Long's equation describes steady-state two-dimensional stratified flow over terrain. Its numerical solutions under various approximations were investigated by many authors. Special attention was paid to the properties of the gravity waves that are predicted to be generated as a result. In this paper we derive a time-dependent generalization of this equation and investigate analytically its solutions under some simplifications. These results might be useful in the experimental analysis of gravity waves over topography and their impact on atmospheric modeling.

Long's equation (Long, 1953, 1954, 1955, 1959) models the flow of inviscid
stratified fluid in two dimensions over terrain. When the base
state of the flow (that is, the unperturbed flow field far upstream) is without
shear the solutions of this equation are in the form of steady lee waves.
Solutions of this equation in various settings and approximations were
studied by many authors (Drazin, 1961; Drazin and Moore, 1967; Durran, 1992;
Lily and Klemp, 1979; Peltier and Clark, 1983; Smith, 1980, 1989; Yih, 1967).
The most common approximation in these studies was to set the
Brunt–Väisälä frequency to a constant or a
step function over the computational domain. Moreover, the values of the
parameters

Long's equation provides also the theoretical framework for the analysis of experimental data (Fritts and Alexander, 2003; Shutts et al., 1988; Vernin et al., 2007; Jumper et al., 2004) under the assumption of shearless base flow. (An assumption which, in general, is not supported by the data.) An extensive list of references appears in Fritts and Alexander (2003), Baines (1995), Nappo (2012) and Yhi (1980).

An analytic approach to the study of this equation and its solutions was initiated recently by the current author (Humi, 2004). We showed that for a base flow without shear and under rather mild restrictions the nonlinear terms in the equation can be simplified. We also identified the “slow variable” that controls the nonlinear oscillations in this equation and using phase averaging approximation derived a formula for the attenuation of the stream function perturbation with height. This result is generically related to the presence of the nonlinear terms in Long's equation. We explored also different formulations of this equation (Humi, 2007, 2009) and the effect of shear on the solutions of this equation (Humi 2006, 2010).

One of the major obstacles to the application of Long's equation in realistic applications is due to the fact that it is restricted to the description of steady states of the flow. It is therefore our objective in this paper to derive a time-dependent generalization of this equation and study the properties of its solutions. The resulting system contains two equations for the time evolution of the density and the stream function. While the equation for the stream function is rather complicated it can be simplified in two instances. The first corresponds to the classical (steady state) Long's equation while the second is time dependent and new (as far as we know). In this paper we explore the properties of the flow in this second case, which might find some applications in the analysis of experimental data about gravity waves (Vernin et al., 2007; Jumper et al., 2004; Nappo, 2012), and its application to atmospheric modeling (Richter et al., 2010; Geller et al., 2013).

The plan of the paper is as follows: in Sect. 2 we derive the time-dependent Long's equation. In Sect. 3 we consider the time evolution and proper boundary conditions on shearless flow over topography. We end with the summary and conclusion in Sect. 4.

In the paper we consider the flow in two dimensions (

One possible interpretation of Eq. (

We can nondimensionalize these equations by introducing

In these new variables Eqs. (

In view of Eq. (

Using this stream function we can rewrite Eq. (

Using

We can suppress

Thus, after all these transformations the system of equations governing
the flow is Eqs. (

To eliminate

To reduce the first and third terms in Eq. (

While Eq. (

However, if

Replacing

To summarize, the equations of the flow in this case are

The reduction of Eq. (

In this section we shall consider the time evolution of a stratified
shearless base flow, viz. a flow which satisfies as

In these limits Eq. (

We now consider perturbations from the (shearless) base flow
described by Eq. (

To find the general form of the solution of these equations we use
Eq. (

Solving Eq. (

A cross section of the perturbation in

The corresponding solution for

Hence, the general solution for

We consider a flow in an unbounded domain over topography with
shape

At the topography we impose the following boundary condition on

A cross section of the perturbation in

To derive the corresponding boundary condition for

As to the boundary condition on

For low lying topography (viz

Steady-state solutions of Long's equation model the vertical structure of plane parallel gravity waves. These solutions are useful, for example, in the parameterizations of unresolved gravity wave drag where the WKBJ (Wentzel–Kramers–Brillouin–Jeffreys) approximation is invoked to describe the time-dependent amplitude spectrum of a packet of gravity waves propagating in a slowly varying background.

The present paper presents an alternative analytical approach to solve (under several restrictions) this and similar time-dependent problems without having to invoke the WKBJ approximation. The analytical insights derived from this approach might be used to complement and verify the numerical results obtained from the WKBJ method.

The author is indebted to the the anonymous referees whose comments helped improve the quality of this paper. Edited by: V. I. Vlasenko Reviewed by: Two anonymous referees