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

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

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

Two examples of bicoherence spectral power as possible
indicator for non-linear interaction. Analyzed data are half-hourly means at
91 km for

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

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

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

Colour coding: percentage of significant bicoherence peaks
for the SDT self-interaction (panels

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

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

Zonal and meridional wind amplitudes of the modelled QDT at 52.5

Modelled monthly mean QTD zonal

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

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