Detrainment Rates Research Articles

Direct Entrainment as a Measure of Dilution

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

Evaluation of a New Approach for Entrainment and Detrainment Rate Estimation

AbstractEntrainment and detrainment rates (ε and δ) constitute the most critical free parameters in mass flux schemes commonly employed for cumulus parameterizations. Recently, Zhu et al. (2021) introduced a new approach that utilizes aircraft observations to simultaneously estimate ε and δ for cumulus clouds, overcoming the limitation of other observation‐based approaches that solely yield ε without offering insights into δ. This study aims to comprehensively evaluate the reliability of this new approach. First, evaluation using an Explicit Mixing Parcel Model demonstrates the capability of the new approach to back‐calculate predetermined ε and δ based on the physical properties before and after the entrainment mixing. Second, evaluation using large‐eddy simulations illustrates that the new approach yields consistent ε and δ profiles compared to the traditional approach. Sensitivity tests indicate a weak sensitivity of the estimated δ with the new approach to the entrained air source. A decrease in the proportion of cloudy air in the assumed detrained air leads to a reduction in the estimated δ, while ε remains unaffected. Finally, the most appropriate assumptions for entrained and detrained air are discussed. Estimating ε for cumulus parameterizations involves acquiring ambient air more than 500 m away from the cloud edge as entrained air. Due to implicit mean field approximations in the traditional approach, determining the optimal assumption for detrained air properties proves challenging. This study confirms the reliability of the new approach in estimating ε and δ, providing confidence in its application to extensive observational data and advancement in parameterization.

Updates in the NCEP GFS PBL and Convection Models with Environmental Wind Shear Effect and Modified Entrainment and Detrainment Rates and Their Impacts on the GFS Hurricane and CAPE Forecasts

Abstract To reduce hurricane intensity bias, the NCEP Global Forecast System (GFS) planetary boundary layer (PBL) and convection schemes have been updated with a new parameterization for environmental wind shear and enhanced entrainment and detrainment rates with increasing PBL or subcloud mean turbulent kinetic energy (TKE) in their updraft and downdraft mass-flux schemes. Tests with the GFS show that the updated schemes significantly reduce the hurricane intensity bias by reducing the momentum transport in the mass-flux schemes. Along with the reduced intensity bias, the hurricane intensity and track errors have also been reduced. On the other hand, to reduce the PBL overgrowth over areas with a higher vegetation fraction or larger surface roughness, the entrainment rate in the PBL mass-flux scheme has also been increased with increasing vegetation fraction or increasing surface roughness. This entrainment rate increase has increased near-surface moisture, and as a result, helped to increase the underestimated convective available potential energy (CAPE) forecasts over the continental United States.

A Shallow‐Deep Unified Stochastic Mass Flux Cumulus Parameterization in the Single Column Community Climate Model

AbstractCumulus parameterization (CP) in state‐of‐the‐art global climate models is based on the quasi‐equilibrium assumption (QEA), which views convection as the action of an ensemble of cumulus clouds, in a state of equilibrium with respect to a slowly varying atmospheric state. This view is not compatible with the organization and dynamical interactions across multiple scales of cloud systems in the tropics and progress in this research area was slow over decades despite the widely recognized major shortcomings. Novel ideas on how to represent key physical processes of moist convection‐large‐scale interaction to overcome the QEA have surged recently. The stochastic multicloud model (SMCM) CP in particular mimics the dynamical interactions of multiple cloud types that characterize organized tropical convection. Here, the SMCM is used to modify the Zhang‐McFarlane (ZM) CP by changing the way in which the bulk mass flux and bulk entrainment and detrainment rates are calculated. This is done by introducing a stochastic ensemble of plumes characterized by randomly varying detrainment level distributions based on the cloud area fraction of the SMCM. The SMCM is here extended to include shallow cumulus clouds resulting in a unified shallow‐deep CP. The new stochastic multicloud plume CP is validated against the control ZM scheme in the context of the single column Community Climate Model of the National Center for Atmospheric Research using data from both tropical ocean and midlatitude land convection. Some key features of the SMCM CP such as it capability to represent the tri‐modal nature of organized convection are emphasized.

What Controls Local Entrainment and Detrainment Rates in Simulated Shallow Convection?

Abstract The lack of consensus as to how entrainment and detrainment must be represented in convection parameterizations highlights the need for in-depth investigations of the processes driving lateral mixing in clouds. Direct estimates of entrainment and detrainment rates are here obtained from high-resolution simulations of shallow cumulus convection using a method that isolates the contributions from various competing processes. Moreover, cloud-averaged entrainment and detrainment rates are computed and correlated with bulk cloud and environmental properties. Detrainment is found to dominate in the middle of the cloud layer, and is locally driven by evaporation along cloud edges. Entrainment and detrainment events also occur due to changes in updraft velocity, but vertical acceleration and deceleration balance each other on average to yield no net entrainment/detrainment. In contrast, entrainment at cloud base is mostly related to condensation in dry updrafts surrounding the clouds, whereas detrainment at the top of the cloud layer is driven by buoyancy reversal. Cloud-averaged entrainment and detrainment rates both correlate preferentially with in-cloud vertical pressure gradients at all altitudes, but not with cloud-core buoyancy. No significant influence of environmental moisture on entrainment/detrainment was found, probably because of the very humid shell surrounding each cloud. Overall, these results suggest that entrainment and detrainment result from two concurring processes: a cloud-scale circulation driven by vertical pressure gradients, and small-scale, turbulent-like motions along the core edges generating mixing and, again, vertical pressure gradients.

Parameterization of Stochastically Entraining Convection Using Machine Learning Technique

AbstractA stochastic mixing model with a machine learning technique is proposed for mass flux convection schemes. The model consists of the stochastic differential equations (SDEs) for the fractional entrainment rate, fractional detrainment rate, fractional dilution rate, and vertical acceleration. Unknowns in SDEs are parameterized using a deep neural network with the inputs of cloud and environment properties. The deep neural network is found to predict entrainment and detrainment rates better than previously proposed parameterizations. The new mixing model is implemented in a unified convection scheme (UNICON) and tested in a single‐column mode for two marine shallow convection cases. It is shown that the simulations with the new mixing model produce realistic mean and variance of various convective updraft properties and that the appropriate amount of stochasticity is generated. Consistently accurate simulations of updraft mass fluxes and moist conserved variables reduce model errors in the original UNICON. Additional sensitivity simulations enabling or disabling the stochasticity in mixing and initialization suggest that most of the cloud variabilities are generated from the mixing process.

Suppressing Grid-Point Storms in a Numerical Forecasting Model

The convective parameterization scheme of the Korean Integrated Model (KIM) is tentatively modified to suppress grid-point storms in the Western Pacific Ocean. The KIM v3.2.11 suffers from the numerical problem that grid-point storms degrade forecasts in the tropical oceans and around the Korean Peninsula. Another convective parameterization scheme, the new Tiedtke scheme, is implemented in the KIM. The artificial storms are suppressed in the test version because the heating and drying tendencies of the new Tiedtke scheme are stronger than those of the default KIM Simplified Arakawa-Schubert (KSAS) scheme. Based on this comparison, the KSAS scheme is modified to strengthen its heating and drying tendencies by reducing the entrainment and detrainment rates. The modified KSAS scheme suppresses grid-point storms and thus decreases grid-scale precipitation in a summertime case simulation. Twenty 10-day forecasts with the default convection scheme (KSAS) and twenty forecasts with the modified scheme are conducted and compared with each other, confirming that the modified KSAS scheme successfully suppresses grid-point storms.

The Grell–Freitas (GF) convection parameterization: recent developments, extensions, and applications

Abstract. Recent developments and options in the GF (Grell and Freitas, 2014; Freitas et al., 2018) convection parameterization are presented. The parameterization has been expanded to a trimodal spectral size to simulate three convection modes: shallow, congestus, and deep. In contrast to usual entrainment and detrainment assumptions, we assume that beta functions (BFs), commonly applied to represent probability density functions (PDFs), can be used to characterize the vertical mass flux profiles for the three modes and use the BFs to derive entrainment and detrainment rates. We also added a new closure for nonequilibrium convection that improved the simulation of the diurnal cycle of convection, with a better representation of the transition from shallow to deep convection regimes over land. The transport of chemical constituents (including wet deposition) can be treated inside the GF scheme. The tracer transport is handled in flux form and is mass-conserving. Finally, the cloud microphysics have been extended to include the ice phase to simulate the conversion from liquid water to ice in updrafts with resulting additional heat release and the melting from snow to rain.

A New Approach for Simultaneous Estimation of Entrainment and Detrainment Rates in Non‐Precipitating Shallow Cumulus

AbstractA new approach is developed for estimating entrainment and detrainment rates in cumulus clouds based on aircraft observations. Equations relating entrainment and detrainment rates to gross entrainment and detrainment are derived. This approach is applied to the Holistic Interactions of Shallow Clouds, Aerosols, and Land‐Ecosystems field campaign, supported by the U.S. Department of Energy's Atmospheric Radiation Measurement program. The results show that both entrainment and detrainment rates decrease with increasing height. Sensitivity tests with different detrained air assumptions yield similar results. The entrainment and detrainment rates can reproduce the cloud thermodynamic variables. Partial correlation analysis indicates that entrainment rate is positively correlated with environmental relative humidity (RH), and detrainment rate is negatively correlated with environmental RH and positively correlated with entrainment rate. This new approach can be applied to other cloud observations to obtain a data set of entrainment and detrainment rates in cumulus clouds.

Unified Entrainment and Detrainment Closures for Extended Eddy-Diffusivity Mass-Flux Schemes.

We demonstrate that an extended eddy‐diffusivity mass‐flux (EDMF) scheme can be used as a unified parameterization of subgrid‐scale turbulence and convection across a range of dynamical regimes, from dry convective boundary layers, through shallow convection, to deep convection. Central to achieving this unified representation of subgrid‐scale motions are entrainment and detrainment closures. We model entrainment and detrainment rates as a combination of turbulent and dynamical processes. Turbulent entrainment/detrainment is represented as downgradient diffusion between plumes and their environment. Dynamical entrainment/detrainment is proportional to a ratio of a relative buoyancy of a plume and a vertical velocity scale, that is modulated by heuristic nondimensional functions which represent their relative magnitudes and the enhanced detrainment due to evaporation from clouds in drier environment. We first evaluate the closures off‐line against entrainment and detrainment rates diagnosed from large eddy simulations (LESs) in which tracers are used to identify plumes, their turbulent environment, and mass and tracer exchanges between them. The LES are of canonical test cases of a dry convective boundary layer, shallow convection, and deep convection, thus spanning a broad rangeof regimes. We then compare the LES with the full EDMF scheme, including the new closures, in a single‐column model (SCM). The results show good agreement between the SCM and LES in quantities that are key for climate models, including thermodynamic profiles, cloud liquid water profiles, and profiles of higher moments of turbulent statistics. The SCM also captures well the diurnal cycle of convection and the onset of precipitation.

A Method for a Direct Measure of Entrainment and Detrainment

AbstractThe entrainment and detrainment rates are important quantities characterizing airmass exchange between clouds and the environment. One of the challenges in calculating the rates is the need to know the velocity vector of the cloud interface in relation to that of the cloud-free air; however, the interface is not well resolved in most cloud model simulations and so the precise value of the vector is not known. Here a new method is described to approximately calculate mass fluxes across the cloud surface in well-resolved simulations of cumulus convection. The method does away with the need to calculate a cloud interface velocity and instead uses gradients of a defined cloud scalar across the cloud interface. As a result, the entrainment and detrainment rates are expressed as an integration over a small region around the cloud interface. The integrand is composed of the total derivative of the cloud scalar. The new method is applied to large-eddy simulations (LES) of a shallow cumulus case and a deep convection case. Compared to a previous method, the approach described here gives 1.5–2 times smaller exchange rates and shows less noise. The smaller exchange rates are explained as the result of differences in how the two methods correct for the advective contribution to variations of cloud volume. Derived two-dimensional distributions of the exchange rates agree well for both methods. Spatial correlation coefficients are about 0.69–0.88 for entrainment and 0.55–0.78 for detrainment.

Parameterization of Entrainment Rate and Mass Flux in Continental Cumulus Clouds: Inference From Large Eddy Simulation

AbstractLarge eddy simulation with contrasting environmental humidity is used to study entrainment and detrainment rates and convective mass flux during the development stages of continental deep cumulus clouds. The cumulus clouds studied here are deeper than marine shallow cumulus or stratocumulus in the published literature. The main objective of the study is to parameterize the entrainment and detrainment rates in monsoon clouds during the break monsoon periods over land, useful for the large‐scale atmospheric models. A systematic decrease in cloud liquid water path is noted in drier environments as clouds become shallower due to the reduction of positive buoyancy in the cloud core. Updraft and downdraft velocities in the subcloud layers are strengthened in the drier environments, which are found to affect in‐cloud updraft strength. A most important finding of the present study is the effect of environmental humidity on the entrainment (and detrainment) rates and the convective mass flux. Many previous studies on this topic reported opposite results. The present study found that decreasing environmental humidity leads to a decrease in both the entrainment rate and the convective mass flux but an increase in the detrainment rate. The relationship of the entrainment parameters with the environmental relative humidity can be used for parameterization in the large‐scale models. The physical mechanism responsible for such response is identified as the strengthening of the subcloud updrafts compensated by the enhanced downdraft into the boundary layer in the drier environments, and this mechanism subsequently led to higher in‐cloud updraft velocity, resulting in lower entrainment rate.

Marine Aerosol Production via Detrainment of Bubble Plumes Generated in Natural Seawater With a Forced‐Air Venturi

AbstractDuring September–October 2016, a marine aerosol generator configured with forced‐air Venturis was deployed at two biologically productive and two oligotrophic regions of the western North Atlantic Ocean to investigate factors that modulate primary marine aerosol (PMA) production. The generator produced representative bubble size distributions with Hinze scales (0.32 to 0.95 mm radii) and void fractions (0.011 to 0.019 Lair Lsw‐1) that overlapped those of plumes produced in the surface ocean by breaking wind waves. Hinze scales and void fractions of bubble plumes varied among seawater hydrographic regions, whereas corresponding peaks and widths of bubble size distributions did not, suggesting that variability in seawater surfactants drove variability in plume dynamics. Peaks in size‐resolved number production efficiencies for model PMA (mPMA) emitted via bubble bursting in the generator were within a narrow range (0.059 to 0.069 μm geometric mean diameter) over wide ranges in subsurface bubble characteristics, suggesting that subsurface bubble size distributions were not the primary controlling factors as was suggested by previous work. Total mass production efficiencies for mPMA decreased with increasing air detrainment rates, supporting the hypothesis that surface bubble rafts attenuate mPMA mass production. Total mass and Na+ production efficiencies for mPMA from biologically productive seawater were significantly greater than those from oligotrophic seawater. Corresponding mPMA number distributions peaked at smaller sizes during daytime, suggesting that short‐lived surfactants of biological and/or photochemical origin modulated diel variability in marine aerosol production.

A Lagrangian convective transport scheme including a simulation of the time air parcels spend in updrafts (LaConTra v1.0)

Abstract. We present a Lagrangian convective transport scheme developed for global chemistry and transport models, which considers the variable residence time that an air parcel spends in convection. This is particularly important for accurately simulating the tropospheric chemistry of short-lived species, e.g., for determining the time available for heterogeneous chemical processes on the surface of cloud droplets. In current Lagrangian convective transport schemes air parcels are stochastically redistributed within a fixed time step according to estimated probabilities for convective entrainment as well as the altitude of detrainment. We introduce a new scheme that extends this approach by modeling the variable time that an air parcel spends in convection by estimating vertical updraft velocities. Vertical updraft velocities are obtained by combining convective mass fluxes from meteorological analysis data with a parameterization of convective area fraction profiles. We implement two different parameterizations: a parameterization using an observed constant convective area fraction profile and a parameterization that uses randomly drawn profiles to allow for variability. Our scheme is driven by convective mass fluxes and detrainment rates that originate from an external convective parameterization, which can be obtained from meteorological analysis data or from general circulation models. We study the effect of allowing for a variable time that an air parcel spends in convection by performing simulations in which our scheme is implemented into the trajectory module of the ATLAS chemistry and transport model and is driven by the ECMWF ERA-Interim reanalysis data. In particular, we show that the redistribution of air parcels in our scheme conserves the vertical mass distribution and that the scheme is able to reproduce the convective mass fluxes and detrainment rates of ERA-Interim. We further show that the estimated vertical updraft velocities of our scheme are able to reproduce wind profiler measurements performed in Darwin, Australia, for velocities larger than 0.6 m s−1. SO2 is used as an example to show that there is a significant effect on species mixing ratios when modeling the time spent in convective updrafts compared to a redistribution of air parcels in a fixed time step. Furthermore, we perform long-time global trajectory simulations of radon-222 and compare with aircraft measurements of radon activity.

Kinematics of local entrainment and detrainment in a turbulent jet

In this paper we investigate the continuous, local exchange of fluid elements as they are entrained and detrained across the turbulent/non-turbulent interface (TNTI) in a high Reynolds number axisymmetric jet. To elucidate characteristic kinematic features of local entrainment and detrainment processes, simultaneous high-speed particle image velocimetry and planar laser-induced fluorescence measurements were undertaken. Using an interface-tracking technique, we evaluate and analyse the conditional dependence of local entrainment velocity in a frame of reference moving with the TNTI in terms of the interface geometry and the local flow field. We find that the local entrainment velocity is intermittent with a characteristic length scale of the order of the Taylor micro-scale and that the contribution to the net entrainment rate arises from the imbalance between local entrainment and detrainment rates that occurs with a ratio of two parts of entrainment to one part detrainment. On average, an increase in local entrainment is correlated with excursions of the TNTI towards jet centreline into regions of higher streamwise momentum, convex surface curvature facing the turbulent side of the jet and along the leading edges of the interface. In contrast, detrainment is correlated with excursions of the TNTI away from the jet centreline into regions of lower streamwise momentum, concave surface curvature and along the trailing edge. We find that strong entrainment is characterised by a local counterflow velocity field in the frame of reference moving with the TNTI which enhances the transport of rotational and irrotational fluid elements. On the other hand, detrainment is characterised by locally uniform flow fields with the local fluid velocity on either side of the TNTI advecting in the same direction. These local flow patterns and the strength of entrainment or detrainment rates are also observed to be strongly influenced by the presence and relative strength of vortical structures which are of the order of the Taylor micro-scale that populate the turbulent region along the jet boundary.

The Interaction Between Boundary Layer and Convection Schemes in a WRF Simulation of Post Cold Frontal Clouds Over the ARM East North Atlantic Site

AbstractThe correct representation of low‐level midlatitude clouds found in the wake of cold fronts strongly relies on the representation of planetary boundary layer (PBL) and convection processes, which are typically parameterized separately in numerical models. Using the Weather Research and Forecasting Model (WRF), this study investigates how distinct pairs of PBL and convection parameterization schemes represent cloud fraction in the post cold frontal region. The simulations focus on the region of the Department of Energy Atmospheric Radiation Measurement (ARM) Eastern North Atlantic observation site in the Azores Islands in the wake of a cold front that passed on 25 December 2015. Different PBL and convection schemes are combined to create 12 distinct configurations. The main differences between the selected physical parameterizations are the strength of vertical mixing and the entrainment. The simulations produce a wide range of cloud fractions, where some configurations significantly underestimate while others clearly overestimate satellite and surface‐based estimates of cloud fraction. A skill score is used to quantitatively assess the performance of each configuration with respect to ground‐based radar data. The key processes that are found to significantly impact the cloud fraction distribution are the strength of the PBL decoupling, the vertical wind shear, entrainment and detrainment rates in shallow convection, and the occurrence of drizzle. This indicates that to successfully simulate post cold frontal clouds, modeled physics must balance strong internal vertical mixing and weak exchange with the free troposphere. For this case study, cloud fraction was more sensitive to the choice of convection scheme than PBL scheme.

Role of Surface Friction on Shallow Nonprecipitating Convection

The role of surface friction on shallow nonprecipitating convection is investigated using a series of large-eddy simulations with varying surface friction velocity and with a cloud identification algorithm. As surface friction intensifies, convective rolls dominate over convective cells and secondary overturning circulation becomes stronger in the subcloud layer, thus transporting more moisture upward and more heat downward between the subcloud and cloud layers. Identifying individual clouds, using the identification algorithm based on a three-dimensional topological analysis, reveals that intensified surface friction increases the number of clouds and the degree of tilting in the downstream direction. Highly intensified surface friction increases wind shear across the cloud base and induces cloud tilting, which leads to a vertically parabolic profile of liquid water mixing ratio instead of the classical two-layer structure (conditionally unstable and trade inversion layers). Furthermore, cloud tilting induces more cloud cover and more cloud mass flux much above the cloud base (e.g., 0.8 < z < 1.2 km), but less cloud cover and less cloud mass flux in the upper cloud layer (e.g., z > 1.2 km) because of increased lateral entrainment rate. Similarly, profiles of directly measured entrainment and detrainment rates show that detrainment in the lower cloud layer becomes smaller with stronger surface friction.

Uncertainties related to the representation of momentum transport in shallow convection

AbstractConvective momentum transport (CMT) has mostly been studied for deep convection, whereas little is known about its characteristics and importance in shallow convection. In this study, CMT by shallow convection is investigated by analyzing both data from large‐eddy simulations (LESs) and reforecasts performed with the Integrated Forecasting System (IFS) of the European Centre for Medium‐Range Weather Forecasts (ECMWF). In addition, the central terms underlying the bulk mass‐flux parametrization of CMT are evaluated offline. Further, the uncertainties related to the representation of CMT are explored by running the stochastically perturbed parametrizations (SPP) approach of the IFS. The analyzed cases exhibit shallow convective clouds developing within considerable low‐level wind shear. Analysis of the momentum fluxes in the LES data reveals significant momentum transport by the convection in both cases, which is directed downgradient despite substantial organization of the cloud field. A detailed inspection of the convection parametrization reveals a very good representation of the entrainment and detrainment rates and an appropriate representation of the convective mass and momentum fluxes. To determine the correct values of mass‐flux and in‐cloud momentum at the cloud base in the parametrization yet remains challenging. The spread in convection‐related quantities generated by the SPP is reasonable and addresses many of the identified uncertainties.

Sampling the Structure of Convective Turbulence and Implications for Grey-Zone Parametrizations

The grey zone of dry convection is the range of scales in which boundary-layer thermals are partly explicitly resolved by numerical weather prediction (NWP) models and partly parametrized. We seek to determine how thermals are divided into subgrid and resolved scales in the grey zone of convective boundary-layer thermals. Reference data for grid-scale and subgrid-scale fields at these resolutions are constructed by filtering 62.5-m large-eddy simulation data. A conditional sampling is adapted to detect subgrid thermals, and is used to characterize the subgrid thermals at several grid spacings in the grey zone. A mass-flux parametrization used in NWP models is compared with the subgrid thermal field. The analysis demonstrates that, although the mass-flux framework is suitable in the grey zone, some assumptions of the mass-flux schemes, usually used at the mesoscale, cannot be made in the grey zone. In particular, the thermal fraction is not small, the resolved vertical velocity is not negligible, the entrainment and detrainment rates depend on the horizontal resolution, the triggering and the closure at the surface are moreover random.

An Analytical Model for Tropical Relative Humidity

Abstract An analytical model is derived for tropical relative humidity using only the Clausius–Clapeyron relation, hydrostatic balance, and a bulk-plume water budget. This theory is constructed for radiative–convective equilibrium and compared against a cloud-resolving model. With some reinterpretation of variables, it can be applied more generally to the entire tropics. Given four variables—pressure, temperature, and the fractional entrainment and detrainment rates—the equations predict the relative humidity (RH) and the temperature lapse rate analytically. The RH is a simple ratio involving the fractional detrainment rate and the water-vapor lapse rate. When integrated upward in height, the equations give profiles of RH and temperature for a convecting atmosphere. The theory explains the magnitude of RH and the “C” shape of the tropospheric RH profile. It also predicts that RH is an invariant function of temperature as the atmosphere warms, and this behavior matches what has been seen in global climate models and what is demonstrated here with cloud-resolving simulations. Extending the theory to include the evaporation of hydrometeors, a lower bound is derived for the precipitation efficiency (PE) at each height: PE > 1 − RH. In a cloud-resolving simulation, this constraint is obeyed with the PE profile taking the shape of an inverted C shape.