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.

Two-dimensional steady-state flow of an isothermal, incompressible stratified fluid over topography is modeled by Long's equation (Long, 1953, 1954, 1955, 1959). A generalization of this equation to three-dimensional flows has appeared in the literature (Akylas and Davis, 2001). However, in the following we restrict our discussion to two dimensions.

Numerical solutions of Long's equation for base flow without shear over simple terrain, which consists of one hill, were derived and analyzed in the literature by several authors (Drazin, 1961; Yih, 1967; Drazin and Moore, 1967; Lily and Klemp, 1979; Smith, 1980, 1989; Peltier and Clark, 1983; Durran, 1992; Smith and Kruse, 2017).

In these studies it was usual to approximate the Brunt–Väisälä frequency by a constant or a step function. In addition, two physical parameters which control the stratification and dispersive effects of the atmosphere were set to zero. Under these approximations, one of the leading second-order derivatives in Long's equation drops out. Moreover, the equation becomes linear (the nonlinear terms disappear). In this singular limit Long's equation reduces to that of a linear harmonic oscillator over the computational domain. The impact of these approximations on the validity of the solution was analyzed in depth in the literature (Smith, 1980, 1989; Peltier and Clark, 1983). These studies demonstrated that these approximations set limits on the physical applicability of these solutions.

Solutions of Long's equation were also used as a framework for the examination and study of experimental data on gravity waves (Shutts et al., 1988, 1994; Fritts and Alexander, 2003; Jumper et al., 2004; Vernin et al., 2007; Richter et al., 2010; Geller et al., 2013). In all of these studies it was assumed that the base flow is shearless. 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 (2004, 2007, 2009, 2010, 2015). It was demonstrated that Long's equation can be simplified for shearless base flow with mild assumptions about the nonlinear terms. In this framework we were able to identify the “slow variable” in Long's equation. This variable controls the emergence of nonlinear oscillations in this equation. In addition we proved the existence of self-similar solutions and derived a formula for the attenuation of the gravity wave amplitude with height. These results follow from the general properties of Long's equation and the nonlinear terms present in this equation.

We considered the effect that shear in the base flow has on the generation of gravity waves and their amplitude in Humi (2006). A new form of Long's equation in which the stream function is replaced by the atmospheric density was derived in Humi (2007). Finally a generalization of Long's equation to time-dependent flows appeared in Humi (2015).

It obvious however that atmospheric flows over topography are not isothermal
in general (see Miglietta and Rotunno, 2014; Richter et al., 2010; Smith and
Kruse, 2017, and their bibliography). With this motivation we derive, in the
first part of this paper, an extension of this equation to include
non-isothermal flows with free convection.

The

The plan of the paper is as follows: in Sects. 2.1 and 2.3 we present an overview of the derivation of the isothermal Long equation and the approximations that are made for its numerical solutions. In Sect. 2.2 we derive the corresponding Long equation for flows with free convection. In Sect. 3 we introduce a transformation which (essentially) linearizes the equation for the perturbation from the base flow. Section 4 discusses the application of this transformation to a flow with shear and presents an analytic solution for this flow. We end with some conclusions in Sect. 5.

In the first part of this section we provide a short overview of the (classical) isothermal Long equation and in the second part we generalize this equation to include free convection.

In two dimensions

In these equations, subscripts denote differentiation with respect to the
subscripted variable,

To non-dimensionalize Eqs. (

Using these new variables, Eqs. (

Equation (

Using this definition of

Using Eq. (

To eliminate

Using Eq. (

As a result we obtain the following equation for

In Eq. (

For example, if we consider a shearless base flow with

It follows from this example that different base
flows at

We consider now a perturbation

When the flow is not isothermal, Eq. (

We can non-dimensionalize these equations using Eq. (

Using Eq. (

The function

For a perturbation

We consider here numerical solutions of Long's equation over an unbounded
domain with a general base flow. The topography of the domain is represented
by a function

To determine the proper boundary condition on

When

When

To compute numerical solutions for the perturbation

In these limits Eq. (

Equation (

The general solution of Eq. (

To satisfy the boundary condition on the terrain, one has to solve the
following integral equation (Drazin, 1961; Lily and Klemp, 1979; Durran,
1992):

To begin with we observe that in Eqs. (

Equation (

Remark: the mathematical “inspiration” for this transformation comes from
somewhat similar transformations which linearize the Ricatti and Burger
equations. From a physical point of view the motivation comes from the desire
to replace the nonlinearities due to the derivatives of

Equations (

Equation (

If

Similar treatment can be applied to Eq. (

To obtain a real solution for

If we substitute

Similarly to Eq. (

The general solution of Eq. (

The radiation boundary condition at

Since the integration in Eq. (

To satisfy boundary condition (

The special case

To examine the application of the formulas derived above, we consider the
flow over a “witch of Agnesi” hill where the height of the topography is
given by

Contour plot of

We consider here a base flow with

Contour plot of

For

Computing numerical solutions for Long's equation has always been a challenge, even in some (singular) limiting cases. In this paper we introduced a transformation of this equation which under mathematically acceptable approximations leads to analytic expressions for the solutions. In particular, these solutions capture the dependence of the wave amplitude on the height.

The paper also provides an extension of Long's equation to the case where the atmospheric flow is not isothermal. This new equation can be solved analytically by the same transformation that is used for Long's equation.

The MATLAB programs used to generate the plots in this paper are available from the author upon request.

The author declares that he has no conflict of interest.

The author would like to thank the editor and referees for their comments during the review process. Edited by: Harindra Joseph Fernando Reviewed by: two anonymous referees