Long's equation in terrain following coordinates

Abstract. Long's equation describes two dimensional stratified atmospheric flow over terrain which is represented by the geometry of the domain. The solutions of this equation over simple topography were investigated analytically and numerically by many authors. In this paper we derive a new terrain following formulation of this equation which incorporates the terrain as part of the differential equation rather than the geometry of the domain. This new formulation enables us to compute analytically steady state gravity wave patterns over complex topography in some limiting cases of the parameters that appear in this equation.