In tropical shallow cumulus clouds latent heat released during water phase transition is an important source of energy driving formation and evolution of cumulus convection. The formulation of phase transition processes (condensation and evaporation) in computer models depends on grid resolution (LES/CRM/NWP) and the method chosen to describe the microphysics (explicit/bin or parameterized/bulk). Obviously, condensation is influenced by microphysics, including aerosols (Kogan and Martin, 1994). The latter effect is often referred to as “convection invigoration”. It is, therefore, important to formulate parameterizations for phase transition processes, especially applicable to cloud resolving (CRM) and NWP models.

In our previous study (Kogan, 2021) we found that a strong correlation exists between integral cloud condensation rate (CR) and integral upward mass flux (MFP). At first sight this result should not be surprising, as mass flux has long been recognized as a major factor affecting cumulus convection (see e.g., Arakawa and Schubert, 1974; Tiedtke, 1989; Suselj et al., 2019). It was also well known that vertical velocity has a strong effect on supersaturation, and, therefore, cloud microphysics (see, e.g., Squires, 1952; Politovich and Cooper, 1988). Nevertheless, the exceptionally high correlation (correlation coefficient R=0.99), and the simple linear relationship between these variables (CR=αlesMFP), was quite remarkable. While the analyzed dataset consisted of more than 2000 clouds in a wide range of sizes and at different stages of their evolution, the slope of the linear fit, αles, remained nearly constant. This result was also obtained in a recent study by Grant et al. (2022) who found a similar linear relationship between condensation and vertical velocity in deep convective clouds.

This study provides theoretical formulation of the CR–MFP relationship based on the cloud drop condensational growth equation and the concept of quasi-steady supersaturation.

Our LES model (SAMBM) employs the dynamical core of the System for Atmospheric Modeling (SAM; Khairoutdinov and Randall, 2003) and the Bulk Microphysics (BM; Kogan, 2013) fine-tuned for shallow Cu convection. The observations from the RICO field campaign (vanZanten et al., 2011) were used for initialization of the LES simulations conducted in a rather large 50.0×50.0×4 km3 domain (500×500×100 grid points).

Over the course of the simulation from 8 to 32 h, we selected 2031 clouds by applying a threshold of liquid water path > 20 g m-2. In order to facilitate the analysis of the complex system of clouds at various stages of their development, the dataset was sorted out by cloud top height and divided into four groups G1–G4, each of which condenses approximately equal amount of water vapor per second. The groups G1–G2 represent clouds mostly in the growing stage, while groups G3–G4 contain mature and decaying clouds.

The analysis of the LES dataset described in Kogan (2021) revealed a remarkably strong correlation between the condensation rate and the upward mass flux. The scatter plot in Fig. 1 shows the CR–MFP dependence for the whole dataset, e.g., for clouds of all sizes and all stages of their evolution.

The data clearly shows a perfect linear relationship (R=0.99) between the condensation rate (CR≡dqldt) and upward mass flux (ρaW) which can be expressed as: 1dqldt=αlesρaW where αles=2.06×10-6 [m-1]. Here liquid water content ql and air density ρa are in kg m-3, vertical velocity W in m s-1. The subscript “les” signifies that coefficient αles is obtained from LES data, as opposite to the similar coefficient αqs derived from theory (see below).

The coefficients αles for individual cloud groups slightly differ from the coefficient for the whole dataset (Fig. 2). They are 4 %–5 % higher for small clouds (2.17 for G1 and 2.13 for G2), while lower for larger clouds (2.052 for G3 and 2.049 for G4 clouds). Obviously, using the specific latent heat constant, one can relate formulation (Eq. 1) to the latent heat released during phase transitions.

The SAMBM model used in our study is a so-called two and half moment model, i.e., it employs three prognostic moments for cloud water variables (cloud water content, qc, cloud drop concentration, nc, and cloud integral radius, Ri), as well as two moments for rain water variables (rain water content, qr, and rain drop concentration, nr) (Khairoutdinov and Kogan, 1999). The use of an additional cloud moment, the integral radius Ri=∫rn(r)dr (n(r) is the drop size distribution function), is essential, as it allows explicit calculation of condensation rates. Specifically, using the condensational growth equation for cloud drop with radius r 2drdt=G(T,P)Sr and having Ri as a prognostic variable, we can explicitly calculate the rate of change of cloud liquid water content: 3dqldt=4πρwG(TP)SRi. Here S is supersaturation, ρw and ρa are water and air density, respectively. The coefficient G(TP) is a weak function of temperature T and pressure P (e.g., Politovich and Cooper, 1988).

In order to solve Eq. (3), we first calculate changes in temperature T and water vapor content qv due to advection and turbulent mixing. The corresponding intermediate values of T* and qv* are used to calculate supersaturation S, and, based on it, condensation/evaporation rates according to Eq. (3). This splitting of dynamical and microphysical terms is analogous to a Lagrangian air parcel model where the two terms in the supersaturation equation (Eq. 4) account for dynamical ascent/descent, as well as latent heat release (e.g., Squires, 1952; Paluch and Knight, 1984). 4dSdt=A1W-A2SRi In Eq. (4) coefficients A1 and A2 are weak functions of temperature T and pressure P (see, e.g., Politovich and Cooper, 1988; Pinsky et al., 2013). As was shown in many studies of supersaturation in clouds (e.g., Paluch and Knight, 1984; Cooper, 1989; Korolev and Mazin, 2003; Pinsky et al., 2013; Siebert and Shaw, 2017), for times larger than phase relaxation time, the Eq. (4) has an asymptotic quasi-steady solution for S: 5Sqs=A1WA2Ri. The quasi-steady supersaturation is an important characteristic of the condensation process, as it corresponds to an equilibrium between contributions to the supersaturation equation from dynamical and microphysical processes. It is appropriate, therefore, to measure the actual cloud supersaturation in units of Sqs: 6S=κSqs. Under adiabatic conditions for times larger than phase relaxation time (e.g. Paluch and Knight, 1984; Khain et al., 2000; Pinsky et al., 2022) the supersaturation reaches its `quasi-steady” value. In this case the kappa factor κ=1. In general, due to mixing and entrainment, the value of κ may deviate from 1. Using LES model, the spatial distribution of κ can be evaluated according to Eq. (9) below, and, thus, κ may serve as a measure of entrainment/mixing in various cloud regions.

Substituting Eq. (6) into Eq. (3) and using Eq. (5), we can rewrite Eq. (3) as: 7dqldt=4πρwGA1A2κW=κA1A3ρaW=καqsρaW where coefficient A3 is expressed through A2 and G. In Eq. (7) we introduced coefficient 8αqs=A1A3 which is a weak function of temperature T and pressure P (Pinsky et al., 2013). Similar to Eq. (3) formulation (Eq. 7) demonstrates that CR is a linear function of upward mass flux. By comparing the LES derived formulation (Eq. 3) with the theoretical formulation (Eq. 7), one can obtain κ as: 9κ=αles/αqs. In order to evaluate αqs which is a function of temperature T and pressure P, we first calculate at each vertical level the values of T and P as horizontal averages over the cloudy areas, and then obtain αqs using Eq. (8), and expressions for A1 and A3. The values of αles are calculated as slopes of linear fits either to the full LES dataset (αqs=2.06), or to each individual group subset (2.17, 2.13, 2.052, and 2.049 for G1–G4 groups, respectively, see Fig. 2).

The vertical profile of κ is shown in Fig. 3 for two times in cloud system evolution. The left panel refers to an early cloud system formation (t=8 h) when only small clouds in G1/G2 groups were formed. The right panel refers to t=24 h when the cloud system has already been fully developed with clouds present in all groups.

As Fig. 3 shows, in the early formed clouds at t=8 h the supersaturation is, on average, smaller by 3 %–5 % than its quasi-steady value (Fig. 3a). This is the result of small values of integral radius Ri at the early stage of cloud formation, which combined with growing updrafts leads to large quasi-steady supersaturations (Eq. 5). At the mature stage Ri is larger, resulting in the decrease of Sqs. The decrease in Sqs is especially pronounced below cloud tops where updrafts approaching the inversion layer are weak.

While all these factors may qualitatively explain the increase of κ with height, the more accurate profile of κ may be obtained by accounting for vertical variation of αles. The profiles shown in Fig. 3 consider only constant with height integral values of αles; i.e., averaged over the whole cloud. Our preliminary results from the study of vertically dependent CR and MFP variables, showed that the coefficient αles slightly increases with height. As the result, the qualitative behavior of κ with height will remain the same.

Data from LES simulations of shallow cumulus clouds demonstrated a nearly perfect correlation between Condensation Rate (CR) and upward/Plus Mass Flux (MFP).