Dynamics of a two-dimensional ribbon of shallow water on an f-plane

An analytical solution is derived for the motion of an infinitely long fluid ribbon of shallow water, with parabolic depth and linear velocity distribution. The solution is of intrinsic dynamical interest and also offers a “bench-mark” for testing numerical methods which handle the intersection of a free surface with a horizontal boundary. The development is expressed in terms of the translation and deformation of the hump. The former component is determined uniquely by the initial velocity at the symmetry axis and the externally imposed uniform pressure gradient. On the other hand, the deformation component depends on the initial shear and divergence, but is unaffected by a uniform external pressure force. Relative to its axis, the fluid ribbon alternately expands and contracts, with a pulsation frequency (≥ the inertial frequency f) that is a function of a Rossby, and a rotational Froude number, defined in terms of the initial velocity, and height field.