One- and two-layer models, with dynamically active temperature in the
upper layer, are of interest in the construction and analysis of
simple coupled atmosphere-ocean models. In this talk, I shall discuss
the fundamentals of the dynamics of a simple one-layer model, with
dynamically-active temperature, interacting with topography H(x,y),
forced by a classical double-gyre wind stress and a thermal forcing
that leads to a cold Northern gyre and a warm Southern gyre. The work
is a natural continuation of Salmon and Ford (J. Mar. Res., 1995), but
here I relax the assumption, made in that paper, of a steady-state
depth-independent temperature T that is a single-valued function of
the streamfunction . This enables us to consider the dynamics on
a continental shelf that arise is such simple models when a cold
Northern gyre and a warm Southern gyre meet, in which internal thermal
boundary layers necessarily arise.
In the limit of a small temperature contrast dT between the Northern and Southern gyres, the entire flow can be solved as a regular perturbation problem in dT about the state of a homogeneous fluid (dT = 0). Howver, the perturbation structure is valid for only very small temperature contrasts dT between the gyres. As dT is increased the internal boundary layer becomes nonlinear, and the streamfunction and temperature must be solved simultaneously. For moderate values of dT the nonlinearity is weak in the sense that the location of the internal boundary layer is known, and hence the streamfunction can be determined from T via a linear equation.
The weakly non-linear internal boundary layer solution can be used to help construct a fully-nonlinear boundary-layer solution, where the location of the boundary layer must be determined as part of the solution, and in the interpretation of a fully-nonlinear solutions obtained without reduction to boundary-layer analysis.
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12th Conference on Atmospheric and Oceanic Fluid Dynamics