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Vertical ascent due to gravity waves (GW) represents one important formation mechanism for cirrus clouds, i.e., clouds consisting solelyof ice crystals. Further, GWs can substantially influence the microphysical cloud properties and cloud life cycle, which are crucialfor the radiative impact of the cirrus clouds. Here we investigate the interactions between high-, mid- or low-frequency GWs and cirrusclouds in the tropopause region. Utilizing asymptotic analysis reduced equations are derived for the self-consistent description of thecirrus dynamics forced by a monochromatic GW. This allows for the construction of prototype parameterization of the number nucleated icecrystals. The GW-cirrus interactions are studied further in cloud resolving large eddy simulation and the results are used to evaluatethe asymptotic parameterization.

The Multi-Scale Gravity Wave Model (MS-GWaM) uses raytracing-based modelling that supports transient gravity-wave parametrisation. The state-of-the-art implementation of MS-GWaM in the upper-atmosphere ICON model solves the raytracing equations in three dimensions and accounts for background (non-orographic) and convective gravity-wave sources. Our work extends the capabilities of MS-GWaM to include orography gravity-wave sources, and we present methods and preliminary results towards this goal. Specifically, we first determine the spectral representation of the topography in each ICON grid cell via Fourier fitting. We then apply linear theory to obtain a representation of the bottom boundary for the raytracer. Finally, the bottom boundary serves as an initial condition for the raytracer-based parametrisation of orographic gravity waves. Preliminary results indicate that a judicious setup of this bottom boundary allows for an optimal tradeoff between computational efficiency and a sufficiently accurate representation of the underlying topography.

Using a 3-dimensional (3D) Lagrangian ray-tracing approach, a realistic gravity-wave (GW) parameterization has been developed: Multi-Scale Gravity Wave Model (MS-GWaM). It is a unique and useful tool to simulate and study the 3D transient dynamics of GWs and their interactions with various meteorological phenomena. We implement MS-GWaM into the ICOsahedral Non-hydrostatic model (ICON) and conduct a simulation of the quasi-biennial oscillation (QBO). A particular focus of the study is on the effect of oblique propagation and transient dynamics of GWs on the simulated QBO dynamics. Source spectra of GWs in MS-GWaM are calculated online using the convective latent heat modeled by ICON's cumulus parameterization, as the QBO dynamics is sensitive to the source spectra. In the GW-source scheme a tuning parameter, the areal fraction of convective cells in a model grid cell, is used to produce a reasonable QBO period. In the simulation result, the amplitude of the QBO is realistic in the middle stratosphere but underestimated in the lower stratosphere. The easterly QBO wind is shorter in height than that of the observed QBO. It is notable that in the solstice seasons when the convective activity is maximal off the equator (~8°), GWs with relatively large horizontal wavelengths tend to propagate equatorward from the active convection region. This oblique propagation leads to an effective coupling between the GWs and the QBO. Another simulation is performed using the same experimental setup except that in MS-GWaM the horizontal propagation is neglected and the steady-state assumption is used, as in conventional GW parameterizations. The QBO in this simulation exhibits a large difference from that using the 3D transient MS-GWaM: The QBO has a much longer period (~4 years) and its easterly phase descends a bit less. QBO jets tend to form off the equator in the solstice seasons, centered on the latitudes of active convection.

Internal gravity waves (IGWs) are important distributors of energy and momentum in a stratified atmosphere. While most IGWs are presumably excited at lower altitudes their effects are most important between the upper troposphere and the mesopause (~85km). During propagating - both in the vertical and the horizontal - nonlinear IGWs can exert a wave drag on the large-scale winds, interact with the large-scale potential temperature, and influence transport and mixing of atmospheric constituents such as aerosols or greenhouse gases.In state-of-the art weather and climate prediction models subgrid-scale IGWs are typically parameterized neglecting both the horizontal wave propagation (single-column assumption), the transient wave behavior including its effect on wave-mean-flow interactions (steady-state assumption) as well as time dependent wave generation. While being computationally efficient the missing physics, however, may lead to model errors and inaccurate predictions under varying boundary conditions. The potential importance of the horizontal wave propagation and wave transience has been shown in various theoretical, numerical and experimental studies.The transient Multi Scale Gravity Wave Model (MS-GWaM) - implemented in the high-top model UA-ICON - aims to improve these shortcomings by allowing for transient and three dimensional wave propagation. The parameterization is based on a multi scale WKBJ analysis of the compressible atmosphere and includes various non-orographic wave sources, non-dissipative wave-mean-flow interactions as well as wave breaking. Internally, the parameterized gravity waves are treated as Lagrangian volumes with the dynamics prescribed by the well known gravity wave modulation equations. A suitable projection method of wave properties onto the unstructured model grid facilitates the calculation of wind and temperature tendencies. What is more, an efficient parallelization of the ray-tracing scheme allows for simulations in reasonable computation times, being much faster than corresponding wave resolving runs.While satisfactorily reproducing the observed zonal-mean wind and potential temperature climatology the model results reveal new insight into the detail of the role of IGWs in the atmosphere. In particular, probability density functions of wave momentum fluxes exhibit the typical observed long tails (i.e. wave intermittency) which cannot be reproduced with steady-state parameterizations. Moreover, the three dimensional distribution of wave momentum and wave action fluxes differ greatly when relaxing the single-column assumption. As an example the well known three dimensional refraction of IGWs into polar jets can be shown.

The Brewer-Dobson circulation (BDC) is a meridional circulation mainly driven through mean-flow forcing by Rossby-wave breaking that is, however, also influenced by gravity waves (GW). As some part of GW dynamics is subgrid-scale in numerical climate and weather prediction models, an appropriate wave parameterization and later an accurate separation of the drivers are necessary to understand the impact of the subgrid-scale GWs on the BDC.Polichtchouk et al. (2018) explored the sensitivity of BDC to the parameterized GW drag by separating the drivers of the residual circulation using the downward-control principle. Even in relatively high-resolution simulations, the parameterized drags are found to significantly contribute to the BDC in the lower stratosphere, especially for northern hemisphere winter-pole downwelling.We parameterize subgrid-scale GWs in the Lagrangian ray-tracing scheme Multi-Scale Gravity-Wave Model (MS-GWaM) (Bölöni et al. 2021; Kim et al. 2021). MS-GWaM includes interactions between the GW field and the resolved flow, which are neglected in the traditional GW parameterization schemes. It has been implemented successfully into of the Icosahedral Non-hydrostatic (ICON) model in its upper-atmosphere configuration. Using the MS-GWaM output, we diagnose the sensitivity of the BDC using the downward-control principle.

In a companion paper, the Multiscale Gravity Wave Model (MS-GWaM) has been introduced and its application to a global model as a transient subgrid-scale parameterization has been described. This paper focuses on the examination of intermittency of gravity waves (GWs) modeled by MS-GWaM. To introduce the variability and intermittency in wave sources, convective GW sources are formulated, using diabatic heating diagnosed by the convection parameterization, and they are coupled to MS-GWaM in addition to a flow-independent source in the extratropics accounting for GWs due neither to convection nor to orography. The probability density function (PDF) and Gini index for GW pseudomomentum fluxes are assessed to investigate the intermittency. Both are similar to those from observations in the lower stratosphere. The intermittency of GWs over tropical convection is quite high and is found not to change much in the vertical direction. In the extratropics, where nonconvective GWs dominate, the intermittency is lower than that in the tropics in the stratosphere and comparable to that in the mesosphere, exhibiting a gradual increase with altitude. The PDFs in these latitudes seem to be close to the lognormal distributions. Effects of transient GW–mean-flow interactions on the simulated GW intermittency are assessed by performing additional simulations using the steady-state assumption in the GW parameterization. The intermittency of parameterized GWs over tropical convection is found to be overestimated by the assumption, whereas in the extratropics it is largely underrepresented. Explanation and discussion of these effects are included.

Current gravity wave (GW) parameterization (GWP) schemes are using the steady-state 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 steady-state assumption, nondissipative wave–mean-flow 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 (MS-GWaM) has been implemented into a state-of-the-art weather and climate model. In this implementation, MS-GWaM leads to a zonal-mean circulation that agrees well with observations and increases GW momentum-flux intermittency as compared with steady-state GWPs, bringing it into better agreement with superpressure balloon observations. Transient effects taken into account by MS-GWaM are shown to make a difference even on monthly time scales: in comparison with steady-state 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 MS-GWaM suggests that small-scale GWs play an important role down to horizontal and vertical wavelengths of 50 km (or even smaller) and 200 m, respectively.

Gravity Waves (GW) alter the propagation path of acoustic energy in the middle atmospheric waveguide and complexify the large scale picture where infrasound (IS) propagation is mainly driven by the seasonal changes in stratospheric winds. Thus, GW affect the detection capability of the IS station network of the International Monitoring System (IMS) established to monitor the compliance with the Comprehensive Nuclear-Test-Ban Treaty (CTBT). Atmospheric models explicitly resolving a part of the GW spectrum are relevant tools to be considered for investigating the effect of GW on infrasound propagation, given increasing computing means made available by HPC facilities. Parabolic equation simulations allow accounting for the partial reflections induced by GW. They can be used to quantify the impact of GW on infrasound transmission loss, for instance. Here, we use atmospheric specification fields obtained in the framework of the Dynamics of the Atmospheric General Circulation Modeled on Nonhydrostatic Domains (DYAMOND). DYAMOND is an international project, initiated by the Max Planck Institute for Meteorology (MPIM) and the University of Tokyo. It describes a framework for the intercomparison of high-resolution global models. It mainly focuses on the troposphere, but some models were run with a high enough top so that GW are resolved up to the stratosphere. Lidar observations are used to validate the model at Observatoire de Haute Provence (France) and we investigate the potential energy of GW activity across the IMS. By filtering out small-scale perturbations (GW) in atmospheric specifications and comparing parabolic equation simulations with and without GW, respectively, we quantify the impact of GW on the main atmospheric waveguide. We focus on the transmission loss derived at the surface, and more particularly in the shadow zones, for different national or IMS infrasound stations during the (northern hemisphere) winter.

Subgrid-scale internal gravity waves (IGWs) are important distributors of energy in a stratified atmosphere. While they are mostly excited at lower altitudes their effects are most important between the upper troposphere to the mesopause (~85km). During propagation–both in the vertical and the horizontal–nonlinear IGWs can exert a wave drag on the mean winds, interact with the mean potential temperature, and mix atmospheric tracers such as aerosols or greenhouse gases. In state-of-the art weather prediction models IGWs are typically parametrized using the single-column and the steady-state assumptions. These parametrizations take into account dissipative effects of IGWs but neglect their horizontal propagation and all of their transient interaction mechanisms such as direct wave-mean-flow interactions. However, the latter have been shown to contribute to IGW dynamics in various idealized studies. Here we present advances of the use of the transient Multi Scale Gravity Wave Model (MS-GWaM) in the upper atmosphere model UA-ICON. Based on Lagrangian ray-tracing the parametrization includes various non-orographic wave sources, transient propagation in both the horizontal and vertical directions, direct wave-mean-flow interactions and wave breaking. The resulting setup satisfactorily reproduces the observed mean-wind and potential temperature climatology and already shows promising insights into the details of the role of IGWs in the atmosphere.

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−1) 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. Plain Language Summary: The variability of the tropical atmosphere at altitudes of about 18–40 km is dominated by a large-amplitude long-term oscillation of wind, the quasi-biennial oscillation, which has a broad impact on the climate and seasonal forecasting. This oscillation is known to be driven by various types of atmospheric waves with multiple spatial scales. Using a numerical model, this study reports a process of interaction between those waves on different scales, which has not been illuminated before. The result implies a potential importance of this process in the progression of the quasi-biennial oscillation. Proper model representations of these multiscale waves and tropical convection are required to simulate this process.

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.

The potential temperature is a widely used quantity in atmospheric science since it corresponds to the entropy and is conserved for adiabatic changes of dry air. As such, it is routinely employed in applications ranging from atmospheric dynamics to transport modeling. The common formula to compute the potential temperature is based on the assumption of a constant specific heat capacity for the dry air, even though the latter is known to vary with temperature. We re-derive the (dry air) potential temperature for a recent temperature-dependent formulation of the specific heat capacity of dry air. The result is expected to provide values which are much closer at the true entropy value (expressed as a temperature) and hence serves as the reference potential temperature. However, its computation is less straightforward compared to the classical one, motivating the development of efficient approximations. Moreover, similarities and differences are discussed between the newly derived reference potential temperature and the classical one based on a constant specific heat capacity. The new reference shows different values and vertical gradients, in particular in the stratosphere and above. Applications of the new reference potential temperature are discussed in the context of common computations in the atmospheric sciences, including the potential vorticity or diabatic heating rates.

The differentially heated rotating annulus is a classic experiment used for the examination of circulation patterns and waves in the atmosphere. In particular, by choosing an atmosphere-like experimental setup that allows the buoyancy frequency to become larger than the Coriolis parameter, it provides a useful tool to study the generation mechanism of spontaneous gravity wave (GW) emission in jet-front systems. Recently, with the aim to gain better understanding about the conditions for the spontaneous generation of GWs, Rodda et al. (2020) compared experimental data with results from numerical simulations and found differences in the GW signal most likely due to the model's treatment of boundary conditions. The aim of the present study is to improve the consistency between the model and experiment and to investigate the effect of the lateral and upper boundary conditions on GW generation and propagation in an atmosphere-like configuration of the annulus. More precisely, we implement the corresponding lateral and surface heat fluxes, air-temperature variations, as well as evaporation at the upper boundary condition into the numerical model and examine the characteristics of the observed GW signals, which are identified by the horizontal divergence field. Our systematic analysis may serve as a basis for subsequent research on the spontaneous GW generation mechanism, following the overarching objective to develop a parameterization scheme for GWs emitted from jets and front. References:Rodda, C., S. Hien, U. Achatz, and U. Harlander, 2020: A new atmospheric-like differentially heated rotating annulus configuration to study gravity wave emission from jets and fronts. Exp. Fluids 61, 2. https://doi.org/10.1007/s00348-019-2825-z

The potential temperature is a widely used quantity in atmospheric science since it is conserved for air's adiabatic changes of state. Its definition involves the specific heat capacity of dry air, which is traditionally assumed as constant. However, the literature provides different values of this allegedly constant parameter, which are reviewed and discussed in this study. Furthermore, we derive the potential temperature for a temperature-dependent parameterization of the specific heat capacity of dry air, thus providing a new reference potential temperature with a more rigorous basis. This new reference shows different values and vertical gradients in the upper troposphere and the stratosphere compared to the potential temperature that assumes constant heat capacity. The application of the new reference potential temperature to the prediction of gravity wave breaking altitudes reveals that the predicted wave breaking height may depend on the definition of the potential temperature used.

Orography induced gravity waves are investigated in a Multi Scale Gravity Wave Model (MS-GWaM) over idealized topography. MS-GWaM is a prognostic gravity-wave model, which parametrizes both the propagation and dissipation of subgrid-scale GWs. It is a Lagrangian ray-tracer model, which applies WKB-theory and calculates the propagation of ray volumes in spectral space. Its novelty is that not only the dissipative effect, but also the non-dissipative effects due to direct wave-mean flow interaction are captured. In our conceptual studies we investigate mountain wave generation, which is induced via a time-dependent large-scale wind encountering a prescribed topography. The framework used in our experiments is the PincFloit model, which integrates the pseudo-incompressible equations. We use it both in low resolution with MS-GWaM and in high resolution LES mode as a wave resolving reference. In the reference LES simulations the idealized topography, a mountain chain, is represented with an immersed boundary method. In the MS-GWaM experiments there is no resolved topography, but its effect is modelled as a lower boundary condition. The lower boundary condition is represented by initializing ray volumes with wave number and wave action density depending on the mountain characteristics and the large scale wind speed, based on the assumption that the flow follows the terrain. In the wave resolving reference experiments the flow does not strictly follow the terrain, but other instabilities (rotor formation, boundary layer separation) arise around the mountains. These processes decrease the available momentum transported by GWs, which was initially not accounted for in MS-GWaM. Thus an overestimation of wind deceleration was found in the MS-GWaM parametrization compared to the wave resolving simulation. To correct for this overestimation, an effective mountain height is introduced into Ms-GWaM, which is calculated by a scaling function between mountain height and flow properties using the Froude number.

The aim of the presented work is to improve the parameterization of subgrid-scale gravity wave (GW) effects on the resolved flow in atmospheric models in a large altitude range from the upper troposphere to the mesopause (~85km). State of the art GW parameterization schemes are using the steady-state approximation for the wave field and therefore assume an instantaneous GW propagation neglecting direct interactions between the GW field and the resolved flow within the whole altitude range mentioned above. As such, these schemes rely on dissipative processes - GW breaking and critical layer filtering - as the only mechanism to accelerate/decelerate the resolved flow. In contrast to this, by dropping the steady-state assumption a contribution to the mean-flow forcing emerges in the form of direct GW-mean-flow interactions. Several idealized studies show that, besides dissipative effects, direct GW-mean-flow interactions contribute to GW dynamics in an important extent (Bölöni et al., 2016, J. Atmos. Sci.}, 73, 4833-4852, Wilhelm et al., 2018, J. Atmos. Sci., 75, 2257-2280, Wei et al., 2019, J. Atmos. Sci., 76, 2715-2738). This motivates the implementation of a transient GW model (MS-GWaM: Multi Scale Gravity Wave Model) to UA-ICON, the upper atmosphere version of ICON (Borchert et al., 2019, Geosci. Model Dev., 12, 3541-3569) which does not rely on the steady-state assumption and thus includes direct GW-mean-flow interactions. As a reference and a representative of currently available GW parameterization schemes a steady-state version of MS-GWaM (ST-MS-GWaM) has been implemented to UA-ICON as well, which shares the treatment of all possible components (wave sources and wave saturation scheme) with the transient MS-GWaM scheme and differs from it "only" in the treatment of propagation, i.e. excluding direct GW-mean-flow interactions and thus transience. Both MS-GWaM and ST-MS-GWaM reproduce the observed wind and temperature climatology (e.g. URAP data: Swinbank, R. and D. A. Ortland, 2003, J. Geophys. Res., 108, D19, 4615) reasonably well, but the transient propagation makes a robust difference in the circulation in perpetual runs. The transient propagation in MS-GWaM substantially contributes to an increase of the GW intermittency in the whole altitude range, giving a better comparison with super-pressure balloon observations (e.g. Hertzog et al., 2012, J. Atmos. Sci., 69, 3433-3448), whereas the lack of transience prevents any occurrence of higher GW momentum flux values than the launch magnitude itself. This is explained by the fact that the direct GW-mean-flow interactions involve a highly transient evolution of the wave action and the vertical group velocity, which often leads to increased pseudo-momentum fluxes as compared to the launch magnitude.

Inertia–gravity waves (IGWs) emitted from jets and fronts are ubiquitous in the atmosphere and have a significant impact on atmospheric processes (Plougonven and Zhang, 2014). Since the mechanism responsible for the spontaneous emission of IGWs during the evolution of an initially balanced flow remain poorly understood, their representation in numerical weather prediction models is challenging (de la Cámara and Lott, 2015). Better understanding of this IGW source mechanism based on idealized numerical simulations is crucial to improve the accuracy of the forecasts. In this study, idealized baroclinic-wave life cycle experiments on the f-plane are performed to investigate spontaneous emission, using a finite-volume solver for the pseudo-incompressible equations (Rieper et al., 2013). In particular, the implementation of a semi-implicit time stepping scheme, along the lines of Smolarkiewicz and Margolin (1997) and Benaccio and Klein (2019), but adjusted to our staggered grid, permits longer simulation runs with much larger domains. A novelty is the implementation of a simple Newtonian heating function based on Held and Suarez (1994), which is used for forcing a baroclinically unstable temperature profile and allows the background state to vary in time (O’Neill and Klein, 2014). The results of the model with semi-implicit time stepping scheme will be documented and compared to an explicit Runge-Kutta scheme. The analysis may serve as a basis for the development and validation of a parameterization scheme for GWs emitted from jets and fronts. References: Benaccio, T., and R. Klein, 2019: A semi-implicit compressible model for atmospheric flows with seamless access to soundproof and hydrostatic dynamics. Mon. Wea. Rev., 147, 4221-4240.de la Cámara, A., and F. Lott, 2015: A parameterization of gravity waves emitted by fronts and jets. Geophys. Res. Lett., 42, 2071-2078.Held, I.M., and M.J. Suarez, 1994: A Proposal for the Intercomparison of the Dynamical Cores of Atmospheric General Circulation Models. Bull. Amer. Meteor. Soc., 75, 1825-1830.O’Neill, W.P., and R. Klein, 2014: A moist pseudo-incompressible model. Atmos. Res., 142, 133-141. Plougonven R., and F. Zhang, 2014: Internal gravity waves from atmospheric jets and fronts. Rev. Geophys., 52, 33-76.Rieper, F., Hickel, S., and U. Achatz, 2013: A conservative integration of the pseudo-incompressible equations with implicit turbulence parameterization. Mon. Wea. Rev., 141, 861-886. Smolarkiewicz, P.K., and L.G. Margolin, 1997: On forward-in-time differencing for fluids: an Eulerian/semi-Langrangian nonhydrostatic model for stratified flows. Atmosphere-Ocean, 35, 127- 152.

Internal gravity waves are a well known mechanism of energy transport in stratified fluids such as the atmosphere and the ocean. Their abundance and importance for various geophysical processes like ocean mixing and momentum deposition in atmospheric jets are widely accepted. While resonant wave-wave interactions of monochromatic disturbances have received intensive study, little work has been done on triad interactions between wave trains that are modulated by a variable mean flow.Using the method of multiple scale asymptotics we consider a weakly non-linear Boussinesq WKBJ theory for interacting gravity wave trains propagating through a finite amplitude background flow. Consequently the wave trains are allowed to spectrally pass through resonance conditions and exchange energy when sufficiently close to resonance. We find a global optimal threshold for the deviation from resonance and derive a corresponding parametrization for the triad interaction applicable to ray tracing schemes.We test the theory with idealized simulations in which two wave trains generate a third by passing through resonance in a sinusoidal background shear flow with varying vertical scales. Comparing WKBJ simulations with wave resolving large eddy simulations we find qualitative and quantitative agreement. Furthermore we assess the impact of the strength of the modulation as well as the effect of the wave amplitudes on the energy exchange between the interacting wave triad.

The intermittency of gravity waves (GWs) is investigated using Multi-Scale Gravity Wave Model (MS-GWaM) implemented in the upper-atmosphere extension of ICON model. The intermittency of GWs is originated from that of wave sources but altered during propagation of the waves. Conventional GW parametrization (GWP), which diagnoses vertical profiles of GW properties under the steady-state assumption, can take into account the source intermittency if the GWP employs flow-dependent sources, while it cannot present the change of intermittency by transient evolutions of GWs. MS-GWaM is a prognostic model that explicitly solves the evolution of positions of waves (as well as their wavenumbers and amplitudes) in time and thus capable of describing the intermittency change. In order to include the source intermittency and variability, we couple the convective source, as diagnosed by subgrid-scale cumulus parametrization in ICON, to MS-GWaM, based on an analytic formulation of GW response to this source. In addition to this, a spatio-temporally uniform, persistent source is prescribed in the extratropics to take into account other non-orographic sources. Orographic sources are currently not used. The GW intermittency is measured by the Gini index, and is found to be quite high in the tropics, compared to that in the extratropics. In both regions, the index has similar values to those obtained from superpressure balloon observations reported in previous studies. A control experiment is performed using GWP based on the steady-state assumption, but coupled to the same wave sources, to assess the effects of transient modelling using MS-GWaM on the simulated intermittency. From comparison to the control experiment, the intermittency is found to increase largely for GWs from the uniform source but to decrease for convective GWs by the transient modelling.

Many subgrid-scale (SGS) parameterizations in climate models contain empirical parameters and are thus data dependent. In particular, it is not guaranteed that the SGS parameterization still helps the model to produce the correct climate projection in the presence of an external perturbation (e.g., because of climate change). Therefore, a climate dependence of tuning parameters is proposed, using the fluctuation–dissipation theorem (FDT). The FDT provides an estimation of the changes in the statistics of a system, caused by a small external forcing. These estimations are then used to update the SGS parameterization. This procedure is tested for a toy atmosphere given by a quasigeostrophic three-layer model (QG3LM). We construct a low-order climate model for this toy atmosphere, based on a reduced number of its empirical orthogonal functions (EOFs), equipped with either an empirical deterministic or an empirical stochastic SGS parameterization. External forcings are considered that are either a local anomalous heat source in the extratropics or a global dynamical forcing represented by individual EOF patterns. A quasi-Gaussian variant of the FDT is able to successfully update the SGS parameterization leading to an improvement in both amplitude and correlation between the low-order climate model and the QG3LM, in case of a perturbed system. The stochastic closure exhibits nearly no improvement compared to the deterministic parameterization. The application of a more sophisticated non-Gaussian FDT algorithm (i.e., the blended short-time/quasi-Gaussian FDT) yields only marginal improvement over the simple quasi-Gaussian FDT.

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 subgrid-scale 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 present-day limited-area weather forecast models are an interesting example. A large-amplitude WKB theory, allowing for a mesoscale unbalanced flow, is then formulated, based on multiscale asymptotic analysis utilizing a proper scale-separation 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 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 submesoscale-resolving simulations of the resonant radiation of mesoscale inertia GWs by a horizontally as well as vertically confined submesoscale GW packet.

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.

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.

The last decade has seen the success of stochastic parameterizations in short-term, medium-range and seasonal forecasts: operational weather centers now routinely use stochastic parameterization schemes to better represent model inadequacy and improve the quantification of forecast uncertainty. Developed initially for numerical weather prediction, the inclusion of stochastic parameterizations not only provides better estimates of uncertainty, but it is also extremely promising for reducing longstanding climate biases and relevant for determining the climate response to external forcing. This article highlights recent developments from different research groups which show that the stochastic representation of unresolved processes in the atmosphere, oceans, land surface and cryosphere of comprehensive weather and climate models (a) gives rise to more reliable probabilistic forecasts of weather and climate and (b) reduces systematic model bias. We make a case that the use of mathematically stringent methods for the derivation of stochastic dynamic equations will lead to substantial improvements in our ability to accurately simulate weather and climate at all scales. Recent work in mathematics, statistical mechanics and turbulence is reviewed, its relevance for the climate problem demonstrated, and future research directions outlined., Comment: 26 pages, 15 figures. Final published version

Significant inertia-gravity wave activity has been frequently observed in the vicinity of jet and front systems in the atmosphere. Although many studies have established the importance of these non-orographic sources, the mechanisms responsible for spontaneous wave emissions are still not fully understood. The complexity of the three-dimensional flow pattern and distribution of the sources over large areas point towards the need for laboratory experiments and idealised numerical simulations. These will help understand the correct interpretation of the fundamental dynamical processes in a simplified, but yet realistic flow. In this study, we emphasise the importance of using set-ups of the differentially heated rotating annulus experiment with a ratio between the buoyancy frequency N, and the Coriolis parameter f larger than one to investigate atmosphere-like emission of gravity waves from baroclinic jets. Indeed, in the atmosphere $$N/f\sim 100$$, but for table-top size experiments this ratio is smaller than one, resulting in an unfavourable condition for the propagation of gravity waves. For this reason, we offer a newly built laboratory experiment supported by numerical simulations that allow $$N/f>1$$. The conditions for gravity wave emission in this new configuration are examined in detail, and the first evidence of IGWs is reported. Moreover, we compare numerical simulations and experimental data focusing on the variations of the temperature T, and its effects on the buoyancy frequency N. It becomes clear, that despite the fact the global structure and baroclinic instability characteristics are very similar, the model and experiment show deviations in N with implications for gravity wave emission. Due to the complex horizontal structure of N, where the largest values occur along the baroclinic jet axis, the inertia-gravity waves in the experiment are observed to be trapped.

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.