Articles | Volume 24, issue 4
Nonlin. Processes Geophys., 24, 727–735, 2017
Nonlin. Processes Geophys., 24, 727–735, 2017

Research article 05 Dec 2017

Research article | 05 Dec 2017

Analytic solutions for Long's equation and its generalization

Mayer Humi Mayer Humi
  • Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609, USA

Abstract. Two-dimensional, steady-state, stratified, isothermal atmospheric flow over topography is governed by Long's equation. Numerical solutions of this equation were derived and used by several authors. In particular, these solutions were applied extensively to analyze the experimental observations of gravity waves. In the first part of this paper we derive an extension of this equation to non-isothermal flows. Then we devise a transformation that simplifies this equation. We show that this simplified equation admits solitonic-type solutions in addition to regular gravity waves. These new analytical solutions provide new insights into the propagation and amplitude of gravity waves over topography.

Short summary
Deriving a generalization of Long's equation to non-isothermal flow shows that Long's equation has (approximate) soliton-like solutions, provides a transformation that linearizes Long's equation (and analytic solutions), and provides analytic solutions for a base flow with shear.