A Geostationary Satellite-Based Approach to Estimate Convective Mass Flux and Revisit the Hot Tower Hypothesis

Abstract

This study aims to revisit the classic “hot tower” hypothesis proposed by Riehl and Simpson (Malkus) in 1958 and revisited in 1979. Our investigation centers on the convective mass flux of hot towers within the tropical trough zone, using geostationary (GEO) satellite data and an innovative analysis technique, known as ML16, which integrates various data sources, including hot tower heights, ambient profiles, and a plume model, to determine convective mass flux. The GEO-based ML16 approach is evaluated against collocated ground-based radar wind profiler observations, showing broad agreement. Our GEO-based estimate of hot tower convective mass flux, 2.8 × 10¹¹–3.4 × 10¹¹ kg s⁻¹, is similar to the revisited estimate in Riehl and Simpson (1979), 2.6–3.0 × 10¹¹ kg s⁻¹. Additionally, our analysis gives a median count of around 550 hot towers with a median size of about 11 km, in contrast to the previous estimates of 1600–2400 hot towers, each characterized by a fixed size of 5 km. We discuss the causes of these discrepancies, emphasizing the fundamental differences between the two approaches in characterizing tropical hot towers. While both approaches have various uncertainties, the evidence suggests that greater credibility should be placed on results derived from direct satellite observations. Finally, we identify future opportunities in Earth Observations that will provide more accurate measurements, enabling further evaluation of the role played by tropical hot towers in mass transport.

Article Highlights

• Using geostationary satellite observations and an observation-constrained plume model, tropical hot towers are identified, and their convective mass fluxes are estimated

• Results are evaluated with collocated A-Train and ground-based estimates, showing broad agreement

• Estimates of convective mass flux, mean size, and total number of hot towers are compared with those made in Riehl and Malkus (1958) and Riehl and Simpson (1979)

1 Introduction

The “hot tower” hypothesis was originally introduced by Riehl and Malkus (1958) to clarify the process of heat transport from the surface to the upper troposphere within the tropical trough zone, providing an explanation for the vertical movement of air against energy gradient to reach high altitudes. At that time, the term “hot tower” had not yet been coined; the authors used the terms “undilute cloud towers” and “cumulonimbus chimneys.”

Subsequent studies revised the undilute requirement for cloud towers defined as deep convective clouds originating in the planetary boundary layer and extending near the upper tropospheric outflow layer, allowing for some entrainment of ambient air (Zipser 2003; Fierro et al. 2009), although the requirement remains in Riehl (1979) revision.

Nevertheless, the core idea of the hypothesis remains: Vertical heat transport, required for energy balance of the tropical trough zone, is achieved through hot towers and not through gradual, large-scale ascent of air masses, which fails to overcome the mid-tropospheric energy minimum. Furthermore, through energy balance and continuity considerations, Riehl and Malkus (1958) estimated the mass flux in undilute hot towers, and, with assumptions about the sizes and vertical velocities within cloud towers, they postulated that a total of 1500–5000 active hot towers embedded within the tropical trough zone maintain the tropical energy balance. A later study by Riehl (1979) revised the hot tower count to 1600–2400.

The “hot tower” concept and subsequent inferences about the properties of tropical deep convective cloud were ground-breaking achievements at a time when global observations were scarce. Nevertheless, any quantitative conclusion about mass and energy transport by hot towers in those early studies should be considered tentative. Indeed, Riehl and Malkus (1958) acknowledged that “lack of [global] data prevents a definitive treatment” and their attempt should be treated as “an initial study proceeding a much more extensive analysis.” Presently, with the wide availability of satellite observations of clouds and precipitation, alongside the development of new analysis techniques in recent years (Masunaga 2022; Luo et al. 2022), it is of interest to revisit the subject. The primary objective of this current paper is to reassess the “hot tower” hypothesis using modern satellite data, particularly from geostationary satellites, and hone in on a key component of the hypothesis—convective mass flux in hot towers.

Convective mass flux is a fundamental parameter for characterizing convective clouds. It controls the amount of heating and drying imposed by convection on the environment (Yanai et al. 1973; Arakawa and Schubert 1974). A large number of global climate model (GCM) cumulus parameterization schemes are based on the concept of convective mass flux (Arakawa 2004). Despite its foundational significance, measuring convective mass flux remains a serious challenge, primarily due to the difficulty of obtaining vertical air motion within intense convective clouds. Historically, storm-penetrating aircraft have been used to directly measure the vertical air velocity inside convective cores (Byers and Brahma 1948; LeMone & Zipser 1980a, 1980b; Lucas et al. 1994). To increase spatial coverage and to address safety concerns, high-altitude aircraft have been deployed to fly above convective storms to remotely retrieve convective air motions using Doppler radars (e.g., Heymsfield et al. 2010). Similarly, ground-based Doppler radars and wind profilers have also been utilized to estimate convective vertical velocities (Giangrande et al. 2013, 2016; Kumar et al. 2015; May and Rajopadhyaya 1999; North et al. 2017; Wang et al. 2019, 2020; Williams 2012).

While aircraft and ground-based observations offer valuable insights into convective mass flux, their scope is limited in both spatial extent and temporal duration. Satellite observations capture information on much larger spatial and temporal scales, but no current satellite is designed to measure the vertical air motions inside convective clouds. To fill the vacuum, Masunaga and Luo (2016; hereinafter ML16) introduced a novel, satellite-based approach, which blends information across scales to globally characterize convective mass flux. Initially, this technique was applied to A-Train satellite data (ML16; Jeyaratnam et al. 2020). Comparisons of ML16’s mass flux computations with collocated ground-based radar wind profiler observations reveal broad agreement (Jeyaratnam et al. 2020). A-Train observations restrict the sampled cases in the tropics to approximately 1:30 AM and 1:30 PM local time. This temporal sampling does not capture the full diurnal variations in convection, which is particularly pronounced over tropical and subtropical land areas (e.g., Bowman et al. 2005; Liu and Zipser 2008). Moreover, the limited temporal sampling makes it difficult to obtain a comprehensive depiction of cloud life cycle. A solution to these challenges lies in geostationary satellites (GEOs), which offer continuous tracking of convective clouds, especially those in low- and mid-latitudes, from inception to dissipation. In addition, GEO platforms capture diurnal variations by providing high-temporal-resolution global observations.

In this study, we apply the ML16 approach to GEO observations, transitioning it away from A-Train-based retrieval. Section 2 describes the data and methodology, followed by an assessment against earlier estimates utilizing A-Train data and ground-based radar wind profiler observations (Sect. 3). Section 4 revisits the “hot tower” hypothesis discussed in Riehl and Malkus (1958; hereafter RM58) and Riehl (1979; hereafter RS79), focusing on convective mass flux in hot towers, as well as the size and counts of the hot towers in tropical trough zones. Some initial results about diurnal variations are also presented (Sect. 5). Finally, we discuss limitations, future improvements, and potential applications.

2 Data and Methodologies

The primary data used in this work are infrared (IR) brightness temperatures (BTs), precipitation rates, and atmospheric profiles of temperature and moisture. The IR BT and precipitation rates were used to identify hot towers or deep convective cores, while the collocated atmospheric profiles of temperature and moisture were used to drive a convective plume model and compute vertical velocities of hot towers, as part of the ML16 approach. (Refer to Sect. 2.4 for details.) IR BT and vertical temperature profiles are also used to help constrain the plume model outputs toward an optimal estimate of convective mass flux.

2.1 Brightness Temperatures and Precipitation Rates

IR BTs are obtained from the NCEP/CPC’s Global Merged IR V1 data (MERGIR) (Janowiak et al. 2017; https://disc.gsfc.nasa.gov/datasets/GPM_MERGIR_1/summary). The MERGIR data combine IR brightness temperatures from all available operational GEO satellites, providing continuous global coverage from 60° S to 60° N with a horizontal resolution of about 4 km and a temporal resolution of 30 min. Precipitation rates are taken from the IMERG dataset, with half-hourly temporal resolution and 0.1° × 0.1° spatial resolution (Huffman et al. 2019).

2.2 Ambient Profiles of Temperature and Moisture

Two datasets provided temperature and moisture information in this study. The primary profiles come from ECMWF Reanalysis v5 (ERA5) data (Hersbach et al. 2023) with 37 pressure levels between 1000 and 1 hPa, temporal resolution of one hour, and spatial resolution of 0.25° × 0.25°. To test the sensitivity of our results to the ambient profile data, we also use the Modern-Era Retrospective analysis for Research and Applications data (MERRA-2; Global Modeling and Assimilation Office (GMAO), Goddard Earth Sciences Data and Information Services Center (GES DISC), n.d.) in certain sections of the paper and compare results with those obtained using ERA5. MERRA-2 data are provided at 41 pressure levels between 1000 and 0.3 hPa. The temporal and spatial resolutions are 3 h and 0.625° × 0.5°, respectively.

2.3 Identifying Hot Towers

Hot towers, or deep convective cores, usually ascend above nearby cirrus anvils, forming a dome-like shape which appear as local minima in IR BT images (Machado and Rossow 1993; Takahashi and Luo 2012, 2014). They produce substantial surface precipitation (Takahashi et al. 2021; Pfister et al. 2022). Consequently, our definition of hot towers involves a combined analysis of IR BTs and surface precipitation rates.

To identify hot towers in GEO IR BT data, we first apply temperature thresholds of 208 K and 215 K to filter for clouds at or above the height of tropical anvils. We chose the 208K threshold partly because it has been used in several studies to isolate cold tropical clouds (Mapes and Houze 1992; Chen et al. 1996). We select the slightly higher temperature of 215 K to test the sensitivity of our vertical mass flux estimates to the choice of brightness temperature threshold, while still restricting to the tall hot towers that RM58/RS79 consider. While these temperatures might seem somewhat subjective, they correspond to the altitude of 13–14 km, typical heights of cirrus anvils in tropical regions (Takahashi and Luo 2012). Hence, towers extending above these heights are likely dome-shaped deep convective cores or hot towers.

Given the cloudy region at or below tropical anvil cloud temperatures, we identify local minima in the filtered scene using the determinant of the Hessian, a metric that quantifies the curvature of two-dimensional images (Lindeberg 1998). This metric provides the concavity of each BT pixel, making it straightforward to isolate local minima. We refine the resulting blobs by approximating each of them as an ellipse (Machado et al. 1998). Figure 5 in the Appendix provides some examples of the hot towers identified.

We then filter the BT local minima to keep only the ones with surface precipitation exceeding criteria for deep convection, following Pfister et al. (2022). Their study analyzed collocated CloudSat/CALIPSO radar/lidar profiles, IR BTs, and surface precipitation rates (from combined radar/microwave/IR TRMM 3B42) and established region-by-region precipitation rate thresholds for tropical deep convection. (Refer to Table 1 in Pfister et al. 2022 for details.)

Table 1 Convective mass flux at 500 hPa averaged within a latitudinal belt extending from the trough to a distance of 10° on the winter side. DJF is December, January, February, and JJA is June, July, August. Units: kg s⁻¹

2.4 ML16 Applied to Geostationary Satellite Observations

ML16 is a hybrid approach which blends information across scales, leveraging plume model computations, to estimate convective mass flux. Here, we provide a concise overview of the method.

To apply the ML16 method to GEO data, we start by identifying deep convective cores (O(1 km)) in GEO data, as described in Sect. 2.3, and calculating the height of those cores. The cloud-top height is found by matching the cloud-top temperature with the collocated temperature profile from ERA5 or MERRA-2. While the use of IR BT underestimates cloud-top height due to the non-blackbody effect (Minnis et al. 2008; Sherwood et al. 2004), the bias is generally negligible for deep convective cores. Unlike thin cirrus and cirrus anvils whose cloud tops are “fuzzy,” deep convective cores contain “packed” cloud tops that emit radiation in the IR, almost like blackbodies. Using collocated CloudSat radar, CALIPSO lidar, and MODIS IR TB observations, Wang et al. (2014) demonstrated that the difference between the physical cloud-top (determined from CloudSat and CALIPSO vertical profiles) and the IR emission level is, on average, approximately 200 m.

For each hot tower or deep convective core identified, the ML16 method is used to compute a vertical velocity (w_c) profile. Only upward motion is quantified in this method, so w_c does not include any contribution from downdrafts. ML16 involves using ambient temperature and moisture profiles representative of the hot tower’s environment (O(10 km)–O(100 km)) to drive a convective plume model. (For details on the plume model equations, see Appendix 1.) The model generates a range of in-cloud vertical velocity (w_c) profiles, each corresponding to a prescribed turbulent entrainment rate—the most important, yet unknown, parameter in the model. The plume model outputs are then constrained using a Bayesian weighting procedure, with cloud-top height serving as the primary constraint for plume model simulations, leading to an optimal estimate of the w_c profile. (For details on the plume model equations and implementation flowchart, see Appendix 1.)

Finally, convective mass flux M_c is calculated in a region on the order of O(100 km) (similar to the current GCM grids) as M_c = σρw_c, where σ represents the fractional coverage of the convective core, defined as the convective core area divided by the grid area, and ρ is the air density. When multiple cores exist within a single grid box, the total convective mass flux is calculated as the sum of all the individual mass fluxes.

3 Comparisons with Previous Studies and Sensitivity Tests

In previous work, estimates of convective mass flux based on A-Train observations were evaluated using collocated ground-based radar wind profiler observations made by the DOE ARM program at Manacapuru, Brazil, during the “Observations and Modeling of the Green Ocean Amazon 2014–2015” (GoAmazon 2014/15; Martin et al. 2016) field campaign. Comparisons reveal broad agreements between ML16’s computations of convective vertical velocity and mass flux and the radar wind profiler’s estimates (Jeyaratnam et al. 2020). It should be noted that, unlike validating satellite retrieval of precipitation, which can be achieved by matching satellite observations with certain “ground truth” (usually radar or rain gauge observations) pixel-by-pixel, there is no such procedure for evaluating convective mass flux due to lack of “ground truth” observations. A meaningful comparison is only possible in a statistical sense, as is done in Jeyaratnam et al. (2020). Consequently, our estimate of convective mass flux through the ML16 approach should represent the mass flux from an ensemble of convective clouds within a mesoscale region, rather than depicting mass transport by an individual cumulonimbus. This concept is similar to GCM’s cumulus parameterization. Following the approach in Jeyaratnam et al. (2020), we compile statistics of convective mass flux based on GEO data at the site of the GoAmazon 2014/15 field campaign.

Figure 1 shows the mean vertical profiles of w_c and M_c computed within a 0.5° × 0.5° domain surrounding the GoAmazon site from 2014 to 2015. These means are derived from scenes when deep convection occurs. The domain analyzed and methodology used were chosen to match the approach of Jeyaratnam et al. (2020). Comparing our current results with Fig. 3 in Jeyaratnam et al. (2020), which is reproduced in the left panels of Fig. 1, we first note that w_c and M_c are broadly comparable in magnitude between the GEO-based estimates and those based on A-Train and radar wind profiler observations.

Fig. 1

Mean vertical profiles of convective vertical velocity (w_c; upper panels) and convective mass flux (M_c; lower panels) computed within a 0.5° × 0.5° domain surrounding the GoAmazon site from 2014–2015. The left two panels are adopted from Jeyaratnam et al. (2020), showing the estimates from ground-based radar wind profiler (RWP; black) and satellite estimates based on A-Train data (ML16; red). The two right columns show results from the current study. The middle column is for IR BT threshold of 208 K, and the right column is for IR BT threshold of 215 K. Triangles and dots represent results based on ERA5 and MERRA-2, respectively. Solid and dashed lines represent results based on hot tower selections with and without surface rainfall filtering, respectively

We tested the sensitivity of the mean vertical velocity and mass flux to choices made in the IR BT threshold, rain filter, and temperature and moisture profile data. The choice of different IR BT thresholds and the inclusion or exclusion of a rain filter influences the mean convective area, resulting in differences of less than 7%. Different ambient profile data affect w_c primarily through the plume model simulations. The main differences occur above 400 hPa, where MERRA-2 produces a w_c that is weaker by approximately 30–40% compared to ERA5. These sensitivity tests provide an estimate of the uncertainties for our GEO-based determination of convective mass flux in tropical hot towers. There is slightly better agreement between ground-based wind profiler observations and results obtained using the IR BT threshold of 208 K. Hence, we use this threshold value in the remainder of the paper.

4 Revisiting the “Hot Tower” Hypothesis

Figure 2 shows the global distribution of convective mass flux in hot towers at 500 hPa for two solstice seasons (DJF and JJA). It should be noted that the mass fluxes depicted here are specifically linked to hot towers as defined in this study, namely, deep convective cores that reach altitudes exceeding 13–14 km. Shallower convection is not included. To create this figure, mass fluxes within grid boxes of size 1° × 1° are computed at hourly intervals for each MERGIR image in 2018–2019. The years 2018–2019 were chosen to reduce missing data biases that are present over the West Pacific in 2015–2016. The plots show the mean mass flux across all scenes within a season, including instances where deep convection does not occur. We focus on 500 hPa to facilitate a comparison with RM58 and RS79, as their hot tower statistics were derived from mass flux at this level. Across the globe, the geographical distributions of convective mass flux in hot towers closely match the locations where deep convective clouds are prevalent in each season. Prior studies have extensively documented the climatology and variability of these tropical deep convective regions using various satellite-based methodologies, such as the analysis of infrared images (Gettelman et al. 2002; Rossow and Pearl 2007), precipitation radar (Liu et al. 2007), and cloud radar observations (Takahashi et al. 2017), and the subject is being expanded upon in a related paper by Pilewskie et al. (2024). In this study, we introduce convective mass flux within hot towers, contributing to an enhanced understanding of the roles played by tropical deep convection, particularly its impact on heat and moisture budget. As demonstrated by Yanai et al. (1973), the apparent convective heating and drying of hot towers are proportional to convective mass flux.

Fig. 2

Global distribution of convective mass flux in hot towers at 500 hPa for two solstice seasons based on ML16-GEO. The upper panel shows DJF, and the lower panel shows JJA. The red lines show the latitude corresponding to the “center of mass” of convective mass flux at each longitude

To compare our estimates with those in RM58 and RS79, we employ a coordinate system that follows the ITCZ. In their studies, RM58 and RS79 used data available at the time to define a constant trough line at about 10° N in northern summer; convective mass flux and hot tower statistics were estimated for a latitudinal belt extending from the trough to a distance of 10° on the winter side (i.e., toward north during the northern hemispheric winter and toward south during the northern hemispheric summer). Following their approach, we define the tropical trough position as the “center of mass” of the convective mass flux for each longitude using the formula:

$$ {\text{Trough}}\left( {{\text{lon}}} \right) = \frac{{\sum\nolimits_{{{\text{lat}}}} {M_{{{\text{lat}},{\text{lon}}}} \times {\text{lat}}} }}{{\sum\nolimits_{{{\text{lat}}}} {M_{{{\text{lat}},{\text{lon}}}} } }} $$

Here, Trough (lon) is the latitude (in degrees) of the equatorial trough as a function of longitude, and M_lat,lon is the mean mass flux as a function of latitude and longitude. The red lines in Fig. 2 show the calculated trough lines or Trough (lon). Notably, the trough lines closely follow the meander of the ITCZ. (Note that the term “ITCZ” was not referenced in RM58; this concept might not have been fully established back then due to a lack of a comprehensive global view of the Earth.)

Table 1 shows our GEO-based estimates of the mean convective mass transport within hot towers at 500 hPa for the 10°-belt on the winter side of the tropical trough for the years 2018–2019. We present the total mass transport (units of kg s⁻¹) within the latitudinal belt and not the mass flux (units of kg m⁻² s⁻¹) because this is the quantity that RM58/RS79 estimated. The selection of hot towers was based on an IR BT threshold of 208 K. Results are presented for two solstice seasons and for the months of February and August. These months (February and August) are chosen to follow the procedure employed in RM58/RS79, which focused on the months during which the position of the equatorial trough reaches its extremes. By doing so, the authors could assume that the ocean heat storage term in their energy budget analysis was negligible, thereby allowing them to neglect this uncertain value. RM58 and RS79 further combined results from February and August to improve the statistics of the sparsely available data at the time. This is acceptable because the winter side of the 10°-trough belt is mostly symmetric for February and August. Moreover, RM58 and RS79 estimated both total convective mass flux and convective mass flux attributed to updrafts, with the latter being more relevant to the ML16 estimates.

According to Table 1, the GEO-based estimate of convective mass transport at 500 hPa (2.8–3.4 × 10¹¹ kg s⁻¹) is similar to the mass transport predicted in RS79 (2.6–3.0 × 10¹¹ kg s⁻¹), although it lies on the upper end of the predicted range of values. Because RS79 presents amended values based on satellite data available at the time, we consider this value to likely be closer to the true mean mass transport in the ITCZ than RM58 (which gives 1.8 × 10¹¹ kg s⁻¹). Considering the ML16 and RM58/RS79 methods represent fundamentally different approaches to estimating convective mass flux, it is remarkable that both methods lead to similar values of mass transport. The RM58/RS79 method implements a “top-down” approach, where constraints on energy balance define vertical motion. The estimation of total vertical mass transport in the tropical trough zone is derived indirectly. In contrast, the ML16 method, applied to GEO data, functions as a “bottom-up” approach. It utilizes satellite observations of clouds and precipitation to define the locations and sizes of hot towers; plume model simulations driven by ambient profiles and constrained by satellite observations provide their vertical velocities, which are directly used to estimate convective mass flux. Both methods possess their intrinsic uncertainties. The main source of uncertainty in the RM58/RS79 approach lies in the accuracy of radiative fluxes within and at the top of the atmosphere, as well as surface latent and sensible heat fluxes. Knowledge of these energy fluxes at a global scale was understandably limited during the 1950s and even the 1970s. For example, the top-of-the-atmosphere radiative flux used in RM58 was only about 50% of its modern value (Hartmann 2016). Conversely, the ML16 approach depends on the proper identification of convective cores and accurate estimates of their vertical velocities.

In Sect. 3, we assessed GEO-based estimates by comparing them with those presented in Jeyaratnam et al. (2020) at the GoAmazon site. Our findings indicate that at 500 hPa, our estimates of convective mass flux are around 10–20% larger. If we interpret this as the overall bias assessment of the GEO-based estimate of convective mass flux, it suggests that convective mass flux in hot towers is slightly overestimated in our study. Nevertheless, the absence of global direct measurements of vertical air motions inside convective clouds hinders our ability to draw a definitive conclusion. Future Earth Observation missions with a focus on cloud dynamics are expected to provide more accurate global observations to revisit and refine our understanding of this subject.

A difference between ML16 method and the hot tower hypothesis comes from the different ways that entrainment is treated. In RM58/RS79, the authors emphasize that cloud towers are undilute. Conversely, the ML16 method solves an entraining and detraining plume model, which means that all results are presented for dilute cloud towers. Entrainment of ambient air impacts cumulus growth and vertical motion, so there will inevitably be some differences between the mass transport estimated by the ML16 method and RM58/RS79.

Another potential source of uncertainty arises from our reliance on GEO-based local minima in IR BTs to detect the presence of hot towers above the anvils. It is possible that the convective core within the cloud, where maximum mass flux occurs, might have a different size than the section rising above the anvils. Updrafts may be smaller than the overshooting cloud area, which could explain why our estimate of vertical mass transport is skewed to the upper end of RS79’s estimate. Depending on observations of clouds to make inferences about updrafts may lead to a systematic bias. Vertical profiles of the convective cores obtained through radar measurements would provide insight into this potential uncertainty. In this regard, several studies based on CloudSat radar data (Takahashi et al. 2017, 2023 and Pilewskie et al. 2024) consistently offer estimates of hot tower size that are similar to the GEO-based results presented here. This consistency suggests that utilizing above-anvil IR BT features provides a reasonable estimate of the size of hot towers. In our ongoing research, we use a large-eddy simulation in an Observing System Simulation Experiment (OSSE) framework to further investigate this potential uncertainty.

Figure 3 presents the statistics for the number of hot towers within the tropical trough zone and their sizes. Following RM58 and RS79, the hot tower size is defined as the square root of the hot tower area. Figure 3 shows a median count of approximately 550 hot towers and a median size of around 11 km. In comparison, RM58 and RS79 provided an estimate of 1500–5000 and 1600–2400 hot towers, respectively, each with a fixed length of 5 km. As mentioned earlier, RM58 and RS79 employed a “top-down” approach, deducing the number of hot towers entirely from their mass flux estimate and a provisional value for hot tower size (5 km), without using any global cloud observations, due to their unavailability at that time. Both the provisional hot tower size and the assumption that the size stays fixed greatly influence the number of towers that Riehl and Simpson estimate. Our GEO-based results (not shown) reveal that the distribution of convective cores sizes follows a power-law, in line with data from aircraft measurements (LeMone and Zipser 1980a, b). The parameters in the power-law remain unchanged with temperature threshold and season. By assuming that the total area of convective cores within the ITCZ amounts to 0.1% of the area of the ITCZ, the integral of the convective core size distribution can be constrained. Using the constrained size distribution to compute the number of convective cores, we find a value of roughly 600, matching our direct measurements. If, however, we assume that all convective cores have a fixed area of 25 km², we find around 1800 cores, similar to what Riehl and Simpson found. Therefore, the factor of 3 difference between the number of cores that we report and the one found in RM58/RS79 could be explained by the assumption about the size distribution of convective cores. Because the ML16 approach leverages satellite observations, more credibility should be attributed to the results derived from this method. It directly observes and derives statistics from cloud observations, providing a more robust foundation for the analysis.

Fig. 3

Box-and-whisker plots showing the distribution of the number of hot towers cores within the tropical trough zone (left) and the core size (right)

It should be noted that the GEO-based analysis of hot tower counts and sizes in this study comes with inherent biases and uncertainties. One factor contributing to these biases and uncertainties is the resolution of the GEO data, which is 4 km. This resolution may potentially result in an overestimation of hot tower sizes, as any core smaller than 4 km would be assumed to be 4 km. There is some evidence from aircraft-based measurements suggesting that the cores of many tropical deep convective clouds are smaller than our estimated value of 11 km (LeMone and Zipser 1980a, b; Heymsfield et al. 2010). However, aircraft-based measurements likely introduce a bias toward weaker convection due to safety concerns, and weaker convection generally features smaller cores, as indicated by multiple years of CloudSat radar observations (Takahashi et al. 2017). Surveys conducted by space-borne cloud radar suggest that the median diameter of deep convection cores is approximately 10–15 km (Takahashi et al. 2017, 2023). A related paper on hot towers by Pilewskie et al. (2024), which primarily used CloudSat radar observations, also arrived at an estimate of hot tower size of 11 km.

5 Diurnal Variations

An important advantage of GEO satellites over polar-orbiting counterparts is their capability to observe the complete diurnal cycle. We present a brief exploration of diurnal variations in convective mass flux. In Fig. 4, the two-dimensional histogram illustrates diurnal changes in convective mass flux at 500 hPa over land and ocean, with overlaid curves indicating mean values. These plots show the diurnal cycle of convective mass flux when convection occurs, omitting all grid boxes that do not contain convective cores. Across the ocean, there is only a marginal diurnal change in convective mass flux, with a minor peak in the early morning. Over land, the diurnal cycle is prominent, showing well-defined minima in the early morning and maxima in the late afternoon. The mean convective mass flux over land is approximately twice as much during the late afternoon compared to the early morning.

Fig. 4

Distribution of convective mass flux at 500 hPa as a function of local time. Color represents relative occurrence frequency (normalized to 1 at each local hour), and the black lines show the mean. Statistics are collected separately for the ocean (left) and the land (right)

These diurnal variations in convective mass flux generally follow the diurnal pattern of convective cloud activities, which have been well documented in prior studies analyzing IR data (Hendon and Woodberry 1993; Machado and Rossow 1993) and TRMM precipitation radar observations (Bowman et al. 2005; Liu and Zipser 2008). However, to our knowledge, this is the first explicit estimation of convective mass flux throughout the entire diurnal cycle. Considering that convective mass flux is a critical parameter controlling the impact of convection on heat and moisture budgets, detailing its diurnal variations is an important initial step toward examining how convection affects the surrounding environment on a diurnal time scale. It is important to note that Fig. 4 represents the average across the whole tropics. The diurnal variation in tropical convection exhibits substantial regional differences, which we intend to investigate in more detail in a future study.

6 Summary and Discussions

The “hot tower” hypothesis is a crucial concept in tropical meteorology that was originally introduced by RM58 and RS79. It explains how heat is transported from Earth’s surface to the upper troposphere in the tropical trough zone, bypassing the mid-tropospheric minimum of total energy. RM58 and RS79 deduced mass fluxes associated with the hot towers and estimated their total numbers based on constraints of energy balance, a remarkable achievement at a time when no global observations of clouds were available. Nevertheless, these estimates should be considered tentative, as cautioned by the authors. This study aims to reevaluate the “hot tower” hypothesis, focusing on convective mass flux, using geostationary (GEO) satellite data and an innovative analysis technique, known as ML16. This satellite-based method integrates various data sources, including hot tower heights, ambient profiles, and a plume model, to determine convective mass flux globally.

The ML16 approach applied to A-Train data was previously evaluated against collocated ground-based radar wind profiler observations during the GoAmazon field campaign, showing overall favorable comparisons. This study expands on the evaluation by comparing the GEO-based implementation of the ML16 method with earlier estimates at the same GoAmazon site. Both vertical velocities and convective mass fluxes are broadly comparable in magnitude among these different estimates. The use of GEO-based observations provides an opportunity to investigate the entire diurnal cycle of hot towers, briefly explored in this paper, with a more comprehensive analysis reserved for a future study. Sensitivity tests are conducted to quantify the uncertainties arising from choices regarding IR BT thresholds, rain filter, and ambient profile data. The largest uncertainty occurs above 400 hPa, where MERRA-2 produces vertical velocities approximately 30–40% weaker compared to those from ERA5.

In accordance with RM58 and RS79, we define the tropical trough zone as a latitudinal belt extending from the trough line to a distance of 10° on the winter side. Comparisons show that our GEO-based estimate of convective mass flux for hot towers within this tropical trough zone, which ranges between 2.8 × 10¹¹ kg s⁻¹ and 3.4 × 10¹¹ kg s⁻¹, is similar to the estimate of RS79, which was around 2.6–3.0 × 10¹¹ kg s⁻¹. Meanwhile, our analysis gives a median count of approximately 550 hot towers within the tropical trough zone with a median size of about 11 km. This contrasts with the estimates provided by RM58 and RS79, which indicated a range of 1500–5000 and 1600–2400 hot towers, respectively, each characterized by a fixed size of 5 km.

It is crucial to highlight the fundamental differences between the ML16 and RM58/RS79 approaches in characterizing tropical hot towers. The RM58/RS79 determines convective mass flux of the hot towers based on energy balance constraints, whereas the ML16 method employs satellite cloud observations to explicitly identify deep convective towers and then utilizes plume model computations, constrained by observations, to derive their vertical velocities and mass fluxes. The RM58/RS79 method takes a “top-down,” indirect approach in characterizing hot towers, while the ML16 method adopts a “bottom-up,” direct approach. Concerning the counts and sizes of hot towers, more credibility should be placed on results derived from direct satellite observations. While the use of GEO IR data may introduce some uncertainties, comparisons with prior surveys of tropical deep convective cores conducted by space-borne cloud radar show broad consistency, suggesting that the GEO-based analysis is reliable.

An aspect of RM58/RS79 that is missing from our analysis is downdrafts. The plume model we solve does not include downdrafts and instead represents convection as a cylinder of upward moving air, a model similar to what RM58/RS79 proposed in their hot tower hypothesis. In reality, convection is made up of updrafts and downdrafts at scales similar to or smaller than the convective cores we observe. It is likely that the hot towers that we observe in GEO contain some fraction of downdrafts, leading us to overestimate convective area coverage. While the original ML16 paper provides a way to estimate downdrafts as the residual between total mass flux and updraft mass flux, we focus only on updrafts in this study because the hot tower hypothesis is mainly concerned with upward moving air.

While both the ML16 approach using GEO data and RS79 give similar estimates of convective mass flux associated with updrafts, the lack of direct measurements of vertical air motions inside convective clouds leaves room for uncertainties and hampers our ability to draw a definitive conclusion. Nevertheless, our current study represents a significant first step toward utilizing modern-day satellite observations, along with a new analysis technique, to quantify the role played by tropical hot towers in mass transport. This is a crucial issue, as convective mass flux controls the effects of convective clouds on energy and water budgets. Looking ahead, it is anticipated that future Earth Observation mission, particularly those focused on cloud dynamics and vertical mass fluxes (e.g., EarthCARE, Illingworth et al. 2015; INCUS, van den Heever et al. 2023; C2OMODO, Brogniez et al. 2022; and future AOS, aos.gsfc.nasa.gov), will be instrumental in providing more accurate data to address this question.

Appendix 1: Numerical Method and Implementation Flowchart

The ML16 method combines remote sensing observations with plume model computations to estimate the vertical velocity and convective mass flux in a procedure shown on the flowchart in Fig. A1. Broadly, the method involves identifying convective cores in geostationary satellite data as local minima in brightness temperature, solving a one-dimensional plume model for a range of entrainment rates, and finally constraining the plume model outputs using the observed cloud-top height and cloud-top buoyancy.

Figure A.1

Flowchart of the ML16 method applied to GEO data. (a) Convective cores are identified as local minima in brightness temperature in geostationary satellite images. (b) Atmospheric sounding information, which is collocated with the convective cores, drives a plume model. (c) The plume model outputs a range of possible buoyancy profiles for a range of entrainment rates. (d) Using a Bayesian weighting procedure, we select the plume that best models the observed cloud-top height and buoyancy. (e) Identify the best-fit vertical velocity profile by the same Bayesian weight as selected from step (d)

The system of differential equations that make up the plume model used in the ML16 method is summarized in Eqs. (1)–(7).

Table 2 Variable definitions for the plume model equations

Table2 defines all of the variables used in these equations. For clarity, z-dependence of variables has been explicitly written except where it is obvious. These equations provide the vertical rate of change of the entrainment, vertical velocity, temperature, and water vapor and water/ice mixing ratios within a cloud. It assumes all cloud properties are horizontally homogeneous.

We solve this system of equations by initializing each function at the lifting condensation level (LCL) and stepping forward in height using a standard finite difference method. Ambient temperature, moisture, and geopotential profiles are used to compute the ambient moist static energy. At the LCL, the in-cloud temperature, density, and relative humidity are assumed to be equal to the ambient ones at the same height. The initial vertical velocity is 1.5 m/s. The model assumes that the pressure difference between the environment and the cloud is small, so the ambient pressure is used wherever pressure is referenced. All of the observables on the right side of the equations are known (from either the initial conditions or the previous integration step) except for $\frac{{dT}_{c}}{dz}$, the in-cloud temperature lapse rate. We deal with this numerically, guessing an initial value of $\frac{{dT}_{c}}{dz}\hspace{0.17em}$= − 0.1 K km⁻¹ and checking for consistency with the rest of the equations, that is, we use this preliminary lapse rate to compute the in-cloud MSE and ${T}_{c}$ one step above the current height, checking whether our initial guess of $\frac{{dT}_{c}}{dz}$ agrees with the finite difference derivative of ${T}_{c}$. If the difference between the finite difference derivative and the initial guess is smaller than some tolerance, then we accept this value and move on. If not, we update the guess with the finite difference derivative and try again until the value converges. When convergence is reached, $\frac{{dT}_{c}}{dz}$ follows moist adiabat, as expected for a saturated environment inside clouds.

The system of equations is solved for turbulent entrainment rates ranging from 0 to 0.4 km⁻¹ to obtain a range of possible plumes, following ML16. The optimal plume is the one with cloud-top height and cloud-top buoyancy closest to one obtained from GEO observations. We estimate the cloud-top buoyancy using a mean climatological buoyancy profile, which defines the cloud-top buoyancy as a function of cloud-top height.

The equations presented here are written in a form somewhat different from the ones originally presented in ML16. We have combined and reformulated expressions to rewrite the total entrainment rate for the $\frac{dM}{dz}>0$ case in terms of the buoyancy, density, and vertical velocity. This step offers an advantage because if we assume that $\frac{dM}{dz}>0$ at the LCL, a reasonable assumption for the deep convective clouds that we model, we can compute the total entrainment rate directly, simplifying the numerical procedure (Table 2 and Fig. A1).

$$ \frac{1}{M\left( z \right)}\frac{{{\text{dM}}}}{{{\text{dz}}}} = \smallint \left( z \right) - \delta \left( z \right) $$

(1)

$$ \frac{{dw_{c} }}{dz} = \frac{1}{{w_{c} \left( z \right)}}\left( {a_{B} B\left( z \right) - \smallint \left( z \right)\left( {w_{c} \left( z \right)} \right)^{2} } \right) $$

(2)

$$ \varepsilon (Z) = \left\{ \begin{aligned} & { { \frac{1}{2}\left( {\epsilon_{T} + \frac{{a_{B} B\left( z \right)}}{{w_{c} \left( z \right)^{2} }} + \frac{1}{\rho \left( z \right)}\frac{d\rho }{{dz}}} \right)} } \, {for\frac{dM}{{dz}} \ge 0} \\& \,\,\,\, {\epsilon_{T} } \, {for\frac{dM}{{dz}} < 0} \\ \end{aligned} \right. $$

(3)

$$ \frac{{dq_{vc} }}{dz} = \varepsilon \left( {\frac{1}{p\left( z \right)}\frac{{de_{s} }}{{dT_{c} }}\frac{{dT_{c} }}{dz} - e_{s} \left( {T_{c} } \right)\frac{1}{{p\left( z \right)^{2} }}\frac{dp}{{dz}}} \right) $$

(4)

$$ \frac{{dq_{i} }}{dz} = \frac{1}{2}\left( {\frac{{dq_{w} }}{dz}\left( {1 - \tanh \left( {\frac{{T_{c} \left( z \right) - T_{0i} }}{{dT_{i} }}} \right)} \right) + \frac{{w_{c} \left( z \right)}}{{dT_{i} }}{\text{sech}}^{2} \left( {\frac{{T_{c} \left( z \right) - T_{0i} }}{{dT_{i} }}} \right)\frac{{dT_{c} }}{dz}} \right) $$

(5)

$$ \begin{aligned} \frac{{dq_{w} }}{dz} & = - \varepsilon \left( z \right)q_{w} \left( z \right) - \left( {\frac{{dq_{vc} }}{dz} + \varepsilon \left( z \right)\left( {q_{vc} \left( z \right) - q_{va} \left( z \right)} \right)} \right) \\ & - \frac{1}{{\tau_{auto} w_{c} \left( z \right)}}\left( {q_{w} \left( z \right) - q_{w,crit} } \right)H\left( {q_{w} \left( z \right) - q_{w,crit} } \right) \\ \end{aligned} $$

(6)

$$ \frac{{dh_{c} }}{dz} = L_{i} \frac{{dq_{i} }}{dz} - \varepsilon \left( {h_{c} \left( z \right) - L_{i} q_{i} \left( z \right) - h_{a} \left( z \right)} \right) $$

(7)