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The interaction between locally monochromatic finite-amplitude mesoscale waves, their nonlinearly induced higher harmonics, and a synoptic-scale flow is reconsidered, both in the tropospheric regime of weak stratification and in the stratospheric regime of moderately strong stratification. A review of the basic assumptions of quasi-geostrophic theory on an f-plane yields all synoptic scales in terms of a minimal number of natural variables, i.e. two out of the speed of sound, gravitational acceleration and Coriolis parameter. The wave scaling is defined so that all spatial and temporal scales are shorter by one order in the Rossby number, and by assuming their buoyancy field to be close to static instability. WKB theory is applied, with the Rossby number as scale separation parameter, combined with a systematic Rossby-number expansion of all fields. Classic results for synoptic-scale-flow balances and inertia-gravity wave (IGW) dynamics are recovered. These are supplemented by explicit expressions for the interaction between mesoscale geostrophic modes (GM), a possibly somewhat overlooked agent of horizontal coupling in the atmosphere, and the synoptic-scale flow. It is shown that IGW higher harmonics are slaved to the basic IGW, and that their amplitude is one order of magnitude smaller than the basic-wave amplitude. GM higher harmonics are not that weak and they are in intense nonlinear interaction between themselves and the basic GM. Compressible dynamics plays a significant role in the stratospheric stratification regime, where anelastic theory would yield insufficient results. Supplementing classic derivations, it is moreover shown that in the absence of mesoscale waves quasi-geostrophic theory holds also in the stratospheric stratification regime.

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... A complementary approach is Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) theory (Bretherton, 1966;Grimshaw, 1975;Achatz et al., 2010;Achatz et al., 2017) which, instead of considering continuous wave spectra, describes the development of locally monochromatic GW fields which feature a nearly discrete spectrum. Moreover, the WKBJ approach takes into account nonlinear interactions between GWs and a spatially and temporally varying mean flow. ...

... In general, wave modulation by a variable background stratification or a sheared mean flow are equally important in the atmosphere (cf. Achatz et al., 2017). However, we restrict the analysis to the case of Boussinesq dynamics with a constant background stratification and zero rotation for the sake of simplicity. ...

... The initial wave amplitudes are taken to be a given fraction, , with respect to the static F I G U R E 3 Ratios of wave energies corresponding of the solutions the simplified system (Equations 71-73) and the parametrized system (76-78) after the interaction. Here, the background shear is set to z u = 2 ∕40,000 s −1 , and the parametrization constant is ≡ 1. (a), (b), and (c) are associated with j = 1, j = 2, and j = 3, respectively instability criterion (e.g., Achatz et al., 2017). In particular ...

Motivated by the question of whether and how wave‐wave interactions should be implemented into atmospheric gravity‐wave parameterizations, the modulation of triadic gravity‐wave interactions by a slowly varying and vertically sheared mean‐flow is considered for a non‐rotating Boussinesq fluid with constant stratification. An analysis using a multiple‐scales WKBJ expansion identifies two distinct scaling regimes, a linear off‐resonance regime, and a non‐linear near‐resonance regime. Simplifying the near‐resonance interaction equations allows for the construction of a parametrization for the triadic energy exchange which has been implemented into a one‐dimensional WKBJ ray‐tracing code. Theory and numerical implementation are validated for test cases where two wave trains generate a third wave train while spectrally passing through resonance. In various settings, of interacting vertical wavenumbers, mean‐flow shear, and initial wave amplitudes, the WKBJ simulations are generally in good agreement with wave‐resolving simulations. Both stronger mean‐flow shear and smaller wave amplitudes suppress the energy exchange among a resonantly interacting triad. Experiments with mean‐flow shear as strong as in the vicinity of atmospheric jets suggest that internal gravity wave dynamics are dominated in such regions by wave modulation. Yet, triadic gravity‐wave interactions are likely to be relevant in weakly sheared regions of the atmosphere.

... As will be shown below this is at least justified if the large-scale flow is in geostrophic and hydrostatic balance. The direct scheme does not rely on any balance assumption with regard to the large-scale flow, and the large-scale flow is forced by anelastic momentum-flux convergence in the momentum equation, an elastic term also in the momentum equation, and entropy-flux convergence in the entropy equation, as given by Grimshaw (1975) and Achatz et al. (2017). All present-day operational IGW parameterizations represent, one way or other, simplified versions of the pseudomomentum approach, where the vertical gradient of pseudomomentum-flux convergence forces the resolved flow, when wave dissipation occurs (Fritts and Alexander 2003;Kim et al. 2003), and neither elastic nor thermal effects are taken into account. ...

... For an explanation of the theoretical underpinnings of the two respective approaches we follow the presentation of Achatz et al. (2017) where, expanding on previous work by Grimshaw (1975), the theory is discussed mostly in nondimensional form. We translate the essentials into dimensional form and choose, for easier tractability, a heuristic formulation. ...

... We translate the essentials into dimensional form and choose, for easier tractability, a heuristic formulation. For all mathematical details, the reader is referred back to Achatz et al. (2017). ...

This paper compares two different approaches for the efficient modeling of subgrid-scale inertia–gravity waves in a rotating compressible atmosphere. The first approach, denoted as the pseudomomentum scheme, exploits the fact that in a Lagrangian-mean reference frame the response of a large-scale flow can only be due to forcing momentum. Present-day gravity wave parameterizations follow this route. They do so, however, in an Eulerian-mean formulation. Transformation to that reference frame leads, under certain assumptions, to pseudomomentum-flux convergence by which the momentum is to be forced. It can be shown that this approach is justified if the large-scale flow is in geostrophic and hydrostatic balance. Otherwise, elastic and thermal effects might be lost. In the second approach, called the direct scheme and not relying on such assumptions, the large-scale flow is forced both in the momentum equation, by anelastic momentum-flux convergence and an additional elastic term, and in the entropy equation, via entropy-flux convergence. A budget analysis based on one-dimensional wave packets suggests that the comparison between the abovementioned two schemes should be sensitive to the following two parameters: 1) the intrinsic frequency and 2) the wave packet scale. The smaller the intrinsic frequency is, the greater their differences are. More importantly, with high-resolution wave-resolving simulations as a reference, this study shows conclusive evidence that the direct scheme is more reliable than the pseudomomentum scheme, regardless of whether one-dimensional or two-dimensional wave packets are considered. In addition, sensitivity experiments are performed to further investigate the relative importance of each term in the direct scheme, as well as the wave–mean flow interactions during the wave propagation.

... Let us summarize the results of the previous section. The gross wave-Froude number depending on the wave itself due to the induced mean flow is specified in terms of (34) and (35) by ...

... In an envisaged companion paper we want to extend our investigations to the stability of gravity waves governed by three-dimensional modulation equations including the Coriolis force. The basis for such a study was already founded in [35]. ...

This study investigates strongly nonlinear gravity waves in the compressible atmosphere from the Earth's surface to the deep atmosphere. These waves are effectively described by Grimshaw's dissipative modulation equations which provide the basis for finding stationary solutions such as mountain lee waves and testing their stability in an analytic fashion. Assuming energetically consistent boundary and far-field conditions , that is no energy flux through the surface, free-slip boundary, and finite total energy, general wave solutions are derived and illustrated in terms of realistic background fields. These assumptions also imply that the wave-Reynolds number must become less than unity above a certain height. The modulational stability of admissible, both non-hydrostatic and hydrostatic, waves is examined. It turns out that, when accounting for the self-induced mean flow, the wave-Froude number has a resonance condition. If it becomes 1/ √ 2, then the wave destabilizes due to perturbations from the essential spectrum of the linearized modulation equations. However, if the horizontal wavelength is large enough, waves overturn before they can reach the modulational stability condition.

... This is conditioned on a manageable separation of the flow and its dynamical equations into balanced and unbalanced parts. For this reason we here restrict ourselves to linear balance conditions and a determination of the balanced flow from the inversion of linear potential vorticity (PV), as is strictly appropriate in the limit of a small Rossby number (Charney 1948;Hoskins, McIntyre & Robertson 1985;Pedlosky 1987;Achatz et al. 2017). This approach is supplemented by the extraction of balanced vertical motion and horizontal divergence by the application of the omega equation. ...

... In our configuration of the differentially heated rotating annulus (see § 2.1) the Rossby number is small (Ro < 1) in most locations. As shown by Bühler & McIntyre (2005) in the Lagrangian mean and by Achatz et al. (2017) in the Eulerian perspective, in that limit IGWs contribute to the nonlinear part of PV, while the linear part is determined exclusively by a geostrophically and hydrostatically balanced component, as also in quasi-geostrophic theory (Charney 1948;Pedlosky 1987;Vallis 2006). Moreover, as can be verified from their polarisation relations, linear IGWs have no linear PV (Phillips 1963;Mohebalhojeh & Dritschel 2001;Smith & Waleffe 2002). ...

The source mechanism of inertia–gravity waves (IGWs) observed in numerical simulations of the differentially heated rotating annulus experiment is investigated. The focus is on the wave generation from the balanced part of the flow, a process presumably contributing significantly to the atmospheric IGW field. Direct numerical simulations are performed for an atmosphere-like configuration of the annulus and possible regions of IGW activity are characterised by a Hilbert-transform algorithm. In addition, the flow is separated into a balanced and unbalanced part, assuming the limit of a small Rossby number, and the forcing of IGWs by the balanced part of the flow is derived rigorously. Tangent-linear simulations are then used to identify the part of the IGW signal that is rather due to radiation by the internal balanced flow than to boundary-layer instabilities at the side walls. An idealised fluid set-up without rigid horizontal boundaries is considered as well, to investigate the effect of the identified balanced forcing unmasked by boundary-layer effects. The direct simulations of the realistic and idealised fluid set-ups show a clear baroclinic-wave structure exhibiting a jet–front system similar to its atmospheric counterparts, superimposed by four distinct IGW packets. The subsequent tangent-linear analysis indicates that three wave packets are radiated from the internal flow and a fourth one is probably caused by boundary-layer instabilities. The forcing by the balanced part of the flow is found to play a significant role in the generation of IGWs, so it supplements boundary-layer instabilities as a key factor in the IGW emission in the differentially heated rotating annulus.

... The horizontal grid spacing is 160 km, and the vertical spacing is 700 m in the stratosphere. Instead of the operational GW parameterization of this model, we use a prognostic parameterization, the Multi-Scale Gravity Wave Model (MS-GWaM), which predicts the time evolution of GW action density field in positionwavenumber phase space (Achatz et al., 2017;Bölöni et al., 2021;Muraschko et al., 2015). A detailed description of MS-GWaM and its application to ICON is provided in Bölöni et al. (2021). ...

A general circulation model is used to study the interaction between parameterized gravity waves (GWs) and large-scale Kelvin waves in the tropical stratosphere. The simulation shows that Kelvin waves with substantial amplitudes (~10 m/s) can significantly affect the distribution of GW drag by modulating the local shear. Furthermore, this effect is localized to regions above strong convective organizations that generate large-amplitude GWs, so that at a given altitude it occurs selectively in a certain phase of Kelvin waves. Accordingly, this effect also contributes to the zonal-mean GW drag, which is large in the middle stratosphere during the phase transition of the quasi-biennial oscillation (QBO). Furthermore, we detect an enhancement of Kelvin-wave momentum flux due to GW drag modulated by Kelvin waves. The result implies an importance of GW dynamics coupled to Kelvin waves in the QBO progression.

... This phenomenon in uences the breaking height as the stability depends sensitively on amplitude. Therefore, we continue by investigating an extended set of modulation equations in section 3 which agrees with the inviscid pseudo-incompressible regime [1,2,5]. Here, the background density is an explicit function of height. ...

We apply spectral stability theory to investigate nonlinear gravity waves in the atmosphere. These waves are determined by modulation equations that result from Wentzel-Kramers-Brillouin theory. First, we establish that plane waves, which represent exact solutions to the inviscid Boussinesq equations, are spectrally stable with respect to their nonlinear modulation equations under the same conditions as what is known as modulational stability from weakly nonlinear theory. In contrast to Boussinesq, the pseudo-incompressible regime does fully account for the altitudinal varying background density. Second,we show for the first time that upward-traveling non-plane wave fronts solving the inviscid nonlinear modulation equations, that compare to pseudo-incompressible theory, are unconditionally unstable. Both inviscid regimes turn out to be ill-posed as the spectra allow for arbitrarily large instability growth rates. Third, a regularization is found by including dissipative effects. The corresponding nonlinear traveling wave solutions have localized amplitude. As a consequence of the nonlinearity, envelope and linear group velocity, as given by the derivative of the frequency with respect to wavenumber, do not coincide anymore. These waves blow up unconditionally by embedded eigenvalue instabilities but the instability growth rate is bounded from above and can be computed analytically. Additionally, all three types of nonlinear modulation equations are solved numerically to further investigate and illustrate the nature of the analytic stability results.

... As well as improving parametrizations of sources for internal waves, the representation of their propagation should be improved. Idealized theoretical models and simulations suggest that the effect of time-transient background winds, wave energetics, and lateral propagation of the waves as well as weakly nonlinear effects acting upon moderately large-amplitude waves may need to be incorporated into the next generation of parametrization schemes [6,37,45,[122][123][124][125]. Further corresponding indirect observational evidence is provided again from radiosonde balloons, showing a conspicuous dependence of the intermittency of internal-wave momentum fluxes on the large-scale wind strength that parametrizations cannot reproduce [120]. ...

Imbalance refers to the departure from the large-scale primarily vortical flows in the atmosphere and ocean whose motion is governed by a balance between Coriolis, pressure-gradient and buoyancy forces, and can be described approximately by quasi-geostrophic theory. Imbalanced motions are manifest as internal gravity waves which can extract energy from these geophysical flows but which can also feed energy back into this motion. Capturing the physics underlying these mechanisms is essential to understand how energy is transported from large geophysical scales ultimately to microscopic scales where it is dissipated. In the atmosphere it is also necessary for understanding momentum transport and its impact upon the mean wind and current speeds. During a February 2018 workshop at the Banff International Research Station (BIRS), atmospheric scientists, physical oceanographers, physicists and mathematicians gathered to discuss recent progress in understanding these processes through interpretation of observations, numerical simulations and mathematical modelling. The outcome of this meeting is reported upon here.

... If the atmospheric mesoscales are characterized as a relatively passive gap between two highly variable regimes, then it is not surprising that it has proved difficult to identify a single dominant mechanism leading to a universal 25/3 energy spectrum. It may be that alternative theoretical frameworks are required, for example, one that explicitly models the dynamics of scale interactions rather than treating them in a purely statistical manner (Achatz et al. 2017). ...

Research on the mesoscale kinetic energy spectrum over the past few decades has focused on finding a dynamical mechanism that gives rise to a universal spectral slope. Here we investigate the variability of the spectrumusing 3 years of kilometer-resolution analyses fromCOSMOconfigured forGermany (COSMO-DE). It is shown that the mesoscale kinetic energy spectrum is highly variable in time but that a minimum in variability is found on scales around 100 km. The high variability found on the small-scale end of the spectrum (around 10km) is positively correlated with the precipitation rate where convection is a strong source of variance. On the other hand, variability on the large-scale end (around 1000 km) is correlated with the potential vorticity, as expected for geostrophically balanced flows. Accordingly, precipitation at small scales is more highly correlated with divergent kinetic energy, and potential vorticity at large scales is more highly correlated with rotational kinetic energy. The presented findings suggest that the spectral slope and amplitude on the mesoscale range are governed by an ever-changing combination of the upscale and downscale impacts of these large- and small-scale dynamical processes rather than by a universal, intrinsically mesoscale dynamical mechanism.

... Wave field (Achatz et al., 2017) ...

The aim of the presented work is to improve the parametrization of subgrid-scale gravity wave (GW) drag on the resolved flow of climate and numerical-weather-prediction models. Current GW parametrization schemes are using the steady-state approximation for the wave field and therefore assume an instantaneous GW propagation neglecting direct interactions between the GWs and the resolved flow in the course of the propagation. As such these schemes rely on wave breaking as the only mechanism to exert a drag on the
resolved flow. Theory shows that dropping the steady-state assumption leads to non-linear GW-meanflow interactions (further on direct GW-meanflow interaction) where the meanflow is forced even in the absence of wave breaking, whereas the meanflow in turn modulates the smaller scale wave field due to wind shear and stratification gradients (Achatz et al., 2017). In idealized simulations it indeed turns out that by applying a transient GW model (i.e. by dropping the steady-state assumption) the contribution of direct GW-meanflow interaction to the GW drag can be as important as that of wave breaking (Bölöni et al., 2016). This motivates the implementation of a transient GW model (further on named MS-GWaM: Multi Scale Gravity Wave Model) to a state-of-the-art global circulation model (GCM) enabling to evaluate the consequences of direct GW-meanflow interactions in a realistic atmospheric circulation. The GCM in which MS-GWaM has been implemented is the Icosahedral Nonhydrostatic Model (ICON) developed jointly by the German Weather Service and the Max-Planck Institute for Meteorology. MS-GWaM in ICON runs stably and provides
substantially different GW drag in comparison with the benchmark steady-state parametrization available in the model. Seasonal simulations with ICON-MS-GWaM provide reasonable zonal mean middle atmospheric circulation and temperature structures in comparison with climatology such as SPARC observations and the HAMMONIA GCM.

... In this regard Achatz et al. (2010Achatz et al. ( , 2017 showed that the consistency between the scale asymptotics of the Euler equations and the pseudo-incompressible equations also holds for hydrostatic gravity waves. Bölöni et al. (2016) investigated the non-hydrostatic modulation equations numerically applying a ray-tracer method. ...

Wentzel–Kramers–Brillouin theory was employed by Grimshaw ( Geophys. Fluid Dyn. , vol. 6, 1974, pp. 131–148) and Achatz et al. ( J. Fluid Mech. , vol. 210, 2010, pp. 120–147) to derive modulation equations for non-hydrostatic internal gravity wave packets in the atmosphere. This theory allows for wave packet envelopes with vertical extent comparable to the pressure scale height and for large wave amplitudes with wave-induced mean-flow speeds comparable to the local fluctuation velocities. Two classes of exact travelling wave solutions to these nonlinear modulation equations are derived here. The first class involves horizontally propagating wave packets superimposed over rather general background states. In a co-moving frame of reference, examples from this class have a structure akin to stationary mountain lee waves. Numerical simulations corroborate the existence of nearby travelling wave solutions under the pseudo-incompressible model and reveal better than expected convergence with respect to the asymptotic expansion parameter. Travelling wave solutions of the second class also feature a vertical component of their group velocity but exist under isothermal background stratification only. These waves include an interesting nonlinear wave–mean-flow interaction process: a horizontally periodic wave packet propagates vertically while draining energy from the mean wind aloft. In the process it decelerates the lower-level wind. It is shown that the modulation equations apply equally to hydrostatic waves in the limit of large horizontal wavelengths. Aside from these results of direct physical interest, the new nonlinear travelling wave solutions provide a firm basis for subsequent studies of nonlinear internal wave instability and for the design of subtle test cases for numerical flow solvers.

... Large-amplitude mesoscale gravity waves, which can originate from a variety of processes and often travel large distances before dissipating (e.g., Achatz et al. 2017), have also been extensively studied and remain difficult to forecast using currently available conventional surface weather observations and numerical guidance. The movement, amplification, and decay of such features through generally stable environments has often been a focus for research (Bosart and Seimon 1988;Crook 1988;Ramamurthy et al. 1993;Zhang et al. 2001;Plougonven and Zhang 2014). ...

Mesoscale convective phenomena induce pressure perturbations that can alter the strength and magnitude of surface winds, precipitation, and other sensible weather which, in some cases, can inflict injuries and damage to property. This work extends prior research to identify and characterize mesoscale pressure features using a unique resource of 1-Hz pressure observations available from the USArray Transportable Array (TA) seismic field campaign.
A two-dimensional variational technique is used to obtain 5 km surface pressure analysis grids every 5 min from 1 March – 31 August 2011 from the TA observations and gridded surface pressure from the Real Time Mesoscale Analysis over a swath of the central United States. Band-pass filtering and feature tracking algorithms are employed to isolate, identify, and assess prominent mesoscale pressure perturbations and their properties. Two case studies, the first involving mesoscale convective systems and second a solitary gravity wave, are analyzed using additional surface observation and gridded data resources. Summary statistics for tracked features during the period reviewed indicate a majority of perturbations last less than 3 h, produce maximum perturbation magnitudes between 2-5 hPa, and move at speeds ranging from 15-35 m s⁻¹. The results of this study combined with improvements nationwide in real-time access to pressure observations at sub-hourly reporting intervals highlight the potential for improved detection and nowcasting of high-impact mesoscale weather features.

Parameterizations of subgrid-scale gravity waves (GWs) in atmospheric models commonly involve the description of the dissipation of GWs. Where they dissipate, GWs have an increased effect on the large-scale flow. Instabilities that trigger wave breaking are an important starting point for the route to dissipation. Possible destabilizing mechanisms are numerous, but the classical vertical static instability is still regarded as a key indicator for the disposition to wave breaking. In this work, we investigate how the horizontal variations associated with a GW could alter the criterion for static instability. To this end, we use an extension of the common parcel displacement method. This three-dimensional static stability analysis predicts a significantly larger range of instability than does the vertical static stability analysis. In this case, the Lindzen-type saturation adjustment to a state of marginal stability is perhaps a less suitable ansatz for the parameterization of the GW breaking. In order to develop a possible ansatz for the GW dissipation due to three-dimensional instability, we apply the methods of irreversible thermodynamics, which are embedded in the Gibbs formalism of dynamics. In this way, the parameterization does not only satisfy the second law of thermodynamics, but it can also be made consistent with the conservation of energy and further (non-)conservation principles. We develop the parameterization for a discrete spectrum of GW packets. Offline computations of GW drag and dissipative heating rates are performed for two vertical profiles of zonal wind and temperature for summer and winter conditions from CIRA data. The results are compared to benchmarks from the literature.

We study the stratified gas in a rapidly rotating centrifuge as a model for the Earth's atmosphere. Based on methods of perturbation theory, it is shown that in certain regimes, internal waves in the gas centrifuge have the same dispersion relation to leading order as their atmospheric siblings. Assuming an air filled centrifuge with a radius of around 50 cm, the optimal rotational frequency for realistic atmosphere-like waves is around 10 000 revolutions per minute. Using gases of lower heat capacities at constant pressure, such as xenon, the rotational frequencies can be even halved to obtain the same results. Similar to the atmosphere, it is feasible in the gas centrifuge to generate a clear scale separation of wave frequencies and therefore phase speeds between acoustic waves and internal waves. In addition to the centrifugal force, the Coriolis force acts in the same plane. However, its influence on axially homogeneous internal waves appears only as a higher-order correction. We conclude that the gas centrifuge provides an unprecedented opportunity to investigate atmospheric internal waves experimentally with a compressible working fluid.

The model Internal Wave Dissipation, Energy and Mixing (IDEMIX) presents a novel way of parameterising internal gravity waves in the atmosphere. IDEMIX is based on the spectral energy balance of the wave field and has previously been successfully developed as a model for diapycnal diffusivity, induced by internal gravity wave breaking in oceans. Applied here for the first time to atmospheric gravity waves, integration of the energy balance equation for a continuous wave field of a given spectrum, results in prognostic equations for the energy density of eastward and westward gravity waves. It includes their interaction with the mean flow, allowing for an evolving and local description of momentum flux and gravity wave drag. A saturation mechanism maintains the wave field within convective stability limits, and a closure for critical layer effects controls how much wave flux propagates from the troposphere into the middle atmosphere. Offline comparisons to a traditional parameterisation reveal increases in the wave momentum flux in the middle atmosphere due to the mean flow interaction, resulting in a greater gravity wave drag at lower altitudes. Preliminary validation against observational data show good agreement with momentum fluxes.

This study introduces a new computational scheme for the linear evolution of internal gravity wave packets passing over strongly non-uniform stratifications and background flows as found, e.g., near the tropopause. Focusing on linear dispersion, which is dominant at small wave amplitudes, the scheme describes general wave superpositions arising from wave reflections near strong variations of the background stratification. Thus, it complements WKB theory, which is restricted to nearly monochromatic waves but covers weakly nonlinear effects in turn. One envisaged application of the method is the formulation of bottom-of-the-stratosphere starting conditions for ray tracing parameterizations that follow nonlinear gravity wave packets into the upper atmosphere. A key feature in this context is the method’s separation of wave packets into up- and downward-propagating components. The paper first summarizes a multilayer method for the numerical solution of the Taylor–Goldstein equation. Borrowing ideas from Eliassen and Palm (Geophys Publ 22:1–23, 1961), the scheme is based on partitioning the atmosphere into several uniformly stratified layers. This allows for analytical plane wave solutions in each layer, which are matched carefully to obtain continuously differentiable global eigenmode solutions. This scheme enables rapid evaluations of reflection and transmission coefficients for internal waves impinging on the tropopause from below as functions of frequency and horizontal wavenumber. The study then deals with a spectral method for propagating wave packets passing over non-uniform backgrounds. Such non-stationary solutions are approximated by superposition of Taylor–Goldstein eigenmodes. Particular attention is paid to an algorithm that translates wave packet initial data in the form of modulated sinusoidal signals into amplitude distributions for the system’s eigenmodes. With this initialization in place, the state of the perturbations at any given subsequent time is obtained by a single superposition of suitably phase-shifted eigenmodes, i.e., without any time-stepping iterations. Comparisons of solutions for wave packet evolution with those obtained from a nonlinear atmospheric flow solver reveal that apparently nonlinear effects can be the result of subtle linear wave packet dispersion.

As present weather-forecast codes and increasingly many atmospheric climate
models resolve at least part of the mesoscale flow, and hence also internal
gravity waves (GWs), it is natural to ask whether even in such configurations
sub-gridscale GWs might impact the resolved flow, and how their effect
could be taken into account. This motivates a theoretical and numerical investigation
of the interaction between unresolved sub-mesoscale and resolved
mesoscale GWs, using Boussinesq dynamics for simplicity. By scaling arguments,
first a subset of sub-mesoscale GWs that can indeed influence the
dynamics of mesoscale GWs is identified. Therein, hydrostatic GWs with
wavelengths corresponding to the largest unresolved scales of present-day
limited-area weather forecast models are an interesting example. A largeamplitude
WKB theory, allowing for a mesoscale unbalanced flow, is then
formulated, based on multi-scale asymptotic analysis utilizing a proper scaleseparation
parameter. Purely vertical propagation of sub-mesoscale GWs is
found to be most important, implying inter alia that the resolved flow is only
affected by the vertical flux convergence of sub-mesoscale horizontal momentum
at leading order. In turn, sub-mesoscale GWs are refracted by mesoscale
vertical wind shear while conserving their wave-action density. An efficient
numerical implementation of the theory uses a phase-space ray tracer, thus
handling the frequent appearance of caustics. The WKB approach and its
numerical implementation are validated successfully against sub-mesoscale
resolving simulations of the resonant radiation of mesoscale inertia GWs by a
horizontally as well as vertically confined sub-mesoscale GW packet.

This study analyzes in situ airborne measurements from the 2008
Stratosphere–Troposphere Analyses of Regional Transport (START08) experiment
to characterize gravity waves in the extratropical upper troposphere and
lower stratosphere (ExUTLS). The focus is on the second research flight
(RF02), which took place on 21–22 April 2008. This was the first airborne
mission dedicated to probing gravity waves associated with strong
upper-tropospheric jet–front systems. Based on spectral and wavelet analyses
of the in situ observations, along with a diagnosis of the polarization
relationships, clear signals of mesoscale variations with wavelengths
~ 50–500 km are found in almost every segment of the 8 h
flight, which took place mostly in the lower stratosphere. The aircraft
sampled a wide range of background conditions including the region near the
jet core, the jet exit and over the Rocky Mountains with clear evidence of
vertically propagating gravity waves of along-track wavelength between 100
and 120 km. The power spectra of the horizontal velocity components and
potential temperature for the scale approximately between ~ 8 and ~ 256 km display an approximate −5/3 power law in
agreement with past studies on aircraft measurements, while the fluctuations
roll over to a −3 power law for the scale approximately between
~ 0.5 and ~ 8 km (except when this part of
the spectrum is activated, as recorded clearly by one of the flight
segments). However, at least part of the high-frequency signals with sampled
periods of ~ 20–~ 60 s and wavelengths of
~ 5–~ 15 km might be due to intrinsic
observational errors in the aircraft measurements, even though the
possibilities that these fluctuations may be due to other physical phenomena
(e.g., nonlinear dynamics, shear instability and/or turbulence) cannot be
completely ruled out.

With the aim of contributing to the improvement of subgrid-scale gravity wave (GW) parameterizations in numerical-weather-prediction and climate models, the comparative relevance in GW drag of direct GW-mean-flow interactions and turbulent wave breakdown are investigated. Of equal interest is how well Wentzel-Kramer-Brillouin (WKB) theory can capture direct wave-mean-flow interactions, that are excluded by applying the steady-state approximation. WKB is implemented in a very efficient Lagrangian ray-tracing approach that considers wave action density in phase-space, thereby avoiding numerical instabilities due to caustics. It is supplemented by a simple wave-breaking scheme based on a static-instability saturation criterion. Idealized test cases of horizontally homogeneous GW packets are considered where wave-resolving Large-Eddy Simulations (LES) provide the reference. In all of theses cases the WKB simulations including direct GW-mean-flow interactions reproduce the LES data, to a good accuracy, already without wave-breaking scheme. The latter provides a next-order correction that is useful for fully capturing the total-energy balance between wave and mean flow. Moreover, a steady-state WKB implementation, as used in present GW parameterizations, and where turbulence provides, by the non-interaction paradigm, the only possibility to affect the mean flow, is much less able to yield reliable results. The GW energy is damped too strongly and induces an oversimplified mean flow. This argues for WKB approaches to GW parameterization that take wave transience into account.

We derive a wave-averaged potential vorticity equation describing the evolution of strongly stratified, rapidly rotating quasi-geostrophic (QG) flow in a field of inertia-gravity internal waves. The derivation relies on a multiple-time-scale asymptotic expansion of the Eulerian Boussinesq equations. Our result confirms and extends the theory of Bühler & McIntyre (
J. Fluid Mech.
, vol. 354, 1998, pp. 609–646) to non-uniform stratification with buoyancy frequency
$N(z)$
and therefore non-uniform background potential vorticity
$f_{0}N^{2}(z)$
, and does not require spatial-scale separation between waves and balanced flow. Our interest in non-uniform background potential vorticity motivates the introduction of a new quantity: ‘available potential vorticity’ (APV). Like Ertel potential vorticity, APV is exactly conserved on fluid particles. But unlike Ertel potential vorticity, linear internal waves have no signature in the Eulerian APV field, and the standard QG potential vorticity is a simple truncation of APV for low Rossby number. The definition of APV exactly eliminates the Ertel potential vorticity signal associated with advection of a non-uniform background state, thereby isolating the part of Ertel potential vorticity available for balanced-flow evolution. The effect of internal waves on QG flow is expressed concisely in a wave-averaged contribution to the materially conserved QG potential vorticity. We apply the theory by computing the wave-induced QG flow for a vertically propagating wave packet and a mode-one wave field, both in vertically bounded domains.

A finite-volume model of the classic differentially heated rotating annulus experiment is used to study the spontaneous emission of gravity waves (GWs) from jet stream imbalances, which may be an important source of these waves in the atmosphere and for which no satisfactory parameterisation exists. Experiments were performed using a classic laboratory configuration as well as using a much wider and shallower annulus with a much larger temperature difference between the inner and outer cylinder walls. The latter configuration is more atmosphere-like, in particular since the Brunt Vaisfila frequency is larger than the inertial frequency, resulting in more realistic GW dispersion properties. In both experiments, the model is initialised with a baroclinically unstable axisymmetric state established using a two-dimensional version of the code, and a low-azimuthal-mode baroclinic wave featuring a meandering jet is allowed to develop. Possible regions of GW activity are identified by the horizontal velocity divergence and a modal decomposition of the small-scale structures of the flow. Results indicate GW activity in both annulus configurations close to the inner cylinder wall and within the baroclinic wave. The former is attributable to boundary layer instabilities, while the latter possibly originates in part from spontaneous GW emission from the baroclinic wave.

Significance
High- and low-pressure systems, commonly referred to as synoptic systems, are the most energetic fluctuations of wind and temperature in the midlatitude troposphere. Synoptic systems are a few thousand kilometers in scale and are governed by a balance between the pressure gradient force and the Coriolis force. Observations collected near the tropopause by commercial aircraft indicate a change in dynamics at horizontal scales smaller than about 500 km. Smaller-scale fluctuations are shown to be dominated by inertia–gravity waves, waves that propagate on vertical density gradients but are influenced by Earth’s rotation.

Wind forcing of the ocean generates a spectrum of inertia–gravity waves that is sharply peaked near the local inertial (or Coriolis) frequency. The corresponding near-inertial waves (NIWs) are highly energetic and play a significant role in the slow, large-scale dynamics of the ocean. To analyse this role, we develop a new model of the non-dissipative interactions between NIWs and balanced motion. The model is derived using the generalised-Lagrangian-mean (GLM) framework (specifically, the ‘glm’ variant of Soward & Roberts,
J. Fluid Mech.
, vol. 661, 2010, pp. 45–72), taking advantage of the time-scale separation between the two types of motion to average over the short NIW period. We combine Salmon’s (
J. Fluid Mech.
, vol. 719, 2013, pp. 165–182) variational formulation of GLM with Whitham averaging to obtain a system of equations governing the joint evolution of NIWs and mean flow. Assuming that the mean flow is geostrophically balanced reduces this system to a simple model coupling Young & Ben Jelloul’s (
J. Mar. Res.
, vol. 55, 1997, pp. 735–766) equation for NIWs with a modified quasi-geostrophic (QG) equation. In this coupled model, the mean flow affects the NIWs through advection and refraction; conversely, the NIWs affect the mean flow by modifying the potential-vorticity (PV) inversion – the relation between advected PV and advecting mean velocity – through a quadratic wave term, consistent with the GLM results of Bühler & McIntyre (
J. Fluid Mech.
, vol. 354, 1998, pp. 301–343). The coupled model is Hamiltonian and its conservation laws, for wave action and energy in particular, prove illuminating: on their basis, we identify a new interaction mechanism whereby NIWs forced at large scales extract energy from the balanced flow as their horizontal scale is reduced by differential advection and refraction so that their potential energy increases. A rough estimate suggests that this mechanism could provide a significant sink of energy for mesoscale motion and play a part in the global energetics of the ocean. Idealised two-dimensional models are derived and simulated numerically to gain insight into NIW–mean-flow interaction processes. A simulation of a one-dimensional barotropic jet demonstrates how NIWs forced by wind slow down the jet as they propagate into the ocean interior. A simulation assuming plane travelling NIWs in the vertical shows how a vortex dipole is deflected by NIWs, illustrating the irreversible nature of the interactions. In both simulations energy is transferred from the mean flow to the NIWs.

A method of second-order accuracy is described for integrating the equations of ideal compressible flow. The method is based on the integral conservation laws and is dissipative, so that it can be used across shocks. The heart of the method is a one-dimensional Lagrangean scheme that may be regarded as a second-order sequel to Godunov's method. The second-order accuracy is achieved by taking the distributions of the state quantities inside a gas slab to be linear, rather than uniform as in Godunov's method. The Lagrangean results are remapped with least-squares accuracy onto the desired Euler grid in a separate step. Several monotonicity algorithms are applied to ensure positivity, monotonicity and nonlinear stability. Higher dimensions are covered through time splitting. Numerical results for one-dimensional and two-dimensional flows are presented, demonstrating the efficiency of the method. The paper concludes with a summary of the results of the whole series “Towards the Ultimate Conservative Difference Scheme.”

A computational model of the pseudo-incompressible equations is used to probe the range of validity of an extended Wentzel–Kramers–Brillouin theory (XWKB), previously derived through a distinguished limit of a multiple-scale asymptotic analysis of the Euler or pseudo-incompressible equations of motion, for gravity-wave packets at large amplitudes. The governing parameter of this analysis had been the scale-separation ratio \$\varepsilon \$ between the gravity wave and both the large-scale potential-temperature stratification and the large-scale wave-induced mean flow. A novel feature of the theory had been the non-resonant forcing of higher harmonics of an initial wave packet, predominantly by the large-scale gradients in the gravity-wave fluxes. In the test cases considered a gravity-wave packet is propagating upwards in a uniformly stratified atmosphere. Large-scale winds are induced by the wave packet, and possibly exert a feedback on the latter. In the limit \$\varepsilon \ll 1\$ all predictions of the theory can be validated. The larger \$\varepsilon \$ is the more the transfer of wave energy to the mean flow is underestimated by the theory. The numerical results quantify this behaviour but also show that, qualitatively, XWKB remains valid even when the gravity-wave wavelength approaches the spatial scale of the wave-packet amplitude. This includes the prevalence of first and second harmonics and the smallness of harmonics with wave number higher than two. Furthermore, XWKB predicts for the vertical momentum balance an additional leading-order buoyancy term in Euler and pseudo-incompressible theory, compared with the anelastic theory. Numerical tests show that this term is relatively large with up to \$30\hspace{0.167em} \% \$ of the total balance. The practical relevance of this deviation remains to be assessed in future work.

As upward-propagating anelastic internal gravity wave packets grow in amplitude, nonlinear effects de-velop as a result of interactions with the horizontal mean flow that they induce. This qualitatively alters the structure of the wave packet. The weakly nonlinear dynamics are well captured by the nonlinear Schrö dinger equation, which is derived here for anelastic waves. In particular, this predicts that strongly nonhydrostatic waves are modulationally unstable and so the wave packet narrows and grows more rapidly in amplitude than the exponential anelastic growth rate. More hydrostatic waves are modulationally stable and so their am-plitude grows less rapidly. The marginal case between stability and instability occurs for waves propagating at the fastest vertical group velocity. Extrapolating these results to waves propagating to higher altitudes (hence attaining larger amplitudes), it is anticipated that modulationally unstable waves should break at lower al-titudes and modulationally stable waves should break at higher altitudes than predicted by linear theory. This prediction is borne out by fully nonlinear numerical simulations of the anelastic equations. A range of sim-ulations is performed to quantify where overturning actually occurs.

The Boussinesq equations provide a convenient modeling framework for studies of gravity-wave dynamics not affected by the impact of varying density on the wave amplitude. In the atmosphere, however, gravity waves undergo tremendous amplitude growth in their upward propagation since atmospheric density has a very strong vertical dependence. This effect, leading to all kinds of wave instabilities, is decisive for the corresponding wave mean-flow interaction. A study of these processes within the complete Euler equations is complicated by their incorporation of sound waves which might at most be of secondary importance in this context. A way around this is the use of an approximated equation set, filtered of sound waves, but representing the dynamics of gravity waves at good accuracy. Both the classic anelastic equations and the pseudo-incompressible equations offer themselves for this purpose. The question arises which of the two, if any, are consistent with a rigorous multiple-scale asymptotics of gravity-wave dynamics in the atmosphere. Hence, such an asymptotics is used to analyze the Euler equations so that the dynamical situation of a gravity wave (GW) near breaking level is best approximated. A simple saturation argument is used to obtain a potential-temperature wave scale, while linear theory yields from the latter the velocity scale, and the wave Exner pressure scale. It also determines the time scale once the spatial scale has been set. As small expansion parameter the product of vertical wave number and potential temperature scale height is used. It is shown that the resulting equation hierarchy is consistent with that obtained from the pseudo-incompressible equations, both for non-hydrostatic and hydrostatic gravity waves. This gives a mathematical justification for the use of the pseudo-incompressible equations for studies of gravity-wave breaking in the atmosphere. An analogous argument does not seem to exist for the anelastic equations

Gravity waves (GWs) and thermal tides are important phenomena in middle-atmosphere dy-namics. Breaking GWs have a major impact on the mean circulation in the middle atmosphere (MA). Due to the limitations in computational power most complex MA circulation models have to incorporate the effect of unresolved GWs via an efficient parametrization. Typically, these are of vertical column type and ignore horizontal and temporal variations in the background fields. However, highly transient tidal perturbations are always present and dominate diurnal variations in the MA through which the GWs propagate. Even in studies of the interaction between GWs and these thermal tides, a possibly important aspect of tidal dynamics, columnar parametrizations of GWs have been applied which do not account for the time dependence of thermal tides. A ray tracing technique is used to illuminate the impact of horizontal gradients of the back-ground (including the tides) and its time dependence on the propagation and dissipation of GWs. It is shown that tidal transience leads to a modulation of the absolute, or sometimes called ground-based, frequency of slowly propagating GWs. Due to large tidal wind variations in the upper mesosphere most parts of the assumed GW spectrum are slowed down in critical layer type regions. Then, the combined action of horizontal wave number refraction and fre-quency modulation induce changes in the horizontal phase speed which may exceed the initial phase speed by orders of magnitude. The phase speed variations have the tendency to follow the shape of the tidal background wind. This effect leads to less critical layer filtering of GWs and therefore decreased periodic background flow forcing due to momentum flux divergences as compared to a classical vertical column parametrization of instantaneously adjusting GW trains.

The effects of gravity wave saturation on the mean circulation and thermal structure of the middle atmosphere were investigated using quasi-compressible numerical and semianalytic wave action models. The study covered mean flow accelerations associated with the propagation of large amplitude gravity waves and convective adjustments to the gravity wave saturation and without saturation. Large amplitude wave motions, near-linear mean wind accelerations and gravity self-accelerations all produced nonlinear features with or without saturation. The wave packet experienced vertical spreading and a concommitant reduction in the local wave action density. A non-WKB effect was detected insofar as self-acceleration permitted the wave motion to propagate beyond a critical level dislocation. The quasi-linear effects are concluded dominant for middle atmospheric motions in some situations and not amenable to linear modeling.

This paper describes a new computationally efficient, ultrasimple nonorographic spectral gravity wave parameterization model. Its predictions compare favorably, though not perfectly, with a model of gravity wave propagation and breaking that computes the evolution with altitude of a full, frequency- and wavenumber-dependent gravity wave spectrum. The ultrasimple model depends on making the midfrequency (hydrostatic, nonrotating) approximation to the dispersion relation, as in Hines' parameterization. This allows the full frequency-wavenumber spectrum of pseudomomentum flux to be integrated with respect to frequency, and thus reduced to a spectrum that depends on vertical wavenumber m and azimuthal direction ø only. The ultrasimple model treats the m dependence as consisting of up to three analytically integrable segments, or "parts". This allows the total pseudomomentum flux to be evaluated by using analytical expressions for the areas under the parts rather than by performing numerical quadratures. The result is a much greater computational efficiency. The model performs significantly better than an earlier model that treated the m dependence as consisting of up to two parts. Numerical experiments show that similar models with more than three parts using the midfrequency approximation yield little further improvement. The limiting factor is the midfrequency approximation and not the number of parts.

Using a new generalization of the Eliassen-Palm relations, we discuss
the zonal-mean-flow tendency /t due to waves in a stratified, rotating
atmosphere, with particular attention to equatorially trapped modes.
Wave transience, forcing and dissipation are taken into account in a
very general way. The theory makes it possible to discuss the
latitudinal (y) and vertical (z) dependence of /t qualitatively and
calculate it directly from an approximate knowledge of the wave
structure. For equatorial modes it reveals that the y profile of /t is
strongly dependent on the nature of the forcing or dissipation
mechanism. A by-product of the theory is a far-reaching generalization
of the theorems of Charney-Drazin, Dickinson and Holton on the forcing
of /t by conservative linear waves.Implications for the quasi-biennial
oscillation in the equatorial stratosphere are discussed. Graphs of y
profiles of /t are given for the equatorial waves considered in the
recent analysis of observational data by Lindzen and Tsay (1975). The y
profile of uI t for
Rossby-gravity and inertio-gravity modes, in Lindzen and Tsay's
parameter ranges, prove extremely sensitive to whether or not small
amounts of mechanical dissipation are present alongside the
radiative-photochemical dissipation of the waves. The
probable importance of low-frequency Rossby waves for the momentum
budget of the descending easterlies is suggested.

An approximate theory is developed of small-amplitude transient eddies
on a slowly varying time-mean flow. Central to this theory is a flux
MT, which in most respects constitutes a generalization of
the Eliassen-Palm flux to three dimensions; it is a conservable measure
of the flux of eddy activity (for small amplitude transients) and is
parallel to group velocity for an almost-plane wave train. The use of
this flux as a diagnostic of transient eddy propagation is demonstrated
by application of the theory to a ten-year climatology of the Northern
Hemisphere winter circulation. Results show the anticipated
concentration of eddy flux along the major storm tracks.While, in a
suitably transformed system, MT may be regarded as a flux of
upstream momentum, it is not a complete description of the eddy forcing
of the mean flow; additional effects arise due to downstream transience
(i.e., spatial inhomogeneity in the direction of the time-mean flow) of
the eddy amplitudes.The relation between MT and the
`E-vector' of Hoskins et al. is discussed.

Extensions of Weinstock's theory of nonlinear gravity waves and a
parameterization of the related momentum deposition are developed. Our
approach, which combines aspects of Hines' Doppler spreading theory with
Weinstock's theory of nonlinear wave diffusion, treats the low-frequency
part of the gravity wave spectrum as an additional background flow for
higher-frequency waves. This technique allows one to calculate frequency
shifting and wave amplitude damping produced by the interaction with
this additional background wind. For a nearly monochromatic spectrum the
parameterization formulae for wave drag coincide with those of Lindzen.
It is shown that two processes should be distinguished: wave breaking
due to instabilities and saturation due to nonlinear diffusionlike
processes. The criteria for wave breaking and wave saturation in terms
of wave spectra are derived. For a saturated spectrum the power spectral
density's (PSD) dependence S(m) = AN2/m3 is
obtained, where m is the vertical wavenumber and N is the
Brunt-Väisäla frequency. Unlike Weinstock's original
formulation, our coefficient of proportionality A is a slowly varying
function of m and mean wind. For vertical wavelengths ranging from 10 km
to 100 m and for typical wind shears, A varies from one half to one
ninth. Calculations of spectral evolution with height as well as related
profiles of wave drag are shown. These results reproduce vertical
wavenumber spectral tail slopes which vary near the -3 value reported by
observations. An explanation of these variations is given.

An overview of the parameterization of gravity wave drag in numerical weather prediction and climate simulation models is presented. The focus is mainly on understanding the current status of gravity wave drag parameterization as a step towards the new parameterizations that will be needed for the next generation of atmospheric models. Both the early history and latest developments in the field are discussed. Parameterizations developed specifically for orographic and convective sources of gravity waves are described separately, as are newer parameterizations that collectively treat a spectrum of gravity wave motions. Differences in parameterization issues and approaches between the lower and middle atmospheres are highlighted. Various emerging issues are also discussed, such as explicitly resolved gravity waves and gravity wave drag in models, and a range of unparameterized gravity wave processes that may need future attention for the next generation of gravity wave drag parameterizations in models

Durran’s pseudo-incompressible equations are integrated in a mass and momentum conserving way with a new implicit turbulence model. This system is sound-proof, which has two major advantages over fully compressible systems: the CFL condition for stable time advancement is no longer dictated by the speed of sound and all waves in the model are clearly gravity waves (GW) or geostrophic modes. Thus, the pseudo-incompressible equations are an ideal laboratory model for studying GW generation, propagation and breaking. Gravity wave breaking creates turbulence which needs to be parameterised. For the first time the adaptive local deconvolution method (ALDM) for implicit large eddy simulation (LES) is applied to non-Boussinesq stratified flows. ALDM provides a turbulence model that is fully merged with the discretisation of the flux function. In the context of non-Boussinesq stratified flows this poses some new numerical challenges, the solution of which we present in this text. In numerical test cases we show the agreement of the results with the literature (Robert’s hot/cold bubble test case), we present the sensitivity to the model’s resolution and discretisation and demonstrate qualitatively the behaviour of the implicit turbulence model for a 2D breaking gravity wave packet.

Idealized model examples of non-dissipative wave–mean
interactions, using small-amplitude and slow-modulation approximations,
are studied in order to re-examine
the usual assumption that the only important interactions are
dissipative. The results
clarify and extend the body of wave–mean interaction theory on
which our present
understanding of, for instance, the global-scale atmospheric
circulation depends (e.g.
Holton et
al. 1995). The waves considered are either gravity
or inertia–gravity waves.
The mean flows need not be zonally symmetric, but are approximately
‘balanced’ in
a sense that non-trivially generalizes the standard concepts of
geostrophic or higher-order balance at low Froude and/or
Rossby number. Among the examples studied are
cases in which irreversible mean-flow changes, capable of persisting
after the gravity
waves have propagated out of the domain of interest, take place
without any need
for wave dissipation. The irreversible mean-flow changes can be
substantial in certain
circumstances, such as Rossby-wave resonance, in which
potential-vorticity contours
are advected cumulatively. The examples studied in detail use
shallow-water systems,
but also provide a basis for generalizations to more realistic,
stratified flow models.
Independent checks on the analytical shallow-water results are
obtained by using a
different method based on particle-following averages in the sense
of ‘generalized
Lagrangian-mean theory’, and by verifying the theoretical
predictions with nonlinear
numerical simulations. The Lagrangian-mean method is seen to
generalize easily to the
three-dimensional stratified Boussinesq model, and to allow a
partial generalization of
the results to finite amplitude. This includes a finite-amplitude
mean potential-vorticity
theorem with a larger range of validity than had been hitherto recognized.

This paper continues the work started in Part 1 (Reznik, Zeitlin & Ben Jelloul 2001) and generalizes it to the case of a stratified environment. Geostrophic adjustment of localized disturbances is considered in the context of the two-layer shallow-water and continuously stratified primitive equations in the vertically bounded and horizontally infinite domain on the $f$-plane. Using multiple-time-scale perturbation expansions in Rossby number $\hbox{\it Ro}$ we show that stratification does not substantially change the adjustment scenario established in Part 1 and any disturbance of well-defined scale is split in a unique way into slow and fast components with characteristic time scales $f_0^{-1}$ and $(f_0 \hbox{\it Ro})^{-1}$ respectively, where $f_0$ is the Coriolis parameter. As in Part 1 we distinguish two basic dynamical regimes: quasi-geostrophic (QG) and frontal geostrophic (FG) with small and large deviations of the isopycnal surfaces, respectively. We show that the dynamics of the FG regime in the two-layer model depends strongly on the ratio of the layer depths. The difference between QG and FG scenarios of adjustment is demonstrated. In the QG case the fast component of the flow essentially does not ‘feel’ the slow one and is rapidly dispersed leaving the slow component to evolve according to the standard QG equation (corrections to this equation are found for times $t\,{\gg}\, (f_0 \hbox{\it Ro})^{-1}$). In the FG case the fast component is a packet of inertial oscillations produced by the initial perturbation. The space-time evolution of the envelope of inertial oscillations obeys a Schrödinger-type modulation equation with coefficients depending on the slow component. In both QG and FG cases we show by direct computations that the fast component does not produce any drag terms in the equations for the slow component; the slow component remains close to the geostrophic balance. However, in the continuously stratified FG regime, as well as in the two-layer regime with the layers of comparable thickness, the splitting is incomplete in the sense that the slow vortical component and the inertial oscillations envelope evolve on the same time scale.

We present a theoretical study of a fundamentally new wave–mean or wave–vortex interaction effect able to force persistent, cumulative change in mean flows in the absence of wave breaking or other kinds of wave dissipation. It is associated with the refraction of non-dissipating waves by inhomogeneous mean (vortical) flows. The effect is studied in detail in the simplest relevant model, the two-dimensional compressible flow equations with a generic polytropic equation of state. This includes the usual shallow-water equations as a special case. The refraction of a narrow, slowly varying wavetrain of small-amplitude gravity or sound waves obliquely incident on a single weak (low Froude or Mach number) vortex is studied in detail. It is shown that, concomitant with the changes in the waves' pseudomomentum due to the refraction, there is an equal and opposite recoil force that is felt, in effect, by the vortex core. This effective force is called a ‘remote recoil’ to stress that there is no need for the vortex core and wavetrain to overlap in physical space. There is an accompanying ‘far-field recoil’ that is still more remote, as in classical vortex-impulse problems. The remote-recoil effects are studied perturbatively using the wave amplitude and vortex weakness as small parameters. The nature of the remote recoil is demonstrated in various set-ups with wavetrains of finite or infinite length. The effective recoil force ${\bm R}_V$ on the vortex core is given by an expression resembling the classical Magnus force felt by moving cylinders with circulation. In the case of wavetrains of infinite length, an explicit formula for the scattering angle $\theta_*$ of waves passing a vortex at a distance is derived correct to second order in Froude or Mach number. To this order ${\bm R}_V\,{\propto}\,\theta_*$. The formula is cross-checked against numerical integrations of the ray-tracing equations. This work is part of an ongoing study of internal-gravity-wave dynamics in the atmosphere and may be important for the development of future gravity-wave parametrization schemes in numerical models of the global atmospheric circulation. At present, all such schemes neglect remote-recoil effects caused by horizontally inhomogeneous mean flows. Taking these effects into account should make the parametrization schemes significantly more accurate.

An exact and very general Lagrangian-mean description of the back effect of oscillatory disturbances upon the mean state is given. The basic formalism applies to any problem whose governing equations are given in the usual Eulerian form, and irrespective of whether spatial, temporal, ensemble, or ‘two-timing’ averages are appropriate. The generalized Lagrangian-mean velocity cannot be defined exactly as the ‘mean following a single fluid particle’, but in cases where spatial averages are taken can easily be visualized, for instance, as the motion of the centre of mass of a tube of fluid particles which lay along the direction of averaging in a hypothetical initial state of no disturbance.
The equations for the Lagrangian-mean flow are more useful than their Eulerian-mean counterparts in significant respects, for instance in explicitly representing the effect upon mean-flow evolution of wave dissipation or forcing. Applications to irrotational acoustic or water waves, and to astrogeophysical problems of waves on axisymmetric mean flows are discussed. In the latter context the equations embody generalizations of the Eliassen-Palm and Charney-Drazin theorems showing the effects on the mean flow of departures from steady, conservative waves, for arbitrary, finite-amplitude disturbances to a stratified, rotating fluid, with allowance for self-gravitation as well as for an external gravitational field.
The equations show generally how the pseudomomentum (or wave ‘momentum’) enters problems of mean-flow evolution. They also indicate the extent to which the net effect of the waves on the mean flow can be described by a ‘radiation stress’, and provide a general framework for explaining the asymmetry of radiation-stress tensors along the lines proposed by Jones (1973).

We derive an asymptotic model that describes the nonlinear coupled evolution of (i) near-inertial waves (NIWs), (ii) balanced quasi-geostrophic flow and (iii) near-inertial second harmonic waves with frequency near $2f_{0}$ , where $f_{0}$ is the local inertial frequency. This ‘three-component’ model extends the two-component model derived by Xie & Vanneste ( J. Fluid Mech. , vol. 774, 2015, pp. 143–169) to include interactions between near-inertial and $2f_{0}$ waves. Both models possess two conservation laws which together imply that oceanic NIWs forced by winds, tides or flow over bathymetry can extract energy from quasi-geostrophic flows. A second and separate implication of the three-component model is that quasi-geostrophic flow catalyses a loss of NIW energy to freely propagating waves with near- $2f_{0}$ frequency that propagate rapidly to depth and transfer energy back to the NIW field at very small vertical scales. The upshot of near- $2f_{0}$ generation is a two-step mechanism whereby quasi-geostrophic flow catalyses a nonlinear transfer of near-inertial energy to the small scales of wave breaking and diapycnal mixing. A comparison of numerical solutions with both Boussinesq and three-component models for a two-dimensional initial value problem reveals strengths and weaknesses of the model while demonstrating the extraction of quasi-geostrophic energy and production of small vertical scales.

Under assumptions of horizontal homogeneity and isotropy, one may derive relations between rotational or divergent kinetic energy spectra and velocities along one-dimensional tracks, such as might be measured by aircraft. Two recent studies, differing in details of their implementation, have applied these relations to the Measurement of Ozone and Water Vapor by Airbus In-Service Aircraft (MOZAIC) dataset and reached different conclusions with regard to the mesoscale ratio of divergent to rotational kinetic energy. In this study the accuracy of the method is assessed using global atmospheric simulations performed with the Model for Prediction Across Scales, where the exact decomposition of the horizontal winds into divergent and rotational components may be easily computed. For data from the global simulations, the two approaches yield similar and very accurate results. Errors are largest for the divergent component on synoptic scales, which is shown to be related to a very dominant rotational mode...

[1] Atmospheric gravity waves have been a subject of intense research activity in recent years because of their myriad effects and their major contributions to atmospheric circulation, structure, and variability. Apart from occasionally strong lower-atmospheric effects, the major wave influences occur in the middle atmosphere, between ∼ 10 and 110 km altitudes because of decreasing density and increasing wave amplitudes with altitude. Theoretical, numerical, and observational studies have advanced our understanding of gravity waves on many fronts since the review by Fritts [1984a]; the present review will focus on these more recent contributions. Progress includes a better appreciation of gravity wave sources and characteristics, the evolution of the gravity wave spectrum with altitude and with variations of wind and stability, the character and implications of observed climatologies, and the wave interaction and instability processes that constrain wave amplitudes and spectral shape. Recent studies have also expanded dramatically our understanding of gravity wave influences on the large-scale circulation and the thermal and constituent structures of the middle atmosphere. These advances have led to a number of parameterizations of gravity wave effects which are enabling ever more realistic descriptions of gravity wave forcing in large-scale models. There remain, nevertheless, a number of areas in which further progress is needed in refining our understanding of and our ability to describe and predict gravity wave influences in the middle atmosphere. Our view of these unknowns and needs is also offered.
Abstract
[1] Atmospheric gravity waves have been a subject of intense research activity in recent years because of their myriad effects and their major contributions to atmospheric circulation, structure, and variability. Apart from occasionally strong lower-atmospheric effects, the major wave influences occur in the middle atmosphere, between ∼ 10 and 110 km altitudes because of decreasing density and increasing wave amplitudes with altitude. Theoretical, numerical, and observational studies have advanced our understanding of gravity waves on many fronts since the review by Fritts [1984a]; the present review will focus on these more recent contributions. Progress includes a better appreciation of gravity wave sources and characteristics, the evolution of the gravity wave spectrum with altitude and with variations of wind and stability, the character and implications of observed climatologies, and the wave interaction and instability processes that constrain wave amplitudes and spectral shape. Recent studies have also expanded dramatically our understanding of gravity wave influences on the large-scale circulation and the thermal and constituent structures of the middle atmosphere. These advances have led to a number of parameterizations of gravity wave effects which are enabling ever more realistic descriptions of gravity wave forcing in large-scale models. There remain, nevertheless, a number of areas in which further progress is needed in refining our understanding of and our ability to describe and predict gravity wave influences in the middle atmosphere. Our view of these unknowns and needs is also offered.

The interaction between solar tides (STs) and gravity waves (GWs) is studied via the coupling of a three-dimensional ray-tracer model and a linear tidal model. The ray-tracer model describes GW dynamics on a spatially and time dependent background formed by a monthly mean climatology and STs. It does not suffer from typical simplifications of conventional GW parameterizations where horizontal GW propagation and the effects of horizontal background gradients on GW dynamics are neglected. The ray-tracer model uses a variant of Wentzel-Kramers-Brillouin (WKB) theory where a spectral description in position-wavenumber space is helping to avoid numerical instabilities otherwise likely to occur in caustic-like situations. The tidal model has been obtained by linearization of the primitive equations about a monthly mean, allowing for stationary planetary waves. The communication between ray-tracer model and tidal model is facilitated using latitude and altitude-dependent coefficients, named Rayleigh-friction and Newtonian-relaxation, and obtained from regressing GW momentum and buoyancy fluxes against the STs and their tendencies. These coefficients are calculated by the ray-tracer model and then implemented into the tidal model. An iterative procedure updates successively the GW fields and the tidal fields until convergence is reached. Notwithstanding the simplicity of the employed GW source many aspects of observed tidal dynamics are reproduced. It is shown that the conventional ``single-column'' approximation leads to significantly overestimated GW fluxes and hence underestimated ST amplitudes, pointing at a sensitive issue of GW parameterizations in general.

Longitudinal and transverse structure functions, \$D_{ll}=\langle {\it\delta}u_{l}{\it\delta}u_{l}\rangle\$ and \$D_{tt}=\langle {\it\delta}u_{t}{\it\delta}u_{t}\rangle\$, can be calculated from aircraft data. Here, \${\it\delta}\$ denotes the increment between two points separated by a distance \$r\$, \$u_{l}\$ and \$u_{t}\$ the velocity components parallel and perpendicular to the aircraft track respectively and \$\langle \,\rangle\$ an average. Assuming statistical axisymmetry and making a Helmholtz decomposition of the horizontal velocity, \$\boldsymbol{u}=\boldsymbol{u}_{r}+\boldsymbol{u}_{d}\$, where \$\boldsymbol{u}_{r}\$ is the rotational and \$\boldsymbol{u}_{d}\$ the divergent component of the velocity, we derive expressions relating the structure functions \$D_{rr}=\langle {\it\delta}\boldsymbol{u}_{r}\boldsymbol{\cdot }{\it\delta}\boldsymbol{u}_{r}\rangle\$ and \$D_{dd}=\langle {\it\delta}\boldsymbol{u}_{d}\boldsymbol{\cdot }{\it\delta}\boldsymbol{u}_{d}\rangle\$ to \$D_{ll}\$ and \$D_{tt}\$. Corresponding expressions are also derived in spectral space. The decomposition is applied to structure functions calculated from aircraft data. In the lower stratosphere, \$D_{rr}\$ and \$D_{dd}\$ both show a nice \$r^{2/3}\$-dependence for \$r\in [2,20]\ \text{km}\$. In this range, the ratio between rotational and divergent energy is a little larger than unity, excluding gravity waves as the principal agent behind the observations. In the upper troposphere, \$D_{rr}\$ and \$D_{dd}\$ show no clean \$r^{2/3}\$-dependence, although the overall slope of \$D_{dd}\$ is close to \$2/3\$ for \$r\in [2,400]\ \text{km}\$. The ratio between rotational and divergent energy is approximately three for \$r<100\ \text{km}\$, excluding gravity waves also in this case. We argue that the possible errors in the decomposition at scales of the order of 10 km are marginal.

By using the renormalization group (RG) method, the interaction between balanced flows and Doppler-shifted inertia-gravity waves (GWs) is formulated for the hydrostatic Boussinesq equations on the f plane. The derived time-evolution equations [RG equations (RGEs)] describe the spontaneous GW radiation from the components slaved to the vortical flow through the quasi resonance, together with the GW radiation reaction on the large-scale flow. The quasi resonance occurs when the space-time scales of GWs are partially comparable to those of slaved components. This theory treats a coexistence system with slow time scales composed of GWs significantly Doppler-shifted by the vortical flow and the balanced flow that interact with each other. The theory includes five dependent variables having slow time scales: one slow variable (linear potential vorticity), two Doppler-shifted fast ones (GW components), and two diagnostic fast ones. Each fast component consists of horizontal divergence and ageostrophic vorticity. The spontaneously radiated GWs are regarded as superpositions of the GW components obtained as low-frequency eigenmodes of the fast variables in a given vortical flow. Slowly varying nonlinear terms of the fast variables are included as the diagnostic components, which are the sum of the slaved components and the GW radiation reactions. A comparison of the balanced adjustment equation (BAE) by Plougonven and Zhang with the linearized RGE shows that the RGE is formally reduced to the BAE by ignoring the GW radiation reaction, although the interpretation on the GW radiation mechanism is significantly different; GWs are radiated through the quasi resonance with a balanced flow because of the time-scale matching.

The renormalization group equations (RGEs) describing spontaneous inertia-gravity wave (GW) radiation from part of a balanced flow through a quasi resonance that were derived in a companion paper by Yasuda et al. are validated through numerical simulations of the vortex dipole using the Japan Meteorological Agency nonhydrostatic model (JMA-NHM). The RGEs are integrated for two vortical flow fields: the first is the initial condition that does not contain GWs used for the JMA-NHM simulations, and the second is the simulated thirtieth-day field by the JMA-NHM. The theoretically obtained GW distributions in both RGE integrations are consistent with the numerical simulations using the JMA-NHM. This result supports the validity of the RGE theory. GW radiation in the dipole is physically interpreted either as the mountain-wave-like mechanism proposed by McIntyre or as the velocity-variation mechanism proposed by Viudez. The shear of the large-scale flow likely determines which mechanism is dominant. In addition, the distribution of GW momentum fluxes is examined based on the JMA-NHM simulation data. The GWs propagating upward from the jet have negative momentum fluxes, while those propagating downward have positive ones. The magnitude of momentum fluxes is approximately proportional to the sixth power of the Rossby number between 0.15 and 0.4.

[1] The atmospheric gravity wave energy spectra often show power law dependencies with wavenumbers and frequencies. A simple mechanism involving off-resonant scale-separated interactions is proposed for their formation, namely the refraction of the wave packets in pseudorandom shears encountered during their vertical propagation. In the Boussinesq and rotating frame approximation the evolution of the spectral distribution of wave action is calculated within the eikonal formalism, i.e., via the simulation of the ray paths for an ensemble of elementary wave packets. The energy spectra are then easily built from the wave action spectra. Experiments are conducted where wave packets propagate away from Dirac delta function, or spectrally uniform sources at low altitudes, in realistic atmospheric background flows. The energy spectra show dependencies with the vertical wavenumber m and horizontal wavenumber k that are consistent with the most widely recognized empirical spectral models. A specific focus is given on the vertical evolution of the vertical wavenumber spectrum. The spectrum shows an invariant scaling as N2/m3 at large wavenumbers. It possesses a central wavenumber whose value depends on the total wave energy and is controlled by the statistics of the background mean flow. Similarly, the wave packet azimuths show an increasingly strong anisotropy resulting from the wave mean flow interaction at critical levels.

The study of internal gravity waves provides many challenges: they move along interfaces as well as in fully three-dimensional space, at relatively fast temporal and small spatial scales, making them difficult to observe and resolve in weather and climate models. Solving the equations describing their evolution poses various mathematical challenges associated with singular boundary value problems and large amplitude dynamics. This book provides the first comprehensive treatment of the theory for small and large amplitude internal gravity waves. Over 120 schematics, numerical simulations and laboratory images illustrate the theory and mathematical techniques, and 130 exercises enable the reader to apply their understanding of the theory. This is an invaluable single resource for academic researchers and graduate students studying the motion of waves within the atmosphere and ocean, and also mathematicians, physicists and engineers interested in the properties of propagating, growing and breaking waves.

[1] For several decades, jets and fronts have been known from observations to be significant sources of internal gravity waves in the atmosphere. Motivations to investigate these waves have included their impact on tropospheric convection, their contribution to local mixing and turbulence in the upper-troposphere, their vertical propagation into the middle atmosphere and the forcing of its global circulation. While many different studies have consistently highlighted jet exit regions as a favored locus for intense gravity waves, the mechanisms responsible for their emission had long remained elusive: one reason is the complexity of the environment in which the waves appear, another is that the waves constitute small deviations from the balanced dynamics of the flow generating them, i.e., they arise beyond our fundamental understanding of jets and fronts based on approximations that filter out gravity waves. Over the past two decades, the pressing need for improving parameterizations of non-orographic gravity waves in climate models that include a stratosphere has stimulated renewed investigations. The purpose of this review is to presents current knowledge and understanding on gravity waves near jets and fronts from observations, theory and modeling, and to discuss challenges for progress in coming years.

The dynamics of internal gravity waves is modelled using WKB theory in position wavenumber phase space. A transport equation for the phase-space wave-action density is derived for describing one-dimensional wave fields in a background with height-dependent stratification and height- and time-dependent horizontal-mean horizontal wind. The mean wind is coupled to the waves through the divergence of the mean vertical flux of horizontal momentum associated with the waves. The phase-space approach bypasses the caustics problem that occurs in WKB ray-tracing models when the wavenumber becomes a multivalued function of position, such as in the case of a wave packet encountering a reflecting jet or in the presence of a time-dependent background flow. Two numerical models were developed to solve the coupled equations for the wave-action density and horizontal mean wind: an Eulerian model using a finite-volume method, and a Lagrangian “phase-space ray tracer” that transports wave-action density along phase-space paths determined by the classical WKB ray equations for position and wavenumber. The models are used to simulate the upward propagation of a Gaussian wave packet through a variable stratification, a wind jet, and the mean flow induced by the waves. Results from the WKB models are in good agreement with simulations using a weakly nonlinear wave-resolving model as well as with a fully nonlinear large-eddy-simulation model. The work is a step toward more realistic parameterizations of atmospheric gravity waves in weather and climate models.

Principles of fluid dynamics are applied to large-scale flows in the
oceans and the atmosphere in this text, intended as a core curriculum in
geophysical fluid dynamics. Emphasis throughout the book is devoted to
basing scaling techniques and the derivation of systematic
approximations to the equations of motion. The inviscid dynamics of a
homogeneous fluid are examined to reveal the properties of
quasi-geostrophic motion. Attention is given to density stratification
as a basis for potential vorticity dynamics. Discussions are presented
of Rossby waves, inertial boundary currents, the beta-plane, energy
propagation, and wave interaction. Turbulent mixing is mentioned in the
context of large-scale flows. The use of the homogeneous model in
investigating wind-driven ocean circulation is demonstrated, and the
quasi-geostrophic dynamics of a stratified fluid are studied for a flow
on a sphere. Finally, instability theory is exposed as a fundamental
concept for dynamic meteorology and ocean dynamics.

Interactions between waves and mean flows play a crucial role in understanding the long-term aspects of atmospheric and oceanographic modelling. Indeed, our ability to predict climate change hinges on our ability to model waves accurately. This book gives a modern account of the nonlinear interactions between waves and mean flows such as shear flows and vortices. A detailed account of the theory of linear dispersive waves in moving media is followed by a thorough introduction to classical wave–mean interaction theory. The author then extends the scope of the classical theory and lifts its restriction to zonally symmetric mean flows. The book is a fundamental reference for graduate students and researchers in fluid mechanics, and can be used as a text for advanced courses; it will also be appreciated by geophysicists and physicists who need an introduction to this important area in fundamental fluid dynamics and atmosphere-ocean science.

A spectral parameterization of mean-flow forcing due to breaking gravity waves is described for application in the equations of motion in atmospheric models. The parameterization is based on linear theory and adheres closely to fundamental principles of conservation of wave action flux, linear stability, and wave-mean-flow interaction. Because the details of wave breakdown and nonlinear interactions are known to be very complex and are still poorly understood, only the simplest possible assumption is made: that the momentum fluxes carried by the waves are deposited locally and entirely at the altitude of linear wave breaking. This simple assumption allows a straightforward mapping of the momentum flux spectrum, input at a specified source altitude, into vertical profiles of mean-flow force. A coefficient of eddy diffusion can also be estimated. The parameterization can be used with any desired input spectrum of momentum flux. The results are sensitive to the details of this spectrum and also realistically sensitive to the background vertical shear and stability profiles. These sensitivities make the parameterization ideally suited for studying both the effects of gravity waves from unique sources like topography and convection as well as generalized broad input spectra. Existing constraints on input parameters are also summarized from the available observations. With these constraints, the parameterization generates realistic variations in gravity-wave-driven, mean-flow forcing.

Middle atmospheric general circulation models (GCMs) must employ a parameterization for small-scale gravity waves (GWs). Such parameterizations typically make very simple assumptions about gravity wave sources, such as uniform distribution in space and time or an arbitrarily specified GW source function. The authors present a configuration of theWholeAtmosphereCommunity ClimateModel (WACCM) that replaces the arbitrarily specifiedGWsource spectrum with GWsource parameterizations. For the nonorographic wave sources, a frontal system and convectiveGWsource parameterization are used. These parameterizations link GW generation to tropospheric quantities calculated by the GCM and provide a model-consistent GW representation. With the newGWsource parameterization, a reasonable middle atmospheric circulation can be obtained and the middle atmospheric circulation is better in several respects than that generated by a typical GW source specification. In particular, the interannual NH stratospheric variability is significantly improved as a result of the source-oriented GW parameterization. It is also shown that the addition of a parameterization to estimate mountain stress due to unresolved orography has a large effect on the frequency of stratospheric sudden warmings in the NH stratosphere by changing the propagation of stationary planetary waves into the polar vortex.

Ogura and Phillips derived the original anelastic model through systematic formal asymptotics using the flow Mach number as the expansion parameter. To arrive at a reduced model that would simultaneously represent internal gravity waves and the effects of advection on the same time scale, they had to adopt a distinguished limit requiring that the dimensionless stability of the background state be on the order of the Mach number squared. For typical flow Mach numbers of , this amounts to total variations of potential temperature across the troposphere of less than one Kelvin (i.e., to unrealistically weak stratification). Various generalizations of the original anelastic model have been proposed to remedy this issue. Later, Durran proposed the pseudoincompressible model following the same goals, but via a somewhat different route of argumentation. The present paper provides a scale analysis showing that the regime of validity of two of these extended models covers stratification strengths on the order of (hsc/θ)dθ/dz < M2/3, which corresponds to realistic variations of potential temperature θ across the pressure scale height hsc of .
Specifically, it is shown that (i) for (hsc/θ)dθ/dz < Mμ with 0 < μ < 2, the atmosphere features three asymptotically distinct time scales, namely, those of advection, internal gravity waves, and sound waves; (ii) within this range of stratifications, the structures and frequencies of the linearized internal wave modes of the compressible, anelastic, and pseudoincompressible models agree up to the order of Mμ; and (iii) if μ < ⅔, the accumulated phase differences of internal waves remain asymptotically small even over the long advective time scale. The argument is completed by observing that the three models agree with respect to the advective nonlinearities and that all other nonlinear terms are of higher order in M.

Studies on the spontaneous emission of gravity waves from jets, both observational and numerical, have emphasized that excitation of gravity waves occurred preferentially near regions of imbalance. Yet a quantitative relation between the several large-scale diagnostics of imbalance and the excited waves is still lacking.
The purpose of the present note is to investigate one possible way to relate quantitatively the gravity waves to diagnostics of the large-scale flow that is exciting them. Scaling arguments are used to determine how the large-scale flow may provide a forcing on the right-hand side of a wave equation describing the linear dynamics of the excited waves. The residual of the nonlinear balance equation plays an important role in this forcing.

The observational evidence for k to the -5/3 law behavior in the
atmospheric kinetic energy spectrum is reviewed. This evidence includes
the results of atmospheric wind variability studies and the observed
scale dependence of atmospheric dispersion. It is concluded that k to
the -5/3 law behavior for time and space scales greater than those that
can be three-dimensionally isotropic is probably a manifestation of the
two-dimensional reverse-cascading energy inertial range.

An analysis is made of Gage's proposal that the horizontal energy
spectrum at mesoscale wavelengths is produced by upscale energy transfer
through quasi-two-dimensional turbulence. It is suggested that principal
sources of such energy can be found in decaying convective clouds and
thunderstorm anvil outflows. These are believed to evolve similarly to
the wake of a moving body in a stably stratified flow. Following the
scale analysis by Riley, Metcalfe and Weissman it is expected that, in
the presence of strong stratification, initially three-dimensionally
isotropic turbulence divides roughly equally into gravity waves and
stratified (quasi-two- dimensional) turbulence. The former then
propagates away from the generation region, while the latter propagates
in spectral space to larger scales, forming the 5/3 upscale transfer
spectrum predicted by Kraichnan. Part of the energy of the stratified
turbulence is recycled into three-dimensional turbulence by shearing
instability, but the upscale escape of only a few percent of the total
energy released by small-scale turbulence is apparently sufficient to
explain the observed mesoscale energy spectrum of the troposphere. A
close analogy is found between the turbulence-gravity wave exchanges
considered here and the turbulence--wave exchanges discussed by Rhines
and Williams.

This article reviews the methods of wave–mean interaction theory for classical fluid dynamics, and for geophysical fluid dynamics in particular, providing a few examples for illustration. It attempts to bring the relevant equations into their simplest possible form, which highlights the organizing role of the circulation theorem in the theory. This is juxtaposed with a simple account of superfluid dynamics and the attendant wave–vortex interactions as they arise in the nonlinear Schrödinger equation. Here the fundamental physical situation is more complex than in the geophysical case, and the current mathematical understanding is more tentative. Classical interaction theory might be put to good use in the theoretical and numerical study of quantum fluid dynamics.

Scale analysis suggests that use of this "pseudo-incompressible equation' is justified if the Langrangian time scale of the disturbance is large compared with the time scale for sound wave propagation and the perturbation pressure is small compared to the vertically varying mean-state pressure. The mass-balance in the "pseudo-incompressible approximation' accounts for those density perturbations associated (through the equation of state) with perturbations in the temperature field. Density fluctuations associated with perturbations in the pressure field are neglected. The pseudo-incompressible equation is identical to the anelastic continuity equation when the mean stratification is adiabatic. The pseudo-incompressible approximation yields a system of equations suitable for use in nonhydrostatic numerical models. It also permits the diagnostic calculation of the vertical velocity in adiabatic flow, and might also be used to compute the net heating rate in a diabatic flow from extremely accurate observations of the three-dimensional velocity field and very coarse resolution (single sounding) thermodynamic data. -from Author

Conservable quantities measuring ‘wave activity’ are discussed. The equation for the most fundamental such quantity, wave-action, is derived in a simple but very general form which does not depend on the approximations of slow amplitude modulation, linearization, or conservative motion. The derivation is elementary , in the sense that a variational formulation of the equations of fluid motion is not used. The result depends, however, on a description of the disturbance in terms of particle displacements rather than velocities. A corollary is an elementary but general derivation of the approximate form of the wave-action equation found by Bretherton & Garrett (1968) for slowlyvarying, linear waves.
The sense in which the general wave-action equation follows from the classical ‘energy-momentum-tensor’ formalism is discussed, bringing in the concepts of pseudomomentum and pseudoenergy, which in turn are related to special cases such as Blokhintsev's conservation law in acoustics. Wave-action, pseudomomentum and pseudoenergy are the appropriate conservable measures of wave activity when ‘waves’ are defined respectively as departures from ensemble-, space- and time-averaged flows.
The relationship between the wave drag on a moving boundary and the fluxes of momentum and pseudomomentum is discussed.

The interaction between short internal gravity waves and a larger-scale mean flow in the ocean is analysed in the Wkbj approximation. The wave field determines the radiation-stress term in the momentum equation of the mean flow and a similar term in the buoyancy equation. The mean flow affects the propagation characteristics of the wave field. This cross-coupling is treated as a small perturbation. When relaxation effects within the wave field are considered, the mean flow induces a modulation of the wave field which is a linear functional of the spatial gradients of the mean current velocity. The effect that this modulation itself has on the mean flow can be reduced to the addition of diffusion terms to the equations for the mass and momentum balance of the mean flow. However, there is no vertical diffusion of mass and other passive properties. The diffusion coefficients depend on the frequency spectrum and the relaxation time of the internal-wave field and can be evaluated analytically. The vertical viscosity coefficient is found to be vv [approximate, equals] 4 x 103cm2/s and exceeds values typically used in models of the general circulation by at least two orders of magnitude.