Exact nonlinear mountain waves propagating upwards

We derive an exact solution to the nonlinear governing equations for mountain waves in the material (Lagrangian) framework. The explicit specification of the individual particle paths enables a detailed study of the flow consisting of oscillations superimposed on a mean current propagating upwards.


Introduction
Due to air compressibility and stratification, atmospheric gravity waves may propagate vertically as well as horizontally. A common type are mountain waves, generated downwind of a long topographic barrier (e.g. a mountain range) when stable air flow passes over the mountain top as strong winds blow in a direction with a significant vector component perpendicular to the orientation of the mountain range. The orographic lift of the air up the slope is relatively uniform along the rising terrain profile, with velocity variations mainly confined to the upward normal direction, but on passing the crest, a much more complicated perturbed flow pattern emerges downstream, comprising a laminar-flow region of smooth mountain wavesoscillatory changes of atmospheric pressure, air temperature and flow velocity that propagate away on the lee (downwind) side-and also layers with clear air turbulence (erratic air flow with high-frequency oscillations in the absence of any visual clues, such as clouds). The smooth mountain waves are formed near the level of the mountain top but as friction slows the * Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. descending air flow near the ground, rotors-turbulent vortices spinning rapidly about an axis of rotation parallel to the mountain range-are generated in the lower levels of the downdraft. Mountain waves may also propagate above the mountain top level, in which case there is a noticeable amplitude amplification with increasing altitude, owing to the decrease of mean air density. Wave steepening and overturning-in a manner roughly analogous to a breaking ocean wave-might occur aloft, often as these waves enter the stratosphere.
The glossary of meteorology recognises two divisions of mountain waves: vertically propagating waves and trapped lee waves.
Vertically propagating mountain waves occur when above the mountain peak the air temperature and density decrease, and the wind speed does not increase significantly with altitude. These atmospheric waves amplify with height and propagate upward into the lower stratosphere, unless they steepen and break before reaching the tropopause, in which case the generated clear air turbulence is a significant hazard to aviation. Strong vertical currents (in excess of 30 m s −1 ) may also develop. This is indicative of the fact that the vertical velocity component in vertically propagating mountain waves is quite significant, while typically in atmospheric flows it is several orders of magnitude smaller than the horizontal velocity component (see [32]); for example, the field data in [3], gathered from flights over the Apennines in Italy, identifies atmospheric regions without breaking waves at 6-8 km altitude, extending over tens of kilometres, over which the peak-to-peak amplitudes in the vertical and horizontal velocities are both about 5 m s −1 . When sufficient moisture is present in the layer above the mountain top, vertically propagating mountain waves produce fascinating lenticular clouds (see figure 1). Even though lenticular clouds seem to have a stationary position, the air particles are constantly moving through them.
Trapped lee waves occur for moderate wind speeds within a stable layer in the lower troposphere, beneath a thermal inversion and/or if stronger winds blow in the middle and upper troposphere. Blocking or gap flow around the mountain are averted if an approaching air flow has enough momentum, so that a moderate wind is forced upwards, overshooting the mountain peak. The presence of stronger winds aloft prevents an upward wave propagation and, because the wind energy is trapped within the stable layer, these mountain waves persist for long distances downwind (for tens or even hundreds of km downstream), at altitudes close to the mountain peak, whereas near the ground frictional effects alter the regular wave pattern. At the same level in the atmosphere, warm humid air is lighter than warm dry air since a molecule of water vapour weights appreciably less than a molecule of nitrogen or oxygen. Consequently, if humidity is high, trapped mountain waves brought about by anabatic winds (flowing up mountain slopes not covered by snow and warmed during the day, so that the air in contact with them becomes warmer and less dense) and strong upper level winds can be revealed by cap clouds over the mountain peak (see figure 2). As the moisture-packed warmer rising air comes into contact with the chilly layer near the mountain top, cloud droplets form by condensation. The horizontal strong winds over the mountain prevent the upward motion of the hitherto rising air. On the lee side, the air warms up as it descends down the slope and the cloud dissipates. Note that while cap clouds occur directly over a mountain peak, lenticular clouds form on the leeward side of the mountain.
The presence of a thermal inversion above the mountain top-a relatively thin (ranging from a few tens to several hundred metres) but quite long warmer layer aloft-also facilitates the generation of trapped lee waves: the upper thermal inversion layer acts like a lid on the updraft, with gravity the restoring force that pulls a raising air parcel, which at this height is colder and more dense than the surrounding air, back down. In trying to attain an equilibrium position, the air parcel will overshoot and undershoot that natural position each time it is rising or sinking because of its own momentum, thus initiating a buoyancy oscillation that propagates over Figure 1. Spectacular lenticular clouds photographed on 3 January 2014 over the highest peak (3993 m) of the Sangre de Cristo mountains in Northern New Mexico. These lensshaped clouds can be single or stacked all the way up to the tropopause, provided that sufficient moisture is present in the layers above the mountain top. Near the crest of the mountain wave that propagates upward in a tilted direction, the warmer air rises into a colder layer and cloud droplets form by condensation. Subsequently, as the wave crest passes, the droplets descend into warmer air and gradually evaporate in the trough of the wave. This up and down motion of the air does not carry the clouds along with the flow, rather, clouds are continuously formed at the wave crest and eroded as the wave trough approaches. Thus lenticular clouds seem motionless in the sky. It is not uncommon to see them stacked or in an elongated line over the lee side of the mountains, if multiple layers of cold air reach dew point at the crest of downwind mountain waves. Reproduced with permission from NOAA Photo Library, Photographer: Richard Hasbrouck.
hundreds of kilometres on the lee side of the mountain (see [29]). Spectacular undular cloud formations (see figure 3) can form by condensation near the top boundary of the inversion layer: the crests of the undulations rise above the lifting condensation level (the height of cloud base), while the wave troughs, where the clouds erode by evaporation, remain below it. Thermal inversions occuring when a cold air mass (e.g. a shallow layer of polar air moving into lower latitudes) undercuts a warmer air mass often feature a sloping upper boundary, while subsidence inversions, common over the northern continents in winter (when a widespread layer of air descends under a large high-pressure centre, being compressed and heated by the resulting increase in atmospheric pressure) are nearly horizontal.
While most of the mountain waves are invisible, we have discussed some scenarios when their presence is revealed by their interaction with high-altitude clouds. Let us also note the possible visualisation of the ground-level air flow generated by mountain waves on the leeward side of the mountain, comprising turbulent rotor clouds. The most common thermal inversion is due to the cooling of the Earth's surface at night by longwave radiation emission to spacethe ground temperature lowers because of the radiation deficit and the layer of air near the ground cools by conduction, with the transfer of heat energy from the initially warmer air to the cooler ground creating a thermal inversion that is most distinct early in the morning. This thermal inversion is stronger in a mountain valley since once the Sun goes down, the denser cold air located near the mountaintops during the day starts to sink into the valley. If there is ample amount of moisture (as in the valleys of the Blue Ridge Mountains which stretch from Pennsylvania to Georgia across the eastern US), the lower layer becomes saturated and a sea of low-lying clouds rests in the valley at sunrise. Mountain waves can make these clouds swirl around the valley before the air becomes warm enough to evaporate the water droplets.
In this paper we pursue a theoretical investigation of upward-propagating short mountain waves, with wavelengths of the order of 2 km, smaller than current grid spacings in global weather and climate prediction models [31]. Most recent studies of mountain waves are devoted to high-resolution numerical simulations (see the state-of-the-art studies [19,21,22]) that quantify specific flow features. The most fundamental properties of mountain waves are typically examined by means of linear theory for steady-state two-dimensional airflow within the setting of the Boussinesq approximation, assuming the atmosphere to be inviscid and adiabatic (see the discussion in [12,31]). However, the available linear theory considerations are not reached by means of a consistent use of non-dimensionalisation and scaling of the governing equations-some phenomenological simplifications are required (see the discussion in section 4.5). Moreover, given that mountain waves are typically large-amplitude flows, nonlinear effects are quite important. Rather than relying on models that arise either by heuristic simplifications of the governing equations or from ad hoc modelling based on observational data, we advocate a systematic derivation of the nonlinear governing equations for mountain waves from the general equations for inviscid compressible flow (the Euler equation, the equation of mass conservation, the equation of state for the air and the first law of thermodynamics). After recalling the governing equations in section 2, we present an explicit solution from the Lagrangian perspective, by providing in section 3 explicit formulas for the motion of individual air particles in the adiabatic dry flow of an upwards propagating mountain wave. Note that the unwieldiness of the Lagrangian approach compared to the Eulerian viewpoint of fluid flows owes precisely to the richness of its kinematical information [2,13]. In our setting it offers detailed insight into basic features of the flow pattern, revealing its organising structures (see section 4). Our solution can be regarded as an improvement of the results in [23], where exact solutions for incompressible inviscid air flow on an interface between two regions of constant density were obtained. We can accommodate a continuous density variation as well as a tilted direction of wave propagation.

Preliminaries
The scales relevant for mountain waves make the effects of the Earth's rotation and sphericity unimportant and we can therefore use a Cartesian representation of the equations of motion (see [32]). We consider a (right-handed) coordinate system with the x ′ -axis pointing in the horizontal direction of wave propagation, with the horizontal y ′ -axis orthogonal to it and with the z ′ -axis upward.
Using primes for physical/dimensional variables-they will be removed when we nondimensionalize-we denote by u ′ , v ′ , w ′ , the corresponding fluid velocity components. If t ′ stands for time, g ′ ≈ 9.8 m s −2 is the (constant) gravitational acceleration at the Earth's surface, ρ ′ is the air density, T ′ is the (absolute) temperature and P ′ is the atmospheric pressure, the governing equations are the Euler equations (see [20]) coupled with the equation of mass conservation the equation of state for an ideal gas, and the first law of thermodynamics Here R ′ ≈ 287 m 2 s −2 K −1 is the gas constant for dry air, c ′ p ≈ 1000 m 2 s −2 K −1 is the specific heat of dry air at an atmospheric pressure of 1000 mb and Q ′ is the heat-source term.
The inviscid governing equations (1)-(6) model mountain waves moving in a laminar layer of finite depth and propagating in a horizontally-upward tilted direction over large distancesthe inviscid setting being adequate if the mountain is tall and steep enough to stand out of the frictional layer (see [29,35]). Beneath this layer, close to the ground, the atmospheric flow is no longer laminar due to the low-level turbulence that is typically associated with atmospheric rotors-overturning eddies concentrating frictional boundary-layer vorticity lifted off the surface (see [33]). On the other hand, above this laminar-flow layer wave breaking is very likely. Indeed, due to the decreasing air density, even small-amplitude waves in the lower troposphere will eventually become large as they propagate far into the upper atmosphere. The wave properties are considerably altered across the tropopause, with the greater static stability of the stratosphere promoting wave breaking (see the discussion in [30]).
In the study of the governing equations (1)-(6) it suffices to keep track of the velocity field (u ′ , v ′ , w ′ ), of the pressure P ′ and of the density ρ ′ . The ideal gas law (5) then specifies the temperature T ′ and the first law of thermodynamics (6) identifies the associated heat sources. To deal in general with the intricate issue of air moisture, one can argue that what is important is not the water vapour per se, but the availability of a heat source to capture the latent heat released when water droplets are formed (see the discussions in [7][8][9]). However, since the orographic lifting of air particles can be considered a dry adiabatic process outside regions of active precipitation (see [20]), for mountain waves we may set Q ′ ≡ 0 in (6).
Within a few percent, the average density in the lower troposphere is aboutρ ′ ≈ 1 kg m −3 (see [34]). With L ′ = 2 km and U ′ = 20 m s −1 suitable physical scales for short mountain waves, we introduce dimensionless variables t, x, y, z, u, v, w, ρ, P and T by the normalization factors being L ′ /U ′ ≈ 100 s,ρ ′ U ′2 ≈ 10 −2 atm and (U ′2 /R ′ ) ≈ 1 • K. We obtain the nondimensional version of the governing equations (1)- (6): ∂v ∂t where From (11) and (12) we get where denotes the material derivative of the variable of state f. Integrating (13) for each air particle from a state at pressure P and temperature T to a state in which the pressure is P (corresponding to 1000 mb) and the temperature is T, we obtain that the potential temperature T, defined by is constant along particle paths. Note that the first law of thermodynamics expresses the conservation of energy, while the second law of thermodynamics sets limits for the transformations between heat energy and the sum of kinetic and potential energies, stating that the entropy of a system that does not exchange energy or mass with its surroundings can not decrease (see the discussion in [1]). In our setting, the phenomenological aspects of the second law of thermodynamics are addressed since the potential temperature T, defined by (14), may be considered as an alternative variable for entropy (see [11]).
We obtain the leading-order dynamics of two-dimensional mountain waves by imposing the conditions of no flow and no variation along the wavefront, i.e. in the y-direction. The system (7)-(12) then simplifies to For any solution (u, w, P, ρ) to the system (15)- (17), the associated temperature T can be determined from equation (18), with the compatibility equation (19) imposed by the first law of thermodynamics and ensuring that the two-dimensional mountain wave propagation outside regions of active precipitation occurs within the confines of the second law of thermodynamics, as shown by the above discussion of the potential temperature of orographic liftings.

Main result
Within the Lagrangian framework, we now present an explicit solution to the governing equations (15)- (19) for two-dimensional mountain waves by specifying, at time t, the positions of the moving air particles in terms of the labelling variables (a, b) and the parameters k > 0 (wavenumber), c > 0 (wave speed) and U, W ∈ R. The labelling variable a runs over the positive numbers (which parametrize the leeward direction), while the label b ∈ (b 1 , b 0 ) with b 1 < b 0 < 0 captures the height of the layer of laminar flow, with the amplitude of the oscillations of a particle increasing with height. Since we used typical scales in the derivation of the nondimensional system (15)-(17), the most realistic choice is k = 2π (corresponding to unit wavelength), with integer multiples describing shorter waves. The flow pattern (20) represents a height-dependent oscillatory motion superimposed on a uniform current with velocity (U, W), and is obtained by merging two basic types of atmospheric motion-uniform linear and circular. The particle paths encoded by the parametrization (20) are trochoidal (see section 4) and the constraint b 0 < 0 has to be imposed since in the limit b ↑ 0 the vorticity of the flow becomes unbounded (see section 4.3); furthermore, for b > 0, the curves (20) are, at any fixed t, parametrizations of prolate cycloids with self-intersections (see [16]), and thus cannot be particle paths (since a self-intersection point corresponds to different values of the labelling variable a, placing different particles at this location). The restriction on the range of the label b reflects the fact that turbulence is typically encountered beneath and aloft of the laminar flow region governed by the equations (15)- (17), whose wavy lower and upper boundaries correspond to setting b = b 1 and b = b 0 in (20), respectively. Let us also note than an Ansatz of type (20) with U = W = 0 was pioneered two centuries ago in [15] for homogeneous inviscid fluid flows in the context of travelling deep-water waves (see the discussion in [5]), being rediscovered independently afterwards [14,25]. It was adapted to edge waves along a sloping beach and to geophysical ocean flows in equatorial regions and at midlatitudes (see [4,6,10,18,24]) but the more intricate setting of compressible atmospheric flows was not explored hitherto. In this context, let us also point out that a mean background wind is typical for the hotspots of mountain waves-the South American Andes, the European Alps and the mountain ranges of Scandinavia, New Zealand and Antarctica (see [17]). Our aim is to show that, given a density in the form of a decreasing function ρ = ρ(b) and a background wind (U, W), for any wavenumber k > 0 there is a wave speed c(k) > 0 and an associated pressure distribution where P * is a positive constant, such that the velocity field (u, w) determined by (20) solves the system (15)- (17). As we will show in section 4, the monotonicity of ρ(b) implies that the density and the pressure decrease with height. To prove the above claim, let us denote ξ = k(a − ct) and compute Let us now consider a density dependence on the (x, z, t)-variables of the form Since (20) yields this corresponds to a labelling-variables dependence ρ = ρ(ξ, b), with due to (24). For a density of the form ρ = ρ(b) we therefore have which implies (17) since (23) holds. It remains to find the dispersion relation that ensures the validity of (15) and (16) for a pressure distribution of the form (21). With this aim, we first use the fact that the acceleration of a particle with labels (a, b) is the total time-derivative of the velocity vector (u, w), and can thus be computed by taking the time derivative of the components of equation (24): Thus we can rewrite (15) and (16) as We now invoke (22) to express (26) in the equivalent form Expressing ∂ 2 P ∂a∂b from each of the two equations in (27), we obtain the compatibility condition which, given that the density ρ is not constant, yields for c > 0 the dispersion relation If (28) holds, then the system (27) simplifies to and (21) validates the solution. Note that since k = 2π L , where L > 0 is the wavelength, the dispersion relation (28) shows that the wave speed is proportional to the square root of the wavelength. While the dispersion relation (28) differs from that obtained by relying on the linear Boussinesq approximation to the equations of motion (see section 6a in [29]), they both predict that mountain waves propagate faster with increasing wavelength.

Discussion
We now highlight the main features of the nonlinear atmospheric flow pattern defined by (20).

The laminar layer
The waves described by (20) within the Lagrangian framework propagate adiabatically through a laminar layer, whose wavy lower and upper boundaries correspond to the label values b = b 1 and b = b 0 , respectively. Beneath the mountain top is the planetary-boundary layer, where overturning eddies concentrating frictional boundary-layer vorticity (rotors) are indicative of low-level turbulence (see figure 4).

Particle paths
For every fixed label (a, b), the map t → (x(a, b; t), z(a, b; t)) represents the parametrization of a trochoid: an undulation with pronounced crest-trough asymmetry, with wide crests and relatively narrow troughs, in contrast to the sinusoidal wave profiles specific to linear theory (see figure 5). For visualisation it is helpful to note that the projections of a circular helix on a fixed plane give all the possible shapes of trochoids-trochoidal curves are, by the Montucla-Guillery theorem, the shadows of a spring.

Velocity field
As alluded to in the introduction, (24) shows that the fluctuations of the horizontal and vertical velocity components about the constant underlying currents U and W, respectively, are of the same order of magnitude ce kb and increase with height. Note that the underlying current (U, W) has typically both components positive, with the updraft W > 0 due to the upward deflection of the air currents by the mountain range.
The inverse of the Jacobian matrix (22) We can express the vorticity of the flow by means of the chain rule as The above formula shows that the magnitude of the vorticity γ = γ(b) increases with height, since γ(b 1 ) < 0 and Sketch of a typical air flow comprising a vertically propagating mountain wave: the orographic lift along the mountain slope is relatively uniform, the mountain wave moves in a layer of laminar flow, with turbulence typically generated beneath, due to overturning eddies (rotors) above the ground on the lee side, and above, due to wave breaking aloft.
This conclusion agrees with the observation that mountain waves can generate strong vortices at high altitudes. Note that the vorticity γ(b) becomes unbounded in the limit b → 0, a singularity indicative of the likelihood of breaking waves.

Density, pressure and temperature
Since the density ρ = ρ(b) is a decreasing function of the label b, due to (22) and (30) 1 − e kb cos ξ 1 − e 2kb ⩽ 0 , because b < 0. This shows that the density decreases with height. Note also that the dispersion relation (28) and the second relation in (26) ensure that pressure decreases with altitude, due to ∂P ∂z = gρ(b) e kb cos ξ − 1 ⩽ 0 .
Let us also note that the flow is adiabatic since with ρ = ρ(b) and P = P(b), the equation of state (11) ensures T = T (b), so that the temperature of an air parcel does not change during its motion. Consequently, (19) holds.

Comparison with the classical linear theory
We now discuss the added value that the presented exact solution brings to the classical linear theory of mountain waves.
The classical linear theory turns out to be quite successful in a statistical sense but on a caseto-case basis there are still many discrepancies with field data (see [28,29]). These are due  (20) with an updraft (that is, for W > 0), throughout the laminar layer. Each particle oscillates and rises, propagating upwards in a tilted direction. The decrease of air density with height amplifies the amplitude of the oscillations aloft, while in the lower part of the laminar layer (below the level of the mountain top, corresponding to the label b = b 1 ) the oscillations are almost imperceptible, while as one descends further in the planetary boundary layer towards the ground, turbulence develops. The lifting of the particles in the laminar layer is a dry adiabatic process but at the upper wavy boundary of the laminar layer condensation occurs near the wave crest of the particle trajectory depicted in blue (corresponding to the label b = b 0 ). to the fact that the available oscillatory motions, representing small perturbations of simple exact horizontal-flow solutions, are not obtained by means of a systematic asymptotic analysis but rely on arguing that pressure variations are not important in the generation of density anomalies in mountain waves and on using the Boussinesq approximation, whereby density is assumed constant except in the buoyancy term. However, especially in the case of vertically propagating mountain waves (that are not trapped in a shallow layer), the Boussinesq approximation is not very accurate since air is compressible under its own weight. Since the previously known explicit solutions do not comprise waves, the derived characteristics of their small wave-perturbations cannot be expected to provide a reliable standard for comparison. It is only qualitatively that we may expect, and do indeed see, similarities with the exact solution provided in the present paper.
The comprehensive discussions of linear mountain waves in [12,28,29] show that the above two assumptions are uncircumventable cornerstones of the classical linear theory. Rather than rehearse the arguments, we choose what is possibly the simplest physically relevant settingthat of flows reaching a steady state-to illustrate the key strains of thought.
The background state for the classical linear theory of two-dimensional mountain waves is a horizontal hydrostatic flow described by specifying u = u 0 (z), T = T 0 (z), w ≡ 0, with P = p 0 (z) and ρ = ρ 0 (z) determined from the corresponding form    dp 0 dz = −g ρ 0 (z) , of the equations (16) and (18); the equations (17) and (19) always hold for these basic flows. From (31) we derive the first-order linear differential equation whose solution is given by where z 0 is the height at which the reference pressure value P (corresponding to 1000 mb) is attained. With the background state fully determined, one examines the small orographic perturbations by representing each variable by means of an asymptotic expansion in terms of a small parameter ε, assuming that the flow will eventually reach a steady state. Upon substituting (32) into the governing equations (15)- (19), we obtain at leading order O(ε) the following equations for the perturbation quantities Denoting by T 0 = T 0 P p 0 µ the potential temperature of the background flow governed by (31), we now introduce Making the Boussinesq approximation-that the background state has constant density and that the effect of the density variations of the perturbation are only important as they affect the buoyancy (the last term in (34))-simplifies the system (33)- (35) to The equations (44)-(46) are coupled with (43), with ρ 0 constant throughout. Eliminating p from (44) and (45), taking (43) into account and expressing ∂u ∂x and ∂ 2 u ∂x∂z from (46), lead us to a linear partial second-order homogeneous differential equation for w: with the functional coefficient called the Scorer parameter (see [26]). The coefficient S(z) is typically dominated by the first buoyancy term but in regions of strong wind shear (when the wind changes sharply with height, often near the surface and in the vicinity of a jet streak) the second term may become important.
A detailed Fourier analysis of equation (47) leads to a classification of mountain waves into two types-trapped waves and vertically propagating waves-according to the vertical profiles of the background temperature and wind velocity. For reviews of observations and simulations verifying aspects of the linear theory we refer to [3,27,36]. The above considerations show that our approach to derive the exact nonlinear solution differs from that used in the classical linear theory. We avoid making extraneous approximations and the solution is precise and clear in its validity, detail, and structure. While it describes ideal conditions that do not capture all the complexities of observed physical behaviour, the exact solution provides the basis for a perturbation construction: small, general deviations can be superimposed on it and the resulting approximate system can be investigated-this is work in progress.

Conclusions
We have presented an exact solution to the nonlinear governing equations for two-dimensional mountain waves by specifying the individual particle paths. We have investigated the nondimensional governing equations for compressible inviscid flow, without relying upon commonly used simplifying assumptions (like hydrostatic balance or the Boussinesq approximation). While the analysis in the Lagrangian framework is somewhat intricate, the gain consists in unraveling the detailed structure of these adiabatic atmospheric flows. The obtained solution accommodates all the salient features of vertically propagating mountain waves: periodic buoyancy oscillations with amplitudes that increase with height, superimposed on an underlying air current and comprising a mechanism for an increase of the vorticity with altitude.
While one cannot expect the presented solution to reproduce faithfully all the fine detail of the very complicated dynamical processes involved in such flows, is represents a useful starting point for a perturbation analysis. The availability of detailed features for the new exact solution opens up new possibilities for systematic investigations: this exact solution, with a more intricate structure than that of the horizontal hydrostatic flows used as background flows in the classical linear theory, can be prescribed as a background state and the general properties of the evolution of its small perturbations can be readily studied within the framework of linear wave theory. The practical benefit of having an intricate but analytically tractable basic flow is that it offers insight into the dynamics of mountain waves that are not merely perturbations of horizontal hydrostatic flows. Current advances in computing enhance the feasibility of numerical simulations of perturbed flows that can capture a wider range of effects, whose importance can be ascertained by comparison with the obtained exact solution.

Data availability statement
The data that support the findings of this study are openly available at the following URL/DOI: www.weather.gov/mrx/mountainwaves.