Updraft Width Implications for Cumulonimbus Growth in a Moist Marine Environment

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

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Abstract

An idealized large-eddy simulation of a tropical marine cloud population was performed. At any time, it contained hundreds of clouds, and updraft width in shallow convection emerging from a subcloud layer appeared to be an important indicator of whether specific convective elements deepened. In an environment with 80%–90% relative humidity below the 0°C level, updrafts that penetrated the 0°C level were larger at and above cloud base, which occurred at the lifting condensation level near 600 m. Parcels rising in these updrafts appeared to emerge from boundary layer eddies that averaged ∼200 m wider than those in clouds that only reached 1.5–3 km height. The deeply ascending parcels (growers) possessed statistically similar values of effective buoyancy below the level of free convection (LFC) as parcels that began to ascend in a cloud but stopped before reaching 3000 m (nongrowers). The growers also experienced less dilution above the LFC. Nongrowers were characterized by negative effective buoyancy and rapid deceleration above the LFC, while growers continued to accelerate well above the LFC. Growers occurred in areas with a greater magnitude of background convergence (or weaker divergence) in the subcloud layer, especially between 300 m and cloud base, but whether the convergence actually led to eddy widening is unclear.

Significance Statement

Cumulonimbus clouds are responsible for many extreme weather phenomena and are important contributors to Earth’s energy balance. However, the processes leading to the growth of individual clouds are not completely understood nor well-represented in weather prediction models. We find that the clouds containing updrafts that start out wider at early stages of their life cycles grow taller, possibly because they are protected more from drier air outside the cloud than narrow clouds. In addition, this work shows how the initial width of clouds might be related to convergence in the lowest part of the atmosphere, at heights where clouds initially develop. However, meteorologists must be careful not to overinterpret these results because numerical simulations inherently include assumptions that may not reflect reality. This reinforces the need to also observe processes occurring at the scales of individual clouds.

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

An idealized large-eddy simulation of a tropical marine cloud population was performed. At any time, it contained hundreds of clouds, and updraft width in shallow convection emerging from a subcloud layer appeared to be an important indicator of whether specific convective elements deepened. In an environment with 80%–90% relative humidity below the 0°C level, updrafts that penetrated the 0°C level were larger at and above cloud base, which occurred at the lifting condensation level near 600 m. Parcels rising in these updrafts appeared to emerge from boundary layer eddies that averaged ∼200 m wider than those in clouds that only reached 1.5–3 km height. The deeply ascending parcels (growers) possessed statistically similar values of effective buoyancy below the level of free convection (LFC) as parcels that began to ascend in a cloud but stopped before reaching 3000 m (nongrowers). The growers also experienced less dilution above the LFC. Nongrowers were characterized by negative effective buoyancy and rapid deceleration above the LFC, while growers continued to accelerate well above the LFC. Growers occurred in areas with a greater magnitude of background convergence (or weaker divergence) in the subcloud layer, especially between 300 m and cloud base, but whether the convergence actually led to eddy widening is unclear.

Significance Statement

Cumulonimbus clouds are responsible for many extreme weather phenomena and are important contributors to Earth’s energy balance. However, the processes leading to the growth of individual clouds are not completely understood nor well-represented in weather prediction models. We find that the clouds containing updrafts that start out wider at early stages of their life cycles grow taller, possibly because they are protected more from drier air outside the cloud than narrow clouds. In addition, this work shows how the initial width of clouds might be related to convergence in the lowest part of the atmosphere, at heights where clouds initially develop. However, meteorologists must be careful not to overinterpret these results because numerical simulations inherently include assumptions that may not reflect reality. This reinforces the need to also observe processes occurring at the scales of individual clouds.

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

The growth of a cumuliform cloud into deep convection that reaches to or near the tropopause represents an important point in its life cycle. Deep convection is the primary mechanism through which moisture is transported into the upper troposphere, thereby supporting the formation of cirrus clouds that play a vital role in Earth’s energy budget through their impacts on radiative transfer in the atmosphere (Sassen et al. 2009). Furthermore, latent heat release in deep convection influences Earth’s atmospheric circulations. The deepest convection typically contains large vertical velocities (e.g., LeMone and Zipser 1980), has increased production of condensate (Grant et al. 2022), and can contribute to upscale growth of laterally extensive stratiform precipitation regions and cirrus anvil clouds (Houze 2004). Deep convection can develop into supercells or contribute to the growth of squall lines, tropical cyclones, or other laterally extensive mesoscale convective systems. As a result, the numerical modeling community engages in studying the transition of shallow convection into deep convection because the greatest immediate societal impacts linked to clouds (e.g., many types of extreme weather events) often require the growth of convection into deep elements. In the tropics, such convection generally penetrates the 0°C level (often located ∼5 km altitude), therefore meaning that the formation of deep convection also coincides with hydrometeor freezing, which contributes to positive buoyancy (e.g., Raymond and Blyth 1992; Holloway and Neelin 2009; Romps and Kuang 2010a). However, convection that is deep for a particular location—for example, where the tropopause is lower than in the tropics—can also form where the surface temperature is near or below 0°C.

The problem of understanding when and how the transition of shallow convection into deep clouds occurs encompasses at least two related but distinct questions. The first is, “When does a population of shallow convection begin to form deep convection?” This question intends to address the growth of numerous elements in an entire cloud population that may extend over hundreds or thousands of kilometers. It is usually focused on how the atmosphere averaged over the entire area in which the convection occurs changes so that it permits the growth of some of its shallow cumuli into deep convection. Perhaps over areas much larger than a single cloud, the atmosphere becomes moister or more unstable, or solar heating produces sufficiently buoyant boundary layer eddies such that some of them can break through layers of convective inhibition and grow relatively unimpeded thereafter to the tropopause. This question of when a cloud population transitions is important for predicting events such as timing of onset of severe weather on a given day or even larger-scale phenomena like the onset of precipitation in equatorial waves. Increased horizontal resolution in models can improve representation of supercells or squall lines (e.g., Bryan and Morrison 2012; Zhang et al. 2016) and better connection between the atmospheric boundary layer and upper ocean (DeMott et al. 2015; Jiang et al. 2020) has improved representation of variability in tropical precipitation at large spatial scales, so there are means to improve predictability without focusing only on cloud processes. However, poor representations of in-cloud mass flux and entrainment/detrainment remain major problems in operational weather and climate prediction systems that still rely on a cumulus parameterization to substitute for high spatial resolution (Neggers 2015; Peters et al. 2021).

The second question is, “Given a population of visually similar shallow clouds, which specific clouds will grow?” In other words, exactly when and where will shallow convection grow into deep convection? This question is decidedly more concerned with short-term, local impacts of moist convection. For example, given extensive observations, can a few minutes extra lead time for an extreme weather event linked to deep convection (e.g., flash flooding or a tornado) be gained by identifying a cloud or cloud cluster that will grow deep before it actually does? If making such a prediction eventually becomes operational reality, it will first be challenged by insufficient model resolution and limited observations of the lower atmosphere near and below the clouds. In the short term, while such a prediction remains not possible because of these limitations, an increased understanding of the mechanisms involved in the growth of individual clouds—even if reached using numerical models as a tool—might elucidate what types of observations will be most important to pursue to perhaps eventually make short-term forecasting of the evolution of individual convective elements a reality.

The two questions posed above, no matter how different their intentions, are also inextricably linked because a population of cumuli cannot collectively transition without some of its individual elements doing so. Therefore, understanding how to differentiate clouds that will not deepen from clouds that eventually will when the clouds are all in the early stages of their life cycles is necessary for answering both questions.

Over the past couple of decades, an increasingly large body of research has focused on the importance of cloud and updraft width as a crucial factor in determining whether a cloud will grow. Kuang and Bretherton (2006) were among the first to use a large-eddy simulation (LES) with the System for Atmospheric Modeling (SAM; Khairoutdinov and Randall 2003) to conclude that cold pools promoted the development of larger updrafts near cloud base, which resulted in deeper convection. This finding was corroborated around the same time by Khairoutdinov and Randall (2006) and later again by Dawe and Austin (2012), with the latter developing a cloud tracking algorithm to find that cloud height in a population of shallow cumuli was most strongly correlated with cloud area at the lifting condensation level (i.e., cloud base). Presumably, an updraft residing in a wide cloud will be better insulated from mixing with drier and cooler air that is entrained into it (Dawe and Austin 2011; Hannah 2017; Ahmed and Neelin 2021). This idea is supported by numerous modeling studies that conclude that fractional entrainment rates are inversely proportional to cloud width (e.g., Kirshbaum and Grant 2012; Morrison et al. 2020; Peters et al. 2020).

Additional recent studies of individual updrafts and clouds in large-eddy simulations have provided additional evidence that cloud and updraft width are important governors of deep convective onset and have investigated the roles that moisture or shear play in modulating the sensitivity of updraft depth to horizontal cloud size. For example, Rousseau-Rizzi et al. (2017) simulated convection focused along a mesoscale convergence line. They found that convective depth increased as the area of updrafts were larger near cloud base, which happened as convergence magnitude increased. In their model simulations, they also found that narrower convection could grow deep in a moist environment but that the sensitivity of cloud depth to cloud width decreased as environmental moisture increased. Similarly, Morrison et al. (2022) concluded using the Cloud Model 1 (CM1; Bryan and Fritsch 2002) that the width of an updraft in the subcloud layer is a dominant factor in determining whether it grows into deep convection. Furthermore, the size dependence is also regulated by free-tropospheric humidity, such that larger updrafts are required in drier environments. Peters et al. (2022) showed how vertical wind shear impacts the minimum radius required for deep convection to develop. Shear is deleterious to an updraft’s buoyancy if its radius is too small because it enhances a downward dynamic pressure gradient acceleration (Peters et al. 2019; Nelson et al. 2022). However, for sufficiently wide updrafts that can overcome this downward forcing, shear is thought to actually enhance their longevity (Peters et al. 2022). These modeling studies suggest that an environmentally dependent critical updraft radius exists, and only when updrafts exceed this width can they develop into deep convection.1 In particular, Peters et al. (2022) and Morrison et al. (2022) focused on a single convective updraft that is encouraged to form near the model domain center by introducing enhanced surface fluxes into the bottom boundary at that location after introducing horizontally uniform sensible and latent heat fluxes. As such, the convective updrafts studied are highly controlled, which is necessary to conduct the careful experiments needed to investigate relationships of convective growth to factors such as horizontal cloud size, environmental moisture, and vertical wind shear. However, interaction between convective elements regularly occurs in nature and is potentially important for how individual clouds evolve as well.

This article details another study of shallow to deep convective transition in LES. This study focuses on the transition in a moist, tropical, marine environment, and instead of attempting to track individual updrafts, Lagrangian parcel tracking was used to characterize how the properties of fluid changes not only as parcels moved through a cloudy updraft but also before they entered one. Furthermore, no specific forcing was used to promote convection other than random temperature perturbations added to the otherwise uniform initial conditions, and a flux of moist static energy from the ocean to the atmosphere continually supported convection. The differences between deepening and nondeepening parcels in a population of simulated convection that at any point in time contains hundreds of individual updrafts and clouds were recorded. The primary objective of this study is to assess two hypotheses for a population of numerous cumulus and cumulonimbus clouds:

  1. Shallow updrafts that grow deep have larger vertical velocities at cloud base than nongrowing updrafts.

  2. In-cloud accelerations are more positive or less negative for updrafts that grow into deep convection than for nongrowing updrafts.

If the first hypothesis is true, it would suggest that convection that grows deep was simply stronger at its nascent stage, possibly as it ascended from the subcloud layer. If the second hypothesis is true, then something about the environment of growing updrafts or characteristics of the updrafts themselves lead to less downward acceleration or greater upward acceleration being experienced by fluid in the updraft. These hypotheses need not be mutually exclusive. It is possible that deepening updrafts have initially larger vertical velocities and experience less negative acceleration once inside a cloud. As stated above, existing literature points to the width of updrafts as a dominant factor, and this also is a major finding of this study.

2. Model configuration

The cumuliform cloud population was simulated using version 21.0 (cm1r21.0) of CM1. Otherwise, the model setup was identical to that described in Powell (2022), except that “parcel” trajectories were recorded and passive tracers were initialized with a value of 1 kg per kg of dry air at every grid cell at and below 600 m and a value of 0 in all other grid cells. Horizontal grid spacing was 100 m, meaning that much of the inertial subrange of turbulence and energy-containing eddies that can grow into cumuliform convection were resolved. Bryan et al. (2003) also suggest that 100 m grid spacing is sufficient for resolving the salient characteristics of deep convective updrafts. This may not be sufficient, however, to fully resolve shallow convection or to identify differences between deepening and nondeepening convection that may arise from differences in updraft or cloud width similar to the grid spacing. However, further reducing grid spacing for this simulation would be prohibitively expensive as the time of this study. When comparing simulated reflectivity distributions with observed radar observations of convection, Powell (2022) found good agreement below the 0°C level, which would have been unlikely had simulated convection not produced realistic vertical velocities and hydrometeor size distributions. Convection was simulated over a warm tropical ocean with the same moist sounding that Powell (2022) used. It had relative humidity of 80%–90% below the 0°C level and over 80% column-integrated relative humidity. The sounding included almost zero vertical wind shear; the wind profile consisted of a near westerly wind of 1.5–4 m s−1 from just above the surface to 150 hPa. A zoomed-in portion of the sounding between the surface and 700 hPa is shown in Fig. 1. The lifting condensation level (LCL) and level of free convection (LFC) for a hypothetical surface-based parcel in this sounding were 948 and 904 hPa, respectively, corresponding to heights of about 650 and 1050 m. The importance of these heights will be clarified later in the article. Table 1 provides a list of model setup details, including those not explicitly discussed in the text. The namelist.input file utilized for the simulation is included as online supplementary information.

Fig. 1.
Fig. 1.

A zoomed-in view of the sounding from Powell (2022) used to initialize the model, including temperature (red) and dewpoint (green). The dashed black lines indicate the lifting condensation level (LCL) and level of free convection (LFC). Red, blue, and green dashed lines respectively represent dry adiabats, moist adiabats, and lines of constant mixing ratio. Mixing ratio lines are labeled by the green numbers near the top of the chart.

Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0065.1

Table 1.

Setup and physics options used for CM1 simulation.

Table 1.

In CM1, parcel trajectories can be used to follow hypothetical fluid parcels in a Lagrangian manner as the model integrates. The parcels each exist essentially as a point in space, and their locations are not required to conform to the grid of the model domain. Three-dimensional linear interpolation from the staggered wind grid was used at each time step to determine how it was advected to its next location. The parcel trajectories were only passively advected and did not cause any changes to the environment. They were initialized at the center of every grid cell below 1000 m altitude, and they were released at model initialization; 10 240 000 parcels were tracked for the entire 24-h integration period. Parcel locations and three-dimensional model output for the entire domain were recorded every 60 s.

3. Analysis technique

Analyzing parcel trajectories in a Lagrangian sense is necessary to assess how some shallow clouds grow while other nearby ones that might appear the same to the naked eye do not. Previous works like Dawe and Austin (2012) and Rousseau-Rizzi et al. (2017) have developed various tracking algorithms so that cloud or updraft objects could be tracked from one model output file to the next. The main advantage of this approach is that cloud updrafts can be individually tracked with changes to their morphology documented throughout their life cycle. However, the algorithms require numerous subjective decisions about how to define updrafts and clouds and those decisions may change as a function of height. Simply applying recently developed object-tracking software (e.g., Heikenfeld et al. 2019) to large three-dimensional datasets—and specifically to updraft objects—is also extremely difficult because even more so than clouds, updrafts rapidly develop, dissipate, change shape, split, and merge, so they are difficult to define. Using parcel trajectories circumvents many of these problems associated with being forced to define and then keep track of individual updrafts but comes with the limitation that some updrafts might not be sampled, especially after the onset of deep convection when many of them have been transported into the upper troposphere. On the other hand, parcel trajectories are not confined to only updrafts and can easily detail properties of air before it enters an updraft. For this study, initializing parcel trajectories at every grid cell below 1000 m appeared to be sufficient to sample a large number of shallow updrafts and all of the deep ones during the analysis period of the simulation.

After model integration, a subset of parcel trajectories were identified for analysis in cloudy updrafts based on the following criteria:

  1. The parcel must have entered a cloud and remained in a cloud for at least 20 consecutive minutes. For this portion of the study, a cloud was defined as existing in any model grid cell where the sum of cloud water (qc) and ice water (qi) mixing ratios was at least 10−6 kg kg−1.

  2. The parcel’s trajectory in a cloud started no higher than 1000 m.

  3. The parcel’s trajectory inside a cloud topped out at either between 1500 and 3000 m or above 6000 m.

A parcel’s trajectory was deemed to have definitively exited a cloud if, after the ≥20-min period in the cloud, it was outside of a cloud for more than 10 min. If a parcel exited a cloud for less than 10 min before reentry, it was considered to be continuing the same in-cloud trajectory, although it did not necessarily have to be in the same cloud. This would allow a parcel rising near the edge of a cloud to be detrained temporarily (perhaps even into a downdraft) but then be entrained back into an updraft in the same cloud or a nearby cloud, possibly one connected to the first cloud at a higher altitude, soon after. Results were not affected significantly by changing the 20-min time in cloud required unless the time was set to a very long or very short time. Too long a time (e.g., 1 h) would exclude many nongrowing convective elements with shorter life cycles. Too short a time (e.g., 5 min) would cause many very small, short-lived clouds to dominate results. Criterion 2 was imposed so that parcel and updraft characteristics could be documented starting at a low level in the cloud or even below the cloud. The levels chosen in criterion 3 are also somewhat arbitrary. We chose 6000 m as a level just above the 0°C level, and Powell (2022) found that precipitation in this simulation increased rapidly after convection reached this level; 1500–3000 m was chosen because it includes convection that started to grow vertically but stalled out at a level much lower than the 0°C level. Separating the nongrower parcels reaching 1500–3000 m from the grower parcels exceeding 6000 m by a 3000-m-wide range would presumably allow differences between the two categories to be obvious. Rousseau-Rizzi et al. (2017) used a similar separation, classifying convection exceeding 8000 m as deep, and that below 3000 m as shallow.

There was no criterion based on vertical velocity. Therefore, parcels could ascend for a few minutes in an updraft and then get caught in a downdraft inside or near a cloud for a few minutes before getting caught in another updraft and moving higher. The parcels that met the above-listed criteria were then categorized into those that topped out between 1500 and 3000 m (nongrowers), and those in deep growing clouds, which reached at least 6000 m (growers). In this article, the terms “growers” and “nongrowers” refer to the parcel trajectories themselves and not necessarily to the updraft or cloud containing them. Consequently, the cloud or updraft properties presented herein are composited values that were compiled over many different clouds and updrafts. They do not necessarily represent how any individual cloud or updraft actually behaved. Rather they are intended to highlight statistically significant differences between large populations of deepening and nondeepening parcels. Individual parcel trajectories in the same cloud may experience different outcomes. For example, in a cloud containing a vigorous updraft located next to a region of in-cloud weak vertical motions, the parcels in the updraft may become growers, while a nearby parcel embedded in the weak vertical motions may only become a nongrower. When composited over a large number of parcels, statistically significant differences between growers and nongrowers were detected (section 4). Tracking individual cloud objects would allow us to specifically identify how growing and nongrowing clouds themselves may differ instead of just comparing properties of parcels within them. However, as stated above, tracking clouds is difficult. For example, multiple seemingly different clouds near the LCL frequently merge at higher levels, making all of the separated cloud pieces at the LCL part of the same 3D cloud object. This complicates the analysis of how each individual component of the full 3D cloud object evolves. Therefore, this study is limited to only tracking parcel trajectories and comparing the properties of them. From this, we can infer what differences there might be between growing and nongrowing clouds and updrafts.

Since parcel 3D locations were recorded every minute for 24 h, only the part(s) corresponding to when a parcel was in a single cloud were separated for additional analysis. Parcels could potentially be tracked through multiple clouds during the simulation if they were somehow transported back downward after being advected upward within earlier clouds. When this occurred, the chunks of data corresponding to the parcel’s travel through different clouds were analyzed as separate trajectories. Finally, many parcels entered clouds from below the LCL (the subcloud layer). To document properties of the boundary layer experienced by parcels before they entered clouds, data were also recorded for each parcel 30 min prior to entry into a cloud. Of this subset of data, properties of the atmosphere at the parcel location were analyzed only when the parcel was below 600 m.

Powell (2022) found that onset of heavy precipitation (and therefore the proliferation of cold pools in the domain) started around 11 h after initialization. The new CM1 V21.0 simulation follows the same evolution of domain-mean precipitation rate and cloud depth. Therefore, the majority of the analysis focused on parcels that rose in convection between 6 and 11 h after model start to enable examination of differences between growers and nongrowers in a population of isolated convective elements. Analysis of in-cloud parcels between 15 and 20 h was also conducted and while not shown in this article, it supported qualitatively similar conclusions to the ones presented herein.

To calculate most atmospheric properties at parcel locations, three-dimensional linear interpolation was completed using the value of model-derived fields at data points surrounding the 3D location of the parcel. The exceptions to this were when calculating the two-dimensional area of a cloud or updraft containing a parcel at the reported height of the parcel or when computing the horizontal distance of the parcel from the nearest cloud edge. These calculations first required identification of contiguous regions of cloud and updraft at each individual height level in the model. For this part of the study, cloud was identified more stringently as grid cells where qc + qi > 10−4 kg kg−1, and updrafts were defined as grid cells where w > 0.5 m s−1. The higher hydrometeor mixing ratio helped to constrain cloud areas to regions surrounding the strongest updrafts so that the computed cloud and updraft area generally were well-correlated. Any grid cells along the edges of the doubly periodic boundaries were considered contiguous with the adjacent grid cell on the other side of the domain. Every grid cell was assigned two values: One corresponding to the area of the updraft to which it belonged at the height where the cell was located and a second corresponding to the area of the cloud to which it partially constituted. When determining the updraft or cloud area for the cloud or updraft that a parcel was located in, the value(s) at the nearest neighbor cloudy and/or updraft grid cell to the parcel location was used.

Vertical accelerations along parcel trajectories were also recorded. Herein, we approximated the vertical momentum equation, in a Lagrangian sense as if it were following a parcel, such that
DwDt1ρpDzTerm(i)1ρpBz+BTerm(ii),
in which term (i) is the dynamic pressure gradient acceleration (dynamic PGA), and term (ii) is the effective buoyancy, the sum of the buoyancy perturbation pressure gradient acceleration (buoyancy PGA) and Archimedean buoyancy (B) defined using reference variables as their domain-mean values at model start. Momentum advection, diffusion, and impacts of turbulence on Dw/Dt were neglected. Of those three, the latter two were several orders of magnitude smaller than the terms included in Eq. (1). Momentum advection was frequently similar or larger in magnitude than term (i) or (ii). However, the median vertical momentum advection in both growers and nongrowers was 0 at all vertical levels (as in the analysis shown for other terms in section 4); therefore, it was neglected in this article. The sum of dynamic and buoyancy PGA was calculated separately from B during model integration. To calculate effective buoyancy, decomposing the total PGA into its dynamic and buoyancy components was required; this was done using the method of Hastings and Richardson (2016). The effective buoyancies plotted herein are the sum of buoyancy PGA computed by this method and the B calculated by CM1. Dynamic PGAs shown herein are the sum of B and total PGA calculated in CM1 minus the effective buoyancy.

Finally, for the vertical profiles illustrated in this article, parcels were categorized into height bins with 100 m depth (e.g., 0–100 m, 100–200 m). Each time a parcel was located within a bin (possibly multiple times as it traveled through a cloud), it was considered a separate sample for the calculation of medians, means, or 95% confidence intervals. For statistical analysis, this approach could be argued to be flawed because there is some autocorrelation between atmosphere properties following a parcel when considered 60 s apart at nearly the same location. Furthermore, a single updraft and cloud could contain numerous different but closely located parcels at one time. This is one of the disadvantages of the parcel trajectory approach to Lagrangian analysis compared to tracking entire updrafts as a single object. However, properties within turbulent cloudy updrafts change rapidly, so for simplicity, no attempt was made to reduce the sample size to account for any autocorrelation. In other words, each minute that a parcel was in a cloud was considered to be a separate “parcel sample.” Figure 2 depicts the number of in-cloud parcel samples for growers (red) and nongrowers (blue) in the lowest 3000 m between 6 and 11 h after model start. The majority of parcels were never embedded into deep convection. Furthermore, at all heights, the number of parcel samples for nongrowers was about two orders of magnitude larger than for growers. The number of parcel samples peaked around 150 000 for nongrowers between 500 and 1000 m, and about 1800 for growers around 1000 m. About 500 parcel samples were available for analysis even for growers all the way up to 3000 m.

Fig. 2.
Fig. 2.

Number of parcel samples per 100-m-wide height bin for grower (red; top abscissa) and nongrower (blue; bottom abscissa) parcels.

Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0065.1

4. Comparing updrafts and clouds containing grower and nongrower parcels

a. In-cloud properties

The cloud population in the model started out as numerous shallow convective elements that developed and then dissipated over the course of less than 1 h. Starting around 6 h after model start, an increasing number of shallow elements began to reach and/or penetrate the 0°C level and precipitate. Since this study is primarily concerned with the growth of shallow convection into deep convection without the influence of mesoscale processes such as cold pools, the analysis presented herein was stopped just before heavy precipitation became widespread in the model around 11 h. However, qualitatively similar results were obtained when analyzing in-cloud parcels later in the simulation when clustered deep convection was prevalent.

First, in-cloud vertical velocity and accelerations were examined. Figure 3 illustrates vertical profiles of the 95% confidence interval of median values of these quantities for growers (red) and nongrowers (blue). The 95% confidence intervals were computed using a bootstrapping method: At each height bin, 100 parcel samples were chosen at random out of all parcel samples in each height bin, and the median was computed. The process was repeated at each height bin 1000 times to produce 1000 sample medians, and the innermost 95% of those 1000 sample medians is denoted by shaded regions. No data are plotted below the LCL (600 m) because only in-cloud parcels were included. When considered over many parcels that presumably occurred in various parts of a cloud, it is presumed that the profiles roughly represent the composite properties of growers and nongrowers. Of course, as explained in section 5, individual parcels may experience significantly different conditions from the composite median or mean.

Fig. 3.
Fig. 3.

Composite vertical profiles up to 3000 m of median (a) vertical velocity, (b) effective buoyancy, (c) dynamic pressure gradient acceleration, and (d) hydrometeor loading in clouds containing growers (red) and nongrowers (blue). The shaded region denotes the 95% confidence interval of the median.

Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0065.1

Below 1400 m, vertical velocity between growers and nongrowers was indistinguishable (Fig. 3a). For both types of parcels, its median was near 0.8 m s−1 near cloud base. It stayed near this value for nongrowers, while it actually decreased to near 0.5 m s−1 for growers right at the LFC near 1000 m. Between the LFC and about 1400 m, the median vertical velocity for both increased to 1.0–1.5 m s−1. Above 1400 m, growers experienced a more gradual increase in median vertical velocity, with a 95% confidence interval ranging from 1.0 to 3.0 m s−1 at 3000 m. In contrast, near 1500 m, the median vertical velocity for nongrowers quickly decreased, then more steadily decreased to zero by 3000 m. The steep drop off in nongrower w is seen at 1500 m because that height is the lower bound for maximum altitude considered for nongrower parcels in this study (section 3). It goes to zero at 3000 m because that is the maximum height allowed for nongrowers. Despite these features caused by the arbitrary decision of what was a nongrower, the fact that growers and nongrowers are virtually indistinguishable in the lowest 800 m of clouds yet experienced opposite signed accelerations above 1400 m is clear.

To further demonstrate that the distributions of vertical velocity just above cloud base were essentially identical between growers and nongrowers, the entire distributions of in-cloud parcel vertical velocity at the 700 m height level are illustrated in Fig. 4 for growers and nongrowers. Each distribution is illustrated as a histogram of normalized frequency separated into w bins with width of 0.1 m s−1, such that the sum of each histogram equals 1. The nongrower distribution is a bit smoother, likely because of the larger parcel sample size (Fig. 2), but otherwise the distributions appear virtually indistinguishable. While growers had a slightly larger frequency of vertical velocities exceeding 1.5 m s−1, they were still relatively uncommon, with each w bin occurring in no more than 2% of growers. They also had a higher frequency of negative vertical velocities than nongrowers, and the peak of the growers’ w distribution was around 0.2 m s−1 compared to 0.4 m s−1 for nongrowers. Both the similarity in the w distributions and the slight offset in the distribution peaks are consistent with the median w at 700 m shown in Fig. 3a, where the 95% confidence intervals overlapped but were slightly offset, with growers tending toward lower median w.

Fig. 4.
Fig. 4.

Histogram of normalized frequency of in-cloud parcel vertical velocity at 700 m height for growers (red) and nongrowers (blue).

Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0065.1

These results are supported by examining the components of acceleration. Figure 3b depicts median effective buoyancy. Like vertical velocity, it was statistically the same up to about 1400 m. Above 1400 m, the median effective buoyancy for growers generally remained between 0.0025 and 0.01 m s−2 up to 3000 m, but for nongrowers, it became negative around 1600 m, and was increasingly negative as height increased, reaching decelerations as large as −0.0075 m s−2 by 3000 m for the highest reaching nongrowers. The median hydrometeor loading component of the effective buoyancy is depicted in Fig. 3d. This term also statistically overlaps between growers and nongrowers up to 1200 m. Above 1200 m, the loading experienced by nongrowers remained about −0.005 m s−2 and increased to between −0.01 and −0.015 m s−2 for growers. This means that the remaining component of the effective buoyancy, which is directly impacted by the thermodynamic profile and consists of contributions from temperature and humidity perturbations as well as buoyancy pressure vertical gradient acceleration combined, was larger than the values shown in Fig. 3b and was approximately 0.02–0.03 m s−2. For nongrowers, the median nonloading component of the effective buoyancy was positive up to about 2500 m (i.e., where the effective buoyancy was greater than the loading). At this height, the effective buoyancy also reached −0.005 m s−2, meaning that the nonloading components of effective buoyancy was zero.

The 95% confidence interval on median dynamic pressure gradient acceleration (Fig. 3c) was generally wider. It ranged from just below 0 to about 0.002 m s−2 from cloud base to 3000 m for nongrowers. The median profile for growers overlapped the nongrower profile at all altitudes shown, but it clearly skewed toward more negative values above the LFC. Above about 1400 m, the median effective buoyancy was larger in magnitude than and opposite in sign from median dynamic PGA. Therefore, the total acceleration felt in the clouds above the LFC was largely governed by effective buoyancy, which was partially balanced by dynamic PGA to lower the net acceleration. However, especially below the LFC, median effective buoyancy and dynamic PGA were much closer in magnitude with 95% confidence intervals overlapping. For example, for growers, both the median effective buoyancy and dynamic PGA between cloud base and the LFC were approximately 0–0.002 m s−2. Therefore, the dynamic PGAs reinforced buoyancy driven accelerations in lower portions of clouds.

It is possible that eddies that grew into deep convection might simply have started off with larger temperatures or humidities than other eddies, making them slightly more buoyant at low levels and more likely to grow into deep convection. Effective buoyancy in Fig. 3b does not support this hypothesis, and Fig. 5 further suggests that at least temperature does not play a role in distinguishing growers from nongrowers. In Fig. 5, the temperature and specific humidity at a parcel location is compared to a “background” value. Any way of choosing the background value is arbitrary, but in this case, it has been computed as the mean temperature or humidity in noncloudy points (using the 10−6 kg kg−1 for identifying cloud) within a 25 × 25 grid cell (2500 m × 2500 m) box at a single height level surrounding and including the grid cell that contains a parcel. Qualitatively, results were not sensitive to increasing or decreasing the length of both sides of the box by up to at least five cells. Temperature perturbations are known to dominate buoyancy in the atmosphere, so it is unsurprising that the median profile of parcel minus background potential temperature (Fig. 5a) had a shape similar to that of effective buoyancy (Fig. 3b). The 95% confidence intervals overlapped below 1500 m, and the median parcel potential temperature, as expected, was lower than the median background temperature below the LFC at 1000 m (i.e., many parcels experienced convective inhibition between the LCL and LFC). Between 1500 and 3000 m, growers were 0.2–0.6 K warmer than surrounding noncloudy air, but the temperature along nongrower trajectories actually decreased relative to its surroundings above 1500 m. In fact, at 3000 m, the median temperature at the parcel location was about 0.2 K cooler than surrounding noncloudy air. This is consistent with the negative effective buoyancy experienced by nongrowers at this altitude.

Fig. 5.
Fig. 5.

As in Fig. 3, but for (a) the parcel temperature relative to the background temperature and (b) the parcel specific humidity relative to its background value. Calculation of background values is described in the text.

Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0065.1

In contrast to temperature, there were statistically significant differences in median specific humidity below about 1200 m. In Fig. 5b, smaller values on the abscissa denote moister surroundings for a parcel. Growers occurred in moister surroundings; the parcel humidity was about 0.6–0.8 kg kg−1 moister than the surrounding noncloudy area, while for nongrowers, the parcel surroundings were relatively drier with height up to 1200 m, where the parcel was roughly 0.8–1.0 kg kg−1 moister than its surroundings. This could imply that the clouds and updrafts containing growers were more likely to ascend because they occurred in moister environments at low levels. Median passive tracer concentrations (Fig. 6) may support this notion. Tracer values were statistically overlapped between 800 and 1400 m for growers and nongrowers, but were larger for nongrowers (0.66 compared to 0.61 for growers) at 550 m. Then at 1200 m, the median tracer value was 0.4 both growers and nongrowers. In other words, the fluid at the parcels’ locations at 1200 m consisted of 40% that originated below 600 m at model start and 60% above 600 m. Comparing passive tracer values at 550 and 1200 m indicates that nongrowers and growers respectively experienced a 39% and 33% reduction in passive tracer concentration between cloud base and 1200 m. Between 1500 and 3000 m, growers were 1.0–1.4 kg kg−1 moister than surrounding noncloudy air (Fig. 5b), suggesting they were diluted less than nongrowers, which were only about 0.8 kg kg−1 moister. Figure 6 also shows that above 1200 m, nongrower parcels were diluted more than grower parcels. Specifically, at 1500 m, the median tracer value was 0.28 for nongrowers and 0.35 for growers, respectively representing a 30% and 13% decrease relative to median values just 300 m below. Then at 3000 m, the median tracer value was 0.07 for nongrowers and 0.15 for growers, respectively representing further 75% and 57% reductions in the median tracer value from 1500 to 3000 m. While we cannot estimate fractional entrainment rates from just this analysis, it is clear that clouds and updrafts containing nongrower parcels were diluted more than the grower parcels. While subgrid scale or numerical mixing likely impacted dilution of parcel buoyancy in the model, they are not separately calculated herein.

Fig. 6.
Fig. 6.

As in Fig. 3, but for the value of the passive tracer expressed as a fraction of its value during model initialization at grid cells below 1500 m.

Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0065.1

Figure 7 illustrates vertical profiles of quantities related to cloud width and highlights one of the key points of this article. At all heights at and above cloud base, the areas of updrafts and clouds containing parcels were larger for growers. Figure 7a shows that median updraft area (defined as the contiguous region including a parcel with w > 0.5 m s−1; see section 3) containing nongrowers was 0.1–0.2 km2 throughout the cloud. For updrafts containing growers, while there was a small decrease in median area between the LCL and the LFC, the median updraft size increased above the LFC to between 0.6 and 1.3 km2 at 3000 m. Differences in width between clouds containing growers and nongrowers echoes the difference in updraft width. Clouds containing growers started out larger than clouds containing nongrowers, and the median width of clouds containing growers increased faster in the lowest 3000 m than for clouds with nongrowers (Fig. 7b). Unlike the other quantities investigated above, the difference in median updraft and cloud area between those holding growers and nongrowers was statistically significant at every height. Presumably then, grower parcels would be more protected from dilution via entrainment of noncloudy air because the clouds they were located in were wider. Figure 7c indicates that this was not entirely the case however, particularly below the LFC. Both growers and nongrowers were separated from a cloud “edge” (defined as the 10−4 kg kg−1 contour) by only about 100 m (a single grid cell) between cloud base and 1100 m. However, as clouds grew above 1500 m, the median distance between grower parcels and cloud edge increased to 200–250 m. This may be consistent with the similar reductions in median tracer values experienced by growers and nongrowers below 1200 m followed by a bigger difference in dilution between growers and nongrowers above 1200 m. However, separating the impacts of cloud and updraft horizontal size, parcel position within a cloud, and environment moisture differences is not attempted here. That grower and nongrower parcels had statistically similar median distances from a cloud edge below 1300 m may seem paradoxical given that the clouds containing growers were larger. A limited manual analysis of randomly selected individual parcel trajectories suggest that this may have happened because there was no preference for where in a cloud a grower parcel was located. In other words, a grower parcel was not more likely to be positioned in the center of a cloud than along an edge. Nongrowers were also equally likely to be positioned near the center or edge of a cloud object. Even along the edge of a cloud, grower parcels could conceivably have experienced less dilution because they were protected more from dilution on at least one side by the wider cloud in which they were located. However, this is currently only a speculative hypothesis.

Fig. 7.
Fig. 7.

As in Fig. 3, but for (a) the area of an updraft containing a parcel, (b) the area of a cloud containing a parcel, and (c) the distance of a parcel from the nearest cloud edge.

Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0065.1

Two-dimensional slices in Fig. 8 (for growers) and Fig. 9 (for nongrowers) offer a different way of viewing the mean cloud and updraft structure at different heights above the LCL surrounding parcels’ locations. Unlike the median profiles shown above, the mean composite slices more heavily weigh the largest values making up the composite, so in general clouds and updrafts appear larger (with w higher in magnitude) than the median profiles suggested. In both Figs. 8 and 9, shading in the top panels [(a)–(e)] depicts vertical velocity, while shading in the bottom panels [(f)–(j)] illustrates total vertical acceleration [the left-hand side of Eq. (1)]. The black line in each panel is the 10−4 kg kg−1 qc contour, and each panel shows slices at 1000–1100, 1500–1600, 2000–2100, 2500–2600, and 3000–3100 m. At every model output time, and for each grower and nongrower parcel within the levels listed above, w, qc, and Dw/Dt in a 1200 × 1200 m2 area surrounding the grid cell containing the parcel was recorded; Dw/Dt was computed as the sum of the two terms in Eq. (1). Each slice seen in Figs. 8 and 9 then denotes the composite mean of all individual 1200 × 1200 m2 slices at a different level. Therefore, the parcel location in these figures is always the origin, located at the center of each slice. Because these slices are composites surrounding various parcels, they do not represent the evolution of individual clouds and updrafts. Rather they simply represent the properties of clouds or updrafts surrounding parcels at various heights. For example, clouds and updrafts containing growers were wider at 3000–3100 m than at 1000–1100 m. This could happen because some individual clouds and updrafts actually widened as they ascended. Visually, Fig. 8 gives that impression. However, it could also happen because closely spaced clouds near cloud base were more likely to connect at higher heights and collectively produce a wider contiguous cloud area. In other words, there were many cases when a contiguous cloudy region at 3000 m was vertically connected to what looked like several individual clouds at the LCL. It is also important to remember that although these figures depict parcels as being located in the center of roughly circular clouds, this is an artifact of compositing. As stated above, grower and nongrower parcels were just as likely to move along any edge (i.e., downwind side, upwind side, or elsewhere) of a cloud as they were its center. Averaged over many parcels, that makes the parcel appear to occur in the center of the composite cloud and updraft.

Fig. 8.
Fig. 8.

Composited horizontal slices of (a)–(e) mean vertical velocity and (f)–(j) mean Dw/Dt surrounding grower parcels within 1200 m of a parcel and at model height levels of 1000–1100, 1500–1600, 2000–2100, 2500–2600, and 3000–3100 m. The parcel location is set at the origin of each slice. The black outline in each slice denotes the 10−4 kg kg−1 contour of cloud water (qc).

Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0065.1

Fig. 9.
Fig. 9.

As in Fig. 8, but for nongrowers. No slice is shown at 3000–3100 m because, by definition, nongrowers did not reach this height.

Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0065.1

With the above caveats in mind, some important differences between growers and nongrowers are still obvious. Surrounding growers, the mean area near the center of the updraft with w > 2 m s−1 (dark blue) increased between 1000 and 3100 m (Figs. 8a–e) from 0 to about 0.2 km2. The area containing upward vertical velocity encompassed most of the composited cloud. Possibly as a result of this larger size, positive vertical accelerations were present at 1000–2100 m, with a broad area of Dw/Dt ≥ 0.003 m s−2, and maximum accelerations reaching as high at 0.007 m s−2. At 2500–2600 and 3000–3100 m, downward accelerations were present in parts of the composite cloud (Figs. 8i–j). Although not shown here, above 3000 m, even the median parcel w in growers decreased a little above 3000 m as more cloudy updrafts experienced deceleration.

The composited clouds seen in Fig. 9 starkly contrast those seen in Fig. 8. For composited clouds containing nongrowers, both cloud and updraft areas were obviously smaller at 1000–1100 m. Cloud with nongrowers appeared to be embedded in regions of weak upward motion (w ≤ 0.2 m s−1) at and below 2100 m oriented in a generally zonal (shear-parallel) strip outside the cloud, which can be seen as a faint blue area outside the cloud in Figs. 9a–e. While the composited cloud was larger at 2000–2100 m than at 1000–1100 m, the composite updraft weakened and appeared to shrink in area above 1500 m until it was nearly nonexistent at 2500–2600 m. Since the maximum altitude permitted for nongrowers was 3000 m (section 3), no horizontal slice is shown at 3000–3100 m. The shrinking and weakening of the composite cloud is consistent with the vertical accelerations and tracer values computed in cloud (Figs. 3 and 6). While upward acceleration of no more than 0.003 m s−2 was present at 1000–1100 m, increasing negative accelerations were present throughout the composite cloud at 1500–2600 m, which quickly destroyed updrafts containing nongrowers. It probably also destroyed updrafts containing an unknown number of growers before they later ascended in a different updraft, an idea that will be considered more in section 5. In other words, the composite in Fig. 8 contains some clouds and updrafts that look more like what is depicted in Fig. 9 because some growers were probably located in nongrowing clouds and updrafts before finally ascending to 6000 m later in a different updraft (although not necessarily a different cloud). Despite this happening, the composite properties of air surrounding grower parcels were clearly different than those surrounding nongrowers.

b. Parcel and updraft properties in the subcloud layer

Between the LCL and LFC, growers and nongrowers were virtually indistinguishable from each other except that the median areal coverage for clouds and updrafts containing growers was noticeably larger, even as low as the cloud base. This encourages an analysis of the structure of eddies that rose from the subcloud layer and developed into clouds. As in the above analysis, parcels that ended up in growing updrafts above the LFC were separated from nongrowers, and the properties of the atmosphere in the boundary layer at these parcel locations were separately computed.

As seen in Figs. 3 and 5, the dynamic and thermodynamic properties of clouds in growers and nongrowers overlapped below the LFC. Were there at least differences in some of these quantities while the parcels were still in the boundary layer? Figure 10, in which all quantities are plotted with the same abscissa limits as in Figs. 37, answers a resounding “no” to that question. Vertical velocity, effective buoyancy, dynamic PGA (Figs. 10a–c), and parcel temperature and specific humidity relative to background values (Figs. 10f–g) were all statistically identical between growers and nongrowers up to 600 m. Of the quantities computed, the only apparent differences were the areal coverages of the updrafts containing parcels (Fig. 10d) and above 300 m, the divergence (Fig. 10e). For this figure, the divergence at the location of a parcel was computed as the average divergence within a 9 × 9 grid cell box centered on the parcel. For growers, the updraft area maximized near 0.4–0.6 km2 at 300–400 m. For nongrowers, it maximized only at 0.2–0.3 km2. Above 100–200 m, the 95% confidence intervals of median updraft areas did not overlap. The 95% confidence interval for grower divergence overlapped with that for nongrowers below 300 m; however, the distribution of divergence was clearly shifted toward more negative values (i.e., more convergence). Growers and nongrowers experienced significantly different divergence starting at 300–400 m. At 400–500 m, growers experienced a median divergence of −7 × 10−5 to 9 × 10−5 s−1 while nongrowers occurred in areas where median divergence was between 1.7 × 10−4 and 3.3 × 10−4 s−1. Therefore, it seems plausible that at least some physical relationship existed between boundary layer convergence and updraft area.

Fig. 10.
Fig. 10.

As in Fig. 3, but below 600 m for parcels located in the subcloud layer before they entered clouds. (a) Vertical velocity, (b) effective buoyancy, (c) dynamic pressure gradient acceleration, (d) area of updraft containing parcel, (e) background divergence as described in the text, (f) parcel temperature relative to the background temperature, and (g) parcel specific humidity relative to the background specific humidity are shown.

Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0065.1

For both growers and nongrowers, median vertical velocity peaked at 0.5 m s−2 at 400–500 m. Median effective buoyancy and dynamic PGA were both nearly 0. Consistent with the buoyancy being nearly the same, there was no difference in median parcel temperature relative to the background. Median specific humidity of growers and nongrowers overlapped in the subcloud layer, although the specific humidity 95% confidence interval for growers actually tended to reach a little lower than for nongrowers. In other words, there is no evidence in this study that boundary layer eddies that are warmer and/or moister than the surrounding environment are required (or even preferred) for deep convection to occur. Whatever unknown process leads to larger boundary layer eddies (possibly related to boundary layer convergence) seems to promote deeper convection.

5. Discussion of caveats: Is updraft width really all that matters?

So far, the results presented seem to make the case that, within an environment known to be sufficiently moist to support deep convection, the horizontal size of the incipient eddy in the boundary layer or near cloud base was the predominant factor in determining whether an updraft and cloud grew into deep convection. However, Fig. 11 suggests a potential counterpoint to this conclusion. It shows the full distributions of updraft area in growers and nongrowers at 300–400 m (Fig. 11a), the height in the subcloud layer where updraft area was maximized, and the lowest height above the LFC where growers and nongrowers experienced significantly different accelerations, at 1500 m (Fig. 11b). In each panel, normalized frequency is shown (similarly to in Fig. 4) in bins with width of 0.1 km2. Why the 95% confidence intervals of the medians were different for nongrowers and growers at both heights is clear. The distribution of updraft area in growers was much wider than for nongrowers, with some updrafts exceeding 1 km2 at both 300–400 m and 1500 m. The mode of the grower distribution was 0.50–0.60 km2 at 300–400 m compared to 0.20–0.30 km2 for nongrowers. At 1500 m, the modal updraft area for growers was 0.2–0.3 km2 for growers compared to less than 0.1 km2 for nongrowers. The particularly interesting result shown in Fig. 11, however, is the overlap between the distributions. For example, at 300–400 m, 25% of grower parcel samples had areas less than or equal to the mode of nongrower parcel samples. In other words, there were grower parcels that were at some point inside small updrafts and nongrower parcels that stalled between 1500 and 3000 m that were resident inside large updrafts. As seen in the previous section, when considering averages over a large number of parcels in a large number of updrafts, horizontal size was an important discriminant between what did and did not grow into deep convection. However, for an individual updraft, horizontal size was seemingly only one factor among many that contributed to its fate.

Fig. 11.
Fig. 11.

Normalized distributions of areas of updrafts containing grower (red) and nongrower (blue) parcels at (a) 300–400 and (b) 1500–1600 m height. Each parcel sample was considered an independent sample when constructing the histograms.

Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0065.1

One possibility is that small updrafts containing growers had large vertical velocities that allowed them to deepen despite potentially increased dilution. Alternatively, perhaps the updrafts containing growers ascended in moister environments and were less susceptible to dilution. To investigate these possibilities, Table 2 shows the 95% confidence interval of Pearson correlation coefficients between updraft area and various quantities both in the subcloud layer and above the LFC for growers only. All correlations shown in Table 2 are no greater than 0.46, and in many cases, are opposite to what we would expect if the above conjectures were true. For example, the maximum correlation between updraft area and vertical velocity was only 0.29–0.43 at 1500–1600 m. Furthermore, the positive correlations imply that smaller growers had weaker vertical velocities. A similar relationship was found between updraft area and either effective buoyancy (Beff) or total vertical acceleration (Dw/Dt). Finally, barely positive relationships between updraft area and parcel temperature or humidity relative to background values (Tdiff or qυdiff) were found. While these results generally support the notion that wider updrafts were less susceptible to dilution and therefore experienced more positive effective buoyancy, they do not support the conjecture that growers in narrow updrafts ascended because they were in stronger updrafts. Instead, many parcels contained within narrow updrafts with weak vertical velocity eventually ascended to 6000 m.

Table 2.

The 95% confidence interval of Pearson correlation coefficients between area of an updraft containing a grower parcel and listed quantities at grower parcel locations.

Table 2.

Apparently, a large number of the narrow grower parcel samples were depicted in Fig. 11 because many grower parcels were counted multiple times in the same height and updraft area bin before they were entrained into updrafts that were wide enough to deepen. For example, in Fig. 11, if a parcel remained between 1500 and 1600 m for 10 min, it would have counted as 10 parcel samples. Conceivably, a parcel could have dwelled within a shallow layer for several minutes within one or more narrow updrafts with weak vertical velocity before eventually rising as part of a wider updraft. Figure 12a supports such an idea. Grower parcels were split into three categories based on how long it took them to reach 6000 m after first being located in a cloud below 1000 m. Fast (red in Fig. 12), medium-fast (black), and slow (blue) growers required, respectively, <30, 30–60, and 60–90 min to move from below 1000 m to above 6000 m. In Fig. 12, large transparent dots denote mean parcel heights and the mean area of the updraft in which they resided in the 30 min prior to reaching 6000 m. Small dots denote the same for medium-fast and slow growers 30–60 min prior to reaching 6000 m. The thick blue dots denote slow growers’ characteristics 60–90 min prior to reaching 6000 m. Each dot represents one minute. The fast growers started out with a mean updraft area of about 0.4 km2 around 700 m, consistent with the median grower updraft area illustrated in Fig. 7a. After remaining below the LFC for about 8 min, the parcels rose steadily to higher levels as the mean area of the updrafts containing them increased. Over about 20 min, the average updraft area in a fast grower nearly tripled from just over 0.4 km2 to nearly 1.2 km2.

Fig. 12.
Fig. 12.

Scatterplots of the mean area of an updraft containing a parcel (abscissa) vs mean parcel height (ordinate). (a) The points are color-coded by how quickly (red: 0–30 min; black: 30–60 min; blue: 60–90 min) a grower parcel ascended from below 1000 to 6000 m. (b) The points are color-coded by the time taken for a nongrower parcel to ascend from below 1000 m to its maximum altitude attained. For both panels, large, transparent dots denote means for the 30 min prior to reaching the indicated height. Small dots indicate means at 30–60 min prior to reaching the height. Large, nontransparent blue dots denote means at 60–90 min prior.

Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0065.1

Interestingly, the medium-fast and slow growers experienced similar rates of deepening over the 30 min immediately prior to the parcels reaching 6000 m, but they took longer to start ascending. Parcels that were in cloud for longer periods of time also started their more rapid ascent from higher up. For example, slow growers slowly rose to around 2000 m for up to 1 h prior to finally ascending to 6000 m more quickly. The grower parcels that were slowest to ascend remained in cloud at average heights of about 1500 m and within updrafts with average area 0.15–0.35 km2 at 60–90 min prior to finally reaching 6000 m. Only about 35 min prior to reaching 6000 m did the median area of updrafts containing the slow grower parcels actually start steadily increasing. At 30–60 min prior to reaching 6000 m, medium-fast grower parcels remained below 1500 m on average and the average area of the updrafts containing these parcels increased to 0.4 km2. Regardless of how long it took for a parcel in any of the categories to ascend from the boundary layer to 6000 m, only after the mean area of updrafts containing the parcels reached 0.4–0.6 km2 did a more rapid pace of lateral and upward growth initiate.

In contrast, the average area of updrafts containing nongrowers never exceeded 0.25 km2. In Fig. 12b, the color and size of dots has the same convention as Fig. 12a, but the time denotes the duration prior to the parcel reaching its maximum height, which was between 1500 and 3000 m. In the 30 min prior to reaching maximum height, the longest-lived nongrower median updraft areas only reached 0.25 km2 before the updrafts contracted (blue dots). Shorter-lived nongrowers were located in even smaller updrafts above 1000 m (red dots). Combined with the result from Table 2 that there is no obvious special characteristic that makes narrow updrafts capable of growing deeper (aside from first growing wider), Fig. 12 supports the hypothesis that updrafts need to reach a certain width before they are resistant enough to dilution to grow into deep convection. In this particular simulation, that appeared to be somewhere between 0.4 and 0.6 km2. From a parcel’s perspective, it could ascend because the updraft in which it is located widens. If so, the results herein do not indicate anything about any variability in cloud/updraft widening as a function of height. In other words, did only the tops of clouds/updrafts widen as they ascended, or did they uniformly widen throughout their vertical extents? A parcel may have also ascended after getting detrained out of one updraft/cloud and into a different, wider one.

Taking the above into consideration, perhaps one of the biggest caveats of this study is that the grower parcels were not separately analyzed during only the period that they were actually ascending to 6000 m. In other words, before grower parcels ascended in a deepening cloud, many growers resided in clouds and updrafts that never even made it above 3000 m. Therefore, many of the vertical profiles for growers in section 4 contain some information from clouds and updrafts that would be considered nongrowers had we successfully tracked clouds and updrafts individually. Perhaps some of the statistical overlap between growers and nongrowers in vertical velocity (Fig. 3a), effective buoyancy (Fig. 3b), parcel temperature (Fig. 5a), tracer value (Fig. 6), and parcel distance from cloud edge (Fig. 7) would not have occurred if we compared nongrowers to growers only in the minutes immediately prior to ascending to 6000 m. To alleviate this concern, w and Beff were compared below the LFC between nongrowers and only the fast growers. Since the fast growers ascended quickly from near cloud base to 6000 m, they are unlikely to have been embedded within updrafts that did not deepen during the time they were analyzed. Figure S1 in the online supplemental material (SM) shows profiles similar to those seen in Figs. 3a and 3b and shows overlap in the median w and Beff between nongrowers and fast growers below the LFC.

While not central to the conclusions herein, section S2 in the SM discusses the possibility that the simulation does not develop boundary layer eddies that are much warmer (i.e., 1 K) than the environment. In summary, Fig. S2 demonstrates how the simulated boundary layer is much more homogeneous than a random and short sample of tropical marine boundary layer temperature observations. The SM argues that the current work (and possibly previous others) cannot yet conclude whether thermodynamic variability in updrafts emerging from the boundary layer plays any role on whether they ultimately deepen. Schemann et al. (2020) lends credence to the idea that the simulation in this article contains a boundary layer that is too thermodynamically homogeneous. They showed that LES initialized with realistic boundary layer heterogeneity outperformed LES with periodic boundary conditions and homogeneous initial conditions at representing cumuliform cloud populations over land. In our simulation, it is feasible—if not likely—that using uniform sea surface temperatures and a horizontally periodic domain contributed to underrepresenting heterogeneity in boundary layer thermodynamic properties relative to what occurs in the real atmosphere.

Also worth considering is that turbulent motions—which are effectively random given our current limited ability to observe or model them—could also influence natural updraft dilution in an unpredictable manner, particularly in the atmospheric boundary layer. For example, a wide updraft might happen to be diluted by apparently random turbulent motions more than surrounding updrafts. This could cause a wide updraft—or at least part of a wide updraft—to not deepen as much as neighboring narrower updrafts that do deepen if they happen to not be strongly diluted. However, the simulation analyzed herein, with its 100 m grid spacing, does not resolve such motions effectively enough to ascertain the role of seemingly random dilution by turbulent motions on the ultimate fate of updrafts emerging from the boundary layer.

6. Conclusions

A population of thousands of shallow cumuli was produced in a simulated low-shear tropical marine environment with column integrated relative humidity of ∼80% using the Cloud Model 1 (CM1). Passively advected parcel trajectories were tracked to provide a Lagrangian perspective on how cloudy updrafts ascending from the atmospheric boundary layer evolved in a variety of clouds. Parcels that entered clouds were placed into one of two categories if they reached 6000 m in cloud (growers) or if they entered cloud but topped out between 1500 and 3000 m (nongrowers). Respectively, the categories served as proxies for deepening convection that penetrated the 0°C level and nondeepening convection that topped out at some low height shortly after passing the level of free convection (LFC). Then, the characteristics of the updrafts and clouds containing growers and nongrowers, as well as the properties of the environment surrounding these parcels, were investigated. This study aimed to determine how the shallow cumuli that grew into deep cumulonimbi differed from the ones that did not grow or how the noncloudy environments of updrafts containing growers were different.

Consistent with prior studies of single convective elements ascending into tropospheres with various moisture and wind profiles (e.g., Morrison et al. 2022; Peters et al. 2022), the likelihood of parcels ascending above the 0°C level increased as the size of the clouds and updrafts they were located in increased (Figs. 79). This is also consistent with prior simulations of both shallow and deep convection in other environments (e.g., Kuang and Bretherton 2006; Dawe and Austin 2012; Rousseau-Rizzi et al. 2017). Furthermore, the clouds containing growers tended to develop from larger subcloud eddies than clouds containing nongrowers. Updrafts containing growers had median areas of 0.4–0.6 km2 below the LFC, while updrafts containing nongrowers had median areas of about 0.2 km2 (Fig. 10d). The larger updraft and cloud area apparently prevented some dilution of updrafts; a positive (although only 0.3–0.4 above the LFC) correlation between updraft area and effective buoyancy was found, and updrafts containing growers had significantly higher effective buoyancy above 1500 m than nongrower updrafts. A brief analysis of passive tracer concentrations indicated that updrafts containing nongrower parcels were diluted more than those containing grower parcels, which is consistent with the size differences found since it is expected that the cores of wider updrafts will be less diluted with surrounding noncloudy air than narrow updrafts. Echoing the hypotheses listed in section 1, the following overarching conclusions were reached regarding this simulation:

  1. Between cloud base and the LFC, the distributions of in-cloud vertical velocities (w) and effective buoyancy (Beff) were similar.

  2. Vertical velocities in the subcloud layer were also not different for updrafts containing growers and nongrowers.

  3. Above the LFC, updrafts containing deeply ascending parcels (growers) had larger effective buoyancy, presumably because of the size differences stated above and the ramifications of these size differences on updraft buoyancy dilution. Dynamic pressure gradient accelerations were not significantly different between updrafts containing growers and nongrowers at any height below 3000 m.

This study’s findings are more nuanced than just the points outlined above, however. For example, the full distributions of updraft areas containing grower parcels and nongrower parcels overlapped significantly (Fig. 11), suggesting—perhaps incorrectly—that there were deepening narrow updrafts. Closer evaluation of the grower parcels contained within narrow updrafts revealed that those parcels remained in cloud at low heights (typically below 2000 m) for several minutes, and in some cases even an hour, before finally beginning to ascend to more rapidly and beyond the 0°C level (Fig. 12). That means that grower parcels were often resident inside narrow updrafts before detraining from the narrow updraft and later becoming entrained into a larger updraft. This left open the possibility that the main reason that w and Beff profiles near cloud base were the same for growers and nongrowers (Fig. 3) was that the grower profiles contained significant amounts of data from when grower parcels were resident in clouds and updrafts containing nongrowers. However, the median w and Beff profiles of only the fast growers, which moved quickly from cloud base to 6000 m (section 5), suggested that these parcels also experienced similar vertical velocities and accelerations as nongrowers (Fig. S1 in supplemental material), which lends more confidence to this article’s central conclusions. Furthermore, below the LFC, the temperature of in-cloud parcels relative to the noncloudy environment was the same regardless of whether the parcel was a grower or nongrower (Fig. 5a). The noncloudy environment appeared to be moister below the LFC surrounding clouds containing grower parcels than those containing nongrower parcels (Fig. 5b), which could indicate that grower parcels at these low levels ascended in an environment that would cause less dilution of buoyancy in updrafts via entrainment. Together, this evidence is supportive of the “nature versus nurture” argument made by Romps and Kuang (2010b), who found that entrainment into cloudy updrafts, and not cloud-base properties, was the most important factor for determining whether a particular cloud would deepen. This appears to be the case for this study as well—at least as far as thermodynamic and kinematic properties of the cloud-base updraft are concerned.

However, there is another important piece to the puzzle to consider: The size of the cloud at cloud base is one of its properties, and the parcels that ascended to 6000 m in this study were located in wider updrafts as low as the top couple hundred meters of the subcloud layer. In other words, to relate to Romps and Kuang (2010b), the size of the cloud at its base (nature) impacted how entrainment and dilution (nurture) ultimately affected it because the initially wide cloud at the LCL was presumably also a wide cloud in the free troposphere above the LFC. To state this idea yet another way, nurture of a cloud updraft is impacted by its nature. Grower parcels only rapidly ascended to 6000 m in this study when the updrafts (defined herein as w > 0.5 m s−1) containing them were at least a certain area, which was about 0.4–0.6 km2. This area corresponds to an updraft width (not radius) of roughly 700–850 m if we assume a circular updraft. Before grower parcels were in an updraft with this area, they would remain below 2000 m. Once the median updraft area exceeded the critical area at cloud base and growers ascended, the growers were resident within updrafts with a median area that ranged from 0.6 to 1.3 km2 at 3000 m, corresponding to a width as large as about 1300 m (Fig. 7). The 700–850 m wide updraft near cloud base is comparable but likely a little smaller than the smallest updraft tested by Morrison et al. (2022), which had a 2 m s−1 contour that was about 700–800 m wide (and presumably a wider spread 0.5 m s−1 contour) in the subcloud layer and expanded as it ascended in an environment with 90% relative humidity (see their Fig. 6). This updraft width also seemed to barely even produce a deep cloud for Morrison et al. (2022), whereas the simulation herein can produce deep convection in a slightly drier environment with relative humidity of about 80% below the 0°C level (see Fig. 3 of Powell 2022). The goal of Morrison et al. (2022) was not to find a critical updraft radius above which convection deepened, but rather to document a dependency of critical updraft radius on free-tropospheric humidity. However, that the simulation in this article can generate deep convection with narrower updrafts in a similarly humid—perhaps slightly drier—environment than in the simulation cited above is an indication that factors other than just tropospheric humidity may impact the threshold cloud width for deepening. For example, although Morrison et al. (2022) used only one thermodynamic sounding for their simulations, they showed theoretically that midtropospheric lapse rate also impacts the combined environmental humidity or cloud updraft width required for deep convection. Specifically, reducing the midtropospheric lapse rate required increased updraft size and/or increased midtropospheric humidity.

As further evidence of how various environmental factors may impact the critical updraft area, the 0.4–0.6 km2 area apparently required for deep convection in this simulation is substantially larger than the updraft area at the LFC that Rousseau-Rizzi et al. (2017) required for convection to reach 6 km. They found that even in a drier sounding than that used herein, convection readily reached 6 km with LFC updraft areas of only 0.25 km2. However, they forced their convection using a line of convergence, and the mean updraft velocity near the ∼1500 m cloud base was ∼4 m s−1. It is possible that this much larger vertical velocity permitted narrower updrafts to grow much deeper than what is seen in this article. Studies such as Rousseau-Rizzi et al. (2017) and Morrison et al. (2022) are contrasted against here primarily to raise the point that differences in critical updraft areas between this article and other studies collectively represent how convection might behave in a subset of possible atmospheric states. Studies such as these reinforce the point that while a critical updraft radius required for deep convection probably exists, it varies as a function of the environment local to the cloud. Therefore, in the real atmosphere, the critical updraft radius probably evolves as the large-scale environment changes as well, and cloud parameterizations in coarse numerical models of the atmosphere should probably reflect this.

Perhaps one of the biggest unresolved questions raised by this study then concerns what controls the subcloud eddy and cloud-base size in the first place. For example, how much does boundary layer heterogeneity impact what exactly the “critical” radius for an individual updraft is? For example, a warmer—or simply more strongly lifted—boundary layer eddy could cause a stronger updraft to develop at cloud base, which could potentially grow into deep convection even if as narrow as the updrafts containing nongrowers in this article. This is essentially the opposite of what was just stated above about nature and nurture though. Kuang and Bretherton (2006) found no differences in the temperature of deepening and nondeepening updrafts below the LFC, and the current study also concludes that the median difference between updraft and environmental temperature in the subcloud layer was almost exactly zero (Fig. 10), which seems to discount the importance of boundary layer thermodynamic heterogeneity in updraft growth. However, the range of boundary layer temperature in this simulation was much lower than what is typically observed over tropical oceans. The boundary layer was likely unrealistically homogeneous—even after cold pools developed (Fig. S2). It is unclear whether the subcloud layers in previous studies contained realistic heterogeneity. Therefore, even though the distributions of vertical velocities of growers and nongrowers in this simulation were virtually identical at cloud base, it is impossible using the current experiment to say whether this result would hold if there were some simulated boundary layer eddies that were significantly warmer (i.e., 1 K) compared to their immediate environment than other relatively cooler updrafts. Thus, this author argues that properly representing boundary layer variability in 3D kinematics and thermodynamic properties in LES is a major issue that should be urgently considered moving forward before we can confidently stand behind any conclusions stating that cloud-base thermodynamic properties or vertical momentum components are truly similar across both deepening and nondeepening updrafts.

Moreover, there are potentially other ways that heterogeneity in the boundary layer could support convection and influence the area required for updrafts to grow deep. For example, Lee et al. (2019) model how land surface heterogeneity can cause mesoscale circulations to develop in light wind scenarios, supporting development of convection over sufficiently large regions with lower than average surface fluxes. Tian et al. (2022) illustrate how the diurnal cycle of where shallow convection preferentially occurs near Tulsa, Oklahoma, is impacted by horizontal variability in land–atmosphere surface fluxes that favors—also under light wind conditions—convective growth in locations where the heterogeneity in surface fluxes induces mesoscale convergence. Over the ocean, it is known that gradients in sea surface temperature are related to low-level convergence on spatial scales down to at least tens of kilometers (e.g., Back and Bretherton 2009; Li and Carbone 2012), so it is tantalizing to hypothesize that the same process could occur on small scales, inducing horizontal variability in the widths of updrafts and favoring their growth in locations where convergence is larger. In this study, there is evidence that at least a relationship exists between convergence and updraft area, but the analysis herein cannot attribute causality. For example, convergence in the area surrounding a grower parcel was larger than that surrounding a nongrower in the subcloud layer above 300 m (Fig. 10e). Furthermore, a positive correlation—albeit a weak one—was found between updraft area and convergence and was roughly 0.2–0.4 between 300 and 1600 m (Table 2). However, it is unclear what caused these eddies to be larger. Perhaps quasi-random boundary layer processes in isolation controlled eddy size at cloud base. Alternatively, an existing cloud in a favorable moist region could have produced stronger subcloud inflow that supported its own widening and deepening. These hypotheses are not addressed by this article and are speculative. Regardless of the exact mechanisms that prefer certain updrafts over others in nature, it is obvious that extensive observations of boundary layer kinematic and thermodynamic structure, coupled with simultaneous measurements of surface properties and the cloud populations above the LCL, will be required to untangle some of the questions raised by numerical simulations. High-resolution model simulations will still be needed to fill in the spatiotemporal gaps left by limited observations, however. LES embedded within a larger mesoscale domain in an effort to make the LES represent the proper heterogeneity in boundary layer structure will be needed. However, the trust of such simulations can only be established if the appropriate observations are available to verify what the model produces. Despite the current challenges of collecting observations at the fine spatial and temporal resolutions required to study clouds and the subcloud turbulent eddies from which they often originate, it is imperative that we use high-resolution numerical simulations as merely a tool to guide future observational strategies and development of operational prediction systems and not simply accept their results as truth.

1

The critical updraft radius is probably related to the idea of atmospheric criticalities associated with precipitation as articulated by Peters and Neelin (2006). Large-scale tropical dynamicists have long thought of this as involving a critical environmental humidity or Archimedean buoyancy (e.g., Bretherton et al. 2004; Ahmed et al. 2020), which are both impacted by cloud width through its impact on entrainment and dilution.

Acknowledgments.

This work was supported partially by the U.S. Department of Energy Atmospheric System Research, an Office of Science Biological and Environmental Research program, under Interagency Agreement 89243021SSC000077; and by the Office of Naval Research Awards N0001421WX01472, N0001422WX1021, and N0001423WX00360. The simulation and computations were executed on the hamming supercomputer located at the Naval Postgraduate School. The author thanks Walter Hannah, Hugh Morrison, and Adam Varble for constructive feedback that helped improved the content in this article and John Peters for reminding the author that CM1 V20.1 contains a nasty bug when passive tracers are activated that causes state variables to be incrementally and unintentionally changed as the model integrates.

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.

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Supplementary Materials

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  • Ahmed, F., and J. D. Neelin, 2021: Protected convection as a metric of dry air influence on precipitation. J. Climate, 34, 38213838, https://doi.org/10.1175/JCLI-D-20-0384.1.

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  • Fig. 1.

    A zoomed-in view of the sounding from Powell (2022) used to initialize the model, including temperature (red) and dewpoint (green). The dashed black lines indicate the lifting condensation level (LCL) and level of free convection (LFC). Red, blue, and green dashed lines respectively represent dry adiabats, moist adiabats, and lines of constant mixing ratio. Mixing ratio lines are labeled by the green numbers near the top of the chart.

  • Fig. 2.

    Number of parcel samples per 100-m-wide height bin for grower (red; top abscissa) and nongrower (blue; bottom abscissa) parcels.

  • Fig. 3.

    Composite vertical profiles up to 3000 m of median (a) vertical velocity, (b) effective buoyancy, (c) dynamic pressure gradient acceleration, and (d) hydrometeor loading in clouds containing growers (red) and nongrowers (blue). The shaded region denotes the 95% confidence interval of the median.

  • Fig. 4.

    Histogram of normalized frequency of in-cloud parcel vertical velocity at 700 m height for growers (red) and nongrowers (blue).

  • Fig. 5.

    As in Fig. 3, but for (a) the parcel temperature relative to the background temperature and (b) the parcel specific humidity relative to its background value. Calculation of background values is described in the text.

  • Fig. 6.

    As in Fig. 3, but for the value of the passive tracer expressed as a fraction of its value during model initialization at grid cells below 1500 m.

  • Fig. 7.

    As in Fig. 3, but for (a) the area of an updraft containing a parcel, (b) the area of a cloud containing a parcel, and (c) the distance of a parcel from the nearest cloud edge.

  • Fig. 8.

    Composited horizontal slices of (a)–(e) mean vertical velocity and (f)–(j) mean Dw/Dt surrounding grower parcels within 1200 m of a parcel and at model height levels of 1000–1100, 1500–1600, 2000–2100, 2500–2600, and 3000–3100 m. The parcel location is set at the origin of each slice. The black outline in each slice denotes the 10−4 kg kg−1 contour of cloud water (qc).

  • Fig. 9.

    As in Fig. 8, but for nongrowers. No slice is shown at 3000–3100 m because, by definition, nongrowers did not reach this height.

  • Fig. 10.

    As in Fig. 3, but below 600 m for parcels located in the subcloud layer before they entered clouds. (a) Vertical velocity, (b) effective buoyancy, (c) dynamic pressure gradient acceleration, (d) area of updraft containing parcel, (e) background divergence as described in the text, (f) parcel temperature relative to the background temperature, and (g) parcel specific humidity relative to the background specific humidity are shown.

  • Fig. 11.

    Normalized distributions of areas of updrafts containing grower (red) and nongrower (blue) parcels at (a) 300–400 and (b) 1500–1600 m height. Each parcel sample was considered an independent sample when constructing the histograms.

  • Fig. 12.

    Scatterplots of the mean area of an updraft containing a parcel (abscissa) vs mean parcel height (ordinate). (a) The points are color-coded by how quickly (red: 0–30 min; black: 30–60 min; blue: 60–90 min) a grower parcel ascended from below 1000 to 6000 m. (b) The points are color-coded by the time taken for a nongrower parcel to ascend from below 1000 m to its maximum altitude attained. For both panels, large, transparent dots denote means for the 30 min prior to reaching the indicated height. Small dots indicate means at 30–60 min prior to reaching the height. Large, nontransparent blue dots denote means at 60–90 min prior.