1. Introduction
Raymond et al. (2015) distinguish two broad categories of tropical disturbances:
In fast-moving disturbances such as equatorial Kelvin waves, mixed Rossby–gravity waves, and inertia–gravity waves, gravity wave dynamics are clearly important.
In slow-moving disturbances such as equatorial Rossby waves, easterly waves, monsoon lows, tropical cyclones, and the Madden–Julian oscillation, strong potential vorticity signatures are evident, which indicates a significant role for balanced dynamics.
The focus of this paper will be on the fast, gravity wave category, specifically on convectively coupled Kelvin waves (CCKWs). According to the tropical wave mode analysis of Wheeler and Kiladis (1999), these account for the second greatest OLR variance after the Madden–Julian oscillation (MJO) and therefore control a substantial fraction of tropical rainfall variability. We will analyze composite structures of tropical CCKW phenomena in terms of physical parameters that Raymond and Fuchs (2007, hereafter RF07) identified as most significant for CCKWs. In this study, we make use of 3D analysis and reanalysis model output, radiosonde data, as well as output from long integrations with the European Centre for Medium-Range Weather Forecasts (ECMWF) model. It is our goal to illustrate the agreement or disagreement between the linear theory of RF07 and observations and then to test the capability of models as well as of reanalyses to manifest observed CCKW characteristics. A broader picture encompasses a similar analysis of all species of tropical disturbance with the Raymond et al. (2015) categorization in mind—a goal we are currently pursuing.
Since Matsuno (1966), we have known that Kelvin waves follow a linear theory in the adiabatic atmosphere, but there is still debate on how these waves couple with convection. There are many linear models that look at CCKWs (e.g., Mapes 2000; Majda and Shefter 2001a,b; Majda et al. 2004; Khouider and Majda 2006a,b, 2008; Fuchs and Raymond 2007; RF07; Kuang 2008a,b; Andersen and Kuang 2008; Fuchs et al. 2012). To parameterize precipitation, Mapes used a convective inhibition (CIN) closure that included a separate accounting of the triggering energy, RF07 and Fuchs et al. (2012) used deep CIN and moisture closures, while the rest of the mentioned models used a variety of schemes including modified convective available potential energy (CAPE) closures in which CAPE is computed only in the lower troposphere. The RF07 model is unique because it can obtain the observed phase speed, growth rate, and vertical structure of CCKWs without prescribing the phase dependence of the vertical heating profile. In this paper we examine several key diagnostic parameters from the RF07 model to assist in understanding the physics behind the CCKW on datasets obtained from 3D analysis and reanalysis model output, radiosonde data, and special runs of the ECMWF model.
Many cloud-resolving model (CRM) studies consider the CCKW, including Grabowski and Moncrieff (2001), Peters and Bretherton (2006), Tulich et al. (2007), Tulich and Mapes (2008), Kuang et al. (2005), and Fuchs et al. (2014). Fuchs et al. (2014) used the CRM of Raymond and Zeng (2005) to identify mechanisms primarily responsible for controlling precipitation in CCKWs and linked them to the linear theory of RF07. They showed that saturation fraction (precipitable water divided by saturated precipitable water) and instability index (an indicator of tropospheric stability) anomalies both lag rainfall and are therefore not the primary causal mechanisms for CCKWs, while the deep convective inhibition (DCIN) decrease and its excursion to negative values led the rainfall, suggesting that the CCKW is controlled by DCIN. The latter finding is in agreement with the linear model of RF07.
Radiosonde observations, though limited, can also provide insight into mechanisms controlling the CCKW. An example of a particularly clean CCKW (Straub and Kiladis 2002) was observed during the Tropical East Pacific Process Study (TEPPS) project (Yuter and Houze 2000). During this project, the Research Vessel Ronald H. Brown was stationed near 8°N, 125°W for approximately 2 weeks in August 1997 and launched six radiosondes per day. RF07 analyzed the radiosonde observations from this project to obtain a time series of convective inhibition and saturation fraction. They showed that the deep convection and resulting precipitation were related to the moistening of the atmosphere but that the onset of precipitation was delayed approximately 1 day from this moistening by the existence of a stable layer. In this case, CIN played a significant role in the timing of the precipitation. A thorough review on CCKWs can be found in Kiladis et al. (2009).
In this paper we will focus on two physical parameters we have found to be important through the linear theory of RF07 and via the CRM experiment of Fuchs et al. (2014): DCIN, which depends on free-tropospheric temperature variations and boundary layer moist entropy, and saturation fraction. The justification for these parameters is as follows:
Deep convection has been shown to occur sporadically in the tropics, while shallow or congestus clouds occur almost continuously in some areas. Raymond (1995) proposed a mechanism whereby deep convective events are controlled by a boundary layer quasi equilibrium existing between the boundary layer and the lowest levels of the free troposphere. Over time scales of 1/2 day, a state of statistical equilibrium occurs whereby boundary layer moist entropy anomalies are soon followed by the vertical mass flux of entropy into and out of the layer, which maintains the equilibrium state. This generally prevents the instability needed to drive a deep convective event such that extra forcing is required. Mapes (2000) posits that one culprit is the gust fronts driven by the deep convection itself. While there is a chicken-and-egg causality issue inherent in this theory, it is nevertheless an idea with considerable empirical support. A truly external agent, however, is the temperature anomaly driven by the forced ascent and descent of a passing atmospheric wave. When the lower free troposphere is adiabatically warmed via wave activity alone, convective inhibition is increased, suppressing the nearly continuous convection. During the next phase of the wave, this inhibiting layer vanishes, allowing parcels with anomalously greater moist entropy to convect beyond the original inhibition layer. Based on in situ observations in the tropical east Pacific from data collected during the East Pacific Investigation of Climate (EPIC2001), Raymond et al. (2003) found that the existence of even a weak stable layer just above the planetary boundary layer (PBL) is sufficient to inhibit the development of deep convection and associated precipitation. This confirms earlier results of Firestone and Albrecht (1986) obtained from dropsonde measurements in the tropical Pacific. Last, the mathematical theory of RF07 as well as the CRM results of Fuchs et al. (2014) suggest that DCIN is the primary mechanism responsible for the onset of the CCKW, though the effects of surface moist entropy fluxes have also been shown to be important in CCKWs (Raymond et al. 2003; Back and Bretherton 2005; Maloney and Esbensen 2005).
Bretherton et al. (2004) analyzed passive microwave satellite observations and found that precipitation is highly correlated with the saturation fraction of the troposphere. Sobel et al. (2004) reached similar conclusions using data taken near Kwajalein Atoll. Results from numerical cloud models also support this conclusion (Lucas et al. 2000; Derbyshire et al. 2004; Raymond and Zeng 2005). The linear model of RF07 shows that a correlation of precipitable water with precipitation is of primary importance for the moisture mode—an instability mechanism linked to the observed correlation between column moisture and rainfall considered by some to be related to the MJO. In contrast, the Kelvin mode is only weakly associated with column moisture changes. Yasunaga and Mapes (2012a) found that precipitable water variations were small and did not exhibit a consistent phase relationship with precipitation in CCKWs compared to that of equatorial Rossby waves and the MJO. In the part of the wavenumber–frequency spectrum containing the greatest CCKW precipitation variance, they found no significant lag between precipitable water and rainfall. Roundy and Frank (2004) also found that precipitable water is only weakly modulated in the CCKW spectrum when compared to spectra for equatorial Rossby waves and the MJO.
2. Linear theory of RF07
We will here briefly review the linear theory of RF07 and also define the convective inhibition and precipitable water indices used in RF07 and in the CRM experiment of Fuchs et al. (2014).
Figure 1 shows the various contributions to the total precipitation as a function of lag time with respect to the precipitation maximum for the eastward-moving convectively coupled gravity mode that maps onto the Kelvin mode in a rotating atmosphere with zonal wavenumber 
Decomposition of precipitation as a function of lag time for (top) the eastward-moving convectively coupled gravity mode with zonal wavenumber 
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
Decomposition of precipitation as a function of lag time for (top) the eastward-moving convectively coupled gravity mode with zonal wavenumber
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
Decomposition of precipitation as a function of lag time for (top) the eastward-moving convectively coupled gravity mode with zonal wavenumber
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
The total precipitation, (6), is given by a thick solid black line, the column moisture contribution 
This effect is illustrated in RF07-NOP2S, where 
RF07 found that the Kelvin mode instability in both RF07-NOP2S and RF07-FULL occurs when DCIN is included in the convective heating closure of the vertically resolved model. Since this mode is unstable when DCIN is the only significant contributor to the heating, it follows that DCIN is a mechanism that can destabilize this wave mode by itself. In contrast, Fuchs and Raymond (2007) found that changes in column-integrated tropospheric moisture acting alone could not destabilize this mode. Since this instability occurs even when DCIN is a function only of the lower-tropospheric temperature anomaly, as in RF07-NOP2S, it follows that the latter is the primary forcing agent of this mode. Last, since the model’s closure mechanism assumes that DCIN modulates rainfall, the resulting Kelvin mode instability is consistent with the idea that DCIN drives the rainfall of CCKWs.
Very little contribution to the precipitation comes from the column moisture perturbation 
We define the pressure limits of the 
3. Data and methodology
We now describe the data and analysis techniques used in our study. The datasets used are summarized in Table 1.
Parameters for datasets used in this study. A dash indicates field is not applicable for this dataset. Each dataset is categorized as either a proxy for precipitation (rain) or as a thermodynamic dataset (thermo). Where two resolutions are shown, a slash indicates both are used; an arrow indicates that the resolution was changed for all regressions. All resolutions refer to the dataset, not to the originating model or sensing instrument. The last column indicates the rain proxy predictor variable(s) used in the linear regressions for each thermodynamic dataset: N (NOAA OLR), T (TRMM 3B42), EO (ECMWF OLR), and ER (ECMWF rainfall rate). The asterisk indicates this variable is only used in the appendix.
a. Data
The radiosonde data used in this study were obtained from the Integrated Global Radiosonde Archive (IGRA). Temperature, pressure, relative humidity, wind speed, and direction are available in this dataset at a time resolution of 
Map of IGRA radiosonde stations (circles) used in this study atop contours of KELVIN filter variance for TRMM-derived precipitation rates (mm day−1) over 1998–2013. The WPAC group circles are filled. Stations that survive the discrimination procedure also have concentric circles. Continents have been outlined for reference.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
Map of IGRA radiosonde stations (circles) used in this study atop contours of KELVIN filter variance for TRMM-derived precipitation rates (mm day−1) over 1998–2013. The WPAC group circles are filled. Stations that survive the discrimination procedure also have concentric circles. Continents have been outlined for reference.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
Map of IGRA radiosonde stations (circles) used in this study atop contours of KELVIN filter variance for TRMM-derived precipitation rates (mm day−1) over 1998–2013. The WPAC group circles are filled. Stations that survive the discrimination procedure also have concentric circles. Continents have been outlined for reference.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
Reanalysis model output is obtained from the ERA-Interim dataset (European Centre for Medium-Range Weather Forecasts 2012) produced by ECMWF (herein ERAI). These data are available on 
Proxies of surface precipitation rate are obtained from the Tropical Rainfall Measuring Mission (TRMM), a joint project of the National Aeronautics and Space Administration (NASA) and the National Space Development Agency of Japan (NASDA). We chose the 3B42 dataset based on spatial coverage and for its thoughtful precipitation estimation algorithm. The 3B42 dataset is comprised of an ensemble of satellite passive microwave and infrared (IR) measurements calibrated to precipitation radar, which are then adjusted to agree with temporal means of ground-based rain gauges. Wherever and whenever the calibrated microwave data are available they are used; otherwise, the calibrated IR is used. The available spatial resolution is 
We also employ output from long integrations with the ECMWF forecast model as well as from several special runs of this model—each with spatial resolution of 
We also employ a set of aquaplanet versions of the ECMWF model. We use low-resolution (
b. Wave filtering
Discrete Fourier transforms (DFT) are performed on each rainfall proxy along the time and longitude dimensions separately at each latitude. Consecutive years are ingested as one time series into the temporal DFT; otherwise, years are processed one at a time. The wavenumber-frequency domain is then reduced to a subset including only the spectrum of interest. We reduce the spectrum using two different filters: a large Kelvin filter window (herein KELVIN), similar to that used in Wheeler and Kiladis (1999), and also a narrow-band filter in frequency and wavenumber (herein MONO) in order to compare with the monochromatic theory of RF07. For KELVIN, a rectangular filter is used to define boundaries in equivalent depth 
c. Linear regression
We found good correspondence in the variables of interest among stations in the WPAC group for IGRA, ERAI, FNL, and T255-OPANA. We found broader correspondence for these locations in T255-LAND. We then averaged each eight together for each variable from the WPAC group to form composites representing the west Pacific for each rainfall proxy and for each wave filter. These composites are given names such as IGRA-NOAA-KELVIN, representing the WPAC group from the IGRA radiosonde dataset, regressed onto NOAA OLR that has been filtered using the KELVIN spectrum. To check the robustness of our wave composites for the IGRA radiosonde dataset, we perform these regressions twice with two independent rainfall proxy datasets: filtered TRMM 3B42 over the interval 1 January 1998–31 December 2011 and filtered NOAA OLR over the interval 1 January 1982–31 December 1997. We repeat this analysis with other IGRA stations around the tropical band but do not analyze them in detail in this work.
To form WPAC group composites from ERAI, FNL, T255-OPANA, and T255-LAND, we use the nearest spatial grid points to each respective WPAC group station to represent members of each composite. For ERAI and FNL, we use the years 2000–13 for both NOAA OLR and TRMM 3B42 rainfall proxies to take advantage of the period for which all these data overlap. For T255-OPANA, we use the years 2008–2013 and form composite regressions using NOAA OLR, TRMM 3B42, and also the ECMWF OLR. For T255-LAND and the ECMWF aquaplanet integrations, we only use the ECMWF OLR, since these models do not assimilate observations. The lack of continents in the aquaplanet models eliminates the focusing effect that gives wave species preferred spatial locations in the real atmosphere, which reduces the signal-to-noise ratio at any given location along the aquaplanet tropical band. We thus choose 25 regularly spaced points along the zonal axis in the respective region of greatest CCKW variance for each model to represent discrete locations of interest. These locations are then averaged together into composites as in the models with orography.
4. Results
In this section we show how the phase relationships and the amplitudes of DCIN and column moisture as well as vertical structures of temperature and zonal wind from radiosonde data, analysis and reanalysis data, and the various ECMWF runs compare to the RF07 linear model theory as well as to each other. Most figures in this section illustrate composites of linear regressions from a set of locations in the west Pacific (IGRA, ERAI, FNL, OPANA, LAND) or else from a set of equidistant locations along the equator (ECMWF aquaplanets). The exception is a scatterplot showing all the IGRA stations used in this study.
a. Radiosonde data, analysis, and reanalysis data
Figure 3 shows the contributions to the precipitation decomposition (i.e., indices from RF07) as a function of lag time for the eastward-moving convectively coupled Kelvin mode based on observations in the WPAC group. The Kelvin-filtered (Fig, 3, middle; KELVIN) and monochromatic wave-filtered (Fig. 3, bottom; MONO) results are shown for regressions onto NOAA OLR over 1982–97 (Fig. 3, left) and onto TRMM 3B42 over 1998–2013 (Fig. 3, right). The RF07 linear model result is placed atop each column to facilitate comparison. The total precipitation is given by a thick solid line, the column moisture contribution 
As in Fig. 1, but for wave composites from radiosonde stations in the west Pacific (i.e., the WPAC group). Results for the (middle) KELVIN and (bottom) MONO filters are shown for (left) NOAA OLR over 1982–97 and (right) TRMM 3B42 over 1998–2013. (top) The full RF07 model plot is shown to facilitate comparison.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
As in Fig. 1, but for wave composites from radiosonde stations in the west Pacific (i.e., the WPAC group). Results for the (middle) KELVIN and (bottom) MONO filters are shown for (left) NOAA OLR over 1982–97 and (right) TRMM 3B42 over 1998–2013. (top) The full RF07 model plot is shown to facilitate comparison.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
As in Fig. 1, but for wave composites from radiosonde stations in the west Pacific (i.e., the WPAC group). Results for the (middle) KELVIN and (bottom) MONO filters are shown for (left) NOAA OLR over 1982–97 and (right) TRMM 3B42 over 1998–2013. (top) The full RF07 model plot is shown to facilitate comparison.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
Figure 3 (middle) shows results from the full Kelvin band filter (KELVIN). Since this filter represents a broad spectrum of frequencies and wavenumbers, the resulting structures are linear combinations of these and thus resemble a wave packet. Figure 3 (bottom) shows results for the monochromatic filter (MONO), which has a narrow frequency band centered at a period of 
The Kelvin band filter shows small differences in period between rainfall proxies, since each projects onto spectral modes differently. This is primarily due to a considerable difference in spatial and temporal resolutions. The Kelvin wave represented by filtered NOAA OLR shows a period of 
The amplitudes of these indices differ between the filtering bands (KELVIN versus MONO) owing to the tapering effect of the wave packet envelope—that is, the linear superposition of a broad spectrum of wavenumbers and frequencies. This point highlights the importance of the monochromatic filter, which allows amplitudes to be more readily compared across the lag axis. We should note, however, that since the MONO filter selects a narrow subset of the spectrum, it is subject to greater variability between datasets (illustrated below). Although amplitudes are distorted in the KELVIN results, the filter is useful for its added statistical robustness as well as for comparing the weight of high- versus low-frequency rain events in the broader KELVIN filter. The general agreement among wave filters suggests they independently illustrate real and salient features of the CCKW.
We now compare the WPAC group results to the linear model of RF07 (Fig. 3, top). The resulting wave period 
A more significant difference arises in the amplitudes of the DCIN indices. In RF07-FULL, the threshold entropy 
Each corresponding index in the WPAC group has amplitude similar to that of RF07-FULL except for 
Figure 4 shows the Kelvin filter results from the ERAI (middle) and the FNL analysis (bottom). These closely resemble the composites from IGRA radiosonde stations. The MONO results for these models (not shown) are also very similar to the corresponding radiosonde composites with matching amplitudes for all datasets and rainfall proxies. The match to ERAI and FNL is not surprising, considering that these models both assimilate data from the WPAC group IGRA stations; although each analysis also depends on a forecast model, which likewise plays an important role in matching the observations.
As in Fig. 3, but for KELVIN composites of (middle) ERAI and (bottom) FNL analysis for the WPAC group. Results are shown for (left) NOAA OLR over 1982–97 and (right) TRMM 3B42 over 1998–2013. (top) Corresponding IGRA radiosonde composites are shown for comparison.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
As in Fig. 3, but for KELVIN composites of (middle) ERAI and (bottom) FNL analysis for the WPAC group. Results are shown for (left) NOAA OLR over 1982–97 and (right) TRMM 3B42 over 1998–2013. (top) Corresponding IGRA radiosonde composites are shown for comparison.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
As in Fig. 3, but for KELVIN composites of (middle) ERAI and (bottom) FNL analysis for the WPAC group. Results are shown for (left) NOAA OLR over 1982–97 and (right) TRMM 3B42 over 1998–2013. (top) Corresponding IGRA radiosonde composites are shown for comparison.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
We also looked at a range of other radiosonde stations around the tropical band to check if the pattern of DCIN minimum preceding the rainfall maximum is robust. Figure 5 shows the number of days the DCIN minimum leads the rainfall maximum as a function of station longitude for KELVIN (top left) and MONO (bottom left) filters. Statistical differences between the rainfall proxies are evident in the plots: TRMM (red crosses) has more scatter—likely an effect of the finer resolution of this dataset, which could lead to greater sensitivity to local convective events not associated with CCKWs. The MONO filter has more scatter than the KELVIN filter, likely owing to the smaller number of wave events entering the composite in that case. The scatter for both rainfall proxies is also partly due to the fact that our limited time resolution gives an uncertainty of ±6 h. Last, there should be some systematic error associated with the rainfall–OLR relationship.
Scatterplots showing lead time of DCIN minimum over precipitation maximum vs IGRA station longitude for the (top) KELVIN and (bottom) MONO filters. Results for NOAA (blue circles) and TRMM (red crosses) are shown for (left) all IGRA stations examined in this study and (right) a subset including only island-based stations where the NOAA and TRMM results agree by better than 1 day.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
Scatterplots showing lead time of DCIN minimum over precipitation maximum vs IGRA station longitude for the (top) KELVIN and (bottom) MONO filters. Results for NOAA (blue circles) and TRMM (red crosses) are shown for (left) all IGRA stations examined in this study and (right) a subset including only island-based stations where the NOAA and TRMM results agree by better than 1 day.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
Scatterplots showing lead time of DCIN minimum over precipitation maximum vs IGRA station longitude for the (top) KELVIN and (bottom) MONO filters. Results for NOAA (blue circles) and TRMM (red crosses) are shown for (left) all IGRA stations examined in this study and (right) a subset including only island-based stations where the NOAA and TRMM results agree by better than 1 day.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
These caveats aside, a majority of the points are above the zero axis in both plots, indicating that for most of the stations, the DCIN minimum precedes or is in phase with the rainfall anomaly. To eliminate potentially erroneous outliers, Fig. 5 (right) shows only island stations where the estimated lead times derived from the independent datasets NOAA OLR and TRMM 3B42 differ by less than 1 day. Stations in Fig. 2 with concentric circles survive this discrimination procedure. Again, results for KELVIN (top right) and for MONO (bottom right) are shown. The reason for this selection is to eliminate stations on large landmasses that might distort the behavior of CCKWs over open ocean and also to eliminate cases where the behavior at a single station during a particular observation interval is dominated by an unusual event. Recall that the two rainfall proxies are not merely independent in the sense that the data originate from different sensing instruments, but also in the sense that they cover distinct time intervals.
Without outliers, the bias toward positive lead times is more evident. In addition, the WPAC group (east of 120°E) stations have robustly positive lead times, with an average near 
The radiosonde stations that show the most robust DCIN–rainfall relationship are therefore those in the northwestern Pacific. Since the CCKW signal is most prominent in that region (Wheeler and Kiladis 1999; Roundy and Frank 2004) the strong consensus on Kelvin-filtered wave phenomena there combined with the theory of RF07 suggests that DCIN control is a causal mechanism for the destabilization and propagation of these waves.
It is also notable that several stations that survive the discrimination procedure for the KELVIN filter lie to the west of the WPAC group and show robustly lagging DCIN. The greatest lag is near the center of the Indian Ocean at Diego Garcia (7.3°S, 72.4°E) and at Seychelles (4.67°S 55.53°E). This is intriguing evidence of the existence of the extra variability in Indian Ocean CCKWs noted by (Roundy 2012a,b). To check if this variability is dependent on the time scale of the wave, we also examined the results of a modified MONO filter using the same wavenumber, but with oscillation period T = 8 days (not shown). We found that the modified filter gave similar phase relationships at Majuro but not at Diego Garcia. While we only examined two stations using the modified filter, this result further supports the notion that a continuum of physical mechanisms exists for CCKWs that propagate at different speeds in the Indian Ocean.
It should be noted here that the lead in DCIN with respect to the rainfall anomaly suggested by these plots may actually be smaller than what is seen in the WPAC group. Since each of our rainfall proxies comprises satellite observations, we are not strictly analyzing the DCIN–rainfall relationship here. If an additional lag exists between rainfall and the proxies used here though, it is likely less than the temporal resolution of our study. Haertel and Kiladis (2004) found that a satellite brightness temperature minimum lagged a budget-derived rainfall anomaly by approximately 4 h in composite 2-day waves derived from the Tropical Ocean Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (see their Fig. 2). In addition, Rickenbach et al. (2008) estimated the lag between rainfall maxima and anvil area maxima for convective towers over Florida to be 1–2 h but also suggested this time could increase for larger systems. Nevertheless, the scatterplots shown in Fig. 5 illustrate the fact that CCKWs have a typical DCIN phase signature, whereby the DCIN anomaly slightly precedes the rainfall anomaly.
Figure 6 shows the vertical structure of the temperature anomaly for the eastward-moving CCKW in the x–z plane for the RF07 model (left panel) and for the WPAC group composite using NOAA OLR (right panel). We use the MONO filter to form composites in this figure to facilitate comparison to the RF07 linear model, and we scale all variables to the same rainfall anomaly at t = 0 days as in the phase plots. The vertical temperature structures are broadly similar in both the characteristic boomerang structure with westward-tilting (toward greater lag times) contours in the low to midtroposphere and an eastward tilt above. While the elbow is higher in the RF07 model than in observations, the overall tilted structure matches the findings from observations (Wheeler et al. 2000; Straub and Kiladis 2002) and from cloud-resolving numerical simulations (Peters and Bretherton 2006; Tulich et al. 2007).
The temperature perturbation (K) vertical structure of the eastward-moving Kelvin mode for (left) the RF07 linear theory and (right) the IGRA WPAC group using the MONO filter and NOAA OLR.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
The temperature perturbation (K) vertical structure of the eastward-moving Kelvin mode for (left) the RF07 linear theory and (right) the IGRA WPAC group using the MONO filter and NOAA OLR.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
The temperature perturbation (K) vertical structure of the eastward-moving Kelvin mode for (left) the RF07 linear theory and (right) the IGRA WPAC group using the MONO filter and NOAA OLR.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
Differences include the sign of the anomaly in the upper troposphere and lower stratosphere, which completes the boomerang shape in the composites, and the lack of surface temperature anomalies in the linear model. The latter deficiency is due to the specified boundary conditions, which could affect the strength of the boundary layer moist entropy anomaly. Furthermore in the model, the maximum temperature anomaly occurs at 8 km, near t = −0.75 days while in the composites this occurs at 10 km, at t = 0 days. There is furthermore a temperature maximum near z = 3.5 km, t = −2 days that is less prominent in the model.
b. ECMWF operational analysis and annual integrations
We now consider the behavior of the RF07 indices in a set of long integrations with the ECMWF model. Figure 7 shows composited indices from the ECMWF operational analysis (T255-OPANA, second row) and the freely running 1-yr integrations with the same model (T255-LAND, third row). The KELVIN composites are in the left column and the MONO composites are in the right column. For comparison, corresponding ERAI results are displayed in the top row. The ERAI reference composites are derived from NOAA OLR, but the TRMM 3B42 results are similar.
As in Fig. 3, but for the (left) KELVIN and (right) MONO results for (second row) ECMWF operational analysis and (third row) ECMWF free-run simulation with orography. (bottom) The 95% confidence intervals for the DCIN estimates in the ECMWF free-run simulation with orography are shown with a slightly broader vertical axis. (top) The corresponding ERAI plots with NOAA OLR (1982–97) are shown for comparison.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
As in Fig. 3, but for the (left) KELVIN and (right) MONO results for (second row) ECMWF operational analysis and (third row) ECMWF free-run simulation with orography. (bottom) The 95% confidence intervals for the DCIN estimates in the ECMWF free-run simulation with orography are shown with a slightly broader vertical axis. (top) The corresponding ERAI plots with NOAA OLR (1982–97) are shown for comparison.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
As in Fig. 3, but for the (left) KELVIN and (right) MONO results for (second row) ECMWF operational analysis and (third row) ECMWF free-run simulation with orography. (bottom) The 95% confidence intervals for the DCIN estimates in the ECMWF free-run simulation with orography are shown with a slightly broader vertical axis. (top) The corresponding ERAI plots with NOAA OLR (1982–97) are shown for comparison.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
These data have half the temporal resolution of ERAI, so features appear more irregular, though T255-OPANA indices seem to match those of ERAI quite well. This is what we expect, since much of the same physics and observational data enter both analyses. It is difficult to compare amplitudes because of the lower resolution of the operational analysis, but the phase relationships match ERAI for both the KELVIN and MONO composites. The DCIN index minimum precedes the rainfall maximum in both cases by approximately 6 h. Recall that this is the lead time that appears for the west Pacific CCKW in the scatterplots of Fig. 5.
In the freely running integrations, however, significant changes occur in the RF07 indices. First, the column moisture anomaly (dotted) falls well past the rainfall anomaly, peaking near lag = +1 day for both wave filters. The threshold entropy anomaly 
The important difference here is that the boundary layer moist entropy anomaly 
Contours of the square root of the variance (standard deviation) of KELVIN-filtered ECMWF OLR for (left) T255-OPANA and (right) T255-LAND. The OLR has been scaled to the rainfall rate (mm day−1). WPAC group stations are marked with circles.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
Contours of the square root of the variance (standard deviation) of KELVIN-filtered ECMWF OLR for (left) T255-OPANA and (right) T255-LAND. The OLR has been scaled to the rainfall rate (mm day−1). WPAC group stations are marked with circles.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
Contours of the square root of the variance (standard deviation) of KELVIN-filtered ECMWF OLR for (left) T255-OPANA and (right) T255-LAND. The OLR has been scaled to the rainfall rate (mm day−1). WPAC group stations are marked with circles.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
To check the statistical significance of these results, we examine the standard deviation of the mean DCIN index across the ensemble of WPAC group locations entering the composite, shown in Fig. 7 (bottom). For each wave filter, the DCIN and 95% confidence intervals (
We now examine the behavior of the ECMWF model in the absence of land with a variety of spatial resolutions. We also run the model with the deep convective parameterization scheme turned off in order to discern its effect upon CCKW signatures.
Plots of RF07 indices versus lag time are shown in Fig. 9. Reference plots from ERAI are placed atop each column for comparison and both KELVIN (left column) and MONO (right column) results are shown. The temporal resolution of the output data in the aquaplanet runs is reduced with respect to the ERAI. The most notable difference between the real-Earth integrations with orography and the aquaplanet models is the reduction in all amplitudes—besides the rainfall—by a factor of 2. This difference may be due to increased OLR variance in this model (see appendix) and is an interesting difference between the ECMWF free run with land (T255-LAND) and those without.
As in Fig. 7, but for ECMWF aquaplanet models.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
As in Fig. 7, but for ECMWF aquaplanet models.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
As in Fig. 7, but for ECMWF aquaplanet models.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
For the lowest-resolution model (
It is interesting to see the effect of the deep convective scheme on this model. Looking at T159-AQUA-DEEP, we see the KELVIN wave packet is narrower than in T159-AQUA-NODEEP. Also, the 
This reduction of the 
Last, we examine an aquaplanet employing the spatial resolution of the current forecast model (
It is worth noting here that part of the difference in wave signatures could result from diminished or even missing spectral peaks within the filtering region. We checked this and found that the spectral peaks in each of the aquaplanet models are considerably more prominent than in the real atmosphere and in T255-LAND. This may explain the small thermodynamic anomalies in the aquaplanet models compared to the standard rainfall anomaly (50 mm day−1) used. That is, if a given DCIN anomaly on the aquaplanet leads to much greater rainfall than in the real atmosphere, a more realistic rainfall event will correspond to a weaker DCIN anomaly in such a model.
The column moisture anomaly 
We now examine the variance in the mean estimates for the aquaplanet indices. Figure 10 shows the DCIN anomaly estimate within 95% confidence intervals for each model shown in Fig. 9 including ERAI. Only the aquaplanets lacking the deep convection scheme show greater uncertainty in the estimates of these curves than does ERAI. Again, there is considerably greater variance in the MONO estimates than in the KELVIN estimates. This supports the veracity of the broad-spectrum anomalies. However, note also that while the monochromatic DCIN estimates have large variance, it is uniform across the phase of the wave. This suggests that there are enough members included in each composite to accurately portray the characteristics of the wave.
As in Fig. 9, but for the 95% confidence intervals for DCIN in ECMWF aquaplanet models. Note that the vertical axes are slightly broader here to allow comparison with T255-LAND in Fig. 7.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
As in Fig. 9, but for the 95% confidence intervals for DCIN in ECMWF aquaplanet models. Note that the vertical axes are slightly broader here to allow comparison with T255-LAND in Fig. 7.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
As in Fig. 9, but for the 95% confidence intervals for DCIN in ECMWF aquaplanet models. Note that the vertical axes are slightly broader here to allow comparison with T255-LAND in Fig. 7.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
Interestingly, the variance of the DCIN estimates decreases significantly when the deep convection scheme is included in the aquaplanet models T159-AQUA and T799-AQUA. This may be due to the fact that deep convective schemes help to quell instabilities in numerical models, which may prevent unusual thermodynamic fluctuations that would artificially increase the variance.
Figure 11 shows the vertical structure of the temperature anomalies for the ECMWF free run with land (T255-LAND) and the highest-resolution aquaplanet model (T1279-AQUA-NODEEP). All composites in this figure result from MONO filter regressions. The corresponding IGRA-NOAA and ERAI-NOAA plots are also shown for comparison. Both ECMWF runs show strong similarity to IGRA, with temperature anomalies centered near 4 and 10 km. While the amplitude of the aquaplanet model is diminished as in the aquaplanet phase plots, it nevertheless has maxima at these levels as well as the observed boomerang shape. In each case, there is a noticeable anomaly at the surface, and the phases of these anomalies are similar. Note that the ERAI anomalies are also diminished compared to IGRA results.
As in Fig. 6, but for (bottom left) the ECMWF free-run simulation with orography and (bottom right) the ECMWF highest-resolution aquaplanet model. (top left) IGRA-NOAA and (top right) ERAI-NOAA results are shown for comparison. All temperature anomalies are regressed onto the respective MONO rainfall proxy.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
As in Fig. 6, but for (bottom left) the ECMWF free-run simulation with orography and (bottom right) the ECMWF highest-resolution aquaplanet model. (top left) IGRA-NOAA and (top right) ERAI-NOAA results are shown for comparison. All temperature anomalies are regressed onto the respective MONO rainfall proxy.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
As in Fig. 6, but for (bottom left) the ECMWF free-run simulation with orography and (bottom right) the ECMWF highest-resolution aquaplanet model. (top left) IGRA-NOAA and (top right) ERAI-NOAA results are shown for comparison. All temperature anomalies are regressed onto the respective MONO rainfall proxy.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
Figure 12 shows the vertical structure of the specific humidity perturbation for the same set of models. In this case, significant differences appear. While IGRA and T255-LAND have strong anomalies near and above the melting level, these anomalies are much weaker in ERAI. Indeed, there are problems with the humidity structure in ERAI owing to biases in the moist physics of that model (Dee et al. 2011) and the fact that the vertical structure of humidity is less constrained than for temperature because of limitations in observational corrections to the short-range forecast. Were the forecast model of ERAI (the operational model in 2006) used for long integrations, only weak CCKW would be produced (Bechtold et al. 2008; Hirons et al. 2013). However, the T255-LAND freely running integrations, employing the more recent model cycle of 2013, also show a deficiency—namely, an overly weak moisture anomaly in the lower troposphere with particularly weak anomalies near the top of the boundary layer. In contrast, IGRA has nearly continuous anomalies over a deep layer extending upward from the boundary layer. Importantly, it is the moisture anomaly in the lower troposphere that is predominantly responsible for the unusual boundary layer moist entropy anomalies 
As in Fig. 11, but for specific humidity anomalies (g kg−1).
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
As in Fig. 11, but for specific humidity anomalies (g kg−1).
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
As in Fig. 11, but for specific humidity anomalies (g kg−1).
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
5. Conclusions
Composite structures of convectively coupled Kelvin waves in the west Pacific and in other tropical locales were analyzed in terms of the linear theory of RF07 using radiosonde data, 3D analysis and reanalysis model output, and various ECMWF model simulations. We used five different thermodynamic data sources (IGRA, FNL, ERAI, ECMWF operational analysis, and freely running 1-yr ECMWF integrations) and four different precipitation proxies (NOAA OLR, TRMM 3B42, ECMWF diagnosed OLR, and ECMWF diagnosed rainfall). The radiosonde composites were constructed using two independent datasets. Derived variables and indices representing deep convective inhibition and precipitable water from these datasets were used to examine the assumptions and conclusions of the RF07 theory.
We find that convective inhibition, represented here by the index DCIN, varies strongly with the CCKW precipitation anomaly. The DCIN anomaly appears to lead the rainfall on the average, but, given uncertainties in the data, the amount of lead time is not clear. The unusual DCIN lag combined with reduced CCKW variance in the ECMWF freely running real-Earth simulation suggests the DCIN lead may be important. The DCIN–rainfall relationship is largely driven by fluctuations in the lower-tropospheric temperature, here represented by the saturated moist entropy. This temperature increases ahead of the rainfall and is closely followed by a boundary layer moist entropy anomaly, shown here to be largely a function of moisture fluctuations. Together, these variations constitute a significant minimum in convective inhibition that leads or is nearly in phase with the CCKW rainfall event. We found this to be true in every dataset and model that we studied in regions where the CCKW variance is large. Since the RF07 linear model results in an unstable CCKW-like mode with realistic vertical structure even when the lower-tropospheric temperature anomaly alone modulates DCIN and thus the convective heating, we conclude that this temperature anomaly is at least as important as the moisture anomaly in driving these waves and perhaps even causes the moisture anomaly. Since the RF07 model produces unstable Kelvin waves even when DCIN leads the rainfall by about 1 h, a small lead in the real atmosphere is significant. The fact that DCIN leads the rainfall in both the RF07 model and in the real atmosphere, combined with the fact that the RF07 model produces unstable Kelvin waves when this is true, suggests that DCIN plays a causal role in driving the deep convective phase of CCKWs.
While the above analysis describes CCKWs in which the variance of filtered rainfall is large, there are interesting differences at other locales that deserve further study. In particular, stations in the Indian Ocean show a different rainfall–DCIN phase relationship that appears to be dependent on the disturbance period. This supports the findings of Roundy (2012a,b) and is further evidence that wave species in the Indian Ocean display a continuum of different phase speeds and physical mechanisms.
The boundary layer moist entropy varies more strongly in the real atmosphere than is predicted by RF07, but given the results of the linear model, it is unclear whether this variable is essential to the destabilization and propagation of CCKWs. The RF07 linear model has an unstable CCKW-like mode even in the absence of surface moisture flux feedback, suggesting the boundary layer moist entropy anomaly near lag = −1 day is not important. However, the contrast between ECMWF aquaplanet simulations with and without parameterized deep convection shows that this variable affects the physical realism of CCKWs. When the deep convective scheme is shut off, convection is suppressed too long in this phase of the wave. Alternatively, Hirons et al. (2013) showed that if the deep convective scheme has weak entrainment, convection is triggered too early in this phase.
Precipitable water variations are considerably smaller than DCIN anomalies during the CCKW passage. We take this to mean that precipitable water has an insignificant effect on CCKWs and vice versa. This result is robust across all of our analyses and is in agreement with Roundy and Frank (2004) and also with Yasunaga and Mapes (2012a,b).
Caveats for this study include the fact that there is considerable random noise in the rainfall proxies and also in the thermodynamic data. This increases the scatter in the DCIN–rainfall relationship. Also, the noted time lag between the actual rainfall event and the OLR anomaly represents a small systematic error in the estimate of the phase relationship between DCIN and rainfall. In addition, the limited temporal resolution of the datasets used here creates uncertainty on the order of ±6 h in the estimates of all phase relationships. Despite these caveats, we find a consistent pattern among a broad variety of datasets: radiosondes, analysis, reanalysis, forecast model output, and even in the cloud-resolving model results of Fuchs et al. (2014).
It is interesting to note that the tilted vertical structure in the RF07 model is a result of including DCIN in the convective closure. Evidence for this is seen in comparing Fig. 5 of RF07, which includes DCIN, with Fig. 4 of Fuchs and Raymond (2007), which does not include DCIN. The tilted structure is also robust to the choice of upper boundary condition, which we have verified by replacing the radiation upper boundary condition solution for one using a rigid lid (not shown). However, the tilted structure in this case neither corresponds to an unstable wave mode nor is unique. In the linear model of RF07, therefore, the radiation upper boundary condition is a necessary, but not sufficient, condition for the westward tilt in an unstable wave mode. In contrast, Kuang (2008a) found a realistic tilted structure in numerical simulations of convectively coupled waves using a CRM linked with a linear wave model even when the linear model was capped by a rigid lid. In addition, Kuang (2008b) found unstable wave modes with realistic vertical structure in a toy linear model capped by a rigid lid. Thus, there is evidence that the lower arm of the characteristic boomerang structure—the westward tilt—in the temperature profile of CCKWs may be robust to the state of the upper boundary.
While the RF07 model assumes a single vertical heating mode, the Kuang (2008b) model uses two heating modes linked by the assumption that midtropospheric humidity modulates the depth of convection. While the latter mechanism may allow the linear model of Kuang (2008b) to exhibit realistic unstable waves beneath a rigid lid, the fact that the RF07 model yields realistic unstable waves when only a single heating mode—and a more realistic upper boundary—are assumed suggests that the RF07 model contains sufficient physics for the destabilization of CCKWs. Since the RF07 model includes only very simple assumptions about the shape and phase of the heating, its realism supports the notion that the tilted structure is an effect of wave, not convective, dynamics.
CCKWs are generally fast-moving disturbances in which wave dynamics are important, while the integrated column water content—important in slow-moving disturbances—does not play a major role. These waves appear to be controlled by changes in convective inhibition primarily due to changes in the buoyancy of air above the PBL. The low-level moisture anomaly that precedes these disturbances is also important as it affects DCIN. The phase relationships between precipitation maxima and anomalous DCIN, as well as the small-amplitude precipitable water anomalies found in west Pacific IGRA radiosonde stations, the FNL analysis, ERAI, the ECMWF operational analysis, and ECMWF aquaplanet simulations strongly support our hypothesis that DCIN is the mechanism that destabilizes CCKWs and is the primary causal mechanism by which CCKWs control convection.
Acknowledgments
We thank Paul Roundy, Zhiming Kuang, and George Kiladis for their helpful reviews of our manuscript as well as for many discussions on the CCKW. We thank Patrick Haertel for discussions on convectively coupled waves in general and for the perusal of TOGA-COARE CCW composites. We also thank Larry Oolman at the University of Wyoming for the use of many radiosonde data and George Kiladis and Maria Gehne at NOAA for providing an interpolated version of the IGRA radiosonde dataset. The TRMM 3B42 data used in this effort were acquired as part of the activities of NASA’s Science Mission Directorate and are archived and distributed by the Goddard Earth Sciences (GES) Data and Information Services Center (DISC) via the Mirador data portal. The SST climatology used in this study was obtained from the LDEO/IRI Data Library. Analysis (FNL) and reanalysis (ERAI) data were obtained from NCAR’s Computational and Information Systems Laboratory (CISL) Research Data Archive. Derived data and coded routines used in this study are available from the corresponding author upon request.
This work was supported by U.S. National Science Foundation Grants ATM1021049 and ATM1342001 and also by the European Commission’s 7th Framework Programme, under Grant Agreement 282672, EMBRACE project.
APPENDIX
Comparison of Diagnostics Using ECMWF OLR and Rainfall
We now examine the differences between the ECMWF OLR and rainfall diagnostics. We found the correlation between these variables to be greatest at a lag of about 5 h, so that the rainfall diagnostic leads the OLR diagnostic by this interval. A lagged relationship makes sense, since the cloud-top temperature anomaly should evolve somewhat later than the mass flux that directly produces surface rainfall.
Figure A1 shows the RF07 indices for the ECMWF operational analysis, T255-OPANA, for both wave filters (KELVIN and MONO) and for both rainfall proxies available in the model (OLR and RAIN). These plots are shown with a larger vertical scale than the earlier plots to accommodate the surface rainfall results. The indices regressed onto the rainfall diagnostic have larger amplitude as a result of the smaller variance of the model rainfall compared to the other rainfall proxies used in this study.
As in Fig. 1, but for wave composites from T255-OPANA when using model (top) OLR and (bottom) rainfall for both the (left) KELVIN and (right) MONO filters. Note that the vertical axes are larger to accommodate the rainfall-regressed cases.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
As in Fig. 1, but for wave composites from T255-OPANA when using model (top) OLR and (bottom) rainfall for both the (left) KELVIN and (right) MONO filters. Note that the vertical axes are larger to accommodate the rainfall-regressed cases.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
As in Fig. 1, but for wave composites from T255-OPANA when using model (top) OLR and (bottom) rainfall for both the (left) KELVIN and (right) MONO filters. Note that the vertical axes are larger to accommodate the rainfall-regressed cases.
Citation: Journal of the Atmospheric Sciences 73, 1; 10.1175/JAS-D-15-0153.1
The ECMWF rainfall standard deviation (
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