High-resolution numerical simulations of non-stationary, nonlinear acoustic–gravity waves (AGWs) propagating upwards from surface wave sources are performed for different temporal intervals relative to activation and deactivation times of the wave forcing. After activating surface wave sources, amplitudes of AGW spectral components reach a quasi-stationary state. Then the surface wave forcing is deactivated in the numerical model, and amplitudes of vertically traveling AGW modes quickly decrease at all altitudes due to discontinuations of the upward propagation of wave energy from the wave sources. However, later the standard deviation of residual and secondary wave perturbations experiences a slower quasi-exponential decrease. High-resolution simulations allowed, for the first time, for the estimation of the decay times of this wave noise produced by slow residual, quasi-standing and secondary AGW spectral components, which vary between 20 and 100

Recently, acoustic–gravity waves (AGWs) are believed to exist almost permanently in the atmosphere (Siefring et al., 2010; Snively et al., 2013; Wei et al., 2015; Lay, 2018; Meng et al., 2019). Observations detect regular AGW presence up to high atmospheric altitudes (e.g., Djuth et al., 2004; Park et al., 2014; Trinh et al., 2018). Modeling of general circulation demonstrated AGW capabilities of transferring energy and momentum from tropospheric wave sources to higher atmospheric levels (e.g., Medvedev and Yiğit, 2019). Nonhydrostatic models of the general circulation of the atmosphere revealed that AGWs permanently exist at all atmospheric heights (e.g., Yiğit et al., 2012).

Many AGWs detected in the atmosphere are excited in the troposphere (Fritts and Alexander, 2003; Snively, 2013; Yiğit and Medvedev, 2014). AGWs can be produced by interactions of winds with mountains (e.g., Gossard and Hooke, 1975), atmospheric jet streams and fronts (e.g., Gavrilov and Fukao, 1999; Dalin et al., 2016), thunderstorms and cumulus clouds (Siefring et al., 2010; Blanc et al., 2014; Lay, 2018), convective regions and shear flows (Townsend, 1966; Fritts and Alexander, 2003; Vadas and Fritts, 2006), typhoons (Wu et al.,2015), volcanoes (De Angelis et al., 2011), waves on the sea surface (Godin et al., 2015), explosions at the Earth's surface (Meng, 2019), earthquakes (Rapoport et al., 2004), tsunamis (Wei at al., 2015), different objects moving in the atmosphere (Afraimovich et al., 2002), big fires, etc. Some AGWs can be generated by mesoscale turbulence in the atmosphere (Townsend, 1965; Medvedev and Gavrilov, 1995). These AGW sources are located mainly at tropospheric heights (Gavrilov and Fukao, 1999; Dalin, 2016).

Most wave sources listed above are non-stationary. They can be activated during initial time intervals, operate for some time and then be deactivated during final time intervals. The initial and final time intervals could be shorter or longer depending on the physical properties of particular wave sources. Non-stationary activating and deactivating wave sources can generate transient AGW pulses propagating upwards from the lower atmosphere, which require their analysis.

High-resolution numerical models are frequently used for studies of meso- and microscale processes in the atmosphere, for example, the Weather Research and Forecasting Nonhydrostatic Mesoscale Model, also known as the North American Mesoscale Model (WRF, 2019), as well as the Regional Atmospheric Modeling System (RAMS) described by Pielke et al. (1992) and other similar models. Direct numerical simulation (DNS) and large-eddy simulation (LES) models (e.g., Mellado, 2018) should be mentioned in this context. Fritts et al. (2009, 2011) used a numerical model of Kelvin–Helmholtz instabilities, AGW breaking and generation of turbulence in atmospheric regions with fixed horizontal and vertical extents. They utilized a Galerkin-type algorithm for turning partial differential equations into equations for spectral series coefficients. Liu et al. (2009) simulated the propagation of atmospheric AGWs and creation of Kelvin–Helmholtz billows. Yu et al. (2017) used a numerical model for AGWs propagating in the atmosphere from tsunamis.

Gavrilov and Kshevetskii (2013) studied nonlinear AGWs with a numerical two-dimensional model, which involved fundamental conservation laws. This model permitted non-smooth solutions of the nonlinear wave equations and gave the required stability of the numerical model (Kshevetskii and Gavrilov, 2005). A respective three-dimensional algorithm was introduced by Gavrilov and Kshevetskii (2014) to simulate nonlinear atmospheric AGWs. Gavrilov and Kshevetskii (2013, 2014) showed that after triggering wave forcing at the lower boundary of the numerical model, initial AGW pulses could reach high atmospheric levels in a few minutes. AGW phase surfaces are quasi-vertical initially, but later they become inclined to the horizon. AGW vertical wavelengths decrease in time and are close to their theoretical predictions after intervals of a few periods of wave forcing.

In this study, using the high-resolution nonlinear wave model developed by Gavrilov and Kshevetskii (2014), we continue simulating transient waves generated by non-stationary AGW sources at the lower boundary and propagating upwards to the atmosphere. The main focus is AGW behavior after deactivations of wave sources in the model. After activating the surface wave source and disappearing initial wave pulses, AGW amplitudes tend to stabilize at all atmospheric altitudes. In this quasi-stationary state, the surface wave forcing is deactivated in the numerical model. After that, amplitudes of traveling AGW modes quickly decrease at all altitudes due to discontinuation of the upward propagation of wave energy from the surface sources. We found, however, that after some time, the standard deviation of residual quasi-standing and secondary wave perturbations experiences a slower exponential decrease with substantial decay times.

These results show that residual and secondary AGW modes produced by transient wave sources can exist for a long time in the stratosphere and mesosphere and form a background level of wave noise there. AGW decay times and their dependences on parameters of the surface wave forcing are estimated for the first time.

In this study, we employed the high-resolution three-dimensional numerical model of nonlinear AGWs in the atmosphere developed by Gavrilov and
Kshevetskii (2014). Currently, this model (called AtmoSym) is available for free online usage (AtmoSym, 2017). The AtmoSym model utilizes the plain
geometry and primitive hydrodynamic three-dimensional equations (Gavrilov and Kshevetskii, 2014):

The AtmoSym model takes into account dissipative and nonlinear processes that accompany AGW propagation. The model is capable of simulating such
complicated processes as AGW instability, breaking and turbulence generation. Dynamical deviations as defined in Eq. (

In our simulations, the wave excitation in Eq. (

The novelty of the present study is deactivating the wave source in Eq. (

Our numerical modeling begins with a steady-state, windless, non-perturbed atmosphere with profiles of background temperature, density, molecular weight
and molecular kinematic viscosity corresponding to January at latitude 50

Spectral density (in relative units) of the surface wave source (Eq.

In this study, we consider AGW modes propagating along the eastward

For parameters of the smoothing factor (Eq.

Figure 2 shows time variations in the standard deviation of wave vertical velocity

Time variations in standard deviations of the wave vertical velocity at different altitudes (marked with numbers) for the steep activation and deactivation of the surface wave source (Eq.

One can see similar increases in

The main goal of this study is the analysis of wave fields remaining after deactivating the surface wave source (Eq.

AGW decay times

For the steep deactivation of the low-amplitude wave source shown in Fig. 2, the decay times in Table 1 are

Contributions may also occur from slow components of the wave source spectrum (see Fig. 1), which can dominate after the recession of faster primary spectral modes. In addition, slow short-wave secondary AGW modes can be produced by nonlinear wave interactions at all stages of high-resolution simulations. The mentioned residual and secondary wave modes can slowly travel to higher atmospheric levels and dissipate there due to increased molecular and turbulent viscosity and heat conductivity, which are small in the lower and middle atmosphere. Therefore, decaying these residual and secondary AGW modes may require substantial time intervals after deactivating wave forcing, as one can see in Fig. 2.

Figure 3 shows the same standard deviation of wave vertical velocity

Same as Fig. 2, but for the sharp wave source activation.

The top right panel of Fig. 3 shows substantial AGW pulses not only at the wave source activation

Figures 2 and 3 represent results for the wave source (Eq.

Ratios

Relative contributions of residual and secondary AGWs can be estimated by the ratio

The simulations described above were made for small-amplitude wave sources (Eq.

Same as Fig. 2, but for the surface wave source (Eq.

Below an altitude of 100

Peculiarities of Fig. 4 for large

Time variations in the wave-induced mean horizontal velocity at different altitudes (marked with numbers) for the steep activations and deactivations of the surface wave source (Eq.

Generation of the wave-induced jet streams was simulated and considered in more detail in our previous papers (Gavrilov and Kshevetskii, 2015;
Gavrilov et al., 2018). Larsen (2000) and Larsen et al. (2005) found frequent high horizontal wind velocities at altitudes near 100

Table 2 represents the ratio

AGW decay times in Table 1 for

To analyze changes in the spatial structure of simulated AGW fields, Figs. 6 and 7 present cross-sections of the field of wave vertical velocity by
the XOZ vertical plane at different time moments during activations and deactivations of the surface wave sources (Eq.

AGW vertical velocity fields at times

Same as Fig. 6, but for time moments after the wave source steep deactivating:

Cross-sections shown in Fig. 7 correspond to time moments after the wave source (Eq.

Figure 7b and c show increasing numbers of small-scale structures, which can be formed by slow short-wave residual wave modes, which appear due to broad wave source spectra in Fig. 1 and due to the generation of secondary waves by nonlinear interactions of primary AGW modes.

The timescale of AGW dissipation in the turbulent atmosphere can be estimated as follows (Gossard and Hooke, 1975):

AGWs propagating in the atmosphere with vertical gradients of the background fields are subject to partial reflections. In particular, strong wave
reflections occur at altitudes of 110–150

The standing AGWs discussed above are composed of the primary wave modes traveling upwards from the surface wave sources
(Eq.

The panels of Fig. 2 for the steep wave source activation and deactivation demonstrate periodical increases and decreases in the residual wave noise standard
deviations (especially at low altitudes), which are superimposed on the exponential decay at

One can raise the question as to the extent to which the results shown in Tables 1 and 2 may depend on so-called “numerical viscosity” caused by mathematical algorithms used in the model. Our model is based on special numerical algorithms accounting for the main conservation laws (Gavrilov and Kshevetskii, 2013, 2014). Therefore, the numerical viscosity is very small. Test simulations showed that in the absence of physical dissipation, wave modes might exist in the model for hundreds of wave periods without noticeable decreases in their amplitudes. In addition, simulated ratios of standard deviations of different components of long-wave fields in the middle atmosphere follow to the polarization relations of conventional theory of nondissipative AGWs (Gavrilov et al., 2015). Therefore, we assume that in the present model, the numerical viscosity is much smaller than molecular and turbulent viscosity and heat conduction, which are involved in the model at all altitudes.

The ratios

Comparisons of the right columns in Table 2 with the same

For the sharp activations and deactivations of the wave sources (Eq.

Substantial numbers of small-scale structures in Fig. 7b and c show increased proportions of wave modes, produced due to high-frequency tails of the
wave forcing spectra in Fig. 1, also due to multiple reflections and nonlinear interactions of these modes. Nonlinear AGW interactions and generations
of secondary waves should be stronger at high altitudes due to increased wave amplitudes (Vadas and Liu, 2013; Gavrilov at al., 2015). Then the
secondary waves can propagate downwards and make small-scale wave perturbations at all atmospheric altitudes (see Figs. 6 and 7). The AGW decay times

Simulations presented in this paper are made for horizontally uniform wave excitation at the ground described by Eq. (

The simulations described above were made for single relatively long AGW spectral components which experience little dissipation in the stratosphere and mesosphere. Real wave fields in the atmosphere are superpositions of a wide range of spectral components generated by a variety of different wave sources. However, after deactivations of wave sources, fast-traveling spectral components disperse to higher altitudes, and short-wave modes are strongly dissipated due to turbulent and molecular viscosity and heat conductivity. Therefore, one may expect that at the final stage of wave disappearance after deactivations of wave forcing, wave fields in the stratosphere and mesosphere should consist of vertically standing, relatively long spectral components, similar to those considered in the present study. These wave fields may contain substantial proportions of residual and secondary wave modes produced by multiple reflections and nonlinear interactions. Such an impression is probably true for the residual wave noise, which may exist for a long time after the wave source deactivation. However, amplitudes of this residual noise become smaller in time, and near active wave sources, amplitudes of generated primary AGWs may far exceed the wave noise.

In this paper, we analyze idealistic cases of long-lived horizontally homogeneous coherent wave sources producing quasi-stationary wave fields in the atmosphere. Such modeling is useful for comparisons of simulated results with standard AGW theories. However, many AGW sources in the atmosphere are local and operate for a short time, which is not enough for developments of steady-state wave fields. Further simulations are required for studying wave decay processes after deactivating such local short-lived wave sources in the atmosphere.

In this study, the high-resolution numerical model AtmoSym is applied for simulating non-stationary, nonlinear AGWs propagating from surface wave
sources to higher atmospheric altitudes. After activating the surface wave forcing and fading away initial wave pulses, AGW amplitudes reach a
quasi-stationary state. Then the surface wave forcing is deactivated in the numerical model, and amplitudes of primary traveling AGW modes quickly
decrease at all altitudes due to discontinuation of wave energy generation by the surface wave sources. However, later the standard deviation of the
residual and secondary wave perturbations produced by slow components of the wave source spectrum, multiple reflections and nonlinear interactions
experiences slower exponential decreases. The decay time of the residual AGW noise may vary between 20 and 100

The high-resolution model of nonlinear AGWs in the atmosphere used is available for online simulations (

SPK participated in the computer code development. AVK prepared background fields for the simulations. NMG made simulations and prepared the initial text of paper, which was edited by all authors.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

AGW numerical simulations were made in the SPbU Laboratory of the Ozone Layer and Upper Atmosphere supported by the Ministry of Science and High Education of the Russian Federation (agreement 075-15-2021-583).

This research has been supported by the Ministry of Science and Higher Education of the Russian Federation (grant agreement 075-15-2021-583) and Russian Science Foundation (grant no. 20-77-10006). Publisher's note: the article processing charges for this publication were not paid by a Russian or Belarusian institution.

This paper was edited by William Ward and reviewed by two anonymous referees.