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Vertical Structure: Baroclinic Quasi-Geostrophic Models
this process allows penetration of the wind-driven circulation below the uppermost layer. In Chapter 4 we discuss an alternative mechanism, ventilation, that
also produces deep motion. The synthesis of these two ideas forms the present
basis of our conceptual understanding of the stratified wind-driven circulation.
We employ a layer model for the circulation to begin with and ignore crossinterface motion. This means that in every layer the fluid of a given density
remains in that layer, and there are thus no sources of fluid entering the layer.
Except for the uppermost layer no layer receives fluid from the upper Ekman
layer. In each layer the fluid recirculates endlessly in the steady state with each
fluid element repeating its round and round trajectory in the gyre an infinite
number of times. This recirculatory aspect of the flow is a crucial feature of the
dynamics of the model. The model confronts directly the issue of how layers
which are not directly forced by Ekman pumping can be set into motion.
Consider the layer equations for either the two-layer [(3.3.1) and (3.3.2)] or
the 2! layer model in the steady state [(3.3.1) and (3.3.3)] in the absence of
cross-interface motion. We can write both steady-state models in a unified
compact form as:
(3.5.1)
and
J(t/12, q2) = J ( t/12, '\1 2 1/12 + py + F2(t/l, - t/12) - G2t/12 + ~~ Zb) (3.5.2)
= -r2 '\1 2 1/12 +curl
to zero (although the term in G2 is actually trivial since its Jacobian with t/1 2 is
identically zero; it is retained, however, for conceptual purposes). If the model
is considered to be a 2! layer model in which the two layers lie over a resting
abyss, the interaction with the bottom is absent, and the bottom friction
parameter r2 and the topographic contribution, Zb, should be set equal to zero
instead.
Over the scale of the gyre motion, L, the relative vorticity can be estimated as:
(3.5.3a)
while the contribution to the potential vorticity of the variation of the layer
thicknesses is estimated as:
(3.5.3b)
and dominates the role of relative vorticity as long as L 2 » L~. This condition
is easily met, and thus in (3.5.1) and (3.5.2) the relative vorticity can be ignored.
Note that in making this approximation we are assuming that the parameter F,
defined by (3.2.7), is much greater than 1. The quasi-geostrophic theory, in
Vertical Structure: Baroclinic Quasi-Geostrophic Models
this process allows penetration of the wind-driven circulation below the uppermost layer. In Chapter 4 we discuss an alternative mechanism, ventilation, that
also produces deep motion. The synthesis of these two ideas forms the present
basis of our conceptual understanding of the stratified wind-driven circulation.
We employ a layer model for the circulation to begin with and ignore crossinterface motion. This means that in every layer the fluid of a given density
remains in that layer, and there are thus no sources of fluid entering the layer.
Except for the uppermost layer no layer receives fluid from the upper Ekman
layer. In each layer the fluid recirculates endlessly in the steady state with each
fluid element repeating its round and round trajectory in the gyre an infinite
number of times. This recirculatory aspect of the flow is a crucial feature of the
dynamics of the model. The model confronts directly the issue of how layers
which are not directly forced by Ekman pumping can be set into motion.
Consider the layer equations for either the two-layer [(3.3.1) and (3.3.2)] or
the 2! layer model in the steady state [(3.3.1) and (3.3.3)] in the absence of
cross-interface motion. We can write both steady-state models in a unified
compact form as:
(3.5.1)
and
J(t/12, q2) = J ( t/12, '\1 2 1/12 + py + F2(t/l, - t/12) - G2t/12 + ~~ Zb) (3.5.2)
= -r2 '\1 2 1/12 +curl
to zero (although the term in G2 is actually trivial since its Jacobian with t/1 2 is
identically zero; it is retained, however, for conceptual purposes). If the model
is considered to be a 2! layer model in which the two layers lie over a resting
abyss, the interaction with the bottom is absent, and the bottom friction
parameter r2 and the topographic contribution, Zb, should be set equal to zero
instead.
Over the scale of the gyre motion, L, the relative vorticity can be estimated as:
(3.5.3a)
while the contribution to the potential vorticity of the variation of the layer
thicknesses is estimated as:
(3.5.3b)
and dominates the role of relative vorticity as long as L 2 » L~. This condition
is easily met, and thus in (3.5.1) and (3.5.2) the relative vorticity can be ignored.
Note that in making this approximation we are assuming that the parameter F,
defined by (3.2.7), is much greater than 1. The quasi-geostrophic theory, in