Effects of Horizontal Geometrical Spreading on the Parameterization of Orographic Gravity Wave Drag. Part II: Analytical Solutions

Effects of horizontal geometrical spreading on the amplitude variation with height of linear three-dimensional hydrostatic orographic gravity waves (OGWs) are quantified via relevant simplifications to the governing transform relations, leading to analytical solutions. The analysis is restricted to elliptical Gaussian obstacles with principal axes aligned parallel and perpendicular to unidirectional shear flow and to vertical displacement and steepness amplitudes, given their relevance to OGW drag parameterizations in global models. Two solutions are derived: a "small l" solution in which horizontal wavenumbers l orthogonal to the flow are taken to be much smaller than those parallel to the flow, and a "single k" solution in which horizontal wavenumbers k parallel to the flow have a single mean value. The resulting analytical relations, valid for arbitrary vertical profiles of upstream winds and stability, depend only on the obstacle's elliptical aspect ratio beta and a normalized height coordinate incorporating wind and stability variations. These analytical approximations accurately reproduce the salient features of the exact numerical transform solutions. These include monotonic decreases with height that asymptotically approach z(-1/2) forms at large z and strong beta dependence in amplitude diminution with height. Steepness singularities close to the surface are shown to be a mathematical consequence of the Hilbert transform approach to deriving complex wavefield solutions. These approximate analytical solutions for horizontal geometrical spreading effects on wave amplitude highlight the importance of this missing physics for current parameterizations of OGW drag and offer an accurate and efficient means of incorporating some of these omitted effects.