Criticality in the Shallow-to-Deep Transition of Simulated Tropical Marine Convection

Scott W. Powell aDepartment of Meteorology, Naval Postgraduate School, Monterey, California

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Abstract

Idealized simulations of tropical, marine convection depict shallow, nonprecipitating cumuli located beneath the 0°C level transitioning into cumulonimbi that reach up to 12 km and higher. The timing of the transition was only weakly related to environmental stability, and 13 of the 15 simulations run with 5 different lapse-rate profiles had rain develop at nearly the same time after model start. The key quantity that apparently controlled deep convective formation was vertical acceleration inside cloudy updrafts between cloud base and the 0°C level. Below a critical value of updraft vertical acceleration, little rainfall occurred. Just as the domain-mean updraft acceleration reached the critical value, the first convection quickly grew to past 12 km altitude. Then, as acceleration increased above the critical value, rain rate averaged in the model domain increased quickly over about a 3-h-long period. The specific value of the critical updraft acceleration depended on how updrafts were defined and in what layer the acceleration was averaged; however, regardless of how criticality was defined, a robust relationship between domain-mean updraft vertical acceleration and rain rate occurred. Positive acceleration of updrafts below the 0°C level was present below 2.75 km and was largest in the 500 m above cloud base. However, the maximum difference between updraft and environmental temperatures occurred between 2 and 3 km. The domain-mean Archimedean buoyancy of updrafts relative to some reference state was a poor predictor for domain-mean rain rate. The exact value of the critical updraft acceleration likely depends on numerous other factors that were not investigated.

Significance Statement

A numerical model is utilized to investigate potential thermodynamic and dynamic quantities related to the growth of cumulus clouds into cumulonimbus clouds over tropical oceans when the atmosphere is sufficiently moist to support rainfall. Archimedean buoyancy alone cannot be used to predict rain rate reliably. Instead the total buoyancy not relative to an arbitrary reference state must be considered. The simulated relationship between total vertical acceleration in updrafts and rain rate was robust. While the processes that control the vertical acceleration remain unclear, our results highlight the importance of observing processes that occur on spatial scales of tens of meters and temporal scales of a few minutes.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Scott W. Powell, scott.powell@nps.edu

Abstract

Idealized simulations of tropical, marine convection depict shallow, nonprecipitating cumuli located beneath the 0°C level transitioning into cumulonimbi that reach up to 12 km and higher. The timing of the transition was only weakly related to environmental stability, and 13 of the 15 simulations run with 5 different lapse-rate profiles had rain develop at nearly the same time after model start. The key quantity that apparently controlled deep convective formation was vertical acceleration inside cloudy updrafts between cloud base and the 0°C level. Below a critical value of updraft vertical acceleration, little rainfall occurred. Just as the domain-mean updraft acceleration reached the critical value, the first convection quickly grew to past 12 km altitude. Then, as acceleration increased above the critical value, rain rate averaged in the model domain increased quickly over about a 3-h-long period. The specific value of the critical updraft acceleration depended on how updrafts were defined and in what layer the acceleration was averaged; however, regardless of how criticality was defined, a robust relationship between domain-mean updraft vertical acceleration and rain rate occurred. Positive acceleration of updrafts below the 0°C level was present below 2.75 km and was largest in the 500 m above cloud base. However, the maximum difference between updraft and environmental temperatures occurred between 2 and 3 km. The domain-mean Archimedean buoyancy of updrafts relative to some reference state was a poor predictor for domain-mean rain rate. The exact value of the critical updraft acceleration likely depends on numerous other factors that were not investigated.

Significance Statement

A numerical model is utilized to investigate potential thermodynamic and dynamic quantities related to the growth of cumulus clouds into cumulonimbus clouds over tropical oceans when the atmosphere is sufficiently moist to support rainfall. Archimedean buoyancy alone cannot be used to predict rain rate reliably. Instead the total buoyancy not relative to an arbitrary reference state must be considered. The simulated relationship between total vertical acceleration in updrafts and rain rate was robust. While the processes that control the vertical acceleration remain unclear, our results highlight the importance of observing processes that occur on spatial scales of tens of meters and temporal scales of a few minutes.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Scott W. Powell, scott.powell@nps.edu

1. Introduction

Satisfactory representation of the formation and maintenance of deep moist convection remains a major challenge for numerical models of the atmosphere. Inherently, any physical model that describes how properties of three-dimensional space evolve either must parameterize or ignore all processes that occur on spatial scales smaller than the model resolution explicitly permits. For the atmosphere, numerical models currently run with horizontal grid spacing as low as approximately 50 cm in direct numerical simulations (Mellado et al. 2018) to up to O(100) km for some climate models. Even the highest resolution models utilized must parameterize the properties of hydrometeor distributions (i.e., microphysical parameterizations) within the discrete clouds that they produce. Coarser models must additionally parameterize cloud populations and their relationships with their surrounding environments (i.e., cumulus parameterizations), and they must also simulate turbulence. Many current weather forecast models now operate at high enough spatial resolution so that cumulus parameterizations are not required; however, structures like updrafts and downdrafts that have spatial scale on O(100) m (Bryan et al. 2003) within individual clouds are not resolved. Similar models are used to produce reanalyses of the atmosphere that are used extensively in the research community, yet these models—and therefore the reanalysis products often inadvertently referred to as “observations”—still suffer from the same limitations that the underlying models contain.

One of the overarching challenges associated with representing moist convection is simulating the distribution and intensity of rainfall. This is particularly problematic considering the primary role of precipitation in Earth’s water cycle and energy balance at up to global and climatic spatial and temporal scales. Precipitation is also closely related to latent heat release in the atmosphere, which impacts the global circulation (Schumacher et al. 2004). Extreme rainfall events produce flooding with potentially major economic impact. Global reanalyses driven by general circulation models, although coupled to a plethora of surface and space-based observations of the atmosphere, struggle to reproduce accurate rainfall quantities (Bosilovich et al. 2008) or the proper spatial distribution of precipitation. The cumulus parameterizations that these models use couple cloud populations to larger-scale temperature and moisture. In nature, cloudy updrafts entrain environmental air that dilutes in-cloud buoyancy, acting to decelerate updrafts. As supported by modeling studies (e.g., Wang and Sobel 2012), this is especially true in the lower troposphere where the saturation deficit can be much larger than aloft, where relatively low temperatures govern maximum saturation deficits observed. As a result, in dry environments, little precipitation occurs because any updrafts that do develop are quickly decelerated to a stop; therefore, vertical mass flux and condensation are extremely limited.

Observations indicate that when column-integrated relative humidity (CRH) in cloudy environments reaches about 70%–80%, precipitation rate averaged over an area begins to rapidly increase as CRH increases (Bretherton et al. 2004; Rushley et al. 2018). Consistent with this, many cumulus parameterizations, such as the Tiedtke (1989) scheme that is used in reanalyses like the European Centre for Medium-Range Weather Forecasts reanalysis version 5 (ERA5; Hersbach et al. 2020), assume that clouds are highly sensitive to low-tropospheric humidity and temperature, and some include a convective “trigger” that initiates convection when a favorable thermodynamic profile is achieved. Such behavior resembles the Peters and Neelin (2006) description of a temperature-dependent critical value of environmental water vapor above which rain rate increased as a power-law relationship. When water vapor reaches this critical value, the transition of convection from shallow cumuli into deep cumulonimbi occurs. During the transition, updrafts break out from beneath the 0°C level and penetrate into the upper troposphere, where moist updrafts containing ice experience larger vertical accelerations than updrafts below the 0°C level (Holloway and Neelin 2009). Thus, the critical value of CRH can be thought of as a tipping point across which convection is able to penetrate through the 0°C level. This assumes that entrainment can be approximated using a deep-inflow assumption, one that observational analysis has shown may be valid (Schiro et al. 2018). At the same time, precipitation rapidly increases. Convective transitions are commonly seen on time scales as short as a few hours; for example, shallow continental cumuli can grow into supercells within a few hours over the central and southern United States. However, similar transitions can also occur in cloud populations over large areas and several days, such as during the onset of the Madden–Julian oscillation (MJO) (Powell and Houze 2013, 2015).

Ahmed and Neelin (2018) reframed the water vapor–precipitation relationship in terms of a quantity that considers both temperature and humidity in the lower troposphere. This quantity, BL, using the nomenclature from Adames et al. (2021), is an acceleration described approximately as
BL=gκL(wBθeBθeL*θeL*wLθeL*θeLθeL*).
Here, θe is equivalent potential temperature, θe* is saturation θe, κL is a virtual correction term [see Eq. (1b) of Adames et al. 2021], subscripts B and L denote pressure-weighted averages in layers that approximately represent the atmospheric boundary layer (B) and lower free troposphere (L), and wB and wL are weights for each term, which Adames et al. (2021) treated as approximately 0.5. They also defined the B and L layers for the deep tropics as 1000–850 and 850–600 hPa, respectively. The first term is analogous to the acceleration imparted by convective available potential energy in the lower troposphere: A warm boundary layer or cool lower free troposphere will increase the mean vertical acceleration of updrafts. The first term also includes the effects of temperature dilution of updrafts. The second term describes the effect of dry environmental air diluting and decelerating buoyantly driven plumes. On average, the relationship between BL and rain rate is robust in observations. Ahmed et al. (2020) describe precipitation as a ramp function such that rain rate is zero below a critical value of BL and a linear function above the same value. They demonstrate that deviations from this ramp function caused by stochastic processes or spatial-based averaging can produce the observed curve that resembles a power-law relationship. The value of BL can be computed using any vertical profile containing temperature and humidity up to 600 hPa, although the layer definitions may differ if the surface is not near 1000 hPa.

Although numerous studies have concluded rainfall is dependent on humidity to first order, a moist atmosphere alone is not sufficient to cause deep convection and heavy rain. In other words, CRH and BL are effective at capturing variability of rain rate across the wide range of humidity or atmospheric stability observed over tropical oceans; however, when the environment is moist enough to support deep convection, the relationships between these variables and rain rate become less well defined. Powell (2019) pointed out using ground-based radar and rawinsonde observations from the deep tropics that rain rate can range from near 0 to over 100 mm day−1 at CRH ≥ 80% on short time scales. This result is corroborated by space-based observations (Rushley et al. 2018). Furthermore, Adames et al. (2021) found similarly wide ranging rain rates above their critical BL value in a combined reanalysis/satellite-derived dataset. Much of this variability is likely associated with the convective life cycle: High rain rates occur when deep convection is prevalent, such as right after a shallow to deep convective transition. Lower rain rates occur after deep convection has moistened and stabilized the atmosphere; then weaker vertical motions and stratiform precipitation dominate. Process-oriented diagnostics such as those developed by Wolding et al. (2020) support this claim as does ground-based radar analysis (Powell 2019). They show that most rain is stratiform when CRH are at its highest values and the atmosphere is becoming more stable. Efforts to represent such upscale growth and “organization” in non-“superparameterized” (i.e., do not contain a cloud-resolving model to resolve cloud statistics per model grid point) cloud parameterizations aim to capture transitions in cloud populations by linking the degree of convective clustering to properties of the cloud environment (e.g., Mapes and Neale 2011; Chen and Mapes 2018), but even if successful, these do not explain the fundamental processes that lead to either less organized, isolated deep cumulonimbi, more organized stratiform, or no precipitation at all.

Kuo et al. (2017) investigated the water vapor–precipitation relationship in the tropics and concluded that the entrainment rate in models greatly impacts the timing of the convective transition such that low entrainment rates permit precipitation to begin before substantial moistening of the atmosphere has occurred. The dependence of convective transition on entrainment rate was echoed for phenomena occurring on longer time scales, such as the MJO (Del Genio et al. 2012). This reinforces the elementary notion that the vertical profile of vertical acceleration and velocity in convective updrafts is essential to the evolution of updrafts because entrainment generally results in dilution of updraft density. However, cumulus parameterizations in general circulation models make numerous unrealistic assumptions about entrainment. Vertical mass fluxes imposed by unrealistic estimates of entrainment in cumulus parameterizations can be significantly different than those depicted in large-eddy simulations (LESs) of buoyant convection (Peters et al. 2021). As a result, cumulus parameterizations misrepresent vertical accelerations and velocities within cumuliform convection and often struggle to adequately predict precipitation rate. This can particularly become a problem in moist environments where precipitation is expected to frequently occur.

Shallow-to-deep convective transition has been studied for several decades—particularly in the tropics where buoyantly driven plume- or thermal-like convection is common. Several quantities have been successfully related to the vertical growth of convection. For example, in observations of east Pacific convection, Raymond et al. (2003) found that the majority of variability in convective depth was attributed to spatiotemporal variability of surface entropy fluxes and temperature anomalies driven by dry adiabatic motions in a climatologically present layer of convective inhibition roughly located between 700 and 850 hPa, which was documented as a layer containing a cloud minimum by Zuidema (1998). Although variability in entropy fluxes played a dominant role in the east Pacific, reduction in “deep convective inhibition” caused by large-scale equatorial waves likely explains some of the variability in convective depth and rainfall in the deep tropics. Indeed, the importance of adiabatic vertical motions has been recently shown to be an important contributor to moistening in disturbances like the MJO (Adames et al. 2021). It is not unreasonable then to suspect that similar processes may control convective transitions at smaller spatial and shorter temporal scales.

In this article, tropical marine convection is modeled in idealized LESs so that cloudy updrafts are coarsely resolved. There are two primary objectives:

  1. Investigate alternative potential quantities other than CRH or BL that have critical values that correspond with formation of deep convection in environments that are already sufficiently moist to support deep convection.

  2. Determine how updraft accelerations relate to rain rate in simulations of tropical marine convection in moist environments.

Our focus on moist environments specifically is motivated by the need to resolve why temperature and humidity over spatial scales much larger than individual clouds are seemingly poor predictors of rain rate when the environment is moist and conditionally unstable. The idealized simulations will be anchored to observed soundings, and cloud structures will be verified using ground-based radar data to provide additional confidence that our simulations represent at least approximately realistic cloud populations.

2. Model

a. Configuration

Cloud populations were simulated using version 20.1 (cm1r20.1) of Cloud Model 1 (Bryan and Fritsch 2002). The domain size was 64 km × 64 km × 20 km. Horizontal grid spacing was 100 m. Vertical grid spacing was 50 m up to 2 km altitude, 250 m above 3.5 km altitude, and linearly varying between 50 and 250 m between 2 and 3.5 km altitude. Doubly periodic boundaries and a constant sea surface temperature of 302 K (29°C) was used. Convection was encouraged by introducing white noise with maximum amplitude of 0.25 K to the potential temperature field in the lowest 500 m of the domain at initialization. Shortwave radiation was turned off. The model was integrated for 24 h, and output written every 10 min was analyzed. Physics options used are listed in Table 1, and the namelist.input file utilized for all of the simulations is included as online supplementary information. No model output written before 3 h after start was analyzed.

Table 1

Physics options used for CM1 simulations.

Table 1

Five different sets of three experiments were conducted. The sounding for our control simulations (Base) is depicted in Fig. 1. The specific humidity in this sounding is the mean during “rainy” periods [see Powell and Houze (2013) for exact definition of “rainy”] during the Dynamics of the Madden–Julian Oscillation (DYNAMO) experiment (Yoneyama et al. 2013). The potential temperature profile was the mean of all soundings launched from 1 October to 31 December 2011 at Gan in Addu City, Maldives. The 0°C level occurred at the model grid point between 5 and 5.25 km altitude. Rawinsonde data were processed and quality controlled according to Ciesielski et al. (2014). The temperature profiles for the soundings used in the other four experiments were perturbed relative to the Base sounding. Each temperature perturbation ΔT corresponded to a hypothetical change in the clear-air vertical velocity profile that caused a change in the adiabatic heating rate such that
ΔTt=SpcpΔω,
in which Sp = ∂s/∂p and s is the dry static energy. The change in the adiabatic heating rate was then integrated over 24 h to determine a total temperature perturbation. The vertical structure of Δω was
Δω(p)=Acos(p550900π),
in which p is pressure (in hPa) and A is the magnitude of Δω at 550 hPa. Above 100 hPa and below 1000 hPa, Δω was set to zero. The five experiments corresponded to different values of A: −0.03, −0.01, 0, +0.01, and +0.03 Pa s−1. These values were selected because they are within the range of variability for wavenumber-1–3 vertical motions associated with the Madden–Julian oscillation (Powell and Houze 2015). Henceforth, each simulation will be referenced by its value of A, except that the A = 0 simulations will be called the Base simulations. Values of ΔT that correspond to A = ±0.03 are displayed in Fig. 2. Δω and ΔT are denoted respectively by solid and dashed lines. The −0.03 sounding (blue) contains a maximum ΔT of −1.5 K at 550 hPa, and the +0.03 sounding (red) is a mirror image. The −0.01 and +0.01 soundings (not plotted) correspond to a maximum ΔT magnitude of 0.5 K. Because ΔT is largest at 550 hPa, negative values for A destabilize the lower troposphere, and positive values stabilize it. Figure 3 depicts the θe and relative humidity (RH) profiles for each sounding. While the temperature profiles were altered, the RH profiles were kept near constant in order to isolate the effect of static stability on convective transition. For each value of A, three simulations were run using a different random arrangement of white noise beneath 500 m. Therefore, output from 15 simulations representing 5 different temperature profiles are included in the analysis herein.
Fig. 1.
Fig. 1.

Sounding for the Base simulation. Red and blue lines respectively denote temperature and dewpoint.

Citation: Journal of the Atmospheric Sciences 79, 7; 10.1175/JAS-D-21-0155.1

Fig. 2.
Fig. 2.

Profiles of perturbation ω (solid) and temperature (dashed) for the −0.03 (blue), Base (black), and +0.03 (red) soundings.

Citation: Journal of the Atmospheric Sciences 79, 7; 10.1175/JAS-D-21-0155.1

Fig. 3.
Fig. 3.

Initial profiles of (left) θe and (right) relative humidity used to force each set of simulations.

Citation: Journal of the Atmospheric Sciences 79, 7; 10.1175/JAS-D-21-0155.1

In-cloud updrafts and connected clear-air updrafts in the subcloud boundary layer were objectively identified using three-dimensional hydrometeor mixing ratio and vertical velocity fields. First, cumuliform clouds were defined as grid points where the sum of cloud water, rain, snow, and graupel mixing ratios exceeded 10−5 kg kg−1. Cloud ice was excluded to avoid undesired classification of anvil clouds or other cirrus clouds not connected to a cumuliform cloud. Then, all in-cloud and clear-air updrafts, including any beneath cloud base, were initially defined as grid points with w > 1 m s−1. Next, only clear-air updrafts that were contiguous in space with the cloudy ≥1 m s−1 updrafts were classified; any remaining clear-air vertical motions exceeding 1 m s−1 were declassified. Then, the remaining clear-air updrafts consequently consisted almost entirely of subcloud-layer upward motion in which clouds were rooted. Finally, any unclassified in-cloud grid points with w > 0.5 m s−1 were also classified as updrafts. However, for part of the analysis (Fig. 6), unclassified grid points with w > 0.1 m s−1 were also classified as updrafts to test the sensitivity of results to the definition of updraft.

b. Comparison of radar reflectivity distributions

Peters et al. (2021) found that updraft mass flux in the Base simulation agreed closely with theoretical estimates. To further validate the model, we compared the distribution of simulated S-band radar reflectivity factor to that observed in convection during DYNAMO. While direct comparison of a more fundamental quantity such as vertical velocity would be preferred to verify that modeled clouds resemble observed ones, they are extremely difficult and expensive to observe in deep convection. However, three-dimensional radar-observed quantities are readily available. A model would likely not be able to reproduce anything close to observed reflectivity distributions if it produced convection that was fundamentally different from that occurring in nature.

For the comparison, radar reflectivity factor observed by the S-PolKa radar located at Addu City between 1 October 2011 and 15 January 2012 was used. The radar was located only a few kilometers from the location where rawinsondes were launched. All available radar data were composited to generate the two-dimensional probability distribution seen as the grayscale shading in the background of Fig. 4. Only echoes identified as “convective,” “mixed,” or “isolated convective core” by the Powell et al. (2016) rain-type classification were included. The modal distribution of reflectivity was between 30 and 40 dBZ from near-surface to about 4 km, where >90% of reflectivity was between 10 and 50 dBZ. Then modal reflectivity decreased with height to about 10–20 dBZ above 10 km, where about 90% of reflectivity was between 5 and 25 dBZ. Simulated modal reflectivity (blue and purple contours) decreased approximately logarithmically (note decibel scale) with height. Between 2 and 3 km height, the most frequent reflectivity observed was between 30 and 35 dBZ, matching the observed distribution.

Fig. 4.
Fig. 4.

Two-dimensional probability distributions of observed (shaded and grayscale) and modeled (contoured and color) S-band reflectivity in convective echoes as a function of reflectivity (binned by 5 dBZ) and altitude (binned by 1 km). Probability distributions are normalized at each vertical level such that the sum of the distribution at each level is 1. The contour interval for the modeled probability is 0.05, and the outer contours denote 0.05.

Citation: Journal of the Atmospheric Sciences 79, 7; 10.1175/JAS-D-21-0155.1

Simulated reflectivity at the model level closest to 2 km was ingested by the Powell et al. (2016) algorithm using the same parameters used to determine rain type from S-PolKa data, and the probability distribution denoted by color contours in Fig. 4 also consists only of “convective,” “mixed,” and “isolated convective core” echoes. Generally, simulated reflectivity distributions were skewed higher than observed distributions. For example, few simulated echoes weaker than 25 dBZ were produced below 3 km, and more frequent 50 dBZ echoes appeared in the model than in observations. Overall, however, there was good agreement between observed and modeled reflectivity below the 0°C level. Above this level, the modeled distribution of reflectivity was skewed much higher than observed echoes. For example, the 90th percentile of reflectivity at 10 km was 40 dBZ in the model but only 25 dBZ in observations. The discrepancy at upper levels may have been due to improper representation of frozen hydrometeor size distributions by the microphysics parameterization, small errors in estimating the temperature-dependent dielectric constant of ice, or the assumption by the reflectivity module in the microphysical parameterization that all hydrometeors are spherical. Such issues are beyond the scope of this article; however, we note that ice cloud is particularly sensitive to the choice of microphysical parameterization. The focus of this article is on the fundamental dynamic processes responsible for convective growth, which should not depend strongly on things like hydrometeor density and size distributions. In general, the simulated and observed reflectivity probability distributions agreed well enough to instill confidence that the clouds generated by the model were realistic even if inaccuracies remain with details in microphysics. Furthermore, although not shown here, the modal distribution for all simulated echoes (including “stratiform” and “isolated convective fringe” in Powell et al. 2016) was between 0 and 10 dBZ, agreeing very well with a composite of all echoes observed by the Ka-band ARM zenith radar (KAZR) located at the rawinsonde launch site. Additionally, the cloud base of simulated convection was between 550 and 600 m, which is realistic compared to cumulus cloud bases observed by KAZR during DYNAMO.

3. Relationship of environmental and updraft properties to rain rate

Temperature and humidity clearly impact convection because they impact updraft vertical velocity through their direct impacts on density and vertical acceleration. However, their exact relationship to rain rate is clearly neither simple nor linear. Deep convection can be suppressed when moisture is ample. Large variability in rain rate produced by a population of clouds can occur for any sufficiently unstable and moist atmosphere. Part of the variability is likely dependent upon properties of the clouds—which themselves may have some nonlinear dependence on thermodynamic properties—and properties of the boundary layer in which the clouds are rooted. For example, updraft mass flux is dependent upon cloud radius via the effects of the latter on entrainment rate. These properties may be controlled by some combination of environmental thermodynamics, boundary layer convergence, and vertical shear (e.g., Hernandez-Deckers and Sherwood 2018; Peters et al. 2021), all of which impact updraft vertical acceleration and velocity.

Figure 5 contains scatterplots of domain-mean rain rate as a function of several modeled quantities. Each dot represents a different output time (every 10 min) and all 15 simulations are combined together into each panel. Each model simulation, containing one of five different temperature profiles but nearly identical relative humidity profiles, represents a possible realistic atmospheric state that is observed over tropical oceans. Domain-mean column relative humidity (Fig. 5a) started at the same value for all simulations. It remained between 86% and 94% throughout each simulation. After deep convection and associated stratiform precipitation developed in the model, the range of rain rates that occurred (4–12+ mm day−1) is reflective of the spread in potential rain rates seen in observations (section 1), although the model did not reproduce the large areas of deep convection and higher rain rates that have been observed in similar environments. Likewise, BL evolved from the initial model state prior to rain occurring (Fig. 5b). The value of BL in the model was computed using the mean profile of T and q through the domain. The 64 km × 64 km domain is comparable to the size of a coarse reanalysis or other global model grid column that might be used to compute BL. After deep convection became present, BL decreased (as will be seen in Fig. 8) as rain rate generally decreased; however, there was much more variability in rain rate than in BL. Again, the range of rain rates produced by the model at values of BL near 0 was wide, ranging from about 4 to 12 mm day−1. In observations, the range of rain rates appears to be dependent on characteristics of convection. The highest rain rates occur when more deep convection is prevalent, but relatively low rain rates can persist in the same environment as deep convection expands laterally into stratiform precipitation that contains weaker vertical motions (Powell 2019).

Fig. 5.
Fig. 5.

Domain-mean rain rate as a function of domain-mean (a) column-integrated relative humidity, (b) BL, (c) updraft w within 450 m of cloud base, (d) updraft buoyancy relative to initial sounding between cloud base and the 0°C level, and (e) updraft Dw/Dt between cloud base and the 0°C level. In (e), the orange dashed line outlines the linear relationship between Dw/Dt and rain rate at high values of Dw/Dt, and an approximate critical Dw/Dt value, following the definition of Ahmed and Neelin (2018) is located where the line intersects the abscissa. Each dot represents a pair of values outputted every 10 min, and results from all 15 experiments are combined. Units (if any) for each abscissa are listed in the respective panel label.

Citation: Journal of the Atmospheric Sciences 79, 7; 10.1175/JAS-D-21-0155.1

More fundamentally, rain rate was dependent upon properties of the cloud updraft itself. It is feasible (at least without disproving it) that updrafts experience a uniform vertical acceleration in the free troposphere throughout time and space, and that the development of deep convection depends on the initial vertical velocity of updrafts in lower parts of clouds before they experience deceleration. In other words, an initially strong updraft may be able to overcome deceleration and still penetrate the 0°C level. Figure 5c illustrates rain rate as a function of domain-mean w averaged in a layer near cloud base, defined here as the layer between 550 and 1000 m and for updrafts with w > 0.5 m s−1. The value of w ranged from 1.1 to 1.5 m s−1 after rain began in the simulations. The relationship between w and rain rate was linear for w up to 1.3 m s−1. However, for larger w, no obvious relationship with rain rate was seen, with rain rate ranging from 3 to 12 mm day−1 for w between 1.3 and 1.5 m s−1. (When combined with Fig. 11, which will be discussed in section 4, this implies that some linear relationship between updraft w beneath 1 km and rain rate did exist prior to the development of convection penetrating to above 12 km.) However, once the deepest convection developed, domain-mean w in this layer mostly remained range-bound between 1.3 and 1.4 m s−1 while rain rate differed significantly. Vertical velocity was averaged over several different layers, but no coherent relationship with rain rate was found for any layer after deep convection first occurred in any simulation.

As opposed to BL, actual vertical acceleration in the updrafts, neglecting friction and sub-gridscale processes and imposing an anelastic approximation, is described by
DwDt1ρpzA+BB,
in which B is Archimedean buoyancy relative to a reference state, ρ is density, p′ is perturbation pressure from hydrostatic balance, and w is vertical velocity. Term A is the vertical pressure gradient acceleration. It corresponds to the force exerted by air that is displaced by rising air and opposes the buoyant force. The dynamic and buoyancy perturbation pressures are included in term A in the analysis herein; however, the dynamic perturbation pressure is generally small for updraft cores in weakly sheared environments (Morrison 2016) such as that in the sounding used to initialize the simulations herein. The vertical advection term in Dw/Dt was the largest of the advective terms in Eq. (4). It was only about 10% of the sum of the right-hand-side terms. However, Jeevanjee and Romps (2015) concluded that inertial forcing (of which −w × Δw is a part) dominated the in-cloud vertical acceleration once precipitation began over maritime oceans.
In CM1, B is defined relative to the initial sounding such that
B=g[θ*θ0+(RυRd1)qυ*qlf],
in which a superscript asterisk denotes deviation from the initial sounding, which is represented by a subscript 0. qυ is the vapor mixing ratio, and qlf is the total liquid plus frozen water mixing ratio. Because B is computed relative to an arbitrary reference state, it does not represent the true buoyant acceleration actually felt by individual updrafts (Doswell and Markowski 2004). Instead the sum of A and B in Eq. (4) is nearly the same as “effective buoyancy,” which is defined by Davies-Jones (2003) as being proportional to the horizontal Laplacian of the local density field. Similar metrics of effective buoyancy have been used by Nugent and Smith (2014) and Jeevanjee and Romps (2016). Together, these two terms account for Archimedean buoyancy as well as the opposing buoyancy pressure gradient acceleration (BPGA). Only the sum of the terms when outputted by CM1 is meaningful because of their dependence on the initial state, which does not represent the unique local environments experienced by each cloud.

In Figs. 5d and 5e, both relative buoyancy and Dw/Dt for in-cloud updrafts with w > 0.5 m s−1 between cloud base at 550 m and 5.25 km altitude are compared to rain rate. This layer is chosen because we presume that the acceleration through the entire layer between cloud base and the 0°C level is important for determining whether an updraft can penetrate into the upper troposphere. A net negative acceleration in this layer would likely prevent most updrafts from growing higher. The cluster of black dots near −0.02 m s−2 in Fig. 5d denotes the similar relative buoyancies experienced by updrafts in all simulations between cloud base and the 0°C level. In other words, modeled rain rate was insensitive to term B in Eq. (4) below the 0°C level. The relationship between domain-mean updraft Dw/Dt beneath the 0°C level (Fig. 5e) was strongest of the five variables displayed or any that were investigated. In Fig. 5e virtually no rain occurred before Dw/Dt reached 0 in any simulation. As Dw/Dt increased above 0, domain-mean rain rate exponentially increased to about 2 mm day−1 for Dw/Dt up to 0.002 m s−2. At larger values of Dw/Dt, rain rate appeared to increase approximately linearly as Dw/Dt increased. Following the method of Ahmed and Neelin (2018), and fitting a line to the scatterplot for large values of Dw/Dt instead of BL, the critical value of domain-mean cloud base to 0°C level Dw/Dt was approximately 0.002 m s−2. For accelerations less than this critical value, updrafts were apparently frequently decelerated to 0 before reaching the 0°C level. Although not shown in Fig. 5, average Dw/Dt in an updraft was negative above 3 km (section 4). Deceleration present in a cumulus cloud between 3 km and the 0°C level must be small enough then so that the cloud can reach the 0°C level given its updraft velocity (which is largely controlled by Dw/Dt below 3 km) when it first experiences the layer of persistent deceleration. This is essentially the same as saying that convective inhibition local to an updraft and dilution of updrafts must be sufficiently reduced for deep convection to develop and high area-averaged rain rates to occur.

The critical value for Dw/Dt is dependent upon both the layer within which updraft Dw/Dt is averaged and the definition of updraft. Figure 6 depicts rain rate dependence on Dw/Dt for different minimum values of in-cloud vertical velocity used to define an updraft and for different layers in which Dw/Dt was calculated. The black dots are the same as in Fig. 5e; however, if in-cloud updrafts are defined as w > 0.1 m s−1 instead of w > 0.5 m s−1, the critical value of Dw/Dt is smaller—less than 0.001 m s−2 in these simulations (red dots). If updrafts are defined as w > 0.1 m s−1 and Dw/Dt is only averaged between cloud base and 1 km altitude (blue dots), then the relationship between rain rate and Dw/Dt appears more like a ramp function than an exponential curve, with a critical Dw/Dt a little greater than 0.004 m s−2. The critical values of Dw/Dt appear to have little dependence on A because all simulations are combined into a single illustration; however, it is plausible—if not likely—that it would have dependence on the relative humidity profile and vertical wind shear, both of which are initialized identically for all 15 simulations. For example, one can imagine that if relative humidity were reduced in the middle troposphere, a larger Dw/Dt in the lower part of the cloud would be required to sustain updrafts while they undergo dilution in order to maintain deep convection. Perhaps if the layer within 1–2 km beneath the 0°C level were dryer, such a strong dependence of rain rate on updraft Dw/Dt in the lowest 500 m of a cloud would not be found because the ultimate fate of a cloud would also be dependent upon dilution occurring at higher altitudes. Such hypotheses could be easily tested but require substantial resources to execute several numerical integrations that are computationally expensive as of when this article was written. The primary conclusion is that Dw/Dt appears to be more fundamental to domain-mean rain rate than CRH, BL, or buoyancy computed relative to a base state without considering the compensating vertical pressure gradient acceleration. Given a relative humidity and wind shear profile such as those prescribed in the initial conditions for each simulation, we can estimate layer-averaged Dw/Dt above which rain rate increases rapidly as Dw/Dt increases.

Fig. 6.
Fig. 6.

Domain-mean rain rate as a function updraft Dw/Dt for updrafts between 550 m (cloud base) and the 0°C level with w > 0.5 m s−1 (black), updrafts between 550 m and 1 km with w > 0.1 m s−1 (blue), and updrafts between 550 m and the 0°C level with w > 0.1 m s−1 (red). Yellow dashed lines denote the linear relationship between Dw/Dt and rain rate at high values of Dw/Dt like the orange line in Fig. 5.

Citation: Journal of the Atmospheric Sciences 79, 7; 10.1175/JAS-D-21-0155.1

4. Convective transitions

Each experiment produced shallow, nonprecipitating convection followed by deep cumulonimbi that grew laterally into small cloud clusters up to 10 km across below 500 hPa. Because boundary conditions were doubly periodic, energy fluxed from the ocean to the atmosphere accumulated in the domain and each simulation ended up in a similar state containing a mixture of convective and stratiform precipitation by 20 h regardless of the initial temperature profile used. Figure 7 depicts the model domain-mean rain rate as a function of time after initialization. Each of the five line plots in Fig. 7a is a composite of the three simulations run with each value of A.

Fig. 7.
Fig. 7.

(a) Time series of model domain-mean rain rate composited over the three simulations run for each value of A listed in the legend. A = 0 for the Base simulation. (b) Time series of domain-mean rate in each individual simulation, with three lines of each color-and-dashed combination in the legend plotted.

Citation: Journal of the Atmospheric Sciences 79, 7; 10.1175/JAS-D-21-0155.1

The solid black line denotes domain-mean rain rate in the Base simulations. Rain in them first occurred after 6 h then increased at an increasing rate through 13 h. The maximum domain-mean rain rate for the Base simulation occurred at 16 h before decreasing but remaining above ∼4 mm day−1 through 24 h. Mean rain rate in the −0.01 (dashed blue) and +0.01 (dashed red) simulations was similar to that in the Base simulations. Rain in the −0.03 simulation composite began slightly sooner than in the other simulations, and on average, the most rapid increase in rain rate occurred 1–2 h earlier than in the other simulations. A greater peak mean rain rate (>8 mm day−1) occurred in the −0.03 simulation composite as well. However, of the three simulations run with A = −0.03, two exhibited rain rate near the mean for the Base simulations, and the earlier increase in rain rate occurred in just one of the three simulations with A = −0.003 (Fig. 7b). Likewise, of the simulations with A = +0.03, two of three demonstrated rain rate increases of similar magnitudes at similar times as the Base simulations. Domain-mean rain rate for all A was bounded between about 4 and 6.5 mm day−1 after 16 h and generally decreased as all the simulations essentially converged to a similar state. In general, increasing lower tropospheric lapse rate (A = −0.03) appeared to increase the chance that rain could start sooner in the model, but overall, there was little sensitivity of rain rate to A. This is possibly consistent with Powell (2019), who found only a weak correlation between low-tropospheric lapse and rain rate after approximately controlling for CRH.

The increases in rain rate for each A corresponded with different values of domain-mean BL as shown in Fig. 8. For the −0.03 simulation, BL peaked near 0.04 m s−2 near the time (∼10 h) rain rate reached 2 mm day−1. Domain-mean peak BL occurred later as A increased, occurring around 12 h after model start on average for the +0.03 runs. Presumably, the atmosphere was then stabilized as deep convection occurred in all simulations and BL decreased. As with rain rate, all simulations converged to the same value of BL by 16 h, which was near 0 after 24 h. The maximum domain-mean BL reached decreased as the stability of the initial sounding and A increased. Therefore, Fig. 8 is another clear indication that rain rate in the model is not necessarily closely related to BL. The increase in rain rate in the different simulations occurred at various values of BL, all of which can occur in various realistic potential atmospheric states represented by the initial soundings. This complicates defining a critical value of BL. Certainly at some value lower than that experienced by any simulation herein (e.g., <−0.10 m s−2) only shallow convection can develop, and little to no rainfall would occur. However, at BL values near or slightly above 0, such as those experienced in our simulations, a wide range of possible rain rates was experienced across the simulations, even after the model produced deep convection. Clearly then, no single critical value of BL signaled transition into deep convection. This is not completely surprising because it only considers environmental temperature and moisture that are representative of conditions outside of clouds. BL does not consider other terms that impact updraft vertical acceleration nor attributes of updrafts like radius (Hernandez-Deckers and Sherwood 2018) that are thought to impact vertical acceleration. Therefore, BL may be better thought of as a proxy for vertical acceleration for use only when limited information is available and used with the caveat that it may not predict rainfall reliably in moist conditions. Once an atmosphere is sufficiently moist and unstable and deep convection with connected stratiform precipitation regions develop, area-averaged rain rate is apparently more dependent upon mesoscale and updraft-scale dynamics than the large-scale thermodynamic profile.

Fig. 8.
Fig. 8.

Time series of domain-mean BL for each experiment.

Citation: Journal of the Atmospheric Sciences 79, 7; 10.1175/JAS-D-21-0155.1

The black line in Fig. 9 denotes the percentage of the domain at 5.5 km altitude—just above the 0°C level—to contain an in-cloud updraft with w > 0.5 m s−1. While the transition from shallow to deep convection occurred over several hours, t = 0 on the abscissa of Fig. 9 is fixed to the time when the first convection extending to near the tropopause occurs in each simulation. This occurred at about the same time that 0.2% of the model domain contained an in-cloud updraft at 5.5 km altitude, and marked the time when a convective cell in the domain first extended upward to around 12 km. One of the simulations with A = −0.03 had this occur only 7.5 h after model start, so the simulation was discarded from analysis for this and subsequent discussion to avoid inclusion of model output from within 3 h of initialization. Including this simulation did not fundamentally alter the results, however. The blue line in Fig. 9 denotes the domain-mean rain rate relative to t = 0, and the shading represents a 95% confidence interval on the mean rain rate that occurred in the 14 remaining simulations using bootstrapping. Rain rate increased in tandem with the fraction of the model domain experiencing updrafts at 5.5 km, with both approximately tripling in the 3 h after in-cloud updraft areal coverage at 5.5 km reached 0.2%. This confirms our initial assumption that the largest rain rates only occurred when convection broke out above the 0°C level. Notably, domain-mean updraft Dw/Dt averaged between 550 m and 5.25 km at t = 0 (red line) was about 0.002 m s−2, which was the critical value shown in Fig. 6 for this layer with cloudy updrafts defined as w > 0.5 m s−2. Although the rate of increase in Dw/Dt decreased after t = 0, rain rate increased at the fastest rate seen throughout the simulations after Dw/Dt exceeded 0.002 m s−2.

Fig. 9.
Fig. 9.

Updraft areal coverage (black), domain-mean rain rate (blue) with 95% confidence interval (shaded), updraft Dw/Dt averaged between 550 m and 5.25 km (red), and the difference between updraft (Tup) and environmental (Tenv) temperature averaged in the same layer (magenta) relative to the first time at which 0.2% of the model domain at 5.5 km altitude contained in-cloud updrafts with w > 0.1 m s−1. The dashed gray line indicates a lag time of zero.

Citation: Journal of the Atmospheric Sciences 79, 7; 10.1175/JAS-D-21-0155.1

Figures 10 and 11 provide more detail about the vertical structure of domain-mean updraft Dw/Dt and w in the simulations relative to t = 0. Two transitions in the height of the deepest convection are seen. The first occurred near t = −3 h, before which only shallow convection containing strong downward acceleration near cloud top was present. It was around this time that Dw/Dt in the 550 m to 5.25 km layer had increased to zero (Fig. 9). The second occurred, by design, at t = 0, when the first convective updrafts reached at least 7 km altitude and extended up to 12 km altitude. After the first deep convective element appeared at t = 0, deep convection and stratiform precipitation regions were present in all simulations. At t = −6 h, positive Dw/Dt was found only up to 1.5 km altitude in updrafts; however, over the subsequent 6 h, the layer containing positive Dw/Dt deepened so that its top reached about 2.75 km. By t = 0, Dw/Dt in the 500 m above cloud base had increased to over 0.006 m s−2. In the hours leading up to t = 0, deceleration above 3 km also gradually decreased in magnitude. Combined with Fig. 9, the largest increase in 550 m–5.25 km Dw/Dt apparently occurred between t = −4 and −2 h.

Fig. 10.
Fig. 10.

Domain-mean updraft Dw/Dt as a function of altitude and time when areal coverage of updrafts at 5.5 km exceeded 0.2% of the domain. Red and blue colors respectively denote positive and negative vertical accelerations, respectively.

Citation: Journal of the Atmospheric Sciences 79, 7; 10.1175/JAS-D-21-0155.1

Fig. 11.
Fig. 11.

As in Fig. 10, but for updraft vertical velocity w.

Citation: Journal of the Atmospheric Sciences 79, 7; 10.1175/JAS-D-21-0155.1

Updraft vertical velocities in convection were maximized, on average, between 2 and 3 km altitude (Fig. 11). However, this value is a composite of the numerous shallow convective elements and relatively few deep convective elements. Visual inspection of the deepest convection in the model output indicated that the largest vertical velocities were located between 6 and 9 km altitude and often exceeded 10 m s−1. Generally, shallow convection that developed prior to t = 0 possessed maximum vertical velocities of >2 m s−1 between 2 and 3 km. The vertical velocities gradually increased in magnitude throughout the simulations. Maximum mean updraft velocity below 3 km increased to >2.5 m s−1 by t = +4 h as updrafts experienced stronger upward acceleration during this time (Fig. 10). It is apparent that as Dw/Dt below 2 km increased, w increased more quickly with height. For example, the minimum altitude of the fourth darkest shaded color in Fig. 11 (about 2 m s−1) was 2 km at t = −4 h. By t = 0, this contour had dropped to near 1.5 km. while mean updraft w at 2 km was around 2.25 m s−1. The negative acceleration persistently present above 3 km through about t = +2 h caused domain-mean updraft w to decrease above 3 km. This mean decrease in w was dominated by updrafts that were not currently part of deep convection. In the strongest convection, although not seen in Fig. 11, updraft velocity continued to increase. This means that Dw/Dt in the strongest updrafts likely is significantly different from the domain mean. Nonetheless, Figs. 5 and 6 suggest that the domain-mean Dw/Dt is still strongly related to domain-mean rain rate. Prior to t = 0, essentially no updrafts with w > 0.5 m s−1 were present above 6 km; however, this abruptly changed at t = 0. The vertical velocity w did not change much above 3 km after t = +2 because mean Dw/Dt in updrafts was near 0 or slightly positive. However, the areal coverage of updrafts at this time reached about 1% and stopped increasing (Fig. 9). After deep convection was present, deceleration was generally ubiquitous above 8 km, with deceleration exceeding 0.01 m s−2 near cloud top. An apparent discontinuity in w is seen after t = 0 around 5 km altitude. This is an artifact of the analysis. Because the analysis includes all updrafts exceeding 0.5 m s−1 that are inside a cloud and spatially contiguous with a stronger core updraft with magnitude exceeding 1 m s−1 (section 2), weak updrafts within laterally expanding stratiform regions are included. This does not as strongly influence the displayed mean w below the 0°C level because typical vertical motions in stratiform areas there are negative.

Wu et al. (2009) concluded that shallow to deep convective transitions occurred as a result of increased relative buoyancy, which was primarily driven by increases in the first term in Eq. (5). In other words, to first order, increases in buoyancy were thought to occur because of increases in updraft temperature relative to environmental temperature. In Fig. 9, we saw that the mean difference between updraft and environmental temperature between 550 m and 5.25 km increases at the same time as mean Dw/Dt in the same layer increases. However, this is misleading. Figure 12a shows how the vertical structure of this temperature difference evolved relative to t = 0. Although not shown in this figure, both domain-mean updraft (Tup) and clear-air environmental (Tenv) temperatures generally first decreased throughout all simulations between cloud base and 5.25 km then increased after deep convection started. Red and blue shading indicate times and altitudes when and where domain-mean updraft temperature (Tup) respectively exceeded or was less than the domain-mean clear-air environmental temperature (Tenv). Several results are noteworthy. The most basic is that updraft temperature is generally up to 0.3 K greater than environmental temperature. At t = −6 h, when only shallow convection was present, the difference was no more than 0.05 K in clouds that seldom exceeded 2 km in altitude. By t = +2 to +4 h, the mean difference had increased to nearly 0.3 K at 2–3 km altitude. ∂(TupTenv)/∂z between cloud base and about 1 km also increased between t = −6 and 0 h, which corresponded with the increase in Dw/Dt in this layer (Fig. 10). While not shown, the maximum magnitudes of both terms A and B in Eq. (4) occurred at the same time in this layer. However, the maximum Dw/Dt did not occur where TupTenv was largest, confirming that relative Archimedean buoyancy was not a good proxy for the effective buoyancy in the simulated updrafts.

Fig. 12.
Fig. 12.

As in Fig. 10, but for the difference between in-updraft and clear-air environmental (a) temperature and (b) specific humidity.

Citation: Journal of the Atmospheric Sciences 79, 7; 10.1175/JAS-D-21-0155.1

Figure 12b illustrates the domain-mean difference between updraft (qup) and environmental (qenv) specific humidity because it also contributes to the buoyancy [Eq. (5)]. Convection tends to moisten the environment prior to formation of deep convection. For example, at 3 km altitude, the difference between updraft and environmental humidity decreased from near 1.4 g kg−1 at t = −4 h to less than 1.0 g kg−1 by t = 0. Therefore, the specific humidity difference between updrafts and the environment could not have contributed to the decrease in deceleration in the 3–4 km layer seen in Fig. 10. Between cloud base and 2 km, where Dw/Dt was positive prior to t = 0, qupqenv changed little. However, after t = 0, the environment began to dry out in the presence of deep convection, particularly below the 0°C level. In the 3–4 km layer, qupqenv increased from near 1.0 g kg−1 to up to 1.2 g kg−1 by t = +6 h. This corresponds to the time and height when and where Dw/Dt was 0–0.002 m s−2 in Fig. 10. Therefore, it is plausible that decreased environmental specific humidity relative to updrafts contributed to the positive Dw/Dt below 5 km after t = +2 h by contributing to B in Eq. (4) without having a fully offsetting change in the BPGA magnitude. This is offered only as a speculative hypothesis as computing a buoyancy budget is not the focus of this article.

The most important conclusion from the analysis of Fig. 12 is that neither the Archimedean buoyancy nor the BPGA alone controlled Dw/Dt. It was the offset between the two terms in Eq. (4) that impacted Dw/Dt, and subsequently, the profile of w. This fact is consistent with the lack of a relationship between B in updrafts beneath the 0°C level and domain-mean rain rate depicted in Fig. 5d.

5. Conclusions

Large-eddy simulations (LESs) of tropical marine convection were conducted using five different initial thermodynamic profiles that possessed nearly identical relative humidity but contained different temperature profiles that could be observed in the real atmosphere if a small perturbation of vertical velocity was integrated over a 24-h period and caused adiabatic warming or cooling. To generate a small ensemble, three simulations for each profile were initialized with different random noise in the subcloud layer. Column-integrated relative humidity (CRH) for each simulation was about 80%, which observations (e.g., Rushley et al. 2018; Powell 2019) indicate can support widespread deep convection but is not sufficient to do so alone. Using this setup, we were able to investigate the role of vertical temperature structure on convection in moist environments while approximately controlling for the known effect of humidity on rainfall. The various profiles corresponded with different lapse rates in the lower troposphere. Not surprisingly, onset of rainfall occurred first, on average, in the most unstable soundings; however, two of the three simulations with the most unstable initial sounding evolved similarly to the less unstable control simulations (Fig. 7). The transition of the cloud population from shallow, nonprecipitating cumuli to predominantly deep convection with attached stratiform precipitation occurred over about 3 h in all simulations. The domain-mean rain rate and BL [Eq. (1)] in these simulations ultimately reached the same values as the most unstable simulation given periodic boundary conditions and identical fixed sea surface temperatures in each (Figs. 7, 8). This implies that only a weak relationship exists between the vertical profile of temperature and area-averaged rain rate, at least on time scales of hours. Other factors such as ocean-atmosphere energy fluxes also play an important role in rainfall variability as seen previously (Raymond et al. 2003), and this probably explains why the relationship between low-tropospheric lapse rate where CRH > 80% and rain rate found by Powell (2019) was weak. The primary conclusion of this study is that area-averaged rain rate in the model was closely linked to values of a quantity that is defined as the sum of buoyancy defined relative to some base state (Archimedean relative buoyancy) and the buoyancy pressure gradient acceleration [Eq. (4)]. This is similar to what prior literature calls “effective buoyancy” (section 1).

The sensitivity of rain rate to thermodynamic quantities such as BL or CRH was also investigated. Once the numerical simulations developed rain—about 6 h after initialization—neither variable, when averaged across the model domain, closely corresponded with rain rate. Considering that each simulated atmosphere was moist to start, this is not surprising given the large spread in rain rates observed in observations at high values of BL or CRH. Instead, the total approximated domain-mean Lagrangian vertical acceleration (Dw/Dt) within cloudy updrafts [Eq. (4)] beneath the 0°C level was closely related to domain-mean rain rate (Fig. 5). The total vertical acceleration consists of buoyancy relative to some arbitrary state—in this case, the initial thermodynamic profile—and the opposing buoyancy pressure gradient acceleration (BPGA). Domain-mean rain rate remained near 0 up to some critical value of domain-mean Dw/Dt inside updrafts. As Dw/Dt in updrafts increased above the critical value, domain-mean rain rate in the model quickly increased as more deep convective elements developed. However, the critical value of Dw/Dt was dependent at least on 1) how an updraft was defined and 2) the layer in which the acceleration was averaged (Fig. 6). For example, defining in-cloud updrafts as areas where w > 0.1 m s−1 and averaging Dw/Dt between cloud base at 550 m and the 0°C level at 5.25 km, a critical value slightly less than 0.001 m s−2 was estimated. When using the same definition for updraft but averaging only between 550 m and 1 km, the estimated critical Dw/Dt exceeded 0.004 m s−2. Numerous other definitions for updraft and layer averages would indubitably yield other critical values for Dw/Dt. The critical value showed little sensitivity to the temperature profile in the initial sounding, however.

Therefore, the values of Dw/Dt required for both any rain and widespread deep convection should not be assumed to be universally applicable. In other words, anything between 0.001 and 0.004 m s−2, for example, is not a “magic number” for deep convection. For this particular model setup, we can simply conclude that 1) critical values of CRH or BL have already been exceeded when the environment is already moist, 2) critical values of Dw/Dt do exist even in moist environments, and 3) the critical values of updraft Dw/Dt depend on how updrafts are defined and in what layer we compute the acceleration. However, in distinctly different environments where w at low levels of cumuliform clouds is different, or the humidity profile differs, or there is more or less vertical wind shear, or the subcloud layer depth is different, for example, the Dw/Dt required for deep convection to occur may also differ. As an example, in an early version of the manuscript for this article, the author ran simulations with a 304.5K sea surface temperature but with the same thermodynamic and wind profiles. The temperature was higher than observed during DYNAMO. The model responded by warming the boundary layer but because the wind was weak, the boundary layer was not moistened much. As a result, cloud base rose to near 1 km. These results are not shown here because they do not depict the observed tropical, marine cloud population over the Indo-Pacific region that often had a cloud base between 500 and 600 m like what is simulated in this article. However, the strong dependence of rain rate on Dw/Dt was still present even in those unrealistic simulations, and those simulations exhibited their own critical Dw/Dt values. The author cautions that the dependence of area-averaged rain rate on Dw/Dt should not be assumed to extend outside of the Indo-Pacific warm pool, where the 0°C level is near 5 km above the ocean. Finally, while our main conclusion is interesting, the practicality of the result currently remains limited because we have no cost-effective way to observe even vertical velocities—let alone vertical accelerations—simultaneously for an entire population of shallow and deep cumuliform clouds. Our best hope currently is to investigate what observable variables appear to most closely relate to Dw/Dt in updrafts and how those relationships might change as the explanatory variables themselves change.

Composited over 14 of the 15 simulations (excluding the leftmost blue line in Fig. 7), the first convection reaching up to 12 km altitude occurred at about the time that 0.2% of the domain at 5.5 km altitude contains an updraft when defined as w > 0.5 m s−1 (Figs. 912). This corresponds with the time that Dw/Dt averaged between 550 m and 5.25 km was ∼0.002 m s−2. A layer of positive Dw/Dt deepened in the hours preceding the first deep convection, extending upward to 2.75 km by the time the first deep convection occurred. The acceleration experienced in the lowest 2 km of updrafts resulted in a maximum vertical velocity between 2 and 3 km altitude (Fig. 11). Until a couple hours after deep convective onset, deceleration of updrafts was common in the 2 km below the 0°C level, and once deep convective updrafts formed, mean deceleration in them occurred above 8 km (Fig. 10). Collectively, the results raise an important question that is left unanswered: What controls the evolution of Dw/Dt below the 0°C level? The numerical simulations show that domain-mean Dw/Dt in updrafts increases in the hours prior to deep convection first developing, but it is unclear what causes Dw/Dt to increase. Figure 9 suggests that the difference between updraft and environmental temperatures is closely related to Dw/Dt. We might expect this to make sense because updraft density decreases as its temperature increases. So if the updraft temperature becomes warmer relative to the environment, then the buoyant acceleration would increase. However, visual comparison of Figs. 10 and 12 indicate that the maximum temperature difference is located at 2–3 km altitude while maximum Dw/Dt was located in the few hundred meters directly above the cloud base, which was near 550 m. Differences in updraft versus environmental specific humidity cannot explain the acceleration prior to deep convection either. In other words, the buoyancy term B in the vertical momentum equation [Eq. (4)] did not correlate well with Dw/Dt in these simulations. The practical ramification of this is that we cannot simply use updraft temperature and environmental temperature and humidity, which are quantities we can observe, to estimate the buoyant acceleration in an updraft. We must also know something about the vertical pressure gradient acceleration as well. Based on the formulation of BPGA in Eq. (4), this implies that we must obtain information about not just the density, but also the vertical gradient of density, inside updrafts.

Of course, the caveats to this study are numerous. Because the simulations are so computationally expensive and require massive amounts of disk storage for output, sensitivity of our results to a variety of factors was not tested. For example, the results might be sensitive to domain size, choice of parameterizations, model time step, boundary conditions, or other factors. Furthermore, the results are based largely on model behavior just 12 h after initialization. The author believes this to be acceptable, however, because the model appeared to first develop a field of uniformly shallow convection that persisted for a few hours in a pseudo-steady state before gradually deepening convection appeared. As such, the numerical results might be interpreted as representing the transition of a quiescent, noncloudy boundary layer into a cumuliform field that later grows vertically as updraft velocities increase, a sequence of events that happens every day somewhere over tropical oceans during the course of several hours. An alternative way of conducting these experiments might be to initialize the model with a sounding that first results in a steady state containing shallow, nonprecipitating convection, then alter the model environment and assess how updrafts evolve from the shallow convective steady state. This would be more computationally expensive, however.

Despite the many open questions regarding the ability to generalize the conclusions, the results do point to a fundamental relationship between an areal mean of individual cloud properties (e.g., updraft acceleration) and rain rate that is not apparent between rain rate and atmospheric quantities that are observed readily and convenient to use, such as vertical profiles of temperature and humidity. This may offer a pathway for improved representation of cloud populations in cumulus parameterizations if the processes that control how updraft acceleration evolves in convective cores over time can be determined using gridscale quantities available to a model. Thus, the author argues that the goal of improving numerical prediction of precipitation must compel intensive, simultaneous observations aimed at processes that occur at spatial scales of tens to hundreds of meters in the boundary layer, shallow convection, and the ocean or land surface beneath from which energy is fluxed to the atmosphere in spatially heterogenous patterns.

While precipitation radars and satellite-based instruments that have been central to extensive observational data collection campaigns of the past several decades can provide larger-scale context for the environment in which updrafts form, more targeted observations are required to elucidate cloud-scale processes and validate conclusions reached by theoreticians and numerical modelers. For example, high-frequency Doppler radars and lidars operating in scanning patterns (Kollias et al. 2014) are capable of obtaining three-dimensional volumes of nonprecipitating cloud properties as well as the kinematic and thermodynamic structure of the clear subcloud layer and clear air surrounding clouds (Weckwerth et al. 2016). Eventually, phased array antennas at high frequency may permit collecting these observations with high temporal resolution. Such observations would be collected within a small area extending only a few kilometers from a high-frequency remote sensing instrument. Collocated vertically pointing Doppler instruments can help collect valuable observations of vertical velocity (e.g., Kollias and Albrecht 2010). Extensive surface-based in situ observing systems can provide details about the ocean or land surface and atmospheric properties at their interface; mobile platforms such as Saildrones (Gentemann et al. 2020) can furthermore provide information about the spatiotemporal variability of these properties. Manned and unmanned aerial platforms, where they can be used, can facilitate observing temperature, humidity, and 3D motions inside and surrounding individual cloud elements, quantities that can be analyzed on their own or used to validate quantities derived from remote sensing instruments. Small aerial platforms can provide information higher in the atmospheric boundary layer than surface based platforms can reach (van den Heever et al. 2021). The combination of existing observational capabilities concentrated within a small area for a long period of time would result in accumulating large datasets from which statistics could be derived. These statistics could help elucidate the most important processes for moist convective evolution at the spatial and temporal scales relevant to short-lived dynamic processes within convection. The resulting datasets would also serve as validation references for the very numerical simulations, like those discussed in this article, that we will continue to use as a substitute for observations that cannot be directly collected.

Acknowledgments.

This work was supported partially by both Office of Naval Research Award N0001421WX01472 and by the U.S. Department of Energy Atmospheric System Research, an Office of Science Biological and Environmental Research program, under Interagency Agreement 89243021SSC000077. The author thanks John Peters for advice on running CM1. Kathleen Schiro and two anonymous reviewers provided very helpful constructive feedback to improve the organization and content of the manuscript.

Data availability statement.

The model output consists of numerous large files that are cumbersome to transfer. However, the namelist and sounding files included in the supplementary information can be readily used to replicate the results when using the model and analysis methods outlined in the main text. High-performance computing resources were provided by the Hamming supercomputer at the Naval Postgraduate School.

REFERENCES

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    • Search Google Scholar
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  • Jeevanjee, N., and D. M. Romps, 2015: Effective buoyancy, inertial pressure, and the mechanical generation of boundary layer mass flux by cold pools. J. Atmos. Sci., 72, 31993213, https://doi.org/10.1175/JAS-D-14-0349.1.

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    • Search Google Scholar
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  • Kollias, P., N. Bharadwaj, K. Widener, I. Jo, and K. Johnson, 2014: Scanning ARM cloud radars. Part I: Operational sampling strategies. J. Atmos. Oceanic Technol., 31, 569582, https://doi.org/10.1175/JTECH-D-13-00044.1.

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    • Search Google Scholar
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    • Search Google Scholar
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Supplementary Materials

Save
  • Adames, Á. F., S. W. Powell, F. Ahmed, V. C. Mayta, and J. D. Neelin, 2021: Tropical precipitation evolution in a buoyancy-budget framework. J. Atmos. Sci., 78, 509528, https://doi.org/10.1175/JAS-D-20-0074.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ahmed, F., and J. D. Neelin, 2018: Reverse engineering the tropical precipitation–buoyancy relationship. J. Atmos. Sci., 75, 15871608, https://doi.org/10.1175/JAS-D-17-0333.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ahmed, F., Á. F. Adames, and J. D. Neelin, 2020: Deep convective adjustment of temperature and moisture. J. Atmos. Sci., 77, 21632186, https://doi.org/10.1175/JAS-D-19-0227.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bosilovich, M. G., J. Chen, F. R. Robertson, and R. F. Adler, 2008: Evaluation of global precipitation in reanalyses. J. Appl. Meteor. Climatol., 47, 22792299, https://doi.org/10.1175/2008JAMC1921.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bretherton, C. S., M. E. Peters, and L. E. Back, 2004: Relationships between water vapor path and precipitation over the tropical oceans. J. Climate, 17, 15171528, https://doi.org/10.1175/1520-0442(2004)017<1517:RBWVPA>2.0.CO;2.

    • Crossref
    • Export Citation
  • Bryan, G. H., and J. M. Fritsch, 2002: A benchmark simulation for moist nonhydrostatic numerical models. Mon. Wea. Rev., 130, 29172928, https://doi.org/10.1175/1520-0493(2002)130<2917:ABSFMN>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bryan, G. H., J. C. Wyngaard, and J. M. Fritsch, 2003: Resolution requirements for the simulation of deep moist convection. Mon. Wea. Rev., 131, 23942416, https://doi.org/10.1175/1520-0493(2003)131<2394:RRFTSO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Chen, B., and B. E. Mapes, 2018: Effects of a simple convective organization scheme in a two-plume GCM. J. Adv. Model. Earth Syst., 10, 867880, https://doi.org/10.1002/2017MS001106.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Chou, M. D., M. J. Suarez, X. Z. Liang, and M. M. H. Yan, 2001: A thermal infrared radiation parameterization for atmospheric studies. NASA Tech. Memo., 68 pp., https://ntrs.nasa.gov/api/citations/20010072848/downloads/20010072848.pdf.

    • Crossref
    • Export Citation
  • Ciesielski, P. E., and Coauthors, 2014: Quality-controlled upper-air sounding dataset for DYNAMO/CINDY/AMIE: Development and corrections. J. Atmos. Oceanic Technol., 31, 741764, https://doi.org/10.1175/JTECH-D-13-00165.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Davies-Jones, R., 2003: An expression for effective buoyancy in surroundings with horizontal density gradients. J. Atmos. Sci., 60, 29222925, https://doi.org/10.1175/1520-0469(2003)060<2922:AEFEBI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Del Genio, A. D., Y. Chen, D. Kim, and M.-S. Yao, 2012: The MJO transition from shallow to deep convection in CloudSat/CALIPSO data and GISS GCM simulations. J. Climate, 25, 37553770, https://doi.org/10.1175/JCLI-D-11-00384.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Doswell, C. A., and P. M. Markowski, 2004: Is buoyancy a relative quantity? Mon. Wea. Rev., 132, 853863, https://doi.org/10.1175/1520-0493(2004)132<0853:IBARQ>2.0.CO;2.

    • Crossref
    • Export Citation
  • Gentemann, C. L., and Coauthors, 2020: Saildrone: Adaptively sampling the marine environment. Bull. Amer. Meteor. Soc., 101, E744E762, https://doi.org/10.1175/BAMS-D-19-0015.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hernandez-Deckers, D., and S. C. Sherwood, 2018: On the role of entrainment in the fate of cumulus thermals. J. Atmos. Sci., 75, 39113924, https://doi.org/10.1175/JAS-D-18-0077.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hersbach, H. , and Coauthors, 2020: The ERA5 global reanalysis. Quart. J. Roy. Meteor. Soc., 146, 19992049, https://doi.org/10.1002/qj.3803.

    • Crossref
    • Export Citation
  • Holloway, C. E., and J. D. Neelin, 2009: Moisture vertical structure, column water vapor, and tropical deep convection. J. Atmos. Sci., 66, 16651683, https://doi.org/10.1175/2008JAS2806.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Jeevanjee, N., and D. M. Romps, 2015: Effective buoyancy, inertial pressure, and the mechanical generation of boundary layer mass flux by cold pools. J. Atmos. Sci., 72, 31993213, https://doi.org/10.1175/JAS-D-14-0349.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Jeevanjee, N., and D. M. Romps, 2016: Effective buoyancy at the surface and aloft. Quart. J. Roy. Meteor. Soc., 142, 811820, https://doi.org/10.1002/qj.2683.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Jiménez, P. A., J. Dudhia, J. F. González-Rouco, J. Navarro, J. P. Montávez, and E. García-Bustamante, 2012: A revised scheme for the WRF surface layer formulation. Mon. Wea. Rev., 140, 898918, https://doi.org/10.1175/MWR-D-11-00056.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kollias, P., and B. Albrecht, 2010: Vertical velocity statistics in fair-weather cumuli at the ARM TWP Nauru Climate Research Facility. J. Climate, 23, 65906604, https://doi.org/10.1175/2010JCLI3449.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kollias, P., N. Bharadwaj, K. Widener, I. Jo, and K. Johnson, 2014: Scanning ARM cloud radars. Part I: Operational sampling strategies. J. Atmos. Oceanic Technol., 31, 569582, https://doi.org/10.1175/JTECH-D-13-00044.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kuo, Y.-H., J. D. Neelin, and C. R. Mechoso, 2017: Tropical convective transition statistics and causality in the water vapor–precipitation relation. J. Atmos. Sci., 74, 915931, https://doi.org/10.1175/JAS-D-16-0182.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Mapes, B., and R. Neale, 2011: Parameterizing convective organization to escape the entrainment dilemma. J. Adv. Model. Earth Syst., 3, M06004, https://doi.org/10.1029/2011MS000042.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Mellado, J. P., C. S. Bretherton, B. Stevens, and M. C. Wyant, 2018: DNS and LES for simulating stratocumulus: Better together. J. Adv. Model. Earth Syst., 10, 14211438, https://doi.org/10.1029/2018MS001312.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Morrison, H., 2016: Impacts of updraft size and dimensionality on the perturbation pressure and vertical velocity in cumulus convection. Part I: Simple, generalized analytic solutions. J. Atmos. Sci., 73, 14411454, https://doi.org/10.1175/JAS-D-15-0040.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Morrison, H., G. Thompson, and V. Tatarskii, 2009: Impact of cloud microphysics on the development of trailing stratiform precipitation in a simulated squall line: Comparison of one- and two-moment schemes. Mon. Wea. Rev., 137, 9911007, https://doi.org/10.1175/2008MWR2556.1.

    • Crossref
    • Export Citation
  • Nugent, A. D., and R. B. Smith, 2014: Initiating moist convection in an inhomogeneous layer by uniform ascent. J. Atmos. Sci., 71, 45974610, https://doi.org/10.1175/JAS-D-14-0089.1.

    • Search Google Scholar
    • Export Citation
  • Peters, J. M., H. Morrison, G. J. Zhang, and S. W. Powell, 2021: Improving the physical basis for updraft dynamics in deep convection parameterizations. J. Adv. Model. Earth Syst., 13, e2020MS002282, https://doi.org/10.1029/2020MS002282.

    • Crossref
    • Export Citation
  • Peters, O., and J. D. Neelin, 2006: Critical phenomena in atmospheric precipitation. Nat. Phys., 2, 393396, https://doi.org/10.1038/nphys314.

  • Powell, S. W., 2019: Observing possible thermodynamic controls on tropical marine rainfall in moist environments. J. Atmos. Sci., 76, 37373751, https://doi.org/10.1175/JAS-D-19-0144.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Powell, S. W., and R. A. Houze Jr., 2013: The cloud population and onset of the Madden-Julian oscillation over the Indian Ocean during DYNAMO-AMIE. J. Geophys. Res. Atmos., 118, 1197911995, https://doi.org/10.1002/2013JD020421.

    • Crossref
    • Export Citation
  • Powell, S. W., and R. A. Houze, 2015: Effect of dry large-scale vertical motions on initial MJO convective onset. J. Geophys. Res. Atmos., 120, 47834805, https://doi.org/10.1002/2014JD022961.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Powell, S. W., R. A. Houze, and S. R. Brodzik, 2016: Rainfall-type categorization of radar echoes using polar coordinate reflectivity data. J. Atmos. Oceanic Technol., 33, 523538, https://doi.org/10.1175/JTECH-D-15-0135.1.

    • Search Google Scholar
    • Export Citation
  • Raymond, D. J., G. B. Raga, C. S. Bretherton, J. Molinari, and C. López-Carrillo, 2003: Convective forcing in the intertropical convergence zone of the eastern Pacific. J. Atmos. Sci., 60, 20642082, https://doi.org/10.1175/1520-0469(2003)060<2064:CFITIC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Rushley, S. S., D. Kim, C. S. Bretherton, and M.-S. Ahn, 2018: Reexamining the nonlinear moisture-precipitation relationship over the tropical oceans. Geophys. Res. Lett., 45, 11331140, https://doi.org/10.1002/2017GL076296.

    • Search Google Scholar
    • Export Citation
  • Schiro, K. A., F. Ahmed, S. E. Giangrande, and J. D. Neelin, 2018: GoAmazon2014/5 campaign points to deep-inflow approach to deep convection across scales. Proc. Natl. Acad. Sci. USA, 115, 45774582, https://doi.org/10.1073/pnas.1719842115.

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  • Fig. 1.

    Sounding for the Base simulation. Red and blue lines respectively denote temperature and dewpoint.

  • Fig. 2.

    Profiles of perturbation ω (solid) and temperature (dashed) for the −0.03 (blue), Base (black), and +0.03 (red) soundings.

  • Fig. 3.

    Initial profiles of (left) θe and (right) relative humidity used to force each set of simulations.

  • Fig. 4.

    Two-dimensional probability distributions of observed (shaded and grayscale) and modeled (contoured and color) S-band reflectivity in convective echoes as a function of reflectivity (binned by 5 dBZ) and altitude (binned by 1 km). Probability distributions are normalized at each vertical level such that the sum of the distribution at each level is 1. The contour interval for the modeled probability is 0.05, and the outer contours denote 0.05.

  • Fig. 5.

    Domain-mean rain rate as a function of domain-mean (a) column-integrated relative humidity, (b) BL, (c) updraft w within 450 m of cloud base, (d) updraft buoyancy relative to initial sounding between cloud base and the 0°C level, and (e) updraft Dw/Dt between cloud base and the 0°C level. In (e), the orange dashed line outlines the linear relationship between Dw/Dt and rain rate at high values of Dw/Dt, and an approximate critical Dw/Dt value, following the definition of Ahmed and Neelin (2018) is located where the line intersects the abscissa. Each dot represents a pair of values outputted every 10 min, and results from all 15 experiments are combined. Units (if any) for each abscissa are listed in the respective panel label.

  • Fig. 6.

    Domain-mean rain rate as a function updraft Dw/Dt for updrafts between 550 m (cloud base) and the 0°C level with w > 0.5 m s−1 (black), updrafts between 550 m and 1 km with w > 0.1 m s−1 (blue), and updrafts between 550 m and the 0°C level with w > 0.1 m s−1 (red). Yellow dashed lines denote the linear relationship between Dw/Dt and rain rate at high values of Dw/Dt like the orange line in Fig. 5.

  • Fig. 7.

    (a) Time series of model domain-mean rain rate composited over the three simulations run for each value of A listed in the legend. A = 0 for the Base simulation. (b) Time series of domain-mean rate in each individual simulation, with three lines of each color-and-dashed combination in the legend plotted.

  • Fig. 8.

    Time series of domain-mean BL for each experiment.

  • Fig. 9.

    Updraft areal coverage (black), domain-mean rain rate (blue) with 95% confidence interval (shaded), updraft Dw/Dt averaged between 550 m and 5.25 km (red), and the difference between updraft (Tup) and environmental (Tenv) temperature averaged in the same layer (magenta) relative to the first time at which 0.2% of the model domain at 5.5 km altitude contained in-cloud updrafts with w > 0.1 m s−1. The dashed gray line indicates a lag time of zero.

  • Fig. 10.

    Domain-mean updraft Dw/Dt as a function of altitude and time when areal coverage of updrafts at 5.5 km exceeded 0.2% of the domain. Red and blue colors respectively denote positive and negative vertical accelerations, respectively.

  • Fig. 11.

    As in Fig. 10, but for updraft vertical velocity w.

  • Fig. 12.

    As in Fig. 10, but for the difference between in-updraft and clear-air environmental (a) temperature and (b) specific humidity.

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