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Global atmospheric models rely on parameterizations to capture the effects of gravity waves (GWs) on middle atmosphere circulation. As they propagate upwards from the troposphere, the momentum fluxes associated with these waves represent a crucial yet insufficiently constrained component. The present study employs three tree‐based ensemble machine learning (ML) techniques to probe the relationship between large‐scale flow and small‐scale GWs within the tropical lower stratosphere. The measurements collected by eight superpressure balloons from the Strateole 2 campaign, comprising a cumulative observation period of 680 days, provide valuable estimates of the gravity wave momentum fluxes (GWMFs). Multiple explanatory variables, including total precipitation, wind, and temperature, were interpolated from the ERA5 reanalysis at each balloon's location. The ML methods are trained on data from seven balloons and subsequently utilized to estimate reference GWMFs of the remaining balloon. We observed that parts of the GW signal are successfully reconstructed, with correlations typically around 0.54 and exceeding 0.70 for certain balloons. The models show significantly different performances from one balloon to another, whereas they show rather comparable performances for any given balloon. In other words, limitations from training data are a stronger constraint than the choice of the ML method. The most informative inputs generally include precipitation and winds near the balloons' level. However, different models highlight different informative variables, making physical interpretation uncertain. This study also discusses potential limitations, including the intermittent nature of GWMFs and data scarcity, providing insights into the challenges and opportunities for advancing our understanding of these atmospheric phenomena. Plain Language Summary: Part of the atmosphere's large‐scale circulation results from motions that are not resolved, or partly resolved, by weather or climate models. These include internal gravity waves, with horizontal scales from a few to hundreds of kilometers. The main sources occur in the troposphere, such as flow over mountains and cloud development. Their three‐dimensional propagation induces major aggregated impacts in the stratosphere and mesosphere, forcing key aspects of the circulation. This forcing is accounted for in climate models by "parameterizations," that mimics the effect of the unresolved waves based on the large‐scale, resolved flow. These parameterizations necessarily retain crude approximations and introduce significant uncertainty in the models. For gravity waves (GWs), sources are a major uncertainty. This study makes use of the high‐altitude balloon campaign Strateole 2 (October 2019–February 2020). Eight balloons circled Earth at heights around 18–20 km, providing unique observations of the GWs. These are used as targets for machine learning (ML) methods that take as inputs the information from outputs of a numerical weather prediction model describing the large‐scale flow. The successes and difficulties of ML provide insights which can guide improvements of parameterizations, such as the most informative large‐scale variables for estimating the unresolved waves. Key Points: Eight superpressure balloons from the Strateole 2 mission provide observations for accurate gravity wave momentum flux (GWMF) estimationThree machine learning (ML) methods are employed to probe the relationship between the GWMFs and ERA5's large‐scale flowsThe most informative large‐scale inputs are provided, along with a discussion of the successes and challenges of ML methods [ABSTRACT FROM AUTHOR]

A theory for flow over gentle hills using a mixing‐length turbulence closure is developed to describe the transition from turbulent orographic form drag to gravity wave drag. It confirms that the first is associated with downstream sheltering, and the second with upstream blocking and strong downslope winds. It shows that the altitude at which the incident flow needs to be taken to calculate the drag is the inner layer scale at which dissipation equilibrates disturbance advection. It also shows that the parameter that controls the transition, here a Richardson number, compares the mountain length with the altitude of the turning points above which the upward‐propagating gravity waves become evanescent. Our solutions are also used to show that the downslope winds penetrate well into the inner layer and that a good fraction of the drag is deposited in the inner layer: all of it in the neutral case, a large fraction in the intermediate cases when there are trapped lee waves, and even in stable situations without trapping part of the gravity wave drag is eroded in the inner layer. Some discussion on how to combine neutral and stratified effects in the parametrization of subgrid scale orography in large‐scale models is given. [ABSTRACT FROM AUTHOR]

The evolution of the two components of the equatorial mountain torque (EMT) applied by mountains on the atmosphere is analyzed in the NCEP reanalysis. A strong lagged relationship between the EMT component along the Greenwich axis [T.sub.M1] and the EMT component along the 90[degrees]E axis TM2 is found, with a pronounced signal on [T.sub.M1] followed by a signal of opposite sign on [T.sub.M2]. It is shown that this result holds for the major massifs (Antarctica, the Tibetan Plateau, the Rockies, and the Andes) if a suitable axis system is used for each of them. For the midlatitude mountains, this relationship is in part associated with the development of cold surges. Following these results, two hypotheses are made: (i) the mountain forcing on the atmosphere is well measured by the regional EMTs and (ii) this forcing partly drives the cold surges. To support these, a purely dynamical linear model is proposed: it is written on the sphere, uses an f-plane quasigeostrophic approximation, and includes the mountain forcings. In this model, a positive (negative) peak in [T.sub.M1] produced by a mountain massif in the Northern (Southern) Hemisphere is due to a large-scale high surface pressure anomaly poleward of the massif. At a later stage, high pressure and low temperature anomalies develop in the lower troposphere east of the mountain, explaining the signal on [T.sub.M2] and providing the favorable conditions for the cold surge development. It is concluded that the EMT is a good measure of the dynamical forcing of the atmospheric flow by the mountains and that the poleward forces exerted by mountains on the atmosphere are substantial drivers of the cold surges, at least in their early stage. Therefore, the EMT time series can be an important diagnostic to assess the representation of mountains in general circulation models. DOI: 10.1175/2010JAS3382.1

The gravity waves (GWs) generated by potential vorticity (PV) anomalies in a rotating stratified shear flow are examined under the assumptions of constant vertical shear, two-dimensionality, and unbounded domain. Near a PV anomaly, the associated perturbation is well modeled by quasigeostrophic theory. This is not the case at large vertical distances, however, and in particular beyond the two inertial layers that appear above and below the anomaly; there, the perturbation consists of vertically propagating gravity waves. This structure is described analytically, using an expansion in the continuous spectrum of the singular modes that results from the presence of critical levels. Several explicit results are obtained. These include the form of the Eliassen--Palm (EP) flux as a function of the Richardson number [N.sup.2]/[[LAMBDA].sup.2], where N is the Brunt--Vaisala frequency and A the vertical shear. Its non-dimensional value is shown to be approximately exp(-[pi]N/[LAMBDA])/8 in the far-field GW region, approximately twice that between the two inertial layers. These results, which imply substantial wave--flow interactions in the inertial layers, are valid for Richardson numbers larger than 1 and for a large range of PV distributions. In dimensional form they provide simple relationships between the EP fluxes and the large-scale flow characteristics. As an illustration, the authors consider a PV disturbance with an amplitude of 1 PVU and a depth of 1 km, and estimate that the associated EP flux ranges between 0.1 and 100 mPa for a Richardson number between 1 and 10. These values of the flux are comparable with those observed in the lower stratosphere, which suggests that the mechanism identified in this paper provides a substantial gravity wave source, one that could be parameterized in GCMs. DOI: 10.1175/2009JAS3134.1

A climatology of the three-dimensional life cycle of the gravest waves in the tropical lower and middle stratosphere is presented. It shows that at periods around 10 days the gravest waves correspond to Kelvin and Rossby--gravity wave packets that substantially affect specific regions in the lower stratosphere. It also shows that the planetary-scale Kelvin waves with zonal wavenumber s = 1 and periods between 10 and 20 days produce a substantial signal. Still at the planetary scales, the climatology also shows that the global planetary Rossby waves with s = 1 and periods around 5 and 16 days have a substantial equatorial signature. This climatology is for all the dynamical fields (horizontal wind, temperature, and geopotential height) and relates the equatorial waves to the equatorial zonal mean flow evolution associated with the quasi-biennial oscillation. The method used to extract the climatology is a composite analysis of the dynamical fields keyed on simple indexes measuring when the waves enter in the stratosphere. For the Kelvin waves, the Rossby--gravity waves, and the 5-and 16-day Rossby planetary waves, these indexes are related to the latitudinal means over the equatorial band of the temperature, the meridional wind, the geopotential height, and the zonal wind respectively. The method is applied first to ERA-40 and then to a simulation done with the LMDz GCM. When compared to the results from ERA-40, this reveals that the LMDz GCM underestimates the Rossby--gravity wave packets and a fraction of the Kelvin wave packets. This deficit is attributed to the fact that the model has a too coarse vertical resolution and an insufficient tropospheric forcing for the horizontal wavenumbers s > 3.

A multi‐model study is carried out to investigate the ability of models to predict the evolution of the quasi‐biennial oscillation (QBO) up to 12 months in advance. All models are initialised from common reanalysis data, and forecasts run for a common set of 30 start dates over 15 years. All models have high skill in predicting the phase evolution of the QBO at 20–30 hPa, with slightly more variable results at higher and lower levels. Other aspects of the predicted QBO are of variable quality, and in some cases are consistently poor. QBO easterlies are too weak in all models at 20–50 hPa, while westerlies can be either too strong or too weak. This results in both a reduced amplitude of the QBO and a westerly bias in zonal‐mean winds, notably at 30 hPa. At 70 hPa models tend to have reduced QBO amplitude and an easterly bias. Despite these failings, a multi‐model ensemble of bias‐ and variance‐corrected forecasts can be used to give accurate and reliable QBO forecasts up to at least a year ahead. Analysis of the zonal momentum budget during the first month of the forecast shows that large‐scale forcing from Eliassen–Palm flux divergence and vertical advection are handled fairly well by the models, although vertical advection terms tend to be weaker than reanalysis estimates. Total tendencies show common errors, suggesting common failings in gravity‐wave drag treatments. Teleconnections from the QBO to Northern Hemisphere winter circulation are also examined, and do not appear to be realistic beyond the first month. Analysis of initialised forecasts is a powerful tool for diagnosing the accuracy of model processes driving the QBO. [ABSTRACT FROM AUTHOR]

We compare the response of the Quasi‐Biennial Oscillation (QBO) to a warming climate in eleven atmosphere general circulation models that performed time‐slice simulations for present‐day, doubled, and quadrupled CO2 climates. No consistency was found among the models for the QBO period response, with the period decreasing by 8 months in some models and lengthening by up to 13 months in others in the doubled CO2 simulations. In the quadrupled CO2 simulations, a reduction in QBO period of 14 months was found in some models, whereas in several others the tropical oscillation no longer resembled the present‐day QBO, although it could still be identified in the deseasonalized zonal mean zonal wind timeseries. In contrast, all the models projected a decrease in the QBO amplitude in a warmer climate with the largest relative decrease near 60 hPa. In simulations with doubled and quadrupled CO2, the multi‐model mean QBO amplitudes decreased by 36 and 51%, respectively. Across the models the differences in the QBO period response were most strongly related to how the gravity wave momentum flux entering the stratosphere and tropical vertical residual velocity responded to the increases in CO2 amounts. Likewise it was found that the robust decrease in QBO amplitudes was correlated across the models to changes in vertical residual velocity, parametrized gravity wave momentum fluxes, and to some degree the resolved upward wave flux. We argue that uncertainty in the representation of the parameterized gravity waves is the most likely cause of the spread among the eleven models in the QBO's response to climate change. [ABSTRACT FROM AUTHOR]

The diurnal and subdiurnal variations of the mass and wind terms of the axial atmospheric angular momentum (AAM) are explored using a 1-yr integration of the Laboratoire de M6t6orologie Dynamique (LMDz) GCM, twelve 10-day ECMWF forecasts, and some ECMWF analysis products. In these datasets, the wind and mass AAMs present diurnal and semidiurnal oscillations for which tendencies far exceed the total torque. In the LMDz GCM, these diurnal and semidiurnal oscillations are associated with axisymmetric (s = 0) and barotropic circulation modes that resemble the second gravest (n = 2) eigensolution of Laplace's tidal equations. This mode induces a Coriolis conversion from the wind AAM toward the mass AAM that far exceeds the total torque. At the semidiurnal period, this mode dominates the axisymmetric and barotropic circulation. At the diurnal period, this n = 2 mode is also present, but the barotropic circulation also presents a mode resembling the first gravest n = 1 eigensolution of the tidal equations. This last mode does not produce anomalies in the mass and wind AAMs. A shallow-water axisymmetric model driven by zonal mean zonal forces, for which the vertical integral equals the zonal mean zonal stresses issued from the GCM, is then used to interpret these results. This model reproduces well the semidiurnal oscillations in mass and wind AAM, and the semidiurnal mode resembling the n = 2 eigensolution that produces them, when the forcing is distributed barotropically in the vertical direction. This model also reproduces diurnal modes resembling the n = l and n = 2 eigensolutions when the forcings are distributed more baroclinically. Among the dynamical forcings that produce these modes of motion, it is found that the mountain forcing and the divergence of the AAM flux are equally important and are more efficient than the boundary layer friction. In geodesy, the large but opposite signals in the mass and wind AAM due to the n = 2 modes can lead to large errors in the evaluation of the AAM budget. The n = 2 responses in surface pressure can affect the earth ellipcity, and the n = 1 diurnal response can affect the geocenter position. For the surface pressure tide, the results suggest that the dynamical forcings of the zonal-mean zonal flow are a potential cause for its s = 0 component.

The backward reflection of a stationary gravity wave (GW) propagating toward the ground is examined in the linear viscous case and for large Reynolds numbers (Re). In this case, the stationary GW presents a critical level at the ground because the mean wind is null there. When the mean flow Richardson number at the surface (J) is below 0.25, the GW reflection by the viscous boundary layer is total in the inviscid limit Re [right arrow] [infinity] The GW is a little absorbed when Re is finite, and the reflection decreases when both the dissipation and J increase. When J > 0.25, the GW is absorbed for all values of the Reynolds number, with a general tendency for the GW reflection to decrease when J increases. As a large ground reflection favors the downstream development of a trapped lee wave, the fact that it decreases when J increases explains why the more unstable boundary layers favor the onset of mountain lee waves. It is also shown that the GW reflection when J > 0.25 is substantially larger than that predicted by the conventional inviscid critical level theory and larger than that predicted when the dissipations are represented by Rayleigh friction and Newtonian cooling. The fact that the GW reflection depends strongly on the Richardson number indicates that there is some correspondence between the dynamics of trapped lee waves and the dynamics of Kelvin-Helmholtz instabilities. Accordingly, and in one classical example, it is shown that some among the neutral modes for Kelvin-Helmholtz instabilities that exist in an unbounded flow when J < 0.25 can also be stationary trapped-wave solutions when there is a ground and in the inviscid limit Re [right arrow] [infinity]. When Re is finite, these solutions are affected by the dissipation in the boundary layer and decay in the downstream direction. Interestingly, their decay rate increases when both the dissipation and J increase, as does the GW absorption by the viscous boundary layer.

A heuristic model is used to study the synoptic response to mountain gravity waves (GWs) absorbed at directional critical levels. The model is a semigeostrophic version of the Eady model for baroclinic instability adapted by Smith to study lee cyclogenesis. The GWs exert a force on the large-scale flow where they encounter directional critical levels. This force is taken into account in the model herein and produces potential vorticity (PV) anomalies in the midtroposphere. First, the authors consider the case of an idealized mountain range such that the orographic variance is well separated between small- and large-scale contributions. In the absence of tropopause, the PV produced by the GW force has a surface impact that is significant compared to the surface response due to the large scales. For a cold front, the GW force produces a trough over the mountain and a larger-amplitude ridge immediately downstream. It opposes somehow to the response due to the large scales of the mountain range, which is anticyclonic aloft and cyclonic downstream. For a warm front, the GW force produces a ridge over the mountain and a trough downstream; hence it reinforces the response due to the large scales. Second, the robustness of the previous results is verified by a series of sensitivity tests. The authors change the specifications of the mountain range and of the background flow. They also repeat some experiments by including baroclinic instabilities, or by using the quasigeostrophic approximation. Finally, they consider the case of a small-scale orographic spectrum representative of the Alps. The significance of the results is discussed in the context of GW parameterization in the general circulation models. The results may also help to interpret the complex PV structures occurring when mountain gravity waves break in a baroclinic environment.

The NCEP-NCAR reanalysis dataset for 1958-97 is used to analyze intraseasonal variations in mountain torques and the large-scale atmospheric circulation patterns associated with them. Spectral analysis of the atmospheric angular momentum (AAM) budget shows that the dominant variations of mountain torque have periodicities near 30 days and shorter, while the dominant AAM variations occur in the 40-60-day band. This difference is due to the 40-60-day AAM variations being primarily related to equatorial processes, while mountain torque variations are associated mostly with extratropical processes. The Northern Hemisphere (NH) mountain torque has enhanced power and significant spectral peaks in the 20-30-day band. The signal in this band accounts for 33% of the NH mountain torque variance, once the seasonal cycle has been removed. Lag composites of the NH 700-hPa geopotential heights based on the 20-30-day mountain torque signal show the latter to be associated with coherent large-scale patterns that resemble low-frequency oscillations identified in this band by previous authors. The composite patterns that are in phase quadrature with the 20-30-day NH mountain torque have a pronounced zonally symmetric component. These patterns are associated with substantial AAM variations, arguably driven by the NH mountain torque in this band. Principal component (PC) analysis of the NH 700-hPa heights shows that, in the 20-30-day band, the mountain torque is also in phase quadrature with the two leading PCs; the first corresponds to changes in the midlatitude jet intensity near the subtropics, while the second corresponds to the Arctic Oscillation. The relationships with AAM of the latter essentially occurs through the mass term. Mountain torques are furthermore, nearly in phase with dominant patterns of low-frequency variability that exhibit substantial pressure gradients across the Rockies and the Tibetan Plateau.

Important aspects of low-frequency variability (LFV) are regional in character, while the mountain torques of the Rockies and the Himalayas evolve quite independently of each other. The hemispheric analysis of Part I is complemented therefore herein by an analysis of the relationships between individual mountain torques and sectorial LFV patterns in the NCEP-NCAR reanalysis. In the 20-30-day band, relationships are found between the Rockies (Himalayas) torque and the dominant patterns of LFV over the Pacific (Eurasia). The composites of the atmospheric flow fields that accompany the Rockies (Himalayas) torque in this band exhibit similarities with known low-frequency oscillations that dominate the Pacific and North American (European and North Atlantic) sectors during certain winters. The composites keyed to the 20-30-day Rockies torque affect the persistent North Pacific (PNP) pattern that controls the extension of the midlatitude jet stream over the eastern Pacific. Furthermore, the unfiltered torques for the Northern Hemisphere (NH) and Rockies anticipate the onset of the two dominant winter Pacific circulation regimes that correlate strongly with the PNP pattern. The composites keyed to the 20-30-day Himalayas torque affect the North Atlantic Oscillation (NAO) pattern, which controls the intensity of the North Atlantic jet stream. Furthermore, the unfiltered torques for the NH and the Himalayas anticipate the breaks of the two dominant winter Atlantic circulation regimes, which correlate strongly with the NAO pattern. These analyses also show that the 20-30-day Rockies (Himalayas) torques produce substantial atmospheric angular momentum (AAM) changes, which are nearly in phase with and larger in amplitude than the AAM changes associated with the midlatitude eastern Pacific (North Atlantic) jet stream variations seen in the composite maps. This result suggests that the Rockies (Himalayas) torque variations drive, at least partially, but actively the changes in the eastern Pacific (North Atlantic) jet stream. These results are consistent with the Himalayas and the Rockies torques contributing separately to changes in the two leading hemispheric EOFs that were described in Part I; the two are associated with a hemispheric index cycle and the Arctic Oscillation, respectively.

A two-dimensional rotating anelastic model is used to analyze the large-scale flow (LSF) response to the breaking of gravity waves (GWs) at critical levels. In the response the balanced part is separated from the inertial oscillations (IOs) and the inertia-gravity waves (IGWs). Interest also focuses on the relative importance of the two, when the regime of the critical levels interaction becomes nonlinear. In the linear periodic case, the balanced response is a mean transverse velocity that equilibrates the wave drag via the Coriolis torque, and the unbalanced one is an IO. Their relative importance is well predicted by a temporal Rossby number associated with the timescale of the GWs forcing onto the mean flow. When the dynamics are nonlinear, the GWs are reflected by the shear layer, affecting the GWs' forcing amplitude. A nonlinear feedback loop also makes the ratio between the IO and the balanced response much larger than in the linear case. In the nonperiodic case, the balanced motion is a growing baroclinic pattern, which results from steering by the shear of the potential vorticity (PV) dipole generated where the GWs break. The unbalanced response consists of IGWs propagating away from the shear layer. Contrary to the periodic case, the ratio between the two is not much affected by nonlinearities, and stays well predicted by a spatial Rossby number associated with the spatial scale of the GWs forcing on the LSF. When this number is near 1, and the interaction nonlinear, the IGWs outside of the shear layer make a substantial fraction of the total wave signal.

The concept of linear mountain waves is generally equated with steady-state stationary waves. This essentially means that the absolute horizontal phase velocity of mountain waves is zero and that their momentum flux profile is independent of height and time in the absence of dissipation and zero-level wind. This paper investigates the generation of linear unsteady mountain gravity waves. The incident flow is transient, starting from zero at a given time and returning to zero after a finite time. The topography is a single horizontal harmonic. The unsteadiness of the waves is due partly to the temporal change of their phase velocity, which takes place during their propagation in the time-dependent mean flow. When the wind ceases, most of the waves present have a phase velocity nearly opposite to the maximum wind. For this reason, mountain waves can propagate through levels of zero mean wind. The transient structure of the wave field also comes from the temporal change of the amplitude of the ground forcing. Moreover, it is the result of the interference between the waves generated while the wind increases and those generated while it decreases. These transient effects naturally lead to z-dependent momentum flux profiles that are analyzed in detail. This paper deals with an analytical model, when the unsteady incident flow is uniform in the vertical direction. When the incident wind varies slowly in time, a comprehensive picture of the wave field is given by an asymptotic expansion of the wave solution. When the mean flow changes rapidly in time and when it varies in the vertical direction, the solution is obtained with a numerical model.

The Stratosphere-troposphere Processes And their Role in Climate (SPARC) Quasi-Biennial Oscillation initiative (QBOi) aims to improve the fidelity of tropical stratospheric variability in general circulation and Earth system models by conducting coordinated numerical experiments and analysis. In the equatorial stratosphere, the QBO is the most conspicuous mode of variability. Five coordinated experiments have therefore been designed to (i) evaluate and compare the verisimilitudeof modelled QBOs under present-day conditions, (ii) identify robustness (or, alternatively the spread/uncertainty) in the simulated QBO response to commonly imposed changes in model climate forcings (e.g., a doubling of CO2 amounts) and, (iii) examine model dependence of QBO predictability. This paper documents these experiments and the recommended output diagnostics. The rationale behind the experimental design and choice of diagnostics is presented. To facilitate scientific interpretation of the results in other planned QBOi studies, consistent descriptions of the models performing each experiment set are given, with those aspects particularly relevant for simulating the QBO tabulated for easy comparison. [ABSTRACT FROM AUTHOR]

The airplane data collected between 4 and 12 km above the Pyrenees during the intensive observation period (IOP) 3 of the Pyrenees Experiment (PYREX) are analyzed again. A spectral analysis of the velocity and potential temperature series shows that the mountain waves are dominated by two oscillations with well-defined horizontal wavenumbers. At nearly all altitudes, at least one among these two oscillations can be extracted: the short oscillation dominates the signal below 6 km and the long one above. These two oscillations contribute to the Reynolds stress below 5 km and not above. Linear steady nondissipative simulations show that the short oscillation is a trapped resonant mode and the long one a leaking, or partially leaking, resonant mode of the background flow. Pseudo-momentum flux budgets show that the short resonant mode only contributes to the Reynolds stress at low level (here below 3 to 4 km typically) while the long one contributes to the Reynolds stress at all levels. At low level, (below 4 to 6 km typically), the long mode can induce a decay of the Reynolds stress amplitude, when it partially leaks toward the stratosphere. Various tests, changing the incident flow profiles within limits provided by the different soundings available this day, reveal, on the one hand, that the above findings are quite robust. On the other hand, they reveal that the resonant modes response is very sensitive to the background flow and orography specifications. In some of the steady linear simulations, the long resonant oscillation has a Reynolds stress that is constant with altitude. In all of them the downwind extent of the lee waves is overestimated and the waves amplitude is too large. To explain these mismatches with the observations, we present simulations that last 3 h only, so the resonant modes patterns are everywhere unsteady. They show that during their build-up phase, all the leaking modes can make the Reynolds stress amplitude decays with altitude at low level (here below 4 to 6 km, typically). At this time, the downstream extent of the waves is also correctly predicted. These linear unsteady simulations also give realistic waves amplitude and Reynolds stress profiles if the mountain is cut off to parameterize nonlinear low-level flow splitting. By using a nonlinear model, the simulated waves are matched to that observed through an adjustment of the parameters of the turbulent diffusion parameterization scheme: with enough dissipation, the model response can become quite realistic. In these nonlinear simulations, the background flow is chosen so that there is only one resonant mode and this mode does not contribute much to the Reynolds stress in the inviscid case. When increasing the mountain height and the dissipation, the overall structure of that mode stays unchanged, and it never contributes much to the Reynolds stress. This indicates that the dissipative and nonlinear processes alone are not likely to produce the observed low-level stress variations associated with the resonant modes.