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 w′∂u0/∂z and w′∂v0/∂z are neglected because of the Wentzel–Kramers–Brillouin or WKB approximation: 1d0u′dt+1ρ0∂p′∂x=fv′,2d0v′dt+1ρ0∂p′∂y=-fu′,3d0w′dt+1ρ0∂p′∂z=gθ′θ0,4d0ρ′dt+∂u′∂x+∂v′∂y+1ρ0∂ρ0w′∂z=0,5d0θ′dt+w′∂θ0∂z=Q′p00pκ=aμ0d0μ′dt,6d0μ′dt+w′∂μ0∂z=S′=-bμ0d0θ′dt. For Q′=0, the dispersion relation for gravity waves results from Eqs. (1) to (5) by introducing sinusoidal perturbations X1′=Xa0⋅exp⁡[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 m12≫1/4H2, 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: 7ωi12=N02kh12+m12f2kh12+m12≈N02kh12m12+f2.

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=-w1′⋅∂θ0/∂z<0, which leads to an initial ozone perturbation μ1′>0 due to the induced increase d0μ1′/dt=-w1′⋅∂μ0/∂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=-w1′⋅∂θ0/∂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 8O3=k2k3M(O2)2J2(O2)J3(O3)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=8.0×10-12⋅exp⁡(-2060/T) cm3 s-1 are chemical reaction rates for ozone production (O+O2+M→O3+M) and ozone loss (O+O3→2O2) (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 9ΔμTμ0=12Δ(k2/k3)(k2/k3)=-122.3T0+2060T02ΔT≡-b0(T0)ΔT. 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)κ⋅d0θ′/dt, the change S′ is given by 10S′=ΔμTΔt=ΔμTΔTΔTΔt=-μ0bd0θ′dt, 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 (b→b⋅hz, 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 ΔQμ≈A⋅Δμ, 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, Q0≈A⋅μ0 and Q′≈A⋅μ′, 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 11Q′=Lm∂Q′∂z≈LmΔQμΔμ∂μ′∂z=τiQ0μ0d0μ′dt, 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.

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+ab)d0θ′dt+w′∂θ0∂z+aμ0∂μ0∂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: 13d0dtgθ0θ′+Nμ2w′=0, with 14Nμ2=N02+Nc2(1+ab), 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′=Xμ0⋅exp⁡[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: 15ωi22=Nμ2kh22+m22f2kh22+m22≈Nμ2kh22m22+f2, 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 w1′N02 leads to dθ2′/dt=dθ1′/dt or -iωi2θ2′=-iωi1θ1′. Furthermore, combining -iωi2θ2′=-Nμ2 w2′ and -iωi1θ1′=-N02w1′ suggests that the amplitude θμ=θμ0⋅exp⁡(z-zs)/2H is stronger than θa=θa0⋅exp⁡(z-zs)/2H by the factor ωi1/ωi2=N02/Nμ2≥1: 16θμ=θa⋅(ωi1/ωi2). 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)/(N02⋅N02), 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∘).

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).

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=θa0⋅exp⁡(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=ps⋅exp⁡(z-zs)/H. Introducing the associated perturbation w1′=-(∂θ0/∂z)-1⋅d0θ1′/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 μ′≈0.2 to 0.5 ppm (Fig. 2c) or μ′/μ0≈5 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′)=aωi1μ1′μ0-1 (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).

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.

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 17Tμn(z)=Ta(z)⋅∏j=1nωi1(zj)ωi2(zj), where the product symbol Πj=1,n denotes the multiplication with ωi1(zj)/ωi2(zj) at each level z1≤zj≤zn. 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 18Tμ(p)=hs(p)⋅Ta(p)+hm(p)⋅Tμn(p), with weighting functions hs=p01.5/(p01.5+pm1.5) and hm=1-hs, where p0 is the background pressure and pm(70∘S)≈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-0.5⋅τ0)/(0.5⋅τ0), 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 Δz=ΔzH 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.

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): 19E=12gN2T′T02. 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.

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/Ta≈1.5 and Eμ/Ea≈2.4 over southern summer polar latitudes towards Tμ/Ta≈1.2 and Eμ/Ea≈1.4 at lower mid-latitudes (40∘ S), and then less rapidly towards Tμ/Ta≈1.1 and Eμ/Ea≈1.2 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.

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 (u′w′) 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 =-ρ0-1∂FGW/∂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.

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=θa0⋅exp⁡(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).