## Abstract

Quasi-equilibrium convective parameterizations share the common assumption that in regions of sustained deep convection rates of change in convective available potential energy (CAPE) are small compared to the magnitude of the large-scale and boundary layer forcings that act to modify CAPE. The more restrictive strict quasi-equilibrium hypothesis (SQE) is that changes in CAPE are dynamically negligible. Under this assumption, tropospheric temperature and thickness variations largely follow a moist adiabat associated with variations in the boundary layer *θ*_{e}. SQE is an attractive simplification for theories of large-scale circulation in the Tropics but has been inadequately tested to date. In this paper, we test the SQE hypothesis over convecting regions of the tropical oceans using microwave sounding unit tropospheric temperature and precipitation fields, along with COADS surface data, on timescales of a month and longer and on space scales ranging upward from *O*(300 km). One prediction of SQE, that *θ*_{e} and vertically averaged temperature (〈*T* 〉) are positively correlated in space and time, is found to hold for month-to-month, seasonal, and interannual timescales, but the proportionality coefficient between 〈*T* 〉 and *θ*_{e} is at most half as large as predicted. This is associated with variations of mean CAPE of as much as 25% over even the largest time and space scales examined.

## 1. Introduction

Cumulus convection plays an important role in both the heat and moisture budgets of the tropical troposphere but occurs on too small a scale to be resolved by general circulation and climate models. The effects of cumulus convection must therefore be parameterized in terms of grid-scale variables, and a number of algorithms have been developed for this purpose. Unfortunately, models have shown sensitivity to the type of convective parameterization that is used (Crum and Stevens 1983; Stark 1976), and there is currently no general agreement on which convective parameterization is “best” suited for any particular space scale.

Different approaches have been used to specify the amount and vertical distribution of convective heating produced by convective parameterizations. In many parameterizations, the amount of convection is adjusted based on the convective available potential energy (CAPE) in the mean atmospheric sounding (e.g., Arakawa and Schubert 1974; Fritsch and Chappell 1980; Emanuel 1991). If convection is sustained, such parameterizations maintain “quasi-equilibrium,” in which convective and large-scale forcings always nearly balance one another so that the rate of change of CAPE remains small compared to the large-scale forcing of CAPE (Arakawa and Schubert 1974). Data from the 1956 Operation Redwing experiment in the Marshall Islands and many more recent field experiments corroborate that quasi-equilibrium holds during periods of sustained convection, at least over the space scales examined [*O*(10^{5} km^{2})].

In quasi-equilibrium, the CAPE itself may still fluctuate substantially. A much stricter simplifying assumption that has been used for studies of large-scale wave phenomena in the Tropics (Emanuel 1987; Neelin et al. 1987; Yano and Emanuel 1991; Emanuel 1993; Emanuel et al. 1994) is that, at all times, the atmospheric virtual temperature profile tracks a moist adiabat tied to the boundary layer air, differing from that profile only in a small and time-independent way. Hence, the change in CAPE resulting from large-scale forcings in regions with sufficient convection is negligible. That is, the convective heating will act to maintain virtual temperature perturbations above the boundary layer along a moist adiabat that is tied to the boundary layer perturbation in equivalent potential temperature (*θ*^{′}_{eb}). We will refer to this assumption as the “strict quasi-equilibrium” (SQE) hypothesis.

When used as part of a convective parameterization, SQE must be supplemented by a scheme for vertical redistribution of moisture. Neelin and Yu (1994) showed that the Betts–Miller convective parameterization (Betts 1986) closely follows SQE for circulations with characteristic timescales of more than a few hours. By reducing the thermal variations of the troposphere to a single mode tied to variations in *θ*^{′}_{eb}, the SQE hypothesis greatly simplifies discussion of the interaction of convection with synoptic and larger-scale tropical dynamics (Neelin and Yu 1994; Emanuel et al. 1994).

Little direct testing of the SQE hypothesis against observations has been carried out to date, especially on larger time and space scales. In this paper, we test the SQE hypothesis on space scales ranging from 250 to 10^{5} km and on monthly and longer timescales using MSU (Microwave Sounding Unit) precipitation and temperature data and COADS (Comprehensive Ocean–Atmosphere Data Set) surface data in the Tropics for the 11-yr period from 1981 to 1991. In addition, we use precipitation rates as a proxy for convective activity and attempt to determine whether the SQE hypothesis holds better when precipitation rate exceeds some threshold. A discussion of the SQE hypothesis and its current observational status is given in section 2, followed by a description of the data used in this study in section 3, analysis of the data in section 4, and a summary of the results in section 5.

## 2. Testing the predictions of SQE

### a. Prior tests of SQE

The SQE hypothesis states that sustained deep convection maintains the atmosphere in a state in which perturbations in virtual temperature [*T*^{ ′}_{v}(*p*)] follow perturbations in the virtual moist adiabat that is tied to *θ*^{′}_{b}. We estimate that the virtual temperature perturbations averaged over the depth of the free troposphere are due 90% or more to temperature and 10% or less to mixing ratio perturbations.^{1} If we neglect the effect of the mixing ratio perturbations and the effect of liquid water loading in convective updrafts, SQE can be stated as

where *θ**^{′}(*p*) is the perturbation saturation equivalent potential temperature of the troposphere above the boundary layer. The SQE hypothesis implies that CAPE remains constant and is roughly equivalent to the assumption of a constant cloud work function for each cloud type in the Arakawa–Schubert convective parameterization.

In reality, CAPE varies in convectively active regions even over large time and space scales, indicating an imperfect correlation between free tropospheric temperature anomalies and *θ*^{′}_{eb}. The SQE hypothesis can still be an accurate simplifying assumption for a dynamical model as long as most of the variance of free tropospheric temperature is related moist adiabatically to *θ*^{′}_{eb}. This can be tested in a vertically integrated sense by measuring variations in CAPE. Over most of the actively convecting Tropics, tropospheric temperature variations of 1–2 K are observed on synoptic and longer timescales. If these variations were entirely uncorrelated with *θ*_{eb}, then CAPE would have to fluctuate by 300–600 J kg^{−1}. Variations of CAPE of 100 J kg^{−1} or less are therefore “dynamically negligible,” while larger variations challenge the SQE hypothesis.

A few studies have examined the variation of CAPE in convectively active parts of the Tropics. Thompson et al. (1979) composited ship soundings of African easterly waves. Figure 1 shows the variation in CAPE and precipitation that they found. The CAPE varies significantly from 750 J kg^{−1} to 1200 J kg^{−1} over one cycle (3–4 days) of the composite wave, in apparent contradiction to SQE. Data from Lin and Johnson (1996) suggest similar variations of CAPE during the ISO in the western Pacific. In both cases, the CAPE is lower during the heavily precipitating phases of the wave than in the lightly precipitating “suppressed” phases. Thus, one SQE-compatible interpretation of these data is that SQE holds in regions of active convection but that the convection in the suppressed phase is too infrequent to maintain SQE. A second interpretation consistent with the spirit of SQE (which we cannot test with our dataset) is that the cloud properties should be calculated for an entraining updraft rather than an undilute updraft. Since relative humidity is in phase with precipitation in both wave types, the reduction of undilute CAPE by the evaporative cooling associated with a given entrainment rate would be larger for the drier environment, allowing the “entraining CAPE” to be approximately constant throughout all phases of the wave.

Statistical studies of tropical soundings have drawn mixed conclusions about SQE. Xu and Emanuel (1989) found that in a large set of western Pacific soundings parcels reversibly lifted from just below cloud base tended to have a small and constant buoyancy throughout the troposphere. Their soundings only had 50-mb vertical resolution, and they did not explicitly calculate frequency distributions of CAPE or correlate CAPE with precipitation or relative humidity. Williams and Renno (1993) examined a suite of tropical soundings from many field experiments. They concluded that CAPE is a strongly increasing function of low-level *θ*_{e}. They presented histograms of CAPE at individual locations that show typical variations in CAPE of well over 1000 J kg^{−1} between soundings at all sites. They include soundings from all times, whether convectively active or not, so their results do not definitively contradict SQE.

In this paper, we will examine the SQE hypothesis over space scales of *O*(300 km) and up, and over timescales of one month and longer using a new approach in which we compare shipboard observations of *θ*_{eb} and satellite retrievals of vertically averaged tropospheric temperature over the tropical oceans.

### b. Datasets

The data used in this study cover the oceanic regions between 30°S and 30°N for the 11 years from 1981 to 1991. The COADS dataset consists of monthly averages of routine synoptic ship reports binned into 2° grid boxes. These data include measurements of SST and surface shipboard) air temperature, specific humidity, pressure, and zonal and meridional winds. Specific humidity can vary substantially in the surface layer, and the measurement heights on different ships are unknown and variable. Another possible systematic error source is deck heating during the daytime. These potential error sources are probably unimportant for our study, since they are unlikely to be correlated with location or month. Because ships seldom stray from shipping lanes, the COADS dataset does not provide uniform data coverage over all the tropical oceans; the central Pacific is particularly underrepresented in the COADS data. However, the coherence of the data between adjacent grid boxes suggests that data coverage over the rest of the oceans is adequate for this study.

The COADS data were interpolated onto 2.5° grid boxes in order to match the grid layout of the MSU dataset. Interpolation onto each 2.5° grid box was done using a simple area-weighted average of the four nearest neighbor 2° grid boxes, with the requirement that at least 80% of the 2.5° grid box be covered by 2° grid boxes with data before interpolation was done. Derived quantities such as *θ*_{eb} were first computed on the 2° boxes and then interpolated to the 2.5° grid boxes.

The MSU data used in this study consists of monthly average MSU channel 2 temperature and the derived precipitation for each month on 2.5° grid boxes over ocean areas. Most of our analysis is performed using monthly anomalies of the temperature from the 11-yr annual cycle, since for most of the study this was what was available to us. Late in the study we did obtain measurements including the annual cycle, so in section 4 we have included a brief analysis of the annual cycle as well.

Channel 2 of the MSU on the TIROS-N polar orbiting satellites measures the brightness temperature at a wavelength of 53.74 GHz, which lies at the edge of an oxygen absorption band. This brightness temperature can be regarded as a weighted vertical average of the atmospheric temperature. Over the oceans, other factors that can affect the brightness temperature are condensate in the form of precipitation-size ice particles; cloud water, which reduces the brightness temperature; and water vapor (Spencer and Christy 1992). Data with anomalously low brightness temperature compared to neighboring data points are assumed to be contaminated by condensate from deep convection and are screened out. A monthly anomaly of 20% in column water vapor changes the channel 2 brightness temperature by only 0.03 K. Thus both these effects contribute negligibly to the monthly anomalies.

The vertical weighting function for MSU channel 2 makes it an excellent proxy for the vertical average of temperature *T*(*p*) between 1000 and 300 mb. Let 〈*T* 〉 denote the monthly anomalies of the vertical average of *T*(*p*) from 1000 to 300 mb. Spencer and Christy (1992) compared monthly MSU channel 2 brightness temperature anomaly with 〈*T* 〉 from radiosonde measurements over Guam and found a correlation coefficient of 0.74 and an rms difference of 0.25°C, which is as good as the correlation and standard error between radiosonde measurements from adjacent tropical sounding stations (Spencer and Christy 1992). Averages over larger regions showed even better agreement. Readers interested in a more thorough discussion of issues involving the physical basis, interpretation, and accuracy of the MSU temperature data are referred to Spencer and Christy (1992).

A threshold test based on the MSU precipitation field is used to identify regions of deep convection. The MSU-derived precipitation is derived using the algorithm described in Spencer (1993). These fields compare reasonably well with other precipitation climatologies such as the Geostationary Operational Environmental Satellite Precipitation Index and the climatology of Legates and Wilmott (1990), but there are some significant local and regional differences, especially in the amplitude of precipitation (Spencer 1993). It is assumed here that the MSU dataset is similar enough to the other precipitation climatologies that the results of this study would not change in a qualitative sense if the other climatologies were used.

### c. Analysis methods

We calculated *θ*_{eb} from the COADS dataset following Bolton (1981), based on the ship-measured temperature, specific humidity, and pressure. It is known that *θ*_{e} is not a constant in the boundary layer but actually varies with height (e.g., Mapes and Houze 1992). It is assumed in this study that variations in *θ*_{eb} are a good proxy for variations in vertically averaged boundary layer *θ*_{e}.

The MSU dataset has few missing observations for the years and spatial coverage included in this study, but the COADS dataset contains large holes in both time and space for which there are no data. To overcome this, the time series were interpolated, but only if at least 85% of the samples in the entire time series were present. If this criteria was met, the missing data for a given month were replaced with the mean of the two adjacent months. If the two adjacent months were not available, the two months adjacent to those were used; the series was discarded if none of these criteria could be met.

The statistical significance of correlation coefficients *R*_{XY} between variables *X* and *Y* used in this study can be estimated as follows. Let *D* be the number of temporal degrees of freedom in the time series. Then, for two normally distributed and uncorrelated time series with a sufficiently large number of temporal degrees of freedom *D* (in practice, 10 or more), the statistic

has a distribution that is approximately standard normal (Hogg and Craig 1978, 303), so that |*W*| > 1.96 for statistical significance at the 95% level using a two-sided test.

Correlations were computed only at locations at which at least 85% of the data was present in both time series, which resulted in time series in which the number of samples, *N,* was at least 112. We chose *D* = *N*/*β*, where *β* is a representative number of months over which the autocorrelation of *θ*_{eb} and 〈*T* 〉 had a significance of 95% or greater. To determine *β*, the *θ*_{eb} and 〈*T* 〉 time series at 30 different locations were autocorrelated over a series of lags from 0 to 14 months. The autocorrelations at each location did not drop below the 95% significance level after the same number of months, and the autocorrelations in some locations did not stay below the 95% significance level after first dropping below that level. Results depended on location, but *β* = 4 was a representative value. The darker shades in Figs. 2–5 indicate correlations of 0.35 or greater, which corresponds to statistical significance at the 95% level using *β* = 4 if data are present in all 132 months.

Most of our analysis was performed with an MSU dataset that only included monthly anomalies of 〈*T* 〉 from the 11-yr average annual cycle. Unless otherwise mentioned, we removed the annual cycle from time series of other variables to create monthly anomalies before computing correlations. Most (50%–70%) of the variance in both time series in convective regions is on short, month-to-month timescales, even when averaged over regions of the width of an ocean basin. As we show in section 4, the correlations between *θ*_{eb} and 〈*T* 〉 in deep convecting regions during an annual cycle have very similar characteristics to the month-to-month and even the interannual variability, so that the impact of removing versus retaining the annual cycle on our results is quite small. In some instances, the time series were subjected to low-pass or high-pass filtering after removing the annual cycle. This was done to help determine the timescales over which the correlations were largest. All of the filtering was done with a two-pole Butterworth filter. For low-pass filtered data, the autocorrelation time will increase, reducing the temporal degrees of freedom. For instance, for *β* = 6, appropriate to a 6-month low-pass filter, only correlations of 0.42 or higher are significant at the 95% level if all data are present. In some cases (section 4), horizontal averaging was done to reduce sampling noise in the time series.

Because the SQE hypothesis is expected to hold only when there is sufficiently widespread precipitation to ensure transport of latent and sensible heat anomalies from the boundary layer to the troposphere, “precipitation windowing” was used in some cases to isolate the data in each time series that was collected when the precipitation rate exceeded some minimum amount *P*_{min}. To determine an appropriate value of *P*_{min} the precipitation time series at all the grid points were concatenated into one series and then ordered from lowest to highest precipitation rate. The resulting series of precipitation was then split into four equal parts, so that the threshold *P*_{min} for each of the four precipitation rate quartiles could be determined. The value of *P*_{min} associated with the highest precipitation rate quartile is 196 mm month^{−1}; the values of *P*_{min} for the other quartiles are, in descending order, 67, 7, and 0 mm month^{−1}.

### d. Tests of SQE using MSU/COADS data

The SQE hypothesis predicts that in convecting regions of the Tropics perturbations in *T*(*p*) should follow the changing moist adiabat associated with perturbations in *θ*_{eb}. Two consequences of SQE we can test with the MSU and COADS data are (i) that *θ*_{eb} and 〈*T* 〉 will move together in lock step, and (ii) what the amplitude of *θ*_{eb} variations will be for given 〈*T* 〉 variations. We will test these predictions on monthly to interannual timescales.

To derive the SQE relationship between *θ*_{eb} and 〈*T* 〉, it is most straightforward to recast SQE in terms of moist static energy *h* = *C*_{p}*T* + *gz* + *Lq,* where *z* is height and *q* is water vapor mixing ratio. In the boundary layer, *h*_{b} ≈ *C*_{p}*θ*_{eb}. An approximate equivalent to (1) is

where *h**(*p*) is the free tropospheric saturated moist static energy and primes indicate perturbations from a reference sounding. Solving for *T* ′ gives

where *T̄*(*p*) is a representative average temperature profile for the Tropics and *q** is the saturation water vapor mixing ratio.

Replacing *T _{b}* and

*θ*with values

_{eb}*T̄*and

_{b}*θ̄*averaged over the Tropics in space and time, and averaging over the 1000–300-mb interval represented by the MSU channel 2 temperature gives

_{eb}where

Equation (6) can be numerically integrated using a representative sounding for the Tropics to compute *T̄*(*p*). For example, using the Jordan (1958) sounding in (6) gives

If we instead allow *T* ′ to follow a reversible moist adiabat, we obtain a somewhat smaller *γ* = 0.31. Equation (7) will be compared to the observed variations of *θ*^{′}_{eb} with 〈*T* ′(*p*)〉. In these comparisons, “*θ*_{esqe}” will be used to refer to the value of *θ*^{′}_{eb} computed from (7) using observed values of 〈*T* ′(*p*)〉 and *γ*_{sqe} = 0.43. The prime notation in (7) will be assumed in the remainder of this paper. Except where noted, *θ _{eb}* will refer to the monthly anomaly from the annual cycle and similarly for 〈

*T*〉.

We have performed two types of analyses. The first, discussed in the remainder of this section, is to look at how strongly *θ _{eb}* and 〈

*T*〉 are correlated in individual grid boxes. This is performed using monthly anomalies of these fields from the annual cycle. The second analysis involves regional averaging of these time series over the convecting regions in four ocean regions. In this analysis, we test the coefficient of proportionality between the fields on intermonthly and interannual timescales and in the mean annual cycle.

## 3. Local correlations of 〈T〉, *θ*_{eb}, and SST

We first examine low-frequency “interannual” variability by applying a 6-month cutoff low-pass filter to the monthly anomalies of *θ _{eb}* and 〈

*T*〉. These low-pass filtered anomalies account for less than 10% of the overall variance in the unfiltered time series. Correlation coefficients were computed for the

*θ*

_{eb}and 〈

*T*〉 time series for each of the 2.5° grid boxes in the Tropics, and the results plotted to produce the correlation map shown in Fig. 2. Recall that regions of darker shading correspond to statistically significant positive correlations; there are no statistically significant negative correlations. The overall strength of the correlation for the Tropics was quantified using an area-averaged correlation coefficient,

*R*

_{oa}, computed by averaging the correlation coefficients from all of the grid boxes. For the correlations in Fig. 2,

*R*

_{oa}= 0.33, somewhat higher than the

*R*

_{oa}= 0.25 achieved when no low-pass filtering is used.

The maximum correlation coefficients are greater than 0.8 and occur on the east and west sides of Central America and in a few locations in the Indian Ocean. Statistically insignificant negative correlations occur north of 20°N in the eastern Pacific and in some parts of the western Pacific and Indian Oceans. Allowing for the vast holes that result due to the sparsity of COADS data, it appears that the correlation is reasonably strong in the Indian and Atlantic Oceans, but less so in the Pacific Ocean. For example, in the west Pacific “warm pool” around 160°E, there seems to be no significant correlation. However, the giant COADS data hole in the Pacific is bisected between ∼30°S, 180° and 10°N, 150°W by a shipping lane along which the correlations are fairly strong and positive. Which, if either, of these two areas may be taken as representative of the Pacific Ocean is unclear.

Figure 3 shows the “intermonthly” correlation map that results when *θ*_{eb} is correlated with 〈*T* 〉 after filtering the data with a high-pass filter using a 6-month cutoff. The value of *R*_{oa} now drops to 0.16, and few of the correlations remain statistically significant. While the overall picture is still one of positive correlations, there are two regions in which the correlation switches from 0.3 or greater to 0 or negative. The first of these is along the east coast of Africa and north into the Arabian Sea, which is a region of very low precipitation along the path of the Indian monsoon winds. The second is in the central Atlantic just north of Guyana and extends northward to about 20°N, where it intersects with the paths of some of the easterly waves that move from Africa to the Caribbean.

It is possible that sampling noise, which is partially removed by low-pass filtering the data, is degrading the high-pass filtered correlations. Spatial averaging of the time series before applying the high-pass filtering removes some of the random noise. It will be shown in section 4 that this does indeed create higher correlations between *θ*_{eb} and 〈*T* 〉.

It is conceivable that the correlation between *θ*_{eb} and 〈*T* 〉 is actually dominated by a correlation between sea surface temperature (SST) and 〈*T* 〉. This will be the case if the relative humidity in the boundary layer is fixed and the surface air temperature is tightly tied to SST. Figures 4 and 5 show correlation maps for SST and 〈*T* 〉 that have been low- and high-pass filtered with 6-month cutoffs. For the low-pass filtered case, *R*_{oa} is 0.27, somewhat lower than the 0.33 correlation between *θ*_{eb} and 〈*T* 〉. The overall correlation is lower for the high-pass filtered case as well (*R*_{oa} = 0.08 vs *R*_{oa} = 0.16 for *θ*_{eb} and 〈*T* 〉). This suggests that the correlation of SST with 〈*T* 〉 reflects their joint correlation to boundary layer characteristics in convecting regions.

Note in Fig. 5 that the correlation of SST with 〈*T* 〉 drops considerably in the Arabian Sea and north of Guyana when the data is high-pass filtered, just as was the case with the high-pass filtered *θ*_{eb} versus 〈*T* 〉 correlation. Indeed, there is a striking similarity in the spatial distribution of the correlation coefficients for SST versus 〈*T* 〉 and *θ*_{eb} versus 〈*T* 〉. Although the correlation between SST and 〈*T* 〉 is smaller than the overall correlation between *θ*_{eb} and 〈*T* 〉, the similarity in the spatial distributions of the correlation coefficients for the two cases suggests that SST is playing a direct role in the *θ*_{eb} versus 〈*T* 〉 correlation, a result that is not surprising given that SST and *θ*_{eb} are connected through surface fluxes of moisture and sensible heat.

We can test the extent of the influence of SST on *θ*_{eb} by correlating these two variables with each other and with the boundary layer specific humidity *q*_{b}. These correlations are shown in Table 1. First, note that *θ*_{eb} and *q*_{b} have a correlation coefficient of 0.99, so it is reasonable to think of *θ*_{eb} as being determined entirely by *q*_{b}. For the low-pass filtered case, the overall correlation coefficient for SST and *θ*_{eb} is 0.60, and for SST and *q*_{b} it is 0.57. Thus SST explains about 36% and 34% of the variance of *q*_{b} and *θ*_{eb}, respectively; the similarity in these numbers is reasonable given the strong correlation of *q*_{b} and *θ*_{eb}.

While a considerable portion of the variability of *θ*_{eb} is explained by its relation to SST, it is clear that there is room for other factors to play a role, such as surface wind speed, convective downdrafts, and entrainment of air into the top of the boundary layer.

### a. Precipitation windowing

So far we have looked only at correlations computed using all of the data regardless of the monthly precipitation rate. Since SQE theory is expected to apply only when there is sufficient precipitation, the previous correlation coefficients *R*_{oa} have been recomputed using the precipitation windowing discussed in section 2. Since precipitation windowing removes many months of data at each grid point, we do not attempt to time filter windowed time series. Table 2 shows the results for each of the four precipitation rate quartiles. There is clearly a gradual increase in the correlation between *θ*_{eb} and 〈*T* 〉, and between SST and 〈*T* 〉, as the threshold precipitation rate *P*_{min} is increased. The value of *R*_{oa} for *θ*_{eb} and 〈*T* 〉 for the case with *P*_{min} = 196 mm month^{−1} is 0.33, which is as large as the value of *R*_{oa} obtained when a 6-month low-pass filter is used with no precipitation windowing. In general, precipitation windowing increases *θ*_{eb} versus 〈*T* 〉 correlations as would be expected if SQE theory is valid and also produces a smaller increase in the correlation between SST and 〈*T* 〉.

The correlation coefficients of *θ*_{eb} and 〈*T* 〉 are greater than 0.5 over much of the Indian, east Pacific, and Atlantic Oceans. The increase in the correlation coefficients when precipitation windowing is used also lends some support to SQE. However, much less of the overall variability of 〈*T* 〉 is actually explained by *θ*_{eb} than one might hope, due in part to the rather low correlations in the portion of the western Pacific that lies east of the Maritime Continent. In the next section, we focus on regional differences using regionally averaged time series.

## 4. Regionally averaged time series

In order to reduce noise in the data associated with random instrument errors and small-scale spatial inhomogeneity, and to test SQE on even larger spatial scales, time series have been constructed by averaging the data over four different regions; the Indian Ocean (20°S–20°N,40°E–100°E), west Pacific (20°S–20°N, 100°E–170°E), east Pacific (20°S–20°N,140°W–100°W and 20°S–10°N,100°W–80°W), and the Atlantic (20°S–20°N,60°W–10°E). The central Pacific has been omitted due to the lack of COADS data there. In addition to spatially averaging over all of the grid points in each of these regions, precipitation windowing in the highest precipitation quartile (*P*_{min} = 196 mm month^{−1}) has also been applied to the data. Precipitation windowing was applied to all of the grid points at a given time, and only those grid points with precipitation rates exceeding *P*_{min} were included in the spatial average. For all variables except 〈*T* 〉, the spatial averaging was done using total quantities. The annual cycle of the resulting time series was separately analyzed. To examine intermonthly and interannual variability, it was removed to produce a monthly anomaly time series. In all four regions, 10% of the variance in 〈*T* 〉 and 30%–40% of the variance in *θ*_{eb} was intermonthly, 30%–40% of the variance in 〈*T* 〉 and 20%–30% of the variance in *θ*_{eb} was in the annual cycle, and 50%–60% of the variance in 〈*T* 〉 and 40% of the variance in *θ*_{eb} was interannual.

### a. Correlations for regionally averaged time series

Before examining the details of the regionally averaged time series of *θ*_{eb}, 〈*T* 〉, and SST, we first show in Figs. 6–9 the correlation coefficients computed using monthly anomaly time series. Each figure intercompares regions with 0-, 6-, or 12-month low-pass filtering, with or without precipitation windowing.

Figure 6 shows the correlations computed for the regionally averaged *θ*_{eb} and 〈*T* 〉 time series. Even the unfiltered, unwindowed correlations are now larger in every region than the largest value of *R*_{oa} obtained using low-pass filtering on the *O*(300 km) scale examined in the previous section. The highest correlation is in the east Pacific (0.71) followed closely by the Indian Ocean (0.69). The correlation in the Atlantic is somewhat smaller (0.58), and the correlation in the west Pacific (0.47) is by far the smallest. When precipitation windowing is applied to the data, the correlation coefficients increase in every region except the Indian Ocean, in which there is a small drop. The increase in the correlation coefficient after precipitation windowing is small in the east Pacific, larger in the Atlantic, and quite substantial in the western Pacific. Note that for all the regions except the west Pacific the correlation coefficient increases considerably as low-pass filtering is applied with increasing low-pass cutoffs. The same is true for the precipitation windowed correlation in the west Pacific, but the unwindowed correlation there is virtually unaffected by the low-pass filtering.

The results for the west Pacific and Atlantic in Fig. 6 are closest to what might be expected if the SQE hypothesis is valid. The linear relation between *θ*_{eb} and 〈*T* 〉 is maintained only if sufficient precipitation is occurring, presumably allowing the clouds to communicate changes in *θ*_{eb} to the troposphere as well as changes in 〈*T* 〉 to the boundary layer via downdrafts. If the west Pacific results in Fig. 6 are recomputed averaging only over “dry” grid boxes with precipitation between 0 and 196 mm month^{−1}, the correlations become small and negative (see Fig. 7). Similar “dry” windowing in the other three regions also results in substantial, though smaller, drops in the correlations between *θ*_{eb} and 〈*T* 〉.

The effect of precipitation windowing on the *θ*_{eb} versus 〈*T* 〉 correlation in the west Pacific is even more dramatic if the west Pacific is redefined to extend from 135°E to 170°E, thereby excluding the Maritime Continent. Table 3 shows the results of correlating *θ*_{eb} and 〈*T* 〉 in the west Pacific with and without the Maritime Continent using various filtering and precipitation windowing strategies. Although the correlations are all smaller when the Maritime Continent is excluded, the increase in the correlations when precipitation windowing is employed is much greater. It is clear that the correlations are most sensitive to precipitation windowing over the warm pool region in the west Pacific.

The results in Figs. 6 and 7 indicate that the SQE assumption will be qualitatively correct in all regions, so long as *P*_{min} = 196 mm month ^{−1} is chosen. In section 3a, it was determined that the correlations between *θ*_{eb} and 〈*T* 〉 tended to be somewhat stronger than the correlations between SST and 〈*T* 〉 but that SST was probably playing a substantial role in determining *θ*_{eb}. Moreover, it was stated that the correlation between SST and 〈*T* 〉 would likely increase at larger time and space scales, over which the atmosphere would be closer to a state of radiative convective equilibrium.

Figure 8 shows the correlation plots for the regionally averaged SST and 〈*T* 〉 time series. The results in this figure indicate that SST and 〈*T* 〉 do not correlate as well as *θ*_{eb} and 〈*T* 〉 in the west Pacific and the Atlantic Ocean on these very large space scales. However, SST and 〈*T* 〉 correlate as well as, or slightly better than, *θ*_{eb} and 〈*T* 〉 in the Indian Ocean and east Pacific. However, even in the latter regions the correlations between *θ*_{eb} and 〈*T* 〉 are very close in size to the SST versus 〈*T* 〉 correlation. As on smaller spatial averaging scales, both unwindowed and windowed *θ*_{eb} and SST are closely related, as shown in Fig. 9. The correlations of SST with *θ*_{eb} are no lower than 0.66 in any of the regions and exceed 0.80 in most cases.

### b. Does SQE predict the correct proportionality coefficient?

Using relation (7), we can quantitatively test the regionally averaged data to see how closely it approximates the SQE hypothesis. Figure 10 shows scatterplots of regionally averaged *θ*_{eb} and 〈*T*〉 for each of the four regions, using precipitation windowing with *P*_{min} = 196 mm month^{−1}, including the seasonal cycle and using no time filtering.

The basin-wide average measurement uncertainties of *θ*_{eb} and 〈*T*〉 are difficult to assess. The regression of 〈*T* 〉 on *θ*_{eb} would be appropriate if *θ*_{eb} had negligible errors relative to its overall variation compared to 〈*T* 〉. The regression of *θ*_{eb} on 〈*T* 〉 would be appropriate if 〈*T* 〉 had much smaller relative errors. If both measurements have comparable relative errors, the slope of the best linear fit would lie between the slopes of the two regression lines we have plotted. The slope of the regression line of 〈*T* 〉 on *θ*_{eb} is 0.11–0.16 for all the regions. The regression of *θ*_{eb} on 〈*T* 〉 yields a larger slope of 0.23–0.29. We conclude that the best fit slope is 0.2 ± 0.1 for all regions.

The above analysis compares overall temporal variations in *θ*_{eb} and 〈*T* 〉, which are mainly associated with intermonthly variability. Figure 11 shows the corresponding scatterplots for the 12 months in the 11-yr average annual cycle. Except for the Atlantic Ocean, the correlations between *θ*_{eb} and 〈*T* 〉 are 0.92–0.94, and the two regression lines for each region have much more similar slopes of 0.25–0.29. In the Atlantic there is somewhat more scatter. The slope is within (but at the high end of) the range suggested by the overall scatterplots.

Lastly, we may ask whether *spatial* variations in *θ*_{eb} and 〈*T* 〉 are similarly correlated. Figure 12 shows their time averages over the four regions. Again, the regions with higher time-mean *θ*_{eb} have higher time-mean 〈*T* 〉, and the slope is 0.15, similar to that found in the temporal scatterplots.

The SQE-predicted slope of 0.43 from (7) is much larger than any of these observed slopes, and even the predicted reversible adiabatic slope of 0.31 is somewhat too large. Equation (7) assumes that an increase in *θ*_{eb} will be accompanied by an increase in 〈*T* 〉 such that CAPE remains constant. Since the observed increases in 〈*T* 〉 are smaller than predicted by (7), the CAPE must on average be increasing when *θ*_{eb} increases. Therefore, CAPE is not remaining constant on monthly timescales even after averaging over large regions (each region covers ∼25 million square kilometers, which is about three times the size of the continental United States).

How large are the changes in CAPE implied by Fig. 10? If 〈*T* 〉 is held constant and *θ*_{eb} is increased by 1 K from 355 K to 356 K, CAPE would increase by ∼300 J kg^{−1}. The slopes of the best fit lines to the observed data in Fig. 10 imply that the change in 〈*T* 〉 is only about a third to a half of that necessary to maintain constant CAPE as required by SQE. Thus a 1 K increase in *θ*_{eb} produces a 150–200 J kg ^{−1} increase in CAPE. CAPE in the Tropics typically lies between 500 and 1500 J kg^{−1}, so a 200 J kg^{−1} fluctuation would represent a 13%–25% change in CAPE. The assumption of constant CAPE made in the SQE hypothesis does not hold, even on the large time and space scales considered here.

Another interesting comparison can be made by constructing a *θ*_{eb} versus 〈*T* 〉 scatterplot for the one-dimensional radiative convective equilibrium (RCE) model described in Brown and Bretherton (1995), which is based on the Emanuel (1991) convective parameterization and a fixed radiative cooling rate. To make such a plot, the RCE model was run out to an approximate equilibrium using a series of different SSTs. The RCE model equilibrates to higher values of *θ*_{eb} and 〈*T* 〉 as SST increases, with 〈*T*′〉/*θ*^{′}_{eb} = *γ* = 0.31. This value is consistent with SQE for the reversible moist adiabat used for lifting undilute updraft air in the Emanuel convective parameterization. It would also be interesting to examine the relationship between 〈*T*′〉 and *θ*^{′}_{eb} in general circulation model output.

### c. Interannual variability in the regional averages

Using MSU channel 2 data, Yulaeva and Wallace (1994) documented the warming of the entire tropical troposphere that slightly lagged ENSO (El Niño–Southern Oscillation) warm events. They also found small SST anomalies in the tropical Atlantic and Indian Oceans that appeared to be associated with ENSO. Convection provides a plausible mechanism for producing this link. The SQE hypothesis suggests that if 〈*T* 〉 rises, *θ*_{eb} will have to rise before convection takes place. We have seen that anomalies of *θ*_{eb} are closely related to SST anomalies, completing the link. The SQE hypothesis does not suggest what surface energy flux imbalance produces the rise of SST in a particular region, only that such a rise is necessary to sustain deep convection in that region.

To test this idea, we focus on the interannual variability in the regional precipitation windowed time series by removing the annual cycle and applying a 12-month low-pass filter to all time series. Figure 13 shows that the dominant signal in 〈*T* 〉 (i.e., *θ _{e}*

_{sqe}=

*γ*

^{−1}

_{sqe}〈

*T*〉) is quite similar in all four regions. The largest 〈

*T*〉 transients occur in 1983–84 and 1987–88. These coincide with the two major well-defined ENSO warm events that occurred during the 1980s. In addition, all four regions show a broad peak in 〈

*T*〉 that begins in mid-1989 and carries on to the end of the time series in 1991; this peak has also been identified as an El Niño signal.

The ENSO signal also shows up in *θ*_{eb} in all four regions in both major warm events, although it is less distinguishable from other variability on shorter timescales. Not surprisingly, the ENSO signal in *θ*_{eb} is strongest in the eastern Pacific, where the amplitude of the SST variations associated with ENSO are strongest (Philander 1990). The amplitudes of *θ _{e}*

_{sqe}variations are approximately half as large as the variations in

*θ*

_{eb}and imply a

*γ*of 0.15–0.2 on interannual timescales in all regions, consistent with our earlier findings for shorter timescales.

The value of *θ _{eRH}* tracks closely with

*θ*in each of the four regions. In addition to documenting the tight coupling between SST and the boundary layer properties, this lends credibility to the

_{eb}*θ*time series, since SST and

_{eb}*θ*are measured independently.

_{eb}A comparison of the 12-month low-pass filtered time series with and without precipitation windowing (not shown) shows that the main effect of precipitation windowing is to improve the match between *θ _{eb}* and 〈

*T*〉 (or

*θ*

_{esqe}) during the warm events. This is particularly evident in the west Pacific, where the correlation between

*θ*and 〈

_{eb}*T*〉 improves markedly for the 1982–83 event when precipitation windowing is applied.

## 5. Summary

The SQE hypothesis predicts that in regions of active deep convection 〈*T* 〉 = *γ**θ*_{eb}, where *γ* ∼ 0.4. Our study tested this hypothesis over the tropical oceans using COADS data to obtain monthly *θ*_{eb}, and monthly MSU data to obtain monthly 〈*T* 〉 over 11 years. To isolate regions of persistent convection, precipitation windowing was used to select only 2.5° × 2.5° grid boxes in which the precipitation rate lay in the highest quartile of the MSU-derived precipitation used for the analysis. We found significant positive correlations in most grid boxes in the regions in which deep convection is most prevalent. The correlations are strongest for variability on timescales exceeding 6 months but are positive even at shorter timescales. The correlations improve when precipitation windowing is used, especially in the west Pacific, supporting the idea that convection acts to couple *θ*_{eb} and 〈*T* 〉. The correlations of *θ*_{eb} and 〈*T* 〉 increase even more if larger areas (about 60°–70° longitude, 40° latitude) are considered.

However, scatterplots of areally averaged observations on intermonthly, seasonal, and interannual timescales, as well as the time-mean spatial variability between regions all imply *γ* ∼ 0.2. This slope is only half that predicted by SQE. These observations imply that CAPE varies by 13%–25%, even on long time and large space scales, contrary to the SQE assumption of constant CAPE. If Williams and Renno’s (1993) results are interpreted in terms of *γ*, they also imply *γ* = 0.1–0.2. This “misplaced” energy could have important consequences in instability studies.

Preliminary results from a similar study using TOGA COARE sounding data (Brown 1994) to examine SQE on timescales of as short as a day indicate that *θ*_{eb} and 〈*T* 〉 still remain positively correlated even on timescales of less than 5 days, but again the ratio of the changes in 〈*T* 〉 to the changes in *θ*_{eb} is much smaller than predicted by SQE. At intermediate timescales of 10–30 days, Brown (1994) did not find a significant correlation between 〈*T* 〉 and *θ*_{eb}.

These results do not fully support the SQE hypothesis on any timescale. However, on interannual timescales, the empirical linear relation between *θ*_{eb} and 〈*T* 〉 does explain a sizeable fraction of the variance of the latter, so it could conceivably be useful for simplified coupled ocean–atmosphere models of the Tropics.

Another important result from this study is that *θ*_{eb} generally correlates somewhat better with 〈*T* 〉 than does SST. This supports the notion that even if SQE is not viable, boundary layer physics plays an important intermediary role in the transport of energy from the sea surface to the free troposphere. This in turn suggests that models of the tropical atmosphere should include time-dependent boundary layer physics and should avoid fixing boundary layer conditions whenever possible. In fact, our results suggest that the moist convective coupling of SST, *θ*_{eb}, and 〈*T* 〉 provides a possible mechanism for tropical Pacific SST variations due to ENSO to affect SST in other ocean basins.

## Acknowledgments

This research was supported by NSF Grants ATM8858846 and ATM9216645. Discussions with Dr. Kerry Emanuel were also very beneficial.

## REFERENCES

## Footnotes

*Corresponding author address:* Dr. Christopher S. Bretherton, University of Washington, Department of Atmospheric Sciences, Box 351640, Seattle, WA 98195-1640.

Email: breth@atmos.washington.edu

^{1}

Lin and Johnson’s (1996) radiosonde time series over the TOGA-COARE IFA suggests that typical monthly anomalies of vertically averaged tropospheric mixing ratio are no larger than 0.5 g kg^{−1}, producing virtual temperature perturbations of 0.1 K. This is 10% of the MSU temperature anomalies.