1. Introduction
A horizontal-scale separation is implied when 
In the tropical troposphere, temperature gradients are small because of the large Rossby radius of deformation (Charney 1963), reflecting the mighty power of gravity to flatten density surfaces efficiently as diabatic effects act more gently to deform them. Temporal changes of temperature are negligible as well, and this combination is referred to as the weak temperature gradient (WTG) approximation (Sobel et al. 2001). Under the WTG approximation, the evolution of CMSE prognosed by Eq. (3) is 
The response to localized heating (like a water-condensation event) in the stratified tropical atmosphere is not confined within the local area near the precipitation, but is extended to a much wider area through the mechanism of equatorial internal waves (Matsuno 1966; Itoh 1977; Gill 1980). The larger-scale motions associated with such waves, in turn, advect water vapor and DSE through Eqs. (1) and (2) in ways that affect the probability of subsequent convection-mediated precipitation. The interplay of these mechanisms makes precipitating convectively coupled tropical disturbances fascinating and challenging to understand and predict.
Unfortunately, Eqs. (1) and (2) are unsatisfactory to work with directly because of the aforementioned large but intimately cancelling terms in the spotty fields of 
In this way, prior studies of the CMSE budget equation have illuminated aspects of many moist tropical weather phenomena, including convectively coupled equatorial wave (CCEW) disturbances (Kiladis et al. 2009) and the Madden–Julian oscillation (MJO; Madden and Julian 1971, 1972). For instance, Kiranmayi and Maloney (2011) examined MSE budgets of the MJO and found significant contributions from horizontal advection in increasing (decreasing) column MSE before (after) peak MJO convection. Sobel et al. (2014) analyzed CMSE over the tropical Indian Ocean and found that radiative feedbacks importantly enhance the MJO’s moist static energy anomalies, while its eastward propagation is associated with advection of MSE. Similar results were also reported over the eastern part of the Maritime Continent by Yokoi and Sobel (2015). Inoue and Back (2015a) explored MSE budgets in rawinsonde array time series data over the western Pacific and found significantly different term balances on different time scales: For higher-frequency variations, vertical advection predominates in explaining CMSE variations, while horizontal advection and physical source terms contribute more to longer time-scale variations. In apparent contrast, studies by Masunaga and L’Ecuyer (2014) and Sumi and Masunaga (2016) emphasized the importance of horizontal as well as vertical MSE advection in budgets of their shorter-time-scale CMSE variations. The overarching point is that the relative sizes of the CMSE budget’s terms are space- and time-scale dependent.
Other CMSE budget investigations have used lagged regression or composite analysis, often with base time series from filtered precipitation data, with signals isolated by masking precipitation variance in the wavenumber–frequency domain and transforming back to physical space (e.g., Kiladis and Weickmann 1992; Bantzer and Wallace 1996; Wheeler et al. 2000; Yasunaga and Mapes 2012). The wavenumber–frequency masking regions (filter boxes) for CCEW anomalies are somewhat ad hoc, but a fairly customary set has evolved through a wide variety of uses in previous investigations and will be utilized later in this study.
In an attempt to synthesize these insights from phenomenon-specific and frequency-specific studies, this study extends CMSE budget analysis to the entire space–time spectral domain for the tropical belt. We isolate precipitation-associated and well-analyzed MSE fluctuations by utilizing cross-spectra between reanalysis-derived estimates of each term in the CMSE budget equation and nearly independently estimated satellite-derived precipitation. From that full-spectral-domain result, we can summarize the results in tables for the entire customary set of filter boxes encompassing the whole family of tropical disturbances. In this way, other CMSE budget studies can at least be emplaced within a common reference standard, even if our present datasets are not definitively accurate—as our analysis of residuals below indicates they are not.
2. Data and methodology
a. Data description
For precipitation estimates, TRMM-3B42, version 7, data are used, spanning the period from 1 January 1998 to 31 December 2013 (Huffman et al. 2007). For dynamical and physical fields, including horizontal and vertical winds, geopotential height, temperature, and specific humidity, our work uses European Centre for Medium-Range Weather Forecasts (ECMWF) interim reanalysis (ERA-Interim, herein ERAI) data (Berrisford et al. 2011). Short-term (6 and 12 h) forecast values from the ERAI dataset were used for model physics scheme–derived source terms such as the latent and sensible heat fluxes at the surface and net radiation at the surface and top of atmosphere. To derive the CMSE budget terms in Eq. (3), vertical integrations were calculated from the surface to 100 hPa. Strictly speaking, the vertically integrated radiative heating rate was calculated as the difference in the radiative flux between the surface and top of the atmosphere fields. In part because of this discrepancy, but also because of other ERAI field and process errors, the data-estimated CMSE budget equation contains residuals, which will be discussed in section 5.
b. Calculation procedure
Space–time power and cross-power spectra were calculated using a standard fast Fourier transform (FFT) algorithm. The calculation procedure is identical to our previous paper (Yasunaga and Mapes 2012) and will be briefly repeated here.
First, precipitation data and estimates of each term in Eq. (3) were partitioned into equatorially symmetric and antisymmetric components for the equatorial band 15°S–15°N. Then, the mean and first three harmonics of the climatological annual cycle were subtracted to yield temporal anomalies. Time series of those anomaly data were divided into 92-day segments, allowing overlaps of about two months to prevent information losses due to windowing of segments. Each segment was detrended and tapered to zero over the first and last nine days. Finally, the power and cross-spectra were computed for each segment and averaged over all segments.
c. Cross-spectra of column-integrated budget equation of MSE
The reason to bring in satellite precipitation (rather than using ERAI precipitation or multiplying ERAI’s own 
d. Interpretation of the cross-spectrum and its relevance to the GMS
The gross moist stability (GMS), which was originally introduced by Neelin and Held (1987), is a measure of MSE transport by airflow per unit of latent energy (vapor) lost to precipitation. A variety of versions of GMS have been defined for data reasons or analyst preference. For instance, entropy rather than MSE may be used as the intensive moist conserved thermodynamic quantity in the numerator, and/or another measure of condensation heating is sometimes used in the denominator (Raymond et al. 2009).
The power spectrum of precipitation, which is the denominator in Eq. (5), is another valid measure for the strength of moist convection at a given wavenumber and frequency. Therefore, the column-integrated MSE advection terms in Eq. (5) 
We reiterate that precipitation variations are governed or explained in this way only indirectly because they are related to CWV variations, which in turn are related to CMSE because WTG holds to a good approximation. However, because of the scale dependence mentioned above, these two logical dependencies are not equally valid, and the nonlinear CWV–P relationship does not imply a constant proportionality coefficient between anomalies, across the entire spectral domain. In the next section, therefore, the relationship between 
3. Results
a. Precipitation versus CMSE (P versus 
Figures 1a and 1b display coherence squared spectra between precipitation and reanalysis CMSE. Although the statistically significant area extends over almost the entire domain, prominent peaks are positioned along with the annotated dispersion curves of the theoretical equatorial waves: n = 1 equatorial Rossby (ERn1) wave, Kelvin (Klvn) wave, mixed Rossby gravity (MRG) wave, n = 0 eastward inertia–gravity (EIGn0) wave, and n = 1, and n = 2 westward inertia–gravity (WIGn1 and WIGn2) waves. Peaks associated with MJO and tropical depression (TD)-type disturbances are also evident. Hereafter, we focus on regions around the peaks of the coherence spectrum between precipitation and CMSE, enclosed with thick solid lines in Figs. 1a and 1b. Labels show the contracted name corresponding to each type of disturbance. There is another significant peak of coherence around wavenumber 3 and period of 1.5–2.5 days. However, it does not represent a CCEW disturbance and is left unenclosed (see section 3c).
(a),(b) Space–time coherence squared spectrum and (c),(d) phase spectrum between precipitation and CMSE; (left) symmetric and (right) antisymmetric components. In (a) and (b), the shading and contour interval is 0.1 with the first contour at 0.02, which is significant at the 99% level with 507 DOF. In (c) and (d), the shading and contour interval is 18°, and positive and negative values (shaded with warm and cool colors) mean the column MSE predates and lags rainfall, respectively. Only the area with coherence squared larger than 0.02 is shaded. Dispersion curves are plotted for Klvn, ERn1, WIGn1, WIGn2, EIGn0, EIGn1, and MRG waves with equivalent depths of 8, 12, 20, 30, 50, and 90 m (corresponding to Klvn wave-phase speeds of about 9, 11, 14, 17, 22, and 30 m s−1, respectively). Six dashed lines for the symmetric components indicate constant phase speeds of 3.0, 4.0, 5.0, 6.0, 7.0, and 8.0 m s−1. The region enclosed with thick solid lines represents boxes to evaluate CMSE budgets for each CCEW (see text).
Citation: Journal of Climate 32, 2; 10.1175/JCLI-D-18-0334.1
Figures 1c and 1d show phase spectra between precipitation and CMSE. The area with coherence squared larger than 0.02, which is significant at the 99% level, is shaded. Positive and negative values of the phase mean that CMSE predates and lags rainfall, respectively (see the appendix for more detail). CMSE lags precipitation around the coherence peak associated with ERn1, Kelvin, MRG, EIGn0 waves, and TD-type disturbances (−26.8° for ERn1, −19.0° for Kelvin, −28.7° for MRG, −7.3° for EIGn0, and −14.1° for TD-type disturbance over the region enclosed with thick solid lines in Figs. 1a,b), while CMSE predates rainfall associated with MJO, EIGn0, and WIG waves (19.3° for MJO, 16.6° for WIGn1, and 4.2° for WIGn2 wave disturbances over the region enclosed with thick solid lines in Figs. 1a,b). The 99% confidence interval for phase difference error is estimated to be about ±14.5° where coherence squared is 0.02 and DOF is 507, so most of the phase differences are statistically distinct from 0. However, we suspect the phase of reanalysis CWV (or CMSE) contains some biases (e.g., Figs. 5 and 6 in Yasunaga and Mapes 2012). Therefore, we still regard it as a valid idea that variations of CMSE and precipitation are basically in phase (at least around the coherent peaks enclosed with thick solid lines in Figs. 1a,b).
An alternate view of the same information is in Fig. 2, which shows real and imaginary components of cross-spectra between precipitation and CMSE. They are normalized by the amplitude of rainfall variation at a given wavenumber and frequency, and, therefore, have the unit of time (hour). The imaginary component of the cross-spectrum (Figs. 2c,d) is smaller than its real component (Figs. 2a,b), reflecting the almost-in-phase relationship we could not reject above. Real components of the cross-spectrum clearly decrease with frequency in Figs. 2a and 2b, meaning that for a given precipitation anomaly, the amplitude of the associated CMSE anomaly is larger for lower-frequency disturbances. This is a familiar result from prior studies (e.g., Mapes et al. 2006) and is consistent with the notion that high-frequency waves are orchestrated by wavelike internal density variations, with only a secondary role for moisture anomalies (e.g., Kuang 2008), while moisture storage and transport is essential to, for instance, the MJO (e.g., Adames and Kim 2016).
(a),(b) Real and (c),(d) imaginary components of cross-power spectrum between precipitation and CMSE normalized by the power spectrum of precipitation. The shading and contour interval is 2.5 h, and positive and negative values are shaded with warm and cool colors, respectively. Only the area with coherence squared larger than 0.02 is shaded.
Citation: Journal of Climate 32, 2; 10.1175/JCLI-D-18-0334.1
Having established the nonconstancy (but almost everywhere positive values) of P–CWV correlations across spectral space, so that 
b. Precipitation versus horizontal advection (P versus 
The coherence squared spectrum between precipitation and column-integrated horizontal advection of MSE (Figs. 3a,b) lacks the CCEW peaks seen above. Instead, the most prominent feature is a gently upward–leftward sloping feature in the “TD type” spectral region, predominantly as positive values in the real and imaginary components (Figs. 3c–f). Since TD-type disturbances generally develop in the intertropical convergence zone (ITCZ) of the Northern Hemisphere, the corresponding signal is split between the symmetric and antisymmetric spectra (right and left columns). The slope of the spectral feature agrees with typical easterly wind speeds of a few meters per second, consistent with horizontal advection of localized anomalies (whose Fourier spatial spectrum is, therefore, broad) by mean winds. In addition, the dominance by positive cross power in the imaginary component (Figs. 3e,f) is also indicative of advectively driven propagation (the advance of temporal phase, in Fourier terms) of CMSE anomalies. These are in accord with common experience for such disturbances, including synoptic disturbances over the Pacific Ocean (e.g., Nitta 1970; Wallace 1971; Reed and Recker 1971; Nitta and Takayabu 1985; Liebmann and Hendon 1990; and many others) as well as African “easterly waves” over the Atlantic Ocean (e.g., Carlson 1969; Frank 1969; Burpee 1972, 1974, 1975; Reed et al. 1977; Nitta 1978; and many others). Despite some differences between basins, the results of Fig. 3 are basically similar if the analysis domain is limited over an Indo-Pacific region (45°–225°E) with warm surface temperature (not shown). On the other hand, the evolution of vorticity, which is one of the essential components of the TD-type mode (e.g., Thorncroft and Hoskins 1994; Sobel and Bretherton 1999; Hall et al. 2006), is beyond the scope of the present MSE-based analysis.
(a),(b) Space–time coherence squared spectrum and (c),(d) real and (e),(f) imaginary components of the cross-power spectrum between precipitation and column-integrated horizontal advection of MSE normalized by the power spectrum of precipitation 
Citation: Journal of Climate 32, 2; 10.1175/JCLI-D-18-0334.1
Another significant (coherence squared > 0.02) area extends upward–rightward from the MJO region, corresponding to an eastward phase speed of about 4.6 m s−1, a speed typical of MJO propagation (e.g., Zhang 2005) and also typical of its westerly wind anomalies. Furthermore, positive imaginary components also indicate that horizontal advection acts to advance the phase of CMSE anomalies associated with MJO, although the real components are negative (meaning that horizontal advection plays a role in damping). Taken together, these results suggest that propagation by horizontal advection of MSE may be one of the essential features that distinguish between dynamical (or equatorial internal density wave) versus nondynamical modes (MJO and TD-type disturbance).
Another interesting feature in Fig. 3a is a peak of coherence near wavenumber 1 (westward) and frequency of 0.2–0.3 cpd, which is not clear in Fig. 1a. Exploring deeper, we found that zonal and meridional winds have barotropic structures at the corresponding wavenumber and frequency and that the horizontal advection of MSE integrated over the troposphere coherently varies with zonal wind in the stratosphere as well as in the troposphere (not shown). Based on previous works (e.g., Hendon and Wheeler 2008; King et al. 2015, 2017), this signal apparently reflects the 5-day Rossby–Haurwitz wave. In fact, Hendon and Wheeler (2008) show that zonal wind associated with the 5-day Rossby–Haurwitz wave lags negative outgoing longwave radiation (OLR) by about 45° and the same-signed phase lag from precipitation (about 70°) is implied at the corresponding wavenumber and frequency in our analysis (not shown, but estimable from Figs. 3c,e). King et al. (2017) found that the 5-day Rossby–Haurwitz wave modulates winds and that precipitation anomalies are strongly correlated with moisture flux convergence in the lower level, but CWV does not show close relationships with those precipitation anomalies, consistent with the absence of this signal in our Fig. 1a.
c. Precipitation versus vertical advection (P versus 
Vertical advection’s cross-spectral coherence (Figs. 4a,b) exhibits clear peaks in almost all modes (MJO, ERn1, Kelvin, MRG, EIGn0, WIGn1, and WIGn2 waves), but with relatively weak coherence in low-frequency and TD-type disturbances. The real component of the cross-spectra (Figs. 4c,d) is universally negative in the high-coherence regions, indicating that vertical advection acts to damp these CMSE anomalies. Absolute values are largest in small-wavenumber regions, perhaps a real effect but perhaps indicating that vertical velocity signals associated with satellite-observed precipitation variations are better analyzed (i.e., captured with greater amplitude by the reanalysis system from observational inputs) at larger scales.
As in Fig. 3, but for the results from cross-power spectrum between precipitation and column-integrated vertical advection of MSE normalized by the power spectrum of precipitation 
Citation: Journal of Climate 32, 2; 10.1175/JCLI-D-18-0334.1
Imaginary components (Figs. 4e,f) are positive for Kelvin, MRG, EIGn0, WIGn1, and WIGn2 waves. This means that vertical advection acts to drive horizontal propagation of such wave disturbances (i.e., it acts to advance temporal phase, in Fourier terms). On the other hand, small positive and negative values are found for the ERn1 wave and MJO, respectively, indicating that vertical advection has small (but slightly advancing and retarding) effects on the migration of those oppositely propagating low-frequency variations, that is, driving both types of anomaly westward.
Another notable feature of Fig. 4a is a fast-propagating signal in the eastward direction (around wavenumber 3 and period of 1.5–2.5 days), which is also confirmed in Fig. 1a. The phase spectrum between precipitation and CMSE in this region (Fig. 1c) indicates that CMSE predates precipitation by a little over 90°. Although variations of the MSE vertical advection and precipitation are almost in phase (phase difference is less than 45°, implied by Figs. 4c,e), there is no corresponding peak in the coherence spectrum between precipitation and column-integrated vertical advection of DSE (not shown). This implies that the upward motion associated with the fast-propagating signal is not accompanied with detectable latent heating, although it advects water vapor and thus 
(a),(c) Space–time coherence squared spectrum between equatorially symmetric precipitation and equatorially symmetric zonal wind at 850 (upper) and 150 hPa (lower) levels. (b),(d) Space–time coherence squared spectrum between equatorially symmetric precipitation and equatorially antisymmetric meridional wind at 850 (upper) and 150 hPa (lower) levels. The contour and shading interval is 0.1 with the first contour at 0.02.
Citation: Journal of Climate 32, 2; 10.1175/JCLI-D-18-0334.1
d. Precipitation versus radiative heating (P versus 
Figures 6a and 6b show coherence squared between precipitation and radiative heating. Coherence peaks are seen in almost all modes (MJO, ERn1, Kelvin wave, TD type, MRG, EIGn0, WIGn1, and WIGn2 waves), similar to that between precipitation and CMSE (Figs. 1a,b), consistent with positive anomalies of CMSE, CWV, and the associated clouds acting to reduce OLR.
As in Fig. 3, but for the results from cross-power spectrum between precipitation and column-integrated radiative heating 
Citation: Journal of Climate 32, 2; 10.1175/JCLI-D-18-0334.1
Positive values of the real component cover almost the entire domain (Figs. 6c,d), indicating that radiation acts to amplify CMSE anomalies associated with any type of disturbance. The value of this positive real component is much greater for low frequencies and large scales. We suspect that precipitation-associated cloudiness may be analyzed too weakly in ERAI data at smaller scales and/or shorter time scales, since cloud is not an assimilated quantity per se but rather depends on assimilated vertical motion as well as moisture. For instance, Fig. 8 of Adames and Kim (2016) indicates from direct satellite observation that OLR per unit precipitation decreases by half (from 0.2 to 0.1) from wavenumber 1 to wavenumber 20 in MJO-related frequencies. Here the comparable slope with wavenumber (albeit of total radiative heating) is much steeper (0.16–0.04). This caveat of our data source for 
The imaginary component is slightly negative almost everywhere, but the absolute value is smaller than that of the real part (Figs. 6e,f). This means that the phase difference is close to 0, that is, that variations of the precipitation and radiative heating are almost in phase, with the negative values indicating a physically plausible slight lag of clouds behind precipitation in convective tropical weather.
e. Precipitation versus surface heat fluxes (P versus H + LE)
Figures 7a and 7b show coherence squared between precipitation and surface heat flux. Coherence and signal strengths are both weak, compared with those of radiative heating (Figs. 6a,b), and peaks are only found around MJO, Kelvin, MRG, and EIGn0 waves. It is interesting that the sign of the real and imaginary components of the cross-spectrum are opposite between the MJO and Kelvin waves (Figs. 7c–f), meaning that surface heat flux amplifies and retards MJO anomalies of CMSE, while it damps and advances Kelvin waves. In the antisymmetric components (right column), surface flux slightly damps (blue area) and drives toward the west both the MRG and EIGn0 wave disturbances (advancing the former but retarding the latter, according to the change of sign in Fig. 7f).
As in Fig. 3, but for the results from cross-power spectrum between precipitation and surface heat flux 
Citation: Journal of Climate 32, 2; 10.1175/JCLI-D-18-0334.1
Another remarkable feature of Fig. 7 is that there is a maximum of coherence around westward wavenumbers from around −1 to −5 and at the 8–15-day period (Fig. 7a), although a corresponding peak is not found in coherence between precipitation and CMSE (Fig. 1a). The real component of the cross-spectrum is large and positive here, while the imaginary component is weaker and negative (Figs. 7c,e), indicative of variations of surface heat flux that are in phase with (or slightly lagging behind) precipitation variations. This peak is also robust even if the analysis domain is limited over the oceanic region (45°–225°E) with warm surface temperature (not shown). Yasunaga et al. (2010) reports that diurnal variations of the sea surface temperature were enhanced in light-wind and clear conditions over the tropical Indian Ocean and suggests that the observed modulation was associated with westerly anomalies with a deep nearly barotropic structure through the troposphere. The coherence peak in Fig. 7a might reflect modulation of surface winds and precipitation by such barotropic disturbances. Further investigations are required to draw firmer conclusions about this intriguing signal.
4. Decomposition of the advection terms
If a total advection effect from Fig. 3 or Fig. 4 is adequately explained by advection of one of the mean 
a. Precipitation versus horizontal advection of decomposed MSEs
Horizontal advection of time-mean MSE 
As in Fig. 3, but for the results from cross-power spectrum between precipitation and column-integrated horizontal advection of time-mean MSE 
Citation: Journal of Climate 32, 2; 10.1175/JCLI-D-18-0334.1
The real component of the cross-spectrum is positive for MJO and ERn1 wave, indicating that advection of time-mean MSE gradients plays a role in amplifying CMSE anomalies associated with MJO and ERn1 wave disturbance (Figs. 8c,d). Meanwhile, the signs of the imaginary component are opposite (positive for MJO and negative for ERn1 wave) in Figs. 8e and 8f, meaning that advection of stationary components in moisture helps both of the MJO and ERn1 wave disturbances migrate eastward. These results are consistent with those obtained by previous theoretical works. Sobel et al. (2001) discuss the importance of the background moisture field to the propagation of low-frequency mode. Adames and Kim (2016) found that horizontal advection of mean moisture by the anomalous winds associated with ERn1, on the eastern side of the precipitation peak within that disturbance, is one of the keys for the eastward migration of that planetary-scale disturbance. Here total advection is found to be of opposite sign to that advection of the long time-mean gradient (noting the opposite signs around ERn1 wave spectral region in Figs. 3e, 8e).
Blue areas just left of center in both Figs. 3c and 8c indicate that the total horizontal advective damping of WIG waves (Fig. 3c) is partly due to advection of stationary (climatological) MSE. Both zonal and meridional winds in the lower troposphere have significant coherence in the corresponding region (e.g., Figs. 5a,b). Since meridional gradients of mean moisture field are generally larger than its zonal gradient, meridional advection might be thought to explain this advective damping of WIG wave disturbances. However, Sumi and Masunaga (2016) claimed that zonal advection of anomalous moisture has larger effects than meridional advection in the WIGn1 wave disturbance. Meanwhile, their filtering box is rather large (cf. our Fig. 1a to Fig. 1 of Sumi and Masunaga 2016), so reconciling this point is not possible without further decompositions of advection, a useful topic for a future study.
Horizontal advection of the remainder pattern 
As in Fig. 3, but for the results from cross-power spectrum between precipitation and column-integrated horizontal advection of the anomaly from time-mean MSE 
Citation: Journal of Climate 32, 2; 10.1175/JCLI-D-18-0334.1
b. Precipitation versus vertical advection of decomposed MSEs
Three terms are involved in decomposed vertical advection. Figure 10 shows cross-spectra when analyzed vertical motion is treated as if it “advects” the whole-tropics time-mean MSE 
As in Fig. 3, but for the results from cross-power spectrum between precipitation and column-integrated vertical advection of time–zonal-mean MSE 
Citation: Journal of Climate 32, 2; 10.1175/JCLI-D-18-0334.1
Vertical advection of spatially patterned but time-mean MSE 
As in Fig. 3, but for the results from cross-power spectrum between precipitation and column-integrated vertical advection of zonal perturbations of the time-mean MSE 
Citation: Journal of Climate 32, 2; 10.1175/JCLI-D-18-0334.1
As in Fig. 3, but for the results from cross-power spectrum between precipitation and column-integrated vertical advection of the anomaly from the time-mean MSE 
Citation: Journal of Climate 32, 2; 10.1175/JCLI-D-18-0334.1
Figures 11 and 12 are both corrections to the fictitious and erroneous vertical advection process in Fig. 10, relative to true advection (Fig. 4). In the real component, both are positive over the entire domain (Figs. 11c,d, 12c,d). Averaging over the whole tropics erroneously decreases and increases the value of moisture compared to its true values in ascending and descending regions, respectively. This leads to underestimated (overestimated) moisture convergence in the moist ascending (dry descending) regions, which makes the damping effect in Figs. 10c and 10d drastically too strong. Figures 11c, 11d, 12c, and 12d are successive partial corrections to that drastic error. This correction could also be understood from consideration of vertical motions in the moist environments around precipitation versus in an average (drier) environment: ascending (descending) regions are in more humid (drier) environments than average in time as well as in space. The relatively larger values in Fig. 12 than Fig. 11 indicate that the humidity of an ascending (or descending) region is due more to temporal than to merely spatial variations. This correction is greatest for low frequencies, which have the largest humidity anomalies per unit precipitation, as discussed around Figs. 2a and 2b. Therefore, positive correlations of the moisture and vertical motion might be essential to such low frequencies and large-scale disturbances. However, the aforementioned (further) decomposition of vertical wind is needed to confirm speculation about interpretation of this positive nonlinear feedback, although the contribution of climatological wind is generally small for vertical advection.
The importance of moisture accession by cumulus congestus clouds and the associated bottom-heavy ascent by their latent heating has been emphasized for the moistening in the MJO developing phase (e.g., Johnson et al. 1999; Kikuchi and Takayabu 2003; Mapes and Bacmeister 2012; Bellenger et al. 2015; and many others). If the wave-phase dependence of top heaviness of ω is misanalyzed in ERAI, this profile-dependent effect might also be misestimated. On the other hand, it should be noted that the assumption of no correlations between cloud-scale horizontal wind and moisture and temperature perturbations in Eq. (3) excludes the effect of the vertical MSE advection by the (model unresolvable) cloud-scale motion.
Imaginary signals in Figs. 11 and 12 are too small to discuss, but are displayed for completeness.
5. Synthesis
To complete our synthesis, Fig. 13 shows the residual term’s cross-spectra in identical form to previous figures. Coherence shows especially significant signals around the Kelvin and EIGn0/MRG wave regions. The residual real component is generally negative except for the relatively high-coherence regions (Kelvin and EIGn0/MRG wave), while the residual imaginary component is systematically positive.
As in Fig. 3, but for the results from cross-power spectrum between precipitation and residual in the column-integrated MSE budget equation.
Citation: Journal of Climate 32, 2; 10.1175/JCLI-D-18-0334.1
This concludes our survey of terms in the CMSE budget of tropical disturbances in spectral space. To summarize them, real and imaginary components of Eq. (5) are summed over the mask area corresponding to each type of the disturbance (annotated in Figs. 1a,b). For clarity, we shall multiply by 100 in the tables below to express these precipitation-associated CMSE tendencies as percentages of total latent heat release at the corresponding wavenumber and frequency.
Table 1 contains the real component, indicative of amplification or damping. All these types of P-correlated CMSE disturbance are amplified by radiation and damped by vertical advection, although the relative contributions are different. Decomposition indicates that the vertical advective damping is much less than it would be if the MSE profile (moisture profile) were the tropical mean, or even the time mean, at each location: In other words, the temporal correlation between moisture and ascent is a first-order positive feedback for all kinds of Fourier disturbances of the observed amplitude. Such quadratic or correlation terms are ignored in linear models, which makes those further from observationally comparable relevance. It should be noted that the advection of wave-scale MSE anomaly by the background or climatological wind is included in our “eddy component,” although the contribution is small for vertical advection.
The real component of the cross-spectrum between precipitation and each term of Eq. (5) summed over the area corresponding to the disturbances associated with ERn1, Klvn, MRG, EIGn0, and WIGn1 and WIGn2 waves as well as the MJO and TD-type mode. The area associated with each disturbance is enclosed with thick solid lines in Figs. 1a and 1b. In the table, bold (italic) text indicates that the sign of the value is statistically significant (insignificant) at the 99% confidence level (DOF 126). All the values are increased by a factor of 100, and the unit is W m−2 (W m−2)−1 × 100.
The imaginary or quadrature component (Table 2) summarizes the effect of each term on propagation (temporal phase advancement) of precipitation-associated CMSE anomalies. Here the residual is larger (on the order of 5% rather than on the order of 1% of the real component, except for the ERn1 and TD mode). These large residuals imply that the reanalysis procedure and/or the underlying model physics may be poorer at propagating the analyzed anomalies than at closing its basic precipitation-associated MSE budget. It should also be noted that residuals are dominantly positive. A positive value of the residual means that phase-advancing or forward-propagation effects by some or all physical terms are underestimated. One candidate for this systematic error could be a poor analysis of vertical velocity. Such a poor analysis could involve the magnitude (too weak or insufficiently precipitation correlated), since the effect of vertical advection is also mostly positive. But it could also involve the profile, for instance, if the analyzed ascent in cloudy disturbances fails to be sufficiently bottom heavy in the front of disturbances and top heavy in the rear, as seen in observational studies on a range of time scales (Mapes et al. 2006).
The residual real component is small in most named wave regions, compared with the residual imaginary component, as seen in Tables 1 and 2. However, it is likely that this residual could also include compensating errors, since it is the sum of all model errors. For instance, we have insinuated that the magnitudes of both vertical velocity and cloudy radiation may be underanalyzed for smaller-scale P-correlated weather features (values too small on the right and left edges of Figs. 4 and 6). Because the sign of these effects systematically cancels, and they are physically linked in the model (which could make their scale-dependent analysis errors similar in magnitude), the small residual in Table 1 may not indicate a lack of problems in the real component. Only additional studies with other reanalysis, radiative heating, and/or precipitation datasets can resolve this doubt, and this caveat should be borne in mind for further discussions below.
a. MJO and ERn1 wave disturbances
The MJO and ERn1 wave filter regions are significantly amplified by radiation (14.3% and 12.1%, respectively) and more weakly by surface heat flux (2.5% and 2.8%, respectively), while they are damped by vertical advection (11.4% and 7.5%, respectively). Although MJO and ERn1 wave disturbances share similar features of the real component, they differ in sign regarding the effect of horizontal advection (−10.1% and 2.9%, respectively). Decomposition indicates that horizontal advection of time-mean MSE tends to amplify MJO and ERn1 wave disturbances, although they are damped by the eddy component. The eddy-damping effects are especially remarkable for MJO, since the contribution is largest among all types of disturbance and is comparable to that of total vertical advection.
The extended cover of clouds in moist regions reduces the longwave emission from Earth to space, causing anomalous heating. Numerous theoretical or observational investigations invoke the cloud radiation feedback to growth of intraseasonal time-scale disturbances like MJO (e.g., Raymond 2001; Bretherton and Sobel 2002; Tian and Ramanathan 2003; Lin and Mapes 2004; Bony and Emanuel 2005; Zurovac-Jevtić et al. 2006; Ma and Kuang 2011; Sobel et al. 2014; Sobel and Gildor 2003; Kim et al. 2015; Adames and Kim 2016). Our results with ERAI radiative heating (Fig. 6) are approximately in quantitative agreement with more direct estimates (Fig. 8 of Adames and Kim 2016) for low-wavenumber and low-frequency disturbances, although P-correlated cloudy radiation anomalies appear to be underanalyzed at smaller scales in the ERAI forecast tendencies dataset, as described in section 3d.
Larger residuals of the imaginary or quadrature component indicate that only a few terms can be considered robust, especially horizontal advection for the MJO and ERn1 wave disturbances, which has a role in leading these disturbances (13.2% and 17.2%, respectively). Decomposition indicates that horizontal advection of time-mean MSE would tend to lead to both MJO and ERn1 wave disturbances to the eastward direction, while horizontal advection of the remainder drives them in different directions (eastward and westward, respectively). The importance of such anomalous eddy components includes any rectified effects of synoptic substructure, as well as the disturbance-scale correlations of flow and moisture, so further decomposition would be required to test whether synoptic eddies may be important to eastward and westward propagations of these planetary-scale MSE + precipitation disturbances (e.g., Maloney and Dickinson 2003; Sobel and Maloney 2012).
The importance of coupling between atmosphere and ocean is a longtime question regarding the destablization of intraseasonal time-scale disturbances (DeMott et al. 2015), the most familiar hypothesis being wind-induced surface heat exchange (WISHE; Emanuel 1987; Neelin et al. 1987). It is widely accepted that coupling improves the fidelity of intraseasonal variations in a model (e.g., Flatau et al. 1997; Waliser et al. 1999; Woolnough et al. 2001; Kemball-Cook et al. 2002; Inness and Slingo 2003; Maloney and Sobel 2004; and many others). On the other hand, recent idealized models suggest that air–sea coupling is not an essential process (e.g., Raymond 2001; Wang and Liu 2011; Sobel and Maloney 2012; Thual et al. 2014; Adames and Kim 2016). Our estimates here support the idea that surface heat flux is of secondary importance to the amplification of MJO, although it is sometimes significant locally (e.g., Dellaripa and Maloney 2015). However, the results here also suggest that surface flux on the west might be important in slowing MJO propagation, an effect emphasized in some early MJO studies (reviewed in Zhang 2005), since its effect on the westward propagation is largest for the MJO (−4.1%), which is at least comparable to the large residuals (6.9%).
b. Kelvin wave and inertia–gravity (EIGn0, WIGn1, and WIGn2) wave disturbances
The Kelvin and EIGn0 wave filter regions are slightly amplified by radiation (4.3% and 2.6% of latent heat release, respectively), while they are significantly damped by vertical advection (−10.8% and −8.4%, respectively) and more weakly by surface heat flux (−2.4% and −2.1%, respectively) and horizontal advection (−2.0% and −1.0%, respectively). Similar features are found in the WIGn1 and WIGn2 wave disturbances (amplification through the radiation process, 1.6% and 1.2%, respectively, and damping through vertical advection, −9.1% and −7.3%, respectively), although the effect of the surface heat flux is opposite (1.2% and 0.8%, respectively) and horizontal advection acts to more significantly damp the disturbances (−4.9% and −2.6%, respectively). Decomposition indicates that the vertical advective damping is enhanced with the tropical mean MSE and that the amplification effect of the nonlinear positive feedback between moisture and vertical motion is more than twice as big as that of radiation processes (the wave-scale MSE anomaly by the background vertical wind is also included in the “nonlinear positive feedback”).
These divergent wave disturbances also share similar characteristics in the imaginary (propagation) component: They are led by vertical advection and retarded by radiation. Compared with the more horizontally advected types of disturbance (MJO, ERn1, and TD-type modes), one remarkable feature is that vertical advection dominates the contribution of the other propagation terms (4.3%, 8.3%, 5.6%, and 6.5% for Kelvin, EIGn0, WIGn1, and WIGn2 wave disturbances, respectively), which can be mostly accounted for by wave-scale self-advection of time–space-mean MSE. However, large residual imaginary components (8.6%, 6.5%, 6.6%, and 5.3% for Kelvin, EIGn0, WIGn1, and WIGn2 wave disturbances) caution against too quantitative a conclusion from these data.
c. TD-type and MRG wave disturbances
Transition from MRG wave disturbances to off-equatorial TD-type disturbances is reported by previous works (e.g., Takayabu and Nitta 1993; Dunkerton and Baldwin 1995; Dickinson and Molinari 2002). The similarity and difference of the cross-spectrum between the two disturbances is, therefore, worth special consideration.
The CMSE anomalies associated with TD-type and MRG filter regions are mildly amplified by radiation (5.5% and 5.8% of latent heat release, respectively), while they are damped by vertical advection (−3.2% and −9.7%, respectively) and effects of the surface heat flux are modest (0.9% and −0.7%, respectively). TD-type disturbances show the weakest damping effect of vertical advection among all types of disturbance. Horizontal advection plays a distinctly different role: It amplifies the TD-type disturbance (5.5%), while the MRG wave disturbance is weakly damped (−0.6%). Decomposition indicates that horizontal advection of the time-mean pattern of MSE plays a role in amplifying the MRG wave disturbance and damping the TD-type mode, respectively, an oblique glimpse of the consequences of their longitude-dependent occurrence patterns. Horizontal advection of remainders from the time-mean MSE is a significant damping for MRG wave disturbances but amplifies the TD-type mode. Further decomposition into mean-flow, wave-scale self-advection, and subwave-scale rectification would be needed to further interpret this difference.
In terms of the propagation (imaginary component), the TD-type and MRG wave disturbances are predominantly led by horizontal (39.1%) and vertical advections (10.2%), respectively. The MRG wave disturbance shares similar characteristics to the gravity wave family of disturbances (led by vertical advection), but leading effects of horizontal advection and surface heat flux are also relatively large. On the other hand, the TD-type disturbance is unique, since horizontal advection can exclusively account for the leading effect. Perhaps in the transition of MRG wave disturbances to TD-type disturbances, the dominant driving force for propagation shifts from vertical advection to horizontal advection, which might be one of the essential features that distinguishes between dynamical and nondynamical modes.
6. Concluding remarks
In the present study, budgets of CMSE were analyzed in zonal wavenumber–frequency space, making use of satellite-derived precipitation to isolate correlated aspects of ERAI reanalysis data. The evolution of CMSE has previously been regressed against (or composited around) filtered precipitation series associated with a particular type of the disturbance to reveal the relative importance of each term in CMSE budget equation to the charging and discharging mechanisms of CMSE. Here we extended that approach to the entire subseasonal spectral domain, utilizing the cross-spectrum between precipitation and each term in MSE budget equation. We believe that this framework can be useful to grasp the different CMSE budget characteristics of different CCEWs at a glance. Despite the elimination of P in CMES budgets [Eq. (3)], this approach connects indirectly to the daydream (or lofty goal) of “precipitation budgets,” as the strong WTG approximation links CMSE to CWV, which is monotonically, if nonlinearly, related to the statistical expectation of precipitation.
If model-forecasted precipitation or column-integrated vertical advection of DSE were used as a measure of convective activity instead of TRMM precipitation, all terms in Eqs. (4) or (5) could be calculated from model outputs only. Indeed, model CMSE itself could be used and the result would simply be its variance budget in the model. However, even state-of-the-art models, and their associated assimilated reanalyses, suffer from biases, especially in moist processes (e.g., Mapes and Bacmeister 2012; Yokoi 2015), so such a study might emphasize artificial characteristics. For this reason, we chose our approach of using almost-independent TRMM-3B42 data. The projection to the more directly observed precipitation is one of the main advantages of the present study.
To explain some essential features of the MJO and CCEWs, theoretical works generally examine necessary conditions for small perturbations to exponentially amplify, assuming a sinusoidal solution in a linearized equation for moisture or MSE or similar variables (e.g., Neelin and Yu 1994; Raymond 2000, 2001; Sobel et al. 2001; Fuchs and Raymond 2002, 2005, 2007; Sobel and Bretherton 2003; Raymond and Fuchs 2007; Sobel and Maloney 2012, 2013; Adames and Kim 2016; and many others). For such an unstable solution to emerge in the model, phase relationships of the perturbation with various terms included in the prognostic equation are the key. The cross-spectrum gives directly useful empirical estimates of these phase differences over a wide range of wavenumbers and frequency and could be a powerful tool for targeting theoretical framings to the phenomena they are meant to elucidate. With this connection to basic understanding, it is hoped that the work reported here may serve as both a data resource and a methodological inspiration for additional observational studies.
Acknowledgments
The results in the present study were obtained using TRMM-3B42 and ERA-Interim. We would like to express our sincere thanks to all concerned in these products. This work was supported by JSPS KAKENHI Grant Numbers JP25400463, JP16KK0095, JP16K05560, and JP16H04048. BEM acknowledges support from NASA Grant NNX15AD11G. Scientific discussions with Drs. George Kiladis, Juliana Dias, and Stefan Tulich and helpful comments from Drs. Tomoki Miyakawa, Noriyuki Nishi, Ángel F. Adames, and an anonymous reviewer are gratefully acknowledged.
APPENDIX
Interpretations of Real and Imaginary Components of the Cross-Spectrum
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