# Search Results

## You are looking at 1 - 10 of 20 items for

- Author or Editor: Ulrich Achatz x

- Refine by Access: All Content x

^{ }

## Abstract

The primary nonlinear dynamics of high-frequency gravity waves (HGWs) perturbed by their most prominent normal modes (NMs) or singular vectors (SVs) in a rotating Boussinesq fluid have been studied by direct numerical simulations (DNSs), with wave scales and values of viscosity and diffusivity characteristic for the upper mesosphere. The DNS is 2.5D in that it has only two spatial dimensions, defined by the direction of propagation of the HGW and the direction of propagation of the perturbation in the plane orthogonal to the HGW phase direction, but describes a fully 3D velocity field. Many results of the more comprehensive fully 3D simulations in the literature are reproduced. So it is found that statically unstable HGWs are subject to wave breaking ending in a wave amplitude with respect to the overturning threshold near 0.3. It is shown that this is a result of a perturbation of the HGW by its leading transverse NM. For statically stable HGWs, a parallel NM has the strongest effect, quite in line with previous results on the predominantly 2D instability of such HGWs. This parallel mode is, however, not the leading NM but a larger-scale pattern, seemingly driven by resonant wave–wave interactions, leading eventually to energy transfer from the HGW into another gravity wave with steeper phase propagation. SVs turn out to be less effective in triggering HGW decay but they can produce turbulence of a strength that is (as that from the NMs) within the range of measured values, however with a more pronounced spatial confinement.

## Abstract

The primary nonlinear dynamics of high-frequency gravity waves (HGWs) perturbed by their most prominent normal modes (NMs) or singular vectors (SVs) in a rotating Boussinesq fluid have been studied by direct numerical simulations (DNSs), with wave scales and values of viscosity and diffusivity characteristic for the upper mesosphere. The DNS is 2.5D in that it has only two spatial dimensions, defined by the direction of propagation of the HGW and the direction of propagation of the perturbation in the plane orthogonal to the HGW phase direction, but describes a fully 3D velocity field. Many results of the more comprehensive fully 3D simulations in the literature are reproduced. So it is found that statically unstable HGWs are subject to wave breaking ending in a wave amplitude with respect to the overturning threshold near 0.3. It is shown that this is a result of a perturbation of the HGW by its leading transverse NM. For statically stable HGWs, a parallel NM has the strongest effect, quite in line with previous results on the predominantly 2D instability of such HGWs. This parallel mode is, however, not the leading NM but a larger-scale pattern, seemingly driven by resonant wave–wave interactions, leading eventually to energy transfer from the HGW into another gravity wave with steeper phase propagation. SVs turn out to be less effective in triggering HGW decay but they can produce turbulence of a strength that is (as that from the NMs) within the range of measured values, however with a more pronounced spatial confinement.

^{ }

## Abstract

The breaking of an inertia–gravity wave (IGW), initiated by its leading normal modes (NMs) or singular vectors (SVs), and the resulting small-scale eddies are investigated by means of direct numerical simulations of a Boussinesq fluid characterizing the upper mesosphere. The focus is on the primary nonlinear dynamics, neglecting the effect of secondary instabilities. It is found that the structures with the strongest impact on the IGW and also the largest turbulence amplitudes are the NM (for a statically unstable IGW) or short-term SV (statically and dynamically stable IGW) propagating horizontally transversely with respect to the IGW, possibly in agreement with observations of airglow ripples in conjunction with statically unstable IGWs. In both cases these leading structures reduce the IGW amplitude well below the static and dynamic instability thresholds. The resulting turbulent dissipation rates are within the range of available estimates from rocket soundings, even for IGWs at amplitudes low enough precluding NM instabilities. Thus SVs can help explain turbulence occurring under conditions not amenable for the classic interpretation via static and dynamic instability. Because of the important role of the statically enhanced roll mechanism in the energy exchange between IGW and eddies, the turbulent velocity fields are often conspicuously anisotropic. The spatial turbulence distribution is determined to a large degree by the elliptically polarized horizontal velocity field of the IGW.

## Abstract

The breaking of an inertia–gravity wave (IGW), initiated by its leading normal modes (NMs) or singular vectors (SVs), and the resulting small-scale eddies are investigated by means of direct numerical simulations of a Boussinesq fluid characterizing the upper mesosphere. The focus is on the primary nonlinear dynamics, neglecting the effect of secondary instabilities. It is found that the structures with the strongest impact on the IGW and also the largest turbulence amplitudes are the NM (for a statically unstable IGW) or short-term SV (statically and dynamically stable IGW) propagating horizontally transversely with respect to the IGW, possibly in agreement with observations of airglow ripples in conjunction with statically unstable IGWs. In both cases these leading structures reduce the IGW amplitude well below the static and dynamic instability thresholds. The resulting turbulent dissipation rates are within the range of available estimates from rocket soundings, even for IGWs at amplitudes low enough precluding NM instabilities. Thus SVs can help explain turbulence occurring under conditions not amenable for the classic interpretation via static and dynamic instability. Because of the important role of the statically enhanced roll mechanism in the energy exchange between IGW and eddies, the turbulent velocity fields are often conspicuously anisotropic. The spatial turbulence distribution is determined to a large degree by the elliptically polarized horizontal velocity field of the IGW.

^{ }

^{ }

## Abstract

This work discusses the formulation and testing of a simplified model of atmospheric dynamics. The model, which has only 200- and 700-mb streamfunctions as its prognostic fields, is designed to have a climate that approximates that of a comprehensive perpetual January general circulation model. Its governing equations are based on a Lorenz-type filtered two-layer model, but its linear terms are replaced by an empirically determined operator; the simplified model is semiempirical. Its basis consists of three-dimensional empirical orthogonal functions that are calculated using a total energy metric. The linear operator is intended to serve as a parameterization of fields, patterns, and dynamics not explicitly represented in the model. The operator is found through an optimization procedure that ensures that the semiempirical model optimally predicts streamfunction tendencies observed to occur in an extended control integration of the general circulation model.

It turns out that a model determined in this way simulates the GCM climatology quite well. The time mean state, time mean transient fluxes, and leading patterns of variability are all very similar to those in the GCM. Notable superiority over the behavior of a standard filtered two-layer model is also found. In order to understand this, calculations are undertaken to identify processes, not explicitly represented in a standard filtered two-layer model, that can be especially well parameterized linearly. Results point to a dynamical balance in the GCM such that deviations of its tendencies from the tendencies given by a standard filtered model are smaller and more nearly a linear function of streamfunction anomaly than are individual terms contributing to the deviations. An analysis of the possibility of reducing the number of basis functions in the semiempirical model shows that, whereas short-time prediction is best for the nontruncated model, in the simulation of climate mean state and transient fluxes the optimum is at rather small pattern numbers (between 30 and 70).

The leading eigenmodes of the empirically determined linear component of the simplified model are found to be nearly neutral.

## Abstract

This work discusses the formulation and testing of a simplified model of atmospheric dynamics. The model, which has only 200- and 700-mb streamfunctions as its prognostic fields, is designed to have a climate that approximates that of a comprehensive perpetual January general circulation model. Its governing equations are based on a Lorenz-type filtered two-layer model, but its linear terms are replaced by an empirically determined operator; the simplified model is semiempirical. Its basis consists of three-dimensional empirical orthogonal functions that are calculated using a total energy metric. The linear operator is intended to serve as a parameterization of fields, patterns, and dynamics not explicitly represented in the model. The operator is found through an optimization procedure that ensures that the semiempirical model optimally predicts streamfunction tendencies observed to occur in an extended control integration of the general circulation model.

It turns out that a model determined in this way simulates the GCM climatology quite well. The time mean state, time mean transient fluxes, and leading patterns of variability are all very similar to those in the GCM. Notable superiority over the behavior of a standard filtered two-layer model is also found. In order to understand this, calculations are undertaken to identify processes, not explicitly represented in a standard filtered two-layer model, that can be especially well parameterized linearly. Results point to a dynamical balance in the GCM such that deviations of its tendencies from the tendencies given by a standard filtered model are smaller and more nearly a linear function of streamfunction anomaly than are individual terms contributing to the deviations. An analysis of the possibility of reducing the number of basis functions in the semiempirical model shows that, whereas short-time prediction is best for the nontruncated model, in the simulation of climate mean state and transient fluxes the optimum is at rather small pattern numbers (between 30 and 70).

The leading eigenmodes of the empirically determined linear component of the simplified model are found to be nearly neutral.

^{ }

^{ }

## Abstract

The problem of nonmodal instabilities of inertia–gravity waves (IGW) in the middle atmosphere is addressed, within the framework of a Boussinesq model with realistic molecular viscosity and thermal diffusion, by singular-vector analysis of horizontally homogeneous vertical profiles of wind and buoyancy obtained from IGW packets at their statically least stable or most unstable horizontal location. Nonmodal growth is always found to be significantly stronger than that of normal modes, most notably at wave amplitudes below the static instability limit where normal-mode instability is very weak, whereas the energy gain between the optimal perturbation and singular vector after one Brunt–Väisälä period can be as large as two orders of magnitude. Among a multitude of rapidly growing singular vectors for this optimization time, small-scale (wavelengths of a few 100 m) perturbations propagating in the horizontal parallel to the IGW are most prominent. These parallel optimal perturbations are amplified by a roll mechanism, while transverse perturbations (with horizontal scales of a few kilometers) are to a large part subject to an Orr mechanism, both controlled by the transverse wind shear in the IGW at its statically least stable altitude, but further enhanced by reduced static stability. The elliptic polarization of the IGW leaves its traces in an additional impact of the roll mechanism via the parallel wind shear on the leading transverse optimal perturbation.

## Abstract

The problem of nonmodal instabilities of inertia–gravity waves (IGW) in the middle atmosphere is addressed, within the framework of a Boussinesq model with realistic molecular viscosity and thermal diffusion, by singular-vector analysis of horizontally homogeneous vertical profiles of wind and buoyancy obtained from IGW packets at their statically least stable or most unstable horizontal location. Nonmodal growth is always found to be significantly stronger than that of normal modes, most notably at wave amplitudes below the static instability limit where normal-mode instability is very weak, whereas the energy gain between the optimal perturbation and singular vector after one Brunt–Väisälä period can be as large as two orders of magnitude. Among a multitude of rapidly growing singular vectors for this optimization time, small-scale (wavelengths of a few 100 m) perturbations propagating in the horizontal parallel to the IGW are most prominent. These parallel optimal perturbations are amplified by a roll mechanism, while transverse perturbations (with horizontal scales of a few kilometers) are to a large part subject to an Orr mechanism, both controlled by the transverse wind shear in the IGW at its statically least stable altitude, but further enhanced by reduced static stability. The elliptic polarization of the IGW leaves its traces in an additional impact of the roll mechanism via the parallel wind shear on the leading transverse optimal perturbation.

^{ }

^{ }

## Abstract

Using a hierarchy of three models of increasing realism and complexity, and expanding on a previous study, optimal perturbations of inertia–gravity wave (IGW) packets are studied with respect to several aspects. It is shown that normal modes are comparatively less able to extract energy from the IGW over finite time due to their time-invariant structure, while singular vectors (SVs) can adjust their dynamical fields flexibly so as to optimize the statically enhanced roll and Orr mechanisms by which they grow. On longer time scales, where the time dependence of the IGW packet precludes a normal-mode analysis, optimal growth is found to further amplify suitable perturbations. The propagation characteristics of these exhibit critical layer interactions for horizontal propagation directions transverse with respect to the IGW, preventing significant vertical propagation, while parallel and obliquely propagating perturbations of sufficiently long horizontal scales are found to radiate gravity waves into altitudes not directly affected by the IGW. The SVs with shorter wavelengths, as found for short optimization times, stay confined via a linear wave duct near the altitude of least static stability where they are excited. At optimization times of the order of the IGW period the leading SVs, with an energy growth by about three orders of magnitude, propagate obliquely, possibly in correspondence to previous results by others from simulations of nonlinear IGW breakdown. The three-dimensional structure of SVs shows an amplitude modulation strictly confining the perturbations also to the horizontal location of least static stability, suggesting a picture of turbulence onset in IGW packets where local patches of growing perturbations initially dominate.

## Abstract

Using a hierarchy of three models of increasing realism and complexity, and expanding on a previous study, optimal perturbations of inertia–gravity wave (IGW) packets are studied with respect to several aspects. It is shown that normal modes are comparatively less able to extract energy from the IGW over finite time due to their time-invariant structure, while singular vectors (SVs) can adjust their dynamical fields flexibly so as to optimize the statically enhanced roll and Orr mechanisms by which they grow. On longer time scales, where the time dependence of the IGW packet precludes a normal-mode analysis, optimal growth is found to further amplify suitable perturbations. The propagation characteristics of these exhibit critical layer interactions for horizontal propagation directions transverse with respect to the IGW, preventing significant vertical propagation, while parallel and obliquely propagating perturbations of sufficiently long horizontal scales are found to radiate gravity waves into altitudes not directly affected by the IGW. The SVs with shorter wavelengths, as found for short optimization times, stay confined via a linear wave duct near the altitude of least static stability where they are excited. At optimization times of the order of the IGW period the leading SVs, with an energy growth by about three orders of magnitude, propagate obliquely, possibly in correspondence to previous results by others from simulations of nonlinear IGW breakdown. The three-dimensional structure of SVs shows an amplitude modulation strictly confining the perturbations also to the horizontal location of least static stability, suggesting a picture of turbulence onset in IGW packets where local patches of growing perturbations initially dominate.

^{ }

^{ }

^{ }

## Abstract

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

## Abstract

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

^{ }

^{ }

## Abstract

In a continuation of previous investigations on deterministic reduced atmosphere models with compact state space representation, two main modifications are introduced. First, primitive equation dynamics is used to describe the nonlinear interactions between resolved scales. Second, the seasonal cycle in its main aspects is incorporated. Stability considerations lead to a gridpoint formulation of the basic equations in the dynamical core. A total energy metric consistent with the equations can be derived, provided surface pressure is treated as constant in time. Using this metric, a reduction in the number of degrees of freedom is achieved by a projection onto three-dimensional empirical orthogonal functions (EOFs), each of them encompassing simultaneously all prognostic variables (winds and temperature). The impact of unresolved scales and not explicitly described physical processes is incorporated via an empirical linear parameterization. The basis patterns having been determined from 3 sigma levels from a GCM dataset, it is found that, in spite of the presence of a seasonal cycle, at most 500 are needed for describing 90% of the variance produced by the GCM. If compared to previous low-order models with quasigeostrophic dynamics, the reduced models exhibit at this and lower-order truncations, a considerably enhanced capability to predict GCM tendencies. An analysis of the dynamical impact of the empirical parameterization is given, hinting at an important role in controlling the seasonally dependent storm track dynamics.

## Abstract

In a continuation of previous investigations on deterministic reduced atmosphere models with compact state space representation, two main modifications are introduced. First, primitive equation dynamics is used to describe the nonlinear interactions between resolved scales. Second, the seasonal cycle in its main aspects is incorporated. Stability considerations lead to a gridpoint formulation of the basic equations in the dynamical core. A total energy metric consistent with the equations can be derived, provided surface pressure is treated as constant in time. Using this metric, a reduction in the number of degrees of freedom is achieved by a projection onto three-dimensional empirical orthogonal functions (EOFs), each of them encompassing simultaneously all prognostic variables (winds and temperature). The impact of unresolved scales and not explicitly described physical processes is incorporated via an empirical linear parameterization. The basis patterns having been determined from 3 sigma levels from a GCM dataset, it is found that, in spite of the presence of a seasonal cycle, at most 500 are needed for describing 90% of the variance produced by the GCM. If compared to previous low-order models with quasigeostrophic dynamics, the reduced models exhibit at this and lower-order truncations, a considerably enhanced capability to predict GCM tendencies. An analysis of the dynamical impact of the empirical parameterization is given, hinting at an important role in controlling the seasonally dependent storm track dynamics.

^{ }

^{ }

## Abstract

A recently developed class of semiempirical low-order models is utilized for the reexamination of several aspects of the complexity and nonlinearity of large-scale dynamics in a GCM. Given their low dimensionality, these models are quite realistic, due to the use of the primitive equations, an efficient EOF basis, and an empirical seasonally dependent linear parameterization of the impact of unresolved scales and not explicitely described processes. Fairly different results are obtained with respect to the dependence of short-term predictability or climate simulations on the number of employed degrees of freedom. Models using 500 degrees of freedom are significantly better in short-term predictions than smaller counterparts. Meaningful predictions of the first 500 EOFs are possible for 4–5 days, while the mean anomaly correlation for the leading 30 EOFs stays above 0.6 for up to 9 days. In a 30-EOF model this is only 6 days. A striking feature is found when it comes to simulations of the monthly mean states and transient fluxes: the 30-EOF model is performing just as well as the 500-EOF model. Since similar behavior is also found in the reproduction of the number and shape of the three significant cluster centroids in the January data of the GCM, one can speculate on a characteristic dimension in the range of a few tens for the large-scale part of the climate attractor. A partial failure diagnosed in the predictability of climate change by our statistical–dynamical models indicates that the employed empirical parameterizations might actually be climate dependent. Understanding their dependence on the large-scale flow could be a prerequisite for applicability to climate change studies. In a further analysis no support is found for the classic hypothesis that the observed cluster centroids, indicating multimodality in the climate statistics, can be interpreted as quasi steady states of the GCM's low-frequency dynamics.

## Abstract

A recently developed class of semiempirical low-order models is utilized for the reexamination of several aspects of the complexity and nonlinearity of large-scale dynamics in a GCM. Given their low dimensionality, these models are quite realistic, due to the use of the primitive equations, an efficient EOF basis, and an empirical seasonally dependent linear parameterization of the impact of unresolved scales and not explicitely described processes. Fairly different results are obtained with respect to the dependence of short-term predictability or climate simulations on the number of employed degrees of freedom. Models using 500 degrees of freedom are significantly better in short-term predictions than smaller counterparts. Meaningful predictions of the first 500 EOFs are possible for 4–5 days, while the mean anomaly correlation for the leading 30 EOFs stays above 0.6 for up to 9 days. In a 30-EOF model this is only 6 days. A striking feature is found when it comes to simulations of the monthly mean states and transient fluxes: the 30-EOF model is performing just as well as the 500-EOF model. Since similar behavior is also found in the reproduction of the number and shape of the three significant cluster centroids in the January data of the GCM, one can speculate on a characteristic dimension in the range of a few tens for the large-scale part of the climate attractor. A partial failure diagnosed in the predictability of climate change by our statistical–dynamical models indicates that the employed empirical parameterizations might actually be climate dependent. Understanding their dependence on the large-scale flow could be a prerequisite for applicability to climate change studies. In a further analysis no support is found for the classic hypothesis that the observed cluster centroids, indicating multimodality in the climate statistics, can be interpreted as quasi steady states of the GCM's low-frequency dynamics.

^{ }

^{ }

## Abstract

The three-dimensionalization of turbulence in the breaking of nearly vertically propagating inertia–gravity waves is investigated numerically using singular vector analysis applied to the Boussinesq equations linearized about three two-dimensional time-dependent basic states obtained from nonlinear simulations of breaking waves: a statically unstable wave perturbed by its leading transverse normal mode, the same wave perturbed by its leading parallel normal mode, and a statically stable wave perturbed by a leading transverse singular vector. The secondary instabilities grow through interaction with the buoyancy gradient and velocity shear in the basic state. Which growth mechanism predominates depends on the time-dependent structure of the basic state and the wavelength of the secondary perturbation. The singular vectors are compared to integrations of the linear model using random initial conditions, and the leading few singular vectors are found to be representative of the structures that emerge in the randomly initialized integrations. A main result is that the length scales of the leading secondary instabilities are an order of magnitude smaller than the wavelength of the initial wave, suggesting that the essential dynamics of the breaking might be captured by tractable nonlinear three-dimensional simulations in a relatively small triply periodic domain.

## Abstract

The three-dimensionalization of turbulence in the breaking of nearly vertically propagating inertia–gravity waves is investigated numerically using singular vector analysis applied to the Boussinesq equations linearized about three two-dimensional time-dependent basic states obtained from nonlinear simulations of breaking waves: a statically unstable wave perturbed by its leading transverse normal mode, the same wave perturbed by its leading parallel normal mode, and a statically stable wave perturbed by a leading transverse singular vector. The secondary instabilities grow through interaction with the buoyancy gradient and velocity shear in the basic state. Which growth mechanism predominates depends on the time-dependent structure of the basic state and the wavelength of the secondary perturbation. The singular vectors are compared to integrations of the linear model using random initial conditions, and the leading few singular vectors are found to be representative of the structures that emerge in the randomly initialized integrations. A main result is that the length scales of the leading secondary instabilities are an order of magnitude smaller than the wavelength of the initial wave, suggesting that the essential dynamics of the breaking might be captured by tractable nonlinear three-dimensional simulations in a relatively small triply periodic domain.

^{ }

^{ }

^{ }

## Abstract

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.

## Abstract

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.