Abstract

Although atmospheric tides driven by solar heating are readily detectable at the earth’s surface as variations in air pressure, their simulations in current coupled global climate models have not been fully examined. This work examines near-surface-pressure tides in climate models that contributed to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change (IPCC); it compares them with tides both from observations and from the Whole Atmosphere Community Climate Model (WACCM), which extends from the earth’s surface to the thermosphere. Surprising consistency is found among observations and all model simulations, despite variation of the altitudes of model upper boundaries from 32 to 76 km in the IPCC models and at 135 km for WACCM. These results are consistent with previous suggestions that placing a model’s upper boundary at low altitude leads to partly compensating errors—such as reducing the forcing of the tides by ozone heating, but also introducing spurious waves at the upper boundary, which propagate to the surface.

1. Introduction

Atmospheric tides are important features of middle- and upper-atmosphere structure and circulation. At the surface, the tides are significant parts of the day–night variations in both climate observations and simulations (Dai and Trenberth 2004; Woolnough et al. 2004). In the tropics—and in midlatitudes after baroclinic waves are removed from consideration—the primary observed day-to-night variation of surface pressure is a semidiurnal (twice a day) cycle despite the obvious diurnal (once a day) cycle of surface temperature. Resolution of this apparent paradox invokes the fact that atmospheric tides are mainly excited above the surface. Thus, wave propagation from the altitudes of tidal excitation down to the surface must be considered in evaluation of surface-pressure variations (Chapman and Lindzen 1970; Hagan et al. 2004).

Solar heating is the primary driver of atmospheric tides. Gravitationally driven atmospheric tides, analogous to ocean tides, are about 20 times weaker than thermally driven atmospheric tides (e.g., Fig. 2L.6 in Chapman and Lindzen 1970). The gravitational tides are distinguishable from thermal tides because they are primarily driven by the moon and thus appear in harmonics of a lunar day, ∼24.8 h. If solar heating is decomposed into Fourier components exp[i( + σΩt)], where λ is longitude, s is zonal wavenumber, t is time, Ω = 2π/(24 h), and σ is normalized frequency, then both diurnal (s = σ = 1) and semidiurnal (s = σ = 2) components migrating with the apparent motion of the sun across the sky are significant. Linear responses to these components of solar hearing are often referred to as “migrating tides.”

Even though most of the solar energy flux absorbed by the earth is deposited near the surface, peak atmospheric heating per unit mass at large scales occurs in the middle atmosphere via absorption of solar ultraviolet (UV) radiation by ozone. Most of the diurnal harmonic of ozone heating goes into a vertically trapped tidal mode (Chapman and Lindzen 1970; Forbes 1995). Diurnal modes that propagate down to the surface do so with vertical wavelengths <30 km, allowing destructive interference in the forcing layer. However, the main semidiurnal-tide mode [Hough function (2, 2)] propagates vertically with wavelength greater than 300 km (e.g., Table 1 in Forbes 1995). Thus, the semidiurnal tide is primarily excited in a broad range of altitudes around 50 km by ozone heating and effectively propagates to the surface. This explanation of the semidiurnal cycle of surface pressure came long after it was observed and only after the thermal structure and wave dynamics of the middle atmosphere were understood (Chapman and Lindzen 1970).

Given this explanation, one would expect to find surface-pressure tides in the output of any atmospheric general circulation model (GCM) that includes the ozone layer. Tides were indeed found by Hunt and Manabe (1968) in a very early GCM with enhanced vertical resolution and domain (18 vertical levels extending to 38-km altitude, comparable to modern climate-oriented GCMs). At the same time, however, Lindzen et al. (1968) pointed out that spurious resonances will occur in a model atmosphere with an artificial “top.” Much later, Zwiers and Hamilton (1986) examined their GCM’s output for tides and concluded that compensating errors were at work: tidal amplitude was diminished by the model’s omission of much of the ozone heating but enhanced by the model’s rigid-lid upper boundary condition, which introduced spurious reflected waves that propagated down to the surface. The result was a fortuitously accurate tide simulation. Recently Hamilton et al. (2008) examined output from the Atmospheric Model for the Earth Simulator (with its top at about 60-km altitude) and concluded that “simulated amplitudes of the semidiurnal oscillation [of surface pressure] have an overall enhancement of 25% over those observed, a deficiency which is reasonably attributable to the effects of the upper boundary condition in the model.”

This study examines the surface-pressure signature of atmospheric tides in GCMs that contributed output to the Fourth Assessment Report (AR4) of the Intergovernmental Panel on Climate Change (IPCC; Solomon et al. 2007). We confine our attention to near-surface pressure because this field is well observed and because the primary database of IPCC AR4 model output (Meehl et al. 2007) contains only near-surface fields in its high-time-frequency output.

The IPCC coupled models were developed for climate change simulations. Diurnal and semidiurnal variations in individual GCMs have been analyzed previously (e.g., Zwiers and Hamilton 1986; Lieberman et al. 1994; Dai and Trenberth 2004; Chang et al. 2008; Hamilton et al. 2008) but a systematic comparison of surface-pressure tides in the IPCC AR4 models has not been done. Although the tides are a linear response to well-known forcing, the limited vertical domain of models may introduce errors in tidal simulations, as noted above. Also—as discussed by Dai and Trenberth (2004) and in section 2 below—the design of most coupled ocean–atmosphere climate models, as well as atmospheric GCMs run with prescribed sea surface temperatures (SSTs) as a lower boundary condition, leads to underpredicted or nonexistent diurnal variations in their SSTs, which can result in reduced diurnal cycles in surface air temperature and pressure over the oceans in these models.

We also examine the tides in output from the Whole Atmosphere Community Climate Model (WACCM). Both the original version, without interactive atmospheric chemistry (WACCM-1; Sassi et al. 2002), and the most recent version, including chemistry (WACCM-3; Richter et al. 2008), are considered. Although it is a full atmospheric GCM including the surface and troposphere, WACCM was developed for studying the middle and upper atmosphere. At these altitudes, tidal oscillations are prominent in the wind and temperature fields from both WACCM and observations (Chang et al. 2008). However, WACCM’s surface output has not been previously examined for the tides.

2. Data sources

The most recent analysis of global observations of surface-pressure tides (Dai and Wang 1999) is used in this study. This analysis included data from weather stations, commercial ships, and ocean buoys during the period 1976–97. High-time-frequency (3-hourly) output from climate models was obtained from readily available sources. WACCM-1 output was downloaded from the National Center for Atmospheric Research (NCAR) community data portal (http://cdp.ucar.edu). WACCM-3 output was provided by the WACCM Working Group of the Community Climate System Model (CCSM) consortium; membership in this and other CCSM Working Groups is open to all interested scientists (see http://www.ccsm.ucar.edu/working_groups/WACCM). IPCC AR4 model output was downloaded from archives of the Coupled Model Intercomparison Project, phase 3 (CMIP3; Meehl et al. 2007). Diurnal and semidiurnal harmonics of the 3-hourly model output were obtained directly by fast Fourier transforms (FFTs) of the time series at each grid point. No interpolation of output from the original model grids was done unless required for statistics comparing different model results and observations. This procedure differs somewhat from the method used by Dai and Wang (1999), who averaged pressure data from 1976 to 1997 at each location to form a composite day–night cycle before fitting diurnal and semidiurnal curves to the 3-hourly composite diurnal data.

Table 1 lists full names and abbreviations and key features of the models examined in this study. All 11 IPCC models that contributed 3-hourly data to CMIP3 are included, except for two models whose 3-hourly output in the database was problematic (e.g., only recorded once every 6 h). Three of the nine models we examined had a simple error in their recorded 3-hourly time coordinate values. After consulting the model developers, we added 1.5 h to both Goddard Institute for Space Studies Models E-H (GISS-EH) and E-R (GISS-ER) times, and we subtracted 4.5 h from the Institute of Numerical Mathematics Coupled Model, version 3.0 (INM-CM3.0), times to correct these errors. Table 1 shows that the IPCC models have top boundaries located between 32- and 76-km altitude, and they place 1–9 levels in the ozone-heating layer. Both WACCM versions have a top at 135-km altitude and place 23 levels in the ozone-heating layer. We define “the ozone-heating layer” as the 30–70-km altitude range, where most of the heating per unit mass occurs in the real world (e.g., Fig. 3.2 in Chapman and Lindzen 1970; it produces the well-known temperature maximum at the stratopause, ∼50-km altitude). As discussed below, ozone heating in the models may occur at lower levels.

Table 1.

Model characteristics and tidal near-surface pressure variations near “Batavia” (now Jakarta; 14°S–2°N, 97.5°–112.5°E).

Model characteristics and tidal near-surface pressure variations near “Batavia” (now Jakarta; 14°S–2°N, 97.5°–112.5°E).
Model characteristics and tidal near-surface pressure variations near “Batavia” (now Jakarta; 14°S–2°N, 97.5°–112.5°E).

All of the IPCC models were driven by incoming solar energy flux at the top of the atmosphere together with observed greenhouse gas and aerosol concentrations for the late twentieth century; they calculated SST and sea ice interactively as part of a coupled ocean–atmosphere GCM. Standard practice in these simulations is to couple the ocean and atmosphere once a day using daily mean values (e.g., Washington et al. 2009), thereby suppressing diurnal variations in SST. Six of the nine models were also driven by observed late-twentieth-century monthly mean SSTs (again without diurnal variations) and sea ice concentration fields imposed as a lower boundary condition; these numerical experiments are named after the Atmosphere Model Intercomparison Project (AMIP; Gates et al. 1999) and here are called AMIP runs. WACCM-3 simulations are AMIP runs, and WACCM-1 simulations are AMIP-like runs forced by a climatological mean seasonal cycle of sea surface temperature (without diurnal variations) and sea ice for the late twentieth century.

CMIP requested that IPCC models provide AMIP 3-hourly output for the year 2000 and coupled ocean–atmosphere 3-hourly output for years 1991–2000 inclusive. INM-CM3.0 however, provided coupled ocean–atmosphere 3-hourly output (as well as AMIP output) only for the year 2000. WACCM-3 provided output for years 1990–2000 inclusive. Therefore, in order to maximize consistency among the different model output datasets, we focus on year 2000 from all of the IPCC models and from WACCM-3. For these models, the results shown below pertain to year 2000 unless noted otherwise. For WACCM-1, results are from an arbitrary “year 1” since this model’s output does not match any particular calendar years. Thus, our model output pertains to a narrower and somewhat later interval of time (only January and July 2000) than do the observations from Dai and Wang (1999), which cover years 1976–97. Interannual variations of the surface-pressure tides are relatively small, however, in contrast to the large interannual variations of the tides in the middle atmosphere (see the appendix).

With the exception of initial screening results shown in Table 1, we applied all FFTs on 32 days of model output (with 24 h/3 h = 8 time points per day) for the months of January or July, with 1-day extensions into adjacent months to produce a number of time points equal to an exact power of 2, thereby saving computer time. Thus, our model-simulated output approximately pertains to January or July, whereas the observed data is compiled for full seasons [i.e., December–February (DJF) and June–August (JJA)].

A far more significant inconsistency arises from a limitation of the CMIP3 database. It contains sea level pressure (PSL) but not surface pressure (PS) in its 3-hourly output, whereas the observations are derived from PS, the more fundamental quantity. In theory a simple formula interconverts PS and PSL (e.g., section 3.2.4 in Wallace and Hobbs 2006) but in practice each model applies its own idiosyncratic procedure, using data that is not generally saved. Therefore it is not possible to convert PSL back to PS in the 3-hourly CMIP3 model output. Both PS and PSL output are available from WACCM-1 at 3-hourly frequency, however. We use this output to assess the magnitude of the problem and to investigate a partial solution.

Figure 1 compares January PS-based tide amplitudes from WACCM-1, the corresponding PSL-based amplitudes, and the same PSL-based amplitudes with areas greater than 1 km in elevation omitted. Table 2 presents associated comparison statistics in both January and July. Aliasing problems are most apparent for the diurnal harmonic, probably because of the strong diurnal cycle of land surface temperature affecting conversion of PS to PSL. For the diurnal tide (Fig. 1a), switching from PS to PSL boosts amplitudes by up to a factor of 6 over land, particularly at high elevations where the concept of equivalent sea level pressure is most dubious. Application of the <1-km constraint removes nearly all of the egregiously high (>360 Pa) PSL-based amplitudes and reduces the global mean overestimate from about a factor of 2 down to 20%–30%. Nevertheless, many land areas remain where PSL-based amplitudes are several times higher than their PS-based counterparts, and the root-mean-square (rms) difference between PS- and PSL-based amplitudes, while reduced from about 95 to 30–40 Pa, is still comparable to the global mean PS-based amplitude (39 Pa for January). For the semidiurnal tide (Fig. 1b), the <1-km constraint removes essentially all of the problematic areas. Under this constraint, statistics comparing PS- with PSL-based semidiurnal amplitudes show global means within 5% of each other, correlations greater than 0.99, and rms differences less than 10% of the global mean.

Fig. 1.

Amplitudes (Pa) of (a) January diurnal harmonics and (b) January semidiurnal harmonics of PS or PSL variations from WACCM-1. Shown for each harmonic are (top) PS variations, (middle) PSL variations, and (bottom) sea level variations with overprinted gray shading for altitudes greater than 1 km. White areas indicate off-scale values (i.e., >360 Pa).

Fig. 1.

Amplitudes (Pa) of (a) January diurnal harmonics and (b) January semidiurnal harmonics of PS or PSL variations from WACCM-1. Shown for each harmonic are (top) PS variations, (middle) PSL variations, and (bottom) sea level variations with overprinted gray shading for altitudes greater than 1 km. White areas indicate off-scale values (i.e., >360 Pa).

Table 2.

Comparison of January and July PS vs PSL tide amplitude maps simulated by WACCM-1. “Unmasked” means that comparison is made over the entire surface area of the earth. “Masked” means that land areas with elevation greater than 1 km are excluded from the comparison.

Comparison of January and July PS vs PSL tide amplitude maps simulated by WACCM-1. “Unmasked” means that comparison is made over the entire surface area of the earth. “Masked” means that land areas with elevation greater than 1 km are excluded from the comparison.
Comparison of January and July PS vs PSL tide amplitude maps simulated by WACCM-1. “Unmasked” means that comparison is made over the entire surface area of the earth. “Masked” means that land areas with elevation greater than 1 km are excluded from the comparison.

In short, for the semidiurnal tide, removing areas less than 1 km in elevation from consideration may remove most of the PS versus PSL inconsistency when comparing IPCC model simulations with observations, while retaining substantial land as well as ocean areas in the comparison. For the diurnal tide, however, this constraint fails to eliminate a number of problematic land points, and the global mean discrepancy is not reduced below 20%–30%. A more severe constraint (perhaps eliminating all land areas from consideration) is needed for accurate comparison of diurnal tide simulations with observations. On the other hand, we wish to make at least some inspection of the model-simulated diurnal tide over land. Also, for WACCM output the PS-based tides are available and there is no obvious reason to impose any geographic constraint. As a simple compromise of these conflicting interests, we place the <1-km constraint on all IPCC PSL-based results below, while for WACCM output we include the full geographic coverage of PS-based results. In the discussion section (section 4) we also present PSL-based WACCM results with the <1-km constraint imposed, allowing a consistent comparison of WACCM’s accuracy with the IPCC models’ accuracy.

3. Results

a. Tidal amplitude

In addition to model characteristics, Table 1 shows results of our first screening test of model output. We took an area average in the vicinity of Jakarta (formerly Batavia) in Indonesia from both the models and observations. The classic observed signature of the tides—shown, for example, in Fig. 1.1 of Chapman and Lindzen (1970)—is a semidiurnal variation of surface pressure at this location. Chapman and Lindzen’s figure shows data taken at Batavia during the first half of November 1919. The amplitude of surface-pressure variation is ∼100 Pa and the phase has maxima at about 1000 and 2200 local solar time (LST). We examined the corresponding half-month in the models (taken from “year 1” in WACCM-3, year 2000 in INM-CM3.0, and year 1991 in all other models) and the September–November (SON) season in the modern observations (Dai and Wang 1999). These observations confirm that the 1919 Batavia data are representative of the tropics in all seasons (see figures discussed below). The modern observations, however, give a smaller semidiurnal harmonic amplitude than older observations do for the full region considered in Table 1 (Haurwitz 1956; see Fig. 2S.3 in Chapman and Lindzen 1970). The smaller amplitude arises from inclusion of more stations in this region.

Table 1 shows that most of the models obtain the correct diurnal harmonic amplitude at Jakarta to within ±20%, but all of them overestimate the amplitude of the semidiurnal harmonic, some by nearly a factor of 2. Thus, the models agree with observations that the semidiurnal harmonic dominates surface-pressure variations, but they overestimate its dominance. The models generally put the maximum near-surface pressure earlier than observed for the diurnal harmonic, but later than observed (by ∼1 h) for the semidiurnal harmonic. (A phase error in the semidiurnal harmonic appears with opposite sign in “classical” linear calculations, probably because they neglect release of latent heat in the tropical atmosphere; Lindzen 1978). Also, results from the coupled ocean–atmosphere twentieth-century experiments exhibit close agreement with results from the AMIP experiments in Table 1. Thus, the surface-pressure tides are not sensitive to SST errors in the coupled ocean–atmosphere simulations or to interannual SST variations. This conclusion is consistent with the classical theory of the migrating tides, although nonmigrating components include sea breezes and low-level jets that are sensitive to surface temperature distributions. Finally, the table shows that for Jakarta—which lies near sea level—there is essentially no difference between amplitudes and phases derived from PS and PSL, as expected.

Figures 2 and 3 map diurnal and semidiurnal tide amplitudes, respectively, from two of the IPCC models, from WACCM-3, and from the observations. To save space, only data for the month of January are shown in the figures. Table 3 presents global statistics for both January and July comparing the same two IPCC models and both WACCM versions with the observations. The two IPCC models selected are INM-CM3.0, which has the lowest model top (32 km) and Centre National de Recherches Météorologiques Coupled Global Climate Model, version 3 (CNRM-CM3), which has the highest model top (76 km). Tide amplitude maps from the other IPCC models (not shown here) have similar appearance. Later sections of this paper will present results from all nine of the IPCC models that provided appropriate 3-hourly output.

Fig. 2.

Amplitudes (Pa) of diurnal harmonics of (top) January PSL variations from two IPCC climate models, (bottom left) January PS variations from WACCM-3, and (bottom right) DJF PS variations from observations. As in Fig. 1, gray shading indicates altitudes greater than1 km for the PSL data; we excluded these areas from further analysis of the IPCC models.

Fig. 2.

Amplitudes (Pa) of diurnal harmonics of (top) January PSL variations from two IPCC climate models, (bottom left) January PS variations from WACCM-3, and (bottom right) DJF PS variations from observations. As in Fig. 1, gray shading indicates altitudes greater than1 km for the PSL data; we excluded these areas from further analysis of the IPCC models.

Fig. 3.

As in Fig. 2, but for the semidiurnal harmonics; note different color scale.

Fig. 3.

As in Fig. 2, but for the semidiurnal harmonics; note different color scale.

Table 3.

Comparison of observed PS tide amplitude with model-simulated tide amplitude from WACCM-1 and WACCM-3 (PS output, unmasked) and from selected CMIP3/IPCC AR4 models (PSL output, masked). “Masked” and “unmasked” are as defined in Table 2.

Comparison of observed PS tide amplitude with model-simulated tide amplitude from WACCM-1 and WACCM-3 (PS output, unmasked) and from selected CMIP3/IPCC AR4 models (PSL output, masked). “Masked” and “unmasked” are as defined in Table 2.
Comparison of observed PS tide amplitude with model-simulated tide amplitude from WACCM-1 and WACCM-3 (PS output, unmasked) and from selected CMIP3/IPCC AR4 models (PSL output, masked). “Masked” and “unmasked” are as defined in Table 2.

Comparison of the WACCM simulations with observations is straightforward because both are available in terms of surface pressure. The resulting statistics are quite similar between the two WACCM versions. From this point of view the most noticeable discrepancy between WACCM and observations is that both model versions underestimate the diurnal tide amplitude over large areas, particularly over the ocean (top panel of Fig. 1a and bottom two panels of Fig. 2) so that the global mean of model/observed amplitude ratio lies in the range 0.6–0.7 (Table 3). As noted in the previous section, standard methods of running most climate models including WACCM underestimate diurnal SST variations. WACCM’s semidiurnal tide amplitude, on the other hand, compares with observations more favorably. The global mean of model/observed semidiurnal amplitude ratio lies in the range 0.9–1 while the simulated global mean (77 Pa) is slightly larger than the observed global mean (73 Pa). The amplitude pattern correlation with observations is 0.6–0.7 for the diurnal harmonic and is greater than 0.9 for the semidiurnal harmonic. The rms difference between WACCM and observed amplitudes is an appreciable fraction of the global mean (∼27 Pa/37 Pa) for the diurnal harmonic and a much smaller fraction of the global mean (∼20 Pa/77 Pa) for the semidiurnal harmonic.

Examination of the top panel of Fig. 1a and the bottom two panels of Fig. 2 also shows that WACCM-1’s excessive values of diurnal amplitude over tropical continents are removed in WACCM-3. Since the diurnal tide over land has a large influence from surface heating, this improvement may be due to changes in the lower-atmosphere climate physics (from the Community Climate Model version 3 in WACCM-1 to the Community Atmosphere Model version 3 in WACCM-3).

For the IPCC models, comparison with observations is not straightforward because—as noted above—the CMIP3 database contains PSL but not PS in its 3-hourly output, whereas the observations are derived only from PS; therefore we consider only areas <1 km in elevation in the IPCC model output. Under this constraint Table 3 shows that for the diurnal harmonic amplitude CNRM-CM3 and INM-CM3.0 attain correlations with observed values that are generally higher than WACCM’s, which do not suffer from the problem of comparing PSL with PS but do suffer the handicap of comparing at all points on the earth’s surface including high elevations. The global means of model/observed diurnal amplitude ratios lie in the range 0.8–1.0 for CNRM-CM3 and INM-CM3.0, higher than for WACCM, but Fig. 2 shows that the two IPCC models achieve this “improvement” by overestimating diurnal tide amplitude in the interiors of tropical continents. The problem is worse for INM-CM3.0, the model with the lowest top (up to a factor of 3 overestimate) than it is for CNRM-CM3.0, the IPCC model with the highest top (up to a factor of 2 overestimate). Much of this discrepancy may be explained by our comparison of PSL-based model output with PS-based observations, however (see Fig. 1 and Table 2). The diurnal amplitudes from the two selected IPCC models are generally underestimated over oceans, as in WACCM’s results.

Turning to the semidiurnal harmonic, the previous section noted that the <1-km elevation constraint removes many of the problematic aspects of comparing PSL-based tide simulations with PS-based observations, and amplitude maps from INM-CM3.0 and CNRM-CM3 show better agreement with both WACCM and observations (Fig. 3). Comparison statistics (Table 3) show global mean IPCC model/observed amplitude ratios in the range 0.91–0.99 and correlation values of ∼0.9, about the same as for WACCM-1 and WACCM-3. The rms differences between the two IPCC models and observations lie in the range 19–24 Pa, compared with 19–22 Pa for WACCM-1 and WACCM-3. The maximum value of semidiurnal amplitude, however, is ∼180–190 Pa for the two IPCC models, compared with 160 Pa for WACCM-3, so WACCM’s maximum is closer to the observed value (140 Pa).

b. Tidal phase

Figures 4 and 5 show times of maximum pressure for the diurnal and semidiurnal harmonics, respectively, in the same models and month (January) as in Figs. 2 and 3. Figures 4 and 5 show that in the tropics, the observed times of maximum are about 0600 LST for the diurnal harmonic and 1000/2200 LST for the semidiurnal harmonic; at higher latitudes the phases are unclear. Simulated diurnal phases from both IPCC models and WACCM-3 are roughly consistent with the observed diurnal phase in the tropics, where amplitude is greatest, but the model-simulated phase varies more smoothly from one location to another than the observed phase. Agreement between the models and observations is more evident for the phase of the semidiurnal tide (Fig. 5), although the observations again exhibit more spatial variation at higher latitudes than the models. It is difficult to determine how much of this discrepancy is due to errors in the models, as opposed to noise in the observations. An unrealistically smooth distribution would suggest underrepresentation of the nonmigrating tides in the GCM calculations. Extracting the migrating components of the tides allows a more precise quantification of their phase, as shown below.

Fig. 4.

Local solar times of maximum near-surface pressure in the diurnal harmonics of (top) January PSL variations from two IPCC climate models, (bottom left) January PS variations from WACCM-3, and (bottom right) DJF PS variations from observations. The clock face dial at the bottom gives the times.

Fig. 4.

Local solar times of maximum near-surface pressure in the diurnal harmonics of (top) January PSL variations from two IPCC climate models, (bottom left) January PS variations from WACCM-3, and (bottom right) DJF PS variations from observations. The clock face dial at the bottom gives the times.

Fig. 5.

As in Fig. 4, but for the semidiurnal harmonics.

Fig. 5.

As in Fig. 4, but for the semidiurnal harmonics.

c. Zonal wavenumber decomposition

Classical linear calculations of the migrating (sun-synchronous) components of the tides provide a good approximation to the observed surface-pressure variations, as well as physical insights not easily obtained from GCMs (Chapman and Lindzen 1970). Therefore, it is worthwhile to isolate the migrating components from the GCM output. The first step in this process is Fourier analysis in the longitude dimension, producing tidal amplitude as a function of latitude and zonal wavenumber s from tidal amplitude as a function of latitude and longitude.

Figures 6 and 7 show results for the diurnal (σ = 1) and semidiurnal (σ = 2) harmonics, respectively. These figures display the same models and month (January) as in Figs. 2 –5. The most obvious feature of Figs. 6 and 7 is the concentration of tidal amplitude in the migrating component, s = σ. This concentration of amplitude is more exaggerated in the models than in the observations. For example, WACCM-3 produces a migrating diurnal tide at the equator that is stronger than its observed counterpart (Fig. 6), even though the overall global mean amplitude of WACCM-3’s diurnal harmonic is less than observed (Fig. 2). Figure 7 indicates that the models overestimate the migrating component of the semidiurnal tide by an even larger factor (about a factor of 2). In addition, Fig. 7 shows a small but distinct stationary (s = 0) semidiurnal tide with multiple peaks in latitude in the model simulations but not in the observations.

Fig. 6.

Fourier-longitude amplitudes (Pa) vs zonal wavenumber and latitude for the diurnal harmonics of (top) January PSL variations from two IPCC climate models, (bottom left) January PS variations from WACCM-3, and (bottom right) DJF PS variations from observations.

Fig. 6.

Fourier-longitude amplitudes (Pa) vs zonal wavenumber and latitude for the diurnal harmonics of (top) January PSL variations from two IPCC climate models, (bottom left) January PS variations from WACCM-3, and (bottom right) DJF PS variations from observations.

Fig. 7.

As in Fig. 6, but for the semidiurnal harmonics.

Fig. 7.

As in Fig. 6, but for the semidiurnal harmonics.

Figure 8 shows diurnal and semidiurnal tidal amplitudes for both January and July as a function of s at the equator from all IPCC models providing appropriate output, from WACCM-3, and from observations. The figure shows that many conclusions drawn from previous figures for INM-CM3.0 and CNRM-CM3 generalize to all the other IPCC models. All models and observations qualitatively agree that the semidiurnal harmonic amplitude is larger than the diurnal harmonic, and that the migrating (s = σ) component dominates both harmonics. Quantitatively, however, the migrating component simulated by every model including WACCM-3 has larger amplitude than observed. WACCM-3’s amplitude is consistently at the lower end—that is, the more accurate end—of the range of model-simulated values. This range varies between about 20% and 100% greater than observed for the diurnal harmonic and between about 70% and 150% greater than observed for the semidiurnal harmonic. As noted above, however, much of the overestimate of diurnal tide amplitude by the IPCC models in this study is due to data limitations, which leads to an inconsistent comparison of PS with PSL. Also, observational uncertainty might reasonably be put at 10%–20%, in which case WACCM and several of the IPCC models would lie within the observational “error bars.”

Fig. 8.

(a) January and (b) July Fourier longitude amplitude at the equator vs zonal wavenumber for (top) diurnal and (bottom) semidiurnal harmonics of all IPCC models providing data, WACCM-3, and observations. As in previous figures, WACCM output and observational data are PS, available IPCC model output is PSL, and IPCC model output for altitudes greater than 1 km was excluded from analysis. Blue-shaded insets show details of peaks in the vicinity of the migrating Fourier-component wavenumbers (red vertical lines).

Fig. 8.

(a) January and (b) July Fourier longitude amplitude at the equator vs zonal wavenumber for (top) diurnal and (bottom) semidiurnal harmonics of all IPCC models providing data, WACCM-3, and observations. As in previous figures, WACCM output and observational data are PS, available IPCC model output is PSL, and IPCC model output for altitudes greater than 1 km was excluded from analysis. Blue-shaded insets show details of peaks in the vicinity of the migrating Fourier-component wavenumbers (red vertical lines).

A second implication of Fig. 8 is that there is little difference between January and July in the equatorial tidal amplitudes, for either the models or the observations.

Figure 9 shows, for the migrating (s = σ) tide components only, phase as a function of latitude. As in Fig. 8, the diurnal and semidiurnal harmonics for both January and July are shown from all available IPCC model output and from WACCM-3. Observations are not shown, however. A more extensive analysis of observations than hitherto published would be necessary to say which of the models provides the most accurate phase in Fig. 9. Recall, however, that the bottom-right panels of Figs. 4 and 5 display the observed local solar times of maximum surface pressure for all zonal wavenumbers combined, which Fig. 8 shows is dominated by the migrating component. Figures 4 and 5 show that in the tropics the observed times of maximum are about 0600 LST for the diurnal harmonic and 1000/2200 LST for the semidiurnal harmonic. Figure 9 shows that diurnal migrating components from all of the models exhibit maximum pressures between 0400 and 0800 LST equatorward of ±30° latitude. Poleward of these latitudes, especially in the winter hemisphere, the phase of the weak diurnal tide is not well defined. The stronger semidiurnal tide exhibits well-defined phases up to at least ±40° latitude in all of the models. Equatorward of these latitudes, the times of semidiurnal maxima lie between about 0930 and 1100/2130 and 2300 LST for all models. Thus, the model-simulated phases are roughly consistent with observations. Note, however, that the differences between different models—up to about 4 h for the diurnal harmonic and 1.5 h for the semidiurnal harmonic—amount to 60° and 45° variations in phase, respectively, so at least some of the models are exhibiting significant phase errors.

Fig. 9.

January and July migrating tide phase vs latitude for (top) diurnal and (bottom) semidiurnal harmonics of all IPCC models providing data and WACCM-3. As in previous figures, WACCM output and observational data are PS, available IPCC model output is PSL, and IPCC model output for altitudes greater than 1 km was excluded from analysis.

Fig. 9.

January and July migrating tide phase vs latitude for (top) diurnal and (bottom) semidiurnal harmonics of all IPCC models providing data and WACCM-3. As in previous figures, WACCM output and observational data are PS, available IPCC model output is PSL, and IPCC model output for altitudes greater than 1 km was excluded from analysis.

One possibly noteworthy feature of the semidiurnal migrating tide in two of the models—the Geophysical Fluid Dynamics Laboratory Climate Model version 2.1 (GFDL-CM2.1) and the Model for Interdisciplinary Research on Climate 3.2, medium-resolution version [MIROC3.2 (medres)]—is a systematic decrease in the time of maximum pressure from the summer hemisphere to the winter hemisphere. Because this feature is determined by the high-latitude behavior of the tides, where the signal is irregular and weak, it seems unlikely that observations can assess its validity.

4. Discussion

Unlike the IPCC climate models, WACCM was developed for middle- and upper-atmospheric research. At these altitudes the atmospheric diurnal tide attains large amplitude (Pirscher et al. 2010) and becomes a significant part of atmospheric motion and temperature variability; here WACCM produces simulations in general agreement with past tidal studies (Chang et al. 2008). WACCM also includes a complete lower atmosphere underneath its middle and upper atmosphere. This study shows that WACCM reproduces the major observed features of the surface-pressure signature of the tides. For the semidiurnal tide—the dominant harmonic in both simulations and observations of surface pressure—WACCM’s global mean amplitude is within ∼10% of observed, its pattern correlation with observations exceeds 0.9, and its rms difference from observations is about 30% of the observed global mean and 15% of the observed maximum value (Table 3; Fig. 3).

Most of the IPCC models exclude at least part of the ozone-absorption altitude range (∼30–70 km) that is thought to be primarily responsible for driving the surface-pressure tide. This does not mean that ozone heating is totally missed by the models, since there are abundant ozone molecules at altitudes less than 30 km to absorb the same UV radiation. Thus, models with low tops could produce a total column-integrated ozone heating in watts per meter squared similar to that in the real world, albeit at lower altitudes. But the same heating in watts per meter squared deposited at lower altitudes implies less heating per unit mass: if the heating occurs where air density is ρH, then the wave amplitude (perturbation temperature, wind velocity, etc.) at this location is proportional to ρH−1. The wave amplitude at the surface is changed by a factor (ρH/ρS)1/2, where ρS is air density at the surface. Multiplying this factor by ρH−1 implies that the wave amplitude at the surface varies as ρH−1/2, so depositing the same heating in watts per meter squared at a lower level (where ρH is larger) diminishes the surface response. This scaling argument is qualitatively consistent with the results of Zwiers and Hamilton (1986). Their GCM placed its top boundary at 10 mb, the same pressure level as the lowest top among the models studied here (INM-CM3.0). As noted by Hamilton et al. (2008), Zwiers and Hamilton’s separate linear calculations imply that wave reflection at their GCM’s top enhanced the semidiurnal tide amplitude by ∼30%, largely canceling a 40% amplitude reduction due to altered ozone heating.

A similar cancelation of errors in low-top models may lead to comparable tide simulations across a wide range of vertical domains. In the present study the PS versus PSL inconsistency between IPCC model output and observations is an important caveat, but for the dominant semidiurnal harmonic, the inconsistency is largely remedied by excluding areas greater than 1 km in elevation (Fig. 1b and Table 2). Under this constraint, both the amplitude maps (Fig. 3) and the comparison statistics (Table 3) imply that the two selected IPCC models (with tops at 32 and 76 km) provide semidiurnal tide simulations comparable in quality to WACCM (with top at 135 km), though not quite as good. The model-observed pattern correlations are all ∼0.9, but the maps show that the two IPCC models make a more severe overestimate than WACCM of the semidiurnal tide maximum near the equator. The semidiurnal phase from the two IPCC models is a fair match to WACCM’s (and to observations equatorward of ±30° latitude; see Fig. 5).

To summarize the surface-pressure signature of atmospheric tides in WACCM-1, WACCM-3, and the IPCC models, Fig. 10 plots the three statistical metrics of model-observed agreement appearing in Table 3 against the two model characteristics appearing in Table 1 (altitude of model top; number of levels with significant ozone absorption of solar radiation in the real world). Thus the figure consists of six frames. In each frame a model-simulated tide harmonic is represented by a single point. WACCM points appear in a single column on the extreme right of each frame, indicating a much higher model top and a much better resolution of ozone absorption than any of the IPCC models. Most points (filled circles) compare PSL-based model tides with PS-based observed tides excluding areas greater than 1 km in elevation. The remaining points (asterisks) show a global comparison of PS-based model tides with PS-based observed tides, which are available only from the two WACCM versions. For WACCM-1, which provided both PS and PSL output at high time frequency, we also show the comparison of PSL-based model tides with PS-based observed tides with areas greater than 1 km in elevation excluded (filled circles in the rightmost columns). These points provide a consistent comparison of WACCM’s accuracy with the IPCC models’ accuracy.

Fig. 10.

Global statistical comparison of model-simulated vs observed tide amplitudes. Each point represents one model-simulated tide harmonic: semidiurnal for January (blue) and July (red) and diurnal for January (brown) and July (green). Filled circles represent comparison of model-simulated PSL tides vs observed PS tides, with elevations greater than 1 km omitted from the comparison. Asterisks represent comparison of model-simulated PS tides (available only from WACCM) vs observed PS tides with the comparison including all the earth’s surface. Three statistics are shown: (top) globally averaged ratio of simulated/observed point values, (middle) spatial correlation of simulated and observed, and (bottom) rms simulated − observed difference. The statistics are plotted as a function of (left) the altitude of each model’s top level and (right) the number of model levels with significant solar radiation absorption by ozone (i.e., in the 30–70-km-altitude range).

Fig. 10.

Global statistical comparison of model-simulated vs observed tide amplitudes. Each point represents one model-simulated tide harmonic: semidiurnal for January (blue) and July (red) and diurnal for January (brown) and July (green). Filled circles represent comparison of model-simulated PSL tides vs observed PS tides, with elevations greater than 1 km omitted from the comparison. Asterisks represent comparison of model-simulated PS tides (available only from WACCM) vs observed PS tides with the comparison including all the earth’s surface. Three statistics are shown: (top) globally averaged ratio of simulated/observed point values, (middle) spatial correlation of simulated and observed, and (bottom) rms simulated − observed difference. The statistics are plotted as a function of (left) the altitude of each model’s top level and (right) the number of model levels with significant solar radiation absorption by ozone (i.e., in the 30–70-km-altitude range).

The most striking feature of Fig. 10 is the absence of any obvious connection between simulation quality, as measured by our comparison statistics, and either 1) the vertical extent of the models or 2) vertical resolution in the altitude range 30–70 km. This result is consistent with the “canceling errors” interpretation of model behavior discussed above. The issue is complicated by the possibility that some GCMs omitting the altitude range where ozone heating occurs in the real world may absorb solar UV radiation lower down, while others may remove the radiation before it reaches the top of the model. Unfortunately the CMIP3/IPCC AR4 database of climate model output does not include heating rates, so there is no simple way to intercompare ozone heating among the IPCC models. Also, modern GCMs generally include parameterizations that reduce wave reflection at the model top. The standard “sponge layer” with a constant time scale for wave damping, however, is less effective for tides than for planetary waves because the latter have longer periods.

In contrast, WACCM with its top at 135 km includes all tidal forcing but produces semidiurnal amplitudes generally lower (i.e., in better accord with observations) than the IPCC models. Evidently any spurious enhancement of the tides by wave reflection at the upper boundary is much reduced in WACCM. Indeed, Fig. 10 shows that the globally averaged ratio of simulated to observed tidal amplitude is about 0.8–1.0 in WACCM but greater than 1 in most of the IPCC models when comparing simulations and observations consistently (i.e., PSL-based simulations versus PS-based observations; filled circles). Figure 10 also shows that the model-observed pattern correlation of tide amplitudes is ∼0.85–0.95 for the semidiurnal harmonic and ∼0.65–0.75 for the diurnal harmonic. This result occurs across the full range of model-top altitudes from 32 km (INM-CM3.0) to 135 km (WACCM).

5. Conclusions

This study documents a comparable quality of surface-pressure tide simulations among nine very different climate models—many of which exclude the altitude range that is most important to real-world tide forcing—and a “whole atmosphere” model that was designed to include the tides. Our results suggest errors that partly cancel in some of the models, at least in terms of the metrics we have chosen. This implication is consistent with previous studies that span the history of GCM simulation (Lindzen et al. 1968, Zwiers and Hamilton 1986, Hamilton et al. 2008) but generally have not influenced the development of climate-oriented GCMs. As such models extend their purview above the troposphere (a necessary condition for full assessment of solar effects on climate), the tides along with other important features of the middle atmosphere must be given more attention.

The cancelation of errors in GCMs with lower tops is not precise, of course. For example, many of the IPCC models overestimate tide amplitudes at the surface. Also, most of the models in Table 1 obtain maximum semidiurnal-tide pressure at local solar times later than observed. (Such phase errors may arise from radiation time steps being much longer than dynamics time steps, leading to inaccurate forcing of model dynamics during the time that radiative heating is held constant.) No doubt more detailed and more physically based comparisons of surface tide simulations and observations would reveal additional distinctions among the IPCC models, and between the IPCC models as a group and WACCM. Inspecting the latitude dependence in terms of modes produced by the classical linear calculations (Hough functions) would provide additional insight, since each of the modes has a different vertical wavelength or e-folding distance. Indeed, plotting the migrating semidiurnal tide amplitudes against latitude for all of the models (not shown here) results in a series of bell-shaped curves symmetric about the equator, consistent with the fundamental semidiurnal Hough function (s = n = 2 in Table 1 and Fig. 7 of Forbes 1995). A more thorough assessment of model behavior in terms of the classical theory might also involve extracting the heating rates out of the GCMs and using them as input to linear calculations, as Zwiers and Hamilton (1986) did with one GCM. Detailed comparison of theory and observations in locally data-dense areas, as done by Hamilton et al. (2008), might also prove useful.

The next generation of the IPCC/CMIP database will include PS rather than PSL in its 3-hourly output (while retaining PSL in daily-mean output for studies of weather systems; see http://cmip-pcmdi.llnl.gov/cmip5/docs/standard_output.pdf). This will allow a much cleaner comparison between climate model simulations and observations of surface atmospheric tides than was possible in the present study. Finally, information about tidal behavior above the surface, even if only for a limited number of models and observations, offers great scope for better understanding the underlying processes, and for the use of classical theory to interpret model results. For example, Pirscher et al. (2010) studied the diurnal tide in the upper troposphere and lower stratosphere, obtaining about 20% greater amplitude from short-term numerical weather predictions compared with more direct observations using Global Positioning System (GPS) radio occultations. This result may be related to our finding that most climate-oriented GCMs overestimate tide amplitude at the surface.

Fig. A1.

The ratio of standard deviation to mean value over all Januaries in WACCM-3-simulated (top) diurnal and (bottom) semidiurnal tide amplitude, for years 1990–2000 inclusive.

Fig. A1.

The ratio of standard deviation to mean value over all Januaries in WACCM-3-simulated (top) diurnal and (bottom) semidiurnal tide amplitude, for years 1990–2000 inclusive.

Acknowledgments

We are grateful to many colleagues for comments on preliminary accounts of these results, in particular to E. C. Cordero and S. Tesfai for correcting the list of model features, and to Robert E. Dickinson for discussion of tide forcing and propagation. We also thank Sabrina Fletcher and Alexandria Ballard for graphic art. We acknowledge the modeling groups, the Program for Climate Model Diagnosis and Intercomparison, and the World Climate Research Program (WCRP) Working Group on Coupled Modelling for their roles in making available the WCRP CMIP3 multi-model dataset. Support of this dataset is provided by the Office of Science, U.S. Department of Energy. This work was performed in part under auspices of the Office of Science, U.S. Department of Energy, by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344, and in part at the National Center for Atmospheric Research, which is sponsored by the National Science Foundation.

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APPENDIX

Interannual Variation of the Surface-Pressure Tide

Interannual variations appear to be small in the GCM surface-pressure tides, especially for the semidiurnal harmonic. Figure A1 shows the ratio of the standard deviation to the mean of January diurnal and semidiurnal tide amplitudes from WACCM-3, taken over all 11 yr provided. For the semidiurnal tide, this ratio is less than 0.05 equatorward of latitudes ±30°, and it only attains values greater than 0.2 poleward of ±60°, where the tidal amplitude is not appreciable (Fig. 3, bottom panel). The observational study of Hamilton and Garcia (1984) is consistent with these model results, obtaining a ratio generally <0.1 for the semidiurnal harmonic from 79 yr of surface barometric data at “Batavia.”

The ratio is greater for the diurnal tide, attaining values ∼1 in many areas poleward of ±60°, but it remains less than 0.4 almost everywhere in the tropics (Fig. A1, top panel). Thus, interannual variations of the diurnal surface-pressure harmonic are nonnegligible, though they are smaller than the interannual variability of diurnal tides in the middle atmosphere.

Footnotes

* The National Center for Atmospheric Research is sponsored by the National Science Foundation.

Corresponding author address: Curt Covey, LLNL Mail Code L-103, 7000 East Avenue, Livermore, CA 94550. Email: covey1@llnl.gov