Ascent from mesoscale features such as mountains and sea breezes locally enhances the organization and intensity of cumulus convection. This forcing differs from synoptic-scale ascent in that it tends to be more rapid, localized, and surface-based. Thus, rather than gradually destabilizing and humidifying the air column, it may suddenly bring the boundary layer directly to saturation. When the background environment is conditionally unstable, this abrupt lifting may initiate vigorous moist convection.
Few modelling studies have explicitly investigated the impact of mesoscale forcing on a resolved cumulus field. Previous research has focused on mesoscale forcing at ‘convection-permitting’ grid spacings of (1 km) (e.g. Kirshbaum and Smith, 2008; Schumacher, 2009; Robinson, et al., 2011), but not at the ‘cloud-resolving’ grid spacings of (100 m) needed to capture internal cloud draughts. The latter approach is very computationally demanding, so it is often restricted to relatively simple, horizontally homogeneous flows. In such flows, it is common to incorporate horizontally homogeneous tendencies to represent large-scale forcings (e.g. Siebesma, et al., 2003), but this assumes a clear scale separation between the physical mechanisms driving ascent and the cloud-scale response. This is not appropriate when the forcings are similar in scale to the clouds themselves. Recent cloud-resolving simulations over relatively large domains have demonstrated that internally generated mesoscale features such as evaporatively driven cold pools and gravity waves are critical for organizing and intensifying moist convection (e.g. Khairoutdinov and Randall, 2006). This suggests that research is also needed to investigate the role of external mesoscale forcings (i.e. those that are not created by the cloud field itself) on the behaviour of cumulus convection.
Mesoscale lifting is important not only for initiating moist convection but also for directly controlling the subsequent cloud dynamics. In large-eddy simulations of trade-wind flow over an idealized tropical island, Kirshbaum and Smith (2009, hereafter KS09) found that the sharp lifting imparted by the narrow ridge increased not only the number of convective cells compared to that in the undisturbed upstream flow, but also their individual vigour and precipitation production. This enhancement occurred despite minimal changes in moist instability, convective inhibition, or unstable-layer depth over the ridge. KS09 attributed this enhancement to the ‘lapse-rate mechanism’: as a partly cloudy layer is bodily lifted, the saturated parcels cool moist adiabatically whereas the dry parcels cool dry adiabatically, which enhances the local horizontal buoyancy gradients compared to convection in a quiescent flow. This allowed the moist convection to intensify in areas of high cloud fraction despite the stabilizing effects of strong compensating descent.
KS09 developed an analytical model to quantify the lapse-rate mechanism where key quantities such as the entrainment rate (represented by a Newtonian damping term) and cloud fraction appeared in the coefficients of an ordinary differential equation. To facilitate analytical solutions, they treated both the cloud fraction and entrainment rate as constants but cautioned that these assumptions may be violated in reality. Indeed, the cloud fraction in their numerical simulations exhibited strong horizontal variability (their Figure 12), which prevented their analytical model from simulataneously representing the convection over the open ocean and over the island. The characteristic cloud sizes also tended to increase over the high terrain, which may have affected the entrainment within the cloud cores. Reduced dilution in wider clouds was found by Khairoutdinov and Randall (2006) to stimulate the transition from shallow to deep convection. Although the fractional entrainment rates calculated by KS09 did not appear to change over the high terrain, the liquid water content within the clouds increased substantially. The latter suggests that the mountain clouds were indeed less diluted by entrainment than those upstream.
As a complement to KS09, this study focuses on the changes in cloud morphology that accompany mesoscale ascent and their impact on the internal cloud properties (e.g. buoyancy, vertical velocity, and liquid-water content) and precipitation production. To this end, we perform large-eddy simulations of shallow, precipitating trade-wind cumuli traversing a steep, narrow mountain barrier representative of the Caribbean Lesser Antilles islands. The numerical set-up, which is described in section 2, is similar to that of KS09 except for some modest improvements. Section 3 presents the numerical simulations and section 4 demonstrates the robustness of the numerical results. Additional idealized simulations are conducted in section 5 that isolate the impacts of cloud horizontal size on cloud dilution, vigour, and precipitation. Section 6 offers physical hypotheses to explain the numerical findings, and section 7 presents the conclusions.
2. Numerical set-up
We use the Bryan cloud model version 13 (Bryan and Fritsch, 2002), a fully nonlinear, compressible, and non-hydrostatic model designed for the explicit simulation of moist convection. Horizontal and vertical advection is performed using a sixth-order centred scheme and a fifth-order scheme with implicit diffusion, respectively. Sixth-order explicit diffusion is applied in the horizontal to minimize spurious grid-scale dynamics. Mixing ratios of all water species, including water vapour (qv), cloud-liquid water (qc), and rain water (qr), are advected using a positive-definite scheme to conserve total water mass. Time integration is performed with a third-order Runge–Kutta scheme. A split time step approach with ten small time steps per large time step is used to maintain the stability of acoustic waves.
Our simulations use slightly more advanced physical parametrization schemes than those of KS09. Whereas KS09 used the basic Kessler scheme for cloud microphysics, we use the two-moment warm-rain scheme of Seifert and Beheng (2001). As in Stevens and Seifert (2008), we use a fixed cloud-droplet concentration of 70cm−3 to represent unpolluted marine air. Subgrid turbulence is parametrized with a 1.5-order TKE scheme, which is more advanced than the first-order Smagorinsky–Lilly scheme used by KS09. These changes result in more realistic precipitation amounts, which previously were strongly overestimated compared to observations.
The model domain is highly asymmetric to capture the dominant physical processes while minimizing computational expense. It is long in the cross-ridge (east–west, or x) direction (Lx = 204.8 km) with open boundary conditions to allow mountain waves to form without major interference from the lateral boundaries. This also provides sufficient time for trade-wind cumuli to develop and reach a quasi-steady state before reaching the island ridge. A 12.8 km wide horizontal Rayleigh damping (or ‘sponge’) layer is also placed at the inflow boundary to limit spurious reflections of perturbation energy). The narrow and periodic along-ridge (north–south, or y) dimension is just wide enough (Ly = 6.4 km) to fit multiple cumuli with characteristic widths of 0.5–2 km. The domain is deep enough (Lz = 12 km) to capture the mountain wave pattern and allow convective towers to ascend into the mid-troposphere. A sponge layer is also used over the uppermost 4 km to limit wave reflections off the rigid model top. The horizontal grid spacing (Δ = 100 m) is typical for simulations of shallow cumulus convection. The vertical grid spacing varies from 50 m over the lowest 4 km to 200 m over the highest 4 km, with a stretching layer between.
Mesoscale ascent is imparted by a Gaussian island ridge with a height of hm = 500 m and half-width of am = 5 km. This is broadly representative of several islands within the Lesser Antilles volcanic chain (e.g. Dominica, Martinique, and Guadeloupe) that are persistently subject to trade-wind flow. The modest ridge height is chosen to minimize the development of a prominent upstream-propagating bore, which in KS09 was found to undesirably disrupt the radiative-convective equilibrium of the upstream flow. The ridge is centred in the western portion of the domain (x0 = 64 km) to provide more time for the shallow convection to mature (here 3 to 4 h) before reaching the island while minimizing interference from the outflow boundary. Because this ridge is meant to represent a generic mesoscale forcing feature, no effort is made to represent the detailed shape of any particular island.
The initial conditions are taken from two intensive observational campaigns over the Carribbean Sea: the Barbados Meteorological and Oceanic Experiment (BOMEX) in 1969 (Holland and Rasmusson, 1973) and the Rain in Cumulus over the Ocean (RICO) experiment in 2004–2005 (Rauber, et al. 2007). The BOMEX conditions are adopted from the model intercomparison of et al. (2003) and the RICO conditions are adopted from the numerical simulations of Stevens and Seifert (2008). Of the two background flows in Stevens and Seifert (2008), we choose the moister flow for its heavier precipitation.
Whereas the BOMEX initial state represents a quasi-equilibrium between cumulus convection, surface fluxes, and large-scale advection, the RICO initial state is out of balance and requires some time to equilibriate. The initial flow was thus integrated for 24 h over a smaller, horizontally uniform domain with dimensions of Lx = Ly = 12.8 km and Lz = 3.5 km and grid spacings of Δx = Δy = 100m and Δz = 50m (using the large-scale advective tendencies and surface fluxes described below). The resulting area-averaged thermodynamic and wind profiles from this simulation (henceforth termed RICO-SMALL) were then used as the horizontally uniform initial state for the larger simulations over complex terrain.
Figure 1 shows the initial vertical profiles for the BOMEX and RICO simulations. Both possess a sub-cloud layer over the lowest 500–600 m, a conditionally unstable ‘cloud’ layer above, and a sharp inversion at the boundary-layer top. Because these profiles were originally specified only up to about 3 km, we extended them up to the stratosphere by fixing the relative humidity and winds to their values at 3 km while setting the temperature lapse rate to moist adiabatic. This highly idealized free-tropospheric representation lacks the westerly vertical shear that often occurs in the subtropics, which may influence the representation of the mountain-wave response. Nonetheless, we have chosen to maintain this approach for simplicity and defer an investigation of the mountain-wave sensitivities to future work.
Large-scale advection, surface fluxes, and radiative cooling rates are also adopted directly from et al. (2003) (BOMEX) and Stevens and Seifert (2008) (RICO). Although we provide a brief overview of these settings here, we refer interested readers to those studies for more quantitative details. Large-scale advective tendencies include subsidence, which is strongest at the inversion level and decays above and below, and horizontal advection of dry air, which is largest at the surface and decays above. These are applied to horizontal wind vectors, potential temperature, and all moisture variables. The sensible (H) and latent (LE) heat fluxes are set to fixed and horizontally uniform values of 10 and 157 W m−2 (BOMEX) and 8 and 191 W m−2 (RICO). Whereas the BOMEX surface fluxes are taken directly from et al. (2003), the RICO values represent domain averages at the end of the RICO-SMALL simulation, which used interactive heat fluxes as specified by Stevens and Seifert (2008). Prescribed clear-air long-wave cooling rates are applied to the boundary layer in both cases to avoid using a full radiation scheme.
The Coriolis force is applied to perturbations from the initial state using a Coriolis parameter of f = 0.376×10−4 s−1 (BOMEX, 15°N) and f = 0.449×10−4 s−1 (RICO, 18°N). For surface friction, a bulk aerodynamical formulation with a drag coefficient of Cd = 1.29×10−3 is used in both cases, which again is uniform over the domain. Although the idealization of horizontally uniform surface roughness and heat flux is clearly violated over land, KS09 found that changes to H, LE, and Cd over steep and narrow islands like the one considered here have only minor effects on the simulated precipitation. Moreover, as our focus is on the basic impacts of lifting rather than the detailed airflow dynamics over real islands, we have opted for simplicity over realism in designing this forcing.
An obvious difference between the two profiles in Figure 1 is that the cloud layer is about 500 m deeper in RICO. Moreover, although the wind magnitudes are similar (decreasing from 8–10 m s−1 at the surface to 3–4 m s−1 aloft), the RICO winds are more northerly. This is partially due to a stronger northerly geostrophic wind component (3.8ms−1 versus 0ms−1 in BOMEX). The RICO low-level flow has also adjusted to surface friction in the RICO-SMALL simulation, which turns the winds counter-clockwise. The flow in the BOMEX simulation quickly undergoes a similar adjustment, developing a near-surface northerly wind component of roughly 2ms−1 (not shown), which brings the two wind profiles into better agreement. Although the soundings in Figure 1 contain a stratospheric layer with high stability, the base of this layer is at 12 km and thus is excluded from the model domain.
To initiate turbulent motions within the boundary layer, uncorrelated fields of uniformly distributed, random perturbations of water-vapour mixing ratio (qv′, peak amplitude 0.05 g kg−1) and potential temperature (θ′, peak amplitude 0.1 K) are added to the initial state over the lowest 1.5 km. These perturbations are continually replenished at x = 192.8 km over the course of the simulation to seed fresh air entering the domain from the inflow boundary.
3. Large-eddy simulations
3.1. Clouds and precipitation
The BOMEX and RICO simulations were both integrated for 6 h and output fields were dumped to file every 10 min. As in the experiments of KS09, both simulations reached a quasi-steady state after about 3 h of time integration. Hence all of the model analysis will be restricted to the t = 3 to 6 h (180 to 360 min) period, which provides a sufficiently large dataset for statistical trends to emerge. As a qualitative depiction of the trade-wind cumuli that develop in the simulations, Figure 2 shows 0.1 g kg−1 isosurfaces of qc at t = 5 h. The two cases are similar in that scattered cumuli develop over the open ocean then increase in number and size over the higher terrain (centred at x0 = 64 km). Whereas the BOMEX case is cloud-free in the lee (x < x0), the RICO case develops a layer of stratus from z = 1.5 to 3 km due to cumulus detrainment over the island. At this time the clouds are generally larger, more organized, and deeper in the RICO case, with two cloud clusters developing over the island.
Figure 3 shows y-averaged qc and liquid-water potential temperature (θl) for the two cases at t = 5 h (the y-averaging lends a denser appearance to the cloud population than in Figure 2). In both simulations, cumuli extend from cloud base into the inversion layer, with a few clouds penetrating above the inversion top. The clouds are generally deeper in the RICO case because of the higher inversion level. Moreover, the higher humidity of the RICO cloud layer (Figure 1) reduces the suppressive effects of entrainment and allows a larger fraction of the cumuli to successfully reach the inversion. As the flow traverses the mountain, the clouds become denser in both cases but not substantially deeper. This is because the inversion does not rise with the terrain –it stays roughly level then plunges toward the surface in the lee, which is reminiscent of the transition to criticality in shallow-water flow (e.g. Smith, et al., 2009).
The generally deeper clouds in the RICO simulation give rise to substantially more precipitation, particularly over the elevated island terrain. Figure 4 compares horizontal profiles of simulated rain rate (P) from these cases to the radar-based climatology of et al. (2009) along a transect crossing the high terrain of Dominica. The observations are binned into three different categories based on the upstream value of P (light: < 2 mm day−1; moderate: 2–10 mm day−1; and heavy: > 10 mm day−1). In agreement with the observations of Holland and Rasmusson (1973), little precipitation develops in the BOMEX simulation, even over the higher terrain. By contrast, in the RICO case P ≈ 2 mm day−1 over the open ocean and increases to around 30 mm day−1 over the island. This falls between the light and moderate cases from et al. (2009), which suggests that the simulated average rain rates are broadly realistic. However, because of the many idealizations invoked in the large-eddy simulations, we do not carry this observational comparison any further.
Figure 3 suggests that the dramatic island precipitation enhancement in the RICO case is not attributable to a deepening of the cloud layer. Hence this enhancement likely arises from some other changes in the properties of these clouds. In the following, we consider the impacts of the island ascent on the cloud coverage, morphology, and internal properties. Although both the BOMEX and RICO simulations are discussed, more attention is devoted to the RICO case due to its larger precipitation production.
3.2. Core profiles
To analyze the internal properties of the simulated clouds, we define two contrasting regions of the flow. The first is the ‘ocean’ region (79 ≤ x ≤ 99 km), a broad upstream area containing mature and quasi-steady cumulus convection that is largely unaffected by the presence of the island ridge. The second is the ‘upslope’ region (64 ≤ x ≤ 69 km), or the portion of the mountain ridge facing the easterly flow that undergoes the strongest forced ascent.
Figure 5(a) compares vertical profiles of buoyant cloud updraught (or ‘core’) fraction for the RICO simulation over the upslope and ocean regions, averaged over 3 to 6 h. As in et al. (2003), cores are defined as gridpoints with positive values of qc, perturbation vertical velocity (w′), and buoyancy (b), where
θv is the density potential temperature
Rd and Rv are the gas constants of dry air and water vapour, and qt is the total water mixing ratio. The perturbation terms in the preceding definitions are defined relative to the local y-average, hence they isolate the convective (rather than the mountain-scale) perturbations. A nearly five-fold difference in core fraction exists just above cloud base and diminishes aloft (Figure 5(a)). Averaged over the nominal cloud-layer boundaries (0.5–2.5 km), the core fraction is 3 to 4 times larger over the upslope than over the ocean. Although this is surely a large enhancement, it falls well short of explaining the 15-fold increase in island precipitation seen in Figure 4.
As found in KS09, the island clouds are not only more numerous but more vigourous and water-rich. This is revealed by conditionally sampled profiles of b, w′, and liquid-water mixing ratio (ql = qc + qr) in Figures 5(b–d). Both b and w′ increase over the island, particularly b over the lower half of the cloud layer and w′ over the upper half. The core ql over the upslope region is almost double that over the ocean region throughout the depth of the cloud layer.
For reference, we overlay adiabatic profiles for mixed-layer sub-cloud parcels lifted through the cloud layer on Figures 5(b–d), which greatly exceed the simulated core profiles even over the island. Entrainment contributes to these differences by diluting the cores with environmental air, but it is not the complete explanation. Another important factor is that some island clouds are not rooted into the sub-cloud layer. As suggested by the θl contours in Figure 3 (and by the mean w profiles in Figure 10a of KS09), the mountain lifting extends well above cloud base, which saturates parcels that are based within the cloud layer. Even if such parcels ascended purely adiabatically, they would still possess much lower b and w′ than the sub-cloud adiabats in Figure 5(b, c).
What physical mechanisms are responsible for enhancing the vigour and water content of the island clouds? As discussed by KS09, the lapse-rate mechanism helps to offset the stabilizing effects of strong compensating descent and maintain vigorous convection in the densely packed island cloud layer. However, the precise enhancement explained by this mechanism is difficult to quantify, because it is mixed with other enhancement mechanisms. An important alternate mechanism relates to the enhanced ql within the island cores (Figure 5(d)) By stimulating droplet growth through collison and coalescence, this enhanced ql significantly increases the precipitation efficiency (PE = P/C, where C is the volume-integrated condensation rate normalized by Ly). In the RICO case, PE triples from 0.04 over the ocean region to 0.12 over the island (54 ≤ x ≤ 74 km). Thus, identifying the cause of the enhanced ql within the island clouds is critical for explaining the strong island precipitation enhancement in the RICO simulation. Note that in the BOMEX case, by contrast, PE is minimal everywhere because the shallower cloud layer does not support significant precipitation.
3.3. Cloud size and entrainment
The larger liquid water supply of the island cores results from a reduction in their dilution through mixing with environmental air. A typical method for quantifying this dilution is through fractional entrainment (ε) profiles, which, following et al. (2003) and et al. (2008), are computed using
where hc(z) is the conditionally averaged moist static energy within the cores and is that of the time- and y-averaged background flow. In both the BOMEX and RICO simulations, the island forcing causes ε to increase below 1 km and decrease above (Figure 6). However, caution must be exercised in interpreting the upslope ε profiles because they are likely contaminated by a non-entrainment-related signal. As previously noted, the deep island ascent leads to the saturation of elevated air parcels, which possess relatively low h. This would produce a negative vertical gradient of hc, and a positive value of ε, even in the absence of entrainment. As a consequence, ε is strongly enhanced lower in the cloud layer where the forced lifting is the strongest. As the forced lifting diminishes aloft, ε better represents the pure effects of entrainment.
In the BOMEX case, the impact of elevated initiation on ε obscures the changes in cloud dilution arising from the bulk lifting. This helps to explain why KS09, who exclusively studied the BOMEX case, concluded that entrainment did not decrease signficantly over the island. However, the deeper RICO cloud layer does show a clear signal of reduced island dilution over the 1–2.5 km layer, where the values of ε over the upslope region are less than half of those over the ocean.
A clearer perspective on entrainment is provided by Paluch diagrams (Paluch, 1979) for the BOMEX and RICO cases in Figure 7. These show the distribution of two conserved variables (here reversible θe and qt) in cloudy gridpoints within the upper half of the cloud layer (z = 1.5km) over 3–6 h. As these variables are only conserved for non-precipitating air parcels, we exclude all gridpoints with P > 1 mm h−1. For clarity we present these diagrams as probability density functions, with darker colours denoting higher densities. The environmental sounding and lines demarking saturation and neutral buoyancy are overlaid for reference. The positions of cloud points on the diagram indicate their degree of dilution; for example, a point possessing cloud-base properties likely ascended from cloud base to the observation level undiluted. A mixture of air from two levels falls roughly along a line connecting those two levels. Hence a quasi-linear distribution suggests a spectrum of mixtures from two distinct levels of the sounding.
In both simulations, the upslope and ocean distributions both broadly fall along a line connecting the sub-cloud level with air from at or above the observation level. The ocean distributions are tightly clustered below the saturation line (Figures 7(b, d)), indicating that most cloudy parcels are heavily entrained and contain relatively low buoyancies and condensed-water amounts. By contrast, the upslope distributions are more uniform from the saturation line down to the surface (Figures 7(a, c)), which indicates a larger proportion of air from lower levels including the sub-cloud layer. These relatively undilute parcels give rise to large values of b and ql when lifted to the observation level.
Evidence of elevated convection is again found over the upslope region. The distribution of cloudy gridpoints over the ocean falls along a tight line connecting the surface to z ≈ 1.5 km, suggesting a spectrum of mixtures of sub-cloud air and air entrained from that level. By contrast, the distributions of cloudy points over the upslope are broader, with their left edges closely following the sounding profile from the surface up to the saturation line. This reflects that the cloud inflow consists of parcels from many different levels of the cloud layer with varying degrees of dilution, which yields a broader spread of θe–qt combinations. Despite this increased elevated inflow, the overall weight of the upslope distributions (as seen by the circled cross symbol) still falls closer to cloud base, reflecting a larger presence of moist, high-entropy air from the sub-cloud layer.
Based on qualitative evidence of cloud enlargement over the island (e.g. Figure 2), we hypothesize that the reduction in island cloud dilution is associated with a systematic increase in the horizontal cloud size. In the classical theory of entraining thermals, the fractional entrainment rate is inversely proportional to the cloud diameter (e.g. Morton, 1957). Although the homogeneous entrainment assumed by this theory is overly simplistic, recent cloud-resolving simulations have reinforced that cloud size is critical for regulating the entrainment process and hence the evolution of convective cells (Khairoutdinov and Randall, 2006). This is because the interiors of wider clouds are better protected from entrained environmental air. In addition, precipitation particles in wider clouds are less likely to exit through the lateral edges (particularly in vertically sheared flows like these), which provides them with more time to accrete cloud droplets during their descent.
Figure 8 presents size spectra at z = 1km for both cores and clouds over 3–6 h. The sizes of these entities are found by determining the area covered by their interconnected gridpoints (Ac) and assuming a circular shape to derive the diameter (). Cores are defined as in section 3.2 and clouds are defined as gridpoints with qc > 0.01g kg−1. Both spectra clearly shift toward larger values over the island (Figure 8(a, b)), confirming the cloud widening under the forced ascent. As shown by the vertical profiles in Figure 9(a, c), the upslope and ocean core/cloud size differences are the largest over the lower half of the cloud layer and gradually diminish aloft. (A physical hypothesis for this vertical contraction in core/cloud size is provided in section 6.3.) The standard deviations of the core/cloud diameters are also much larger over the island (Figure 8(b, d)), reflecting a broadening of the size spectrum.
To illustrate the sensitivity of internal core properties to core size, Figure 10 shows vertical profiles of the maximum core b, w′, and ql, conditionally averaged over different size bins based on statistics over 3–6 h. As the cores become larger, their maximum buoyancy, vigour, and condensed-water content all monotonically increase. The profiles fall progressively closer to the sub-cloud adiabat, indicating less dilution through the entrainment of environmental air. Although the maximum core properties were chosen to show the characteristics of the least dilute air parcels, the same sensitivity to cloud size is found for the averaged core properties (as well as the maximum and averaged cloud properties).
4. Robustness experiments
One may question whether the above results are sensitive to the exact choice of numerical set-up in section 2. Some potential weaknesses with that set-up are that the horizontal resolution of Δ = 100 m may be too coarse for a study of shallow-cumulus morphology (e.g. Brown, 1999), that the domain width Ly = 6.4 km may artificially constrain the size of convective circulations (e.g. de Roode, et al., 2004), and that the cloud properties may be highly sensitive to the subgrid microphysical and turbulence settings and/or surface-flux implementation. To address these issues we have conducted a suite of experiments that are identical to the ‘baseline’ RICO simulation except for the following changes:
1.RICO-DX50: Δx is halved to 50 m.
2.RICO-2LY: Ly is doubled to 12.8 km.
3.RICO-NC140: The cloud-droplet concentration is doubled to 140 cm−3 to represent more polluted air.
4.RICO-KK: The Seifert and Beheng (2001) warm-rain scheme is replaced by the commonly used double-moment scheme of Khairoutdinov and Kogan (2000).
5.RICO-SMAG: The 1.5-order TKE scheme is replaced by a first-order Smagorinsky–Lilly scheme.
6.RICO-IFLX: The fixed surface heat and moisture fluxes are replaced by an interactive bulk aerodynamic scheme with a fixed sea-surface temperature of 299.8 K and a non-dimensional heat exchange coefficient of Ch = 0.0011 (Stevens and Seifert, 2008).
The mean core sizes at z = 1 km , fractional entrainment rates over z = 1 to 2 km , and surface rain rates () in these simulations over 3 to 6 h are compared in Table 1. The values for the island and ocean regions are respectively distinguished by the ‘i’ and ‘o’ subscripts. Because the core-size enhancement is focused over the upslope, we only consider that region in the calculation of and . However, is calculated over the whole island (54 ≤ x ≤ 74 km) because much of the precipitation spills over the crest.
Table 1. Comparisons of the mean core size , mean fractional entrainment rate over z = 1 to 2 km , and the mean precipitation rate for the robustness experiments over 3 to 6 h.
‘i’ and ‘o’ subscripts denote island and ocean regions; definitions of these are given in the text.
As seen from the RICO-2LY, RICO-SMAG, and RICO-IFLX simulations, little to no sensitivity is found to the domain width, subgrid turbulence formulation, or implementation of the surface fluxes. The sensitivities to microphysics settings are also generally weak except for the RICO-NC140 case, where the increased aerosol loading suppresses the precipitation production slightly. This is caused by a reduction in mean cloud-droplet diameter and a reduction in the autoconversion from cloud to rain (e.g. Rogers and Yau, 1989).
The RICO-DX50 simulation exhibits a significant decrease in the mean core size, suggesting that the shallow convection may be under-resolved at Δ = 100 m (Brown, 1999). Inspection of cloud-size distributions (not shown) indicates that this decreased results primarily from an increase in the number of very small cores (D < 0.5 km). By contrast, the total number of larger clouds decreases only slightly. Because smaller clouds tend to be more strongly diluted than larger clouds, their increased prominence increases (Table 1). However, this effect is relatively modest because is naturally weighted toward larger clouds containing more internal gridpoints. Interestingly, the number of very large clouds with D > 1.5 km actually increases over the island in the RICO-DX50 case, which may explain why also increases.
Although this comparison suggests that the cloud sizes and rain rates exhibit some sensivitity to cloud microphysical settings and grid resolution, they also show a robust response to the island forcing. Regardless of the numerical configuration, the forced ascent causes a large (∼ 60%) increase in mean cloud size, a halving of the fractional entrainment rate, and a dramatic (five- to ten-fold) precipitation enhancement. As a consequence, the key result that island lifting simultaneously broadens the cloud-size distribution, reduces the core dilution, and enhances the precipitation rate is strongly reinforced by these experiments.
5. Controlling the cloud size
Although the impact of cloud diameter on entrainment and precipitation production is consistent with expectation, the complexity of the simulations of section 3 renders it difficult to demonstrate a causal link. In particular, some of the enhancement in cloud vigour over the island is likely owing to the lapse-rate mechanism of KS09, which obscures the impacts of cloud widening. To isolate the role of cloud size on cloud dilution and vigour, we conduct additional simulations in which the sizes of the island clouds are explicitly controlled. Because all of these simulations use the same upstream flow and island forcing, the lapse-rate effect is invariant and the differing responses may be attributed purely to differences in the horizontal cloud size. The design of these experiments is similar to that of the RICO simulation except for the following modifications:
1.We switch off all surface fluxes (heat, moisture, and momentum), which eliminates upstream turbulence. The large-scale forcings and Coriolis force are also removed, leaving a balanced upstream flow in the absence of moist convection or frictional drag.
2.Without any need for ocean convection to spin up in the upstream flow, the long upstream fetch of our model domain is no longer needed. We thus move the inflow boundary from x = 204.8 km to x = 153.6 km to save computational expense.
3.Rather than add random perturbations to the initial flow, we add simple two-dimensional sinusoidal fields with a fixed horizontal scale, λ. These perturbations control the scales of the moist convection that develops over the island.
4.We use a uniform background zonal wind speed of U = −6 m s−1, which is representative of the mean zonal wind in the boundary layer of the RICO and BOMEX simulations. The absence of background shear allows the perturbations to remain relatively vertically coherent over time.
The perturbations take the form:
where Qv = 0.2 g kg−1 is large enough to dominate over numerical round-off error but small enough for the upstream flow to remain unsaturated. The perturbations are vertically uniform (i.e. columnar in shape) from the surface up to 1.5 km, above which they are zero. They are added to the initial upstream flow over a region centred at x = 90 km of length Lp = 32 km and width Ly and replenished at intervals of Lp/U, the approximate time required to travel a distance of Lp. Note that θ′ in (5) is designed to exactly cancel the buoyancy signature of qv′ so that θv, as defined in (2), is unchanged by the addition of the perturbations. With identical densities as their surroundings, the perturbations are dynamically passive until reaching saturation over the mountain.
Figure 11 compares qc at t = 5 h for three such experiments with different values of λ: 1.6 km (LAM1.6), 3.2 km (LAM3.2), and 6.4 km (LAM6.4). These cross-sections are shown at z = 1.5 km to avoid shallow, near-surface clouds that develop directly over the island. The regions with qv′ > 0 and θ′ < 0 are the first to saturate in the forced ascent, which give rise to cumuli with diameters on the order of λ/2. The clouds form farther upstream in the LAM1.6 case due to spurious low-amplitude reflections off the inflow boundary, which are more severe at smaller forcing scales. However, because these clouds are only marginally buoyant and not supported by mesoscale ascent over the flat terrain, they remain weak until reaching the island.
The sensitivity of the internal cloud properties to cloud horizontal scale is shown by the conditionally averaged core profiles in Figure 12, which are computed identically to those in Figure 5. However, rather than comparing different regions within the same simulation, we compare the same upslope region for the three different cases (LAM1.6, LAM3.2, and LAM6.4). As shown in Figure 12(a), the core fractions are generally similar, increasing to a peak of ≈ 0.35 at z ≈ 800 m then rapidly decreasing to ≈ 0.1 over the remainder of the cloud layer. The core b, w′, and ql all tend to increase with λ through the majority of the cloud layer and fall progressively closer to the adiabatic profile (Figures 12(b–d)).
As expected, the larger and less diluted clouds generate more precipitation. Figure 13 shows profiles of rain rate from the LAM1.6, LAM3.2, and LAM6.4 simulations, indicating a pronounced increase in both maximum and area-integrated precipitation rate for larger λ. In contrast to the simulated P profiles in Figure 4, these maxima fall over the lee slope rather than directly over the crest. This is because the relatively weak upstream perturbations imposed here require more time to initiate and grow into precipitating cells, which shifts the precipitation distribution downwind.
The results from these idealized simulations, in which the clouds are confined to the mountain and their sizes explicitly controlled, are consistent with those of the BOMEX and RICO simulations, where the cloud widening was a natural response to the the island ascent (for reasons that will be discussed in section 6). Both sets of simulations show clear evidence of reduced dilution, increased vigour, and heavier precipitation within the larger clouds.
6. Mechanisms for cloud broadening
Although the forced island ascent obviously increases the amount of cloud condensation, this does not necessarily imply a change in cloud morphology. Increased condensation could be fully realized through an increase in the number of clouds and/or the updraught ascent rates. In the following, we provide physical hypotheses to explain the cloud broadening over the steep terrain.
6.1. Saturation of impinging water-vapour anomalies
One factor influencing cumulus cloud size is the physical scale of the processes that initiate the clouds. The characteristic scales of these processes differ between the ocean and the island. Over the ocean, where the island-forced ascent is absent, clouds are initiated by sub-cloud updraughts that lift air parcels through cloud base. While some island clouds arise from the same process, others are initiated by the bulk ascent of impinging moisture anomalies (Woodcock, 1960; KS09). As low-level air layers approach saturation over the island, areas with larger water-vapour contents saturate first and rapidly gain buoyancy as they begin to release latent heat. The dominant horizontal scale of these water-vapour anomalies may be significantly larger than that of the vertical-velocity perturbations (de Roode, et al. 2004), which favours the initiation of larger island clouds. Observations from the DOMinica EXperiment (DOMEX 2011) have recently highlighted the presence of broad, neutrally buoyant moisture ‘patches' with characteristic scales of 3–8 km in the sub-cloud layers of trade-wind flows impinging on Dominica (Smith, et al. 2012).
The scale separation between upstream vertical-velocity and water-vapour perturbations is illustrated by two-dimensional power spectra of w′ and qv′ at z = 500m for the RICO simulation in Figure 14. These are computed using a Fast-Fourier Transform over a 6.4×6.4km box centred over the ocean region at x = 89km. The transformed fields are multiplied by their complex conjugate to obtain the power and integrated along circles of constant horizontal wavenumber (, where k and l are the x- and y-wavenumbers) to yield one-dimensional spectra, which are then averaged over 3–6 h. The peak of the w′ and qv′ spectra clearly fall at different horizontal wavelengths (λ = 2π/κ); whereas w′ peaks at λ ≈ 1 km, qv′ peaks at λ ≈ 2 km.
Because the Fourier spectra from the RICO simulation are calculated from aperiodic fields, they likely contain spurious signals arising from the assumed periodicity of the Fourier-transform calculation. Moreover, the spectra may be artificially constrained by the narrow y-dimension and the short (∼ 3–4 h) period of cumulus development between the inflow boundary and the ocean region. To determine whether these spectra are representative of mature cloud fields over larger periodic domains, we overlay on Figure 14 the averaged spectra from the RICO-SMALL simulation (over the entire 12.8×12.8km domain) over its last 6 h. Compared to the RICO case, the spectral peak of w′ is virtually unchanged and that of qv′ is very similar. Thus the scale separation between the upstream w′ and qv′ perturbations in the upstream flow is reasonably well represented in the RICO and BOMEX simulations. This supports the hypothesis that the forced saturation of broad moisture anomalies over the ridge tends to produce larger island clouds.
6.2. Scaling of the TKE budget
The island cloud widening may also be argued from an energetics perspective. Consider the turbulent kinetic energy (TKE) budget
where e is the TKE, u, v, and w are the zonal, meridional, and vertical wind components, p is pressure, and E is the turbulent dissipation rate. These variables have been decomposed into a steady mean term (e.g. ) plus a time-varying perturbation term (e.g. e′ = e′(x,y,z,t)). Each term of this budget is given a unique name and symbol; the terms on the left side are the time tendency (DT) and advection (ADV), respectively, and terms on the right are the buoyancy production (B), shear production in x and y (S = Sx + Sy), transport (T), pressure work (P), and turbulent dissipation (E), respectively. In large-eddy simulations of shallow-cumulus convection, Grant and Brown (1999) and Grant and Lock (2004) found that the two dominant terms of this budget were B and E, the latter of which has a characteristic scale of w∗3/zcld, where w∗ is a characteristic updraught velocity scale (Tennekes and Lumley, 1972). By equating these terms, they derived an expression for w∗,
where CAPE is the Convective Available Potential Energy of the sub-cloud layer and mb is the cloud-base mass flux (m s−1). Applying further dimensional arguments, they found that the dominant terms of the TKE budget can be scaled according to
where zcld is the depth of the cumulus cloud layer. Assuming that each updraught–downdraught circulation occupies a cylindrical volume of diameter zcld, and interpreting mb/w∗ as a fractional area occupied by the updraughts, a characteristic updraught diameter may be written as
The fractional entrainment rate may also be scaled by assuming that the kinetic energy supplied to the entrained air (∼εmbw∗2) equals the turbulent dissipation rate, which gives
where Aε is the fraction of the TKE production available for entrainment and is treated as a constant.
This scaling usefully links the gross characteristics of the cloud layer (mb, CAPE, and zcld) to properties of cumuli within it (D,ε). However, because it was developed for cumulus convection in radiative-convective equilibrium, it may not apply to flows with strong localized forcing. Its applicability to the RICO simultion is assessed by Figure 15(a), which compares horizontal profiles of the full (resolved plus subgrid) B, S, ADV, and E terms, each vertically averaged over the cloud layer (DT, T, and P are omitted because DT is negligable and T and P represent internal TKE redistribution rather than net production or dissipation). Following Grant and Lock (2004), the cloud-layer bottom is defined as the lowest level at which a lifted sub-cloud parcel becomes positively buoyant (based on the time- and y-averaged sounding over 3–6 h). The cloud-layer top is defined as the lowest level at which the dry static stability N2 exceeds 2 × 10−4 s−2. As expected, B and E clearly dominate the TKE budget over the ocean. As the flow ascends the ridge, all terms increase in magnitude, but not at the same location. The trough in E falls downwind of the peak in B by about one half-width (5 km), corresponding to a time lag of 1000 s that approximately matches the cell overturning time-scale (zcld/w∗). Over this region, the dominant balance between B and E is replaced by a three-way balance between a TKE source (B) and two TKE sinks (E and ADV).
Because the balance between B and E is disrupted over the ridge upslope, the ‘equilibrium’ w∗ in (6) is likely to be overestimated. Nonetheless, the similarity theory still impressively quantifies the sharp island-induced changes to the cloud field. When scaled by
the absolute values of B and other significant terms collapse to (1) over a long fetch extending from the ocean to the ridge crest (Figure 15(b)). The ability of the scaling to capture the ten-fold increase buoyancy production over the island suggests that it may help to interpret the dynamics of clouds subject to mesoscale ascent. In this spirit, we combine (6) and (8) to give
Given that (i) mb increases dramatically over the ridge due to the forced uplift through cloud base, (ii) CAPE decreases due to the release of moist instability, and (iii) zcld remains essentially unchanged (Figure 3), this theory predicts D to increase over the ridge, which is consistent with our results (Figure 8). Physically, the increase in cloud size occurs because w∗, which is constrained by the degree of moist instability (CAPE) in (6), is unable to increase proportionately to mb during the island ascent. For the extra cloud-base mass flux to ascend through the cloud layer, the updraught areal coverage must increase. As the number concentration of updraughts is assumed to be controlled by zcld (which does not increase over the island), this must be realized through a widening of the updraughts themselves.
Because is inversely dependent on D2 in (9), the theory also predicts a sharp reduction in cloud dilution during the forced island ascent. Although the total exchange between cloud and environmental air increases in the densely packed island cloud field, the fractional entrainment rate () still decreases. From an energetics perspective, the weakened dilution again relates to the inability of w∗ to increase proportionately to mb over the ridge. For fixed values of and zcld, is proportional to w∗/mb in (9), hence it must decrease over the island. If remained constant, the energy supplied to entrained air (mbw∗2) would increase faster than the turbulent dissipation rate (w∗3/zcld). For a balance to be maintained between these two processes, must decrease.
A comparison between the simulated and theoretical estimates of D and in the BOMEX and RICO cases is presented in Table 2. The simulated values are calculated using the same procedure as in Table 1 (except that in BOMEX is calculated over 1–1.5 km rather than 1–2 km) and the theoretical values are found by diagnosing mb, CAPE, and zcld from the simulations and substituting them into (8) and (9), then averaging these values over the upslope and ocean regions. In computing the theoretical , the arbitrary constant in (9) is set to 0.06.
Table 2. Comparison of simulated values and theoretical estimates of the mean core size and mean fractional entrainment . The latter is computed over z = 1 to 1.5 km in BOMEX and z = 1 to 2 km in RICO, in accordance with their different cloud-layer depths.
‘i’ and ‘o’ subscripts denote island and ocean regions; description of these is given in the text.
In both cases the theory captures the simulated trend for D to increase and to decrease over the island but overpredicts the magnitudes of these changes. This overestimation may stem from the assumption that the number of clouds remains fixed over the island, which forces all of the increased cloud mass to go into increased D. By contrast, both the cloud size and number increase over the island in the simulations (e.g. Figure 2). Moreover, because the theory neglects elevated convective initiation, the values of mb, CAPE, and zcld, which are defined for sub-cloud parcels, may not be fully representative over the island.
The theory also predicts the cores to be wider and less dilute in RICO than in BOMEX. Although Table 2 does show larger D values for the RICO simulation, the differences between the two cases are much less than predicted. Moreover, although decreases substantially as expected in the RICO case, actually increases slightly. While these discrepancies may raise some concerns about the theory, they may also be linked to practical limitations with the simulations. Cores with D ∼ 200 m like those predicted over the ocean in BOMEX cannot be resolved at Δ = 100 m and are thus likely to be aliased to larger sizes. As shown in section 4, even the clouds from the RICO case are inadequately resolved at Δ = 100 m. Much finer grids with Δ < 50 m are thus required to robustly compare D and between these two cases. The results are also sensitive to the method by which the model diagnostics are calculated; if the entire cloud layer was used for (rather than only that above 1 km), the resulting value would actually be slightly lower in RICO than in BOMEX (this is qualitatively apparent from Figure 6).
Although the TKE scaling does not perfectly reproduce all of the simulated cloud properties, it crucially succeeds in capturing the tendency for D () to dramatically increase (decrease) under mesoscale ascent in both the BOMEX and RICO cases. This trend is robust regardless of Δ or any practical choices made in computing D or . Hence the scaling appears to provide a useful conceptual basis for interpretating the response of the simulated cumulus convection to the island-forced ascent.
6.3. Further discussion
What is the relative importance of the two mechanisms for island cloud broadening described in sections 6.1 and 6.2? Although these effects are difficult to quantitatively disentangle, some insight is provided by Figures 9(a, c), which show a rapid contraction of core/cloud size over z = 0.5 to 1 km in both the BOMEX and RICO simulations. This defies the expectation that cloud size should increase with height owing to the increased success of larger, less diluted clouds in penetrating deeper into the cloud layer. We hypothesize that this sharp core/cloud size reduction results from a transition between the two mechanisms above. Near cloud base where the forced lifting is intense, the cloud sizes are strongly controlled by the impinging moisture anomalies, which assume relatively large scales (> 1 km) (Figure 14). As the clouds ascend and the forced ascent diminishes, their horizontal sizes become increasingly controlled by the bulk properties of the cloud layer, which is addressed by the TKE scaling. As shown in Table 2, this favours significantly smaller core/cloud diameters.
Another perspective on the transition between the two mechanisms relates to the degree of convective equilibrium of the turbulent flow. Referring to Figure 15, the forced-saturation hypothesis of section 6.1 is most applicable within the rapidly ascending ‘non-equilibrium’ regions where |E| is much less than |B| (0.5 ≤ (x − x0)/a ≤ 2). This is largely confined to the low-level airflow over the steepest slopes, which experiences the greatest forced ascent. Elsewhere (excluding the non-convective region (x − x0)/a < 0), the scaling of section 6.2 explains the ‘equilibrium’ relationship between mb, D, and . Future research will aim to merge the two physical mechanisms from sections 6.1 and 6.2 into a more complete theory.
We have conducted simulations of trade-wind cumuli impinging on an idealized island ridge to investigate the role of mesoscale ascent on the morphology, vigour, and precipitation production of a shallow cumulus cloud field. Although this represents a natural follow-up to Kirshbaum and Smith (2009) (KS09), it differs in the basic physical mechanism of interest. Whereas KS09 focused on a dynamical enhancement process that we term the ‘lapse-rate’ mechanism, the current study focuses on changes in cloud horizontal size and entrainment that accompany the mesoscale ascent (or the ‘cloud-size’ mechanism). Two shallow-cumulus flows are simulated with slightly different open-ocean characteristics: the first is non-precipitating and the second is lightly precipitating. These flows are based on composite soundings and large-scale forcings from the BOMEX and RICO field programmes. The numerical results are summarized as follows:
Both cases showed a clear enhancement in the areal coverage, buoyancy (b), vertical velocity perturbation (w′), and liquid-water content (ql) of convective cores embedded within the ascending island flow.
The buoyant cores in the ascending airflow were substantially less diluted by environmental air than those over the open ocean. This was linked to a systematic increase in cloud size, which shielded the interior core regions from entrained environmental air. The cloud vigour and liquid-water content both increased dramatically with cloud horizontal area.
Whereas the shallow cloud layer in the BOMEX case did not support significant rainfall, the deeper cloud layer in the RICO case produced light oceanic rainfall (∼ 2mm day−1 from scattered showers) that increased dramatically over the island (∼ 30mm day−1 from more frequent, heavier showers). This precipitation enhancement was associated not with an increase in cloud depth (the inversion base delineating the maximum cloud-top height remained nearly level over the island before plunging in the lee), but with an increase in cloud areal coverage and precipitation efficiency.
The initiation of wider clouds over the steep island slopes was favoured by relatively broad upstream water-vapour anomalies in the sub-cloud layer with dominant horizontal scales of ∼ 1–2 km. These were brought to saturation and gained buoyancy through latent-heat release during the rapid forced ascent.
The preference for larger island clouds was also argued on the basis of a scaling of the TKE budget (Grant and Brown, 1999; Grant and Lock, 2004). A basic constraint on the turbulent vertical velocity scale implies wider clouds and weakened fractional entrainment rates under forced ascent. The trends predicted by this scaling were consistent with those obtained from the simulations.
The mechanisms for convective invigoration described herein may have significant implications for weather and climate prediction. Shallow cumulus and cumulus congestus are poorly resolved even in the most powerful weather forecast models. As a consequence, they are often parametrized using mass-flux-type convection schemes in which fractional entrainment rates represent a large source of uncertainty. The current results suggest that these depend strongly on the rate of mesoscale lifting, an effect that is typically neglected. Further research is needed to better isolate and quantify this mechanism with a view toward incorporating it in future models.
This research was partially supported by National Environment Research Council under grant NE/E016391/1. Simulations were performed through an allocation from the National Center for Atmospheric Science on the HECToR supercomputer service, which is funded by the UK Research Councils. The first author is indebted to Ron Smith, Justin Minder, Rich Rotunno, and Jim Doyle for helpful discussions during the course of this work. We thank two anonymous reviewers for their extremely useful comments, which helped to significantly improve this manuscript.