
follows from (4.11.24) that Vn = 0 at p = PD·
Within the mixed layer the density surfaces are vertical. That is, the density
is equal to its surface value, p.( ¢, 8), throughout the mixed layer. Thus at the
base of the mixed layer the boundary condition on the system ( 4.11.22) is:
z = -hm(¢, 8) at p = p.(¢, 8).
( 4.11.25)
Each of these equations has its layer equivalent. If we take note of (4.3.18),
the layer equation statements of potential vorticity conservation and the
hydrostatic relation between the interface depths and the pressure (4.3.7) are
identical to the finite difference version of (4.11.22a,b) although we have not
yet indicated how q is to be determined in (4.11.22a).
The Sverdrup relation ( 4.3.15) also has an exact equivalent in the
continuous model. To demonstrate this we must start with the mixed layer
and calculate its transport. As opposed to the treatment in the previous section,
we now allow both the mixed layer depth and its density to be functions of
longitude as well as latitude.
Within the mixed layer the density, being independent of depth, allows us
to write the pressure as:
Pm = Ps( ¢, 8) - Psgz
( 4.11.26)
where Ps is the (unknown) surface pressure. This means that the surface
pressure is given by:
Ps = Pm + Psgz = nm.
( 4.11.27)
Therefore nm within the mixed layer is independent of depth and equal to the
surface pressure. By continuity of pressure, at the base of the mixed layer:
n = nm(Ps) at P = Ps
(4.11.28)
251
As a differential equation in p the boundary conditions for the system
(4.11.22) are rather unusual mathematically but rather straightforward
physically. At the base of the thermocline, i.e., at z = -D, the density and
the pressure must be continuous with that of the resting fluid. Thus for each ¢
and e in the gyre, at z = -D, where the density is PD (and which is a function of
horizontal position):
n=nD(p); z=ZD(P) at P=PD·
( 4.11.24)
Of course we do not know ahead of time where the base of the thermocline
is, and therefore where the boundary is in either z or p. That is, we are dealing
in density coordinates with a free boundary problem, and an important part of
the solution is the determination at each latitude and longitude of the density,
PD, which sets the base of the thermocline. Since the density is continuous, the
velocity (but not the vertical shear) must be continuous across the surface
z = -D, and thus the velocity vanishes there as it does in the abyss. Therefore it
follows from (4.11.24) that Vn = 0 at p = PD·
Within the mixed layer the density surfaces are vertical. That is, the density
is equal to its surface value, p.( ¢, 8), throughout the mixed layer. Thus at the
base of the mixed layer the boundary condition on the system ( 4.11.22) is:
z = -hm(¢, 8) at p = p.(¢, 8).
( 4.11.25)
Each of these equations has its layer equivalent. If we take note of (4.3.18),
the layer equation statements of potential vorticity conservation and the
hydrostatic relation between the interface depths and the pressure (4.3.7) are
identical to the finite difference version of (4.11.22a,b) although we have not
yet indicated how q is to be determined in (4.11.22a).
The Sverdrup relation ( 4.3.15) also has an exact equivalent in the
continuous model. To demonstrate this we must start with the mixed layer
and calculate its transport. As opposed to the treatment in the previous section,
we now allow both the mixed layer depth and its density to be functions of
longitude as well as latitude.
Within the mixed layer the density, being independent of depth, allows us
to write the pressure as:
Pm = Ps( ¢, 8) - Psgz
( 4.11.26)
where Ps is the (unknown) surface pressure. This means that the surface
pressure is given by:
Ps = Pm + Psgz = nm.
( 4.11.27)
Therefore nm within the mixed layer is independent of depth and equal to the
surface pressure. By continuity of pressure, at the base of the mixed layer:
n = nm(Ps) at P = Ps
(4.11.28)