Publications within Project 3DMSD
Latest:
 Rodal, Marie, & Schlutow, Mark (2021). Waves in the gas centrifuge: Asymptotic theory and similarities with the atmosphere. Journal of Fluid Mechanics, 928, A17. https://doi.org/10.1017/jfm.2021.811
 Schlutow, Mark and Voelker, Georg S.. "On strongly nonlinear gravity waves in a vertically sheared atmosphere" Mathematics of Climate and Weather Forecasting, vol. 6, no. 1, 2020, pp. 6374. https://doi.org/10.1515/mcwf20200103
Alphabetic order:
 Achatz, U., Ribstein, B., Senf, F., and Klein, R., 2016: The interaction between synopticscale balanced flow and a finiteamplitude mesoscale wave field throughout all atmospheric layers: Weak and moderately strong stratification. Q. J. R. Met. Soc. vol. 143 (2017), pp. 342–361 https://doi.org/10.1002/qj.2926
 Bölöni, G., Kim, Y., Borchert, S., & Achatz, U. (2021). Toward Transient SubgridScale Gravity Wave Representation in Atmospheric Models. Part I: Propagation Model Including Nondissipative Wave–MeanFlow Interactions, Journal of the Atmospheric Sciences, 78(4), 13171338. Retrieved May 10, 2021, https://doi.org/10.1175/JASD200065.1
 Bölöni, G., Ribstein, B., Muraschko, J., Sgoff, C., Wei, J. and U. Achatz 2016: The interaction between atmospheric gravity waves and largescale flows: an efficient description beyond the nonacceleration paradigm. J. Atmos. Sci. vol. 73 (2016), pp. 4833–4852 https://doi.org/10.1175/JASD160069.1 pdf
 Pütz, C., Schlutow, M. and Klein, R.: Initiation of ray tracing models: evolution of smallamplitude gravity wave packets in nonuniform background. Theor. Comput. Fluid Dyn. (2019) 33: 509535; https://doi.org/10.1007/s0016201900504z
 Ribstein, B. and U. Achatz 2016: The interaction between gravity waves and solar tides in a linear tidal model with a 4D raytracing gravitywave parameterization. J. Geophys. Res., 121, https://doi.org/10.1002/2016JA022478
 Ribstein, B., U. Achatz, and F. Senf 2015: The interaction between gravity waves and solar tides: Results from 4D ray tracing coupled to a linear tidal model, J. Geophys. Res. Space Physics, 120, 6795–6817, https://doi.org/10.1002/2015JA021349
 Rodal, Marie, & Schlutow, Mark (2021). Waves in the gas centrifuge: Asymptotic theory and similarities with the atmosphere. Journal of Fluid Mechanics, 928, A17. https://doi.org/10.1017/jfm.2021.811
 Schlutow, Mark and Wahlén, Erik. "Generalized modulation theory for strongly nonlinear gravity waves in a compressible atmosphere" Mathematics of Climate and Weather Forecasting, vol. 6, no. 1, 2020, pp. 97112. https://doi.org/10.1515/mcwf20200105
 Schlutow, Mark and Voelker, Georg S.. "On strongly nonlinear gravity waves in a vertically sheared atmosphere" Mathematics of Climate and Weather Forecasting, vol. 6, no. 1, 2020, pp. 6374. https://doi.org/10.1515/mcwf20200103
 Schlutow, M.; Klein, R.; Achatz, U., 2017: Finiteamplitude gravity waves in the atmosphere: traveling wave solutions, J. Fluid Mech., 826, 1034  1065 https://doi.org/10.1017/jfm.2017.459
 Schlutow, M., Wahlén, E. & Birken, P. (2019). Spectral stability of nonlinear gravity waves in the atmosphere. Mathematics of Climate and Weather Forecasting, 5(1), pp. 1233. Retrieved 20 Sep. 2019, https://doi.org/10.1515/mcwf20190002
 Schlutow, M., 2019: Modulational Stability of Nonlinear Saturated Gravity Waves. J. Atmos. Sci., 76, 3327–3336, https://doi.org/10.1175/JASD190065.1
 Sutherland, B.R., Achatz, U., Caulfield, C. and Klymak, J. M., 2019: Recent progress in modeling imbalance in the atmosphere and ocean. Phys. Rev. Fluids 4, 010501 , January 2019; https://doi.org/10.1103/PhysRevFluids.4.010501
 Völker, G.S., Akylas T.R. and U. Achatz, 2021: An Application of WKBJ Theory for Triad Interactions of Internal Gravity Waves in Varying Background Flows. Quart. J. Roy. Met. Soc., 147, 1112–1134. DOI: 10.1002/qj.3962
 Wei, J., G. Bölöni, and U. Achatz, 2019: Efficient Modeling of the Interaction of Mesoscale Gravity Waves with Unbalanced LargeScale Flows: PseudomomentumFlux Convergence versus Direct Approach. J. Atmos. Sci., 76, 2715–2738, https://doi.org/10.1175/JASD180337.1
 Wilhelm, J., T.R. Akylas, G. Bölöni, J. Wei, B. Ribstein, R. Klein, and U. Achatz, 2018: Interactions between Mesoscale and Submesoscale Gravity Waves and Their Efficient Representation in MesoscaleResolving Models. J. Atmos. Sci., 75, 2257–2280, https://doi.org/10.1175/JASD170289.1
Abstracts:
 The interaction between synoptic‐scale balanced flow and a finite‐amplitude mesoscale wave field throughout all atmospheric layers: weak and moderately strong stratification, Achatz, U., Ribstein, B., Senf, F., and Klein, R, https://doi.org/10.1002/qj.2926
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 (GMs), 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.  Toward Transient SubgridScale Gravity Wave Representation in Atmospheric Models. Part I: Propagation Model Including Nondissipative Wave–MeanFlow Interactions; Bölöni, G., Kim, Y., Borchert, S., & Achatz, U. (2021)., https://doi.org/10.1175/JASD200065.1
Current gravity wave (GW) parameterization (GWP) schemes are using the steadystate assumption, in which an instantaneous balance between GWs and mean flow is postulated, thereby neglecting transient, nondissipative interactions between the GW field and the resolved flow. These schemes rely exclusively on wave dissipation, by GW breaking or near critical layers, as a mechanism leading to forcing of the mean flow. In a transient GWP, without the steadystate assumption, nondissipative wave–meanflow interactions are enabled as an additional mechanism. Idealized studies have shown that this is potentially important, and therefore the transient GWP Multiscale Gravity Wave Model (MSGWaM) has been implemented into a stateoftheart weather and climate model. In this implementation, MSGWaM leads to a zonalmean circulation that agrees well with observations and increases GW momentumflux intermittency as compared with steadystate GWPs, bringing it into better agreement with superpressure balloon observations. Transient effects taken into account by MSGWaM are shown to make a difference even on monthly time scales: in comparison with steadystate GWPs momentum fluxes in the lower stratosphere are increased and the amount of missing drag at Southern Hemispheric high latitudes is decreased to a modest but nonnegligible extent. An analysis of the contribution of different wavelengths to the GW signal in MSGWaM suggests that smallscale GWs play an important role down to horizontal and vertical wavelengths of 50 km (or even smaller) and 200 m, respectively.  The Interaction between Atmospheric Gravity Waves and LargeScale Flows: An Efficient Description beyond the Nonacceleration Paradigm, Bölöni, G., Ribstein, B., Muraschko, J., Sgoff, C., Wei, J. and U. Achatz, https://doi.org/10.1175/JASD160069.1
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 (GMs), 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.  Initiation of ray tracing models: evolution of smallamplitude gravity wave packets in nonuniform background. By Pütz, C., Schlutow, M. and Klein, R; https://doi.org/10.1007/s0016201900504z
This study introduces a new computational scheme for the linear evolution of internal gravity wave packets passing over strongly nonuniform 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 bottomofthestratosphere 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 downwardpropagating 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 nonuniform backgrounds. Such nonstationary 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 phaseshifted eigenmodes, i.e., without any timestepping 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.  The interaction between gravity waves and solar tides in a linear tidal model with a 4D raytracing gravitywave parameterization. Ribstein, B. and U. Achatz 2016, J. Geophys. Res.,121, https://doi.org/10.1002/2016JA022478
Gravity waves (GWs) play an important role in atmospheric dynamics. Due to their short wavelengths, they must be parameterized in current weather and forecast models, which cannot resolve them explicitly. We are here the first to report the possibility and the implication of having an online GW parameterization in a linear but global model that incorporates their horizontal propagation, the effects of transients and of horizontal background gradients on GW dynamics. The GW parameterization is based on a ray‐tracer model with a spectral formulation that is safe against numerical instabilities due to caustics. The global model integrates the linearized primitive equations to obtain solar tides (STs), with a seasonally dependent reference climatology, forced by a climatological daily cycle of the tropospheric and stratospheric heating, and the (instantaneous) GW momentum and buoyancy flux convergences resulting from the ray tracer. Under a more conventional “single‐column” approximation, where GWs only propagate vertically and do not respond to horizontal gradients of the resolved flow, GW impacts are shown to be significantly changed in comparison with “full” experiments, leading to significant differences in ST amplitudes and phases, pointing at a sensitive issue of GW parameterizations in general. In the full experiment, significant semidiurnal STs arise even if the tidal model is only forced by diurnal heating rates. This indicates that an important part of the tidal signal is forced directly by GWs via their momentum and buoyancy deposition. In general, the effect of horizontal GW propagation and the GW response to horizontal large‐scale flow gradients is rather observed in nonmigrating than in migrating tidal components.  The interaction between gravity waves and solar tides: Results from 4D ray tracing coupled to a linear tidal model. Ribstein, B., U. Achatz, and F. Senf, 2015: , J. Geophys. Res. Space Physics, 120, 6795–6817, https://doi.org/10.1002/2015JA021349
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 wave number 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.  Asymptotic theory and similarities with the atmosphere: Rodal, Marie, & Schlutow, Mark (2021). Waves in the gas centrifuge: Journal of Fluid Mechanics, 928, A17. https://doi.org/10.1017/jfm.2021.811
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 atmospherelike 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 higherorder correction. We conclude that the gas centrifuge provides an unprecedented opportunity to investigate atmospheric internal waves experimentally with a compressible working fluid.  Generalized modulation theory for strongly nonlinear gravity waves in a compressible atmosphere: Schlutow, Mark and Wahlén, Erik. https://doi.org/10.1515/mcwf20200105
This study investigates strongly nonlinear gravity waves in the compressible atmosphere from theEarth’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 andtesting their stability in an analytic fashion. Assuming energetically consistent boundary and farfield conditions, that is no energy flux through the surface, freeslip boundary, and finite total energy, general wavesolutions are derived and illustrated in terms of realistic background fields. These assumptions also implythat the waveReynolds number must become less than unity above a certain height. The modulational stability of admissible, both nonhydrostatic and hydrostatic, waves is examined. It turns out that, when accounting for the selfinduced mean flow, the waveFroude number has a resonance condition. If it becomes1/√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 reachthe modulational stability condition.  On strongly nonlinear gravity waves in a vertically sheared atmosphere: Schlutow, Mark and Voelker, Georg S. https://doi.org/10.1515/mcwf20200103
We investigate strongly nonlinear stationary gravity waves which experience refraction due to a thinvertical shear layer of horizontal background wind. The velocity amplitude of the waves is of the same orderof magnitude as the background ow and hence the selfinduced mean ow alters the modulation propertiesto leading order. In this theoretical study, we show that the stability of such a refracted wave depends on theclassical modulation stability criterion for each individual layer, above and below the shearing. Additionally,the stability is conditioned by novel instability criteria providing bounds on the meanow horizontal windand the amplitude of the wave. A necessary condition for instability is that the meanow horizontal wind inthe upper layer is stronger than the wind in the lower layer.  Finiteamplitude gravity waves in the atmosphere: traveling wave solutions. Schlutow, M.; Klein, R.; Achatz, U., 2017, J. Fluid Mech., 826, 10341065 https://doi.org/10.1017/jfm.2017.459
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 nonhydrostatic 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 waveinduced meanflow 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 comoving 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 pseudoincompressible 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–meanflow interaction process: a horizontally periodic wave packet propagates vertically while draining energy from the mean wind aloft. In the process it decelerates the lowerlevel 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. 
Spectral stability of nonlinear gravity waves in the atmosphere,Schlutow, M., Wahlén, E. & Birken, P. (2019), Mathematics of Climate and Weather Forecasting, 5(1), pp. 1233. Retrieved 30 Apr. 2019, https://doi.org/10.1515/mcwf20190002
We apply spectral stability theory to investigate nonlinear gravity waves in the atmosphere. These waves are determined by modulation equations that result from WentzelKramersBrillouin 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 pseudoincompressible regime does fully account for the altitudinal varying background density. Second,we show for the first time that upwardtraveling nonplane wave fronts solving the inviscid nonlinear modulation equations, that compare to pseudoincompressible theory, are unconditionally unstable. Both inviscid regimes turn out to be illposed 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.

Modulational stability of nonlinear saturated gravity waves; Schlutow, M., 0., J. Atmos. Sci. 2019, https://doi.org/10.1175/JASD190065.1
Stationary gravity waves, such as mountain lee waves, are effectively described by Grimshaw’s dissipative modulation equations even in high altitudes where they become nonlinear due to their large amplitudes. In this theoretical study, a waveReynolds number is introduced to characterize general solutions to these modulation equations. This nondimensional number relates the vertical linear group velocity with wavenumber, pressure scale height and kinematic molecular/eddy viscosity. It is demonstrated by analytic and numerical methods that Lindzentype waves in the saturation region, i.e. where the waveReynolds number is of order unity, destabilize by transient perturbations. It is proposed that this mechanism may be a generator for secondary waves due to direct wavemeanflow interaction. By assumption the primary waves are exactly such that altitudinal amplitude growth and viscous damping are balanced and by that the amplitude is maximized. Implications of these results on the relation between meanflow acceleration and wave breaking heights are discussed.
 Recent progress in modeling imbalance in the atmosphere and ocean. Sutherland, B.R., Achatz, U., Caulfield, C. and Klymak, J. M., Phys. Rev. Fluids 4, 010501 , January 2019; https://doi.org/10.1103/PhysRevFluids.4.010501
Imbalance refers to the departure from the largescale primarily vortical flows in the atmosphere and ocean whose motion is governed by a balance of Coriolis, pressuregradient, and buoyancy forces and can be described approximately by quasigeostrophic theory or similar balance models. Imbalanced motions are manifest either as fully nonlinear turbulence or as internal gravity waves which can extract energy from these geophysical flows but which can also feed energy back into the flows. Capturing the physics underlying these mechanisms is essential to understanding 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 modeling. The outcome of this meeting is reported upon here.  An Application of WKBJ Theory for Triad Interactions of Internal Gravity Waves in Varying Background Flows: Völker, G.S., Akylas T.R. and U. Achatz, 2021 DOI: 10.1002/qj.3962
Motivated by the question of whether and how wave–wave interactions should be implemented into atmospheric gravity‐wave parametrizations, 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‐scale WKBJ (Wentzel–Kramers–Brillouin–Jeffreys) expansion identifies two distinct scaling regimes, a linear off‐resonance regime, and a nonlinear 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. However, triadic gravity‐wave interactions are likely to be relevant in weakly sheared regions of the atmosphere.  Efficient modelling of the interaction of mesoscale gravity waves with unbalanced largescale flows: Pseudomomentumflux convergence versus direct approach. Wei, J., G. Bölöni, and U. Achatz, 2019; , https://doi.org/10.1175/JASD180337.1
This paper compares two different approaches for the efficient modelling of subgridscale inertiagravity waves in a rotating compressible atmosphere. The first approach, denoted as pseudomomentum scheme, exploits the fact that in a Lagrangianmean reference frame the response of a largescale flow can only be due to forcing momentum. Presentday gravitywave parameterizations follow this route. They do so, however, in a Eulerianmean formulation. Transformation to that reference frame leads, under certain assumptions, to pseudomomentumflux convergence by which the momentum is to be forced. It can be shown that this approach is justified if the largescale flow is in geostrophic and hydrostatic balance. Otherwise, elastic and thermal effects might be lost. In the second approach, called direct scheme and not relying on such assumptions, the largescale flow is forced both in the momentum equation, by anelastic momentumflux convergence and an additional elastic term, and in the entropy equation, via entropyflux convergence. A budget analysis based on onedimensional 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 highresolution waveresolving simulations as a reference, this study shows conclusive evidence that the direct scheme is more reliable than the pseudomomentum scheme, regardless whether onedimensional or twodimensional 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 wavemeanflow interactions during the wave propagation.  Interactions between Mesoscale and Submesoscale Gravity Waves and Their Efficient Representation in MesoscaleResolving Models. Wilhelm, J., T.R. Akylas, G. Bölöni, J. Wei, B. Ribstein, R. Klein, and U. Achatz, 2018; https://doi.org/10.1175/JASD170289.1
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 subgridscale GWs might impact the resolved flow and how their effect could be taken into account. This motivates a theoretical and numerical investigation of the interactions between unresolved submesoscale and resolved mesoscale GWs, using Boussinesq dynamics for simplicity. By scaling arguments, first a subset of submesoscale GWs that can indeed influence the dynamics of mesoscale GWs is identified. Therein, hydrostatic GWs with wavelengths corresponding to the largest unresolved scales of presentday limitedarea weather forecast models are an interesting example. A largeamplitude WKB theory, allowing for a mesoscale unbalanced flow, is then formulated, based on multiscale asymptotic analysis utilizing a proper scaleseparation parameter. Purely vertical propagation of submesoscale GWs is found to be most important, implying inter alia that the resolved flow is only affected by the vertical flux convergence of submesoscale horizontal momentum at leading order. In turn, submesoscale GWs are refracted by mesoscale vertical wind shear while conserving their waveaction density. An efficient numerical implementation of the theory uses a phasespace ray tracer, thus handling the frequent appearance of caustics. The WKB approach and its numerical implementation are validated successfully against submesoscaleresolving simulations of the resonant radiation of mesoscale inertia GWs by a horizontally as well as vertically confined submesoscale GW packet.