Long's equation in terrain following coordinates

Abstract

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.

The effect of shear on the generation of gravity waves

“…This equation has to be solved numerically (Davis, 1999;Drazin, 1961Drazin, , 1969 subject to the boundary conditions mentioned above. However recently (Humi, 2009) we showed how this problem can be solved analytically using a "terrain following formulation" of Long's equation. It is clear from the form of the general solution given by Eq.…”

“…Applying the boundary conditions atz = 0 yields A(x) = −h(x) − δ N 2 . Using the radiation boundary conditions implies that B(x) = H (A(x)) (see Humi, 2009 for a detailed discussion of this derivation).…”

“…This result is generically related to the presence of the nonlinear terms in Long's equation. New representations of this equation in terms of the atmospheric density and terrain following coordinates were derived in (Humi, 2007(Humi, , 2009). Partial results on the impact that shear can have on the generation and amplitude of gravity waves when the base flow consists of "pure shear" were investigated by us in (Humi, 2006).…”

Interference of lee waves over mountain ranges

“…Quite recently, Humi (2009) derived an approximate analytical model which incorporates the shape of a complex obstacle into the coefficients of the model equation. This formulation of the Long's model uses the transformation of independent variables to the terrain coordinates.…”

Analytic solutions for Long's equation and its generalization

“…However, this assumption is incorrect, in general, and is not justified by the experimental data. (For a comprehensive list of references, see Yih, 1980, Baines, 1995, and Nappo, 2012 A new method to derive analytic solutions of Long's equation was initiated by the present author in Humi (2004Humi ( , 2007Humi ( , 2009Humi ( , 2010Humi ( , 2015. It was demonstrated that Long's equation can be simplified for shearless base flow with mild assumptions about the nonlinear terms.…”