Environmental sensitivities of shallow-cumulus dilution. Part I: Selected thermodynamic conditions

Abstract. Cumulus entrainment, and its consequent dilution of buoyant cloud cores, strongly regulates the life cycle of shallow cumuli yet remains poorly understood. Herein, new insights into this problem are obtained through large-eddy simulations that systematically investigate the sensitivity of shallow-cumulus dilution to cloud-layer relative humidity (RH), cloud- and subcloud-layer depths, and continentality (i.e., the land-ocean contrast). The simulated cloud-core dilution is found to be strongly sensitive to continentality, with fractional dilution rates twice as large over the ocean as over land. Using a similarity theory based on the turbulent-kinetic-energy (TKE) budget, the reduced cloud-core dilution over land is attributed to larger cloud-base mass flux (mb), driven by stronger surface heating and subcloud turbulence. As mb increasies, the fractional dilution rate must decrease to maintain energetic equilibrium. A positive sensitivity is also found to cloud-layer RH, with the core dilution increasing by 25–50 % for a 10 % enhancement in RH. This sensitivity is interpreted using the buoyancy-sorting hypothesis, in that mixtures of cloud and environmental air are more likely to become negatively buoyant and detrain (rather than diluting the cloud core) in drier cloud layers. By contrast, the sensitivities of (marine) shallow-cumulus dilution to cloud- and subcloud-layer depths are weak, with virtually no sensitivity to the former and only a 15 % reduction in dilution for a tripling of the latter (from 250 m to 750 m). These surprisingly weak sensitivities are readily explained by offsetting effects in the TKE similarity theory. Altogether, these experimental findings provide useful, though still incomplete, guidance for flow-dependent shallow-cumulus entrainment formulations in large-scale models.


Florida peninsula, Neggers et al. (2003) found bulk-dilution rates of 1.5-3.0 km −1 both in observations and LES. Although trends cannot be easily inferred from such a small observational sampling, these findings suggest geographic differences in cloud dilution, possibly related to whether the clouds form over land or the ocean. Although a potential sensitivity to continentality was recognized by Brown et al. (2002), it has not yet been explored in detail or physically interpreted.
One of the factors often invoked to explain cumulus dilution rate is horizontal cloud size. Assuming a cylindrical cloud with 70 homogeneous entrainment, the rate of cloud dilution scales inversely with cloud radius (R; e.g., de Rooy et al., 2013). This sensitivity arises because, at a given level, the flux of entrained air scales with the cloud perimeter (∼ R) but the entrained air mixes over the cloud area (∼ R 2 ), implying less dilution at larger R. Although entrainment is far from homogeneous in real clouds, both observational and LES studies indicate weaker dilution for larger cloud horizontal areas (e.g., Squires and Turner, 1962;Dawe and Austin, 2012;Khairoutdinov and Randall, 2006;Kirshbaum and Grant, 2012;Rousseau-Rizzi et al., 2017). 75 Cloud size is determined by numerous factors, one being the cloud-layer depth. In 3D isotropic turbulence, the horizontal size of turbulent eddies broadly scales with the layer depth. Thus, deeper cloud layers may support wider clouds and consequently smaller dilution rates. Using CRM simulations of the diurnal cycle of convection over land, Del Genio and Wu (2010) showed that the dilution rates of shallow cumuli exceeded those for cumulonimbi. While subsequent studies have reached similar conclusions (e.g., Derbyshire et al., 2011;Stirling and Stratton, 2012), little attention has been paid to the sensitivity of cloud 80 size and dilution to cloud-layer depth within the shallow-cumulus regime alone.
The subcloud layer may also influence cloud size, mainly by regulating the size of turbulent eddies that breach the lifting condensation level to form clouds. Cloud-resolving numerical simulations have shown that larger incipient thermals at the level of free convection (LFC) evolve into less dilute and deeper clouds (e.g., Rousseau-Rizzi et al., 2017). One factor regulating initial cloud size is the subcloud layer depth; deeper subcloud layers give rise to wider thermals that tend to initiate wider 85 clouds. To date, however, no study has systematically examined the sensitivity of shallow-cumulus dilution to subcloud-layer depth. Another factor regulating initial cloud size is the characteristic scale of surface heterogeneities. The idealized LES of Rieck et al. (2014) showed that, through their control over cloud size, such heterogeneities also regulated cumulus dilution within the cloud layer.
The above discussion indicates that, while the controls on shallow-cumulus entrainment and dilution are an active area of 90 research, at least some of the environmental sensitivities and corresponding physical explanations of this process are either poorly understood or differ between different studies. Improving this understanding is essential given that the representation of cumulus convection, and particularly the entrainment process, is a major source of global climate sensitivity in those models (e.g., Murphy et al., 2004;Knight et al., 2007;Sanderson et al., 2008;Rougier et al., 2009;Klocke et al., 2011). In both global climate and many weather forecast models, shallow cumuli are parameterized due to insufficient grid resolution, and 95 will continue to be parameterized for some time.
The simplest approach to parameterizing entrainment is to prescribe a fixed entrainment profile within the cloud layer. However, this method neglects the wide range of entrainment/dilution rates found in reality. Raymond and Blyth (1986) and Kain and Fritsch (1990) addressed this limitation by invoking the "buoyancy-sorting" concept, where entrainment and detrainment are determined by the buoyancy of cloud-environmental mixtures. Positively buoyant mixtures are retained within the core and 100 added to the entrainment rate, while negatively buoyant mixtures are expelled from the cloud and contribute to detrainment.
This method links dilution to convective available potential energy (CAPE) and cloud-layer RH but ignores other potentially important parameters like continentality and vertical wind shear. Alternatively, some parameterizations link entrainment to parameterized updraft speed (e.g., Neggers et al., 2002), buoyancy (e.g., von Salzen andMcFarlane, 2002), or both (e.g., Gregory, 2001). Furthermore, while recent entrainment formulations have directly accounted for the cloud-layer RH sensitivity, the sign 105 of this sensitivity differs in different schemes (e.g., Bechtold et al., 2008;Stirling and Stratton, 2012), reflecting the incomplete understanding of this sensitivity.
To improve the representation of convection in global models, the understanding of shallow-cumulus dilution must improve.
The present study focuses on the sensitivities of shallow-cumulus dilution to certain environmental parameters that, as indicated in the above discussion, either remain unclear or have not received adequate attention. These include relative humidity, 110 continentality, subcloud-layer depth, and cloud-layer depth. To help resolve contradictory findings from past studies, we use a consistent LES framework, applied to a broad range of continental and maritime shallow-cumulus cloud fields, to systematically examine and physically interpret each sensitivity. The numerical setup and the different test cases are detailed in Sect. 2, along with a description of the analysis methods. In Sect. 3, the sensitivities of the fractional dilution rate to thermodynamic profiles and the continentality are presented. A discussion of these results is presented in Sect. 4. Section 5 concludes this 115 study.

LES of shallow cumulus cloud fields
For the numerical experiments, we use the Bryan Cloud Model version 17 (CM1; Bryan and Fritsch, 2002), a fully nonlinear, compressible, and nonhydrostatic atmospheric model. A monotonic 5th-order Weighted Essentially Non-Oscillatory (WENO) 120 advection scheme is used for both scalars and velocity vectors. For time integration, a third-order Runge-Kutta scheme with a split time step to account for acoustic waves is employed. All simulations use a f -plane approximation and have periodic horizontal, semi-slip lower, and free-slip upper boundary conditions. A Rayleigh damping zone of 500-m depth immediately below the model top is used to limit the spurious reflection of internal-gravity waves at the upper boundary. Further numerical settings depend on the individual cases and are listed in Table 1. Four different shallow-cumulus ensembles form the basis of this study, covering different environmental conditions in which shallow cumuli develop in nature. Of these cases, two are based on field campaigns that were conducted in maritime environments in the eastern Caribbean Sea, whereas the other two field experiments were conducted in the continental environment at the ARM-SGP central facility in northern Oklahoma. These field campaigns are the Barbados Oceanographic and Meteorological Experiment (BOMEX; Holland and Rasmusson, 1973), the Rain in (Shallow) Cumulus over the Ocean (RICO ;Rauber 130 et al., 2007) study, a LES comparison study of the diurnal cycle of shallow cumulus convection at the ARM-SGP observatory (ARM-SGP; Brown et al., 2002) and the RACORO campaign (also at the SGP observatory; Vogelmann et al., 2012).
Based on the field observations, LES intercomparison studies described in Siebesma et al. (2003) for BOMEX, vanZanten et al. (2011) for RICO, and Brown et al. (2002) for ARM-SGP and an observation-LES study by Endo et al. (2015) for RACORO were undertaken and provide the configurations for our LES experiments (Table 1). All simulations are initialized 135 with vertical profiles of potential temperature (θ), water-vapor mixing ratio (q v ), and horizontal winds (u and v) as prescribed in the above-mentioned studies, along with small-amplitude random perturbations in θ and q v to seed convective motions.
Horizontal grid spacings range from 32 m to 150 m and the vertical grid spacing is 40 m.
Large-scale subsidence, horizontal advection of heat and moisture, longwave cooling, and surface fluxes are adopted directly

Bulk-dilution rate
The simulated fractional dilution rate is calculated following Siebesma and Cuijpers (1995, hereafter SC95), where a formu-150 lation of the prognostic equation of conserved variables such as total water specific humidity (s t ) is utilized. By decomposing the equation for s t into a cloud "core" and an environmental part, SC95 obtained where E is the dilution rate and 'co' denotes the cloud core. The plain overbar is a horizontal domain average and the overbar indexed 'co' is a conditional average over the cloud cores (with respect to the cloud-core average). At a given vertical level, 155 the cloud-core properties are calculated as the conditional average over all horizontal grid points that contain significant liquid water (q c > 0.01 g kg −1 ), are positively buoyant with respect to the horizontal domain average and are ascending, similar to Siebesma et al. (2003). We define the area-averaged cloud-core mass flux as M co ≡ ρa co w co , where ρ is the air density, a co is the cloud-core fraction, and w co is the conditionally averaged cloud-core vertical velocity. The fractional dilution rate is obtained by dividing Eq. (1) by M co (ε SC95 = E/M co ). We calculate ε SC95 at each model output time, over all vertical levels 160 where cloud cores are simulated, to obtain instantaneous vertical ε SC95 profiles.
While entrainment describes the ingestion of environmental air into the cloud, detrainment refers to the expulsion of cloudy air into the environment. The detrainment rate D can be determined by using the obtained dilution rate and the continuity equation (SC95) to give (2)

TKE similarity theory
To gain physical insight into the sensitivities of the diagnosed ε SC95 to environmental conditions, we use the shallow-cumulus similarity theory based on the turbulent-kinetic-energy (TKE) budget of Grant and Brown (1999) and Grant and Lock (2004).
This theory allows for the estimation of ε and other bulk properties based on limited information about the larger-scale en-170 vironment and the subcloud layer. To the extent that it captures the simulated sensitivities of cloud dilution environmental conditions, it can help to determine the underlying physical causes.
For non-precipitating, weakly sheared shallow-cumulus-cloud fields in statistical equilibrium, the ensemble-averaged rate of buoyancy production of TKE (w b ) and the turbulent dissipation (d) dominate the TKE budget where w is vertical velocity and b is buoyancy. Perturbations from the horizontally averaged mean are indicated by primes.
Through a scale analysis, Grant and Lock (2004) where m b is the density-normalized cloud-core-base mass flux, w * is the turbulent vertical-velocity scale, CAPE is the convective available potential energy of the cloud-core layer, and z cld is the cloud-core-layer depth. The nondimensional factor 180 (m b /w * ) 1/2 can be interpreted as a cloud fractional area (Grant and Lock, 2004). The dissipation is scaled as and, using Eqs. (3)-(5), To scale the cloud dilution, two assumptions are made: (i) entrainment is an intermediate process between buoyancy production 185 and turbulent dissipation, and (ii) a fixed fraction (A ε ) of updraft kinetic energy is imparted to the portion of entrained air that 6 https://doi.org/10.5194/acp-2020-336 Preprint. Discussion started: 29 April 2020 c Author(s) 2020. CC BY 4.0 License.
reaches and dilutes the cloud core. The kinetic energy transferred to the core-entrained air, or εm b w * 2 , is then multiplied by the scaling parameter (m b /w * ) 1/2 (like the other TKE source terms above) and equated to A ε d: Substituting Eq. (6) into Eq. (7), we obtain Thus, based on this theory, the fractional dilution rate (ε) increases weakly with CAPE and more strongly decreases with increasing cloud depth and cloud-base mass flux. Estimations of A ε in LES of maritime shallow cumuli have ranged from 0.03-0.06 (Grant and Lock, 2004;Kirshbaum and Grant, 2012). Over a larger sampling of both maritime and continental cases, 195 Although the TKE similarity theory has impressively captured sensitivities of cloud dilution in different studies (Kirshbaum and Grant, 2012;Drueke et al., 2019), it has several limitations of note, one being the assumption of statistical equilibrium.
While few real-world cloud fields strictly meet this condition, the associated errors are small even in rapidly evolving cloud fields (Kirshbaum and Grant, 2012). Furthermore, Drueke et al. (2019) showed that the theory, when used as a simulated clouddilution retrieval, reasonably estimated dilution rates for both maritime and diurnally forced continental cloud ensembles, the 200 latter of which are inherently transient. In addition, this theory does not account for the impacts on environmental RH on cloud dilution, and thus cannot be expected to explain this sensitivity. The assumption of a negligible shear production/loss term also restricts the applicability of this theory to weakly sheared cloud fields. Finally, the two above-mentioned assumptions (that dilution scales with other TKE source terms and A ε is fixed) may limit the accuracy of the resulting dilution estimates.

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To begin, we examine the evolution of seven quantities-sensible and latent heat fluxes (H and LE, respectively), total cloud cover (cc), cloud-base and cloud-top heights (z b and z t , respectively), cloud-base mass flux (m b ), and vertically integrated and horizontally averaged TKE-for one maritime (BOMEX) and one continental (ARM-SGP) case in Fig. 1. For BOMEX, the sensible and latent heat fluxes are constant in time (Fig. 1a) and, after an initial spin-up of about two hours, the cloud ensemble reaches a statistical quasi-steady state. The cloud cover, defined as the fraction of vertical columns containing at least one grid 210 point with q c > 0.01 g kg −1 , maintains a low value of around 10-12 %, and the cloud-base and cloud-top height are roughly constant at ∼0.5 km and ∼2.0 km, respectively. The m b fluctuates around 0.04 m s −1 throughout the simulation, and, after some initial transience, the TKE remains roughly constant at around 350 m 3 s −2 over the last 4 h (left column in Fig. 1). The second maritime case (RICO; not shown) is also characterized by a cloud-base height of 0.5-0.6 km, but the cloud layer is deeper and capped by an inversion based at around 2.0 km, which fosters sufficiently deep convection to produce precipitation The continental ARM-SGP case undergoes a pronounced diurnal cycle (right column in Fig. 1 for ARM-SGP; RACORO is not shown). Clouds initiate in the mid-morning with an increasing total cloud cover that reaches a maximum of about 25 % in the early afternoon and decreases toward the evening (Fig. 1d). Due to the warming, drying, and vertical growth of the subcloud layer, the cloud base rises from initially ∼0.7 km in the mid-morning to ∼1.3 km in the early evening. The cloud-top height 220 reaches a maximum of ∼2.9 km in the late afternoon (Fig. 1f). Unlike the roughly constant m b in BOMEX and RICO, m b is zero for several hours before rising rapidly to a maximum at around 4 h (greatly exceeding the maximum in BOMEX) and then decreasing. The TKE increases more gradually to a maximum about six times that of BOMEX at around 4 h, then decreases.
Each afternoon of the RACORO three-day simulation evolves similarly to that of ARM-SGP, except that different days exhibit different levels of cloudiness due to variations in large-scale forcing (not shown).

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The above four cases capture a range of environmental conditions in which shallow cumuli develop and form the basis of our analysis. However, since several environmental aspects vary between these cases, the attribution of dilution sensitivity to any specific parameter would be difficult with these experiments alone. Therefore, to isolate environmental parameters of interest, we perform additional experiments with initial conditions systematically varied in one or more cases. A summary of the full set of numerical experiments is given in Table 2.  Fig. 1.

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To examine the sensitivity of cloud dilution to cloud-layer RH, the initial profiles of cloud-layer RH for the BOMEX maritime case (Fig. 2a) and the ARM-SGP continental case (Fig. 2b)  In environments with higher RH, fewer entrained parcels become negatively buoyant, and thus a higher fraction of entrained air remains within the cloud core, leading to increased dilution.
The difference between the fractional dilution and detrainment rates (ε − δ) regulates the normalized vertical gradient of the convective mass flux (e.g., Betts, 1975). Since δ generally exceeds ε for all experiments (Figs. 3e and f), this difference 270 is negative and implies a diminishing cloud mass flux with height ( Fig. 4a and c). This difference is maximized in drier environments (consistent with Wang and McFarquhar, 2008), and hence the cumuli remain smaller and narrow more rapidly with height in those cases. This narrowing is demonstrated by profiles of effective cloud radius (R eff ), defined at a given level

275
where N c is the number of clouds at that height and N g the number of horizontal grid points in each cloud ( Fig. 4b and d).
The effective radius is the smallest in the driest versions of both the BOMEX and ARM-SGP cases. The slight increase in cloud-base height in drier environments stems from the mixing of drier cloud-layer air into the subcloud layer. A schematic of the mechanisms underlying the cloud-layer RH sensitivity is given in Fig. 5.

Layer depth 280
To investigate the dependence of ε on the cloud-layer and subcloud-layer depth, we focus exclusively on the nearly statistically steady BOMEX case. The strong diurnal forcing in ARM-SGP renders the layer depths transient and difficult to control.

Cloud-layer depth
We vary the initial cloud-layer depth, (z cld ) 0 , by raising or lowering the base of the trade-wind inversion from the CTRL case with the θ lapse rate held fixed in each layer. Three experiments are conducted, one with a 250-m shallower cloud layer than 285 CTRL (CL-SHAL1), one with a 250-m deeper cloud layer than CTRL (CL-DEEP1), and one with a 500-m deeper cloud layer than CTRL (CL-DEEP2) (Fig. 6a). To maintain quasi-equilibrium, the profiles of the large-scale forcings within the cloud layer (subsidence and longwave radiative cooling) are shortened or lengthened accordingly.
For each experiment, ε SC95 profiles are diagnosed for each ensemble member using Eq. (1) and then averaged over all six ensemble members (Fig. 6b), with the ensemble standard deviation shown by yellow shading. These profiles are similar over 290 the lowermost 500 m of the cloud layer and only diverge above ∼1.2 km, with more rapid vertical decay for the shallower cloud layers. This apparently weak sensitivity is reflected by the nearly constant layer-averaged ε SC95 as (z cld ) 0 doubles from 0.75 m (CL-SHAL1) to 1.5 m in (CL-DEEP2) ( Fig. 7a and c).
To physically interpret the above sensitivity (or lack thereof), we use the TKE similarity theory of Sect.

with constant
A ε = 0.035 (as in Drueke et al., 2019). The value of ε TKE depends on three larger-scale parameters: m b , z cld , and CAPE 295 (Eq. (8)). The m b is evaluated at the lowest height at which q c co > 0.01 g kg −1 , z cld is the depth of the cloud-core layer where q c co > 0.01 g kg −1 , and CAPE is integrated over this same layer. Evaluating Eq. (8) using these parameters reveals a good match to the ε SC95 diagnoses (Fig. 7a). While ε TKE agrees very well with ε SC95 for the CTRL, CL-DEEP1, and CL-DEEP2 experiments, it overestimates the dilution rate for the CL-SHAL1 by ∼0.1 km −1 . While this estimate lies outside the one-standard-deviation range of ε SC95 , the relative error is still less than 10 %.

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As shown in Fig. 7b, m b is similar for the four cases, which is expected because the initial subcloud conditions are held fixed. Similar to the initial range of (z cld ) 0 specified above (0.75-1.5 km), z cld during the 3-h analysis period ranges from ∼0.95 m in CL-SHAL1 to ∼1.65 m in CL-DEEP2 (Fig. 7c). These increases in z cld , which tend to decrease ε TKE in Eq. (8), are approximately offset by increases in CAPE, the cube root of which increases similarly to z cld ( Fig. 7d and e). As a result, ε remains relatively constant as (z cld ) 0 is varied.

Subcloud-layer depth
Analogous to the above experiments, we vary the initial subcloud-layer depths by raising (SCL-DEEP1) or lowering (SCL-SHAL1) the cloud base by 250 m from its value in CTRL, with θ and q v lapse rates again held fixed in each layer (Fig. 8a). To maintain a constant cloud-base RH for all experiments, we increase (SCL-SHAL1) or decrease (SCL-DEEP1) the subcloud q v accordingly. Also, we keep the cloud-layer vertical wind shear fixed at the CTRL value of 1.8 m s −1 km −1 , with shear bases Whereas the diagnosed ε SC95 near cloud base modestly but systematically decreases as the subcloud-layer depth is increased, the value near cloud top is similar between the three cases (Fig. 8b). As a result, the layer-averaged ε SC95 decreases by a total of about 15 % for the near-tripling of the subcloud-layer depth between SCL-SHAL1 and SCL-DEEP1 (Fig. 9a).
As before, we use the TKE theory, as embodied in Eq. (8), to physically interpret the results. The theory accurately captures 320 the modest sensitivity of ε SC95 to subcloud-layer depth (Fig. 9a). Similar to the offsetting tendencies in Sect. 3.2.1, a ∼10 % increase of z cld is compensated by a 10 %-increase of CAPE 1/3 over these three experiments (Figs. 9c, d and e). For its part, the m b tends to increase with subcloud-layer depth (Fig. 9b), likely owing to stronger, less hydrostatic turbulence in deeper subcloud layers (e.g., Tang and Kirshbaum, 2020). With fixed A ε and offsetting effects on z cld and CAPE, the modest increase of m b in deeper subcloud layers leads to modestly smaller ε TKE . An elaboration on the physical link between ε TKE and m b 325 is provided below.

Continentality
Along with the aforementioned sensitivity of ε to RH, a second sensitivity is apparent in Fig. 3 Based on these 18 experiments, a robust sensitivity of ε SC95 to continentality is found, in that the cloud dilution is consis-335 tently larger for the maritime cloud fields than for the continental ones (Fig. 10a). Whereas ε SC95 averages to 0.57 km −1 over the continental cases, it averages to 1.24 km −1 for the maritime cases, and hence ε ocean ≈ 2.2 × ε land . The TKE similarity theory gives similar values, with a mean ε TKE of 1.18 km −1 in the maritime cases and 0.64 km −1 in the continental cases (Fig. 10b). The similarity between the theoretical ε TKE and the model-diagnosed ε SC95 suggest that the TKE theory adequately captures the simulated trends.

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A closer look at CAPE, z cld , and m b again facilitates a physical explanation of the sensitivities of ε TKE and ε SC95 . CAPE and z cld vary significantly between the different experiments, but lack a systematic sensitivity to continentality (Figs. 11a and b). Furthermore, CAPE and z cld compensate each other for all but two experiments when the cloud layer becomes very shallow (Fig. 11c). The m b , on the other hand, is persistently much smaller for the maritime cases than for the continental cases (Fig. 11d). On average, m b for the continental cases is 2.7 times the m b value of maritime cases. This trend is owing 345 to stronger sensible heat fluxes over land (Figs. 1a and f), which drive more intense subcloud turbulence. The more vigorous boundary layer updrafts, in turn, generate larger m b by transporting more kinetic energy across cloud base (Brown et al., 2002, Fig. 11d). When raised to the 2/3 power in Eq. (8), this trend in m b almost fully explains the sensitivity of cloud dilution to continentality.
sides, the entrainment term on the left-hand side exhibits a stronger sensitivity to m b (∼ m 5/3 b ) than does the buoyancy flux/dissipation term on the right-hand side (∼ m 1 b ). For a given z cld , dissipation depends on the turbulent velocity scale w * , which is only a weak function of m b in Eq. (6). While entrainment also depends on w * , it additionally depends directly on m b , as εm b represents the flux of entrained air into the cloud. The stronger sensitivity of the entrainment term to m b implies that, when m b is varied, ε must act to partially offset these changes. Thus, to maintain the energetic scaling between TKE 355 source terms (buoyancy flux and dissipation) and entrainment flux as m b increases, ε must decrease.

Discussion
As discussed in Sect. 1, various studies have found strong correlations between horizontal cloud area and cloud dilution, arguing that wider clouds better protect their inner cores from the suppressive effects of entrainment (Khairoutdinov and Randall, 2006;Kirshbaum and Grant, 2012;Rieck et al., 2014;Rousseau-Rizzi et al., 2017). In the present study, however, this perspective 360 has not yet been taken-only the reverse problem (the impacts of mixing on cloud horizontal area) was considered in Sect. 3.1.
In this section, we evaluate the link between cloud width and cloud dilution in more detail.
In the cloud-layer-depth sensitivity experiments of Sect. 3.2.1, increased cloud-layer depth led to deeper clouds (as expected) but not to increased cloud widths, at least over the bulk of the cloud layer (Fig. 12a). Thus, increased cloud depth does not always correspond to increased cloud width, which reflects the anisotropic nature of turbulence in conditionally unstable layers 365 (where the saturated updrafts are statically unstable and the unsaturated downdrafts are statically stable). Nevertheless, the similar insensitivity of cloud width and cloud dilution to cloud-layer depth may reflect a causal link between these two metrics.
A modest sensitivity of cloud width to subcloud-layer depth was found in Sect. 3.2.2, with R eff at cloud base increasing by around 40 % for a near-tripling of the layer depth. This result suggests that turbulence in the subcloud layer is also anisotropic (though less so than in the cloud layer) and/or that the widest thermals in deeper layers may be suppressed by stronger ad-370 verse vertical pressure gradients (e.g., Tang and Kirshbaum, 2020) and therefore exhibit less penetration into the cloud layer.
However, the latter speculation is contradicted by the shallow-cumulus observations of Lamer and Kollias (2015), where wider subcloud turbulent structures were found to be associated with stronger updrafts. In any case, the decreased cloud dilution in simulations with wider clouds again is consistent with the notion of wider clouds being less diluted.
The cloud-layer RH experiments, however, reveal that cloud size and dilution rate are not always negatively correlated. The 375 clouds that form in the drier environments exhibit smaller R eff but are less diluted than those in moister environments (Figs. 4 and 13). In these cases, the smaller dilution rates in the drier flows are caused by the detrainment of negatively buoyant mixtures of entrained air, and this detrainment leads to a narrowing of the cloud. Thus, while cloud size may play an important role in regulating cloud dilution, it is also a reflection of the dilution and detrainment history of the cloud.
Similarly, cloud size alone is not a reliable indicator of the dilution rate for the different maritime and continental cases 380 (Fig. 14). While the larger clouds in ARM-SGP are less diluted than the shallower clouds in BOMEX, clouds of similar if not larger size in RICO are even more diluted than those in BOMEX (Fig. 10a). The ε SC95 in RICO is the largest of all cases under consideration while the R eff in RICO is comparable to some RACORO cases (Fig. 14). The cloud-size variation of the different RACORO cases (each of the three days is counted as a separate case) is due to the changes in the large-scale forcings.
These findings reinforce that the horizontal cloud size does not single-handedly explain all of the sensitivities of cloud dilution.

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It is important to note that CTRL-HR simulations exhibit systematically smaller cloud than the experiments with lower grid spacings (Hanley et al., 2015).
A more robust physical interpretation of the majority of the cloud-dilution sensitivities evaluated herein is provided by the TKE similarity theory. This theory has been used to gain insight into the weak sensitivity to the cloud-layer depth, the weak dependence on subcloud-layer depth, and the strong sensitivity to continentality. However, the theory was not used to explain 390 the moderate sensitivity to cloud-layer RH because environmental humidity is neglected in the theoretical formulation (Grant and Brown, 1999;Grant and Lock, 2004). Moreover, it cannot explain sensitivities to the vertical wind profile, which, as will be shown in a companion paper, can also strongly impact cloud dilution and detrainment.
The buoyancy-sorting concept provides a useful physical explanation of the sensitivity of cloud dilution to cloud-layer RH found herein, as well as that in the numerical simulations of Stirling and Stratton (2012) and Bera and Prabha (2019) and the 395 observations of Lu et al. (2018). For a given amount of cloud-environmental mixing, a smaller fraction of entrained air becomes negatively buoyant and detrains in moister environments, resulting in more diluted cloud cores. On the other hand, our findings contrast with the conclusion of Derbyshire et al. (2004) andde Rooy et al. (2013) that cloud dilution decreases with increasing cloud-layer RH. However, their simulations explored a very different part of parameter space, with much larger cloud-layer RH variations (25 to 90 %) and much deeper, precipitating convection.

400
The limited range of cloud-layer RH sampled our experiments stemmed from the need to maintain a certain level of consistency between different sensitivity tests. To generate comparable cloud fields in the different experiments, we held the cloudbase RH fixed and only modified the cloud-top RH (with a linear variation of q v in-between). As a result, even large changes in the latter did not greatly change the layer-averaged RH. Therefore, we caution that the positive sensitivity of shallow-cumulus dilution to the small changes in layer-averaged RH found herein may not apply throughout the very large parameter space of 405 atmospheric convection.
Although simulated dilution rates for shallow cumuli tend to be larger than those for deep cumuli (e.g., Khairoutdinov and Randall, 2006;Del Genio and Wu, 2010), the sensitivity of the dilution rate to cloud-layer depth within the shallow cumulus regime was found to be minimal, at least for a fixed cloud-layer lapse rate. Similar to the cloud-layer RH experiments, the small range of cloud-layer depth from 1.1 in CL-SHAL1 to 1.8 km in CL-DEEP2 limits the generality of the result.

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However, extending the experiments to even deeper cloud layers is problematic since the characteristics of the cloud field would change significantly. In particular, deeper clouds would begin to produce substantial precipitation, which could change their morphology and environmental sensitivities (Khairoutdinov and Randall, 2006;Grant, 2007). Similar considerations also explain why the subcloud-layer depth was not changed more substantially in the corresponding experiments.
This study has advanced the physical understanding of cumulus dilution and its sensitivity to environmental conditions using large-eddy simulation. As used herein, the term "dilution" is synonymous to bulk entrainment, where cloud-environmental mixing is inferred through the vertical gradient of conditionally averaged variables within the buoyant cloud core. The term "entrainment" is reserved for all of the air ingested into the cloud, even if the air differs from environmental air and/or does not contribute to cloud-core dilution. For generality and to explore the role of continentality, both maritime and continental shallow-420 cumulus cloud ensembles were simulated. The sensitivity of cloud dilution to four different environmental thermodynamic parameters, including cloud-layer relative humidity (RH), cloud-and subcloud-layer depths, and continentality, was quantified and interpreted. These conditions were chosen because the sensitivities of dilution to them are either uncertain (cloud-layer RH), unexplained (continentality), and/or largely unexamined (cloud-and subcloud-layer depths).
Systematic experiments with different initial thermodynamic profiles revealed a positive correlation between cumulus di-425 lution and cloud-layer RH. Specifically, a 10 %-increase in RH led to a 25 % and 47 % increase in fractional dilution in the maritime BOMEX case and the continental ARM-SGP case, respectively. This finding was explained by the concept of buoyancy sorting (Fig. 4): in drier environments, a larger fraction of entrained air becomes negatively buoyant and detrains, leaving a smaller but less dilute cloud core. This trend is consistent with the observations of Lu et al. (2018)  Moreover, cloud dilution was found to be strongly sensitive to continentality, with fractional dilution rates in maritime clouds about twice those in continental clouds. To explain this sensitivity, a similarity theory of shallow-cumulus transports based on the turbulent-kinetic-energy (TKE) budget was used (Grant and Brown, 1999;Grant and Lock, 2004;Kirshbaum and Grant, 2012). This theory suggests that this sensitivity can be attributed to a larger cloud-base mass flux (m b ) over land, driven by 435 larger surface sensible heat fluxes and more intense subcloud turbulence. From an energetics perspective, this sensitivity arises from the theoretical assumption that the kinetic energy imparted to entrained air scales with the dominant TKE source terms (buoyancy flux and dissipation). The former is more sensitive to the cloud-base mass flux than the latter, and thus, to maintain equilibrium, changes in m b must be partially offset by changes in the cloud dilution rate.
Relative to the sensitivities of cloud dilution to cloud-layer RH and continentality, the sensitivities to both cloud-layer depth 440 (a 2-3 % change in dilution for a doubling of the layer depth) and subcloud-layer depth (a 15 % decrease in dilution for a tripling of the layer depth) were weak. These minimal sensitivities may be surprising given the tendency for simulated cloud dilution to weaken during the transition from shallow to deep convection (e.g., Khairoutdinov and Randall, 2006;Del Genio and Wu, 2010). Using the TKE similarity theory, the weak sensitivity to cloud-layer depth was interpreted to result from offsetting effects of increased moist instability, which tends to increase dilution, and increased cloud depth, which tends to weaken 445 dilution. The slightly stronger sensitivity of cloud dilution to subcloud-layer depth, like the sensitivity to continentality, was dominated by changes in cloud-base mass flux. Deeper subcloud layers produced larger cloud-base mass fluxes, which thereby decreased the dilution rate.
The above physical interpretations do not invoke a causal relationship between horizontal cloud size and cloud dilution, which is a common perspective for explaining entrainment/dilution sensitivities (e.g., Khairoutdinov and Randall, 2006;Kir-450 shbaum and Grant, 2012;Rousseau-Rizzi et al., 2017). In particular, it is often found that wider clouds tend to undergo less dilution than narrower clouds. The cloud-size perspective was found to qualitatively apply to the cloud-layer and subcloudlayer depth experiments, where minimal to slight increases cloud size were accompanied by minimal to slight decreases in dilution. However, this perspective did not apply to the cloud-layer RH or continentality experiments, where the relationship between cloud size and dilution was reversed or inconclusive. Based on these findings, we support the conclusion of Hannah 455 (2017) that the relationship between cloud size and cloud dilution is complex and interactive, and inferences concerning this relationship should be made with care.
Additional steps are needed to strengthen the understanding of shallow-cumulus dilution and improve parameterizations of shallow convection in large-scale models, most of which do not explicitly account for the sensitivities found herein (e.g., Webb et al., 2015;Lu et al., 2016). One important step is to identify and examine the full range of environmental sensitivities of 460 cloud dilution, which may extend beyond the parameters considered to date. As a step in this direction, a companion paper will examine, using a similar LES approach, the impacts of the vertical wind profile on cloud dilution. However, perhaps the most efficient way to identify all of the sensitivities of cloud dilution is through climatological observations of cloud dilution in different locations. Such an ambitious objective may be possible using one or more of the ground-based dilution retrievals evaluated in Drueke et al. (2019), and future efforts are planned to conduct long-term retrievals at multiple ARM observatories.

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Code and data availability. The Bryan Cloud Model (CM1) is available under http://www2.mmm.ucar.edu/people/bryan/cm1/. Primary data and scripts used in the analysis and other supplementary information that may be useful for reproducing the author's work are archived by the Department of Atmospheric and Oceanic Sciences (McGill University) and can be obtained by contacting sonja.drueke@mail.mcgill.ca.
Author contributions. S.D. and D.J.K. developed the scientific question, and S.D. conducted the simulations and carried out the analysis under the supervision of D.J.K. and co-supervision of P.K. S.D. prepared the paper with contributions from D.J.K. and P.K.