1. Introduction

Of the 342 W m⁻² of solar radiation reaching the top of the atmosphere, about 168 W m⁻² are absorbed by the earth’s surface and 67 W m⁻² are absorbed by the atmosphere (43 W m⁻² by water vapor, 14 W m⁻² by ozone, 7 W m⁻² by clouds, and 3 W m⁻² by O₂ and CO₂) (Kiehl and Trenberth 1997). This atmospheric solar heating, combined with upward eddy conduction of heat from the ground, generates internal gravity waves in the atmosphere at periods of the integral fractions of a solar day (primarily at the diurnal and semidiurnal periods). These waves cause regular oscillations in atmospheric wind, temperature, and pressure fields, which are often referred to as atmospheric tides (Wallace and Hartranft 1969; Chapman and Lindzen 1970; Haurwitz and Cowley 1973; Hsu and Hoskins 1989; Whiteman and Bian 1996). Due to the fact that 24 hours are longer than the local pendulum day (the period corresponding to the local Coriolis parameter) poleward of 30° lat, the diurnal tide is incapable of propagating vertically over these regions to reach the ground (Lindzen 1967). Although the diurnal tide is often suppressed at the ground, its amplitude in the upper atmosphere is comparable with, and may be much larger than, the semidiurnal tide.

In the surface pressure field, the semidiurnal oscillation S₂ is thought to predominate over much of the globe, and is regularly distributed in its amplitude (which is about 1 mb in the Tropics and decreases poleward) and phase (S₂ usually peaks 2–3 h before noon and midnight). The classic tidal theory (Chapman and Lindzen 1970) predicts that solar heating through ozone explains about two-thirds of S₂’s amplitude while water vapor and heating from the ground account for the rest. Oscillation S₂ is dominated by its wavenumber 2 mode S²₂ (moving westward at the speed of the mean sun) (Haurwitz and Cowley 1973). In contrast to S₂, the diurnal oscillation S₁ in the surface pressure field is very irregularly distributed and is influenced strongly by landmasses. Water vapor heating provides a much larger contribution to S₁ than ozone heating (Groves and Wilson 1982). Locally, diurnal temperature and wind variations can greatly increase the amplitude of S₁.

Model calculations with more recent ozone and water vapor heating profiles suggest that additional heating is needed in order to explain the observed amplitude of S₁ and S₂ (Groves and Wilson 1982; Braswell and Lindzen 1998). Forbes and Garrett (1979) suggested that nonmigrating tides (which depend on both local time and longitude) could be excited by orographic features and longitudinal variations in water vapor and ozone. Model simulations suggest that the upward sensible heat flux from the ground due to solar heating contributes significantly to the nonmigrating components of S₁ (Tsuda and Kato 1989). The migrating component (which is a function of local time) of S₁ is underestimated by more than one-third by the classic tide theory even with the updated ozone and water vapor heating (Braswell and Lindzen 1998). The latent heating associated with convective precipitation, which has a strong diurnal cycle (Dai et al. 1999a), was found to be important mostly for S₂ (Lindzen 1978). It is suggested that the recently debated anomalous solar absorption by clouds or water vapor, which is not included in the radiation fluxes from Kiehl and Trenberth (1997) cited above, could significantly reduce this gap between the theory-predicted and the observed S₁ (Braswell and Lindzen 1998).

Besides heating, there is also a mass forcing on surface pressure tides resulting from the imbalance between evaporation and precipitation, which is significant globally on seasonal timescales (about 0.2 mb) (van den Dool and Saha 1993; Trenberth and Guillemot 1994). We know that evaporation peaks during the day (at least over the land), while precipitation occurs more frequently during early morning over the ocean (Janowiak et al. 1994) and in the afternoon over many land areas (Dai et al. 1999a). As the sun moves over the oceans (land), global evaporation is likely to increase (decrease). These diurnal variations are likely to induce a diurnal cycle of net water flux at the surface, which could provide an additional forcing for surface pressure tides. However, we are unaware of any studies addressing this mass forcing.

Because the longitude-dependent nonmigrating diurnal tides are nonnegligible at the surface and mixed up with the migrating tides, the migrating tides can be determined only from data around a latitude circle (Lindzen 1979). Most of our observational knowledge of S₁ and S₂ on a global scale comes from the analyses of surface pressure data from about 250 stations done by Haurwitz et al. from the 1950s to the early 1970s (Haurwitz 1956; Haurwitz 1965; Haurwitz and Cowley 1973). A majority of these stations were located in western Europe and North America while only about 50 of them were in the Southern Hemisphere. Most of the open oceans were not covered at all in these analyses. Obviously, the global distributions of S₁ and S₂ and their seasonal variations derived from such a limited spatial coverage should be interpreted with caution. Furthermore, the sparse spatial sampling and coarse grid resolution (15° long by 10° lat) used by Haurwitz and Cowley (1973) make their results of wave components likely to be unreliable, especially at high wavenumbers. Nevertheless, the S₁ and S₂ maps and their main wave components from these analyses have been widely cited and applied in the literature (e.g., Tsuda and Kato 1989;Lindzen 1990; Whiteman and Bian 1996; Deser and Smith 1998; Braswell and Lindzen 1998). Hamilton (1980a) updated Haurwitz’s dataset by adding 35 new stations (only two of them in the Southern Hemisphere). Regional analyses of S₁ and S₂ (e.g., Trenberth 1977; Mass et al. 1991; Kong 1995) revealed more small-scale features. More recent efforts to document global variations of S₁ and S₂ by analyzing the European Centre for Medium-Range Weather Forecasts analysis (Hsu and Hoskins 1989) and National Centers for Environmental Prediction (NCEP) reanalysis (van den Dool et al. 1997) 6-hourly data were hampered by the coarse temporal resolution of the data as well as the deficiencies of the models.

In this study, we update the earlier analyses and document S₁ and S₂ on a global scale by analyzing 3-hourly surface pressure data from 1976 to 1997 from all the land and ocean weather stations included in the Global Telecommunication System (GTS) and the marine reports from the Comprehensive Ocean–Atmosphere Data Set (COADS). There are about 15 000 stations included in GTS, but only 4000–7000 stations report at any given time, which still provides good spatial coverage over all the continents and islands except the Antarctic. The COADS data include most of the reports from commercial ships and various kinds of buoys and provide fairly good coverage of the global oceans during the 1976–97 period. We also analyzed the 3-hourly surface pressure data from 1980 to 1994 from the reanalysis with version 1 of the Goddard Earth Observing System (GEOS-1) Data Assimilation System (DAS) and found that the amplitudes of S₁ and S₂ of the reanalysis data tend to be smaller in high latitudes and larger in low latitudes than observed. We applied harmonic analysis to derive the amplitudes and phases of S₁ and S₂, and zonal harmonic analysis (Haurwitz and Cowley 1973) to decompose S₁ and S₂ into various zonal wave components. We also compared the geographical and seasonal patterns of S₁ and S₂ with those of atmospheric total column ozone, atmospheric precipitable water, and sensible heat flux from the earth’s surface [which is approximated by the mean diurnal range of surface air temperature (DTR)]. We find that spatial variations in the amplitude of S₁ are strongly correlated with DTR, suggesting that sensible heat flux is a major forcing for S₁ over land areas.

In section 2, we describe the datasets used in this study. Harmonic and zonal harmonic analysis methods are described in section 3. The global distributions of S₁ and S₂, their seasonal variations, and zonal wave components are presented and compared with earlier studies in section 4. In section 5 we compare the seasonal and spatial patterns of S₁ and S₂ with those of atmospheric total column ozone, atmospheric precipitable water, and DTR. A summary is given in section 6.

2. Data

The 3-hourly surface pressure data were extracted from the GTS synoptic weather reports archived at the National Center for Atmospheric Research (NCAR) (DS464.0; http://www.scd.ucar.edu/dss/datasets/ds464.0.html). This surface dataset, which covers the time period from July 1976 to April 1997 and has a volume of about 2.5 GB per year, contains 3-hourly (0000, 0300, 0600 UTC, etc.) surface pressure measurements from about 15 000 land and ocean stations and 6-hourly marine reports from commercial ships and buoys. Because the spatial coverage of this dataset is poor over the equatorial Pacific and the southern oceans, we also processed individual marine reports from the COADS dataset (Woodruff et al. 1993) from January 1976 to December 1995 (data after 1995 had not been released at the time of analysis). The COADS reports include hourly data from a small number of buoys and 6-hourly and 3-hourly data from ships and buoys. We used the COADS reports at the GTS 3-hourly reporting times (i.e., 0000, 0300, 0600 UTC, etc.). The number of reports of surface pressure at 0300, 0900, 1500, and 2100 UTC is generally less than that at 0000, 0600, 1200, and 1800 UTC for both the GTS and COADS datasets, especially over the southern oceans in the COADS data.

In contrast to the fixed land stations, the ships and many of the buoys are moving and thus their reports do not have a fixed long–lat location. Thus, we averaged individual marine reports within each 2.5° long by 2° lat box to obtain a mean value for each grid box at each reporting time. Land and fixed ocean stations were analyzed individually for the amplitude and phase of S₁ and S₂, which were then combined with those from the gridded marine data. The amplitude and phase were further gridded onto a 5° long × 4° lat grid using the natural neighbor interpolation method (Watson 1994). The gridded amplitudes and phases were then used in the plotting and zonal harmonic analysis.

Over the southern oceans (i.e., south of about 50°S) and some parts of the central equatorial Pacific, surface pressure data are available only at 0000, 0600, 1200, and 1800 UTC. These 6-hourly marine reports are not sufficient for sampling the semidiurnal cycle. Previous analyses (e.g., Haurwitz and Cowley 1973) and the 3-hourly GEOS-1 reanalysis data (which have a global coverage) indicated that relative to local time the semidiurnal cycle does not change rapidly in the longitudinal direction over the oceans. This allowed us to use reports (after the daily mean is removed) 10° to the east (40 min later) and to the west (40 min earlier) to supplement the 6-hourly data at some (∼15%) of the ocean grid boxes (away from coastlines). This resulted in 12 reports per day (0000, 0040, 0520, 0600, 0640 UTC, etc.) at these ocean boxes. Sampling tests with given diurnal and semidiurnal harmonics showed that the 3-hourly even sampling (i.e., 0000, 0300, 0600 UTC, etc.) and the supplemented 12 reports day⁻¹ sampling are sufficient to exactly reproduce the original harmonics.

The individual observations at each synoptic hour were averaged over each season (DJF = December–February, MAM = March–May, JJA = June–August, and SON = September–November) for each year. For the seasonally averaged values to be used in our harmonic analysis, we required at least seven of the eight synoptic hours to have data for the land stations and eleven of the twelve supplemented synoptic hours to have data for the marine boxes. About 7500 GTS stations meet this requirement with 4 or more yr of data during the 1976–97 period. Of the stations, about 4500 stations have at least 10 yr of data during the period. GTS stations and the ocean grid boxes with less than 4 yr of data were excluded from our analysis. Mass et al. (1991) showed that over the United States using more than 4 yr of data did not change the results of S₁ and S₂. The stations and ocean boxes with sufficient data that were used in our analysis are shown in Fig. 1. It can be seen that the station and marine observations provide good coverage over the globe except for the Antarctic, the Arctic Ocean, and the southern Pacific (south of 52°S and 70°–180°W) where the data are sparse. Given the relatively large-scale nature of S₁ and S₂, the spatial coverage of Fig. 1 should be sufficient for capturing the large-scale (>500 km) features of S₁ and S₂, especially over the Northern Hemisphere.

The annual-mean surface pressure data (expressed as anomalies relative to daily mean values) used to derived annual S₁ and S₂ are shown in Fig. 2. It can be seen that the westward-propagating wavenumber 2 mode predominates over low and middle latitudes but becomes less evident at high latitudes. The anomalies are largest over South America and Africa. Figure 2 is qualitatively consistent with the global patterns in the semidiurnal component of atmospheric pressure velocity ω (Trenberth et al. 1995), assuming that ω represents vertical motion of the air.

The station and marine data of surface pressure potentially contain various errors. For example, the surface pressure data are resolved to the nearest 0.1 mb, which is a substantial fraction of the amplitudes of S₁ and S₂ (about 1 mb at low latitudes and 0.2–0.5 mb at high latitudes). However, the measurement errors are likely to be random and thus are significantly reduced after averaging over the seasons and the 22-yr period. Another error may result from changes in barometers. Cooper (1984) shows that the amplitudes of the pressure tides may change by up to 20% due to changes in mercury barometers mainly because different barometers have different temperature corrections. Both mercury and aneroid barometers are used at land stations and on ships. It will be very difficult, if possible at all, to quantify the systematic errors associated with various types of barometers in all the station and marine records. We will compare our results to those based on the data before the 1970s when mercury barometers were often used. Furthermore, the phase estimated from the relatively coarse temporal sampling (3-h interval) will have an uncertainty up to ±1.5 h. In our analysis temporal averaging and spatial smoothing were applied to emphasize the mean large-scale features. It is likely that the maximum amplitudes shown in this study may be lower than those at individual stations because of the smoothing used in contouring.

Because of its global coverage, especially over the oceans, we also analyzed the 3-hourly averaged surface pressure dataset from the reanalysis with the GEOS-1 DAS (Wu et al. 1997). This monthly dataset covers the period from 1980 to 1994 and is on a 2.5° × 2° global grid. While the data assimilation model incorporates surface observations, the model itself has a significant influence on the assimilated surface pressure field. Since the pressure data are averaged over a 3-h period, the calculated amplitudes of S₁ and S₁ need to be multiplied by a factor (F) to obtain the real amplitudes that one would get from using instantaneous values (such as from the station data). The factor F = ωΔt/[2 sin(ωΔt/2)] where ω = angular frequency and Δt = averaging time period (=3 h), can be easily derived by averaging a sine function over a given period. Here F = 1.0262 for S₁ and 1.1107 for S₂, with ω = 2π/24 for S₁ and 2π/12 for S₂.

3. Analysis methods

We applied harmonic and zonal harmonic analyses to extract the solar diurnal and semidiurnal oscillations from the surface pressure data. We followed the analysis procedure described by Haurwitz and Cowley (1973). The surface pressure data at each location are represented by

View Expanded

where P₀ is the daily mean pressure, A_n is the amplitude (note: the peak-to-peak amplitude is 2A_n), σ_n the phase, t′ local mean solar time (LST) expressed in degrees or radians (i.e., t′ = 2πt₁/24, where t₁ is LST in hours). The 3-hourly station data and the supplemented marine data are insufficient for resolving harmonics with periods shorter than 12 h. Although accurate estimates of these high-order harmonics are not available over most of the globe, some analyses of hourly data showed that these high-order harmonics (primarily S₃) are much (at least a factor of 4–5) smaller than S₁ or S₂ (Trenberth 1977; Cooper 1984; Mass et al. 1991). Therefore, the error induced by the aliasing of the high-order harmonics is likely to be small (⩽10% over most areas) in Eqs. (1) and (2).

The harmonic coefficients a_n and b_n are then interpolated onto a 5° long × 4° lat grid using the natural neighbor interpolation method (Watson 1994). At each latitude θ, the gridded coefficients are then expanded using trigonometric series of the longitude λ (which is zero at Greenwich and increases eastward)

View Expanded

where the k^ν_n, l^ν_n, q^ν_n, and r^ν_n depend on θ. Substituting Eqs. (3) and (4) into (2), replacing t′ with Greenwich mean time (GMT) t = t′ − λ, after some trigonometric transformations (2) becomes (Haurwitz and Cowley 1973)

View Expanded

Waves with wavenumber s are referred to as S^s_n. In particular, waves with s = n travel westward at the speed of the mean sun. When expressed in local mean time t′, these waves are independent of longitude and called migrating tides. The waves with s ≠ n depend on both local time and longitude and are called nonmigrating tides.

The amplitude, A^s_n(θ) = [(a^s_n)² + (b^s_n)²]^1/2, of the waves with large positive or negative wavenumbers is sensitive to the regional features in the gridded a_n and b_n fields. Our tests showed that in general the higher the resolution used in the gridding (which is limited by the spatial sampling of the data), the larger the amplitude at the high wavenumbers. Based on the spatial coverage of the data, we used a resolution of 5° long by 4° lat, which effectively smooths out variations on scales less than about 500 km. A global mean amplitude, A^s_n = Σ_θ [A^s_n(θ) cosθ]/Σ_θ cosθ (Haurwitz and Cowley 1973), is computed for each wavenumber and is used to evaluate which wave components are important for S₁ and S₂ on a global scale.

4. Results

5. Implications for the forcing of S₁ and S₂

Although sensible heating from the ground has been shown to be an important forcing for diurnal nonmigrating tides in model simulations (Tsuda and Kato 1989), it has not been included in classic tidal calculations (e.g., Chapman and Lindzen 1970; Braswell and Lindzen 1998). The fact that the amplitude of S₁ and its seasonal variations are much larger over the land than over the ocean suggests that over land areas S₁ is greatly enhanced by the sensible heating from the ground. Atmospheric total ozone concentrations are fairly uniform zonally over low and middle latitudes (McPeters et al. 1996) and water vapor contents are higher in marine air (Randel et al. 1996). Thus the solar absorption by ozone and water vapor cannot explain the land–sea differences in the amplitude of S₁. On average, the earth’s surface absorbs about half of the incoming solar radiation at the top of the atmosphere (Kiehl and Trenberth 1997). Over land areas, the absorbed solar radiation at the ground creates large sensible heating of the surface air and generates large diurnal ranges of surface air temperature (DTR) (Fig. 17; see also Dai et al. 1999b). On the other hand, because of the unlimited water for evaporation, the large specific heat of water, and the turbulence mixing in surface oceans, solar heating of the ocean surface does not provide strong sensible heating of marine surface air, resulting in much smaller DTR over the oceans than over land areas (Fig. 17). Indeed, the geographic variations of S₁ (Fig. 3) correlate significantly (attained significance level p < 0.1%) with those of DTR (Fig. 17) (spatial correlation coefficient r = 0.61 for DJF, 0.67 for JJA, and 0.65 for annual mean). This is consistent with Mass et al. (1991), who found a correlation of 0.71 between summer diurnal amplitudes of surface pressure and DTR over the United States. Earlier studies also found stronger S₁ in clear sky and large DTR days than in cloudy and low DTR days in New York (Spar 1952b), but insignificant correlation between S₁ and DTR at a Bermuda maritime station (Haurwitz 1955).

Forced by the ozone and water vapor absorption of solar radiation, the classic tidal theory predicts a peak amplitude of 381 μb at the equator for annual S¹₁ (Braswell and Lindzen 1998), which is far below the observed (626 μb) (Table 1). Even with much weaker ground sensible heating than shown in Fig. 17, Tsuda and Kato (1989) showed through model simulations that large nonmigrating diurnal tides can be excited by the differential heating between land and ocean. Our results provide further evidence suggesting that sensible heat from the ground is a major forcing for both the migrating (S¹₁) and nonmigrating (S^s₁, s ≠ 1) diurnal tides. Including this forcing in theoretical calculations should substantially reduce the gap between the theory-predicted and observed diurnal tides.

Although dominated by the migrating mode S²₂, the semidiurnal tide S₂ exhibits considerable regional and zonal variations in its amplitude (cf. Fig. 8). For example, the amplitude of S₂ is smaller over the Atlantic and central Pacific than in other regions of the Tropics. Atmospheric water vapor content is comparable over the equatorial Pacific, Atlantic, and Indian Oceans and is much lower over Africa (Randel et al. 1996), while atmospheric total ozone concentration is fairly uniform over the Tropics (McPeters et al. 1996). Therefore, ozone and water vapor forcing appears unable to explain the variations of S₂ in the Tropics. Clouds, which can reflect (thus reducing the solar radiation reaching the lower troposphere where most of the water vapor absorption occurs) and absorb (thus providing additional forcing for S₂) solar radiation, are variable in space and could be a significant forcing for S₂. Precipitation has large diurnal variations (Dai et al. 1999a). The latent heating in convective precipitation is shown to be a significant forcing for S₂ in the Tropics (Lindzen 1978;Hamilton 1981). It could also induce additional zonal and regional variations in the amplitude of S₂. On the other hand, the sensible heating from the ground appears to have much smaller effects on S₂ than on S₁, as implied by the relatively small land–sea differences of S₂ (cf. Fig. 8). The signal of the influences by clouds and latent heating in S₂ is much weaker than that of the sensible heating in S₁. Therefore, a simple correlation between S₂ and cloudiness or convective precipitation is likely to result in weak relationships.

Figures 6 and 13 show that there is some asymmetry relative to the equator in the migrating tides. For example, the peak amplitude of S¹₁ is about 4 south of the equator. Furthermore, summer S¹₁ is substantially stronger than winter S¹₁ from about 20°N to 50°N. These features are not accounted for by the classic tidal theory with only ozone and water vapor heating (Braswell and Lindzen 1998).

6. Summary

Global surface pressure data of the 1976–97 period from over 7500 stations and COADS marine reports have been analyzed using harmonic and zonal harmonic methods. The spatial coverage of these data is greatly improved over previous similar analyses and is sufficient over most of the globe for capturing large-scale (>500 km) variations. However, the data are still sparse over the Antarctic, the Arctic, and the southern (south of about 50°S) oceans, where our results are less reliable.

We find that the diurnal pressure oscillation S₁ is comparable to the semidiurnal pressure oscillation S₂ in magnitude over much of the globe except for the low-latitude open oceans, where S₂ is about twice as strong as S₁. Over Northern Hemisphere land areas, S₁ is even stronger than S₂ during summer. Our results show that S₁ is more important than previously thought (e.g., Chapman and Lindzen 1970; Lindzen 1990).

Compared with previous studies (Haurwitz 1956; Haurwitz 1965; Haurwitz and Cowley 1973), we find that there is much more variation in S₁ and S₂ on regional and continental scales, and that the nonmigrating tides are considerably stronger than previously reported. Oscillation S₁ is greatly influenced by landmasses and is generally stronger over the land than over the ocean. The highest amplitudes (∼1.3 mb) of S₁ are found over northern South America and eastern Africa close to the equator. Here S₁ is also strong (∼1.1 mb) over high terrain such as the Rockies and the Tibetan Plateau. Over the open oceans, the amplitude of S₁ is about 0.3–0.6 mb. Over land areas, S₁ is strongest in local summer as the solar heating reaches a maximum, whereas its seasonal variation over the oceans is small. At low latitudes S₁ reaches maxima in the morning around 0600–0800 LST, while it peaks in the late morning around 1000–1200 LST at midlatitudes except for eastern Asia and the North Pacific where the maximum is around 0600–0800 LST.

The wavenumber 1 component S¹₁ of S₁ has a global mean amplitude of about 370 μb, compared with the next strongest wave (S²₁) of about 100 μb. The amplitude of S¹₁ decreases poleward rapidly at low latitudes, but its dependence on latitude cannot be adequately represented by simple polynomials of the cosine of latitude.

The largest amplitudes of S₂ (∼1.0–1.3 mb) are found in the Tropics over South America, the eastern and western Pacific, and the Indian Ocean. The amplitude of S₂ is about 0.3–0.5 mb over middle and high latitudes. Land–sea differences at a given latitude in the amplitude of S₂ are much smaller than for S₁. The seasonal variations of S₂ appear to be out of phase with S₁. Specifically, the maximum-amplitude zone of S₂ extends northward to about 20°N in DJF while S₂ is weaker over low latitudes and northern midlatitudes in JJA. Here S₂ peaks around 1000 and 2200 LST at low- and midlatitudes.

Oscillation S₂ is dominated by its migrating (wavenumber 2) mode S²₂, which has a global mean amplitude ranging from 523 (for JJA) to 566 (for DJF) μb, more than five times larger than the next strongest wave (∼100 μb). Here S²₂ peaks in the Tropics (about 958 μb for JJA and 1081 μb for DJF) and decreases rapidly poleward with a slower pace at mid- and high latitudes.

The amplitudes of high wavenumber components decrease gradually with increasing wavenumbers. We did not find any other wave components (except S¹₁ and S²₂) that are substantially stronger than their adjacent ones. Low gridding resolution of S₁ and S₂ tends to reduce the amplitudes at higher wavenumbers and enhance S¹₁ and S²₂.

The fact that the amplitude of S₁ and its seasonal variations are much larger over the land than over the ocean suggests that over land areas S₁ is greatly enhanced by the sensible heating from the ground. The spatial variations of the amplitude of S₁ correlate significantly with those in the diurnal temperature range at the surface, which is a proxy of the surface sensible heating. On the other hand, the land–sea differences of S₂ are relatively small, suggesting that the sensible heating from the ground has a much smaller influence on S₂. Although the amplitude of S₂ generally decreases poleward with largest gradients in the subtropics, there are considerable zonal variations that cannot be explained by zonal variations in atmospheric ozone and water vapor. Other forcings such as those through reflection and absorption of solar radiation by clouds and latent heating associated with convective precipitation are likely to be important for the zonal and regional variations in S₂. There is also some asymmetry relative to the equator in the migrating tides, especially S¹₁, which also implies additional forcings other than the solar heating by atmospheric ozone and water vapor.