| description abstract | This paper presents an exact solution for nonlinear shallow-water waves on an f plane. It is a long wave satisfying the hydrostatic balance. It is also a solitary wave maintained by the balance of a dispersion effect of Coriolis force and the nonlinear effect of advection and finite wave amplitude. It satisfies the conservation of potential vorticity. It has a negative wave height, a negative (positive) relative vorticity in the Northern (Southern) Hemisphere, and a propagation speed smaller than that of the shallow-water wave. It propagates upstream, as a consequence of the conservation of potential vorticity, with a speed of (1 ? |a|/h)[gh(1 ? |a|/h)]1/2, where a is the wave height (a < 0). The trough of the surface elevation is a singular point where the first-order derivative approaches infinity. However, the region of this singularity is very small for such a wave on earth. For instance, with a 1-m amplitude, the horizontal length scale of the wave is several hundred kilometers but the large derivative region is only a fraction of a centimeter. In reality, if this wave exists, friction and surface tension would erase the sharp surface gradient on such a small scale. The wave profile and its horizontal length scale do not depend on water depth at the first-order approximation. The length scale is proportional to the square root of the product of amplitude and gravity and is inversely proportional to the Coriolis parameter, which can be expressed as L ? 10(g|a|)1/2/f. | |
