The stratosphere is part of the Earth's atmosphere, embedded between the troposphere and the mesosphere at an altitude of about 10 to 55 km. It is a stably stratified medium, which enables the propagation of acoustic–gravity waves. Its temperature varies from about 220 K at the lower boundary to about 270 K at the upper boundary. The temperature rises because solar energy is converted into kinetic energy when ozone molecules absorb ultraviolet (UV) radiation, leading to a warming of the stratosphere. The warming of the stratosphere can occur through another mechanism known as sudden stratospheric warming (SSW). This is rapid warming with a temperature increase of several tens of degrees in just a few days .

SSWs are caused by the breaking of planetary-scale (Rossby) waves and gravity waves that propagate upward from the troposphere . The rapid warming and descent of the polar air affect tropospheric weather, shifting jet streams, storm tracks, and the Northern Annular Mode, making cold-air outbreaks over North America and Eurasia more likely . This phenomenon mainly occurs in winter and spring, about six times per decade . SSW events can be divided into major and minor events based on their warming intensity, according to whether an event causes the polar circulation to reverse. Warmings are commonly classified as minor when the zonal-mean 10 hPa meridional temperature gradient between 60 and 90° N reverses and as major when in addition the zonal-mean 10 hPa zonal wind at 60° N reverses . SSWs affect the atmosphere above and below the stratosphere, producing widespread effects on atmospheric chemistry, temperatures, winds, neutral (non-ionized) particles, and electron densities . Therefore, SSWs are the most prominent manifestation of connections between the lower, middle, and upper atmosphere, and a proper and detailed study of such events is important for understanding the interactions between different atmospheric layers . SSWs influence the global meridional residual circulation, and meridional coupling between different latitudes is observed. For example, SSWs influence mesospheric temperatures in the tropics , and they likely also have an effect on the opposite hemisphere .

In this article, the focus is on atmospheric gravity waves (GWs), which are part of acoustic–gravity wave spectra. Namely, it is known that acoustic waves, unlike GWs, are strongly absorbed in the atmosphere . The rate of absorption is proportional to the wave frequency squared. Gravity waves exist over a wide range of horizontal scales and typically have timescales short enough to ignore rotation, heat transfer, and friction . They are usually categorized by their source of origin, which can be orography or synoptic systems such as convection , jets, or fronts . These waves typically propagate from the troposphere through the stratosphere into the mesosphere. With exponential amplitude growth, the gravity waves will have grown so large that they become unstable and break, thereby altering the atmospheric flow by depositing stored momentum and energy . Depending on the phase speed of the waves and the velocity of the background wind, one can define a critical layer where the intrinsic frequency of the waves would approach the inertial frequency and the vertical wavelength would approach zero . If such a critical layer is present, gravity waves will break somewhere below that level and deposit more momentum already in the stratosphere. Dissipating and breaking GWs decelerate the background wind as the momentum forcing and influence planetary waves by either changing the wave guide or generating in situ planetary waves through baroclinic instabilities .

Before the SSW, the stratospheric zonal-mean winds are eastward. They filter out a significant portion of the eastward-directed GWs, favouring the upward propagation of harmonics with phase velocities directed westward. During SSW, the deceleration of the westerly jet in the stratosphere allows more propagation of GWs with eastward phase speeds into the mesosphere, and the resultant eastward gravity wave drag (GWD) induces equatorward mass flow, resulting in the upward motion and adiabatic cooling in the polar mesosphere . The unusually low temperatures at the altitude of the conventional undisturbed polar winter stratopause were linked to this reduced GWD and associated weakening of the descending branch of the mesospheric residual circulation, which normally warms the winter polar stratopause . Polar cap temperatures from the Aura microwave limb sounder (MLS) averaged north of 60° N show a joint occurrence of a warm stratosphere and a cold mesosphere in 71 % of major warmings in 2004–2015 . In their study, analysed 40 years of long-term ERA5 output in order to study the general trends in GW variations before, during, and after the SSW. Their results indicate that although the main driver of SSWs is planetary waves, GWs can contribute to the occurrences and strength of SSWs.

In this article, the impact of stratospheric temperature change on GW characteristics is studied. We analysed the upward propagation of GWs through the Earth's atmosphere, modelled by two different temperature layers separated by a horizontal plane boundary. The analytical equation for the reflection coefficient is derived and applied to the troposphere–stratosphere and stratosphere–mesosphere discontinuities under normal atmospheric conditions and during an SSW event. Two important points can be distinguished: the first is that GWs coming from the troposphere into the stratosphere participate in the generation of SSWs, and the second is that GWs generated in the stratosphere during SSWs also participate in the mesospheric dynamics.

The standard set of hydrodynamic (HD) equations describes the dynamics of adiabatic processes in the neutral atmosphere stratified by the presence of gravity with constant acceleration g = 9.81 ms-2. 1The continuity and ideal gas equation can be written as∂ρ∂t+∇⋅(ρv)=0,p=ρRT,2the momentum equation can be written asρ∂v∂t+v⋅∇v=-∇p+ρg,3and an adiabatic law for a perfect gas can be written as∂p∂t+v⋅∇p=γpρ∂ρ∂t+v⋅∇ρ. Here, R=R0/M is the individual gas constant for molecules with molar mass M, R0 = 8.314 Jmol-1K-1 is the universal gas constant, and γ=cp/cv=(j+2)/j is the ratio of specific heats for gas particles with j=5 degrees of freedom. The physical quantities ρ, p, T, and v have the usual meanings: gas density, pressure, temperature, and velocity.

The considered basic state in the stratified atmosphere is composed of two half spaces with constant sound speeds, separated by a horizontal plane boundary z=0. The two regions are characterized by the neutral atmosphere densities ρ01 and ρ02 adjacent to the lower and upper sides of the boundary z=0. The unperturbed density profile can be expressed as follows: 12ρ0(z)=ρ01e-z/H1,z<0,region (1),ρ0(z)=ρ02e-z/H2,z>0,region (2), where H(n)=vsn2/γg and n=1,2. There is a density, pressure, and temperature jump across z=0. The boundary condition that has to be applied at z=0 in the basic state is the continuity of the unperturbed pressure p0 at z=0 , which yields 13ρ02ρ01=vs12vs22=T1T2=s=const.

The boundary conditions for perturbations are continuity of both the vertical fluid displacement ξz′ and the pressure perturbation p′-gρ0(z)ξz′ at the boundary z=0. Moreover, the energy density of the perturbations has to diminish to zero as |z| tends to infinity.

The harmonic wave, which propagates through regions (1) and (2), does not change its frequency and the horizontal wave vector component Kp, parallel to the boundary z=0. However, the vertical wave vector component Kz has a discontinuity at the boundary z=0, where it changes from Kz1 to Kz2 according to the dispersion equation (Eq. ). We assume that a wave propagates from the lower region (1) upward toward the boundary z=0 and that the waves continuing past it are absorbed with no reflection in the upper region (2). In this case, in the lower region, the perturbations are the superposition of the incident and reflected waves, while in the upper region, there is only the transmitted wave. The reflection coefficient of AGWs is defined as the square of the absolute value of the reflection amplitude. Using dimensionless physical values for brevity, the reflection coefficient can be written as (see details in ) 14R=1-γ21Vh2-1-s2sVh2-1+(s-1)Vh22+γ2Ω2Vv12Vv12Vv22⋅s2(sVh2-1)2-1(Vh2-1)21-γ21Vh2-1-s2sVh2-1+(s-1)Vh22+γ2Ω2Vv12Vv1Vv2⋅ssVh2-1+1Vh2-122+2γΩVv1(Vh2-1)1-γ21Vh2-1-s2sVh2-1+(s-1)Vh21-γ21Vh2-1-s2sVh2-1+(s-1)Vh22+γ2Ω2Vv12Vv1Vv2⋅ssVh2-1+1Vh2-122. Here, Vv1 and Vv2 are the vertical phase velocities of AGWs in regions (1) and (2), respectively, given by the following equations: 15Vv1=ΩKz1=VhΩVh2Ω2-Ωco2-Ω2-ΩBV2 and 16Vv2=ΩKz2=VhΩsVh2Ω2-sΩco2-Ω2-sΩBV2, while Vh is the horizontal phase velocity given by Eq. (). If Vv12 and Vv22 are positive, AGWs propagate through regions (1) and (2), respectively. If Vv12, Vv22<0, these waves are evanescent and not of interest to this study.

The analytical equations derived in Sects.  and  are used to analyse the propagation of GWs and their reflection/transmission properties at the troposphere–stratosphere and stratosphere–mesosphere discontinuities. Gravity waves can reach the stratosphere from below, but they can also be excited in the stratosphere during a minor SSW . This source mechanism to generate GWs is known as spontaneous adjustment . Excited in situ within the stratosphere, GWs can propagate upward toward the mesosphere.

In the stratosphere, at an altitude of about 35 km, a temperature is T = 240 K and γ=1.4, sound velocity is vs=γRT = 310 ms-1, and scale height is H = 7000 m. The Brunt–Väisälä frequency is ωBV=γ-1g/vs = 0.02 s-1. During SSW, the temperature in the stratosphere can rise by more than 25 K, i.e. T = 265 K. Sound velocity is now vs = 326 ms-1, scale height is H = 7738 m, and the Brunt–Väisälä frequency is lower than before SSW, i.e. ωBV = 0.019 s-1.

SSWs trigger a chain of events that lead to anomalies in the stratosphere and thus to anomalies in the adjacent layers – the troposphere and mesosphere. Stratospheric anomalies are caused mainly by wave forcing from the dense troposphere. Two types of waves that play an important role in the stratospheric variability are gravity waves and planetary (Rossby) waves. Gravity waves considered in this article exist in a stably stratified atmosphere. Their characteristics and reflection/transmission properties in the Earth's and solar atmosphere are described in the scientific literature . Gravity waves that propagate from the troposphere to the stratosphere affect the generation of SSW . The reflection coefficient shown in Fig.  indicates that GWs with a small frequency of Ω<0.2, i.e. ω < 0.009 s-1, about 2 times smaller than the Brunt–Väisälä frequency ωBV = 0.021 s-1, can penetrate the stratosphere and influence its dynamics. found that these GWs can contribute to the occurrences of SSWs up to 30 %. This result is confirmed in the works of and . During SSW, the temperature in the stratosphere increases by several tens of degrees. Figure  shows that SSW events prevent GW propagation from the troposphere toward the stratosphere, which is consistent with known scientific results . Gravity waves with the reflection coefficient R<0.4 have a frequency of Ω<0.1, i.e. ω < 0.005 s-1, and a horizontal phase velocity of 0.3<Vh<0.9 or 90ms-1<vh<267ms-1. These waves are the best candidates for the transition from the troposphere to the stratosphere. Note that the frequency range for GW transmission is reduced from ω < 0.009 s-1 in the no-SSW case to ω < 0.005 s-1 in the SSW case. This means that the frequency band for GW transmission from the troposphere to the stratosphere is narrower. The same conclusion can be drawn for the horizontal phase velocity since its value in the no-SSW case is 29.7 ms-1 < vh < 267 ms-1, while in the SSW case its value is 90 ms-1 < vh < 267 ms-1.

The inhibition of GWs propagating upward from the troposphere to the stratosphere (Fig. ) and the causal absence of gravity wave breaking in the mesosphere explain the mesospheric cooling during an SSW event . Moreover, the mesospheric wind changes are related to the ways that the stratosphere influences the filtering of GWs . Therefore, the state of the stratosphere is important for the propagation of GWs in the upper atmosphere. It varies when the SSW starts. While an increase in the reflection coefficient at the troposphere–stratosphere discontinuity was expected, Figs.  and  show the decrease in the reflection coefficient for GWs at the stratosphere–mesosphere discontinuity, which requires an explanation. We believe that the generation of GWs in the stratosphere, in situ, during SSW increases the possibility that these waves penetrate the mesosphere. This could be the reason for the lower reflection coefficient compared to the case without SSW. Although GWs generated in the stratosphere contribute to the dynamics and temperature of the mesosphere, they cannot compensate for the strong reflection of the GWs generated in the troposphere at the troposphere–stratosphere discontinuity; see Fig. . The result is a detected mesospheric cooling. This cooling is the strongest for the GWs with Ω>0.2 or ω > 0.008 s-1 and with Vh<0.1 or vh < 32.6 ms-1 because these waves have the lowest chance of crossing the stratosphere–mesosphere discontinuity and entering the mesosphere; see Fig. . This is in agreement with the strongest mesospheric cooling found in .

The stratopause is the boundary between the stratosphere and the mesosphere at an altitude of about 55 km . It is characterized by a reversal of the atmospheric lapse rate . The beginning of the SSW is characterized by the rapid descent of the stratopause and surrounding warm layer into the stratosphere, associated with warming that is characteristic of SSW. The stratopause reaches its lowest altitude at around 30 km . Above the descended stratopause, the atmosphere experiences a dramatic cooling of about 30 K at an altitude of 50 km, parallel to stratospheric warming . In this article, the stratopause is assumed to be a plane boundary between the stratosphere and the mesosphere. Its altitude is not relevant for the results obtained in the analysis, since the results depend only on the temperature ratio, i.e. the values of the parameter s. These values are computed assuming a temperature increase of 25 K in the middle stratosphere at an altitude of about 35 km and a temperature decrease of 25 K in the lower mesosphere at an altitude of about 50 km. This is in accordance with the aforementioned scientific literature.

Disruption of the polar vortex during the SSW events allows cold air to descend from the stratosphere to the troposphere and moves it from the pole to the mid-latitudes. These changes affect the climate and may lead to a dramatic decrease in temperature in northern Europe . This confirms the existence of the two-way stratospheric–tropospheric dynamical coupling . In addition, SSW-induced temperature changes can modify chemical reaction rates, which is particularly important for upper-stratospheric ozone .

Changes in the stratosphere are also caused by solar activity. Namely, in the Earth's atmosphere, solar spectral irradiance (SSI) forcing plays a key role as the main driver in the so-called top-down mechanism . This mechanism originates in the stratosphere, where UV radiation modulates local radiative heating at the tropical stratopause and ozone chemistry. In addition, the SSI directly impacts the UV photolysis of O₂, an important source of ozone in the stratosphere. The potential drop/rise in the solar UV activity can substantially affect the ozone layer, which in turn affects stratospheric temperature, circulation, tropospheric climate, and the UV intensity reaching the ground . In the upper stratosphere, satellite observations show an increase in temperature of 1–2 K from the solar minimum to solar maximum activity during the 11-year solar cycle . Note that this temperature increase is much smaller than the temperature increase of several tens of kelvins during SSW that occurs in a few days. Therefore, the analysis presented in this article is not applicable to stratospheric temperature changes during the 11-year solar cycle.

Disturbances in the stratosphere and changes in GW propagation during SSW events affect the electron concentration in the lower ionosphere. Namely, in the presence of GWs, the electron concentration becomes time dependent, and this influences the reflection of very low frequency waves (VLFs), as studied in and , with consequences for telecommunications and navigation. It appears that the SSWs can be considered within the framework of the atmosphere–ionosphere system .

SSWs have a long-lasting effect within the stratosphere, as well as an impact on the adjacent troposphere and mesosphere. SSWs impact the tropospheric circulation, confirming the existence of the stratospheric–tropospheric dynamical coupling . During the SSW events, the reflection coefficient for GWs at the troposphere–stratosphere discontinuity increases significantly (Fig. ). This filtration of GWs has a major impact on mesospheric dynamics because generation of GWs in the stratosphere during SSW cannot compensate for the reduction in the GWs from the troposphere. Therefore, during SSW we have the following two accompanied processes – stratospheric warming and mesospheric cooling. Gravity waves are the coupling mechanism between these two processes. We used HD equations and temperature as the main parameters to derive the dispersion equation for GWs and their reflection coefficient. An increase in the reflection coefficient at the troposphere–stratosphere discontinuity, i.e. an increase in downward GW fluxes, can be used to predict SSW events, as done in . Detailed knowledge of how stratospheric anomalies influence tropospheric weather will open the door to improved climate models and forecasts. The effects of SSWs on the upper atmosphere will enable scientists to improve space weather forecasting and especially to determine day-to-day variability in the ionosphere . The physical processes that contribute to the variability of the Earth's atmospheric layers also operate in other planetary atmospheres and define their dynamics and energy budgets. Therefore, the information obtained from this study about the coupling between Earth's atmospheric layers may be applicable to the atmospheres of other planets.