Articles | Volume 22, issue 16
Atmos. Chem. Phys., 22, 10425–10441, 2022
https://doi.org/10.5194/acp-22-10425-2022
Atmos. Chem. Phys., 22, 10425–10441, 2022
https://doi.org/10.5194/acp-22-10425-2022
Research article
16 Aug 2022
Research article | 16 Aug 2022

Ozone–gravity wave interaction in the upper stratosphere/lower mesosphere

Ozone–gravity wave interaction in the upper stratosphere/lower mesosphere
Axel Gabriel Axel Gabriel
  • Department Modelling of Atmospheric Processes, Leibniz-Institute of Atmospheric Physics (IAP) at the University of Rostock e.V., Kühlungsborn, Germany

Correspondence: Axel Gabriel (gabriel@iap-kborn.de)

Abstract

The increase in amplitudes of upward propagating gravity waves (GWs) with height due to decreasing density is usually described by exponential growth. Recent measurements show some evidence that the upper stratospheric/lower mesospheric gravity wave potential energy density (GWPED) increases more strongly during the daytime than during the nighttime. This paper suggests that ozone–gravity wave interaction can principally produce such a phenomenon. The coupling between ozone-photochemistry and temperature is particularly strong in the upper stratosphere where the time–mean ozone mixing ratio decreases with height. Therefore, an initial ascent (or descent) of an air parcel must lead to an increase (or decrease) in ozone and in the heating rate compared to the environment, and, hence, to an amplification of the initial wave perturbation. Standard solutions of upward propagating GWs with linear ozone–temperature coupling are formulated, suggesting amplitude amplifications at a specific level during daytime of 5 % to 15 % for low-frequency GWs (periods ≥4 h), as a function of the intrinsic frequency which decreases if ozone–temperature coupling is included. Subsequently, the cumulative amplification during the upward level-by-level propagation leads to much stronger GW amplitudes at upper mesospheric altitudes, i.e., for single low-frequency GWs, up to a factor of 1.5 to 3 in the temperature perturbations and 3 to 9 in the GWPED increasing from summer low to polar latitudes. Consequently, the mean GWPED of a representative range of mesoscale GWs (horizontal wavelengths between 200 and 1100 km, vertical wavelengths between 3 and 9 km) is stronger by a factor of 1.7 to 3.4 (2 to 50 J kg−1, or 2 % to 50 % in relation to the observed order of 100 J kg−1, assuming initial GW perturbations of 1 to 2 K in the middle stratosphere). Conclusively, the identified process might be an important component in the middle atmospheric circulation, which has not been considered up to now.

1 Introduction

Atmospheric gravity waves (GWs), with horizontal wavelengths of 100 to 2000 km, are produced in the troposphere and propagate vertically through the stratosphere and mesosphere, where gravity wave breaking processes are important drivers of the middle atmospheric circulation (e.g., Andrews et al., 1987; Fritts and Alexander, 2003). Usually, upward propagating GWs are described by sinusoidal wave perturbations in a slowly varying background flow of an exponentially growing amplitude with height due to decreasing density (ez/2H, where H is the scale height). Recently, Baumgarten et al. (2017) found some evidence that the growth of the GW amplitudes between the middle stratosphere and upper mesosphere might be stronger during the daytime than during nighttime. The aim of the present paper is to examine whether ozone–gravity wave interaction can principally produce such an amplification.

Seasonal variations of gravity wave potential energy density (GWPED) have been derived based on satellite data or lidar measurements (e.g., Geller et al., 2013; Ern et al., 2004, 2018; Kaifler et al., 2015; Baumgarten et al., 2017). At summer middle and polar latitudes, the order of the monthly mean GWPED increases from approximately 1 J kg−1 in the middle stratosphere (30–40 km) to 10 J kg−1 in the lower mesosphere (50–60 km) and 100 J kg−1 in the upper mesosphere (80–90 km), with usual initial GW perturbations in the middle stratosphere in the order of about 1 to 2 K, and wave periods primarily between 4 to 10 h (e.g., Kaifler et al., 2015; Baumgarten et al., 2017, 2018; Ern et al., 2018). Generally, the GW sources in the middle stratosphere are weaker, but the relative increase in the GWPED between the middle stratosphere and upper mesosphere are much stronger in summer than in winter, including a less pronounced seasonal cycle in the upper mesosphere than in the levels below. This is primarily due to the seasonal change in critical level filtering of the GWs by the zonal wind (e.g., Kaifler et al., 2015; Ern et al., 2018), but also due to specific GWs generated by convection and propagating towards polar latitudes (Chen et al., 2019), or to additional sources of GWs in the mesosphere, independent of the GWs at lower levels (Reichert et al., 2021). Recently, model simulations with resolved GWs suggested multistep vertical coupling processes, producing such secondary GWs as a result of dissipating primary GWs, which can strongly enhance the GW amplitudes in the upper mesosphere (e.g., Becker and Vadas, 2018; Vadas et al., 2018; Vadas and Becker, 2018). However, the potential role of daytime–nighttime differences in the increase in GW amplitudes with height have been considered only very sparsely up to now.

Baumgarten et al. (2017) derived monthly means of the GWPED from full-day lidar temperature measurements at northern mid-latitudes (54 N, 12 E), and found a stronger relative increase between 35 and 40 km and between 55 and 60 km for full-day than nighttime observations during summer months, but less pronounced differences during winter. For example, for July, the GWPED at 55–60 km show values of about 1×10-2 J m−3 (or 10 J kg−1) for full-day measurements but about 0.5×10-2 J m−3 for nighttime only (or 0.2 but 0.1 J m−3, if the measured temperature fluctuations are vertically filtered for vertical wavelengths Lm<15 km), where the GWPED at 35–40 km remains nearly unchanged, indicating a difference between full-day and nighttime values by a factor of about 2. Generally, measurements of the mesospheric GWPED are much more uncertain during summer than winter months (e.g., Kaifler et al., 2015; Ehard et al., 2015; Baumgarten et al., 2017), and the signal-to-noise ratio of the lidar measurements is not as good during the daytime than during nighttime (e.g., Rüfenacht et al., 2018), which can stimulate some doubt on the reliability of the daytime–nighttime differences derived from these specific measurements. In addition, taking the potential uncertainties of the analyzing methods into account (i.e., the temporal filtering methods used for the measured time series), Baumgarten et al. (2017) speculated that a change in the phase of long periodic waves (e.g., diurnal and semidiurnal tides) could change the filtering conditions for GWs. However, Baumgarten et al. (2017) conclusively assumed that the detected daytime–nighttime differences are of true geophysical origin, where an unequivocal explanation of this phenomenon remained open. Considering also that the full-day observations of Baumgarten et al. (2018) during May 2016 showed pronounced GW activity, particularly at altitudes between 42 and 50 km where the coupling between ozone and temperature is particularly strong, it seems to be worthwhile to examine whether ozone–gravity wave interaction could principally lead to such daytime–nighttime differences in the GW amplitudes. This must then also lead to a potential effect on the differences in the GWPED between polar day and polar night. The examination of the present paper is based on standard equations describing upward propagating GWs in a constant background flow, excluding other processes controlling the GWPED variability, to provide a clear understanding and quantification of the potential effect, which cannot be achieved based on observational data analysis or comprehensive model calculations alone.

The coupling of temperature and ozone is particularly strong in the upper stratosphere due to the short photochemical lifetime of ozone (e.g., Brasseur and Solomon, 1995). Linear relationships for a change in the heating rate due to a change in ozone, and a change in photochemistry due to a change in temperature, were derived from basic theory or satellite observations, and have been introduced in standard equations of stratospheric dynamics to examine the effects on the stratospheric circulation, planetary-scale wave patterns, and equatorial Kelvin waves (Dickinson, 1973; Douglass et al., 1985; Froidevaux et al., 1989; Cordero et al., 1998; Cordero and Nathan, 2000; Nathan and Cordero, 2007; Ward et al., 2000; Gabriel et al., 2011a). Large-scale ozone-dynamic coupling processes also show significant effects in numerical weather prediction (NWP) or general circulation models (GCMs) (Cariolle and Morcrette, 2006; Gabriel et al., 2007, 2011b; Gillet et al., 2009; Waugh et al., 2009; McCormack et al., 2011; Albers et al., 2013). However, possible effects of mesoscale ozone–gravity wave interaction in the upper stratosphere/lower mesosphere (USLM) have not been considered up to now.

The basic idea of the present paper can be summarized as follows: in the USLM, the time–mean ozone mixing ratio μ0(z) decreases with height (μ0/z<0). Therefore, at a specific level in the ULSM, an ascending air parcel initially forced by an upward propagating sinusoidal GW pattern (i.e., the wave crest with vertical velocity perturbation w>0) must lead to an increase (μ/t>0) by both transport (because -wμ0/z>0) and photochemistry (because the temperature-dependent ozone production increases in the case of adiabatic cooling), and, hence, in the heating rate (Q(μ)>0), comparable to the latent heat release in the troposphere in the case of condensation. Then, the induced perturbation (Δθ>0, where θ is potential temperature) reinforces the initial ascent, where the lapse rate ((θ0+Δθ)/z<θ0/z) decreases (z= constant), suggesting an effective ozone adiabatic lapse rate in the upper stratosphere comparable to the moist adiabatic lapse rate in the troposphere. Analogously, a descending air parcel (the wave trough where w<0) leads to a decrease (μ/t<0) and a corresponding change (Q(μ)<0), reinforcing the initial descent. Overall, this process must lead to a significant amplification of the initial GW amplitude at this level, and, hence, to a successive amplification of the amplitude during the upward level-by-level propagation through the ULSM.

In Sect. 2, standard equations for GWs in a zonal mean background flow with and without linearized ozone–temperature coupling are formulated to quantify the amplitude amplification at a specific level (or altitude) and latitude. Then, in Sect. 3, the cumulative amplitude amplification during the propagation through the USLM is derived, based on an idealized approach of the upward level-by-level propagation of GWs with specific horizontal and vertical wavelengths. Section 4 concludes with a summary and discussion.

https://acp.copernicus.org/articles/22/10425/2022/acp-22-10425-2022-f01

Figure 1(a–c) Zonal and monthly mean background, (a) temperature T0, (b) ozone mixing ratio O3 (the dashed line denotes where O3/z=0), and (c) ozone heating rate Q0, January 2001, extracted from a simulation with the circulation and chemistry model HAMMONIA; (d–f) amplification factors, (d) 1+ab and (e) N02/Nμ2 for a GW with horizontal and vertical wavelengths Lk=500 km and Lm=5 km, and (f) N02/Nμ2 for a GW with Lk=800 km and Lm=3 km; shaded areas denote the latitudes where the amplification is limited by the length of daylight (τi>τday).

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2 Ozone–gravity wave interaction

In the following section, ozone–gravity wave interaction is analyzed based on standard equations describing GWs in a background atmosphere, where the solutions are illustrated for southern summer conditions. The background is prescribed by monthly and zonal mean temperature T0, ozone μ0, and short-wave heating rate Q0 of January 2001 (Fig. 1a–c), derived from a simulation with the high-altitude general circulation and chemistry model HAMMONIA (details of the model are given by Schmidt et al., 2010). The heating rate Q0 (Fig. 1c) is primarily due to the absorption of solar radiation by ozone, and largely agrees with southern summer solar heating rates derived from satellite measurements by Gille and Lyjak (1986) but with somewhat smaller maximum values (in the order of ∼10 %). Figure 1c shows that Q0 is particularly strong in the USLM where μ0/z<0 (the dashed line in Fig. 1b indicates μ0/z=0). The HAMMONIA model includes 119 layers up to 250 km with increasing vertical resolution between ∼0.7 km in the middle stratosphere and ∼1.4 km in the middle mesosphere, with a horizontal resolution of 3.75. In the following section, this grid is used to illustrate the analytic solutions of upward-propagating GWs.

2.1 Amplification of GW amplitudes at a specific level

2.1.1 Basic equations

Following Fritts and Alexander (2003), we consider standard Eqs. (1)–(5) describing GW propagation in a background flow, with linear GW perturbations T, θ, u, v, w, p, and ρ (T is temperature; θ=T (p00/p)κ is potential temperature;, p(z) is pressure; p00=1000 hPa; z is altitude; u, v, and w are zonal, meridional, and vertical wind perturbations, respectively; and p and ρ are the perturbations in pressure and density, respectively). Additionally, we include an ozone-dependent heating rate perturbation Q(μ) in the potential temperature equation (Eq. 5) and Eq. (6) for the ozone perturbation μ with a temperature-dependent perturbation in ozone photochemistry S(T), where a(ϕ,z)>0 and b(ϕ,z)>0 are linear coupling parameters as a function of latitude ϕ and altitude z specified below, ρ0(z)=ρ00exp-(z-z0)/H is background density, H∼7 km is scale height, ρ00 is a reference value at altitude z0, u0 is a zonal mean background wind, d0/dt=/t+u0/x+v0/y, where /x and /y denote the derivations in longitude and latitude, g is the gravity acceleration, and f is the Coriolis parameter; the background shear terms wu0/z and wv0/z are neglected because of the Wentzel–Kramers–Brillouin or WKB approximation:

(1)d0udt+1ρ0px=fv,(2)d0vdt+1ρ0py=-fu,(3)d0wdt+1ρ0pz=gθθ0,(4)d0ρdt+ux+vy+1ρ0ρ0wz=0,(5)d0θdt+wθ0z=Qp00pκ=aμ0d0μdt,(6)d0μdt+wμ0z=S=-bμ0d0θdt.

For Q=0, the dispersion relation for gravity waves results from Eqs. (1) to (5) by introducing sinusoidal perturbations X1=Xa0exp[i(k1x+l1y+m1z-ω1t)]exp(z-zs)/2H, where X1 denotes the perturbation quantities, Xa0 the initial amplitude at altitude zs at the lower boundary of the upper stratosphere, exp(z-zs)/2H the exponential growth of the amplitude due to decreasing density, k1 and l1 the horizontal and meridional wave number, m1<0 the vertical wave number for upward propagating GWs with |m1|=2π/Lm1 and vertical wavelength Lm1, and ω1 the frequency (here, the subscript 1 denotes the solutions for Q=0). We focus on horizontal and vertical wavelengths Lh1≥50 km and Lm1≤15 km, where kh1=2π/Lh1 is the horizontal wave number given by kh1=(k12+l12)1/2, therefore (1+kh12/m12)1. Compressibility effects due to the vertical change in background density are excluded assuming m121/4H2, which is valid for vertical wavelengths Lm≤30 km. Then, the dispersion relation for the intrinsic frequency ωi1=ω1-k1u0 is given for the frequency range N02>ωi12>f2, where N02=(g/θ0)θ0/z denotes the Brunt–Vaisala frequency:

(7) ω i 1 2 = N 0 2 k h 1 2 + m 1 2 f 2 k h 1 2 + m 1 2 N 0 2 k h 1 2 m 1 2 + f 2 .

2.1.2 Ozone–temperature coupling

For specifying the parameter b, we consider the vertical ascent w1>0 in the wave crest of an initial sinusoidal GW perturbation, related to an adiabatic cooling term d0θ1/dt=-w1θ0/z<0, which leads to an initial ozone perturbation μ1>0 due to the induced increase d0μ1/dt=-w1μ0/z>0 via transport, and to a change in ozone photochemistry described by S(T1) (for the descent w1<0 in the wave trough, the formulations are analogous but with μ1<0 and d0θ1/dt=-w1θ0/z>0). In the USLM region, ozone is very short-lived and approximate in photochemical equilibrium (Brasseur and Solomon, 1995), i.e., for pure oxygen chemistry it is approximately given by

(8) O 3 = k 2 k 3 M ( O 2 ) 2 J 2 ( O 2 ) J 3 ( O 3 ) 1 / 2 ,

where J2(O2) and J3(O3) are photo-dissociation rates, and k2=6.0×10-34(300/T)2.3 cm6 s−1 and k3=8.0×10-12exp(-2060/T) cm3 s−1 are chemical reaction rates for ozone production (O+O2+MO3+M) and ozone loss (O+O32O2) (Appendix C of Brasseur and Solomon, 1995; Table 2 of Schmidt et al., 2010). Accordingly, following Brasseur and Solomon (1995), a relative change in ozone ΔμT/μ0=ΔO3/O3 due to a change in temperature ΔT is given by

(9) Δ μ T μ 0 = 1 2 Δ ( k 2 / k 3 ) ( k 2 / k 3 ) = - 1 2 2.3 T 0 + 2060 T 0 2 Δ T - b 0 ( T 0 ) Δ T .

Then, defining b=b0(p/p00)κ and introducing a total temperature change ΔT/Δt within a background flow described by d0T/dt=(p/p00)κd0θ/dt, the change S is given by

(10) S = Δ μ T Δ t = Δ μ T Δ T Δ T Δ t = - μ 0 b d 0 θ d t ,

which is the right-hand term of Eq. (6). Overall, the initial ascent w1>0 leads to an increase in ozone via transport, and the related adiabatic cooling to an increase in ozone because of the induced change S>0. Analogously, the initial descent w1<0 leads to a decrease in ozone via transport and an induced change S<0. The height-dependence of b is specified by considering that the ozone photochemistry of the USLM region is related to the spatial structure of Q0, which is characterized by a Gaussian-type height-dependence centered at the maximum of Q0 and a rapid decrease with latitude in the extratropical winter hemisphere (see Fig. 1c). Therefore, b is multiplied with the normalized factor hz=Q0/Q00, where Q00 is the averaged profile of Q0 over the summer hemisphere (bbhz, where hz(z)≈1 in the summer upper stratosphere at the altitude where Q0 reach maximum values). A similar approach of Gaussian-type height-dependence in ozone–temperature coupling was successfully used by Gabriel et al. (2011a) to analyze observed planetary-scale waves in the ozone distribution.

Following previous works (e.g., Cordero and Nathan, 2000; Cordero et al., 1998; Nathan and Cordero, 2007; Ward et al., 2000; Gabriel et al., 2011a), the sensitivity of the upper stratospheric heating rate to a change in ozone is approximately described by the linear approach ΔQμAΔμ, where A=A(ϕ,z) is a time-independent linear function. If we assume the same sensitivity for both the slowly varying background and the mesoscale GW perturbation propagating within the background flow, Q0Aμ0 and QAμ, we may write ΔQμ/Δμ=Q0/μ0=Q/μ. At a specific altitude z or pressure level p(z), we consider a GW perturbation over the vertical scale of a vertical wavelength, Δz=Lm. Then, considering that μ/z=imμ=(τi/Lm)(-iωiμ) with τi=2π/ωi, the first-order heating rate perturbation is given by

(11) Q = L m Q z L m Δ Q μ Δ μ μ z = τ i Q 0 μ 0 d 0 μ d t ,

which is the right-hand side of Eq. (5) when defining a0=τiQ0 and a=a0(p00/p)κ. Except in polar summer regions, the effect of Q is limited by the length of daytime (here denoted by τday) in case of large wave periods. Therefore, we set the time increment to τi=τday in the case of τi>τday, which reduces the effect of Q during the time period of 24 h (e.g., τi≤12 h over the Equator). Overall, reassuming an initial ascent w1>0, the induced increase in ozone μ>0 at a pressure level p(z) leads to a heating rate perturbation Q>0 at this level counteracting to the initial adiabatic cooling and therefore reinforcing the initial ascent. Analogously, an initial descent w1<0 is reinforced by inducing a perturbation Q<0.

Note here that the use of Δz=Lm in Eq. (11) provides a suitable measure of the effect of ozone–temperature coupling on the GW amplitudes at a specific level over the vertical distance Lm. It is also possible to set a smaller vertical scale Δz<Lm leading to smaller values QΔz=(Δz/Lm)Q at a specific level, where Δz denotes, for example, the distances of a vertical grid used in a numerical model. This modification does not change the effect over the vertical distance Lm but it provides better vertical resolution when calculating the cumulative amplitude amplification during the upward level-by-level propagation, particularly in the case of small vertical wavelengths or small vertical group velocities, as described in the next subsection.

2.1.3 Amplification of GW amplitudes at a specific level

The parameterizations of Q and S provide a useful modification of the potential temperature tendency when introducing d0μ/dt of Eq. (6) into (Eq. 5):

(12) ( 1 + a b ) d 0 θ d t + w θ 0 z + a μ 0 μ 0 z = 0 .

Here, the amplification factor 1+ab (with ab>0) describes the feedback of the GW-induced ozone perturbation to the change in potential temperature, and θ0/z+(a/μ0)μ0/z an ozone adiabatic lapse rate which is – in the USLM region – smaller than θ0/z because of μ0/z<0. Alternatively, we may write:

(13) d 0 d t g θ 0 θ + N μ 2 w = 0 ,

with

(14) N μ 2 = N 0 2 + N c 2 ( 1 + a b ) ,

where Nc2=(g/θ0)(a/μ0)μ0/z. As with the lapse rate, Nμ2 is smaller than N02 because Nc2<0 and (1+ab)>1. If ozone–temperature coupling becomes weak, below and above the USLM region, Nμ2 converges to N02.

Analogous to the standard solution given above, we introduce sinusoidal GW perturbations of the form X2=Xμ0exp[i(k2x+l2y+m2z-ω2t)]exp(z-zs)/2H in Eqs. (1)–(4) and (13) (here, the subscript 2 denotes the solutions with ozone–gravity wave coupling) which leads to the modified dispersion relation:

(15) ω i 2 2 = N μ 2 k h 2 2 + m 2 2 f 2 k h 2 2 + m 2 2 N μ 2 k h 2 2 m 2 2 + f 2 ,

where ωi2=ω2-k2u0 and kh2=(k22+l22)1/2.

Equation (13) provides an evident measure of the amplification of a GW amplitude at a specific altitude z or pressure level p(z). On the one hand, introducing the same initial adiabatic potential temperature perturbation dθ1/dt, either with or without ozone–temperature coupling, leads to w2=w1(N02/Nμ2). Consistently, introducing the same initial perturbation w1N02 leads to dθ2/dt=dθ1/dt or -iωi2θ2=-iωi1θ1. Furthermore, combining -iωi2θ2=-Nμ2 w2 and -iωi1θ1=-N02w1 suggests that the amplitude θμ=θμ0exp(z-zs)/2H is stronger than θa=θa0exp(z-zs)/2H by the factor ωi1/ωi2=N02/Nμ21:

(16) θ μ = θ a ( ω i 1 / ω i 2 ) .

Overall, the introduced process of ozone–temperature coupling leads to a decrease in the GW frequency and a corresponding amplification in the GW amplitude described by the factor ωi1/ωi2 or N02/Nμ2. Note that the vertical variations in N02 could affect the increase in amplitude with height, particularly in the summer upper mesosphere. Therefore, N02 is vertically averaged over the USLM region (from 30 to 0.03 hPa, or ∼25 to ∼70 km altitude) to focus on the effects of ozone–gravity wave interaction only. Moreover, note that the relation ωi1/ωi2=N02/Nμ2 implies not only a change in amplitude but also a slight change in the relation of horizontal and vertical wave numbers described by (kh2/m2)=(Nμ2/N02)(kh1/m1)+f2(Nμ2-N02)/(N02N02), i.e., a slight change in the direction of upward propagating GWs which is perpendicular to the angle α of the phase lines defined by cos(α)=±(kh/m). However, as illustrated in the following section, ozone–gravity wave interaction is particularly relevant for a range of wavelengths and periods where the induced changes in α are very small (for Lm1/Lkh1<0.05, or wave periods τi>2 h, the change in α is less than 0.0001).

2.1.4 Examples of the amplification of GW amplitudes at specific levels

Figure 1d–f shows the factor 1+ab and the quotient N02/Nμ2 for a GW with horizontal and vertical wavelengths Lk=500 km and Lm=5 km, and the quotient N02/Nμ2 for a GW with Lk=800 km and Lm=3 km. In the first example, the factor 1+ab (Fig. 1d) contributes to the amplification of the GW amplitude at a specific level by up to 6 %–8 %, and the overall factor (N02/Nμ2=(1+ab)N02/(N02+Nc2) (Fig. 1e) by up to 8 %–12 % (including a decrease in the lapse rate of up to 3 % described by (N02+Nc2)/N02, not shown here). The second example (Fig. 1f) shows that the factor N02/Nμ2 is larger in the case of larger horizontal and smaller vertical wavelengths, reaching amplifications of up to 12 %–14 % (shaded areas denote the latitudinal range where the amplification is reduced due to the length of daytime, i.e., where τi>τday).

https://acp.copernicus.org/articles/22/10425/2022/acp-22-10425-2022-f02

Figure 2Changes due to ozone–temperature coupling at a specific level induced by an initial GW perturbation with horizontal and vertical wavelengths Lk=500 km and Lm=5 km, and with exponential increase in amplitude with height (initial temperature amplitude Ta(zs)=1 K at zs≈35 km; p=6.28 hPa). (a) Change in ozone due to vertical transport, (b) change in ozone due to photochemistry, (c) total change in ozone, (d) relative change in ozone, (e) change in the heating rate, and (f) change in the temperature perturbation.

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For illustration of the induced change in ozone at a specific level (Fig. 2a–d), we assume an initial GW perturbation θ1 with exponentially growing amplitude θa=θa0exp(z-zs)/2H, with an initial temperature amplitude Ta0 of 1 K at zs≈35 km (ps=6.28 hPa) increasing to ∼8 K at z≈65 km (p=0.1 hPa). In the present paper, we formulate the solutions for pressure levels p, i.e., the initial perturbation is alternatively described by θa=θa0(ps/p)1/2, assuming p=psexp(z-zs)/H. Introducing the associated perturbation w1=-(θ0/z)-1d0θ1/dt in Eq. (6) leads to d0μ1/dt=[(μ0/z)/(θ0/z)-bμ0]d0θ1/dt, and, considering d0μ1/dt=-iωi1μ1 and d0θ1/dt=-iωi1θ1, to an initial ozone perturbation μ1=θ1[(μ0/z)/(θ0/z)-bμ0]. For the example of the ascent (w1>0) shown in Fig. 2, we set θ1<0, leading to μ1>0. For Lk=500 km and Lm=5 km, the contributions μ(TR)=θ1[(μ0/z)/(θ0/z)] (related to transport; Fig. 2a) and μ(CH)=-bμ0θ1 (related to S; Fig. 2b) sum up to a total change of μ0.2 to 0.5 ppm (Fig. 2c) or μ/μ05 to 10 % (Fig. 2d) in the USLM region, where the feedback to the heating rate is particularly strong.

The related change in the heating rate at a specific level (Fig. 2e) is given by comparing Eq. (5) with and without ozone–temperature coupling. Assuming that the same initial ascent or adiabatic cooling as above leads to (w2-w1)(θ0/z)=Q(μ1), or, when introducing w2=(ωi1/ωi2)w1 to Q(μ1)=(ωi1/ωi2-1)(-ωi1θ1)=aωi1μ1μ0-1 (where Q(μ1)>0 in case of w1>0), Fig. 2e shows that Q(μ1) reach values of 0.15 K h−1 over the tropics and 0.25 K h−1 at southern summer polar latitudes. Then, consistent with Eq. (16), we yield θ2-θ1=(ωi1/ωi2-1)θ1 where (ωi1/ωi2-1)=-aμ0-1[(μ0/z)/(θ0/z)-bμ0] for the change in the potential temperature perturbation, i.e., changes in temperature of 0.2–0.3 K in the USLM region (Fig. 2f). In summary, analogously considering the corresponding change for the descent, we yield an increase in the amplitude of the oscillating GW pattern at a specific level by up to 5 %–10 % in ozone and 0.2–0.3 K in temperature.

For other initial wavelengths (or associated frequencies), the latitude-height dependence is very similar to those shown in Figs. 1d–f and 2, whereas the magnitude of the amplification factor ωi1/ωi2 becomes smaller in the case of increasing vertical and decreasing horizontal wavelengths, or decreasing frequencies, as illustrated in Fig. 3 for an altitude where ωi1/ωi2 reach maximum values (1.156 hPa or ≈47 km altitude). Figure 3a shows values of ωi1/ωi2>1.02 for wave periods of τi>2 h steadily increasing with an increasing initial period up to values between 1.14 and 1.15. This value is limited, on the one side, because of the increasing duration of nighttime with latitude towards equatorial and northern winter regions (denoted by shaded areas), and, on the other side, because of the increasing Coriolis force in southern summer middle and polar regions (i.e., because of ωi12>f2).

https://acp.copernicus.org/articles/22/10425/2022/acp-22-10425-2022-f03

Figure 3Amplification factor ωi1/ωi2 at a level of the maximum values of ωi1/ωi2 (1.156 hPa) illustrating the decrease in the intrinsic frequency with (ωi2) compared to without (ωi1) ozone–temperature coupling (compare with Fig. 1e–f), (a) latitudinal distribution of ωi1/ωi2 as a function of the initial wave period τi [in h], and (b–c) dependence of ωi1/ωi2 on the horizontal and vertical wavelengths Lk and Lm [in km] at (b) 70 S and (c) 10 S. Shaded areas show where the amplification is limited by the length of daytime (τi>τday).

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Consistently, the amplification factor increases with decreasing vertical and increasing horizontal wavelengths (Fig. 3b and c show examples for 70 and 10 S), where the values are limited by the length of daytime in the case of small relations Lm/Lk denoting the conditions where τi>τday (Fig. 3c, shaded area). Figure 3 also indicates that the examples with Lk=500 km and Lm=5 km (Figs. 1e; 2) and Lk=800 km and Lm=3 km (Fig. 1f) represent scales where ozone–gravity wave interaction is particularly efficient.

Overall, Figs. 1d–f, 2 and 3 illustrate the amplification of GW amplitudes at a specific level and a specific time. As far as the GWs are continuously propagating upward through several levels where ωi1/ωi2-1>0, the amplification will be successively reinforced at each level. This cumulative amplification can lead to much stronger GW amplitudes at upper mesospheric altitudes in the case with ozone–gravity wave interaction than in the case without, as demonstrated in the next subsection.

https://acp.copernicus.org/articles/22/10425/2022/acp-22-10425-2022-f04

Figure 4Illustration of the successive amplification of GW amplitudes during the upward level-by-level propagation. (a) Amplification factor ωi1/ωi2 at 70 S for GW with horizontal wavelength Lk=500 km and vertical wavelength Lm=5 km (solid red line), and, for comparison, Lm=3 (dashed) and Lm=9 km (dotted). (b) Temperature amplitudes for GW with Lk=500 km and Lm=5 km, depicting the initial perturbation Ta (blue) and the successively amplified amplitudes Tμj(zj)|j=1,n (light blue towards red; here, n=8 for Lm=5 km). (c) Same as (b) but for the relative amplitudes Tμj(zj)|j/Ta (solid lines) together with the profiles of the previous level multiplied by ωi1/ωi2 (i.e., Tμj-1(zj-1)(ωi1/ωi2), dashed lines) and a fitted approach Tμ (thick solid red line, defined by Eq. 18). (d) Same as (a) for the case Lk=500 km and Lm=5 km but at 10 S including the limitation due to the length of nighttime conditions. (e) Relative values Tμ/Ta at 70 S for different horizontal (red: Lk=500 km, purple: Lk=800 km) and vertical (dashed: Lm=3, solid: Lm=5 km, dotted: Lm=9 km) wavelengths. (f) Same as (e) but for the relative values Eμ/Ea of the related gravity wave potential energy density (GWPED, defined by Eq. 19).

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2.2 Upward propagating GWs in a background flow

2.2.1 Level-by-level amplification of GW amplitudes

In the following section, a solution for the cumulative amplification during the vertical level-by-level propagation is derived, excluding – to a first guess – other effects like small-scale diffusion, wave-breaking processes, interaction of GWs with atmospheric tides, or so-called secondary GWs. Following the Huygens principle, each point of a propagating wave front at a specific level is the source of a new wave at this level, i.e., a single upward propagating GW, which is amplified at a level zj−1, is the initial perturbation amplified at the next level zj. For illustration (Fig. 4a–c), we choose an initial GW with horizontal and vertical wavelengths Lm=500 km and Lm=5 km as above, where the vertical distance between the levels zj−1 and zj is set by the initial vertical wavelength Δz=Lm. First, we focus on polar latitudes during southern polar summer (70 S) with daytime conditions only. Thereafter we consider the modification for middle and equatorial latitudes where GWs with weak vertical group velocities propagate through USLM during both daytime and nighttime.

For orientation, Fig. 4a shows the profiles ωi1/ωi2 for Lk=500 km and Lm=5 km at 70 S (solid), and, for comparison, for Lm=3 (dashed) and Lm=9 km (dotted), indicating the altitude range where ozone–temperature coupling is relevant (note that the depicted distance of pressure levels approximately represents a 5 km distance in altitude). Beginning with a first level at zs≈35 km (6.28 hPa), the wave propagates through eight layers between ≈35 and ≈70 km (0.06 hPa) where the amplification of the amplitude is relevant. At each of these levels, denoted by zj=zs+(j-1)Δz (j=1, n; here n=8), the amplitude at zj will be amplified by the factor ωi1(zj)/ωi2(zj) at zj. Starting with an exponentially growing amplitude Ta(z)=Ta(zs)exp(z-zs)/2H (where we set Ta(zs)=1 K again), we yield a new amplitude Ta1(z1)=Ta(z1)ωi1(z1)/ωi2(z1) at the level z1, defining a new exponentially growing amplitude Tμ1(z)=Ta1(z1)exp(z-z1)/2H. We then yield Ta2(z2)=Tμ1(z2)ωi1(z2)/ωi2(z2) at the level z2, defining Tμ2(z)=Ta2(z2)exp(z-z2)/2H, and so forth. Finally, the amplitude at the level zn in the middle mesosphere is described by

(17) T μ n ( z ) = T a ( z ) j = 1 n ω i 1 ( z j ) ω i 2 ( z j ) ,

where the product symbol Πj=1,n denotes the multiplication with ωi1(zj)/ωi2(zj) at each level z1zjzn. As mentioned above, the solutions are calculated on pressure levels, i.e., z represents the geopotential height, and the vertical distance Δz between the levels is given by Δz=-(ρ0g)-1Δp=-H(T0)(Δp/p), where H(T0)=g/(RT0) is the height-dependent scale height defined by the background. Note that using a constant scale height H0=7 km instead of H(T0) only leads to second-order changes in the cumulative amplitude amplification (the sensitivity test is described below in Sect. 2.2.4), because H(T0) only varies slightly in the USLM region (between ∼7.5 km at summer stratopause altitudes and ∼6.5 km at 70 km).

Figure 4b shows the initial amplitude Ta (blue line) and the series of the successively amplified amplitudes Tμ1, Tμ2, …, Tμn (from light blue towards red line). Figure 4c shows the related series of constant relative values Tμ1/Ta, Tμ2/Ta, …, Tμn/Ta, starting at the level zj (solid lines) together with the previous values starting at zj−1, multiplied by the factor ωi1/ωi2 (dotted lines), illustrating the successively increasing growth of the amplitude during the upward level-by-level propagation. Finally, the amplitudes converge to Tμn(z) when reaching the upper mesosphere, where Tμn(z) is stronger than Ta(z) by a factor of ∼1.47. Figure 4c also shows the fitted relative increase in the amplitude Tμ/Ta (thick red line) describing the continuous change in the growth rate of the amplitude, where Tμ(z), or Tμ(p), is defined by

(18) T μ ( p ) = h s ( p ) T a ( p ) + h m ( p ) T μ n ( p ) ,

with weighting functions hs=p01.5/(p01.5+pm1.5) and hm=1-hs, where p0 is the background pressure and pm(70S)0.96 hPa the level of the maximum of ωi1/ωi2 (note that the height of this maximum is slightly decreasing from pm≈0.89 hPa over the South Pole to pm≈1.3 hPa over the Equator).

For middle and equatorial latitudes, daytime–nighttime conditions are considered by setting the amplification factor to Fd=ωi1/ωi2 during daytime but to Fd=1 during nighttime over the vertical wave propagation distance of 1 full day. In detail, we define the parameter Lday=(τday-0.5τ0)/(0.5τ0), where τ0=24 h and τday is the duration of daytime within 24 h at the latitude ϕ (with Lday=1 during polar summer and Lday=0 at the Equator). Further, considering the vertical group velocity cgz=ωi1/m1=-(ωi1/m1)(ωi12-f2)/ωi12 (with initial frequency ωi1 and vertical wavelength m1 as a first guess), the sinusoidal wave propagation structure between the middle stratosphere and middle mesosphere is described by Lcgz=cos(2πτ0(z-zm)/cgz) changing periodically between 1 and −1 over one wavelength, where z and zm are given in kilometers and cgz in kilometers per hour, and where Lcgi=1 at the level pm, or altitude zm(pm). Then, the combined parameter Ld=Lday+Lcgi separates the vertical propagation distance into daytime and nighttime fractions by defining a constant value Cd=1 in the case of Ld>1 and Cd=0 in the case of Ld≤1, where the factor Fd=1+Cd((ωi1/ωi2)-1) provides Fd=ωi1/ωi2 in the case of daytime and Fd=1 in the case of nighttime.

As an example, Fig. 4d shows the profile of the resulting amplification factor Fd at 10 S for a GW with Lk=500 km and Lm=5 km as above, with an associated vertical group velocity cgz of about 7 km per 12 h, illustrating that we define Fd(zj)=ωi1(zj)/ωi2(zj), where zj is located in the daytime region (red) but Fd(zj)=1, where zj is located in the nighttime region (blue). The indicated vertical wave propagation distance during daytime increases towards southern summer polar latitudes but decreases towards northern winter polar latitudes. Note that for vertical wavelengths examined in the present paper (Lm≤15 km), a vertical shift of the phase – as defined by the altitude zm in the definition of Lcgz – does not have a significant impact on the cumulative amplification of the GW amplitudes because of the Gaussian-type structure of the profile of Fd=ωi1/ωi2, which has been verified by several test calculations with levels other than pm, or altitudes other than zm.

In the following section, the fitted profiles Tμ are used for further examinations with different horizontal and vertical wavelengths, where the vertical level-by-level amplification is calculated by using the distances ΔzzH of the vertical grid of HAMMONIA instead of Δz=Lm. This includes a smaller amplification factor Fω=ωi1/ωi2 over the vertical distance ΔzH because of the smaller heating rate perturbation QΔzH=(ΔzH/Lm)Q (see Eq. 11 and the related discussion). However, the resulting differences in the amplification at a specific level over the vertical distance Lm are nearly the same, except for some small differences of less than 0.5 % due to the different vertical resolution (i.e., Fω(Δz=Lm)1+(Fω(Δz=ΔzH)-1)(Lm/ΔzH)). Additionally, the resulting cumulative amplification in the upper mesosphere remains nearly unchanged (Tμn(Δz=Lm)Tμnh(Δz=ΔzH), where nh is the number of the HAMMONIA levels in the USLM), where small differences between Tμnh and Tμn of less than 10 % occur only at middle and equatorial latitudes in the case of small vertical wavelengths (or small vertical group velocities) when considering the vertical propagation during both daytime and nighttime described below.

2.2.2 Cumulative amplitude amplifications for representative examples

Figure 4e illustrates the dependence of the amplitude amplification on the horizontal and vertical wavelengths Lk and Lm at 70 S, where it is not affected by nighttime conditions. In comparison to the example of Lk=500 km and Lm=5 km leading to a cumulative amplification of ∼1.47 (solid red line), a larger vertical wavelength of Lm=9 km leads to a smaller value of ∼1.15 (dotted red line), but a smaller vertical wavelength of Lm=3 km leads to a larger value of ∼2.27 (dashed red line), because the induced increase in the ozone perturbation μ produces a heating rate perturbation Q within a shorter (in the case of Lm=9 km) or larger (in the case of Lm=3 km) time increment τi. For the same reason, the amplification is generally larger if the horizontal wavelength Lk is larger, e.g., in the case of Lk=800 km, the final amplification in the upper mesospheric amplitudes amounts to ∼1.22 for Lm=9 km (dotted purple line), ∼1.63 for Lm=5 km (solid purple line), and ∼2.56 for Lm=3 km (dashed purple line).

The related GWPED (here denoted by E) is derived following Kaifler et al. (2015):

(19) E = 1 2 g N 2 T T 0 2 .

Introducing T=T2 and N=Nμ, or T=T1 and N=N0, leads to the case with (Eμ) or without (Ea) ozone–gravity wave interaction. Figure 4f shows the relative amplitudes Eμ/Ea related to Fig. 4e. In the case of Lk=500 km (red lines), the final amplifications reach values of ∼1.32 for Lm=9 km (dotted), ∼2.17 for Lm=5 km (solid), and ∼5.21 for Lm=3 km (dashed), and in case of Lk=800 km (purple lines) values of ∼1.50 for Lm=9 km (dotted), ∼2.70 for Lm=5 km (solid), and ∼6.62 for Lm=3 km (dashed). Overall, these factors provide a first-order estimate of the effect of ozone–gravity wave coupling at 70 S during polar summer, i.e., in case of large horizontal (≥500 km) and small vertical (≤5 km) wavelengths, we find cumulative amplifications in the upper mesosphere in the order of ∼1.5 to ∼2.5 in the temperature perturbations and in the order of ∼3 to ∼7 in the related GWPED.

https://acp.copernicus.org/articles/22/10425/2022/acp-22-10425-2022-f05

Figure 5Cumulative amplifications of the GW amplitude during the upward level-by-level propagation for a GW with Lk=500 km and Lm=5 km, (a) Cumulative increase in the temperature amplitudes described by Tμ/Ta. (b) Related increase in the gravity wave potential energy density (GWPED) described by Eμ/Ea; background conditions: January 2001.

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2.2.3 Cumulative amplitude amplifications depending on latitude

For the GW with Lk=500 km and Lm=5 km, Fig. 5 shows the latitudinal dependence of the cumulative amplifications of the temperature perturbation (indicated by Tμ/Ta, Fig. 5a) and the related GWPED (indicated by Eμ/Ea, Fig. 5b). The values decrease from Tμ/Ta1.5 and Eμ/Ea2.4 over southern summer polar latitudes towards Tμ/Ta1.2 and Eμ/Ea1.4 at lower mid-latitudes (40 S), and then less rapidly towards Tμ/Ta1.1 and Eμ/Ea1.2 at 20 N. Overall, although the amplification of the GW amplitudes decreases rapidly with the decrease in the length of daytime, it is still quite strong at mid-latitudes.

https://acp.copernicus.org/articles/22/10425/2022/acp-22-10425-2022-f06

Figure 6Cumulative amplification of the GW amplitudes similar to that in Fig. 5, but at upper mesospheric levels (0.01 hPa, ∼80 km) for different horizontal and vertical wavelengths Lk (red: 500 km, purple: 800 km) and Lm (dotted: 9 km, solid: 5 km, dashed: 3 km), (a) Tμ/Ta, (b) Eμ/Ea.

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Figure 6 shows the relations Tμ/Ta (Fig. 6a) and Eμ/Ea (Fig. 6b) at upper mesospheric levels (0.01 hPa, ∼80 km) for different horizontal and vertical wavelengths as used for Fig. 4e and f. For both Lk=500 km (red) and Lk=800 km (purple), the amplifications of the temperature perturbations and of the related GWPED are strongest for Lm=3 km (dashed lines), at polar latitudes with values between 2.5 and 3 in Tμ/Ta and between 7 and 9 in Eμ/Ea, and at middle and equatorial latitudes between 1.5 and 1.8 in Tμ/Ta and between 2.4 and 3.5 in Eμ/Ea. These values decrease with increasing vertical wavelength, i.e., when changing Lm=5 km (solid lines) or Lm=9 km (dotted lines) roughly to 1.7 or 1.25 in Tμ/Ta and ∼3.0 or ∼1.5 in Eμ/Ea at polar latitudes, and roughly to ∼1.25 or ∼1.2 in Tμ/Ta and ∼1.5 or ∼1.25 in Eμ/Ea at middle and equatorial latitudes. Overall, for the mesoscale GWs with small vertical and large horizontal wavelengths, the cumulative amplifications due to ozone–gravity wave coupling leads to much stronger amplitudes at upper mesospheric altitudes during daytime than during nighttime, in the GW perturbations by a factor between ∼1.5 at summer mid-latitudes and ∼3 for polar daytime conditions, and in the GWPED by a factor between ∼3 at summer mid-latitudes and ∼9 for polar daytime conditions.

Note that vertical momentum flux terms FGW=ρ0 (uw) can be derived from local profiles T if the background is known, i.e., by FGW=ρ0E (k m−1) (Ern et al., 2004). Therefore, the amplification of the GW amplitudes must lead to the same amplification of the flux term FGW and, if the GWs do not break at lower levels, of the associated gravity wave drag GWD =-ρ0-1FGW/z in the upper mesosphere, suggesting an important effect of ozone–gravity wave interaction on the meridional mass circulation, particularly at polar latitudes. However, more detailed investigations need extensive numerical model simulations with a spectrum of resolved GWs, which is beyond the scope of the present paper.

2.2.4 Sensitivity to varying conditions

In the following section, we estimate the sensitivity of the GW amplitude amplification on nonlinear processes and background conditions which could modulate the first-guess results described above. For example, the decrease in the frequency towards ωi2<ωi1 includes a slight decrease in the vertical group velocity towards cgz2<cgz1, which can additionally strengthen the process of amplitude amplification because the wave propagates somewhat more slowly through the ULSM region. However, this effect is at least 1 order smaller than the first-order process described above, as derived from test calculations including this effect. For example, for Lk=500 km and Lm=5 km, cgz2 is smaller than cgz1 by 15 %–20 % at southern summer polar latitudes and 5 %–10 % at middle and equatorial latitudes. Subsequently, at a specific level, the amplification factor Fd(cgz2) is stronger than Fd(cgz1) by 2 %–3 % at polar latitudes and less than 1 % at middle and equatorial latitudes. Including this change in the successive level-by-level propagation leads to a weak successive increase in the cumulative amplifications by ∼5 % at 1 hPa to ∼10 % at 0.01 hPa at polar summer latitudes, and by only ∼1 % at 1 hPa to ∼2 % at 0.01 hPa at middle and equatorial latitudes.

We also estimate the sensitivity of the amplitude amplification on the ozone background μ0, considering the observed long-term changes in upper stratospheric ozone in the order of up to −8 % per decade (e.g., Sofieva et al., 2017; WMO, 2018), and the uncertainty in the maximum of the heating rate Q0, which is smaller in the used HAMMONIA data in the order of ∼10 % compared to those derived from satellite measurements, as mentioned above. In the case of a 10 % reduction in ozone, the cumulative amplification in the upper mesospheric GW amplitudes is weaker by about 5 % for the example with Lm=5 km and 10 % for Lm=3 km (i.e., at 70 S, we yield a cumulative amplification of ∼1.4 to ∼2.25 instead of ∼1.5 to ∼2.5), and the related amplification of the GWPED is weaker by about 10 % for Lm=5 km and 20 % for Lm=3 km (at 70 S, a cumulative amplification of ∼2.7 to ∼7.2 instead of ∼3 to ∼9). Analogously, in the case of an increase in Q0 by 10 %, the cumulative amplification is stronger by 5 % or 10 % in the GW amplitudes and by 10 % or 20 % in the related GWPED amplitudes.

Another question arises about the sensitivity of the effect of ozone–gravity wave coupling to atmospheric tides or the diurnal cycle in stratospheric ozone, which are planetary-scale processes changing the background conditions for the propagation of the mesoscale GW perturbations. For example, Schranz et al. (2018) observed stronger amplitudes in upper stratospheric ozone during daytime than during nighttime in the order of 5 % (summer solstice) to 8 % (May). Such a difference would correspond to a change in the cumulative amplification of the upper mesospheric GW amplitudes or GWPED in the order of 5 % to 10 % or 10 % to 20 %, as follows from the sensitivity of the effect on the prescribed long-term change in stratospheric ozone derived above.

Baumgarten and Stober (2019) derived amplitudes of tides in the order of up to 0.5 K in the middle stratosphere (∼35 km) increasing up to 2 K at ∼50 km and ∼4 K at 70 km, which would correspond to a change in the lapse rate in the order of up to 0.1 K km−1, or in the Brunt-Vaisala frequency N02 in the order of 1 %. As follows from Eq. (14), a change in the amplification factor Fd=N02/Nμ2 due to a relative change ΔN02/N02 is given by the factor [1+(ΔN02/N02)]/[1+(ΔN02/N02)(N02/Nμ2)(1+ab)-1]. Therefore, for wavelengths Lk≥500 km and Lm≤5 km, a relative increase (decrease) of 1 % in N02 would lead to a relative decrease (increase) in the amplification factor of up to 0.035 % at stratopause altitudes, which is much less than the effects of the changes in the vertical group velocity or in ozone described above. Moreover, even if a relative change ΔN02/N02 would be much larger (10 %–50 %), it does not change the amplification factor of a specific level by more than 1 %–3 %, and, hence, the cumulative amplification of the GW amplitudes in the upper mesosphere by more than 5 %–10 %.

Assuming exponential growth of the amplitudes (e(z-zs)/2H) between two levels, the usual approach of a constant scale height (e.g., H∼7 km) instead of a height-dependent scale height H(T0)=g/(RT0) can principally lead to significant differences in the GWPED profiles (e.g., Reichert et al., 2021). For estimating the relevance of a change in H on the cumulative amplitude amplification, the solutions are also calculated for an initial GW perturbation θa=θa0exp(z-zs)/2H with a prescribed scale height H0=7 km instead of θa=θa0(ps/p)1/2, and a related vertical distance Δz=-H0(Δp/p) instead of Δz=-H(T0)(Δp/p) (note that H(T0) varies in the USLM region between ∼7.5 km at summer stratopause altitudes and ∼6.5 km at 70 km). Compared to the values shown in Figs. 5 and 6, the cumulative amplification of the upper mesospheric GW amplitudes is weaker by about 5 % (Lm=5 km) to 10 % (Lm=3 km) over the southern summer polar latitudes and weaker by about 1 % (Lm=5 km) to 3 % (Lm=3 km) at summer mid-latitudes. Correspondingly, the related GWPED values are weaker by about 7.5 % (Lm=5 km) to 20 % (Lm=3 km) over the southern summer polar latitudes, and 1.5 % (Lm=5 km) to 5 % (Lm=3 km) at summer mid-latitudes. Overall, these differences are smaller than the first-order effect of ozone–gravity wave coupling by approximately 1 order, where the use of H(T0) instead of H0 at the levels of relevant amplification leads to somewhat stronger amplitude amplifications, particularly over the southern summer polar latitudes, because of the difference between the high background temperatures in the summer stratopause region and the low background temperatures in the summer mesosphere (see Fig. 1a).

2.2.5 Potential effect on mean GW amplitudes

In the following section, the potential effect of ozone–gravity wave interaction is estimated for an average over a representative range of 16 different mesoscale GW events (horizontal wavelengths: 200, 500, 800, and 1100 km, vertical wavelengths: 3, 5, 7, and 9 km; see, for comparison, the amplification factor as function of wavelengths shown in Fig. 3b, c). Although these settings are idealistic, the results provide a first-guess quantification of the potential effect on time–mean GWPED values usually derived from measurements, where several different GWs contribute to the analyzed temperature fluctuations derived from the detected temperature profiles.

https://acp.copernicus.org/articles/22/10425/2022/acp-22-10425-2022-f07

Figure 7Similar to Fig. 6 but for both relative and absolute changes in mean values averaged over 16 representative mesoscale GW events with different horizontal and vertical wavelengths Lk and Lm (Lk: 200, 500, 800, and 1100 km, Lm: 3, 5, 7, and 9 km), (a) relative change in temperature amplitude Tm/Ta (solid red line; dashed lines denote the standard deviation), (b) absolute change TmTa for the case of an initial temperature perturbations Ta0 of 1 K (orange line) and 2 K (purple line) in the middle stratosphere, (c) and (d) same as (a) and (b) but for the GWPED (i.e., for Em/Ea and EmEa).

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Figure 7 illustrates both the relative and absolute changes in the resulting mean upper mesospheric GW temperature amplitudes (Fig. 7a, b) and in the mean GWPED (Fig. 7c, d). The relative increase in the mean temperature amplitude (Fig. 7a, solid red line) is stronger by a factor increasing from about 1.3 (±0.1) at summer low and middle latitudes up to 1.7 (±0.2) at summer polar latitudes (values in brackets denote 1 standard deviation). This corresponds to a stronger increase from about 7 K (±2 K) up to 17.5 K (±4.5 K) in the case of an initial GW perturbation of 1 K in the middle stratosphere (at 6.28 hPa or ≈35 km) (Fig. 7b, solid orange line), and from about 14 K (±4 K) up to 35 K (±9 K) in the case of an initial GW perturbation of 2 K (Fig. 7b, solid purple line).

The relative increase in the mean GWPED (Fig. 7c, solid red line) is stronger by a factor increasing from about 1.7 (±0.2) at summer low and middle latitudes up to 3.4 (±0.8) at summer polar latitudes. This corresponds to a stronger increase in the absolute GWPED values from about 2 J kg−1 (±0.5 J kg−1) at summer low and middle latitudes up to 12 J kg−1 (±3 J kg−1) at summer polar latitudes in the case of an initial GW perturbation of 1 K at 35 km (Fig. 7d, solid orange line), and from about 8 J kg−1 (±2 J kg−1) up to 48 J kg−1 (±0.5 J kg−1) in the case of an initial GW perturbation of 2 K (Fig. 7d, solid purple line).

In summary, we find an absolute increase in the order of 7 to 35 K in the mean GW temperature amplitudes and 2 to 50 J kg−1 in the mean GWPED values, assuming usual initial GW perturbations in the order of 1 to 2 K in the middle stratosphere, where the effect is particularly large during polar daytime conditions. Note that, assuming exponential growth with height only, this potential effect can be much larger in the case of stronger initial amplitudes in the middle stratosphere (the absolute changes in the temperature amplitudes increase linearly and those in the GWPED values quadratically with increasing initial GW perturbations at 35 km) and in specific geographical regions or time periods where primary GWs with large horizontal and small vertical wavelengths are excited (e.g., where Lk≥800 and Lm≤3 km). However, the GWs with very large amplitudes might dissipate by nonlinear wave-breaking processes before reaching the upper mesosphere.

3 Summary and conclusions

The present paper shows that ozone–gravity wave interaction in the upper stratosphere/lower mesosphere (USLM) leads to a stronger increase in gravity wave (GW) amplitudes with height during daytime than during nighttime, particularly during polar summer. The results include information about both the amplification of the GW amplitudes at a specific level and the cumulative increase in the amplitudes during the upward level-by-level propagation of the wave from middle stratosphere to upper mesosphere.

In a first step, standard equations describing upward propagating GWs with and without linearized ozone–gravity wave coupling are formulated, where an initial sinusoidal GW perturbation in the vertical ozone transport and temperature-dependent ozone photochemistry produces a heating rate perturbation as a function of the initial intrinsic frequency, which determines the duration of the perturbation at a specific level over the distance of the initial vertical wavelength. The solution reveals an amplification of the ascending and descending perturbations of the sinusoidal GW pattern at this level, i.e., a decrease in the intrinsic frequency due to both the induced changes in the lapse rate (or Brunt–Vaisala frequency) and the positive feedback of the coupling on the initial GW perturbation, and an associated increase in the GW amplitude by a factor ωi1/ωi21 defined by the relation of the intrinsic frequencies without (ωi1) and with (ωi2) ozone–gravity wave coupling. This amplitude amplification is dependent on the horizontal and vertical wavelengths, Lk and Lm, where the effect is most efficient for GWs with Lk≥500 km and Lm≤5 km, or initial frequencies τi≥4 h, representing mesoscale GWs forced by cyclones or fronts, or by the orography of mountain ridges like the Rocky Mountains, Andes, or Norwegian Caledonides. For southern summer conditions, strongest amplitude amplifications at specific levels of about 5 %–15 % over the perturbation distance of one vertical wavelength are located near the stratopause, with peak values over the Equator and over summer polar latitudes.

In a second step, an analytic approach of the upward level-by-level propagation of the GW perturbations with and without ozone–gravity wave interaction reveals the cumulative amplitude amplification, where the wave is propagating upward with the vertical group velocity defined by the initial GW parameters, and where daytime–nighttime conditions at middle and equatorial latitudes are considered. Representative examples with different initial wavelengths illustrate that the successive increase in both the GW amplitudes and the related gravity wave potential energy density (GWPED) converge to much stronger amplitudes in the upper mesosphere during daytime than during nighttime. This effect strongly decreases with latitude between summer polar and mid-latitudes because of the decrease in the length of daytime, nearly constant at equatorial latitudes, and decreasing again with latitude towards insignificant values in the winter extratropics.

In summary, the strongest impact of ozone–gravity wave interaction is found for wave periods ≥4 h (related to the wavelengths Lk≥500 km and Lm≤5 km), i.e., in a range of wave periods usually observed at summer middle and polar latitudes. For prescribed single GWs with large horizontal wavelengths (500 to 800 km) and small vertical wavelengths (3 to 5 km), the upper mesospheric GW temperature amplitudes are stronger by a factor between 1.25 and 1.75 at summer low and middle latitudes and between 1.5 and 3 at summer polar latitudes, and the corresponding GWPED by a factor between 1.5 and 3.5 and between 3 and 9. For a representative range of 16 different mesoscale GW events (Lk between 200 and 1100 km, Lm between 3 and 9 km), the mean temperature amplitudes are stronger by a factor between 1.3 at summer low and middle latitudes to 1.7 at summer polar latitudes, e.g., stronger by about 7 to 17.5 K (or 14 to 35 K) in the case of an initial GW perturbation of 1 K (or 2 K) in the middle stratosphere (at ∼35 km). The corresponding relative increase in the mean GWPED is stronger by a factor between 1.7 at summer low and middle latitudes and 3.4 at summer polar latitudes, e.g., for the same example as above, stronger by about 2 to 12 J kg−1 (or 8 to 48 J kg−1). These values range in the order between 2 % and 50 % of the observed order of the mean upper-mesospheric GWPED amplitudes (100 J kg−1). These absolute differences can be larger in the case of stronger initial perturbations in the middle stratosphere, or in specific geographical regions or time periods where primary GWs with large horizontal and small vertical wavelengths (e.g., where Lk≥800 km and Lm≤3 km) are excited. However, the GWs with very large amplitudes might dissipate by nonlinear wave-breaking processes before reaching the upper mesosphere. Overall, these values result from an idealistic approach and cannot entirely explain the details of specific measurements. Nevertheless, they provide a first-guess quantification of the potential effect of ozone–gravity wave interaction on the GW amplitudes.

The variety of horizontal and vertical wavelengths used in the present paper are representative of mesoscale GWs in the USLM region. Observations suggest not only characteristic vertical wavelengths of GWs between ∼2–5 km in the lower stratosphere increasing to ∼10–30 km in the upper mesosphere, but also the existence of large vertical wavelengths greater than 10 km in the ULSM region, particularly above convection in equatorial regions or over southernmost Argentina (e.g., Alexander, 1998; McLandress et al., 2000; Fritts and Alexander, 2003; Alexander and Holton, 2014; Hocke et al., 2016; Baumgarten et al., 2018; Reichert et al., 2021). The results of the present paper suggest that the effect of ozone–gravity wave coupling decreases with increasing vertical wavelengths Lm≥9 km, but strongly increases with decreasing vertical wavelengths Lm≤5 km. The latter could lead to more pronounced GW breaking and dissipation processes in the upper stratosphere during daytime than during nighttime, and – subsequently – to more prominent GWs with larger vertical wavelengths of Lm≥5 km, which would be consistent with the observed GW characteristics at these altitudes presented by Baumgarten et al. (2018).

As mentioned in the introduction, the measurements of Baumgarten et al. (2017) show some evidence that the increase in the GWPED values with height is stronger during full-daytime than during nighttime by a factor of about 2, or, roughly assuming a 2:1 relation of daytime and nighttime (16 h daytime and 8 h nighttime) for high summer mid-latitudes, stronger during daytime than during nighttime by a factor of about 2.5. For comparison, the estimated effect of ozone–temperature coupling for these latitudes (factor of 1.7) is somewhat smaller and would lead to an increase in the nighttime GWPED in the order of ∼50 % (0.7:1.5) of the observed increase. Conclusively, although the difference derived by Baumgarten et al. (2017) might be uncertain as mentioned in the introduction, and although the approach of the present paper cannot entirely explain the details of specific local measurements during a specific time period, the comparison confirms that ozone–gravity wave interaction might be able to produce significant daytime–nighttime differences in the GW amplitudes at high summer mid-latitudes.

Current state-of-the-art general circulation models (GCMs) usually use a variety of prescribed tropospheric sources and tuning parameters in the gravity wave drag (GWD) parameterizations, forcing the middle atmospheric circulation (e.g., McLandress et al., 1998; Fritts and Alexander, 2003; Garcia et al., 2017), where the extreme low temperatures observed in the summer upper mesosphere provide an important benchmark for the quality of the upwelling branch and the associated adiabatic cooling produced by the models. Including ozone–gravity wave interaction into the GCMs might lead to a substantial improvement of the used GWDs and the associated processes driving the summer mesospheric circulation, because the related increase in the GWPED must lead to a similar increase in the vertical momentum flux term determining the GWD. However, the incorporation of ozone–gravity wave interaction into a state-of-the-art GCM using a GWD, or into a numerical model with resolved GWs, needs extensive test simulations, which is beyond the scope of the present paper.

In particular, current GCMs indicate significant changes in the time–mean circulation of the upper mesosphere due to the stratospheric ozone loss over Antarctica during southern spring and early summer via the induced changes in the GWD (Smith et al., 2010; Lossow et al., 2012; Lubi et al., 2016). Long-term changes in upper stratospheric ozone of up to −8 % per decade, derived from satellite measurements (e.g., Sofieva et al., 2017; WMO, 2018), could also affect the mesospheric circulation in the stratosphere and mesosphere by modulating the GW amplitudes and, hence, the GWD. Based on the idealized approach of the present paper, we estimate the sensitivity of the amplification of the GW amplitudes in the upper mesosphere on changes in the ozone background μ0 and the ozone-related heating rate Q0(μ0), revealing that, for horizontal and vertical wavelengths Lk≥500 km and Lm≤5 km, a change of ±10 % in μ0 or Q0 results in a change of ±10 % to ±20 % in the upper mesospheric GWPED. Conclusively, the summer mesospheric upwelling might be much more sensitive to the long-term changes in upper stratospheric ozone as has been suggested by the GCMs up to now.

In the approach of the present paper, the variations due to the diurnal cycle in stratospheric ozone and atmospheric tides are excluded to examine the potential effect of ozone–gravity wave interaction as clearly as possible, based on standard equations describing upward propagating GWs in a constant background. On the one hand, these variations can principally modulate the effect of ozone–gravity wave coupling by changing the planetary-scale background conditions for the propagation of the mesoscale GWs. Assuming – to a first order – linear modulations in the background ozone and background lapse rate according to observed diurnal or tidal variations, the sensitivity calculations of the present paper suggest that the related modulations in the amplitude amplification are smaller than the effect of ozone–gravity wave coupling by approximately 1 order. Further test calculations have shown that the use of a height-dependent scale height H(T0) instead of a constant scale height H0 at the levels of relevant amplification leads to stronger amplitude amplifications, particularly over the southern summer polar latitudes, because of the high temperatures in the stratopause region and the very low temperatures in the upper mesosphere, where the related differences are also smaller than the first-order process (e.g., in the GWPED, for vertical wavelengths between Lm=5 km and Lm=3 km, between about 7.5 % to 20 % at summer polar latitudes and less than 5 % at summer mid-latitudes).

On the other hand, short-term fluctuations in the balanced zonal and meridional winds due to atmospheric tides can principally lead to changes in the upward GW propagation characteristics, and, hence, to significant daytime–nighttime differences in the growth of the GW amplitudes with height, including nonlinear feedbacks of the propagating mesoscale GWs to the short-term balanced flow components. Further, multistep vertical coupling processes producing secondary GWs in the mesosphere could depend on daytime–nighttime conditions or tidal variations, which could also produce significant daytime–nighttime differences in the growth of the GW amplitudes with height. Considering the remarkably strong effect of ozone–gravity wave coupling suggested by the present paper, we may speculate that it significantly affects these possible changes in the GW amplitudes due to short-term fluctuations in the balanced winds or multistep vertical coupling. However, an unequivocal quantification of the effects of these processes and the involved nonlinear interactions of the daytime–nighttime differences in the GWPED needs much more investigations, e.g., based on extensive GW resolving model simulations with interactive ozone photochemistry, which is beyond the scope of the present paper.

The results of the present paper might stimulate further daytime–nighttime observations of GW activity, particularly at specific measurement sites where the GWs are usually characterized by specific horizontal and vertical wavelengths, e.g., downwind of specific mountain ridges (east of Rocky Mountains, Southern Andes or Norwegian Caledonides), which could be helpful to better understand how ozone–gravity wave coupling is operating in situ.

Data availability

Background data and programs visualizing the presented analytic solutions are available upon request from the author.

Competing interests

The contact author has declared that none of the authors has any competing interests.

Disclaimer

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

Acknowledgements

The author thanks Hauke Schmidt (Max Planck Institute for Meteorology (MPI-Met), Hamburg) for providing HAMMONIA background data. Thanks also to two reviewers for critical comments.

Financial support

The publication of this article was funded by the Open Access Fund of the Leibniz Association.

Review statement

This paper was edited by Mathias Palm and reviewed by two anonymous referees.

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Recent measurements show some evidence that the amplitudes of atmospheric gravity waves (horizontal wavelengths of 100–2000 km), which propagate from the troposphere (0–10 km) to the stratosphere and mesosphere (10–100 km), increase more strongly with height during daytime than during nighttime. This study shows that ozone–temperature coupling in the upper stratosphere can principally produce such an amplification. The results will help to improve atmospheric circulation models.
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