# Effective Emission Heights of Various OH Lines From X-shooter and SABER
Observations of a Passing Quasi-2-Day Wave

###### Abstract

Chemiluminescent radiation of the vibrationally and rotationally excited OH
radical, which dominates the nighttime near-infrared emission of the Earth’s
atmosphere in wide wavelength regions, is an important tracer of the chemical
and dynamical state of the mesopause region between 80 and 100 km. As
radiative lifetimes and rate coefficients for collision-related transitions
depend on the OH energy level, line-dependent emission profiles are expected.
However, except for some height differences for whole bands mostly revealed by
satellite-based measurements, there is a lack of data for individual lines. We
succeeded in deriving effective emission heights for 298 OH lines thanks to
the joint observation of a strong quasi-2-day wave (Q2DW) in eight nights in
2017 with the medium-resolution spectrograph X-shooter at the Very Large
Telescope at Cerro Paranal in Chile and the limb-sounding SABER radiometer on
the TIMED satellite. Our fitting procedure revealed the most convincing
results for a single wave with a period of about 44 h and a vertical
wavelength of about 32 km. The line-dependent as well as altitude-resolved
phases of the Q2DW then resulted in effective heights which differ by up to 8
km and tend to increase with increasing vibrational and rotational excitation.
The measured dependence of emission heights and wave amplitudes (which were
strongest after midnight) on the line parameters implies the presence of a
cold thermalized and a hot non-thermalized population for each vibrational
level.

JGR: Atmospheres

Institut für Physik, Universität Augsburg, Augsburg, Germany Deutsches
Fernerkundungsdatenzentrum, Deutsches Zentrum für Luft- und Raumfahrt,
Weßling-Oberpfaffenhofen, Germany Institut für Astro- und Teilchenphysik,
Universität Innsbruck, Innsbruck, Austria Instituto de Astronomía, Universidad
Católica del Norte, Antofagasta, Chile

Stefan<EMAIL_ADDRESS>
X-shooter-based intensities of 298 OH lines from eight nights show a strong
Q2DW in southern summer 2017

Fits of the Q2DW phase in the X-shooter data and SABER-based OH emission
profiles were used to derive effective OH emission heights

The line-dependent wave amplitudes confirm the presence of cold and hot OH
populations for each vibrational level

## Plain Language Summary

Hydroxyl (OH) is an important molecule in the Earth’s atmosphere at altitudes
between 80 and 100 km. It is the main source of atmospheric nighttime
radiation in the near-infrared wavelength range and is therefore a valuable
tracer of the chemistry and dynamics at high altitudes. The emission spectrum
consists of various lines which are related to different levels of vibration
and rotation. Although the vertical emission distribution should depend on the
given line due to differences in the deactivation of the corresponding energy
levels, the line-specific details have been uncertain until now. We have
succeeded in deriving effective time-averaged emission heights for 298 OH
lines based on the combination of ground-based line-resolved and space-based
height-resolved observations of a very strong rising wave with a period close
to 2 days and a relatively short vertical wavelength in eight nights in 2017.
The resulting heights (obtained via the line-dependent wave phases) differ by
up to 8 km and generally increase with higher molecular vibration and
rotation. They are valuable for ground-based studies of other waves and
contribute (combined with conclusions from the wave amplitudes) to a better
understanding of the internal processes in OH molecules.

##  1 Introduction

The nocturnal atmospheric emission of wide wavelength regions in the near-
infrared is dominated by chemiluminescent radiation of various roto-
vibrational bands up to the vibrational level $v=9$ of the electronic ground
state of the hydroxyl (OH) radical [<]e.g.,¿meinel50,noll15,rousselot00.
Excited OH is mostly produced by the reaction of hydrogen with ozone [Bates
Nicolet (1950)]. The nightglow emission essentially originates from altitudes
between 80 and 100 km, and is therefore an important tracer of the chemistry
and dynamics in the Earth’s mesopause region. Rocket flights showed typical
peak heights of about 87 km and layer widths of about 8 km [Baker Stair
(1988)]. Moreover, OH emission profiles have frequently been observed by limb-
sounding instruments in space [<]e.g.,¿baker07,dodd94,savigny12,wuest20,yee97.
Rare ground-based altitude measurements relied on the identification of the
same emission features in imaging instruments at different sites
[<]e.g.,¿kubota99,moreels08. yu17 combined wind observations with a meteor
radar and a Fabry-Perot interferometer focusing on OH to estimate peak heights
of the radiation. Finally, rough estimates are also possible by means of the
well-established negative relation between effective emission height and
emission intensity [<]e.g.,¿garcia17,liu06,mulligan09,savigny15,yee97, which
can be explained by the larger change of the layer profile at lower altitudes
due to the stronger relative variations of atomic oxygen (which is required
for the ozone production). The published relations had to be calibrated with
satellite observations.

The hydrogen–ozone reaction mainly produces OH populations in high $v$ between
7 and 9 [<]e.g.,¿adler97. In contrast, measured populations increase with
decreasing $v$ [<]e.g.,¿takahashi81,cosby07,noll15. This discrepancy is caused
by the step-by-step relaxation process due to the radiation of photons and
collisions with other atmospheric constituents [<]e.g.,¿adler97,xu12,noll18b.
As the radiative lifetimes and rate coefficients for the collisions with
different species depend on the OH level, the resulting emission profiles for
individual lines should differ. Models show an increase of the peak emission
height with increasing $v$
[<]e.g.,¿adler97,makhlouf95,mcdade91,savigny12,xu12. The spread between high
and low $v$ may amount to several kilometers with possibly larger height
shifts for low $v$. Collisions of OH with atomic oxygen appear to be
especially crucial for the $v$-dependent discrepancies [von Savigny . (2012)].
The impact of rotational energy on the altitude distribution has been modeled
quite rarely. dodd94 discussed the differences between populations with
rotational quantum numbers $N$ up to 7 and those with $N\geq 11$. Density
profiles for the larger $N$ appear to peak higher with similar altitude
differences as obtained for the comparison of low and high $v$. Moreover, the
density peaks seem to show less variation for larger $N$ with respect to
changes in $v$. noll18b presented modeling results for $v=9$ and found maximum
$N$-dependent height differences between 1.5 and 2.8 km depending on the
uncertain rate coefficients for collisions with atomic oxygen.

Measurements of the impact of the OH energy state on the height distribution
were successfully performed with space-based limb sounding. The studies mostly
focused on the comparison of the emission profiles of a few roto-vibrational
bands [Noll . (2016), Sheese . (2014), von Savigny Lednyts’kyy (2013), von
Savigny . (2012)]. The results suggest typical height changes for $\Delta v=1$
of about 0.4 to 0.5 km. These differences can vary by a significant fraction
of the values. They tend to increase with lower peak altitude and higher
atomic oxygen concentration [von Savigny Lednyts’kyy (2013)]. The former can
change by several kilometers forced by waves on different time scales and the
general circulation [<]e.g.,¿liu06,nikoukar07,noll16,teiser17,yee97. The
majority of the variability is found below the emission peak [Nikoukar .
(2007)], where the atomic oxygen density profile is particularly steep. In
contrast to whole roto-vibrational bands, there is a lack of studies of
individual lines with different $N$. The only noteworthy data known to us are
related to observations with the Cryogenic InfraRed Radiance Instrumentation
for Shuttle (CIRRIS 1A) Michelson interferometer onboard Space Shuttle
Discovery for three days in 1991 [Dodd . (1994)]. The measured limb profiles
comprise lines of several $v$ with $N\leq 4$ as well as purely rotational
lines with $N$ around 15 and 30. However, the interpretation of the data
required sophisticated modeling.

There is an alternative approach to estimate effective emission heights of OH
lines. Perturbations passing the mesopause region will produce characteristic
patterns in the OH intensity time series that will be shifted in time
depending on the altitude of the emission [Noll . (2015), Schmidt . (2018)].
In this way, the relative layering of different emissions can be obtained from
ground-based observations of individual lines. Nevertheless, the derivation of
absolute heights will also need information on the vertical propagation of the
perturbation through the OH emission layer as can be measured by satellite-
based limb-sounding instruments. As the satellites relevant for nightglow
observations have nearly Sun-synchronous orbits
[<]e.g.,¿russell99,savigny12,yee97, the minimum time scale of the variations
needs to be of the order of days for a promising combination of the profile
data with ground-based spectra of OH lines in order to link wave phases with
altitudes in the same region.

The so-called quasi-2-day wave (Q2DW) can achieve very high amplitudes in the
mesopause region at low to middle latitudes. In particular, strong Q2DWs occur
in the Southern Hemisphere for several weeks in the summer months January and
February [<]e.g.,¿ern13,gu19,tunbridge11,walterscheid15. The period is close
to 2 days but can vary from 42 to 54 h [Gu . (2019)]. For periods very close
to 48 h, there can be phase locking with tidal modes [Walterscheid Vincent
(1996)]. Southern Q2DWs usually show a dominating westward-moving longitudinal
pattern with a zonal wavenumber of 3 (W3), which can be accompanied by other
modes with different periods, zonal wavenumbers, and propagation directions
[<]e.g.,¿he21,pedatella12,tunbridge11. Q2DWs are regarded to belong to the
class of Rossby-gravity waves [<]e.g.,¿salby81. Their genesis is probably
related to baroclinic/barotropic instabilities in the summertime easterly
zonal wind jet in the lower mesosphere [Plumb (1983)], which seem to be linked
to increased gravity-wave drag [Ern . (2013)]. The amplitudes maximize at the
OH emission heights and can reach more than 10 K in temperature [Gu . (2019),
Tunbridge . (2011)] and 50 m s-1 in wind [Limpasuvan . (2005), Wu . (1993)].
Between the Q2DW source region and these altitudes at low to middle latitudes,
the propagation perpendicular to the zonal component is effectively upward.
The situation is different in the lower thermosphere well above the OH layer,
where a northward meridional flow tends to be dominant in the Southern
Hemisphere [Yue . (2012)]. At OH emission heights, vertical wavelengths are
mostly longer than 20 km and are often even beyond 100 km [Huang . (2013),
Reisin (2021)]. The impact of a Q2DW on OH emission was investigated by
pedatella12 based on space-based OH profile data for January 2006. The peak
emission rate varied by a factor of about 4. This large effect appears to be
mostly related to Q2DW-induced variations of the atomic oxygen concentration
at the OH emission heights. The concentration can also be modulated by tides
and gravity waves.

We have succeeded in deriving effective emission heights for about 300 OH
lines by means of the combination of ground-based spectroscopic and space-
based height-resolved observations of a strong Q2DW in January to February
2017. The OH line intensities originate from near-infrared spectra of the
medium-resolution spectrograph X-shooter [Vernet . (2011)] of the Very Large
Telescope (VLT) of the European Southern Observatory (ESO) at Cerro Paranal in
Chile (24.6∘ S, 70.4∘ W). Scans of the OH emission profiles in two filter
bands at about 1.6 and 2.1 $\mu$m are related to measurements with the
Sounding of the Atmosphere using Broadband Emission Radiometry (SABER)
instrument onboard the Thermosphere Ionosphere Mesosphere Energetics Dynamics
(TIMED) satellite [Russell . (1999)]. In the following section 2, we will
describe the data sets for both instruments. Then, we will present our
approach to fit the observed wave (section 3). Our results will be shown in
section 4, which also includes our estimates of line-dependent effective
emission heights (section 4.3) and a brief discussion of another Q2DW in 2019
(section 4.4). Finally, we will draw our conclusions (section 5).

##  2 Data

###  2.1 X-shooter

The X-shooter medium-resolution echelle spectrograph [Vernet . (2011)] mounted
at an 8 m telescope of the VLT can simultaneously observe a very wide
wavelength range from 0.3 to 2.5 $\mu$m covered by three spectroscopic arms
with separate optical components and detectors. Since the strongest OH bands
for all upper vibrational levels $v^{\prime}$ from 2 to 9 are found in the
near-infrared regime [<]e.g.,¿rousselot00, we focus our analysis on the
correspondingly named NIR arm, which extends from 1.0 to 2.5 $\mu$m. For this
arm and scientific targets, the projected slit width can vary between 0.4 and
1.5′′ on the sky, which corresponds to a spectral resolving power between
12,000 and 3,500. The projected slit length is fixed to 11′′.

The spectra used originate from the ESO Science Archive Facility. This study
only uses a small fraction of the data that were processed by us. The whole
sample comprises about 90,000 spectra in the NIR arm with an exposure time of
at least 10 s that were taken between October 2009 and September 2019. The
basic processing of the raw echelle spectra was performed by means of the
official reduction pipeline [Modigliani . (2010)] in version v2.6.8 and
calibration data preprocessed by ESO [<]see also¿unterguggenberger17. The
resulting two-dimensional (2D) wavelength-calibrated sky spectra were
subjected to post-processing optimized for the retrieval of airglow emission.
First, the 2D spectra were averaged along the spatial direction in order to
obtain one-dimensional (1D) spectra. Then, the separation of sky emission and
astronomical target performed by the pipeline was improved. For this goal, we
extracted residual sky in the 2D object spectrum from the third of the slit
positions with the lowest integrated counts in the whole wavelength range.
After the minimization of noise by smoothing, we added the resulting 1D mean
spectrum to the sky spectrum. This approach was particularly useful for bright
star-like astronomical targets.

Although the pipeline produces flux-calibrated spectra, we performed this
processing step independently in order to minimize the errors by means of a
consistent approach for the entire data set. For the NIR arm, pipeline-
processed spectra of the relatively bright spectrophotometric standard stars
EG 274 and LTT 3218 [Moehler . (2014)] were corrected for telluric absorption
using a model-based fitting approach [Smette . (2015), Kausch . (2015)] and
then compared to the theoretical spectral energy distributions in order to
derive individual response curves that indicate the wavelength-dependent
instrumental quantum efficiency. As the resulting response curves for both
stars did not exactly match, we lowered the quantum efficiency of the EG 274
curves (by about 3% in a wide wavelength range) for a better consistency.
Thus, remaining systematic uncertainties in the absolute flux calibration for
clear sky conditions mainly depend on the quality of the reference spectrum
for LTT 3218. As the response curves are relatively uncertain at wavelengths
longer than 1.9 $\mu$m [<]see¿noll15, we used spectra of so-called telluric
standard stars with well-known spectral energy distributions (Rayleigh–Jeans
law) for atmospheric absorption measurements to derive a general correction
function. In order to further improve the quality, master response curves for
10 time periods with lengths between 9 and 15 months were created including
only the best-quality data and involving a final smoothing procedure. The
splitting of the periods is usually at New Year but also considers a change in
the calibration products in January 2013 and a recoating of the main mirror in
December 2016. An analysis of the variability of the flux-calibrated star
spectra points to relative uncertainties of only 2 to 3% up to 2.1 $\mu$m in
the NIR arm. This is a clear improvement in comparison to a maximum difference
in the response of about 12%. The final set of master response curves was
applied to the 1D sky spectra. A minor fraction of the spectra were taken with
a so-called $K$-blocking filter [Vernet . (2011)], which reduces the useful
wavelength range to wavelengths shorter than 2.1 $\mu$m. These spectra had to
be calibrated with a 1.3 times higher response in order to correctly consider
differences in the calibration data.

The measurement of the line intensities was performed in two steps. First, the
underlying continuum was determined. After various tests, our procedure used
the first quintile of the pixel intensities (cut at 20%) in wavelength ranges
with a relative width of 0.008 to derive the continuum at the corresponding
central positions. The width was increased by the factors 5.0 and 2.5 around
the dense roto-vibrational O2 emission bands at about 1.27 and 1.58 $\mu$m,
respectively [<]e.g.,¿rousselot00. It was multiplied by a factor of 0.4 beyond
2.08 $\mu$m in order to avoid issues with the steeply increasing continuum due
to thermal radiation of the telescope. After subtraction of the resulting
continuum, the residual flux was integrated in specific wavelength ranges
depending on the central wavelength of the OH $\Lambda$ doublet, separation of
both components [<]e.g.,¿noll20, and slit width. As an X-shooter profile of an
unresolved line doublet (which is true in most cases) can be approximated by
the combination of a fixed Gaussian and a slit-dependent boxcar, the
integration limits had been optimized to assure relatively stable distances to
the positions marking half peak intensity. For example, the size of these
margins amounts to about 60% of the full integration range minus the $\Lambda$
doublet separation for the most frequently used slit with a width of 0.9′′.
The separation of the integration limits was sufficiently wide to avoid
significant flux losses by uncertainties in the wavelength calibration
(fraction of a pixel). Positions of suitable lines were taken from brooke16.
Important selection criteria were clear detectability (at least for long
exposures), the lack of blends with other emission lines, a smooth continuum
without strong absorption features, and high atmospheric transmission. In the
end, the selection procedure (which also included a check for outliers in the
scientific analysis) resulted in 298 OH $\Lambda$ doublets from 14 bands,
which cover upper vibrational levels $v^{\prime}$ between 2 and 9 and upper
rotational levels $N^{\prime}$ up to 15.

The measured line intensities were corrected for the van Rhijn effect [van
Rhijn (1921)], i.e. the projected layer width, assuming a reference altitude
of 87 km. The correction of this effect is important as there is a strong
variation in the zenith angles of astronomical observations. We also
considered the absorption of line emission by molecules in the lower
atmosphere in a similar way as decribed by noll15. The approach involves the
calculation of line-specific transmissions based on the zenith angle, Cerro
Paranal atmospheric data [Noll . (2012)], the Line-By-Line Radiative Transfer
Model [<]LBLRTM,¿clough05, and the assumption of purely Doppler-broadened OH
lines for a temperature of 190 K at brooke16 wavelengths. Transmissions for
entire $\Lambda$ doublets were derived by means of weights for the individual
components that were taken from the branch-specific OH level population fits
of noll20. Reference line transmissions were calculated for zenith and a
typical amount of precipitable water vapour (PWV) of 2.5 mm. Doublets with
reference transmissions lower than 70% were not considered for the final line
set (average of 96%). Differences in the optical depth in the spectroscopic
data set were calculated depending on the most crucial parameters, zenith
angle and PWV. For the latter, accurate measurements with a Low Humidity And
Temperature PROfiler (L-HATPRO) microwave radiometer are available at Cerro
Paranal [Kerber . (2012)]. However, as the L-HATPRO data set starts in 2014
and shows several gaps afterwards, we mostly used these data to calibrate
intensity ratios of lines with low and high transmission in the bands OH(2-0)
and OH(6-4) to estimate PWV values for each spectrum [<]cf.¿xu20. Up to very
high L-HATPRO PWVs of about 16 mm, the approach works quite well with a mean
offset of -0.1 mm and a standard deviation of 0.6 mm. With an upper cut at 6
mm (which still represents about 90% of the data), the systematic shift is
negligible and the scatter is only half as large.

The quality of the measurement of OH line intensities strongly varies from
spectrum to spectrum due to strong changes in spectral resolution, exposure
time, atmospheric water vapour, and especially contamination by (spatially
extended) astronomical targets that can have a different effect on each line.
For this reason, we performed a line-specific selection of suitable
observations. Starting with a preselected set of 88,481 useful spectra, we
carried out an iterative $\sigma$-clipping procedure for each line. This
outlier rejection involved limits with respect to the underlying continuum,
the intensity error, and the intensity itself. The error was estimated from
the deviation of the residual continuum at both margins of the integration
range from the zero line. It was made sure that it was always lower than 10%
for the selected measurements. As a result, the line-specific selection rates
varied between 69.4% and 99.8% with a mean value of 94.1% for wavelengths
shorter than 2.1 $\mu$m. In order to handle the wide range of exposure times
between 10 s and 2.5 h and the corresponding impact on the data quality, we
did not use the individual measurements for the analysis. Instead, we divided
the 10 years into bins of 30 min, which are sufficiently short for the
analysis of Q2DWs. Each bin contains the line-specific spectra with matching
central time. The effective intensities were calculated weighted by the
exposure time. In the subsequent analysis, we only considered those bins with
a summed exposure time of at least 10 min. The average numbers of the selected
bins were 19,480 and 17,001 for lines at wavelengths shorter and longer than
2.1 $\mu$m. A lower minimum exposure time of 5 min did not significantly
change the results of the analysis but increased the scatter.

The resulting binned time series were inspected with respect to Q2DW features.
The focus was on the months January and February. The X-shooter data set shows
a strongly varying time coverage due to the different astronomical observing
programs and the usual mounting of two to three instruments at the same
telescope, which can cause gaps of the order of weeks. As a consequence, we
clearly identified Q2DW events only in the years 2017 and 2019. Moreover, only
fractions of the lifetime of a wave were well covered by X-shooter
observations. The good intervals are eight nights from 26 January to 3
February 2017 and seven nights from 11 to 18 January 2019. The corresponding
maximum numbers of bins of 30 min are 88 and 89, respectively. As the sample
of selected spectra is different for each line, up to three bins from 2017
were lost for line wavelengths below 2.1 $\mu$m. At longer wavelengths, where
30 OH lines are affected by the optional $K$-blocking filter, the bin numbers
were 78 and 35 for the years 2017 and 2019, respectively. The comparison of
Q2DW results of lines with the same upper levels but different bin numbers did
not show any significant discrepancies for the data from 2017. However, the
lines at long wavelengths had to be excluded from the analysis of the data
from 2019. The drop from 89 to 35 bins was too large.

Our discussion of the results in section 4 will focus on the event in 2017
since only these data were suitable to estimate effective emission heights of
OH lines, i.e. the main goal of this study. The issues related to the data
from 2019 will briefly be described in section 4.4.

###  2.2 SABER

For better understanding the Q2DW-related X-shooter data and adding important
altitude-dependent information, we also considered limb-sounding data (version
v2.0) of the SABER radiometer on TIMED [Russell . (1999)]. The observing
channels centered on 1.64 and 2.06 $\mu$m essentially cover emission of the
OH(4-2) and OH(5-3) bands as well as the OH(8-6) and OH(9-7) bands [Baker .
(2007)]. Consequently, the effective upper vibrational levels of both channels
are about 4.6 and 8.3 [Noll . (2016)]. We used the so-called “unfiltered”
products, i.e. the given volume emission rates (VERs) had been corrected for
the missing line emission of the stated OH bands in the wavelength ranges
covered by the channels [Mlynczak . (2005)]. For some checks, we also
considered VER profiles of the channel at about 1.27 $\mu$m, which mostly
comprises the O2(a-X)(0-0) emission band [<]e.g.,¿noll16,rousselot00.
Moreover, we used the kinetic temperature products, which are based on CO2
observations at 15 $\mu$m combined with modeling [Dawkins . (2018), Remsberg .
(2008)]. All profiles were interpolated to match the same vertical grid with a
step size of 0.2 km. The natural vertical resolution of about 2 km for the
relevant altitude range smoothes the profiles, but estimates of the peak
altitude are possible with much higher accuracy. For the two OH channels, we
also calculated vertically integrated VERs, which should correlate with the
summed X-shooter line intensities in the same wavelength range. The
integration was limited to altitudes that did not deviate more than 15 km from
the mean of the two heights with half peak emission, which was often several
hundred meters higher than the emission peak.

As the X-shooter data just revealed clear Q2DW events in 2017 and 2019
(section 2.1), we only selected the archived SABER data from January and
February of both years. The spatial selection was performed for a latitude
band centered on Cerro Paranal with a full width of 10∘. In terms of the
longitude, the width was 20∘. As the satellite moves during single scans, the
reference selection coordinates were taken at 87 km. The size of the
geographical area is a compromise between minimization of spatial effects and
maximization of the sample size [<]see¿[and the discussion in section
4.2]noll16. For the period from 26 January to 3 February 2017, the described
limits resulted in 44 profiles. As the nearly Sun-synchronous orbit of TIMED
precesses with a period of about 60 days [Russell . (1999)], the available
coverage of local time (LT) is very limited for our study. In the case of the
eight nights from 2017, the coverage is restricted to LTs of about 21:00 and
about 04:00. Each time window includes half of the profiles. For the seven
nights from 11 to 18 January 2019, only 19 profiles with LTs close to 23:00
from the descending node of the TIMED orbit could be used. Measurements at
about 06:00 could not be taken as the Sun was above the horizon. The latter
significantly changes the OH emission with respect to total intensity and
effective layer height [<]e.g.,¿yee97.

##  3 Methods

For the derivation of effective emission heights, we need to fit the Q2DW in
the time series of X-shooter-based OH line intensities (section 2.1) and
broad-band SABER emission rates (section 2.2) with a suitable wave model. In
the end, it is important that the resulting effective wave phases for the
various OH lines can be linked to altitudes via a SABER-based monotonic
phase–height relation with a significant range of phases for the same wave
period. Moreover, the model needs to be sufficiently robust to produce
consistent fits for all OH lines and all relevant altitude levels. As long as
these conditions are fulfilled, it is not essential that the model is
particularly accurate with respect to the real wave properties, which could be
quite complex. Even a model with very different wave parameters might work
(see section 4.3). We therefore applied a simple periodic cosine function.
However, we did not use the entire data set for the fits as it turned out that
especially the important data from 2017 showed a strong dependence of the wave
amplitude on local time, which is probably related to significant interactions
between the Q2DW and tides (see section 4.1). In order to avoid a line-
dependent systematic bias, we fitted only data points with similar LT and
combined the reliable fits with respect to the wave phases afterwards. The
fitting of the LT dependence of the amplitude by a second wave did not work as
the variability pattern could not be reproduced by a simple trigonometric
function. Motivated by previous results on Q2DWs (see section 1), we also
tested a model with two Q2DW components using the whole data set. This model
was only partly able to fit the time series. Moreover, the best wave
parameters were not convincing as very similar amplitudes and vertical
wavelengths were needed for components with different periods (above and below
48 h) and opposite vertical propagation directions. The introduction of
additional wave components could improve the fits. However, the increased
number of fit parameters could negatively affect the robustness of the fits
and the derivation of useful phases for the emission height estimates.

For our final wave fits, we used the fit formula

$f(t,t_{\mathrm{LT}})=c(t_{\mathrm{LT}})\left(a(t_{\mathrm{LT}})\cos\left(2\pi\left(\frac{t}{T}\,–\,k\,\Delta\lambda\,-\,\phi\right)\right)+1\right),$
(1)

where $t$ is the time of the observation relative to a reference. For the Q2DW
in 2017, we used 30 January 2017 12:00 LT (mean solar noon) at Cerro Paranal,
which is in the middle of the time interval. For 2019, the reference was 15
January 2019 12:00 LT. The period of the wave is $T$ and a relative amplitude
in the range from 0 to 1 is given by $a$. The latter depends on the LT-related
data selection, which is marked by $t_{\mathrm{LT}}$. This also applies to the
factor $c$, which can alternatively be interpreted as an additive constant. As
we divided all data by the average of the respective times series before the
fit, the product $c{\cdot}a$ represents the wave amplitude relative to the
sample mean. Moreover, $c$ values clearly different from 1 indicate additional
variability beyond the Q2DW and/or limitations in the assumption of a cosine
function. The latter is not unlikely in the view of the huge amplitudes found
(see section 4). The height-dependent phase $\phi$ is defined relative to $T$,
i.e. it is a fraction without unit. As the temporal term in the formula is
positive, the minus sign in front of $\phi$ is needed for the correct
definition [<]cf.¿forbes95. The phase is given for the longitude of Cerro
Paranal $\lambda_{\mathrm{CP}}$ relative to the full circle. Deviations
$\Delta\lambda$ from $\lambda_{\mathrm{CP}}$ cause an influence of the zonal
wavenumber $k$ on the fit. Westward wave propagation is marked by negative
$k$. As the dominating Q2DW mode in the southern hemisphere is W3 (see section
1), we used $k=-3$. This choice was confirmed by additional fits with free $k$
of a SABER sample without longitude and LT restriction, which also agreed with
the SABER-based results of gu19 for 2017. For the local X-shooter data, we
always set $\Delta\lambda=0$. For the fits of the satellite-based SABER data,
the zonal term is a small but significant correction with average positive and
negative $\Delta\lambda$ of about 0.014 (i.e. 5∘) for 2017 and about 0.012 for
2019. The fitting was performed with a least-squares algorithm involving
bounds (e.g. for excluding negative amplitudes). After some tests with $T$ as
a free fit parameter, we decided to make separate fits for a grid of periods
$T$ from 40 to 60 h with a step size of 1 h. This approach significantly
improved the robustness of the fits and made sure that the resulting wave
phases for the different OH emissions referred to the same $T$. Consequently,
the only free parameters for each fit were $c$, $a$, and $\phi$.

Table 1: Derivation of optimum Q2DW phase relative to a period of 44 h at 30 January 2017 12:00 LT at Cerro Paranal for X-shooter-based relative OH(4-2)P1(1) intensities LT bin | Sample | Phase | Errora | Weightb
---|---|---|---|---
$20-21$ | 6 | 0.733 | 0.131 | 0.000
$21-22$ | 11 | 0.847 | 0.554 | 0.000
$22-23$ | 8 | 0.426 | 0.369 | 0.000
$23-24$ | 11 | 0.406 | 0.104 | 0.000
$00-01$ | 12 | 0.406 | 0.049 | 0.127
$01-02$ | 13 | 0.381 | 0.031 | 0.201
$02-03$ | 14 | 0.388 | 0.022 | 0.285
$03-04$ | 12 | 0.423 | 0.016 | 0.387
$04-05$ | 1 | 0.000 | 9.999 | 0.000
Average | 51 | 0.403 | 0.018 | 1.000
aFit uncertainty for LT bins and weighted
standard deviation of hour-specific phases in
the case of the weighted average.
bWeight of zero for LT bins with less than
10 data points and/or a phase error more
than 5 times higher than the minimum.

For the X-shooter data consisting of relative intensities of 30 min bins for
each line, the fitting procedure included three steps. First, we used all data
of each line for a rough initial estimate of the phase $\phi$. In the next
step, we performed separate fits for each hour of the night with at least five
available bins. The hour-dependent fits were then used to derive an optimum
phase for all data. Table 1 illustrates this procedure for
OH(4-2)P1($N^{\prime}=1$), a period of 44 h, and the year 2017\. As the wave
amplitude strongly depended on LT in 2017 (section 4.1), we only considered
the four to five 1 h intervals with the most reliable phases for each line,
which essentially excluded the evening data. In the case of 2019, six to eight
intervals could be used. Intervals with less than nine 30 min bins were never
included. Line-dependent differences in the number of bins were only observed
for intervals that were usually rejected. The optimum phase was calculated by
weighted averaging using the inverse of the phase error of the fits of the
selected LT intervals as weights (Table 1). Assuming the existence of a unique
$\phi$ for the considered data, its uncertainty was derived from the weighted
standard deviation of the phases of the individual fits. In the final step of
the fit procedure, the hour-dependent fits were repeated with the phase fixed.
In this way, the parameters $c$ and $a$ and their uncertainties were derived
for each hour.

The fitting of the SABER data could be simplified. The derivation of an
optimum phase was not necessary as only one LT range for 2017 provided
reasonable fits due to the LT dependence of the wave amplitude (see section
4.2). For 2019, only one nighttime interval was available (section 2.2). As an
alternative fitting approach for 2017, we fitted all data simultaneously but
with two additional fit parameters in order to scale $c$ and $a$ depending on
the half of the night. The results were almost identical but with slightly
higher uncertainties. Therefore, we preferred the separate fits with the
morning data as the reference for the phase derivation as in the case of the
X-shooter data. In general, we fitted the vertically integrated VERs as well
as the VER profiles from the two OH-related channels. For the latter, we first
rebinned the profile data with a step size of 1 km. Moreover, we usually
started the fitting procedure at an altitude of 87 km, where the OH emission
is relatively strong. Thereafter, we used the resulting fit parameters as
start values for the adjacent altitudes. The results for the latter were then
taken for the next layers. In this way, we also obtained reasonable fits for
heights with very weak OH emission (see section 4.2). Altitudes between 79 and
99 km were considered.

##  4 Results

###  4.1 OH Lines

Figure 1: X-shooter-based OH line intensity time series for the local time
period from 26 January to 3 February 2017 given as deviation in days from mean
solar noon at Cerro Paranal on 30 January. Circles show the intensities of
OH(4-2)P1(1) (a) and OH(4-2)P1(14) (b) relative to the mean of the considered
30 min bins. The local times of the latter are emphasized by different colors.
The Q2DW fits with a fixed period of 44 h, LT-dependent amplitudes, and line-
specific phases are marked by solid lines. Daytime data gaps are bridged by
dotted lines.

In this section, we discuss the results of the Q2DW fits of the X-shooter-
based OH line measurements for the wave event in 2017. Figure 1 shows examples
of the corresponding time series of 30 min bins (section 2.1) for the
$\Lambda$ doublets OH(4-2)P1(1) (a) and OH(4-2)P1(14) (b), which only differ
by the rotational quantum number. However, the difference in the rotational
energy of 3,239 cm-1 is the largest of the entire set of 298 lines. Hence, the
variability patterns should deviate. The inspection of the figure confirms
this assumption. Deviations can be observed in the day-to-day development of
the maximimum and minimum relative intensities and the local times showing
these extreme values. Nevertheless, the basic pattern is similar. There are
alternating nights with low and high relative intensities, which clearly
points to the presence of a Q2DW. The apparent amplitudes are large. For
OH(4-2)P1(1), the maximum-to-minimum ratio is 6.1. The standard deviation
relative to the mean for the 88 bins indicates 0.45. For OH(4-2)P1(14), the
corresponding values for the 86 available bins are slightly lower (4.4 and
0.36, respectively).

Figure 2: Derivation of the most likely period for the Q2DW in 2017 based on
the root mean square of the relative intensity fit (circles, left axis) and
the phase error relative to the period (squares, right axis) averaged for all
298 OH lines.

Figure 1 also displays our best fits with LT-dependent amplitudes for both OH
emissions. The fits were performed for all LT bins except for 04:00 to 05:00,
which includes only a single observation (Table 1). The fixed period for all
LT bins was set to 44 h. This decision is justified by Figure 2, which shows
the root mean square (rms) for the differences between fit and observed data
relative to the line-specific mean intensity averaged for all 298 lines as a
function of the period grid from 40 to 60 h. Moreover, the average phase error
relative to the period (Table 1) is displayed for the same set of lines and
periods. The minimum values of both indicators (0.15 for the rms and 0.015 for
the phase uncertainty) are clearly located at 44 h. The accuracy of this
period, which is limited by the step size of 1 h, is highly sufficient for the
estimation of emission heights as we will discuss in section 4.3. Moreover,
note that the scatter in the periods from the minimum rms derived for the
individual lines is fairly small (63% with 44 h and 33% with 45 h) despite
significant differences in the time series. Q2DW periods below 48 h appear to
be rather typical in regions close to Cerro Paranal as long-term OH
observations at El Leoncito (31.8∘ S, 69.3∘ W) indicate [Reisin (2021)].
Figure 2 reveals that periods close to 2 full days are not able to reproduce
the change of the variability pattern from night to night shown in Figure 1.
In particular, the small apparent amplitudes of the first two nights for
OH(4-2)P1(1) suggest that the maximum and minimum were outside the range of
local times covered by the X-shooter data, i.e. too late in the morning. With
a period distinctly smaller than 48 h, it was then possible to observe the
extreme intensities afterwards at nighttime. Therefore, the time series is
just long enough to robustly derive a wave period. The variability pattern of
OH(4-2)P1(14) is more regular at the beginning with high deviations from the
mean intensity, which excludes that the Q2DW was significantly weaker in the
first nights as the consideration of only the P1(1) data could suggest. The
differences between both lines are obviously the result of shifts in the
positions of the extremes. Consequently, both emissions need to show different
Q2DW phases. Our fits revealed phases of 0.403 and 0.229 at the reference time
for the lines with $N^{\prime}=1$ and 14, respectively. The resulting phase
difference is much larger than the mean uncertainty reported above (which
should be even smaller for phase comparisons). Hence, the wave of 2017 is
obviously suitable to safely separate lines based on their fitted phases.

Figure 3: Dependence of fitted Q2DW parameters on local time for OH(4-2)P1(1)
(circles) and OH(4-2)P1(14) (squares). The abscissa shows the mean local time
for nighttime intervals with a width of 1 h. Results are provided for the
amplitude $c{\cdot}a$ (solid lines) and the additive constant $c$ (dashed
lines). In all cases, the plotted values are relative to the mean line
intensity for the Q2DW-related sample. The mean fit uncertainties for
$c{\cdot}a$ and $c$ for data points with non-zero weight for the phase
derivation (Table 1) are 0.09 and 0.06, respectively. The legend also provides
the effective phases $\phi$ relative to the period of 44 h for both OH lines
(cf. Table 1).

The results for the two example $\Lambda$ doublets are further analyzed in
Figure 3, which shows the amplitude, i.e. the product of the fit parameters
$c$ and $a$ relative to the mean intensity of the time series, as a function
of the mean local time for intervals with a width of 1 h. The wave amplitudes
of both emissions strongly depend on local time. There is no detection of a
Q2DW before 22:00. Afterwards the amplitudes increase and then become
relatively stable. This development is faster for OH(4-2)P1(14), where a
shallow maximum with an amplitude of 0.46 is visible between 0:00 and 01:00.
The maximum for OH(4-2)P1(1) is significantly higher and amounts to 0.74. It
is present between 01:00 and 02:00. These remarkable structures cannot be
explained by a single wave with fixed amplitude. A model with multiple wave
components would certainly better reproduce the patterns, but at least a model
with two components did not return reliable wave properties (section 3). The
persistent lack of wave-like variability at the beginning of the nights is
hard to explain in this way. Hence, our assumption of LT-dependent Q2DW
amplitudes, which leads to the narrow peaks of the fit models in Figure 1,
appears to be the most promising approach for achieving a satisfying agreement
with the observed time series. This model requires that the physical
conditions that control the wave propagation and/or the sensitivity of the OH
emission for Q2DWs change depending on the time of the day. Nonlinear
interaction with especially the diurnal and semidiurnal tides [<]e.g.,¿palo99
could explain this behavior. In this context, phase locking, which would
significantly weaken the diurnal tide [Walterscheid Vincent (1996), Hecht .
(2010), Walterscheid . (2015)], did not seem to be active at Cerro Paranal
during the considered time interval as periods very close to 48 h cannot
explain the time series. The LT dependence of the Q2DW amplitude in OH
emission might also be affected by the negative nocturnal trend of atomic
oxygen (produced at daytime) that would always be present without vertical
dynamics especially at the lowest OH emission altitudes [Marsh . (2006)].

Figure 3 also shows the constant (or factor) $c$ relative to the mean
intensity. For an undisturbed cosine oscillation centered on the mean
intensity of the time series, $c$ should be close to 1 (see section 3). In
reality, the values range between 0.7 and 1.3. They cause that the maximum
$c{\cdot}a$ are not located at the end of the night as in the case of $a$ (not
shown). Although the joint fitting of $c$ and $a$ might lead to some
systematics due to possible degeneracies, asymmetries in very large intensity
variations that cause larger deviations during the crest of the wave are more
likely. Moreover, there are also variations in the nocturnal OH intensity
without a wave. An inspection of the entire X-shooter data set suggests that
this is probably a minor effect as the climatological changes for the
investigated times in southern summer are of the order of only 10%
[<]cf.¿noll17. The intensity even tends to decrease with increasing LT in
contrast to our results for $c$.

Figure 4: Maximum amplitude $c{\cdot}a$ relative to the mean (a) and phase
$\phi$ relative to the period at 30 January 12:00 LT at Cerro Paranal (b) for
all 298 OH lines used for the fit of the Q2DW event in 2017. The abscissa
shows the energy of the upper level of the transition minus the lowest energy
for the corresponding vibrational state $v^{\prime}$. The latter is given by
colored numbers. The representative amplitude (phase) uncertainties are about
0.021 (0.005), 0.023 (0.008), and 0.038 (0.017) for energy differences below
400 cm-1, between 400 and 800 cm-1, and above 800 cm-1, respectively.

Next, we discuss the whole set of 298 lines. Figure 4a shows the line-specific
maximum of the amplitude $c{\cdot}a$ of all hour intervals as a function of
the energy of the upper state of the transition relative to the lowest energy
of the corresponding vibrational level $v^{\prime}$. The reference states are
characterized by a rotational quantum number $N^{\prime}=1$ and the electronic
substate $\mathrm{X}^{2}\mathrm{\Pi}_{3/2}$ ($F^{\prime}=1$) [<]e.g.,¿noll20,
i.e. the energy difference equals 0 cm-1 for P${}_{1}(1)$ and Q${}_{1}(1)$
lines. If we focus on these lines, the figure reveals a clear increase of the
amplitude for decreasing $v^{\prime}$ with values between 0.56 for
$v^{\prime}=9$ and 0.74 for $v^{\prime}=3$. This trend is highly significant
as the uncertainties derived from lines with the same upper level are only
about 0.02. The amplitude differences for adjacent vibrational levels are
larger for high $v^{\prime}$. For increasing rotational energy, we find an
increase for all vibrational levels which is more pronounced for low
$v^{\prime}$. Up to about 500 cm-1, the increase ranges from 0.12 for
$v^{\prime}=9$ to 0.23 for $v^{\prime}=2$. For the lowest vibrational levels,
this rise results in enormous amplitudes of more than 90% of the sample mean
intensity. At higher rotational energies, the amplitudes decrease again. This
trend appears to start slightly later for lower $v^{\prime}$, but not later
than about 700 cm-1. The drop of the amplitude is much stronger for lower
$v^{\prime}$. For the highest $N^{\prime}$, the amplitude appears to be
independent of the vibrational level and is distinctly smaller than for low
$N^{\prime}$. The mean amounts to about 0.48 for energy differences $\Delta
E^{\prime}$ larger than 1300 cm-1.

The complex pattern in Figure 4a needs an explanation. As we can expect that
the emission altitudes increase with higher $v^{\prime}$ and $N^{\prime}$
[<]e.g.,¿adler97,dodd94,noll18b,savigny12, lower amplitudes with increasing
quantum numbers imply a decrease of the wave amplitude with increasing
altitude. Such a trend is consistent with the dependence of the amplitude on
$v^{\prime}$ for $\Delta E^{\prime}$ up to at least 1,000 cm-1 and the
comparison of amplitudes for very low and very high $N^{\prime}$. An issue
seems to be that the highest amplitudes are located at intermediate rotational
energies. However, this feature can be explained by means of the structure of
the OH rotational level population, which can be characterized by a cold and a
hot population for each $v^{\prime}$ [Cosby Slanger (2007), Kalogerakis .
(2018), Kalogerakis (2019), Noll . (2020), Oliva . (2015)]. While the
temperature of the cold population, which dominates the emission at low
$N^{\prime}$, is consistent with the ambient temperature, i.e. about 190 K on
average at Cerro Paranal [Noll . (2016), Noll . (2020)], the hot population
shows apparent temperatures between about 700 K for $v^{\prime}=9$ [Noll .
(2020)] and about 12,000 K for $v^{\prime}=2$ [Oliva . (2015)]. Such extreme
values reflect significant deviations from the local thermodynamic equilibrium
(LTE), which are related to the non-LTE nascent population of the
hydrogen–ozone reaction [Llewellyn Long (1978)] as well as an insufficient
frequency of collisions that thermalize the rotational level population
compared to $v$-changing collisions and the radiation of airglow photons [Noll
. (2018)]. An inspection of the combined populations now shows that the
contributions of the cold and the hot component are of the same order at
similar $\Delta E^{\prime}$ as those where we find the maximum amplitudes in
Figure 4a [Noll . (2020)]. Consequently, the latter can be explained by an
increased sensitivity of the corresponding OH lines to the Q2DW due to the
strong impact of the wave-induced variation of the ambient temperature
(affecting the cold population) on the rotational energy where cold and hot
populations have similar contributions. Changes in the cold component are more
crucial because of its much steeper decline with increasing energy. Also note
that the relative contribution of the hot population is lower than a percent
for $v^{\prime}\leq 6$ at 0 cm-1 [Noll . (2020), Oliva . (2015)]. The rapid
growth of this fraction with increasing $\Delta E^{\prime}$ can then explain
the increase of the Q2DW amplitudes until the maximum values at 500 to 700
cm-1. Moreover, the drop of the amplitudes marks the energies where the
contribution of the cold population becomes minor. Hence, the highest
$N^{\prime}$ show a pure hot population, which can obviously be described by a
single amplitude. The lack of a dependence of the amplitude on $v^{\prime}$
suggests that the $v^{\prime}$-specific hot components are linked and might be
represented by a single population (at least in part). The structure of the
full roto-vibrational populations, which show connections between high
$N^{\prime}$ for adjacent $v^{\prime}$ [Cosby Slanger (2007), Noll . (2020)],
seems to confirm this interpretation.

Q2DW phases relative to the period of 44 h at 30 January 2017 12:00 LT are
shown for all 298 OH $\Lambda$ doublets in Figure 4b. In general, there is a
clear decrease of the phase with increasing $v^{\prime}$ and $N^{\prime}$. For
the levels with $F^{\prime}=1$ and $N^{\prime}=1$, the phase ranges from 0.436
for $v^{\prime}=2$ to 0.333 for $v^{\prime}=9$, which is distinctly more than
the uncertainty of 0.005 derived from lines with the same upper levels. For
higher $N^{\prime}$, the phases decrease almost linearly and similar for all
$v^{\prime}$ with shifts of about 0.04 up to about 500 cm-1. The trend may
even hold until about 1,500 cm-1 but with a decreasing difference between low
and high $v^{\prime}$. For the highest $v^{\prime}$, there is a flattening of
the phase change, which might even stop at the end. However, this remains
questionable due to the relatively small number of lines and relatively high
uncertainties. The average phase for $\Delta E^{\prime}\geq 2,000$ cm-1 is
0.23, which is about 0.2, i.e. almost 9 h, lower than the maximum shown by
OH(2-0)P1(1). This relatively large spread is promising in terms of a
sensitive derivation of effective emission heights for various OH emission
lines (see section 4.3).

###  4.2 OH Emission Profiles

Figure 5: SABER-based vertically integrated OH VERs for the local time period
from 26 January to 3 February 2017 given as deviation in days from mean solar
noon at Cerro Paranal on 30 January. The results for the OH-related channels
centered on 1.6 $\mu$m (a) and 2.1 $\mu$m (b) are shown relative to the mean
of the considered 44 data points (circles). The local times at the reference
coordinates of the observations are emphasized by different colors. The Q2DW
fits for the two LT ranges are marked by crosses.

In order to link wave phases and heights, we need altitude-resolved OH
emission measurements. The required data were obtained by the two OH channels
of the limb-scanning SABER radiometer (section 2.2). For a comparison with the
X-shooter-based time series, Figure 5 shows vertically integrated VERs of the
channels OH(1.6 $\mu$m) (a) and OH(2.1 $\mu$m) (b) relative to the mean of the
44 measurements that are representative of the region around Cerro Paranal
during the eight nights in 2017 covered by the X-shooter data set. Both
channels show variability patterns that are consistent with a strong Q2DW for
the 22 data points taken at about 04:00 LT, whereas the data related to about
21:00 LT do not display clear wave features. These results agree with those of
the X-shooter data set in terms of the pronounced time dependence of the wave
amplitude (Figure 3). The maximum-to-minimun ratios and standard deviations
relative to the mean are 7.4 and 0.46 for all OH(1.6 $\mu$m) data and 6.6 and
0.42 for all OH(2.1 $\mu$m) data. The smaller values for the latter are
consistent with the decrease of the wave amplitude with increasing
$v^{\prime}$ for the X-shooter data (Figure 4). Remember that the effective
$v^{\prime}$ are about 4.6 and 8.3 for the two SABER OH channels (section
2.2). The maximum-to-minimum ratios are higher than the values of 6.1 for
OH(4-2)P1(1) and especially 4.4 for OH(4-2)P1(14) (section 4.1). The standard
deviations are very similar to the value of 0.45 for the line with
$N^{\prime}=1$ but higher than the result of 0.36 for the line with high
$N^{\prime}$. These findings agree well as the integrated emission of the
SABER channels is dominated by lines with small $N^{\prime}$. Consequently,
SABER and X-shooter data show a consistent picture of the Q2DW event in 2017.
Differences in the measurement approaches and sample properties do not appear
to have a significant impact.

Figure 5 also shows our fits of the two LT ranges based on the approach
described in section 3. Compared to the 04:00 LT data that we exclusively used
for the phase derivation, the amplitude $c{\cdot}a$ for the 21:00 LT data was
significantly smaller. The ratios for the 21:00 and 04:00 amplitudes were
about 0.09 for OH(1.6 $\mu$m) and about 0.20 for OH(2.1 $\mu$m). Thus, the fit
of the evening measurements shows only very small Q2DW-related variations.
Moreover, the corresponding factors for $c$ were 0.86 and 0.87, i.e. the
intensity level for the evening data was lower on average. For the subsequent
analysis, we only focused on the morning data.

Figure 6: Derivation of the most likely period for the morning data of the
Q2DW in 2017 based on the root mean square of the relative intensity fit
(circles, left axis) and the phase error relative to the period (squares,
right axis) for the SABER OH channel centered on 2.1 $\mu$m.

The fitting was performed for the same wave period as in the case of the
X-shooter data, i.e. 44 h (section 4.1). Using the same period is important
for the derivation of the effective emission heights. Nevertheless, we checked
a wide range of periods in terms of the fit quality. The results for OH(2.1
$\mu$m) and the 04:00 LT data are shown in Figure 6. The corresponding plot
for OH(1.6 $\mu$m) is very similar. The rms relative to the mean indicates a
clear minimum of 0.13 at 43 h. The phase error derived from the fit also shows
a minimum there (0.013), although it is less pronounced. Consequently, the
SABER-related fit returns the best result for a period which is only 1 h lower
than for the X-shooter data. This shift can also be observed for the maximum
phase uncertainty (48 vs. 49 h). In the view of the various differences
between both data sets, the deviation is small. It is not crucial for the
estimates of the line-specific emission altitudes. A comparison of the results
for 43 and 44 h did not indicate significant deviations with respect to the
general uncertainties (section 4.3). In the following, we will focus on the
results for 44 h, which represents the best period from the X-shooter data
set, which is much larger than the SABER data set in terms of OH emission
features and observing times.

Interestingly, the period can change significantly if the geographical area is
not restricted to locations close to Cerro Paranal. Our additional check of
the SABER data without longitude and LT limits (cf. section 3) revealed a most
likely period of 49 h for the two OH channels. This result is in good
agreement with the findings of gu19, who reported a period of 48 h for the
Q2DW in 2017 based on the kinetic temperatures from SABER. We repeated this
fit and found the same. For the global fits, we preferentially used the
combined evening and morning data, i.e. we neglected a possible LT dependence
of the amplitude. This setting resulted in better fits, which is also in
contrast to our experience with the data restricted to the region around Cerro
Paranal. Moreover, an inspection of the wave properties depending on the
longitude showed that the South American sector exhibited the clearest Q2DW
variability pattern. In other regions, the wave was much less pronounced. This
observation may explain why gu19 only identified an intermediate wave
amplitude of 10 K at 82.5 km for the event in 2017. The latitude- and time-
resolved SABER-based results for a fixed period of 48 h from xiong18 show
amplitudes slightly below 10 K at 84 km at about 25∘ S for the relevant time
interval at the end of January and beginning of February. We find about 18 K
around Cerro Paranal in the morning. In addition, the local times with the
strongest variations changed depending on longitude, which indicates why
global fits without the corresponding scaling factors worked better. This
complex variability pattern suggests that the Q2DW is significantly disturbed
by interactions with other planetary waves, tides, gravity waves, and/or the
mean flow. For OH emission from SABER observations, pedatella12 already found
a complex longitude dependence of the Q2DW-related variability due to
contributions of nonmigrating tides, stationary planetary waves, and secondary
waves that are preferentially generated by the nonlinear interaction of the
Q2DW and migrating tides. The latter supports our assumption that such
interactions were important for the Q2DW event in 2017 with a pronounced LT
dependence of the measured amplitude.

Figure 7: Vertical VER profiles for the Q2DW event in 2017 from the SABER OH
channels centered on 1.6 $\mu$m (a) and 2.1 $\mu$m (b). As described in the
legend, both subfigures show two individual profiles with dates separated by
about 24 h (crosses and plus signs), mean profiles (circles) for the evening
(upper peak) and morning (lower peak), as well as standard deviation (SD)
profiles (diamonds) for the evening (upper peak) and morning (lower peak). For
the calculation of the curves representing the evening data, one profile with
unusually large VERs below 80 km had been excluded.

We now turn to the discussion of the height-dependent impact of the Q2DW.
Figure 7 shows different kinds of emission profiles for the two OH channels
centered on 1.6 $\mu$m (a) and 2.1 $\mu$m (b). For each channel, two
individual profiles from 31 January and 1 February 2017 at about 21:00 LT are
displayed. The huge differences between both profiles with a time difference
of 1 day demonstrate the implications for the OH emission by a large amplitude
Q2DW. On 31 January, the OH(1.6 $\mu$m) profile peaked at a very low altitude
of 82 km with an enormous VER of 81 nW m-3. In contrast, the emission almost
vanished completely at this altitude on the next day. As only the emission at
the highest altitudes remained relatively similar, the peak moved upward by 11
km to 93 km, where the VER was only 8 nW m-3. The situation for OH(2.1 $\mu$m)
is very similar with the peaks just 1 km higher. It is not clear whether the
VER minimum at 89 km on 1 February is real as it cannot be seen in the
corresponding data from OH(1.6 $\mu$m).

Figure 7 also shows mean and standard deviation profiles for the two LT
regimes of about 21:00 and 04:00 in the eight selected nights. The evening
data indicate typical mean profiles with respect to the long-term averages for
Cerro Paranal [Noll . (2017)]. The centroid altitudes of the plotted profiles
of both channels are 88.1 and 89.5 km, respectively. The standard deviation
profiles of the 21:00 LT data only indicate relatively weak variations. The
peaks of these profiles are about 1 to 2 km lower than for the mean profiles.
These shifts are expected due to the steepening of the atomic oxygen gradient
with decreasing height [<]e.g.,¿smith10, which makes the lower altitudes more
sensitive to vertical transport of this important gas for the OH production
and subsequent relaxation and destruction processes. The strong oxygen-induced
emission variations at low altitudes lead to a significant perturbation of the
mean profile for the morning data. The latter therefore extends to lower
heights (with a 3 km lower peak) and gets more broadened for both channels.
The peak of the standard deviation profile moves in a similar way. The
emission variability is much larger than in the case of the evening data (as
expected). Note that the mean centroid emission altitudes of the individual
profiles for 04:00 LT of 88.1 km for OH(1.6 $\mu$m) and 89.1 km for OH(2.1
$\mu$m) are very similar to those of the 21:00 LT data (88.0 km and 89.5 km)
since the large differences in the VERs which affect the plotted mean profiles
do not matter here. Nevertheless, the larger impact of the Q2DW on the morning
data can be recognized by the standard deviation of the individual centroid
altitudes of about 2.3 km for both OH channels, which is distinctly higher
than about 0.6 km for the evening data (if one outlier is excluded).

Figure 8: Fitted phase relative to the period of 44 h as a function of height
for the SABER product kinetic temperature (stars) and the VERs of O2(a-X)(0-0)
at 1.27 $\mu$m (diamonds), OH(1.6 $\mu$m) (squares), and OH(2.1 $\mu$m)
(circles). The time series comprised the 22 limb scans taken at about 04:00 LT
in eight nights of 2017. The reference time for the plotted phases was 30
January 12:00 LT at Cerro Paranal. Fit uncertainties, which are useful for
relative quality comparisons, are indicated by dotted lines. Moreover, the
plot shows the resulting regression line for a linear fit of the OH(2.1
$\mu$m) phases in the altitude range between 80 and 97 km.

With the knowledge of the height-dependent response of the OH emission on the
passing Q2DW, we now discuss the wave phases as a function of altitude. Figure
8 shows the phase functions between 79 and 99 km from fits of the morning data
with a period of 44 h as described in section 3 for both OH channels. The
phases are given for 30 January 2017 12:00 LT at Cerro Paranal. At least for
the altitude range between 80 and 90 km, there is a clear trend of decreasing
phase with increasing height for the OH emissions at about 1.6 and 2.1 $\mu$m.
For 80 to 89 km, both curves are almost identical with a mean absolute
difference of only 0.005. However, the discrepancy is rapidly growing above 89
km. While the trend continues for OH(2.1 $\mu$m), OH(1.6 $\mu$m) shows a
complex behavior with increasing and decreasing phase. The latter is not
trustworthy. As expected from the ratio of standard deviation to mean profiles
in Figure 7, the fitted wave amplitudes relative to the mean rapidly decrease
with increasing altitude. From 80 to 93 km, $c{\cdot}a$ drops from 1.18 to
0.19 for OH(2.1 $\mu$m). However, this is still moderate compared to a
decrease from 1.23 to 0.04 for OH(1.6 $\mu$m). Consequently, it appears that
the wave-induced variability in the emission at about 1.6 $\mu$m was too low
to track the wave in a reliable way (cf. evening data in Table 1 and Figure
3), whereas it was still sufficient for OH(2.1 $\mu$m). Note that the latter
emission peaks higher in the atmosphere (Figure 7).

In order to check this interpretation, we also fitted the emission of
O2(a-X)(0-0) at about 1.27 $\mu$m, which is also observed by SABER. The
profile of this airglow emission strongly changes during the night [Noll .
(2016)] due to the decay of an excited population originating from ozone
photolysis at daytime [<]e.g.,¿lopez89. However, in the second half of the
night, the emission distribution is similar to the one of OH. The fits only
show a relatively weak decrease of the wave amplitude by a factor 2 over the
entire plotted altitude range. Hence, the resulting phases in Figure 8 appear
to be reliable. They clearly support the trend found for OH(2.1 $\mu$m). This
result is further confirmed by fits of SABER-based kinetic temperature
profiles [Dawkins . (2018)], where the corresponding phase profile is also
plotted. The temperature fits are relatively robust since the wave amplitude
remained relatively high in the entire studied altitude regime (at least 11 K
up to 98 km with maximum values of about 18 K at 82 km and 21 K at 94 km). The
shape of the atomic oxygen profile does not matter for the temperature.

In conclusion, we have strong hints for a monotonically decreasing phase with
increasing height over the entire altitude range relevant for OH. This
implication is ideal for the estimate of effective emission heights. Moreover,
the earlier maxima at higher altitudes indicate that the wave was rising. This
interpretation is also supported by test fits of the diurnal tide with the
same fitting algorithm [<]cf., e.g.,¿griffith21. Consequently, the wave
propagation is consistent with the assumed origin of Q2DWs in the lower
mesosphere [<]e.g.,¿ern13. The results suggest that the OH-relevant wave
phases are best described by the profile fits for OH(2.1 $\mu$m). There is an
almost linear relation between phase and height over a wide altitude range. We
found that the optimimum interval for a linear regression extends from 80 to
97 km. Then, we obtain a very high coefficent of determination r2 of 0.995.
The inverse slope of the regression line corresponds to the vertical
wavelength $\lambda_{\mathrm{z}}$ of the wave. It amounts to $31.7\pm 0.6$ km,
which is near the peak of the distribution of W3 Q2DW wavelengths derived by
huang13 based on SABER data. If we only consider the altitude range up to 89
km, where OH(1.6 $\mu$m) can also be used, it is almost the same
$\lambda_{\mathrm{z}}$ but with a larger uncertainty ($31.9\pm 1.6$ km). For
OH(1.6 $\mu$m), we then obtain $33.6\pm 1.8$ km, which agrees within the
regression uncertainties.

###  4.3 Effective OH Emission Heights

Combining the line-specific effective phases from section 4.1 with the slope
of the relation between phase and height for OH(2.1 $\mu$m) from section 4.2
for the same wave period of 44 h allowed us to estimate the effective emission
heights of 298 OH lines. For a direct conversion, it is just necessary to
subtract the phase for each line from the intercept at 0 km of 3.027 and to
multiply the result with the vertical wavelength of 31.74 km (see section
4.2). However, our calculation was more complex as we also considered possible
phase deviations due to the differences in the lines of sight as well as
geographical and temporal distributions of the X-shooter and SABER
measurements for the Q2DW event in 2017. In order to estimate this effect, we
used the fitted effective phases of 0.388 and 0.345 for the vertically
integrated VERs of the OH channels centered on 1.6 and 2.1 $\mu$m,
respectively, and compared them with X-shooter-based effective phases for sets
of OH lines that are representative of these channels. We weighted the phases
for individual lines as shown in Figure 4b by the product of the measured line
intensity and the channel-specific transmission [Baker . (2007)] at the line
position. Moreover, we checked the impact of the fact that our line sample is
not complete (section 2.1). We found that the effective $v^{\prime}$ of our
line mixes did not change for OH(1.6 $\mu$m) but deviated by $+0.08$ for
OH(2.1 $\mu$m) compared to the reference values of 4.57 and 8.29 by noll16.
However, lowering the contribution of the $v^{\prime}=9$ lines to get a match
in the effective $v^{\prime}$ for OH(2.1 $\mu$m), i.e. 8.29, did not affect
the effective phases, which turned out to be 0.379 and 0.327 for the X-shooter
data. Consequently, the SABER phases appear to be shifted by about 0.0135 on
average. We subtracted this value from the intercept of the regression line
before we calculated the effective emission heights. In terms of altitude,
this shift corresponds to a change of $-0.43$ km with an uncertainty of 0.13
km derived from the difference between the results for both channels.

For the estimate of the height uncertainties, we also checked whether our
simple linear regression analysis without the consideration of the height-
dependent fit quality (Figure 8) could cause systematic phase offsets. For
this purpose, we calculated a residual phase deviation from the differences
between measured phase and regression line at all heights weighted by the
wave-induced variability. The resulting height uncertainty only amounts to
0.11 km thanks to the convincing fit of the OH(2.1 $\mu$m) profile data.
Concerning the X-shooter-related uncertainties, we used the representative
phase uncertainties 0.005, 0.008, and 0.017 (cf. caption of Figure 4), which
are based on phase differences for OH lines with the same upper level for the
energy differences $\Delta E^{\prime}$ below 400 cm-1, between 400 and 800
cm-1, and above 800 cm-1, respectively. Combined with the small uncertainties
reported above, the effective absolute height errors resulted in about 0.24,
0.30, and 0.55 km, respectively.

The effective heights for the 298 OH $\Lambda$ doublets range from 81.8 to
89.7 km with an average of 84.88 km and a standard deviation of 1.43 km.
Compared to the uncertainties, the line-specific differences are highly
significant. The given altitudes are representative of the maximum emission
variability induced by the Q2DW in the eight investigated nights of 2017. This
can be illustrated for the vertically integrated VERs of the SABER channels at
1.6 and 2.1 $\mu$m, for which we derived effective heights of 83.75 and 85.13
km based on the already stated phases. The profile plots for about 04:00 LT in
Figure 7 indicate that these wave-related heights are between the peak at 83
km and the centroid altitude at 84.6 and 85.6 km of the standard deviation
profiles of the two channels.

If the evening data of both instruments had resulted in reliable wave fits, we
would probably have obtained significantly higher altitudes in agreement with
the standard deviation profiles of these data. Hence, the resulting heights of
our approach depend on the wave properties and the state of the background
atmosphere during the analyzed time interval. Therefore, it is desirable to
also provide heights for the studied OH lines which are representative of
longer time scales. Moreover, there should be a closer relation to the peak or
centroid emission altitudes that are usually used in the literature,
especially with respect to studies of OH rotational temperatures as indicators
of the effective ambient temperature for the OH emission layer. The relevant
heights for temperature and intensity variations can differ by several
kilometers [<]e.g.,¿swenson98. For example, we find for the ratio of the
intensities of OH(3-1)P1(1) and OH(3-1)P2(2), which are often used for
rotational temperature studies [<]e.g.,¿beig03,schmidt13,noll15, a phase shift
of $-0.13$ compared to the mean phase for both lines that corresponds to an
altitude difference of $+3.8$ km with an uncertainty of several hundred meters
(including a possible small discrepancy in the phase–height relations of
temperature and OH intensity as indicated by Figure 8). Note that the general
use of temperatures instead of intensities for height estimates is not
possible as rotational temperatures show much higher measurement uncertainties
and could vary in a different way than the satellite-based kinetic
temperatures due to the non-LTE contributions which increase with increasing
$v^{\prime}$ and $N^{\prime}$ [Noll . (2020)]. For the derivation of the
reference heights, we considered the 14-year averages of the centroid
altitudes of $87.81\pm 0.02$ km for OH(1.6 $\mu$m) and $89.20\pm 0.02$ km for
OH(2.1 $\mu$m) from noll17 that were calculated based on 4,496 SABER profiles
taken close to Cerro Paranal. These values are about 4.06 and 4.07 km higher
than our results for the Q2DW phase fits but they are close to the averages of
the individual centroid altitudes during the investigated event (section 4.2).
The large difference can therefore be mainly explained by the strongly
increasing VERs for emission profiles with lower peaks and the resulting
impact on the vertical variability distribution. It is promising that the
shifts are almost the same for both OH channels as it implies that the Q2DW-
based effective height differences between lines are also representative of
the long-term averages of the centroid emission heights. At least for lines
with low $N^{\prime}$ that mainly contribute to the SABER VERs, the deviations
should be much smaller than the stated uncertainties of a few hundred meters.
Consequently, we can also provide effective OH emission heights for average
conditions at Cerro Paranal by shifting all line-specific heights by $+4.07$
km. The resulting mean height for all 298 $\Lambda$ doublets is 88.95 km,
which is in between the centroid altitudes for the reference profiles of both
SABER OH channels.

Our height estimates are based on a model that assumes a single wave with a
period of 44 h and an amplitude depending on local time. In order to increase
the confidence in the corresponding results, it is important to know how
changes in this model affect the derived emission heights. As already
mentioned in section 4.2, we could have also used a period of 43 h as
indicated by the SABER-based fit results in Figure 6. In comparison to 44 h,
we obtained a mean height for the 298 lines which is 0.09 km higher before and
0.02 km lower after the shift. The standard deviation did not change, i.e. it
is 1.43 km. Hence, the impact of a period change by 1 h is negligible. We also
performed a test for an extremely different period of 56 h, which marks a
secondary minimum of the phase error in Figure 6 and results in a downward-
propagating Q2DW with a $\lambda_{z}$ of $53.9\pm 2.3$ km for OH(2.1 $\mu$m).
Nevertheless, the changes of the two mean heights were only $+0.21$ and
$-0.17$ km and the standard deviation just decreased by $0.08$ km. This
promising result demonstrates that even an unrealistic period can lead to
reliable heights as long as a sufficiently linear phase–height relation with a
significant spread of phases is present. For this reason, periods around 50 h
do not work for the analyzed Q2DW as they mark the reversal of the vertical
propagation direction, which leads to very long $\lambda_{z}$. In section 3,
we have already discussed a two-wave model with fixed amplitudes that we
checked as an alternative but resulted in unlikely wave parameters and low-
quality fits. Nevertheless, even such a model appears to provide useful
information on the OH height distribution. Focusing on the standard deviation
of the effective heights of all lines, the best-fitting waves with periods of
43 and 51 h and opposite vertical propagation directions returned 2.18 and
0.72 km. Both values show a large discrepancy but the average is 1.45 km,
which is very close to the value for our preferred model of 1.43 km.

Figure 9: Final effective heights for the considered 298 OH lines derived from
a combination of fits of X-shooter line measurements and SABER VER data of the
OH channel centered on 2.1 $\mu$m for the Q2DW event in 2017. The wave-related
effective heights were shifted upward by 4.07 km to be representative of the
mean SABER-related centroid altitudes for Cerro Paranal from noll17. The
abscissa shows the energy of the upper level of the transition minus the
lowest energy for the corresponding vibrational state $v^{\prime}$. The latter
is given by colored numbers. The representative height uncertainties are about
0.24, 0.30, and 0.55 km for energy differences below 400 cm-1, between 400 and
800 cm-1, and above 800 cm-1, respectively.

With the confirmation of the robustness of our results, we now show the
distribution of the reference heights for all investigated OH lines in Figure
9. The altitudes range from 85.9 to 93.8 km, i.e. the maximum difference is
almost 8 km and therefore of the same order as the width of the full OH
emission layer [<]e.g.,¿baker88. The detailed structure of the plotted data
distribution was already discussed in terms of the wave phases in section 2.1.
The distribution is just inverted compared to the phases in Figure 4b. The
effective emission heights increase for higher vibrational and rotational
excitations as expected. Focusing on the levels with $F^{\prime}=1$ and
$N^{\prime}=1$, the altitude difference between $v^{\prime}=9$ (89.14 km) and
2 (85.89 km) amounts to 3.26 km, which is about 0.47 km for a difference of 1
in $v^{\prime}$ on average. This result agrees very well with the
corresponding values found in previous studies of a few OH bands related to
satellite data [Noll . (2016), Sheese . (2014), von Savigny Lednyts’kyy
(2013), von Savigny . (2012)] and ground-based data combined with a sodium
lidar [Schmidt . (2018)]. The height difference for $\Delta v^{\prime}=1$
decreases with increasing vibrational excitation. Our analysis revealed 0.59
km for $v^{\prime}\leq 4$ but 0.29 km (i.e. the half) for $v^{\prime}\geq 7$.
This behavior agrees qualitatively with the modeling results of savigny12 and
is obviously caused by the $v^{\prime}$-dependent Einstein and collisional
rate coefficients. As the previously published results refer to emissions of
bands instead of lines, we also checked the change of the height differences
with increasing rotational energy. Taking only data with $\Delta E^{\prime}$
between 400 and 600 cm-1 ($N^{\prime}$ between 4 and 6) as in section 4.1, we
found almost the same difference between $v^{\prime}=9$ and 2 (3.20 km) but at
altitudes that are about 1.1 km higher. The change of the differences for
$\Delta v^{\prime}=1$ with $v^{\prime}$ also agrees within the uncertainties.
Consequently, the results for entire bands should agree with those of the
significantly contributing brighter lines of low to intermediate $N^{\prime}$.

For the highest rotational levels, we do not see a clear dependence of the
effective emission altitudes on $v^{\prime}$. The differences appear to
decrease with increasing $N^{\prime}$. As the nearly linear height increase
with $\Delta E^{\prime}$ seems to flatten (especially for higher
$v^{\prime}$), there might even be a nearly constant effective emission height
for the highest $N^{\prime}$. This behavior was already modeled by dodd94.
However, quantitatively, there are some discrepancies. Our mean for all lines
with $\Delta E^{\prime}\geq 2,000$ cm-1 is 92.3 km. If we assume that this
height is representative of all $v^{\prime}$, we estimate
$v^{\prime}$-specific maximum changes compared to $\Delta E^{\prime}=0$ cm-1
between 3.2 and 6.4 km. dodd94 only modeled 0 to 2 km and a reference height
for the highest $N^{\prime}$ of 89 km. Based on the data set of noll17 that we
also used for the derivation of our reference heights, noll18b modeled
rotational level populations with a focus on $v^{\prime}=9$. For two models
with very different rate coefficients for collisions of OH with atomic oxygen,
the maximum height changes resulted in 1.5 and 2.8 km and a maximum emission
height of almost 92 km in the latter case. If we estimate the height changes
from the plotted data for $v^{\prime}=9$, we obtain a rough lower limit of 1.5
to 2 km. These values might be better for a comparison since $v^{\prime}=9$
levels with $\Delta E^{\prime}\geq 2,000$ cm-1 would be far beyond the
exothermicity limit of the hydrogen–ozone reaction [Cosby Slanger (2007), Noll
. (2018)] of about 1,070 cm-1. In any case, the model results appear to match
the correct order of magnitude, although the uncertainties are large. The
model of noll18b also predicts a flattening of the height increase for high
$N^{\prime}$. Moreover, this model gives nearly constant heights for the
lowest $N^{\prime}$. The latter is something which we do not find in our
analysis that suggests a linear increase due to the growing contribution of
the hot population. The nearly constant height for the OH lines with the
highest $\Delta E^{\prime}$ of all $v^{\prime}$ also seems to be supported by
the measured wave amplitudes. The interesting levels are occupied by a pure
hot population [Noll . (2020)], which appears to show a nearly constant
amplitude for the Q2DW in 2017 (Figure 4). Moreover, the SABER-based fits of
the profiles indicate a steep gradient of the amplitude at the heights
relevant for OH (section 4.2). Hence, an almost constant amplitude would
require a relatively narrow altitude distribution for the OH lines dominated
by the hot population. For lower $N^{\prime}$, the interpretation of the wave
amplitudes is more difficult due to the mixing of cold and hot populations
with different altitude distributions.

###  4.4 Impact of Time Range

Figure 10: Maximum amplitude $c{\cdot}a$ relative to the mean (a) and phase
$\phi$ relative to the period at 15 January 12:00 LT at Cerro Paranal (b) for
all 270 OH lines used for the fit of the Q2DW event in 2019. The plot is
similar to Figure 4. Figure 11: Fitted phase relative to the period of 44 h as
a function of height for the SABER product kinetic temperature (stars) and the
VERs of O2(a-X)(0-0) at 1.27 $\mu$m (diamonds), OH(1.6 $\mu$m) (squares), and
OH(2.1 $\mu$m) (circles). The time series comprised the 19 limb scans taken at
about 23:00 LT in seven nights of 2019. The reference time for the plotted
phases was 15 January 12:00 LT at Cerro Paranal. Fit uncertainties, which are
useful for relative quality comparisons, are indicated by dotted lines.
Moreover, the plot shows the resulting regression line for a linear fit of the
OH(2.1 $\mu$m) phases in the altitude range between 80 and 93 km.

Our OH height estimates are based on eight nights of a single Q2DW. Although
they appear to be reliable according to the discussion in section 4.3, the
analysis of another wave event would be a good quality check. As already
described in section 2.1, we were able to identify another Q2DW in the
X-shooter data of seven nights from 11 to 18 January 2019. We analyzed the
corresponding line measurements in the same way as for the Q2DW in 2017.
However, we only considered lines with wavelengths shorter than 2.1 $\mu$m
because of too few spectra taken without a $K$-blocking filter (section 2.1).
The resulting fits showed that a period of 44 h also appears to be the best
choice. On the other hand, the LT dependence of the wave amplitude was very
different from the curves in Figure 3. The variation was much smaller. The
maximum and minimum $c{\cdot}a$ values only differed by a factor of 2 for the
two example lines. Moreover, the highest amplitudes were reached between 22:00
and midnight, which corresponds to a shift of $-2$ h compared to the data from
2017. While the maximum relative amplitude for OH(4-2)P1(14) did not change
much (0.49 vs. 0.46 for 2017), it was significantly lower for OH(4-2)P1(1)
(0.31 vs. 0.74 for 2017). As a consequence, hot populations obviously showed a
stronger response to the Q2DW than cold populations, which is the opposite
situation compared to 2017. This reversal was also seen for the dependence of
the amplitude on $v^{\prime}$ for low $N^{\prime}$ (Figure 10a). Nevertheless,
the impact of the mixing of both populations at intermediate rotational
energies (400 to 800 cm-1) was similar. The related lines showed the largest
amplitudes, although only values up to about 0.62 were found. In conclusion,
the two-population model appears to be confirmed by the data from 2019.
However, the Q2DW was weaker and showed different amplitude relations, which
suggests that the properties of such waves are highly variable due to changes
in their generation, propagation, and interaction with the background
atmosphere (including other waves). This high amount of variability had
already been observed before [<]e.g.,¿ern13,gu19,tunbridge11.

For our purpose, it is important that a wave can be used for estimates of
effective emission heights. Hence, the phase relations are crucial.
Unfortunately, the pattern for our set of OH lines was completely different
from the situation in 2017 (Figure 4b). A clear $v^{\prime}$ dependence was
not found and there was no monotonic decrease of the phase with increasing
$v^{\prime}$ (Figure 10b). Instead, the phase even increased up to about 600
cm-1. The expected behavior was only present for higher energies. An important
detail for the explanation of this unexpected structure is the maximum range
of phases, which amounts to only 0.041. The standard deviation was 0.007.
Consequently, the phase was almost constant. We also fitted the available
SABER data for a better understanding. For the seven nights in 2019, we could
use 19 profiles, all taken at about 23:00 LT (section 2.2). The profile fits
for both OH channels agree quite well with the X-shooter data. There were only
small phase changes without clear direction (Figure 11). For OH(2.1 $\mu$m),
we found a maximum phase difference of only 0.13 for the height interval
between 80 and 93 km, which minimizes the deviation of the fitted phases from
a linear relation. The resulting regression line is almost vertical with a
highly uncertain wavelength $\lambda_{z}$ of $280\pm 240$ km. The rising wave
turns into a descending one with $\lambda_{z}=300\pm 210$ km for a fit up to
97 km as in the case of 2017 (Figure 8). Hence, the propagation direction
remains unclear. The long wavelength may explain the reduced dependence of the
wave amplitude on the OH line parameters (partly related to the emission
height) and local time. The latter might point to a less efficient interaction
of the Q2DW with the migrating diurnal tide, which has a relatively short
$\lambda_{z}$ [<]e.g.,¿forbes95 that is similar to the about 32 km for our
best fit of the Q2DW in 2017.

huang13 investigated $\lambda_{z}$ of southern W3 Q2DWs based on SABER
temperature data from 2002 to 2011 and found a wide range of possible values
from above 10 km to beyond 100 km at an altitude of 85 km. As other studies
using different techniques also found a high variability for low-to-middle
southern latitudes in the mesopause region [Ern . (2013), Guharay . (2013),
Reisin (2021)], the differences in $\lambda_{z}$ for the analyzed Q2DWs in
2017 and 2019 do not appear to be uncommon. For a strong event in southern
summer 2002 to 2003, huang13 also investigated the change of the wavelength
over the lifetime of the wave of several weeks. They found a trend of
decreasing $\lambda_{z}$ and less variability with increasing age. As we
investigated data from 11 to 18 January in 2019 but from 26 January to 3
February in 2017, this trend would be consistent with our wave fits. However,
as the Q2DWs can be quite different from year to year, a convincing check
would need a detailed wave analysis over the entire lifetime for the different
years. In any case, the properties of the Q2DW in the selected time interval
in 2019 were not suitable for our phase-sensitive investigation.

##  5 Conclusions

In our study, we derived reference centroid emission heights for average
conditions of various individual OH lines for the first time. This success
required the combination of OH line intensities from ground-based spectra
taken with the astronomical X-shooter spectrograph at Cerro Paranal in Chile
and space-based limb sounding of emission profiles in the two OH-related
channels of the SABER radiometer on TIMED. Moreover, we benefited from the
observation of a strong quasi-2-day wave (Q2DW) with both instruments in eight
nights at the beginning of 2017. For the region around Cerro Paranal and the
given observing period (mainly limited by the X-shooter data coverage), our
wave fits of both data sets (separated depending on local time) with a cosine
function revealed a most likely period of 44 h and vertical wavelength of
about 32 km (based on the SABER channel centered on 2.1 $\mu$m), which makes
this wave event very suitable for phase-sensitive investigations. The
amplitudes strongly varied. Our fits revealed particularly high amplitudes up
to almost 100% of the mean intensity for emission lines related to
intermediate rotational energy between 400 and 800 cm-1 and low vibrational
upper level $v^{\prime}$, the second half of the night, and altitudes below
the emission peak. The high values for lines with intermediate rotational
energies indicate an amplification of the variation due to the mixing of cold
(thermalized) and hot (non-thermalized) OH rotational level populations, which
maximizes for the stated energy range. The local time dependence (with no wave
detection at the beginning of the night) suggests that the Q2DW was strongly
affected by the changing diurnal atmospheric conditions (e.g. by tides). Apart
from the altitude dependence of the intrinsic amplitude of the upward-
propagating wave, the OH emission should also be affected by the increasing
relative atomic oxygen variability with decreasing height. Furthermore, the
wave properties significantly depend on the selection of the geographical area
and the time range. Another Q2DW which was present in the X-shooter data of
2019 indicated a vertical wavelength being too long to provide sufficient
phase sensitivity for our purpose.

From the effective wave phases of each line measured by X-shooter and the
relation between phase and altitude from the height-resolved fits of the SABER
profiles, we first estimated effective emission heights that are
representative of the altitudes with the strongest wave amplitudes during the
studied eight nights in 2017. For OH(2.1 $\mu$m), the phase change between 80
and 97 km was almost perfectly linear, which allowed us to derive reliable
heights without ambiguities. The resulting heights of the 298 investigated OH
emissions cover a range of about 8 km with an average of 84.9 km. Lines with
higher $v^{\prime}$ and/or rotational upper level $N^{\prime}$ show higher
effective altitudes. At low rotational energies, the height increase appears
to be almost linear, whereas lines with high $N^{\prime}$ indicate a
flattening of the trend and a decreasing difference between different
$v^{\prime}$. The latter could imply the presence of a universal hot
population. Finally, we derived line-specific reference altitudes that are
representative of the long-term centroid heights at Cerro Paranal. In
combination with results for both SABER OH channels from a previous study, we
found that a fixed positive shift of about 4.1 km is obviously sufficient for
our line set. The resulting heights therefore range from 85.9 to 93.8 km with
an average of 88.9 km.

Our results may provide important constraints for a better modeling of the
layering of OH emission. Moreover, the line-dependent heights could be used to
study other wave events where suitable profile data are not available. In
particular, gravity waves with their too short periods for repeated satellite
observations may constitute an appealing target. Finally, our approach could
also be applied to other airglow emissions. The X-shooter spectra contain
various candidates. Consequently, the results of our study and possible future
applications are quite promising with respect to a better understanding of the
chemistry and dynamics of the Earth’s mesopause region.

## Open Research

The basic X-shooter data for this project originate from the ESO Science
Archive Facility at http://archive.eso.org and are related to different
observing programs. In particular, raw NIR-arm spectra taken between 26
January and 3 February 2017 and between 11 and 18 January 2019 were processed
(using the corresponding calibration data) and then analyzed. This project
made also use of SABER v2.0 limb-sounding products at http://saber.gats-
inc.com from January and February of the years 2017 and 2019. Both archives
can be accessed after registration.

The input data for the fitting procedure described in section 3 and the
results as shown in the figures are available at the public repository Zenodo
[Noll . (2022)]. In detail, the data release includes time series of the
intensities of the investigated OH lines measured in the X-shooter spectra for
the analyzed periods binned in 30 min steps. The corresponding SABER profiles
for the two OH channels as well as the vertically integrated OH emissions are
also provided as time series. Moreover, there are tables with the plot data of
the 11 figures. For the Q2DW from 2019, where only two figures are included in
the paper, some additional tables similar to the ones for the Q2DW from 2017
are considered.

###### Acknowledgements.

Stefan Noll is financed by the project NO 1328/1-3 of the German Research
Foundation (DFG). We thank Holger Winkler from Universität Bremen for his
contribution to the discussion and Sabine Möhler from ESO for her support with
respect to the X-shooter calibration data. Moreover, we are grateful to the
three anonymous reviewers for their valuable comments.

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