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.
Introduction
Atmospheric gravity waves (GWs), with horizontal wavelengths of 100 to
2000 km, are produced in the troposphere and propagate vertically through
the stratosphere and mesosphere, where gravity wave breaking processes are important drivers of the middle atmospheric circulation (e.g., Andrews et al., 1987; Fritts and Alexander, 2003). Usually, upward propagating GWs are
described by sinusoidal wave perturbations in a slowly varying background
flow of an exponentially growing amplitude with height due to decreasing
density (∼ez/2H, where H is the scale height). Recently,
Baumgarten et al. (2017) found some evidence that the growth of the GW
amplitudes between the middle stratosphere and upper mesosphere might be stronger during the daytime than during nighttime. The aim of the present paper is to examine whether ozone–gravity wave interaction can principally produce such an amplification.
Seasonal variations of gravity wave potential energy density (GWPED) have
been derived based on satellite data or lidar measurements (e.g., Geller et
al., 2013; Ern et al., 2004, 2018; Kaifler et al., 2015; Baumgarten et al., 2017). At summer middle and polar latitudes, the order of the monthly mean GWPED increases from approximately 1 J kg-1 in the middle
stratosphere (30–40 km) to 10 J kg-1 in the lower mesosphere (50–60 km) and 100 J kg-1 in the upper mesosphere (80–90 km), with usual initial GW perturbations in the middle stratosphere in the order of about 1 to 2 K, and wave periods primarily between 4 to 10 h (e.g., Kaifler et al., 2015; Baumgarten et al., 2017, 2018; Ern et al., 2018). Generally, the GW sources in the middle stratosphere are weaker, but the relative increase in the GWPED between the middle stratosphere and upper mesosphere are much stronger in summer than in winter, including a less pronounced seasonal cycle in the upper mesosphere than in the levels below. This is primarily due to the seasonal change in critical level filtering of the GWs by the zonal wind (e.g., Kaifler et al., 2015; Ern et al., 2018), but also due to specific GWs
generated by convection and propagating towards polar latitudes (Chen et
al., 2019), or to additional sources of GWs in the mesosphere, independent
of the GWs at lower levels (Reichert et al., 2021). Recently, model
simulations with resolved GWs suggested multistep vertical coupling
processes, producing such secondary GWs as a result of dissipating primary
GWs, which can strongly enhance the GW amplitudes in the upper mesosphere
(e.g., Becker and Vadas, 2018; Vadas et al., 2018; Vadas and Becker, 2018). However, the potential role of daytime–nighttime differences in the increase in GW amplitudes with height have been considered only very sparsely up to now.
Baumgarten et al. (2017) derived monthly means of the GWPED from full-day
lidar temperature measurements at northern mid-latitudes (54∘ N,
12∘ E), and found a stronger relative increase between 35 and 40 km and between 55 and 60 km for full-day than nighttime observations during summer months,
but less pronounced differences during winter. For example, for July, the
GWPED at 55–60 km show values of about 1×10-2 J m-3 (or 10 J kg-1) for full-day measurements but about 0.5×10-2 J m-3 for nighttime only (or 0.2 but 0.1 J m-3, if the measured temperature fluctuations are vertically filtered for vertical wavelengths Lm<15 km), where the GWPED at 35–40 km remains nearly unchanged, indicating a difference between full-day and nighttime values by a factor of about 2. Generally, measurements of the mesospheric GWPED are much more uncertain during summer than winter months (e.g., Kaifler et al., 2015; Ehard et al., 2015; Baumgarten et al., 2017), and the signal-to-noise ratio of the lidar measurements is not as good during the daytime than during nighttime (e.g., Rüfenacht et al., 2018), which can stimulate some doubt on the reliability of the daytime–nighttime differences derived from these specific measurements. In addition, taking the potential uncertainties of the analyzing methods into account (i.e., the temporal filtering methods used for the measured time series), Baumgarten et al. (2017) speculated that a change in the phase of long periodic waves (e.g., diurnal and semidiurnal tides) could change the filtering conditions for GWs. However, Baumgarten et al. (2017) conclusively assumed that the detected daytime–nighttime differences are of true geophysical origin, where an unequivocal explanation of this phenomenon remained open. Considering also that the full-day observations of Baumgarten et al. (2018) during May 2016 showed pronounced GW activity, particularly at altitudes between 42 and 50 km where the coupling between ozone and temperature is particularly strong, it seems to be worthwhile to examine whether ozone–gravity wave interaction could principally lead to such daytime–nighttime differences in the GW amplitudes. This must then also lead to a potential effect on the
differences in the GWPED between polar day and polar night. The examination
of the present paper is based on standard equations describing upward
propagating GWs in a constant background flow, excluding other processes
controlling the GWPED variability, to provide a clear understanding and
quantification of the potential effect, which cannot be achieved based on
observational data analysis or comprehensive model calculations alone.
The coupling of temperature and ozone is particularly strong in the upper
stratosphere due to the short photochemical lifetime of ozone (e.g.,
Brasseur and Solomon, 1995). Linear relationships for a change in the
heating rate due to a change in ozone, and a change in photochemistry due to
a change in temperature, were derived from basic theory or satellite
observations, and have been introduced in standard equations of
stratospheric dynamics to examine the effects on the stratospheric
circulation, planetary-scale wave patterns, and equatorial Kelvin waves
(Dickinson, 1973; Douglass et al., 1985; Froidevaux et al., 1989; Cordero et
al., 1998; Cordero and Nathan, 2000; Nathan and Cordero, 2007; Ward et al., 2000; Gabriel et al.,
2011a). Large-scale ozone-dynamic coupling processes also show significant
effects in numerical weather prediction (NWP) or general circulation models (GCMs)
(Cariolle and Morcrette, 2006; Gabriel et al., 2007, 2011b; Gillet et al.,
2009; Waugh et al., 2009; McCormack et al., 2011; Albers et al., 2013).
However, possible effects of mesoscale ozone–gravity wave interaction in the
upper stratosphere/lower mesosphere (USLM) have not been considered up to
now.
The basic idea of the present paper can be summarized as follows: in the
USLM, the time–mean ozone mixing ratio μ0(z) decreases with
height (∂μ0/∂z<0). Therefore, at a
specific level in the ULSM, an ascending air parcel initially forced by an
upward propagating sinusoidal GW pattern (i.e., the wave crest with vertical
velocity perturbation w′>0) must lead to an increase
(∂μ′/∂t>0) by both transport
(because -w′∂μ0/∂z>0)
and photochemistry (because the temperature-dependent ozone production
increases in the case of adiabatic cooling), and, hence, in the heating rate
(Q′(μ′)>0), comparable to the latent heat
release in the troposphere in the case of condensation. Then, the induced
perturbation (Δθ′>0, where θ is
potential temperature) reinforces the initial ascent, where the lapse rate
(∂(θ0+Δθ′)/∂z<∂θ0/∂z) decreases (∂z= constant), suggesting an effective ozone adiabatic lapse rate in the upper stratosphere comparable
to the moist adiabatic lapse rate in the troposphere. Analogously, a descending air parcel (the wave trough where w′<0) leads to a decrease (∂μ′/∂t<0) and a corresponding change (Q′(μ′)<0), reinforcing the initial descent. Overall,
this process must lead to a significant amplification of the initial GW
amplitude at this level, and, hence, to a successive amplification of the
amplitude during the upward level-by-level propagation through the ULSM.
In Sect. 2, standard equations for GWs in a zonal mean background flow
with and without linearized ozone–temperature coupling are formulated to
quantify the amplitude amplification at a specific level (or altitude) and
latitude. Then, in Sect. 3, the cumulative amplitude amplification during the propagation through the USLM is derived, based on an idealized approach of the upward level-by-level propagation of GWs with specific horizontal and
vertical wavelengths. Section 4 concludes with a summary and discussion.
(a–c) Zonal and monthly mean background, (a) temperature T0,
(b) ozone mixing ratio O3 (the dashed line denotes where ∂O3/∂z=0), and (c) ozone heating rate Q0, January 2001,
extracted from a simulation with the circulation and chemistry model
HAMMONIA; (d–f) amplification factors, (d)1+ab and (e)N02/Nμ2 for a GW with horizontal
and vertical wavelengths Lk=500 km and Lm=5 km, and (f)N02/Nμ2 for a GW with
Lk=800 km and Lm=3 km; shaded areas denote the latitudes where
the amplification is limited by the length of daylight (τi>τday).
Ozone–gravity wave interaction
In the following section, ozone–gravity wave interaction is analyzed based on standard equations describing GWs in a background atmosphere, where the solutions are illustrated for southern summer conditions. The background is prescribed by monthly and zonal mean temperature T0, ozone μ0,
and short-wave heating rate Q0 of January 2001 (Fig. 1a–c), derived
from a simulation with the high-altitude general circulation and chemistry
model HAMMONIA (details of the model are given by Schmidt et al., 2010). The
heating rate Q0 (Fig. 1c) is primarily due to the absorption of solar
radiation by ozone, and largely agrees with southern summer solar heating
rates derived from satellite measurements by Gille and Lyjak (1986) but with
somewhat smaller maximum values (in the order of ∼10%). Figure 1c
shows that Q0 is particularly strong in the USLM where ∂μ0/∂z<0 (the dashed line in Fig. 1b indicates ∂μ0/∂z=0). The HAMMONIA model includes 119 layers up to 250 km with increasing
vertical resolution between ∼0.7 km in the middle stratosphere and
∼1.4 km in the middle mesosphere, with a horizontal resolution of
3.75∘. In the following section, this grid is used to illustrate the
analytic solutions of upward-propagating GWs.
Amplification of GW amplitudes at a specific levelBasic equations
Following Fritts and Alexander (2003), we consider standard Eqs. (1)–(5) describing GW propagation in a background flow, with linear GW perturbations T′, θ′, u′, v′, w′, p′, and ρ′ (T′ is
temperature; θ′=T′ (p00/p)κ is
potential temperature;, p(z) is pressure; p00=1000 hPa; z is altitude;
u′, v′, and w′ are zonal, meridional, and vertical
wind perturbations, respectively; and p′ and ρ′ are the perturbations in pressure and density, respectively). Additionally, we include an ozone-dependent heating rate perturbation Q′(μ′) in the potential temperature equation (Eq. 5) and Eq. (6) for the ozone perturbation μ′ with a temperature-dependent perturbation in ozone photochemistry S′(T′), where a(ϕ,z)>0 and b(ϕ,z)>0 are linear coupling parameters as a function of latitude ϕ and
altitude z specified below, ρ0(z)=ρ00exp-(z-z0)/H is background density, H∼7 km is scale height, ρ00 is a reference value at altitude z0, u0 is a zonal mean
background wind, d0/dt=∂/∂t+u0∂/∂x+v0∂/∂y, where ∂/∂x and ∂/∂y denote the derivations in longitude and
latitude, g is the gravity acceleration, and f is the Coriolis parameter; the background shear terms 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:
ωi12=N02kh12+m12f2kh12+m12≈N02kh12m12+f2.
Ozone–temperature coupling
For specifying the parameter b, we consider the vertical ascent w1′>0 in the wave crest of an initial sinusoidal GW
perturbation, related to an adiabatic cooling term d0θ1′/dt=-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
O3=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
ΔμTμ0=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
S′=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
Q′=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.
Amplification of GW amplitudes at a specific level
The parameterizations of Q′ and S′ provide a useful
modification of the potential temperature tendency when introducing
d0μ′/dt of Eq. (6) into (Eq. 5):
(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:
d0dtgθ0θ′+Nμ2w′=0,
with
Nμ2=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:
ωi22=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′=-Nμ2w2′ 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:
θμ=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∘).
Examples of the amplification of GW amplitudes at specific levels
Figure 1d–f shows the factor 1+ab and the quotient N02/Nμ2 for a GW with horizontal and vertical
wavelengths Lk=500 km and Lm=5 km, and the quotient
N02/Nμ2 for a GW with
Lk=800 km and Lm=3 km. In the first example, the factor 1+ab
(Fig. 1d) contributes to the amplification of the GW amplitude at a
specific level by up to 6 %–8 %, and the overall factor
(N02/Nμ2=(1+ab)⋅N02/(N02+Nc2) (Fig. 1e) by up to 8 %–12 % (including a
decrease in the lapse rate of up to 3 % described by (N02+Nc2)/N02, not shown here). The second example (Fig. 1f) shows that the factor N02/Nμ2 is
larger in the case of larger horizontal and smaller vertical wavelengths,
reaching amplifications of up to 12 %–14 % (shaded areas denote the
latitudinal range where the amplification is reduced due to the length of
daytime, i.e., where τi>τday).
Changes due to ozone–temperature coupling at a specific level
induced by an initial GW perturbation with horizontal and vertical
wavelengths Lk=500 km and Lm=5 km, and with exponential
increase in amplitude with height (initial temperature amplitude
Ta(zs)=1 K at zs≈35 km; p=6.28 hPa). (a) Change in ozone due to vertical transport, (b) change in ozone due to photochemistry, (c) total change in ozone, (d) relative change in ozone, (e) change in the
heating rate, and (f) change in the temperature perturbation.
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).
Amplification factor ωi1/ωi2 at a level
of the maximum values of ωi1/ωi2 (1.156 hPa)
illustrating the decrease in the intrinsic frequency with (ωi2) compared to without (ωi1) ozone–temperature coupling (compare
with Fig. 1e–f), (a) latitudinal distribution of ωi1/ωi2 as a function of the initial wave period τi [in h],
and (b–c) dependence of ωi1/ωi2 on the horizontal
and vertical wavelengths Lk and Lm [in km] at (b) 70∘ S
and (c) 10∘ S. Shaded areas show where the amplification is
limited by the length of daytime (τi>τday).
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.
Illustration of the successive amplification of GW amplitudes
during the upward level-by-level propagation. (a) Amplification factor
ωi1/ωi2 at 70∘ S for GW with
horizontal wavelength Lk=500 km and vertical wavelength Lm=5 km (solid red line), and, for comparison, Lm=3 (dashed) and
Lm=9 km (dotted). (b) Temperature amplitudes for GW with
Lk=500 km and Lm=5 km, depicting the initial perturbation
Ta (blue) and the successively amplified amplitudes Tμj(zj)|j=1,n (light blue towards red; here, n=8 for
Lm=5 km). (c) Same as (b) but for the relative amplitudes Tμj(zj)|j/Ta (solid lines) together with the profiles
of the previous level multiplied by ωi1/ωi2 (i.e.,
Tμj-1(zj-1)⋅(ωi1/ωi2), dashed
lines) and a fitted approach Tμ (thick solid red line, defined by
Eq. 18). (d) Same as (a) for the case Lk=500 km and Lm=5 km
but at 10∘ S including the limitation due to the length of
nighttime conditions. (e) Relative values Tμ/Ta at
70∘ S for different horizontal (red: Lk=500 km, purple:
Lk=800 km) and vertical (dashed: Lm=3, solid: Lm=5 km, dotted: Lm=9 km) wavelengths. (f) Same as (e) but for the
relative values Eμ/Ea of the related gravity wave potential
energy density (GWPED, defined by Eq. 19).
Upward propagating GWs in a background flowLevel-by-level amplification of GW amplitudes
In the following section, a solution for the cumulative amplification during the vertical level-by-level propagation is derived, excluding – to a first guess – other effects like small-scale diffusion, wave-breaking processes, interaction of GWs with atmospheric tides, or so-called secondary
GWs. Following the Huygens principle, each point of a propagating wave
front at a specific level is the source of a new wave at this level, i.e., a
single upward propagating GW, which is amplified at a level zj-1, is
the initial perturbation amplified at the next level zj. For
illustration (Fig. 4a–c), we choose an initial GW with horizontal and
vertical wavelengths Lm=500 km and Lm=5 km as above, where the
vertical distance between the levels zj-1 and zj is set by the
initial vertical wavelength Δz=Lm. First, we focus on polar
latitudes during southern polar summer (70∘ S) with daytime
conditions only. Thereafter we consider the modification for middle and equatorial latitudes where GWs with weak vertical group velocities propagate through USLM during both daytime and nighttime.
For orientation, Fig. 4a shows the profiles ωi1/ωi2 for Lk=500 km and Lm=5 km at 70∘ S (solid),
and, for comparison, for Lm=3 (dashed) and Lm=9 km
(dotted), indicating the altitude range where ozone–temperature coupling is
relevant (note that the depicted distance of pressure levels approximately represents a 5 km distance in altitude). Beginning with a first level at
zs≈35 km (6.28 hPa), the wave propagates through eight layers between ≈35 and ≈70 km (0.06 hPa) where the amplification of the amplitude is relevant. At each of these levels, denoted by
zj=zs+(j-1)⋅Δz (j=1, n; here n=8), the amplitude
at zj will be amplified by the factor ωi1(zj)/ωi2(zj) at zj. Starting with an exponentially growing amplitude Ta(z)=Ta(zs)⋅exp(z-zs)/2H (where we set Ta(zs)=1 K again), we yield a new amplitude
Ta1(z1)=Ta(z1)⋅ωi1(z1)/ωi2(z1) at the level z1, defining a
new exponentially growing amplitude Tμ1(z)=Ta1(z1)⋅exp(z-z1)/2H. We then yield Ta2(z2)=Tμ1(z2)⋅ωi1(z2)/ωi2(z2) at
the level z2, defining Tμ2(z)=Ta2(z2)⋅exp(z-z2)/2H, and so forth. Finally, the amplitude at the level zn
in the middle mesosphere is described by
Tμn(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
Tμ(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.
Cumulative amplitude amplifications for representative examples
Figure 4e illustrates the dependence of the amplitude amplification on the
horizontal and vertical wavelengths Lk and Lm at 70∘ S,
where it is not affected by nighttime conditions. In comparison to the
example of Lk=500 km and Lm=5 km leading to a cumulative
amplification of ∼1.47 (solid red line), a larger vertical wavelength
of Lm=9 km leads to a smaller value of ∼1.15 (dotted red line), but a smaller vertical wavelength of Lm=3 km leads to a larger value
of ∼2.27 (dashed red line), because the induced increase in the ozone
perturbation μ′ produces a heating rate perturbation Q′
within a shorter (in the case of Lm=9 km) or larger (in the case of
Lm=3 km) time increment τi. For the same reason, the
amplification is generally larger if the horizontal wavelength Lk is
larger, e.g., in the case of Lk=800 km, the final amplification in the
upper mesospheric amplitudes amounts to ∼1.22 for Lm=9 km
(dotted purple line), ∼1.63 for Lm=5 km (solid purple line),
and ∼2.56 for Lm=3 km (dashed purple line).
The related GWPED (here denoted by E) is derived following Kaifler et al. (2015):
E=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.
Cumulative amplifications of the GW amplitude during the upward
level-by-level propagation for a GW with Lk=500 km and Lm=5 km, (a) Cumulative increase in the temperature amplitudes described by
Tμ/Ta. (b) Related increase in the gravity wave potential
energy density (GWPED) described by Eμ/Ea; background
conditions: January 2001.
Cumulative amplitude amplifications depending on latitude
For the GW with Lk=500 km and Lm=5 km, Fig. 5 shows the
latitudinal dependence of the cumulative amplifications of the temperature
perturbation (indicated by Tμ/Ta, Fig. 5a) and the related
GWPED (indicated by Eμ/Ea, Fig. 5b). The values decrease from
Tμ/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.
Cumulative amplification of the GW amplitudes similar to that in Fig. 5, but at upper mesospheric levels (0.01 hPa, ∼80 km) for different
horizontal and vertical wavelengths Lk (red: 500 km, purple: 800 km)
and Lm (dotted: 9 km, solid: 5 km, dashed: 3 km), (a)Tμ/Ta, (b)Eμ/Ea.
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.
Sensitivity to varying conditions
In the following section, we estimate the sensitivity of the GW amplitude amplification on nonlinear processes and background conditions which could modulate the first-guess results described above. For example, the decrease
in the frequency towards ωi2<ωi1 includes
a slight decrease in the vertical group velocity towards cgz2<cgz1, which can additionally strengthen the process of amplitude
amplification because the wave propagates somewhat more slowly through the
ULSM region. However, this effect is at least 1 order smaller than the
first-order process described above, as derived from test calculations
including this effect. For example, for Lk=500 km and Lm=5 km,
cgz2 is smaller than cgz1 by 15 %–20 % at southern summer
polar latitudes and 5 %–10 % at middle and equatorial latitudes. Subsequently, at a specific level, the amplification factor
Fd(cgz2) is stronger than Fd(cgz1) by 2 %–3 % at
polar latitudes and less than 1 % at middle and equatorial latitudes. Including this change in the successive level-by-level propagation leads
to a weak successive increase in the cumulative amplifications by ∼5 % at 1 hPa to ∼10 % at 0.01 hPa at polar summer latitudes, and
by only ∼1 % at 1 hPa to ∼2 % at 0.01 hPa at middle and equatorial latitudes.
We also estimate the sensitivity of the amplitude amplification on the ozone
background μ0, considering the observed long-term changes in upper
stratospheric ozone in the order of up to -8 % per decade (e.g., Sofieva
et al., 2017; WMO, 2018), and the uncertainty in the maximum of the heating
rate Q0, which is smaller in the used HAMMONIA data in the order of
∼10% compared to those derived from satellite measurements, as
mentioned above. In the case of a 10 % reduction in ozone, the cumulative amplification in the upper mesospheric GW amplitudes is weaker by about
5 % for the example with Lm=5 km and 10 % for Lm=3 km
(i.e., at 70∘ S, we yield a cumulative amplification of ∼1.4
to ∼2.25 instead of ∼1.5 to ∼2.5), and the related
amplification of the GWPED is weaker by about 10 % for Lm=5 km and
20 % for Lm=3 km (at 70∘ S, a cumulative amplification of
∼2.7 to ∼7.2 instead of ∼3 to ∼9). Analogously, in the
case of an increase in Q0 by 10 %, the cumulative amplification is
stronger by 5 % or 10 % in the GW amplitudes and by 10 % or 20 % in the related GWPED amplitudes.
Another question arises about the sensitivity of the effect of ozone–gravity
wave coupling to atmospheric tides or the diurnal cycle in stratospheric
ozone, which are planetary-scale processes changing the background
conditions for the propagation of the mesoscale GW perturbations. For
example, Schranz et al. (2018) observed stronger amplitudes in upper
stratospheric ozone during daytime than during nighttime in the order of 5 %
(summer solstice) to 8 % (May). Such a difference would correspond to a
change in the cumulative amplification of the upper mesospheric GW
amplitudes or GWPED in the order of 5 % to 10 % or 10 % to 20 %, as
follows from the sensitivity of the effect on the prescribed long-term
change in stratospheric ozone derived above.
Baumgarten and Stober (2019) derived amplitudes of tides in the order of up
to 0.5 K in the middle stratosphere (∼35 km) increasing up to 2 K at
∼50 km and ∼4 K at 70 km, which would correspond to a change in
the lapse rate in the order of up to 0.1 K km-1, or in the
Brunt-Vaisala frequency N02 in the order of 1 %. As
follows from Eq. (14), a change in the amplification factor
Fd=N02/Nμ2 due to a
relative change ΔN02/N02 is
given by the factor [1+(ΔN02/N02)]/[1+(ΔN02/N02)(N02/Nμ2)(1+ab)-1]. Therefore, for wavelengths Lk≥500 km and Lm≤5 km, a relative increase (decrease) of 1 % in
N02 would lead to a relative decrease (increase) in the
amplification factor of up to 0.035 % at stratopause altitudes, which is
much less than the effects of the changes in the vertical group velocity or
in ozone described above. Moreover, even if a relative change ΔN02/N02 would be much larger
(10 %–50 %), it does not change the amplification factor of a
specific level by more than 1 %–3 %, and, hence, the cumulative
amplification of the GW amplitudes in the upper mesosphere by more than 5 %–10 %.
Assuming exponential growth of the amplitudes (∼e(z-zs)/2H)
between two levels, the usual approach of a constant scale height (e.g.,
H∼7 km) instead of a height-dependent scale height
H(T0)=g/(RT0) can principally lead to significant differences in
the GWPED profiles (e.g., Reichert et al., 2021). For estimating the
relevance of a change in H on the cumulative amplitude amplification, the
solutions are also calculated for an initial GW perturbation θa=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).
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.
Similar to Fig. 6 but for both relative and absolute changes in mean values averaged over 16 representative mesoscale GW events with different horizontal and vertical wavelengths Lk and Lm (Lk:
200, 500, 800, and 1100 km, Lm: 3, 5, 7, and 9 km), (a) relative change
in temperature amplitude Tm/Ta (solid red line; dashed lines
denote the standard deviation), (b) absolute change Tm-Ta for
the case of an initial temperature perturbations Ta0 of 1 K (orange
line) and 2 K (purple line) in the middle stratosphere, (c) and (d) same as (a) and (b) but for the GWPED (i.e., for Em/Ea and Em-Ea).
Figure 7 illustrates both the relative and absolute changes in the resulting
mean upper mesospheric GW temperature amplitudes (Fig. 7a, b) and in the
mean GWPED (Fig. 7c, d). The relative increase in the mean temperature
amplitude (Fig. 7a, solid red line) is stronger by a factor increasing
from about 1.3 (±0.1) at summer low and middle latitudes up to 1.7 (±0.2) at summer polar latitudes (values in brackets denote
1 standard deviation). This corresponds to a stronger increase from about 7 K (±2 K) up to 17.5 K (±4.5 K) in the case of an initial GW
perturbation of 1 K in the middle stratosphere (at 6.28 hPa or ≈35 km) (Fig. 7b, solid orange line), and from about 14 K (±4 K) up to
35 K (±9 K) in the case of an initial GW perturbation of 2 K (Fig. 7b, solid purple line).
The relative increase in the mean GWPED (Fig. 7c, solid red line) is
stronger by a factor increasing from about 1.7 (±0.2) at summer low and middle latitudes up to 3.4 (±0.8) at summer polar latitudes. This
corresponds to a stronger increase in the absolute GWPED values from about 2 J kg-1 (±0.5 J kg-1) at summer low and middle latitudes up to
12 J kg-1 (±3 J kg-1) at summer polar latitudes in the case of an
initial GW perturbation of 1 K at 35 km (Fig. 7d, solid orange line), and
from about 8 J kg-1 (±2 J kg-1) up to 48 J kg-1 (±0.5 J kg-1) in the case of an initial GW perturbation of 2 K (Fig. 7d, solid purple line).
In summary, we find an absolute increase in the order of 7 to 35 K in the
mean GW temperature amplitudes and 2 to 50 J kg-1 in the mean
GWPED values, assuming usual initial GW perturbations in the order of 1 to
2 K in the middle stratosphere, where the effect is particularly large
during polar daytime conditions. Note that, assuming exponential growth
with height only, this potential effect can be much larger in the case of
stronger initial amplitudes in the middle stratosphere (the absolute changes
in the temperature amplitudes increase linearly and those in the GWPED values quadratically with increasing initial GW perturbations at 35 km) and
in specific geographical regions or time periods where primary GWs with
large horizontal and small vertical wavelengths are excited (e.g., where
Lk≥800 and Lm≤3 km). However, the GWs with very large
amplitudes might dissipate by nonlinear wave-breaking processes before reaching the upper mesosphere.
Summary and conclusions
The present paper shows that ozone–gravity wave interaction in the upper
stratosphere/lower mesosphere (USLM) leads to a stronger increase in gravity wave (GW) amplitudes with height during daytime than during nighttime,
particularly during polar summer. The results include information about both
the amplification of the GW amplitudes at a specific level and the
cumulative increase in the amplitudes during the upward level-by-level propagation of the wave from middle stratosphere to upper mesosphere.
In a first step, standard equations describing upward propagating GWs with
and without linearized ozone–gravity wave coupling are formulated, where an
initial sinusoidal GW perturbation in the vertical ozone transport and
temperature-dependent ozone photochemistry produces a heating rate
perturbation as a function of the initial intrinsic frequency, which
determines the duration of the perturbation at a specific level over the
distance of the initial vertical wavelength. The solution reveals an
amplification of the ascending and descending perturbations of the
sinusoidal GW pattern at this level, i.e., a decrease in the intrinsic frequency due to both the induced changes in the lapse rate (or
Brunt–Vaisala frequency) and the positive feedback of the coupling on the
initial GW perturbation, and an associated increase in the GW amplitude by a factor ωi1/ωi2≥1 defined by the relation of
the intrinsic frequencies without (ωi1) and with (ωi2) ozone–gravity wave coupling. This amplitude amplification is
dependent on the horizontal and vertical wavelengths, Lk and Lm,
where the effect is most efficient for GWs with Lk≥500 km and
Lm≤5 km, or initial frequencies τi≥4 h,
representing mesoscale GWs forced by cyclones or fronts, or by the orography
of mountain ridges like the Rocky Mountains, Andes, or Norwegian Caledonides.
For southern summer conditions, strongest amplitude amplifications at
specific levels of about 5 %–15 % over the perturbation distance of
one vertical wavelength are located near the stratopause, with peak values
over the Equator and over summer polar latitudes.
In a second step, an analytic approach of the upward level-by-level
propagation of the GW perturbations with and without ozone–gravity wave
interaction reveals the cumulative amplitude amplification, where the wave
is propagating upward with the vertical group velocity defined by the
initial GW parameters, and where daytime–nighttime conditions at middle and equatorial latitudes are considered. Representative examples with different initial wavelengths illustrate that the successive increase in both the GW
amplitudes and the related gravity wave potential energy density (GWPED)
converge to much stronger amplitudes in the upper mesosphere during daytime
than during nighttime. This effect strongly decreases with latitude between
summer polar and mid-latitudes because of the decrease in the length of
daytime, nearly constant at equatorial latitudes, and decreasing again with
latitude towards insignificant values in the winter extratropics.
In summary, the strongest impact of ozone–gravity wave interaction is found
for wave periods ≥4 h (related to the wavelengths Lk≥500 km and Lm≤5 km), i.e., in a range of wave periods usually observed
at summer middle and polar latitudes. For prescribed single GWs with large horizontal wavelengths (500 to 800 km) and small vertical wavelengths (3 to
5 km), the upper mesospheric GW temperature amplitudes are stronger by a
factor between 1.25 and 1.75 at summer low and middle latitudes and between 1.5 and 3 at summer polar latitudes, and the corresponding GWPED by a factor between 1.5
and 3.5 and between 3 and 9. For a representative range of 16 different mesoscale GW events (Lk between 200 and 1100 km, Lm between 3 and 9 km), the
mean temperature amplitudes are stronger by a factor between 1.3 at summer
low and middle latitudes to 1.7 at summer polar latitudes, e.g., stronger by about 7 to 17.5 K (or 14 to 35 K) in the case of an initial GW perturbation
of 1 K (or 2 K) in the middle stratosphere (at ∼35 km). The
corresponding relative increase in the mean GWPED is stronger by a factor
between 1.7 at summer low and middle latitudes and 3.4 at summer polar latitudes, e.g., for the same example as above, stronger by about 2 to 12 J kg-1 (or 8 to 48 J kg-1). These values
range in the order between 2 % and 50 % of the observed order of the mean upper-mesospheric GWPED amplitudes (100 J kg-1). These absolute
differences can be larger in the case of stronger initial perturbations in the middle stratosphere, or in specific geographical regions or time periods
where primary GWs with large horizontal and small vertical wavelengths
(e.g., where Lk≥800 km and Lm≤3 km) are excited.
However, the GWs with very large amplitudes might dissipate by nonlinear wave-breaking processes before reaching the upper mesosphere. Overall, these
values result from an idealistic approach and cannot entirely explain the
details of specific measurements. Nevertheless, they provide a first-guess
quantification of the potential effect of ozone–gravity wave interaction on
the GW amplitudes.
The variety of horizontal and vertical wavelengths used in the present paper
are representative of mesoscale GWs in the USLM region. Observations
suggest not only characteristic vertical wavelengths of GWs between ∼2–5 km in
the lower stratosphere increasing to ∼10–30 km in the upper
mesosphere, but also the existence of large vertical wavelengths greater
than 10 km in the ULSM region, particularly above convection in equatorial
regions or over southernmost Argentina (e.g., Alexander, 1998; McLandress et
al., 2000; Fritts and Alexander, 2003; Alexander and Holton, 2014; Hocke et al., 2016; Baumgarten et
al., 2018; Reichert et al., 2021). The results of the present paper suggest
that the effect of ozone–gravity wave coupling decreases with increasing
vertical wavelengths Lm≥9 km, but strongly increases with
decreasing vertical wavelengths Lm≤5 km. The latter could lead to more pronounced GW breaking and dissipation processes in the upper
stratosphere during daytime than during nighttime, and – subsequently – to more prominent GWs with larger vertical wavelengths of Lm≥5 km, which would be consistent with the observed GW characteristics at these altitudes presented by Baumgarten et al. (2018).
As mentioned in the introduction, the measurements of Baumgarten et al. (2017) show some evidence that the increase in the GWPED values with height is stronger during full-daytime than during nighttime by a factor of about 2, or,
roughly assuming a 2:1 relation of daytime and nighttime (16 h daytime
and 8 h nighttime) for high summer mid-latitudes, stronger during
daytime than during nighttime by a factor of about 2.5. For comparison, the estimated effect of ozone–temperature coupling for these latitudes (factor of 1.7) is
somewhat smaller and would lead to an increase in the nighttime GWPED in the order of ∼50% (0.7:1.5) of the observed increase. Conclusively,
although the difference derived by Baumgarten et al. (2017) might be
uncertain as mentioned in the introduction, and although the approach of the
present paper cannot entirely explain the details of specific local
measurements during a specific time period, the comparison confirms that
ozone–gravity wave interaction might be able to produce significant
daytime–nighttime differences in the GW amplitudes at high summer
mid-latitudes.
Current state-of-the-art general circulation models (GCMs) usually use a
variety of prescribed tropospheric sources and tuning parameters in the
gravity wave drag (GWD) parameterizations, forcing the middle atmospheric circulation (e.g., McLandress et al., 1998; Fritts and
Alexander, 2003; Garcia et al., 2017), where the extreme low temperatures
observed in the summer upper mesosphere provide an important benchmark for
the quality of the upwelling branch and the associated adiabatic cooling
produced by the models. Including ozone–gravity wave interaction into the
GCMs might lead to a substantial improvement of the used GWDs and the
associated processes driving the summer mesospheric circulation, because the
related increase in the GWPED must lead to a similar increase in the
vertical momentum flux term determining the GWD. However, the incorporation
of ozone–gravity wave interaction into a state-of-the-art GCM using a GWD, or into a numerical model with resolved GWs, needs extensive test simulations, which is beyond the scope of the present paper.
In particular, current GCMs indicate significant changes in the time–mean
circulation of the upper mesosphere due to the stratospheric ozone loss over
Antarctica during southern spring and early summer via the induced changes
in the GWD (Smith et al., 2010; Lossow et al., 2012; Lubi et al., 2016).
Long-term changes in upper stratospheric ozone of up to -8% per decade, derived from satellite measurements (e.g., Sofieva et al., 2017; WMO, 2018), could also affect the mesospheric circulation in the stratosphere and
mesosphere by modulating the GW amplitudes and, hence, the GWD. Based on the
idealized approach of the present paper, we estimate the sensitivity of the
amplification of the GW amplitudes in the upper mesosphere on changes in the
ozone background μ0 and the ozone-related heating rate
Q0(μ0), revealing that, for horizontal and vertical
wavelengths Lk≥500 km and Lm≤5 km, a change of ±10% in μ0 or Q0 results in a change of ±10% to
±20% in the upper mesospheric GWPED. Conclusively, the summer
mesospheric upwelling might be much more sensitive to the long-term changes
in upper stratospheric ozone as has been suggested by the GCMs up to now.
In the approach of the present paper, the variations due to the diurnal
cycle in stratospheric ozone and atmospheric tides are excluded to examine
the potential effect of ozone–gravity wave interaction as clearly as possible,
based on standard equations describing upward propagating GWs in a constant
background. On the one hand, these variations can principally modulate the
effect of ozone–gravity wave coupling by changing the planetary-scale
background conditions for the propagation of the mesoscale GWs. Assuming –
to a first order – linear modulations in the background ozone and
background lapse rate according to observed diurnal or tidal variations, the
sensitivity calculations of the present paper suggest that the related
modulations in the amplitude amplification are smaller than the effect of
ozone–gravity wave coupling by approximately 1 order. Further test
calculations have shown that the use of a height-dependent scale height
H(T0) instead of a constant scale height H0 at the levels of
relevant amplification leads to stronger amplitude amplifications,
particularly over the southern summer polar latitudes, because of the high temperatures in the stratopause region and the very low temperatures in the upper
mesosphere, where the related differences are also smaller than the
first-order process (e.g., in the GWPED, for vertical wavelengths between
Lm=5 km and Lm=3 km, between about 7.5 % to 20 % at summer polar latitudes and less than 5 % at summer mid-latitudes).
On the other hand, short-term fluctuations in the balanced zonal and
meridional winds due to atmospheric tides can principally lead to changes in
the upward GW propagation characteristics, and, hence, to significant
daytime–nighttime differences in the growth of the GW amplitudes with
height, including nonlinear feedbacks of the propagating mesoscale GWs to
the short-term balanced flow components. Further, multistep vertical
coupling processes producing secondary GWs in the mesosphere could depend on
daytime–nighttime conditions or tidal variations, which could also produce
significant daytime–nighttime differences in the growth of the GW
amplitudes with height. Considering the remarkably strong effect of
ozone–gravity wave coupling suggested by the present paper, we may speculate
that it significantly affects these possible changes in the GW amplitudes
due to short-term fluctuations in the balanced winds or multistep vertical
coupling. However, an unequivocal quantification of the effects of these
processes and the involved nonlinear interactions of the daytime–nighttime
differences in the GWPED needs much more investigations, e.g., based on
extensive GW resolving model simulations with interactive ozone
photochemistry, which is beyond the scope of the present paper.
The results of the present paper might stimulate further daytime–nighttime
observations of GW activity, particularly at specific measurement sites where
the GWs are usually characterized by specific horizontal and vertical
wavelengths, e.g., downwind of specific mountain ridges (east of Rocky
Mountains, Southern Andes or Norwegian Caledonides), which could be helpful
to better understand how ozone–gravity wave coupling is operating in
situ.
Data availability
Background data and programs visualizing the presented analytic solutions are available upon request from the author.
Competing interests
The contact author has declared that none of the authors has any competing interests.
Disclaimer
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Acknowledgements
The author thanks Hauke Schmidt (Max Planck Institute for Meteorology (MPI-Met), Hamburg) for providing HAMMONIA
background data. Thanks also to two reviewers for critical comments.
Financial support
The publication of this article was funded by the Open Access Fund of the Leibniz Association.
Review statement
This paper was edited by Mathias Palm and reviewed by two anonymous referees.
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