Topographies lacking tidal conversion

The consensus is that in a stratified sea a classical model of tidal flow over irregular but smooth topography necessarily leads to the generation of internal tides, regardless of the shape of the topography. This is referred to as tidal conversion. Here it is shown, however, that there exists a large class of topographies for which there is neither tidal conversion nor any scattering of incident internal waves. This result is obtained in a uniformly stratified, rigid-lid sea using a barotropic tide that, owing to its large horizontal scale, is supposed to be simply a mass-conserving, periodic back-and-forth flow. The baroclinic response at the tidal frequency is, upon non-dimensionalizing and stretching of coordinates, determined by a standard hyperbolic boundary value problem (BVP). We here solve this hyperbolic BVP by mapping a domain of complicated, yet a priori unknown shape, onto a uniform-depth channel for which the same hyperbolic problem is known to display neither conversion of the barotropic tide nor scattering of internal wave modes. The map achieving this is required to satisfy hyperbolic Cauchy-Riemann equations, defined as analogues of the Cauchy-Riemann equations that are used in solving elliptic problems. Mapping the rigid-lid surface in the original Cartesian frame onto a rigid-lid surface in the transformed frame, this map is solved in terms of one arbitrary function. Each particular function defines a new topographic shape that can be computed a posteriori. The map is unique provided the Jacobian of transformation does not vanish, which is guaranteed for subcritical bottom topography, whose slope is everywhere less than that of the characteristics. For topographies that can thus be mapped onto a channel, tidal conversion and scattering are absent. Examples discussed include the (classical) wedge, a (near-Gaussian) ridge, a continental slope and (near) sinusoidal topographies.