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ABSTRACT
The general problem of an ocean on a rotating sphere is considered. The governing equations for an inviscid, incompressible fluid, written in spherical coordinates that are fixed at a point on the rotating Earth, together with the free surface and rigid bottom boundary conditions, are introduced. An exact solution of this system is presented; this describes a steady flow that is moving only in the azimuthal direction, with no variation in this direction. However, this azimuthal velocity component has an arbitrary variation with depth (i.e., radius), and so, for example, an Equatorial Undercurrent (EUC) can be accommodated. The pressure boundary condition at the free surface relates this pressure to the shape of the surface via a Bernoulli relation; this provides the constraint on the existence of a solution, although the restrictions are somewhat involved in spherical coordinates. To examine this constraint in more detail, the corresponding problems in model cylindrical coordinates (with the equator "straightened" to become a generator of the cylinder), and then in the tangent-plane version (with the β-plane approximation incorporated), are also written down. Both these possess similar exact solutions, with a Bernoulli condition that is more readily interpreted in terms of the choices available. Some simple examples of the surface pressure, and associated surface distortion, are presented. The relevance of these exact solutions to more complicated, and physically realistic, flow structures is briefly mentioned.
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1. Introduction
Exact solutions in fluid mechanics are quite rare and, typically, they describe extreme and ideal conditions that do not correspond with the complexities of observed physical behavior. For example, we have a number of exact solutions for viscous flow (e.g., plane Poiseuille flow, Stokes' unsteady problems, and many more; see Drazin and Riley 2007) and the classical inviscid nonlinear surface wave discovered by Gerstner (see Gerstner 1809; Constantin 2011). In these examples, and most others associated with exact solutions, it is possible to contrive a situation that comes close to the physical scenario relevant to the solution (see the discussion in Constantin 2012,2014; Henry 2013; Hsu 2014). However, this is usually far removed from the flows that we encounter in nature, for example, in the atmosphere, the oceans, rivers, pipes, and so on. Although very restricted in any...