Decay times of atmospheric acoustic-gravity waves after deactivation of wave forcing

. 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/deactivation times of 10 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 slower quasi-exponential decrease. High-resolution simulations allowed, for the first time, estimating the decay times of this wave noise 15 produced by slow residual, quasi-standing and secondary AGW spectral components, which vary between 20 and 100 hrs depending on altitude and the rate of wave source activation/deactivation. The standard deviations of the wave noise are larger for the case of sharp activation/deactivation of the wave forcing compared to the steep processes. These results show that transient wave sources may create long-lived wave perturbations, which can form a background level of wave noise in the atmosphere. This should be taken into account in parameterizations of atmospheric AGW impacts.

from the surface sources. We found, however, that after some time, the standard deviation of residual quasi-standing and secondary wave perturbations experiences more slow exponential decrease with substantial decay times. 65 These results show that residual and secondary AGW modes produced by transient wave sources can exist for 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.

Numerical model
In this study, we employed the high-resolution three-dimentional numerical model of nonlinear AGWs in the atmosphere 70 developed by Gavrilov and Kshevetskii (2014). Currently, this model (called as 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): 75 where t is time; p, ρ, T are pressure, density and temperature, respectively; v β are velocity components along the coordinate axes x β ; σ iβ is the viscous stress tensor; g is the acceleration due to gravity; c p is the specific heat capacity at constant pressure; R is the atmospheric gas constant; ε is the specific heating rate; d/dt = ∂/∂t+v β ∂/∂x β ; repeating Greek indexes assume summation. Quantities σ iβ and ε in Eq. (1) contain stresses and heating rates produced by molecular viscosity and heat conductivity (see details in Gavrilov and Kshevetskii, 2014). After numerical integration of Eq. (1), dynamical 80 deviations (marked with primes below) from stationary background values p 0 , ρ 0 , T 0 and v i0 are calculated: ′ . (2) The AtmoSym model takes into account dissipative and nonlinear processes accompanied AGW propagation. The model is capable to simulate such complicated processes as AGW instability, breaking and turbulence generation. Dynamical deviations (2) describe both wave perturbations and modifications of background fields due to momentum and energy 85 exchange between dissipating AGWs and the atmosphere. The background temperature T 0 (z) is obtained from the semiempirical NRLMSISE-00 atmospheric model (Picone et al., 2001). Background dynamic molecular viscosity, µ 0 , and heat conductivity, κ 0 , are estimated using the Sutherland's formulae (Kikoin, 1976): where γ is the heat capacity ratio, Pr is the Prandtl number. The AtmoSym model involves also the mean turbulent thermal 90 conductivity and viscosity having maxima about 10 m 2 s −1 in the boundary layer and in the lower thermosphere, and a broad minimum up to 0.1 m 2 s -1 in the stratosphere (Gavrilov and Kshevetskii, 2014). The upper boundary conditions at z = h have the following form (Kurdyaeva et al., 2018): where indices 1 and 2 correspond to horizontal directions, w = v 3 is vertical velocity. Conditions (4) may cause reflections of 95 AGWs coming from below. The upper boundary at the present study is set at h = 600 km, where molecular viscosity and heat conductivity are very high and reflected waves are strongly dissipated. Sensitivity tests reveal that the impact of conditions at the upper boundary (4) is negligible at altitudes z < h -2H, where H is the atmospheric scale height. Therefore, at altitudes of the middle atmosphere analyzed in this paper, the influence of the upper boundary conditions (4) could be negligible. The lower boundary conditions at the Earth's surface have the following form (see Kurdyaeva et al., 2018): 100 where W 0 and σ have sense of the amplitude and frequency, is the position vector in the horizontal plane, k 1 and k 2 are the wavenumbers along horizontal axes x 1 and x 2 , respectively. The last relation for the surface vertical velocity in (5) serves as the source of plane AGW modes in the AtmoSym model. Such plane modes can represent spectral components of tropospheric dynamical processes. Their effects can be approximated by 105 appropriate sets of effective spectral components of vertical velocity at the lower boundary (Townsend, 1965(Townsend, , 1966. Along horizontal axes x 1 and x 2 , one can assume periodicity of wave fields where F denotes any of simulated hydrodynamic quantities, L 1 = n 1 λ 1 and L 2 = n 2 λ 2 are horizontal dimensions of the analyzed atmospheric region; λ 1 = 2π/k 1 и λ 2 = 2π/k 2 are wavelengths along axes x 1 and x 2 , respectively; n 1 and n 2 are 110 integers. In our simulations, the wave excitation (5) is activated at the moment t = t a and then its amplitude W 0 does not change for some time. One should expect that at small amplitudes of wave source (5), the numerical solutions in the lower and middle atmosphere should tend at to a steady-state plane AGW modes corresponding to the traditional linear theory (e.g., Gossard and Hooke, 1975). Gavrilov and Kshevetskii (2015) showed good agreement of ratios of simulated amplitudes of 115 different wave fields with polarization relations of linear AGW theory (Gossard and Hooke, 1975) at at altitudes up to 100 km.
The novelty of the present study is deactivating the wave source (5) at some moment t = t d after reaching the described  (5) could create an initial AGW pulse, which can reach high altitudes in a few minutes. To control the rate of the 120 wave source activation/deactivation, in the present simulations, we multiply the surface vertical velocity in (5) by a function where s a and s d are constants.

Results of numerical simulations
Our numerical modeling begins from steady state windless non-perturbed atmosphere with profiles of background 125 temperature, density, molecular weight and molecular kinematic viscosity corresponding to January at latitude 50° N at medium solar activity according to the NRLMSISE00 model (Picone et al., 2001), which are presented in Figure 1 of the paper by Gavrilov et al. (2018).
In this study, we consider AGW modes propagating along the eastward axis x and use horizontal dimension of considered atmospheric region to be equal to the circle of latitude at 50°N, which is L x ≈ 27000 km. At horizontal boundaries of this 130 circle of latitude, we use periodical boundary conditions (6). Representing the circle of latitude by a rectangle area assumes fixed L x at all altitudes, while in spherical coordinates L x is increasing in altitude. However, the differences in L x at altitudes of the middle atmosphere do not exceed 2%. Modeling was performed with the surface wave source (5) for AGW modes having amplitudes W 0 = 0.01 -0.1 mm/s and horizontal phase speeds c x ~ 50 -200 m/s. The number of wave periods along the circle of latitude is taken to be n 1 = 32. This corresponds the horizontal wavelength λ x = L x /n 1 ≈ 844 km and AGW 135 periods τ = λ x /c x ~ 4.7 -1.2 hr for the specified above range of c x values. The horizontal grid spacing of the numerical model is ∆x = λ x /16 and the time step of calculations was automatically adjusted to ∆t ≈ 2.9 s. The vertical grid of the model covers altitudes up to h = 600 km and contains 1024 non-equidistant nodes. Vertical spacing varies between 12 m and 3 km from the lower to the upper boundary, so about 70% grid nodes are located in the lower and middle atmosphere.
For parameters of the smoothing factor (7) in the present simulations, we take t a = 10 5 s ≈ 28 hr and t d = 4×10 5 s ≈ 110 hr, 140 and consider steep AGW source activating and deactivating with s a = s d = 3.3×10 4 s ≈ 9 hr and sharp triggering at s a = s d = 0.3 s. The shape of smoothing factor (7) influences the spectrum of the surface wave source in the model. Figure 1 shows spectra of the sinusoidal source (5) with wave period τ = 2π/ σ = 2 hr, which were calculated using 20-hour running time intervals corresponding to the phases of activation, activated state and deactivation of the wave source (5) with the mentioned above "steep" and "sharp" values of s a and s d in Eq. (7). Comparisons of solid and dashed lines in Figure 1 show 145 that the sharp activation and deactivation of the wave source decreases the spectral density at frequency of the main spectral https://doi.org/10.5194/acp-2021-824 Preprint. Discussion started: 26 October 2021 c Author(s) 2021. CC BY 4.0 License. maximum. However, the sharp triggering considerably increases the high-frequency part of the wave spectra in Figure 1, which means larger proportions of acoustic waves generated by quickly varying wave sources in the atmosphere.  (7).  Figure 2 show moments t = t a ≈ 28 hr and t = t d ≈ 110 hr of the surface wave source activation and deactivation in (7). The bottom left panel of Figure 2 for the Earth's surface shows that the wave source amplitude increases steeply at t < t a , maintains constant at t a < t < t d and steeply decreases to zero at t > t d in accordance with (7). 160 Similar increases in δw during the activation interval t < t a one can see at all altitudes in Figure 2. At altitudes higher than 60 km noisy components are noticeable in Figure 2 at t < t a , which can be produced by acoustic components of the wave source spectrum shown in Figure 1a. However for the steep smoothing factor q(t) in (7) this acoustic noise is substantially smaller than the wave amplitudes at t > t a at all altitudes. In Figure 2, one can see later transition to a quasi-stationary wave regime with steady amplitudes at higher altitudes compared to that at the Earth's surface. This reflects a time delay τ e ~ z/c z 170 required for the main modes of internal gravity waves (IGWs) to propagate from the surface to altitude z with the mean vertical group velocity c z ~ λ z /τ, where λ z and τ are the mean vertical wavelength and wave period, respectively. For the shown in Figure 2 wave excitation (5) with λ x = 844 km and c x = 50 m/s, using the traditional theory of AGWs (e.g., Gossard and Hook, 1975), one can estimate τ ~ 4.7 hr, λ z ~ 15 km and t e ~ (6.7 -13.3)τ ~ 31 -62 hr for z = 100 -200 km. This corresponds to the time delays between the moments t a and achieving quasi-stationary amplitudes at different altitudes in 175

Steep wave source triggering
The main goal of this study is the analysis of wave fields after deactivating of the surface wave source (5), which was made applying (7) with t d ≈ 110 hr and s d ≈ 9 hr for the steep triggering shown in Figure 2. One can see that after the wave source deactivation, AGW amplitudes start to decrease from their quasi-stationary values at all altitudes with time delays t e discussed above. Just after the wave forcing deactivating, δw decreases relatively fast similarly to the decrease in the wave 180 source amplitude in the bottom left panel of Figure 2. This may reflect disappearing of fast traveling AGW modes due to discontinuities of their generation after the wave forcing deactivation. However, later, at t > 170 hr all panels in Figure 2 demonstrate slower δw decreases, which can be approximated by exponential curves δw ~ exp(-t/τ 0 ) with the decay times τ 0 presented in Table 1for  For the steep deactivation of the low-amplitude wave source shown in Figure 2, the decay times in Table 1 are τ 0 ~ 17 -98 hr depending on altitude, which is much larger than the time scale of the steep deactivation s d ≈ 9 hr. Such slow decay 190 rates may be caused by partial reflections of the wave energy making vertically quasi-standing AGW modes (see section 4). Contributions may also occur from slow components of wave source spectrum (see Figure 1), which can dominate after the recession of faster primary spectral modes. In addition, slow shortwave AGW modes can be generated by nonlinear wave interactions at all stages of high-resolution simulations. Mentioned quasi-standing, residual and secondary wave modes can slowly travel to higher atmospheric levels and dissipate there due to increased molecular and turbulent viscosity and heat 195 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 Figure 2. The right top panel of Figure 3 shows substantial AGW pulses not only at the wave source activation t a , but also at the moment of wave source sharp deactivation t d , when δw values have additional maxima at high altitudes. Stronger AGW 210 pulses caused by the sharp wave source activation/deactivation increase proportions of slow quasi-standing, residual and secondary wave components after turning off the wave forcing in the AtmoSym model. Therefore exponential decays of δw start earlier and are more pronounced in Figure 3 than those in the respective panels of Figure 2. AGW decay times τ 0 corresponding to the exponential approximations in Figure 3 for the sharp wave source activation are given in the left column of Table 1 (5) with c x = 50 m/s. Table 1 contains also the decay times for the wave excitation with c x = 100 m/s. Respective primary AGWs have larger vertical wavelengths and should experience smaller molecular and turbulent dissipation in the atmosphere. For the steep activation/deactivation of the wave source (5)  220 with small amplitude W 0 = 0.01 mm/s, Table 1 reveals larger values of τ 0 for AGWs with c x = 100 m/s compared to those with c x = 50 m/s. Therefore, smaller dissipation of the faster AGW modes corresponds to longer time for their decay, especially at altitudes below 100 km. For the sharp activation/deactivation of the wave source at W 0 = 0.01 mm/s, the left columns of Table 1 shows approximately equal τ 0 values for waves with c x = 50 m/s and c x = 100 m/s. Relative contributions of residual and secondary AGWs can be estimated by the ratio δw/W 0 at the beginning of the 225 exponential tails in Figures 2 and 3 at t = 170 hr, which is presented in Table 2. For the steep wave source activation/deactivation at s a = s d ≈ 9 hr in (7) and W 0 = 0.01 mm/s in (5), Table 2 shows smaller ratios of the residual wave noise at altitudes below 200 km for the wave forcing with c x = 100 m/s compared to c x = 50 m/s. At the sharp wave source activation/deactivation at s a = s d ≈ 0.3 s in (7), the ratios of residual waves are larger at all altitudes compared to the steep case in Table 2. For the wave forcing (5) with c x = 100 m/s, the ratios are comparable or smaller at altitudes below 150 km 230 and larger above 150 km compared to the case of c x = 50 m/s. Larger ratios of residual and secondary waves at sharp wave source triggering in Table 2 may explain generally larger AGW decay times τ 0 in the left columns of Table 1 for W 0 = 0.01 mm/s, as far as the dissipation of stronger wave noise may require longer time intervals.

Larger amplitude wave sources
Described above simulations were made for small-amplitude wave sources (5) Figure 4 are first decreasing relatively fast due to the discontinuing generation of primary AGW modes at the lower boundary. At t > 150 -170 hr, more slow decays of residual 245 and secondary wave modes occur at all altitudes in Figure 4 with decay times τ 0 listed in Table 1 for the steep and sharp wave source activation/deactivation.  In Figure 5, one can see substantial u 0 rises at altitudes above 100 km during the wave source operation. Rising u 0 decreases the AGW intrinsic frequency and vertical wavelength (e.g., Gossard and Hook, 1975). This may increase wave dissipation due to molecular viscosity and heat conductivity leading to the gradual decrease in AGW amplitude in the right top panel of Figure 4 in the time interval between t a and t d . The rate of u 0 weakening after the wave source deactivation decreases slowly 265 in time in the right top panel of Figure 5, so that the wave-induced horizontal winds are still substantial after hundreds wave source periods at high altitudes. An interesting feature is an increase in u 0 at t > t d in the panel of Figure 5 for z = 100 km.
This shows that residual and secondary AGWs slowly traveling upwards from below can produce substantial wave accelerations of the mean flow for long time after deactivations of the surface wave sources. Table 2 represents the ratio δw/W 0 at the moment t ≈ 170 hr for larger amplitude surface wave sources (5), which may 270 characterize a proportion of residual and secondary waves after disappearing the fast traveling modes of the wave excitation.
At steep wave source activations/deactivations with s a = s d ≈ 9 hr, Table 2 demonstrates approximately same δw/W 0 values below altitude of 100 km and generally smaller values at higher altitudes for W 0 = 0.1 mm/s as compared with W 0 = 0.01 mm/s, if one considers columns for fixed c x at different W 0 values. This may be caused by the discussed above transfer of wave energy to wave-induced jets, which can provide also larger reflections and dissipation of wave components with larger 275 amplitudes.
AGW decay times in Table 1 for W 0 = 0.1 mm/s at altitudes below 100 km are generally larger for the sharp wave source triggering (s a = s d ≈ 0.3 s) than those for the steep triggering (s a = s d ≈ 9 hr) similar to the case of smaller wave source amplitude discussed in section 3.2. At high altitudes in Table 1 for W 0 = 0.1 mm/s, wave decay times for the sharp wave source deactivating become smaller, than those for the steep triggering.  (7). Figure 6a shows that after dispersion and dissipation of the initial AGW pulse just after the wave forcing activation time, t a ≈ 28 hr, wave fronts become inclined to the horizon. This behavior is characteristic for the main IGW mode with period τ ~ 4.7 hr, which is dominating in the spectrum of the wave source having c x = 50 m/s similar to Figure 1a. In the middle and at the end of quasi-stationary intervals shown in Figures 2 -290 4, the inclined wave fronts in Figures 6b and 6c expand to the entire considered XOZ region and wave amplitudes become larger compared to Figure 6a. Cross-sections shown in Figure7 correspond to time moments after the wave source (5) deactivation at t d ≈ 110 hr. Figure   7a shows that just after turning off the wave source, the inclined fronts are destroyed, first, in the lower atmosphere. Above altitude 50 km, the wave field structure in Figure 7a is still similar to Figures 6b and 6c. Later, in Figures 7b and 7c, wave amplitudes become smaller, especially at low and high altitudes. Therefore, maximum AGW amplitudes in Figure 7c are 300 located at altitudes 80 -120 km. This explains the growing wave-induced horizontal velocity at altitude 100 km after the wave source deactivation in the respective panel of Figure 5. At heights below 50 km in Figure 7, directions of wave front inclinations to the horizon are opposite to those in Figure 6. This reveals existence of downward traveling IGW modes in the stratosphere and troposphere after deactivations of the surface wave sources. Such modes could be produces by partial reflections of primary upward traveling IGWs at higher atmospheric levels (see section 4). 305 Figures 7b and 7c show increasing amounts of small-scale structures, which can be formed by slow shortwave residual wave modes, which appear due to broad wave source spectra in Figure 1  The time scale of AGW dissipation in the turbulent atmosphere can be estimated as follows (Gossard and Hook, 1975) where K z is the total vertical coefficient of turbulent and molecular viscosity and heat conductivity. For the main primary AGW modes simulated in this study and having λ z ~ 15 -30 km (see section 3.1), τ d ~ 10 3 -10 5 hr at altitudes below 100 km. These values are much larger than the AGW decay times τ 0 in Table 1. Therefore, attenuations of primary AGW modes in the middle atmosphere shown in Figures 2 -7 after deactivations of the surface wave forcing cannot be explained by direct 315 turbulent and molecular dissipations.
AGWs propagating in the atmosphere with vertical gradients of the background fields are subjects to partial reflections. In particular, strong wave reflections occur at altitudes 110 -150 km, where large vertical gradients of the mean temperature exist (e.g., Yiğit and Medvedev, 2010;Walterscheid and Hickey, 2011;Gavrilov and Kshevetskii, 2018). Partial reflections of wave energy propagating upwards from the wave sources before their deactivations may produce vertically standing 320 waves in the middle atmosphere. Simulations by Gavrilov and Yudin (1987) showed that the standing-wave ratio for IGW amplitudes might reach 0.4 at altitudes below 100 km. After deactivations of wave sources, vertically traveling AGW modes propagate quickly upwards and dissipate at higher atmospheric altitudes. This gives fast decreases in AGW amplitudes at all heights in Figures 2 and 4 just after the wave source deactivations. After disappearing fast traveling modes, residual quasistanding AGWs produced by partial reflections may form long-lived wave structures in the atmosphere shown in Figures 2 -325 7.
The standing AGWs discussed above are composed of the primary wave modes traveling upwards from the surface wave sources (5) and downward propagating waves reflected at higher atmospheric levels. After the wave source deactivations, the reflected downward waves propagate to the Earth's surface and create wave fronts at low altitudes in Figure 7, which are inclined to the horizon in directions opposite to the fronts of primary AGWs shown in Figure 6. These downward traveling 330 waves are reflected from the ground and propagate upwards back to the middle atmosphere. Kurdyaeva et al. (2018) showed that such AGW reflections from the ground could be equivalent to additional wave forcing at the lower boundary, which is still effective after deactivations of primary surface wave sources. Upward traveling from the ground and reflected again at higher altitudes waves can maintain quasi-standing AGW structures for long time (see Figure 7). As far as wave reflections are partial, portions of wave energy can for long time propagate to higher altitudes and dissipate there. This can explain 335 relatively large AGW decay times τ 0 in the lower and middle atmosphere shown in Figures 2 -4 and in Table. 1. Even after substantial time from the wave source turning off, AGW structures in Figures 7b and 7c at altitudes above 50 km are still similar to those shown in Figure 6 during active wave forcing. Table 2 Table 1 reveal larger decay times τ 0 of waves with c x = 100 m/s due to their larger vertical wavelength and smaller dissipation in the middle atmosphere.

Shown in
Comparisons of the right columns in Table 2 with the same c x and different W 0 show that values of δw/W 0 for each c x are 345 approximately equal at altitudes below 60 km and become smaller at higher altitudes for larger amplitude wave sources. This may reflect larger transfers of AGW energy to wave-induced jet streams and to secondary nonlinear modes produced by larger-amplitude waves. Respective right columns of Table 1 show higher decay times τ 0 of larger-amplitude wave noise corresponding to W 0 = 0.1 mm/s at altitudes higher 100 km. This noise can be maintained for long time by wave energy fluxes propagating with stronger residual and secondary waves from the middle atmosphere to higher altitudes. 350 For the sharp activations/deactivations of the wave sources (5), the left columns of Table 2 show values of δw/W 0 , which are much larger compared to respective right columns for the steep wave forcing triggering. These ratios are less dependent on the speed and amplitude of simulated AGWs and could be connected with wave pulses produced by sharp activations/deactivations of the wave sources (see spectra in Figure 1). AGW decay times for the sharp triggering in respective left columns of Table 1 are also less dependent on wave parameters. 355 Substantial amounts of small-scale structures in Figures 7b and 7c shows increased proportions of wave modes, produced due to high-frequency tails of the wave forcing spectra in Figure 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 Figures 6 and 7). The AGW 360 decay times τ 0 in Table 1 are generally larger for longer AGW modes with c x = 100 m/s. This may be explained by their smaller dissipation due to turbulent and molecular viscosity and heat conductivity in the atmosphere. Due to small coefficients of turbulent dissipation in the stratosphere and mesosphere, maximum AGW decay times in Table 1 exist at altitudes 30 -100 km. Quasi-standing and secondary AGWs may exist there for several days after deactivations of the wave forcing. Wave energy can slowly penetrate upwards from the stratosphere and mesosphere and maintain a background level 365 of AGW activity at higher altitudes. Figure 7c reveals that after 10 days of simulations, largest amplitudes of the residual wave field exist at altitudes 70 -110 km. It is enough for creations of wave accelerations, which can act and modify the mean velocity at altitudes near 100 km for the long time after the wave source deactivations (see respective panels of Figure   5).
Described above simulations were made for single relatively long AGW spectral components, which experience small 370 dissipation in the stratosphere and mesosphere. Real wave fields in the atmosphere are superpositions of 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 disappearing after deactivations of wave forcing, wave fields in the stratosphere and mesosphere should consist of quasi-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.
In this paper, we analyzed 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 short time, which is not enough 380 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.

Conclusion
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 385 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 more slow exponential decreases. The decay time of the residual AGW noise may vary between 20 390 and 100 hr, having maxima in the stratosphere and mesosphere. Standard deviations of the residual AGWs in the atmosphere are much larger at sharp activations/deactivations of the wave forcing compared to the steep processes. These results show that transient wave sources in the lower atmosphere could create long-lived residual and secondary wave perturbations in the middle atmosphere, which can slowly propagate to higher altitudes and form a background level of wave noise for time intervals of several days after deactivations of wave sources. Such behavior should be taken into account in 395 parameterizations of AGW impacts in numerical models of dynamics and energy of the middle atmosphere.

Data availability.
Used high-resolution model of nonlinear AGWs in the atmosphere is available for online simulations (see the reference AtmoSym, 2017). The computer code can be also available under the request from the authors.