The Influence of Successive Thermals on Entrainment and Dilution in a Simulated Cumulus Congestus

Daniel H. Moser Department of Atmospheric Sciences, University of Illinois at Urbana–Champaign, Urbana, Illinois

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Sonia Lasher-Trapp Department of Atmospheric Sciences, University of Illinois at Urbana–Champaign, Urbana, Illinois

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Abstract

Cumulus clouds are frequently observed as comprising multiple successive thermals, yet numerical simulations of entrainment have not investigated this level of detail. Here, an idealized simulated cumulus congestus consisting of three successive thermals is used to analyze and understand their role in maintaining the high liquid water content in the core of the cloud, which past 1D modeling studies have suggested can ultimately determine its ability to precipitate. Entrainment and detrainment are calculated directly at the edge of the cloud core at frequent time intervals. Entrainment maxima occur at the rear of the toroidal circulation associated with each thermal and thus are transient features in the lifetime of multithermal clouds. The evolution of the least diluted parcels within each thermal shows that the entrainment rates alone cannot predict the erosion of the high liquid water content cores. A novel analysis of samples of entrained and detrained air within each successive thermal illustrates tendencies for even positively buoyant air, containing condensate, to be entrained by later thermals that rise in the wakes of their predecessors, limiting their dilution. The later thermals can achieve greater depths and produce precipitation when a single thermal could not. Future work is yet needed to evaluate the generality of these results using multiple clouds simulated in different environments with less-idealized modeling frameworks. Implications for current cumulus parameterizations are briefly discussed.

© 2017 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 e-mail: Daniel H. Moser, dmoser2@illinois.edu

Abstract

Cumulus clouds are frequently observed as comprising multiple successive thermals, yet numerical simulations of entrainment have not investigated this level of detail. Here, an idealized simulated cumulus congestus consisting of three successive thermals is used to analyze and understand their role in maintaining the high liquid water content in the core of the cloud, which past 1D modeling studies have suggested can ultimately determine its ability to precipitate. Entrainment and detrainment are calculated directly at the edge of the cloud core at frequent time intervals. Entrainment maxima occur at the rear of the toroidal circulation associated with each thermal and thus are transient features in the lifetime of multithermal clouds. The evolution of the least diluted parcels within each thermal shows that the entrainment rates alone cannot predict the erosion of the high liquid water content cores. A novel analysis of samples of entrained and detrained air within each successive thermal illustrates tendencies for even positively buoyant air, containing condensate, to be entrained by later thermals that rise in the wakes of their predecessors, limiting their dilution. The later thermals can achieve greater depths and produce precipitation when a single thermal could not. Future work is yet needed to evaluate the generality of these results using multiple clouds simulated in different environments with less-idealized modeling frameworks. Implications for current cumulus parameterizations are briefly discussed.

© 2017 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 e-mail: Daniel H. Moser, dmoser2@illinois.edu

1. Introduction

Since the early thermal studies of the 1950s, it has been suggested that cumulus clouds may result from a series of rising thermals (or pulses) after separating from a heated boundary layer (Scorer and Ludlam 1953; Malkus and Scorer 1955; Harrington 1958; Scorer 1957; Woodward 1959). The transient, pulsating nature of clouds by successive thermals has been documented in modern aircraft observations (Blyth and Latham 1993; Damiani et al. 2006) and numerical simulations (Carpenter et al. 1998; Zhao and Austin 2005; Heus et al. 2009). The potential for earlier thermals to moisten their local environment has also long been recognized (Scorer and Ludlam 1953; Saunders 1961). Successive thermals can rejuvenate the cloud updraft, causing a long-term increase in the maximum updraft speed, cloud water content, and cloud-top height and may aid the transition from shallow to deep convection (Kirshbaum 2011). Cloud-top height and radar reflectivity have been observed to increase when thermals ascend through the remnants of their predecessors (French et al. 1999). Successive thermals are prominent throughout the development stages of storms (e.g., Dye et al. 1983; Miller et al. 1983; Raymond and Blyth 1989) and can influence their spatial charge distributions (Mitzeva et al. 2003). The proliferation of ice and graupel has also been associated with successive thermals (Blyth and Latham 1993).

The evolution of an individual cloud is largely dependent on the growth and decay of individual thermals and the microphysical processes occurring within them that modified by entrainment. Entrainment of environmental air into a cumulus cloud via a primary thermal circulation is well established (e.g., Scorer 1957; Woodward 1959; MacPherson and Isaac 1977; Blyth et al. 1988; Stith 1992; Carpenter et al. 1998; Blyth et al. 2005; Damiani et al. 2006). Entrainment limits the cloud vertical extent and its longevity and thus, over a multitude of clouds, influences the vertical transport of heat, moisture, and momentum into the upper levels of the troposphere, influencing the climate, large-scale dynamics, and atmospheric radiation budgets. On smaller temporal and spatial scales, entrainment and mixing produces variability in supersaturation, drop size distributions, and thus liquid water content, affecting for example the efficiency of the warm-rain process (Cooper et al. 2013) or ice processes resulting from riming (Cooper and Lawson 1984). Its importance to cloud strength, longevity, and precipitation have promoted much study and debate regarding the underlying mechanism (e.g., Warner 1970; Simpson 1971; Paluch 1979; Raymond and Wilkening 1982; Blyth et al. 1988; Taylor and Baker 1991; Siebesma 1998; Blyth et al. 2005; Heus et al. 2008).

Past modeling studies have illustrated the potential importance of successive thermals in enhancing precipitation processes but have been limited by artificial, 1D modeling frameworks. Mason and Jonas (1974) selectively placed a second parcel into a 1D parcel model to represent a successive thermal and forced the second parcel to mix (in a prescribed manner) with the first thermal as it collapsed. The resulting droplet size distribution was argued to qualitatively resemble the broad, bimodal distributions measured in situ by Warner (1969). Roesner et al. (1990) extended this 1D modeling approach beyond two thermals, included collision–coalescence of drops, and concluded that a sequence of thermals was capable of producing raindrops when a solitary thermal was not. Blyth and Latham (1997) used a prescribed multithermal 1D model framework to conclude that the replenishment of liquid water by successive thermals had the greatest impact on ice production, and thus precipitation, in the observed cumuli. While the applicability of these 1D models was questioned in part for having artificial, unrealistic interaction between the successive thermals (Warner and Mason 1975; Hobbs and Rangno 1998), they emphasize the potential importance of successive thermals to precipitation.

No modeling study has examined entrainment within distinct successive thermals and carefully analyzed the dynamics and thermodynamics of the process to assess their role in maintaining high liquid water content cores in the clouds. Fields of cumulus clouds simulated with cloud-resolving models have often been used in studies of entrainment and detrainment (Siebesma and Cuijpers 1995; Heus et al. 2008; Dawe and Austin 2013) and can be useful for statistical analyses of the bulk cloud properties. However, such temporal and spatial averaging misses the transient nature of successive thermals within each cloud in order to better determine cloud feedbacks on the large-scale environment. Simulations of one or a few clouds (e.g., Carpenter et al. 1998; Blyth et al. 2005; Zhao and Austin 2005; Heus et al. 2009; Yeo and Romps 2013), while very detailed, have not analyzed successive thermals. The identification of different thermals within a single cloud can be a labor-intensive task that is not easily automated.

Recent advances in the direct calculation of entrainment (e.g., Romps 2010; Dawe and Austin 2011a, hereafter DA11) have opened new opportunities for its quantification in successive thermals. Historically, entrainment and detrainment were calculated from numerical simulations using bulk methods (Tiedtke 1989; Schumann and Moeng 1991; Siebesma and Cuijpers 1995; Siebesma 1996), where the exchange between cloud and environmental air is inferred by tracing the change in some conserved thermodynamic scalar variable (such as total water or liquid water potential temperature) within the cloud or its environment. Direct methods, on the other hand, calculate entrainment and detrainment directly from the resolved local cloud velocity fields and have found rates twice as strong as bulk rates (Romps 2010; DA11). This disparity highlights the importance of the local entrained air on influencing the dilution of the cloud. Much of the entrained air into the buoyant updraft cloud cores shared properties of the moist cloudy shell surrounding the cores, which ultimately bias bulk calculations toward weaker rates (Dawe and Austin 2011b). Yeo and Romps (2013) calculated that over half of all entrained parcels in their simulated deep convective cloud were associated with “recycled” parcels that were previously detrained from the cloud and had higher equivalent potential temperatures than the environment. As shown by these studies, the moist cloud shell certainly influences dilution of the cloud by entrainment, but how this aspect might differ for successive thermals requires investigation.

The potential importance of successive thermals in cumuli is well recognized, but basic knowledge that ties together entrainment, dilution, and successive thermals is noticeably lacking. Because of the slow and tedious nature of manually and carefully identifying individual thermals, only a single cloud simulation is examined here. Entrainment and detrainment are quantified using the direct calculation method of DA11 in order to provide the most detail over the small temporal and spatial scales represented in the 3D model. While the magnitudes of entrainment and dilution rates are of direct interest for many problems in atmospheric science, the emphasis of the current study is on comparing those rates calculated for a single thermal versus successive thermals in a relative sense. Future work with a large domain will address a comparison of similar simulations with observations of mixed-phase convective precipitation from the Convective Precipitation Experiment (COPE; Leon et al. 2016), held in southwestern England in summer 2013.

2. Methods

a. Model description

The Straka Atmospheric Model (Straka and Anderson 1993; Gilmore et al. 2004) is a 3D cloud-resolving model that has been used to reliably study entrainment in nonprecipitating cumulus at scales (≤50 m) within the inertial subrange (Carpenter et al. 1998; Blyth et al. 2005; Cooper et al. 2013). Here, a gridpoint spacing of 50 m was also used in all directions over the entire domain (8.0 km in the horizontal and 7.5 km in the vertical) with a 0.75-s time step. The model solves the time-variant, nonhydrostatic quasi-compressible form of the Navier–Stokes equations as described by Anderson et al. (1985), using supercompressibility and time splitting to treat acoustic waves. Prognostic equations are explicitly solved for the three velocity components, u, υ, and w; perturbation pressure p′; potential temperature θ; and subgrid-scale turbulent kinetic energy e. A single-moment, bulk microphysics parameterization scheme [based on Lin et al. (1983)] predicts six mixing ratios: water vapor, cloud water, rain, cloud ice, aggregates, graupel, and hail; it is described in detail by Gilmore et al. (2004). Sedimentation of all hydrometeors was turned off unless otherwise noted in order to track the influence of entrainment alone upon depletion of the hydrometeors inside the cloud.

b. Simulation setup

The computational domain was initialized from a sounding taken over southwestern England (Fig. 1) that supported shallow convection (up to ~4.5 km) that eventually precipitated heavily. The directional wind shear was weak up to this height in the observed sounding and was made unidirectional to simplify the analysis. The wind shear below cloud base was also eliminated (i.e., winds had constant direction and magnitude) to promote wide thermals.

Fig. 1.
Fig. 1.

Model “environment” conditions modified from an atmospheric sounding sampled near a line of cumulus congestus over southwestern England.

Citation: Journal of the Atmospheric Sciences 74, 2; 10.1175/JAS-D-16-0144.1

The cloud forcing method is identical to that introduced by Klaassen and Clark (1985) and used by Carpenter et al. (1998). Sensible heating is applied to the lowest levels of the domain, starting from the surface value of 50 W m−2 and decreasing by an e-folding length α of 300 m so that heating is approximately zero at cloud base. A Gaussian heating function was also applied at the center of the domain, where the peak heating (450 W m−2) occurred, decreasing with distance, at a standard deviation σG of 2000 m. Approximately 50% of all heating occurs within a 1000-m radius from the heating center, initially producing a cloud of updraft σG. This main Gaussian heating function linearly increased for 15 min, after which the cloud was formed, was then sustained at its maximum strength for 5 min, and then reduced to zero within 1 min (total forcing time of 21 min).

Turbulence is a prerequisite for simulating realistic cumulus congestus. Consistent with the method of Carpenter et al. (1998), turbulence was initiated in the environment by forcing four “priming” thermals (produced by Gaussian heating functions with σG = 1000 m and peak values of 350 W m−2) throughout the domain, before the central Gaussian heating function was initiated that produced the main cloud for analysis. One hour of simulation time proved adequate to saturate all scales of motion with turbulence [i.e., the horizontally averaged value of ∂TKE(z)/∂t ≈ 0], after which time the main Gaussian heating function was activated as described above.

c. Quantification of dilution

In this study, dilution refers to the reduction of the total water mass (water vapor, cloud water, and ice combined, which would otherwise be conserved in an adiabatic parcel lacking precipitation) resulting from the incorporation of entrained air into the cloud. Here, the change in total water mixing ratio qT (i.e., the sum of the mixing ratios of water vapor, cloud water, cloud ice, snow, and graupel) with height of the least diluted parcel (LDP) contained within each thermal is tracked, which is essentially a dilution rate expressed per unit distance instead of time. These parcels are responsible for the maximum cloud-top height, are regions of precipitation initiation/production, and contain the maximum buoyancy and vertical velocity.

The LDP was defined as the (100 m)3 volume having the highest mean qT following Zhao and Austin (2005) and was tracked manually through the simulation. The locations of thermals were determined by regions of strong vorticity (associated with the toroidal circulation), large buoyancy perturbations, maxima in the vertical buoyancy fluxes, and relatively high adiabatic LWC fractions plotted at 30-s intervals. The cap and rear of each thermal were recorded, the latter recognizable from the strong inflow of the toroidal circulation, and within this depth the maximum mean qT over any (100 m)3 volume was plotted as a function of height. The location of the LDP within a thermal could be then identified as a distinct maximum in the qT curve, always occurring above a local minimum where the entrained air at the rear of the circulation had depleted qT. In the early stages of each thermal, there was frequently a region of undiluted (i.e., adiabatic) air from cloud base extending up into each thermal. The location of the LDP in these cases was selected to be the highest extent of this undiluted region. Typically, the LDP was located within 200 m of the thermal cap.

d. Quantification of entrainment and detrainment

Entrainment here is calculated from the inflow of air into the cloud core. The rates of mass exchange with the air outside the core, and the properties of the entrained and detrained air, are sensitive to the definition of the cloud core. The cloud core was defined as the surface containing vertical velocity exceeding 1 m s−1 and the sum of cloud water and ice mixing ratios exceeding 1 g kg−1, and thus the cloud core can be located within thermals and in the space between successive thermals. These criteria are more restrictive than those used, for example, by Romps (2010), who restricted the cloud water only to 10−2 g kg−1, in order to quantify entrainment into the high-LWC regions deeper inside the cloud here. Less restrictive definitions may increase entrainment and detrainment rates.

The subgrid tetrahedronal interpolation method of DA11 was implemented to identify the cloud-core surface for computing entrainment and detrainment during the simulation. Following their method, entraining and detraining fluxes through the core surface were computed for an Arakawa C grid using
e1
where ρ is the moist air density (kg m−3); V is the (often partial) volume of the grid cell containing the cloud core; u, υ, and w are the velocity components of the flow (m s−1); and W is the surface area of the cloud core (m2) contained in the grid cell. The algorithm divides each grid cell into 48 tetrahedrons, some of which are completely occupied by the core and some only partially or not at all. For tetrahedrons only partially containing the core, the point of intersection of the core surface along the legs of the tetrahedron is linearly interpolated between vertices that satisfy the criteria of being within the core and those that are not. The volume of all tetrahedrons containing the cloud core is then summed to determine V, and the core surface area is found by summing all W values, which are the areas of the tetrahedron facets outlining the core volume. Following Romps (2010) and DA11, for each grid cell, net positive values of Eq. (1) are considered entrainment E (kg m−3 s−1) and net negative values are detrainment D (kg m−3 s−1).

Fractional entrainment ε and detrainment δ are often examined in large-eddy simulations of cumulus fields to scale the horizontal mass fluxes with the vertical mass fluxes of the updrafts. Fractional entrainment (detrainment) is defined as E/M (D/M), where M is the fractional vertical mass flux of the cloud core, equivalent to ρacw, where ρ is the air density of the cloud core, w is the core updraft speed, and ac is the fraction of the domain horizontal area that contains cloud core. Similar calculations are also shown here to relate to past studies, but ε and δ values near cloud top become artificially exaggerated because of the small sampling volumes and weak vertical mass fluxes and thus are less reliable there than the E and D vertical profiles.

3. Cloud development as a sequence of thermals

Vertical cross sections of the fraction of the adiabatic liquid water content (ALWC) through the center of the cloud (Fig. 2) illustrate the progression of its dilution by entrainment (sedimentation of all hydrometeors was turned off). After 5 min from the cloud inception (Fig. 2a), little entrainment had occurred, as shown by the majority of the cloud having in excess of 90% ALWC. Two large eddies evident by the wind velocity vectors and slightly lower ALWC fraction on the left side of the cloud have begun to introduce dry air inward. By this time, the surface heating had been applied for 20 min and was transitioning into a 1-min cool-down period, after which the surface heat flux was turned off.

Fig. 2.
Fig. 2.

Zoomed-in vertical cross sections of the fraction of TWC/ALWC (filled contours; scale in upper right) with in-plane velocity vectors, plotted at every other grid point, overlaid at times noted. Colored boxes indicate position of identified thermals discussed in the text.

Citation: Journal of the Atmospheric Sciences 74, 2; 10.1175/JAS-D-16-0144.1

Five minutes later (Fig. 2b), cloud-top height had increased to 2.5 km. Strong entrainment and mixing at the sides of the cloud near cloud top, coinciding with a large toroidal circulation (mostly enclosed in the blue box and more dominant on the left side) depleted the total condensed water content (TWC) to 40% of the ALWC in some places. Other eddies are present along the edges of the cloud farther below (e.g., near 1.5 km and near cloud base) but do not mix dry air as far into the interior of the cloud. The rear inflow of the toroidal circulation near 2 km is introducing dry air deeply into the cloud core, substantially diluting it there compared to lower altitudes. This region is directly below a region of maximum vertical velocity and buoyancy (Fig. 3, blue curve). The location of the toroidal circulation near the ascending cloud top with respect to the locations of maximum w and the domain-averaged vertical kinematic flux of the perturbation virtual potential temperature , as well as the disappearance of undiluted parcels near the cloud top, agree with the schematic model of a single rising thermal [cf. Fig. 6.5 in Trapp (2013)].

Fig. 3.
Fig. 3.

The maximum updraft speed and over the computational domain at 10 (blue), 12.5 (green), and 17.5 min (red).

Citation: Journal of the Atmospheric Sciences 74, 2; 10.1175/JAS-D-16-0144.1

The cloud top continued to ascend, reaching 3.0-km height by 12.5 min (Fig. 2c), and the first thermal (blue box; here denoted thermal 1) began to tilt to the right as it encountered stronger environmental winds. The toroidal circulation continued to erode and dilute the thermal (Fig. 2d). A second ascending thermal was located behind thermal 1 (Fig. 2c, red box, here denoted thermal 2). Meanwhile the first thermal climbed to a maximum height of 3.4 km, where entrainment and mixing diminished most of its buoyancy and condensate. By 15 min (Fig. 2d), thermal 1 was no longer buoyant, and its remaining cloudy elements descended and detrained, settling at a height near 3.0 km, the location of a stable layer in the sounding.

Thermal 2 ascended just below thermal 1. Two maxima in the vertical profiles at 12.5 min are evident (Fig. 3, right; green lines) one associated with thermal 1 centered at a height of 2.6 km and the other with thermal 2, near 2.0 km. By 15 min (Fig. 2d), thermal 2 had diverted around a negatively buoyant “hole” and continued to ascend through some remnants of thermal 1 at 3.0 km. Above 3.2 km, thermal 2 encountered pristine environmental air (Fig. 2e), where the maximum TWC started to decline. The circulation on the right-hand side had widened, encompassing nearly the entire remnants of thermal 1, as the cloud ascended into a layer of stronger environmental winds. Mixing of the entrained air caused rapid cooling of the cloud top, eroding the remaining positive buoyancy in thermal 2, but it still ascended to 4.0 km (Fig. 2f) because of its positive vertical momentum. After reaching its apex, the cloud top began to collapse and descend. The nonbuoyant cloudy remnants of thermal 2, like those from thermal 1, descended and detrained into a shallow layer at a height of 3.0 km.

A third thermal (thermal 3) appeared at 1.8-km height, below thermal 2 (Fig. 2d), and is shown enclosed by a yellow box in Fig. 2e. Unlike thermal 2, thermal 3 was not ascending in close proximity behind its predecessor; there was a 1.2-km gap between the second and third thermals. The longer delay between successive thermals was due to the weakening residual surface heat perturbation at this time. The maximum at this time (Fig. 3, right; red line) associated with thermal 3 (located near 2.2 km) was weaker compared to the earlier thermals ascending through this altitude. Like the previous two thermals, thermal 3 also exhibited a toroidal circulation where the majority of its entrainment occurred initially. It only ascended to 3.2 km, however, where it encountered a broad region of negatively buoyant, descending cloud air associated with the collapsing cloud top after 20 min (not shown). Incapable of pushing through the collapsing thermal 2, thermal 3 stalled and fell apart. By this time, the surface heat perturbation had diminished to the point where no additional thermals were generated.

Figure 4 (solid lines) summarizes the time evolution of the cloud top, maximum updraft and downdraft speeds, and hydrometeor water contents. Cloud-base height was 0.8 km with a temperature of 9.6°C. In time, as the cloud ascended through a more unstable layer, the maximum updraft speed peaked at 13 m s−1 (Fig. 4b). By 18 min, the cloud reached a maximum height of 4.0 km (approximately −10°C) (Fig. 4c), nearly as high as the equilibrium level for an undiluted parcel (4.5 km). By 17 min (Fig. 4a), the maximum TWC reached 3.6 g m−3 and consisted of mainly cloud water (80%) and some graupel (20%) produced from freezing and subsequent riming of rain. The ALWC was 4.4 g m−3 at 4.0-km height.

Fig. 4.
Fig. 4.

Time series of the (a) maxima of TWC, LWC, and rain and graupel mass content, (b) maximum updraft and downdraft speeds, and (c) cloud-top height from the simulated cloud for simulations with hydrometeor sedimentation disabled (solid lines) and enabled (dashed lines).

Citation: Journal of the Atmospheric Sciences 74, 2; 10.1175/JAS-D-16-0144.1

In an extra simulation when hydrometeors are allowed to sediment (Fig. 4; dashed lines), the LWC becomes markedly less as falling rain and graupel scavenge more cloud water (Fig. 4a), and this illustrates the importance of successive thermals to the development of precipitation. Thermal 1 creates very little rain or graupel, while the second and third thermals help to produce an order of magnitude more. Between 10 and 15 min, some raindrops (dashed red line) initially formed in thermal 1, but rain production was boosted between 15 and 17 min by the reintroduction of high LWC into the cloud by thermal 2. Some of this rain was even transported by the rejuvenated updraft to colder regions, where it subsequently froze into graupel (dashed purple line between 17 and 19 min), and eventually fell back into warmer air and melted into rain. Raindrops were then again reintroduced into the high LWC of the ascending thermal 3, creating the most rain in the cloud lifetime at 20 min. This scenario demonstrates the efficacy of successive thermals to magnify precipitation production when a single thermal may be incapable of producing much at all. The key mechanism is the repeated replenishment of the liquid water in the updraft by successive thermals, which is in turn partly controlled by the entrainment into each. Here, the focus is on understanding the differences in entrainment resulting from these successive thermals that govern the production of the high-LWC core. To simplify the analysis, the simulation without sedimentation of hydrometeors (Fig. 4, solid lines) is used hereinafter to isolate the influence of entrainment apart from scavenging of cloud water by falling precipitation.

4. Direct calculation of entrainment and detrainment

To study the evolution of entrainment in the context of multiple successive thermals, it must be diagnosed over short time scales and tracked in time. Entrainment and detrainment into or out of the cloud core were evaluated at every 0.75-s time step following the method outlined in section 2d, averaged over 30-s periods, and horizontally averaged to produce mean vertical profiles (Fig. 5). For reference, the corresponding fractional values are also shown (Fig. 6). In all but Fig. 5f, where the updraft has died, the inflow of boundary layer air into the cloud base appears as strong “entrainment” at 1.1 km (where the 1 g kg−1 cloud-core definition begins); it is shown only for perspective. Here too ε and δ values near the cloud base in Fig. 6f, like those at cloud top, become artificially exaggerated because of the weakened vertical mass flux and thus are less reliable than the E and D vertical profiles there.

Fig. 5.
Fig. 5.

Directly calculated rates of entrainment E (blue lines) and detrainment D (dashed green lines) at the cloud-core surface at times noted.

Citation: Journal of the Atmospheric Sciences 74, 2; 10.1175/JAS-D-16-0144.1

Fig. 6.
Fig. 6.

As in Fig. 5, but for fractional rates of entrainment ε (blue line) and detrainment δ (dashed green line).

Citation: Journal of the Atmospheric Sciences 74, 2; 10.1175/JAS-D-16-0144.1

a. Thermal 1

At 5 min (Fig. 5a) and above the artificial maximum at cloud base, entrainment magnitudes E were typically below 0.1 × 10−4 kg m−3 s−1 (fractional rate ε less than 1.0 km−1 in Fig. 6a). This entrainment was ineffective in diluting the cloud core, as nearly adiabatic parcels were present over all levels within the cloud (Fig. 2a). Thermal 1’s toroidal circulation required some time/vertical distance to form, as it was not yet evident at 5 min (Fig. 2a), but by 10 min (Fig. 2b) it was clear in the flow velocities. This development was reflected in E and ε (Figs. 5b, 6b); local maxima increased to 0.5 × 10−4 kg m−3 s−1 and 1.6 km−1, respectively. The maximum E (Figs. 5b, 6b) occurred at the rear of the main thermal circulation near 2.0-km height.

b. Thermal 2

By 12.5 min, the toroidal circulation of thermal 2 gained strength and appeared as a second entrainment maximum between 2.0 and 2.5 km (Figs. 5c, 6c) with a value of 1.0 × 10−4 kg m−3 s−1 (ε = 2.1 km−1). It was also collocated with the rear inflow of its toroidal circulation. Thermal 1 continued to ascend with E and ε values half as large. By 15 min (Figs. 5d, 6d), thermal 2 entrained at a similarly high rate (maximum E = 1.0 × 10−4 kg m−3 s−1; maximum ε = 2.6 km−1) as the thermal ascended through and above the remnants of thermal 1 into drier air. Once its buoyancy began to decrease at 17 min, the maximum E fell below 0.5 × 10−4 kg m−3 s−1, and ε decreased to 1.1 km−1 (not shown).

c. Thermal 3

By 20 min, the toroidal circulation of thermal 3 (visible in Fig. 2e) has developed, and the maximum E (Fig. 5e) was 0.6 × 10−4 kg m−3 s−1 at 2.2 km, less than the strongest rates of thermal 2 but still producing a large ε of 2.6 km−1 (Fig. 6e). Since ε scales with the fractional vertical mass flux, thermal 3 entrained strongly for its weaker vertical mass flux, with ε rates reaching as high as those for thermal 2. The surface heating had been off for 14 min by this time, limiting the maximum buoyancy of thermal (Fig. 3, right, red line), which contributed to a weaker updraft. By 22.5 min (Fig. 5f), the inflow into the base of the cloud had nearly ceased, and E had waned over all levels of the cloud, now dominated by detrainment. Large ε values were sporadic through the cloud depth (Fig. 6f), but these were likely spurious, as the maximum fractional vertical mass flux had decreased to about 10% of its previous peak value, thus making the determination of the core volume questionable and tending to artificially exaggerate the fractional rates, as also found by DA11.

5. Dilution of initial and successive thermals

A thermal is composed of many parcels, which experience various levels of dilution from entrainment. The LDP within a thermal not only controls its maximum height and updraft speed, but also the development of precipitation, as discussed earlier in section 3. The LDP within each thermal was tracked with height, as described in section 2c, and their eventual dilution is shown in Fig. 7.1 All of the thermals contained undiluted parcels up to 2.4-km altitude, near the distance above cloud base where the toroidal circulation could be first identified visually. The LDP for thermal 2 was tracked separately for two different phases: thermal 2a denotes the time while it ascended in the wake and remnants of thermal 1, and thermal 2b denotes when it emerged from those remnants and was entraining pristine environmental air.

Fig. 7.
Fig. 7.

Change in total water mixing ratio qT and fraction of adiabatic water content (TWC/ALWC) with altitude for the least-diluted parcel within each thermal. Dilution rate (DqT/Dz) quantified as the inverse slope of regression lines as labeled for thermals 1, 2a, and 2b. Legend also lists the maximum directly calculated fractional entrainment rate εmax for each thermal.

Citation: Journal of the Atmospheric Sciences 74, 2; 10.1175/JAS-D-16-0144.1

Once above 2.4-km altitude, qT of the LDP of thermal 1 decreased up to its maximum height of 3.1 km at a rate of 0.8 g kg−1 km−1 (inverse slope of best-fit line in Fig. 7). The LDP in thermal 2a ascended nearly undiluted (dilution rate of only 0.2 g kg−1 km−1), as far as 3.2 km, until emerging from the remnants of thermal 1. Figure 8 is a horizontal cross section through the center of thermal 2a (containing a high fraction of ALWC but also identified through its positive buoyancy; not shown) as it ascended through the wake of thermal 1 at a height of 2.7 km, 14 min after the cloud formed. It is surrounded by cloudy air left in the wake of thermal 1, which is entrained by the main toroidal circulation. Once thermal 2 reached a height of 3.2 km (now called thermal 2b), it began to emerge from the detritus of thermal 1 and encountered pristine environmental air, diluting quickly at a rate of 1.8 g kg−1 km−1 despite the maximum entrainment rate decreasing at this time to 0.5 × 10−4 kg m−3 s−1 (ε = 1.1 km−1) from 1.0 × 10−4 kg m−3 s−1 (ε = 2.6 km−1). Although thermal 1 had also climbed through pristine air, its rate of dilution was slightly weaker than that of thermal 2b because the latter was entraining drier air above 3 km (Fig. 1). The dilution of thermal 3 was initially weaker at 0.6 g kg−1 km−1 but increased to 3.4 g kg−1 km−1 as it encountered the collapsing cloud top.

Fig. 8.
Fig. 8.

Horizontal cross section at 14 min at the altitude of the rear of the second thermal, showing the fraction of ALWC at a height of 2.7 km, with in-plane wind vectors overlaid. The core of the second thermal is contained mainly within the 0.6 shaded contour, surrounded by the cloudy remnants of the first thermal.

Citation: Journal of the Atmospheric Sciences 74, 2; 10.1175/JAS-D-16-0144.1

From this analysis, it is clear that the maximum entrainment rate is not the sole control on the rate of dilution of a thermal, as shown in some past studies, but here the analysis is expanded to investigate effects on successive thermals. A summary comparison of dilution rates and fractional entrainment rates ε among the thermals is shown in the second and fourth columns of Table 1, along with the corresponding mean radii of a sphere enclosing each thermal in the fifth column. While the greatest ε occurred for thermal 2a, its dilution rate was much smaller as it ascended through the remnants of thermal 1. Its width was the greatest of all the thermals, but the high value of ε calculated at the core surface clearly shows that the core was not protected from entrainment by this greater width. Later, when it had ascended into pristine environmental air (thermal 2b), its dilution rate was over twice that of thermal 1 (that had also interacted with unperturbed environmental air). One might assume its dilution was aided by its smaller width, and yet its ε was lower. Its faster dilution was due to the entrainment of much drier environmental air compared to thermal 1. In contrast, thermal 3 initially had a lower dilution rate, owing to its ascent in the wake of its predecessor, but later the very high rate of dilution (also listed in Table 1) corresponds to the time of the collapse of thermal 2 upon it, which with subsequent mixing quickly decreased the properties of its LDP.

Table 1.

Summary of the entrained-air analysis for each thermal.

Table 1.

6. Characteristics of the entrained air

The properties of the air entrained into, and detrained out of, the cloud core can be estimated for each thermal using a slightly modified version of Eq. (1) that characterizes the air within the mass fluxes into each grid cell at the isosurface defining the cloud core. For each grid cell, the mean properties (qT, TWC, and the virtual potential temperature θυ) of the entrained air were calculated as an average of the properties of each nearest gridcell neighbor upstream from the entraining grid cell, weighted by the entrained mass flux:
e2
where a is the scalar value at the center of the grid cell of interest, aup is the scalar value of the neighboring upstream grid cell (in x, y, or z directions) that was being entrained, ρ is the moist air density, and W is the surface area of the core on the faces of the grid cell (found using the same tetrahedronal interpolation method as used earlier). Equation (2) is applied only at grid cells containing portions of the core surface (still defined as having a vertical velocity exceeding 1 m s−1 and a condensed water mixing ratio exceeding 1 g kg−1), which changes location in time. To better understand the dilution of the LDP for each thermal, only grid cells at the vertical levels encompassing the LDP and the immediate five levels (250 m) below (to also capture air entrained at the rear of the toroidal circulation) were analyzed.
The calculations were run at each time step for 2 min during the middle of the time period when the LDP of each thermal was tracked; entrainment/detrainment resulted in O(104) “samples” of qT, TWC, and θυ. The samples were then placed into 100 bins to create a probability density function (PDF) for each variable, either entrained into the core or detrained from the core. A PDF of these same variables over all grid cells within the entire core (not just at the core edge) was also constructed over the same time period and altitudes. The calculated qT values were normalized (denoted as ) using
e3
where is the undiluted cloud-base value of qT (8.6 g kg−1), and is the environmental qT at the sample altitude. Thus, represents unperturbed environmental air at the sampling level, and a value of represents undiluted cloud-base air. The values of TWC were normalized by the ALWC at the sampling level, and the values of θυ were converted to perturbation values by subtracting the sample value from the environmental value at the sampling level, to track the total amount of condensate and buoyancy within the samples, respectively. A summary of some statistics comparing similar aspects of these distributions is given in Table 1. The shapes of the resulting distributions are frequently similar to a normal distribution, and thus the standard deviation σ is used to quantify their variability.

a. Thermal 1

The PDF of entrained air (red) for thermal 1 (Fig. 9, top left) is quite broad (σ = 0.18), with parcels exhibiting a variety of values. About half of all the entrained samples contained , and a quarter had values less than 0.10, indicating that many entrained samples were mostly composed of environmental air. Very few samples (17%) had values that overlapped (pink and purple bins) with air originating from the core (white bins) or air that was detrained from it (blue bins). The shapes of the TWC/ALWC distributions (Fig. 9, top center) were nearly identical to those for . Despite conducting this sampling well inside the cloud, at the core edge, purely cloud-free samples are common (~25%) and result from the strong entrainment at the rear of the thermal that introduced dry air into the center of the thermal, visible in Fig. 2c. Most of the entrained air (85%; Fig. 9, top right) was negatively buoyant , having cooled by evaporating some cloudy air before being introduced into the core, as also noted in the trajectory analysis of Yeo and Romps (2013).

Fig. 9.
Fig. 9.

Probability density functions of entrained, detrained, and core samples of (left) , (center) TWC/ALWC, and (right) for (top) thermal 1 between 11.5 and 13.5 min and (bottom) thermal 2a (while ascending in the wake of thermal 1) between 13.5 and 15.5 min.

Citation: Journal of the Atmospheric Sciences 74, 2; 10.1175/JAS-D-16-0144.1

b. Thermals 2a and 2b

The entrained samples of thermal 2a frequently had values (Fig. 9, bottom left) noticeably greater than for thermal 1; the probability of entraining low values of (<0.25) were 15% and 59% for thermals 2a and 1, respectively. The entrainment of pristine, environmental air was nearly nonexistent in thermal 2a. Rather, a strong mode was located at with a weaker mode located at . The former is closely aligned with the modes of the core and detrainment PDFs, indicating a significant amount of entrained samples had qT values identical to the core, which would not contribute toward its dilution. Fewer of the entrained samples contained no condensate (3%; Fig. 9, bottom center), as opposed to thermal 1 (25%), and far fewer (26%) held TWC/ALWC fractions less than 0.25. In strong contrast to thermal 1, less than half (40%) of the entrained parcels were negatively buoyant (as compared to 85% in thermal 1). The other mode at did not overlap with the core PDF and likely represents the aged, decaying cloudy remnants of thermal 1. Overall, it is clear that the entrainment of low-, low-TWC, and negatively buoyant mixtures was significantly reduced as thermal 2a climbed through the detritus of thermal 1, resulting in slower dilution despite its higher rate of entrainment, as summarized in Table 1.

The air entrained into thermal 2b (Fig. 10, top) that was ascending in the pristine environmental air contrasts sharply with that surrounding thermal 2a and is qualitatively similar to that entrained into thermal 1 (Fig. 9, top). The probability of entraining samples with more than doubled (Table 1). The variability of the entrained air increased from σ = 0.17 to 0.23. The air entrained into the core overlapped less with air detrained from the core (33%) compared to thermal 2a (49%), resulting in a larger fraction of the entrained air being effective in diluting the cloud core. Most (86%) of the air entrained into the core was again negatively buoyant (Fig. 10, top right) after strong evaporation of condensate, which was also evident from the number of entrained samples containing no condensate (22%; Fig. 10, top center).

Fig. 10.
Fig. 10.

As in Fig. 9, but for (top) thermal 2b (after emerging from the wake of thermal 1) between 16.5 and 18.5 min and (bottom) thermal 3 (while ascending in the wakes of the previous thermals) between 19.5 and 21.5 min.

Citation: Journal of the Atmospheric Sciences 74, 2; 10.1175/JAS-D-16-0144.1

c. Thermal 3

The collapse of the cloud top onto the third thermal resulted in the dilution of thermal 3 by a different mechanism. The PDF analysis is thus shown only for the time before the effects of the collapsing cloud top were realized. During this time, thermal 3 (Fig. 10, bottom) also ascended into the wake of its predecessor and entrained moistened, cloudy air similar to that of thermal 2a, but exhibiting less variability in of entrained air (σ = 0.10) and rarely entraining air with (Table 1). Interestingly, the mode is located near 0.45, lower than the mode of 0.6 for thermal 2, indicating the entrained air was not as moist for thermal 3, and likely owing to the longer delay between thermal 2 and thermal 3. Similar to thermal 2a, only 36% of the entrained air was negatively buoyant, indicating minimal evaporative cooling. As a result, most entrained parcels had TWC/ALWC fractions around 0.45. Despite similar calculated maximum values of entrainment, mean radii, and depth of ascent to thermal 1 (Table 1), thermal 3 diluted slowly as a result of ascending in the wake of thermal 2.

7. Summary and discussion

The entrainment, detrainment, and dilution of three thermals occurring in succession in a fully 3D simulated cumulus congestus were analyzed individually, and in detail, to understand and illustrate the maintenance of high-LWC cores in such clouds, which can significantly increase precipitation formation. Past 1D modeling studies artificially prescribed such a succession to show its potential impact on precipitation formation and thus could not investigate the dilution mechanism realistically. Other, more recent modeling studies have noted that clouds modify their environment and thus entrain air that can be quite different than that present farther from the cloud edge but have not studied how this modification changes with the passage of successive thermals.

This study has shown the following:

  • Each thermal consists of a large toroidal circulation, and entrainment rates calculated directly, at the edge of the cloud core, show they are maximized at its rear inflow. Entrainment maxima are thus a transient feature over the lifetime of a multithermal cloud.

  • These transient entrainment maxima in a multithermal cloud affect the dilution of the high-LWC core of the cloud, and, as found in other recent studies (not focused on multiple thermals), entrainment rates are not the sole determinant on the rate of dilution of the high-LWC core. The least-diluted parcel was tracked in each thermal as it ascended to estimate dilution rates. Dilution rates are higher when a given thermal ascends through undisturbed environmental air and lower when thermals rise through the wakes of previous thermals, despite sometimes having stronger calculated entrainment rates. Past studies have explained this effect resulting from entrainment of the “moist shell” around an individual cloud, rather than pristine environmental air, but this effect is even more pronounced in successive thermals.

  • Using a unique PDF analysis of air entrained and detrained into the core of each thermal, such “shells” are even more altered by the passage of previous thermals. The entrained air is not only moister in the wakes of previous thermals but can also contain high amounts of condensate (depending on the time interval between thermals) and can be positively buoyant, both aspects only contributing minimally toward diluting the cloud core. Entrainment of this “less erosive” air by thermals traveling through the wakes of their predecessors allows the high-LWC cores in clouds to persist, making them capable of producing rain and graupel when a single thermal could not, as hypothesized by previous 1D modeling studies.

A significant limitation of this study is that the analysis is applied to a single simulated cloud, but this was necessary because of the detailed, labor-intensive analysis of manually identifying the thermals and their LDPs. A large parameter space yet needs to be explored, including different thermodynamic environments, different wind shear profiles, and the effects of precipitation. The relative timing between successive thermals also needs further examination: as shown here, the forceful collapse of a rigorous thermal can destroy a subsequent thermal ascending beneath it and hints at the possibility of cyclic behavior in thermal development. Perhaps using new proposed algorithms to automate the identification of separate thermals in simulations of cloud fields (e.g., Dawe and Austin 2012; Sherwood et al. 2013; Heus and Seifert 2013; Park et al. 2016), studies of ensembles of clouds over a broad parameter space could evaluate the generality of the present results. Future simulations of successive thermals using multimoment bulk or bin microphysical schemes are also required to revisit the influence of successive thermals on precipitation, given the computational advances since the earlier modeling studies nearly 40 years ago. A forthcoming study expanding on the current simulation to include a system of precipitating clouds observed in southwestern England will apply the techniques used here to further understand how precipitation processes might have been enhanced by successive thermals active on that day.

While it is beyond the scope of the current study to investigate the implications of these results to current cumulus parameterizations used in larger-scale models, future exploration does appear warranted. Many clouds consist of multiple thermals, and these entities have been shown not only to produce transient, discrete maxima in entrainment, but also to provide a means to maintain more buoyancy and higher LWC in the cores of later thermals during the cloud lifetime. The undiluted parcels found throughout the lower half of the cloud for much of its lifetime, and undiluted ascent at higher altitudes later, differ from those predicted by time-averaged entrainment and detrainment profiles typically calculated from bulk estimation methods and employed in most cumulus parameterizations. While mean fractional entrainment rates can predict the dilution of the mean properties of the cloud core, they cannot represent the least diluted parcels [noted by Siebesma and Cuijpers (1995)] that ultimately govern the maximum cloud-top height, updraft speeds, and LWC. Some attempt to include the diversity in convective updrafts, which could result in part from successive thermals, is currently being made by introducing stochastic entrainment events over a field of cumuli into parameterizations (e.g., Romps and Kuang 2010; Sušelj et al. 2013; Romps 2016). It is unclear at this point how important representation of the details of multiple thermals is for larger-scale models, however. The “average” response of an entraining plume model over multiple thermals may be sufficient when the atmosphere is strongly unfavorable for deep convection. When it is marginally favorable, however, neglecting enhanced buoyancy and high-LWC cores supplied by successive thermals may preclude the initiation of deeper convection and ostensibly higher precipitation rates.

Acknowledgments

Dr. Jerry Straka generously supplied this version of the cloud model. Dr. Alan Blyth provided valuable feedback in the initial stage of the project. All computations were run on the high-performance computing system Yellowstone (ark:/85065/d7wd3xhc), provided by NCAR’s Computational and Information Systems Laboratory, sponsored by the National Science Foundation. The thermodynamic sounding used to initialize the model was collected during the Convective Precipitation Experiment (COPE). Support for this study was from NSF Award AGS-1230292. This work was initiated when both authors were affiliated with Purdue University.

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  • Sušelj, K., J. Teixeira, and D. Chung, 2013: A unified model for moist convective boundary layers based on a stochastic eddy-diffusivity/mass-flux parameterization. J. Atmos. Sci., 70, 19291953, doi:10.1175/JAS-D-12-0106.1.

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  • Zhao, M., and P. H. Austin, 2005: Life cycle of numerically simulated shallow cumulus clouds. Part II: Mixing dynamics. J. Atmos. Sci., 62, 12911310, doi:10.1175/JAS3415.1.

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1

Dilution rates are based on the decrease in scalar quantities (here qT) with ascent distance (or time), which can be affected by the parameterization for subgrid-scale turbulent mixing. The parameterization used in the Straka model is from Klemp and Wilhelmson (1978), where the turbulent kinetic energy dissipation coefficient Ce and the turbulent Prandtl number Pr are prescribed. Dilution rates were recalculated for other simulations run using coefficients spanning the ranges recommended by Deardorff (1980) for stably stratified conditions (Ce = 0.20, Pr = 1.00) and turbulent, convective conditions (Ce = 0.90, Pr = 0.33). Resulting changes in the dilution rates were negligible, indicating that resolved eddy motions are responsible for the dilution rates shown in Fig. 7.

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    • Search Google Scholar
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  • Sušelj, K., J. Teixeira, and D. Chung, 2013: A unified model for moist convective boundary layers based on a stochastic eddy-diffusivity/mass-flux parameterization. J. Atmos. Sci., 70, 19291953, doi:10.1175/JAS-D-12-0106.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Taylor, G. R., and M. B. Baker, 1991: Entrainment and detrainment in cumulus clouds. J. Atmos. Sci., 48, 112121, doi:10.1175/1520-0469(1991)048<0112:EADICC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Tiedtke, M., 1989: A comprehensive mass flux scheme for cumulus parameterization in large-scale models. Mon. Wea. Rev., 117, 17791800, doi:10.1175/1520-0493(1989)117<1779:ACMFSF>2.0.CO;2.

    • Crossref
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  • Trapp, R. J., 2013: Mesoscale-Convective Processes in the Atmosphere. Cambridge University Press, 377 pp.

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    • Crossref
    • Search Google Scholar
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  • Warner, J., 1970: On steady-state one-dimensional models of cumulus convection. J. Atmos. Sci., 27, 10351040, doi:10.1175/1520-0469(1970)027<1035:OSSODM>2.0.CO;2.

    • Crossref
    • Search Google Scholar
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    • Crossref
    • Search Google Scholar
    • Export Citation
  • Woodward, B., 1959: The motion in and around isolated thermals. Quart. J. Roy. Meteor. Soc., 85, 144151, doi:10.1002/qj.49708536407.

  • Yeo, K., and D. M. Romps, 2013: Measurement of convective entrainment using Lagrangian particles. J. Atmos. Sci., 70, 266277, doi:10.1175/JAS-D-12-0144.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zhao, M., and P. H. Austin, 2005: Life cycle of numerically simulated shallow cumulus clouds. Part II: Mixing dynamics. J. Atmos. Sci., 62, 12911310, doi:10.1175/JAS3415.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Fig. 1.

    Model “environment” conditions modified from an atmospheric sounding sampled near a line of cumulus congestus over southwestern England.

  • Fig. 2.

    Zoomed-in vertical cross sections of the fraction of TWC/ALWC (filled contours; scale in upper right) with in-plane velocity vectors, plotted at every other grid point, overlaid at times noted. Colored boxes indicate position of identified thermals discussed in the text.

  • Fig. 3.

    The maximum updraft speed and over the computational domain at 10 (blue), 12.5 (green), and 17.5 min (red).

  • Fig. 4.

    Time series of the (a) maxima of TWC, LWC, and rain and graupel mass content, (b) maximum updraft and downdraft speeds, and (c) cloud-top height from the simulated cloud for simulations with hydrometeor sedimentation disabled (solid lines) and enabled (dashed lines).

  • Fig. 5.

    Directly calculated rates of entrainment E (blue lines) and detrainment D (dashed green lines) at the cloud-core surface at times noted.

  • Fig. 6.

    As in Fig. 5, but for fractional rates of entrainment ε (blue line) and detrainment δ (dashed green line).