High‐Order Harmonics of Thermal Tides Observed in the Atmosphere of Mars by the Pressure Sensor on the InSight Lander

Thermal tides are atmospheric planetary‐scale waves with periods that are harmonics of the solar day. In the Martian atmosphere thermal tides are known to be especially significant compared to any other known planet. Based on the data set of pressure timeseries produced by the InSight lander, which is unprecedented in terms of accuracy and temporal coverage, we investigate thermal tides on Mars and we find harmonics even beyond the number 24, which exceeds significantly the number of harmonics previously reported by other works. We explore comparatively the characteristics and seasonal evolution of tidal harmonics and find that even and odd harmonics exhibit some clearly differentiated trends that evolve seasonally and respond to dust events. High‐order tidal harmonics with small amplitudes could transiently interfere constructively to produce meteorologically relevant patterns.


Introduction
Atmospheric tides are a natural response of planetary atmospheres to the periodic forcing exerted whether by gravity (gravitational tides), or insolation (thermal tides).They consist of planetary-scale oscillation modes with periods that are harmonics of the forcing period (the solar rotation period, in the case of thermal tides) and integer zonal wavenumbers.A tide with an harmonic number equal to the zonal wavenumber has a phase speed equal to the apparent speed of the Sun, and is called a migrating tide.Other tides with various phase speeds are called nonmigrating tides.A detailed explanation of migrating and non-migrating tides can be found for example, in Forbes et al. (2020).The classical theory of atmospheric tides can be found in Chapman and Lindzen (1969).
Most studies focus on tidal harmonics S1 and S2 (i.e., harmonics 1 and 2, we will use this notation from now onwards), given that low order harmonics are much stronger compared to higher ones.Some detailed studies have also analyzed S3 and S4 on Earth (e.g., Moudden & Forbes, 2013;Smith et al., 2004) and reported them in Venus and in Mars (Guzewich et al., 2016;Peralta et al., 2012).Higher order harmonics have been detected on Earth (up to S12; He et al., 2020;Hedlin et al., 2018;Hupe et al., 2018;Sakazaki & Hamilton, 2020) and on Mars (up to S6; Sánchez-Lavega et al., 2023).In addition to it, recent studies based on simulations (Lian et al., 2023;Wilson & Murphy, 2015;Wilson et al., 2017) have found harmonics up to S6 and S7 on Mars, and they indicate that such harmonics are dominated by migrating modes.On Earth, other mechanisms in addition to solar insolation have been observed to contribute to S3 and S4: nonlinear interaction between tides (e.g., Teitelbaum et al., 1989), and interaction between tides and gravity waves (e.g., Geißler et al., 2020), other references can be found in Pancheva et al. (2021).
On Mars, due to the low thermal inertia of the atmosphere, thermal tides are stronger in relation to the atmospheric thickness than in any other planet of the solar system (Barnes et al., 2017); they are a key aspect of the general circulation, and they react strongly to the presence of dust (Leovy & Zurek, 1979;Ordóñez-Etxeberria et al., 2019;Viúdez-Moreiras et al., 2020), leading to feedback that could play a key role in the establishment of large dust events (Barnes et al., 2017).Many authors have explored thermal tides on Mars based on surface stations (Guzewich et al., 2016;Hess et al., 1977;Sánchez-Lavega et al., 2023;Viúdez-Moreiras et al., 2020;this work), satellites (e.g., Fan et al., 2022;Forbes et al., 2020;Guerlet et al., 2023;López-Valverde et al., 2023;Whiters et al., 2011), andsimulations (e.g., Guerlet et al., 2023;Wilson & Hamilton, 1996).Due to their lower sensibility compared to ground stations, remote sensing instruments from satellites can only detect the stronger low-order tidal harmonics.As a counterpart, a single station on the ground can observe the daily cycle at a single location, which is insufficient to separate modes with different wavenumbers.Therefore, a given harmonic number observed by a ground station is actually a mix of tidal modes with different zonal wavenumbers.This is why a network of ground stations on Mars would be especially useful to disentangle the global structure of thermal tides (e.g., Wilson & Kahre, 2022).
In this study, we benefit from the unprecedented accuracy (50 mPa) and temporal coverage of the data set produced by the pressure sensor (PS) on the InSight lander (Banfield et al., 2019;Banfield et al., 2020;Spiga et al., 2018) to investigate thermal tides on Mars from Ls (Solar Longitude) 304°in MY34 (Martian Year 34), to Ls 20°in MY36.This interval corresponds to sols 15-824 of the InSight mission.Our results support the existence of tidal modes beyond S24.We comparatively analyze the characteristics and seasonal evolution of such harmonics.Results are discussed in relation to dust climatologies produced by Montabone et al. (2015Montabone et al. ( , 2020)), and using the nomenclature for annually repeating dust events (A, B, C) proposed by Kass et al. (2016).The main dust events that took place during the acquisition of this data set are: dust event C in MY34 (sols 40-80; C34 from now onwards; analyzed in detail by Viúdez-Moreiras et al., 2020), regional off-season dust event in MY35 (sols 180-220; R), dust event A in MY35 (sols 550-650; A35), and dust event C in MY35 (sols 700-740; C35).
For convenience, we use a nomenclature for seasons centered on equinoxes and solstices (with a shift of 45°in Solar Longitude (Ls) from the conventional meteorological seasons).In the context of this paper, the northward equinox season refers to Ls 315°-45°, northern solstice season refers to Ls 45°-135°, southward equinox season refers to Ls 135°-215°, and southern solstice season refers to Ls 225°-315.

Results
In this section we discuss the number of tidal harmonics present in the data set (Section 2.1), show the overall differences between even and odd harmonics (Section 2.2), explore the seasonal evolution of the diurnal pressure cycle driven by the ensemble of tidal harmonics (Section 2.3), and look in detail at the properties of separate harmonics (Section 2.4).We use Numpy (Harris et al., 2020) and Scipy (Virtanen et al., 2020) for our computations.

High-Order Tidal Harmonics in the InSight PS Data Set
We compute a periodogram (as implemented by Virtanen et al. (2020)) on the longest interval with continuous InSight measurements available.We show this periodogram in Figure 1.
Clear power spectral density peaks over the estimated noise level coincident with expected tidal harmonics are present up to S26, being the absence of a peak for S23 the only exception.Other tidal harmonics might be present beyond S26, clear peaks are S31, S32, S37, S38, and even S43, but the absence of other harmonics and the lower fraction of sols in which such harmonics display amplitudes over the instrumental noise level (5% for S43) prevent us from considering those as robustly detected harmonics.
We computed periodograms for other long intervals (Figure S2 in Supporting Information S1).S23, absent in Figure 1, is clearly present in part of them.The periodogram for the interval spanning from sol 37 to sol 66 is particularly interesting, because C34 dust event took place during those sols, and this boosted the amplitude of solar harmonics (see Sections 2.3 and 2.4); such periodogram is much more noisy than the one shown in Figure 1, but it displays coincident peaks with much shorter periods, even beyond harmonic number 48.For the remainder of this paper we only discuss those harmonics robustly detected up to S26.Do these high-order solar harmonics correspond to actual thermal tides?This question can be raised from the mathematical and from the physical point of view.From the mathematical point of view, it can be argued that transient daily repeating non-tidal pressure patterns repeating at the same Local Time (LT) in consecutive sols could induce high-order solar harmonics in Fourier analysis.From the physical point of view, our analysis of high-order harmonics is limited by the fact that we rely on a single station, and therefore it is not possible to fully confirm that these harmonics are part of globally coherent tidal modes.In order to further investigate the presence of these harmonics in the data set, we computed the phase of the anomaly of pressure at specific ranges of the spectrum corresponding to tidal harmonics beyond S12 and beyond S24.The results are presented in Figure S1 of Supporting Information S1, and they reveal that wave-like patterns at those frequencies repeat similarly in different sols at most LTs (especially in some seasons, including the period of sols 490-567 corresponding to Figure 1).This shows that the measured signals inducing peaks at high-order harmonics in our periodograms are not transient daily repeating patterns, but they are present most of the time and repeat consistently in different sols, which is what we would expect if these signals corresponded to the interference of tidal harmonics.Therefore, within the limitation that we relay on a single station, our analysis is compliant with the interpretation of these high-order solar harmonics as thermal tides.
The gravitational tidal force of Phobos might produce gravitational tides on the atmosphere of Mars (as the Moon does on Earth).The orbital period of Phobos relative to a fixed point on the surface of Mars is around 40,000 s, but the corresponding peak is not present in our periodograms.

Differences Between Even and Odd Harmonics
When we analyze the properties of all these tidal harmonics as computed for every sol using the Fast Fourier Transform (FFT; Cooley & Tukey, 1965), we notice systematic differences between even and odd harmonics (excluding S1, which is peculiar due to its mix of migrating and non-migrating components and its vertical structure).These differences are noticeable in Figure 2a, where odd harmonics systematically exhibit smaller amplitudes between S3 and S13. Figure 2b is an example of the contribution of even and odd harmonics to the diurnal cycle of pressure; shadowed areas represent the pressure anomaly produced by even and odd harmonics, we find that the contribution of odd harmonics to the diurnal cycle is much smaller than that of even harmonics.Figure 2c shows that the relative contribution of odd harmonics is larger during the northern solstice season.However, it falls abruptly with the starting of the southward equinox season (Ls 135°), when global dust content starts to rise.There is a soft increase from Ls 240°to Ls 360°.And shorter boosts take place during dust events: there are clear peaks coincident with C34, R, and C35 dust events.
Figure 2a also shows that the averaged amplitudes of even and odd modes in logarithmic scale fall in a clear linear trend between S2 and S14.The equivalent figures for individual sols (Movie S3) exhibit variations in the amplitudes of harmonics, and the linear trend only arises clearly when averaging some number of sols.The seasonality of this linear trend is further investigated in Figure 2d.We see that even modes follow this trend basically always, while odd modes present many exceptions.

Seasonal Evolution of the Ensemble of Tidal Modes
Figure 3 is a convenient representation of the seasonal evolution of daily pressure cycle as modeled by the ensemble of tidal harmonics.It represents the variation of absolute pressure over the diurnal cycle for every sol of the data set.
Red and blue horizontal bands are the most evident feature in this figure, these bands represent daily repeating tidal patterns and correspond to the parts of the diurnal cycle when the absolute pressure is increasing (red) or decreasing (blue).Prominent bands are present after sunrise and after sunset, they correspond to the bumps in the pressure timeseries usually observed at 8-20 LT, studied by Wilson et al. (2017) and Yang et al. (2023).We note that the trends during the equinox seasons, when the sun is crossing the equator in opposite directions, are well symmetric, especially in the evening, after sunset.
In addition to the smooth seasonal evolution, other patterns at shorter temporal scales are present in this figure in coincidence with dust events C34, C35, and R.During these events, the derivatives of pressure reach higher values, corresponding to larger amplitudes of tidal modes; and the daily cycle of pressure advances, corresponding to a shift in the phases.Unfortunately, a gap without observations during peak of A35 prevents us from extracting clear conclusions about the effects of that event.
Due to their lower amplitudes (see Figure 2a), high-order harmonics are underrepresented in Figure 3.However, the phases of high-order harmonics over S12 can be observed in Figure S1 of Supporting Information S1, and there is an obvious correlation in their seasonal evolution and that apparent here in Figure 3.This evidences that Geophysical Research Letters 10.1029/2023GL107674 high-order harmonics evolve with a certain degree of synchronization with low-order harmonics, which is also evident in the next subsection.

Seasonal Evolution of Individual Tidal Modes
Each tidal harmonic is characterized by an amplitude and a phase that evolve over time.The daily values of these parameters are computed using FFT and are represented for tidal harmonics up to S12 in Figure 4. We analyze these results in terms of seasonal variations, which are more or less coincident with the equinox and solstice seasons; and transient variations, which happen at shorter timescales of dozens of sols or less and are more likely connected to dust events or other transient meteorological phenomena.Seasonal variations are in some cases quite sudden, this is especially the case for S3 and S4 in the boundaries of the northern solstice season, and that is likely connected to the strong seasonal changes in the boundaries of that season previously shown in Figure 3.
Harmonics of order higher than S3 tend to converge into overall similar seasonal trends that are in part different for even and odd harmonics.Even harmonics display larger amplitudes and delayed phases during the equinox seasons compared to the solstice seasons.Wilson et al. (2017) also found in their simulations enhanced amplitudes for S4-S6 around equinoxes.Odd harmonics also display a typically delayed phase during equinox seasons, but their amplitude variations happen at shorter timescales (dozens of sols) and are in some cases more likely connected to transient phenomena (e.g., dust storms).
Harmonics S1-S3 exhibit their own patterns, with seasonal variations and large transient variations in response to the main dust events.The anti-correlation in the amplitudes of S3 and S4 observed by Guzewich et al. (2016) and Sánchez-Lavega et al. ( 2023) is also present in our results.
Transient dust events C34, R, A35, and C35 leave in some cases very strong footprints both in amplitudes and phases, usually in the form of peaks that only last a few sols compared to the longer length of such dust events.Amplitudes usually increase in response to dust events, but in some cases they decrease; this is especially clear in S4-S6 during dust events R and C35 respectively.
The distribution of phases exhibits different levels of randomness that are different in different tidal harmonics and seasons.It tends to be more random in odd and higher order harmonics, and in solstice seasons (northern solstice: S5, S10, S12; southern solstice: S7, S9, S11).The equivalent graphs for harmonics higher than S12 are available in Figure S4 of Supporting Information S1, and they show that phases look more and more random for those higher order harmonics.

Summary and Perspectives
Our analysis supports that high-order tidal harmonics (even beyond S24; Section 2.1) are present in the data set acquired by the PS on InSight.The finding of a large number of harmonics and the excellent temporal coverage of this data set have enabled us to explore the seasonal evolution of: (a) The differences between even and odd harmonics (Sections 2.2 and 2.4), (b) The ensemble of thermal tides in the daily cycle of pressure (Section 2.3), (c) The individual tidal harmonics (Section 2.4).
We analyzed seasonal variations in terms of seasons centered on equinoxes and solstices, and our results adapt very well to them, with S3 and S4 experiencing sudden changes exactly in the limits of such seasons.The low thermal inertia of the Martian atmosphere makes it respond more quickly to changes in insolation, and that is likely the reason why this definition of seasons centered in equinoxes and solstices adapts so good to certain observations of the Martian atmosphere.The average amplitude of tidal harmonics tends to fall exponentially between S2 and S14, with different rates for even and odd harmonics (Figure 2a); the latter exhibit systematically smaller amplitudes.These trends undergo a seasonal evolution, possibly connected to the evolution of thermal forcing depending on insolation and aerosols distribution.Theoretical and modeling studies are needed to further investigate the processes leading to these trends.High order and odd harmonics tend to exhibit more random variations in their phase and amplitude, their smaller amplitudes probably make them more susceptible to slight changes in the atmosphere, and therefore they might respond more easily to the presence of aerosols.The seasonally larger amplitude of even harmonics around equinoxes matches with the finding of the same trend for S4-S6 in simulations (Wilson et al., 2017).
The existence of high-order harmonics on Mars that repeat similarly in different sols, probably with some degree of interannual repeatability, could have consequences on the mesoscale and the microscale in the form of daily repeating patterns of subtle variations of meteorological variables that in some cases could trigger daily repeating phenomena.This is especially the case because a large number of low amplitude harmonics with similar frequencies can transiently interfere constructively to produce signals with larger amplitudes (as also suggested for Earth by Hedlin et al. (2018)), and such signals could even be confused with gravity waves (a possibility that we will explore as part of a separate work).
In any case, the present paper opens a new window for theoretical works and modeling to explore features reported here (e.g., differences between even and odd harmonics), check whether models reproduce the observed behavior of high-order harmonics, use such models to learn more about these harmonics, determine the contribution of different mechanisms to the different tidal harmonics (e.g., Geißler et al., 2020), and explore the possible consequences and effects of high-order tidal harmonics, on Mars and maybe on other planets as well.When it comes to observations, many aspects of the InSight data set remain to be explored, and other missions currently operating on Mars can try to find high-order harmonics, and the coordinated analysis of data from those missions can contribute to investigate their global structure.This work enhances the interest to include pressure sensors with improved accuracy in future landers, and to deploy a network to make simultaneous observations.la Recherche (ANR-19-CE31-0008-08 MAGIS) and CNES.All co-authors acknowledge NASA, Center National d′Études Spatiales (CNES) and its partner agencies and institutions (UKSA, SSO, DLR, JPL, IPGP-CNRS, ETHZ, IC, and MPS-MPG), and the flight operations team at JPL, CAB, SISMOC, MSDS, IRIS-DMC, and PDS for providing InSight data.The members of the InSight engineering and operations teams made the InSight mission possible and their hard work and dedication is acknowledged here.We thank Lucas Lange for his useful comments.And we thank John Wilson and an anonymous reviewer for their excellent work reviewing this paper, which largely contributed to improve it.JHB expresses his support to UN Secretary General António Guterres in his calls to stop violations of Human Rights and International Law, and to act against climate change.JHB expresses his support to scientist colleagues on trial and in jail across the world for their non-violent actions against climate inaction.https:// scientistrebellion.org/scientists-on-trial/.

Figure 1 .
Figure 1.Periodogram showing power spectral density (PSD) as function of wave period computed on an interval with continuous coverage over 77 sols (sols 490-567).Bottom graph is simply a continuation of the top one.Harmonics of the Mars mean solar rotation period (∼88,775 s) are indicated by green vertical lines with the harmonic number written in the upper axis.The percentage of sols within this interval in which the amplitude of each harmonic is over instrumental noise level is indicated in the top of the graphs.Brown curve represents the average PSD, red line represents an indicative estimation of the noise level (see Text S2 in Supporting Information S1 for details).Yellow shadowing starting after harmonic number 26 represents a part of the periodogram where some PSD peaks coincident with harmonics are coincident but it is not clear if they are actual tidal harmonics.The correction proposed by Lange et al. (2022) for the pressure sensor data set does not produce any significant change in this periodogram.

Figure 2 .
Figure 2. Differences between even and odd (excluding S1) harmonics.(a) Daily amplitude of tidal modes averaged over a whole year, error bars represent the standard deviation.(b) Example of contribution of even and odd harmonics to the diurnal cycle of pressure, corresponding to sol 129.(c) Relative weight of the contribution of even and odd harmonics as function of the sol number, measured as the ratio between the filled areas in panel (a).(d) Pearson correlation coefficient for the linear regression of the amplitude of even and odd modes in logarithmic scale between S2 and S14.Each linear fit has been computed for the average amplitude of 5 sols, a value of 1 means perfect linear correlation.Vertical lines in panels (b and c) represent the limits between meteorological seasons, solstice seasons are shadowed in pale yellow.

Figure 3 .
Figure 3. Climatology of the diurnal variation of pressure (dP/dt) in units of Pa/h.High frequency signals (with periods below 3,700 s) have been removed from the original signal before computing the derivative.Sunrise and sunset are indicated by white curves around 6-18 LT (Local True Solar Time).The global dust content as computed from the climatologies of Montabone et al. (2015, 2020) is represented in the bottom panel, and dust events are indicated.Seasons are delimited by vertical lines and solstice seasons are shadowed in pale yellow in the bottom (dust) panel.

Figure 4 .
Figure 4. Amplitude (top graph in each panel; blue) and phase (bottom graph in each panel; green; expressed as Local Time of one maximum) of individual tidal harmonics as a function of the sol number up to S12, which is enough to visualize the main trends found.Points below the instrumental noise level (0.05 Pa) are represented in dark red.Solstice seasons are shadowed in pale yellow.In phase graphs, the vertical axis contains all the range of possible variability except for S1, S2, S3, S4, and S6, which are very stable and their range has been constrained to appreciate better their subtle patterns within their small range of variation; this is indicated by orange horizontal lines in the top and bottom limits of the graph.