As in Part 1 of this study, we conduct LES of shallow-cumulus ensembles using the Bryan Cloud Model version 17 CM1;. In LES mode, CM1 accurately reproduces the findings from past LES inter-comparison studies of shallow cumuli . To examine diverse cloud fields, we consider one maritime case and one continental case, the former based on the LES inter-comparison study of the Barbados Oceanographic and Meteorological Experiment (BOMEX) by , and the latter based on the LES inter-comparison of shallow cumuli at the US Atmospheric Radiation Measurement (ARM) Southern Great Plains (SGP) observatory in Oklahoma .

The model configuration is similar to that in , with a monotonic fifth-order weighted essentially non-oscillatory (WENO) advection scheme for both scalars and velocity; a third-order Runge–Kutta time-differencing scheme; and periodic horizontal, semi-slip lower, and free-slip upper boundary conditions as well as an f-plane approximation. The Coriolis force is applied to wind perturbations from the initial, geostrophic profile. A horizontal grid spacing of 32 m is used to adequately resolve the turbulent circulations of interest. The BOMEX horizontal domain size of 6.4×6.4 km2 is left unchanged from , while the ARM-SGP domain size is doubled from 6.4×6.4 km2 in to 12.8×12.8 km2 to capture the larger-scale circulations of this cloud field. For each experiment, an ensemble of six members is conducted, each with a different field of small-amplitude random perturbations added to the initial potential temperature and water-vapor mixing ratio fields. The results presented for each case are averaged over this ensemble.

We conduct various idealized experiments to quantify the impacts of the initial wind profile on the cloud dilution. To examine the impacts of cloud-layer vertical shear, the first set of experiments (CL-SHR) initializes zero wind in the subcloud layer and positive, linear westerly vertical shear in the cloud layer (Fig. ). While the absence of subcloud winds differs from the standard configurations of these cases, it limits the development of subcloud vertical shear that, as will be seen, may indirectly affect cloud dilution. To ensure that the shear layer is fully contained within the cloud layer, the shear base is placed at 720 m in BOMEX and 1000 m in ARM-SGP (dashed lines in Fig. ). Zonal vertical shears ranging from 0 to 9 ms-1km-1 (CTRL to S9), in increments of 3 ms-1km-1, are applied from the shear base to the domain top. For BOMEX, we also include an experiment with vertical wind shear of 1.8 ms-1km-1, matching that of (Table ).

In a second set of experiments (WIND), we evaluate the sensitivity of simulated cloud dilution to vertically uniform zonal geostrophic winds of magnitude U. In line with the prevailing wind directions at the two locations, we consider easterly winds in BOMEX and westerly winds in ARM-SGP (Table ). The winds increase from zero (CTRL) up to 10 ms-1 (U10; dashed lines in Fig. a–d). Finally, to examine the impacts of subcloud geostrophic vertical shear on cloud dilution, a third suite of experiments vary the near-surface shear (SCL-SHR). These profiles are identical to those in WIND except for having zero surface wind and a layer of linear zonal shear over the lowest 250 m (dashed lines in Fig. e and g). Individual simulations from this suite of experiments are named based on their shear magnitude; for example, the case of 40 ms-1km-1 of near-surface shear is named US40.

Due to the short durations of active cloud development (<6 h) in the BOMEX and ARM-SGP simulations, we do not apply any forcings to maintain the wind profile at its initial values. The wind profiles thus vary with time, mainly through the action of subcloud and cloud-layer vertical mixing. Nevertheless, as will be seen, the qualitative differences between the various cases are maintained throughout the simulations, although slightly reduced over time (see solid lines in Figs. –). To determine whether sensitivities to cross-grid flow like those highlighted by affect our model results, we have compared various runs with a fixed and a translating grid (at the approximate average speed of the cloud-layer flow). The differences between these runs were minimal, suggesting that such effects are not significant for our model configuration. Therefore, for consistency, all simulations described herein use a stationary grid.

The maritime BOMEX CTRL case represents a typical trade-wind cloud field, except for the lack of ambient winds. The surface heat fluxes, large-scale advection and subsidence tendencies, and simulated convection come into balance to yield a statistically quasi-steady flow over 3–6 h . The cloud base stays at roughly 500 m throughout the simulation, and the cloud top extends above the base of the trade-wind inversion at 1.5 km Fig. 1 of. A horizontal cross section of w at the midpoint of the subcloud layer and a time of 4 h shows a cellular turbulence pattern with variations on a broadly similar scale (∼500 m) as the subcloud-layer depth (Fig. a). The small-scale subcloud turbulence gives rise to small active cumuli with mean radii at the level of free convection (RLFC) of 80 m (Fig. c). Active clouds are defined as clouds possessing a positively buoyant and ascending internal core, and a circular cloud shape is used to infer RLFC based on the horizontal area occupied by the cloud.

The continental ARM-SGP CTRL case, in contrast, exhibits a time-evolving cloud field forced by the diurnal cycle of the surface heat fluxes . Shallow cumuli first initiate at about 11:00 local solar time (LST) and dissipate by around 20:00 LST. Over that time, the cloud base rises from 0.6 to 1.3 km . As shown by the w cross section at the midpoint of the subcloud layer at 13:15 LST (Fig. b), the deeper subcloud layer in ARM-SGP gives rise to larger horizontal circulations than in BOMEX. As a result, RLFC in ARM-SGP is larger (Fig. d), with an averaged value (207 m) more than double that of BOMEX. The cloud droplet number concentration is smaller in the maritime BOMEX experiments (100 cm-3) than in the continental ARM-SGP simulations (250 cm-3).

Based on the time evolution of the BOMEX and ARM-SGP simulations, we define analysis periods to be used for the detailed calculations to follow. These periods are selected to avoid model spin-up or cloudless intervals, thus focusing on the well-developed turbulent cloud fields of interest. The quasi-stationarity of the BOMEX case permits the use of a relatively long 3 h averaging period, covering 3–6 h. Due to the diurnal evolution of the cloud field in the continental ARM-SGP experiments, a shorter averaging time of 1 h is used, running from 14:00–15:00 LST. Unless otherwise specified, all calculations herein are conducted during these analysis periods.

Over the course of the BOMEX CL-SHR simulations, the shear base lowers from its initial value (720 m) down to about 500 m due to cloud-layer vertical mixing. As a result, the cloud base and shear base nearly coincide over the analysis period (solid lines in Fig. a and b). The flow remains predominately westerly with a weak northerly component in the cloud layer. Similarly, in the ARM-SGP CL-SHR simulations, the shear base over the analysis period roughly coincides with the cloud base at 1050 m (Fig. c and d). Although the subcloud winds remain weak, they turn more northerly due to larger surface drag, enhanced turbulent mixing, and a stronger Coriolis force. In both cases, the cloud-layer shear weakens modestly over time (Fig. a and c). In the most extreme S9 case, the cloud-mass-flux-weighted shear weakens to 6.4 ms-1km-1 (BOMEX) and 6.9 ms-1km-1 (ARM-SGP) over the analysis period, a reduction of around 25 %. Thus, despite the gradually weakening shear, the two sets of simulations exhibit comparable shear magnitudes throughout.

As the cloud-layer shear is increased, the subcloud layer is minimally affected, with similar w fields in the CTRL and S9 cases (cf. Figs. a–d and a and b). In the cloud layer, the clouds widen with increasing shear, a signal that extends down to the LFC. The RLFC distributions shift toward larger scales in both BOMEX and ARM-SGP (Fig. c and d) but much more so in BOMEX. The mean RLFC increases by 27 % in BOMEX, compared to 14 % in ARM-SGP.

Although not shown for brevity, the simulated cumuli respond to the imposed vertical shear in expected ways: (i) they tilt downshear with height e.g.,; (ii) they develop pressure-anomaly dipoles straddling the clouds along the shear axis, with high pressure upshear and low pressure downshear e.g.,; and (iii) their buoyant cloud cores shift to the upshear side of the cloud e.g.,. Moreover, the cloud-top heights decrease under stronger vertical shear, suggesting a shear-induced cloud suppression. Compared to the CTRL experiments, the cloud-top height decreases by 15 %–20 % in the S9 versions of the BOMEX and ARM-SGP cases (not shown).

Diagnosis of the simulated bulk fractional entrainment rate, or simply the “dilution rate”, follows the formulation of hereafter SC95. This quantity measures how much pure environmental air would need to be entrained to achieve the simulated core dilution. Contrary to direct entrainment calculations, it does not quantify the actual mixing across the cloud perimeter. Dilution is evaluated based on the budget of the total water specific humidity (st): Estenv-stco=Mco∂stco∂z+∂acoρw′st′‾co∂z1+acoρ∂stco∂t-acoρ∂st‾∂tforcing, where E represents the bulk entrainment rate. The subscript “co” denotes conditional averages within cloud cores and “env” indicated the averages over the environment, defined as all non-core regions. The core mass flux (Mco) is defined as Mco=ρacowco, where ρ is the air density, aco the cloud-core fraction, and w the vertical velocity. The plain overbar denotes the horizontal domain average, while the overbar indexed “co” represents the conditional average of the fluctuations of the cloud cores with respect to the cloud-core average. The right-hand side of Eq. () considers the changes to the conserved st with height (z) and in time (t), along with changes to st owing to vertical turbulent fluxes and large-scale forcings. To obtain the bulk dilution rate (ε), we divide E by Mco. Instantaneous vertical profiles of ε are calculated at each model output time over the full horizontal domain, at all vertical levels where cloud-core grid points are found. To compare ε across the different simulations, we perform some averaging to obtain representative values. First, 15 min running averages are calculated from the 5 min model output data, from which bulk cloud-layer averages are computed over the central 50 % of the cloud layer. Both the vertical profiles and the bulk values are then averaged over the analysis period.

The ε profiles thus obtained increase monotonically with vertical wind shear (Fig. ). This sensitivity originates at cloud base, increases to a maximum near the cloud-layer midpoint, and decreases rapidly near cloud top. Averaged over the central 50 % of the cloud layer, ε in the BOMEX S9 experiment (2.6 km-1) is about 50 % larger than in the CTRL simulation (1.7 km-1). This sensitivity is comparable but slightly weaker (∼40%) in ARM-SGP.

The initially uniform zonal velocity profiles from the WIND simulations evolve into vertically varying profiles with strong low-level vertical shear. At analysis time, the zonal wind is strongly forward-sheared near the surface, exhibits a nearly constant value within the central part of the subcloud layer, and then exhibits additional forward shear extending into the lower cloud layer (Fig. a and c). The Coriolis force induces a cyclonic turning of these frictionally decelerated winds, leading to a meridional wind component that is maximized within the subcloud layer (Fig. b and d). As with the CL-SHR experiments, the stronger surface drag and Coriolis force in ARM-SGP leads to stronger frictional deceleration and cyclonic wind turning than in the corresponding BOMEX simulations.

In BOMEX U10, the low-level shear organizes the subcloud flow into longitudinal bands, or rolls, aligned with the low-level winds (Fig. a), which contrasts with the more disorganized convection in ARM-SGP U10 (Fig. b). These differing responses likely stem from the different surface heating rates in the two cases. The larger surface buoyancy flux in ARM-SGP yields a more buoyancy-dominated convective boundary layer (CBL), which overwhelms any shear-induced turbulence organization. The potential for convective rolls may be assessed based on the Monin–Obukhov length (L), where L represents the height at which buoyancy dominates over shear in the production of turbulent kinetic energy (TKE): 2L=-θ‾vu′w′‾2+u′w′‾20.75kgw′θv′‾, where k=0.4 is the von Kármán constant; g is the gravitational acceleration; θv is the virtual potential temperature; u, v, and w are x, y, and z wind components; primes denote perturbations from a temporal or spatial average (denoted by overbars); and all quantities are evaluated at the surface e.g.,. Negative L corresponds to CBLs, with smaller magnitudes reflecting more dominant buoyancy production.

Taking zi as the subcloud-layer depth and evaluating L at the surface, and summing resolved and subgrid fluxes in Eq. (), we obtain -zi/L=1.6 and -zi/L=9.5 for the BOMEX and ARM-SGP U10 cases, respectively. The smaller BOMEX value is more favorable for rolls, falling into the 0≤-zi/L≤4.5 range reported by as conducive for roll development. This roll organization at larger U in BOMEX is associated with increased subcloud length scales: the wavelength of the spectral peak in the subcloud kinetic-energy spectrum increases by about 110 % from CTRL to U10 (Fig. c) as the mean RLFC increases by ∼70% (Fig. e). This systematic increase in cloud size with subcloud shear is consistent with the conclusion by about vertical shear and updraft width in supercells. For ARM-SGP, the subcloud length scales also increase, but by a much smaller amount (20 %; Fig. d), and RLFC increases minimally (Fig. f).

The BOMEX ε decreases with increasing U over most of the cloud layer, except near cloud base (Fig. a). Averaged over the central 50 % of the cloud layer, ε decreases by 25 % from CTRL to U10. In contrast, the ARM-SGP ε changes minimally across the experiments (Fig. b), with the cloud-layer-averaged ε decreasing by only 7 % from CTRL to U10. In both BOMEX and ARM-SGP, the ε sensitivity depends on height, with the dominant trend lying in the mid-to-upper cloud layer and non-systematic variations in the lower cloud layer.

The only initial difference between the WIND and SCL-SHR experiments is the latter's geostrophic subcloud wind shear. Although vertical mixing modifies the wind profiles over time, they maintain stronger vertical wind shear in the subcloud and lower cloud layer than the corresponding WIND cases (Fig. e–h). The strengthened cloud-layer shear is most pronounced in the BOMEX US40 case, where the shear extends up to 1.5 km, compared to only 1 km in the BOMEX U10 case. As in the WIND experiments, the subcloud shear tends to organize the subcloud turbulence into shear-parallel rolls in BOMEX (but not in ARM-SGP), with even larger increases in the mean of the cloud-size distribution (not shown). Again, ε generally decreases with increasing winds, with a larger cloud-layer-averaged decrease in BOMEX (22 %) than in ARM-SGP (13 %) between the CTRL and US40 cases (Fig. c and d).

The ε sensitivity in BOMEX is characterized by a positive trend over 0.5–0.8 km that reverses to a negative trend above (Fig. c). This feature may be owing to a combination of two effects, the first being the positive sensitivity of ε to vertical shear established in the CL-SHR experiments. As U increases, so does the lower-cloud-layer shear in SCL-SHR, which may yield the positive sensitivity of ε to U over the lower cloud layer (Fig. c). A second effect is the possibility of elevated cloud initiation by vertically propagating internal gravity waves. As previously noted, the SCL-SHR flows exhibit wider subcloud updrafts and sharper cloud-base shears as U is increased (Fig. e). Both factors favor vertically propagating waves via the “obstacle effect” e.g.,, in which wave disturbances are forced by airflow over clouds penetrating into the cloud layer. Compared to CTRL, the US40 case exhibits a much wider distribution of cloud-base heights over 0.5–0.8 km (not shown). Such heterogeneity in cloud-base height complicates the interpretation of ε, for reasons outlined in . Namely, vertical variations in conditionally averaged core conserved properties cannot be unambiguously attributed to cloud dilution, because they can also be explained by variations in the source layers of different clouds. Thus, the trend in ε over 0.5–0.8 km may be more reflective of variations in cloud-base height than of variations in cloud dilution.

Although the WIND and SCL-SHR experiments reach a common conclusion that subcloud-layer shear tends to decrease cloud-layer dilution, the consideration of both sets of simulations aids physical interpretation. For one thing, it shows that geostrophic shear is not required to realize this trend; even frictionally induced shear at the surface, which is present in all flows, suffices. Also, as mentioned above, the SCL-SHR experiments reveal an effect that was absent in WIND: a systematic enhancement in lower-cloud-layer dilution. Attribution of this effect to the enhanced lower-cloud-layer shear is facilitated by the WIND experiments, where the lower-cloud-layer shear and dilution vary much less between the different cases.

As mentioned in Sect. , the conditionally averaged cloud-core w (or wco) may influence ε by controlling the timescale over which ascending clouds or cloud cores are exposed to environmental air e.g.,. Although such a one-way causal sensitivity between wco and ε likely oversimplifies their relationship, the correlation between wco and ε still merits examination. In the CL-SHR experiments, wco at cloud base is larger in ARM-SGP (2.6 ms-1) than in BOMEX (1.7 ms-1) (Fig. a and b). After a brief decrease between cloud base and the LFC, wco rebounds to maxima near cloud top of 2.5–5 ms-1 in ARM-SGP and 1–2.5 ms-1 in BOMEX. The larger wco in ARM-SGP is owing to stronger surface heating, which drives stronger subcloud turbulence and cloud-base updrafts, in conjunction with larger cloud-core buoyancy bco (Fig. b and d), which enhances vertical motions above the LFC.

Cloud-layer shear induces a systematic reduction in wco in the CL-SHR experiments, which is expected given the tendency of this shear to tilt and weaken cumulus updrafts e.g.,. Near cloud base, where wco is dominated by subcloud momentum, the differences between the various cases are small. These differences increase with height to a maximum near the cloud tops. Averaged over the central 50 % of the cloud layer, BOMEX exhibits a 47 % decrease in wco between CTRL and S9, compared to a 27 % decrease in ARM-SGP. Hence, increased cloud-layer shear is associated with decreased wco and larger ε, consistent with the findings of .

For the BOMEX WIND and SCL-SHR experiments, wco is again largest in the CTRL cases and decreases with increasing U (Fig. c and e). Cloud-layer averages of wco decrease by 8 % and 27 %, respectively, between the CTRL and the end members of each suite (U10 and US40). By contrast, the corresponding ARM-SGP experiments are nearly insensitive to U, with only a 7 % and 6 % decrease in cloud-layer-averaged wco. Unlike the CL-SHR experiments where wco and ε varied inversely, ε decreases with decreasing wco in these experiments. Thus, wco cannot be the only factor regulating the simulated dilution rate. Although the plot of wco-1 against ε for all simulations indicates a large correlation coefficient (R=0.93), the differing trends of the CL-SHR and WIND/SCL-SHR experiments are obvious (Fig. a). All continental experiments are more resilient to subcloud- and cloud-layer wind shear and show weaker sensitivities to the imposed changes in geostrophic winds, particularly in the WIND and SCL-SHR experiments. When only the maritime experiments are considered, the correlation coefficient between ε and wco is substantially reduced (R=0.65). Thus, while wco strongly influences (and/or is influenced by) ε, other factor(s) must also be important.

To investigate the processes regulating wco, we use the core-averaged w equation in CM1 following: 3DwDtco=-cpθρ∂π′∂zco+bco-εwwco21-aco, where cp is the specific heat of dry air at constant pressure, θρ is the density potential temperature, π is the Exner function and π′ its perturbation relative to the horizontal average, εw is the fractional entrainment rate of w, and the effects of subgrid turbulent mixing are neglected. The dominant terms on the right side of Eq. () are the first two (pressure gradient and buoyancy) e.g.,, and we henceforth neglect the entrainment term because it has been found to be small e.g.,.

The expected tendency for cloud-layer shear to enhance the adverse vertical perturbation pressure gradient (or VPPGco) is reproduced in the CL-SHR experiments but more strongly so in BOMEX than in ARM-SGP (Fig. a and c). To interpret why the BOMEX VPPGco is more sensitive to the shear, we use the linear theory of shallow convection in , who found that the VPPG-induced updraft suppression depends on the cloud width and layer depth (Appendix ). Because narrower and taller clouds are more tilted by the shear than are wider, shallower clouds, they experience a larger VPPGco enhancement with increasing shear. The convective growth rate (σ) calculated using the linear theory for both BOMEX and ARM-SGP is more than halved between the CTRL and S9 cases. The marginal reduction owing to the shear may be measured by the ratio of the growth rates in the S9 and CTRL cases, which is smaller for BOMEX (0.33) than for ARM-SGP (0.45), suggesting greater shear-induced suppression for the narrower clouds in BOMEX. Although these differences in σ are not dramatic, they lead to large differences over time because σ is an exponential argument. For example, over a 10 min period representing the growing phase of a shallow cumulus e.g.,, the theoretical shear-induced reduction in w becomes twice as large in BOMEX as in ARM-SGP.

The second important term in Eq. () is the cloud-core buoyancy, which is highly sensitive to lateral entrainment e.g.,. argued that a faster ascending core experiences less entrainment and, hence, maintains larger buoyancy, which further accelerates its ascent. Although bco and wco both decrease with increasing vertical shear in the CL-SHR experiments, the bco sensitivity is comparatively modest and of similar strength in BOMEX and ARM-SGP (Fig. b and d). Thus, while both of the dominant terms in Eq. () tend to suppress wco in shear flows, the VPPGco term largely explains the contrasting sensitivities of wco to cloud-layer shear in BOMEX and ARM-SGP (Fig. a and b).

The adverse VPPGco also strengthens with increasing U across the BOMEX WIND and SCL-SHR experiments, at a magnitude comparable to that in the CL-SHR experiments (Fig. e and i). This result contrasts sharply with the minimal corresponding variations in ARM-SGP (Fig. g and k). Given that the VPPGco sensitivity in CL-SHR was attributed to the prescribed cloud-layer shear, it is fair to wonder if the VPPGco sensitivity in BOMEX is owing to the strong lower-cloud-layer shear that develops in the WIND and SCL-SHR suites (Fig. a–d). While this shear likely plays an important role in enhancing the VPPGco in the lower cloud layer, it gradually decays with height above cloud base. However, the VPPGco sensitivity extends throughout the cloud layer, suggesting that the lower-cloud-layer shear is not the sole cause.

Another mechanism behind the VPPGco sensitivities in the BOMEX WIND and SCL-SHR experiments is the associated sensitivity of bco to the background winds (Fig. f and j). In both sets of experiments, the maximum bco increases with U. To relate this sensitivity to VPPGco, we turn to the diagnostic decomposition of the Boussinesq pressure equation e.g.,, which may be written 4cpθρ0∇2π′=-∇⋅u⋅∇u︸∇2pd′+∂b∂z︸∇2pb′, where θρ0 is a reference value of θρ, u=(u,v,w), and pb′ and pd′ denote the buoyancy and dynamic pressure perturbation components, respectively. Away from solid boundaries, the above may be roughly simplified as 5π′∝-∇2pd′-∂b∂z. Neglecting the impacts of the pd′ term, larger vertical gradients in bco are associated with larger adverse VPPGco, with lower π′ below the level of maximum buoyancy and higher π′ above it. This implies that the stronger VPPGco sensitivity in the BOMEX WIND and SCL-SHR experiments (relative to the corresponding ARM-SGP experiments) is, in part, associated with the larger bco that develops at larger U. The cause of this wind-induced increase in bco is examined in Sect. .

The sensitivities of wco in the WIND and SCL-SHR experiments thus appear to be driven by variations in both the VPPGco and bco terms in Eq. (). While VPPGco exhibits a comparable decrease with increasing U as in the CL-SHR experiments, offsetting variations in bco lead to a muted sensitivity of wco (Fig. e, f, i, and j and Fig. c and e). The sensitivities of both wco and ε in these experiments are much stronger in BOMEX than in ARM-SGP. Unlike in the CL-SHR experiments, these differences cannot be attributed to differential effects on vertical shear on cloud tilting because the cloud-layer shear is too weak. A physical explanation for this behavior is thus required, and one will be provided in Sect. .

The analysis in Sect.  showed that wco often correlates negatively with ε, which may be explained by the role of wco in regulating the timescale over which ascending thermals are exposed to environmental air. However, the fact that wco did not exclusively control ε (Fig. a) suggests that other factors are needed to explain the ε sensitivities. One such factor is the nature of the entrained air in the core shell. As the fraction of environmental air within the shell decreases, it becomes less efficient at diluting the cloud core, for all else being equal. Following , we assume that entrained air is drawn from the cloud-core shell. At each vertical level, this shell is defined following as all non-core grid points immediately adjacent to core points, whether they are saturated or not. We further assume that core–shell air can be expressed as a linear mixture of core and environmental air at that level. Any conserved variable (e.g., the total water specific humidity, st) may thus be written 6(st)sh=μ(st)co+(1-μ)(st)en, where “en” and “sh” respectively denote the environment and cloud-core shell, and μ is the mixing fraction, or the fraction of cloud-core air within the cloud-core shell. Over all the simulations conducted herein, ε tends to increase with μ-1, with a good correlation between them (Fig. b).

For the CTRL simulations, the cloud-layer-averaged μ is 0.53 for BOMEX and 0.70 for ARM-SGP (Fig. a). Thus, a given entrainment flux yields less core dilution in ARM-SGP than in BOMEX. The imposed cloud-layer vertical wind shear in the CL-SHR experiments has a negligible effect on μ in ARM-SGP, with a marginal increase of only 0.5 % between the CTRL and S9 experiments. In contrast, BOMEX shows a more substantial 15 % corresponding increase (0.61), indicating a shift toward less dilute cloud-core shells. Despite the stronger increase in μ in BOMEX across the CL-SHR experiments, μ is always larger in ARM-SGP (Fig. a). This effect, along with the universally larger wco in ARM-SGP, largely explains why ε is always smaller in ARM-SGP than in BOMEX.

The BOMEX core shells also become less diluted as U is increased in the WIND simulations, with μ increasing from 0.53 in CTRL to 0.60 in U10 (Fig. b). This increase is the largest between the CTRL and U2.5 cases and ultimately levels off between the U5 and U10 cases. A similar lack of sensitivity between the U5 and U10 cases is apparent in the wco, VPPGco, bco, and ε profiles (Fig. c, Fig. e and f, and Fig. a, respectively). As in the CL-SHR experiments, μ varies minimally in the corresponding ARM-SGP experiments, with an increase of only 4 % from CTRL to U10. Whereas the addition of subcloud geostrophic shear in the SCL-SHR experiments has a minimal additional impact on μ in ARM-SGP (Fig. c), it leads to a stronger and more systematic increase in μ in BOMEX, to a value of 0.68 in US40. Although the variations in μ across the various suites of simulations are modest, they suffice to explain the notable variations in bco in the BOMEX WIND and SCL-SHR experiments (Fig. b, f, and j). This is shown by a simple entraining parcel calculation that explicitly accounts for the cloud-core shell (Appendix ).

What controls the sensitivity of μ in BOMEX to the initial wind profile, and why is this sensitivity lacking in ARM-SGP? These questions are addressed by looking farther afield than just the immediate core-adjacent grid points that constitute the cloud-core shell. To this end, we define a wider region surrounding the core as the “cloud-core margin”, over which μ falls from its in-core values of approximately unity down to 0.5. This margin, which encapsulates the cloud-core shell, can be viewed as a finite-width halo of mixed air surrounding the core that shields it from pure environmental air. The width of this margin is henceforth denoted Rm.

At each vertical level, we define Rm as the distance from the core edge to the nearest grid point where μ≤0.5. This quantity is evaluated separately along both coordinate axes to compare the along- and cross-wind directions. The values of Rm thus presented are averaged over the central 50 % of the cloud layer (Fig. ). While Rm is nearly axisymmetric for the CTRL cases (orange markers in Fig. ), it develops anisotropy in the sensitivity tests. In the BOMEX CL-SHR experiments, Rm grows in all directions but to the largest degree (120 %) on the downshear side of the cloud cores (Fig. a). In ARM-SGP, Rm only grows noticeably on the downshear side, by around 50 % (Fig. b). The downshear widening of Rm is consistent with the formation of a humid downshear wake e.g.,.

In the BOMEX WIND experiments, Rm again grows in all directions as U is increased (Fig. c) but to a slightly lesser degree than in the corresponding CL-SHR experiments. Moreover, the core-margin expansion is maximized on the northern and western flanks of the cores, as opposed to the east side in the CL-SHR experiments, due to the weak east-southerly shear that develops in the lower cloud layer (Fig. a and b). For the ARM-SGP WIND experiments, Rm again undergoes less variation than in the corresponding BOMEX experiments, with the largest expansion on the downshear (southeasterly) side of the cores (Fig. d). In the SCL-SHR experiments, Rm shows an even stronger sensitivity to U in BOMEX, while it remains virtually unchanged from the WIND experiments in ARM-SGP (Fig. e and f).

In absolute terms, Rm averaged over all directions is about 60 % smaller in the CTRL BOMEX case than in the CTRL ARM-SGP case (Fig. a–c). The cloud cores in BOMEX thus have narrower buffer zones surrounding them, which increases their exposure to environmental air. Wider core margins tend to exhibit larger μ because, as the transition from core to environmental air becomes more gradual, the air immediately adjacent to the core becomes more core-like. For the three sets of experiments, the ARM-SGP Rm is generally less sensitive to changes in the wind profile than the corresponding BOMEX value. Whereas the former exhibits a maximum increase of 17 % for the CL-SHR experiments, BOMEX exhibits a maximum increase of 153 % for the SCL-SHR experiments.

In general, Rm correlates well with μ (cf. Figs. a–c and ), with the lone exception being the ARM-SGP CL-SHR experiments, where a 17 % enhancement in Rm does not coincide with increased μ. We hypothesize that two factors combine to regulate Rm: the turbulence intensity within the cloud margin, which determines the lateral eddy diffusion rate, and wco, which determines the diffusion timescale. These two effects may be combined into a nondimensional number r=TKECL1/2/wco, where TKECL is the cloud-layer averaged TKE. Larger r implies increased turbulent diffusion within the core margin, which enhances Rm.

For each suite of simulations (e.g., BOMEX WIND), the relative variations in r align well with corresponding variations in Rm (Figs. g–i and a–c), again with the exception of the ARM-SGP CL-SHR experiments, where r increases more rapidly than Rm across the suite of experiments. Note that variations in r between different suites of experiments (e.g., BOMEX CL-SHR versus ARM-SGP CL-SHR) must be multiplied by a relevant length scale to permit direct comparison to the dimensional Rm. The most appropriate length scale is the mean core radius or the length scale of maximum TKECL, both of which are about twice as large in ARM-SGP than in BOMEX (Sect. ). With this factor taken into account, the variations in r become even more consistent with those in Rm.

Using r to help interpret variations in Rm, we attribute the increased Rm in the CL-SHR experiments to a joint decrease in wco (Fig. a and b) and an increase in TKECL (Fig. d). The wco effect was explained in detail in Sect. , and the TKE effect is a direct result of enhanced turbulent shear production. Although the same basic trend holds in BOMEX and ARM-SGP, the sensitivity is stronger in BOMEX due to its stronger shear-induced suppression of wco, as well as the minimal changes in μ across the ARM-SGP simulations (Fig. a). These results suggest that wco has competing effects on cloud dilution. A decreased wco lengthens the exposure of cloud cores to their environment, but it also favors a wider core margin that better shields the cores from their environment. In CL-SHR, the former effect dominates over the latter, and ε increases.

The r and Rm trends across the WIND experiments differ between BOMEX and ARM-SGP, with ARM-SGP showing minimal changes due to the joint invariance of wco and TKECL (Figs. d and e). In BOMEX, by contrast, wco decreases slightly as TKECL increases across the WIND experiments (Figs. c and e), leading to a substantial increase in r and Rm (Fig. b and h). These trends are amplified in the SCL-SHR experiments (Fig c and i), where r and Rm rapidly increase owing to a large decrease in wco combined with a large increase in TKECL (Figs. e and f).

To explain the mechanisms causing the variations in Rm across the WIND experiments, the subcloud dynamics must be considered. Due to surface friction, larger U leads to enhanced subcloud vertical shear, which gives rise to increased subcloud TKE and length scales. The larger subcloud thermals, in turn, initiate larger clouds, and RLFC increases by 73 % and 13 % across the BOMEX and ARM-SGP experiments, respectively (Fig. e and f). The fractional increases in low-level TKE in Fig. a and b are also much larger in BOMEX (50 %) than in ARM-SGP (10 %–20 %), due to the larger baseline TKE in the strongly heated ARM-SGP case. Vertical transport of this enhanced subcloud TKE, combined with the weak shear that forms in the lower cloud layer, leads to enhanced TKECL, particularly in BOMEX (Fig. e). Correspondingly, r and RLFC increase by a larger fraction in BOMEX than in ARM-SGP (Fig. h), a trend that strengthens in the SCL-SHR experiments (Fig. a and b).

The two competing impacts of reduced wco are again active in the WIND experiments, but in this case ε decreases with increasing U (Fig. a), suggesting that the buffering effect of the wider core margin dominates over the diluting effect of a longer core-exposure timescale. For ARM-SGP, on the other hand, the changes in wco and μ are both small, leading to only a minimal decrease in ε (Fig. b). The SCL-SHR experiments again exhibit similar trends in ε as those in the WIND experiments (Fig. c and d). In BOMEX SCL-SHR, larger increases in subcloud TKE and TKECL lead to even larger increases in μ. However, the stronger corresponding reduction in wco counters this effect to yield a similar ε trend as that in BOMEX WIND.

Returning to the CL-SHR experiments, the positive sensitivity of ε to vertical wind shear differs from , , and , who all found minimal corresponding sensitivities. These differences may be explained by a combination of factors. Because evaluated ε based on the vertical mass flux profile alone, they neglected the important role of detrainment in shaping that profile. Although calculated ε using a rigorous method (SC95), they used geostrophic shear profiles extending over both the subcloud and cloud layers. Given that cloud-layer shear and subcloud shear have opposing effects on ε, it is possible that these two effects largely canceled out. Similar to , used vertically constant shear profiles in their LES study. Furthermore, they employed the simpler “bulk-plume” method to ε, which neglects two of the terms in the SC95 formulation . More difficult to reconcile is the recent observational finding from that retrieved ε does not vary systematically with cloud-layer shear, in oceanic or continental locations. It is possible that offsetting effects between subcloud and cloud-layer shear also occur in reality and/or that the differences between geostrophic winds (used herein) and full winds used in could explain these differences.

Following from the results presented above, we have developed an empirical relationship for ε that takes the two key controls on cloud dilution identified herein into account. These controls are the “core-exposure effect” regulated by wco and the “core–shell dilution effect” (i.e., the amount of dilution per unit of entrainment) determined by μ. As seen in Fig. , these two quantities vary roughly inversely with ε, which guides the form of the empirical function. Based on all the experiments conducted herein, we propose the following empirical function: 7εDKK=wcoαμβ+γ, with α=-1.14, β=-1.84, and γ=-0.2. Calculated for CL-SHR, WIND, and SCL-SHR experiments, εDKK approximates the simulated ε very well (R=0.99; Fig. ). Thus, wco and μ can explain nearly all variation in ε found in this study. However, wco, μ, and ε are highly inter-dependent, in that wco and μ both regulate ε and are influenced by it. In addition to ε, wco and μ also depend on other processes, namely the vertical perturbation pressure gradient and buoyancy for wco and cloud-layer turbulence and wco for μ. The interrelationship of wco, μ, and ε is complicated and demands further analysis. However, such an investigation is beyond the scope of this study and is deferred to future work.