2 Homogeneous Models of the Ocean Circulation
2.1 Introduction
The Sverdrup balance, which was derived in Chapter 1 for a baroclinic flow
that does not interact with the bottom, is also valid for a homogenous
(constant density) ocean if the bottom is flat, and if the bottom friction is small
in the sense described in Section 1.4. Thus both a stratified and a homogeneous
model would have the same Sverdrup transport for the same wind stress. This
has encouraged the belief that the homogenous model, more tractable than a
general stratified model because its horizontal velocity is independent of depth,
might adequately represent at least the vertical average of the actual
circulation. Although this might be true in the interior of the ocean under
the conditions that the Sverdrup theory holds, it is less likely to be true in any
region, such as the western boundary layer, where nonlinear effects are
probably important. The vertical average of quadratic products representing
nonlinear processes in the equations of motion does not equal the product of
the vertical average of each of the terms. Hence even for the vertical average of
the circulation the homogeneous model is expected to be seriously inaccurate.
We are interested in the homogeneous model less because we expect
it to be able to simulate accurately any particular aspect of the acutal circulation than because it presents in a more tractable form some of the same
important dynamical issues as the general baroclinic problem. In particular, we
are interested in understanding the relationship between the interior circulation
and the dynamics of the western boundary current. Can the Sverdrup interior
really always be joined to a western boundary current? There are, as we shall
see in investigating this central question, some surprising links between the
detailed dynamics of the western boundary current and the overall circulation
pattern, even in the interior.
One of the most vexing problems in theoretical oceanography is the specification of a useful model for the dissipation of energy and vorticity, i.e., the
parameterization of small-scale turbulent mixing unresolved by our explicit,
large-scale dynamical equations. The hope has often been expressed that when
the circulation is vigorous enough that nonlinearity is dominant, the effect of
small-scale dissipation of unresolved motions is negligible for the dynamics, or
at least unimportant, qualitatively. Confidence in the inconsequential nature of
2.1 Introduction
The Sverdrup balance, which was derived in Chapter 1 for a baroclinic flow
that does not interact with the bottom, is also valid for a homogenous
(constant density) ocean if the bottom is flat, and if the bottom friction is small
in the sense described in Section 1.4. Thus both a stratified and a homogeneous
model would have the same Sverdrup transport for the same wind stress. This
has encouraged the belief that the homogenous model, more tractable than a
general stratified model because its horizontal velocity is independent of depth,
might adequately represent at least the vertical average of the actual
circulation. Although this might be true in the interior of the ocean under
the conditions that the Sverdrup theory holds, it is less likely to be true in any
region, such as the western boundary layer, where nonlinear effects are
probably important. The vertical average of quadratic products representing
nonlinear processes in the equations of motion does not equal the product of
the vertical average of each of the terms. Hence even for the vertical average of
the circulation the homogeneous model is expected to be seriously inaccurate.
We are interested in the homogeneous model less because we expect
it to be able to simulate accurately any particular aspect of the acutal circulation than because it presents in a more tractable form some of the same
important dynamical issues as the general baroclinic problem. In particular, we
are interested in understanding the relationship between the interior circulation
and the dynamics of the western boundary current. Can the Sverdrup interior
really always be joined to a western boundary current? There are, as we shall
see in investigating this central question, some surprising links between the
detailed dynamics of the western boundary current and the overall circulation
pattern, even in the interior.
One of the most vexing problems in theoretical oceanography is the specification of a useful model for the dissipation of energy and vorticity, i.e., the
parameterization of small-scale turbulent mixing unresolved by our explicit,
large-scale dynamical equations. The hope has often been expressed that when
the circulation is vigorous enough that nonlinearity is dominant, the effect of
small-scale dissipation of unresolved motions is negligible for the dynamics, or
at least unimportant, qualitatively. Confidence in the inconsequential nature of