190
Theory of the Ventilated Thermocline
We assume, again for simplicity, that the eastern boundary is a line of
constant longitude on which the geostrophic zonal velocity must vanish. Our
principal assumption is that the motion is adiabatic and frictionless so that
potential vorticity is conserved where the flow is not directly exposed to Ekman
pumping. Thus at any location the governing equations are:
(a) the conservation of potential vorticity where the layer is not exposed to
Ekman pumping, or, for steady flow:
( 4.4.1)
(b) the Sverdrup balance for layers I and 2:
2
P "L- Vnhn = fwE
(4.4.2)
n=I
or, equivalently, (4.3.15).
At any location if there are, in the general case, N - M layers in motion,
there are N - M - 1 of them for which the law of conservation of potential
vorticity applies, and the remaining Sverdrup relation fills out our system
yielding N - M equations for the N - M unknown interface depths. Once these
are known, geostrophy (4.3.11) and the hydrostatic relation (4.3.10) determine
the velocity field completely. The essential problem is determining the potential
vorticity functions Qn which a priori are arbitrary functions. These are determined completely at the outcrop lines by the process of subduction in which
southward moving heavier fluid at the surface is forced to slide under the fluid
of the lighter layer whose edge is pinned to its surface position by the
specification of the surface density field, i.e., by the specification of the position
of the outcrop line. It is important to realize that specifying the outcrop line
does nothing more than specify the surface density. It does not determine the
density structure below the surface.
Single-Layer Region
We start our analysis of the motion in the region (}z ~ () ~ eo, i.e., in the
latitude band between the northern boundary of the gyre and the first outcrop
line south of the northern boundary where f = fz. In this region layer I is
absent. Layer 2 is the only moving layer, and it is directly forced by Ekman
pumping. There is no layer in this region in which potential vorticity is
conserved and the only nontrivial equation is the Sverdrup relation which
reduces to:
( 4.4.3)
Theory of the Ventilated Thermocline
We assume, again for simplicity, that the eastern boundary is a line of
constant longitude on which the geostrophic zonal velocity must vanish. Our
principal assumption is that the motion is adiabatic and frictionless so that
potential vorticity is conserved where the flow is not directly exposed to Ekman
pumping. Thus at any location the governing equations are:
(a) the conservation of potential vorticity where the layer is not exposed to
Ekman pumping, or, for steady flow:
( 4.4.1)
(b) the Sverdrup balance for layers I and 2:
2
P "L- Vnhn = fwE
(4.4.2)
n=I
or, equivalently, (4.3.15).
At any location if there are, in the general case, N - M layers in motion,
there are N - M - 1 of them for which the law of conservation of potential
vorticity applies, and the remaining Sverdrup relation fills out our system
yielding N - M equations for the N - M unknown interface depths. Once these
are known, geostrophy (4.3.11) and the hydrostatic relation (4.3.10) determine
the velocity field completely. The essential problem is determining the potential
vorticity functions Qn which a priori are arbitrary functions. These are determined completely at the outcrop lines by the process of subduction in which
southward moving heavier fluid at the surface is forced to slide under the fluid
of the lighter layer whose edge is pinned to its surface position by the
specification of the surface density field, i.e., by the specification of the position
of the outcrop line. It is important to realize that specifying the outcrop line
does nothing more than specify the surface density. It does not determine the
density structure below the surface.
Single-Layer Region
We start our analysis of the motion in the region (}z ~ () ~ eo, i.e., in the
latitude band between the northern boundary of the gyre and the first outcrop
line south of the northern boundary where f = fz. In this region layer I is
absent. Layer 2 is the only moving layer, and it is directly forced by Ekman
pumping. There is no layer in this region in which potential vorticity is
conserved and the only nontrivial equation is the Sverdrup relation which
reduces to:
( 4.4.3)