1. Introduction
To parameterize convection for large-scale models and turbulence for finer-scale models correctly, it is important to understand dynamics driving mixing between the edges of convective clouds and their environment in the best possible way. However, a precise picture of the mixing characteristics is still missing and studies of fundamental understanding and parameterization of entrainment/detrainment in convective clouds remain an active field of research (De Rooy et al. 2013).
In deep convective clouds, Verrelle et al. (2017) have shown by using large-eddy simulation (LES) at 50-m resolution that the main subgrid-scale turbulence occurs in the updraft core. They also demonstrated that the commonly used eddy-diffusivity turbulence approach in mesoscale models1 underestimated the thermal production of subgrid turbulent kinetic energy (TKE) and did not enable the countergradient structures (Deardorff 1966) to be reproduced. In contrast, the approach proposed by Moeng (2014), parameterizing the subgrid vertical thermodynamical fluxes in terms of horizontal gradients of resolved variables, reproduced these characteristics and limited the overestimation of vertical velocity. Strauss et al. (2019), considering the same LES resolution, generalized the results to the vertical and horizontal heat and moisture fluxes over the entire cloud life cycle. This new type of turbulence closure is beginning to attract attention in the scientific community (Hanley et al. 2019).
Pioneers in idealized simulation of cumulus clouds with a very fine-scale resolution were Klaassen and Clark (1985) and Grabowski and Clark (1991, 1993a), who studied the development of instabilities on the edges of cumulus clouds but two of their three studies were in 2D. They depicted instabilities of dynamic origin with a low impact of environmental wind shear (Grabowski and Clark 1993b). Following Bryan et al. (2003), a phenomenon occurring in the atmosphere is correctly represented by an LES when the mesh size is smaller by a factor of approximately 100 than its characteristic scale. A cumulus congestus with a diameter of a few kilometers is therefore adequately simulated by an LES employing a resolution of 50 m. However, the large eddies observed on the edges of cumulus clouds with diameters ranging from about 100 to 500 m (Damiani et al. 2006) necessitate finer grids considering the effective resolution. Stevens et al. (2005) showed that a grid resolution on the order of 10 m was necessary to obtain converging statistics in shallow cumulus clouds. A sensitivity test to resolutions ranging from 10 to 80 m horizontally and from 5 to 40 m vertically showed that new scales of motion appeared with each doubling of the resolution and that the finest-grid simulation resolved the eddies responsible for lateral and penetrative entrainment. In the same way, Matheou et al. (2011) obtained statistical convergence for a resolution of 20 m in a case of nonprecipitating cumulus. Their study revealed negative buoyancy surfaces surrounding simulated cumulus on horizontal cross sections. Those surfaces significantly increased with a doubling of the resolution from 40 to 20 m.
Hoffmann et al. (2014) combined observations from the Cloud, Aerosol, Radiation and Turbulence in the Trade Wind Regime over Barbados (CARRIBA) campaign, LES at 5-m resolution and direct numerical simulations (DNS) to analyze entrainment and mixing in shallow cumulus clouds and to measure the shortcomings of each approach. They showed that evaporative cooling effects and dissipation rates were correctly reproduced by the LES, while the intermittency of small-scale turbulence was not well resolved. Idealized DNS studies of cumulus cloud edges should indeed be a way to study mixing on cumulus edges with a very good precision, as already done to study evaporative cooling effects (Abma et al. 2013) or microphysical considerations (Kumar et al. 2012, 2013, 2017; Kumar et al. 2014). However, the limitations in computing power only allow these DNS to be performed on a piece of cloud edge suited for the study of very small-scale mixing. Therefore, an LES of 5-m resolution around a cumulus congestus appears to be a good compromise to study turbulence effects on the edges of convective clouds by simulating the entire cloud.
To parameterize convection in large-scale models,2 an ensemble of convective clouds can be represented as an entraining plume rising through the atmosphere (e.g., Tiedtke 1989; Kain and Fritsch 1990; Lin and Arakawa 1997a,b). Considering an ensemble of clouds (e.g., Arakawa and Schubert 1974) instead of a single cloud is justified by the stochastic nature of entrainment in cumulus clouds (e.g., Lin and Arakawa 1997b; Romps and Kuang 2010; Nie and Kuang 2012). However, some difficulties are encountered in the parameterization of entrainment in these plumes (e.g., Romps 2010; Hannah 2017). More theoretical studies focused on the best possible representation of cloud physics lean toward a model of a rising bubble leaving behind and detraining into a turbulent wake as first proposed by Blyth et al. (1988). Such a cloud presents a high pressure area near its top that stands relatively undiluted and contains a toroidal circulation detraining on its sides as it entrains near its rear (Zhao and Austin 2005; Moser and Lasher-Trapp 2017). Strong observational evidence of the toroidal circulations (Damiani et al. 2006; Damiani and Vali 2007; Wang and Geerts 2015) reinforces this view of cloud dynamics.
Around cumulus clouds, a large-scale compensating subsidence in the environment occurs as a result of a dry adjustment process mediated by internal gravity waves that extends up to Rossby radius away from the updraft (Arakawa and Schubert 1974; Bretherton and Smolarkiewicz 1989). In addition, very close to the cloud, a subsiding shell was observed by Jonas (1990) and Rodts et al. (2003). From aircraft measurements, Perry and Hobbs (1996) and Lu et al. (2003) found halos of enhanced humidity in the clear air surrounding cumulus clouds, broader on their downshear side and generally confined below any stable layer. Heus and Jonker (2008) using numerical simulations highlighted a buoyancy reversal (Siems et al. 1990) that appeared to be spatially correlated to the subsiding shell, thus described as thermally driven and fueled by evaporative cooling. This view is contested by Park et al. (2017), who showed in their simulated trade wind cumuli case that evaporative cooling is not the primary factor driving subsiding shells and accompanied buoyancy reversal, but instead vertical convective mixing. Glenn and Krueger (2014) displayed evidences of the existence of subsiding shells for deep convective clouds in a numerical simulation. Abma et al. (2013) and Nair et al. (2020) performed DNS of idealized cloud edges in which effects of evaporative cooling spontaneously generated self-similar, Reynolds number independent subsiding shells. Interpretations of their results were found consistent with fine-scale observations (Katzwinkel et al. 2014). In summary, these studies suggest that while evaporative cooling appears capable of initiating downdrafts at the cloud edges, it is not yet clear if it is a necessary condition.
The importance of evaporative cooling at cloud top in producing penetrating downdrafts (Squires 1958) is still uncertain. Some theories presented entrainment as a thermally driven process in which evaporative cooling played an important role (Squires and Turner 1962; Paluch 1979; Emanuel 1994). According to these theories, the cloud is entraining dry air at its top, which, by causing the evaporation of cloudy air, created downdrafts penetrating deep inside the cumulus, altering its dynamics. This vision of entrainment has been strongly contested (Grabowski and Clark 1993a; Heus et al. 2008) and the state of the art is leaning toward a lateral entrainment whose impact in terms of evaporative cooling is still poorly known. The flaws in buoyancy sorting models, widely used in convection parameterization schemes noted by De Rooy et al. (2013), also suggest this theoretical approach may be inadequate.
In this context, the objective of this article is to better document and understand the characteristics of the fine-scale dynamics of a cumulus congestus with a focus on subgrid and resolved turbulence aspects. Another goal is to understand how fine-scale evaporative cooling effects impact the downdrafts located in the subsiding shell and if they promote penetrative downdrafts from the cloud top.
For this purpose, a giga-large-eddy simulation of a cumulus congestus is performed with a 5-m resolution using a downscaling method. When the convective development is well established, dynamics and turbulence are investigated employing a partition that allows to distinguish the center of the cloud from its inner and outer edges. New insights that come with the fine resolution about the subsiding shell and about the importance of the toroidal circulation for the exchanges between the cloud and its environment are presented. Evaporative cooling is turned off, and the impact on cloud dynamics and vortices is examined, using potential temperature and vertical velocity budgets.
The simulation carried out in this study is introduced in section 2. Section 3 presents the simulated cloud dynamics and a first attempt to characterize turbulence. Section 4 introduces a partition between the cloud interior, inner edges, outer edges, and the far environment employed to compare distributions in the different regions. Section 5 focuses on physical processes on the edges of the simulated cloud attempting to compare the results of the simulation with other published results. In section 6, the impact of evaporative cooling on the cloud dynamics and turbulence are investigated. For this purpose, 2 min of the simulation are redone by canceling the cooling effects related to water phase changes. Section 7 presents a qualitative study of interactions between the cloud and its environment by focusing on individual eddies, before we conclude in section 8.
2. Presentation of the simulation
The model used to perform the simulation is the atmospheric nonhydrostatic MesoNH model (Lafore et al. 1997; Lac et al. 2018), which is based on the anelastic approximation of the pseudoincompressible system of Durran (1989). It uses the C grid of Arakawa for spatial discretization, with Cartesian coordinates. A fourth-order centered advection scheme is used for the momentum components, with the explicit Runge–Kutta method (RKC4) as a time scheme. Fourth-order numerical diffusion operator is applied for the wind to damp the numerical energy accumulation in the shortest wavelengths. The piecewise parabolic method advection scheme (Colella and Woodward 1984) is applied for the other components with a forward-in-time integration. These numerical schemes allow a fine effective resolution of 5Δx (Ricard et al. 2013). Potential temperature (noted θ) and mixing ratios (noted r) of hydrometeors and water vapor are the thermodynamical prognostic variables. The ICE3 mixed-phase one-moment scheme (Pinty and Jabouille 1998) is used for the microphysics considering three solid [ice crystal (ri), snow (rs), and graupel (rg)] and two liquid [cloud droplets (rc) and raindrops (rr)] hydrometeor species. The turbulence scheme is a 1.5-order closure scheme (Cuxart et al. 2000, hereinafter called CBR) employed in its complete three-dimensional formulation. It uses a prognostic equation for the TKE and a K-gradient formulation for the parameterized turbulent fluxes. The system is closed with the Deardorff (1972) mixing length, which is expressed as
A first coarser simulation has been performed to produce a field of cumulus clouds. The domain on which this parent simulation was produced extended horizontally over 40 km × 40 km and 20 km vertically with a 50-m horizontal resolution and a 5-m vertical resolution under the altitude of 5 km with a progressive stretching above, which stopped when a 200-m resolution was reached (slightly above z = 6.4 km). The time step was 0.125 s during 80 min. The lateral boundary conditions were cyclic. The initial conditions coming from Weisman and Klemp (1982) were convectively unstable with a surface mixing ratio of 16 g kg−1. To introduce heterogeneity into the planetary boundary layer, Gaussian white noise was added to the potential temperature between the ground and a height of 1 km. The wind shear was moderate with a quarter-circle hodograph: u and υ changed from 0 to 6.4 m s−1 between the ground and a height of 2 km. Above 2 km, υ remained constant as u increased from 6.4 to 8 m s−1 up to a height of 6 km. Above 6 km, the mean horizontal wind remained constant (u = 8 m s−1 and υ = 6.4 m s−1).
From this first simulation, a downscaling approach has been applied, by performing a refined simulation on a cubic domain of 5 km per side with a 5-m resolution vertically and horizontally, resulting in a giga-LES. The nested domain was chosen around a cumulus congestus between the 55th and 60th minutes of the parent simulation. The downscaling framework consisted in using initial and lateral boundary conditions from the parent simulation as coupling files with open lateral boundary conditions refreshed every 10 s and temporal interpolation in between. This refined simulation is the one considered in this study (hereinafter referred to as the CTRL simulation).
Time series of the mean values of vertical velocity, cloud water mixing ratio, subgrid TKE and total TKE (sum of resolved and subgrid TKE) over the cloudy area are displayed in Fig. 1 as a function of the altitude for the duration of the refined simulation. The resolved TKE is computed as
Temporal evolution averaged over the cloudy area (with an rc + ri ≥ 10−6 kg kg−1 threshold) as a function of altitude for (a) the subgrid TKE (m2 s−2), (b) the total TKE (m2 s−2), (c) the vertical velocity (m s−1), and (d) the cloud water mixing ratio (kg kg−1). The instant of 4 min is marked with a black dashed line.
Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-20-0386.1
The convective cloud is well present from the start of the simulation and a cloudy area extends between 900 m and 4.5 km. A characteristic threshold of 10−6 kg kg−1 for the cloud water mixing ratio is used throughout this study to define the cloudy areas. The subgrid TKE (Fig. 1a) begins to increase after approximately 100 s corresponding to the spinup period. The total TKE (Fig. 1b) shows larger values, which indicates that resolved part dominates over subgrid part. Vertical velocity and cloud water mixing ratio (Figs. 1c,d) gradually increase as the cloud rises and develops. The altitudes with strong values of the TKE are correlated with the altitudes of strong vertical velocity. The highest values of the cloud water mixing ratio are located near the top of the cloud. We consider that after 4 min the spinup period is largely finished at very fine-scale resolution and that the convective cloud has a steady growth trend, as the cloud top linearly rises, which can be considered as representative of the development of the cumulus cloud. The instant of 4 min has been selected for the diagnoses presented in this article and most of the figures that are displayed correspond to that time.
The CTRL simulation has also been used to produce synthetic images of cloud radiances through a path-tracing Monte Carlo algorithm, underlying the realistic appearance of the simulated cloud (Villefranque et al. 2019).
3. Presentation of the simulated cumulus congestus
a. Dynamics
The vertical cross section of vertical velocity displayed in Fig. 2a reveals a wide updraft occupying most of the volume of the cloud and intensifying with altitude. It peaks at 3.5 km, approximately 500 m below the cloud top. The color scale used in this figure is deliberately made asymmetrical to highlight downdrafts. Between z = 3.25 km and z = 3.75 km, the strongest downdrafts are visible as they encompass the vertical cloud–environment interface. These downdrafts are characteristic of the toroidal circulation at cumulus top that will be discussed in section 5. Around the rising cloud, the environmental air collapses with low values of negative vertical velocity relative to the rate of ascent of the cloud. This large volume of slowly sinking air is significant in terms of mass flux (see section 4).
Vertical cross sections at y = 2.25 km [along the dashed line represented in (e) and (f)] at t = 4 min of (a) the vertical velocity (m s−1), (b) the buoyancy (m s−2), (c) the cloud water mixing ratio (kg kg−1), and (d) the precipitating water mixing ratio (kg kg−1). Also shown are horizontal cross sections at (e) z = 2.5 and (f) z = 3.5 km [along the dashed lines represented in (a); the line at z = 1.5 km is used in Fig. 9, below] of the vertical velocity (m s−1). The black isolines correspond to the cloud contour plotted as rc + ri = 10−6 kg kg−1. The blue outlines in (e) and (f) delimit the subdomain where the statistical analysis is computed.
Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-20-0386.1
Figure 2b displays the buoyancy at the same time computed as
Figure 2c displays the cloud water mixing ratio in the liquid phase as no ice is present. It increases with the altitude with strong gradients at the cloud-top interface. Rainwater mixing ratio is displayed in Fig. 2d. At this time, the cloud precipitates from z = 4 km, near its top below the highly buoyant core, to z = 1 km. Graupel and snow mixing ratios are negligible.
At 2.5 km height, the cloud is irregularly shaped with an elongation in the wind direction (Fig. 2e). Strong downdrafts originating from smaller clouds located near the lower lateral boundary of the refined subdomain are visible, but they will not be taken into account in the following statistical analysis because they are not associated with the main cumulus congestus. Through the center of the toroidal circulation at a height of 3.5 km, (Fig. 2f) displays strong vertical velocity gradients inside the cloud near its edges and a negative vertical velocity halo around its entire periphery. At the lower altitude of 2.5 km (Fig. 2e) this type of halo does not appear.
b. Turbulence
The high resolution of the LES offers the possibility to use the resolved flow to determine turbulence statistics. The subgrid-scale turbulence represents, by design, a very small fraction of the total turbulence. Figure 3 displays vertical cross sections at y = 2.25 km of the resolved TKE, the ratio of the subgrid TKE to the total TKE, and of the dynamical and thermal productions of subgrid and resolved TKE. If the subgrid turbulence represents only a small amount of the total turbulence of the cloud, it is more present at the edges. The resolved TKE is strongly correlated with vertical velocity. Its strongest values are reached above 2.5 km and its weakest ones in the downshear side of the cloud at low altitudes. Weaker values of the resolved TKE are also present outside the cloud near its edges especially in the downshear regions below 2.5 km and around its top due to the disturbance of the environment by the rising cloud top. The points where the resolved TKE is below the threshold of 0.5 m2 s−2 are masked.
Vertical cross sections at y = 2.25 km of (a) the resolved TKE (m2 s−2), (b) the ratio of the subgrid TKE to the total TKE above a threshold of 0.5 m2 s−2, (c) the thermal production of subgrid TKE (m2 s−3), (d) the dynamical production of subgrid TKE (m2 s−3), (e) the thermal production of resolved TKE (m2 s−3), and (f) the dynamical production of resolved TKE (m2 s−3).
Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-20-0386.1
The same mask as for the total TKE (above 0.5 m2 s−2) is applied for the vertical cross section of the ratio of the subgrid to the total TKE (Fig. 3b). High values of this ratio are characteristic of small eddies that are poorly resolved (when subgrid TKE is nonnegligible relative to total TKE) or of weak turbulence (when the total TKE is low). While it is weak in the cloud interior, highest values are obtained on the edges of the cumulus congestus especially in the downshear region corresponding to the turbulent wake, which can be explained by a lower presence of resolved large eddies in these areas. This ratio is commonly used as a measurement of the relevance of the LES employing the threshold of 0.2 to determine if most of the energetic eddies are resolved. As a few points only exceed this characteristic threshold, it demonstrates that a resolution of 5 m is well adapted to resolve the main part of the turbulence on the edges of a cumulus congestus. Using a coarse-graining approach (e.g., Bogenschutz et al. 2010; Honnert et al. 2011), the subgrid TKE reaches the level of the resolved TKE for a resolution around 200 m, indicating that around this scale half the turbulence is resolved and half is subgrid (not shown). A quantitative analysis of the subgrid-scale turbulence will be introduced in section 4.
The main production terms of the subgrid TKE budget, presented in Figs. 3c and 3d, show that the dynamical production is much greater than the thermal production by several orders of magnitude. The thermal production of subgrid TKE is especially aligned with cloud top and edges, as it is related to the vertical gradients of virtual potential temperature that are strong near horizontal interfaces as discussed in Klaassen and Clark (1985).
Because subgrid thermal production is negligible, the very fine-scale turbulence appears to be weakly dependent on buoyancy gradients. However, the high concentration of thermal production over the cloud edges justifies an interest for the thermal instabilities for larger scales since the ratio of the thermal production to the dynamical production of turbulence in convective clouds depends on the scale considered (Verrelle et al. 2017) and since the resolved thermal production largely dominates the resolved dynamical production (Figs. 3e,f). At 5-m resolution, the thermal production is largely resolved (a factor of 1000 between the subgrid part and resolved part) while the dynamical production remains largely subgrid (a factor of 10 between the resolved part and subgrid part). Overall, the total (i.e., resolved and subgrid) thermal production dominates by a factor of 2 over the total dynamical production. Using a coarse-graining method on the LES fields as in Verrelle et al. (2017), we can clearly see how the production of subgrid TKE depends on the spatial scale: at very fine scale the production is almost only of dynamic origin, then the thermal part increases progressively to become of the same order toward the 250-m resolution (Fig. 4).
Mean vertical profiles in the cloud of thermal (TP; orange lines) and dynamical production (DP; blue lines) of subgrid TKE (m2 s−3) for (a) the LES at 5 m and for coarse graining at (b) 50, (c) 100, and (d) 250 m.
Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-20-0386.1
To summarize, considering subgrid-scale turbulence, it has been shown that a 5-m resolution is sufficient to resolve most of the eddies and thus, to be in LES regime over the edges of a cumulus congestus. The ratio of subgrid TKE to the total TKE is the strongest on the edges of the cloud. Dynamical production dominates over thermal production for the subgrid TKE but overall, the thermal production of the resolved TKE is larger than the dynamical production. These results are in agreement with Verma (2019).
4. Partition of the cloud and its environment
To quantitatively characterize the edges of the simulated cloud with regard to the general structure of a cloud, a partition is employed. The partition distinguishes the cloud inner and outer edges from the cloud interior and the far environment (Fig. 5). Sensitivity tests on the width of the inner edges have been performed: 50- and 100-m-wide inner edges lead to similar interpretation of the results. We will keep a 50-m inner edge in the following sections. The small pieces of cumulus are considered as inner edges. The width selected for the outer edges is 150 m because 50 m would be too narrow to capture the dynamics of the subsiding shell and a larger extension would be too wide to study the near-cloud dynamics. In this section, special attention is paid to the subsiding shell and any downdraft located inside the cloud using vertical mass-flux profiles, thermodynamical diagrams, and statistical distributions.
Vertical cross sections of the vertical velocity (m s−1) at y = 2.25 km using different masks to illustrate (a) the cloud interior beyond 50 m from the edge, (b) the 50-m-wide inner edge, (c) the 150-m-wide outer edge, and (d) the far environment beyond the 150-m-wide outer edge.
Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-20-0386.1
a. Mass flux
Vertical profiles of the mass flux (kg s−1) in the different areas of the cloud and the environment: the cloud mass flux (gray line), the environmental mass flux (black line), the mass flux in the 50-m-wide inner edge (blue line), and the mass flux in 100-, 150-, 200-, 300-, 400-, and 500-m-wide outer edges (colored lines) for the (a) CTRL and (b) NOCOOL simulations. Also shown are thermodynamic diagrams comparing the nonprecipitating total water rc + rυ (kg kg−1) and the buoyancy b (m s−2) at z = 3.5 km over the subdomain plotted in Fig. 2f for the (c) CTRL and (d) NOCOOL simulations. The color of the dots is indicative of the different areas of the partition; the larger dots indicate the center of mass of each area. The dashed line corresponds to the environmental curve, with crosses indicating the altitudes of 3 and 4 km. The indications about the green and black continuous lines and associated dots are given in the text.
Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-20-0386.1
The environment (ENV) is characterized by a subsiding mass flux (black line) that approximately compensates its ascending counterpart (gray line) as the mean vertical profiles of cloudy and environmental mass fluxes have similar shapes with opposed variations. The mass flux obtained in a 150-m-wide outer edge represents less than half of the descending mass flux of the environment at low altitude and less than one-third of it at high altitude (pink line). Thus, over z = 2.3 km, the portion of the subsiding shell lying within the first 150 m out of the cloud is therefore responsible for less than half of the compensation of the upward mass flux. A 300-m-thick outer edge contains half of the descending mass flux while a 500-m-thick outer edge contains approximately the two-thirds of it. It is interesting to compare these results with those obtained by Jonker et al. (2008) for a large number of shallow cumulus clouds. They calculated that 50% of the cloud mass flux was compensated in the 150 m that followed the frontier between the cloud and its environment and that 90% of it was compensated in a range of 400 m. As the cumulus congestus simulated here presents a much larger radius than the clouds simulated in Jonker et al. (2008) with typical radii of several hundred meters, we find that a larger fraction of the mass flux is compensated close to the cloud (in proportion to its size) for deeper cumulus clouds. Our study also suggests that a part of the mass-flux compensation is associated to a hill-vortex circulation while another part (which is less important but far from being negligible) comes from downdrafts occurring near the cloud edges. This is in line with the study of Katzwinkel et al. (2014), which distinguished an inner shell with negative buoyancy and strongly negative vertical velocity and an outer shell with a neutral buoyancy and less negative vertical velocity.
We also investigate the importance of the air located inside the cloud, near its edges (which might have mixed with environmental air) in the upward mass flux. The mass flux computed in a 50-m inner edge corresponds to the blue line in Fig. 6a. The inner edge contains a large amount of the total cloud mass flux at low altitudes due to the split nature of the cloud far below the highly buoyant core, while at the altitudes where the highly buoyant core occupies most of the cloud, the edges have a lower contribution to the upward mass flux.
b. Thermodynamical diagram
Thermodynamic diagrams similar to those presented in Park et al. (2017), here comparing the buoyancy and the nonprecipitating water mixing ratio (rnp = rυ + rc) at z = 3.5 km, are displayed in Fig. 6c, with the color indicating the area of the partition to which each point belongs. The diagram displays a hook shape that has been interpreted by Park et al. (2017) as a signature for vertical convective mixing (due to the overturning of vertical eddies). The points belonging to the far environment (beyond 150 m from a cloudy point) have low values of rnp and near-zero values for buoyancy, and are fairly homogeneous, while the points located in the 150-m outer edges have statistically lower values of buoyancy, most of them negative, and higher values of rnp but less than the threshold of saturation associated with the transition toward the cloudy points. In-cloud points have higher values of rnp and buoyancy due to condensation. Expectedly, the most intense values are obtained in the cloud interior. The environmental curve has been plotted as the black dotted line. The points located outside the cloud grossly lie along that curved line meaning that those points correspond to parcels originating from higher or lower altitudes and brought to the considered altitude by convective mixing. The offset from this line could be explained by the turbulent mixing that takes place between neighboring parcels. This constitutes a first element in the analysis that will be developed in section 6b.
The points located inside the cloud lie along a line interpreted as displaying the cumulative effects of convective mixing and condensational heating. Those two lines cross around rnp = 6 g kg−1 and b = −0.03 m2 s2, highlighting a buoyancy reversal that appears to be mostly located in the inner and outer edges. This buoyancy reversal exists over the edges of the cumulus congestus for any altitude (not shown).
c. Statistical distributions
To characterize the dynamical properties of the cloud and its environment in more detail, Fig. 7 displays distributions of the vertical velocity and buoyancy for the four different regions of the partition. Figure 8 displays the distributions of turbulence related quantities for the cloud interior and the interior 50 m from the cloud edge. The distributions focus on the main cumulus congestus corresponding to the subdomain indicated in Figs. 2e and 2f excluding the small clouds located in the inflow near the domain boundaries.
Vertical distributions of (left) the vertical velocity (m s−1) and (right) the buoyancy (m s−2) for (a),(b) the cloud interior, (c),(d) the 50-m inner edge, (e),(f) the 150-m outer edge, and (g),(h) the far environment at t = 4 min. The vertical mean profiles are plotted on the left of the distributions, and the mean distribution obtained by averaging occurrences for all altitudes in each bin is plotted at the bottom of the panels. These statistics have been computed in the subdomain plotted in Figs. 2e and 2f.
Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-20-0386.1
Vertical distributions at t = 4 min for (left) the cloud interior and (right) the 50-m inner edge of (a),(b) the thermal and (c),(d) the dynamical production of subgrid TKE (m2 s−3), (e),(f) the ratio of the subgrid TKE to the total TKE, and (g),(h) the logarithmic distribution of the anisotropic ratio for the points where the subgrid TKE is superior to a threshold of 10−3 m2 s−2 computed as
Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-20-0386.1
In the cloud, both the buoyancy and vertical velocity reach their strongest positive values near the cloud top (Figs. 7a,b). This is in line with the schematic thermal model of Blyth et al. (1988) and Zhao and Austin (2005). However, the positive values of the vertical speed and buoyancy obtained over the whole vertical extent indicate that the cloud is continuously alimented with warm, ascending air from below its base in the manner of a turbulent plume. Besides, the strong, continuous supply to the cloud core coming from lower altitudes and the highly turbulent nature of both the cloud core and its surroundings (Figs. 2f, 6b) highlight strong interactions between these two regions. This is reminiscent of Squires and Turner (1962), who considered that a bubble representation of a cloud was adequate for shallow cumulus while a plume representation was more judicious for cumulonimbus as the cumulus congestus simulated here would be somewhere in between.
The frequencies of occurrence of negative vertical velocity are more important in the 50-m inner edge (Fig. 7c) that in the cloud interior (Fig. 7a) at all altitudes. This means that many eddies or thermally induced downdrafts are present in the rather thin edges introduced here while the interior of the cloud mostly rises. Near the cloud top, stronger negative values of the vertical velocity are visible relative to other altitudes. They are not confined to the 50-m inner edge as their signature is also present on the cloud interior distribution. Those can be related to the toroidal circulation visible in Fig. 2a.
High frequencies of occurrence for negative buoyancy, with absolute values increasing with the altitude, are visible on the distribution of the inner edges (Fig. 7d). They are smaller in the cloud interior (Fig. 7b), which means that negative buoyancy is mainly confined to the inner edges.
The distributions for the outer edge have a more symmetric Gaussian shape with less dispersion. For the vertical velocity distribution (Fig. 7e), the Gaussian shape is centered around zero near the cloud base. Above it, the median value becomes progressively negative as the altitude increases until the cloud top above which the frequencies become exclusively positive. There is a strong negative peak around z = 3.5 km coinciding with the toroidal circulation. The buoyancy distribution (Fig. 7f) follows the same pattern but with less dispersion in particular for positive buoyancy. Another difference is that the most negative values are reached above the cloud top, which is consistent with Fig. 2b. This ascending area with negative buoyancy is driven by the higher pressure there due to the push of the rising cloud.
The far environment (Figs. 7g,h) is characterized by a very tight Gaussian distribution around zero reflecting a more homogeneous atmosphere except in the boundary layer marked by an alternation between updrafts and downdrafts. Negative values between 1 and 2 km are due to the remnants of smaller clouds to the side of the main cloud.
As snapshots in Fig. 3 have shown that small eddies are mainly concentrated near the cloud edges, a more quantitative characterization of turbulence is provided with the distributions presented in Fig. 8 for the inner edges and the cloud interior. They confirm that at 5-m resolution, the magnitude of the thermal production of subgrid turbulent kinetic energy is less than that of dynamical production by a factor of 200. Characteristics of the thermal production distributions are nevertheless interesting insofar as the thermal production and destruction are tangibly stronger in the 50-m-wide inner edges that they are in the cloud interior. It is also the case for horizontal thermodynamical fluxes (not shown). Again, it is not surprising because those are computed with the rnp and θl (liquid–ice potential temperature) gradients, which are strong at the interface (see appendix).
On the contrary, the dynamical production of subgrid TKE is slightly stronger in the cloud interior than near its edges, which is not surprising as velocity gradients are stronger near the main updraft. Distributions for the cloud interior and the inner edges have the same shapes with a gradual increase of the spread with altitude until a strong peak slightly above z = 3.5 km, which corresponds to the location of the center of the toroidal circulation, and a decrease farther up.
The ratio of the subgrid TKE to the total TKE is more important on the inner edges than in the cloud interior, which quantitatively demonstrates the result discussed in the previous section. The relative importance of fine-scale turbulence is potentially greater on the edges of the cloud than in the cloud interior. The vertical distribution of the ratio is approximately the same for the cloud interior and the inner edges. Its strongest values are encountered in the upper two-thirds of the cloud.
The subgrid anisotropy ratio, which is the ratio between horizontal (along y here) and vertical wind speed subgrid variances, is displayed in Figs. 8g and 8h for the cloud interior and the 50-m inner edges. Distributions are Gaussian with a median centered on 1. Occurrences for ratios greater than 2 (or respectively less than 0.5) are rare, which means that few eddies are more than twice as wide (respectively high) as they are high (respectively wide). Isotropy for subgrid eddies is statistically well obtained and no important difference appears between the cloud interior and the inner edges.
To summarize this section, a buoyancy reversal occurs on the edges of the simulated cumulus congestus. Such a phenomenon has already been noted in fine-resolution simulations (Matheou et al. 2011) but was studied less in detail. Negative values of vertical velocity and buoyancy associated with turbulent effects and/or convective mixing are generally confined to the vicinity of the cloud–environment interface. As discussed in the previous section, the subgrid turbulence represents a greater proportion of total turbulence on the edges. It is mainly produced by dynamical effects despite thermal production being stronger near the interface than in the cloud core. It is statistically isotropic with similar distributions of the anisotropic ratio between the edges and the cloud interior. We find the signature of well-known phenomena associated with cloud dynamics such as the toroidal circulation or the subsiding shell. The simulated subsiding shell is significant in terms of mass flux with a characteristic width proportionally smaller to what was obtained around smaller clouds. In the next section, the toroidal circulation and the subsiding shell will be examined more in detail with an emphasis on the new insights provided by this study.
5. Physical and dynamical processes on the edges of the cloud
a. Characterization of the subsiding shell
Horizontal profiles along the x coordinate of liquid cloud water mixing ratio rc, vertical velocity w, buoyancy b, and the resolved TKE are displayed in Fig. 9 for the altitudes of 1.5, 2.5, and 3.5 km in the plane of the cross sections displayed in Fig. 2a. The shaded area emphasizes the cloudy parts. The profiles can be compared with those presented in Heus and Jonker (2008) (simulations of a large sample of shallow cumulus) and in Katzwinkel et al. (2014) (very fine-scale measurements on a large sample of shallow cumulus). For this perspective and in order to keep the information about the fine-scale processes on the edges of the cloud, all of the variability of the fields is presented to be similar to the aircraft measurements. In the case of an actively growing cloud, Katzwinkel et al. (2014) obtained, at 100 m below the cloud tops, an inner shell width of 7 m, an outer shell width of 60 m and characteristic values for vertical velocity and buoyancy represented by 10th percentiles of w10% = −1.7 m s−1 and b10% = −0.008 m s−2. The profiles obtained here display higher values (−6 m s−1 and −0.02 m s−2) and larger widths for the inner and outer shells (approximately 50 m and 500 m in the upshear region for z = 3.5 km), which is not surprising since shallower cumulus clouds were considered in Katzwinkel et al. (2014).
Horizontal profiles at t = 4 min along the lines indicated in Fig. 2a for (a),(e),(i) the cloud water mixing ratio (g kg−1), (b),(f),(j) the vertical velocity (m s−1), (c),(g),(k) the buoyancy (m s−2), and (d),(h),(l) the resolved TKE (m2 s−2) at y = 2.25 km and z = (top) 3.5, (middle) 2.5, and (bottom) 1.5 km. The shaded areas correspond to the cloudy region above the rc + ri ≥ 10−6 kg kg−1 threshold.
Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-20-0386.1
The minimum values for vertical velocity are obtained near the cloud edges. In Heus and Jonker (2008), enhanced positive vertical velocity is found in the cloud upshear region while the subsiding shell is wider in downshear regions. This result is well reproduced in the present study, particularly at the lower two levels shown for enhanced positive vertical velocity, and at the highest level shown for the wider subsiding shell in the downshear region. A negative buoyancy peak is visible in the upshear regions (Figs. 9c,g,k, left sides) at the interface. It is very thin and close to the cloud just outside of it. It is wider in the downshear regions. This peak indicates that the resolution is fine enough to better account for evaporative cooling effects at the cloud edges by resolving finer-scale turbulent mixing in line with Hoffmann et al. (2014). If the transect is selected closer to the cloud top as those presented in Katzwinkel et al. (2014), the obtained buoyancy peak is wider due to the buoyancy reversal at the cloud top (Fig. 2b). Cloud mixing ratio has strong gradients near the interface except at z = 1.5 km for the downshear side. The linear evolution of rc at low altitudes is a signature of a turbulent wake more marked in the downshear regions while the updraft tends to coincide with the upshear interface.
Although the resolved TKE profiles have nonzero values in the regions where downdrafts occur, a sharp drop is still noted relative to the in-cloud value. The latter is less important in the downshear regions of the profiles at z = 2.5 km and z = 1.5 km that correspond to the turbulent wake.
No sign of strong turbulence effects in the outer edges of the cloud is noticed on the horizontal profiles, distributions (not shown), or vertical cross sections even though the observational study of Gerber et al. (2005) described a strongly turbulent subsiding shell.
b. Toroidal circulation eddies
A toroidal structure similar to those observed in Damiani et al. (2006), Damiani and Vali (2007), and Wang and Geerts (2015) is clearly visible near the cloud top during the five minutes of the refined simulation. Streamlines for the wind fluctuations (for which the mean horizontal wind of the subdomain has been subtracted at each altitude) are displayed in a vertical cross section in Fig. 10a. Eddies following the characteristics of toroidal circulation are visible at the very top of the cloud, but other large eddies are also present on lower edges (e.g., at y = 1.5 km, z = 2.5 km or y = 4 km, z = 3 km). However, the mean vorticity computed with a 500-m box average (not shown) on the downshear eddy is 0.012 s−1, which is weaker (in absolute value) than what is obtained for the toroidal circulation eddies, which is around 0.035 s−1. Instabilities also develop along the upshear interface at lower altitudes but with smaller size (approximately 100–200 m, e.g., at x = 2.2 km, z = 2.6 km). As in Zhao and Austin (2005), the perturbation pressure is positive at the top of the toroidal circulation eddies and negative in their center and lower part (Fig. 10b). The enstrophy (defined as the square of the vorticity modulus) is very heterogeneous and emphasizes the presence of numerous small-scale structures (Fig. 10c). Low values are present in the outer edge and stronger values remain close to the cloud–environment interface.
Vertical cross sections at y = 2.25 km for (a) the streamlines for the w and
Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-20-0386.1
Figures 10d–f display the vertical and horizontal resolved turbulent fluxes of nonprecipitating water (rc + rυ) and the u′w′ eddy covariance. The resolved vertical flux of nonprecipitating total water mixing ratio is strongly positive in the cloudy updraft and slightly positive in the downdrafts associated with the toroidal circulation. For the resolved vertical flux of liquid potential temperature, the signs are opposite (not shown). This can be understood intuitively as nonprecipitating water mixing ratio is larger in the cloud than in the environment and liquid potential temperature is weaker in the cloud than in the environment. Therefore, the horizontal resolved fluxes of the discussed quantities are positive or negative depending on the direction of the u velocity associated with the toroidal circulations eddies; u′w′ follows the same pattern. As these resolved eddies represent important carriers for the exchange of heat, moisture, and momentum between the cloud and its environment, those results are meaningful concerning the importance of correctly parameterizing the contribution of toroidal circulation eddies in kilometer-scale models.
The results are consistent with the picture of cumulus clouds addressed by Zhao and Austin (2005), which depicted an ascending cloud top preserving a strong buoyancy by shedding weakly buoyant parcels near its top and whose upper part stands relatively undiluted. The part below the ascending core entrains and detrains in equivalent and homogeneous ways (profiles of bulk estimates, not shown) as in De Rooy et al. (2013).
Well-established toroidal circulation and subsiding shell are generated. Their characteristics are consistent with the size of the simulated cumulus in previous studies. The fine resolution allows one to begin representing fine-scale evaporative cooling effects in the subsiding shell, but it does not question the prime importance of the toroidal circulation in the exchanges even though smaller resolved eddies also participate in the mixing.
6. Impact of evaporative cooling
To measure the importance of evaporative cooling on the dynamics of the cloud and its subsiding shell, a restart of the refined simulation is performed during 2 min starting from the second minute where cooling effects related to changes of state (essentially evaporation of cloud water or rain) are canceled at each time step. The results with the modification (NOCOOL simulation) and without the modification (CTRL simulation) are compared at t = 4 min.
a. Mass flux
In the NOCOOL simulation, both positive and negative mass fluxes are enhanced suggesting that convection is attenuated by evaporative cooling effects (Fig. 6b). These results are different from those of Park et al. (2017), who obtained more intense updrafts and associated turbulent mixing with evaporative cooling and showed that the latter increases the mass flux in the peripheral region of the cloud core. However, Park et al. (2017) suppressed evaporative cooling during a long time, considered smaller clouds and used a coarser resolution (20 m horizontally and 25 m vertically), so it is difficult to determine where the differences come from.
For the subsiding shell, the compensation of the positive mass flux by the environment occurs farther away from the cloud in the NOCOOL simulation with a weaker downward mass flux in the 100-m-wide outer edge, so the subsiding shell has a weaker contribution to the environmental compensation when the evaporative cooling is suppressed. Two-thirds of the upward mass flux are still compensated in a 500-m-wide outer edge (above z = 2.5 km), and the large-scale compensating subsidence is more intense. The 50-m inner edge also includes a larger portion of the positive mass flux.
b. Thermodynamical diagram
Another difference between our results and those of Park et al. (2017) is the impact of the suppression of evaporative cooling on the shape of the thermodynamic diagram (Fig. 6d). The cloud interior and the far environment signatures are similar to those obtained with the CTRL simulation while the points belonging to the inner and outer edges become widespread with more positive values of buoyancy as well as higher mixing ratios in the outer edge.
The points deviating from the hook shape appear to be lying along mixing lines connecting parcels located inside the cloud to parcels located in the environment. Park et al. (2017) explained the shape of the diagram with convective mixing and condensation. The results presented in Fig. 6d suggest that, for a larger cloud as considered here, turbulent mixing also plays a role. It is intriguing that the mixing lines do not appear as clearly as for the diagram obtained for the CTRL simulation. We make the hypothesis that evaporative cooling shifts the mixed parcels close to the hook shape and this is tested by computing the evaporative cooling resulting from a mixing event between a parcel originating from the cloud and another one originating from the environment. A line (in green) has been drawn as an example on the diagram in Fig. 6c between the centers of mass of the points located in the inner edge and in the outer edge. The characteristics of the parcel that would be obtained for an isentropic mixing between two equal volumes of air representative of the inner and outer edges are selected then the shift in buoyancy associated with evaporative cooling effects is computed (black line). The resulting point happens to fall near the convective mixing line (Fig. 6c). This does not imply that evaporative cooling systematically replaces mixed parcels on the hook shape but does imply that it is possible to find configurations for which this happens.
c. Statistical distributions
Distributions similar to those presented in Fig. 7 but for the NOCOOL simulation are displayed in Fig. 11. In comparing Figs. 11d and 6d, it is seen that most of the negative and weakly positive buoyancy frequencies are suppressed for the inner edge. The distribution of buoyancy on the outer edges displays less negative values between z = 2.5 km and z = 4 km and a reinforcement of positive values at all altitudes. The increase is particularly marked around 3.25 km altitude where the base of the toroidal circulation is located. Vertical velocity is also impacted, as negative values are less frequent for the inner edges, the minima are smaller in terms of absolute value and the vertical means are globally greater. The distribution for the outer edges remains centered around zero, but it is slightly more spread, which is consistent with the intensification of the convective circulation obtained for the NOCOOL simulation (Fig. 6b). The cloud interior is also impacted but to a lesser extent. The far environment is unchanged and remains characterized by a relatively homogeneous atmosphere slightly subsident and neutral in terms of buoyancy.
As in Fig. 7, but for the NOCOOL simulation.
Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-20-0386.1
d. Budgets
To evaluate the relative contribution of the different terms of the potential temperature equation, budgets cumulated over 10 s (between 3 min 50 s and 4 min) are presented in Fig. 12 for the CTRL simulation. In particular, the evaporation–condensation term (evaporation process is suppressed in the NOCOOL experiment) is compared with the other terms. The cloud contours correspond to the position of the cloud at 4 min.
Vertical cross sections at y = 2.25 km of the potential temperature budget terms (K) accumulated over 10 s at t = 4 min for the CTRL simulation: (a) vapor condensation–cloud water evaporation, (b) rain evaporation, (c) advection, and (d) sum of the turbulent diffusion and dissipation terms.
Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-20-0386.1
The evaporation–condensation term is displayed in Fig. 12a. The contribution of the condensational heating is strong and associated with the updraft. Consequently, it does not allow any determination of evaporative cooling inside the cloud or at its top. Nevertheless, evaporative cooling effects are visible at the edge of the cloud at all altitudes. As they are weak relative to the condensational heating, the color scale is made asymmetric. Some evaporative cooling effects are obtained on the lateral edges of the cloud especially at the altitudes of the toroidal circulation. The downshear region of the cloud associated with the turbulent wake is also characterized by strong evaporative cooling effects.
The second term of the potential temperature budget relative to phase changes is due to rain evaporation. It is mostly present below the downshear region of the tilted cloud but it also appears below large eddies on the cloud edge including those associated with the toroidal circulation. The cooling by rain evaporation is weak relative to evaporative cooling of liquid cloud water. The cooling contribution of these two first terms corresponds to the part of the budget that is removed in the NOCOOL simulation.
The strongest term of the potential temperature budget is the advection term (Fig. 12c). It is mostly negative in the cloud interior above 2.5 km and positive along the cloud–environment interface at the top of the cloud, which is consistent with the cloud elevation. The subgrid turbulent term (due to turbulent diffusion and dissipation) makes a mainly positive contribution (Fig. 12d) that is weak in comparison with the resolved transport term.
The budget terms for the vertical velocity are displayed in Fig. 13 to illustrate the different contributions for the evolution of vertical velocity and in particular of subsidence. The pressure term is displayed in Fig. 13a. It is negative in the cloud near its top, opposing its ascent. Above the cloud top, it is slightly positive and negative in the rest of the environment. An overlook of all the terms of the budget indicates that this term is the main contribution for the supply of the subsidence far from the cloud–environment interface.
Vertical cross sections at y = 2.25 km of the vertical velocity budget terms (m s−1) accumulated over 10 s at t = 4 min: (a) the pressure perturbation gradient, (b) advection, (c) buoyancy for the CTRL simulation, and (d) buoyancy for the NOCOOL simulation.
Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-20-0386.1
The second term presented in Fig. 13b is the advection term, which is the strongest contribution to the budget. It is mainly negative in the upshear region of the cloud due to the advection of the updraft by the mean environmental wind. It is strongly positive in the highly buoyant core near the cloud top, which can be associated with a supply of vertical momentum coming from the lower altitudes. It opposes the negative gradient perturbation pressure term at the top of the cloud allowing the core to elevate. As the cloud is actively rising, the intensity of the vertical advection term due to condensational heating is stronger than the barrier of the gradient perturbation pressure. Consequently, the updraft is broad with a comparatively small toroidal circulation. Far from the cloud–environment interface, the advection term opposes the setting up of the large-scale compensating subsidence in the upshear region while reinforcing it in the downshear region. Near the cloud, it is a strong contribution to the subsiding shell as negative values of vertical velocity are associated with eddies located at the cloud–environment interface (Figs. 5c and 10a).
The term in Fig. 13c is characteristic of the impact of buoyancy (including evaporative cooling) on vertical velocity. As for the term associated to changes of water state in Fig. 12, the color scale is made asymmetric. This term makes a positive contribution over almost the entire cloud interior and it is the most intense in the ascending core. Negative contributions, which result from mixing with environmental air and subsequent evaporative cooling, are the strongest in the turbulent wake in the downshear region, which is in agreement with halos of enhanced humidity observed by Perry and Hobbs (1996) and Lu et al. (2003) on the downshear sides of cumulus clouds. The same buoyancy term in the NOCOOL simulation (Fig. 13d) displays an intensification of positive contributions, particularly near the edges, while negative contributions at the edges and in the turbulent wake are attenuated. The contribution of evaporative cooling on the velocity budget at the top of the cloud cannot be dissociated from the large negative contribution due to the resistance of the environmental air.
In this section, the differences with an alternative simulation where evaporative cooling effects were suppressed have been studied. The suppression of evaporative cooling disturbs the buoyancy reversal that settles on the edges of the cloud, which has an impact on the subsiding shell. The latter is also dynamically influenced by advection effects, in particular by eddies at the cloud–environment interface. The downdrafts of the subsiding shell appear to be both driven by evaporative cooling effects and by the dynamics of vortical structures, with effects more strongly related to vortices in the upshear regions and near the top of the cloud, and effects more strongly related to evaporative cooling in the downshear regions. In the next section, individual eddies will be studied qualitatively for both simulations in order to understand more precisely the effects of the evaporative cooling.
7. Discussion on instabilities
In this last section, the circulation of individual eddies is examined, comparing the CTRL and the NOCOOL simulations. The clouds in the two simulations are very similar except for small changes in the structure of the cloud–environment interface.
For both simulations, one of the eddies associated with the toroidal circulation, displayed in Figs. 14a and 14d, has a diameter of approximately 500 m. Its center is located near the interface. In the CTRL simulation, the streamlines indicate that air enters the cloud below the protuberance associated with the upper part of the cloud and exits the cloud above it. It is the same for the NOCOOL simulation, but increased positive vertical velocity is noted, which is consistent with the enhancement of the general circulation obtained in this alternative simulation. In the vicinity of the interface, a general decrease in buoyancy and more negative values characterize the CTRL simulation (Fig. 14c vs Fig. 14f). This is also consistent with what has been obtained in the experiment of Gorska et al. (2014). This phenomenon takes place almost all along the cloud–environment interface (Fig. 12a) and it is due to small-scale mixing at the edges. This is not present in the NOCOOL simulation, by construction.
Vertical cross sections of an eddy of toroidal circulation in the upshear region of the cloud for (a)–(c) the CTRL simulation and (d)–(f) the NOCOOL simulation for (left) vertical velocity (m s−1) and streamlines, (center) cloud water mixing ratio rc (kg kg−1), and (right) buoyancy b (m s−2). The black isolines correspond to the cloud contour plotted as rc + ri = 10−6 kg kg−1.
Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-20-0386.1
In the CTRL simulation, the entrainment caused by the toroidal circulation eddy induces a decrease in buoyancy and cloud water mixing ratio near the base of this circulation. This entrainment is associated with a deeper penetration on both fields in the NOCOOL simulation while the drop in buoyancy is more important at this location for the CTRL simulation. The buoyancy term is negative at the base of the toroidal circulation eddy in CTRL (Fig. 14c), which is a signature of the strong evaporative cooling associated with engulfment by this vortex. This effect does not appear in the NOCOOL simulation (Figs. 14e,f). In the CTRL simulation, the entrained air impacts the altitudes located directly below those at which it is introduced, while in the NOCOOL simulation, the entrained air is more constrained by the toroidal circulation.
Eddies closer to the summit are now explored. Emanuel (1981) argued that cumulus top being naturally unstable, a mix of environmental and cloudy air would be negatively buoyant at any altitude and that therefore cumulus clouds should contain a large amount of small penetrative downdrafts, which would not be explicitly resolved unless a very fine resolution is used.
No downdrafts capable of traveling over long distances (>200–300 m) are simulated in Heus et al. (2008) for shallow convective clouds with a 50-m resolution or in Böing et al. (2014) for deep convective clouds with a slightly coarser resolution. However, penetrative downdrafts may appear for a cloud reaching its level of neutral buoyancy (Carpenter et al. 1998). The cloud simulated here does not reach its level of neutral buoyancy but the fine resolution could allow to produce penetrative downdrafts due to processes that were not resolved in coarser simulations.
A visual inspection of the horizontal cross sections of vertical velocity at the cloud top displayed in Fig. 15 reveals several downdrafts included in the cloud or creating discrete holes in it (e.g., x = 2.30 km, y = 2.05 k; x = 2.95 km, y = 2.70 km; x = 3.05 km, y = 1.95 km). They are rare and never penetrate deeply inside the cloud. Systematically associated with negative buoyancies (not shown), they disappear or are attenuated in the NOCOOL simulation (Fig. 15b). In CTRL they might therefore be caused by instabilities for which evaporative cooling plays a role.
(top) Horizontal cross sections and (top middle) vertical cross sections along the A–B line of the vertical velocity (m s−1) near the cloud top for (a),(c) the CTRL and (b),(d) the NOCOOL simulations at t = 4 min. Also shown are vertical cross sections of rc (kg kg−1) along the C–D line at (e) y = 2.15 km and t = 4 min and (f) y = 2.35 km and t = 4 min 40 s for the CTRL simulation. The black isolines correspond to the cloud contour plotted as rc + ri = 10−6 kg kg−1.
Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-20-0386.1
A vertical cross section of one of these downdrafts indicates that the cloud penetration by environmental air is associated with two counterrotating vortices with characteristic sizes less than 100 m (Fig. 15c). The instability introduces negatively buoyant environmental air into the cloud by forming a tongue comparable in length to the diameter of the eddies associated with the instability. The extremity of this tongue is mixed with the underlying cloudy air, slowing its ascent. This instability is strongly attenuated in the NOCOOL simulation (Fig. 15d).
It is difficult to determine if the entrainment event is caused by evaporative cooling or if evaporative cooling is merely a consequence of the entrainment caused by the instability as hypothesized by Grabowski and Clark (1991). An analysis using Lagrangian tracers statistically confirms that the subsidences do not penetrate deeply into the cloud (not shown). The intensity of the ascending core, strongly buoyant at the top of the cloud as shown by the distributions (Figs. 7a–d) with high values of cloudy water, opposes this penetration.
On the top part of the cloud, eddies with diameters ranging between 50 and 200 m develop in CTRL (Figs. 15e,f, indicated by arrows). Their rotation is opposite to that of the vortex ring as in the experiments of Gorska et al. (2014), who related those eddies to Kelvin–Helmholtz instabilities due to the horizontal outflow near the top of the cloud. Other types of eddies are also resolved. For example, small eddies appear in the turbulent wake (not shown), which is the seat of strong evaporative cooling effects (Figs. 12a,b and 13c).
The results displayed in this section concern the new insights on cloud-edge instabilities. They suggest that entrainment is both associated with small-scale mixing at the interface and engulfment by large eddies. The fate of engulfed air appears to be modified by evaporative cooling effects but no penetrative downdrafts traveling over long distances (more than 200 m) are simulated for this growing cumulus congestus.
8. Conclusions
A very fine resolution (5 m) has been used to simulate a cumulus congestus and its environment during 5 min with a downscaling method to characterize the turbulent eddies on its edges. This resolution can represent processes that have been little studied through numerical simulations and to obtain new insights on processes that have been studied with coarser resolutions. First, the simulated congestus is consistent with recent studies concerning cumulus cloud dynamics. The main characteristics of the cumulus are shown in Fig. 16, giving an overview of the dynamics associated with the simulated growing cloud. The toroidal circulation (Damiani et al. 2006) and the subsiding shell (Heus and Jonker 2008) are obtained even though an even finer resolution may be necessary to represent certain physical processes impacting the latter with more precision. To study turbulence, dynamical, and thermodynamical effects on the edges of the cloud, a partition has been defined. This partition differentiates the cloud interior, its inner and outer edges and the environment. A classical threshold on the liquid water and ice mixing ratios (10−6 kg kg−1) is employed to define the cloud–environment interface. This definition is suited for the study of the turbulent/nonturbulent interface associated with the cloud dynamics as this interface encompasses the cloud–environment one very closely (e.g., da Silva et al. 2014; Mellado 2017). The turbulence is significantly lower in the outer edges than in the inner edges thus defined and the gradients of the resolved variables are strong at the interface. The fine-scale turbulence is well resolved by the simulation with a ratio between the subgrid TKE and the total TKE less than 0.2, while it is stronger on the edges of the cloud due to the stronger presence of unresolved eddies. At 5-m resolution, the dynamical production of subgrid turbulence clearly dominates over the thermal production in agreement with Verma (2019). The subgrid thermal production is stronger close to the interface than in the cloud interior. The thermal production becomes predominant at resolved scales, the tipping point between thermal and dynamical productions occurring around the 250-m scale.
Synthesis of the different dynamical characteristics of the cumulus congestus. The colors represent different areas of vertical velocity and buoyancy (red: w ≥ 0.1 m s−1 and b ≥ 0.065 m s−2; cyan: w ≤ −0.1 m s−1; green: w > 0 m s−1 and b ≤ −0.0025 m s−2; white: other cloud points; yellow: other environmental points). The brown isolines correspond to the cloud contour plotted as rc + ri = 10−6 kg kg−1.
Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-20-0386.1
The key features of a turbulent wake and an ascending cloud top are very well resolved but the continuous nature of the distributions depicts a cloud also sharing characteristics of a plume probably due to its size. The cumulus congestus seems to be composed of several thermals that come together to form a more powerful lift, which compares well to the thermal chain concept as described by Morrison et al. (2020). It is worth mentioning that the entrainment and mixing effects obtained for this congestus cloud with only a single simulation can be strongly influenced by the chosen environment, which in this case is very humid. Further studies would be needed to test sensitivity to environmental conditions like wind shear or humidity, as the latter will modify the evaporative cooling. The importance of the toroidal circulation for the resolved turbulent fluxes and thus for the exchanges between the cloud and its environment is highlighted showing the necessity to parameterize the effects of these eddies correctly in kilometer-scale models.
The subsiding shell is consistent with previous studies both qualitatively and statistically. The evaporative cooling effects immediately outside the cloud edge are better resolved than in most studies and impact the cloud dynamics by reducing buoyancy and assisting downdrafts. The evaporative cooling effects thus contribute to the subsiding shell even though this latter is also dynamically influenced by advection effects, in particular by eddies at the cloud–environment interface.
Moreover, a buoyancy reversal is clearly obtained on the edges of the cumulus congestus generalizing the results obtained by Park et al. (2017) to deeper clouds and precising those of Matheou et al. (2011). As a consequence of the different experimental conditions (size of the cloud, resolution, and duration of the simulation) when compared with the study of Park et al. (2017), different conclusions are obtained concerning the nature of this reversal: in our study evaporative cooling effects mitigate convection and have a stronger impact on the buoyancy reversal at the edges. A partition of the cloud and its environment shows that occurrences of negative buoyancy and significant negative vertical velocity are common in the vicinity of the cloud–environment interface but scarcer in the cloud interior (at more than 50–100 m from the environment) or in the far environment (at more than 150 m from the cloud). Concerning the cloud exterior, these results are in agreement with Glenn and Krueger (2014) that also investigated cloud-edge downdrafts in an LES of deep convection but with an horizontal resolution of 100 m.
As in previous studies, evaporative cooling effects appear to be associated with small-scale mixing alongside the cloud–environment interface but some evaporative cooling inside the cloud is also indicative of large-scale engulfment due to eddies with a diameter of a few tens to a few hundreds of meters on the edges of the cloud, explicitly simulated in this simulation. Further study would be necessary to precisely determine the distribution of these effects.
An alternative simulation where evaporative cooling is suppressed during 2 min shows that evaporative cooling has an impact on the buoyancy reversal due to the mixing. The suppression of evaporative cooling removes most of the negative values of buoyancy and increases the positive ones close to the cloud–environment interface. The vertical velocity is also increased in the inner edges. The interior of the cloud, for its part, is mostly indirectly impacted by the suppressed evaporative cooling, inducing an increase of the vertical mass flux and the subsequent compensation in the environment.
A qualitative study of an eddy of the toroidal circulation indicates that eliminating evaporative cooling appears to increase dilution by entrainment by allowing dry air to penetrate deeper into the cloud. Evaporative cooling causes the entrained air to plunge to lower altitudes while remaining close to the cloud–environment interface. The determination of the final fate of the cooled air would for instance necessitate the use of Lagrangian trajectories.
A few downdrafts are simulated at the cloud top. They are sometimes associated with intrusions of dry air, but none appear to sink more than around 100 m into the cloud, which is in agreement with what Heus et al. (2008) obtained with a coarser resolution. The intensity of the ascending core, strongly buoyant at the top of the cloud opposes this penetration. It is not possible here to determine if the cloud-top downdrafts arise from instabilities directly caused by evaporative cooling (Squires and Turner 1962; Emanuel 1981) or from dynamical instabilities to be posteriorly reinforced downward by evaporative cooling (Grabowski and Clark 1993a).
The nonlinear interactions between the different scales of motion that occur on the edges of cumulus clouds are complex and this article is an attempt in understanding which processes, scales and structures are predominant for entrainment and detrainment in deep convective clouds. An interesting sequel could be to quantify the relative importance of engulfment and small-scale mixing over different locations at the interface. It is worth mentioning that the entrainment and mixing effects obtained for this congestus cloud with only a single simulation can be strongly influenced by the chosen environment. Further studies would be needed to test sensitivity to environmental conditions like wind shear or humidity, as the latter will modify the evaporative cooling. Shear effects need to be further investigated, as Lasher-Trapp et al. (2021) found that the entrainment is stronger in the sheared clouds in a growing stage of a supercell thunderstorm but Peters et al. (2019) have shown that it is not enhanced by wind shear on moist thermals. Another important limitation of this study is the use of a single-moment microphysical scheme. The use of a two-moment scheme would make it possible to better take into account some interaction processes between turbulence and microphysics such as the activation of droplets, the degree of heterogeneity of the mixing, considering some observation studies as Burnet and Brenguier (2007), for instance. The impact of radiative cooling at cloud tops should also be estimated. It would also be interesting to see how the results could be generalized to deeper convective clouds involving a mixed microphysics.
Nonhydrostatic models resolving small-scale mesoscale circulations such as cumulus convection and sea-breeze circulations with horizontal scales on the order of a few kilometers to 100 m and a regional domain of a typical size of a few kilometers to a few thousand kilometers.
Hydrostatic models with horizontal resolutions as small as about 10 km over a global domain or with a size of several thousand kilometers.
Acknowledgments.
The authors thank Jean-Luc Redelsperger and Antoine Verrelle for their valuable and helpful comments. The authors thank Walter Hannah and two other anonymous reviewers whose comments helped to improve the paper. This work was supported by the French National Research Programme LEFE of the National Institute for Universe Sciences (INSU) through the TurbDeepCloud project.
Data availability statement.
The data created or used during this study are available by writing to the corresponding author at CNRM.
APPENDIX
Thermal and Dynamical Productions of Turbulence
The resolved productions can be obtained directly by computing the resolved fluxes as the product of the fluctuations (being defined as the difference with the horizontal average over the domain).
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