1 Introduction

Gravity waves in the atmosphere (often called buoyancy waves in this context) play an important rȏle in many of the dynamical processes that we observe and hope to understand. Although these waves are generated in the lower-to-middle regions of the troposphere, often by flow over mountains, their effects extend into the stratosphere and beyond; see [1, 2]. Furthermore, they are generally accepted as being significant in the momentum and energy exchanges that contribute to global circulation patterns in the middle and upper atmosphere. This motion, in turn, modifies the thermal structure of the atmosphere; see [3, 4]. Thus these waves cannot be ignored in any reasonable attempt to model the climate–currently of considerable interest–although the standard approach is to treat such contributions via suitable parameterisations of their effects (see [5]). For the interested reader, the general properties of gravity waves, and their various rȏles, are described and discussed in any good text; see, for example, [6,7,8]. Notwithstanding the considerable relevance and importance of gravity waves, our presentation here has a rather more limited objective. We will be driven, in the main, by the application of classical (applied) mathematical methods of fluid dynamics to this phenomenon, aiming to put the theory of gravity waves on a mathematically firm foundation. The familiar and standard approach to this (and many other problems in the oceanic and atmospheric sciences) is to write down a suitable system of model equations, usually based on appropriate physical principles. Typically, however, this development is combined with the use of ad hoc assumptions, perhaps, for example, ignoring the effects of rotation and invoking the Boussinesq approximation. There might also be simplifying assumptions related to the thermodynamic properties of the background state of the atmosphere. This is not the approach of the mathematical fluid dynamicist, who follows an alternative route; we explain the underlying philosophy.

Thus we start with the complete set of governing equations for a viscous, compressible fluid, coupled to a suitable version of the first law of thermodynamics and an equation of state; it is this set which constitutes a model in our approach. The technique that we adopt–the standard one in this type of study–is to find a suitable non-dimensionalisation and scaling that captures the essence of the flow configuration, but imposes minimal assumptions, aiming to retain as many of the physical attributes as possible, but always following a precise and well-defined procedure. The nett result is to produce a (reduced) system of equations which are often amenable to analysis and are based on specific approximations. This system necessarily possesses a strong pedigree and a mathematically robust structure. Furthermore, the errors in the formulation are precisely defined and so, if higher-order terms are required, they are readily accessible. But this leaves us with the all-important question: what should be the basis for the approximation?

It has been demonstrated, for both the oceans ([9, 10]) and the atmosphere ([11, 12]) (and combined in [13]), that a single over-arching assumption is sufficient to produce a system of equations which retain all the essential physics and which can be usefully analysed (and often solved). This simply requires the development of equations based on the thin-shell approximation that represents the atmosphere (or ocean) as a thin layer on the surface of a nearly-spherical Earth. It is then straightforward to define a set of (independent) parameters which, each defined to be O(1) in terms of the underlying limiting process (represented by \(\varepsilon \to 0\): the thin shell), encompass all the physical attributes of the flow. In particular, this enables us to retain the effects of the Earth’s rotation as well as the consequences of using spherical geometry (which is usually only slightly simplified by invoking the thin-shell approximation, but importantly so). The intention here, therefore, is to adapt this approach to provide a discussion of gravity waves in the atmosphere, although this does involve a new and additional complication. Independently of the background-flow properties and flow configuration, we have initial (and/or boundary) data which give rise to the gravity wave; this necessarily introduces an extra parameter (α, measuring the amplitude of this wave, for example). The procedure then involves the perturbation of the background state, using the parameter α, followed by applying the thin-shell approximation (\(\varepsilon \to 0\)) to each term arising at O(\(\alpha^{n}\)), n = 0, 1, 2…, although we will construct the details only for the O(1) and O(α) terms. The result is to produce the thin-shell approximation, retaining all the essential physics of the flow, of both the background state of the atmosphere and of the gravity wave. It is clear that this is not the usual route followed in the standard work on gravity waves in the atmosphere, and so we must expect some important differences to arise. Nevertheless, it behoves us to relate to these familiar ideas in every way that we can. The plan for this paper is therefore quite clear.

Firstly, we provide the full set of governing equations for a compressible gas, with variable viscosity, written in rotating, spherical coordinates. However, for simplicity, we do not include the correction for the shape of the Earth’s geoid. As shown in [11], this simply provides a small correction to the background state and is uncoupled from the O(ε) terms. So with our intention of retaining only the leading-order description (in terms of the thin shell) of both the background state and its wave perturbation, we omit this contribution here. (It is certainly tiresome to include the geoid correction, and it makes for a cumbersome presentation, but it is readily accessible if required.) This system of governing equations is then described on the basis of an asymptotic expansion, retaining O(1) and O(α) terms only, each for arbitrary ε. The next stage involves the non-dimensionalisation of the background state–the O(1) terms–and then, separately, the O(α) problem is non-dimensionalised; this second procedure differs from the first, not least because the former is steady and the latter is unsteady. The two systems of equations, at O(1) and O(α), are then approximated by invoking the thin-shell property (\(\varepsilon \to 0\)), providing a coherent and consistent development for the background state and the gravity wave. Finally we analyse the reduced problem that we have derived and relate the results to existing theories; we find that our systematic use of non-dimensionalisation and scaling produces some differences when compared to the standard description.

2 Governing equations

We introduce a set of (right-handed) spherical coordinates \((\phi ,\theta ,r^{\prime})\): \(r^{\prime}\) is the distance (radius) from the centre of the sphere, θ (with \(- {\pi \mathord{\left/ {\vphantom {\pi 2}} \right. \kern-\nulldelimiterspace} 2} \le \theta \le {\pi \mathord{\left/ {\vphantom {\pi 2}} \right. \kern-\nulldelimiterspace} 2}\)) is the polar angle and ϕ (with \(0 \le \phi < 2\pi\)) is the azimuthal angle, i.e. the angle of longitude. The unit vectors in this \((\phi ,\theta ,r^{\prime})\)-system are \(({\mathbf{e}}_{\phi } ,{\mathbf{e}}_{\theta } ,{\mathbf{e}}_{r} )\), respectively, and the corresponding velocity components are \((u^{\prime},v^{\prime},w^{\prime})\;( \equiv {\mathbf{u^{\prime}}})\); \({\mathbf{e}}_{\phi }\) points from West to East, and \({\mathbf{e}}_{\theta }\) from South to North (see Fig. 1). (We use primes, throughout the formulation of the problem, to denote physical (dimensional) variables; these will be removed when we non-dimensionalise.) The \((\phi ,\theta ,r^{\prime})\)-system defines points on the stationary sphere, other than at the two poles where the unit vectors are not well-defined. In this coordinate system, but now written in the rotating frame, we have the Navier–Stokes equation for a compressible fluid and with a dynamic eddy viscosity which depends on only \(r^{\prime }\):

$$ \begin{aligned}& \rho^{\prime } \frac{{Du^{\prime } }}{{Dt^{\prime } }} + \frac{{\rho^{\prime } }}{{r^{\prime } }}\left( { - u^{\prime } v^{\prime } \,\tan \theta + u^{\prime } w^{\prime } ,\,u^{{\prime }{2}} \tan \theta + v^{\prime } w^{\prime } , - u^{{\prime }{2}} - v^{{\prime }{2}} } \right) \\ &\quad + 2\Omega^{\prime } \rho^{\prime } \left( { - v^{\prime } \sin \theta + w^{\prime } \cos \theta ,u^{\prime } \sin \theta , - u^{\prime } \cos \theta } \right) + r^{\prime } \rho^{\prime } \Omega^{{\prime }{2}} \left( {0,\sin \theta \cos \theta , - \cos^{2} \theta } \right) \\ &\quad = - \nabla^{\prime } p^{\prime } + \rho^{\prime } \left(0,0 - g^{\prime } \frac{{R^{{\prime }{2}} }}{{r^{{\prime }{2}}}}\right) + \mu^{\prime } \nabla^{{\prime }{2}}u^{\prime } + \frac{{d\mu^{\prime } }}{{dr^{\prime } }}r^{\prime } \frac{\partial }{{\partial r^{\prime } }}\left( {\frac{{u^{\prime } }}{{r^{\prime } }}} \right). \\ \end{aligned} $$
(1)

where \(p^{\prime}(\phi ,\theta ,r^{\prime},t^{\prime})\) is the pressure in the fluid, \(\rho^{\prime}(\phi ,\theta ,r^{\prime},t^{\prime})\) the density and \(\mu^{\prime}(r^{\prime})\) the dynamic eddy viscosity; \(t^{\prime}\) is time. We have introduced \(\Omega^{\prime} \approx 7 \cdot 29 \times 10^{ - 5} \;{\text{rad}}\;{\text{s}}^{ - 1}\), the constant rate of rotation of the Earth, and \(g^{\prime} \approx 9 \cdot 81\;{\text{m}}\;{\text{s}}^{ - 2}\) the acceleration of gravity at the surface of the Earth (with \(R^{\prime}\) an average radius of the Earth). The additional notation used here is familiar:

$$ \frac{{{\text{D}}\;\;}}{{{\text{D}}t^{\prime}}} \equiv \left( {\frac{\partial \,}{{\partial t^{\prime}}} + \frac{{u^{\prime}}}{{r^{\prime}\cos \theta }}\frac{\partial \;}{{\partial \phi }} + \frac{{v^{\prime}}}{{r^{\prime}}}\frac{\partial \;}{{\partial \theta }} + w^{\prime}\frac{\partial \;}{{\partial r^{\prime}}}} \right),\,\nabla^{\prime} \equiv \left( {\frac{1}{{r^{\prime}\cos \theta }}\frac{\partial \;}{{\partial \phi }},\frac{1}{{r^{\prime}}}\frac{\partial \;}{{\partial \theta }},\frac{\partial \;}{{\partial r^{\prime}}}} \right) $$
Fig. 1
figure 1

The spherical coordinate system, where θ is the polar angle measured from the Equator, ϕ the azimuthal angle and \(r^{\prime}\) the distance from the origin

Full size image

and

$$ \nabla^{{\prime}{2}} \equiv \frac{{\partial^{2} \,}}{{\partial r^{{\prime}{2}} }} + \frac{2}{{r^{\prime}}}\frac{\partial \,}{{\partial r^{\prime}}} + \frac{1}{{r^{{\prime}{2}} }}\left( {\frac{1}{{\cos^{2} \theta }}\frac{{\partial^{2} \,}}{{\partial \phi^{2} }} + \frac{{\partial^{2} \,}}{{\partial \theta^{2} }} - \tan \theta \frac{\partial \,}{{\partial \theta }}} \right). $$

In Eq. 1 we have retained, for ease of presentation, only those viscous terms that contribute to the dominant behaviours (as \(\varepsilon \to 0\)) in the background state and in the description of the wave component, as will become clear later. (The complete prescription for the viscous terms can be found in [11].) We also require the equation of mass conservation

$$ \frac{{{\text{D}}\rho^{\prime}}}{{{\text{D}}t^{\prime}}} + \rho^{\prime}\left[ {\frac{1}{{r^{\prime}\cos \theta }}\left( {\frac{{\partial u^{\prime}}}{\partial \phi } + \frac{\partial \,}{{\partial \theta }}\left( {v^{\prime}\cos \theta } \right)} \right) + \frac{1}{{r^{{\prime}{2}} }}\frac{\partial \,}{{\partial r^{\prime}}}\left( {r^{{\prime}{2}} w^{\prime}} \right)} \right] = 0, $$
(2)

together with the two equations describing the thermodynamics of the atmosphere:

$$ p^{\prime} = \rho^{\prime}\Re^{\prime}T^{\prime}, $$
(3)

the equation of state, where \(T^{\prime}\) is the (absolute) temperature and \(\Re^{\prime} \approx 287\;{\text{m}}^{{2}} {\text{s}}^{ - 2} {\text{K}}^{ - 1}\) is the gas constant, and the first law in the form

$$ c^{\prime}_{p} \frac{{{\text{D}}T^{\prime}}}{{{\text{D}}t^{\prime}}} - \kappa^{\prime}{\kern 1pt} \nabla^{{\prime}{2}} T^{\prime} - \frac{1}{{\rho^{\prime}}}\frac{{{\text{D}}p^{\prime}}}{{{\text{D}}t^{\prime}}} = Q^{\prime}({\mathbf{x^{\prime}}},t^{\prime}), $$
(4)

where \(c^{\prime}_{p} \approx 10^{3} \;{\text{m}}^{2} {\text{s}}^{ - 2} {\text{K}}^{ - 1}\) is the constant specific heat of predominantly dry air at constant pressure and \({{\kappa^{\prime}} \mathord{\left/ {\vphantom {{\kappa^{\prime}} {c^{\prime}_{p} }}} \right. \kern-\nulldelimiterspace} {c^{\prime}_{p} }} \approx 2 \times 10^{ - 5} \;{\text{m}}^{2} {\text{s}}^{ - 1}\) is the constant thermal diffusivity; \(Q^{\prime}\) is a general heat-source term.

This system of Eqs.  1,  2,  3 and  4, constitutes the model which is at the heart of our analysis. The procedure hereafter uses non-dimensionalisation and scaling techniques, within the thin-shell approximation, aiming to minimise the assumptions and thereby retain all the physical properties of the flow. Furthermore, without prescribing or assuming them beforehand, we will be able to identify all the ingredients of the flow that appear at the same order (and so cannot be arbitrarily set aside). The development therefore requires us, first, to determine the background state of the atmosphere; we then superimpose on this a suitable wave perturbation.

3 Background state

Although a long-term aim of this work is to derive a reliable theory for waves that propagate in an arbitrary direction, in an atmosphere with a corresponding general background flow (wind), we limit the discussion in this initial investigation. (A study of nonlinear wave propagation in an arbitrary direction within the atmosphere, using the thin-shell approximation, can be found in [14], which shows the complexities that are involved.) So here we consider a background atmosphere which incorporates a steady flow in only the zonal direction (so \({\mathbf{u^{\prime}}} \equiv (u^{\prime},0,0)\)); the waves that we introduce later will also propagate with a horizontal component in only the zonal direction (all of which corresponds to the typical geometry used in most conventional descriptions of gravity waves). With these observations in mind, our background state is described by the system of equations derived from (1,  2,  3, 4, 5 and 6):

$$ \begin{aligned} &\rho^{\prime}\frac{{u^{\prime}}}{{r^{\prime}\cos \theta }}\frac{\partial \;}{{\partial \phi }}(u^{\prime},0,0) + \frac{{\rho^{\prime}}}{{r^{\prime}}}(0,u^{{\prime}{2}} \tan \theta , - u^{{\prime}{2}} ) \\ &\qquad + 2\Omega^{\prime}\rho^{\prime}(0,u^{\prime}\sin \theta , - u^{\prime}\cos \theta ) + r^{\prime}\rho^{\prime}\Omega^{{\prime}{2}} (0,\sin \theta \cos \theta , - \cos^{2} \theta ) \\ &\quad = - \nabla^{\prime}p^{\prime} + \rho^{\prime}\left( {0,0, - g^{\prime}\frac{{R^{{\prime}{2}} }}{{r^{{\prime}{2}} }}} \right) + \mu^{\prime}\nabla^{{\prime}{2}} (u^{\prime},0,0) + \frac{{{\text{d}}\mu^{\prime}}}{{{\text{d}}r^{\prime}}}r^{\prime}\frac{\partial \;}{{\partial r^{\prime}}}\left( {\frac{{(u^{\prime},0,0)}}{{r^{\prime}}}} \right) \\ \end{aligned} $$
(5)

with

$$ \frac{{u^{\prime}}}{{r^{\prime}\cos \theta }}\frac{{\partial \rho ^{\prime}}}{{\partial \phi }} + \frac{{\rho ^{\prime}}}{{r^{\prime}\cos \theta }}\frac{{\partial u^{\prime}}}{{\partial \phi }} = 0 \,\,\,,{\text{i}}.{\text{e}}.,\frac{{\partial \;}}{{\partial \phi }}(u^{\prime}\rho ^{\prime}) = 0, $$
(6)

and

$$ p^{\prime} = \rho^{\prime}\Re^{\prime}T^{\prime}, $$
(7)
$$ c^{\prime}_{p} \frac{{u^{\prime}}}{r^{\prime}\cos \theta }\frac{{\partial T^{\prime}}}{\partial \phi } - \kappa^{\prime}{\kern 1pt} \nabla^{{\prime}{2}} T^{\prime} - \frac{1}{{\rho^{\prime}}}\frac{{u^{\prime}}}{r^{\prime}\cos \theta }\frac{{\partial p^{\prime}}}{\partial \phi } = Q^{\prime}({\mathbf{x^{\prime}}},t^{\prime}). $$
(8)

We now introduce a non-dimensionalisation defined by

$$ \left. \begin{gathered} \quad \quad r^{\prime} = (1 + \varepsilon z)R^{\prime},\;u^{\prime} = \overline{U^{\prime}}U(\phi ,\theta ,z;\varepsilon ) \hfill \\ \rho^{\prime} = \overline{\rho }^{\prime}{\kern 1pt} \Delta (\phi ,\theta ,z;\varepsilon ),\;\mu^{\prime} = \overline{\mu }^{\prime}m(z),\;p^{\prime} = P^{\prime}\Pi (\phi ,\theta ,z;\varepsilon ) \hfill \\ \end{gathered} \right\}, $$
(9)

where \(\varepsilon = {{H^{\prime}} \mathord{\left/ {\vphantom {{H^{\prime}} {R^{\prime}}}} \right. \kern-\nulldelimiterspace} {R^{\prime}}}\), and \(H^{\prime}\) is the maximum thickness of the troposphere (no more than about 16 km), \(\overline{U^{\prime}}\) is a suitable speed scale that represents both the moving atmosphere (the wind) and the speed of propagation, and we choose the pressure scale \(P^{\prime} = \overline{\rho }^{\prime}(R^{\prime}\Omega^{\prime})^{2}\) (\(\approx 1\;{\text{atm}}\), the natural choice for the background state of the atmosphere). We have also used an average density of the atmosphere, \(\overline{\rho }^{\prime}\), and an average coefficient of dynamic viscosity, \(\overline{\mu }^{\prime}\); the temperature is written as \(T^{\prime} = ({{P^{\prime}} \mathord{\left/ {\vphantom {{P^{\prime}} {\overline{\rho }^{\prime}\Re^{\prime}}}} \right. \kern-\nulldelimiterspace} {\overline{\rho }^{\prime}\Re^{\prime}}}){\rm T} = {{(R^{{\prime}{2}} \Omega^{{\prime}{2}} } \mathord{\left/ {\vphantom {{(R^{{\prime}{2}} \Omega^{{\prime}{2}} } {\Re^{\prime}}}} \right. \kern-\nulldelimiterspace} {\Re^{\prime}}}){\rm T}\). This normalisation factor, \({{R^{{\prime}{2}} \Omega^{{\prime}{2}} } \mathord{\left/ {\vphantom {{R^{{\prime}{2}} \Omega^{{\prime}{2}} } {\Re^{\prime}}}} \right. \kern-\nulldelimiterspace} {\Re^{\prime}}}\), is approximately 800° K, so that the temperature in the troposphere ranges from about \({\rm T} = 0 \cdot 36\) down to \({\rm T} = 0 \cdot 27\). We see that ε is our fundamental thin-shell parameter and so \(\varepsilon \to 0\) will provide the basis for the over-arching approximation that we impose. It then follows that (5, 6, 7 and 8) can be written as

$$ \frac{1}{{\omega^{2} }}\Delta U\frac{\partial U}{{\partial \phi }} = - \frac{\partial \Pi }{{\partial \phi }} + \frac{(1 + \varepsilon z)\cos \theta }{{\hat{R}_{e} }}\left\{ {m\nabla^{2} U + (1 + \varepsilon z)\frac{{{\text{d}}m\;}}{{{\text{d}}z}}\frac{\partial \;}{{\partial z}}\left( {\frac{U}{1 + \varepsilon z}} \right) + {\text{O}}(\varepsilon^{2} )} \right\}; $$
(10)
$$ \frac{\partial \Pi }{{\partial \theta }} = - \Delta \tan \theta \left\{ {(1 + \varepsilon z)\cos \theta + \frac{U}{\omega }} \right\}^{2} ; $$
(11)
$$ \frac{\partial \Pi }{{\partial z}} = \frac{{\varepsilon {\kern 1pt} \Delta }}{1 + \varepsilon z}\left\{ {(1 + \varepsilon z)\cos \theta + \frac{U}{\omega }} \right\}^{2} - g\frac{\Delta }{{(1 + \varepsilon z)^{2} }}; $$
(12)
$$ \frac{\partial \;}{{\partial \phi }}(\Delta U) = 0; $$
(13)
$$ \Pi = \Delta {\rm T}; $$
(14)
$$ \frac{U}{(1 + \varepsilon z)\cos \theta }\left\{ {c_{p} \frac{{\partial {\rm T}}}{\partial \phi } - \frac{1}{\Delta }\frac{\partial \Pi }{{\partial \phi }}} \right\} - \kappa \nabla^{2} {\rm T} = Q, $$
(15)

where

$$ \omega = \frac{{R^{\prime}\Omega^{\prime}}}{{\overline{U^{\prime}}}},\,g = \frac{{g^{\prime}H^{\prime}}}{{(R^{\prime}\Omega^{\prime})^{2} }} \approx 0 \cdot 72,\,Q = \frac{{Q^{\prime}}}{{\overline{U^{\prime}}R^{\prime}\Omega^{{\prime}{2}} }}, $$
$$ c_{p} = \frac{{c^{\prime}_{p} }}{{\Re^{\prime}}} \approx 5 \cdot 25,\,\kappa = \frac{{\kappa^{\prime}R^{\prime}}}{{\overline{U^{\prime}}H^{{\prime}{2}} \Re^{\prime}}} \approx 10^{ - 5} ,\,\hat{R}_{e} = \frac{{\overline{\rho }^{\prime}\overline{U^{\prime}}R^{\prime}}}{{\overline{\mu }^{\prime}}}\left( {\frac{{H^{\prime}\Omega^{\prime}}}{{\overline{U^{\prime}}}}} \right)^{2} , $$

and the error (in (10)) has been indicated. However, it is more convenient (and perhaps more natural) to define the Reynolds number in this context as

$$ R_{e} = \frac{{\overline{\rho }^{\prime}\Omega^{\prime}H^{{\prime}{2}} }}{{\overline{\mu }^{\prime}}} \approx 10^{5} , $$

which corresponds precisely to that used in both [11] and [12], allowing a direct comparison with this earlier work (and this is also the best choice for what follows in Sect. 4). It should be noted that the specific values of the parameters quoted here are irrelevant and are provided for information only: we are developing an asymptotic theory based on the limiting process \(\varepsilon \to 0\). Also, we observe that \(\Omega^{\prime}R^{\prime} \approx 467\) m/s and so any realistic value for \(\overline{U^{\prime}}\) shows that ω is very large; indeed, the most appropriate definition is

$$ \omega = \frac{{R^{\prime}\Omega^{\prime}}}{{\overline{U^{\prime}}}} = \frac{{R^{\prime}}}{{H^{\prime}}}\frac{{H^{\prime}\Omega^{\prime}}}{{\overline{U^{\prime}}}} = \frac{\Omega }{\varepsilon }\,{\text{keeping}}\,\Omega = \frac{{H^{\prime}\Omega^{\prime}}}{{\overline{U^{\prime}}}} = {\text{O}}(1), $$
(16)

and then

$$ \hat{R}_{e} = R_{e} \frac{{R^{\prime}\Omega^{\prime}}}{{U^{\prime}}} = R_{e} \frac{\Omega }{\varepsilon }. $$
(17)

We observe that \(\Omega^{\prime}H^{\prime} \approx 1 \cdot 2\;{\text{m/s}}\) and that the phase speeds for gravity waves are typically no more than about 10 m/s (which, in turn, is usually somewhat greater than the background wind speed; see [15, 16]). We therefore conclude that the contribution to the background state from any underlying zonal flow is small (in terms of ε) although, as we will see later, this wind component in our model for the atmosphere does appear at leading order in the description of the wave propagation. Equations 10, 11, 12, 13, 14 and 15 therefore provide a description of our chosen background state of the atmosphere, being generated by the leading terms in the expansion based on an amplitude parameter. This system, with the inclusion of the complete prescription of the viscous term, could be analysed without invoking the thin-shell approximation, but that, we suggest, is not a useful exercise in the context of the atmosphere that envelopes the Earth.

4 Wave propagation

We return to our original equations,(1,  2,  3, 4, and seek an asymptotic solution expressed in the dimensional form

$$ s^{\prime}(\phi ,\theta ,r^{\prime},t^{\prime};\varepsilon )\sim S^{\prime}(\phi ,\theta ,r^{\prime};\varepsilon ) + \alpha s^{\prime}_{0} (\phi ,\theta ,r^{\prime},t^{\prime};\varepsilon ),\,\alpha \to 0, $$
(18)

for each of \(u^{\prime},v^{\prime},w^{\prime},p^{\prime},\rho^{\prime},T^{\prime}\) and \(Q^{\prime}\), where the leading term in each expansion, namely \(S^{\prime}\), is the background state described by the system of Eqs. (5, 6, 7 and 8) (and in non-dimensional form in 10, 11, 12, 13, 14 and 15), and α is an (independent, nondimensional) amplitude parameter. The first stage in our presentation of a theory for gravity waves is to formulate the general problem at O(α), i.e. the equations that describe the general, linearised wave theory. In the context of our systematic approach, these equations should be recorded before we proceed; they are as follows

$$ \begin{aligned} &\Delta^{\prime}\frac{{\partial u^{\prime}_{0} }}{{\partial t^{\prime}}} + \frac{{\rho^{\prime}_{0} }}{{r^{\prime}\cos \theta }}U^{\prime}\frac{{\partial U^{\prime}}}{\partial \phi } + \Delta^{\prime}\Bigg\{ {\frac{1}{{r^{\prime}\cos \theta }}\frac{\partial }{{\partial \phi }}(U^{\prime}u^{\prime}_{0} ) + \frac{{v^{\prime}_{0} }}{{r^{\prime}}}\frac{{\partial U^{\prime}}}{\partial \theta }} + w^{\prime}_{0} \frac{{\partial U^{\prime}}}{{\partial r^{\prime}}}\\&\qquad + \frac{{U^{\prime}}}{{r^{\prime}}}\left( {w^{\prime}_{0} - v^{\prime}_{0} \tan \theta } \right)+ 2\Omega^{\prime}\Delta^{\prime}\left( {w^{\prime}_{0} \cos \theta - v^{\prime}_{0} \sin \theta } \right) \Bigg\} \hfill \\&\quad = - \frac{1}{{r^{\prime}\cos \theta }}\frac{{\partial p^{\prime}_{0} }}{\partial \phi } + \mu^{\prime}\nabla^{{\prime}{2}} u^{\prime}_{0} + r^{\prime}\frac{{{\text{d}}\mu^{\prime}}}{{{\text{d}}r^{\prime}}}\frac{\partial \;}{{\partial r^{\prime}}}\left( {\frac{{u^{\prime}_{0} }}{{r^{\prime}}}} \right); \hfill \\ \end{aligned} $$
(19)
$$ \begin{aligned} &\Delta^{\prime}\left( {\frac{{\partial v^{\prime}_{0} }}{{\partial t^{\prime}}} + \frac{{U^{\prime}}}{{r^{\prime}\cos \theta }}\frac{{\partial v^{\prime}_{0} }}{\partial \phi }} \right) + \frac{{\rho^{\prime}_{0} }}{{r^{\prime}}}U^{{\prime}{2}} \tan \theta + 2\frac{{\Delta^{\prime}}}{{r^{\prime}}}U^{\prime}u^{\prime}_{0} \tan \theta + 2\Omega^{\prime}\left( {\Delta^{\prime}u^{\prime}_{0} + \rho^{\prime}_{0} U^{\prime}} \right)\sin \theta \\ &\quad + r^{\prime}\Omega^{{\prime}{2}} \rho^{\prime}_{0} \sin \theta \cos \theta = - \frac{1}{{r^{\prime}}}\frac{{\partial p^{\prime}_{0} }}{\partial \theta } + \mu^{\prime}\nabla^{{\prime}{2}} v^{\prime}_{0} + r^{\prime}\frac{{{\text{d}}\mu^{\prime}}}{{{\text{d}}r^{\prime}}}\frac{\partial \;}{{\partial r^{\prime}}}\left( {\frac{{v^{\prime}_{0} }}{{r^{\prime}}}} \right); \\ \end{aligned} $$
(20)
$$ \begin{aligned} &\Delta^{\prime}\left( {\frac{{\partial w^{\prime}_{0} }}{{\partial t^{\prime}}} + \frac{{U^{\prime}}}{{r^{\prime}\cos \theta }}\frac{{\partial w^{\prime}_{0} }}{\partial \phi }} \right) - \frac{{\rho^{\prime}_{0} }}{{r^{\prime}}}U^{{\prime}{2}} \tan \theta - 2\frac{{\Delta^{\prime}}}{{r^{\prime}}}U^{\prime}u^{\prime}_{0} - 2\Omega^{\prime}\left( {\Delta^{\prime}u^{\prime}_{0} + \rho^{\prime}_{0} U^{\prime}} \right)\cos \theta \\ &\quad - r^{\prime}\Omega^{{\prime}{2}} \rho^{\prime}_{0} \cos^{2} \theta = - \frac{{\partial p^{\prime}_{0} }}{{\partial r^{\prime}}} - g^{\prime}\frac{{R^{{\prime}{2}} }}{{r^{{\prime}{2}} }}\rho^{\prime}_{0} + \mu^{\prime}\nabla^{{\prime}{2}} w^{\prime}_{0} + r^{\prime}\frac{{{\text{d}}\mu^{\prime}}}{{{\text{d}}r^{\prime}}}\frac{\partial \;}{{\partial r^{\prime}}}\left( {\frac{{w^{\prime}_{0} }}{{r^{\prime}}}} \right); \\ \end{aligned} $$
(21)
$$ \begin{aligned} &\frac{{\partial \rho^{\prime}_{0} }}{{\partial t^{\prime}}} + \frac{{U^{\prime}}}{{r^{\prime}\cos \theta }}\frac{{\partial \rho^{\prime}_{0} }}{\partial \phi } + \frac{{u^{\prime}_{0} }}{{r^{\prime}\cos \theta }}\frac{{\partial \Delta^{\prime}}}{\partial \phi } + \frac{{v^{\prime}_{0} }}{{r^{\prime}}}\frac{{\partial \Delta^{\prime}}}{\partial \theta } + w^{\prime}_{0} \frac{{\partial \Delta^{\prime}}}{{\partial r^{\prime}}} + \frac{{\rho^{\prime}_{0} }}{{r^{\prime}\cos \theta }}\frac{{\partial U^{\prime}}}{\partial \phi } \\ &\quad + \Delta^{\prime}\left\{ {\frac{1}{{r^{\prime}\cos \theta }}\left( {\frac{{\partial u^{\prime}_{0} }}{\partial \phi } + \frac{\partial \;}{{\partial \theta }}(v^{\prime}_{0} \cos \theta )} \right) + \frac{1}{{r^{{\prime}{2}} }}\frac{\partial \;}{{\partial r^{\prime}}}\left( {r^{{\prime}{2}} w^{\prime}_{0} } \right)} \right\} = 0; \\ \end{aligned} $$
(22)
$$ p^{\prime}_{0} = \Re^{\prime}(\rho^{\prime}_{0} {\rm T}^{\prime} + \Delta^{\prime}T^{\prime}_{0} ), $$
(23)
$$ \begin{aligned} &c^{\prime}_{p} \left( {\frac{{\partial T^{\prime}_{0} }}{{\partial t^{\prime}}} + \frac{{U^{\prime}}}{{r^{\prime}\cos \theta }}\frac{{\partial T^{\prime}_{0} }}{\partial \phi } + \frac{{u^{\prime}_{0} }}{{r^{\prime}\cos \theta }}\frac{{\partial {\rm T}^{\prime}}}{\partial \phi } + \frac{{v^{\prime}_{0} }}{{r^{\prime}}}\frac{{\partial {\rm T}^{\prime}}}{\partial \theta } + w^{\prime}_{0} \frac{{\partial {\rm T}^{\prime}}}{{\partial r^{\prime}}}} \right) - \kappa^{\prime}\nabla^{{\prime}{2}} T^{\prime}_{0} \\ &\quad - \frac{1}{{\Delta^{\prime}}}\left( {\frac{{\partial p^{\prime}_{0} }}{{\partial t^{\prime}}} + \frac{{U^{\prime}}}{{r^{\prime}\cos \theta }}\frac{{\partial p^{\prime}_{0} }}{\partial \phi }} \right) + \frac{{\rho^{\prime}_{0} }}{{\Delta^{{\prime}{2}} }}\frac{{U^{\prime}}}{{r^{\prime}\cos \theta }}\frac{{\partial \Pi^{\prime}}}{\partial \phi } \\ &\quad - \frac{1}{{\Delta^{\prime}}}\left( {\frac{{u^{\prime}_{0} }}{{r^{\prime}\cos \theta }}\frac{{\partial \Pi^{\prime}}}{\partial \phi } + \frac{{v^{\prime}_{0} }}{{r^{\prime}}}\frac{{\partial \Pi^{\prime}}}{\partial \theta } + w^{\prime}_{0} \frac{{\partial \Pi^{\prime}}}{{\partial r^{\prime}}}} \right) = q^{\prime}_{0} . \\ \end{aligned} $$
(24)

These equations are, respectively, the three components of the Navier–Stokes equation, the equation of mass conservation, the equation of state and the first law; throughout, we have used primes, in conjunction with our notation for the background state, to indicate dimensional (physical) variables. Thus Eqs. 5, 6, 7, 8 and 19, 20, 21, 22, 23 and 24 together describe the background state of the atmosphere and its perturbation (which, perforce, contains the details of any wave propagation). It is these two systems that are to be solved in order to provide a comprehensive description of gravity waves in the atmosphere, all as derived from our equations that model, in its entirety, the atmosphere (with the proviso that we have used a simplified version of the viscous terms, but this is easily remedied, as required). Clearly, this is a considerable undertaking and so, to proceed, we now invoke the thin-shell approximation–and nothing more–to produce a manageable set of equations based on a well-defined limiting process. We will comment on the validity of this asymptotic approach later.

We expect that the perturbed state–the looked-for wave element (a wave packet)–will appear only in a suitable neighbourhood of some \(\phi = \phi_{0}\); we set

$$ \phi - \phi_{0} = \beta \,\Phi , $$
(25)

where \(\beta (\varepsilon )\) is to be chosen. So, equivalently, we are seeking a solution in the form

$$ s^{\prime}(\phi ,\theta ,r^{\prime},t^{\prime};\varepsilon )\sim S^{\prime}(\phi ,\theta ,r^{\prime};\varepsilon ) + \alpha s^{\prime}_{0} (\Phi ,\phi ,\theta ,r^{\prime},t^{\prime};\varepsilon ), $$

which is expressed in terms of multiple-scales (because \(s^{\prime}_{0}\) will also depend on ϕ via the forcing provided by \(S^{\prime}\)); we note that the derivative in ϕ then transforms according to

$$ \frac{\partial \;}{{\partial \phi }} \to \frac{1}{\beta }\frac{\partial \;}{{\partial \Phi }} + \frac{\partial \;}{{\partial \phi }}, $$

although it turns out that the particular case of interest here does not require the use of multiple scales, as we shall see. Furthermore, we require a suitable non-dimensionalisation of these equations, and foremost is the need to introduce a time scale; we set

$$ t^{\prime} = \tau^{\prime}t, $$

where the scale \(\tau^{\prime}\) is to be chosen in order to produce the appropriate balance that describes wave propagation. However, it is seen directly that we cannot otherwise use the non-dimensionalisation that we introduced for the background state (as given in (9)); indeed, we find that we must set

$$ p^{\prime}_{0} = \overline{\rho }^{\prime}\overline{U^{\prime}}^{2} p_{0} $$

for consistency, and then we must also treat the perturbation density differently:

$$ \rho^{\prime}_{0} = \hat{\rho }^{\prime}\rho_{0} , $$

for some suitable scale density \(\hat{\rho }^{\prime}\). Thus the complete non-dimensionalisation of the O(α) problem is

$$ \left. \begin{gathered} r^{\prime} = (1 + \varepsilon z)R^{\prime},\;t^{\prime} = \tau^{\prime}t,\;(u^{\prime}_{0} ,v^{\prime}_{0} ,w^{\prime}_{0} ) = \overline{U^{\prime}}(u_{0} ,v_{0} ,w_{0} ), \hfill \\ \rho^{\prime}_{0} = \hat{\rho }^{\prime}\rho_{0} ,\;\mu^{\prime} = \overline{\mu }^{\prime}m(z),\;p^{\prime}_{0} = \overline{\rho }^{\prime}\overline{U}^{{\prime}{2}} p_{0} ,\;T^{\prime}_{0} = \frac{{\overline{U}^{{\prime}{2}} }}{{\Re^{\prime}}}T_{0} \hfill \\ \end{gathered} \right\} $$
(26)

where no changes are required to the velocity or the viscosity scales. Now the equation of state for the perturbation, (23), becomes

$$ p_{0} = \frac{{\hat{\rho }^{\prime}}}{{\overline{\rho }^{\prime}}}\left( {\frac{{R^{\prime}\Omega^{\prime}}}{{U^{\prime}}}} \right)^{2} \rho_{0} {\rm T} + \Delta T_{0} ; $$

but we have already seen that \({{R^{\prime}\Omega^{\prime}} \mathord{\left/ {\vphantom {{R^{\prime}\Omega^{\prime}} {U^{\prime}}}} \right. \kern-\nulldelimiterspace} {U^{\prime}}}\) is large–indeed, we have set \({{R^{\prime}\Omega^{\prime}} \mathord{\left/ {\vphantom {{R^{\prime}\Omega^{\prime}} {U^{\prime} = {\text{O}}(\varepsilon^{ - 1} )}}} \right. \kern-\nulldelimiterspace} {U^{\prime} = {\text{O}}(\varepsilon^{ - 1} )}}\)–and so we choose

$$ \hat{\rho }^{\prime} = \left( {\frac{{U^{\prime}}}{{R^{\prime}\Omega^{\prime}}}} \right)^{2} \overline{\rho }^{\prime}; $$
(27)

that is, the scaling on the two temperatures (and correspondingly on the two pressures)–both background and perturbation–forces a scaled density-perturbation, associated with the passage of the wave, which is small relative to the background density.

The equation of motion, written as components in (19, 20 and 21), shows that the balance necessary to produce a description of wave propagation requires the choice

$$ \tau^{\prime} = \frac{{R^{\prime}}}{{\overline{U}^{\prime}}}\beta , $$

and then the balance between the terms \({{\partial w_{0} } \mathord{\left/ {\vphantom {{\partial w_{0} } {\partial \Phi }}} \right. \kern-\nulldelimiterspace} {\partial \Phi }}\) and \({{\partial p_{0} } \mathord{\left/ {\vphantom {{\partial p_{0} } {\partial z}}} \right. \kern-\nulldelimiterspace} {\partial z}}\) in the nondimensional version of Eq. 21 gives

$$ \beta (\varepsilon ) = \varepsilon , $$
(28)

and so we have

$$ \phi - \phi_{0} = \varepsilon {\kern 1pt} \Phi \,\text{and}\,\tau^{\prime} = \frac{{H^{\prime}}}{{\overline{U}^{\prime}}}. $$
(29)

The first expression in (29) describes the size of the region (in the azimuthal direction) which accommodates the wave packet (using \(\Phi = {\text{O}}(1)\)), showing that the horizontal scale is the same as the vertical scale of the troposphere. The second shows that the time scale associated with the wave motion is about 25 min (based on \(H^{\prime} = 15\) km and \(\overline{U}^{\prime} = 10\) m/s) which is a reasonable measure for the period of gravity waves, being neither seconds nor hours.

Finally, we use the non-dimensionalisation and scalings given in (25, 26, 27, 28 and 29) to find the non-dimensional versions of the perturbation equations, (19, 20, 21, 22 and 23), and then retain only the leading order in ε; we obtain

$$ \begin{aligned} \Delta & \left\{ {\frac{{\partial u_{0} }}{\partial t} + \frac{U}{\cos \theta }\frac{{\partial u_{0} }}{\partial \Phi } + w_{0} \frac{\partial U}{{\partial z}} + 2\Omega (w_{0} \cos \theta - v_{0} \sin \theta )} \right\} \\ & = - \frac{1}{\cos \theta }\frac{{\partial p_{0} }}{\partial \Phi } + \frac{\Omega }{{R_{e} }}\left\{ {\frac{\partial \;}{{\partial z}}\left( {m\frac{{\partial u_{0} }}{\partial z}} \right) + \frac{m}{{\cos^{2} \theta }}\frac{{\partial^{2} u_{0} }}{{\partial \Phi^{2} }}} \right\}; \\ \end{aligned} $$
(30)
$$ \Delta \left\{ {\frac{{\partial v_{0} }}{\partial t} + \frac{U}{\cos \theta }\frac{{\partial v_{0} }}{\partial \Phi } + 2\Omega u_{0} \sin \theta } \right\} = \frac{\Omega }{{R_{e} }}\left\{ {\frac{\partial \;}{{\partial z}}\left( {m\frac{{\partial v_{0} }}{\partial z}} \right) + \frac{m}{{\cos^{2} \theta }}\frac{{\partial^{2} v_{0} }}{{\partial \Phi^{2} }}} \right\}; $$
(31)
$$ \begin{aligned} \Delta \left\{ {\frac{{\partial w_{0} }}{{\partial t}} + \frac{U}{{\cos \theta }}\frac{{\partial w_{0} }}{{\partial \Phi }} - 2\Omega u_{0} \cos \theta } \right\} & = - \frac{{\partial p_{0} }}{{\partial z}} - g\rho _{0} \\ & + \frac{\Omega }{{R_{e} }}\left\{ {\frac{{\partial \;}}{{\partial z}}\left( {m\frac{{\partial w_{0} }}{{\partial z}}} \right) + \frac{m}{{\cos ^{2} \theta }}\frac{{\partial ^{2} w_{0} }}{{\partial \Phi ^{2} }}} \right\}; \\ \end{aligned} $$
(32)
$$ \frac{\partial \;}{{\partial \Phi }}(\Delta u_{0} ) + \frac{\partial \;}{{\partial z}}(\Delta w_{0} \cos \theta ) = 0; $$
(33)
$$ p_{0} = \rho_{0} {\rm T} + \Delta T_{0} . $$
(34)

The viscous terms appear at this order because the associated Reynolds number is

$$ \frac{{\overline{\rho }^{\prime}\overline{U}^{\prime}H^{\prime}}}{{\overline{\mu }^{\prime}}} = \frac{{R_{e} }}{\Omega }, $$
(35)

but we could still impose large Reynolds number, if that is convenient; we are presenting these equations with \(R_{e} = {\text{O}}(1)\). The first law, generated from Eq. (24), is written in more detail; this becomes

$$ \begin{aligned} &\Omega^{2} \left( {c_{p} L({\rm T}) - \frac{1}{\Delta }L(\Pi )} \right) + \varepsilon^{2} \left( {\frac{\partial \;}{{\partial t}} + \frac{U}{(1 + \varepsilon z)\cos \theta }\frac{\partial \;}{{\partial \Phi }}} \right)\left( {c_{p} T_{0} - \frac{{p_{0} }}{\Delta }} \right) \\ &\quad - \varepsilon^{3} \kappa \left( {\frac{{\partial^{2} T_{0} }}{{\partial z^{2} }} + \frac{1}{{\cos^{2} \theta }}\frac{{\partial^{2} T_{0} }}{{\partial \Phi^{2} }}} \right) + \varepsilon^{3} \frac{{\rho_{0} }}{{\Delta^{2} }}\frac{U}{\cos \theta }\frac{\partial \Pi }{{\partial \phi }} + {\text{O}}(\varepsilon^{4} ) = q_{0} , \\ \end{aligned} $$
(36)

where

$$ L \equiv w_{0} \frac{\partial \;}{{\partial z}} + \varepsilon \left\{ {\frac{{u_{0} }}{(1 + \varepsilon z)\cos \theta }\frac{\partial \;}{{\partial \phi }} + \frac{{v_{0} }}{1 + \varepsilon z}\frac{\partial \;}{{\partial \theta }}} \right\} $$

and \(q_{0} (\Phi ,\phi ,\theta ,z;\varepsilon )\) is a general, nondimensional heat-source term appropriate to this order of approximation in ε.

5 Interpretation, analysis and solutions

Before we examine the two systems of reduced equations that we have derived, for the background atmosphere and for the propagation of waves, we should briefly comment on the validity of the asymptotic approach that we have adopted. It is altogether unrealistic to expect that we can produce a definitive statement–let alone a proof–of the asymptotic correctness of our approximation. The complexity of the problem and, of course, that of the underlying set of governing equations for the fluid, precludes this (except, perhaps, for very idealised flows with simple geometry). On the other hand, there is a long line of successes, stretching over more than a century, that attest to the efficacy of these methods in the study of mathematical fluid dynamics. This is not as mathematically encouraging as we would wish, but we are able to be more precise in the context of the current analysis, suggesting that we can have confidence in the results presented here.

There are two main areas where the asymptotic validity might be in question, leading to a breakdown or failure of uniformity. The first involves the nature of the wave itself; provided that the waves do not steepen, and so remain sufficiently smooth, then no breakdown on this score will arise. We will therefore seek wave solutions for which no such problem occurs which will, in part, require that suitable initial data is assumed. The determination of scales on which nonlinearity and steepening might be important (and so may be a significant contribution to this type of study) is left for a further investigation. The second possible difficulty is more convincingly described and explained. Non-uniformities will certainly arise if z or Φ are allowed to increase without bound, but these difficulties are avoided if we impose some reasonable constraints. Thus we limit the application to the troposphere–in any event, this is the only layer of the atmosphere that we consider–where \(0 \le z \le 1\): any non-uniformities associated with terms such as \((\varepsilon z)^{n}\), n = 0, 1, 2,…, for \(z > 1\), cannot occur. Similarly, we consider wave packets that are restricted in length, ensuring that \(\Phi = {\text{O}}(1)\) and no larger. We therefore proceed with the understanding that steepening is avoided, \(0 \le z \le 1\) and \(\Phi = {\text{O}}(1)\).

The obvious and natural approach to the discussion of any mathematical problem concerning the atmosphere is to input the heat sources, \(Q^{\prime}({\mathbf{x^{\prime}}},t^{\prime})\) in Eq. 4, and then aim to solve for \({\mathbf{u^{\prime}}},\;p^{\prime},\;\rho^{\prime}\;{\text{and}}\;T^{\prime}\)(given a suitable model for the dynamic eddy viscosity). However, the nature of the heat sources, and how to model them sufficiently accurately, are not well-understood; see [17]. We therefore tackle this problem from a different direction. Essentially, we develop a solution of interest (based on the type of solution that we are seeking and on what is observed) and then use the first law of thermodynamics to provide a description of the heat sources required to drive and maintain the motion. (Further details of these ideas can be found in [11, 12].) We therefore proceed with the aim of constructing appropriate solutions for the background atmosphere and for the waves, and then interpreting the first law to produce suitable heat sources for the motion. As it happens, the solutions of specific interest here do not require or involve a heat source to drive or maintain the motion.

5.1 Background state of the atmosphere

From Eqs. 10, 11, 12, 13, 14 and 15, with ( 17), we can find a suitable solution for the background state of the atmosphere, with an associated wind. In this initial phase of the investigation, we choose a wind represented by

$$ U = U_{0} (1 + \varepsilon z)\cos \theta , $$
(37)

where \(U_{0}\) is a constant, this form being essentially a constant-speed flow because the scenario that we have in mind is of a gravity wave propagating in a fairly narrow vertical band, in some small neighbourhood around a chosen, fixed meridional angle, θ0 (although these two conditions are not imposed). Consistent with this interpretation, we observe that \(U\sim U_{0} \cos \theta\) as \(\varepsilon \to 0\) and, in our development, θ plays the rȏle of a parameter throughout. More general solutions are possible and accessible; this is an avenue that might be explored in a future study. So using (37), we have a solution with an overall error O(\(\varepsilon^{3}\)), which is independent of ϕ and is valid for arbitrary m(z):

$$ \Pi = \Pi_{0} {\rm T}^{\gamma } \,\,\text{and}\,\,\Delta = \Pi_{0} {\rm T}^{\gamma - 1} $$
(38)

where \(\Pi_{0}\) and \(\gamma \;( > 1)\) are constants, with

$$ {\rm T} = {\rm T}_{0} + \frac{1}{\gamma }\left\{ {\frac{1}{2}\left( {1 + \varepsilon z} \right)^{2} \left( {1 + \varepsilon \frac{{U_{0} }}{\Omega }} \right)\cos^{2} \theta - \frac{gz}{{1 + \varepsilon z}}} \right\}, $$
(39)

where \({\rm T}_{0} (\theta )\) is a constant at fixed θ. This solution recovers that described in [11, 12] when we impose \(\varepsilon \to 0\) and choose \(\gamma = c_{p}\) (a particular choice that we shall comment on later), for we then obtain

$$ {\rm T} = {\rm T}_{0} (\theta ) + \frac{1}{{c_{p} }}\left( {\frac{1}{2}\cos^{2} \theta - gz} \right). $$
(40)

The leading-order contribution to the heat source, generated by the temperature profile (39), is then \(Q \equiv {\text{O}}(\varepsilon )\); in the case of (40), this gives the familiar result that the only heat transferred to the atmosphere is up from the surface of the Earth, by virtue of \({{\partial {\rm T}} \mathord{\left/ {\vphantom {{\partial {\rm T}} {\partial z}}} \right. \kern-\nulldelimiterspace} {\partial z}} = - {g \mathord{\left/ {\vphantom {g {c_{p} }}} \right. \kern-\nulldelimiterspace} {c_{p} }}\).

5.2 The wave perturbation

The equations that describe the leading-order wave perturbation can now be analysed, after an update based on the solution that we have introduced for the background state. In particular we introduce U (from (37)) and, because there is no dependence on ϕ, the perturbation equations depend on Φ but not also on ϕ, i.e. the use of multiple scales is now redundant. Thus from (30, 31, 32, 33 and 34) we obtain

$$ \begin{aligned} \frac{{\partial u_{0} }}{\partial t} & + U_{0} \frac{{\partial u_{0} }}{\partial \Phi } + 2\Omega (w_{0} \cos \theta - v_{0} \sin \theta ) \\ & = - \frac{1}{\Delta \cos \theta }\frac{{\partial p_{0} }}{\partial \Phi } + \frac{\Omega }{{R_{e} \Delta }}\left\{ {\frac{\partial \;}{{\partial z}}\left( {m\frac{{\partial u_{0} }}{\partial z}} \right) + \frac{m}{{\cos^{2} \theta }}\frac{{\partial^{2} u_{0} }}{{\partial \Phi^{2} }}} \right\}; \\ \end{aligned} $$
(41)
$$ \frac{{\partial v_{0} }}{\partial t} + U_{0} \frac{{\partial v_{0} }}{\partial \Phi } + 2\Omega u_{0} \sin \theta = \frac{\Omega }{{R_{e} \Delta }}\left\{ {\frac{\partial \;}{{\partial z}}\left( {m\frac{{\partial v_{0} }}{\partial z}} \right) + \frac{m}{{\cos^{2} \theta }}\frac{{\partial^{2} v_{0} }}{{\partial \Phi^{2} }}} \right\}; $$
(42)
$$ \begin{aligned} \frac{{\partial w_{0} }}{{\partial t}}{\text{ }} + U_{0} \frac{{\partial w_{0} }}{{\partial \Phi }} - 2\Omega u_{0} \cos \theta & = - \frac{1}{\Delta }\frac{{\partial p_{0} }}{{\partial z}} - g\frac{{\rho _{0} }}{\Delta }{\text{ }} \\ & + \frac{\Omega }{{R_{e} \Delta }}\left\{ {\frac{{\partial \;}}{{\partial z}}\left( {m\frac{{\partial w_{0} }}{{\partial z}}} \right) + \frac{m}{{\cos ^{2} \theta }}\frac{{\partial ^{2} w_{0} }}{{\partial \Phi ^{2} }}} \right\}; \\ \end{aligned} $$
(43)
$$ \frac{\partial \;}{{\partial \Phi }}(\Delta u_{0} ) + \frac{\partial \;}{{\partial z}}(\Delta w_{0} \cos \theta ) = 0; $$
(44)
$$ p_{0} = \rho_{0} {\rm T} + \Delta T_{0} . $$
(45)

Finally, the first law of thermodynamics, (36), simplifies to give

$$ \begin{gathered} \Omega^{2} \left( {c_{p} w_{0} \frac{{\partial {\rm T}}}{\partial z} - \frac{1}{\Delta }w_{0} \frac{\partial \Pi }{{\partial z}}} \right) + \varepsilon^{2} \left( {\frac{\partial \;}{{\partial t}} + U_{0} \frac{\partial \;}{{\partial \Phi }}} \right)\left( {c_{p} T_{0} - \frac{{p_{0} }}{\Delta }} \right) \hfill \\ - \varepsilon^{3} \kappa \left( {\frac{{\partial^{2} T_{0} }}{{\partial z^{2} }} + \frac{1}{{\cos^{2} \theta }}\frac{{\partial^{2} T_{0} }}{{\partial \Phi^{2} }}} \right) + {\text{O}}(\varepsilon^{4} ) = q_{0} , \hfill \\ \end{gathered} $$
(46)

which, as outlined earlier, can be used to identify any heat sources associated with the motion, if that is appropriate. This is now rewritten in terms of the chosen background state of the atmosphere given in (38) and (39):

$$ \begin{gathered} \Omega^{2} (c_{p} - \gamma )w_{0} \frac{{\partial {\rm T}}}{\partial z} + \varepsilon^{2} \left( {\frac{\partial \;}{{\partial t}} + U_{0} \frac{\partial \;}{{\partial \Phi }}} \right)\left( {c_{p} T_{0} - \frac{{p_{0} }}{\Delta }} \right) \hfill \\ - \varepsilon^{3} \kappa \left( {\frac{{\partial^{2} T_{0} }}{{\partial z^{2} }} + \frac{1}{{\cos^{2} \theta }}\frac{{\partial^{2} T_{0} }}{{\partial \Phi^{2} }}} \right) + {\text{O}}(\varepsilon^{4} ) = q_{0} , \hfill \\ \end{gathered} $$

and the balance that is required for gravity waves requires the choice

$$ \gamma = c_{p} \left( {1 + \varepsilon^{2} \Gamma } \right), $$

for constant Γ = O(1); thus we obtain

$$ \left( {\frac{\partial \;}{{\partial t}} + U_{0} \frac{\partial \;}{{\partial \Phi }}} \right)\left( {c_{p} T_{0} - \frac{{p_{0} }}{\Delta }} \right) - c_{p} \Gamma \Omega^{2} w_{0} \frac{{\partial {\rm T}}}{\partial z} + {\text{O}}(\varepsilon ) = q_{0} . $$
(47)

At this stage of the calculation, we should make a few important observations, particularly in relation to rȏle of Eq. 47. In the case \(\Gamma = 0\) (and so \(\gamma = c_{p}\)), we see that any heating associated with points (particles) that move in the atmosphere (induced by the passage of the wave) vanishes altogether at this order–a familiar result. On the other hand, this contribution in Eq. 47 must balance–in the absence of any heat sources, so \(q_{0} \equiv 0\)–the variation of the potential temperature as it is convected in the flow. Our nondimensional version of the potential temperature is

$$ \left( {{\rm T} + \alpha T_{0} } \right)\left( {\Pi + \alpha p_{0} } \right)^{{ - {1 \mathord{\left/ {\vphantom {1 {c_{p} }}} \right. \kern-\nulldelimiterspace} {c_{p} }}}} \sim {\rm T}\Pi^{{ - {1 \mathord{\left/ {\vphantom {1 {c_{p} }}} \right. \kern-\nulldelimiterspace} {c_{p} }}}} \left\{ {1 + \frac{\alpha }{{c_{p} {\rm T}}}\left( {c_{p} T_{0} - \frac{{p_{0} }}{\Delta }} \right)} \right\}\,\,as\,\,\alpha \to 0, $$

confirming that the perturbation potential temperature is proportional to

$$ c_{p} T_{0} - \frac{{p_{0} }}{\Delta }. $$

Further, we have, for \(\Pi = \Pi_{0} {\rm T}^{\gamma }\) (see 38), that the leading-order contribution to the potential temperature is proportional to

$$ {\rm T}^{{1 - {\gamma \mathord{\left/ {\vphantom {\gamma {c_{p} }}} \right. \kern-\nulldelimiterspace} {c_{p} }}}} = {\rm T}^{{ - \varepsilon^{2} \Gamma }} , $$

recovering the familiar result that the potential temperature is constant if \(\gamma = c_{p}\) but, significantly, this cannot be the case for the existence of gravity waves.

We note that our version of the perturbation equations has a number of important features. Firstly, the general scaling-approach that we have adopted has allowed the inclusion of the dominant viscous terms, although these could be ignored (at this order) if we invoke the additional requirement that \(R_{e} \to \infty\). However, our overall aim has been to show how all the physical attributes of the flow can be included using a systematic approach, so we will retain these terms and use them to investigate (albeit superficially) how they affect the propagation of waves. Secondly, we have shown that the familiar Boussinesq approximation–often invoked as an ad hoc ingredient–follows directly by using a careful and consistent scaling and non-dimensionalisation methods. Thus, although the density perturbation generated by the passage of the wave is very small (see (27)), it is not altogether absent at this order; it does appear, but only in association with the gravity term in the dynamical equations. It is then coupled to the other thermodynamic quantities, at this same order, by the appropriate version of the equation of state (see ( 45)). Indeed, this scaling then shows that changes in density, pressure and temperature are all the same size (in terms of ε); this conclusion is rather at variance with standard theories (which balance density and temperature changes), but the effect on the waves is only fairly minor, as we shall see. Finally, our introduction of the parameter \(\Omega = {{H^{\prime}\Omega^{\prime}} \mathord{\left/ {\vphantom {{H^{\prime}\Omega^{\prime}} {U^{\prime}}}} \right. \kern-\nulldelimiterspace} {U^{\prime}}}\) which, reasonably, must be treated as O(1) in this context, confirms that the effects of rotation (Coriolis terms) cannot be ignored in the discussion of gravity waves; this contribution is traditionally omitted in these studies. Further, a consequence of including the Earth’s rotation is that the velocity component in the meridional direction, \(v_{0}\), also plays a rȏle; see ( 41) and ( 42). The standard equations that are typically used to examine buoyancy waves are recovered from ours simply by setting Ω = 0 in (41, 42, 43, 44 and 45).

It is unrealistic to expect that we can find a complete solution of the system  (41, 42, 43, 44 and 45), with ( 47), which describes the propagation of gravity waves. However, we are able to make a number of observations and, in particular, relate our version to that usually presented in the literature. Before we start, we should emphasise that our system of equations has been derived solely by invoking the thin-shell approximation, thereby retaining all the physical attributes of the flow. Thus, although we may ignore the viscous contribution, because we could impose the additional requirement that \(R_{e} \to \infty\), the same cannot be said of the contribution that arises by working in the rotating frame. These terms necessarily must be regarded as providing an O(1) contribution at this level of approximation, for any realistic choice of speed scale, \(U^{\prime}\). So, in order to produce some useful and relevant observations, we must introduce some reasonable simplifications. We will examine two specific scenarios: firstly, we consider the inviscid case with, in addition, the assumption of constant coefficients; secondly, we produce a solution which retains the viscous contribution, but simplifies other elements of the flow.

5.2.1 Inviscid waves

The construction of a single equation, having imposed \(R_{e} \to \infty\), that describes the propagation of waves is somewhat tiresome, but altogether routine. So, rather than present a sequence of intermediate steps, we carefully describe the procedure:

  • Eliminate \(T_{0}\) between (45) and (47) to produce (A);

  • Eliminate \(\rho_{0}\) between (A) and (43) to produce (B);

  • Eliminate \(p_{0}\) between (B) and (41) to produce (C);

  • Eliminate \(v_{0}\) between (C) and (42).

This leaves a single equation in \(u_{0}\) and \(w_{0}\), at which stage we introduce a stream function, ψ, using (44):

$$ u_{0} = \frac{1}{\Delta }\frac{\partial \psi }{{\partial z}},\,\,\,w_{0} = - \frac{1}{\Delta \cos \theta }\frac{\partial \psi }{{\partial \Phi }}, $$

which leads to

$$ \begin{aligned} &\left( {\frac{\partial \;}{{\partial t}} + U_{0} \frac{\partial \;}{{\partial \Phi }}} \right)^{2} \left\{ {\frac{{\partial^{2} \psi }}{{\partial \Phi^{2} }} + \left( {\frac{{\partial^{2} \psi }}{{\partial z^{2} }} + \frac{{g(c_{p} - 1)}}{{c_{p} {\rm T}}}\frac{\partial \psi }{{\partial z}}} \right)\cos^{2} \theta } \right\} \\ & \quad+ \Omega^{2} \left( {\frac{{\partial^{2} \psi }}{{\partial z^{2} }} + \frac{{g(c_{p} - 1)}}{{c_{p} {\rm T}}}\frac{\partial \psi }{{\partial z}}} \right)\sin^{2} (2\theta ) + \Omega^{2} \frac{g\Gamma }{{c_{p} {\rm T}}}\frac{{\partial {\rm T}}}{\partial z}\frac{{\partial^{2} \psi }}{{\partial \Phi^{2} }} \\ &\quad - \frac{{2g\Omega (c_{p} - 1)\cos^{2} \theta }}{{c_{p} {\rm T}}}\left( {\frac{\partial \;}{{\partial t}} + U_{0} \frac{\partial \;}{{\partial \Phi }}} \right)\frac{\partial \psi }{{\partial \Phi }} + \frac{g\Delta \cos \theta }{{c_{p} {\rm T}}}\frac{{\partial q_{0} }}{\partial \Phi } = 0. \\ \end{aligned} $$
(48)

It is immediately evident that Eq. (48) possesses a number of novel features. First, and perhaps foremost, the effects of using a rotating frame are evident in the terms involving the parameter Ω (but with \(\Omega^{2} \Gamma\) treated separately, as we describe later). Second, we have terms in \({{\partial \psi } \mathord{\left/ {\vphantom {{\partial \psi } {\partial z}}} \right. \kern-\nulldelimiterspace} {\partial z}}\) which are associated with the more familiar \({{\partial^{2} \psi } \mathord{\left/ {\vphantom {{\partial^{2} \psi } {\partial z^{2} }}} \right. \kern-\nulldelimiterspace} {\partial z^{2} }}\) term; these new terms have been generated by our non-dimensionalisation and scaling which have shown that changes in pressure, density and temperature all arise at the same asymptotic order. In addition, there is the expected appearance of θ (simply playing the rȏle of a parameter here), which arises from the use of spherical coordinates and, finally, we can admit an external forcing by virtue of the heat-source term \((q_{0} )\).

The terms in \({{\partial \psi } \mathord{\left/ {\vphantom {{\partial \psi } {\partial z}}} \right. \kern-\nulldelimiterspace} {\partial z}}\), that appear in (48), are readily accommodated by writing

$$ \psi = \Psi (\Phi ,Z,t)\,{\text{with}}\,Z = \frac{1}{g}\left( {\frac{{c_{p} }}{{c_{p} - 2}}} \right){\rm T}^{{2 - c_{p} }} , $$

at fixed θ, to give

$$ \frac{{\partial^{2} \psi }}{{\partial z^{2} }} + \frac{{g(c_{p} - 1)}}{{c_{p} {\rm T}}}\frac{\partial \psi }{{\partial z}} = {\rm T}^{{2(c_{p} - 1)}} \frac{{\partial^{2} \Psi }}{{\partial Z^{2} }}, $$

which is proportional to

$$ Z^{{{{2(c_{p} - 1)} \mathord{\left/ {\vphantom {{2(c_{p} - 1)} {(2 - c_{p} )}}} \right. \kern-\nulldelimiterspace} {(2 - c_{p} )}}}} \frac{{\partial^{2} \Psi }}{{\partial Z^{2} }}. $$

However, the conventional interpretation of this equation, for gravity waves, is to seek a harmonic travelling-wave solution, keeping all the coefficients constant; this is equivalent to assuming that we are in some small neighbourhood of a fixed z. In order to provide a comparison with existing theories, we will follow this same route. (Clearly, an interesting and important challenge is to examine Eq. 48, written consistently in terms of Z, but that is left for a future investigation.) Thus we seek a solution of

$$ \begin{gathered} \left( {\frac{\partial \;}{{\partial t}} + U_{0} \frac{\partial \;}{{\partial \Phi }}} \right)^{2} \left\{ {\frac{{\partial^{2} \Psi }}{{\partial \Phi^{2} }} + {\rm T}^{{2(c_{p} - 1)}} \frac{{\partial^{2} \Psi }}{{\partial Z^{2} }}\cos^{2} \theta } \right\} \hfill \\ + \Omega^{2} {\rm T}^{{2(c_{p} - 1)}} \frac{{\partial^{2} \Psi }}{{\partial Z^{2} }}\sin^{2} (2\theta ) + \Omega^{2} \frac{g\Gamma }{{\rm T}}\frac{{\partial {\rm T}}}{\partial z}\frac{{\partial^{2} \Psi }}{{\partial \Phi^{2} }} \hfill \\ - \frac{{2g\Omega (c_{p} - 1)\cos^{2} \theta }}{{c_{p} {\rm T}}}\left( {\frac{\partial \;}{{\partial t}} + U_{0} \frac{\partial \;}{{\partial \Phi }}} \right)\frac{\partial \Psi }{{\partial \Phi }} + \frac{g\Delta \cos \theta }{{c_{p} {\rm T}}}\frac{{\partial q_{0} }}{\partial \Phi } = 0 \hfill \\ \end{gathered} $$
(49)

in the form

$$ \Psi = \Psi_{0} {\text{e}}^{{{\text{i}}(l\Phi + kZ - \varpi t)}} , $$

where l, k and \(\varpi\) are real constants, and \(\Psi_{0}\) is a complex-valued constant; we write \({{\partial {\rm T}} \mathord{\left/ {\vphantom {{\partial {\rm T}} {\partial z}}} \right. \kern-\nulldelimiterspace} {\partial z}} = - {g \mathord{\left/ {\vphantom {g {c_{p} }}} \right. \kern-\nulldelimiterspace} {c_{p} }}\) (see (40)) but T is otherwise treated as a constant, and there is no forcing (\({{\partial q_{0} } \mathord{\left/ {\vphantom {{\partial q_{0} } {\partial \Phi = 0}}} \right. \kern-\nulldelimiterspace} {\partial \Phi = 0}}\)). Then (49) gives the dispersion relation

$$ \begin{aligned} & (\varpi - U_{0} l)^{2} \left( {l^{2} + {\text{T}}^{{2(c_{p} - 1)}} k^{2} \cos ^{2} \theta } \right) - (\varpi - U_{0} l)\frac{{2g\Omega l(c_{p} - 1)\cos ^{2} \theta }}{{c_{p} {\text{T}}}} \\ & \quad = \Omega ^{2} k^{2} {\text{T}}^{{2(c_{p} - 1)}} \sin ^{2} (2\theta ) + \frac{{g^{2} \Gamma }}{{c_{p}^{2} {\text{T}}}}\Omega ^{2} l^{2} , \\ \end{aligned} $$
(50)

and so

$$ \begin{aligned} \varpi & = U_{0} l + \frac{{2g\Omega l(c_{p} - 1)\cos^{2} \theta }}{{c_{p} {\rm T}\left( {l^{2} + {\rm T}^{{2(c_{p} - 1)}} k^{2} \cos^{2} \theta } \right)}} \\ & \pm \sqrt {\frac{{\Omega^{2} N^{2} l^{2} + \Omega^{2} k^{2} {\rm T}^{{2(c_{p} - 1)}} \sin^{2} (2\theta )}}{{\left( {l^{2} + {\rm T}^{{2(c_{p} - 1)}} k^{2} \cos^{2} \theta } \right)}} + \left\{ {\frac{{2g\Omega l(c_{p} - 1)\cos^{2} \theta }}{{c_{p} {\rm T}\left( {l^{2} + {\rm T}^{{2(c_{p} - 1)}} k^{2} \cos^{2} \theta } \right)}}} \right\}^{2} } , \\ \end{aligned} $$
(51)

where we have written

$$ N^{2} = \frac{{g^{2} \Gamma }}{{c_{p} {\rm T}}}, $$
(52)

which requires that \(\Gamma > 0\). Here, N is a nondimensional Brunt-Väisälä frequency, and its connection with the familiar (dimensional) Brunt-Väisälä frequency is readily demonstrated. First we note that

$$ N^{2} = - \Gamma g^{\prime}\left( {\frac{{H^{\prime}}}{{R^{\prime}\Omega^{\prime}}}} \right)^{2} \frac{1}{{{\rm T}^{\prime}}}\frac{{\partial {\rm T}^{\prime}}}{{\partial z^{\prime}}}, $$

where the primes, as earlier, denote dimensional variables. We must now compare this with \({{\partial^{2} } \mathord{\left/ {\vphantom {{\partial^{2} } {\partial t^{{\prime}{2}} }}} \right. \kern-\nulldelimiterspace} {\partial t^{{\prime}{2}} }}\) (dimensional), which gives

$$ \frac{{\partial^{2} \;}}{{\partial t^{{\prime}{2}} }}\sim - \varepsilon^{2} \Omega^{2} \Gamma g^{\prime}\left( {\frac{{U^{\prime}}}{{H^{\prime}\Omega^{\prime}}}} \right)^{2} \frac{1}{{{\rm T}^{\prime}}}\frac{{\partial {\rm T}^{\prime}}}{{\partial z^{\prime}}} = N^{{\prime}{2}} , $$

where \(N^{{\prime}{2}} = g^{\prime}\frac{\partial \;}{{\partial z^{\prime}}}\left\{ {\ln \left( {{\rm T}^{{ - \varepsilon^{2} \Gamma }} } \right)} \right\};\)

thus our nondimensional N recovers the familiar expression for \(N^{\prime}\), when written in terms of our expression for the potential temperature.

Finally, we observe that the frequency, \(\varpi\), of the harmonic wave (given in (51)), subsumes the standard expression for atmospheric gravity waves. To see this, first keep \(\Omega N\) fixed (because we have just seen that \(\Omega N\) produces \(N^{\prime}\)) and then set \(\Omega = 0\) (no Coriolis terms); this gives

$$ \varpi = U_{0} l \pm \frac{\Omega Nl}{{\sqrt {l^{2} + {\rm T}^{{2(c_{p} - 1)}} k^{2} \cos^{2} \theta } }}. $$

Now the adjustment that is provided by the term \({\rm T}^{{2(c_{p} - 1)}}\) (which appears in our asymptotic version of the problem) is accommodated by defining \({\rm T}^{{c_{p} - 1}} k\cos \theta\) as the wave number in the vertical direction, and then we obtain the result often quoted in the standard texts; see, for example, [6]. This, albeit simplistic, discussion of our equation that describes the propagation of gravity waves in the atmosphere is, arguably, the most useful check that we can perform. However, because we have retained the dominant viscous terms, we will carry out a similar calculation and so investigate how these terms contribute to the solution.

5.2.2 Waves with viscosity

For this calculation, we follow the same philosophy as above, i.e. we seek a harmonic-wave solution, having introduced Z and then treating T as constant but, for simplicity (in terms of algebra and presentation) we will ignore the rotation (Coriolis) terms and assume constant eddy viscosity (m). With these simplifications in mind, we find that

$$ \frac{\partial \;}{{\partial z}}\left( {m\frac{\partial \;}{{\partial z}}} \right) + \frac{1}{{\cos^{2} \theta }}\frac{{\partial^{2} \;}}{{\partial \Phi^{2} }} \equiv - \frac{m}{{\cos^{2} \theta }}\left( {l^{2} + {\rm T}^{{2(c_{p} - 1)}} k^{2} \cos^{2} \theta } \right), $$

for a solution which is proportional to \(\exp [{\text{i}}(l\Phi + kZ - \varpi t)]\). Then, with the Coriolis terms ignored, the dispersion relation gives

$$ \varpi = U_{0} l \pm \sqrt {\frac{{l^{2} \Omega^{2} N^{2} }}{\Lambda } - \left( {\frac{\Omega m}{{2R_{e} \Delta \cos^{2} \theta }}\Lambda } \right)^{2} } - {\text{i}}\frac{\Omega m}{{2R_{e} \Delta \cos^{2} \theta }}\Lambda , $$
(53)

where

$$ \Lambda = l^{2} + {\rm T}^{{2(c_{p} - 1)}} k^{2} \cos^{2} \theta . $$

The harmonic wave, with this frequency function, (53), therefore decays in t > 0 because

$$ I_{m} (\varpi ) < 0. $$

The rest of the expression in (53) is real for sufficiently large \(R_{e}\) (and, with our definitions, \(R_{e}\) is typically about \(10^{5}\)). We conclude that the inclusion of the viscous terms produces the expected effect–decay in time–as the gravity wave propagates through the atmosphere (in the absence of forcing).

6 Discussion

Buoyancy waves play an important rȏle in any studies of the motion of the atmosphere, and so their properties need to be analysed and understood. However, there seems to be no derivation of the relevant system of equations based on the general equations for a compressible, viscous fluid. The main thrust of our presentation has been to remedy this situation by extending the work described in [11, 12] to gravity waves in the atmosphere. To this end, we have shown–rather surprisingly, perhaps–that it requires only the imposition of the thin-shell approximation in order to make headway. This is the natural and obvious way forward, because the atmosphere constitutes a spherical shell and so, in a careful treatment, we must ensure that we correctly and accurately represent this property. In addition, the waves are propagating in a rotating frame and so, again, care must be taken to accommodate this aspect of the flow environment. The upshot is that, although some properties of the solution confirm the familiar theories, other elements, usually ignored, must be included in a consistent theory.

The fundamental and underlying approximation of a thin shell enables us to produce a consistent transition from spherical coordinates to (locally) rectangular Cartesian coordinates. However, as we have shown, the non-dimensionalisation that is associated with this approximation is critical to a proper asymptotic description. We have seen that there is a necessary requirement to use different speed scales for the velocity (which then impacts on the corresponding definitions for pressure and temperature) for the background state of the atmosphere and for its perturbation. So we use speed scales \(R^{\prime}\Omega^{\prime}\) and \(U^{\prime}\), and then introduce the ratio \({{U^{\prime}} \mathord{\left/ {\vphantom {{U^{\prime}} {H^{\prime}\Omega^{\prime}}}} \right. \kern-\nulldelimiterspace} {H^{\prime}\Omega^{\prime}}}\); indeed, although it would hide some of the detail, we could choose \(U^{\prime} = H^{\prime}\Omega^{\prime}\) as the second speed scale. The consequences of these choices lead to some important observations about the flow.

One of the familiar modelling assumptions in the description of these flows is the Boussinesq approximation. This states that density changes (due to the passage of the wave) are important in the gravity term, but nowhere else in the flow dynamics. Our non-dimensionalisation and scaling confirm this, even though the resulting density ratio, wave/background, is very small. However, this same density property shows that changes in pressure, density and temperature, associated with the passage of the wave, are necessarily all the same size; this is at variance with the usual modelling assumptions. This might, at first sight, seem to pose a serious technical problem in the description of the wave, but this additional complication is easily remedied by using Z \(\left( { = \left( {{{c_{p} } \mathord{\left/ {\vphantom {{c_{p} } {g(c_{p} - 2)}}} \right. \kern-\nulldelimiterspace} {g(c_{p} - 2)}}} \right){\rm T}^{{2 - c_{p} }} } \right)\) rather than z as the appropriate variable in the vertical direction. So, although the assumption that changes in pressure are much smaller than those in density and temperature is not correct, this is not critical to the development. Further, when we invoke the ad hoc condition that changes in z (or Z) in the coefficients can be ignored then, for z close to some \(z = z_{0}\), the argument of the harmonic solution becomes no more than a phase shift, combined with suitably relabelled wave number that is associated with the z variable, precisely as in the standard theories.

There are two aspects to the representation of the correct geometrical configuration: that for the thin shell on the spherical Earth and for the rotation of the Earth. Although the development, based on the spherical geometry, is essential in a careful treatment of the problem, the requirement of the thin shell, combined with a limited extent in the azimuthal direction, shows that the underlying geometry is evidenced only by the parametric dependence on θ, to leading order in ε. Indeed, we have found that the horizontal scale is the same as the vertical scale (i.e. both measured by ε), which then forces a time scale that is measured in minutes. However, this development necessarily leads to a complication which is not part of the standard theories of gravity waves.

Our non-dimensionalisation and scaling, which aims to capture all the essential features of the atmosphere and the wave, also show that the Coriolis terms cannot be ignored. These terms appear at the same order as all the terms that typically contribute to the propagation of linear waves. That such terms cannot be ignored is plain when we note, in Eq. 49, that Ω is associated with both the Coriolis contribution and the Brunt-Väisälä frequency: we cannot ignore one at the dispense of the other. It is evident, and expected, that these additional terms complicate the dispersion relation, (50), and its resulting solution, (51). These two results have been written down here on the basis that we could ignore the variation of the coefficients in our wave equation, but this was simply a device to check against existing and standard results. (We will return to this issue shortly.) We may observe from (51) that the effects of rotation are important for wave numbers \(k \to \infty\), l fixed, but unimportant for \(l \to \infty\), k fixed. On the other hand, with the appropriate simplification–no rotation–the standard result is recovered. Finally, we have shown that a suitable definition of the Reynolds number, treated as O(1) as \(\varepsilon \to 0\), ensures that viscosity plays no rȏle in the dominant description of the background state of the atmosphere with a wind, but is important in the dominant description of the wave motion. All the above has provided an overview of the how our approach relates to the more familiar versions of this problem; we now give a critical assessment of the new aspects that we have presented.

The development that we have followed has shown how it is possible to start from a general state of the atmosphere, with a wind, which provides the background in which the waves propagate. Although the solution chosen here was rather simple in its structure, other choices are possible, either involving more general velocity profiles for the zonal wind, or more complicated flow systems in general. Critically, it is the use of the thin-shell approximation which makes this approach manageable although, whatever we choose to use, it must feed into the wave-propagation problem that is generated at O(α). Thus we have put in place a procedure for describing a suitable model of the atmosphere that is sufficiently accurate and also amenable to a careful mathematical treatment. The corresponding situation for the wave component, on the other hand, is rather more involved.

The derivation of a linear wave equation describing the propagation of gravity waves is, on the face of it, fairly straightforward, but the solution of the resulting equation is far from routine. First, the choice of the background state of the atmosphere, with a wind included, produces an equation with variable coefficients–and our version presented here is one of the simplest that can be derived. It is clear that a comprehensive discussion of the gravity wave requires a suitable, and mathematically correct, solution of Eq. 48. This is a considerable challenge and, although what we have presented here indicates that we are on the right track, more work is needed to understand all the properties of this equation and their relevance to the propagation of gravity waves. Further, the inclusion of the contribution from the Coriolis terms to the wave propagation requires us to ascertain what effect this has on the properties of the wave, and whether these are observed. One aspect not explored here is the construction of waves (which are not solutions of (48) with no forcing) for which we can interpret the heat source associated with the passage of the wave. Indeed, as we have mentioned earlier, this is the most convenient path to tread when heat-forcing is included; to solve for the wave solution, given some appropriate forcing, is not a realistic way forward because the nature of the heating is not well-understood.

In conclusion, this presentation has placed the theory of gravity waves propagating in the atmosphere on a sound mathematical basis. This has involved, in part, the need to construct a careful and carefully-defined asymptotic procedure, which makes clear the route that must be taken to generate a suitable theory, but we can go further. So, apart from the need for a systematic study of our wave equation (as mentioned above), the way is open for more extensive investigations. These should include: the construction of a different background state of the atmosphere, with a realistic wind which blows from an arbitrary direction; associated with this, a wave that is propagating in a general direction over the Earth and, last but not least, to show how the waves are related to a mechanism that generates them, e.g. from flow over a mountain range. There is, we submit, still much to do.