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**Atmospheric Chemistry and Physics**
An interactive open-access journal of the European Geosciences Union

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**Research article**
09 Jan 2020

**Research article** | 09 Jan 2020

Modelled effects of temperature gradients and waves on the hydroxyl rotational distribution in ground-based airglow measurements

^{1}Norwegian University of Science and Technology (NTNU), Trondheim, 7491, Norway^{2}Birkeland Centre for Space Science (BCSS), Department of Physics and Technology, University of Bergen, Norway

^{1}Norwegian University of Science and Technology (NTNU), Trondheim, 7491, Norway^{2}Birkeland Centre for Space Science (BCSS), Department of Physics and Technology, University of Bergen, Norway

**Correspondence**: Christoph Franzen (franzen.christoph@rwth-aachen.de)

**Correspondence**: Christoph Franzen (franzen.christoph@rwth-aachen.de)

Abstract

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Spectroscopy of the hydroxyl (OH) airglow has been a commonly used way to remotely sense temperatures in the mesopause region for many decades. This technique relies on the OH rotational state populations to be thermalized through collisions with the surrounding gas into a Boltzmann distribution characterized by the local temperature. However, deviations of the rotational populations from a Boltzmann distribution characterized by a single temperature have been observed and attributed to an incomplete thermalization of the OH from its initial, non-thermodynamic-equilibrium distribution. Here we address an additional cause for the apparent amount of excess population in the higher rotational levels of the OH airglow brought about by integrating these OH emissions through vertical gradients in the atmospheric temperature. We find that up to 40 % of the apparent excess population, currently attributed to incomplete thermalization, can be due to the vertical temperature gradients created by waves. Additionally, we find that the populations of the different upper vibrational levels are affected differently. These effects need to be taken into account in order to assess the true extent of non-thermodynamic-equilibrium effects on the OH rotational populations.

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Franzen, C., Espy, P. J., and Hibbins, R. E.: Modelled effects of temperature gradients and waves on the hydroxyl rotational distribution in ground-based airglow measurements, Atmos. Chem. Phys., 20, 333–343, https://doi.org/10.5194/acp-20-333-2020, 2020.

1 Introduction

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The hydroxyl (OH) airglow has been employed for many years for remote sensing of the mesosphere and lower thermosphere (MLT) region, an example of which may be found in Smith et al. (2010). The 8 km thick airglow layer is created at about 90 km altitude (Baker and Stair, 1988; Xu et al., 2012) by the highly exothermic reduction of ozone:

$$\begin{array}{}\text{(R1)}& \mathrm{H}+{\mathrm{O}}_{\mathrm{3}}\to {\mathrm{OH}}^{\ast}+{\mathrm{O}}_{\mathrm{2}}(\mathrm{5.3}\times {\mathrm{10}}^{-\mathrm{19}}\mathrm{J}).\end{array}$$

The excess heat of reaction, $\sim \mathrm{5.3}\times {\mathrm{10}}^{-\mathrm{19}}\mathrm{J}$, produces the OH^{∗} in excited
vibrational quantum levels of ${v}^{\prime}=\mathrm{6}$–9 (e.g. Mlynczak and Solomon, 1993).
Subsequent radiative cascading and collisional-deactivation produces
OH^{∗} in all vibrational levels ≤9. Radiative deactivation can
occur between any two vibrational quantum levels, but transitions with
Δ*v*=2 are preferred (Langhoff et al., 1986).

The excess energy of Reaction (1) also creates rotational excitation within
the OH^{∗} molecule in addition to the vibrational excitation. The
nascent rotational population for high rotational levels, *J*^{′}, shows a
distribution characteristic of a temperature far above the local atmospheric
temperature. Llewellyn et al. (1978) reported a nascent temperature of
760 K for ${v}^{\prime}=\mathrm{9}$, whereas others report temperatures as high as 9000–10 000 K (Oliva et al., 2015; Kalogerakis et al., 2018). Low
rotational levels (with *N*≤4) with energy separations less than kT, the
amount typically exchanged during collisions, have been observed to have
efficient energy transfer in the thermalization process (Maylotte et al.,
1972; Polanyi and Sloan, 1975; Polanyi and Woodall, 1972). Thus, emission
from these states has been observed to be characterized by a single-temperature Boltzmann distribution (Harrison et al., 1970, 1971; Pendleton et al., 1993; Perminov et al., 2007; Sivjee et al.,
1972; Sivjee and Hamwey, 1987). However, emission observed from the higher
rotational levels (*N* > 4), where the energy separation exceeds
kT, has indicated an anomalous, non-thermalized population that cannot be
described using the same Boltzmann temperature that characterizes the lower
rotational levels (Cosby and Slanger, 2007; Dodd et al., 1994;
Kalogerakis, 2019; Noll et al., 2015; Pendleton et al., 1989, 1993). In keeping with the terminology employed by
Pendleton et al. (1993), we refer to the condition where the
population can be described by a single-temperature Boltzmann distribution
as local thermodynamic equilibrium (LTE), whereas non-local thermodynamic
equilibrium (NLTE) is used when the population departs from that
distribution. Work is currently underway to use observations of the excess
populations in the high rotational levels of the OH airglow to determine
state-to-state quenching coefficients and to understand the thermalization
process in OH (Kalogerakis et al., 2018). The
term LTE as used here is not technically correct as it does not account for
radiative effects (i.e. the emission of airglow photons) on the
rotational–vibrational level population distribution. Instead it relates
only to the collisional distribution of the rotational levels being
characterized by the temperature of the surrounding gas, as has been done in
Pendleton et al. (1993).

Here we examine the effects of temperature gradients in the OH emission
region on the resulting vertically integrated spectrum of the Meinel Δ*v*=2 sequence. To achieve this, model work was executed, where the model
assumes that for each vibrational level, the rotational population
distribution of the OH is in LTE at every altitude. The emission in each
rotational line is then integrated vertically. We find that, even if the OH
rotational levels are in strict LTE with the surrounding atmosphere, the
temperature gradients through the OH emission region will create apparent
excess emission in the higher OH rotational lines. Here we calculate the
apparent excess population relative to the Boltzmann population expected
using the temperature determined by the population of rotational levels with
*N*≤4. This excess population can be incorrectly interpreted as due to
NLTE effects, affecting the subsequent calculations of the thermalization
process. The deviations in the inferred populations from a
single-temperature Boltzmann distribution are compared with observations
that include both NLTE and temperature gradient effects. This comparison is
made for realistic atmospheric temperature profiles that have been perturbed
with realistic atmospheric gravity waves in order to help quantify the true
NLTE content needed to construct a quantitative picture of OH
thermalization. Recent measurements indicate that the Boltzmann distribution
of the rotational levels may be characterized by a temperature that is
higher than that of the surrounding gas due to incomplete thermalization
(Noll et al., 2018). However, the purpose of this paper is to
show that, even if complete thermalization with the surrounding gas takes
place, ground-based measurements integrating through temperature gradients
within the OH layer will not see a rotational population described by a
Boltzmann distribution characterized by a single temperature.

Figure 1 shows an example of the OH airglow volume emission rate (VER) and a
temperature profile, measured from the Sounding of the Atmosphere using
Broadband Emission Radiometry (SABER) instrument aboard the NASA
Thermosphere Ionosphere Mesosphere Energetics and Dynamics (TIMED) satellite
(Mlynczak, 1997; Russell et al., 1999). This specific measurement is a
zonal mean and monthly average from July 2016 between the latitudes of 20
and 30^{∘} N. The OH VER from the vibrational levels 8 and 9 is
shown in red in the figure, while the black curve shows the temperature
profile. It can be seen that the temperature is not constant through the OH
layer. In this example the atmospheric temperature changes by over 10 K
through the layer. This observed behaviour is similar to other observations,
for example from French and Mulligan (2010), who
compared TIMED/SABER observations with ground-based observations.

Waves will exacerbate this effect by perturbing both the OH VER and changing the temperature gradient. Thus, the rotational level population distribution of the OH, even if thermalized at each altitude, will have different temperatures at each of those altitudes. Any instrument that integrates through the OH layer will therefore not see rotational line emission resulting from a single, average temperature but from the whole span of temperatures present in the layer.

2 Method

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We utilize a steady-state model of the OH VER, described below, to synthesize individual synthetic rotational spectra at 1 km intervals from 74 to 110 km. The model assumes that for each vibrational level, the rotational level population distribution of the OH is in LTE (i.e. can be described by a single-temperature Boltzmann distribution) with the local temperature at each altitude. Each rotational line is integrated in altitude to give the net spectrum that would be observed by an instrument integrating through the layer. The distribution of emission in the rotational lines is then used to infer the population of the OH rotational levels, allowing us to quantify the portion of the inferred excess population in the upper levels that is due to the temperature gradients across the OH layer.

The atmospheric background temperature and concentrations of N_{2},
O_{2}, H and O are taken from the US Naval Research Laboratory's mass
spectrometer and incoherent scatter radar model (NRLMSISE-00)
(Picone et al., 2002). The steady-state ozone
concentration is then calculated from balancing the production and loss
processes. The production mechanism is

$$\begin{array}{}\text{(R2)}& \mathrm{O}+{\mathrm{O}}_{\mathrm{2}}+\mathrm{M}\to {\mathrm{O}}_{\mathrm{3}}+\mathrm{M},\end{array}$$

where M is a reaction mediator. The temperature-dependent rate coefficients for this reactions are taken from the International Union for Pure and Applied Chemistry (IUPAC) Gas Kinetic Database (Atkinson et al., 2004). Loss processes include losses due to O via

$$\begin{array}{}\text{(R3)}& \mathrm{O}+{\mathrm{O}}_{\mathrm{3}}\to {\mathrm{2}\mathrm{O}}_{\mathrm{2}},\end{array}$$

using the reaction rate coefficient of Sander et al. (2003).
The loss of O_{3} to atomic hydrogen,

$$\begin{array}{}\text{(R4)}& \mathrm{H}+{\mathrm{O}}_{\mathrm{3}}\to {\mathrm{OH}}^{\ast}+\mathrm{O},\end{array}$$

was also used to calculate the production rate of OH^{∗} for each
vibrational level, using the reaction rate coefficient (Sander
et al., 2003). Due to the exothermicity of Reaction (4), vibrational levels
from ${v}^{\prime}=\mathrm{6}$ to 9 can be populated. The production of each vibrational level
OH^{∗}(v') is calculated using the branching ratios from
Sander et al. (2019). Collisional loss for each OH${}^{\ast}\left({v}^{\prime}\right)$ vibrational level was calculated for collisions with O, O_{2} and
CO_{2} using the rate coefficients of Dodd et al. (1991), Knutsen et al. (1996), Dyer et al. (1997) and Chalamala and Copeland (1993). The model
assumes quenching to the ground vibrational state, known as “sudden death”
for the O, and stepwise quenching by one vibrational unit for the O_{2}
and the CO_{2} (McDade and Llewellyn, 1987). N_{2} is not
considered as a quencher in this model. The rate coefficient for OH
quenching with N_{2} is small, and Knutsen et al. (1996) were
only able to provide an upper limit. Since the O_{2}∕N_{2} mixing ratio
is nearly constant up to the turbopause, the O_{2} is 3 times more
effective at quenching than the N_{2}. Thus, neglecting the N_{2} is
well within the uncertainty of the O_{2} rate coefficient and does not
significantly affect the altitude distribution of the OH. The relative shape
and peak height of the altitude profiles of the individual vibrational
levels agree closely with those of the more sophisticated model of
Adler-Golden (1997). As previously mentioned, the rotational
population distribution within this vibrational level is taken to be a
Boltzmann distribution characterized by the local temperature at this
altitude.

The total radiative loss from each vibrational level is given by
${N}_{{v}^{\prime}}\left(z\right)\cdot {A}_{{v}^{\prime}}$, where ${N}_{{v}^{\prime}}\left(z\right)$ is the concentration of the hydroxyl
*v*^{′} vibrational level at altitude *z* and ${A}_{{v}^{\prime}}$ is its inverse lifetime,
calculated from (Langhoff et al., 1986). The total VER of
any *v*^{′} to *v*^{′′} vibrational transition is then given by
${V}_{{v}^{\prime}{v}^{\prime \prime}}\left(z\right)\phantom{\rule{0.25em}{0ex}}=\phantom{\rule{0.25em}{0ex}}{A}_{{v}^{\prime}{v}^{\prime \prime}}\phantom{\rule{0.25em}{0ex}}\cdot \phantom{\rule{0.25em}{0ex}}{N}_{{v}^{\prime}}\left(z\right)$, where ${A}_{{v}^{\prime}{v}^{\prime \prime}}$ is the transition
probability for the vibrational transition from *v*^{′} to *v*^{′′} that is calculated from
the Einstein coefficients from Langhoff et al. (1986).
Although transition probabilities from a number of different studies differ
from those of Langhoff et al. (1986), the integration over
the rotational distribution, required to obtain the vibrational population,
is relatively insensitive to the choice of transition probabilities. This
radiative cascade into lower vibrational levels then acts as an additional
production term for levels below ${v}^{\prime}=\mathrm{9}$. Balancing these production and loss
terms at each height yields the steady-state concentration of each OH*
vibrational level as a function of height, ${N}_{{v}^{\prime}}\left(z\right)$. Such a model has been
shown to fit observations, for example TIMED/SABER (Xu et al.,
2012). The application of this model to the simulation of a ground-based
measurement under the influence of a realistic temperature gradient is
undertaken as described below.

As mentioned above, the background atmospheric temperature profile can be perturbed by waves. We therefore include wave-induced temperature perturbations in the model. Following Holton (1982), we limited the wave growth with altitude to maintain temperature gradients to below the dry adiabatic lapse rate. The wave has the form

$$\begin{array}{}\text{(1)}& {A}_{\mathrm{w}}\left(z\right)\cdot \mathrm{cos}\left({\displaystyle \frac{\mathrm{2}\mathit{\pi}z}{{\mathit{\lambda}}_{\mathrm{w}}}}+{\mathit{\phi}}_{\mathrm{w}}\right),\end{array}$$

where *ϕ*_{w} is the wave's phase, and *A*_{w}(*z*) is a function of altitude
so as not to exceed the dry adiabatic lapse rate. The lower edge of the
model is at 74 km altitude. The wave amplitude as a function of altitude is
shown in Fig. 2 for the case of a wave with an amplitude of 10 K at 74 km
altitude and a vertical wavelength of 20 km. This example is given for an
isothermal atmosphere of 200 K (Fig. 2a, dashed black line). The wave grows in
amplitude with altitude to conserve energy (dashed blue line), but at regions
where the lapse rate exceeds the dry adiabatic lapse rate (here between 90
and 95 km, and between 110 and 115 km), the wave loses energy and the
amplitude decreases (Holton, 1982). The breaking wave is shown in red in
Fig. 2a. Figure 2b shows the instantaneous lapse rate (change in
temperature with altitude) of the non-breaking wave (dashed blue line) and the
breaking wave (red), which never crosses the dry adiabatic lapse rate of 10 K km^{−1} (dashed black line). Figure 2c shows the amplitude of the non-breaking wave
(dashed blue line), which increases exponentially, and for the breaking wave (red),
which decreases at the altitudes where the wave dissipates energy.

The background atmosphere mixing ratios from the NRLMSISE-00 model were also perturbed using the gravity wave polarization relations from Vincent (1984). This perturbed background atmosphere and this temperature profile were then used in the steady-state model for the OH to yield a new, wave-perturbed ${N}_{{v}^{\prime}}\left(z\right)$ for the analysis.

The VER of a rotational transition for the *J*^{′} to *J*^{′′} state from the upper state,
*v*^{′}, of the *v*^{′} to *v*^{′′} vibrational band is given by

$$\begin{array}{}\text{(2)}& {V}_{v{}^{\prime}v{}^{\prime \prime}J{}^{\prime}J{}^{\prime \prime}}\left(z\right)={N}_{v{}^{\prime},J{}^{\prime}}\left(z\right)\cdot {A}_{v{}^{\prime}v{}^{\prime \prime},\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}J{}^{\prime}J{}^{\prime \prime}},\end{array}$$

where ${A}_{{v}^{\prime}{v}^{\prime \prime},{J}^{\prime}{J}^{\prime \prime}}$ is the transition probability for the *J*^{′} to *J*^{′′} rotational
transition in the *v*^{′} to *v*^{′′} vibrational transition. Due to vibrational–rotational
coupling, these coefficients are specific to each vibrational transition.
${N}_{{v}^{\prime},{J}^{\prime}}\left(z\right)$ is the population of the upper rotational level, *J*^{′}, in the upper
vibrational level, *v*^{′}. We concentrate in this paper on the 3∕2 electronic
subset of the OH airglow and drop the spin–orbit splitting quantum number *F*
in all equations for readability. Assuming that collisions have thermalized
the closely spaced rotational levels with the surrounding gas at temperature
*T*, their population may be described using a Boltzmann distribution written
as

$$\begin{array}{}\text{(3)}& {N}_{v{}^{\prime},J{}^{\prime}}\left(z\right)={\displaystyle \frac{{N}_{v{}^{\prime}}\left(z\right)\cdot \mathrm{2}\left(\mathrm{2}J{}^{\prime}+\mathrm{1}\right)}{{Q}_{\mathrm{R}}\left(T\left(z\right)\right)}}\cdot \mathrm{exp}\left({\displaystyle \frac{-{E}_{v{}^{\prime},J{}^{\prime}}}{{k}_{\mathrm{B}}\cdot T\left(z\right)}}\right),\end{array}$$

where ${N}_{{v}^{\prime}}\left(z\right)$ is the total population of the *v*^{′} vibrational level at altitude
*z* calculated from the model, ${E}_{{v}^{\prime},{J}^{\prime}}$ is the energy of the *J*^{′} rotational
level, the factor $\mathrm{2}(\mathrm{2}{J}^{\prime}+\mathrm{1})$ is the degeneracy of that level (including Λ-doubling) and *Q*_{R}(*T*(*z*)) is the rotational partition function (Herzberg,
1950). *k*_{B} is the Boltzmann constant.

Using the rotational transition probabilities of Rothman et al. (2013),
the rotational line VERs are calculated within a single *v*^{′} to *v*^{′′} transition
assuming that the OH is in LTE for the rotational level populations with the
surrounding gas at each altitude and therefore follows a Boltzmann
distribution of population characterized by the local temperature. The VER
of each rotational line of a given vibrational transition is integrated
through the layer from 74 km through to 110 km to give the intensity of the
line, ${I}_{{v}^{\prime}{v}^{\prime \prime},{J}^{\prime}{J}^{\prime \prime}}$. This results in a net spectrum of low rotational
lines whose intensities are enhanced relative to the mean temperature of the
emission region when emitted from the cooler regions. Accompanying this are
high rotational lines whose intensities are enhanced relative to the mean
temperature of the emission region when emitted from the warm regions. Each
of these is weighted by the VER of the vibrational transition at each
altitude, ${N}_{{v}^{\prime}}\left(z\right)\phantom{\rule{0.25em}{0ex}}\cdot \phantom{\rule{0.25em}{0ex}}{A}_{{v}^{\prime}{v}^{\prime \prime}}$.

As an example, the OH model was used to create a synthetic spectrum of the
(7,4) rotational–vibrational band, assuming LTE for the rotational level
populations at every altitude level, for conditions for mid-July for a
mid-latitude region. Specifically, the data presented are for Boulder,
Colorado (40.0^{∘} N; 105.6^{∘} W), to make a direct comparison
with the findings from Pendleton et al. (1993). The background
temperature profile (black line) and the (7,4) VER variation with altitude
(red line) are shown in Fig. 3a. These are perturbed by the wave with an
initial amplitude of 10 K at 74 km altitude, which has grown to about 30 K
at 90 km altitude, and a vertical wavelength of 30 km, shown in the dashed blue line.
Waves with similar amplitudes have been observed at these altitudes
(Picard et al., 2004).

The resulting distribution of rotational line intensities as a function wavelength for two altitudes is shown in Fig. 3b and c. The net spectrum as observed by a ground-based instrument is analysed, as detailed below, to examine the influence of this high rotational level tail on the fitted temperature.

After integration through the layer, the relative population of the *J*^{′} state
relative to the lowest rotational energy level, ${E}_{{v}^{\prime}}$, is now given in
terms of the line intensity by

$$\begin{array}{}\text{(4)}& \begin{array}{rl}{\displaystyle \frac{{I}_{v{}^{\prime},v{}^{\prime \prime},J{}^{\prime}J{}^{\prime \prime}}}{\mathrm{2}\left(\mathrm{2}J{}^{\prime}+\mathrm{1}\right)\cdot {A}_{v{}^{\prime}v{}^{\prime \prime},J{}^{\prime}J{}^{\prime \prime}}}}& ={Q}_{\mathrm{R}}\left(T\right){N}_{v{}^{\prime}}\\ & \cdot \mathrm{exp}\left({\displaystyle \frac{-\left({E}_{v{}^{\prime},J{}^{\prime}}-{E}_{v{}^{\prime}}\right)}{{k}_{\mathrm{B}}T}}\right),\end{array}\end{array}$$

where *N**v*^{′} is the integral of the vibrational population ${N}_{{v}^{\prime}}\left(z\right)$ over altitude,
and *T* is the effective rotational temperature of the altitude-integrated
spectrum. From this, the observed line intensities from rotational levels of
known quantum number, energy, and transition probability may be used to
define the relative total population and temperature using

$$\begin{array}{}\text{(5)}& \begin{array}{rl}\mathrm{ln}& \left({\displaystyle \frac{{I}_{v{}^{\prime},v{}^{\prime \prime},J{}^{\prime}J{}^{\prime \prime}}}{\mathrm{2}\left(\mathrm{2}J{}^{\prime}+\mathrm{1}\right)\cdot {A}_{v{}^{\prime}v{}^{\prime \prime},J{}^{\prime}J{}^{\prime \prime}}}}\right)=\mathrm{ln}\left({Q}_{\mathrm{R}}\left(T\right){N}_{v{}^{\prime}}\right)\\ & -{\displaystyle \frac{\mathrm{1}}{T}}\left[{\displaystyle \frac{\left({E}_{v{}^{\prime},J{}^{\prime}}-{E}_{v{}^{\prime}}\right)}{{k}_{\mathrm{B}}}}\right].\end{array}\end{array}$$

Figure 4 shows the result in terms of Eq. (9) of integrating each rotational line in the (7,4) Meinel band for the atmospheric perturbation of a wave with an amplitude of 30 K at 90 km altitude and 30 km vertical wavelength. This is the same wave as shown in Fig. 3. Following Pendleton et al. (1993), fitting a temperature to the lowest three rotational levels of the OH spectrum yields the dashed red curve shown in Fig. 4. It may be seen that the lowest three rotational levels are well characterized by a single, Boltzmann rotational temperature, ${T}_{\mathrm{1},\mathrm{3}}=\mathrm{155.8}\pm \mathrm{0.8}$ K, as has been observed in nightglow spectra (Espy and Hammond, 1995; Franzen et al., 2017; Harrison et al., 1970; Noll et al., 2015). However, there is excess emission in the higher rotational lines which could be interpreted as populations exceeding that expected from a thermalized Boltzmann distribution. This apparent excess population occurs even though the OH distribution was constrained to be a Boltzmann distribution with a single temperature at each altitude.

A non-linear formulation can be used in order to characterize this excess population. This non-linear fit is of the form

$$\begin{array}{}\text{(6)}& \begin{array}{rl}\mathrm{ln}& \left({\displaystyle \frac{{I}_{v{}^{\prime},v{}^{\prime \prime},J{}^{\prime}J{}^{\prime \prime}}}{\mathrm{2}\left(\mathrm{2}J{}^{\prime}+\mathrm{1}\right)\cdot {A}_{v{}^{\prime}v{}^{\prime \prime},J{}^{\prime}J{}^{\prime \prime}}}}\right)=\mathrm{ln}\left({Q}_{\mathrm{R}}\left(T\right){N}_{v{}^{\prime}}\right)\\ & -{\displaystyle \frac{\mathrm{1}}{T}}\left[{\displaystyle \frac{\left({E}_{v{}^{\prime},J{}^{\prime}}-{E}_{v{}^{\prime}}\right)}{{k}_{\mathrm{B}}}}\right]+{\displaystyle \frac{\mathit{\beta}}{{T}^{\mathrm{2}}}}{\left[{\displaystyle \frac{\left({E}_{v{}^{\prime},J{}^{\prime}}-{E}_{v{}^{\prime}}\right)}{{k}_{\mathrm{B}}}}\right]}^{\mathrm{2}}.\end{array}\end{array}$$

Here *β* is a non-linearity parameter, which is a free parameter in the
non-linear fit. This non-linear fit is also presented in Fig. 4 as a solid
blue line, showing that the intensity in rotational lines as high as ${J}^{\prime}=\mathrm{9.5}$ is now fitted. The retrieved temperature, ${T}_{\mathrm{NL}}=\mathrm{154.9}\pm \mathrm{0.1}$ K, is the same as *T*_{1,3} within the fitting uncertainties, with a *β*
factor of 0.019.

So far, we have considered only a single wave with one given amplitude,
wavelength and phase as an example. Different wave amplitudes *A*_{w} will,
of course, change the populations seen in Fig. 4. In the limiting case,
where *A*_{w}=0, the original background temperature profile is obtained.
Different wavelengths *λ*_{w} will also change the shape of
the atmospheric temperature profile. Short *λ*_{w} waves can change
the temperature toward both higher and lower temperatures within the OH
layer. Thus, their effect can be small when integrated over the whole layer.
Longer *λ*_{w} waves, especially with *λ*_{w} on the order of,
or longer than, the thickness of the OH layer, can introduce a temperature
change that only warms one end of the layer and cools the opposite end. When
such gradients reinforce the background temperature change with altitude,
they can change the total integral over the OH layer substantially, as seen
in Fig. 4. Lastly, the wave phase *φ*_{w} can change the influence of
the wave on the total integral of the OH layer. The same wave would have the
opposite temperature perturbation if *φ*_{w} was shifted by half a
wavelength. In this case, the wave would work against the natural
temperature gradient of the background atmosphere without a wave, resulting
in a smaller effect than that shown in Fig. 4.

All three wave parameters – *A*_{w}, *λ*_{w} and *φ*_{w} –
should therefore be considered when modelling different waves to examine
their influence on the total integrated OH spectrum. In this research, the
three parameters were adjusted in equidistant steps. The amplitude *A*_{w}
was varied between 0 and 40 K at an altitude of 90 km, spanning a range of
previously observed semi-diurnal tide amplitudes (Hagan et al., 1999;
Oberheide et al., 2011; Picard et al., 2004; She et al., 2002; Shepherd and
Fricke-Begemann, 2004; Zhang et al., 2006). The vertical wavelength *λ*_{w} was varied between 2 and 80 km, spanning the range of gravity
waves, tides and planetary waves (Davis et al.,
2013). The phase *φ*_{w} was varied between 0 and 2*π*.

3 Results and discussion

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Figure 4 shows that a large atmospheric temperature gradient can produce
higher populations in the higher *J*^{′} rotational lines than would be expected
from a strict LTE fit with only one effective temperature. We now want to
quantify how large this apparent excess population can become and compare it
to the study by Pendleton et al. (1993). An ensemble of waves
was simulated as described above. The apparent excess population and the
non-linearity parameter, *β*, were calculated for each wave-perturbed
temperature background profile.

Figure 5 shows two different wave scenarios for the (7,4) transition. The red plot (with the axis on the left-hand side) shows a no-wave scenario, where the atmospheric background temperature profile from the NRLMSISE-00 model is used. The apparent excess population is the ratio between the intensity of a rotational line integrated in altitude and the intensity of that line predicted by a Boltzmann distribution fitted to the distribution of integrated line intensities of the lowest three rotational lines using a single, effective temperature. Thus, an apparent excess population of 1 is the same population as that predicted from a single-temperature Boltzmann distribution fitted to the lowest three rotational lines. Similarly, an apparent excess population of 2 is a population twice as large as that predicted from this single-temperature Boltzmann distribution. It is clear that the first three lines with ${J}^{\prime}\le \mathrm{3.5}$ can be characterized by a single, effective Boltzmann temperature. However, all higher lines show populations in excess of that expected from a single effective temperature. The effective overpopulation approaches approximately 1.12 for ${J}^{\prime}=\mathrm{9.5}$, even though the OH molecule is thermalized with the local temperature at every altitude.

The blue bars (with the plot on the right side) show a scenario with a wave
perturbing the atmosphere. This specific wave has an amplitude *A*_{w}=30 K at 90 km altitude, *λ*_{w}=30 km and *φ*_{w}=4.9 rad. This
would be a large amplitude gravity wave but still within the range of
atmospheric tidal observations. Although different *J*^{′} levels are affected
differently, the general shape is similar to that observed with the no-wave
scenario. However, the magnitude of the effect is much greater. The first
three lines show the populations expected in a single-temperature LTE case,
while higher *J*^{′} levels yield increasingly higher apparent excess populations.
For the highest line considered in this paper with ${J}^{\prime}=\mathrm{9.5}$, the apparent
excess population is twice that expected for a Boltzmann distribution with a
single effective temperature, despite the OH being in LTE for the rotational
level populations at every altitude.

These results can be compared to the findings from Pendleton
et al. (1993) in their Fig. 16 for the (7,4) transition above Boulder,
Colorado, during midsummer, the same season and transition as presented
here. These results are shown as grey bars in Fig. 5a and b. Although
the measurement from Pendleton et al. (1993) included NLTE
effects, the overall shape of the distribution is similar to the LTE for the
rotational level population simulations presented here. When comparing
these results, note that Pendleton et al. (1993) use the
lower-state quantum number *N*, which corresponds to our ${J}^{\prime}+\mathrm{1}/\mathrm{2}$. The first three
rotational levels in this study and in Pendleton et al. (1993)
have populations that follow a Boltzmann distribution characterized by a
single temperature, and above that, there is excess population. The
difference between the absolute numbers we observe and those of
Pendleton et al. (1993) indicates the portion of the NLTE that
might be due to the temperature gradient effects observed here. For example,
Pendleton et al. (1993) reported an apparent excess population
of around a factor of 2 for the *P*(*N*=7) (i.e. ${J}^{\prime}=\mathrm{6.5}$) line. However, for the
two cases presented here in Fig. 5, there is less apparent excess
population. While the no-wave scenario yields an apparent excess population
of 1.02, a background profile with a wave yields 1.17 times the population.
These numbers mean that up to about 17 % of the effect
Pendleton et al. (1993) observed could be due to wave activity
and not NLTE effects. Thus, the effect of the atmospheric temperature
background has to be considered in addition to NLTE effects whenever the
populations inferred from integrated airglow observations of high *J*^{′} lines are
to be used in kinetic thermalization studies.

This calculation of the apparent excess population can now be repeated for
waves of different amplitudes, wavelengths and phases. Figure 6 shows the
non-linearity in terms of the *β* value of the temperature fit (from
Eq. 10) and the corresponding apparent excess population of the sixth
rotational line with ${J}^{\prime}=\mathrm{6.5}$ (*N*=7), the highest line reported by
Pendleton et al. (1993) in their NLTE study of the OH. The
figure shows results related to the (7,4) vibrational transition, which is
the same transition that Pendleton et al. (1993) used. Figure 6a shows that for vertical wavelengths above about 20 km, the non-linearity
of the temperature fit increases with wave strength, approaching 2 % of
the linear temperature variation (see Eq. 10). The maximum
non-linearity is observed at vertical wavelengths of about 20 km for waves
with an amplitude below 10 K, while for waves with an amplitude of 40 K
the maximum non-linearity is observed at vertical wavelengths around 40 km.
For shorter vertical wavelengths, the non-linearity is smaller (at about 0.5 %) and decreases with increasing wave amplitude. Figure 6b shows the
apparent excess populations for a wave with a phase that yields the highest
apparent excess population for a given wave amplitude and wavelength. There
are small but observable effects in the limiting case of no waves
(background atmosphere; see Fig. 5a), and the apparent excess population
increases for longer wavelengths and stronger waves similarly to the
non-linearity of the fit. Extreme waves with 40 K amplitude at an altitude
of 90 km can cause up to 1.3 apparent excess populations in the ${J}^{\prime}=\mathrm{6.5}$ line.

While Fig. 6 shows the (7,4) transition, Fig. 7 shows the same analysis for
the commonly observed (3,1) transition. Both the *β* value and the
apparent excess population look qualitatively similar to the (7,4) band, but
the effect of waves on the (3,1) transition is about 30 % stronger. That
means that the non-linearity exceeds 2.5 % and the apparent excess
population of the ${J}^{\prime}=\mathrm{6.5}$ level is up to 1.4 for the largest waves shown
here.

Figures 6 and 7 show the results for the phase that created the largest apparent excess population. Hence, the phase is not constant for each point in these two figures but rather varies to show the largest apparent excess population for the wave amplitude and wavelength in question. The mean effect of all different phases simulated is independent of the transition and varies between 20 % and 40 % of the maximum effect presented in Figs. 6 and 7.

Tests showed that the difference in the apparent excess population between
the (3,1) and (7,4) bands is unlikely to be due to the altitude separation
of the different vibrational levels in the OH airglow layer (von
Savigny et al., 2012). Repeating the analysis and weighting the ${v}^{\prime}=\mathrm{3}$ and 7
levels with the same VER profile yields essentially the same result, as did
performing the analysis in an unperturbed isothermal background temperature
profile. Instead, the difference in the apparent excess population between
the (3,1) and (7,4) bands is likely due to the compressed rotational energy
structure of the higher vibrational levels that lie closer to the
dissociation limit. This compression of the rotational energy levels is due
to the increased moment of inertia and hence the reduced rotational
constant, associated with the larger average inter-nuclear distance of the
higher vibrational levels. Thus a given *J*^{′} level in a low vibrational state
will have more rotational energy than one in a high vibrational state. Thus,
for a given temperature, higher rotational levels will be thermally
populated in the higher vibrational levels. These thermally populated higher
rotational levels then make the perturbing effects of waves relatively less
important.

Figure 8 illustrates the dependency of the amount of apparent excess population on the different vibrational upper levels at different altitudes. For consistency, the apparent excess population of the ${J}^{\prime}=\mathrm{6.5}$ upper level is again shown. All data points are for the wave presented above as an example, with an amplitude of 30 K at 90 km altitude and a vertical wavelength of 30 km.

The smallest upper vibrational level of ${v}^{\prime}=\mathrm{2}$ shows an apparent excess population of about 1.28, while the highest upper vibrational level of ${v}^{\prime}=\mathrm{9}$ only shows an apparent excess population of about 1.08.

4 Conclusions

Back to toptop
Spectroscopic observations of the OH airglow have been commonly used to measure temperatures in the MLT. The OH radiates over an extended, Chapman-like layer that extends over several kilometres, over which the temperature is changing. Strong waves perturbing the MLT can make this change in temperature within the OH layer substantial. The simulations executed here show that these temperature profiles can create an apparent non-thermal population of the rotational levels in a given Meinel (v', v”) band. Even though the simulations calculated the rotational population distribution of the OH to be in LTE with the surrounding gas at every altitude, the integrated intensities of the higher rotational lines indicate an apparent excess population that could be misinterpreted as contributing to the NLTE effects previously reported (Cosby and Slanger, 2007; Noll et al., 2015; Pendleton et al., 1993).

We have shown in this work that the influence of a wave with an amplitude of 30 K and a vertical wavelength of 30 km at an altitude of 90 km can produce an apparent excess population of 1.12; i.e. they can explain 12 % of the effects previously ascribed to NLTE by Pendleton et al. (1993) for the (7,4) transition. Larger waves cause an apparent excess population of up to 1.3 and can therefore explain up to 30 % of these effects. Smaller waves can also have an apparent excess population of up to 1.1, and their impact cannot be ignored. Other transitions with lower vibrational quantum numbers show even higher apparent excess populations caused by this temperature-variation effect. We conclude that it is necessary to consider the temperature profile in order to infer OH rotational level population distributions from ground-based airglow observations.

Data availability

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Data availability.

The SABER data presented in Fig. 1 are available under http://saber.gats-inc.com/ (last access: 8 January 2020). No other data were used in this paper.

Author contributions

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Author contributions.

CF modified the chemical model with dynamic inputs, ran the model and interpreted the model output. CF also wrote the paper. PJE supported the analysis process and the writing of the paper. REH gave input on the interpretation and the writing of the paper.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

This work was supported by the Research Council of Norway (CoE) under contract 223252/F50. We thank Halvor Borge, who undertook some preliminary studies on this research during his master thesis at NTNU in the spring semester of 2018. We also thank Kate Faloon for her work on the chemical model.

Financial support

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Financial support.

This research has been supported by the Research Council of Norway (grant no. 223252/F50).

Review statement

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Review statement.

This paper was edited by William Ward and reviewed by two anonymous referees.

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Short summary

Ground-based observations of the hydroxyl (OH) airglow have indicated that the rotational energy levels may not be in thermal equilibrium with the surrounding gas. Here we use simulations of the OH airglow to show that temperature changes across the extended airglow layer, either climatological or those temporarily caused by atmospheric waves, can mimic this effect for thermalized OH. Thus, these must be considered in order to quantify the non-thermal nature of the OH airglow.

Ground-based observations of the hydroxyl (OH) airglow have indicated that the rotational energy...

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