Haurwitz solutions of the nonlinear shallow-water equations for small froude number
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Summary
In this note we find special solutions of the nonlinear shallow-water equations. From the first integrals of the potential vorticity and energy equations for steady flow we derive a single equation in the streamfunction. In the limiting case of very small Froude number, that equation has no solutions corresponding to gravity waves. Under a nonlinear transformation of dependent variable, it becomes a linear equation whose solutions are related to Haurwitz's solutions for nondivergent barotropic flow in spherical surfaces. The distinguishing feature of these solutions is that the streamlines coincide with contours of the free surface: thus, although the height of the free surface varies, the motions of the fluid are horizontal, and the flow is nondivergent.
The solutions are easily modified to correspond to Rossby waves propagating eastward or westward without change of shape.
Keywords
Climate Change Waste Water Water Management Vorticity Water PollutionPreview
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