# Forcing mechanisms of the 6 h tide in the mesosphere/lower thermosphere

### Christoph Geißler

### Friederike Lilienthal

### Amelie Krug

Solar tides such as the diurnal and semidiurnal tide, are
forced in the lower and middle atmosphere through the diurnal cycle of solar
radiation absorption. This is also the case with higher harmonics like the
quarterdiurnal tide (QDT), but for these also non-linear interaction of
tides such as the self-interaction of the semidiurnal tide, or the
interaction of terdiurnal and diurnal tides, are discussed as possible
forcing mechanism. To shed more light on the sources of the QDT, 12 years of
meteor radar data at Collm (51.3^{∘} N, 13^{∘} E) have been
analyzed with respect to the seasonal variability of the QDT at 82–97 km
altitude, and bispectral analysis has been applied. The results indicate
that non-linear interaction, in particular self-interaction of the
semidiurnal tide probably plays an important role in winter, but to a lesser
degree in summer. Numerical modelling of 6 h amplitudes qualitatively
reproduces the gross seasonal structure of the observed 6 h wave at Collm.
Model experiments with removed tidal forcing mechanisms lead to the
conclusion that, although non-linear tidal interaction is one source of the
QDT, the major forcing mechanism is direct solar forcing of the 6 h tidal
components.

The mesosphere and lower thermosphere (MLT) dynamics are strongly influenced by atmospheric waves, including the solar tides with periods of a solar day and its harmonics. Their wind amplitudes usually maximize around 100–120 km. At these heights, their amplitudes are comparable with the mean wind. Thus, the solar tides are an intrinsic part of the global circulation and more accurate knowledge of tides leads to a better understanding of the wind fields in the MLT in general. Shorter period tides often have smaller amplitudes, so that in the past especially the diurnal tide (DT), the semidiurnal tide (SDT), and also the terdiurnal tide (TDT) have been considered in investigations. The quarterdiurnal tide (QDT), however, although it also forms an integral part of the middle and upper atmosphere dynamics, has attained much less attention, mainly due to its small amplitude in the MLT. Near the surface the 6 h oscillation at times can be a major wave component as seen e.g. in barographic records (e.g., Warburton and Goodkind, 1977), but the 6 h amplitude in the MLT is generally substantially smaller than the one of the DT, SDT, and TDT. Consequently, only few attempts to determine the MLT QDT characteristics from radar or satellite observations have been made so far, and very few studies included the modelling of the QDT global structure and its sources.

Considerable QDT amplitudes have been reported by Walterscheid and Sivjee (1996, 2001)
in the high-latitude winter, but they concluded that these were
not migrating but zonally symmetric tides. From medium frequency radar winds
over Adelaide, Australia and Davis, Antarctica, Kovalam and Vincent (2003)
found signatures of 6 and 8 h tides, but belonging to a wavenumber 1 tide,
so that they concluded that these oscillations are not thermally forced but
possibly owing to non-linear interactions. Smith et al. (2004) investigated
the QDT over Esrange, Sweden, and found that the QDT wind amplitudes on a
monthly average may exceed 5 m s^{−1} at 97 km altitude, and that they maximize
in winter. Smith et al. (2004) also performed numerical simulations that
revealed that much of the wintertime QDT is forced by the 6 h harmonic of
solar heating, but without direct forcing the tide still appears and also
maximizes in winter. Without direct solar forcing, the nonmigrating tidal
modes became proportionally larger. Jacobi et al. (2017) used Collm (51.3^{∘} N,
13.0^{∘} E) meteor radar data. Their observed amplitudes and their
seasonal cycle were similar to the ones presented by Smith et al. (2004).
Comparison with radar observations at Obninsk, Russia, also indicated that
most of the QDT signal at midlatitudes is probably due to its migrating
components. Liu et al. (2006) noted a 6 h signature in medium frequency
radar data over Wuhan, China, but mainly in their upper height gates above
90 km. They found from bispectral analyses that there are indications for
non-linear interaction of tides as a possible forcing mechanism, but only in
the upper height gates.

The 6 h harmonics of ozone heating rates have been calculated from Aura/MLS
observations by Xu et al. (2012), who noted that the main 6 h forcing
during solstice is in the winter hemisphere. Xu et al. (2014) analyzed
nonmigrating tides from TIMED/SABER satellite observations. They confirmed
earlier results that the QDT is largest in winter, and found indications
that the nonmigrating QDT is likely to be forced by non-linear interaction
between the DT and TDT, while the interaction between stationary planetary
waves and the QDT is weak, likely because of the small amplitudes of the
migrating QDT. In a further study, Liu et al. (2015), again using
TIMED/SABER data, analyzed the migrating QDT between 50^{∘} S and
50^{∘} N in the middle atmosphere. From their analyses they
considered both direct heating and tidal interaction as possible sources of
the QDT. Azeem et al. (2016) analyzed observation from RAIDS/NIRS
instruments, which confirmed solar heating as main source for the 6 h tide.
But also other sources like non-linear interaction between tides could not
be excluded.

To summarize, to date there are rather few analyses of the QDT both locally and on a global scale, and in particular the forcing mechanisms of the QDT are still unclear and should be investigated further. Therefore, in the following we use the Collm data presented by Jacobi et al. (2017) and apply bispectral analysis to obtain indicators for possible forcing through non-linear interaction. In addition, we use a mechanistic numerical model and analyze the most likely forcing mechanism for the QDT through removing either solar heating or non-linear interaction for the model

The horizontal winds over Collm (51^{∘} N, 13^{∘} E) have been
estimated from measurements obtained by a SKiYMET meteor radar, which is
operated on 36.2 MHz since summer 2004. Details of the radar and the radial
wind determination principle can be found in Jacobi (2012) and Stober et al. (2012).
An update of the radar was performed in 2015, but without change of
the transmit frequency (Stober et al., 2017). The individual meteor trail
reflection heights vary between about 75 and 110 km, with a maximum meteor
count rate around 90 km (e.g., Stober et al., 2008). The data are binned in
6 different not overlapping height gates, which are centred at 82, 85, 88,
91, 94, and 98 km. Meteors show a vertical distribution with
increasing/decreasing count rates with height below/above 90 km, so that the
nominal heights do not necessarily correspond to the mean heights.
Therefore, below/above 90 km mean heights tend to be higher/lower than
nominal heights and in particular the real mean height of the uppermost
height gate is closer to 97 rather than 98 km. For the other height gates,
the difference between real and nominal height is small (Jacobi, 2012).

Individual radial winds calculated from the meteors are collected to form half-hourly mean values using a least-squares fit of the horizontal wind components to the raw data under the assumption that vertical winds are small (Hocking et al., 2001). Tidal wind parameters at each height gate have been calculated by applying a least-squares regression analysis of one month of either zonal or meridional half-hourly horizontal winds on a model wind field including mean wind and tidal oscillations (Jacobi et al., 2017).

Bispectral analysis has proved to be a powerful tool to detect quadratic
phase coupling between 3 frequencies. A bispectrum *B*(*ω*_{1},*ω*_{2}) is nonzero for triplets of frequencies ${\mathit{\omega}}_{\mathrm{1}},{\mathit{\omega}}_{\mathrm{2}},{\mathit{\omega}}_{\mathrm{3}}$, when one of them is the sum or difference of the
others, and there is phase coherence. Since non-linear interaction between
tidal components result in exactly such a phase and frequency relationship,
peaks in the bispectra can be used to detect possible non-linear
interaction. Note however, that bispectra do not provide a real proof of
non-linear interaction, because phase and frequency triplets could
accidentally arise due to other reasons. The bispectrum may be written as:

with the third-order cumulant

where *v*(*k*) are the individual detrended wind values and *E* is the
expectancy value (Nikias and Raghuveer, 1987). For calculating the
bispectral estimate of a finite record, the datasets are subdivided into *K*
records with *M* samples. *K* and *M* should be as large as possible to reduce the
large variance of the bispectral estimate and the phase coherence, which
occurs by chance. For short data sets overlapping windows should be used to
get enough segments (Lii and Helland, 1981). We used records of four days,
i.e. *M*=192 half-hourly means, with a 50 % overlap, resulting in *K*=14 for one month (13 for February). A Hanning window was used. We apply the
squared bicoherence spectrum *b*^{2}(*ω*_{1},*ω*_{2}) after
Kim and Powers (1978) to estimate the fraction of power that is generated by
quadratic phase coupling. It uses the power spectrum *P*(*ω*) for
normalization and is suitable to examine frequency triplets, which show
peaks in the power or amplitude spectrum:

Two examples of zonal wind bicoherence spectra are shown in Fig. 1, one of them indicating a possible self-interaction of the SDT (Fig. 1a), the other one indicating coupling of the TDT and DT, and both resulting in a 6 h component.

In order to estimate, during which month of the year and at which altitude significant non-linear interaction is likely to occur, we estimated the 95 % significance level of bispectral peaks after Haubrich (1965) using ${b}^{\mathrm{2}}\ge \mathrm{6}/\mathrm{dof}$ as a conservative estimate (Elgar and Guza, 1985), with the degrees of freedom dof = 2 K. Actually, overlapping the records influences dof, which can be used to balance the effect of windowing the data records before the analysis (Emery and Thomson, 2001). Then, for each month of the year and for each height gate we calculated the number of years when we found significant spectral peaks for possible TDT/DT interaction and SDT self-interaction. Dividing this number by 12 resulted in a percentage of years, so that this shows how frequently there is an indication for significant non-linear interaction at this height and month. Figure 2 shows the percentage of years with significant peaks indicating self-interaction of the SDT (upper row) and interaction of the DT and TDT (lower row) in colour coding. In addition, the 12-year mean monthly mean tidal amplitudes are shown as contours. During winter, significant SDT self-interaction is frequently seen especially in the upper height gates (Fig. 2a, b). This generally coincides with maxima of the QDT, but also of the SDT which maximizes during winter (e.g. Jacobi, 2012). A secondary seasonal maximum of possible significant SDT self-interaction is found during autumn equinox, when the SDT also has a maximum (Jacobi, 2012). Significant bispectral peaks from interaction of the DT and TDT are less frequent (Fig. 2c, d). This kind of interaction has more effect during spring and autumn, when both the TDT and the DT have their maxima (Jacobi, 2012). The spring maximum of TDT/DT interaction is connected with a secondary maximum of QDT amplitudes, which is, however, only weakly visible during autumn also.

To conclude, there is indication for significant non-linear interaction both of the SDT and of the TDT/DT, and these are more frequent during the months when the respective tidal components have large amplitudes. During these months maxima of the QDT are observed as well.

Since bispectral analysis neither provides a real proof for non-linear
interaction nor gives quantitative measures how strong such a wave forcing
would be, we performed numerical experiments using the Middle and Upper
Atmosphere Model (MUAM, Pogoreltsev et al., 2007) to analyze the influence
of different QDT forcing terms like direct solar heating and non-linear
interaction at the latitude of the Collm radar observations. MUAM is a
non-linear primitive equation mechanistic model of the atmospheric
circulation with a resolution of 5^{∘} × 5.625^{∘} in the
horizontal. In 56 vertical layers it extends up to a log-pressure height *z* of
160 km, which is given by $z=H\mathrm{ln}({p}_{\mathrm{0}}/p)$ with *p*_{0}=1000 hPa and a
scale height *H*=7 km. For time integration, a Matsuno (1966) scheme has
been applied with a time step of 225 s. The model does not account for
tropospheric effects such as orography or latent heat release. Therefore,
the zonal mean temperature below 30 km is nudged with monthly mean
ERA-Interim temperature fields in order to provide realistic dynamical
features in the lower atmosphere.

Heating of the atmosphere due to absorption of solar radiation by water vapour, carbon dioxide, ozone, oxygen and nitrogen is introduced in the model via a radiation parameterization after Strobel (1981) (see also Fröhlich et al., 2003). The ozone and water vapour fields are prescribed as zonal means, so that mainly migrating tides are reproduced in the model. Infrared cooling of carbon dioxide is parameterized after Fomichev et al. (1998), while ozone infrared cooling in the 9.6 µm band is calculated after Fomichev and Shved (1985). Gravity waves in the middle atmosphere are parameterized according to a linear scheme as described by Fröhlich et al. (2003).

Zonal and meridional wind amplitudes of the modelled QDT at 52.5^{∘} N
(the nearest grid point to the Collm latitude) have been obtained from a
frequency-wavenumber analysis of the migrating tides from one month of
modelled data at each latitude and height level separately. They are shown
in Fig. 3a, b. As in the observations, the meridional component is somewhat
smaller than the zonal one. The zonal and meridional amplitudes maximize
during February/March and November. Smaller maxima show up in May and July.
The meridional component shows maxima also in February and March, as well as
in October and November, but also during the summer months. The modelled
amplitudes are generally smaller than the observed ones. This may be due to
the fact that we only analyze migrating tides in the model. It has to be
mentioned, however, that Jacobi et al. (2017) did not find evidence for a
strong nonmigrating QDT so that their contribution to the total amplitude
should not be as large as the difference between model and observations. The
used model version does not include latent heat release in the troposphere,
which also forces tidal components, and this may explain part of the reduced
amplitudes. The observed seasonal cycle of the observed tide, however, is
reproduced in general, except for another minimum near winter solstice,
which is only weakly seen in the observations (Fig. 2, and Figs. 7 and 8 of
Jacobi et al., 2017). The observed spring maxima are shifted a bit towards
the summer.

To analyze the contribution of solar and non-linear forcing on the QDT amplitudes at higher midlatitudes, we removed the zonal wavenumber 4 component, which in the present configuration of the model is equivalent to the migrating QDT, from either (i) the non-linear terms of the prognostic equations or (ii) from the solar heating. The method of removing wavenumber components in the different forcing terms has been described in Lilienthal et al. (2018). Note that we only modelled migrating tidal components, so our results cannot be compared, e.g., with those of Xu et al. (2014). As Fig. 3c, d shows, the effect of removing non-linear terms is small and the QDT amplitudes are only weakly reduced. Partly, e.g. during midsummer, the amplitude increases slightly when non-linear forcing is removed. This is most likely an effect of destructive interference of the tidal components forced through absorption of solar radiation and by non-linear interaction. As Fig. 3e, f show, the remaining amplitudes after removing solar heating are small. This seasonal distribution is different from the observed one. An existence of non-linear forcing can be seen in December and January below 100 km, during the other months this is only the case at altitudes above 100 km.

Bispectral analysis of observed MLT winds at Collm indicate that non-linear interaction of tides may play a role in forcing the QDT. The major effect seen in the radar observations is due to self-interaction of the SDT, while interaction of the DT and TDT contribute to the QDT in spring and autumn. However, the fact that there are indications for non-linear interaction does not necessarily mean that the resulting QDT amplitudes are really strong. Indeed, MUAM model experiments show that, although there exists possible non-linear forcing of the QDT through tidal interaction, the resulting amplitudes are small, while the quarterdiurnal component of solar heating is the dominant forcing mechanism of the migrating QDT at the Collm latitude.

Of course the observations and model results cannot be compared directly.
The main difference is that local radar measurements deliver the full
amplitude, i.e. migrating and nonmigrating tides together, and separating is
not possible from one observation alone. Furthermore, the MUAM model only
provides reduced amplitudes, and does, for example, not show components due
to latent heat release. In addition, the latitudinal resolution of the model
is 5^{∘}, but the meridional structure of the QDT and its forcing is
rather complex (e.g. Smith et al., 2004; Xu et al., 2012) so that a higher
meridional model resolution may provide more accurate results.

The results presented here are all based on monthly mean analyses. Tides, however, are known to vary also at the day-to-day time scale, e.g. through their interaction with planetary waves. Therefore, it is possible that at time scales shorter than one month, the SDT, TDT, or DT are increased and non-linear interactions leading to a QDT signature would be stronger in relation to solar forcing. While analyzing this is beyond the scope of this paper and would require a modified modelling approach including e.g. the analysis of planetary waves, a more comprehensive analysis of the QDT forcing at short time scales is certainly worthwhile and should be performed in further studies. Finally, MUAM is a global model and thus global results of QDT forcing can be obtained, but satellite observations will be necessary for validation of the results. Therefore, in future analyses, we plan to extend the model analysis and use QDT amplitude distributions from GPS radio occultations (e.g. Arras and Wickert, 2017) for validiation.

Collm radar wind data are available from the corresponding author upon request.

MUAM model code is available from the corresponding author upon request. Bispectral analysis was performed using the Higher Order Spectrum Estimation python toolkit, © 2015 synergetics.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Kleinheubacher Berichte 2017”. It is a result of the Kleinheubacher Tagung 2017, Miltenberg, Germany, 25–27 September 2017.

This study has been supported by Deutsche Forschungsgemeinschaft through
grant JA 836/34-1.

Edited by: Ralph Latteck

Reviewed by: two anonymous referees

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