Organization Development in Precipitating Shallow Cumulus Convection: Evolution of Turbulence Characteristics

Oumaima Lamaakel aDepartment of Mechanical Engineering, University of Connecticut, Storrs, Connecticut

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Georgios Matheou aDepartment of Mechanical Engineering, University of Connecticut, Storrs, Connecticut

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Abstract

Horizontal organization or mesoscale variability is an important mechanism in the interaction of the boundary layer with the large-scale conditions. The development of organization in a precipitating cumulus trade wind boundary layer is studied using large-eddy simulations with extensive horizontal domains, up to 160 × 160 km2 and fine grid resolution (40 m). The cloud fields vary between different computational domain sizes. Mean profiles and vertical velocity statistics do not vary significantly, both with respect to the domain size and when large-scale organization develops. Turbulent kinetic energy (TKE) rapidly increases when organization develops. The increase of TKE is attributed to the horizontal component, whereas the vertical velocity variance does not change significantly. The large computational domains blend the boundary between local convective circulations and mesoscale horizontal motions leading to the dependence of horizontal TKE on the LES domain size. Energy-containing horizontal length scales are defined based on the premultiplied spectra. When large-scale organization develops, the premultiplied spectra develop multiple peaks corresponding to the characteristic horizontal scales in the boundary layer. All flow variables have a small length scale of 1–2 km, which corresponds to local convective motions, e.g., updrafts and cumulus clouds. Organization development creates additional larger length scales. The growth rate of the large length scale is linear and it is about 3–4 km h−1, which agrees well with the growth rate of the cold pool radii. A single energy containing length scale is observed for vertical velocity for the entire run (even after organized convection develops) that is fairly constant with height.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Georgios Matheou, matheou@uconn.edu

Abstract

Horizontal organization or mesoscale variability is an important mechanism in the interaction of the boundary layer with the large-scale conditions. The development of organization in a precipitating cumulus trade wind boundary layer is studied using large-eddy simulations with extensive horizontal domains, up to 160 × 160 km2 and fine grid resolution (40 m). The cloud fields vary between different computational domain sizes. Mean profiles and vertical velocity statistics do not vary significantly, both with respect to the domain size and when large-scale organization develops. Turbulent kinetic energy (TKE) rapidly increases when organization develops. The increase of TKE is attributed to the horizontal component, whereas the vertical velocity variance does not change significantly. The large computational domains blend the boundary between local convective circulations and mesoscale horizontal motions leading to the dependence of horizontal TKE on the LES domain size. Energy-containing horizontal length scales are defined based on the premultiplied spectra. When large-scale organization develops, the premultiplied spectra develop multiple peaks corresponding to the characteristic horizontal scales in the boundary layer. All flow variables have a small length scale of 1–2 km, which corresponds to local convective motions, e.g., updrafts and cumulus clouds. Organization development creates additional larger length scales. The growth rate of the large length scale is linear and it is about 3–4 km h−1, which agrees well with the growth rate of the cold pool radii. A single energy containing length scale is observed for vertical velocity for the entire run (even after organized convection develops) that is fairly constant with height.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Georgios Matheou, matheou@uconn.edu

1. Introduction

Flow organization is a key feature of turbulent flows. Organization typically refers to large-scale coherent motions, where “large” length scales have comparable sizes to the flow mean profiles, e.g., the boundary layer depth. Specifically, convection organization in the atmospheric boundary layer can be partitioned into (i) vertically coherent motions, which form the updrafts and downdrafts (e.g., Schmidt and Schumann 1989; Neggers et al. 2019), and (ii) the horizontal organization of the boundary layer flow, including the horizontal spatial distribution of the convective elements, such as cloud streets and cellular patterns (e.g., Savic-Jovcic and Stevens 2008; Minor et al. 2011; Dagan et al. 2018; Anurose et al. 2020; Saggiorato et al. 2020; Stevens et al. 2020).

In atmospheric convection, vertical organization accounts for nearly all of mass and momentum vertical transport. The importance of organization with respect to the flow transport properties has been recognized early on—at least for the vertical coherent motions—and had led to the mass flux approach in convective parameterization formulations (e.g., Ooyama 1971; Arakawa and Schubert 1974; Arakawa 2004; Soares et al. 2004; Kuang and Bretherton 2006).

Whereas vertical coherence is primarily influenced by the local conditions (e.g., surface fluxes, vertical stability) horizontal organization is modulated by the large-scale environment and additional processes, most notably cloud microphysics and precipitation (Zuidema et al. 2012; Jeevanjee and Romps 2013; Spill et al. 2019; Vogel et al. 2020, 2021). Accordingly, the horizontal organization or mesoscale variability is an important mechanism in the interaction of the boundary layer with the large-scale conditions. Recently, a renewed interest in mesoscale organization of shallow convection is facilitated by a combination of factors, which include the increase of observational datasets, the development of cloud pattern objective classification methods, and the need to understand the response of the boundary layer to climate-scale warming. The development of cloud-pattern taxonomies (e.g., Stevens et al. 2020; Janssens et al. 2021) reduces the infinite natural variability of turbulence to a few dimensions and enables investigations of the associations between convection patterns and large-scale conditions (Bony et al. 2020; Schulz et al. 2021; Narenpitak et al. 2021).

Horizontal organization modulates the convective environment and has the potential to alter vertical transport (e.g., Bretherton et al. 2005; Muller and Held 2012; Smalley and Rapp 2020). In general, the three-dimensional turbulence in the boundary layer has the potential to link horizontal and vertical organization. This relation between horizontal and vertical coherent motions is the overarching question in the present investigation. The characteristics of horizontal–vertical coherence have implications on how clouds respond to climate-scale warming (Bony et al. 2020) and the formulation of convection parameterizations. To date, limited consideration is given to the effects of organization in parameterization formulation, which implies a limited dependence between horizontal and vertical flow coherence. Relatively few convection parameterizations have been proposed to account for large-scale horizontal organization, e.g., the models of Qian et al. (1998) and Grandpeix and Lafore (2010) for deep convection. A closely related question is as follows: What is the time scale (or growth rate) of development of convective organization in the boundary layer? This is also a parameterization-relevant question because a time adjustment period is needed for the turbulent flow to respond to changes in the large-scale forcing.

The dynamic time-evolving nature of the boundary layer, which is perhaps overlooked in static satellite images (Stevens et al. 2020; Janssens et al. 2021), is central in the present investigation. A high-resolution modeling approach using large-eddy simulation (LES) modeling is followed because three-dimensional time-resolved data are needed to quantify the evolution of convective organization. The representative case of Rain in Cumulus over the Ocean (RICO) (Rauber et al. 2007; van Zanten et al. 2011) is used to study organization development in a precipitating cumulus trade wind boundary layer.

The outline of the study is as follows. The model formulation and the simulation cases are documented in section 2. The results are discussed in section 3. First, the evolution of cloud organization is presented, followed by physical-space turbulent flow statistics. In the second part of the results, flow correlations are discussed and horizontal length scales are defined and analyzed. Finally, conclusions are summarized in section 4.

2. Methodology

The LES model of Matheou and Chung (2014) is used. The model numerically integrates the conservation equations for mass, momentum, liquid water potential temperature θl, and total water mixing ratio qt on an f plane. Cloud microphysics are modeled using the two-moment bulk warm-rain parameterization of Seifert and Beheng (2001). The buoyancy-adjusted stretched-vortex subgrid-scale (SGS) model (Chung and Matheou 2014) is used to account for the effects of the unresolved turbulence. The fourth-order centered fully conservative scheme of Morinishi et al. (1998) adapted for the anelastic approximation is used to approximate the momentum and scalar advection terms. A monotone flux-limited discretization is used for the microphysical advection terms to ensure water-mass conservation. The derivatives in the SGS model are estimated using second-order centered differences. A third-order Runge–Kutta method is used for time integration (Spalart et al. 1991). A Rayleigh damping layer is used at the top 0.5 km of the domain to limit gravity wave reflection.

The simulations correspond to the RICO case specification. Following van Zanten et al. (2011), the initial sounding has a trade wind cumulus atmospheric boundary layer structure, the flow is driven by a constant in time geostrophic wind profile, the effect of the large-scale environment (subsidence and advection) and clear air radiative cooling are included as source terms in the θl and qt equations, and surface fluxes are dynamically computed based on a constant sea surface temperature (SST) using bulk aerodynamic formulas [van Zanten et al. 2011, Eqs. (1)–(4)].

All simulations are carried out in a doubly periodic domain in the horizontal directions. Three simulations with horizontal domain sizes Lx = Ly = 40.96, 81.92, and 163.84 km are carried out, where x and y coordinates are along the zonal and meridional directions. The vertical domain height is Lz = 5 km. All simulations have uniform and isotropic grid spacing Δx = Δy = Δz = 40 m. The horizontal domain size is the only difference in the present simulations. The domain sizes are larger than van Zanten et al. (2011), which used domains of 202 × 4 km3. The simulations are performed in the Galilean frame translating with approximately the domain-mean wind [−6, −4] m s−1 (Matheou and Lamaakel 2021). The simulations are run for 30 h, instead of 24 h in van Zanten et al. (2011), to study the development of convection in the large domains. Table 1 summarizes the setup of the three LES runs. An additional control simulation that does not include the process of precipitation is described in the appendix.

Table 1

Summary of the large-eddy simulations. The simulations differ in the area of the horizontal domain. The number of grid points in the meridional (x) and zonal (y) directions are Nx = Ny, the number of vertical grid points is Nz, the grid spacing is Δx = Δy = Δz, Lx = Ly is the horizontal domain extend, and Lz the height of the computational domain.

Table 1

Validation of the present LES model for the RICO case is presented in Matheou and Chung (2014). Also, the model produces cloud features (i.e., patterns and their horizontal dimensions) in agreement with observations (Rauber et al. 2007; Nuijens et al. 2009; Snodgrass et al. 2009) and other model results (van Zanten et al. 2011; Seifert and Heus 2013; Li et al. 2014). As discussed in Matheou and Chung (2014), the model generates grid-independent results at the present Δx = 40 m grid resolution, therefore grid resolution is not a parameter in the present study. Further, the LES model was successfully used in several previous studies spanning a diverse set of meteorological conditions (Matheou et al. 2011; Inoue et al. 2014; Matheou and Bowman 2016; Matheou 2016; Thorpe et al. 2016; Matheou 2018; Matheou and Teixeira 2019; Jongaramrungruang et al. 2019; Couvreux et al. 2020).

3. Results

a. Cloud field organization

The horizontal periodicity of the computational domain artificially imposes a maximum wavelength or length scale. The effects of domain size on convection organization are shown in Fig. 1. The largest domain size (Fig. 1c) includes features that are larger than the Lx = 40.96 and 81.96 km domains. Figure 2 shows the time history of the organization development in the large domain. Figure 2 shows cloud liquid water path (LWP) at 2-h intervals in t = 16–30 h, which effectively visualizes the cloud field. Up to t = 16 h the boundary layer is composed of scattered cumulus. The development of precipitation after t = 16 h results in the generation of cold pools, density currents driven by precipitation evaporative cooling below cloud base (e.g., Moncrieff and So 1989; Xu and Moncrieff 1994). Multiple cold pools are observed which grow in area and reach sizes comparable to the large domain run by t = 28 h. Cold pool interactions form complex structures at t = 30 h.

Fig. 1.
Fig. 1.

Cloud liquid water path (LWP) at t = 24 h for the three LES domains. All computational domains are periodic in the horizontal directions.

Citation: Journal of the Atmospheric Sciences 79, 9; 10.1175/JAS-D-21-0334.1

Fig. 2.
Fig. 2.

Cloud liquid water path at 2-h intervals from t = 16–30 h for run C. Panels show the development of large-scale cloud organization in the initially homogeneous cloud field. Axes are as in Fig. 1.

Citation: Journal of the Atmospheric Sciences 79, 9; 10.1175/JAS-D-21-0334.1

As discussed in the following sections, kinetic energy and scalar variance are transferred from small to large length scales in the boundary layer. The smaller domain sizes artificially terminate the evolution of this inverse energy cascade by limiting the maximum length scale. In other words, smaller domains do not allow the creation of large structures and force the convection to attain an organization state that “fits” into the domain. Note, that the presently smallest domain size is twice as large as the domain size of the model intercomparison study (van Zanten et al. 2011).

b. Turbulent flow statistics

The LWP contours of Figs. 1 and 2 show quite large sensitivity of the horizontal convection organization to domain size. In this section, the effects of horizontal organization on the turbulent flow statistics are quantified. Figure 3 shows time traces of vertically integrated turbulent kinetic energy,VTKE(t)=(1/2)0Lzρu2+υ2+w2dz, cloud liquid water path (suspended condensate), rainwater path (RWP), cloud top zc, cloud base zb, inversion height zi, fraction of cloud cover cc, and surface precipitation rate. The angle brackets denote the instantaneous horizontal average. The inversion height zi is defined as the height of the maximum θl gradient. Cloud base and cloud top are defined as the minimum and maximum heights with liquid water mixing ratio ql > 10−5 kg kg−1.

Fig. 3.
Fig. 3.

Time traces of vertically integrated turbulent kinetic energy (VTKE), cloud liquid water path (LWP), rainwater path (RWP), cloud-base height (zb), cloud-top height (zc), inversion height (zi), cloud cover (cc), and surface precipitation rate for the three LES domains. The legend corresponds to the horizontal domain length.

Citation: Journal of the Atmospheric Sciences 79, 9; 10.1175/JAS-D-21-0334.1

The traces of Fig. 3 corroborate the picture of the LWP plots in Figs. 1 and 2: For t < 16 h the boundary layer has short horizontal length scales and it is accurately captured by all domain sizes, because all traces are statistically identical. For t > 16 h, the flow rapidly becomes more energetic with a large increase of VTKE. The smallest domain size shows differences after t = 16 h and the intermediate domain size departs from the run C trace at about t = 20 h. Run C traces appear to “saturate” (i.e., stop increasing) at about t = 24 h. Because a larger-domain simulation is not available, it cannot be inferred from the present results if the behavior of run C traces for t > 24 h is because of inadequate domain size or cold pool interactions that can restrict the evolution of newly formed structures.

After hour 25, run A traces exhibit large swings because energy is accumulated in a small domain and then released as a burst of convective activity. Small domain sizes can induce artificial unsteadiness in convection.

Figure 4 shows profiles at t = 24 h for the three runs. Profiles are calculated using instantaneous horizontal averages (no time averaging), except the wq¯t flux which is additionally time averaged in a 10-min interval to create smoother curves. The profiles help place into context the results shown in Figs. 13. The profiles of the prognostic variables (wind, θ, and qt) are essentially identical for all domains (the profiles are the same at all times, even though only t = 24 h is shown). Accordingly, the inversion height zi in Fig. 3 is the same for all runs.

Fig. 4.
Fig. 4.

Profiles of u-component wind, potential temperature θ, total water mixing ratio qt, cloud liquid water mixing ratio ql, turbulent kinetic energy (TKE), horizontal component of TKE, vertical velocity variance, and vertical total water flux at t = 24 h for the three LES domains. Profiles are horizontal averages except the total water flux which is additionally averaged in time for a 10-min interval.

Citation: Journal of the Atmospheric Sciences 79, 9; 10.1175/JAS-D-21-0334.1

Interestingly, the same mean flow corresponds to multiple instances of horizontal organization. There is an inherent difficulty in predicting horizontal variability and organization from the mean state. This is a feature of turbulent flows dating back to the structure of shear flows and the (surprising at the time) discovery of large-scale turbulence organization (Brown and Roshko 1974). The effects of organization are shown in the second-order flow statistics: covariances and TKE.

The differences in TKE between the different domain sizes are attributed to the horizontal component and are concentrated at the top part of the cloud layer near the inversion. No significant differences are observed in the vertical velocity variance, which suggests that the character of vertically coherent motions is not altered by the change of horizontal organization in the present case. This conclusion is further discussed in the next section.

Another aspect of the differences in the traces of Fig. 3 is the depth of the cloud layer, which is almost 1 km deeper in run C compared to run A at about t = 24 h. The TKE, ql, and wq¯t profiles of Fig. 4 show that the run C curves have small extensions above 2.5 km. The area fraction of cloud tops with zc > 2.5 km is very small. The cloud-top height distribution is quantified in the normalized histograms of Fig. 5. At t = 22 h (Fig. 5b) the zc distribution of run C has a “fatter” tail compared to the other computational domains. At t = 16 h, all domains have identical zc distributions, whereas at t = 30 h, there is large variability because of the different states of the cloud field.

Fig. 5.
Fig. 5.

Distributions of cloud-top height in the three LES domains. Panels correspond to distributions at t = 16, 22, and 30 h. The vertical line denotes the inversion height zi.

Citation: Journal of the Atmospheric Sciences 79, 9; 10.1175/JAS-D-21-0334.1

Even though cloud tops reach up to 3.5 km (Fig. 3), Figs. 4 and 5 show that most of the cloud and turbulence is confined below zi. Thus, zi in Fig. 3 is more representative of the growth of the boundary layer compared to zc.

The results of Figs. 35 suggest three regimes of boundary layer evolution, excluding the initial model spinup for t < 2 h, in agreement with previous studies (e.g., Stevens and Seifert 2008; Matheou et al. 2011; Anurose et al. 2020). In the first regime t < 16 h a nonprecipitating cumulus-topped boundary layer forms. In the second regime, 16 < t < 25 h, precipitation develops, which generates horizontal organization. Also, the growth rate of zi decreases compared to the first regime. In the last regime, t > 25 h, convection transitions to a statistical steady state. All present simulations are not in steady state, which is achieved for t > 30 h (Seifert et al. 2015; Anurose et al. 2020). Moreover, in model setups with more realistic representations of radiative forcing, because of the observed differences in the cloud structure with respect to domain size, particularly differences in stratiform cloud cover (see Fig. 1), it is possible that each domain will reach a different statistically steady state, see discussion in Seifert et al. (2015).

The analysis in the next section compares flow statistics at times t = 16, 22, and 30 h to representatively sample the three boundary layer evolution regimes.

c. Length scales and spectra

To determine the growth rate of horizontal organization development, suitable turbulent-flow length scales must be defined. Because the boundary layer is assumed horizontally homogeneous, there is no physical horizontal length scale that is externally imposed. In other words, there are no boundary conditions to force a specific horizontal flow pattern. However, horizontal statistical homogeneity does not imply that there are no horizontal length scales, as can be readily observed in Fig. 2. Seifert and Heus (2013), following de Roode et al. (2004), used the horizontal length scale corresponding to 1/3 of the resolved scale qt variance. Presently, we focus on defining physical length scales to guide the analysis. Unlike the method of de Roode et al. (2004), the present method does not depend on the total variance, i.e., it is independent of the domain size. Moreover, as in the previous sections, the statistics in the current and next section correspond to the entire convection ensemble. Cumulus characteristics vary with respect to location in the LES domain as shown in Fig. 2 and as discussed in other studies (Li et al. 2014).

In a spatially statistically homogeneous three-dimensional turbulent flow the two-point correlation is defined as (Batchelor 1953, p. 24)
Rij(r,t)=ui(r,t)uj(x+r,t),
where the angle brackets denote a spatial averaging over coordinate x. In the present case, Rij does not depend on the spatial horizontal coordinates (x, y) and is a function of time, height, and the separation vector r. The present analysis utilizes only part of the full Rij tensor. The averaging is performed on horizontal planes and Rij is normalized by Rij(0,t)=uiuj to form the longitudinal and transverse two-point correlation functions
fu(rx,z,t)=R11(rx,z,t)uu,
gw(rx,z,t)=R33(rx,z,t)ww.
The subscripts denote the prognostic variable. The variance in the denominator of (2) and (3) depends on z and t. The integral length scale is the integral of the correlation function, Lu(z,t)=0fu(rx,z,t)drx, which can be used to define a large or “outer” flow scale (Batchelor 1953, p. 105).

Figure 6 shows correlation functions for u, w, qt, and θl. The plots show correlations taken in the zonal direction x, that is r = [rx, 0, 0] in (1). Even though the flow is not strictly horizontally isotropic, there is no dominant directional organization in the boundary layer. Correlations in the meridional direction y are similar to rx and conclusions do not depend on the choice of horizontal direction. The panels of Fig. 6 show correlation functions at different times (t = 16, 22, and 30 h) and heights (z = 0.3 km about half-height in the subcloud layer, z = 0.6 km somewhat above cloud base, z = 1.2 km about the midheight of the cloud layer, and z = 2 km near the inversion height zi).

Fig. 6.
Fig. 6.

Two-point correlations at different heights and times for the large domain run Lx = 160 km (run C). Each row corresponds to a different variable. Columns correspond to different times. Lines in each panel correspond to different heights.

Citation: Journal of the Atmospheric Sciences 79, 9; 10.1175/JAS-D-21-0334.1

In Fig. 6, only gw exhibits the classical turbulent flow behavior with the correlation approaching zero for rx > 1 km at all heights and times. Also, gw shows small variation with height, compared to the other correlations in Fig. 6.

Before organization develops, t = 16 h in Fig. 6, fu, fq, and fθ, have narrow peaks near rx = 0 and are nearly zero for rx > 10 km. Small dependence with z is also observed in fq and fθ. Development of organization results in broader peaks for fu, fq, and fθ and larger differences with respect to z. After large-scale organization development, fu, fq, and fθ do not approach zero for large rx. Near the inversion zi the correlation functions of u, qt, and θl have relatively large values for the entire rx range. As shown in Fig. 2, convection generates few large structures that “fill” the entire domain. The random nature of the horizontal flow at large distances is lost and large structures affect the flow in the entire domain. Similar features in the two-point correlations functions of w and θ are observed in LES of emerging mesoscale cellular convection patterns (Schröter et al. 2005).

In the present LES, the integral length scale is not a reliable measure of a horizontal length scale because the correlation distances are very large compared to the computational domain and the two-point correlation function does not always approach zero for large rx. Therefore, an alternative method is used to define horizontal length scales. We consider one-dimensional spectra along the longitudinal direction on horizontal planes, similar to the averaging used to compute the correlation functions. For scalar variables, u, and w the zonal x direction is used, and for υ the meridional y direction is used. Figures 7 and 8 show spectra of u, w, θl, and qt at t = 16 h and t = 22 h, respectively. Two types of spectra are shown, the normalized one-dimensional spectral function, e.g., Φuu(kx)uu1, and the premultiplied spectrum kxΦuu(kx)uu1. The premultiplied spectrum is proportional to the power in a logarithmic band centered at kx (Bullock et al. 1978; Jiménez 1998). Accordingly, local maxima can be used to identify energy-containing structures. All premultiplied spectra curves are smoothed by using a Gaussian filter to make their shape easier to discern. The filtering is performed in logarithmic space,
Φ¯(k)=Φ(k)g(logklogk)dk,
where the filter kernel is g(x)=10/πexp(10x2).
Fig. 7.
Fig. 7.

One-dimensional spectra along the x direction of u-component of wind, vertical velocity w, liquid water potential temperature θl, and total water mixing ratio qt at t = 16 h and z = 0.36 km. (top) The normalized spectral function Φ(k) and (bottom) the premultiplied spectra kΦ(k). Curves correspond to the different LES domains as in Fig. 3. Straight lines correspond to k−5/3 scaling. The x axis is converted to length scale to help in the interpretation of the spectra.

Citation: Journal of the Atmospheric Sciences 79, 9; 10.1175/JAS-D-21-0334.1

Fig. 8.
Fig. 8.

One-dimensional spectra along the x direction of u-component of wind, vertical velocity w, liquid water potential temperature θl, and total water mixing ratio qt at t = 22 h and z = 0.36 km. (top) The normalized spectral function Φ(k) and (bottom) the premultiplied spectra kΦ(k). Curves correspond to the different LES domains as in Fig. 3. Straight lines correspond to k−5/3 scaling. The x axis is converted to length scale to help in the interpretation of the spectra.

Citation: Journal of the Atmospheric Sciences 79, 9; 10.1175/JAS-D-21-0334.1

Figures 7 and 8 show spectra for all three computational domains in the subcloud layer, where the flow is continuously turbulent. Figures 7 and 8 correspond to the spectral view of the computational domain sensitivity and effects of organization. The abscissa in the spectra plots is converted to length l = 2π/kx to help the interpretation of the results.

Similar to previous results, at t = 16 h spectra in the three domains are nearly identical in the range of overlap. Premultiplied spectra exhibit a single peak at about the same length scale ∼ 1.5 km. Spectra of w, θl, and qt have subranges with constant-exponent scaling. Only Φww exhibits a subrange with ∼k−5/3. Scalar variables have “shallower” scalings with exponents near −1.4. Similar characteristics are observed in stratocumulus-topped boundary layers (Matheou 2018). The qt spectra of the RICO case in Schemann et al. (2013) have exponents −1.7 and −4.6 at low and high wavenumbers, respectively, but a direct comparison with the present results is not possible because in Schemann et al. (2013) the spectra are averaged both in height and time. The classical Kolmogorov–Obukhov scaling does not apply to the present case because the flow is not isotropic and θl and qt are active scalars. Moreover, as shown by the correlation functions (Fig. 6), the flow is not strictly statistically horizontally homogeneous.

At t = 22 h, Φuu, Φθθ, and Φqq differ in the large-scale (low wavenumber) region with respect to the domain size. Organization leads to an increase in the energy content of the large scales, as expected. The increase in spectral energy at large scales between t = 16 and 22 h is larger and the range of scales broader than the LES of Bretherton and Blossey (2017, their Fig. 16), which does not include cloud arc and cold pool structures. Moreover, the premultiplied spectra develop multiple peaks. Figure 9 shows premultiplied spectra at t = 22 h at four heights. The length scale of the w premultiplied spectra peak does not change with height, which is the key result of Fig. 9. The other spectra have larger variations with height but the peak locations do not depend significantly with height, particularly for the shortest-length-scale peak. The premultiplied spectra of w and LWP of Schröter et al. (2005) show single peaks during their 12.5-h-long LES run, without any significant variation of the location of the peak. A direct comparison with the LES of Schröter et al. (2005) is challenging because their case has large anisotropy as cloud streets evolve toward an approximately uniform cellular convection pattern.

Fig. 9.
Fig. 9.

One-dimensional premultiplied spectra along the x direction at t = 22 h at four heights for the large-domain run.

Citation: Journal of the Atmospheric Sciences 79, 9; 10.1175/JAS-D-21-0334.1

We use the length scales corresponding to the peak of the premultiplied spectrum lpeak to determine the energy containing structures. As discussed in Jiménez (1998), lpeak does not necessarily correspond to the integral length scale Lu. For specific flows, lpeak can be related to Lu as originally proposed by von Kármán (1948), see also discussion in Trush et al. (2020). The present analysis follows similar methods applied to turbulent wall-bounded flows (e.g., Jiménez 1998; del Álamo et al. 2004). The spectra in the subcloud layer at z = 0.3 km are used to derive the length scales of the energy containing structures because the flow is continuously turbulent, thus avoiding any complications from large-scale intermittent turbulence.

Figure 10 shows the time evolution of length scales lu, lυ, lw, lq, and lθ, where the subscript corresponds to the prognostic variable. The largest simulation domain is used in the length scale analysis. When the premultiplied spectra have multiple peaks, multiple symbols are plotted in Fig. 10 for the same time (e.g., two filled circles at t = 20 h). Also plotted in Fig. 10 are the radii of the first two cold pools rcp1 and rcp2. The cold pool radii are determined by fitting a circle through the arcs of qt structures at z = 0.3 km. Because of the large variability of the length scales, the ordinate of the lu, lυ, lq, and lθ plots is shown in logarithmic scale, but the growth rate of the length scales is linear.

Fig. 10.
Fig. 10.

Time evolution of horizontal length scales in the atmospheric boundary layer. Panels show the evolution of the energy-containing horizontal length scales of (a) wind components lu and lυ, (b) vertical velocity lw, (c) liquid water potential temperature lθ, and (d) total water mixing ratio lq. Also shown in (d) are the radii of two cold pools rcp1 and rcp2. The development of organization creates a wide range of scales, thus a logarithmic scale is used for the y axis. Lines denote growth rates of 3 and 4 km h−1, which appear curved on the linear–logarithmic axes.

Citation: Journal of the Atmospheric Sciences 79, 9; 10.1175/JAS-D-21-0334.1

For the duration of the simulation there is only a single vertical velocity length scale lw and its value remains relatively constant lw ≈ 1.5 km for t > 5 h. The horizontal wind components develop additional length scales after horizontal organization develops. Similar to lw, a smaller energy containing length scale is present for u and υ that is about 1.5–3 km for the duration of the simulation. The smaller length scale of θl and qt does not register at a few times for t > 20 h because the spectra do not have a local maximum, even though a “bump” is still present in these locations, see Figs. 9c and 9d.

Figure 10d includes the evolution of the radii of the first two cold pools that form in the LES rcp1 and rcp2. The initial formations have radius of about 10 km and grow to nearly fill the entire computational domain (Fig. 2). The growth rates of rcp1 and rcp2 agree well with lq. Also, the length scale lq and the cold pool radius are in good agreement, which might be a fortuitous coincidence in the present LES. Similar to other turbulent flow scalings, the length scales of Fig. 10 quantify the flow up to a proportionality constant of order unity. The cold pools are generated at different times but grow with similar rates (Fig. 10d) independently, which suggests that the run C domain is sufficiently large to capture the development of horizontal organization in the flow.

The growth rate of energy containing structures length scales is 3–4 km h−1. Most of the data in Fig. 10 fall within this range (the constant slope lines dl/dt appear curved on the linear–logarithmic plot). The present growth-rate estimates are larger than previous simple models. For instance, the present growth rates are about 50% larger than the estimates in Romps and Jeevanjee (2016). Further, in the present LES, cold pools and horizontal length scales exhibit a sustained nearly linear growth rate from 10 to 100 km. Some of the discrepancies between the present results and the density current model of Romps and Jeevanjee (2016) can be attributed to the dynamic nature of convection at the density current front.

d. Convective updraft characteristics

A repeated observation in the analysis of the LES output is the insensitivity of the vertical coherent motions to horizontal organization. The insensitivity of local convective motions is present both when the creation of large scales is suppressed in the small computational domain sizes and when comparing different times in the convection evolution of the largest domain run. In this section we further explore the evolution of local convection characteristics by comparing three different times in the largest domain run C. The covariance decomposition method of Chinita et al. (2018) is used to compute updraft area fraction αu, downdraft area fraction αd, and mean updraft vertical velocity wu at t = 16, 22, and 30 h.

Following Chinita et al. (2018), the joint probability density functions (JPDs) of wθl and wqt constructed from the LES output are decomposed into local mixing and coherent motion parts. A joint Gaussian distribution is used to model the local mixing JPD component. The complement of the joint Gaussian JPD represents the covariance part that is attributed to coherent vertical motions. The coherent part can be further divided into updraft and downdraft parts based on the sign of vertical velocity. The JPD decomposition is performed at each model level. The method has a single parameter: the fraction p of the total JPD used to fit the joint Gaussian, which is expressed as a probability (i.e., the integral of JPD) centered at the JPD peak. In the present analysis p = 0.3 is used. The method is not sensitive to the choice of p value. A relatively large p value is presently used to generate smoother curves.

Figures 11 and 12 show updraft and downdraft area fractions and mean updraft vertical velocity using wθl and wqt JPDs, respectively. The profiles are not smooth because each level is treated independently and small differences in the joint Gaussian approximation part of the total JPD cause variations of the statistics. The large differences in Fig. 12 for z > 2 km with respect to time can be attributed to the change in the character of convection when penetrative updrafts rise in the stable inversion layer after organization develops. In the subcloud layer (z < 0.5 km), most of the variations with respect to time are observed in the downdraft area fraction αd, which is consistent with the characteristics of area increasing cold pools ad is larger at t = 22 and 30 h compared t = 16 h. Overall, in agreement with the mean profiles of Fig. 4, horizontal organization mostly modifies the convection characteristics in the upper part of the boundary layer, whereas mean convective updraft characteristics near the surface show smaller variations for αu and wu.

Fig. 11.
Fig. 11.

Convection characteristics based on the wθl joint distribution as a function of height for the large-domain run at three times. Panels show updraft area fraction αu, downdraft area fraction αd, and mean updraft vertical velocity wu.

Citation: Journal of the Atmospheric Sciences 79, 9; 10.1175/JAS-D-21-0334.1

Fig. 12.
Fig. 12.

Convection characteristics based on the wqt joint distribution as a function of height for the large-domain run at three times. Panels show updraft area fraction αu, downdraft area fraction αd, and mean updraft vertical velocity wu.

Citation: Journal of the Atmospheric Sciences 79, 9; 10.1175/JAS-D-21-0334.1

The results of Figs. 11 and 12 correspond to the entire boundary layer ensemble. The spatial variability of convection in the RICO case is discussed in Li et al. (2014), who found that updrafts and clouds forming in the downwind side of cold pools differ from convection outside cold pools. In general, the present results are consistent with the conclusions of Li et al. (2014). Some of the present observations, such as higher cloud tops in organized states (Fig. 3d), were also observed in Li et al. (2014): “cold pool-influenced updrafts tend to exceed the other updrafts.”

4. Conclusions

The development of horizontal organization in a precipitating shallow cumulus trade wind boundary layer is investigated using large-eddy simulations of the conditions observed during the Rain in Cumulus over the Ocean (RICO) field campaign (Rauber et al. 2007; van Zanten et al. 2011). The development of horizontal organization is studied by considering simulations in different computational domains and analysis of turbulence statistics in the largest domain run.

The development of precipitation results in the formation of cold pools that grow and fill the computational domain. The largest domain simulation has horizontal dimensions of 163.84 × 163.84 km2. Smaller domain sizes (81.92 × 81.92 km2 and 40.96 × 40.96 km2) artificially constraint the largest structures and are used to investigate the effects of organization on flow statistics. Even though the cloud fields vary significantly between the different computational domain sizes, some flow statistics show small sensitivity to horizontal organization. Mean profiles of u, υ, θl, qt, ql, and vertical velocity statistics do not vary significantly, both with respect to the domain size and when organization develops in the largest domain simulation. Organization results in deeper clouds, almost 1-km-higher tops than the inversion height. However, the impact of a few very high cumulus tops on the overall boundary layer is very small.

Turbulent kinetic energy (TKE) rapidly increases when convection organization develops. The increase of TKE is attributed to the horizontal component, whereas vertical velocity variance does not change significantly. Further, most of the effects of organization are observed at the top half of the boundary layer, near and above the inversion. The creation of large horizontal scales of u and υ and thermodynamic variables results in long horizontal distance two-point correlations. Particularly, for t > 24 h and within the cloud layer many two-point correlations have values > 0.2 at distances larger than 20 km.

Because flow variables do not always decorrelate at large distances, the premultiplied spectra are used to define energy-containing horizontal length scales. When large-scale organization forms, the premultiplied spectra develop multiple peaks, which is not commonly observed in classical shear-driven turbulent flows (Jiménez 1998; del Álamo et al. 2004). The time evolution of the energy-containing length scales is used to understand the organization characteristics and estimate the growth rate of horizontal organization. The length scale evolution and cold pool radii are shown in Fig. 10, which is the key result of the present study. All flow variables have a small length scale of 1–2 km for the entire simulation, which corresponds to convective motions, e.g., updrafts and cumulus clouds. Organization development creates an additional larger length scale for θl and qt. Up to two additional length scales are observed in the horizontal wind components. The growth rate of the large length scale is in the range 3–4 km h−1, which agrees well with the growth rate of the cold pool radii. A single energy containing length scale is observed for vertical velocity for the entire run (i.e., even after organized convection develops) that is fairly constant with height.

The overall view emerging from the synthesis of results is that horizontal organization does not fundamentally modify the local convective elements. Horizontal organization primarily rearranges randomly scattered convective elements (i.e., scattered cumulus to cloud arcs in the present case), thus making the convection somewhat more efficient (cloud tops penetrating higher in the inversion layer) and generating stronger horizontal cloud circulations near the inversion (larger horizontal TKE), with associated large stratiform cloud anvils.

The present results may help explain the relative success of many convection parameterizations that do not account for large horizontal features, e.g., cold pools. At the same time, the variation of some quantities with respect to domain size and horizontal organization challenges the comparison between LES and convection parameterizations. For instance, which of the present results a successful parameterization must capture? The present large domains blend the boundary between local convection and mesoscale horizontal motions. This is illustrated in the dependence of horizontal TKE on the LES domain size.

Even though the present LESs agree with observations and previous model results (Rauber et al. 2007; Nuijens et al. 2009; Snodgrass et al. 2009; van Zanten et al. 2011) the simulations have two main limitations. First, the approximation of a uniform large-scale forcing over a relatively large area (>100 km). This approximation allows for well-characterized conditions, but in the real atmosphere the geostrophic wind, subsidence, and SST vary with location (e.g., George et al. 2021). For instance, a nonuniform geostrophic wind may result in modulation of growth rate of cold pools and anisotropic horizontal energy transfer to the large scales. Second, the very simple radiative cooling parameterization used in the present case is appropriate for scattered short-lived cumulus, but it does not account for the radiative effects of stratiform anvils and any potential radiative feedback effects.

Acknowledgments.

This work was funded by the National Science Foundation via Grant AGS-1916619. The research presented in this paper was supported by the systems, services, and capabilities provided by the University of Connecticut High Performance Computing (HPC) facility. Figures were created with Matplotlib (Hunter 2007).

Data availability statement.

The large-eddy simulation model computer code and model output data are available at https://cfd.engr.uconn.edu.

APPENDIX

Nonprecipitating Simulation

A control simulation was carried out to confirm that the rapid growth rate of the horizontal length scales is because of the development of precipitation and the associated cold pool structures. The simulation has the same grid spacing and domain size as run B, the medium-domain-size run, but formation of precipitation is suppressed in the run, similar to Matheou et al. (2011). That is, all water condensate is assumed suspended.

Figure A1 shows LWP from the nonprecipitating simulation at t = 16, 22, and 30 h. For the duration of the present simulations when precipitation is suppressed, the cloud field is composed of relatively small scattered cumulus without any significant large-scale organization (cf. Fig. 2). Figure A2 shows the evolution of the energetic length scales for the nonprecipitating run. Figure A2 is a counterpart of Fig. 10 and information from the latter, such as the cold pool radii and constant growth rate slopes, is retained. All variables exhibit a single premultiplied spectra peak which corresponds to a single length scale (some of the undulations of lu cause the algorithm to register two peaks at t = 30 h). The length scale is nearly constant for all variables and ranges in 1–3 km, similar to the smaller scale in the precipitating runs. The results of Fig. A2 suggest that the additional premultiplied spectra peaks are the result of precipitation-induced large-scale organization.

Fig. A1.
Fig. A1.

Liquid water path for the nonprecipitating run at t = 16, 22, and 30 h.

Citation: Journal of the Atmospheric Sciences 79, 9; 10.1175/JAS-D-21-0334.1

Fig. A2.
Fig. A2.

Time evolution of horizontal length scales in the simulation without precipitation. Panels show the evolution of the energy-containing horizontal length scales of (a) wind components lu and lυ, (b) vertical velocity lw, (c) liquid water potential temperature lθ, and (d) total water mixing ratio lq. Also shown in (d) are the radii of two cold pools rcp1 and rcp2. Unlike the precipitating cases, variables exhibit a single length scale which is nearly constant for the duration of the simulation.

Citation: Journal of the Atmospheric Sciences 79, 9; 10.1175/JAS-D-21-0334.1

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