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Tsunami-generated linear acoustic–gravity waves in the atmosphere with altitude-dependent vertical stratification and horizontal background winds are studied with the long-term goal of real-time tsunami warning. The initial-value problem is examined using Fourier–Laplace transforms to investigate the time dependence and to compare the cases of anelastic and compressible atmospheres. The approach includes formulating the linear propagation of acoustic–gravity waves in the vertical, solving the vertical displacement of waves and pressure perturbations numerically as a set of coupled ODEs in the Fourier–Laplace domain, and employing den Iseger’s algorithm to carry out a fast and accurate numerical inverse Laplace transform. Results are presented for three cases with different atmospheric and tsunami profiles. Horizontal background winds enhance wave advection in the horizontal but hinder the vertical transmission of internal waves through the whole atmosphere. The effect of compressibility is significant. The rescaled vertical displacement of internal waves at 100-km altitude shows an arrival at the early stage of wave development due to the acoustic branch that is not present in the anelastic case. The long-term displacement also shows an O(1) difference between the compressible and anelastic results for the cases with uniform and realistic stratification. Compressibility hence affects both the speed and amplitude of energy transmitted to the upper atmosphere because of fast acoustic waves.

A multilayer Fourier method is developed for modeling gravity waves above some altitude at which the vertical velocity is given as a boundary condition. The method is described in a general context in which the altitude and time variation of Fourier components for fixed horizontal wavenumbers is specified by a linear homogeneous partial differential equation (PDE) of any order, with coefficients varying with altitude. The coefficients are required to meet certain conditions so that causality is preserved and upgoing and downgoing modes can be defined. It is shown that causality is not preserved unless dissipative modes are included. An imaginary frequency shifting technique is introduced to allow upgoing and downgoing modes to be defined in certain situations that would otherwise be problematic. The pervasive problem of numerical swamping, endemic to multilayer approaches to viscous and thermally diffusive gravity-wave problems, is solved using a scattering matrix method, related to a method from seismology, which differs fundamentally from earlier gravity-wave solution methods cited herein. The method is applied to gravity waves in the linear, nonhydrostatic, anelastic, viscous, and thermally diffusive case with vertically varying background winds, in addition to certain limiting cases. The PDE in altitude and time is obtained from a dispersion relation that includes odd powers of the vertical wavenumber, making necessary the aforementioned imaginary frequency shifting technique.

In previous work (Knight et al., 2019) a multilayer Fourier method was introduced for obtaining solutions for a partial differential equation (PDE) associated with a dispersion relation for linear gravity waves in a vertically varying, anelastic, nonhydrostatic, viscous, and thermally diffusive atmosphere over some altitude range, given boundary conditions that allow waves to enter at the lower boundary and exit at the top. The previous work described an exact solution method for the dispersion-relation PDE using all wave modes. The current work describes a type of approximate solution, based on a single upgoing gravity-wave mode for each Fourier component, which is computationally cheaper and simpler to implement. Two different approaches to defining the single-mode solution are presented, one based on the Wentzel–Kramers–Brillouin (WKB) method and the other based on a simplified version of the multilayer approximation. The two approaches are shown to be mathematically equivalent. The accuracy of the single-mode approximation is investigated through comparison with the all-mode solution for a realistic example in which gravity waves generated by a convective storm in the troposphere propagate into the upper thermosphere and for which the single-mode approximation gives reasonably accurate results. It is necessary to include imaginary frequency shifts in some Fourier components in order for the different types of modes (e.g., upgoing vs. downgoing and gravity-wave vs. dissipative) to be well-defined in the presence of molecular viscosity. A systematic approach to determining the smallest possible imaginary frequency shift for any given Fourier component is introduced. The new method efficiently suppresses a specific type of numerical artifact associated with large imaginary frequency shifts.

An approximate solution for the steady linear internal wave field generated by a localized, horizontally moving source in a thermocline is derived using ray and WKB (Wentzel–Kramers–Brillouin) theory. The waves are assumed to be steady in a reference frame moving with the source velocity. This solution is shown to agree in the limit of propagating waves with an existing solution that is expressed in terms of an eigenfunction expansion. The WKB method is shown to reproduce all the factors in the eigenfunction solution, not just the eigenfunctions themselves. It also reveals the physical significance of the terms in the eigenfunction solution. Furthermore, the WKB solution suggests an alternate way to compute the wave field numerically that is computationally more efficient.

On 14 July 2014 during the Deep Propagating Gravity Wave Experiment (DEEPWAVE), aircraft remote sensing instruments detected large-amplitude gravity wave oscillations within mesospheric airglow and sodium layers at altitudes z ~ 78–83 km downstream of the Auckland Islands, located ~1000 km south of Christchurch, New Zealand. A high-altitude reanalysis and a three-dimensional Fourier gravity wave model are used to investigate the dynamics of this event. At 0700 UTC when the first observations were made, surface flow across the islands’ terrain generated linear three-dimensional wave fields that propagated rapidly to z ~ 78 km, where intense breaking occurred in a narrow layer beneath a zero-wind region at z ~ 83 km. In the following hours, the altitude of weak winds descended under the influence of a large-amplitude migrating semidiurnal tide, leading to intense breaking of these wave fields in subsequent observations starting at 1000 UTC. The linear Fourier model constrained by upstream reanalysis reproduces the salient aspects of observed wave fields, including horizontal wavelengths, phase orientations, temperature and vertical displacement amplitudes, heights and locations of incipient wave breaking, and momentum fluxes. Wave breaking has huge effects on local circulations, with inferred layer-averaged westward flow accelerations of ~350 m s−1 h−1 and dynamical heating rates of ~8 K h−1, supporting recent speculation of important impacts of orographic gravity waves from subantarctic islands on the mean circulation and climate of the middle atmosphere during austral winter.

Tsunami-generated acoustic–gravity waves have been observed to propagate in the atmosphere up to the ionosphere, where they have an impact on the total electron content. The authors simulate numerically the propagation of two-dimensional linear acoustic–gravity waves in an atmosphere with vertically varying stratification and horizontal background winds. The authors’ goal is to compare the difference in how much energy reaches the lower ionosphere up to an altitude of 180 km, where the atmosphere is assumed to be anelastic or fully compressible. The authors consider three specific atmospheric cases: a uniformly stratified atmosphere without winds, an idealized case with a wind jet, and a realistic case with an atmospheric profile corresponding to the 2004 Sumatra tsunami. Results show that for the last two cases, the number and height of turning points are different for the anelastic and compressible assumptions, and the net result is that compressibility enhances the total transmission of energy through the whole atmosphere.

Mesospheric bores have been observed and measured in the mesopause region near 100-km altitude, where they propagate horizontally along a duct of relatively strong density stratification. Here, a weakly nonlinear theory is developed for the description of these mesospheric bores. It extends previous theories by allowing internal gravity wave radiation from the duct into the surrounding stratified regions, which are formally assumed to be weakly stratified. The radiation away from the duct is expected to be important for bore energetics. The theory is compared with a numerical simulation of the full Navier–Stokes equations in the Boussinesq approximation. Two initial conditions are considered. The first is a solitary wave solution that would propagate without change of form if the region outside the duct were unstratified. The second is a sinusoid that evolves into an undular bore. The main conclusion is that, while solitary waves and undular bores decay by radiation from the duct, they can survive as significant structures over sufficiently long periods (~100 min) to be observable.

Expressions are derived for the wave steepness η z of mountain waves, defined as the altitude ( z ) derivative of the vertical displacement η . The derivation begins with a known Fourier integral for η . The altitude derivative of the Fourier integral for η introduces a singularity in the integrand that is not absolutely integrable at any altitude. It is seen, nonetheless, that the altitude derivative can be moved under the integral sign for altitudes above the ground, but not at the ground, for which a special expression is found relating wave steepness to the mountain height function. It is shown that the upwind limit of wave steepness at the ground is zero, but the downwind limit is nonzero in general, and an expression for the downwind limit in terms of the mountain height function is given. It is also found that the imaginary part of complex wave steepness diverges approaching the ground, and an asymptotic formula is given. The results are applied to three mountain topography shapes: elliptical Gaussian, circular with algebraic decay, and infinite ridge.

Effects of horizontal geometrical spreading on the amplitude variation with height of linear three-dimensional hydrostatic orographic gravity waves (OGWs) are quantified via relevant simplifications to the governing transform relations, leading to analytical solutions. The analysis is restricted to elliptical Gaussian obstacles with principal axes aligned parallel and perpendicular to unidirectional shear flow and to vertical displacement and steepness amplitudes, given their relevance to OGW drag parameterizations in global models. Two solutions are derived: a “small l” solution in which horizontal wavenumbers l orthogonal to the flow are taken to be much smaller than those parallel to the flow, and a “single k” solution in which horizontal wavenumbers k parallel to the flow have a single mean value. The resulting analytical relations, valid for arbitrary vertical profiles of upstream winds and stability, depend only on the obstacle’s elliptical aspect ratio β and a normalized height coordinate incorporating wind and stability variations. These analytical approximations accurately reproduce the salient features of the exact numerical transform solutions. These include monotonic decreases with height that asymptotically approach z−1/2 forms at large z and strong β dependence in amplitude diminution with height. Steepness singularities close to the surface are shown to be a mathematical consequence of the Hilbert transform approach to deriving complex wavefield solutions. These approximate analytical solutions for horizontal geometrical spreading effects on wave amplitude highlight the importance of this missing physics for current parameterizations of OGW drag and offer an accurate and efficient means of incorporating some of these omitted effects.

Numerical transform solutions for hydrostatic gravity waves generated by both uniform and sheared flow over elliptical obstacles are used to quantify effects of horizontal geometrical spreading on amplitude evolution with height. Both vertical displacement and steepness amplitudes are considered because of their close connections to drag parameterizations in weather and climate models. Novel diagnostics quantify the location and value of the largest wavefield amplitudes most likely to break at each altitude. These horizontal locations do not stray far from the obstacle peak even at high altitudes. Resulting vertical profiles of wave amplitude are normalized to remove density and refraction effects, thereby quantifying the horizontal geometrical spreading contribution, currently absent from parameterizations. Horizontal geometrical spreading produces monotonic amplitude decreases with height through wave-action conservation as waves propagate into progressively larger horizontal areas. Accumulated amplitude reductions are appreciable for all but the most quasi-two-dimensional obstacles with long axes orthogonal to the flow, and even these are impacted appreciably if the obstacle is rotated by more than 20°–30°. Profiles are insensitive to the obstacle’s functional form but vary strongly in response to changes in its horizontal aspect ratio. Responses to background winds are captured by a vertical coordinate transformation that remaps profiles to a universal form for a given obstacle. These results show that horizontal geometrical spreading has comparable or larger effects on wave amplitudes as the refraction of vertical wavenumbers and thus is important for accurate parameterizations of wave breaking and drag.

A vertical eigenfunction equation is solved to examine the partial reflection and partial transmission of tsunami-generated gravity waves propagating through a height-dependent background atmosphere from the ocean surface into the lower thermosphere. There are multiple turning points for each vertical eigenfunction (at least eight in one example), yet the wave transmission into the thermosphere is significant. Examples are given for gravity wave propagation through an idealized wind jet centered near the mesopause and through a realistic vertical profile of wind and temperature relevant to the tsunami generated by the Sumatra earthquake on 26 December 2004.

The Deep Propagating Gravity Wave Experiment (DEEPWAVE) project was conducted over New Zealand and the surrounding regions during June and July 2014, to more fully understand the generation, propagation, and effects of atmospheric gravity waves. A large suite of instruments collected data from the ground to the upper atmosphere (~100 km), with several new remote-sensing instruments operating on board the NSF Gulfstream V (GV) research aircraft, which was the central measurement platform of the project. On 14 July, during one of the research flights (research flight 23), a spectacular event was observed as the GV flew in the lee of the sub-Antarctic Auckland Islands (50.7°S). An apparent "ship wave" pattern was imaged in the OH layer (at ~83.5 km) by the Utah State University Advanced Mesospheric Temperature Mapper and evolved significantly over four successive passes spanning more than 4 h. The waves were associated with orographic forcing generated by relatively strong (15-20 m/s) near-surface wind flowing over the rugged island topography. The mountain wave had an amplitude T_ ~ 10 K, a dominant horizontal wavelength ~40 km, achieved a momentum flux exceeding 300 m2 s-2, and eventually exhibited instability and breaking at the OH altitude. This case of deep mountain wave propagation demonstrates the potential for strong responses in the mesosphere arising from a small source under suitable propagation conditions and suggests that such cases may be more common than previously believed. © 2016. American Geophysical Union. All Rights Reserved.

Mesoscale model simulations of representative trade winds impinging upon the Big Island of Hawaii are diagnosed for their mountain-wave characteristics by coupling a mesoscale model to a Fourier method. Localized phase-averaged wave momentum fluxes are calculated, which facilitates the study of wave generation from fine-scale topographic features. We find that the wave momentum fluxes are dominated by forcing from subsidiary topographic peaks, with the broader island topography controlling flow splitting and lee vortex generation. Waves also arise at the far northern and southern extremities of the island by acceleration of split flow. The strength of the local momentum fluxes proves to be sensitive to a small change in the incident flow direction. Areally integrated fluxes (wave drag) align closely with the incident flow direction and are an order of magnitude smaller than linear predictions and an order of magnitude larger than corresponding dividing streamline predictions. We briefly discuss the relevance of these results to the parameterization of subgrid-scale mountain-wave drag in climate and weather models.

Fully nonlinear mesoscale model simulations are used to investigate the momentum fluxes of gravity waves that emerge at a “far-field” height of 6 km from steady unsheared flow over both an axisymmetric and elliptical obstacle for nondimensional mountain heights ĥm = Fr−1 in the range 0.1–5, where Fr is the surface Froude number. Fourier- and Hilbert-transform diagnostics of model output yield local estimates of phase-averaged momentum flux, while area integrals of momentum flux quantify the amount of surface pressure drag that translates into far-field gravity waves, referred to here as the “wave drag” component. Estimates of surface and wave drag are compared to parameterization predictions and theory. Surface dynamics transition from linear to high-drag (wave breaking) states at critical inverse Froude numbers Frc−1 predicted to within 10% by transform relations. Wave drag peaks at Frc−1 < ĥm ≲ 2, where for the elliptical obstacle both surface and wave drag vacillate owing to cyclical buildup and breakdown of waves. For the axisymmetric obstacle, this occurs only at ĥm = 1.2. At ĥm ≳ 2–3 vacillation abates and normalized pressure drag assumes a common normalized form for both obstacles that varies approximately as ĥm−1.3. Wave drag in this range asymptotes to a constant absolute value that, despite its theoretical shortcomings, is predicted to within 10%–40% by an analytical relation based on linear clipped-obstacle drag for a Sheppard-based prediction of dividing streamline height. Constant wave drag at ĥm ∼ 2–5 arises despite large variations with ĥm in the three-dimensional morphology of the local wave momentum fluxes. Specific implications of these results for the parameterization of subgrid-scale orographic drag in global climate and weather models are discussed.

A Fourier method is used to model mountain waves that have nearby turning points in a wind jet. In Fourier space, the propagation equations are solved by ray theory. To correct for the ray singularity at a turning point without time-consuming special-function evaluations, the ray solution is linearly interpolated across the breakdown region. The Fourier solutions for the spatial wavefield are compared with mesoscale model simulations in two cases: two-dimensional flow over idealized topography with uniform stratification and a sech-squared wind profile and three-dimensional flow over the island of Jan Mayen with stratification and wind profiles taken from radiosonde measurements. The latter case reveals the partial transmission of trapped mountain waves into the stratosphere.

Swath-scanned thermal radiances from the Advanced Microwave Sounding Units (AMSU-A) on the NOAA and Aqua satellites are used to image the horizontal temperature structure of a long-wavelength mountain wave that formed over southern Scandinavia on 14 January 2003. Data from all six stratospheric channels show this wave propagating through the full depth of the stratosphere. In channels 9–11 (altitudes ∼20–90 hPa) the imaged wave has a phase structure broadly consistent with a stationary wave radiated by the southeastward flow over and above the quasi-elliptical terrain of southern Norway. Channel 12 radiances at ∼10 hPa, however, show a remarkable abrupt change in imaged wave structure: the horizontal wavelength contracts, phase lines rotate anticlockwise by 30° –40° , and peak activity migrates to the south to lie over Denmark and northern Germany. Similar structure persists in channels 13 and 14 (at ∼2–5 hPa) with progessively increasing radiance amplitudes. These features are stable over the ∼10 hours of AMSU-A measurements from five separate overpasses, and are validated against independent radiances acquired by the Atmospheric Infrared Sounder (AIRS) and retrieved AIRS/AMSU-A temperature profiles from the Aqua overpass. This change at the channel 11/12 interface coincides with an onset of anticlockwise rotation (backing) and intensification of background stratospheric winds with height. Fourier-ray and spatial ray modelling incorporating these directionally sheared winds and simplified orographic forcing reproduce the salient features of the observations, but only after wave-induced temperature perturbations have been converted to channel radiances using a forward model. The differential visibility of various components of this three-dimensional mountain wave to the AMSU-A channel weighting functions has a first-order impact on the observations at heights above 10 hPa. Once that is factored in, the combined observations and modelling provide direct experimental support for the Shutts model's predictions of how backing wind vectors affect the vertical evolution of three-dimensional mountain waves. Implications of these observations for orographic gravity wave drag parametrization are briefly discussed. Copyright © 2007 Royal Meteorological Society

A time-dependent generalization of a Fourier-ray method is presented and tested for fast numerical computation of high-resolution nonhydrostatic mountain-wave fields. The method is used to model mountain waves from Jan Mayen on 25 January 2000, a period when wavelike cloud banding was observed long distances downstream of the island by the Advanced Very High Resolution Radiometer Version 3 (AVHRR-3). Surface weather patterns show intensifying surface geostrophic winds over the island at 1200 UTC caused by rapid eastward passage of a compact low pressure system. The 1200 UTC wind profiles over the island increase with height to a jet maximum of ∼60–70 m s−1, yielding Scorer parameters that indicate vertical trapping of any short wavelength mountain waves. Separate Fourier-ray solutions were computed using high-resolution Jan Mayen orography and 1200 UTC vertical profiles of winds and temperatures over the island from a radiosonde sounding and an analysis system. The radiosonde-based simulations produce a purely diverging trapped wave solution that reproduces the salient features in the AVHRR-3 imagery. Differences in simulated wave patterns governed by the radiosonde and analysis profiles are explained in terms of resonant modes and are corroborated by spatial ray-group trajectories computed for wavenumbers along the resonant mode curves. Output from a nonlinear Lipps–Hemler orographic flow model also compares well with the Fourier-ray solution horizontally. Differences in vertical cross sections are ascribed to the Fourier-ray model’s current omission of tunneling of trapped wave energy through evanescent layers.

The Fourier-ray method involves ray tracing in a Fourier-transform domain. The ray solutions are then Fourier synthesized to produce a spatial solution. Here previous steady-state developments of the Fourier-ray method are extended to include a transient source of mountain waves. The method is illustrated with an initial value problem in which the background flow is started abruptly from rest and then maintained at steady velocity. The resulting wave transience is modeled in a simple way. All rays that radiate from the mountain, including the initial rays, are assigned the full amplitude of the longtime steady-state solution. Time dependence comes in through the changing position of the initial rays. This is sufficient to account for wave transience in a test case, as demonstrated by comparison with simulations from a mesoscale numerical model.

A method is developed to describe the linear internal wavefield generated by an oscillating source in horizontally moving, depth-dependent background. Ray theory is used to approximate the vertical eigenfunctions. A spatial solution is then obtained by inverse Fourier transform. This is a practical approach with a more convenient range of validity than the method of spatial ray tracing. The solutions for a stationary source and for a vertically oscillating source in a vertically sheared background current are given for illustration.

Several recent studies of internal gravity waves have been expressed in a Lagrangian reference frame, motivated by the observation that in this frame the dispersion relation then excludes the Eulerian Doppler shifting term due to a background flow. Here the dispersion relation in a Lagrangian reference frame is explicitly derived for a background flow and background density that are slowly varying with respect to the waves, but are otherwise arbitrary functions of space and time. Two derivations are given, both yielding the same result. The first derivation involves a transformation of the dispersion relation from Eulerian to Lagrangian coordinates, while the second derivation involves a wave-packet analysis of the equations of motion directly in Lagrangian coordinates. The authors show that, although the Eulerian Doppler shifting term is removed from the dispersion relation by the transformation of the frequency when passing from an Eulerian to a Lagrangian reference frame, a dependence on the b...

▪ Abstract We review the use of ray models for internal waves, particularly formulations for calculating wave amplitudes along the ray. These are expressed in spatial, wave number, and phase-space coordinates. The choice of formulation affects not only the difficulty of the calculations for rays and caustics but also the degree to which the waves satisfy slowly varying assumptions. We describe several examples taken from atmospheric and oceanic applications that illustrate the variety of options for ray models.

A previously derived approximation to the standard Fourier integral technique for linear mountain waves is extended to include nonhydrostatic effects in a background flow with height-dependent wind and stratification. The approximation involves using ray theory to simplify the vertical eigenfunctions. The generalization to nonhydrostatic waves requires special treatment for resonant modes and caustics. Resonant modes are handled with a small amount of damping, and caustics are handled with a uniformly valid approximation involving the Airy function. This method is developed for both two- and three-dimensional flows, and its results are shown to compare well with an exact analytical result for two-dimensional mountain waves and with a numerical simulation for two- and three-dimensional mountain waves.

The ray solution for stationary hydrostatic mountain waves has a singularity along the vertical axis directly over the mountain. We use Maslov's method to improve the ray prediction. The ray solution is determined in the wave-number domain and is then mapped by inverse Fourier transform to give a spatial description of the wave field that approximates the linear solution over the mountain and elsewhere. We develop Maslov's method for a medium with a height-dependent mean wind and mean buoyancy, and we compare it with the traditional transform method and with the results of a numerical simulation. Copyright © 2002 Royal Meteorological Society

Wave propagation from a localized source into a non-uniform medium is analysed using a hybrid method. This consists of an integral representation in the near-field and a numerical ray tracing in the far-field. The ray tracing includes the calculation of wave-action density and is initialized from the stationary-phase approximation to the near-field integral. The method is tested on the problem of mountain waves in a turning wind, as studied by Shutts. A ray calculation with rotation added to the Shutts model is also presented and is interpreted with a simple analytical model in which the horizontal divergence of rays balances the vertical convergence of rays. In the example presented here, the inclusion of rotation speeds up the approach to wave-breaking amplitudes.

This paper deals with the numerical verification of the theory developed by Derzho and Grimshaw (DG) (1997, Phys. Fluids 9(11), 3378–3385) regarding solitary waves in stratified fluids with recirculation regions. The Boussinesq approximation is made and the stratification is chosen such that the Brunt-Vaisala frequency differs only slightly from uniform stratification. To establish the consistency of the numerical scheme the usual KdV and mKdV solutions are tested first and then the solutions obtained by DG are considered. It is found that these waves remain of permanent form and are stationary when viewed at their corresponding phase speed. The recirculation region remains stagnant to first order as predicted by DG.

Ray theory is used to study the refraction of short oceanic internal waves by a spectrum of large amplitude inertia waves superimposed on a weakly sheared steady current. The results suggest that the steady current has a significant cumulative effect on short-wave propagation over the timescale of a few inertia periods. The strength of ray convergence is also computed, as this affects short-wave amplitudes. Typically we find weak ray convergence and much slower growth toward instability with increasing vertical wavenumber than in a steady-shear critical-layer model.

The stability of continuously stratified vortices with large displacement of isopycnal surfaces on the f-plane is examined both analytically and numerically. Using an appropriate asymptotic set of equations, we demonstrated that sufficiently large vortices (i.e. those with small values of the Rossby number) are unstable. Remarkably, the growth rate of the unstable disturbance is a function of the spatial coordinates. At the same time, the corresponding boundary-value problem for normal modes has no smooth square-integrable solutions, which would normally be regarded as stability.We conclude that (potentially) stable vortices can be found only among ageostrophic vortices. Since this assumption cannot be verified analytically due to complexity of the primitive equations, we verify it numerically for the particular case of two-layer stratification.

Present Doppler-spreading models for short internal waves in the atmosphere and the ocean either neglect the time dependence of the background long-wave shear entirely, or ignore time-dependent effects in their parameterization of dissipation. We use ray and numerical simulations to examine the Doppler spreading of a short internal-wave packet by an inertia-wave packet. The results are enough to show that time dependence in the long-wave shear can make a significant difference to short-wave behavior, and will need to be taken into account in future modelling efforts.

We use a high-resolution spectral numerical scheme to solve the two-dimensional equations of motion for the flow of a uniformly stratified Boussinesq fluid over isolated bottom topography in a channel of finite depth. The focus is on topography of small to moderate amplitude and slope and for conditions such that the flow is near linear resonance of either of the first two internal wave modes. The results are compared with existing inviscid theories: the steady hydrostatic analysis of Long (1955), time-dependent linear long-wave theory, and the fully nonlinear, weakly dispersive resonant theory of Grimshaw & Yi (1991). For the latter, we use a spectral numerical technique, with improved accuracy over previously used methods, to solve the approximate evolution equation for the amplitude of the resonant mode. Also, we present some new results on the modal similarity of the solutions of Long and of Grimshaw & Yi. For flow conditions close to linear resonance, solutions of Grimshaw & Yi's evolution equation compare very well with our fully nonlinear numerical solutions, except for very steep topography. For flow conditions between the first two resonances, Long's steady solution is approached asymptotically in time when the slope of the topography is sufficiently small. For steeper topography, the flow remains unsteady. This unsteadiness is manifested very clearly as periodic oscillations in the drag, which have been observed in previous numerical simulations and tow-tank experiments. We explain these oscillations as mainly due to the internal waves that according to linear theory persist longest in the neighbourhood of the topography.

Large altitude asymptotic approximations are derived for vertical displacements due to mountain waves generated by hydrostatic wind flow over arbitrary topography. This leads to new asymptotic analytic expressions for wave-induced vertical displacement for mountains with an elliptical Gaussian shape and with the major axis oriented at any angle relative to the background wind. The motivation is to understand local maxima in vertical displacement amplitude at a given height for elliptical mountains aligned at oblique angles to the wind direction, as identified in Eckermann et al. [“Effects of horizontal geometrical spreading on the parameterization of orographic gravity-wave drag. Part 1: Numerical transform solutions,” J. Atmos. Sci. 72, 2330–2347 (2015)]. The standard stationary phase method reproduces one type of local amplitude maximum that migrates downwind with increasing altitude. Another type of local amplitude maximum stays close to the vertical axis over the center of the mountain, and a new generalized stationary phase method is developed to describe this other type of local amplitude maximum and the horizontal variation of wave-induced vertical displacement near the vertical axis of the mountain in the large altitude limit. The new generalized stationary phase method describes the asymptotic behavior of integrals where the asymptotic parameter is raised to two different powers (1/2 and 1) rather than just one power as in the standard stationary phase method. The vertical displacement formulas are initially derived assuming a uniform background wind but are extended to accommodate both vertical shear with a fixed wind direction and vertical variations in the buoyancy frequency.

[1] The Vadas-Fritts ray-tracing model for convectively generated gravity waves is analyzed using the stationary phase approximation and is interpreted in terms of a ray Jacobian approximated by the density of rays. The Vadas-Fritts model launches rays from the convective source region, with initial conditions for the ray-tracing deduced from a near-field integral representation. In the far-field the rays are binned in space-time grid cells. The contribution of each ray to the spatial wave amplitude is determined by its spectral amplitude and by the local density of rays within the grid cells. The present analysis accomplishes two things. First, the stationary phase analysis gives the formal initial conditions for the ray-tracing, which mostly agree with the Vadas-Fritts initialization but also suggest some refinements. Secondly, the Jacobian and ray-density analysis shows how the Vadas-Fritts model can be generalized to follow a beam of rays with a single moving grid cell.

We use numerical simulations based on the shallow water equations for a homogeneous fluid on an f-plane to study the evolution of a cyclonic eddy over a topographic slope. The results are compared with analogous simulations using the nondivergent approximation in order to determine the effect of the free-surface vortex stretching terms. The simulations use a geometry and parameters related to the experimental results of Carnevale et al. (1991). We find that this vortex stretching produces a significant reduction in the eddy speed. Our numerical results are supported by theoretical analyses, largely set in the quasi-geostrophic context.

Using the nondivergent barotropic equation on an f plane, the evolution of an eddy placed on a topographic slope is studied. The theoretical analysis is in two pans. First, using the linearized equations appropriate for weak eddies it is shown that the eddy will disperse into propagating coastally trapped waves. Second, using the conservation of potential vorticity, a theory is developed to describe the initial movement of the eddy. A set of numerical results that generally confirm the theoretical predictions is then described. Finally, the authors' model is compared with experimental results obtained by Carnevale and coauthors.

We show high-resolution satellite observations of mountain wave events in the stratosphere above South Georgia Island in the remote southern Atlantic Ocean and compute the wave momentum fluxes for these events. The fluxes are large, and they imply important drag forces on the circulation. Small island orography is generally neglected in mountain wave parameterizations used in global climate models because limited model resolution treats the grid cell containing the island as ocean rather than land. Our results show that satellite observations can be used to quantitatively constrain mountain wave momentum fluxes, and they suggest that mountain waves from island topography may be an important missing source of drag on the atmospheric circulation.

Results are presented from a numerical model for barotropic, nondivergent flow in a region with the typical geometry and topography for a strait connecting two ocean basins. The study is motivated by questions concerning the transmission, or generation, of energy through, or in, Bass Strait. The numerical model is a spectral collocation method developed specifically to handle regions with steep and complex topography. Two basic problems are considered, these being the forcing of flow in the strait by a coastally-trapped wave incident at one side, and the forcing by wind stress in the strait, either stationary, or steadily propagating. For the first problem we find that the transmission of energy is enhanced when the boundary conditions allow an oscillatory flux in the strait, as opposed to the case when no such flux is permitted. Mode 2 is more readily transmitted than mode 1 but this is largely counterbalanced by its greater decay due to friction. For the second problem when the forcing is due t...

We describe the high-resolution spectral modelling of nondivergent barotropic linearized flow over steep irregular topography. We use collocation to evaluate spatial derivatives in the barotropic vorticity equation, and a spectral multigrid technique to accelerate the iterative solution of the vorticity—stream function relation. The computational domain is a rectangular channel, which can be conformally mapped into more interesting shapes, as we also discuss. A Fourier-series representation is used in the (periodic) direction parallel to the walls of the channel, and a sine series in the cross-channel direction. For much of the paper we concentrate on the numerical techniques, though results are provided, including an application to the Bass Strait region of southeast Australia.

The atmosphere s response to the total solar eclipse of 4 December 2002 is studied using a prototype high-altitude global numerical weather prediction model (NOGAPS-ALPHA). Local reductions in solar ultraviolet (UV) radiation during the eclipse are estimated using astronomical calculations of umbral and penumbral surface trajectories and observed solar limb darkening at ~ 200-300 nm. In NOGAPS-ALPHA these UV eclipse shadows yield stratospheric radiative cooling rate footprints peaking near 27 K day 1, a value 2 3 times larger than assumed in previous modeling. Difference fields between NOGAPS-ALPHA runs with and without this eclipse forcing reveal vertically deep middle atmospheric responses, with three-dimensional horizontal structures very similar to the large-scale bow-wave response first proposed by Chimonas (1970). Such structure appears clearly only at later times when total eclipses have abated and gravity waves generated in the stratosphere have had time to propagate vertically. Bow-wave amplitudes and direct thermal cooling responses are both small (]1 K for temperature and ]2 3 m s 1 for horizontal winds), contradicting some rocketsonde measurements that suggest much larger responses near 50 60 km altitude. We also find clear evidence of a bow-wave-like response in the model s surface pressure fields, with an amplitude 0.1 0.5 hPa, while surface air temperatures in NOGAPS-ALPHA show 4 K cooling over Africa during the eclipse. Both findings are consistent with surface atmospheric data acquired during previous eclipse passages.

The refraction of a spectrum of short internal waves by a packet of inertia waves is considered. It is found that a short inertia-wave packet produces a larger net change in a spectrum of short waves than a long inertia-wave packet, provided the short waves are trapped at some time during the interaction. Specifically, an incident narrow-banded short-wave spectrum broadens more quickly and more fully when refracted by the short inertia-wave packet. The results are explained in terms of a frequency conservation law, which constrains short-wave change in the limit of a long inertia-wave packet.

[1] The Cryogenic Infrared Spectrometers and Telescopes for the Atmosphere (CRISTA) instrument measured stratospheric temperatures and trace species concentrations with high precision and spatial resolution during two missions. The measuring technique is infrared limb-sounding of optically thin emissions. In a general approach, we investigate the applicability of the technique to measure gravity waves (GWs) in the retrieved temperature data. It is shown that GWs with wavelengths of the order of 100–200 km horizontally can be detected. The results are applicable to any instrument using the same technique. We discuss additional constraints inherent to the CRISTA instrument. The vertical field of view and the influence of the sampling and retrieval imply that waves with vertical wavelengths ∼3–5 km or larger can be retrieved. Global distributions of GW fluctuations were extracted from temperature data measured by CRISTA using Maximum Entropy Method (MEM) and Harmonic Analysis (HA), yielding height profiles of vertical wavelength and peak amplitude for fluctuations in each scanned profile. The method is discussed and compared to Fourier transform analyses and standard deviations. Analysis of data from the first mission reveals large GW amplitudes in the stratosphere over southernmost South America. These waves obey the dispersion relation for linear two-dimensional mountain waves (MWs). The horizontal structure on 6 November 1994 is compared to temperature fields calculated by the Pennsylvania State University (PSU)/National Center for Atmospheric Research (NCAR) mesoscale model (MM5). It is demonstrated that precise knowledge of the instrument's sensitivity is essential. Particularly good agreement is found at the southern tip of South America where the MM5 accurately reproduces the amplitudes and phases of a large-scale wave with 400 km horizontal wavelength. Targeted ray-tracing simulations allow us to interpret some of the observed wave features. A companion paper will discuss MWs on a global scale and estimates the fraction that MWs contribute to the total GW energy (Preusse et al., in preparation, 2002).

Ray calculations are used to examine short-wave propagation through an inertial current. The results exhibit differences from theories of short-wave refraction based on steady shears. Caustics take the place of critical layers as the locations of ray focusings and form for short waves at all intrinsic frequencies and at all phases of the inertial oscillation. Conversely, short waves that refract to, and remain at, high vertical wavenumber can escape instability altogether. Similarities with the steady-shear model of Hartman (1975) and contrasts with the Taylor-Goldstein model are emphasised. Checks on the validity of the slowly-varying approximation are also made.

Ray-tracing models of short, oceanic internal waves commonly predict the existence of caustics, where neigh-boring rays cross. Here we present a linear model of short-wave refraction that yields caustics for the short waves at all intrinsic frequencies and vertical wavenumbers. The caustics form as the short waves are focused by a single progressive inertial wave that is modeled first as infinitely sinusoidal and then as a localized packet. The solutions are obtained by a combination of analytical and numerical techniques. A boundary-layer matching gives a convenient expression for the maximum amplitude near the caustic, while the more robust, uniformly valid solution indicates that the boundary-layer results are accurate for realistically large inertial shears only at very close distance to the caustic. Analytic solutions allow parameter dependences to be studied in detail. Implications for wave breaking are also drawn: in this model, the changes in short-wave amplitude (corrected to remove the ...

The interaction between a wave packet of small-amplitude short internal waves, and a finite-amplitude inertial wave field is described to second order in the short-wave amplitude. The discussion is based on the principle of wave action conservation and the equations for the wave-induced Lagrangian mean flow. It is demonstrated that as the short internal waves propagate through the inertial wave field they generate a wave-induced train of trailing inertial waves. The contribution of this wave-induced mean flow to the total energy balance is described. The results obtained here complement the finding of Broutman & Young (1986) that the short internal waves undergo a net change in energy after their encounter with the inertial wave field.

An analytical model is developed for nondivergent barotropic flow in a rectangular region, with depth contours parallel to the longshore boundary. The model is linear, with flow forced by wind stress and subject to dissipation by bottom stress. The friction due to the bottom stress is parameterized by a single friction coefficient, which represents a gross decay time for the whole region. Analytical solutions are developed in detail for a steady or periodic wind stress, which is independent of the cross-shelf coordinate. Results are presented for a range of values of the region dimension and the friction coefficient. Comparisons are made between the analytical model and recent observations of the wind-driven flow in the northern Great Barrier Reef.