Abstract
Starting from the Euler equation expressed in a rotating frame in spherical coordinates, coupled with the equation of mass conservation and the appropriate boundary conditions, a thin-layer (i.e. shallow water) asymptotic approximation is developed. The analysis is driven by a single, overarching assumption based on the smallness of one parameter: the ratio of the average depth of the oceans to the radius of the Earth. Consistent with this, the magnitude of the vertical velocity component through the layer is necessarily much smaller than the horizontal components along the layer. A choice of the size of this speed ratio is made, which corresponds, roughly, to the observational data for gyres; thus the problem is characterized by, and reduced to an analysis based on, a single small parameter. The nonlinear leading-order problem retains all the rotational contributions of the moving frame, describing motion in a thin spherical shell. There are many solutions of this system, corresponding to different vorticities, all described by a novel vorticity equation: this couples the vorticity generated by the spin of the Earth with the underlying vorticity due to the movement of the oceans. Some explicit solutions are obtained, which exhibit gyre-like flows of any size; indeed, the technique developed here allows for many different choices of the flow field and of any suitable free-surface profile. We comment briefly on the next order problem, which provides the structure through the layer. Some observations about the new vorticity equation are given, and a brief indication of how these results can be extended is offered.
Highlights
Many of the surface currents in our oceans link up to form gyres, which are large areas of water flowing in a roughly circular pattern and which tend to dominate the central regions of the open oceans; see figure 1 and the NASA web page http://oceanmotion.org/html/background/ wind-driven-surface.htm
The largest are the gyres in the Pacific Ocean and in the Atlantic Ocean with, in addition, the Indian Ocean gyre; there are a number of smaller gyres scattered across our oceans, e.g. the Beaufort Gyre in the Canada basin off the coast of Alaska
That gyres exist as a natural consequence of the general equations of motion, with clear and well-defined properties, is, we suggest, an important observation, even though a number of less critical physical characteristics may have to be suppressed in the development
Summary
Many of the surface currents in our oceans link up to form gyres, which are large areas of water flowing in a roughly circular pattern and which tend to dominate the central regions of the open oceans; see figure 1 and the NASA web page http://oceanmotion.org/html/background/ wind-driven-surface.htm. The circulation of a gyre is controlled by three factors: the global wind patterns—the most significant contributor, the rotation of the Earth, and the presence of landmasses and bottom topography They are, generally, very stable, with predictable properties and extent; floating debris often remains in a gyre for decades, and for that which reaches the centre, there is no escape. There is, considerable variation, from gyre to gyre, in the strength, width and effective depth of the surface currents that drive the gyre This general structure of a rotating flow, with little vertical motion, is the familiar pattern exhibited by a vortex, which is the most natural model to use as a basis for a mathematical discussion of these phenomena; e.g. The assumption of a thin layer of fluid, which is naturally coupled with weak vertical motion, is needed in order to make headway; this is the problem that we describe in detail here
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