Under consideration for publication in J. Fluid Mech. 1 Resonance of long waves generated by storms obliquely crossing shelf topography in a rotating ocean S. THIEBAUT† AND R. VENNELL Ocean Physics Group, Department of Marine Science, University of Otago, Dunedin 9054, New Zealand (Received 17 December 2010) The oceanic forced wave beneath a moving atmospheric disturbance is amplified by Proudman resonance. When modified by the Earth’s rotation this classical resonance only occurs if the disturbance time scale is smaller than the inertial period. With or without Coriolis effects free transients generated by storm forced waves obliquely cross- ing step changes in water depth at particular angles are shown to resonate by exciting a range of barotropic free waves. Rotationally influenced slow atmospherically forced waves crossing a vertical coast at a critical angle lead to a form of sub-critical resonance, which occurs only when the component of the disturbances’ phase velocities along the coast matches that of a free Kelvin wave (KW). In a rotating ocean transients generated by disturbances crossing a step at a particular angle are shown to excite a free double Kelvin wave (DKW). This new type of resonance only occurs for sufficiently large steps and disturbances with time scale greater than the inertial period. A storm crossing a step shelf can result in the excitation of an infinite set of edge-waves, a single KW, a unique DKW and a first-mode continental shelf wave, depending on the topography and the disturbance time scale, translation speed and incident angle. The study of resonances and wave mode excitations generated by storms crossing a coast or a continental shelf may contribute to understanding how a particular combination of the storm character- istics can result in destructive coastal events with time scales encompassing the typical meteotsunami period band (tens of minutes) and storm surges with periods of several hours or days. 1. Introduction The forced wave generated by a moving atmospheric disturbance can be amplified by the well-known Proudman resonance when the translation speed of the disturbance ap- proaches the shallow water wave speed (i.e. at critical translation speed) (Lamb 1932; Proudman 1953). A disturbance travelling parallel to the coast over a linearly sloping bottom can also lead to a large forced ocean wave amplification due to Greenspan res- onance (Greenspan 1956). This is due to the excitation of one of the coastally trapped edge-wave modes, and is large at critical disturbance speeds (Greenspan 1956). Garrett (1970) showed how free waves can be generated when a fast super-critical atmospheric disturbance crosses a step. He noted the possibility of typically slow moving weather sys- tems crossing ridges and creating free waves, commenting that they would be too small to be detected, though Vennell (2007) (hereafter V07) showed that transient free waves of significant amplitudes can be generated by small fast-moving storms moving across shelves and ridges. Vennell (2010) (hereafter V10) showed that a sub-critical resonance † Email address for correspondence: thise135@student.otago.ac.nz Journal of Fluid Mechanics, 682, 261 -288 (2011) uncorrected proof