**Research article**| 16 Aug 2022

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

Axel Gabriel

**Axel Gabriel**Axel Gabriel

- Department Modelling of Atmospheric Processes, Leibniz-Institute of Atmospheric Physics (IAP) at the University of Rostock e.V., Kühlungsborn, Germany

- 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)

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

Received: 20 Dec 2021 – Discussion started: 03 Jan 2022 – Revised: 11 Jul 2022 – Accepted: 21 Jul 2022 – Published: 16 Aug 2022

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.

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 ($\sim {e}^{z/\mathrm{2}H}$, 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 $\mathrm{1}\times {\mathrm{10}}^{-\mathrm{2}}$ J m^{−3} (or 10 J kg^{−1}) for full-day measurements but about 0$.5\times {\mathrm{10}}^{-\mathrm{2}}$ 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 *L*_{m}<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 *μ*_{0}(*z*) decreases with
height ($\partial {\mathit{\mu}}_{\mathrm{0}}/\partial z<\mathrm{0}$). 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 ${w}^{\prime}>\mathrm{0}$) must lead to an increase
($\partial {\mathit{\mu}}^{\prime}/\partial t>\mathrm{0}$) by both transport
(because $-{w}^{\prime}\partial {\mathit{\mu}}_{\mathrm{0}}/\partial z>\mathrm{0}$)
and photochemistry (because the temperature-dependent ozone production
increases in the case of adiabatic cooling), and, hence, in the heating rate
(${Q}^{\prime}\left({\mathit{\mu}}^{\prime}\right)>\mathrm{0}$), comparable to the latent heat
release in the troposphere in the case of condensation. Then, the induced
perturbation ($\mathrm{\Delta}{\mathit{\theta}}^{\prime}>\mathrm{0}$, where *θ* is
potential temperature) reinforces the initial ascent, where the lapse rate
($\partial ({\mathit{\theta}}_{\mathrm{0}}+\mathrm{\Delta}{\mathit{\theta}}^{\prime})/\partial z<\partial {\mathit{\theta}}_{\mathrm{0}}/\partial z$) decreases ($\partial z=$ 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 ${w}^{\prime}<\mathrm{0}$) leads to a decrease ($\partial {\mathit{\mu}}^{\prime}/\partial t<\mathrm{0}$) and a corresponding change (${Q}^{\prime}\left({\mathit{\mu}}^{\prime}\right)<\mathrm{0}$), 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.

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 *T*_{0}, ozone *μ*_{0},
and short-wave heating rate *Q*_{0} 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 *Q*_{0} (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 *Q*_{0} is particularly strong in the USLM where $\partial {\mathit{\mu}}_{\mathrm{0}}/\partial z<\mathrm{0}$ (the dashed line in Fig. 1b indicates $\partial {\mathit{\mu}}_{\mathrm{0}}/\partial z=\mathrm{0}$). 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; ${\mathit{\theta}}^{\prime}={T}^{\prime}$ (${p}_{\mathrm{00}}/p{)}^{\mathit{\kappa}}$ is
potential temperature;, *p*(*z*) is pressure; *p*_{00}=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 ${Q}^{\prime}\left({\mathit{\mu}}^{\prime}\right)$ 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 $a(\mathit{\varphi},z)>\mathrm{0}$ and $b(\mathit{\varphi},z)>\mathrm{0}$ are linear coupling parameters as a function of latitude *ϕ* and
altitude *z* specified below, ${\mathit{\rho}}_{\mathrm{0}}\left(z\right)={\mathit{\rho}}_{\mathrm{00}}{\mathrm{exp}}^{-(z-z\mathrm{0})/H}$ is background density, *H*∼7 km is scale height, *ρ*_{00} is a reference value at altitude *z*_{0}, *u*_{0} is a zonal mean
background wind, ${d}_{\mathrm{0}}/\mathrm{d}t=\partial /\partial t+{u}_{\mathrm{0}}\partial /\partial x+{v}_{\mathrm{0}}\partial /\partial y$, where $\partial /\partial x$ and $\partial /\partial y$ denote the derivations in longitude and
latitude, *g* is the gravity acceleration, and *f* is the Coriolis parameter; the background shear terms ${w}^{\prime}\partial {u}_{\mathrm{0}}/\partial z$ and ${w}^{\prime}\partial {v}_{\mathrm{0}}/\partial z$ are neglected because of the
Wentzel–Kramers–Brillouin or WKB approximation:

For ${Q}^{\prime}=\mathrm{0}$, the dispersion relation for gravity waves results from Eqs. (1) to (5) by introducing sinusoidal perturbations ${X}_{\mathrm{1}}^{\prime}={X}_{\mathrm{a}\mathrm{0}}\cdot \mathrm{exp}\left[i\right({k}_{\mathrm{1}}x+{l}_{\mathrm{1}}y+{m}_{\mathrm{1}}z-{\mathit{\omega}}_{\mathrm{1}}t\left)\right]\cdot {\mathrm{exp}}^{(z-zs)/\mathrm{2}H}$, where ${X}_{\mathrm{1}}^{\prime}$ denotes the
perturbation quantities, *X*_{a0} the initial amplitude at altitude *z**s* at
the lower boundary of the upper stratosphere, ${\mathrm{exp}}^{(z-zs)/\mathrm{2}H}$ the
exponential growth of the amplitude due to decreasing density, *k*_{1} and
*l*_{1} the horizontal and meridional wave number, *m*_{1}<0 the
vertical wave number for upward propagating GWs with
$\left|{m}_{\mathrm{1}}\right|=\mathrm{2}\mathit{\pi}/{L}_{\mathrm{m}\mathrm{1}}$ and vertical wavelength *L*_{m1}, and
*ω*_{1} the frequency (here, the subscript 1 denotes the solutions
for ${Q}^{\prime}=\mathrm{0}$). We focus on horizontal and vertical wavelengths
*L*_{h1}≥50 km and *L*_{m1}≤15 km, where ${k}_{\mathrm{h}\mathrm{1}}=\mathrm{2}\mathit{\pi}/{L}_{\mathrm{h}\mathrm{1}}$ is the horizontal wave number given by
${k}_{\mathrm{h}\mathrm{1}}=({k}_{\mathrm{1}}^{\mathrm{2}}+{l}_{\mathrm{1}}^{\mathrm{2}}{)}^{\mathrm{1}/\mathrm{2}}$,
therefore ($\mathrm{1}+{k}_{\mathrm{h}\mathrm{1}}^{\mathrm{2}}/{m}_{\mathrm{1}}^{\mathrm{2}})\approx \mathrm{1}$. Compressibility effects due to the vertical change in background
density are excluded assuming ${m}_{\mathrm{1}}^{\mathrm{2}}\gg \mathrm{1}/\mathrm{4}{H}^{\mathrm{2}}$, which is valid for vertical wavelengths
*L*_{m}≤30 km. Then, the dispersion relation for the intrinsic
frequency ${\mathit{\omega}}_{i\mathrm{1}}={\mathit{\omega}}_{\mathrm{1}}-{k}_{\mathrm{1}}{u}_{\mathrm{0}}$ is given for
the frequency range ${N}_{\mathrm{0}}^{\mathrm{2}}>{\mathit{\omega}}_{i\mathrm{1}}^{\mathrm{2}}>{f}^{\mathrm{2}}$, where ${N}_{\mathrm{0}}^{\mathrm{2}}=(g/{\mathit{\theta}}_{\mathrm{0}})\cdot \partial {\mathit{\theta}}_{\mathrm{0}}/\partial z$ denotes the Brunt–Vaisala frequency:

### 2.1.2 Ozone–temperature coupling

For specifying the parameter *b*, we consider the vertical ascent ${w}_{\mathrm{1}}^{\prime}>\mathrm{0}$ in the wave crest of an initial sinusoidal GW
perturbation, related to an adiabatic cooling term ${d}_{\mathrm{0}}{\mathit{\theta}}_{\mathrm{1}}^{\prime}/\mathrm{d}t=-{w}_{\mathrm{1}}^{\prime}\cdot \partial {\mathit{\theta}}_{\mathrm{0}}/\partial z<\mathrm{0}$, which leads to an initial ozone perturbation ${\mathit{\mu}}_{\mathrm{1}}^{\prime}>\mathrm{0}$ due to the induced increase ${d}_{\mathrm{0}}{\mathit{\mu}}_{\mathrm{1}}^{\prime}/\mathrm{d}t=-{w}_{\mathrm{1}}^{\prime}\cdot \partial {\mathit{\mu}}_{\mathrm{0}}/\partial z>\mathrm{0}$ via transport, and to a change in ozone photochemistry
described by ${S}^{\prime}\left({T}_{\mathrm{1}}^{\prime}\right)$ (for the descent ${w}_{\mathrm{1}}^{\prime}<\mathrm{0}$ in the wave trough, the formulations are analogous but
with ${\mathit{\mu}}_{\mathrm{1}}^{\prime}<\mathrm{0}$ and ${d}_{\mathrm{0}}{\mathit{\theta}}_{\mathrm{1}}^{\prime}/\mathrm{d}t=-{w}_{\mathrm{1}}^{\prime}\cdot \partial {\mathit{\theta}}_{\mathrm{0}}/\partial z>\mathrm{0}$). 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

where *J*_{2}(O_{2}) and *J*_{3}(O_{3}) are photo-dissociation rates,
and ${k}_{\mathrm{2}}=\mathrm{6.0}\times {\mathrm{10}}^{-\mathrm{34}}\cdot (\mathrm{300}/T{)}^{\mathrm{2.3}}$ cm^{6} s^{−1}
and ${k}_{\mathrm{3}}=\mathrm{8.0}\times {\mathrm{10}}^{-\mathrm{12}}\cdot \mathrm{exp}(-\mathrm{2060}/T)$ cm^{3} s^{−1} are chemical reaction rates for ozone production (O+O_{2}+M→O_{3}+M) and ozone loss (O+O_{3}→2O_{2}) (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 $\mathrm{\Delta}{\mathit{\mu}}_{T}/{\mathit{\mu}}_{\mathrm{0}}=\mathrm{\Delta}$O_{3}/O_{3} due to a change in temperature Δ*T* is given by

Then, defining $b={b}_{\mathrm{0}}\cdot (p/{p}_{\mathrm{00}}{)}^{\mathit{\kappa}}$ and introducing a
total temperature change $\mathrm{\Delta}T/\mathrm{\Delta}t$ within a background flow
described by ${d}_{\mathrm{0}}{T}^{\prime}/\mathrm{d}t=(p/{p}_{\mathrm{00}}{)}^{\mathit{\kappa}}\cdot {d}_{\mathrm{0}}{\mathit{\theta}}^{\prime}/\mathrm{d}t$, the change *S*^{′} is given by

which is the right-hand term of Eq. (6). Overall, the initial ascent
${w}_{\mathrm{1}}^{\prime}>\mathrm{0}$ leads to an increase in ozone via transport,
and the related adiabatic cooling to an increase in ozone because of the
induced change ${S}^{\prime}>\mathrm{0}$. Analogously, the initial descent
${w}_{\mathrm{1}}^{\prime}<\mathrm{0}$ leads to a decrease in ozone via transport and
an induced change ${S}^{\prime}<\mathrm{0}$. The height-dependence of *b* is
specified by considering that the ozone photochemistry of the USLM region is
related to the spatial structure of *Q*_{0}, which is characterized by a
Gaussian-type height-dependence centered at the maximum of *Q*_{0} and a rapid decrease with latitude in the extratropical winter hemisphere (see Fig. 1c). Therefore, *b* is multiplied with the normalized factor
$hz={Q}_{\mathrm{0}}/{Q}_{\mathrm{00}}$, where *Q*_{00} is the averaged profile of *Q*_{0}
over the summer hemisphere ($b\to b\cdot hz$, where *h**z*(*z*)≈1 in
the summer upper stratosphere at the altitude where *Q*_{0} 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 $\mathrm{\Delta}{Q}_{\mathit{\mu}}\approx A\cdot \mathrm{\Delta}\mathit{\mu}$, where $A=A(\mathit{\varphi},z)$ 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, ${Q}_{\mathrm{0}}\approx A\cdot {\mathit{\mu}}_{\mathrm{0}}$ and ${Q}^{\prime}\approx A\cdot {\mathit{\mu}}^{\prime}$, we may write $\mathrm{\Delta}{Q}_{\mathit{\mu}}/\mathrm{\Delta}\mathit{\mu}={Q}_{\mathrm{0}}/{\mathit{\mu}}_{\mathrm{0}}={Q}^{\prime}/{\mathit{\mu}}^{\prime}$. At a
specific altitude *z* or pressure level *p*(*z*), we consider a GW perturbation
over the vertical scale of a vertical wavelength, Δ*z*=*L*_{m}.
Then, considering that $\partial {\mathit{\mu}}^{\prime}/\partial z=im{\mathit{\mu}}^{\prime}=({\mathit{\tau}}_{i}/{L}_{\mathrm{m}})\cdot (-i{\mathit{\omega}}_{i}{\mathit{\mu}}^{\prime})$ with ${\mathit{\tau}}_{i}=\mathrm{2}\mathit{\pi}/{\mathit{\omega}}_{i}$, the first-order
heating rate perturbation is given by

which is the right-hand side of Eq. (5) when defining *a*_{0}=*τ*_{i}*Q*_{0} and $a={a}_{\mathrm{0}}\cdot ({p}_{\mathrm{00}}/p{)}^{\mathit{\kappa}}$. Except in
polar summer regions, the effect of *Q*^{′} is limited by the length of
daytime (here denoted by *τ*_{day}) in case of large wave periods.
Therefore, we set the time increment to *τ*_{i}=*τ*_{day} in the
case of *τ*_{i}>*τ*_{day}, which reduces the effect
of *Q*^{′} during the time period of 24 h (e.g., *τ*_{i}≤12 h over the Equator). Overall, reassuming an initial ascent
${w}_{\mathrm{1}}^{\prime}>\mathrm{0}$, the induced increase in ozone ${\mathit{\mu}}^{\prime}>\mathrm{0}$ at a pressure level *p*(*z*) leads to a heating rate
perturbation ${Q}^{\prime}>\mathrm{0}$ at this level counteracting to the
initial adiabatic cooling and therefore reinforcing the initial ascent.
Analogously, an initial descent ${w}_{\mathrm{1}}^{\prime}<\mathrm{0}$ is reinforced by
inducing a perturbation ${Q}^{\prime}<\mathrm{0}$.

Note here that the use of Δ*z*=*L*_{m} 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 *L*_{m}. It is also possible to set a smaller vertical scale Δ*z*<*L*_{m}
leading to smaller values ${Q}_{\mathrm{\Delta}z}^{\prime}=(\mathrm{\Delta}z/{L}_{\mathrm{m}})\cdot {Q}^{\prime}$ at a specific level, where Δ*z* 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
*L*_{m} 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
${d}_{\mathrm{0}}{\mathit{\mu}}^{\prime}/\mathrm{d}t$ of Eq. (6) into (Eq. 5):

Here, the amplification factor 1+*a**b* (with *a**b*>0) describes the
feedback of the GW-induced ozone perturbation to the change in potential
temperature, and $\partial {\mathit{\theta}}_{\mathrm{0}}/\partial z+(a/{\mathit{\mu}}_{\mathrm{0}})\cdot \partial {\mathit{\mu}}_{\mathrm{0}}/\partial z$ an ozone adiabatic lapse rate which is – in
the USLM region – smaller than $\partial {\mathit{\theta}}_{\mathrm{0}}/\partial z$
because of $\partial {\mathit{\mu}}_{\mathrm{0}}/\partial z<\mathrm{0}$. Alternatively,
we may write:

with

where ${N}_{c}^{\mathrm{2}}=(g/{\mathit{\theta}}_{\mathrm{0}})\cdot (a/{\mathit{\mu}}_{\mathrm{0}})\cdot \partial {\mathit{\mu}}_{\mathrm{0}}/\partial z$. As with the lapse rate, ${N}_{\mathit{\mu}}^{\mathrm{2}}$ is smaller than ${N}_{\mathrm{0}}^{\mathrm{2}}$ because ${N}_{c}^{\mathrm{2}}<\mathrm{0}$ and $(\mathrm{1}+ab)>\mathrm{1}$. If ozone–temperature coupling becomes weak, below and above the USLM region, ${N}_{\mathit{\mu}}^{\mathrm{2}}$ converges to ${N}_{\mathrm{0}}^{\mathrm{2}}$.

Analogous to the standard solution given above, we introduce sinusoidal GW perturbations of the form ${X}_{\mathrm{2}}^{\prime}={X}_{\mathit{\mu}\mathrm{0}}\cdot \mathrm{exp}\left[i\right({k}_{\mathrm{2}}x+{l}_{\mathrm{2}}y+{m}_{\mathrm{2}}z-{\mathit{\omega}}_{\mathrm{2}}t\left)\right]\cdot {\mathrm{exp}}^{(z-zs)/\mathrm{2}H}$ 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:

where ${\mathit{\omega}}_{i\mathrm{2}}={\mathit{\omega}}_{\mathrm{2}}-{k}_{\mathrm{2}}{u}_{\mathrm{0}}$ and ${k}_{h\mathrm{2}}=({k}_{\mathrm{2}}^{\mathrm{2}}+{l}_{\mathrm{2}}^{\mathrm{2}}{)}^{\mathrm{1}/\mathrm{2}}$.

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
$\mathrm{d}{\mathit{\theta}}_{\mathrm{1}}^{\prime}/\mathrm{d}t$, either with or without ozone–temperature
coupling, leads to ${w}_{\mathrm{2}}^{\prime}={w}_{\mathrm{1}}^{\prime}\cdot ({N}_{\mathrm{0}}^{\mathrm{2}}/{N}_{\mathit{\mu}}^{\mathrm{2}})$. Consistently,
introducing the same initial perturbation ${w}_{\mathrm{1}}^{\prime}{N}_{\mathrm{0}}^{\mathrm{2}}$ leads to $\mathrm{d}{\mathit{\theta}}_{\mathrm{2}}^{\prime}/\mathrm{d}t=\mathrm{d}{\mathit{\theta}}_{\mathrm{1}}^{\prime}/\mathrm{d}t$ or $-i{\mathit{\omega}}_{i\mathrm{2}}{\mathit{\theta}}_{\mathrm{2}}^{\prime}=-i{\mathit{\omega}}_{i\mathrm{1}}{\mathit{\theta}}_{\mathrm{1}}^{\prime}$. Furthermore, combining
$-i{\mathit{\omega}}_{i\mathrm{2}}{\mathit{\theta}}_{\mathrm{2}}^{\prime}=-{N}_{\mathit{\mu}}^{\mathrm{2}}$
${w}_{\mathrm{2}}^{\prime}$ and $-i{\mathit{\omega}}_{i\mathrm{1}}{\mathit{\theta}}_{\mathrm{1}}^{\prime}=-{N}_{\mathrm{0}}^{\mathrm{2}}{w}_{\mathrm{1}}^{\prime}$ suggests that the amplitude
${\mathit{\theta}}_{\mathit{\mu}}={\mathit{\theta}}_{\mathit{\mu}\mathrm{0}}\cdot {\mathrm{exp}}^{(z-zs)/\mathrm{2}H}$ is
stronger than ${\mathit{\theta}}_{a}={\mathit{\theta}}_{\mathrm{a}\mathrm{0}}\cdot {\mathrm{exp}}^{(z-zs)/\mathrm{2}H}$ by
the factor ${\mathit{\omega}}_{i\mathrm{1}}/{\mathit{\omega}}_{i\mathrm{2}}={N}_{\mathrm{0}}^{\mathrm{2}}/{N}_{\mathit{\mu}}^{\mathrm{2}}\ge \mathrm{1}$:

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 ${\mathit{\omega}}_{i\mathrm{1}}/{\mathit{\omega}}_{i\mathrm{2}}$ or
${N}_{\mathrm{0}}^{\mathrm{2}}/{N}_{\mathit{\mu}}^{\mathrm{2}}$. Note that the vertical
variations in ${N}_{\mathrm{0}}^{\mathrm{2}}$ could affect the increase in amplitude
with height, particularly in the summer upper mesosphere. Therefore,
${N}_{\mathrm{0}}^{\mathrm{2}}$ 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
${\mathit{\omega}}_{i\mathrm{1}}/{\mathit{\omega}}_{i\mathrm{2}}={N}_{\mathrm{0}}^{\mathrm{2}}/{N}_{\mathit{\mu}}^{\mathrm{2}}$ implies not only a change in amplitude but also a slight
change in the relation of horizontal and vertical wave numbers described by
$({k}_{h\mathrm{2}}/{m}_{\mathrm{2}})=({N}_{\mathit{\mu}}^{\mathrm{2}}/{N}_{\mathrm{0}}^{\mathrm{2}})({k}_{\mathrm{h}\mathrm{1}}/{m}_{\mathrm{1}})+{f}^{\mathrm{2}}({N}_{\mathit{\mu}}^{\mathrm{2}}-{N}_{\mathrm{0}}^{\mathrm{2}})/({N}_{\mathrm{0}}^{\mathrm{2}}\cdot {N}_{\mathrm{0}}^{\mathrm{2}})$, 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 $\mathrm{cos}\left(\mathit{\alpha}\right)=\pm ({k}_{h}/m)$. 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 ${L}_{\mathrm{m}\mathrm{1}}/{L}_{kh\mathrm{1}}<\mathrm{0.05}$, or wave
periods *τ*_{i}>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+*a**b* and the quotient ${N}_{\mathrm{0}}^{\mathrm{2}}/{N}_{\mathit{\mu}}^{\mathrm{2}}$ for a GW with horizontal and vertical
wavelengths *L*_{k}=500 km and *L*_{m}=5 km, and the quotient
${N}_{\mathrm{0}}^{\mathrm{2}}/{N}_{\mathit{\mu}}^{\mathrm{2}}$ for a GW with
*L*_{k}=800 km and *L*_{m}=3 km. In the first example, the factor 1+*a**b*
(Fig. 1d) contributes to the amplification of the GW amplitude at a
specific level by up to 6 %–8 %, and the overall factor
$({N}_{\mathrm{0}}^{\mathrm{2}}/N{\mathit{\mu}}^{\mathrm{2}}=(\mathrm{1}+ab)\cdot {N}_{\mathrm{0}}^{\mathrm{2}}/({N}_{\mathrm{0}}^{\mathrm{2}}+{N}_{c}^{\mathrm{2}})$ (Fig. 1e) by up to 8 %–12 % (including a
decrease in the lapse rate of up to 3 % described by $({N}_{\mathrm{0}}^{\mathrm{2}}+{N}_{c}^{\mathrm{2}})/{N}_{\mathrm{0}}^{\mathrm{2}}$, not shown here). The second example (Fig. 1f) shows that the factor ${N}_{\mathrm{0}}^{\mathrm{2}}/{N}_{\mathit{\mu}}^{\mathrm{2}}$ 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 *τ*_{i}>*τ*_{day}).

For illustration of the induced change in ozone at a specific level (Fig. 2a–d), we assume an initial GW perturbation ${\mathit{\theta}}_{\mathrm{1}}^{\prime}$ with
exponentially growing amplitude ${\mathit{\theta}}_{a}={\mathit{\theta}}_{\mathrm{a}\mathrm{0}}\cdot {\mathrm{exp}}^{(z-zs)/\mathrm{2}H}$, with an initial temperature amplitude *T*_{a0} of 1 K at
*z**s*≈35 km (*p**s*=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 ${\mathit{\theta}}_{a}={\mathit{\theta}}_{\mathrm{a}\mathrm{0}}\cdot (ps/p{)}^{\mathrm{1}/\mathrm{2}}$, assuming
$p=ps\cdot {\mathrm{exp}}^{(z-zs)/H}$. Introducing the associated perturbation
${w}_{\mathrm{1}}^{\prime}=-(\partial {\mathit{\theta}}_{\mathrm{0}}/\partial z{)}^{-\mathrm{1}}\cdot {d}_{\mathrm{0}}{\mathit{\theta}}_{\mathrm{1}}^{\prime}/\mathrm{d}t$ in Eq. (6) leads to ${d}_{\mathrm{0}}{\mathit{\mu}}_{\mathrm{1}}^{\prime}/\mathrm{d}t=\left[\right(\partial {\mathit{\mu}}_{\mathrm{0}}/\partial z)/(\partial {\mathit{\theta}}_{\mathrm{0}}/\partial z)-b{\mathit{\mu}}_{\mathrm{0}}]\cdot {d}_{\mathrm{0}}{\mathit{\theta}}_{\mathrm{1}}^{\prime}/\mathrm{d}t$, and, considering ${d}_{\mathrm{0}}{\mathit{\mu}}_{\mathrm{1}}^{\prime}/\mathrm{d}t=-i{\mathit{\omega}}_{i\mathrm{1}}{\mathit{\mu}}_{\mathrm{1}}^{\prime}$ and ${d}_{\mathrm{0}}{\mathit{\theta}}_{\mathrm{1}}^{\prime}/\mathrm{d}t=-i{\mathit{\omega}}_{i\mathrm{1}}{\mathit{\theta}}_{\mathrm{1}}^{\prime}$, to an initial
ozone perturbation ${\mathit{\mu}}_{\mathrm{1}}^{\prime}={\mathit{\theta}}_{\mathrm{1}}^{\prime}\cdot \left[\right(\partial {\mathit{\mu}}_{\mathrm{0}}/\partial z)/(\partial {\mathit{\theta}}_{\mathrm{0}}/\partial z)-b{\mathit{\mu}}_{\mathrm{0}}$]. For the example of the ascent
(${w}_{\mathrm{1}}^{\prime}>\mathrm{0}$) shown in Fig. 2, we set ${\mathit{\theta}}_{\mathrm{1}}^{\prime}<\mathrm{0}$, leading to ${\mathit{\mu}}_{\mathrm{1}}^{\prime}>\mathrm{0}$. For
*L*_{k}=500 km and *L*_{m}=5 km, the contributions ${\mathit{\mu}}^{\prime}\left(TR\right)={\mathit{\theta}}_{\mathrm{1}}^{\prime}\cdot \left[\right(\partial {\mathit{\mu}}_{\mathrm{0}}/\partial z)/(\partial {\mathit{\theta}}_{\mathrm{0}}/\partial z\left)\right]$ (related to transport; Fig. 2a) and ${\mathit{\mu}}^{\prime}\left(CH\right)=-b{\mathit{\mu}}_{\mathrm{0}}{\mathit{\theta}}_{\mathrm{1}}^{\prime}$ (related to *S*^{′}; Fig. 2b) sum up to a total change of ${\mathit{\mu}}^{\prime}\approx \mathrm{0.2}$ to 0.5 ppm (Fig. 2c) or ${\mathit{\mu}}^{\prime}/{\mathit{\mu}}_{\mathrm{0}}\approx \mathrm{5}$ 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
(${w}_{\mathrm{2}}^{\prime}-{w}_{\mathrm{1}}^{\prime}\left)\right(\partial {\mathit{\theta}}_{\mathrm{0}}/\partial z)={Q}^{\prime}({\mathit{\mu}}_{\mathrm{1}}^{\prime})$, or, when introducing ${w}_{\mathrm{2}}^{\prime}=({\mathit{\omega}}_{i\mathrm{1}}/{\mathit{\omega}}_{i\mathrm{2}})\cdot {w}_{\mathrm{1}}^{\prime}$ to
${Q}^{\prime}\left({\mathit{\mu}}_{\mathrm{1}}^{\prime}\right)=({\mathit{\omega}}_{i\mathrm{1}}/{\mathit{\omega}}_{i\mathrm{2}}-\mathrm{1})(-{\mathit{\omega}}_{i\mathrm{1}}{\mathit{\theta}}_{\mathrm{1}}^{\prime})=a{\mathit{\omega}}_{i\mathrm{1}}{\mathit{\mu}}_{\mathrm{1}}^{\prime}{\mathit{\mu}}_{\mathrm{0}}^{-\mathrm{1}}$ (where ${Q}^{\prime}\left({\mathit{\mu}}_{\mathrm{1}}^{\prime}\right)>\mathrm{0}$ in case of ${w}_{\mathrm{1}}^{\prime}>\mathrm{0}$),
Fig. 2e shows that ${Q}^{\prime}\left({\mathit{\mu}}_{\mathrm{1}}^{\prime}\right)$ 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 ${\mathit{\theta}}_{\mathrm{2}}^{\prime}-{\mathit{\theta}}_{\mathrm{1}}^{\prime}=({\mathit{\omega}}_{i\mathrm{1}}/{\mathit{\omega}}_{i\mathrm{2}}-\mathrm{1})\cdot {\mathit{\theta}}_{\mathrm{1}}^{\prime}$ where $({\mathit{\omega}}_{i\mathrm{1}}/{\mathit{\omega}}_{i\mathrm{2}}-\mathrm{1})=-a{\mathit{\mu}}_{\mathrm{0}}^{-\mathrm{1}}\left[\right(\partial {\mathit{\mu}}_{\mathrm{0}}/\partial z)/(\partial {\mathit{\theta}}_{\mathrm{0}}/\partial z)-b{\mathit{\mu}}_{\mathrm{0}}]$ 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 ${\mathit{\omega}}_{i\mathrm{1}}/{\mathit{\omega}}_{i\mathrm{2}}$ becomes smaller in the case of increasing vertical and decreasing horizontal wavelengths, or decreasing frequencies, as illustrated in Fig. 3 for an altitude where ${\mathit{\omega}}_{i\mathrm{1}}/{\mathit{\omega}}_{i\mathrm{2}}$ reach
maximum values (1.156 hPa or ≈47 km altitude). Figure 3a shows
values of ${\mathit{\omega}}_{i\mathrm{1}}/{\mathit{\omega}}_{i\mathrm{2}}>\mathrm{1.02}$ for wave
periods of *τ*_{i}>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 ${\mathit{\omega}}_{i\mathrm{1}}^{\mathrm{2}}>{f}^{\mathrm{2}}$).

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 ${L}_{\mathrm{m}}/{L}_{\mathrm{k}}$
denoting the conditions where *τ*_{i}>*τ*_{day}
(Fig. 3c, shaded area). Figure 3 also indicates that the examples with
*L*_{k}=500 km and *L*_{m}=5 km (Figs. 1e; 2) and *L*_{k}=800 km and *L*_{m}=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 ${\mathit{\omega}}_{i\mathrm{1}}/{\mathit{\omega}}_{i\mathrm{2}}-\mathrm{1}>\mathrm{0}$, 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.

## 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 *z*_{j−1}, is
the initial perturbation amplified at the next level *z*_{j}. For
illustration (Fig. 4a–c), we choose an initial GW with horizontal and
vertical wavelengths *L*_{m}=500 km and *L*_{m}=5 km as above, where the
vertical distance between the levels *z*_{j−1} and *z*_{j} is set by the
initial vertical wavelength Δ*z*=*L*_{m}. 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 ${\mathit{\omega}}_{i\mathrm{1}}/{\mathit{\omega}}_{i\mathrm{2}}$ for *L*_{k}=500 km and *L*_{m}=5 km at 70^{∘} S (solid),
and, for comparison, for *L*_{m}=3 (dashed) and *L*_{m}=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
*z**s*≈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
${z}_{j}=zs+(j-\mathrm{1})\cdot \mathrm{\Delta}z$ (*j*=1, *n*; here *n*=8), the amplitude
at *z*_{j} will be amplified by the factor ${\mathit{\omega}}_{i\mathrm{1}}\left({z}_{j}\right)/{\mathit{\omega}}_{i\mathrm{2}}\left({z}_{j}\right)$ at *z*_{j}. Starting with an exponentially growing amplitude ${T}_{\mathrm{a}}\left(z\right)={T}_{\mathrm{a}}\left(zs\right)\cdot {\mathrm{exp}}^{(z-zs)/\mathrm{2}H}$ (where we set *T*_{a}(*z**s*)=1 K again), we yield a new amplitude
${T}_{\mathrm{a}\mathrm{1}}\left({z}_{\mathrm{1}}\right)={T}_{\mathrm{a}}\left({z}_{\mathrm{1}}\right)\cdot {\mathit{\omega}}_{i\mathrm{1}}\left({z}_{\mathrm{1}}\right)/{\mathit{\omega}}_{i\mathrm{2}}\left({z}_{\mathrm{1}}\right)$ at the level *z*_{1}, defining a
new exponentially growing amplitude ${T}_{\mathit{\mu}\mathrm{1}}\left(z\right)={T}_{\mathrm{a}\mathrm{1}}\left({z}_{\mathrm{1}}\right)\cdot {\mathrm{exp}}^{(z-z\mathrm{1})/\mathrm{2}H}$. We then yield ${T}_{\mathrm{a}\mathrm{2}}\left({z}_{\mathrm{2}}\right)={T}_{\mathit{\mu}\mathrm{1}}\left({z}_{\mathrm{2}}\right)\cdot {\mathit{\omega}}_{i\mathrm{1}}\left({z}_{\mathrm{2}}\right)/{\mathit{\omega}}_{i\mathrm{2}}\left({z}_{\mathrm{2}}\right)$ at
the level *z*_{2}, defining ${T}_{\mathit{\mu}\mathrm{2}}\left(z\right)={T}_{\mathrm{a}\mathrm{2}}\left({z}_{\mathrm{2}}\right)\cdot {\mathrm{exp}}^{(z-z\mathrm{2})/\mathrm{2}H}$, and so forth. Finally, the amplitude at the level *z*_{n}
in the middle mesosphere is described by

where the product symbol ${\mathrm{\Pi}}_{j=\mathrm{1},\phantom{\rule{0.125em}{0ex}}n}$ denotes the multiplication with
${\mathit{\omega}}_{i\mathrm{1}}\left({z}_{j}\right)/{\mathit{\omega}}_{i\mathrm{2}}\left({z}_{j}\right)$ at each level
${z}_{\mathrm{1}}\le {z}_{j}\le {z}_{n}$. As mentioned above, the solutions
are calculated on pressure levels, i.e., *z* represents the geopotential
height, and the vertical distance Δ*z* between the levels is given by
$\mathrm{\Delta}z=-({\mathit{\rho}}_{\mathrm{0}}g{)}^{-\mathrm{1}}\mathrm{\Delta}p=-H({T}_{\mathrm{0}})\cdot (\mathrm{\Delta}p/p)$, where $H\left({T}_{\mathrm{0}}\right)=g/\left(R{T}_{\mathrm{0}}\right)$ is the height-dependent
scale height defined by the background. Note that using a constant
scale height *H*_{0}=7 km instead of *H*(*T*_{0}) only leads to second-order
changes in the cumulative amplitude amplification (the sensitivity test is
described below in Sect. 2.2.4), because *H*(*T*_{0}) 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 *T*_{a} (blue line) and the series of the successively amplified amplitudes *T*_{μ1}, *T*_{μ2},
…, *T*_{μn} (from light blue towards red line). Figure 4c shows the related series of constant relative values ${T}_{\mathit{\mu}\mathrm{1}}/{T}_{\mathrm{a}}$,
${T}_{\mathit{\mu}\mathrm{2}}/{T}_{\mathrm{a}}$, …, ${T}_{\mathit{\mu}n}/{T}_{\mathrm{a}}$, starting at the
level *z*_{j} (solid lines) together with the previous values starting at
*z*_{j−1}, multiplied by the factor ${\mathit{\omega}}_{i\mathrm{1}}/{\mathit{\omega}}_{i\mathrm{2}}$ (dotted
lines), illustrating the successively increasing growth of the amplitude
during the upward level-by-level propagation. Finally, the amplitudes
converge to *T*_{μn}(*z*) when reaching the upper mesosphere, where *T*_{μn}(*z*) is stronger than *T*_{a}(*z*) by a factor of ∼1.47. Figure 4c
also shows the fitted relative increase in the amplitude ${T}_{\mathit{\mu}}/{T}_{\mathrm{a}}$ (thick red line) describing the continuous change in the growth rate of the
amplitude, where *T*_{μ}(*z*), or *T*_{μ}(*p*), is defined by

with weighting functions
$hs={p}_{\mathrm{0}}^{\mathrm{1.5}}/({p}_{\mathrm{0}}^{\mathrm{1.5}}+{p}_{\mathrm{m}}^{\mathrm{1.5}})$ and $hm=\mathrm{1}-hs$,
where *p*_{0} is the background pressure and ${p}_{\mathrm{m}}\left(\mathrm{70}{}^{\circ}\phantom{\rule{0.125em}{0ex}}\mathrm{S}\right)\approx \mathrm{0.96}$ hPa the level of the maximum of ${\mathit{\omega}}_{i\mathrm{1}}/{\mathit{\omega}}_{i\mathrm{2}}$ (note that the height of this maximum is slightly decreasing from
*p*_{m}≈0.89 hPa over the South Pole to *p*_{m}≈1.3 hPa over the Equator).

For middle and equatorial latitudes, daytime–nighttime conditions are considered by setting the amplification factor to ${F}_{\mathrm{d}}={\mathit{\omega}}_{i\mathrm{1}}/{\mathit{\omega}}_{i\mathrm{2}}$ during daytime but to *F*_{d}=1 during
nighttime over the vertical wave propagation distance of 1 full day. In
detail, we define the parameter ${L}_{\mathrm{day}}=({\mathit{\tau}}_{\mathrm{day}}-\mathrm{0.5}\cdot {\mathit{\tau}}_{\mathrm{0}})/(\mathrm{0.5}\cdot {\mathit{\tau}}_{\mathrm{0}})$, where *τ*_{0}=24 h
and *τ*_{day} is the duration of daytime within 24 h at the
latitude *ϕ* (with *L*_{day}=1 during polar summer and *L*_{day}=0
at the Equator). Further, considering the vertical group velocity
${c}_{gz}=\partial {\mathit{\omega}}_{i\mathrm{1}}/\partial {m}_{\mathrm{1}}=-({\mathit{\omega}}_{i\mathrm{1}}/{m}_{\mathrm{1}})\cdot ({\mathit{\omega}}_{i\mathrm{1}}^{\mathrm{2}}-{f}^{\mathrm{2}})/{\mathit{\omega}}_{i\mathrm{1}}^{\mathrm{2}}$ (with initial frequency *ω*_{i1} and vertical
wavelength *m*_{1} as a first guess), the sinusoidal wave propagation structure between the middle stratosphere and middle mesosphere is described
by ${L}_{cgz}=\mathrm{cos}(\mathrm{2}\mathit{\pi}{\mathit{\tau}}_{\mathrm{0}}\cdot (z-{z}_{m})/{c}_{gz})$
changing periodically between 1 and −1 over one wavelength, where *z* and
*z*_{m} are given in kilometers and *c*_{gz} in kilometers per hour, and where
*L*_{cgi}=1 at the level *p*_{m}, or altitude *z*_{m}(*p*_{m}). Then, the
combined parameter ${L}_{\mathrm{d}}={L}_{\mathrm{day}}+{L}_{cgi}$ separates the vertical
propagation distance into daytime and nighttime fractions by defining a constant value *C*_{d}=1 in the case of *L*_{d}>1 and *C*_{d}=0
in the case of *L*_{d}≤1, where the factor ${F}_{\mathrm{d}}=\mathrm{1}+{C}_{\mathrm{d}}\cdot \left(\right({\mathit{\omega}}_{i\mathrm{1}}/{\mathit{\omega}}_{i\mathrm{2}})-\mathrm{1})$ provides ${F}_{\mathrm{d}}={\mathit{\omega}}_{i\mathrm{1}}/{\mathit{\omega}}_{i\mathrm{2}}$ in the case of daytime and *F*_{d}=1 in the case of
nighttime.

As an example, Fig. 4d shows the profile of the resulting amplification
factor *F*_{d} at 10^{∘} S for a GW with *L*_{k}=500 km and
*L*_{m}=5 km as above, with an associated vertical group velocity *c*_{gz}
of about 7 km per 12 h, illustrating that we define
${F}_{\mathrm{d}}\left({z}_{j}\right)={\mathit{\omega}}_{i\mathrm{1}}\left({z}_{j}\right)/{\mathit{\omega}}_{i\mathrm{2}}\left({z}_{j}\right)$, where
*z*_{j} is located in the daytime region (red) but *F*_{d}(*z*_{j})=1,
where *z*_{j} 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 (*L*_{m}≤15 km), a vertical shift of the phase – as defined
by the altitude *z*_{m} in the definition of *L*_{cgz} – does not have a
significant impact on the cumulative amplification of the GW amplitudes
because of the Gaussian-type structure of the profile of ${F}_{\mathrm{d}}={\mathit{\omega}}_{i\mathrm{1}}/{\mathit{\omega}}_{i\mathrm{2}}$, which has been verified by several test
calculations with levels other than *p*_{m}, or altitudes other than
*z*_{m}.

In the following section, the fitted profiles *T*_{μ} are used for further examinations with different horizontal and vertical wavelengths, where the vertical level-by-level amplification is calculated by using the distances
Δ*z*=Δ*z*_{H} of the vertical grid of HAMMONIA instead of
Δ*z*=*L*_{m}. This includes a smaller amplification factor
${F}_{\mathit{\omega}}={\mathit{\omega}}_{i\mathrm{1}}/{\mathit{\omega}}_{i\mathrm{2}}$ over the vertical distance
Δ*z*_{H} because of the smaller heating rate perturbation ${Q}_{\mathrm{\Delta}zH}^{\prime}=(\mathrm{\Delta}{z}_{H}/{L}_{\mathrm{m}})\cdot {Q}^{\prime}$ (see Eq. 11
and the related discussion). However, the resulting differences in the amplification at a specific level over the vertical distance *L*_{m} are
nearly the same, except for some small differences of less than 0.5 % due to
the different vertical resolution (i.e., ${F}_{\mathit{\omega}}(\mathrm{\Delta}z={L}_{\mathrm{m}})\approx \mathrm{1}+\left({F}_{\mathit{\omega}}\right(\mathrm{\Delta}z=\mathrm{\Delta}{z}_{H})-\mathrm{1})\cdot ({L}_{\mathrm{m}}/\mathrm{\Delta}{z}_{H})$). Additionally, the resulting
cumulative amplification in the upper mesosphere remains nearly unchanged
(${T}_{\mathit{\mu}n}(\mathrm{\Delta}z={L}_{\mathrm{m}})\approx {T}_{\mathit{\mu}nh}(\mathrm{\Delta}z=\mathrm{\Delta}{z}_{H})$, where *n**h* is the number of the HAMMONIA levels in the
USLM), where small differences between *T*_{μnh} and *T*_{μn} 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 *L*_{k} and *L*_{m} at 70^{∘} S,
where it is not affected by nighttime conditions. In comparison to the
example of *L*_{k}=500 km and *L*_{m}=5 km leading to a cumulative
amplification of ∼1.47 (solid red line), a larger vertical wavelength
of *L*_{m}=9 km leads to a smaller value of ∼1.15 (dotted red line), but a smaller vertical wavelength of *L*_{m}=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 *L*_{m}=9 km) or larger (in the case of
*L*_{m}=3 km) time increment *τ*_{i}. For the same reason, the
amplification is generally larger if the horizontal wavelength *L*_{k} is
larger, e.g., in the case of *L*_{k}=800 km, the final amplification in the
upper mesospheric amplitudes amounts to ∼1.22 for *L*_{m}=9 km
(dotted purple line), ∼1.63 for *L*_{m}=5 km (solid purple line),
and ∼2.56 for *L*_{m}=3 km (dashed purple line).

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

Introducing ${T}^{\prime}={T}_{\mathrm{2}}^{\prime}$ and *N*=*N*_{μ}, or ${T}^{\prime}={T}_{\mathrm{1}}^{\prime}$ and *N*=*N*_{0}, leads to the case with (*E*_{μ}) or
without (*E*_{a}) ozone–gravity wave interaction. Figure 4f shows the
relative amplitudes ${E}_{\mathit{\mu}}/{E}_{\mathrm{a}}$ related to Fig. 4e. In the case of
*L*_{k}=500 km (red lines), the final amplifications reach values of ∼1.32 for *L*_{m}=9 km (dotted), ∼2.17 for *L*_{m}=5 km (solid),
and ∼5.21 for *L*_{m}=3 km (dashed), and in case of *L*_{k}=800 km
(purple lines) values of ∼1.50 for *L*_{m}=9 km (dotted), ∼2.70 for
*L*_{m}=5 km (solid), and ∼6.62 for *L*_{m}=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.

### 2.2.3 Cumulative amplitude amplifications depending on latitude

For the GW with *L*_{k}=500 km and *L*_{m}=5 km, Fig. 5 shows the
latitudinal dependence of the cumulative amplifications of the temperature
perturbation (indicated by ${T}_{\mathit{\mu}}/{T}_{\mathrm{a}}$, Fig. 5a) and the related
GWPED (indicated by ${E}_{\mathit{\mu}}/{E}_{\mathrm{a}}$, Fig. 5b). The values decrease from
${T}_{\mathit{\mu}}/{T}_{\mathrm{a}}\approx \mathrm{1.5}$ and ${E}_{\mathit{\mu}}/{E}_{\mathrm{a}}\approx \mathrm{2.4}$ over
southern summer polar latitudes towards ${T}_{\mathit{\mu}}/{T}_{\mathrm{a}}\approx \mathrm{1.2}$ and
${E}_{\mathit{\mu}}/{E}_{\mathrm{a}}\approx \mathrm{1.4}$ at lower mid-latitudes (40^{∘} S),
and then less rapidly towards ${T}_{\mathit{\mu}}/{T}_{\mathrm{a}}\approx \mathrm{1.1}$ and ${E}_{\mathit{\mu}}/{E}_{\mathrm{a}}\approx \mathrm{1.2}$ 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 ${T}_{\mathit{\mu}}/{T}_{\mathrm{a}}$ (Fig. 6a) and ${E}_{\mathit{\mu}}/{E}_{\mathrm{a}}$ (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 *L*_{k}=500 km (red) and *L*_{k}=800 km (purple), the
amplifications of the temperature perturbations and of the related GWPED are
strongest for *L*_{m}=3 km (dashed lines), at polar latitudes with values
between 2.5 and 3 in ${T}_{\mathit{\mu}}/{T}_{\mathrm{a}}$ and between 7 and 9 in ${E}_{\mathit{\mu}}/{E}_{\mathrm{a}}$, and at middle and equatorial latitudes between 1.5 and 1.8 in ${T}_{\mathit{\mu}}/{T}_{\mathrm{a}}$ and between 2.4 and 3.5 in ${E}_{\mathit{\mu}}/{E}_{\mathrm{a}}$. These values decrease with increasing
vertical wavelength, i.e., when changing *L*_{m}=5 km (solid lines) or
*L*_{m}=9 km (dotted lines) roughly to ∼1.7 or ∼1.25 in
${T}_{\mathit{\mu}}/{T}_{\mathrm{a}}$ and ∼3.0 or ∼1.5 in ${E}_{\mathit{\mu}}/{E}_{\mathrm{a}}$ at
polar latitudes, and roughly to ∼1.25 or ∼1.2 in ${T}_{\mathit{\mu}}/{T}_{\mathrm{a}}$ and ∼1.5 or ∼1.25 in ${E}_{\mathit{\mu}}/{E}_{\mathrm{a}}$ 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 *F*_{GW}=*ρ*_{0} (${u}^{\prime}{w}^{\prime}$) can be derived from local profiles *T*^{′}
if the background is known, i.e., by ${F}_{\mathrm{GW}}={\mathit{\rho}}_{\mathrm{0}}E\cdot $ (k m^{−1})
(Ern et al., 2004). Therefore, the amplification of the GW amplitudes must
lead to the same amplification of the flux term *F*_{GW} and, if the GWs do
not break at lower levels, of the associated gravity wave drag
GWD $=-{\mathit{\rho}}_{\mathrm{0}}^{-\mathrm{1}}\partial {F}_{\mathrm{GW}}/\partial z$ 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 *ω*_{i2}<*ω*_{i1} includes
a slight decrease in the vertical group velocity towards *c*_{gz2}<*c*_{gz1}, 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 *L*_{k}=500 km and *L*_{m}=5 km,
*c*_{gz2} is smaller than *c*_{gz1} by 15 %–20 % at southern summer
polar latitudes and 5 %–10 % at middle and equatorial latitudes. Subsequently, at a specific level, the amplification factor
*F*_{d}(*c*_{gz2}) is stronger than *F*_{d}(*c*_{gz1}) 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 *μ*_{0}, 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 *Q*_{0}, 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 *L*_{m}=5 km and 10 % for *L*_{m}=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 *L*_{m}=5 km and
20 % for *L*_{m}=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 *Q*_{0} 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 ${N}_{\mathrm{0}}^{\mathrm{2}}$ in the order of 1 %. As
follows from Eq. (14), a change in the amplification factor
${F}_{\mathrm{d}}={N}_{\mathrm{0}}^{\mathrm{2}}/{N}_{\mathit{\mu}}^{\mathrm{2}}$ due to a
relative change $\mathrm{\Delta}{N}_{\mathrm{0}}^{\mathrm{2}}/{N}_{\mathrm{0}}^{\mathrm{2}}$ is
given by the factor [$\mathrm{1}+(\mathrm{\Delta}{N}_{\mathrm{0}}^{\mathrm{2}}/{N}_{\mathrm{0}}^{\mathrm{2}})]/[\mathrm{1}+(\mathrm{\Delta}{N}_{\mathrm{0}}^{\mathrm{2}}/{N}_{\mathrm{0}}^{\mathrm{2}})({N}_{\mathrm{0}}^{\mathrm{2}}/{N}_{\mathit{\mu}}^{\mathrm{2}})(\mathrm{1}+ab{)}^{-\mathrm{1}}$]. Therefore, for wavelengths *L*_{k}≥500 km and *L*_{m}≤5 km, a relative increase (decrease) of 1 % in
${N}_{\mathrm{0}}^{\mathrm{2}}$ 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 $\mathrm{\Delta}{N}_{\mathrm{0}}^{\mathrm{2}}/{N}_{\mathrm{0}}^{\mathrm{2}}$ 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 ($\sim {e}^{(z-zs)/\mathrm{2}H}$)
between two levels, the usual approach of a constant scale height (e.g.,
*H*∼7 km) instead of a height-dependent scale height
$H\left({T}_{\mathrm{0}}\right)=g/\left(R{T}_{\mathrm{0}}\right)$ 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 ${\mathit{\theta}}_{a}={\mathit{\theta}}_{\mathrm{a}\mathrm{0}}\cdot {\mathrm{exp}}^{(z-zs)/\mathrm{2}H}$ with a prescribed scale
height *H*_{0}=7 km instead of ${\mathit{\theta}}_{a}={\mathit{\theta}}_{\mathrm{a}\mathrm{0}}\cdot (ps/p{)}^{\mathrm{1}/\mathrm{2}}$, and a related vertical distance $\mathrm{\Delta}z=-{H}_{\mathrm{0}}\cdot (\mathrm{\Delta}p/p)$ instead of $\mathrm{\Delta}z=-H\left({T}_{\mathrm{0}}\right)\cdot (\mathrm{\Delta}p/p)$ (note that *H*(*T*_{0}) 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 % (*L*_{m}=5 km) to 10 % (*L*_{m}=3 km) over the southern summer
polar latitudes and weaker by about 1 % (*L*_{m}=5 km) to 3 % (*L*_{m}=3 km) at summer mid-latitudes. Correspondingly, the related GWPED values are
weaker by about 7.5 % (*L*_{m}=5 km) to 20 % (*L*_{m}=3 km) over the southern summer polar latitudes, and 1.5 % (*L*_{m}=5 km) to 5 % (*L*_{m}=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*(*T*_{0}) instead of *H*_{0} 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.

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
*L*_{k}≥800 and *L*_{m}≤3 km). However, the GWs with very large
amplitudes might dissipate by nonlinear wave-breaking processes before reaching the upper mesosphere.

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 ${\mathit{\omega}}_{i\mathrm{1}}/{\mathit{\omega}}_{i\mathrm{2}}\ge \mathrm{1}$ defined by the relation of
the intrinsic frequencies without (*ω*_{i1}) and with (*ω*_{i2}) ozone–gravity wave coupling. This amplitude amplification is
dependent on the horizontal and vertical wavelengths, *L*_{k} and *L*_{m},
where the effect is most efficient for GWs with *L*_{k}≥500 km and
*L*_{m}≤5 km, or initial frequencies *τ*_{i}≥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 *L*_{k}≥500 km and *L*_{m}≤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 (*L*_{k} between 200 and 1100 km, *L*_{m} 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 *L*_{k}≥800 km and *L*_{m}≤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 *L*_{m}≥9 km, but strongly increases with
decreasing vertical wavelengths *L*_{m}≤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 *L*_{m}≥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 *μ*_{0} and the ozone-related heating rate
*Q*_{0}(*μ*_{0}), revealing that, for horizontal and vertical
wavelengths *L*_{k}≥500 km and *L*_{m}≤5 km, a change of ±10 *%* in *μ*_{0} or *Q*_{0} 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*(*T*_{0}) instead of a constant scale height *H*_{0} 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
*L*_{m}=5 km and *L*_{m}=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.

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

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

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

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.

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

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

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