1. Introduction
Deep convection is a critically important atmospheric process that causes high-impact weather, humidifies the stratosphere, and regulates the atmospheric circulation and Earth’s hydrologic cycle. Because of its very turbulent and complex nature, however, this process remains challenging to understand and predict. This challenge arises in part from a lack of understanding of the parameters controlling deep convection initiation (DCI) (e.g., Crook 1996; Weckwerth and Parsons 2006; Bennett et al. 2011). DCI requires a conditionally unstable atmosphere and that air parcels reach their level of free convection, above which latent heat release generates positive buoyancy that drives further ascent. However, these conditions are not sufficient because mitigating factors such as entrainment and adverse vertical pressure gradients limit the growth of nascent cumuli (e.g., Houston and Niyogi 2007). In recent decades, various modeling studies have outlined additional mechanisms that may regulate DCI, which may be subdivided into two categories: nature and nurture (Romps and Kuang 2010; Dawe and Austin 2012). The former corresponds to initial properties of newly saturated air near cloud base, while the latter corresponds to the environmental conditions subsequently experienced by the ascending cloud.
Among the nurture parameters, the cloud-layer lapse rate determines whether cumuli can maintain sufficient buoyancy through latent heating to overcome the loss of buoyancy from entrainment (e.g., Houston and Niyogi 2007). Another important parameter is the midlevel moisture content: using cloud-resolving models (CRMs) and single-column models, Derbyshire et al. (2004) found that, within a given conditionally unstable environment, ambient midlevel humidity regulates the likelihood of DCI and the intensity of moist convection. This sensitivity is owing to the suppressive effects of entrainment (viz., dilution and evaporation of buoyant cloud cores), which increase in drier environments. In CRM simulations of DCI over a mountain–plain solenoidal circulation, Kirshbaum (2011) found that precursor clouds can facilitate DCI by providing a locally saturated midlevel environment that protects successor clouds from their otherwise dry surroundings. Detrainment from an ensemble of cumulus congestus can also facilitate DCI by gradually moistening the midlevel flow (Waite and Khouider 2010), but this process may be less important than large-scale forcing on diurnal time scales (Hohenegger and Stevens 2013). These three moisture-related effects can be categorized as follows: (i) initial moisture content, (ii) local moistening from individual clouds, and (iii) mean moistening from a cloud ensemble. The latter two effects differ from each other in that (ii) requires interaction between successive clouds while (iii) does not.
Romps and Kuang (2010) defined nature as the thermodynamic and kinematic conditions of newly saturated cloud parcels. In their large-eddy simulations (LESs) of shallow convection, they found that near-cloud-base parcel properties were largely uncorrelated with properties of the same parcels just a few hundred meters above. Thus, from a parcel perspective, nature appears secondary to nurture in regulating the properties of developing cumuli. Dawe and Austin (2012), by contrast, defined nature as the properties of newly saturated thermals. A key difference between this definition—which is adopted herein—and that of Romps and Kuang (2010) is that it also encompasses initial morphological characteristics of cloud entities, in particular their horizontal areas near cloud base. In their LESs of shallow convection, Dawe and Austin (2012) and Kirshbaum and Grant (2012) both found the initial cloud area to strongly influence the properties of the cumuli in the upper cloud layer, including their ultimate depth. This sensitivity stems from the fact that wider and thus more voluminous clouds tend to undergo less internal dilution for a given (absolute) entrainment rate than narrower clouds. Thus, from the perspective of whole thermals, nature may actually be more important than nurture.
The above findings for simulated shallow convection appear to carry over to simulated deep convection. As in Romps and Kuang (2010), Böing et al. (2012) found that the large differences in the properties of shallow and deep cloud parcels aloft could not be explained by differences in these same parcels near cloud base. On the other hand, consistent with Dawe and Austin (2012), Khairoutdinov and Randall (2006) found the transition from shallow to deep convection to coincide with a widening of the cumulus cloud bases and a reduction of cloud dilution. They argued that these wider clouds stemmed from the formation of precipitation and its attendant subcloud cold pools, which gave rise to wider boundary layer thermals. Heterogeneities in topography may also give rise to wider cumuli that support DCI. In simulations of DCI over idealized patches of surface heterogeneity, Rieck et al. (2014) found that the sizes of the largest cumuli tended to scale with the patch size. Kang and Bryan (2011) also found DCI to be promoted by mesoscale heterogeneities in land cover but focused on processes by which these heterogeneities eliminate CIN rather than the detailed internal properties of the resulting clouds.
The notion that the widening of incipient cloudy thermals is decisive for DCI demands reinforcement because it is largely based on diagnostic analyses (e.g., the cloud depth at a given time is correlated to cloud-base core sizes at that same time). A more causal and convincing conclusion could be obtained from a Lagrangian analysis that tracks cumuli (and their internal properties) from their boundary layer roots to their ultimate decay aloft, as was done by Dawe and Austin (2012) for shallow convection. The current study pursues this end using a new thermal-tracking algorithm suited for this purpose.
Thermal-tracking algorithms are gaining in popularity as the use of CRMs and large-eddy simulations becomes more widespread (Dawe and Austin 2012; Sherwood et al. 2013; Terwey and Rozoff 2014; Romps and Charn 2015). Such algorithms permit the correlation of various nature and nurture parameters (e.g., initial core size, local perturbation moistening) with the maximum depth ultimately reached by the cloud. They thus can provide valuable insights into the conditions that allow some clouds to ascend deeper than others. Such algorithms have been applied to various aspects of the moist convection problem, but never before to improving the understanding of DCI. The thermal-tracking algorithm developed herein enables investigation of several plausible mechanisms for DCI in a consistent framework.
Moreover, unlike past studies that considered horizontally homogeneous surface heating (e.g., Khairoutdinov and Randall 2006; Waite and Khouider 2010), this study considers DCI over a local heat source, which gives rise to a thermally driven mesoscale convergence line. Such convergence lines are common triggers for DCI over land and have been used in nowcasting thunderstorm initiation for decades (e.g., Wilson and Mueller 1993). In Colorado, for example, radar observations suggest that over 70% of DCI is forced by boundary layer convergence lines (Wilson and Schreiber 1986). The current approach thus addresses a meteorologically important problem and permits comparison between the behaviors of cumuli in forced versus unforced environments.
An explanation of the methodology follows in section 2, with the thermal-tracking method in section 2c. The general results and the mechanistic analyses are presented in sections 3 and 4, respectively. Section 5 discusses and interprets the results, and section 6 presents the conclusions.
2. Methodology
a. Model setup
The simulations use CM1 (release 17; Bryan and Fritsch 2002), a 3D, Eulerian, nonhydrostatic, finite-difference model with a third-order Runge–Kutta time-split explicit primary solver (Klemp and Wilhelmson 1978). Two attractive attributes of CM1 are that it considers dissipative heating and a more accurate representation of the moist thermodynamic equation than comparable models. An LES configuration is used with the Morrison double-moment cloud microphysics scheme (Morrison et al. 2005) and a 1.5-order turbulent kinetic energy (TKE) scheme for subgrid turbulence. The simulations use fifth-order advection in both the horizontal and vertical, with positive-definite advection of moisture variables. The Coriolis force is neglected for simplicity.
The simulations are designed to produce clouds that gradually deepen and, in some cases, transition to deep convection. In addition to its meteorological relevance (as discussed in section 1), a convergence-line forcing helps to control DCI and study its underlying mechanisms. In particular, it increases the possibility of cumuli interacting and detraining aloft to add significant midlevel moisture over a limited spatial region. The nominal domain size is 120 × 180 × 20 km3 with an x-grid spacing of 250 m within the central 60 km that gradually stretches to 750 m at the domain edges, along with a uniform 250-m y grid spacing. All simulated clouds develop within the central 60-km region and, thus, are represented at 250-m grid spacing in both directions. The vertical grid spacing is 100 m from the surface up to 6 km and then stretches to a maximum of 400 m at 12 km and remains fixed at 400 m up to the model top at 20 km.
The above grid spacings are the minimum that could be afforded for numerous large-domain simulations. According to Bryan et al. (2003), a horizontal grid spacing of Δ ≈ 100 m is necessary to resolve scales down to the inertial subrange, where subgrid turbulence parameterizations are properly formulated. They found Δ = 250 m adequate to model deep convection, but a smaller Δ may be needed to adequately capture the full DCI process, which also involves shallow cumuli and cumulus congestus. Although Blossey et al. (2009) found that Δ = 200 m was adequate to represent shallow convection, the necessary grid resolution to obtain robust results depends on the experimental setup and is thus generally uncertain. To address this uncertainty, section 5d performs sensitivity tests that double and halve the value of Δ.
The simulations are initialized with a Weisman–Klemp type sounding (Weisman and Klemp 1982) with a boundary layer water-vapor mixing ratio of 13 g kg−1, a surface temperature of 27°C, and a tropopause temperature of −60°C. The control (CTRL) sounding has a mean-layer (0–500 m) convective available potential energy (CAPE) of 1157 J kg−1 and a convective inhibition (CIN) of 50 J kg−1 (Fig. 1b), which is representative of a moderately unstable Great Plains environment (e.g., Soderholm et al. 2014). To prevent large-scale saturation over the domain, the initial relative humidity (RH) with respect to liquid is restricted to a maximum of 0.9. The large initial CIN suppresses cumulus convection everywhere except directly over the convergence line. It also limits vertical growth of the cumuli over the convergence line and, thus, provides an extended period of cumulus development prior to DCI. The ambient winds are zero at all levels and initial random perturbations of maximum amplitude 1 K are added to all grid points to seed convective motions.
Skew T–logp initial soundings of background temperature (thick blue), background dewpoint (dashed blue), and ascent temperature (thick black) for (a) LCAPE (CAPE = 957 J kg−1), (b) CTRL (CAPE = 1157 J kg−1), and (c) HCAPE (CAPE = 1357 J kg−1) environments.
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0221.1
b. Experiments
With the CTRL case as a reference, sensitivity experiments are conducted to isolate the response of the cloud field to modest changes in initial midlevel conditions. In doing so, these experiments directly address the nurture question. The nature question is addressed simultaneously by statistical analysis of the cloud properties of each simulation using the thermal-tracking algorithm.
1) RH sensitivity test
To determine the impact of midlevel humidity on DCI, two initial RH profiles are created that differ from the CTRL case (Fig. 2). In the Moist profile, the RH aloft is increased to a value that lies halfway between the CTRL value and unity, with a linear transition over 3–6 km. In the Dry profile, the RH is reduced by an equivalent amount. Unlike Derbyshire et al. (2004), these RH profiles vary with height and are designed to only marginally differ from the CTRL sounding. The RH modifications cause slight changes in CAPE (<1%) because of the associated changes in ambient virtual temperature. However, these small CAPE changes are presumed to be negligible compared to those associated with the RH.
Initial RH profiles for the RH sensitivity tests.
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0221.1
2) CAPE sensitivity test
Cumuli in more strongly conditionally unstable environments develop larger buoyancies and thus face less resistance to vertical ascent. Houston and Niyogi (2007) tested the impacts of cloud-layer lapse rates on DCI by systematically adjusting the lapse rate to modify the parcel buoyancy profile, while holding CAPE and CIN fixed. In a similar fashion, we assess the impact of moist instability on DCI by holding CIN fixed but varying CAPE by ±200 J kg−1. This is achieved by modifying the temperature profile from the LFC to the level of neutral buoyancy (LNB) while adjusting the qυ profile to keep the RH unchanged. The high CAPE [HCAPE; low CAPE (LCAPE)] temperature profile increases (decreases) the adiabatic parcel buoyancy with respect to the CTRL values (Figs. 1a and 1c). This buoyancy is adjusted only between the LFC and the LNB so that the heights of these levels remain fixed to the CTRL values, with linear transitions to a maximum modification halfway between the two. Hence, only the convecting layer between the LFC and LNB is modified.
3) Midlevel line-relative wind sensitivity test
While a zero-wind environment represents a reasonable starting point, it is unrealistic. This sensitivity test adds light midlevels winds in the zonal (cross line) direction, which may impact the mechanisms of successive-cumuli interaction (Kirshbaum 2011) and of progressive environmental moistening (Waite and Khouider 2010), the former by increasing the separation between clouds and the latter by transporting away the midlevel RH anomaly above the convergence line. These experiments retain the surface forcing of the CTRL case as well as zero winds over 0–3 km, the latter to ensure that the dynamical response to the heating remains similar. The zonal wind increases linearly over 3–6 km to a value U and is held at U aloft. The first case (U25) sets U = 2.5 m s−1 and the second (U50) sets U = 5 m s−1.
c. Thermal tracking
1) Tracking criteria
Our simple tracking algorithm aims to capture the full life cycle of developing cumuli, from their roots near the LCL to their uppermost heights. Existing convective thermal-tracking algorithms use methods such as streamfunctions (calculated explicitly or derived from theoretical models) to define a boundary in space that is tracked in time (Romps and Charn 2015; Sherwood et al. 2013) or define a “thermal core” as the maximum velocity cell of every updraft at every vertical level (e.g., Terwey and Rozoff 2014). For our study, the streamfunction method has the inconvenience of not tracking the thermals to their greatest heights and the maximum velocity method has the inconvenience of not allowing a complete representation of the cloud thermodynamic properties. Since DCI involves both shallow and deep stages of the cloud life cycle, it is useful to define tracking criteria that vary with height. All of the grid points meeting these criteria are considered to be part of a convective thermal. The points are then connected in space to form a cluster that is tracked in time (based on overlap between consecutive times) to form the thermal’s history. In the end, each such time sequence is attributed a unique identity. The model output is written to file in time steps of 1.5 min so that, at a characteristic thermal ascent rate of 3 m s−1 (Hernandez-Deckers and Sherwood 2016), thermals with depths of ~250 m or greater can be tracked in time.
Among other things, this tracking algorithm will allow us to evaluate Kirshbaum’s (2011) hypothesis that local perturbation moisture can promote DCI. It will also permit evaluation of the hypothesis that cloud-top height is partly determined by the core size near cloud base of (Khairoutdinov and Randall 2006; Rochetin et al. 2014). Using the tracking algorithm, we will analyze the impact of the initial thermal horizontal area on the final height reached by the cloud. This capability represents an upgrade over previous studies that established only simultaneous relations between cloud area and cloud depth and will, thus, sharpen the focus on the nature aspect of DCI. Also, the capacity to compare multiple plausible mechanisms for DCI using a consistent framework permits an assessment of each mechanism’s relative contributions to DCI.
2) Algorithm
This algorithm is based on both the vertical velocity w and the sum of the cloud water and ice mixing ratios (qc + qi), the latter separating moist updrafts from dry boundary layer updrafts. We first define minimal criteria for a grid cell to qualify as a thermal. Consistent with Romps and Charn (2015), values of 1 m s−1 for vertical velocity and 10−2 g kg−1 for qc + qi are chosen. However, such uniform thresholds are insufficient to track the full cloud development. The tracking suffers if these criteria are too stringent to detect nascent or decaying thermals or too weak to distinguish two close thermals with intersecting boundaries. To reduce such effects, the criteria are permitted to vary with height and are based on the global statistics of the CTRL case, in the form of vertical profiles of horizontally averaged w and qc + qi. The averages are conditional, restricted to just the cells qualifying for the minimal updraft requirements stated above, at each output time. Of course, taking an average rather than a percentile threshold is arbitrary, but it is attractively simple and, as will be seen, allows for full life-cycle detection and few boundary interactions (see section 5c). We then time average these profiles over the duration of the CTRL simulation to give w and qc + qi tracking profiles that apply to the entire simulation. These profiles are relatively strict at low to middle levels (Fig. 3), which helps to separate the many closely spaced thermals at these levels, and less strict aloft, allowing the tracking to continue into the decay phase. To provide a consistent representation of the thermal sizes and depths between simulations, the tracking profiles of the CTRL case are applied to all cases, rendering the method ideally suited for our sensitivity tests that do not vary greatly from CTRL.
Threshold (a) w and (b) qc + qi profiles used in tracking algorithm.
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0221.1
A few logical conditions are imposed to address the unavoidable cloud-boundary interactions (merging and splitting). First, if a cloud splits while shallow (below 3 km), the higher cloud keeps its original identity so that it can be traced back to the LCL. The lower cloud, on the other hand, is assigned a new identity and is assumed to have formed where the splitting occurred. Second, if a cloud splits when the tops of both components are above 3 km, it is considered to have split during deep ascent and both components keep the original identity. Finally, if two clouds merge at any point during their life cycle, the newly formed cluster is assigned the identity of the stronger of the two components prior to the merger (the one with the larger w). This stronger cloud is considered more appropriate since it is likely to reach deeper and its evolution is less likely to be modified by the merging process. The influences of merging and splitting on the tracking results are analyzed in section 5c.
Figure 4 shows a sample of the tracking in the CTRL case, taken at domain x centerline at 4.5-min intervals. The clouds are tracked continuously, without identity confusion and with a realistic shape, from their shallow stages up to their decay.
Cloud tracking along the domain x centerline. Thick black lines represent simulated qc + qi contours of 0.01, 0.05, and 0.1 g kg−1. Each colored area represents the extent of a detected thermal. The color of each thermal is held fixed in time to visualize the evolution of each tracked entity.
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0221.1
3. Results overview
We begin with a broad overview of the evolution of clouds and other key parameters in every simulation described in section 2b. Figure 5 shows the y-maximum qc + qi time averaged over three different periods and corresponding y-averaged wind vectors for the CTRL case. As the sensible heating increases, two fronts form on either side of the heating zone that migrate toward the center (Fig. 5a). When these two fronts collide at the x centerpoint (Fig. 5b), a stronger boundary layer updraft develops that forces the first thermals past the LFC. The circulation further strengthens and generates larger clouds, which reach maximum heights at around 0600 (Fig. 5c). Clouds reaching above 8 km are typically heavily precipitating and thus termed “deep” while clouds that fail to reach 3 km are nonprecipitating and thus termed “shallow.”
Contours of y-maximum qc + qi (filled grayscale contours; g kg−1) and y-averaged wind vectors (blue arrows). The former are averaged in time over the time interval on each panel while the latter are taken at the centerpoint of the time interval. Wind vectors with speeds smaller than 0.25 m s−1 are omitted to focus on the mesoscale boundary layer circulation.
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0221.1
As shown by the time series in Figs. 6a and 6c, boundary layer destabilization over the heat source gives rise to shallow cumuli prior to 0300 despite minimal surface convergence. Frontal collision at the x centerline at around 0330 causes this convergence to increase sharply, generating an organized updraft with a mean y-averaged intensity of 2.4 m s−1 below cloud base. This collision forces the first thermals past the LFC, leading to an abrupt deepening of the cloud field (Fig. 6c). The cloud height then increases nonmonotonically to reach near the LNB at around 0630 and then retreats at later times. Meanwhile, the tropospheric relative humidity increases with the detrainment of midlevel clouds, particularly after the peak forcing at 0600 (Fig. 6d). The frontal collision is also apparent in the CAPE (CIN) time series, which reach local maxima (minima) at that time, then change only gradually for the remainder of the simulation (Fig. 6b).
Time series of (a) surface convergence magnitude, averaged in y along the convergence line and in x over the central 1 km of the domain; (b) pseudoadiabatic mean-layer (500 m) CAPE and CIN over the central 1 km of the convergence line; (c) cloud top, corresponding to the maximum height at which qc + qi exceeds 0.1 g kg−1, with LNB overlaid in black; and (d) midlevel RH calculated by volume-weighted average over the central 5 km of the domain and between heights of 3 and 8 km for control, moist, and dry cases.
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0221.1
a. Sensitivity to initial midlevel humidity
Figure 6c shows that, in general, a higher initial midlevel RH increases the maximum cloud-top height. Whereas clouds in the Dry and CTRL cases do not reach 8 km until about 0530, those in the Moist case become deep almost immediately after the frontal collision at 0330. The RH also increases over the duration of each simulation, but less so in the moister cases where the saturation deficit is small. Convergence briefly decreases in the moist simulation following the peak convection period (Fig. 6a), reflecting the development of cold pools propagating laterally away from the centerline. The thermals still overshoot the LNB after this occurs because the cold pools continue to provide robust forcing even after the main convergence line weakens.
b. Sensitivity to CAPE
Because of the prescribed changes in the CAPE sensitivity experiments, the CAPE time series are initially offset from each other by 200 J kg−1 (Fig. 7b). Nonetheless, these cases undergo similar CAPE variations over the course of the day. Figure 7c shows that the HCAPE clouds become deep directly after the frontal collision, behaving similarly to the Moist clouds. The midlevel RH maximum at around 0700 is largest for the HCAPE case (Fig. 7d), because the steeper cloud-layer lapse rate allows more clouds to reach the midlevels. The LCAPE case undergoes the slowest transition to deep convection and its cloud tops are the shallowest of the three cases.
As in Fig. 6, but for the CTRL, LCAPE, and HCAPE simulations.
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0221.1
c. Sensitivity to line-relative winds
In both the U25 and U50 cases, the midlevel moisture anomaly that develops over the convergence line as a result of cumulus detrainment is transported downwind, which all but eliminates the local midlevel moistening (Fig. 8b). The clouds in both of these cases reach similar depths at similar times, with the maximum cloud-top heights (~9 km) lower than that of the CTRL case (~10 km). None of these clouds, however, ever reach the LNB despite reaching 8 km earlier than in the CTRL case. The CAPE, CIN, and convergence time series for these cases are nearly identical to those of the control case (not shown). Overall, the marginal differences between the U25 and U50 cases are smaller than those between the CTRL and U25 cases. Thus, the elimination of the midlevel moist anomaly, which is already nearly complete in the U25 case, appears to explain most (but not all) of the differences in cloud depth between these simulations.
As in Figs. 6c and 6d, but for the CTRL, U25, and U50 simulations.
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0221.1
4. Thermal-tracking analysis
To interpret the development of convection within a given simulation, as well as the sensitivities exhibited by the different simulations, we apply the thermal-tracking algorithm to relate various geometric, kinematic, and thermodynamic properties of the thermals to their ultimate depth.
a. General role of core size
Khairoutdinov and Randall (2006) and more recent studies (e.g., Dawe and Austin 2012; Rieck et al. 2014) suggest that core horizontal area near cloud base regulates the subsequent cloud vertical development. Thus, we begin by calculating the effective horizontal thermal area (the ratio of its volume to its depth) near cloud base, specifically the moment if/when its calculated centroid reaches the LFC at ~2.05 km (ALFC). Because of limited output time resolution, the centroids of some clouds jump over the 2.05-km level between output time steps. Therefore, thermals with centroids that momentarily lie one vertical level above or below 2.05 km are also admitted.
Before analyzing the effects of ALFC in detail, we first inspect the processes regulating it. Following Rieck et al. (2014), who studied the thermal size distribution owing to a given mesoscale forcing, Fig. 9 shows a scatterplot of ALFC as a function of surface convergence, the latter averaged over the full domain in y and over the central 1 km of the domain in x, in the CTRL case. We average the convergence in y rather than computing it locally to evaluate the mesoscale (as opposed to microscale) controls on the core-size distribution. The correlation is lagged to account for the ascent of surface parcels to the LFC at a mean boundary layer updraft velocity of 2.4 m s−1. The largest initial thermals coincide with the maximum surface convergence at 0600. The maximum core sizes produced prior to heavy precipitation (green dots) are just as large as those produced after heavy precipitation (blue and red dots). Because the core-size distribution does not change radically after the onset of heavy precipitation and the formation of attendant subcloud cold pools, the mesoscale convergence line appears to control the core-size distribution throughout the simulation. All cases generate broadly similar size distributions as CTRL except for Moist (and, to a lesser extent, HCAPE), where the onset of heavy precipitation causes a noticeable enhancement in the peak values of ALFC (not shown).
CTRL case ALFC as a function of y-averaged horizontal convergence at the domain centerline. Green dots are thermals originating before the rain rate exceeds 10 mm h−1 (corresponding to the 40-dBZ reflectivity threshold), blue dots originate after the 10 mm h−1 threshold and before 0600, and red dots originate after 0600.
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0221.1
Insight into the impacts of initial core size is provided by scatterplots of ALFC versus Hmax (Figs. 10a–c), where the latter is defined as the highest point in the thermal’s history where 
Scatterplots of final cloud height and initial area for (a) Dry, (b) CTRL, and (c) Moist cases. The plots are fitted with second-order polynomial (thick black line). The markers are color coded as in Fig. 9.
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0221.1
Because Fig. 9 showed that the ALFC distribution is sensitive to mesoscale convergence magnitude, Fig. 10 implies that the cloud-top height is also highly sensitive to the mesoscale convergence magnitude. The maximum values of the ALFC distribution shift monotonically upward from 0300 to 0600 and downward after 0600, following the prescribed changes in surface forcing (not shown). As a result, the clouds do not reach 8 km at 0430, when CIN and CAPE reach their minimum and maximum (see Fig. 6b). Rather, they ascend the deepest just past the time of the maximum forcing at 0600, once sufficiently large cores have developed.
Figure 11b shows profiles of mean maximum buoyancy B of the tracked thermals binned with respect to ALFC in CTRL. For each thermal, a vertical profile of its maximum buoyancy at each vertical level during its life cycle is created, where buoyancy is 
(a)–(c) Maximum buoyancy, (d)–(f) updraft velocity, and (g)–(i) total water content profiles binned with respect to ALFC for the Dry, CTRL, and Moist cases.
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0221.1
The sensitivity of B to ALFC in Fig. 11b is likely explained by entrainment. While the environmental mass ingested by the cloud roughly scales with the cloud surface area, the associated dilution roughly scales with inverse cloud volume, which causes the fractional entrainment to roughly scale as the inverse cloud radius (e.g., Kirshbaum and Grant 2012). This allows larger thermals to maintain positive b up to higher levels because they are less diluted or shed negatively buoyant mixtures through detrainment. While the B profiles become increasingly adiabatic with increasing ALFC, the realized buoyancy is still small compared to the pseudoadiabatic profiles. Altogether, these findings strengthen the hypothesized link between core area, cloud-layer properties, and cloud-top height (Khairoutdinov and Randall 2006; Kirshbaum and Grant 2012; Schlemmer and Hohenegger 2014) in a more causal Lagrangian framework.
Profiles of maximum w (W) and 
b. Sensitivity to midlevel humidity
1) Sensitivity to initial 
Figures 10a–c present ALFC versus Hmax scatterplots for the experiments varying midlevel moisture (hence, 
The increased scatter in Moist (Fig. 10) suggests that the overall correlation between initial thermal size and cloud-top height weakens as the environment becomes more favorable for deep convection. In the Dry case, only seven thermals reach deep levels (>8 km) owing to their general inability to withstand the effects of entrainment. By contrast, 67 thermals reach deep levels in the moist case, including some with very small ALFC (~0.1 km2).
The differences in midlevel 
2) Sensitivity to mean perturbation moistening
The mean perturbation moistening 
Figures 12a and 12b decouples these two effects by creating two subsets of points based on their value of 
Scatterplots of final cloud height and initial area, separated into upper (gray) and lower (white) quartiles of (a),(b) 
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0221.1
The enhancement in the heights reached by the smaller cores at larger 
These results are partially consistent with Waite and Khouider (2010), who illustrated the positive impact of 
3) Sensitivity to local perturbation moistening
The local perturbation moistening 
Large 
c. Sensitivity to cloud-layer lapse rate
Increasing CAPE systematically increases 
Scatterplots of (a)–(c) final cloud height and initial area and (d)–(f) buoyancy profiles binned with respect to ALFC for LCAPE, CTRL, and HCAPE cases. Markers in (a)–(c) are color coded as in Fig. 9 and plots are fitted with second-order polynomial (thick black line). Initial core-size bins in (d)–(f) are defined as in Fig. 11.
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0221.1
Given the less favorable environment for deep convection in the LCAPE case, large initial thermals are required for clouds to reach past 8 km, and only seven do so. On the other hand, the HCAPE case exhibits a steep slope and wide scatter similar to the Moist case. As ALFC increases past 0.2 km2, many thermals approach the LNB and the sensitivity to ALFC weakens. In this case a full 55 thermals ascend deep, including numerous ones with small ALFC. For smaller thermals, the slope of the best-fit curve in the CTRL case lies in between that of the LCAPE and HCAPE cases (Figs. 13a–c). In contrast, for larger thermals it is steeper than that of the HCAPE case because the sensitivity to ALFC persists up to the largest sizes.
d. Sensitivity to midlevel line-relative winds
Increasing the background line-relative wind tends to suppress vertical cloud development, causing the slopes of the scatterplots of ALFC versus Hmax to become gradually shallower (Figs. 14a–c). Although the B profiles are similar for the CTRL, U25, and U50 cases, the latter two return to zero slightly faster after reaching their maxima at ~4 km, particularly at intermediate sizes (Figs. 14d–f). One explanation for this slightly reduced buoyancy aloft is that the midlevel perturbation moistening is eliminated by advection by the line-relative winds (Fig. 8b). However, since this moisture is already almost completely removed in the U25 case, this mechanism cannot explain the further reduced cloud-top heights in the U50 case. We thus speculate that the midlevel vertical shear also reduces the simulated cloud buoyancy (e.g., Hill 1968; Markowski and Richardson 2010), particularly in the U50 case. While vertical shear can also organize deep convection into supercells or multicells (Weisman and Klemp 1982), such organization would require much stronger shear than that considered here.
Scatterplots of (a)–(c) final cloud height and initial area and (d)–(f) buoyancy profiles binned with respect to ALFC for CTRL, U25, and U50 cases. Markers in (a)–(c) are color coded as in Fig. 9 and plots are fitted with second-order polynomial (thick black line). Initial core-size bins in (d)–(f) are defined as in Fig. 11.
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0221.1
e. Statistical significance tests
Thus far, differences in the ALFC and Hmax relationship between different simulations have been assessed using qualitative comparisons of scatterplots and quadratic fit curves. To better quantify these differences, we statistically compare the distributions of Hmax for five different ALFC bins between different simulations in Fig. 15. Statistical significance is evaluated using the Wilcoxon test, which determines whether two random variables are drawn from continuous distributions with the same medians (Hamill 1999). These analyses indicate that, for the extreme members of the RH and line-relative-wind sensitivity tests, differences in Hmax for the three largest ALFC bins are significant at the 95% confidence level (Figs. 15b,c). For the CAPE sensitivity tests, these differences are significant for the four largest bins (Fig. 15a).
Statistical analysis of differences in median Hmax in different simulations, with sampling binned according to ALFC. Each panel compares a different pair of simulations, with × indicating statistically significant differences between the simulations for that size bin. Each data point represents the median value from that bin, and error bars indicate the edges of the second or third quartiles of each distribution. Dashed lines indicate the boundaries of each ALFC bin.
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0221.1
The statistical significance of differences between Hmax samples within different ALFC bins in each simulation were also assessed. The medians of these samples, which generally increase with ALFC, are significantly different from each other at the 95% confidence level for the smallest four bins in all but two cases (dry and U50), where they are significantly different for the smallest three bins. Altogether, this analysis lends quantitative support to two key findings: (i) the ultimate cloud-top height of moist thermals increases with the initial core size and (ii) for a given initial core size, the ultimate cloud-top height is sensitive to modest changes in the midlevel environment.
5. Discussion
a. Initial core size
Thus far, ALFC has been used as the primary representation of a thermal’s nature. However, other initial properties such as w, b, and qT also correlate with Hmax (not shown). Why, then, do we choose ALFC? It is because, as the initial level is lowered from the LFC down to just above the LCL (1.75 km), only ALFC and w retain an obvious correlation with Hmax, with ALFC exhibiting the strongest correlation of all (not shown). During ascent from 1.75 km to the LFC, the correlation between thermodynamic properties (b, qT) and cloud geometry (ALFC, Hmax) increases rapidly, likely because the larger cores are less diluted by entrainment. Thus, these experiments suggest that ALFC is the dominant nature parameter, controlling both the thermodynamic properties within the cloud layer and the cloud-top height.
b. Nature versus nurture
What roles do nature and nurture play in controlling cloud-top height over a mesoscale convergence line? Our most robust finding is that, within a given simulation, thermals with larger initial horizontal areas tend to ascend deeper. Looking across the simulations, the size required for thermals to reach upper levels increases as the initial environment becomes less favorable for deep convection (e.g., the gentler ALFC–Hmax slope in the dry, LCAPE, and U50 cases). Thus, for given environmental conditions (nurture), thermals must exhibit a minimum initial core size (nature) to ascend deeply.
As found by Rieck et al. (2014) and shown in Fig. 9, the stronger the low-level forcing, the larger the initial thermals, which implies a greater likelihood of deep convection. Thus, a causal relation between low-level forcing and DCI can be drawn: stronger forcing creates larger cores, which tend to ascend deeper. Table 1 clearly shows that not only do more thermals reach deep under more favorable conditions but also that a larger proportion of the deep thermals have ALFC < 0.3 km2. Thus, initial core size becomes less important as the environment becomes more predisposed to deep convection. Consistent with the findings of Dawe and Austin (2012) for shallow convection, neither nature nor nurture is solely responsible for DCI; the depth of convection depends on a combination of initial properties (viz., ALFC) and environmental conditions from the LFC to the LNB.
Number of thermals reaching a cloud-top height of 8 km for all simulations. The first row is the number of thermals with initial area less than 0.3 km2 and the second row is the number of thermals with initial area greater than 0.3 km2.
In Khairoutdinov and Randall (2006), despite the presence of a favorable environment (initial CAPE = 1600 J kg−1, CIN = 15 J kg−1, and reasonably moist midlevel flow), horizontally homogeneous surface forcing and consequently weak low-level forcing prevented DCI until stronger forcing was established by cold pools. Waite and Khouider (2010), by contrast, found that midlevel moistening by shallow convection and cumulus congestus is required to provide a sufficiently favorable environment for deep convection. Our experiments have shown that, at least in the more marginal cases (dry, CTRL, and LCAPE), diurnal midlevel moistening by detraining congestus tends to systematically raise cloud-top height by as much as 1 km (Fig. 12). However, compared to the dominance of ALFC, cumulus moistening (both mean and local) has only a minor impact. Although we speculate that core size is generally more important than diurnal moistening for convection initiation, future experiments that systematically vary the strength and scale of the subcloud forcing would be useful to evaluate the robustness of this result.
As expected for entraining cumuli, Bmax attained by the thermals is generally far less than the adiabatic value. As a consequence, most thermals do not reach the LNB. In light of this finding, the Houston and Niyogi (2007) method of increasing the lapse rate of the cloud bearing layer while keeping CAPE constant can alternatively be seen as redistributing buoyancy downward into the cloud layer, which increases the “realized” CAPE of the ascending clouds. It simply makes more potential energy available to thermals as they struggle to ascend through the midtroposphere. Because smaller initial thermals are the most suppressed by entrainment, these thermals are most likely to be influenced by such a vertical buoyancy redistribution.
c. Further analysis of the tracking algorithm
The chosen approach of using fixed core-averaged profiles from the CTRL case as the thresholds in the thermal-tracking algorithm [section 2c(1)] is somewhat arbitrary and thus merits some critical evaluation. However, choosing a vertically uniform threshold [e.g., w ≥ 1 m s−1 and qc + qi ≥ 10−2 g kg−1, as was used by Romps and Charn (2015)] is equally arbitrary. Moreover, given that all the cases herein share the same LCL, LFC, and LNB, the use of fixed thresholds across all simulations is a reasonable choice.
Table 2 shows the numbers of the tracked clouds and the merging and splitting statistics for the control, dry, and moist cases. “Number thermals” is the total number of thermals tracked in the simulation, “low ratio” is the fraction of the thermals with centroids that cross the LFC (2.05 km) during their ascent, and “intercept ratio” is the fraction of those thermals that are considered in the analysis. The latter quantity is reduced by clouds whose centroids jump over the LFC between output time intervals or trace clouds that originate aloft. “Merge,” “split high,” and “split low” are flag numbers for mergers and splits. The most important such interaction is the merging because it causes the loss of the identity of one of the two thermals. Fortunately, removing the flagged merging thermals from the scatterplots did not at all affect the interpretation of the results (not shown). Moreover, even though the tracking criteria for all simulations are based on the CTRL case, the merging and splitting statistics do not change systematically from one case to another. However, there is a general increase in these interactions in the cases with the most intense convection (Moist and HCAPE), where more thermals reach the mid- to upper levels and are thus more likely to interact.
Tracking algorithm statistics. Quantities in the top row are defined in the text.
Over the course of this study, various tracking methods were tested, ranging from vertically uniform thresholds to time-varying thresholds using running-window averages that differed for each simulation. If the vertically uniform threshold was set too low, the algorithm could not distinguish low-level thermals and exhibited a much higher amount of cloud interactions, undermining the tracking effectiveness. If set too high, the algorithm would not capture the clouds’ formation and decay phases. On the other hand, the time-varying thresholds captured the full cloud life cycle well and limited the cloud-boundary interactions but complicated the analysis of tracking bias and the comparisons between different simulations. Overall, while testing and comparing various different tracking methods, the key results of the study remained robust (with the few exceptions just noted).
d. Sensitivity to horizontal grid spacing
As mentioned in section 2, computational constraints limited Δ to 250 m for most experiments, which may be insufficient to properly represent the full DCI process. To evaluate the robustness of the foregoing results to Δ, we perform two additional versions of the Dry case, one in which Δ is halved to 125 m (Dry125) and one in which Δ is doubled to 500 m (Dry500). Similar experiments were also performed using the Moist case as a reference, with similar results (not shown). To save expense in the Dry125 case, the domain width in the y direction is reduced by a factor of 3.
Relative to the baseline Dry case, the ALFC distributions shift toward smaller sizes in the Dry125 case and larger sizes in the Dry500 case (Figs. 16a,b). While the strong correlation between ALFC and Hmax from the Dry case (Fig. 10a) is retained in the Dry125 case, this correlation weakens dramatically in the Dry500 case, with much larger scatter around the quadratic fit curve and a sharply reduced goodness of fit. These differing sensitivities to ALFC are reinforced in the corresponding B profiles, where the Dry125 case retains a clear sensitivity to ALFC while the Dry500 case does not (Figs. 16c,d).
As in Figs. 10a and 11a, but for the (a),(c) Dry125 and (b),(d) Dry500 simulations.
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0221.1
Additional analysis of these simulations reveals that the model representation of the low-level convergence line, ALFC, and in-cloud entrainment are all sensitive to Δ (not shown). Given computational and space constraints, a thorough analysis of these contributions to Hmax is deferred to a subsequent, dedicated study. Nonetheless, these preliminary comparisons imply that while the quantitative relationship between ALFC and Hmax depends on Δ, the qualitative importance of initial core size (and, hence, nature) on cloud evolution does not, provided that Δ is sufficiently small. This point is supported by the Δ = 100-m LES experiments of Khairoutdinov and Randall (2006) and Kirshbaum and Grant (2012), which revealed qualitatively similar sensitivities of cumulus growth to low-level core size as those found herein.
6. Conclusions
The understanding of deep convection initiation (DCI) remains limited owing to the many parameters that influence cumulus development and the chaotic behavior of the process itself. This study has investigated the relative importance of “initial” properties of ascending moist thermals just above cloud base (nature) and environmental parameters during ascent (nurture) on the depth reached by cumuli, using idealized simulations of moderately unstable environments with large CIN. In these simulations, convection was forced by a sharp mesoscale convergence line driven by localized diurnal heating. The dominant nature parameter was found to be the initial core horizontal area at the LFC (~2 km) and the ascent parameters were initial and perturbation midlevel moisture (the latter due to cumulus detrainment), temperature lapse rate (CAPE), and midlevel line-relative winds transverse to the convergence line. These factors were related to the depth of convection using a new thermal-tracking algorithm.
In strongly inhibited and moderately unstable environments like those simulated herein, no single initial parameter or ascent condition is sufficient to ensure DCI. Rather, a combination of sufficient cloud-layer lapse rate, midlevel moisture, and initial core size is necessary. Generally, a decrease in one of these parameters must be compensated by an increase of another for DCI to occur. Increased initial core size, midlevel lapse rate, and midlevel humidity both increase core buoyancy within the cloud layer and thus increase the likelihood of DCI. Increased cross-line background wind suppresses cores by eliminating the accumulation of midlevel moisture above the convergence line and possibly also increasing entrainment (owing to the associated vertical shear). The impact of local or mean moistening by detraining cumuli was smaller than that of initial core size but still significant; diurnal midlevel humidification systematically enhanced the depth of smaller cores by up to 1 km.
Using a Lagrangian framework, our findings strengthen the hypothesis, previously derived from diagnostic analyses of simulations with horizontally homogeneous surface forcing (e.g., Khairoutdinov and Randall 2006), that larger cores at cloud base tend to develop larger buoyancies within the cloud layer and ascend deeper. However, in the presence of a mesoscale convergence zone, evaporative cold pools are not necessary to create these large cores—the externally forced convergence can be equally effective in creating large cores. Consistent with Rieck et al. (2014), the strength of the subcloud forcing controls the initial core-size distribution and, in turn, the depth reached by convective towers. Moreover, in contrast to convection in horizontally homogeneous environments, even very weak transverse midlevel winds may strongly inhibit cloud development over mesoscale convergence lines.
Acknowledgments
The authors are grateful to George Bryan for freely providing the CM1 code. Funding for this study was provided by the Natural Sciences and Engineering Research Council (NSERC) Discovery Grant NSERC/RGPIN 418372-12. Numerical simulations were performed on the Guillimin supercomputer at McGill University, under the auspices of Calcul Québec and Compute Canada. The authors are grateful for the insightful and constructive comments of three anonymous reviewers.
REFERENCES
Bennett, L. J., and Coauthors, 2011: Initiation of convection over the Black Forest mountains during COPS IOP15a. Quart. J. Roy. Meteor. Soc., 137, 176–189, doi:10.1002/qj.760.
Blossey, P. N., C. S. Bretherton, and M. C. Wyant, 2009: Subtropical low cloud response to a warmer climate in a superparameterized climate model. Part II: Column modeling with a cloud resolving model. J. Adv. Model. Earth Syst., 1 (8), doi:10.3894/JAMES.2009.1.8.
Böing, S. J., H. J. J. Jonker, A. P. Siebesma, and W. W. Grabowski, 2012: Influence of the subcloud layer on the development of a deep convective ensemble. J. Atmos. Sci., 69, 2682–2698, doi:10.1175/JAS-D-11-0317.1.
Bryan, G. H., and J. M. Fritsch, 2002: A benchmark simulation for moist nonhydrostatic models. Mon. Wea. Rev., 130, 2917–2928, doi:10.1175/1520-0493(2002)130<2917:ABSFMN>2.0.CO;2.
Bryan, G. H., J. C. Wyngaard, and J. M. Fritsch, 2003: Resolution requirements for the simulation of deep moist convection. Mon. Wea. Rev., 131, 2394–2416, doi:10.1175/1520-0493(2003)131<2394:RRFTSO>2.0.CO;2.
Crook, N. A., 1996: Sensitivity of moist convection forced by boundary layer processes to low-level thermodynamic field. Mon. Wea. Rev., 124, 1767–1785, doi:10.1175/1520-0493(1996)124<1767:SOMCFB>2.0.CO;2.
Dawe, J. T., and P. H. Austin, 2012: Statistical analysis of an LES shallow cumulus cloud ensemble using a cloud tracking algorithm. Atmos. Chem. Phys., 12, 1101–1119, doi:10.5194/acp-12-1101-2012.
Derbyshire, S. H., I. Beau, P. Bechtold, J.-Y. Grandpeix, J.-M. Piriou, J.-L. Redelsperger, and P. M. M. Soares, 2004: Sensitivity of moist convection to environmental humidity. Quart. J. Roy. Meteor. Soc., 130, 3055–3079, doi:10.1256/qj.03.130.
Durran, D. R., 1999: Numerical Methods for Wave Equations in Geophysical Fluid Dynamics. Springer-Verlag, 465 pp.
Hamill, T., 1999: Hypothesis tests for evaluating numerical precipitation forecasts. Wea. Forecasting, 14, 155–167, doi:10.1175/1520-0434(1999)014<0155:HTFENP>2.0.CO;2.
Hernandez-Deckers, D., and S. C. Sherwood, 2016: A numerical investigation of cumulus thermals. J. Atmos. Sci., 73, 4117–4135, doi:10.1175/JAS-D-15-0385.1.
Hill, G. E., 1968: On the orientation of cloud bands. Tellus, 20, 132–137, doi:10.1111/j.2153-3490.1968.tb00357.x.
Hohenegger, C., and B. Stevens, 2013: Preconditionning deep convection with cumulus congestus. J. Atmos. Sci., 70, 448–464, doi:10.1175/JAS-D-12-089.1.
Houston, A. L., and D. Niyogi, 2007: The sensitivity of convective initiation to the lapse rate of the active cloud-bearing layer. Mon. Wea. Rev., 135, 3013–3032, doi:10.1175/MWR3449.1.
Kang, S.-L., and G. H. Bryan, 2011: A large-eddy simulation study of moist convection initiation over heterogeneous surface fluxes. Mon. Wea. Rev., 139, 2901–2917, doi:10.1175/MWR-D-10-05037.1.
Khairoutdinov, M., and D. Randall, 2006: High-resolution simulation of shallow-to-deep convection transition over land. J. Atmos. Sci., 63, 3421–3436, doi:10.1175/JAS3810.1.
Kirshbaum, D. J., 2011: Cloud-resolving simulations of deep convection over a heated mountain. J. Atmos. Sci., 68, 361–378, doi:10.1175/2010JAS3642.1.
Kirshbaum, D. J., and A. L. M. Grant, 2012: Invigoration of cumulus cloud fields by mesoscale ascent. Quart. J. Roy. Meteor. Soc., 138, 2136–2150, doi:10.1002/qj.1954.
Klemp, J. B., and R. B. Wilhelmson, 1978: The simulation of three-dimensional convective storm dynamics. J. Atmos. Sci., 35, 1070–1096, doi:10.1175/1520-0469(1978)035<1070:TSOTDC>2.0.CO;2.
Markowski, P., and Y. Richardson, 2010: Mesoscale Meteorology in Midlatitudes. Wiley, 430 pp.
Morrison, H., J. A. Curry, and V. I. Khvorostyanov, 2005: A new double-moment microphysics parameterization for application in cloud and climate models. Part I: Description. J. Atmos. Sci., 62, 1665–1677, doi:10.1175/JAS3446.1.
Rieck, M., C. Hohenegger, and C. C. van Heerwarden, 2014: The influence of land surface heterogeneities on cloud size development. Mon. Wea. Rev., 142, 3830–3846, doi:10.1175/MWR-D-13-00354.1.
Rochetin, N., F. Couvreux, J.-Y. Grandpeix, and C. Rio, 2014: Convection triggering by boundary layer thermals. Part I: LES analysis and stochastic triggering formulation. J. Atmos. Sci., 71, 496–514, doi:10.1175/JAS-D-12-0336.1.
Romps, D. M., and Z. Kuang, 2010: Nature versus nurture in shallow convection. J. Atmos. Sci., 67, 1655–1666, doi:10.1175/2009JAS3307.1.
Romps, D. M., and A. B. Charn, 2015: Sticky thermals: Evidence for a dominant balance between buoyancy and drag in cloud updrafts. J. Atmos. Sci., 72, 2890–2901, doi:10.1175/JAS-D-15-0042.1.
Schlemmer, L., and C. Hohenegger, 2014: The formation of wider and deeper clouds as a result of cold-pool dynamics. J. Atmos. Sci., 71, 2842–2858, doi:10.1175/JAS-D-13-0170.1.
Sherwood, S. C., D. Hernandez-Deckers, and M. Colin, 2013: Slippery thermals and the cumulus entrainment paradox. J. Atmos. Sci., 70, 2426–2442, doi:10.1175/JAS-D-12-0220.1.
Soderholm, B., B. Ronalds, and D. J. Kirshbaum, 2014: The evolution of convective storms initiated by an isolated mountain ridge. Mon. Wea. Rev., 142, 1430–1451, doi:10.1175/MWR-D-13-00280.1.
Terwey, W. D., and C. M. Rozoff, 2014: Objective convective updraft identification and tracking: Part 1. Structure and thermodynamics of convection in the rainband regions of two hurricane simulations. J. Geophys. Res. Atmos., 119, 6470–6496, doi:10.1002/2013JD020904.
Waite, M. L., and B. Khouider, 2010: The deepening of tropical convection by congestus preconditioning. J. Atmos. Sci., 67, 2601–2615, doi:10.1175/2010JAS3357.1.
Weckwerth, T. M., and D. B. Parsons, 2006: A review of convection initiation and motivation for IHOP 2002. Mon. Wea. Rev., 134, 5–22, doi:10.1175/MWR3067.1.
Weisman, M. L., and J. B. Klemp, 1982: The dependence of numerically simulated convective storms on vertical wind shear and buoyancy. Mon. Wea. Rev., 110, 504–520, doi:10.1175/1520-0493(1982)110<0504:TDONSC>2.0.CO;2.
Wilson, J. W., and W. E. Schreiber, 1986: Initiation of convective storms at radar-observed boundary-layer convergence lines. Mon. Wea. Rev., 114, 2516–2536, doi:10.1175/1520-0493(1986)114<2516:IOCSAR>2.0.CO;2.
Wilson, J. W., and C. K. Mueller, 1993: Nowcast of thunderstorms initiation and evolution. Wea. Forecasting, 8, 113–131, doi:10.1175/1520-0434(1993)008<0113:NOTIAE>2.0.CO;2.
