We report on the characteristics of tidal variations in the Martian lower atmosphere (<45 km) using the Mars Express (MEX) Planetary Fourier Spectrometer (PFS) temperature data for about three Martian years (between the ends of MY26 and MY29). The PFS data, which widely cover local time, enable us to investigate diurnal variations in the atmospheric temperature at various altitudes. We focus on diurnal variations in the atmospheric temperature and on longitudinal temperature variability in a fixed local time frame. We find that the latitudinal and diurnal variations at 0.52 mbar (∼25 km) during the dust-clear period (Ls = 30°–60°) are consistent with general characteristics presented by previous numerical simulations. The characteristics of the diurnal variations as a function of altitude in the tropics are also explained as results from the propagation of the migrating diurnal tide. The longitudinal temperature variability in the dayside (14.36–14.94 LT) equatorial regions (10°S–5°S) near the northern summer solstice (Ls= 76°–83°) in MY28 are investigated. The longitudinal temperature structure has two local maxima at 2.85 mbar (∼10 km) but is relatively uniform at 0.52 mbar. We find that the wave-3 structure is apparent at 0.11 mbar (∼40 km) in the present case. This structure would be strongly dependent on activities of the atmospheric waves, e.g., the diurnal Kelvin wave 2 (DK2).
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 Thermal tides play an important role in determining the general circulation, thermal structure and vertical coupling between the lower and the upper atmosphere of terrestrial planets. They are expected to be stronger on Mars than on Earth, because the Martian atmosphere is very thin (only about 0.6% as thick as the Earth’s atmosphere). The thermal tides are classified into two groups. One is the migrating tides, which propagate westward with the apparent solar motion, and are generated by cyclic heating due to absorption of the solar radiation. The other is the non-migrating tides, which propagate westward or eastward independent of the apparent solar motion, and are excited by the interaction of the migrating solar forcing with non-uniform components such as topography and radiative processes by dust [Forbes et al., 2002].
 The migrating tides have been investigated mainly by GCM studies [e.g., Wilson and Hamilton, 1996; Wilson and Richardson, 2000; Takahashi et al., 2006]. Wilson and Richardson  studied the equatorial temperature variations as a function of local time and altitude (<∼30 km) for three dust conditions. Takahashi et al.  investigated the vertical and latitudinal structure of the migrating diurnal tide (<∼90 km) under low dust condition for two seasonal periods. Although observations of diurnal variations in atmospheric quantities (pressure, temperature, density, and wind) are essential for understanding the migrating tides in the Martian atmosphere, there have been only a few studies [e.g., Wilson and Richardson, 2000; Banfield et al., 2000]. This is because major orbiter observations were restricted to two fixed local times constrained by the sun-synchronous orbits, such as the Mars Global Surveyor (MGS) observations during the mapping phase.Banfield et al. identified the migrating diurnal and semidiurnal tides from the limited MGS TES data during pre-mapping phase. The 15μm channel (T15) observations by the Viking Infrared Thermal Mapper (IRTM) provided the atmospheric temperature at 0.5 mbar (∼25 km) with broad coverage of local time over two Martian years. However, Wilson and Richardson  pointed out that the original T15temperatures were significantly contaminated by the surface radiance, particularly during the dust-clear periods.
 The Mars Express (MEX) Planetary Fourier Spectrometer (PFS) has observed the lower atmosphere and the surface of Mars for about three Martian years. The PFS has broadly covered local time, latitude, longitude, and solar longitude Lsdue to the non sun-synchronous orbit of the MEX. In particular, the local time coverage of the PFS data has a great advantage in studying the diurnal variations. The PFS data also enable us to investigate the atmospheric temperature variations at various altitudes, which is the advantage over the Viking IRTM data. In this paper, we investigate the tidal variations in the Martian lower atmosphere at various altitudes up to ∼45 km during the dust-clear period from the PFS data. We focus on diurnal variations in the atmospheric temperature and on longitudinal temperature structure in a fixed local time frame.
2. Data Set
 The PFS was mainly designed for monitoring the 3-D temperature field in the lower atmosphere, the surface temperature, the abundances of minor constituents, and the optical properties of aerosols. The PFS is a Fourier interferometer which covers the infrared spectral range of 250–8200 cm−1 in two channels: the long wavelength channel (LWC) and the short wavelength channel (SWC). For both channels, the apodized spectral resolution is ∼2 cm−1 and the sampling step is ∼1 cm−1. The instrument and its radiometric performance are described in detail by Formisano et al.  and Giuranna et al. [2005a, 2005b].
 The temperature profile (from the surface to ∼45 km) was retrieved from nadir observations of the 667 cm−1 CO2 band by the LWC. The vertical resolution of temperature profile is ∼10 km. The random error in temperature profile caused by noise equivalent radiation (NER) is estimated at ∼2 K (15–20 km above the surface), increasing up to ∼7 K (∼45 km) [Grassi et al., 2005b]. Note that some spectra were not retrieved when the brightness temperature of the incident radiation exceeded the detector temperature (i.e., during dust storm periods). The temperature profiles have been used to study the thermal structure over Olympus Mons [Grassi et al., 2005a; Wolkenberg et al., 2010], the Hellas basin [Grassi et al., 2007], and the south polar cap [Giuranna et al., 2008].
 In this study, we use a total of 335,563 profiles. These data encompass the entire period of the MEX orbits 10 to 7118, which corresponds to about three Martian years between the ends of MY 26 (Ls = 331°) and MY 29 (Ls = 306°).
3.1. Diurnal Variations in the Atmospheric Temperature
 Because the coverage of local time by the PFS observations is strongly correlated with Ls over the course of the mission, we cannot meaningfully investigate diurnal variations by using temperature data for a single Martian year, regardless of which seasonal period we choose. Therefore, we determine a seasonal period that shows little seasonal and interannual changes and then average the data in this period over about three Martian years.
Figure 1a shows the diurnally and longitudinally averaged atmospheric temperature at 0.52 mbar (∼25 km) as a function of Ls and latitude over about three Martian years (from late MY26 to late MY29). The temperature are averaged into bins with a size of 6° in Ls and 6° in latitude. At this altitude, the global warming and cooling of the atmosphere in response to the change in solar insolation between aphelion (Ls = 71°) and perihelion (Ls = 251°) is clearly seen. In addition, the temperature near the northern winter solstice (Ls= 270°) varies greatly from year to year in response to dust conditions. At most seasons, the high-temperature regions appear in both the northern and southern mid-latitudes. These high temperatures would be caused both by direct solar heating and by adiabatic heating in the descending branch of the Hadley circulation [e.g.,Wilson, 1997; Forget et al., 1999].
 To clarify the seasonal and interannual variability of the Martian atmosphere, Figure 1b shows the globally averaged atmospheric temperature at 0.52 mbar as a function of Ls. For latitude, the data are weighted by the cosine of latitude (proportional to the surface area) and then averaged. For Ls, the data are binned in the same manner as for Figure 1a. In the aphelion season (Ls= 0°–180°), the temperature shows good year-to-year repeatability. Specifically, the seasonal and interannual temperature differences forLs = 30°–60° are less than ∼2.5 K, which is of the same order as the retrieval error. In the perihelion season (Ls = 180°–360°), while the increasing trend in temperature for Ls= 180°–220° also shows good year-to-year repeatability, the temperatures forLs = 220°–360° are highly variable. The interannual variability obtained from the PFS data shows trends similar to those obtained from the TES data for MY24–26 [Smith, 2004]. Wolkenberg et al.  evaluated the PFS and the TES atmospheric temperatures for the overlapping period from Ls = 337° in MY26 to Ls = 77° in MY27 and concluded that good agreement was found at 0.5 mbar in the regions from 40°N to 40°S. The detailed comparison of atmospheric temperature obtained by these instruments is described by Wolkenberg et al. .
 Based on Figure 1, we average the data for Ls = 30°–60° over about three Martian years. Figure 2 shows the latitudinal and diurnal temperature variations at three different altitudes (2.85, 0.52, and 0.11 mbar) for Ls= 30°–60°. The data are binned every 6° for latitude and every 1 hour for local time. At 2.85 mbar (∼10 km), the temperature distribution is asymmetric about the equator, especially in high latitudes, because of the obliquity of the rotation axis (∼25°). The high-temperature regions (>205 K) appear in the late afternoon in both the tropics and the mid-latitudes. At 0.52 mbar (∼25 km), the high-temperature regions (∼170 K) are also seen in the northern and southern mid-latitudes in the late afternoon, while the temperatures in the tropics (12°S–12°N) are lower than those in the mid-latitudes by 10–15 K. In the tropics, the temperatures in the morning are higher than those in the late afternoon by ∼10 K, which is not seen in the temperature distribution at 2.85 mbar. At 0.11 mbar (∼40 km), the temperatures in the southern mid-latitudes are somewhat higher than those in the northern mid-latitudes.Forget et al.  simulated that the southern polar warming appeared at 40–80 km in early northern spring (Ls= 0°–30°) which would be primarily caused by adiabatic heating in the subsiding branch of the Hadley circulation. The latitudes where high-temperature regions appear correspond to their results. The low-temperature regions (∼140 K) in the tropics occur in the early afternoon at this altitude.
3.2. Longitudinal Temperature Structure in a Fixed Local Time Frame
 The tidal variation in the atmospheric temperature T at certain latitude and a given longitude λ can be expressed as follows [Forbes et al., 2002]:
where t is time, Ω is the planetary rotation rate, s denotes the zonal wave number (s = 0 for zonally uniform, s > 0 for westward propagating wave, s < 0 for eastward propagating wave), n denotes the subharmonic of a solar day (e.g., n = 1 for the diurnal tide, n = 2 for the semidiurnal tide), Tn,s is the amplitude, and ϕn,s is the phase. In equation (1), we use t = tLT − λ/Ω, where tLT is local time.
 In a fixed local time frame, equation (1) can be further simplified by setting the observed wave number k = ∣s − n∣:
where Tk and ϕk denote the observed amplitude and phase, respectively.
 Assuming that longitudinal variations in the temperature at each altitude can be represented by k in the range 0–4 (with k = 0 for constant temperature), we can obtain Tk and ϕk for each wave component from equation (2). This method of analysis has been applied previously to radio science data [e.g., Cahoy et al., 2006]. In order to investigate the longitudinal structure as a function of altitude during the dust-clear period, we use temperature profiles in the narrow ranges ofLs, local time, and latitude in which temperature profiles are obtained with good longitudinal coverage.
Figure 3 shows the longitudinal temperature variations at three different altitudes in the dayside (14.36–14.94 LT) equatorial region (10°S–5°S) near the northern summer solstice (Ls = 76°–83°) in MY28. The solid curve is the least squares fit expressed by equation (2). The dashed curves are confidence intervals (1-σ) of the fit. The amplitudes and phases of wave components are summarized in Table 1. Note that only non-migrating tides can produce the longitudinal variations in a fixed local time frame. At 2.85 mbar (∼10 km), the maximum temperature difference for longitude exceeds 20 K. The two temperature maxima at around 55°E and 255°E correspond to the regions of high surface topography [Smith et al., 1999]. The wave-2 component is the strongest, followed by wave-1 and 4 components, the amplitudes of which are similar to each other. At 0.52 mbar (∼25 km), the longitudinal variation is relatively uniform compared with that at 2.85 mbar. No wave components are significant. At 0.11 mbar (∼40 km), the wave-3 structure is significantly apparent. Although not shown here, this wave-3 structure is also clearly seen above an altitude of 0.3 mbar (∼30 km). The phase of the wave-3 component moves eastward by ∼15° as pressure decreases from 0.3 mbar (∼30 km) to 0.04 mbar (∼45 km).
Table 1. The Amplitudes and Phases of Wave Components Obtained from the Least Squares Fit
The 1-σ uncertainties of parameters are also presented.
Constant temperature (K)
Wave-1 amplitude (K)
Wave-2 amplitude (K)
Wave-3 amplitude (K)
Wave-4 amplitude (K)
Wave-1 phase (°)
Wave-2 phase (°)
Wave-3 phase (°)
Wave-4 phase (°)
4. Summary and Discussion
 Taking advantage of broad local time coverage of the PFS data with sufficient vertical resolution, we first succeeded in clarifying the latitudinal and diurnal temperature variations at three distinct altitudes during the dust-clear period (Ls = 30°–60°). These variations at 0.52 mbar (∼25 km) shown in Figure 2b are found to be similar in distribution to the simulated T15 temperatures at Ls = 37° in Figure 19d of Wilson and Richardson . That is, the clear phase separation in tidal responses between the tropics and mid-latitudes predicted by the GCM simulations and tidal theory for the dust-clear periods can be confirmed by averaging the PFS data over about three Martian years. This result observationally supports the hypothesis ofWilson and Richardson  that the Viking IRTM T15data are significantly contaminated by the surface radiance, particularly during dust-clear periods. The temperature maxima that occur at the mid-latitudes in the late afternoon (15–18 LT) are slightly higher in the PFS results than in the simulated ones by ∼5 K. This may be attributed to the difference in sensing altitude between the PFS data and the simulatedT15 channel or the difference of dust conditions.
Takahashi et al.  performed the GCM simulations to investigate the latitudinal and vertical structure of the migrating diurnal tide for the northern spring equinox (Ls = 343°–15°). Takahashi et al. [2006, Figure 3b] showed that the phases of temperature component of the migrating diurnal tide were almost constant below ∼40 km in both the northern and southern mid-latitudes; that is, the temperature maxima in these regions occur at 14–15 LT. The observed temperature maxima (late afternoon) in both the northern and southern mid-latitudes at 0.52 mbar (∼25 km) and 2.85 mbar (∼10 km) shown inFigures 2b and 2c, respectively, correspond to the simulation results. The temperature maxima in the mid-latitudes below 0.52 mbar (∼25 km) are caused by direct heating due to absorption of solar radiation by atmospheric molecules and aerosols. In contrast, the phase in the tropics decreases with increasing altitude, as shown byTakahashi et al. [2006, Figures 3b and 5], which means local time at temperature maximum shifts toward morning with increasing altitude. In the simulation results, the temperature maxima are seen in late afternoon and the early morning at 2.85 mbar (∼10 km) and 0.52 mbar (∼25 km), respectively. The temperature minimum is also seen in near midday at 0.11 mbar (∼40 km). The observed diurnal variations at three different altitudes in the tropics, as shown in Figure 2, approximately correspond to those predicted by the GCM simulations.
 In Figure 3, we showed the longitudinal temperature variability in a fixed local time in the equatorial region (10°S–5°S) near the northern summer solstice (Ls= 76°–83°) in MY28. The longitudinal structure varies significantly with altitude. That is, the longitudinal structure at 2.85 mbar (∼10 km) has two local maxima due to the topography but that at 0.52 mbar (∼25 km) is relatively uniform. The longitudinal structures above 0.3 mbar (∼30 km) are dominated by wave-3 component. This wave-3 structure is also found in the other latitude regions (5°S–0°S and 5°S–10°S).
 There are two possible tidal modes that can be responsible for the wave-3 structure [Wilson, 2000, 2002; Forbes et al., 2002; Banfield et al., 2003; Withers et al., 2003; Cahoy et al., 2006]. One is the diurnal Kelvin wave 2 (DK2) that is an eastward propagating diurnal tide with zonal wave number 2 (n = 1, s = −2) and the other is the semidiurnal Kelvin wave 1 (SK1) that is an eastward propagating semidiurnal tide with zonal wave number 1 (n = 2, s= −1). DK2 is mainly excited by the migrating tide interacting with the wave-3 component of topography. SK1 is also excited by similar processes. DK2 and SK1, which have long vertical wavelengths of ∼10 and ∼14 scale heights, respectively [Withers et al., 2003], can propagate into the upper atmosphere and produce the longitudinal variability of densities at aerobraking altitudes. The wave-3 structure is largely attributed to DK2 in the tropics and to SK1 in the mid- and high latitudes [Wilson, 2002; Withers et al., 2003; Bougher et al., 2004].
 The diurnal Kelvin wave 1 (DK1), which causes the wave-2 structure and is in near-resonance in the Martian atmosphere, was identified from the TES data [Wilson, 2000; Banfield et al., 2003]. From vertically-resolved TES data,Banfield et al. concluded that the wave-2 structure caused mainly by DK1 was consistently apparent at altitudes above 0.8 mbar through a Martian year (MY24) and there was little change in the amplitude and phase throughout MY24. In the present case, however, the amplitude of wave-2 component above 0.3 mbar (∼30 km) is unexpectedly small. Such small amplitude of wave-2 component was also found in the upper atmosphere (∼110 km) from the MEX SPICAM pressure profiles [Withers et al., 2011]. The absence of wave-2 component found in both the lower and the upper atmosphere is a problem to be solved in future.
 We have reported the first results of tidal variations in the Martian lower atmosphere from the MEX PFS temperature data. In order to clarify characteristics of thermal tides through comparison with those obtained by other observations and numerical studies, as the next step, a statistical analysis with the PFS data will be necessary for understanding seasonal and latitudinal dependence of the tidal variations in the Martian lower atmosphere.
 T. M. Sato is supported by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science (JSPS). This work is supported by a Grant-in-Aid for Scientific Research (22340142) from the Tohoku University Global COE program titled “Global Education and Research Center for Earth and Planetary Dynamics”. Operations of the PFS and a part of the PFS science are funded by Mars Express Phase E2 project of the ASI.
 The Editor thanks Stephen Bougher and Paul Withers for their assistance in evaluating this paper.