1. Introduction

Over the tropical oceans, averaged over a region several hundred kilometers on a side, evaporation is expected to be roughly proportional to wind speed because near-surface humidity and temperature are relatively uniform, even as wind speed and area-averaged precipitation rate fluctuate greatly from day to day (Raymond et al. 2003; Sobel et al. 2004; Johnson and Lin 1997; Bretherton et al. 2004). Higher wind speeds are associated with more evaporation and latent heat flux from the ocean to the atmosphere. In regions of persistent deep tropical convection, it has been suggested that “convection follows the winds,” that is, in a region and period of increased mean surface winds, rainfall is enhanced because of increased surface latent heat fluxes (Raymond et al. 2003). In the east Pacific, using dropsonde data taken during the Eastern Pacific Investigation of Climate (EPIC), Raymond et al. (2003) found that day-to-day variations in surface entropy fluxes (mainly associated with evaporation) computed from dropsonde measurements and averaged over a 400 km × 400 km region were highly correlated with precipitation. However, this idea has never been tested using a large observational dataset, like that provided by passive microwave sensor satellite retrievals of wind speed and precipitation. In this paper, we look for relationships between area-averaged, satellite-derived wind speed and precipitation throughout the Pacific intertropical convergence zone (ITCZ) and compare our findings to some simple conceptual models of tropical oceanic convection and its relationship to large-scale circulations.

Many convective closures utilize the idea of a statistical “quasi equilibrium” on larger-than-convective (but still subdaily) time scales, between the large-scale processes that promote convective instability and the instability being removed by convection (e.g., Arakawa and Schubert 1974; Raymond 1995). Other “moist adjustment” schemes relax the atmospheric temperature and moisture profiles to specified vertical structures when convective instability is diagnosed (e.g., Betts 1986). In either case, we anticipate that in regions of persistent convection there will likely be statistical correlations between “forcings” such as evaporative moistening of the planetary boundary layer (PBL) and rainfall. These correlations can provide useful insights into the convective dynamics and serve as useful tests for large-scale models and their embedded convective parameterizations.

Evaporation alone introduces little water when compared to the daily precipitation amounts typically observed in mesoscale convective complexes. Instead, the main immediate moisture source for the precipitation is moisture convergence. This suggests that a “convergence feedback” is occurring, in which small increases in evaporation drive much larger increases in moisture convergence and precipitation.

Here, we use 4 yr of passive microwave satellite data to test daily wind speed–precipitation correlations in the Pacific ITCZ. To test the robustness of the wind speed–precipitation correlation, we also use several months of ground-based radar-derived area-averaged precipitation measurements from Kwajalein Island in the central Pacific ITCZ, as well as vector wind retrievals from QuikSCAT. Then we investigate whether data from two reanalyses capture the observed wind speed–precipitation correlation: the 40-yr European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analysis (ERA-40) and the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis. The reanalysis data are then further used to explore possible mechanisms for convergence feedback, using column-integrated moist static energy (MSE) budgets.

2. Data description

We use a methodology similar to that used by Bretherton et al. (2004, hereafter BPB04), who describe simple relationships between humidity and precipitation over tropical ocean regions. As in BPB04, we obtained 0.25° gridded retrievals (retrieval algorithm version 5.0) of precipitation, water vapor path, and surface wind speed (normalized to 10 m) from Remote Sensing Systems (http://www.ssmi.com/) for all satellite passes of the Special Sensor Microwave Imager (SSM/I) and Tropical Rainfall Measuring Mission (TRMM) Microwave Imager (TMI) that occurred during 1998 through 2001. The retrieval algorithm, described in detail in Wentz (1997) and Wentz and Spencer (1998), uses brightness temperatures observed at 19.35, 22.235, 37, and 85 GHz to simultaneously estimate rain rate, water vapor path, near-surface wind speed, and cloud liquid water. For the analysis, the gridded data were first averaged over all available passes (typically 2–6), each day at each grid column, and then averaged to a coarser 2.5° grid, as described in BPB04. To maintain consistency with SSM/I retrievals, a potentially more accurate wind speed estimate for TMI overpasses that also uses the TMI 11-GHz channel was not used; instead the Wentz algorithm based on the 37-GHz channel for wind speed retrieval was used for both TMI and SSM/I. No correction was made to account for possible biases associated with uneven sampling of the diurnal cycle. For validation of the SSM/I wind speeds, we use QuikSCAT SeaWinds gridded daily vector wind retrievals, also obtained from Remote Sensing Systems.

From the Data Support Section of the NCAR, we obtained ERA-40 (23 pressure levels, 2.5° grid; see online at http://www.ecmwf.int) temperature, wind speeds, humidity, precipitation, surface latent and sensible heat fluxes, and net radiative fluxes at the surface and top of the atmosphere. From the Climate Diagnostics Center (http://www.cdc.noaa.gov), we also obtained comparable NCEP–NCAR reanalysis data (17 pressure levels, 2.5° grid; Kalnay et al. 1996). From reanalysis temperatures, we computed daily average saturation water vapor path at each point following BPB04.

3. Microwave-derived relationships between precipitation and wind speed

We examined the relationship between 5° × 5° daily averaged wind speed and precipitation (from the SSM/I and TMI) at 10°N at several longitudes that are in the Pacific ITCZ for much of the year. First, four adjacent 2.5° × 2.5° grid points were averaged into a 550 km × 550 km box at each longitude. Then all 4 yr of wind speed and precipitation data (1461 days) were sorted by wind speed and subdivided into bins containing 60 gridpoint days of data (24 bins at each location). Within each such bin, the mean, as well as the 25th, 50th, and 75th percentiles of the 60 precipitation estimates were calculated; 5° × 5° boxes were used instead of 2.5° × 2.5° boxes to average out some of the random measurement error in the SSM/I precipitation retrievals, but our results are not sensitive to this choice. Figure 1 illustrates this analysis in the western Pacific warm pool (at 10°N, 160°E) and in the eastern Pacific ITCZ (at 10°N, 95°W). In the western Pacific warm pool (Fig. 1a), no relationship between wind speed and rainfall is visible. However, in the eastern Pacific (Fig. 1b), there tends to be more rainfall on days with stronger winds. The correlation coefficient of daily wind speed with daily rainfall is 0.32, which is highly statistically significant. The sample size of 1461 corresponds to 570 degrees of freedom (taking into account the lag-1 autocorrelation in time), so a correlation coefficient of 0.05 is statistically significant at the 99.95% level. However, on many high wind speed days there is no precipitation.

A complicating factor in this analysis is that the precipitation belts vary with the seasonal cycle. Only when convection is easily and frequently initiated do we expect to see a correlation between wind speed and precipitation. Outside deep convective regions, high wind speeds still enhance surface moisture fluxes, but this moisture is stored and advected downstream rather than being rained out locally. BPB04 found that “column relative humidity,” r, defined as water vapor path (WVP) divided by saturation WVP, is strongly correlated with daily precipitation. When r is low, it does not rain, even when winds are strong. A single exponential fit curve, with near doubling of expected precipitation for each 4% increase in r, is adequate to describe the SSM/I-derived r precipitation correlation in all tropical ocean regions.

To control for these seasonal variations at each location, we repeated the analysis of Fig. 1, isolating only those days in the 4-yr record with r greater than 0.75, to focus only on those times when it is likely to rain. From Fig. 4 of BPB04, at r = 0.75, the 25th, 50th, and 75th percentiles of SSM/I-derived rainfall are 2, 5, and 12 mm day⁻¹; the true rainfall distribution at this r is likely more narrowly peaked (BPB04) and nearly excludes nonraining days. Figure 2 shows that with this r threshold, there is a remarkable linear correlation between wind speed and precipitation at all ITCZ locations examined. As in Fig. 1, at each location the wind speed and precipitation data were sorted by wind speed and subdivided into binds of 60 gridpoint days. On days with stronger winds, there tends to be more rainfall. Even in the eastern Pacific, the correlation is substantially stronger than in Fig. 1, and the quartiles plotted show many fewer high wind speed days during which there is no precipitation.

In the ERA-40 and NCEP–NCAR reanalysis (averaged daily over 2.5° × 2.5° centered on the same locations as the wind speed and precipitation), the increase in evaporation that occurs as wind speed rises from 4 to 8 m s⁻¹ averages less than 3 mm day⁻¹ (over the gridpoint days used in Fig. 2), which is small compared with the observed increases in precipitation. This is consistent with a convergence feedback.

Wind speed explains a moderate amount of the daily variability in precipitation, as is visible in the correlation coefficients and quartiles plotted in Fig. 2. The trends are highly statistically significant at all plotted locations because of the large amount of data (266–545 days with r > 0.75). A correlation of 0.13 is statistically significant at the 99.95% level at all locations (degrees of freedom range from 166 to 460 taking into account the lag-1 autocorrelation). The correlation and the regression slope in Fig. 2 vary spatially, with the highest correlation and the steepest slope (greatest increase in precipitation for a given increase in wind speed) east of 120°W. There is also a strong correlation in the western Pacific (160°W), but the slope is not as steep as in the eastern Pacific. In the central Pacific, the correlations are not as strong, and the slopes are less steep.

One possible explanation for the spatial variations in the wind speed–precipitation relationship is that moisture advection could be varying. Reanalyses (discussed further in section 6) suggest that moisture advection is generally a drying effect in the ITCZ. Thus, increased dry advection in higher wind conditions could counterbalance the increases in evaporation. Another possibility is that the amount of induced moisture convergence could vary in association with the mean thermodynamic and vertical motion profiles. In section 6, we will investigate these ideas using reanalysis data.

4. Validation using KWAJEX data

There are a number of uncertainties involved in the retrieval of precipitation estimates from the SSM/I on a daily time scale. Inherent in the empirical Wentz algorithm are random errors in retrieving precipitation associated with ambiguities in mapping brightness temperature to rainfall. There are also errors associated with horizontally inhomogeneous precipitation being treated as uniform on retrieval scales, even though there is a nonlinear relationship between brightness temperature and precipitation. Even if retrievals were perfect, convection evolves over the course of a day and one may ask whether two–six retrievals averaged over a 5° box are an adequate sample.

To address these concerns, BPB04 describe a detailed comparison between the daily SSM/I data and an 18-month sample (July–December 1999–2001) of a Kwajalein radar–derived estimate of daily rainfall area averaged over a circle of 240-km radius. The SSM/I wind speed data and the column relative humidity data were averaged over the two closest 2.5° × 2.5° grid boxes to Kwajalein (8.72°N, 167.73°E), centered at 8.75°N, 168.75°W and 8.75°N, 166.25°W.

We calculated the distribution of this Kwajalein radar-derived rainfall proxy, binned by wind speed, for days when column relative humidity is above 0.75 (shown in Figs. 5a,c). The correlation of QuikSCAT vector mean winds with the radar rainfall proxy (Fig. 5c) is similar to the wind–precipitation relationship derived using only the passive microwave imagers (Fig. 5b). A correlation coefficient of 0.12 is statistically significant at the 99% level given the sample size (270 days, 180 degrees of freedom). The weaker relationship between the QuikSCAT vector mean winds and the SSM/I–TMI precipitation (Fig. 5d) may be due to a combination of the smaller sample size (as mentioned above, only 1–2 QuikSCAT satellite passes occur each day compared to two–six passive microwave passes) and nonconcurrent sampling by the two satellites. Figure 5 supports the nature of the passive-microwave-derived relationships between wind speed and precipitation. A natural extension of this work would be to perform a similar analysis using the TRMM precipitation radar combined with simultaneous TMI-derived wind speeds and column relative humidity.

5. Comparison of passive microwave wind speed–precipitation correlation with reanalyses

A remaining question is whether the documented wind speed–precipitation correlation is a feature that is generally captured in the ERA-40 and NCEP–NCAR reanalysis and if so, whether the reanalyses can help us better understand it. We first test the extent to which these reanalyses reproduce the observed correlation between wind speed and precipitation at high column relative humidity r.

ERA-40 water vapor path is very closely correlated (correlation coefficient is 0.95 using 1998–2001 data in Pacific at 10°N) with SSM/I–TMI retrievals (from which ERA-40 assimilates retrievals of WVP and wind speed), but ERA-40 has substantially more high r occurrences than are observed. NCEP–NCAR WVP is less strongly correlated (correlation coefficient is 0.70) with the SSM/I–TMI product (which NCEP–NCAR reanalysis does not assimilate), and r tends to be lower than observed. Although ERA-40 has a wider range of column relative humidities and more precipitation than NCEP–NCAR reanalysis, both reanalyses have similar relationships between r and precipitation, which are qualitatively, but not quantitatively similar to the SSM/I-derived relationship (not shown).

Since the two reanalyses have similar r-precipitation relationships, we test for a wind speed–precipitation correlation as in Fig. 4. Typical examples from the NCEP–NCAR reanalysis and ERA-40 of the distribution of precipitation binned by wind speed on days with r ≈ 0.75 are shown in Fig. 6. ERA-40 has a only a very weak correlation between wind speed and precipitation in the eastern Pacific (Fig. 6a) though the correlation is higher in the western Pacific (correlation coefficient is 0.40). This is somewhat surprising since ERA-40 assimilates both the SSM/I–TMI water vapor path and wind speed. ERA-40 and SSM/I wind speed only have a correlation coefficient of 0.60, so the SSM/I wind speed is clearly used only as a weak constraint on the ERA-40. NCEP–NCAR reanalysis does not assimilate the SSM/I–TMI data, but it shows a stronger, more consistent wind speed–precipitation correlation (Fig. 6b) when binned by r. The discrepancies in the wind speed–precipitation relationship between observations and the two reanalyses suggest that this relationship is a useful test of the parameterization of moist convective processes in large-scale models.

6. Interpretation of wind speed–precipitation correlation

In this section, we proceed from the hypothesis that the observed wind speed–precipitation correlation is the result of a convergence feedback in which small evaporation anomalies drive larger precipitation anomalies (fed by moisture convergence) and explore possible mechanisms for the convergence feedback.

While the main moisture source for precipitation is moisture convergence, this does not mean moisture convergence should be regarded as a conceptually separate external forcing that “causes” precipitation. Because of vertical moisture stratification in the atmosphere, the horizontal convergence at lower levels and divergence aloft associated with upward motion naturally induce moisture convergence. Hence, moisture convergence can be viewed as a response to latent heating due to deep convection. Since latent heating occurs as a result of moisture convergence and condensation, this chain of logic is circular. A different framework is required to understand when and where convection occurs within large-scale regions of mean convergence like the ITCZ.

One perspective on the problem is through a column-integrated moist static energy budget (schematic shown in Fig. 7). To a first approximation, column moist static energy is not changed by convective circulations within the column itself. Hence, time-mean sources of tropospheric column-integrated moist static energy must be balanced by advective export of moist static energy (Neelin and Held 1987). Two-layer and idealized models of the tropical circulation (e.g., Neelin and Held 1987; Zebiak 1986; Neelin and Zeng 2000; Sobel and Bretherton 2000) have suggested that evaporation (or more generally, local sources of troposphere-integrated moist static energy) and precipitation (associated with the export of moist static energy) may drive much larger anomalies in precipitation, mainly through a “vertical MSE advection” feedback mechanism. Increased evaporation moistens the PBL, promoting conditional instability and more convection, and adding MSE to the atmosphere column. The enhanced deep convection diverges the added MSE through divergence of air with higher MSE in the upper troposphere and convergence of air with slightly lower MSE in the lower and midtroposphere. The enhanced convection also increases moisture convergence, allowing a large precipitation response. In this view, the degree to which convection is enhanced depends on the “gross moist stability,” defined as the net export of MSE per unit of net upward convective mass flux; a smaller gross moist stability requires more convective enhancement to export a given amount of column MSE. Sobel and Bretherton, however, noted that the relationship between evaporation and precipitation is not convincingly borne out in observations and suggested that horizontal moisture advection might also be playing a role in determining the distribution of precipitation over the warm sea surface temperature regions.

To test the hypothesis that local sources of MSE could enhance convection by causing advective export of MSE by divergent circulations, we computed moist energy budget terms from ERA-40 and the NCEP–NCAR reanalysis. Although an individual realization of reanalysis data may have substantial errors, we hope that by statistically analyzing the large datasets (4 yr of daily data at 2.5° resolution), we can extract some useful information. The reanalyses are subject to possible model biases and have substantially more uncertainty than exists in the observed wind speed–precipitation relationship, but comparison of the two reanalyses may at least suggest a range of plausible values for individual budget terms.

At each pressure level provided to us in each reanalysis, storage, horizontal advection, and vertical advection were computed from both reanalyses, corresponding to the first four terms in the budget equation for MSE (h):

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The fifth and sixth terms describe turbulent transport of h and the radiative flux divergence, respectively, and S_h includes small source terms due to the nonconservation of h in moist thermodynamic processes, particularly involving ice. We vertically integrate (1), assuming no top-of-atmosphere turbulent fluxes (brackets indicate mass-weighted vertical integrals of storage and horizontal and vertical advection):

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We also extracted net column-integrated radiative cooling ΔF_rad and surface MSE fluxes F^h_turb(0) from the reanalyses. Here, R_h is a residual due to small vertically averaged source terms <S_h>, discretization errors due to the limited resolution of the reanalysis output, and mean drifts of the model from reality. If the MSE budget is consistent, R_h should be small compared to the dominant budget terms.

Column-integrated budget terms from (2) were geographically combined, as in Fig. 4, binned by column relative humidity, and then within each column relative humidity bin, binned by wind speed with a bin size of 60 gridpoint days. A caveat to the analysis, as applied to vertical advection, is that model vertical motion is correlated strongly with precipitation, and reanalysis precipitation is not necessarily accurate, or even strongly correlated with wind speed, as Fig. 6a shows. Hence, vertical h advection and its trends with wind speed may be quite inaccurate; we might hope for horizontal h advection to be more reliable. For both NCEP–NCAR and ERA-40 budgets, r and wind speed–binned budget residuals R_h are wind speed independent, averaging less than 40 W m⁻¹ at all wind speeds, and surface heat fluxes are well correlated with wind speed as expected. Storage is also small in the mean, and ΔF_rad is only weakly wind speed dependent.

The two reanalyses give surprisingly different estimates of horizontal advection (mainly associated with differences in moisture advection). Figure 8 shows examples of column-integrated horizontal and vertical advection as well as evaporation as a function of wind speed in the east and west Pacific ITCZ regions we previously analyzed. Column relative humidities in the example shown are approximately 0.75, but the nature of the result is not sensitive to this choice. In the two ERA-40 cases, as wind speed increases, horizontal dry advection increases at a similar rate to the evaporation increase (about 10 W m⁻² per 1 m s⁻¹ increase in wind speed). The result of these canceling effects is that increases in wind speed do not generally feed more moist static energy into the tropospheric column. Thus there is no need for vertical advection to close the MSE budget, and in fact ERA-40-predicted vertical advection is comparatively small at all wind speeds.

In the NCEP–NCAR reanalysis, horizontal advection is a weaker function of wind speed in both regions. In the western Pacific, dry advection is wind speed independent, and vertical advective export of MSE balances the evaporation increases in higher wind speed conditions. In the NCEP–NCAR east Pacific case, horizontal and vertical advection both contribute roughly equally to balancing the evaporative source. Both NCEP–NCAR cases are compatible with the vertical MSE advection feedback mechanism, in contrast to the ERA-40.

Since the ERA-40 is incompatible with the vertical MSE advection mechanism for convergence feedback, we also consider two other related mechanisms that do not require column-integrated vertical MSE advection to balance surface fluxes.

Raymond (1995) suggested a related theory for a relationship between evaporation and rainfall, “boundary layer quasi-equilibrium” (BLQ), where on larger-than-convective time scales, boundary layer MSE remains roughly constant through a balance between evaporative fluxes generating MSE and evaporatively cooled downdrafts from deep convection flushing low MSE air into the boundary layer (Raymond 1995; Raymond et al. 2003). A simple schematic of this theory is shown in Fig. 9. In this case, the term “boundary layer” must be extended above cloud base since shallow cumulus circulations are responsible for substantial mixing on these time scales. For the purposes of this paper, we follow Raymond et al. (2003) and view this balanced shallow mixing layer as extending roughly up to 850 mb.

As with the vertical MSE advection mechanism, horizontal MSE advection is considered secondary in this balance. A substantial amount of horizontal MSE advection occurs above 850 mb, so it is natural to ask whether the evaporative moistening dominates over the dry advection within the boundary layer in both the ERA-40 and NCEP–NCAR reanalysis, even if this is not the case for the whole troposphere. If this were the case, both reanalyses could be consistent with BLQ.

An attraction of the BLQ theory is that it could also explain why the slope of the wind speed–precipitation correlation is greater for higher-column relative humidity conditions. The dominant boundary layer MSE source is assumed to be evaporation E, balanced by an MSE sink M_dΔh, from deep convective low h downdrafts. MSE storage, horizontal advection, and radiation vertically integrated over the boundary layer are considered to be less important. Here, Δh is the typical MSE deficit of downdraft air compared to the boundary layer, and M_d is the downdraft mass flux; M_d is assumed proportional to updraft mass flux M_u, which in turn is proportional to precipitation. Hence the dominant MSE balance in the boundary layer is

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While we do not know the exact behavior of c = M_d/P (the downdraft mass associated with a given amount of precipitation), we expect that in a moister environment, both c and Δh would decrease since there would be less evaporative cooling surrounding updrafts. Hence updraft mass (and precipitation) would need to increase to sustain a given E. This implies P/E larger at high r, as observed.

To test the BLQ theory, we calculated moist static energy budgets (using the reanalyses) for the 850 mb to surface layer in each column and again stratified the data by r and wind speed. BLQ would predict higher wind speeds to consistently be associated with significantly stronger evaporation than horizontal MSE advection in the boundary layer. However, the relationship between wind speed and boundary layer horizontal MSE advection in both the ERA-40 and NCEP–NCAR reanalysis is similar to the column-integrated horizontal advection shown in Fig. 8. Thus, the boundary layer horizontal MSE advection in NCEP–NCAR reanalysis is compatible with BLQ. As in the full tropospheric column, in ERA-40, increased low-level dry advection occurring at higher wind speeds cancels the evaporation increases, so net boundary layer moistening associated with increases in wind speed is very small. Hence, the ERA-40 is not compatible with BLQ.

We propose a third conceptual model for the wind speed–precipitation feedback, which we cannot test using the reanalyses, and which is a variation on BLQ. It posits that wind-induced evaporation increases the shallow cumulus population (Fig. 10). Each shallow cumulus updraft has a chance of deepening into a precipitating system that depends mainly on ambient humidity. Thus, both stronger winds and increased column relative humidity lead to more precipitation, even if there is no net vertical advection of MSE in the column average or below 850 mb. This hypothesis would perhaps best first be tested using simulations with a cloud-resolving model.

7. Conclusions

Four years of passive microwave satellite retrievals from the SSM/I and TMI were used to look at the relationship between daily wind speed and precipitation. At high-column relative humidities (conditions under which deep convection is likely to occur), there is a significant correlation between wind speed and precipitation. The slope of the wind speed–precipitation correlation varies geographically, and rapidly increases in moister conditions. Surface wind speed explains a small fraction of the daily rainfall variability but provides useful insight into how surface forcing modulates tropical convection and also provides an interesting test on large-scale forecast models in the Tropics. QuikSCAT data show that the correlation is not associated with mesoscale gustiness induced by convective systems. Hence, the higher surface fluxes are likely causing the incidence of deep convection to increase.

Other data sources were analyzed to show that the result is robust. Area-averaged precipitation estimates derived from a radar at Kwajalein Island were compared with the microwave precipitation estimates, and 2.5° vector mean winds computed from QuickSCAT were compared with the SSM/I- and TMI-derived wind speeds. In all cases, a wind speed–precipitation correlation is observed in moist conditions.

Physically, higher wind speeds promote more evaporation, which destabilizes the boundary layer and can trigger deep convection. However, quantification of this argument proves unexpectedly subtle. The observed increases in precipitation are much greater than evaporation changes associated with the increased wind speed, so we deduce that a convergence feedback is occurring. Using MSE budgets constructed from the ERA-40 and NCEP–NCAR reanalysis, we tested several mechanisms for a convergence feedback. Tests were inconclusive since in ERA-40 (but not in NCEP–NCAR) low-level horizontal advection is comparable to evaporation, so there is not a consistent buildup of column, or even boundary-layer-integrated moist static energy that must be vented by deep convection.

It would be interesting to study these mechanisms in a somewhat more idealized situation such as a large-domain Cloud-Resolving Model (CRM) simulation, where data assimilation techniques and observational errors would not complicate the analysis. One could also test the “convective initiation” mechanism in this framework. More generally, forays into analyzing column moist static energy budgets have led the authors to further interest in studying the extent to which reanalyses and CRM simulations support current theories about column moist static energy budgets and gross moist stability.