Effects of Horizontal Geometrical Spreading on the Parameterization of Orographic Gravity Wave Drag. Part I: Numerical Transform Solutions

Abstract

umerical transform solutions for hydrostatic gravity waves generated by both uniform and sheared flow over elliptical obstacles are used to quantify effects of horizontal geometrical spreading on amplitude evolution with height. Both vertical displacement and steepness amplitudes are considered because of their close connections to drag parameterizations in weather and climate models. Novel diagnostics quantify the location and value of the largest wavefield amplitudes most likely to break at each altitude. These horizontal locations do not stray far from the obstacle peak even at high altitudes. Resulting vertical profiles of wave amplitude are normalized to remove density and refraction effects, thereby quantifying the horizontal geometrical spreading contribution, currently absent from parameterizations. Horizontal geometrical spreading produces monotonic amplitude decreases with height through wave-action conservation as waves propagate into progressively larger horizontal areas. Accumulated amplitude reductions are appreciable for all but the most quasi-two-dimensional obstacles with long axes orthogonal to the flow, and even these are impacted appreciably if the obstacle is rotated by more than 20°?30°. Profiles are insensitive to the obstacle?s functional form but vary strongly in response to changes in its horizontal aspect ratio. Responses to background winds are captured by a vertical coordinate transformation that remaps profiles to a universal form for a given obstacle. These results show that horizontal geometrical spreading has comparable or larger effects on wave amplitudes as the refraction of vertical wavenumbers and thus is important for accurate parameterizations of wave breaking and drag.