Editorial Type:
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Article Type:
Research Article

Direct Entrainment as a Measure of Dilution

Gunho (Loren) Oh aDepartment of Earth, Ocean and Atmospheric Sciences, University of British Columbia, Vancouver, British Columbia, Canada

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https://orcid.org/0009-0002-2916-0350
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Philip H. Austin aDepartment of Earth, Ocean and Atmospheric Sciences, University of British Columbia, Vancouver, British Columbia, Canada

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Online Publication:
14 Oct 2024
Print Publication:
01 Oct 2024
DOI:
https://doi.org/10.1175/JAS-D-24-0018.1
Page(s):
1783–1797
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Abstract

The turbulent transport of mass, energy, moisture, and momentum between the clouds and the surrounding environment plays a central role in determining the vertical structure of the troposphere. This article investigates the connection between net entrainment, defined here as the net transport of air into individual clouds, and net dilution, defined as the tendencies of passive tracers such as static energy or total water mixing ratio. Entrainment and detrainment rates for 2.6 × 106 individual cloud samples are obtained from a large-eddy simulation of shallow convective boundary layer atmosphere that explicitly calculates the turbulent fluxes across the cloud boundaries. The equations describing the tendencies of cloud mass and tracer concentrations are derived as a function of directly calculated entrainment and detrainment rates of the individual clouds, and used to calculate net entrainment and dilution rates. Directly calculated net entrainment and dilution rates agree well with cloud mass and tracer tendencies and give a dilution time scale of 13 min. In contrast, the traditional bulk-plume approximation overestimates the effect of entrainment and detrainment on the dilution of cloud field properties due to the differential tracer transport through the moist shell surrounding the cloud. Using direct measures of entrainment and detrainment for individual clouds separates different processes that influence the turbulent mass transport between the clouds and their environment.

© 2024 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Loren Oh, loh@eoas.ubc.ca

Abstract

The turbulent transport of mass, energy, moisture, and momentum between the clouds and the surrounding environment plays a central role in determining the vertical structure of the troposphere. This article investigates the connection between net entrainment, defined here as the net transport of air into individual clouds, and net dilution, defined as the tendencies of passive tracers such as static energy or total water mixing ratio. Entrainment and detrainment rates for 2.6 × 106 individual cloud samples are obtained from a large-eddy simulation of shallow convective boundary layer atmosphere that explicitly calculates the turbulent fluxes across the cloud boundaries. The equations describing the tendencies of cloud mass and tracer concentrations are derived as a function of directly calculated entrainment and detrainment rates of the individual clouds, and used to calculate net entrainment and dilution rates. Directly calculated net entrainment and dilution rates agree well with cloud mass and tracer tendencies and give a dilution time scale of 13 min. In contrast, the traditional bulk-plume approximation overestimates the effect of entrainment and detrainment on the dilution of cloud field properties due to the differential tracer transport through the moist shell surrounding the cloud. Using direct measures of entrainment and detrainment for individual clouds separates different processes that influence the turbulent mass transport between the clouds and their environment.

© 2024 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Loren Oh, loh@eoas.ubc.ca
Keywords: Clouds; Convective clouds; Convective-scale processes; Mass fluxes/transport; Large eddy simulations

1. Introduction

Cumulus clouds play a fundamental role in governing the global weather and climate through vertical transport of energy and moisture. Warm, moist air ascends from the relatively warm surface of the Earth, eroding atmospheric stratification and transferring heat and moisture to the upper atmosphere. This becomes the fuel that drives the hydrological cycle of the atmosphere (Arakawa 2004). The clouds also reflect, absorb, and emit radiation and act as one of the main feedback mechanisms of the global climate (Ceppi et al. 2017). Over much of the tropical and subtropical oceans, it takes the form of shallow cumulus clouds, which are small, short-lived clouds that occur within the atmospheric boundary layer.

Despite the relatively small scale in both space and time, cumulus convection is crucially important for global climate. The ubiquity of shallow clouds over the global ocean makes them the main source of energy and momentum in the atmospheric boundary layer, and the preconditioning of the atmosphere feeds the development of deep convection. The current generation of global climate and weather prediction models have recently employed grid spacings O(10) km (Roberts et al. 2020). However, an explicit modeling of moist convection requires a minimum grid spacing of O(100)m for deep convection (Bryan et al. 2003) and O(10)m for shallow convection (Matheou et al. 2011; Sato et al. 2017, 2018) to achieve statistical convergence. A cumulus cloud, which can stretch over a few kilometers, is driven by turbulent eddies that are at least two orders of magnitude smaller (Bryan et al. 2003), and the characteristic structures of moist convection cannot be explicitly represented at coarser resolutions.

A variety of cumulus parameterization schemes have been developed to represent the effects of moist convection in large-scale models, with mass-flux schemes with bulk-plume approximations (Tiedtke 1989; Nordeng 1994; Gregory and Rowntree 1990; Kain and Fritsch 1990) among the most widely used for shallow convection. These parameterizations begin with the tracer continuity equation as follows:
Dt(ρϕ)=∂∂t(ρϕ)+∇h⋅(ρϕu)+∂∂z(ρϕw)=F,
where ϕ is the tracer mixing ratio, ρ is the air density (kg m−3), u is the horizontal velocity vector of the air (m s−1), w is the vertical velocity (m s−1), and F represents all sources and sinks for ϕ.

Equation (1) describes the rate of changes in some cloud tracer concentrations Dt(ρϕ) for each large-eddy simulation (LES) model grid cell. To estimate this rate for the entire cloud field, bulk-plume approximations make a simplifying assumption that all clouds in the model domain can be represented by a single contiguous bulk plume, whose size is defined by the cloud fraction σ (Betts 1973). Because of this assumption, this framework is called the bulk-plume approach.

Integrating Eq. (1) over this bulk plume using the Leibnitz theorem for the first and the third terms and the divergence theorem for the horizontal divergence term yields (Siebesma and Cuijpers 1995)
Dt⟨ρϕ⟩=∂∂t(ρσ⟨ϕ⟩)+1A∮i n⋅(u−ui)ϕ dl+∂∂z(ρσ⟨ϕw⟩)=⟨F⟩,
where A is the area of the bulk plume, u is the air velocity (m s−1), ui is the velocity of the cloud interface (m s−1), n is a normal vector pointing directly outwards from the cloud interface, and ⟨⟩ is the average over the cloud field, which defines the properties of the bulk plume.
The second term in Eq. (2) represents the tracer flux across the plume boundary. Then, we can define the flux into the plume as entrainment and out of the cloud region as detrainment, which yields
e=−1A∮n⋅(u−ui)<0ρn⋅(u−ui) dl,
d=+1A∮n⋅(u−ui)>0ρn⋅(u−ui) dl,
where e and d denote entrainment and detrainment rates (kg m−3 s−1), respectively.
We can further introduce flux-weighted tracer concentrations for the entrained air ϕe = ⟨eϕ⟩/⟨e⟩ and ϕd = ⟨dϕ⟩/⟨d⟩ for the detrained air. Then, the second term in Eq. (2) represents the net flux of tracers across the cloud boundary in terms of entrainment and detrainment rates. That is,
Dt⟨ρϕ⟩=∂∂t(ρσ⟨ϕ⟩)+eϕe−dϕd+∂∂z(ρσ⟨ϕw⟩)=⟨F⟩.
To calculate the dilution rate for the cloud field according to Eq. (5), estimates of entrainment and detrainment rates e and d, as well as the properties of the entrained and detrained air ϕe and ϕd, respectively, need to be obtained. Because the mixing between the cloud field and the environment is difficult to quantify, the mean cloud field assumption is made to simplify the turbulent mass exchange processes. This approach was pioneered by Siebesma and Cuijpers (1995) who quantified bulk-plume entrainment and detrainment rates based on the budget equations of cloud field tracers and validated their accuracy using the results from a large-eddy simulation. Here, the cloud field, represented by a single bulk plume, mixes directly with the environment. The dry air from the environment replaces cloudy air through entrainment, and the cloudy air replaces dry environmental air through detrainment.
The mean cloud field assumption reduces the number of unknowns in Eq. (5) by making a simplifying assumption about the dilution of the bulk plume due to entrainment and detrainment. That is,
eϕe−dϕd≈ebϕ¯−db⟨ϕ⟩,
where eb and db represent the bulk-plume estimates of entrainment and detrainment rates (kg m−3 s−1), respectively, and the overbar denotes the average over the environmental region. The mean cloud field assumption then allows the bulk-plume approach to describe the net dilution of the cloud field in terms of entrainment and detrainment rates by assuming that the properties of the entrained and detrained air can be inferred from the bulk properties of the cloud field and the environment.
Based on these assumptions, Siebesma and Cuijpers (1995) estimated eb and db by representing turbulent fluxes of the bulk plume and the environment separately as follows:
⟨w′ϕ′⟩=⟨wϕ⟩−⟨w⟩⟨ϕ⟩,
w′ϕ′¯=wϕ¯−w¯ ϕ¯,
which can be substituted into the vertical divergence term in Eq. (1) separately for the bulk plume and the environment. Then, applying these approximations for the bulk plume and the environment yields
ρ∂σ⟨ϕ⟩∂t=−∂M⟨ϕ⟩∂z+ebϕ¯−db⟨ϕ⟩−∂ρσ⟨w′ϕ′⟩∂z︸⟨Fturb⟩+ρσ(∂ϕ¯∂z)forcing︸ρσ⟨F⟩,
ρ∂(1−σ)ϕ¯∂t=+∂Mϕ¯∂z−ebϕ¯+db⟨ϕ⟩−∂ρ(1−σ)w′ϕ′¯∂z︸Fturb¯+ρ(1−σ)(∂ϕ¯∂z)forcing︸ρ(1−σ)F¯,
where M = σρ〈w⟩ is the vertical mass flux, averaged over the bulk plume.
Using mass continuity and isolating the bulk-plume entrainment and detrainment terms, we obtain the expressions for the bulk-plume entrainment and detrainment rates eb and db, respectively,
eb(ϕ¯−⟨ϕ⟩)=+ρσ∂⟨ϕ⟩∂t+M∂⟨ϕ⟩∂z+⟨Fturb⟩−ρσ⟨F⟩,
db(ϕ¯−⟨ϕ⟩)=−ρ(1−σ)∂ϕ¯∂t+M∂ϕ¯∂z−Fturb¯+ρ(1−σ)F¯,
which includes two corrections to the derivation by Siebesma and Cuijpers (1995): the signs of turbulent fluxes have been corrected and db in Eq. (12) is now correctly expressed as turbulent fluxes of environmental properties (Siebesma and Holtslag 1996).

Numerical methods to directly calculate Eqs. (3) and (4) have been introduced by Romps (2010) and Dawe and Austin (2011a). Romps (2010) solved this problem by integrating cloud fluxes over time for each grid point, and Dawe and Austin (2011a) performed tetrahedral interpolation of each model grid cell in order to calculate the changes in the subgrid-scale cloud surface.

However, directly calculated entrainment and detrainment rates e and d have been found to be at least twice as large as the bulk-plume estimates eb and db, respectively [Eqs. (11) and (12)]. Observational studies and numerical simulations have attributed the presence of moist shell surrounding the cloud region as the source of this discrepancy (Jonas 1990; Heus and Jonker 2008; Romps 2010; Dawe and Austin 2011b). Because the bulk-plume approximations ignore the role of the moist shell, eb and db do not reflect the actual transport of air mass across the cloud boundaries.

An alternative approach to resolve this discrepancy has been introduced by Dawe and Austin (2011b). Instead of estimating entrainment and detrainment rates based on the mean cloud field assumption [Eqs. (11) and (12)], the effect of the moist shell is ignored. That is, shell-corrected entrainment and detrainment rates eϕ and dϕ, respectively, are assumed to satisfy the mean cloud field assumption as follows:
eϕϕ¯−dϕ⟨ϕ⟩=eϕe−dϕd,
which leads to the definition of shell-corrected entrainment and detrainment rates:
eϕ=e−(eϕe−ϕ¯⟨ϕ⟩−ϕ¯+d⟨ϕ⟩−ϕd⟨ϕ⟩−ϕ¯),
dϕ=d−(eϕe−ϕ¯⟨ϕ⟩−ϕ¯+d⟨ϕ⟩−ϕd⟨ϕ⟩−ϕ¯),
but eϕ and dϕ, however, are still dependent on the mean cloud field assumption and are not meant to reflect the physical tendencies of cloud properties; the main purpose of the shell correction method is to simulate the mean cloud field assumption while preserving correct rates for net entrainment e − d and net dilution eϕe − dϕd. As a result, eϕ and dϕ cannot be used as a measure of the actual turbulent mass transport between individual clouds and the surrounding environment.

Another approach to diagnose the effect of the moist shell during turbulent mass exchange processes has been introduced by Hannah (2017), who introduced the concept of proportional dilution, or proportional changes in mean cloud tracer concentrations due to entrainment and detrainment as well as the relative dilution effects of entrained and detrained air, which accounts for the effect of the moist shell. The rate of proportional dilution for a modeled cloud field has been tested using a series of warm bubble simulations (Hannah 2017).

Despite these efforts, an accurate mathematical framework that can be used to describe the dynamics and thermodynamics of the clouds is still missing. The bulk-plume approach, while useful for convective parameterization schemes in large-scale models of the atmosphere, relies on simplifying assumptions about the cloud field. Given that, the primary purpose of this paper is to evaluate the assumptions used in bulk-plume estimates of entrainment and detrainment rates by directly calculating the changes in cloud mass Dt(ρA) and tracer concentrations Dt(ρϕA) from the LES model output. This allows us to examine if the assumptions made in bulk-plume estimates realistically represent the physical mechanisms that govern the mixing processes between individual clouds and the surrounding environment, which is a step toward an improved convective parameterization scheme that can realistically represent individual processes contributing to the dilution of clouds.

In this paper, we perform direct calculations of entrainment and detrainment rates based on a large-eddy simulation of a cloud field. Following Romps (2010), we derive a set of mathematical expressions for the changes in cloud properties as a function of entrainment and detrainment rates and further introduce the concept of cloud dilution (section 2). This is followed by the description of the numerical model and the sampling method (section 3). Then, changes in mean cloud field properties are examined from the LES model results. By evaluating traditional bulk-plume estimates and comparing them to the directly calculated rates, we test the validity of the assumptions used in bulk-plume approximations (section 4). Tendencies of individual cloud properties are then evaluated based on the proposed formulations (section 5). We discuss the physical constraints to define the actual changes in cloud properties in section 6 and conclude this paper in section 7.

2. Theoretical background

a. Governing equations

Prognostic equations for (quasi-)conserved variables during moist convection, such as the total water content during shallow convection, have been established for both bulk-plume estimates (Siebesma and Cuijpers 1995) and direct calculations of entrainment and detrainment rates (Romps 2010; Dawe and Austin 2011a), but these cloud budget equations have yet to be verified at the scale of individual clouds. Following Romps (2010), we present a formal definition of mass exchange processes describing the changes in individual cloud mass and cloud tracer concentrations. Direct calculation of entrainment and detrainment rates allows us to determine local distributions of changes in three-dimensional cloud boundaries, and by integrating the turbulent fluxes across the cloud boundaries, we can tract the changes in the properties of individual clouds.

Here, we are interested in the behavior of arbitrary regions of cloudy air C, defined as a set of contiguous points that contain cloudy air. Following Siebesma and Cuijpers (1995), these points are defined as positively buoyant (B > 0), upward-moving (w > 0) points containing condensed liquid water (ql > 0). These are typically referred to as cloud cores, but we will refer to cloud cores simply as clouds.

Individual cloud properties can be defined as a generic cloud tracer ϕ. To study the changes in individual cloud properties, values of ϕ are summed over each contiguous cloud region. This summation over the ith cloud region Ci containing a set of N points Ci={(x1,y1),(x2,y2),…,(xN,yN)} can be defined as
⟨ϕ⟩i≜∑(xn,yn)∈CiNϕ(xn,yn).
A mathematical formulation of turbulent mass exchange processes between an arbitrary region and its surroundings requires a definition of that region, which in this case is a contiguous area containing cloudy air. Following Romps (2010), we define the activity field A as follows:
A(x,y)={1,if (x,y)∈C0otherwise,
for an arbitrary cloud region C.
Then, changes in cloud properties in terms of a generic tracer ϕ can be described by its total derivative. For an arbitrary scalar field S(x, t), its total derivative is defined as
DtS=DDtS=∂∂tS+∇⋅(Su),
which can be used to describe the changes in cloud mass Dt(ρA) and tracer concentration Dt(ρϕA) pertaining to the ith cloud region Ci, defined as Ci={(x1,y1),(x2,y2),…,(xN,yN)|A=1} for N contiguous grid cells within that cloud region.
Following Romps (2010), the tendencies of cloud mass Dt(ρA) and tracer concentration Dt(ρϕA) for the ith cloud region Ci can be expressed numerically as a function of entrainment and detrainment rates as follows:
DDt⟨ρA⟩i=∂∂t⟨ρA⟩i+∇⋅⟨ρuA⟩i=⟨e⟩i−⟨d⟩i,
DDt⟨ρϕA⟩i=∂∂t⟨ρϕA⟩i+∇⋅⟨ρϕuA⟩i=⟨eϕ⟩i−⟨dϕ⟩i+⟨Sϕ⟩i,
where Sϕ is the sum of all nonadvective and nondiffusive sources and sinks of the cloud tracer ϕ. The definition of Sϕ is dependent on the numerical implementation of the tetrahedral interpolation scheme, where advective and diffusive fluxes need to be calculated to directly calculate entrainment and detrainment rates. Given that, in the case of weakly precipitating moist convection during BOMEX, the only significant source of the total water content qt is precipitation.

Ultimately, we are interested in the tendencies of the average cloud tracer concentration ϕc=⟨ρϕA⟩i/⟨ρA⟩i; it is useful to diagnose the changes in ϕc for individual clouds, irrespective of changes in cloud size. This allows for a calculation of the changes in the average cloud field properties and account for the effect of the changes in cloud size. The tendencies of ϕc have been examined earlier by Hannah (2017), but the derivation is dependent on the assumption that the mass-flux-weighted properties of a cloud are the same as mass-weighted properties, and as a result, the proposed rate of dilution cannot fully describe the tendencies of ϕc.

Here, we propose a derivation for the rate of individual cloud dilution in terms of ϕc without employing such approximations. That is,
Dtϕc=DDtϕc=DDt⟨ρϕA⟩i⟨ρA⟩i,
which can be expanded using the quotient rule as follows:
DDtϕc=DDt⟨ρϕA⟩i⟨ρA⟩i,
=1⟨ρA⟩i2{⟨ρA⟩i(DDt⟨ρϕA⟩i)−⟨ρϕA⟩i(DDt⟨ρA⟩i)},
=1⟨ρA⟩i{(DDt⟨ρϕA⟩i)−⟨ρϕA⟩i⟨ρA⟩i(DDt⟨ρA⟩i)}.
Here, we can replace the two material derivative terms with Eqs. (19) and (20) as follows:
DDt⟨ρϕA⟩i⟨ρA⟩i=1⟨ρA⟩i{(DDt⟨ρϕA⟩i)−ϕc(DDt⟨ρA⟩i)},
=1⟨ρA⟩i{(⟨eϕ⟩i−⟨dϕ⟩i+⟨ρSϕ⟩i)−ϕc(⟨e⟩i−⟨d⟩i)},
=1⟨ρA⟩i{−⟨e⟩i(ϕc−ϕe)+⟨d⟩i(ϕc−ϕd)+⟨Sϕ⟩i},
where
ϕc=⟨ρϕA⟩i⟨ρA⟩i,
ϕe=⟨eϕ⟩i⟨e⟩i, and
ϕd=⟨dϕ⟩i⟨d⟩i
represent the average, entrained, and detrained tracers for the individual clouds, respectively.
However, as proposed by Hannah (2017), clouds will dilute toward an equilibrium with the surrounding environment; hence, changes in ϕc need to be adjusted by the anomaly ϕ′=ϕ−ϕ¯. Normalizing Eqs. (24) and (27) by the mean anomaly ⟨ρϕ′A⟩i/⟨ρA⟩i yields
⟨ρA⟩i⟨ρϕ′A⟩iDDtϕc=1⟨ρϕ′A⟩i{(DDt⟨ρϕA⟩i)−ϕc(DDt⟨ρA⟩i)},
=1⟨ρϕ′A⟩i{−⟨e⟩i(ϕc−ϕe)+⟨d⟩i(ϕc−ϕd)+⟨ρSϕ⟩i},
which has units of per second (s−1) and is the inverse of the dilution time scale, which is a measure of how rapidly the cloud is being diluted toward an equilibrium with its surroundings.

Equations (31) and (32) describe the proportional changes in ϕc, including the contributions from the terms that have previously been ignored, such as nonadvective and nondiffusive sources and sinks ⟨Sϕ⟩i and the horizontal divergence term ∇xy⋅⟨ρϕA⟩i. The inclusion of Sϕ indicates that Eqs. (31) and (32) remain true for any cloud tracer ϕ where all sources and sinks for the tracer can be identified. This allows us to quantify the relative contributions of all dynamic and thermodynamic processes to cloud mass and tracer concentrations, for example, in the presence of strong wind shear (which will affect the horizontal divergence term ∇xy⋅⟨ρϕA⟩i) or vigorous precipitation (which will increase the precipitative falloff in ⟨Sϕ⟩i).

Equations (31) and (32) represent two different numerical techniques to calculate the proportional changes of ϕc within individual clouds. In the following sections, the results from the LES model run will be used to validate the equations for turbulent mass exchange processes introduced here.

b. Entrainment or dilution?

In the previous section, we have defined the rate of dilution as the rate of change in the cloud tracer concentration ϕ due to (net) entrainment. In the literature, the rate of entrainment is often used to refer to that of dilution and is especially the case for most convective parameterization schemes currently in use, which are based on a simple entraining plume model (Betts 1975). Entrainment and detrainment rates, however, can be calculated explicitly as the rate at which the cloud volume changes (Siebesma and Holtslag 1996).

Equation (19) defines the cloud mass entrainment and detrainment rates by the conservation of cloud mass ⟨ρA⟩i for the ith cloud region. On the other hand, we refer to dilution as the changes in tracer concentrations ⟨ρϕA⟩i or, more plainly, the physical properties of the cloud. It is perspicuous to think of the rate dilution in terms of, for example, the total water specific humidity as the cloud is being diluted due to mixing with the dry environment, or vertical velocity as an ascending air parcel gets slowed down due to mixing with downdrafts. However, it is not intuitive, and perhaps even misleading, to refer to entrainment as dilution; it is not cloud mass, but its properties, that is being diluted as a result of turbulent mixing processes.

For this reason, bulk-plume approximations measure the effect of entrainment and detrainment on cloud dilution but not necessarily the entrainment and detrainment rates. Likewise, studies employing numerical techniques such as the finite-difference method to estimate the changes in cloud tracer concentrations are in fact estimating the rate of dilution (Wang 2020; Savre 2022) but not the rates of entrainment and detrainment.

These definitions are further motivated by an observation that the rate of dilution cannot easily be decomposed. That is,
⟨eϕ⟩i≠⟨e⟩i⟨ϕ⟩i,
⟨dϕ⟩i≠⟨d⟩i⟨ϕ⟩i,
due to the locality of entrainment. At the scale of individual clouds, the rate of entrainment must be considered separately from that of dilution. This is true for any linear operator ⟨⟩i over individual cloud regions. The bulk-plume approximation considers entrainment and detrainment rates to be turbulent fluxes of the entire cloud field, and the mean cloud field assumption approximates the properties of the entrained and detrained air from those of the dry environment and the mean cloud field, respectively. This may be useful in inferring the turbulent fluxes of the cloud field as a whole, but they do not reflect the physical processes governing the turbulent mass transfer between the clouds and their surroundings.

Given that, we will refer to mass entrainment ⟨e⟩i and detrainment ⟨d⟩i to describe the changes in cloud mass Dt⟨ρA⟩i [Eq. (19)] and dilution ⟨eϕ⟩i and concentration ⟨dϕ⟩i to describe the changes in tracer concentration Dt⟨ρϕA⟩i [Eq. (20)]. Likewise, changes in the average tracer concentration can be referred to as the average dilution rate [Eq. (24)] and changes in the proportional tracer concentration with respect to the tracer anomaly as the proportional dilution rate [Eq. (31)].

3. Methodology

a. LES model setup

A high-resolution LES of boundary layer shallow cumulus clouds using an LES model called the system for atmospheric modeling (SAM; Khairoutdinov and Randall 2003) has been used for this study. The model run has been initialized with observations from Barbados Oceanographic and Meteorological Experiment (BOMEX; Holland and Rasmusson 1973; Nitta and Esbensen 1974).

The LES model is based on SAM, version 6.11.8, where we have added the entrainment module, which implements the tetrahedral interpolation scheme introduced by Dawe and Austin (2011a). In addition to bugfixes since version 6.8.2 used by Dawe and Austin (2011a), it includes the effects of precipitation and phase changes governed by the microphysics scheme. This allows us to extend the definition of the cloud tracer ϕ to those that are not necessarily conserved during moist convection, as long as all sources and sinks for ϕ can be quantified. Here, the total water content qt = qυ + ql is used as the cloud tracer ϕ. Direct calculations using the temporal integration method by Romps (2010) has also been tested, and the results are equivalent to those from the tetrahedral interpolation scheme.

The model run has been performed over a 38.4 km × 12.8 km × 42 km domain. The model grid size is set to 25 m horizontally, and 256 vertical levels are defined such that the vertical grid size is 25 m up to 3 km, well above the inversion layer at 1.6 km, increasing exponentially until it reaches the domain top. The calculations have been performed for 24 simulation hours, although the first 12 h of the simulation are dedicated to initialization and spinup and not used for the analysis. The simulation has been performed with a 1-s temporal resolution, and three-dimensional output fields have been generated at 60-s intervals. Shortwave and longwave radiative fluxes and heating effects are governed by the Rapid Radiative Transfer Model (RRTM) (Clough et al. 2005; Iacono et al. 2008; Blossey et al. 2013). The radiation module is coupled with a two-moment microphysics scheme (Morrison et al. 2005a,b; Morrison and Gettelman 2008). The cloud droplet number concentration (CDNC) was set to 120 cm−3 based on the value introduced by Rasmussen et al. (2002) for maritime convection.

b. Cloud sampling

Dynamic and thermodynamic fields from the large-eddy simulation were sampled conditionally according to a sampling criterion by Siebesma and Cuijpers (1995). As previously mentioned, a cloud is defined as buoyant (B > 0), upward-moving (w > 0), regions of air containing condensed liquid water (ql > 0). Horizontally contiguous regions of the conditionally sampled clouds are then isolated at every model altitude, each forming a cloud sample. The cloud samples are identified using a numerical algorithm that checks the (horizontal) connectivity of each cloudy cell to all its neighboring cells on a two-dimensional plane.

Each cloud sample is obtained by adding all relevant dynamic and thermodynamic values of the corresponding region of cloudy air, with the exception of entrainment. Because the tetrahedral interpolation scheme calculates subgrid-scale entrainment rates, subgrid-scale cloud regions (tetrahedra) are often found just outside the resolved cloudy cells. The sampling algorithm resolves this issue by performing an additional nearest-neighbor lookup for local entrainment rates once individual cloud regions have been defined, which gives a ≈20% increase in the total entrainment rate for each cloud sample. This additional step has been suggested by Dawe and Austin (2013). Individual cloud samples are also tracked in time based on a cloud tracking algorithm based on Dawe and Austin (2012), which is used to calculate the changes in cloud properties over time.

All temporal and spatial derivatives have been calculated by the second-order central finite-difference method, and tendencies in cloud tracer concentrations due to entrainment and detrainment are obtained independently by the direct calculation method based on the tetrahedral interpolation scheme introduced by Dawe and Austin (2011a).

A total of roughly 2.6 × 106 unique cloud samples have been generated from the BOMEX LES model run, after filtering out samples containing less than two grid cells. This corresponds to the summation operation ⟨⟩i for the ith cloud region C for i = 1, 2, …, 2.6 × 106. To avoid oversampling, we randomly select 60% of all cloud samples using a random number generator for each time step. This allows us to obtain a more statistically meaningful set of cloud samples and avoid sampling too many cloud samples that are connected vertically within the same cloud column.

4. Mean cloud field

We begin the analysis of the high-resolution LES output with an investigation of the vertical profiles of entrainment and detrainment rates for the mean cloud field. Turbulent mass flux profiles during BOMEX have been studied previously (Siebesma and Cuijpers 1995; Dawe and Austin 2011a,b; Romps 2010), with the focus on the entrainment and detrainment profiles of the mean cloud field. Here, bulk-plume estimates eb and db [Eqs. (11) and (12)] and shell-corrected entrainment and detrainment rates eϕ and dϕ [Eqs. (14) and (15)] will be compared to directly calculated entrainment and detrainment rates e and d for the mean cloud field as well as the actual changes in mean cloud field tracers from the LES model run.

Mean cloud field properties have been obtained by taking an average over the entire cloud field. The mean cloud field can be defined as a horizontal (slab-)average over all model grid cells within the cloud region. All calculations are performed over the slab-averaged cloud fields, and we introduce the summation over the entire cloud region as follows:
⟨ϕ⟩≜1N∑(xn,yn)∈Cϕ(xn,yn),
where C is the set of all model grid points in the model domain Ci={(x1,y1),(x2,y2),…,(xN,yN)|A=1} that contain cloudy air and N is its size.

Figure 1 shows vertical profiles of mass entrainment (left panel) and detrainment (right panel) rates for the mean cloud field. Directly calculated entrainment and detrainment rates (blue line) are roughly twice as large as bulk-plume approximations (dashed green line), and these distributions closely match those from previous studies of shallow cumulus convection during BOMEX (Siebesma and Cuijpers 1995; Romps 2010; Dawe and Austin 2011a). Shell-corrected entrainment and detrainment rates (orange line) are similar in shape but are consistently smaller. Considering that these rates have been obtained by adjusting the directly calculated entrainment and detrainment rates by the mean cloud field approximation, these rates reflect the bulk-plume assumption but have been artificially lowered to produce the correct rate of changes in cloud mass ⟨e⟩ − ⟨d⟩.

Fig. 1.
Fig. 1.

Vertical profiles of (left) cloud mass entrainment and (right) detrainment rates based on direct calculations using tetrahedral interpolation (⟨e⟩ and ⟨d⟩; blue lines), shell correction method (⟨eϕ⟩ and ⟨dϕ⟩; orange lines), and bulk-plume approximation (eb and db; dashed green lines), for the mean cloud field, averaged over the entire model run.

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-24-0018.1

To examine the actual changes in cloud mass Dt⟨ρA⟩, Eq. (19) has been calculated explicitly from the LES model output. The left-hand side has been obtained by the second-order central finite-difference method, and the right-hand side has been obtained by mass entrainment and detrainment rates (Fig. 2). The right-hand side of Eq. (19) calculates Dt⟨ρA⟩ in terms of entrainment and detrainment rates, which can be calculated by the direct calculation method [Eqs. (3) and (4), blue curve] and bulk-plume approximations [Eqs. (11) and (12), dashed green curve]. The shell correction method [Eqs. (14) and (15)] is not included here as the shell-corrected net entrainment rates ⟨eϕ⟩ − ⟨dϕ⟩, by definition, must be the same as the directly calculated net entrainment rates. That is, ⟨eϕ⟩ − ⟨dϕ⟩ = ⟨e⟩ − ⟨d⟩.

Fig. 2.
Fig. 2.

Vertical profiles of the net cloud mass entrainment Dt⟨ρA⟩ obtained from direct calculations using the tetrahedral interpolation scheme ⟨e⟩ − ⟨d⟩ (blue line), finite-difference method (orange line), and bulk-plume estimates eb − db (dashed green line), for the mean cloud field, averaged over the entire model run.

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-24-0018.1

Figure 2 shows vertical distributions of cloud field net entrainment rates based on direct calculations ⟨e⟩ − ⟨d⟩ (blue curve), total derivative of cloud mass Dt⟨ρA⟩ (orange curve), and bulk-plume estimates eb − db (dashed green curve). The direct calculation method accurately reproduces the modeled tendencies of cloud mass. On the other hand, the magnitude of net entrainment rates from the bulk-plume approximation eb − db is consistently larger than the total derivative of cloud mass Dt⟨ρA⟩ from the LES model results. The relative error between the two methods can be quantified by the mean percentage error (MPE). The MPE is 18.3% for direct calculations and 77.9% for bulk-plume approximations.

Likewise, Eq. (20) can be examined by calculating the left-hand side of the equation using second-order central finite difference (Dt⟨ρA⟩, orange curve) and the right-hand side using the tetrahedral interpolation (⟨eϕ⟩ − ⟨dϕ⟩, blue curve) and bulk-plume approximation (ebϕ¯−dbϕc, dashed green line). The MPE value is 8.1% for direct calculations and 129% for the bulk-plume approximations.

Equations (19) and (20) have been calculated to obtain the changes in cloud mass Dt⟨ρA⟩ (Fig. 2) and tracer concentrations Dt⟨ρϕA⟩ (Fig. 3), and directly calculated entrainment and detrainment rates ⟨e⟩ and ⟨d⟩ are shown to be an accurate measure of changes in cloud mass and tracer concentrations. Figures 2 and 3 confirm that the numerical techniques used to calculate turbulent fluxes in Eqs. (19) and (20) yield the same tendencies of cloud mass and tracer concentrations. Conversely, bulk-plume estimates of entrainment and detrainment eb and db overestimate the effect of entrainment on both the changes in cloud mass and tracer concentrations.

Fig. 3.
Fig. 3.

Vertical profiles of the net dilution Dt⟨ρϕA⟩ for the total water content qt from direct calculations ⟨eϕ⟩ − ⟨dϕ⟩ (blue line), finite-difference method (orange line), and bulk-plume estimates ebϕ¯−dbϕc (dashed green line), for the mean cloud field, averaged over the entire model run.

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-24-0018.1

5. Individual cloud samples

In the previous section, we have shown that directly calculated entrainment and detrainment rates can be used to accurately determine the changes in cloud mass [Eq. (19)] and tracer concentrations [Eq. (20)] of the mean cloud field. To this end, we use the conditional sampling method in section 3b to obtain entrainment and detrainment profiles for the individual cloud regions. In this section, local distributions of directly calculated rates of entrainment and detrainment will be used to evaluate how the properties of the individual clouds change due to turbulent mixing processes. Furthermore, directly calculated entrainment and detrainment are also used to obtain a size-independent measure of dilution for individual clouds [Eq. (32)].

As described in section 3b, 2.6 × 106 individual cloud samples have been generated for all contiguous cloud regions in the domain. Figure 4 shows the distributions of individual cloud area (left panel) and vertical velocity (right panel). Small clouds dominate the statistical distribution, and there is large variability in size, especially near the cloud base at around 600 m above the ground. The average vertical velocity increases with altitude, except at the cloud top where the clouds accelerate and dissipate. The variability in vertical velocity remains large and shows no correlation with height; individual clouds ascend at difference rates regardless of their sizes. The results are consistent with previous studies involving the evolution of mean cloud field during BOMEX (Siebesma and Cuijpers 1995; Siebesma et al. 2003; Dawe and Austin 2013).

Fig. 4.
Fig. 4.

Vertical profiles of (left) cloud area (blue line) and (right) average vertical velocity (orange line) of individual clouds, averaged over all cloud samples. Individual cloud samples are shown as dots, where 400 points have been randomly sampled for visualization. The shaded region represents one standard deviation of the corresponding distribution.

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-24-0018.1

Figure 5 shows vertical distributions of cloud mass entrainment and detrainment rates ⟨e⟩i and ⟨d⟩i, obtained by finding the total entrainment (left panel) and detrainment (right panel) rates over the individual cloud regions. The vertical profile of the total cloud mass entrainment rate ⟨e⟩i is similar to that of cloud size (Fig. 4, left panel) in that both remain relatively constant across the cloud layer; larger clouds entrain more air as the cloud perimeter increases proportionally with cloud size (Dawe and Austin 2013). The total detrainment rate ⟨d⟩i, however, increases rapidly across the inversion layer, where upward-moving, buoyant parcels dissipate.

Fig. 5.
Fig. 5.

As in Fig. 4, but for vertical profiles of the directly calculated (left) entrainment rate ⟨e⟩i (red line) and (right) detrainment rate ⟨d⟩i (purple line) integrated over each individual cloud region.

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-24-0018.1

Figures 4 and 5 show that the underlying variability in the properties of individual clouds is largely hidden when average cloud field properties are examined; the variability in individual cloud tendencies is large (Fig. 4), and no discernible correlation can be seen with the individual cloud entrainment and detrainment rates (Fig. 5).

Figure 6 shows vertical profiles of the net entrainment rate based on directly calculated entrainment and detrainment rates ⟨e⟩i − ⟨d⟩i (blue line) and total derivative of individual cloud mass Dt⟨ρA⟩i (orange line) and horizontal divergence ∇xy⋅⟨ρA⟩i (dashed green line), averaged over all cloud samples at different altitudes. Vertical profiles of ⟨e⟩i − ⟨d⟩i [right-hand side of Eq. (19)] closely match the observed tendencies in cloud mass from the LES model output, calculated as the total derivative in cloud mass Dt⟨ρA⟩i [left-hand side of Eq. (19)] using the finite-difference method. The contribution to cloud mass tendencies from the horizontal divergence term ∇xy⋅⟨ρA⟩i remains negligible except near the cloud top, where the clouds disperse.

Fig. 6.
Fig. 6.

The distribution of changes in cloud mass Dt⟨ρA⟩i for the ith cloud region directly calculated based on the tetrahedral interpolation scheme ⟨e⟩i − ⟨d⟩i (blue dots) and vertical profiles of the total derivative Dt⟨ρA⟩i calculated directly by the tetrahedral interpolation scheme [right-hand side of Eq. (19); blue line], finite-difference method [left-hand side of Eq. (19); orange line], averaged horizontally over all cloud samples. The horizontal divergence term ∇xy⋅⟨ρA⟩i has also been calculated separately (green dashed line). Individual cloud samples are represented by blue dots, where 400 cloud regions have been randomly sampled for visualization, and the shaded region represents one standard derivation of the distributions of individual cloud samples.

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-24-0018.1

Likewise, vertical profiles of the proportional dilution rate based on direct calculations of entrainment and detrainment rates ⟨eϕ⟩i − ⟨dϕ⟩i (blue line) and total derivative of cloud tracer concentration Dt⟨ρϕA⟩i (orange line) are shown in Fig. 7 [Eq. (20)]. The two independently calculated tendencies shown in Fig. 7 closely resemble each other, and the contribution by the horizontal divergence in the cloud tracer concentration ∇xy⋅⟨ρϕA⟩i (dashed green line) remains negligible except above the inversion layer.

Fig. 7.
Fig. 7.

As in Fig. 6, but for changes in the cloud tracer concentration Dt⟨ρϕA⟩i directly calculated by the tetrahedral interpolation scheme ⟨eϕ⟩i − ⟨dϕ⟩i (blue curve) and the finite-difference method (orange curve) from Eq. (20).

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-24-0018.1

Figures 6 and 7 show that the numerical methods used to calculate the changes in individual cloud mass [Eq. (19); Fig. 6] and tracer concentrations [Eq. (20); Fig. 7] are accurate at the scale of individual clouds. The variability in vertical distributions of individual cloud mass and tracer concentrations can be attributed to turbulent fluxes due to entrainment and detrainment, which indicates that in the absence of strong wind shear or vigorous precipitation, directly calculated entrainment and detrainment rates can precisely determine the changes in cloud mass ∇xy⋅⟨ρA⟩i as net entrainment and tracer concentration Dt⟨ρϕA⟩i as net dilution.

The effect of horizontal divergence terms in Eqs. (19) and (20) (green lines) remains small except above the inversion layer at 1.6 km above the ground, where it accounts for roughly 10% of the total flux. Likewise, temporal changes in individual cloud mass ∂t⟨ρA⟩i and tracer concentrations ∂t⟨ρϕA⟩i [first terms in Eqs. (19) and (20)] are at least two orders of magnitude smaller (not shown). Given that, it is possible to obtain an accurate estimate of the net cloud mass entrainment Dt⟨ρA⟩i and net dilution Dt⟨ρϕA⟩i solely from the vertical divergence term. Similarly, nonadvective and nondiffusive source and sink terms ⟨Sϕ⟩i behave much like the horizontal divergence terms in that it remains small except above the inversion layer and accounts for about 10% of the total flux (not shown). As the clouds ascend through the inversion layer and reach the top of the cloud layer, precipitation occurs when the clouds expand and dissipate rapidly. For the BOMEX case, the only significant source of ⟨Sϕ⟩i is precipitation, and its contribution to the changes in cloud mass and tracer concentrations is limited to roughly 10% of the total flux above the inversion layer. Still, these terms are necessary for an accurate calculation of the proportional dilution rate [Eqs. (31) and (32)], which is calculated as a difference between large turbulent fluxes; relative differences between individual terms in Eqs. (31) and (32) are less than 5%.

The total derivatives in Eq. (31) have been obtained by the second-order central finite-difference method, and turbulent mass fluxes in Eq. (32) have been directly calculated by the tetrahedral interpolation scheme. Figure 8 shows the result of these calculations. Equation (31), based on the finite-difference method, yields an average proportional dilution rate of −0.001 27 ± 0.0026 (s−1), and Eq. (32), based on the direct calculation method, gives −0.001 31 ± 0.0010 (s−1). The variability in individual cloud dilution rates in Eq. (31) is much larger than that in Eq. (32), mainly because total derivative terms are calculated by the finite-difference method, and the time step is limited by the 60-s sampling rate, whereas turbulent fluxes in Eq. (32) have been obtained by the tetrahedral interpolation scheme applied at every model time step of 1 s. Although vertical distributions of Eqs. (31) and (32) closely match each other, the direct calculation method based on the tetrahedral interpolation scheme [Eq. (32)] provides a more accurate and numerically stable way to determine the correct proportional dilution rate for the individual clouds.

Fig. 8.
Fig. 8.

The distribution of proportional dilution rates calculated by the finite-difference method (blue dots) and the corresponding vertical profiles of the mean proportional dilution rate calculated by the finite-difference method [Eq. (31); orange line] and by tetrahedral interpolation scheme [Eq. (32); blue line] averaged horizontally over all cloud samples. Individual cloud samples are represented by blue dots, where 400 cloud regions have been randomly sampled for visualization. The shaded region represents one standard deviation of the directly calculated proportional dilution term [Eq. (32)].

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-24-0018.1

Inverting the mean proportional dilution rates gives the dilution time scale τ representing the time a parcel takes to become completely diluted by the inclusion of the dry air from its surroundings. Here, the dilution time scale is roughly 12.8 min for Eq. (31), based on the second-order central finite-difference method, and 13.1 min for Eq. (32), based on the tetrahedral interpolation scheme, indicating that on average, shallow cumulus clouds dilute in roughly 13 min. This is longer than the conventional eddy turnover time scale of 5 min, previously reported by Neggers et al. (2002), but close to the observed lifetime of simulated shallow cumulus clouds (Naumann and Seifert 2016; Romps et al. 2021) of roughly 10–15 min. Shallow cumulus clouds, on average, reach the top of the convective layer before they become significantly diluted due to the presence of the moist shell.

Changes in cloud properties relative to its size are small; the difference between the tendencies of cloud mass and cloud tracer concentration (Figs. 6 and 7) is less than 5%. Still, the finite-difference method [Eq. (31); orange line] and the direct calculation methods [Eq. (32); blue line] yield proportional dilution rates for individual clouds that are within 3% of each other, although the variability in the distribution of the proportional dilution rates is much larger for the finite-difference method.

As a preliminary examination of the correlation between the proportional dilution rate and individual cloud properties, we performed the kernel density estimate (Parzen 1962; Rosenblatt 1956) on individual cloud samples. Figure 9 shows the correlation between the proportional dilution rate and cloud area (left panel) and between the proportional dilution rate and vertical velocity (right panel) for individual clouds based on probability density function (PDF). The PDF of the proportional dilution rate shows large variability (Fig. 9, left panel), and no correlation can be observed; the time it takes for a cloud to fully homogenize with its surroundings is unrelated to its size. The PDF of proportional dilution rates can be better expressed as a function of the average vertical velocity wc=⟨ρwA⟩i/⟨ρA⟩i (Fig. 9, right panel), though the variability still remains large. There is a general trend in the PDF of proportional dilution rates as a function of wc where slowly ascending clouds take longer to fully become diluted, but individual cloud dilution rates vary significantly. A group of clouds ascending at the same rate can have vastly different dilution time scales.

Fig. 9.
Fig. 9.

PDFs showing the correlation (left) between log10 of the cloud area (m2) and proportional dilution rate (s−1) and (right) between individual cloud average vertical velocity (m s−1) and proportional dilution rate (s−1) for individual cloud samples. Individual cloud samples are represented by blue dots, where 400 points have been randomly sampled for visualization.

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-24-0018.1

As shown in Fig. 8, representing the changes in individual cloud tracer concentrations ∂t⟨ρϕA⟩i depends on accurate measures of entrainment and detrainment rates e and d, as well as the properties of the entrained and detrained air ϕe and ϕd [Eqs. (31) and (32)]. To examine the properties of the entrained and detrained air during the dilution of individual clouds, we plotted the average properties of individual cloud samples against the properties of entrained air ϕe (blue dots) and detrained air ϕd (orange dots) as well as the properties of the large-scale atmosphere ϕ¯ (green dots) in Fig. 10. Here, the properties of the entrained and detrained air have been plotted against the average property ϕc of the corresponding cloud region.

Fig. 10.
Fig. 10.

The distribution of the total water content qt (kg kg−1) within individual cloud samples ϕc in relation to the average total water content of the large-scale atmosphere ϕ¯ (green dots), entrained air ϕe (blue dots), and detrained air ϕd (orange dots). Each dot represents the property of an individual cloud region Ci for an ith cloud. For the sake of visualization, we randomly sampled 400 points from the distributions of qt, ϕc, and ϕd each. The black line represents a set of points where ϕc = ϕ{e,d}.

Citation: Journal of the Atmospheric Sciences 81, 10; 10.1175/JAS-D-24-0018.1

The property of the air being detrained ϕd remains close to the mean cloud property ϕc. The property of the entrained air ϕe deviates from ϕc as the parcel ascends but still remains much closer to ϕc than ϕ¯. When a parcel mixes with the environment during its ascent, ϕe becomes drier with altitude, leading to faster dilution at higher altitudes, but rapid detrainment of cloudy air (cf. Fig. 4) counteracts the effect of entrainment. As previously observed, this is due to the existence of moist shell (Dawe and Austin 2011b; Hannah 2017). Mixing does not occur directly between the cloud and the environment, but there exists a shell of moist air shrouding the cloud region that forms when the moist air detrains from the cloud. Yeo and Romps (2013) found that from a high-resolution simulation of individual cumulus clouds using Lagrangian particles, the moist shell mixes with the dry environment to become relatively drier than the cloud, and the air is recycled when it gets entrained back into the cloud from the moist shell. Within the cloud layer, the properties of the shell can be estimated by a linear mixing model (Drueke et al. 2021), but the relationship does not hold near the cloud top where the variability in ϕe increases (Fig. 10). According to Heus et al. (2008) and Yeo and Romps (2013), this appears to be because the source of the moist air near the top of the cloud layer is partly the recycled air from the cloud itself as well as subsidence from previous convective activities.

The vertical distribution of the proportional dilution rate is not well correlated to that of net entrainment. This explains why clouds can be entraining rapidly while being mostly undiluted; entrainment generally leads to dilution, but it cannot be considered to be the sole measure of dilution.

6. Discussion

We have used the results from a high-resolution LES model run to calculate the tendencies of cloud mass ⟨ρA⟩ (Fig. 2) and tracer concentrations ⟨ρϕA⟩ (Fig. 3). These tendencies are then compared to the net entrainment rate (Fig. 2) and net dilution rate (Fig. 3) based on the bulk-plume approach and the direct calculation method. The results show that the changes in both cloud mass and tracer concentrations can be accurately determined by directly calculated entrainment and detrainment rates, while the bulk-plume estimates overestimate the effect of entrainment.

This discrepancy comes from the mean cloud field assumption; the atmosphere is decomposed into the active cloud field and the dry environment. The air being entrained into the cloud field ϕe has the properties of the large-scale atmosphere ϕ¯, and the air being detrained from the cloud field ϕd has the properties of the mean cloud field based on a simplified model where the cloud field as a whole mixes directly with the dry environment. The mean cloud field assumption is represented explicitly in the bulk-plume dilution rate ebϕ¯−dbϕc [Eq. (6)]. However, as Eq. (20) suggests, tendencies of cloud tracer concentrations depend not only on the rate of turbulent transport but also on the properties of the entrained and detrained air ϕe and ϕd. As shown in Fig. 10, the values of ϕe and ϕd are much closer to each other than assumed by the mean cloud field approximation. Because the turbulent mixing processes in the bulk-plume approach need to incorporate both the physical transport of air and an adjustment for the mean cloud field approximation (i.e., the difference between the properties of the large-scale atmosphere ϕ¯ and the air immediately adjacent to the cloud), eb and db are dependent on the particular choice of (quasi-)conserved tracers ϕ and especially their correlation with the actual rate of mass transport.

Shell-corrected entrainment and detrainment rates eϕ and dϕ produce the correct changes in cloud mass and tracer concentrations but still suffer from the mean cloud field approximation not properly reflecting the properties being exchanged during the turbulent mixing processes. The efficacy of these methods, nonetheless, is that they provide an implicit measure of mean cloud field dilution rates at the expense of introducing nonphysical rates of entrainment and detrainment.

To further examine the accuracy of the direct calculation method at the scales of individual clouds, we calculated the changes in cloud mass ⟨ρA⟩i (net entrainment rate; Fig. 6) and tracer concentrations ⟨ρϕA⟩i (net dilution rate; Fig. 7) from 2.6 × 106 cloud samples obtained from the LES model run. Likewise, the proportional dilution rates based on the direct calculation method [Eq. (32)] can also accurately reproduce the proportional changes in cloud mass and tracer concentrations in 2.6 × 106 cloud samples. The results confirm that the direct calculation method used to calculate the entrainment and detrainment rates provides an accurate measure of these tendencies at the scale of individual clouds; that is, net entrainment and dilution rates based on the direct calculation method show much less variability than the finite-difference method.

The only physical constraints on entrainment and detrainment rates are given by Eqs. (19) and (20). That is, as long as the net entrainment ⟨e⟩ − ⟨d⟩ matches the change in cloud mass Dt⟨ρA⟩, and ⟨eϕ⟩ − ⟨dϕ⟩ + ⟨Sϕ⟩ matches the change in tracer concentration Dt⟨ρϕA⟩, the entrainment and detrainment rates can be considered to be true. In that sense, bulk-plume estimates are less accurate as the net bulk-plume entrainment rate eb − db cannot reproduce the changes in cloud mass (Fig. 2) and the net bulk-plume dilution rate ebϕ¯−dbϕc cannot reproduce the changes in cloud tracer concentrations (Fig. 3). These rates are based on the mean cloud field assumption that mass exchange processes occur directly between the cloud field and the dry environment. Here, noncloudy air is directly replaced by cloudy air as a result of entrainment, leading to a systemic underestimation of entrainment and detrainment rates and an overestimation of the effects of entrainment and detrainment on the changes in cloud mass and tracer concentrations.

Directly calculated entrainment and detrainment rates (Romps 2010; Dawe and Austin 2011a), on the other hand, can be considered a more physical representation of entrainment and detrainment rates based on the mathematical formulations presented here. The rates of entrainment and detrainment in these calculations are determined by the value of e − d for each grid cell; that is, local entrainment and detrainment rates are determined by net entrainment:
e=e−dif (e−d)>0,
d=e−dif (e−d)<0,
for every model grid cell with nonzero net entrainment rates. Because only the net change in cloud surface can be observed and the cloud surface cannot be expanding and shrinking at the same time, a measure of entrainment based on the net change in the cloud surface [Eqs. (3) and (4)] is the physical representation of entrainment and detrainment processes, and for this reason, the direct calculation method for entrainment and detrainment rates e and d provides an accurate representation of the turbulent mass exchange between the clouds and the atmosphere, as shown in Figs. 5 and 6, which can be used to diagnose the correct tendencies in cloud mass Dt⟨ρA⟩ and tracer concentrations Dt⟨ρϕA⟩.

An implicit measure of the cloud dilution rate, nevertheless, may still be useful if one aims to construct a simplified model of convection, while incorporating the effect of moist shell. An accurate measure of dilution must satisfy Eqs. (19) and (20) and account for the properties of the entrained and detrained air. This is the main goal of the shell-correction method proposed by Dawe and Austin (2011b). Shell-corrected entrainment and detrainment rates eϕ and dϕ are obtained by combining Eqs. (19) and (20) and satisfy mass continuity, which makes them a true measure of turbulent mass fluxes. However, while the shell correction method can correctly diagnose the changes in cloud mass and tracer concentrations, they do not have any physical significance due to the mean cloud field assumption.

As such, a simple parameterization model for moist convection that can diagnose the tendencies of cloud mass and tracer concentrations still remains difficult to achieve. The (proportional) rate of dilution depends not only on entrainment and detrainment rates but also on the local properties of the entrained and detrained air (Fig. 10), which cannot be resolved by the large-scale models. Nor can the rate of dilution be expressed as a function of cloud size (Fig. 9, left panel). Individual cloud vertical velocity is a better predictor of proportional dilution rate (Fig. 9, right panel). Still, the variability in the probability distributions given in Fig. 9 remains large; clouds ascending at similar speeds can still exhibit a wide range of dilution rates. As such, more work needs to be done to examine the factors driving the variability in the distribution of individual cloud properties.

7. Conclusions

We introduced numerical techniques that can explicitly evaluate the exact tendencies of individual cloud mass and tracer concentrations at the scale of individual clouds. From the LES model run, 2.6 × 106 cloud samples were retrieved and used to verify the representation of turbulent mass transport based on directly calculated entrainment and detrainment rates [Eqs. (3) and (4)]. The tendencies of cloud mass (Fig. 6) and tracer concentrations (Fig. 7) from the LES model run have been found to be consistent with the proposed formulation of turbulent mass exchange processes.

These fluxes were also used to verify traditional bulk-plume approximations to entrainment and detrainment rates eb and db [Eqs. (11) and (12)], where the cloud budget equations are constructed for a single bulk plume representing all clouds in the domain. The discrepancy in entrainment and detrainment rates between the two methods has also been investigated, whose source is attributed to the mean cloud field approximation. Because of this approximation, bulk-plume estimates overestimate the changes in cloud mass (Fig. 2) and tracer concentrations (Fig. 3).

We further examined the rate of proportional dilution of individual clouds and found that the average dilution time scale, which describes the time a cloud takes to be fully diluted, is τ ≈ 13 min. The relatively slow rate of dilution, despite strong entrainment, is attributed to the presence of moist shell modulating turbulent mixing processes. Previously detrained moist air is not immediately abandoned but often recycled (Yeo and Romps 2013). While the existence of moist shell has been previously observed (Jonas 1990; Rodts et al. 2003), the high-resolution LES model allowed us to precisely evaluate its contribution to cloud budget equations.

Direct calculations of entrainment and detrainment rates may not be immediately useful to convective parameterization schemes where entrainment and detrainment rates are diagnosed based on traditional bulk-plume approximations, and simply replacing bulk-plume estimates with directly calculated values will not produce the correct rate of dilution. Nevertheless, establishing a set of equations that determine the correct changes in cloud mass and tracer concentrations is a step toward a better understanding of the dynamic and thermodynamic processes that govern moist convection. The set of equations introduced in this paper allows for a physical interpretation of the individual factors that can influence the changes in cloud mass (in terms of cloud mass entrainment and detrainment) and cloud tracer concentration (in terms of cloud dilution). For example, in the case of precipitating deep convection, the role of Sϕ may become prominent. Likewise, the presence of a strong vertical wind shear can alter the distribution of vertical and horizontal divergence terms in Dtϕc, which can in turn be used to investigate the effect of wind shear on entrainment and detrainment. Directly calculated entrainment and detrainment rates have been shown to produce the correct rate of mass transport and can serve as a basis for the objective of developing more accurate parameterization schemes for large-scale models of the atmosphere.

A preliminary examination of the correlation between the proportional dilution rate and cloud size found no significant correlation (Fig. 9, left panel), which indicates that the time a cloud takes to become fully diluted due to the turbulent mixing processes does not depend on the size of the cloud. The mean cloud vertical velocity (Fig. 9, right panel) is a better predictor of the proportional dilution rate, although the variability remains large. More work is needed to determine the factors that drive the individual cloud variability and to improve our understanding of the physical mechanisms of entrainment and detrainment as well as dilution, which can be used to construct an improved cloud parameterization scheme that can better represent the dynamics of moist convection.

Acknowledgments.

We thank Marat Khairoutdinov for making SAM available. We would also like to thank Peter Blossey for helping us implement the radiative transfer model and resolve issues with SAM.

Data availability statement.

The data from the LES model run are available by request from the authors. The LES model used in this paper, as well as the exact model parameters, is publicly available at https://github.com/lorenghoh/sam_loh. Jupyter notebooks including the numerical analysis are also publicly available at https://github.com/lorenghoh/entrainment.

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  • Betts, A. K., 1975: Parametric interpretation of trade-wind cumulus budget studies. J. Atmos. Sci., 32, 1934–1945, https://doi.org/10.1175/1520-0469(1975)032<1934:PIOTWC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Blossey, P. N., and Coauthors, 2013: Marine low cloud sensitivity to an idealized climate change: The CGILS LES intercomparison. J. Adv. Model. Earth Syst., 5, 234–258, https://doi.org/10.1002/jame.20025.

    • Search Google Scholar
    • Export Citation

  • Bryan, G. H., J. C. Wyngaard, and J. M. Fritsch, 2003: Resolution requirements for the simulation of deep moist convection. Mon. Wea. Rev., 131, 2394–2416, https://doi.org/10.1175/1520-0493(2003)131<2394:RRFTSO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Ceppi, P., F. Brient, M. D. Zelinka, and D. L. Hartmann, 2017: Cloud feedback mechanisms and their representation in global climate models. Wiley Interdiscip. Rev.: Climate Change, 8, e465, https://doi.org/10.1002/wcc.465.

    • Search Google Scholar
    • Export Citation

  • Clough, S. A., M. W. Shephard, E. J. Mlawer, J. S. Delamere, M. J. Iacono, K. Cady-Pereira, S. Boukabara, and P. D. Brown, 2005: Atmospheric radiative transfer modeling: A summary of the AER codes. J. Quant. Spectrosc. Radiat. Transfer, 91, 233–244, https://doi.org/10.1016/j.jqsrt.2004.05.058.

    • Search Google Scholar
    • Export Citation

  • Dawe, J. T., and P. H. Austin, 2011a: Interpolation of LES cloud surfaces for use in direct calculations of entrainment and detrainment. Mon. Wea. Rev., 139, 444–456, https://doi.org/10.1175/2010MWR3473.1.

    • Search Google Scholar
    • Export Citation

  • Dawe, J. T., and P. H. Austin, 2011b: The influence of the cloud shell on tracer budget measurements of LES cloud entrainment. J. Atmos. Sci., 68, 2909–2920, https://doi.org/10.1175/2011JAS3658.1.

    • Search Google Scholar
    • Export Citation

  • Dawe, J. T., and P. H. Austin, 2012: Statistical analysis of an LES shallow cumulus cloud ensemble using a cloud tracking algorithm. Atmos. Chem. Phys., 12, 1101–1119, https://doi.org/10.5194/acp-12-1101-2012.

    • Search Google Scholar
    • Export Citation

  • Dawe, J. T., and P. H. Austin, 2013: Direct entrainment and detrainment rate distributions of individual shallow cumulus clouds in an LES. Atmos. Chem. Phys., 13, 7795–7811, https://doi.org/10.5194/acp-13-7795-2013.

    • Search Google Scholar
    • Export Citation

  • Drueke, S., D. J. Kirshbaum, and P. Kollias, 2021: Environmental sensitivities of shallow-cumulus dilution – Part 2: Vertical wind profile. Atmos. Chem. Phys., 21, 14 039–14 058, https://doi.org/10.5194/acp-21-14039-2021.

    • Search Google Scholar
    • Export Citation

  • Gregory, D., and P. R. Rowntree, 1990: A mass flux convection scheme with representation of cloud ensemble characteristics and stability-dependent closure. Mon. Wea. Rev., 118, 1483–1506, https://doi.org/10.1175/1520-0493(1990)118<1483:AMFCSW>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Hannah, W. M., 2017: Entrainment versus dilution in tropical deep convection. J. Atmos. Sci., 74, 3725–3747, https://doi.org/10.1175/JAS-D-16-0169.1.

    • Search Google Scholar
    • Export Citation

  • Heus, T., and H. J. J. Jonker, 2008: Subsiding shells around shallow cumulus clouds. J. Atmos. Sci., 65, 1003–1018, https://doi.org/10.1175/2007JAS2322.1.

    • Search Google Scholar
    • Export Citation

  • Heus, T., G. van Dijk, H. J. J. Jonker, and H. E. A. Van den Akker, 2008: Mixing in shallow cumulus clouds studied by Lagrangian particle tracking. J. Atmos. Sci., 65, 2581–2597, https://doi.org/10.1175/2008JAS2572.1.

    • Search Google Scholar
    • Export Citation

  • Holland, J. Z., and E. M. Rasmusson, 1973: Measurements of the atmospheric mass, energy, and momentum budgets over a 500-kilometer square of tropical ocean. Mon. Wea. Rev., 101, 44–55, https://doi.org/10.1175/1520-0493(1973)101<0044:MOTAME>2.3.CO;2.

    • Search Google Scholar
    • Export Citation

  • Iacono, M. J., J. S. Delamere, E. J. Mlawer, M. W. Shephard, S. A. Clough, and W. D. Collins, 2008: Radiative forcing by long-lived greenhouse gases: Calculations with the AER radiative transfer models. J. Geophys. Res., 113, D13103, https://doi.org/10.1029/2008JD009944.

    • Search Google Scholar
    • Export Citation

  • Jonas, P. R., 1990: Observations of cumulus cloud entrainment. Atmos. Res., 25, 105–127, https://doi.org/10.1016/0169-8095(90)90008-Z.

    • Search Google Scholar
    • Export Citation

  • Kain, J. S., and J. M. Fritsch, 1990: A one-dimensional entraining/detraining plume model and its application in convective parameterization. J. Atmos. Sci., 47, 2784–2802, https://doi.org/10.1175/1520-0469(1990)047<2784:AODEPM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Khairoutdinov, M. F., and D. A. Randall, 2003: Cloud resolving modeling of the ARM summer 1997 IOP: Model formulation, results, uncertainties, and sensitivities. J. Atmos. Sci., 60, 607–625, https://doi.org/10.1175/1520-0469(2003)060<0607:CRMOTA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Matheou, G., D. Chung, L. Nuijens, B. Stevens, and J. Teixeira, 2011: On the fidelity of large-eddy simulation of shallow precipitating cumulus convection. Mon. Wea. Rev., 139, 2918–2939, https://doi.org/10.1175/2011MWR3599.1.

    • Search Google Scholar
    • Export Citation

  • Morrison, H., and A. Gettelman, 2008: A new two-moment bulk stratiform cloud microphysics scheme in the Community Atmosphere Model, version 3 (CAM3). Part I: Description and numerical tests. J. Climate, 21, 3642–3659, https://doi.org/10.1175/2008JCLI2105.1.

    • Search Google Scholar
    • Export Citation

  • Morrison, H., J. A. Curry, and V. I. Khvorostyanov, 2005a: A new double-moment microphysics parameterization for application in cloud and climate models. Part I: Description. J. Atmos. Sci., 62, 1665–1677, https://doi.org/10.1175/JAS3446.1.

    • Search Google Scholar
    • Export Citation

  • Morrison, H., J. A. Curry, M. D. Shupe, and P. Zuidema, 2005b: A new double-moment microphysics parameterization for application in cloud and climate models. Part II: Single-column modeling of arctic clouds. J. Atmos. Sci., 62, 1678–1693, https://doi.org/10.1175/JAS3447.1.

    • Search Google Scholar
    • Export Citation

  • Naumann, A. K., and A. Seifert, 2016: Recirculation and growth of raindrops in simulated shallow cumulus. J. Adv. Model. Earth Syst., 8, 520–537, https://doi.org/10.1002/2016MS000631.

    • Search Google Scholar
    • Export Citation

  • Neggers, R. A. J., A. P. Siebesma, and H. J. J. Jonker, 2002: A multiparcel model for shallow cumulus convection. J. Atmos. Sci., 59, 1655–1668, https://doi.org/10.1175/1520-0469(2002)059<1655:AMMFSC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Nitta, T., and S. Esbensen, 1974: Heat and moisture budget analyses using BOMEX data. Mon. Wea. Rev., 102, 17–28, https://doi.org/10.1175/1520-0493(1974)102<0017:HAMBAU>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Nordeng, T.-E., 1994: Extended versions of the convective parametrization scheme at ECMWF and their impact on the mean and transient activity of the model in the tropics. ECMWF Tech. Memo. 206, 42 pp., https://doi.org/10.21957/e34xwhysw.

  • Parzen, E., 1962: On estimation of a probability density function and mode. Ann. Math. Stat., 33, 1065–1076, https://doi.org/10.1214/aoms/1177704472.

    • Search Google Scholar
    • Export Citation

  • Rasmussen, R. M., I. Geresdi, G. Thompson, K. Manning, and E. Karplus, 2002: Freezing drizzle formation in stably stratified layer clouds: The role of radiative cooling of cloud droplets, cloud condensation nuclei, and ice initiation. J. Atmos. Sci., 59, 837–860, https://doi.org/10.1175/1520-0469(2002)059<0837:FDFISS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Roberts, M. J., and Coauthors, 2020: Impact of model resolution on tropical cyclone simulation using the HighResMIP–PRIMAVERA multimodel ensemble. J. Climate, 33, 2557–2583, https://doi.org/10.1175/JCLI-D-19-0639.1.

    • Search Google Scholar
    • Export Citation

  • Rodts, S. M. A., P. G. Duynkerke, and H. J. J. Jonker, 2003: Size distributions and dynamical properties of shallow cumulus clouds from aircraft observations and satellite data. J. Atmos. Sci., 60, 1895–1912, https://doi.org/10.1175/1520-0469(2003)060<1895:SDADPO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Romps, D. M., 2010: A direct measure of entrainment. J. Atmos. Sci., 67, 1908–1927, https://doi.org/10.1175/2010JAS3371.1.

  • Romps, D. M., R. Öktem, S. Endo, and A. M. Vogelmann, 2021: On the life cycle of a shallow cumulus cloud: Is it a bubble or plume, active or forced? J. Atmos. Sci., 78, 2823–2833, https://doi.org/10.1175/JAS-D-20-0361.1.

    • Search Google Scholar
    • Export Citation

  • Rosenblatt, M., 1956: Remarks on some nonparametric estimates of a density function. Ann. Math. Stat., 27, 832–837, https://doi.org/10.1214/aoms/1177728190.

    • Search Google Scholar
    • Export Citation

  • Sato, Y., S.-i. Shima, and H. Tomita, 2017: A grid refinement study of trade wind cumuli simulated by a Lagrangian cloud microphysical model: The super-droplet method. Atmos. Sci. Lett., 18, 359–365, https://doi.org/10.1002/asl.764.

    • Search Google Scholar
    • Export Citation

  • Sato, Y., S.-i. Shima, and H. Tomita, 2018: Numerical convergence of shallow convection cloud field simulations: Comparison between double-moment Eulerian and particle-based Lagrangian microphysics coupled to the same dynamical core. J. Adv. Model. Earth Syst., 10, 1495–1512, https://doi.org/10.1029/2018MS001285.

    • Search Google Scholar
    • Export Citation

  • Savre, J., 2022: What controls local entrainment and detrainment rates in simulated shallow convection? J. Atmos. Sci., 79, 3065–3082, https://doi.org/10.1175/JAS-D-21-0341.1.

    • Search Google Scholar
    • Export Citation

  • Siebesma, A. P., and J. W. M. Cuijpers, 1995: Evaluation of parametric assumptions for shallow cumulus convection. J. Atmos. Sci., 52, 650–666, https://doi.org/10.1175/1520-0469(1995)052<0650:EOPAFS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Siebesma, A. P., and A. A. M. Holtslag, 1996: Model impacts of entrainment and detrainment rates in shallow cumulus convection. J. Atmos. Sci., 53, 2354–2364, https://doi.org/10.1175/1520-0469(1996)053<2354:MIOEAD>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Siebesma, A. P., and Coauthors, 2003: A large eddy simulation intercomparison study of shallow cumulus convection. J. Atmos. Sci., 60, 1201–1219, https://doi.org/10.1175/1520-0469(2003)60<1201:ALESIS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Tiedtke, M., 1989: A comprehensive mass flux scheme for cumulus parameterization in large-scale models. Mon. Wea. Rev., 117, 1779–1800, https://doi.org/10.1175/1520-0493(1989)117<1779:ACMFSF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Wang, Z., 2020: A method for a direct measure of entrainment and detrainment. Mon. Wea. Rev., 148, 3329–3340, https://doi.org/10.1175/MWR-D-20-0046.1.

    • Search Google Scholar
    • Export Citation

  • Yeo, K., and D. M. Romps, 2013: Measurement of convective entrainment using Lagrangian particles. J. Atmos. Sci., 70, 266–277, https://doi.org/10.1175/JAS-D-12-0144.1.

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  • Arakawa, A., 2004: The cumulus parameterization problem: Past, present, and future. J. Climate, 17, 2493–2525, https://doi.org/10.1175/1520-0442(2004)017<2493:RATCPP>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Betts, A. K., 1973: Non-precipitating cumulus convection and its parameterization. Quart. J. Roy. Meteor. Soc., 99, 178–196, https://doi.org/10.1002/qj.49709941915.

    • Search Google Scholar
    • Export Citation

  • Betts, A. K., 1975: Parametric interpretation of trade-wind cumulus budget studies. J. Atmos. Sci., 32, 1934–1945, https://doi.org/10.1175/1520-0469(1975)032<1934:PIOTWC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Blossey, P. N., and Coauthors, 2013: Marine low cloud sensitivity to an idealized climate change: The CGILS LES intercomparison. J. Adv. Model. Earth Syst., 5, 234–258, https://doi.org/10.1002/jame.20025.

    • Search Google Scholar
    • Export Citation

  • Bryan, G. H., J. C. Wyngaard, and J. M. Fritsch, 2003: Resolution requirements for the simulation of deep moist convection. Mon. Wea. Rev., 131, 2394–2416, https://doi.org/10.1175/1520-0493(2003)131<2394:RRFTSO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Ceppi, P., F. Brient, M. D. Zelinka, and D. L. Hartmann, 2017: Cloud feedback mechanisms and their representation in global climate models. Wiley Interdiscip. Rev.: Climate Change, 8, e465, https://doi.org/10.1002/wcc.465.

    • Search Google Scholar
    • Export Citation

  • Clough, S. A., M. W. Shephard, E. J. Mlawer, J. S. Delamere, M. J. Iacono, K. Cady-Pereira, S. Boukabara, and P. D. Brown, 2005: Atmospheric radiative transfer modeling: A summary of the AER codes. J. Quant. Spectrosc. Radiat. Transfer, 91, 233–244, https://doi.org/10.1016/j.jqsrt.2004.05.058.

    • Search Google Scholar
    • Export Citation

  • Dawe, J. T., and P. H. Austin, 2011a: Interpolation of LES cloud surfaces for use in direct calculations of entrainment and detrainment. Mon. Wea. Rev., 139, 444–456, https://doi.org/10.1175/2010MWR3473.1.

    • Search Google Scholar
    • Export Citation

  • Dawe, J. T., and P. H. Austin, 2011b: The influence of the cloud shell on tracer budget measurements of LES cloud entrainment. J. Atmos. Sci., 68, 2909–2920, https://doi.org/10.1175/2011JAS3658.1.

    • Search Google Scholar
    • Export Citation

  • Dawe, J. T., and P. H. Austin, 2012: Statistical analysis of an LES shallow cumulus cloud ensemble using a cloud tracking algorithm. Atmos. Chem. Phys., 12, 1101–1119, https://doi.org/10.5194/acp-12-1101-2012.

    • Search Google Scholar
    • Export Citation

  • Dawe, J. T., and P. H. Austin, 2013: Direct entrainment and detrainment rate distributions of individual shallow cumulus clouds in an LES. Atmos. Chem. Phys., 13, 7795–7811, https://doi.org/10.5194/acp-13-7795-2013.

    • Search Google Scholar
    • Export Citation

  • Drueke, S., D. J. Kirshbaum, and P. Kollias, 2021: Environmental sensitivities of shallow-cumulus dilution – Part 2: Vertical wind profile. Atmos. Chem. Phys., 21, 14 039–14 058, https://doi.org/10.5194/acp-21-14039-2021.

    • Search Google Scholar
    • Export Citation

  • Gregory, D., and P. R. Rowntree, 1990: A mass flux convection scheme with representation of cloud ensemble characteristics and stability-dependent closure. Mon. Wea. Rev., 118, 1483–1506, https://doi.org/10.1175/1520-0493(1990)118<1483:AMFCSW>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Hannah, W. M., 2017: Entrainment versus dilution in tropical deep convection. J. Atmos. Sci., 74, 3725–3747, https://doi.org/10.1175/JAS-D-16-0169.1.

    • Search Google Scholar
    • Export Citation

  • Heus, T., and H. J. J. Jonker, 2008: Subsiding shells around shallow cumulus clouds. J. Atmos. Sci., 65, 1003–1018, https://doi.org/10.1175/2007JAS2322.1.

    • Search Google Scholar
    • Export Citation

  • Heus, T., G. van Dijk, H. J. J. Jonker, and H. E. A. Van den Akker, 2008: Mixing in shallow cumulus clouds studied by Lagrangian particle tracking. J. Atmos. Sci., 65, 2581–2597, https://doi.org/10.1175/2008JAS2572.1.

    • Search Google Scholar
    • Export Citation

  • Holland, J. Z., and E. M. Rasmusson, 1973: Measurements of the atmospheric mass, energy, and momentum budgets over a 500-kilometer square of tropical ocean. Mon. Wea. Rev., 101, 44–55, https://doi.org/10.1175/1520-0493(1973)101<0044:MOTAME>2.3.CO;2.

    • Search Google Scholar
    • Export Citation

  • Iacono, M. J., J. S. Delamere, E. J. Mlawer, M. W. Shephard, S. A. Clough, and W. D. Collins, 2008: Radiative forcing by long-lived greenhouse gases: Calculations with the AER radiative transfer models. J. Geophys. Res., 113, D13103, https://doi.org/10.1029/2008JD009944.

    • Search Google Scholar
    • Export Citation

  • Jonas, P. R., 1990: Observations of cumulus cloud entrainment. Atmos. Res., 25, 105–127, https://doi.org/10.1016/0169-8095(90)90008-Z.

    • Search Google Scholar
    • Export Citation

  • Kain, J. S., and J. M. Fritsch, 1990: A one-dimensional entraining/detraining plume model and its application in convective parameterization. J. Atmos. Sci., 47, 2784–2802, https://doi.org/10.1175/1520-0469(1990)047<2784:AODEPM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Khairoutdinov, M. F., and D. A. Randall, 2003: Cloud resolving modeling of the ARM summer 1997 IOP: Model formulation, results, uncertainties, and sensitivities. J. Atmos. Sci., 60, 607–625, https://doi.org/10.1175/1520-0469(2003)060<0607:CRMOTA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Matheou, G., D. Chung, L. Nuijens, B. Stevens, and J. Teixeira, 2011: On the fidelity of large-eddy simulation of shallow precipitating cumulus convection. Mon. Wea. Rev., 139, 2918–2939, https://doi.org/10.1175/2011MWR3599.1.

    • Search Google Scholar
    • Export Citation

  • Morrison, H., and A. Gettelman, 2008: A new two-moment bulk stratiform cloud microphysics scheme in the Community Atmosphere Model, version 3 (CAM3). Part I: Description and numerical tests. J. Climate, 21, 3642–3659, https://doi.org/10.1175/2008JCLI2105.1.

    • Search Google Scholar
    • Export Citation

  • Morrison, H., J. A. Curry, and V. I. Khvorostyanov, 2005a: A new double-moment microphysics parameterization for application in cloud and climate models. Part I: Description. J. Atmos. Sci., 62, 1665–1677, https://doi.org/10.1175/JAS3446.1.

    • Search Google Scholar
    • Export Citation

  • Morrison, H., J. A. Curry, M. D. Shupe, and P. Zuidema, 2005b: A new double-moment microphysics parameterization for application in cloud and climate models. Part II: Single-column modeling of arctic clouds. J. Atmos. Sci., 62, 1678–1693, https://doi.org/10.1175/JAS3447.1.

    • Search Google Scholar
    • Export Citation

  • Naumann, A. K., and A. Seifert, 2016: Recirculation and growth of raindrops in simulated shallow cumulus. J. Adv. Model. Earth Syst., 8, 520–537, https://doi.org/10.1002/2016MS000631.

    • Search Google Scholar
    • Export Citation

  • Neggers, R. A. J., A. P. Siebesma, and H. J. J. Jonker, 2002: A multiparcel model for shallow cumulus convection. J. Atmos. Sci., 59, 1655–1668, https://doi.org/10.1175/1520-0469(2002)059<1655:AMMFSC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Nitta, T., and S. Esbensen, 1974: Heat and moisture budget analyses using BOMEX data. Mon. Wea. Rev., 102, 17–28, https://doi.org/10.1175/1520-0493(1974)102<0017:HAMBAU>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Nordeng, T.-E., 1994: Extended versions of the convective parametrization scheme at ECMWF and their impact on the mean and transient activity of the model in the tropics. ECMWF Tech. Memo. 206, 42 pp., https://doi.org/10.21957/e34xwhysw.

  • Parzen, E., 1962: On estimation of a probability density function and mode. Ann. Math. Stat., 33, 1065–1076, https://doi.org/10.1214/aoms/1177704472.

    • Search Google Scholar
    • Export Citation

  • Rasmussen, R. M., I. Geresdi, G. Thompson, K. Manning, and E. Karplus, 2002: Freezing drizzle formation in stably stratified layer clouds: The role of radiative cooling of cloud droplets, cloud condensation nuclei, and ice initiation. J. Atmos. Sci., 59, 837–860, https://doi.org/10.1175/1520-0469(2002)059<0837:FDFISS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Roberts, M. J., and Coauthors, 2020: Impact of model resolution on tropical cyclone simulation using the HighResMIP–PRIMAVERA multimodel ensemble. J. Climate, 33, 2557–2583, https://doi.org/10.1175/JCLI-D-19-0639.1.

    • Search Google Scholar
    • Export Citation

  • Rodts, S. M. A., P. G. Duynkerke, and H. J. J. Jonker, 2003: Size distributions and dynamical properties of shallow cumulus clouds from aircraft observations and satellite data. J. Atmos. Sci., 60, 1895–1912, https://doi.org/10.1175/1520-0469(2003)060<1895:SDADPO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Romps, D. M., 2010: A direct measure of entrainment. J. Atmos. Sci., 67, 1908–1927, https://doi.org/10.1175/2010JAS3371.1.

  • Romps, D. M., R. Öktem, S. Endo, and A. M. Vogelmann, 2021: On the life cycle of a shallow cumulus cloud: Is it a bubble or plume, active or forced? J. Atmos. Sci., 78, 2823–2833, https://doi.org/10.1175/JAS-D-20-0361.1.

    • Search Google Scholar
    • Export Citation

  • Rosenblatt, M., 1956: Remarks on some nonparametric estimates of a density function. Ann. Math. Stat., 27, 832–837, https://doi.org/10.1214/aoms/1177728190.

    • Search Google Scholar
    • Export Citation

  • Sato, Y., S.-i. Shima, and H. Tomita, 2017: A grid refinement study of trade wind cumuli simulated by a Lagrangian cloud microphysical model: The super-droplet method. Atmos. Sci. Lett., 18, 359–365, https://doi.org/10.1002/asl.764.

    • Search Google Scholar
    • Export Citation

  • Sato, Y., S.-i. Shima, and H. Tomita, 2018: Numerical convergence of shallow convection cloud field simulations: Comparison between double-moment Eulerian and particle-based Lagrangian microphysics coupled to the same dynamical core. J. Adv. Model. Earth Syst., 10, 1495–1512, https://doi.org/10.1029/2018MS001285.

    • Search Google Scholar
    • Export Citation

  • Savre, J., 2022: What controls local entrainment and detrainment rates in simulated shallow convection? J. Atmos. Sci., 79, 3065–3082, https://doi.org/10.1175/JAS-D-21-0341.1.

    • Search Google Scholar
    • Export Citation

  • Siebesma, A. P., and J. W. M. Cuijpers, 1995: Evaluation of parametric assumptions for shallow cumulus convection. J. Atmos. Sci., 52, 650–666, https://doi.org/10.1175/1520-0469(1995)052<0650:EOPAFS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Siebesma, A. P., and A. A. M. Holtslag, 1996: Model impacts of entrainment and detrainment rates in shallow cumulus convection. J. Atmos. Sci., 53, 2354–2364, https://doi.org/10.1175/1520-0469(1996)053<2354:MIOEAD>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Siebesma, A. P., and Coauthors, 2003: A large eddy simulation intercomparison study of shallow cumulus convection. J. Atmos. Sci., 60, 1201–1219, https://doi.org/10.1175/1520-0469(2003)60<1201:ALESIS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Tiedtke, M., 1989: A comprehensive mass flux scheme for cumulus parameterization in large-scale models. Mon. Wea. Rev., 117, 1779–1800, https://doi.org/10.1175/1520-0493(1989)117<1779:ACMFSF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Wang, Z., 2020: A method for a direct measure of entrainment and detrainment. Mon. Wea. Rev., 148, 3329–3340, https://doi.org/10.1175/MWR-D-20-0046.1.

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  • Yeo, K., and D. M. Romps, 2013: Measurement of convective entrainment using Lagrangian particles. J. Atmos. Sci., 70, 266–277, https://doi.org/10.1175/JAS-D-12-0144.1.

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  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    Vertical profiles of (left) cloud mass entrainment and (right) detrainment rates based on direct calculations using tetrahedral interpolation (⟨e⟩ and ⟨d⟩; blue lines), shell correction method (⟨eϕ⟩ and ⟨dϕ⟩; orange lines), and bulk-plume approximation (eb and db; dashed green lines), for the mean cloud field, averaged over the entire model run.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    Vertical profiles of the net cloud mass entrainment Dt⟨ρA⟩ obtained from direct calculations using the tetrahedral interpolation scheme ⟨e⟩ − ⟨d⟩ (blue line), finite-difference method (orange line), and bulk-plume estimates eb − db (dashed green line), for the mean cloud field, averaged over the entire model run.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    Vertical profiles of the net dilution Dt⟨ρϕA⟩ for the total water content qt from direct calculations ⟨eϕ⟩ − ⟨dϕ⟩ (blue line), finite-difference method (orange line), and bulk-plume estimates ebϕ¯−dbϕc (dashed green line), for the mean cloud field, averaged over the entire model run.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    Vertical profiles of (left) cloud area (blue line) and (right) average vertical velocity (orange line) of individual clouds, averaged over all cloud samples. Individual cloud samples are shown as dots, where 400 points have been randomly sampled for visualization. The shaded region represents one standard deviation of the corresponding distribution.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    As in Fig. 4, but for vertical profiles of the directly calculated (left) entrainment rate ⟨e⟩i (red line) and (right) detrainment rate ⟨d⟩i (purple line) integrated over each individual cloud region.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    The distribution of changes in cloud mass Dt⟨ρA⟩i for the ith cloud region directly calculated based on the tetrahedral interpolation scheme ⟨e⟩i − ⟨d⟩i (blue dots) and vertical profiles of the total derivative Dt⟨ρA⟩i calculated directly by the tetrahedral interpolation scheme [right-hand side of Eq. (19); blue line], finite-difference method [left-hand side of Eq. (19); orange line], averaged horizontally over all cloud samples. The horizontal divergence term ∇xy⋅⟨ρA⟩i has also been calculated separately (green dashed line). Individual cloud samples are represented by blue dots, where 400 cloud regions have been randomly sampled for visualization, and the shaded region represents one standard derivation of the distributions of individual cloud samples.

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    As in Fig. 6, but for changes in the cloud tracer concentration Dt⟨ρϕA⟩i directly calculated by the tetrahedral interpolation scheme ⟨eϕ⟩i − ⟨dϕ⟩i (blue curve) and the finite-difference method (orange curve) from Eq. (20).

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    The distribution of proportional dilution rates calculated by the finite-difference method (blue dots) and the corresponding vertical profiles of the mean proportional dilution rate calculated by the finite-difference method [Eq. (31); orange line] and by tetrahedral interpolation scheme [Eq. (32); blue line] averaged horizontally over all cloud samples. Individual cloud samples are represented by blue dots, where 400 cloud regions have been randomly sampled for visualization. The shaded region represents one standard deviation of the directly calculated proportional dilution term [Eq. (32)].

  • <sc>Fig</sc>. 9.
    View in gallery
    Fig. 9.

    PDFs showing the correlation (left) between log10 of the cloud area (m2) and proportional dilution rate (s−1) and (right) between individual cloud average vertical velocity (m s−1) and proportional dilution rate (s−1) for individual cloud samples. Individual cloud samples are represented by blue dots, where 400 points have been randomly sampled for visualization.

  • <sc>Fig</sc>. 10.
    View in gallery
    Fig. 10.

    The distribution of the total water content qt (kg kg−1) within individual cloud samples ϕc in relation to the average total water content of the large-scale atmosphere ϕ¯ (green dots), entrained air ϕe (blue dots), and detrained air ϕd (orange dots). Each dot represents the property of an individual cloud region Ci for an ith cloud. For the sake of visualization, we randomly sampled 400 points from the distributions of qt, ϕc, and ϕd each. The black line represents a set of points where ϕc = ϕ{e,d}.

Fig. 1.

Vertical profiles of (left) cloud mass entrainment and (right) detrainment rates based on direct calculations using tetrahedral interpolation (⟨e⟩ and ⟨d⟩; blue lines), shell correction method (⟨eϕ⟩ and ⟨dϕ⟩; orange lines), and bulk-plume approximation (eb and db; dashed green lines), for the mean cloud field, averaged over the entire model run.
Fig. 2.

Vertical profiles of the net cloud mass entrainment Dt⟨ρA⟩ obtained from direct calculations using the tetrahedral interpolation scheme ⟨e⟩ − ⟨d⟩ (blue line), finite-difference method (orange line), and bulk-plume estimates eb − db (dashed green line), for the mean cloud field, averaged over the entire model run.
Fig. 3.

Vertical profiles of the net dilution Dt⟨ρϕA⟩ for the total water content qt from direct calculations ⟨eϕ⟩ − ⟨dϕ⟩ (blue line), finite-difference method (orange line), and bulk-plume estimates ebϕ¯−dbϕc (dashed green line), for the mean cloud field, averaged over the entire model run.
Fig. 4.

Vertical profiles of (left) cloud area (blue line) and (right) average vertical velocity (orange line) of individual clouds, averaged over all cloud samples. Individual cloud samples are shown as dots, where 400 points have been randomly sampled for visualization. The shaded region represents one standard deviation of the corresponding distribution.
Fig. 5.

As in Fig. 4, but for vertical profiles of the directly calculated (left) entrainment rate ⟨e⟩i (red line) and (right) detrainment rate ⟨d⟩i (purple line) integrated over each individual cloud region.
Fig. 6.

The distribution of changes in cloud mass Dt⟨ρA⟩i for the ith cloud region directly calculated based on the tetrahedral interpolation scheme ⟨e⟩i − ⟨d⟩i (blue dots) and vertical profiles of the total derivative Dt⟨ρA⟩i calculated directly by the tetrahedral interpolation scheme [right-hand side of Eq. (19); blue line], finite-difference method [left-hand side of Eq. (19); orange line], averaged horizontally over all cloud samples. The horizontal divergence term ∇xy⋅⟨ρA⟩i has also been calculated separately (green dashed line). Individual cloud samples are represented by blue dots, where 400 cloud regions have been randomly sampled for visualization, and the shaded region represents one standard derivation of the distributions of individual cloud samples.
Fig. 7.

As in Fig. 6, but for changes in the cloud tracer concentration Dt⟨ρϕA⟩i directly calculated by the tetrahedral interpolation scheme ⟨eϕ⟩i − ⟨dϕ⟩i (blue curve) and the finite-difference method (orange curve) from Eq. (20).
Fig. 8.

The distribution of proportional dilution rates calculated by the finite-difference method (blue dots) and the corresponding vertical profiles of the mean proportional dilution rate calculated by the finite-difference method [Eq. (31); orange line] and by tetrahedral interpolation scheme [Eq. (32); blue line] averaged horizontally over all cloud samples. Individual cloud samples are represented by blue dots, where 400 cloud regions have been randomly sampled for visualization. The shaded region represents one standard deviation of the directly calculated proportional dilution term [Eq. (32)].
Fig. 9.

PDFs showing the correlation (left) between log10 of the cloud area (m2) and proportional dilution rate (s−1) and (right) between individual cloud average vertical velocity (m s−1) and proportional dilution rate (s−1) for individual cloud samples. Individual cloud samples are represented by blue dots, where 400 points have been randomly sampled for visualization.
Fig. 10.

The distribution of the total water content qt (kg kg−1) within individual cloud samples ϕc in relation to the average total water content of the large-scale atmosphere ϕ¯ (green dots), entrained air ϕe (blue dots), and detrained air ϕd (orange dots). Each dot represents the property of an individual cloud region Ci for an ith cloud. For the sake of visualization, we randomly sampled 400 points from the distributions of qt, ϕc, and ϕd each. The black line represents a set of points where ϕc = ϕ{e,d}.
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