Responses of Shallow Cumulus Convection to Large-scale ...

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Responses of Shallow Cumulus Convection to Large-scale

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Temperature and Moisture Perturbations: a comparison of

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large-eddy simulations and a convective parameterization based

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on stochastically entraining parcels

Ji Nie

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∗

Department of Earth and Planetary Sciences, Harvard University, Cambridge, Massachusetts

Zhiming Kuang

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Department of Earth and Planetary Sciences, and School of Engineering and Applied Sciences Harvard University, Cambridge, Massachusetts

∗

Corresponding author address: Department of Earth and Planetary Sciences, Harvard University, Cam-

bridge, Massachusetts E-mail: [email protected]

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ABSTRACT

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Responses of shallow cumuli to large-scale temperature/moisture perturbations are examined

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through diagnostics of large-eddy-simulations (LES) of the undisturbed Barbados Oceano-

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graphic and Meteorological Experiment (BOMEX) case and a stochastic parcel model. The

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parcel model reproduces most of the changes in the LES-simulated cloudy updraft statistics

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in response to the perturbations. Analyses of parcel histories show that a positive temper-

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ature perturbation forms a buoyancy barrier, which preferentially eliminates parcels that

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start with lower equivalent potential temperature or have experienced heavy entrainment.

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Besides the amount of entrainment, the height at which parcels entrain is also important in

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determining their fate. Parcels entraining at higher altitudes are more likely to overcome the

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buoyancy barrier than those entraining at lower altitudes. Stochastic entrainment is key for

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the parcel model to reproduce the LES results. Responses to environmental moisture pertur-

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bations are quite small compared to those to temperature perturbations, because changing

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environmental moisture is ineffective in modifying buoyancy in the BOMEX shallow cumulus

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case.

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The second part of the paper further explores the feasibility of a stochastic-parcel-based

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cumulus parameterization. Air parcels are released from the surface layer and tempera-

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ture/moisture fluxes effected by the parcels are used to calculate heating/moistening ten-

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dencies due to both cumulus convection and boundary layer turbulence. Initial results show

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that this conceptually simple parameterization produces realistic convective tendencies and

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also reproduces the LES-simulated mean and variance of cloudy updraft properties, as well

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as the response of convection to temperature/moisture perturbations.

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1. Introduction

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Shallow cumulus convection plays important roles in the large-scale atmospheric circula-

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tion. By enhancing vertical transport of heat and moisture, shallow cumuli regulate the sur-

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face fluxes and maintain the thermodynamic structures over the vast subtropical trade wind

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region, a region that also provides the inflow for the deep convective intertropical conver-

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gence zone. It has long been recognized that representations of shallow cumulus convection

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in large-scale models significantly impact the resulting circulation (e.g. Tiedtke 1989). More

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recently, it is further suggested that, because of its abundance, the shallow cumuli are a lead-

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ing factor in determining the cloud-climate feedback (Bony et al. 2004; Bony and Dufresne

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2005). It is therefore important to understand the dynamics of shallow cumulus convection

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and to better parameterize it in global climate models or general circulation models (GCM).

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Our strategy to better understand shallow cumuli is to look at how they respond to

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small perturbations to their large-scale environment. This approach has been applied previ-

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ously to deep convection (Mapes 2004; Kuang 2010; Tulich and Mapes 2010; Raymond and

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Herman 2011). Kuang (2010) was able to determine the responses of convection (using a

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cloud-system-resolving model) to a sufficiently complete set of perturbations in their large-

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scale environment, and use these responses to approximate the behavior of convection near

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a reference state. A range of interesting behaviors were found, such as stronger sensitivity

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of convection to temperature perturbations in the lower troposphere than those in the up-

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per troposphere (Kuang 2010; Tulich and Mapes 2010; Raymond and Herman 2011). The

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physical processes behind the responses however are not yet fully understood (Kuang 2010;

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Tulich and Mapes 2010). In this study, we shall use shallow non-precipitating convection,

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the dynamics of which are simpler without the many complicating processes associated with

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precipitation, as a starting point to understand and model the physical processes behind the

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responses of cumulus convection to large-scale temperature and moisture perturbations.

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Despite its relative simplicity, shallow cumulus convection involves interactions among a

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number of subcomponents, such as the subcloud layer, whose thermodynamic properties and 2

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turbulence statistics, together with the strength of convective inhibition, set the properties

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and the amount of cloudy updrafts at the cloud base; the interactions between clouds and

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their environment and within the clouds themselves, which determine the evolution of cloudy

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updrafts as they rise from the cloud base; and the fate of these cloudy updrafts as they

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penetrate into the inversion layer. These interactions are reflected in the construct of many

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contemporary parameterizations (e.g. Bretherton et al. 2004; Neggers and Siebesma 2002).

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In this paper, we will focus on perturbations in the cloud layer (above the cloud base but

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below the inversion), thus emphasizing the evolution of cloudy updrafts as they rise from

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the cloud base. Being able to better study individual processes in isolation is an advantage

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of looking at responses to small perturbations, as the other subcomponents can be regarded

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as mostly unchanged.

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Besides changes in domain heating and moistening rates in response to temperature and

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moisture perturbations, we place special emphases on changes in the statistical distributions

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of cloudy updrafts. While many of the current shallow schemes are bulk schemes (e.g.

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Kain and Fritsch 1990; Bretherton et al. 2004), it is important to capture the statistical

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distribution of the cloudy updrafts in order to better simulate microphysics and chemistry

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beyond the goal of simulating heating and moistening tendencies. This is similar to efforts of

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probability distribution function (PDF) based parameterizations (e.g. Lappen and Randall

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2001; Larson et al. 2002; Golaz et al. 2002).

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This paper has two parts. In the first half, we aim to understand changes in the statistical

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distributions of cloudy updrafts in response to the temperature/moisture perturbations.

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We will use extensive diagnostics of large-eddy-simulations (LES) with the aid of a tracer-

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encoding technique. We will then use a stochastic parcel model (SPM) to help interpret

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the results of the LES. Section 2 provides brief introductions to the models and the tracer-

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encoding technique, as well as an overview of the experiments. As we will show in section

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3, the SPM reproduces many of the features of the LES-simulated shallow convection and

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its responses to temperature and moisture perturbations. We then investigate the evolution

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history of the parcels in the SPM to identify the physical processes behind the responses

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(section 3).

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In the second half of the paper, we develop the SPM further into a parameterization of

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shallow cumulus convection, which differs from the previous parameterizations based on bulk

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entraining/detraining plume models (e.g. Yanai et al. 1973; Kain and Fritsch 1990; Bechtold

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et al. 2001; Bretherton et al. 2004) or ensemble plume/parcel models (e.g. Arakawa and

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Schubert 1974; Neggers and Siebesma 2002; Cheinet 2003), but is closer in spirit to episodic

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mixing models of Raymond and Blyth (1986) and Emanuel (1991). We will introduce the

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parameterization framework and present simulation results from the parameterization and

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compare them with the LES results, including the responses to temperature and moisture

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perturbations (section 4). The main conclusions are then summarized and discussed in the

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last section.

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2. Experimental overview

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a. Models

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The LES model used here is Das Atmosph¨ arische Modell (DAM; Romps 2008). It is a

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three-dimensional, finite-volume, fully compressible, nonhydrostatic, cloud-resolving model.

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We use a tracer encoding technique to infer the history of cloudy updrafts (Romps and

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Kuang 2010b,a). Two artificial tracers, purity and an equivalent potential temperature

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tracer (θe,tracer ) are released at a specified height (tracer release height, Zrelease ). At and

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below Zrelease , the purity tracer is set to 1 and θe,tracer of a grid point is set to its equivalent

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potential temperature θe . The two tracers are advected in the same way by the model

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velocity field, and both are set to zero for grid points not in the vicinity of cloudy updrafts

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(cloudy updrafts are defined here as gridpoints with liquid water content ql greater than 10−5

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kg kg−1 and vertical velocity w greater than 0.5 m s−1 ). A grid point is considered to be in

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the vicinity of cloudy updrafts if it is within a 3 grid point distance from any cloudy updrafts. 4

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By reading the value of these tracers at some level above Zrelease , one can infer some aspects

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of the updraft’s history: the purity tracer estimates the total amount of environmental

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air entrained, while the ratio of θe,tracer and purity gives the updraft’s equivalent potential

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temperature θe value at Zrelease . We shall refer to this ratio as “θe,ed ”, which stands for

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encoded equivalent potential temperature. The readers are referred to the above references

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for some examples of the usages of the tracer encoding technique (e.g. section 2 of Romps

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and Kuang (2010b)).

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We shall also use the stochastic parcel model developed in Romps and Kuang (2010b,a)

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to help understand the results from the LES. This model integrates the prognostic equa-

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tions of parcel properties (height z(τ ), volume V(τ ), temperature T(τ ), water vapor content

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qv (τ ), liquid-water content ql (τ ), and vertical velocity w(τ )) forward in time (τ ), given the

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environmental soundings (temperature Ten (z) and moisture qv,en (z), where the subscript

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denotes an environmental variable) and the initial conditions of the parcels. In this paper,

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we set the drag coefficient to zero so that updraft velocity is affected only by buoyancy and

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entrainment. This is done here solely for simplicity, although there have been arguments

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made for clouds being slippery thermals that experience little drag (Sherwood et al. 2010).

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The parcel model uses a stochastic entrainment scheme. More specifically, for a moving

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parcel, the probability that it entrains after a time step ∆τ is P = ∆τ |w|/λ,

en

(1)

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where λ is the e-folding entrainment distance. If entrainment occurs, the ratio of the en-

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trained environmental air mass to the parcel’s mass follows the probability function −σ ln(r),

(2)

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where σ is the mean ratio of the entrained mass and r is a random number between 0 and

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1. Because entrainment is stochastic, a large number of parcels are required to cover the full

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range of entrainment scenarios in order to obtain statistically significant results.

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b. Experimental design

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The shallow cumulus system that we study here is a well-studied oceanic trade cumulus

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case from the Barbados Oceanographic and Meteorological Experiment (BOMEX). The LES

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runs have a horizontal and vertical resolution of 25m, with a horizontal domain size of

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12.8km × 12.8km and a vertical extent of 3km. It is initialized using the initial profiles

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of temperature, water vapor and horizontal velocities described in Siebesma and Cuijpers

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(1995), with some small random noises. The radiative cooling and other large-scale forcing

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are also prescribed as in Siebesma and Cuijpers (1995). A bulk parameterization with a drag

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coefficient of 1.4 × 10−3 is used to calculate the surface fluxes off the 300.4K ocean surface.

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The first 3 hours of the model run are discarded as spin-up. Over the following 2 hours,

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snapshots are saved every 10 minutes. We then add a temperature or moisture perturbation

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to each of these snapshots and use them as initial conditions for the perturbed runs. In this

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way, we produce an ensemble of control and perturbed runs with different initial conditions.

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We then diagnose the differences between the ensemble averages of the control runs and the

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perturbed runs.

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Three representative perturbations are studied: a 0.5K temperature anomaly centered at

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987.5m (the T987.5 case); a 0.5K temperature anomaly centered at 1262.5m (the T1262.5

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case); and a 0.2 g kg−1 moisture anomaly centered at 987.5m (the Q987.5 case). The

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anomalies added are horizontally uniform and Gaussian-shaped in height with a half width

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of 75m. The tails of the Gaussian-shaped anomaly are truncated at levels more than 200m

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away from the center. As an example, Fig. 1a shows the positive temperature anomaly for

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the T987.5 case. The anomalies are sizable but the responses are still in the linear regime

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for the most part. Our choice of the size of the perturbations was constrained by the desire

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to obtain statistically robust results (given the computational constraints) and to limit the

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extent of nonlinearity. For each case, we have done both runs with positive perturbations and

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runs with negative perturbations. Although there is some nonlinearity (described below), the

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responses to positive and negative perturbations are mostly similar but with opposite signs. 6

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The results shown in next section are the averages of 6 members with positive perturbations.

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There is one complication with calculating convective responses to perturbations in DAM:

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as the temperature and moisture perturbations disrupt the hydrostatic balance, the model

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adjusts to regain hydrostatic balance. The main effect of this adjustment is an instantaneous

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adiabatic cooling of the layer after the positive temperature and moisture anomalies are

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added. For temperature perturbations, this cooling is negligible. However, as we will show in

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section 3d, convective heating responses to moisture perturbations are remarkably small and

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the temperature change associated with this hydrostatic adjustment becomes noticeable. We

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ran DAM in a single column setting to calculate the cooling from this hydrostatic adjustment

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and shall correct for this effect in the results that we present.

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We bin the cloudy updrafts based on their purity and θe,ed to produce two dimensional

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(2D) distributions of their properties as functions of those two variables. Taking the T987.5

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case for example, in the LES, the purity and the θe tracers are released at Zrelease = 762.5m

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(For the T1262.5 case, Zrelease = 1037.5m, and for the Q987.5 case, Zrelease = 762.5m). We

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then sample cloudy updrafts at Zsample = 1212.5m (For the T1262.5 case, Zsample = 1487.5m,

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and for the Q987.5 case, Zsample = 1212.5m) for the 1 hour period following the introduction

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of the anomaly. We then calculate the total mass flux, the mean w and θe of all cloudy

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updrafts inside each bin defined by the purity and θe,ed values. These 2D distributions (Fig.

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2a-c) provide information on the history of the updrafts traveling through the layer between

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Zrelease and Zsample . The sampling period of the first hour is long enough to give statistically

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significant results but not too long for the added anomaly to have evolved heavily through

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convective adjustment. During this period, the properties of the cloudy updrafts at Zrelease

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are roughly the same for the control runs and the perturbed runs. Therefore, differences in

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the distributions at Zsample (Fig. 3a-c) give responses of cloudy updrafts to the introduced

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temperature anomaly.

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We compute the same diagnostics for the SPM. The parcels in the SPM are viewed as

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analogs of the cloudy updraft grid points in the LES. Millions (1 millions in this study) of

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parcels are released at Zrelease in the SPM. Two conditions need to be specified for the SPM:

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the initial conditions at Zrelease and environmental profiles between Zrelease and Zsample . To

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assign them appropriate initial conditions, we sample 100 LES snapshots to get a 3D PDF

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of the cloudy updrafts at Zrelease as functions of w, T , and qt . The PDFs at Zrelease from the

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LES are found to be similar between the control and the perturbed runs. In the SPM, we

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shall use the PDFs from the LES control runs for both the control and the perturbation runs.

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The initial conditions of the parcels are drawn randomly from this 3D PDF. This procedure

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ensures that the parcels in the SPM have the same statistical properties as updrafts in the

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LES at Zrelease . For the control run of the SPM, the environmental sounding is the ensemble

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mean sounding of the LES control runs. For the perturbed runs of the SPM, we add the

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same perturbations as in the LES perturbation runs. We also follow the same sampling

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processes at Zsample . In the SPM, the purity of a parcel is the ratio of its mass at Zrelease

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to its mass at zsample , and parcels’ initial θe at Zrelease takes the place of θe,ed in the LES. In

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the figure axes, we call it “initial θe ” to remind readers that it is SPM-generated results.

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Because the pair of parameters (λ, σ) mainly control the entrainment process, we briefly

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discuss how they are chosen. Romps and Kuang (2010b) surveyed a large range of com-

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binations of λ and σ. They defined an objective function based on mass flux agreement

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between the LES and the SPM and searched the best fitting λ and σ, which are λ = 226 m

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and σ = 0.91, that minimizes the objective function. They also note that there is a valley

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in the [λ, σ] space, where pairs of [λ, σ] give very similar values of the objective function.

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For pairs of parameters that are along this valley, although the total amount of mass flux is

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similar, the distribution of mass flux in terms of purity is quite different. For example, the

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parameters used in Romps and Kuang (2010b) allow too many undiluted updrafts compared

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to the distributions from our current 25 meter resolution run (Fig. 2a), even though their

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parameters are consistent with the total amount of mass flux and purity. We have chosen λ

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= 125 m and σ = 0.32 to give a mass flux distribution as a function of purity (Fig. 2d) that

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matches the distribution from the current LES simulations (Fig. 2a).

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3. Response to Temperature and Moisture Perturbations a. Responses in the T987.5 case

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For a temperature perturbation centered at 987.5 meters, the amplitude of the initially

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added temperature anomaly decreases by a factor of 2 in about 30 minutes (Fig. 1b), a

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result of the convective adjustment process. In addition, there is slight warming in the trade

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inversion around 1600m. Moisture responses (Fig. 1c) show that layers below 987.5m expe-

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rience moistening and layers above experience drying. The moistening propagates downward

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towards the surface with time. The basic features of the responses can be understood in

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terms of inhibition of cloudy updrafts by the added temperature anomaly. The inhibition

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causes the region of the initial temperature perturbation to cool. The reduced penetrative

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entrainment in the inversion layer also leads to the warming near 1600m. Furthermore, the

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enhanced detrainment in and below the region of the temperature perturbation leads to the

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moistening, while the reduced detrainment above leads to the drying. The temperature per-

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turbations that we add (with a peak amplitude of 0.5K and a half width of 75m) represent

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a change in stratification that is comparable to the background stratification at this height.

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This weakens the vertical stratification substantially over the upper half of the perturbation,

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leading to a local overturning circulation that gives rise to the dipole response around 1100m

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in the moisture field (Fig. 1c) over the first half hour. For negative perturbation cases, the

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dipole response in the moisture field is found over the lower half of the introduced pertur-

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bation. It is certainly desirable to remove these dipole responses with perturbations that

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perturb the stratification less strongly. Given our desire to have the perturbations relatively

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localized in height so that we perturb one shallow layer at a time, weaker perturbations

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in stratification require the use of smaller amplitude perturbations. We have performed an

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experiment with the perturbation amplitude halved to 0.25K. The dipole structure in the

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moisture field centered at 1100m is no longer present. We have also confirmed that the 9

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results that we present below for the 0.5K perturbation runs hold in the 0.25K perturbation

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run as well except that it is considerably noisier with the smaller perturbation. One could

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certainly use a large number of ensemble members to improve the signal to noise ratio for

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the 0.25K case. However, because of the substantial computational cost involved, we have

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opted to present results from the 0.5K runs.

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We now investigate changes in the statistics of cloudy updrafts. The mass flux distri-

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bution of the LES control run (Fig. 2a) is mostly located between 347.5K and 349.5K

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on the θe,ed axis, reflecting variations of updrafts’ properties at Zrelease . The mass flux is

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mostly located between 0.2 and 1.0 on the purity axis, while the maximum lies around 0.45.

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This indicates that most updrafts mix with environmental air when they go across the layer

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between Zrelease and Zsample . However, there are some undiluted updrafts with purity close

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to 1 (Here “undiluted” is relative to Zrelease not to the cloud base). The SPM-generated

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mass flux distributions (Fig. 2d) show general similarities. We find that the LES mass flux

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distribution has a narrower range in θe,ed than that of the SPM. The narrowing is due to

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in-cloud mixing in the LES, which homogenizes the initial identities of the air that makes

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up the updrafts. In our SPM, in-cloud mixing is not included at the moment for simplicity.

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In-cloud mixing is a process that we would like to add into the SPM in the future, perhaps

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following the approach of Krueger et al. (1997).

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The distributions of the LES simulated w and θe are plotted in Fig. 2b.c. Both θe and

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w show a tilted structure with an increasing gradient in the upper-right direction. This

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is simply because parcels that initially have higher θe,ed or entrain less environmental air

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(thus have higher purity) will end up with higher θe and also achieve higher w values due

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to stronger buoyancy acceleration and less slowdown by entrainment. We have also plotted

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the same distributions for total water content qt and buoyancy b. They show similar tilted

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patterns as θe (figures are not shown).

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The SPM-generated w and θe distributions (Fig. 2e-f) show a similar gradient. Closer

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inspection shows that the agreement in w is not as good as that in θe . More specifically,

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the constant w contours of the LES distribution are almost vertical in regions of high purity

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(Fig. 2b), indicating little dependence of w on the encoded θe . For high encoded-θe values,

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the dependence of w on purity is also somewhat weakened (constant w contours being more

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horizontal). We speculate that the discrepancy between the LES and the SPM is because

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of the intra-cloud interaction between updrafts. In our LES simulations, the resolution is

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relatively high and clouds are well resolved. The cloudy parcels can exchange momentum

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through pressure gradient force in addition to actual mixing of fluids. This could cause

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momentum exchange between the most active cloud cores (upper-right corner in Fig. 2b)

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and their surrounding cloudy parcels (upper-left and lower-right corner), while keeping their

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thermodynamic properties such as θe unchanged. The same figures for runs with the same

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settings but with a resolution of 100m × 100m × 50m are plotted in Fig. 2g-i. The w

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distribution of the low resolution runs is more similar to the SPM results and to that of

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θe . We speculate that in the lower resolution runs, clouds are less well resolved so that the

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disparity in the intra-cloud homogenization of momentum and thermodynamic properties

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is reduced. To test this idea, we calculate the correlation between w and θe for the cloudy

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updrafts. Using 60 snapshots from high resolution runs and also 60 snapshots from the

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low resolution runs, we found that the overall correlation between w and θe of the cloudy

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updrafts is 0.57 for the high resolution runs, while for the low resolution runs, it is 0.75. This

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is inline with the results that w and θe have more similar patterns in low resolution runs than

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in high resolution runs. We have further separated the w and θe variations into intra-cloud

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and inter-cloud components. The correlation for inter-cloud variations of w and θe are 0.62

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for both the high and low resolution runs, while the correlation for intra-cloud variations is

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0.54 for the high resolution runs, significantly smaller than that of the low resolution runs

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(0.78). While these results are inline with our argument, more detailed studies are clearly

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needed to fully understand the differences seen between the high and low resolution runs.

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Note also that in the low resolution runs, the peak mass flux has a higher purity than that

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of the higher resolution runs (0.6 versus 0.45), indicating reduced entrainment because of

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the lower resolution.

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Changes in above statistics in the perturbed run as compared to the control run are

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plotted in Fig. 3. Fig. 3a shows the fractional change in cloudy updraft mass flux in response

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to the perturbation. It is clear that updrafts with low initial θe and those that entrain heavily

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(thus have low purity) are preferentially removed by the temperature perturbation. The mass

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flux of updrafts with high initial θe and those less diluted by entrainment are less affected.

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The contours of constant fractional change in mass flux are almost along the contours of θe

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(Fig. 2c), indicating the controlling influence of buoyancy (for saturated air, θe is a good

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proxy for buoyancy): the temperature anomaly forms a buoyancy barrier that preferentially

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inhibits updrafts with low buoyancy.

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Responses in w (Fig. 3b) are negative over most regions, mainly because the temperature

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perturbation decreases the convective available potential energy (CAPE) of the updrafts.

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The SPM-generated w responses also show the dominant effects of the CAPE decrease. The

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smaller decrease in w for updrafts with lower purity is because parcels that entrain heavily

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gain additional buoyancy from the environmental temperature anomaly. A secondary factor

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is the reference state w. As changes in CAPE affect w2 , a higher reference state w implies a

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smaller change in w for the same change in CAPE (Fig. 2e). In regions with strong fractional

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decrease in mass flux (i.e. the lower-left corner), changes in the height of entrainment events

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also contribute to the smaller w decrease, as will be discussed in section 3b. The less negative

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patch in the upper-right part of the LES-simulated w responses is not captured by the SPM

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and is not understood. We speculate that the buoyancy barrier associated with the positive

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temperature perturbation may inhibit intra-cloud momentum transport and lead to this less

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negative patch. More studies into this behavior are clearly needed. Intriguingly, this less

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negative patch is not seen in the lower resolution LES simulations (not shown).

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Cloudy updrafts with high θe,ed or high purity (the upper-right part of the PDF), in

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which there is little mass flux change, show a slight increase in their θe due to entrainment of

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warmer environmental air (Fig. 3c). On the other hand, updrafts with lower θe,ed and lower

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purity (the lower-left part of the PDF), in which there is a significant fractional decrease in

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mass flux, show a decrease in their θe (Fig. 5c,f). The reason for this will be discussed in

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section 3b.

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The responses of SPM-generated statistics (Fig. 3d-f) are generally similar to those of

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the LES. The LES responses are generally smaller than the SPM responses because the

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introduced temperature anomaly in the LES decays significantly over the 1-hour sampling

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period (Fig. 1b). In the SPM, we have kept the temperature anomaly constant. We have

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also performed SPM experiments where the evolving soundings of the LES perturbation runs

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were used. It reduced the SPM responses and brought their amplitudes to agreement with

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those of the LES.

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The stochastic entrainment process is key for the SPM to match the LES results. To

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highlight this point, we designed another experiment where we run the SPM with constant

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entrainment of  = 1.8 × 10−3 m−1 instead of the stochastic entrainment. We choose this

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constant entrainment rate to give a similar amount of overall entrainment as in the stochastic

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entrainment case. Other settings, such as initial conditions of the parcels and the environ-

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ment soundings are unchanged. Because for the constant entrainment case, the purity of

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parcels sampled at Zsample is the same, we only plot mass flux distribution as a function of

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θe,ed . The PDFs of the SPM with constant entrainment, the SPM with stochastic entrain-

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ment, and the LES are shown in Fig. 4 (summing Fig. 2a/d along the purity axis gives the

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black line in Fig. 4c/b. ). With the constant entrainment, the fate of a parcel is determined

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by its initial conditions. With the temperature perturbation, there is a threshold of around

340

348.5K in terms of the initial θe . Mass fluxes of updrafts with initial θe below the threshold

341

are totally cut off, while mass fluxes with initial θe above the threshold are not affected at

342

all. The threshold has a finite width because the initial conditions of the updrafts have vari-

343

ations in w, which are not reflected on the initial θe axis. On the other hand, the decrease

344

of LES mass flux is over almost all ranges of θe,ed , although the mass flux with lower θe,ed

345

decreases more. The LES results are much more similar to the SPM results with stochastic

13

346

entrainment (Fig. 4b). This analysis indicates that stochastic entrainment is essential in the

347

parcel model and more realistic than constant entrainment.

348

b. The height of entrainment

349

We now analyze the parcels’ evolution history in the SPM to understand the decreases

350

of θe and their collocation with decreases of mass flux in the lower-left half of Fig.3 c,f. It

351

turns out that decreases of mass flux and composite θe are due to one single mechanism.

352

We plot the trajectories of randomly selected parcels reaching Zsample with a purity of

353

0.6 in the phase space of Z and purity (Fig. 5). In other words, we are looking at one

354

thin band of the distribution around a purity of 0.6 in Fig. 3d. The purities of parcels

355

at Zrelease = 762.5m are 1 by design. A sudden decline of purity indicates an entrainment

356

event. In the control runs, entrainment events are roughly uniformly distributed in height.

357

However, in the perturbed runs, most parcels reaching Zsample experience entrainment at

358

relatively high altitudes. Since the entrainment probability functions are the same, it implies

359

that in the perturbed case, the parcels that entrain heavily at lower heights cannot penetrate

360

the perturbation layer to be sampled.

361

The above analysis indicates that the height at which an air parcel entrains is important

362

in determining its fate. To further illustrate this point, we perform the following experiment.

363

We release a parcel at Zrelease = 762.5m with initial conditions of T = 293.67K, qt = 16.82

364

g kg−1 and w =1.32 m s−1 , which are the mean properties of the cloudy updrafts at Zrelease

365

in the LES. While traveling toward Zsample , this parcel entrains only once. The single en-

366

training event will dilute the parcel to a purity of 0.6. We release the same parcel a number

367

of times and each time have the parcel entrain at a different height between Zrelease and

368

Zsample , separated by an interval of 30m. This experiment eliminates variations in the initial

369

conditions. It also simplifies the stochastic entrainment treatment by emphasizing only one

370

aspect: for a parcel that entrains the same amount of environmental air, it may entrain at

371

different heights. We plot the trajectories of these parcels in the phase space of Z and w 14

372

(Fig. 6). The uppermost line is the trajectory of an undiluted parcel. A sudden decline in

373

w indicates an entrainment event because the parcel entrains environmental air with zero w.

374

For the control run, if the parcel entrains below about 850m, it becomes negatively buoyant

375

and also descends. If it entrains above 850m, it is temporarily negatively buoyant immedi-

376

ately after entrainment. However, after continued ascent by inertia for a certain distance,

377

it becomes positively buoyant and accelerates upward. The critical height separating rising

378

and descending trajectories is 850m for the control run. With the temperature perturbation,

379

this critical height is higher than that of the control run, reaching about 1037.5m (Fig. 6b).

380

Thus, in the perturbed runs, parcels that entrain at lower altitudes are filtered out. The

381

remaining parcels preferentially entrain at higher altitudes where the environmental θe is

382

lower. As a result, the updrafts’ θe sampled at Zsample decreases. This is why in Fig. 4c the

383

composite θe decreases in regions with significant decreases in mass flux. This preference

384

for parcels that entrain at higher altitudes also contribute to the smaller w decrease in the

385

lower-left part of Fig3 b,e. This is because parcels that entrain at higher altitudes enjoy

386

undiluted buoyancy acceleration over a longer distance, thus attain higher kinetic energy.

387

While parcels that entrain at higher altitudes also lose more kinetic energy because they

388

have greater w at the time of entrainment, this effect is weaker compared to the buoyancy

389

effect. Thus the preferential elimination processes described above also have the effect of in-

390

creasing the composite updrafts’ w. This effect is most significant where there is substantial

391

fractional mass flux decrease. Our analysis shows that in the region with fractional mass

392

flux decreases over 60%, this elimination process has an effect on w that is nearly equal to

393

the effect of additional buoyancy gained by entraining warmer environmental air.

394

The above analysis is only for parcels with one set of initial conditions and purity. One

395

can do similar calculations for parcels with different initial conditions and purities. If the

396

temperature perturbation can effectively lift the critical height, then the mass flux and θe

397

of the composite updrafts reaching Zsample will decrease, while the decrease in w of the

398

composite updrafts reaching Zsample will be smaller.

15

399

c.

Responses in the T1262.5 case

400

We have performed the same analyses as in the previous subsection for the T1262.5 case.

401

While for the T987.5 case, we ran the simulation for 3 hours to provide a sense of longer time

402

evolution of the anomalies, because of limited computational resource, we ran the T1262.5

403

case and the Q987.5 case described in the next subsection only for 1 hour, which is the period

404

over which we sample the statistics of the cloudy updrafts.

405

The evolution of the large-scale environment is shown in Fig. 7. Its responses are

406

generally similar to the T987.5 case: strong local cooling, moistening below and drying

407

above the perturbation. The dipole response in the moisture field in the upper half of the

408

temperature perturbation seen in the T987.5 case is not found here (Fig. 7b) because the

409

background stratification is stronger at 1262.5m.

410

The responses of the distributions of cloudy updraft properties and the comparison be-

411

tween the LES and the SPM (Fig. 8) also share many similarities with the T987.5 case, which

412

indicates that the mechanisms discussed in the T987.5 case also operate in the T1262.5 case.

413

The main difference between these two cases is that the decay of the initially imposed

414

anomaly is slower in the T1262.5 case than in the T987.5 case. It shows that responses

415

of cumulus convection to temperature perturbations at higher altitudes are weaker, similar

416

to what was found in deep convection (Kuang 2010; Tulich and Mapes 2010; Raymond

417

and Herman 2011). It is presumed here that this is because the background (control run)

418

liquid potential temperature θl flux convergence at 1262.5m is smaller than at 987.5m so

419

that changes in the θl flux convergence caused by the temperature perturbation at 1262.5m

420

are also smaller than at 987.5m. On the other hand, the background qt flux convergence

421

at 1262.5m is of a similar magnitude as that at 987.5m, so the moisture responses in the

422

T1262.5 case (Fig. 7b) is comparable to that in the T987.5 case (Fig. 1c).

16

423

d.

Responses in the Q987.5 case

424

The responses to moisture perturbations are quite different from those to temperature

425

perturbations. The added moisture anomaly is also damped as expected (Fig. 9b). However,

426

the temperature responses are remarkably small, of the order of 10−3 K. Note that we have

427

corrected for the effect of the hydrostatic adjustment (which produces a negative temperature

428

anomaly with a peak amplitude of 9.5 × 10−3 K and the same shape as the added moisture

429

anomaly) as discussed in section 2b. The temperature in the perturbed layer increases with

430

time, while there is cooling centered around 1500m in the inversion layer. These changes

431

are consistent with an enhancement of cloudy updrafts by the added moisture anomaly,

432

which warms the perturbed layer and causes the cooling near 1500m through penetrative

433

entrainment in the inversion layer.

434

Statistics of cloudy updrafts again show general agreement between the LES and the

435

SPM. Both LES and SPM show that mass flux in the low purity and low θe,ed region increases

436

while mass flux in the other regions is mostly unchanged. It indicates that a more moist

437

environment benefits less buoyant updrafts by increasing their θe . The w distribution of

438

the SPM shows an increase over a tilted band with purity around 0.5. Over the area with

439

purity close to 1, w is decreased because the moisture perturbation increases environmental

440

buoyancy slightly, which decreased CAPE for undiluted parcels. The w signal of the LES

441

is relatively noisy. It also shows increases in the region with medium purity values, similar

442

to that of the SPM. There is also a hint of w increase near purity of 0.8, which is not

443

captured by the SPM. The w increase in the region with medium purity values is because of

444

the entrainment of higher θe air, which boosts the parcel/updraft buoyancy and increases its

445

vertical velocity. This effect is significant only for parcels/updrafts that experience significant

446

entrainment. For parcels that entrain too heavily (i.e. those with the lowest purity values),

447

however, the same effect of entrainment height selection discussed in section 3b comes into

448

play: with a positive moisture perturbation to the environment, parcels/updrafts that entrain

449

at lower altitudes and could not reach the sampling height can now reach it. These parcels 17

450

have had undiluted buoyancy acceleration over shorter distances and thus lower vertical

451

velocities, weighing down the average w of parcels with low purities. For both LES and

452

SPM, θe shows general increases over all the regions because the moisture anomaly increases

453

environmental θe . For updrafts with lower purity, the increase in θe is larger because these

454

are parcels that entrain more.

455

The reason that responses in mass flux and heating are very small for the moisture

456

perturbation is that moisture anomalies are inefficient in changing either the environmental

457

air’s or the updrafts’ buoyancy. Note that a 0.2g/kg specific humidity change in similar

458

to a 0.5K temperature in terms of the change to the equivalent potential temperature.

459

To illustrate this point, we plot the mixing diagrams (as in e.g. Bretherton et al. 2004)

460

of a typical parcel with environmental soundings of the control, T987.5 and Q987.5 cases

461

at the 987.5m height (Fig. 11). The parcel has T = 293.67K, qt = 16.82 g kg−1 at

462

Zrelease =762.5m, and is taken to rise undiluted to 987.5m then mix with the environment.

463

With the temperature anomaly, the environmental density decreases significantly so that

464

almost all mixtures are negatively buoyant. However, differences between the mixing lines

465

of the control and Q987.5 cases are very small. The environmental density is decreased

466

slightly with the moisture anomaly, shown as the slight descent of non-mixed points with

467

χ = 0 (χ is the fraction of environmental air in the mixture). The χ value corresponding

468

to neutral buoyancy only shifts rightward slightly with the moisture anomaly. Comparing

469

with temperature anomalies, moisture anomalies are not efficient in changing updrafts’ fate

470

in the BOMEX case.

18

471

472

473

4. A Shallow Convective Parameterization Based On Stochastically Entraining Parcels a. Constructing the parameterization scheme

474

The basic function of a convective parameterization is to use large-scale variables to es-

475

timate tendencies of mass, heat, moisture, momentum and other tracers due to convective

476

motions. A convective parameterization typically contains the following two key compo-

477

nents: The first is the determination of cloud base conditions. The second is a cloud model

478

that describes the evolution of clouds as they rise above the cloud base, a key being their

479

interactions with the environmental air.

480

A variety of approaches have been used for the cloud model, including, for example, a

481

bulk constant entrainment plume or an ensemble of plumes with fixed entrainment rates

482

(e.g. Simpson 1971; Arakawa and Schubert 1974; Tiedtke 1989; Bechtold et al. 2001). Some

483

parameterizations include some coupling between entrainment and updrafts, an example

484

being the buoyancy sorting approach (Raymond and Blyth 1986; Kain and Fritsch 1990;

485

Bretherton et al. 2004). The stochastic entrainment method used in this study provides

486

another approach to represent the mixing processes. This method explicitly simulates the

487

stochastic nature of mixing, specified through the two probability functions (equation (1)

488

and (2)) described earlier in section 2.

489

The determination of the cloud base conditions is what we need to extend the SPM into

490

a parameterization. We shall follow a treatment similar to Cheinet (2003) and represent

491

convective transport in the subcloud layer and that in the cloud layer within a single frame-

492

work. By doing so, we are assuming that fluxes in the subcloud layer are dominated by

493

surface generated eddies and the subcloud layer turbulent fluxes can also be parameterized

494

by the stochastic parcel model.

495

We shall release parcels directly from the near surface layer (the lowest model level Zs ).

496

The statistical distributions of these parcels’ initial conditions (m, T, qv , w at Zs ), where m 19

497

is the mass of the parcel, will be determined by the surface fluxes (sensible heat flux FH

498

and latent heat flux FL ) and surface layer statistics. The treatment follows closely that of

499

Cheinet (2003) and is described below for completeness.

500

Variations in T , qv , and w near the surface are approximated by Gaussian distributions.

501

In addition, both measurements and LES diagnostics show that near the surface, w, T and

502

qv are strongly correlated with each other. Let CXY be the correlation coefficient between

503

X and Y. We treat CwT , Cwq , CT q as given parameters. Assuming that w follows a Gaussian distribution s with a zero mean and a standard

504

505

deviation of σw : s= √

w2 1 exp(− 2 ). 2σw 2πσw

(3)

506

σw is specified using formula (A3, A4) in the appendix of Cheinet (2003). T can be described

507

as σT T = T¯ + CwT w + x, σw

(4)

508

where T¯ is the mean environmental temperature at Zs . σT is the standard deviation of

509

temperature. x represents a white noise and is independent of w with a standard deviation

510

of σx . From the definition of σT and CwT , we have 2 σT2 = CwT σT2 + σx2 ,

(5)

511

Z σw σT CwT =

FH w(T − T¯)sdw = . ρ¯

(6)

512

In the above equation, the density of each parcel is approximated by the environmental mean

513

density ρ¯ in calculating FH . The same approximation is applied later in equation (9). Given

514

surface sensible heat flux FH , σT and σx can be calculated from equations (5) and (6). The

515

formula of qv is similar to that of T, qv = q¯v +

σq Cwq w + Cxq x + y, σw

(7)

516

where q¯v is the mean environmental specific humidity. σq is the standard deviation of qv .

517

y is a white noise independent of both w and x. In addition, from the definition of σq and 20

518

Cwq , we have 2 2 2 σq2 = Cwq σq2 + Cxq σx + σy2 , Z FL . σw σq Cwq = w(qv − q¯v )sdw = ρ¯

519

520

(8) (9)

From equations (4) and (7), we have CqT

Cxq σx2 = CwT Cwq + . σT σq

(10)

521

σq , σy and Cxq can then be solved from equations (8-10). As a summary, given inputs FH ,

522

FL and parameters σw , CwT , Cwq , CT q , the statistic distributions of the initial conditions are

523

determined.

524

525

In the parameterization, w is discretized into N1 bins between 0 and ασw (we use a α of 3, and the truncated tail is sufficiently small) : wi =

iασw , i = 1, ..., N1 , N1

(11)

α (iα)2 exp(− ). 2N12 2πN1

(12)

526

si = √ 527

si can also be viewed as the fractional area occupied by the parcels belonging to the ith bin.

528

The mass of parcels in the ith bin that cross the lowest model level above the surface over

529

area A during ∆t is Mi = ρ × velocity × area × time = ρwi si A∆t.

(13)

530

We further divide the air mass of each bin into N2 parcels equally. So the total number of

531

parcels released is N1 × N2 and the mass of each parcel is mi =

ρsi wi A∆t . N2

(14)

532

The reason to divide each bin is because entrainment in this parameterization is a stochastic

533

process. It requires the number of parcels to be large enough to ensure statistical stability.

534

We will find later that the factor A∆t is cancelled in the calculation of convective tendencies.

21

535

In our method, because si decays exponentially as i2 increases, the total mass of parcels

536

with large w is much smaller than that of parcels with smaller w. By using the same N2

537

for all vertical velocity bins, the implied mass per parcel is much smaller for the high w

538

bins. One could carefully divide the ith bin into N2,i parcels, so that each parcel has a

539

mass that is close to the mass of air blobs in the real atmosphere (or the LES). However, in

540

that case, we will find the number of low w parcels to be much larger than the number of

541

high w parcels, and most of the computational resources will be spent on the low w parcels,

542

which are less important in the convective mass flux calculation, instead of the high w

543

parcels, which will be underrepresented given computational constraints. Furthermore, the

544

implied mass per parcel is inconsequential in our scheme because drag force is ignored in this

545

parameterization, so that the size of a parcel does not affect its evolution and the outputs

546

of the parameterization. In other words, we set all N2,i to the same value for computational

547

economy and do not suggest that there is certain relationship between the size of the parcel

548

and its vertical velocity.

549

With the initial statistical distributions in place, parcels are drawn randomly from this

550

distribution and released from the lowest model layer. These parcels will rise, mix with the

551

environment, oscillate around their neutral buoyancy level, and eventually come to rest. The

552

evolution history of the parcels are solved by integrating the prognostic equations of parcel

553

properties, with the entrainment processes specified as in section 2a and in Romps and Kuang

554

(2010b,a). Because the mixing processes in the subcloud layer are different from those in the

555

cloud layer, a different set of entrainment parameters [λ, σ] are used in the subcloud layers,

556

and tuned to give satisfactory results. Since we focus on shallow convection, precipitation

557

and ice processes are turned off.

558

The temperature and moisture fluxes affected by the parcels are used to estimate the

559

convective heating and moistening tendencies. Let the environmental soundings (θl,en , qt,en )

560

be specified on discrete levels Z. (We choose θl as the prognostic temperature variable.) The

561

fluxes are defined on half levels Zh . The transports (fluxes times area times time) carried by

22

562

these parcels across a certain level can easily be obtained by summing over all parcels that

563

cross this level:

mass transports =

X

mk ,

(15)

564

θl transports =

X

mk θl,k ,

(16)

qt transports =

X

mk qt,k ,

(17)

565

566

where k is the index of parcels that cross this level over a time interval ∆t. For parcels

567

crossing this level from above, their transports should be marked as negative.

568

When the parcels rise, the environmental air subsides to compensate for the mass trans-

569

port by these parcels and ensures that there is no net mass accumulation. The subsiding air

570

also transports heat and moisture: compensating mass transports = −

X

mk ,

(18)

571

compensating θl transports = −

X

mk θl,dn ,

(19)

compensating qt transports = −

X

mk qt,dn ,

(20)

572

573

where θl,dn and qt,dn are the mean θl and qt for the compensating subsidence. Here we ap-

574

proximate the properties of the compensating subsidence by its mean values, thus neglecting

575

transport due to variations within the compensating subsidence. This is a good approxima-

576

tion for the cloud layer (Siebesma and Cuijpers 1995), but is less accurate in the subcloud

577

layer. θl,dn is estimated as θl,dn

P θl,en − sk θl,k P = , 1 − sk

(21)

578

and the same for qt,dn . The fractional area sk occupied by a parcel at an interface averaged

579

over a unit time is: sk =

mk . ρk wk A∆t

(22)

580

P The total fractional area occupied by convective updrafts ( sk ) is significant in the subcloud

581

layer and negligible in the cloud layer. 23

582

The convective heating and moistening tendencies are the vertical convergence of the net

583

fluxes (sum of the parcel transports and the compensating transports, then normalized by

584

area A and time period ∆t) :

585

P P ∂θl,en ∂( (mk θl,k ) − mk θl,dn ) 1 ρ¯ =− ∂t ∂z A∆t

(23)

P P ∂qt,en ∂( (mk qt,k ) − mk qt,dn ) 1 ρ¯ =− ∂t ∂z A∆t

(24)

From Eq. (14), we see that area A and the time interval ∆t will be cancelled out in Eqs.

586

(23) and (24).

587

b. The BOMEX run

588

We test the parameterization by running it in the BOMEX setting. The BOMEX initial

589

soundings, large-scale forcing and surface fluxes for the parameterization run are the same

590

as Siebesma and Cuijpers (1995). The vertical levels are from 80m to 3000m with a spacing

591

of 160m. The environmental soundings are adjusted every ∆t = 60s. The time integration

592

scheme of equation (23-24) is forward Euler. The fluxes (16-17) are calculated by sampling

593

the properties of parcels at half levels. When calculating the compensating fluxes (19-20),

594

the minmod flux limiter scheme (Durran 1999) is used. The parameterization is integrated

595

for 3 hours.

596

Parameters for the surface initial conditions are CwT = 0.58, CqT = 0.55 (which are

597

the same as Cheinet (2003) and Stull (1988), although we use T instead of θv in these

598

correlation coefficients), and Cwq = 0.63 (which comes from our LES diagnoses). We find

599

that the parameterization results are not sensitive to those correlation coefficients over a

600

large range. By setting N1 = 15, N2 = 10, the output is already statistically steady. Runs

601

with much larger N1 and N2 do not alter the results.

602

We use the same entrainment parameters (λ = 125m, σ = 0.32) as described in section

603

2. We find that the parameterization gives nearly equally good performance for large ranges 24

604

of [λ, σ] along the bottom of the valley of the objective function in Fig. 7 of Romps and

605

Kuang (2010b). Closer to the constant entrainment limit, however, the results degrade, and

606

the intra-cloud variations (see later in Fig. 14) are severely underestimated. The cloud base

607

is around 600m in the BOMEX case. For this first study, the subcloud layer entrainment

608

parameters are specified as [λsbc = 30m, σsbc = 0.06], which give fairly good results. In the

609

future, efforts are needed to better constrain [λsbc , σsbc ].

610

The initial sounding and the sounding after 3hours are shown in Fig. 12a,b.. Although

611

there are some drifts in the mean state, the convective tendency generally balances the large-

612

scale forcing. The fluxes given by the parameterization (Fig. 12c,d.) are broadly similar to

613

the LES results (Siebesma and Cuijpers 1995; Cheinet 2004), except our parameterization

614

over-predicts both the negative θl flux and the positive qt flux near 1300m.

615

We now sample the mean properties of “active cloudy air” for both the LES and the

616

parameterization. Results using two definitions of “active cloudy air” are shown: one is

617

cloudy updraft air as in section 3 (parcels/grids with ql > 10−5 kg kg−1 and w > 0.5

618

m s−1 ), the other is cloud core with the same definition as Cheinet (2004) (parcels/grids

619

with ql > 10−5 kg kg−1 , w > 0 m s−1 and positively buoyant). The results show that

620

the parameterization generally reproduces the results of LES with both definitions (Fig.

621

13). The exception is the cloud core w in the inversion. We have also examined the mean

622

properties of cloudy air (saturated), or active cloud core with other definitions, the results

623

all show agreement between the LES and the parameterization except for w in the inversion.

624

Besides the mean values of these properties, the standard variations are also examined

625

(Fig. 14). Once again, except layers that are in the inversion (above 1500m), the parame-

626

terization and the LES results match well. The discrepancy in w between the LES and the

627

parameterization in the inversion layer is not fully understood and requires further inves-

628

tigation. One possible reason for the decrease in the LES simulated w could be enhanced

629

wave drag. As parcels penetrate into the inversion layer, a region of strong stratification and

630

thus higher buoyancy frequency, they can become more effective in exciting gravity waves

25

631

and therefore become subject to enhanced gravity wave drag. Such processes are absent in

632

the current parameterization. However, we note that Warren (1960) provided wave drag

633

solutions that we can incorporate into our parameterization in the future.

634

c. Response of the parameterization to temperature and moisture perturbations

635

Another important aspect of the parameterization is to capture the response of convection

636

to changes in the large-scale environment. To this end, we have computed the linear response

637

functions (hereafter as the LRF matrix) of this parameterization to a full set of temperature

638

and moisture perturbations. The LRF matrix of the parameterization is calculated by simply

639

adding, one at a time, temperature/moisture anomalies in each layer of the mean sounding.

640

The anomalous heating and moistening tendencies from the parameterization form the LRF

641

matrix. A comparison of the full LRF matrices from the parameterization and from the LES

642

will be reported in a separate paper. Here we shall only discuss the three perturbed cases

643

(T987.5, T1262.5, Q987.5) described in Section 3.

644

645

With the LRF matrix, M , we can compute the evolution of any perturbations by integrating the equation dX = M X, dt

(25)

646

where X is the state vector (temperature and moisture profile) (Kuang 2010). The time

647

evolution of the sounding anomalies for the T987.5, T1262.5, and Q987.5 cases are shown in

648

Fig. 15, and should be compared with LES results shown in Fig. 1, Fig. 7 and Fig. 9. Note

649

that results in Fig. 15 are for an integration over 3 hours. The agreement is quite good in

650

terms of both the pattern and the magnitudes of the responses. The warming seen in the

651

parameterization in response to the moisture perturbation however, appears stronger than

652

that in the LES.

26

653

5. Conclusions and Discussions

654

In this paper, the responses of a shallow cumulus ensemble to large-scale temperature

655

and moisture perturbations are investigated using an LES and a stochastic parcel model.

656

We have further introduced a parameterization of shallow cumulus convection (including

657

subcloud layer turbulence) based on the stochastic parcel model.

658

The main findings are:

659

1. The SPM in general reproduces the LES responses to large-scale temperature and

660

moisture perturbations, not only in terms of the domain mean heating and moistening ten-

661

dencies, but also in terms of changes in the statistics of the cloudy updrafts. The stochastic

662

entrainment scheme in the SPM is key for the SPM to match the LES results. It suggests

663

that the stochastic entrainment approach is a good way of representing the mixing process

664

in simple models.

665

2. There are however some discrepancies in the w field between the SPM and LES

666

responses, suggesting the treatment of momentum evolution in the SPM might be overly

667

simplistic.

668

669

3. A positive temperature perturbation to the environmental sounding forms a buoyancy barrier that inhibits cloudy updrafts that have lower initial θe or entrain heavily.

670

4. For parcels that have the same amount of entrainment, the height at which parcels

671

entrain is important in deciding their fate. Parcels entraining at higher altitudes are more

672

likely to survive the buoyancy barrier and vice versa.

673

674

5. Convective heating responses to moisture perturbation above the cloud base are quite small for a shallow cumulus regime like BOMEX.

675

6. A parameterization based on the stochastic parcel model gives promising results in

676

terms of both the simulated mean state and the simulated responses to temperature and

677

moisture perturbations.

678

We argue that an important advantage of the SPM and the parameterization based

679

on it is that they explicitly include the inhomogeneity of cloudy air associated with the 27

680

stochastic mixing process. Although other ensemble plume/parcel model also contains some

681

inhomogeneity of cloudy air at the same height, the inhomogeneity is only introduced through

682

variations in the cloud base conditions (e.g. Neggers and Siebesma 2002). The present

683

approach includes the variations introduced by the stochastic nature of mixing, which was

684

shown to be the main cause of inhomogeneity in cloudy air (Romps and Kuang 2010b).

685

Capturing this inhomogeneity is important in order to better simulate microphysics and

686

chemistry beyond the goal of simulating the heating and moistening tendencies. Compared

687

to the assumed PDF approach (e.g. Lappen and Randall 2001; Larson et al. 2002; Golaz et al.

688

2002), the present approach may be viewed as a Monte-Carlo version of the PDF approach;

689

while it is somewhat more expensive, it can be more general and versatile in dealing with

690

different PDFs.

691

Certain treatments in the present parameterization are chosen for simplicity and could

692

and should be improved in the future. For example, the current neglect of momentum drag

693

on the parcels, including the lack of wave drag, is clearly unrealistic. Notwithstanding the

694

need to balance simplicity and realism, effects of gravity wave drag as described in Warren

695

(1960) should be explored. The lack of inter-parcel interaction is also an idealization based

696

on a limiting scenario, bulk plume being the opposite extreme of instant inter-parcel homog-

697

enization. In-cloud mixing of parcels and/or momentum exchanges without actual mixing

698

of parcels could be added in the spirit of Krueger et al. (1997). We have limited ourselves

699

to shallow non-precipitation cumuli in the current parameterization and have neglected pre-

700

cipitation processes. How to extend the current model to include the additional processes

701

brought about by precipitation so that it can serve as a unified parameterization for both

702

shallow and deep convection is a research question for future studies.

703

Acknowledgments.

704

We thank David Romps very much for valuable discussions and help with the DAM and

705

SPM models at the initial stage of the project. This research was partially supported by the 28

706

Office of Biological and Environmental Research of the U.S. Department of Energy under

707

Grant DE-FG02-08ER64556 as part of the Atmospheric Radiation Measurement Program

708

and NSF Grants ATM-0754332 and AGS-1062016. The Harvard Odyssey cluster provided

709

much of the computing resources for this study.

29

710

711

REFERENCES

712

Arakawa, A. and W. H. Schubert, 1974: Interaction of a cumulus cloud ensemble with the

713

large-scale environment, part i. J. Atmos. Sci., 31, 674–701.

714

Bechtold, P., E. Bazile, F. Guichard, P. Mascart, and E. Richard, 2001: A mass-flux con-

715

vection scheme for regional and global models. Quart. J. Roy. Meteor. Soc., 127, 869

716

–886.

717

Bony, S. and J.-L. Dufresne, 2005:

Marine boundary layer clouds at the heart of

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tropical cloud feedback uncertainties in climate models. Geophys. Res. Lett, 32,

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doi:10.1029/2005GL023 851.

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721

Bony, S., J.-L. Dufresne, H. L. Treut, J.-J. Morcrette, and C. Senior, 2004: On dynamic and thermodynamic components of cloud changes. Climate Dynamics, 22, 71–86.

722

Bretherton, C. S., J. R. Mccaa, and H. Grenier, 2004: A new parameterization for shal-

723

low cumulus convection and its application to marine subtropical cloud-topped boundary

724

layers. part i: Description and 1d results. Mon. Wea. Rev., 132, 864–882.

725

Cheinet, S., 2003: A multiple mass-flux parameterization for the surface-generated convec-

726

tion. a multiple mass-flux parameterization for the surface-generated convection. part i:

727

Dry plumes. J. Atmos. Sci., 60, 2313–2327.

728

729

730

731

Cheinet, S., 2004: A multiple mass flux parameterization for the surface-generated convection. part ii: Cloudy cores. J. Atmos. Sci., 61, 1093–1113. Durran, D. R., 1999: Numerical methods for wave equations in geophysical fluid dynamics. Springer, 267 pp.

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736

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738

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Emanuel, K. A., 1991: A scheme for representing cumulus convection in large-scale models. J. Atmos. Sci., 48, 2313–2335. Golaz, J.-C., V. E. Larson, and W. R. Cotton, 2002: A pdf-based model for boundary layer clouds. part i: Method and model description. J. Atmos. Sci., 59, 3540–3551. Kain, J. S. and M. Fritsch, 1990: A one-dimensional entraining/detraining plume model and its application in convective parameterization. J. Atmos. Sci., 47, 2784 –2802. Krueger, S. K., C.-W. Su, and P. A. Mcmurtry, 1997: Modeling entrainment and finescale mixing in cumulus clouds. J. Atmos. Sci., 54, 2697–2712.

740

Kuang, Z., 2010: Linear response functions of a cumulus ensemble to temperature and

741

moisture perturbations and implication to the dynamics of convectively coupled waves. J.

742

Atmos. Sci., 67, 941–962.

743

Lappen, C.-L. and D. A. Randall, 2001: Toward a unified parameterization of the boundary

744

layer and moist convection. part i: A new type of mass-flux model. J. Atmos. Sci., 58,

745

2021–2036.

746

Larson, V. E., J.-C. Golaz, and W. R. Cotton, 2002: Small-scale and mesoscale variability

747

in cloudy boundary layers: Joint probability density functions. J. Atmos. Sci., 59, 3519–

748

3539.

749

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751

752

753

754

Mapes, B. E., 2004: Sensitivities of cumulus-ensemble rainfall in a cloud-resolving model with parameterized large-scale dynamics. J. Atmos. Sci., 61, 2308–2317. Neggers, R. A. J. and A. P. Siebesma, 2002: A multiparcel model for shallow cumulus convection. J. Atmos. Sci., 59, 1655–1668. Raymond, D. J. and A. M. Blyth, 1986: A stochastic mixing model for nonprecipitating cumulus clouds. J. Atmos. Sci., 43, 2708–2718.

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756

757

758

759

760

761

762

Raymond, D. J. and M. J. Herman, 2011: Convective quasi-equilibrium reconsidered. Submitted to Journal of Advances in Modeling Earth Systems. Romps, D. M., 2008: The dry-entropy budget of a moist atmosphere. J. Atmos. Sci., 65, 3779–3799. Romps, D. M. and Z. Kuang, 2010a: Do undiluted convective plumes exist in the upper tropical troposphere? J. Atmos. Sci., 67, 468–484. Romps, D. M. and Z. Kuang, 2010b: Nature versus nurture in shallow convection. J. Atmos. Sci., 67, 1655–1666.

763

Sherwood, S., M. Colin, and F. Robinson, 2010: A revised conceptual model of cumulus

764

clouds as thermal vortices. Eos, Trans. Amer. Geophys. Union, (Fall Meeting Suppl.)

765

abstract A24C-04.

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769

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774

775

776

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Siebesma, A. P. and J. W. M. Cuijpers, 1995: Evaluation of parametric assumptions for shallow cumulus convection. J. Atmos. Sci., 52, 650–666. Simpson, J., 1971: On cumulus entrainment and one-dimensional models. J. Atmos. Sci., 28, 449–455. Stull, R. B., 1988: An Introduction to Boundary Layer Meteorology. Kluwer Academic, 670 pp. Tiedtke, M., 1989: A comprehensive mass flux scheme for cumulus parameterization in large-scale models. Mon. Wea. Rev., 117, 1779–1800. Tulich, S. N. and B. E. Mapes, 2010: Transient environmental sensitivities of explicitly simulated tropical convection. J. Atmos. Sci., 67, 923–940. Warren, F. W. G., 1960: Wave resistance to vertical motion in a stratified fluid. J. Fluid Mechanics, 7, 209–229. 32

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Yanai, M., S. Esbensen, and J.-H. Chu, 1973: Determination of bulk properties of tropical cloud clusters from large-scale heat and moisture budgets. J. Atmos. Sci., 30, 611– 627.

33

780

781

List of Figures 1

(a): The initial temperature anomaly of the T987.5 case. The heights of

782

Zrelease and Zsample are also marked. Evolutions of (b) temperature and (c)

783

moisture anomalies after the initial temperature perturbation is introduced.

784

2

(a) - (c): Cloudy updraft statistics of the LES control run: (a) mass flux, (b) w

785

and (c) θe shown as functions of purity and θe,ed . The mass flux distributions

786

are normalized to range from 0 to 1. (d) - (f) are same as (a) - (c) but for

787

results from the SPM control run. (g) - (i) are same as (a) - (c) but for results

788

from a lower resolution LES (shown are averages over 30 ensemble members).

789

3

cent (a), and changes in w (b) and θe (c) in the T987.5 case as functions

791

of purity and θe,ed . The background black contours are the LES control run

792

mass flux plotted in Fig2. a. (d) - (f) same as (a) - (c) but for the SPM.

793

The background black contours are the SPM control run mass flux plotted in

794

Fig2. d. Color values outside the lowest black contour are zeroed out. 4

stant entrainment and (b) stochastic entrainment. (c) is the LES-generated

797

distributions. The control runs are in black, the perturbed runs are in red

798

and the differences are in blue. 5

40

The trajectories of parcels with purity between 0.58 and 0.62 at Zsample .

800

Each line represents one trajectory. (a) is for the control run and (b) is for

801

the perturbed run. For each run, only 12 trajectories, which are randomly

802

chosen, are plotted for readability.

803

6

39

The SPM-generated distribution of mass flux as a function of θe,ed with (a) con-

796

799

38

(a)-(c): Color indicates LES-simulated fractional changes in mass flux in per-

790

795

37

41

The trajectories of parcels that entrain the same amount of environmental air

804

but at different heights. The blue dots indicate negative buoyancy while red

805

dots indicate positive buoyancy. (a) is the control run and (b) is the perturbed

806

run.

42 34

807

7

Same as Fig. 1b and c, but for the T1262.5 case.

43

808

8

Same as Fig. 3, but for the T1262.5 case.

44

809

9

Same as Fig. 1b and c, but for the Q987.5 case.

45

810

10

Same as Fig. 3, but for the Q987.5 case.

46

811

11

Buoyancies, divided by gravitational acceleration, (i.e. minus of the fractional

812

density anomaly or b) of a typical cloudy updraft parcel after mixing different

813

fractions (χ) of environmental air at 987.5m. A value of b greater than 0

814

indicates positive buoyancy. The black line uses the environmental sounding

815

from the control runs, while the red line uses the T987.5 sounding and the

816

green line uses the Q987.5 sounding.

817

12

(a) The initial θl profile (blue) and the profile after a 3 hours parameterization

818

run (red). (b) is the same as (a) but for the qt profile. (c) the parameterization

819

generated θl flux averaged between 1.5 to 3 hours of the parameterization run.

820

(d) is the same as (c) but for the qt flux.

821

13

updrafts. Solid lines are LES results and dot-dash lines are results from the

823

parameterization. The blue curves are for cloudy updrafts, while the red

824

curves are for the cloud cores. See text for definition. In (a) and (b), the

825

black dash lines are the environmental profiles and the black solid lines are

826

the adiabatic profiles from the cloud base. 14

48

The mass flux weighed properties ((a) θl , (b) qt , (c) ql and (d) w) of cloudy

822

827

47

49

The standard deviation among modelled cloudy updrafts for (a) θl (b) qt (c)

828

ql and (d) w. Solid lines are LES results and dot-dash lines are results from

829

our parameterization. The blue curves are for cloudy updrafts, while the red

830

curves are for the cloud cores.

50

35

831

15

Evolutions of (left column) temperature and (right column) moisture anoma-

832

lies after the initial perturbation is introduced as simulated using the linear

833

response functions of the parameterization. The first, second and third rows

834

are for the T987.5, T1262.5, and Q987.5 cases, respectively.

36

51

(b)

1800

1600

1600

Z(m)

z(m)

1200 1000

1000 800

800

Zrelease

600

0.05

1400

0.05 0.1

0.25

0.1 0.15

0.1 0.05

0.05

0.3 0.25 0.2 0.15 0.1

0.1 0.2 0.15

0.05

0.05

1200

0.05

0.1

0.15 0.1

0.15

0.1

200

0.1

0.2

0.3

T(K)

0.4

0.5

0

−0.02

0.14

−0.1

0.14 0.1

0.06

400

−0.05

0.02 0.06

0.1

−0.15

−0.2

200

0

0

−0.06

0.1

0.06

0.02

600

0.05

0.02 0.06

0

−0.1

−0.06

0.1

0.05

02

200

−0.02 0.02 0.06

−0.14

−0.1 −0.02

−0.02

0.1

0

−0.1

−0.1 −0.06 −0.18 −0.22 −0.18 −0.14

0.

400

1000

−0.02

−0.06 −0.14

0.02

800

0.05

600

400

0

1400

0.2

0.1

−0.02

1600

.06 −0

1200

1800

0.05

−0.02

Zsample

1400

qv(g/kg)

2000

0.02

1800

(c)

T(K)

2000

−0.06

T pert.

2000

z(m)

(a)

0.5

1

1.5

time(hr)

2

2.5

3

0

0

−0.25 0.5

1

1.5

time(hr)

2

2.5

3

Fig. 1. (a): The initial temperature anomaly of the T987.5 case. The heights of Zrelease and Zsample are also marked. Evolutions of (b) temperature and (c) moisture anomalies after the initial temperature perturbation is introduced.

37

encoded θe

0.6

349

0.5

348.5

0.4

348

347 346.5

0.2

0.4

0.6

purity

0.8

1

350.5 350

348

0.2

347

0.1

346.5

0.9

351

0.8

350.5

0.7

350

348.5

347.5 347

(g) 351

0.6

purity mass flux

0.8

1

0.1

350

0.5

348.5

0.4

348

0.3

347.5

347

0.6

purity

0.8

1

2.5 2

347.5

346.5

347

1.5 0.2

0.4

0.6

purity

0.8

1

346

349

345 344 343 342 0.4

1

0.6

purity θ (K)

0.8

1

e

348

349.5

346

349

344

348.5 348

342

347.5 347 346.5

341

350

350.5 350

3.5

348

0.1

348

349.5

351

4

348.5

0.2

349

(i)

3

350

350

346.5 0.2

4.5

349

e

347

1

350

347 0.4

purity w(m/s)

1

347.5

1 0.8

purity θ (K)

0.8

348

1.5

0.6

0.6

350.5

2

0.4

0.4

348.5

2.5

349.5

346.5

0.2

3.5 3

340 0.2

351

4

350.5

encoded θe

0.6

349

342

(f )

4.5

351

0.7

346.5

5

349

(h)

348

347

1

w(m/s)

346.5 0.2

0.8

349.5

purity

0.8

347

0.9

350.5

0.6

347.5

0.2 0.4

0.4

348

0.3

344

348.5

1 0.2

348.5

0.4

348

346

349

347.5

349.5

0.5

348

350 349.5

1.5

347.5

0.6

349

encoded θe

348.5

initial θe

initial θe

349.5

346.5 0.2

2

(e)

mass flux

351

349

350

350.5

2.5

349.5

0.3

347.5

(d)

350

0.7

θe(K)

351

encoded θe

350

(c) 3

350.5

0.8

349.5

w(m/s)

351

inital θe

350.5

(b)

e

0.9

encoded θ

mass flux

351

encoded θe

(a)

340 0.2

0.4

0.6

purity

0.8

1

Fig. 2. (a) - (c): Cloudy updraft statistics of the LES control run: (a) mass flux, (b) w and (c) θe shown as functions of purity and θe,ed . The mass flux distributions are normalized to range from 0 to 1. (d) - (f) are same as (a) - (c) but for results from the SPM control run. (g) - (i) are same as (a) - (c) but for results from a lower resolution LES (shown are averages over 30 ensemble members).

38

348.5 348

−45

348.5

0.4

351

0.6

purity mass flux

0.8

1

(e) 15

350.5

349

−30

348.5

−45

348

−60

347.5

−75

347 0.4

0.6

purity

0.8

1

−90

0.4

0.6

purity w(m/s)

0.8

1

350

349

−0.3

348.5

−0.4

348

−0.5

347.5

−0.6

347 346.5 0.2

(f )

−0.2

0.4

0.6

purity

0.8

1

−0.12

348

346.5

−0.15 −0.18 −0.21 0.2

0.4

0.6

purity θ (K)

0.8

1

−0.24

e

351

0.1

350.5 350

−0.1

349.5

−0.09

347

−0.35

0

−0.06

349

347.5

0.1

350.5

−15

349.5

0.2

−0.03

348.5

−0.3

351

0

350

346.5 0.2

346.5

−90

0

349.5

−0.25

e

0.2

−0.2

347

0.03

350

−0.15

348

0.06

350.5

−0.1

347.5

−75

347

initial θe

349

−60

347.5

(d)

−30

e

351

−0.05

349.5

encoded θe

e

encoded θ

349

θ (K)

(c)

0

350

−15

349.5

0.05

350.5

0

350

w(m/s)

351

encoded θe

350.5

346.5

(b)

15

0

349.5

initial θe

mass flux

351

initial θ

(a)

−0.1

349

348.5

−0.2

348

−0.3

347.5

−0.4

347 346.5 0.2

0.4

0.6

purity

0.8

1

−0.5

Fig. 3. (a)-(c): Color indicates LES-simulated fractional changes in mass flux in percent (a), and changes in w (b) and θe (c) in the T987.5 case as functions of purity and θe,ed . The background black contours are the LES control run mass flux plotted in Fig2. a. (d) - (f) same as (a) - (c) but for the SPM. The background black contours are the SPM control run mass flux plotted in Fig2. d. Color values outside the lowest black contour are zeroed out.

39

(a)

351

ctl T pert. dif

350.5 350

initial θe

349.5 349 348.5 348 347.5 347 346.5 346

(b)

0

0.1

0.2

0.3

0.4

0.5

0.6

mass flux

0.7

0.8

0.9

351

1

ctl T pert. dif

350.5 350

initial θe

349.5 349 348.5 348 347.5 347 346.5 346

(c)

0

0.1

0.2

0.3

0.4

0.5

0.6

mass flux

0.7

0.8

351

0.9

1

ctl T pert. dif

350.5 350

encoded θe

349.5 349 348.5 348 347.5 347 346.5 346

0

0.1

0.2

0.3

0.4

0.5

0.6

mass flux

0.7

0.8

0.9

1

Fig. 4. The SPM-generated distribution of mass flux as a function of θe,ed with (a) constant entrainment and (b) stochastic entrainment. (c) is the LES-generated distributions. The control runs are in black, the perturbed runs are in red and the differences are in blue.

40

(a)

(b)

parcel trajectories in purity and z space

1.1

0.9

0.9

0.8

0.8

purity

1

purity

1

0.7

0.7

0.6

0.6

0.5

0.5

0.4 750

parcel trajectories in purity and z space

1.1

800

850

900

950

1000

z

1050

1100

1150

1200

0.4 750

800

850

900

950

1000

z

1050

1100

1150

1200

Fig. 5. The trajectories of parcels with purity between 0.58 and 0.62 at Zsample . Each line represents one trajectory. (a) is for the control run and (b) is for the perturbed run. For each run, only 12 trajectories, which are randomly chosen, are plotted for readability.

41

(a)

3

red : positive buoyant

3

blue: negative buoyant

2.5

2

2

1.5

1.5

1

0.5

0

0

−0.5

−0.5 800

850

900

950

1000

z(m)

1050

1100

1150

1200

red : positive buoyant

blue: negative buoyant

1

0.5

−1 750

parcel trajectories in w and z space

3.5

w(m/s)

w(m/s)

2.5

(b)

parcel trajectories in w and z space

3.5

−1 750

800

850

900

950

1000

z(m)

1050

1100

1150

1200

Fig. 6. The trajectories of parcels that entrain the same amount of environmental air but at different heights. The blue dots indicate negative buoyancy while red dots indicate positive buoyancy. (a) is the control run and (b) is the perturbed run.

42

T(K)

(a)2000

0.45

qv(g/kg)

(b)2000

0.1 1800

0.35

1400

0.05 0.1 0.15 0.2 0.3

1200

0.4 0.30.35 0.2 0.25 0.1 0.05

0.25

1000

0.05 0.1 0.15 0.2

0.25

0.3 0.25 0.2 0.15 0.1 0.05

0.05 0.1 0.15 0.2

0.2 0.15 0.1 0.05

0.25

0.3 0.25 0.2

0.4

0.5

time(hr)

0.6

0.7

0.8

−0.03

−0.01

1

0 0.

0.03

−0.03

−0.05

−0.07

−0.05 −0.05 −0.03 −0.01 0.01 0.03 0.050.07

0.05

0.01

0.09

0.03

0.08

−0.05 −0.07 −0.07 −0.03 −0.01 0.01 0.03 0.05 0.07 0.09 0.11 0.07 0.01

0.02 0

−0.06 −0.08 0

0.1

0.2

0.3

0.4

0.5

time(hr)

0.6

Fig. 7. Same as Fig. 1b and c, but for the T1262.5 case.

43

0.04

−0.04

200 0

0.9

0.06

−0.02

600 400

0.05

200

0.3

1 −0.0

−0.01 −0.03

800

0.1

400

0.2

1400 1200

0.15

600

0.1

−0.01

1000

800

0

1600

z(m)

z(m)

1600

0

1800

0.4

0.7

0.8

0.9

350

−0.05

349

348

−0.1

348

347

−0.15

346

−0.2

encoded θe

encoded θe

−30

347

−45

346

−60

345 344 0.2

0.4

(d)

0.6

purity mass flux

0.8

1

0.2

0.6

purity w(m/s)

0.8

1

−30

347

−45

346

−60

345

−75

344 0.4

0.6

purity

0.8

1

−90

−0.35

0.1

350

348

347

−0.3

346

−0.4

−0.6 0.6

purity

0.8

1

−0.1 −0.15 −0.2 −0.25 −0.3 −0.35 0.4

(f )

−0.2

−0.5

−0.05

0.2

348

0.4

0

344

349

0.2

0.05

345

−0.1

344

0.1

346

350

345

0.6

purity θe(K)

0.8

1

0 −0.1

347

−0.2

346

−0.3

345

−0.4

344 0.2

−0.4

0.1

0.4

Fig. 8. Same as Fig. 3, but for the T1262.5 case.

44

0.15

347

0

349

initial θe

initial θe

0.4

(e)

−15

348

0.2

−90

0

349

−0.3

344

15

350

−0.25

345

−75

θe(K)

0

349

−15

348

(c)

0.05

350

0

349

w(m/s)

(b)

15

encoded θe

mass flux 350

initial θe

(a)

0.6

purity

0.8

1

−0.5

(a)

T(K)

2000

−3

x 10

(b)

4

1800 −0.001

1600

−0.003 −0.00 5 −0.003 −0.001 0.001

1400

−0.003 −0.005 −0.007 −0.005 −0.003 −0.001 0.001

1600

2

1000

0.001

800

0.005

0.001

600

0.01

1200

−2

0.01 0.03 0.05 0.07 0.09 0.11 0.15 0.17 0.13

z(m)

00 0.

z(m)

3

0.00

1000

0.01 0.03 0.05 0.07 0.09 0.11

0.13

0.15 0.11 0.13 0.09 0.07 0.05 0.03 0.01

800

0.003

−4

400

0.14

0.01

1400 0

0.003

0.003 0.005 0.005 0.003

0.16

1800

1

1200

−0.001

qv(g/kg)

2000

0.01 0.01 0.03 0.05 0.07 0.09 0.11

0.13

0.1

0.08

0.11 0.09 0.07 0.05 0.03 0.01

0.06

600 0.04

400

−6

0.02

200

200 −8 0

0

0.1

0.2

0.3

0.4

0.5

time(hr)

0.6

0.7

0.8

0.12

0

0.9

0

0.1

0.2

0.3

0.4

0.5

time(hr)

0.6

Fig. 9. Same as Fig. 1b and c, but for the Q987.5 case.

45

0.7

0.8

0.9

0

350

20

e

349.5

0.05

350

0.04

349.5

0.03

349

15

349

0.02

348.5

10

348.5

0.01

348

5

347.5

0

347 346.5

(d)

0.2

0.4

351

0.6

purity mass flux

0.8

1

−5

40

−0.03

346.5

351

e

20

349 348.5

10

348 347.5

0

347 0.4

0.6

purity

0.8

1

−10

1

0.12

350.5

0.09

350

0.06

349.5

0.03

349

0

(f )

−0.09

347.5

347

−0.12

347

1

0.8

1

0.32 0.28 0.24 0.2 0.16

348.5

0.12

348

346.5 0.2

Fig. 10. Same as Fig. 3, but for the Q987.5 case.

46

0.6

purity θe(K)

349

347.5

0.8

0.4

349.5

−0.06

0.6

0.2

350

348

purity

0.02

351

−0.03

0.4

0.04

350.5

348.5

346.5 0.2

0.06

348

346.5

0.8

0.08

348.5

347

0.6

0.1

349

347.5

purity w(m/s)

0.12

349.5

−0.02 0.4

0.14

350

347 0.2

0.16

350.5

−0.01

initial θe

349.5

0

351

347.5

30

350

346.5 0.2

348

(e)

350.5

initial θ

350.5

θe(K)

(c)

0.06

encoded θe

25

w(m/s)

351

e

350.5

encoded θ

(b)

30

initial θ

mass flux

351

encoded θe

(a)

0.08 0.04 0.4

0.6

purity

0.8

1

−3

2

x 10

ctl T pert. Q pert.

1.5

1

0.5

b

0

−0.5

−1

−1.5

−2

−2.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fraction of environmental air ! Fig. 11. Buoyancies, divided by gravitational acceleration, (i.e. minus of the fractional density anomaly or b) of a typical cloudy updraft parcel after mixing different fractions (χ) of environmental air at 987.5m. A value of b greater than 0 indicates positive buoyancy. The black line uses the environmental sounding from the control runs, while the red line uses the T987.5 sounding and the green line uses the Q987.5 sounding.

47

(b)

(a) 3000

(c)

3000

ini. 3 hrs

(d)

3000

ini. 3 hrs

3000

2500

2500

2500

2000

2000

2000

2000

1500

1500

1500

1500

1000

1000

1000

1000

500

500

500

500

z

2500

0

300

305

θl (K)

310

315

0

0.005

0.01

0

0.015

qt (kg/kg)

−30

−20

−10

0

θl flux(W/m2)

10

0

0

50

100

qt flux(W/m2)

150

Fig. 12. (a) The initial θl profile (blue) and the profile after a 3 hours parameterization run (red). (b) is the same as (a) but for the qt profile. (c) the parameterization generated θl flux averaged between 1.5 to 3 hours of the parameterization run. (d) is the same as (c) but for the qt flux.

48

(b)

2000

LES para.

LES para.

299

300

301

θl(K)

302

303

500 10

1000

12

14

16

q (g/kg)

500

18

t

LES para.

1500

z

1000

298

2000

LES para.

1500

z

1000

500

2000

1500

z

1500

(d)

(c)

2000

z

(a)

1000

0

0.5

1

1.5

q (g/kg) l

2

2.5

500

0

1

2

3

w(m/s)

4

5

Fig. 13. The mass flux weighed properties ((a) θl , (b) qt , (c) ql and (d) w) of cloudy updrafts. Solid lines are LES results and dot-dash lines are results from the parameterization. The blue curves are for cloudy updrafts, while the red curves are for the cloud cores. See text for definition. In (a) and (b), the black dash lines are the environmental profiles and the black solid lines are the adiabatic profiles from the cloud base.

49

(b)

2000

0.5

1

θl std (K)

1.5

500

1000

0

0.5

1

qt std(g/kg)

1.5

2000

LES para.

1500

z

1000

0

LES para.

1500

z

1000

500

2000

LES para.

1500

z

1500

(d)

(c)

2000

LES para.

z

(a)

500

1000

0

0.2

0.4

ql std(g/kg)

0.6

500

0

0.5

1

1.5

w std(m/s)

2

Fig. 14. The standard deviation among modelled cloudy updrafts for (a) θl (b) qt (c) ql and (d) w. Solid lines are LES results and dot-dash lines are results from our parameterization. The blue curves are for cloudy updrafts, while the red curves are for the cloud cores.

50

T(K)

q(g/kg)

2000

2000 0.3

0.1 0.05 0.15 0.3 0.25 0.2 0.15 0.1

1000

0.2 0.15

0.05

0

500

0.05 1

2

0

3

2000

−0.05 −0.1

0

1

2

3

2000

.3 00.25 0.1

1000 500

0.05

0.1

1500

0.25

0.25

0.2 0.15

0.2

0

0.15 0

0

z(m)

0.15 0.2

0.1

0.1

−0.03 −0.06 −0. 12 −0.09

0.3

0.05

−0.15

0.05

1500

z(m)

0

−0.15

0 0

1000

0.1

0

500

0.05

−0.06 −0.09 −0.12 8 −0.15 −0.1 −0.12 −0.09 −0.06 −0.03 −1.1102e−16 0.03 0.06 0.09 0.06 0.03 0.06

−0.03

1500

0.25

z(m)

z(m)

1500

−0.09 −0.06 −0.03 −1.1102e−16 0.03 0.06 0.09

1000

0 −0.05

0.06

0.03

−0.1

500

0.05 0

1

2

z(m)

1500 0 1000

0.004

2 −0.01

−0.004

0

3

2000

500

−0.15

0

−0.008 −0.004 0.008 0.012 0.02 0.016 0.008 0.012 0.004

0.02

2000

0.01

1500

0

z(m)

0

0

0

1

2

3

0.1

0. 02

0.02

0.04 0.10.06 0.1 2 0.08 0.04 0.02

1000

0.05

500

0

−0.01

0

0 0

0

1

2

0

3

time(hr)

0

1

2

3

time(hr)

Fig. 15. Evolutions of (left column) temperature and (right column) moisture anomalies after the initial perturbation is introduced as simulated using the linear response functions of the parameterization. The first, second and third rows are for the T987.5, T1262.5, and Q987.5 cases, respectively.

51