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## Abstract

Using Albrecht's model, approximate analytical formulas are found for the dependence of the steady-State mean thermodynamic structure of a partly cloudy convective marine boundary layer on external parameters. Our goals are 1) to understand the physical factors that influence the vertical profiles of mean relative humidity, temperature, and fractional cloudiness within the cloud layer using the model to gain insight into the strato-cumulus-trade cumulus transition in the subtropical trade wind regime, and 2) to understand the sensitivity of the model to tunable internal parameters. The model, a prototype for bulk models of trade cumulus boundary layer, consists of a well-mixed subcloud layer topped by a cumulus layer and a sharp trade inversion. In the simplest formulation discussed here, precipitation is ignored and simple parameterizations for radiative cooling and fractional cloudiness are used.

The analytical approximation agrees well with exact steady-state numerical solutions of Albrecht's model. The cloud-base and trade-inversion heights are not strongly dependent on adjustable parameters within the cloud model and are largely determined by bulk balances of radiative fluxes, surface fluxes, and subsidence in a manner similar to the more empirical model of Betts and Ridgway. The cloud-layer sounding and the cloud fraction are affected by external parameters only through changes in the cloud-base latent heat flux and the cloud thickness. The cloud fraction is quite sensitive to two tunable internal constants in the cloud model that affect rates of cloud entrainment and detrainment, respectively. For most choices of SST and upper-air conditions, these constants can be tuned to produce either a mainly saturated (stratocumulus-like) cloud layer or a trade cumulus-like layer with no environmental saturation. The sensitivity of cloud fraction to SST and mean subsidence is explored for two choices of these constants and the effect of unsteadiness due to downstream changes in external conditions are considered.

## Abstract

Using Albrecht's model, approximate analytical formulas are found for the dependence of the steady-State mean thermodynamic structure of a partly cloudy convective marine boundary layer on external parameters. Our goals are 1) to understand the physical factors that influence the vertical profiles of mean relative humidity, temperature, and fractional cloudiness within the cloud layer using the model to gain insight into the strato-cumulus-trade cumulus transition in the subtropical trade wind regime, and 2) to understand the sensitivity of the model to tunable internal parameters. The model, a prototype for bulk models of trade cumulus boundary layer, consists of a well-mixed subcloud layer topped by a cumulus layer and a sharp trade inversion. In the simplest formulation discussed here, precipitation is ignored and simple parameterizations for radiative cooling and fractional cloudiness are used.

The analytical approximation agrees well with exact steady-state numerical solutions of Albrecht's model. The cloud-base and trade-inversion heights are not strongly dependent on adjustable parameters within the cloud model and are largely determined by bulk balances of radiative fluxes, surface fluxes, and subsidence in a manner similar to the more empirical model of Betts and Ridgway. The cloud-layer sounding and the cloud fraction are affected by external parameters only through changes in the cloud-base latent heat flux and the cloud thickness. The cloud fraction is quite sensitive to two tunable internal constants in the cloud model that affect rates of cloud entrainment and detrainment, respectively. For most choices of SST and upper-air conditions, these constants can be tuned to produce either a mainly saturated (stratocumulus-like) cloud layer or a trade cumulus-like layer with no environmental saturation. The sensitivity of cloud fraction to SST and mean subsidence is explored for two choices of these constants and the effect of unsteadiness due to downstream changes in external conditions are considered.

## Abstract

No abstract available.

## Abstract

No abstract available.

## Abstract

In Part I, an idealized model of nonprecipitating moist convection in a shallow conditionally unstable layer of viscous and diffusive air between two parallel plates was introduced, and the “linear” instability of an exactly saturated static state maintained by diffusion was investigated. If there are initially many clouds, the “linear” theory predicted that weaker clouds are suppressed by the subsidence warming and drying from the ever-growing stronger clouds, and the average cloud spacing becomes arbitrarily large as time goes on. Each growing cloud is surrounded by compensating subsidence, which decreases away from the cloud with a characteristic decay scale *R _{s}
*, the subsidence radius, which can be understood from gravity wave arguments.

In Part II, fields of finite amplitude clouds are considered. An asymptotic analysis is performed in which the moist Rayleigh number *N _{c}
*

^{2}exceeds by only a small amount μ the value

*N*

_{ c0}

^{2}necessary for the onset of convection. This leads to a nonlinear set of “cloud field equations” which predict how the amplitudes and positions of all the clouds evolve in time. These equations predict a minimum stable cloud spacing λ

_{ c }≈

*R*log(μ

_{s}^{−1}). If the cloud spacing λ < λ

_{ c }, slight differences in the strengths of neighboring clouds increase until the weaker clouds are suppressed. Unevenly spaced clouds drift until they become evenly spaced, ultimately resulting in a steady field of identical clouds with uniform spacing λ > λ

_{ c }.

Numerical experiments with dry stability *N _{d}
* =

*N*corroborate the conclusions from the cloud field equations when

_{c}*N*

_{c}^{2}/

*N*

_{ c0}

^{2}is less than ten. As

*N*

_{c}^{2}increases, the numerically determined λ

_{ c }. becomes approximately 1.8

*R*≈ 1.8

_{s}*N*. There is a second threshold spacing λ

_{d}_{ t }≈ 1.6

*N*<λ

_{d}_{ c }not predicted by the asymptotic theory, below which a field of identical growing clouds is transient. This leads to two types of cloud field evolution. If

*N*

_{c}^{2}/

*N*

_{ c0}

^{2}is less than 10, all initial conditions lead to steady uniformly spaced fields of identical clouds. If

*N*

_{c}^{2}/

*N*

_{ c0}

^{2}is on the order of 10 or larger, a field of clouds initiated by horizontally homogeneous random buoyancy perturbations rapidly grows. While it is growing the subsidence radius around each cloud remains O(1); clouds are quite closely spaced. As the clouds mature,

*R*increases rapidly to

_{s}*N*. The clouds are spaced much closer than λ

_{d}_{ t }apart, so they all dissipate. If the initial conditions are less random, however, so that a few widely spaced clouds break out first, these clouds inhibit the convection which later grows around them, ultimately become steady and drift toward a uniform spacing. In both cases there is no tendency toward cloud clustering.

The steady cloud fields predicted by the model are probably never realized in the atmosphere due to other physical processes such as boundary layer forcing or precipitation, which favor small cloud spacings despite the large Rayleigh number. The primary conclusion that one can draw from the model is that compensating motions in the cloud layer are always competing with these other processes, tending to increase the spacing between convective clouds as subsidence-induced warming and drying suppresses the weaker circulations.

## Abstract

In Part I, an idealized model of nonprecipitating moist convection in a shallow conditionally unstable layer of viscous and diffusive air between two parallel plates was introduced, and the “linear” instability of an exactly saturated static state maintained by diffusion was investigated. If there are initially many clouds, the “linear” theory predicted that weaker clouds are suppressed by the subsidence warming and drying from the ever-growing stronger clouds, and the average cloud spacing becomes arbitrarily large as time goes on. Each growing cloud is surrounded by compensating subsidence, which decreases away from the cloud with a characteristic decay scale *R _{s}
*, the subsidence radius, which can be understood from gravity wave arguments.

In Part II, fields of finite amplitude clouds are considered. An asymptotic analysis is performed in which the moist Rayleigh number *N _{c}
*

^{2}exceeds by only a small amount μ the value

*N*

_{ c0}

^{2}necessary for the onset of convection. This leads to a nonlinear set of “cloud field equations” which predict how the amplitudes and positions of all the clouds evolve in time. These equations predict a minimum stable cloud spacing λ

_{ c }≈

*R*log(μ

_{s}^{−1}). If the cloud spacing λ < λ

_{ c }, slight differences in the strengths of neighboring clouds increase until the weaker clouds are suppressed. Unevenly spaced clouds drift until they become evenly spaced, ultimately resulting in a steady field of identical clouds with uniform spacing λ > λ

_{ c }.

Numerical experiments with dry stability *N _{d}
* =

*N*corroborate the conclusions from the cloud field equations when

_{c}*N*

_{c}^{2}/

*N*

_{ c0}

^{2}is less than ten. As

*N*

_{c}^{2}increases, the numerically determined λ

_{ c }. becomes approximately 1.8

*R*≈ 1.8

_{s}*N*. There is a second threshold spacing λ

_{d}_{ t }≈ 1.6

*N*<λ

_{d}_{ c }not predicted by the asymptotic theory, below which a field of identical growing clouds is transient. This leads to two types of cloud field evolution. If

*N*

_{c}^{2}/

*N*

_{ c0}

^{2}is less than 10, all initial conditions lead to steady uniformly spaced fields of identical clouds. If

*N*

_{c}^{2}/

*N*

_{ c0}

^{2}is on the order of 10 or larger, a field of clouds initiated by horizontally homogeneous random buoyancy perturbations rapidly grows. While it is growing the subsidence radius around each cloud remains O(1); clouds are quite closely spaced. As the clouds mature,

*R*increases rapidly to

_{s}*N*. The clouds are spaced much closer than λ

_{d}_{ t }apart, so they all dissipate. If the initial conditions are less random, however, so that a few widely spaced clouds break out first, these clouds inhibit the convection which later grows around them, ultimately become steady and drift toward a uniform spacing. In both cases there is no tendency toward cloud clustering.

The steady cloud fields predicted by the model are probably never realized in the atmosphere due to other physical processes such as boundary layer forcing or precipitation, which favor small cloud spacings despite the large Rayleigh number. The primary conclusion that one can draw from the model is that compensating motions in the cloud layer are always competing with these other processes, tending to increase the spacing between convective clouds as subsidence-induced warming and drying suppresses the weaker circulations.

## Abstract

A defining feature of moist convection is latent heating. A simple, mathematically tractable but thermody-namically reasonable Kuo-type model is developed to isolate some important effects of latent heating on the structure and organization of moist convection. Convection in a shallow, unsheared layer of viscous moist air between two rigid horizontal plates is examined. Unlike previous analytical work, a realistic thermodynamic equation is used, based on the assumption that no precipitation falls out of saturated air. This assumption isolates reversible latent heating from the complicating effects of precipitation. The crucial step is to express the buoyancy of moist air as a simple function of adiabatically conserved, linearly mixing properties of the air; this function is different in saturated than in unsaturated air.

In Part I, the new model is used to find analytical solutions for infinitesimal motions in a conditionally unstable, exactly saturated atmosphere. As in previous work, the most unstable circulation is an isolated, cylin-drical, single updraft cloud, surrounded by an infinite expanse of subsiding clear air. In contrast to earlier work, strong downdrafts occur *inside* the cloud near its edge. In a separate note, it will be shown that all growing circulations of infinitesimal amplitude are station-no “linear” wave-CISK is possible.

The most important prediction is that subsidence decays exponentially away from the cloud in a horizontal distance *R _{s}
*. A simple approximate formula for

*R*in terms of the growth rate, viscosity, and Coriolis parameter is derived and rationalized. The prediction of infinite cloud spacing will be resolved by a theory of finite-amplitude convection developed analytically in Part II and numerically in Part III, which predicts a finite minimum cloud spacing related to

_{s}*R*and the strength of convection.

_{s}## Abstract

A defining feature of moist convection is latent heating. A simple, mathematically tractable but thermody-namically reasonable Kuo-type model is developed to isolate some important effects of latent heating on the structure and organization of moist convection. Convection in a shallow, unsheared layer of viscous moist air between two rigid horizontal plates is examined. Unlike previous analytical work, a realistic thermodynamic equation is used, based on the assumption that no precipitation falls out of saturated air. This assumption isolates reversible latent heating from the complicating effects of precipitation. The crucial step is to express the buoyancy of moist air as a simple function of adiabatically conserved, linearly mixing properties of the air; this function is different in saturated than in unsaturated air.

In Part I, the new model is used to find analytical solutions for infinitesimal motions in a conditionally unstable, exactly saturated atmosphere. As in previous work, the most unstable circulation is an isolated, cylin-drical, single updraft cloud, surrounded by an infinite expanse of subsiding clear air. In contrast to earlier work, strong downdrafts occur *inside* the cloud near its edge. In a separate note, it will be shown that all growing circulations of infinitesimal amplitude are station-no “linear” wave-CISK is possible.

The most important prediction is that subsidence decays exponentially away from the cloud in a horizontal distance *R _{s}
*. A simple approximate formula for

*R*in terms of the growth rate, viscosity, and Coriolis parameter is derived and rationalized. The prediction of infinite cloud spacing will be resolved by a theory of finite-amplitude convection developed analytically in Part II and numerically in Part III, which predicts a finite minimum cloud spacing related to

_{s}*R*and the strength of convection.

_{s}## Abstract

Integral methods are used to show that in a simple model of nonprecipitating, moist convection, no small-amplitude propagating or oscillatory, two- or three-dimensional convective instabilities can grow from a quiescent basic state. This result holds for an inviscid layer with vertically varying stratification (of either finite or infinite depth) and for a uniformly stratified viscous layer of finite depth. The implications for the theory of wave-CISK are discussed.

## Abstract

Integral methods are used to show that in a simple model of nonprecipitating, moist convection, no small-amplitude propagating or oscillatory, two- or three-dimensional convective instabilities can grow from a quiescent basic state. This result holds for an inviscid layer with vertically varying stratification (of either finite or infinite depth) and for a uniformly stratified viscous layer of finite depth. The implications for the theory of wave-CISK are discussed.

## Abstract

The fastest growing modes in Emanuel's recent model of inviscid precipitating convection in a saturated atmosphere with no shear have large horizontal wavenumbers, permitting a simple analytical analysis of their phase speed, growth rate, and modal structure as functions of the model parameters. It is found that the fastest growing modes are always localized near the ground, are slightly sloped, and grow almost as fast as if precipitation were to fall instantly to the ground. In any wind profile with weak unidirectional shear, rolls aligned along the shear vector grow fastest.

## Abstract

The fastest growing modes in Emanuel's recent model of inviscid precipitating convection in a saturated atmosphere with no shear have large horizontal wavenumbers, permitting a simple analytical analysis of their phase speed, growth rate, and modal structure as functions of the model parameters. It is found that the fastest growing modes are always localized near the ground, are slightly sloped, and grow almost as fast as if precipitation were to fall instantly to the ground. In any wind profile with weak unidirectional shear, rolls aligned along the shear vector grow fastest.

## Abstract

The authors investigate the hypothesis that horizontal moisture advection is critical to the eastward propagation of the Madden–Julian oscillation (MJO). Consistent diagnostic evidence has been found in recent MJO-permitting global models viewed from the moisture-mode dynamical paradigm. To test this idea in a causal sense, tropical moisture advection by vorticity anomalies is artificially modulated in a superparameterized global model known to produce a realistic MJO signal. Boosting horizontal moisture advection by tropical vorticity anomalies accelerates and amplifies the simulated MJO in tandem with reduced environmental gross moist stability. Limiting rotational horizontal moisture advection shuts the MJO down. These sensitivities are robust in that they are nearly monotonic with respect to the control parameter and emerge despite basic-state sensitivities favoring the opposite response. Speedup confirms what several diagnostic lines of evidence already suggest—that anomalous moisture advection is fundamental to MJO propagation. The rotational component is shown to be especially critical. Amplification further suggests it may play a role in adiabatically maintaining the MJO.

## Abstract

The authors investigate the hypothesis that horizontal moisture advection is critical to the eastward propagation of the Madden–Julian oscillation (MJO). Consistent diagnostic evidence has been found in recent MJO-permitting global models viewed from the moisture-mode dynamical paradigm. To test this idea in a causal sense, tropical moisture advection by vorticity anomalies is artificially modulated in a superparameterized global model known to produce a realistic MJO signal. Boosting horizontal moisture advection by tropical vorticity anomalies accelerates and amplifies the simulated MJO in tandem with reduced environmental gross moist stability. Limiting rotational horizontal moisture advection shuts the MJO down. These sensitivities are robust in that they are nearly monotonic with respect to the control parameter and emerge despite basic-state sensitivities favoring the opposite response. Speedup confirms what several diagnostic lines of evidence already suggest—that anomalous moisture advection is fundamental to MJO propagation. The rotational component is shown to be especially critical. Amplification further suggests it may play a role in adiabatically maintaining the MJO.

## Abstract

A new general purpose boundary layer parameterization that permits realistic treatment of stratocumulus-capped boundary layers (SCBLs) with coarse vertical resolution is described. It combines a 1.5-order turbulent closure model with an entrainment closure at the boundary layer top. Three different implementations of the entrainment closure, in which the boundary layer height is respectively prognosed, reconstructed from thermodynamic values at the grid points, or restricted to lie on a flux level of the host model grid, are tested in a single-column modeling framework at both fine and coarse vertical resolution. The first two approaches permit a stratocumulus top and base to lie between grid levels and evolve continuously with time, but are more complicated to implement in a three-dimensional model.

The model performs very well in cases of dry convection, whatever the inversion implementation and the vertical resolution. With 15-mb or better vertical resolution, all approaches properly simulate mixing in SCBLs, including daytime cloud thinning and a transition to decoupling and conditional instability as SST increases. With coarser resolution, details of the implementation influence the simulated cloud thickness, which is systematically underestimated with the restricted inversion approach. A method for computing vertical advective fluxes at the boundary layer top that explicitly accounts for the inversion is presented; an essential component of the reconstructed inversion implementation, this vertical advection scheme also improves SCBL simulation at low resolution with a restricted inversion. For comprehensive simulation of boundary layer convection, this scheme should be coupled with a parameterization of shallow cumulus convection; this will be described in a forthcoming paper.

## Abstract

A new general purpose boundary layer parameterization that permits realistic treatment of stratocumulus-capped boundary layers (SCBLs) with coarse vertical resolution is described. It combines a 1.5-order turbulent closure model with an entrainment closure at the boundary layer top. Three different implementations of the entrainment closure, in which the boundary layer height is respectively prognosed, reconstructed from thermodynamic values at the grid points, or restricted to lie on a flux level of the host model grid, are tested in a single-column modeling framework at both fine and coarse vertical resolution. The first two approaches permit a stratocumulus top and base to lie between grid levels and evolve continuously with time, but are more complicated to implement in a three-dimensional model.

The model performs very well in cases of dry convection, whatever the inversion implementation and the vertical resolution. With 15-mb or better vertical resolution, all approaches properly simulate mixing in SCBLs, including daytime cloud thinning and a transition to decoupling and conditional instability as SST increases. With coarser resolution, details of the implementation influence the simulated cloud thickness, which is systematically underestimated with the restricted inversion approach. A method for computing vertical advective fluxes at the boundary layer top that explicitly accounts for the inversion is presented; an essential component of the reconstructed inversion implementation, this vertical advection scheme also improves SCBL simulation at low resolution with a restricted inversion. For comprehensive simulation of boundary layer convection, this scheme should be coupled with a parameterization of shallow cumulus convection; this will be described in a forthcoming paper.

## Abstract

Decoupling during the “Lagrangian” evolution of a cloud-topped boundary layer advected equatorward by the trade winds in an idealized eastern subtropical ocean is studied using a mixed-layer model (MLM). The sea surface temperature is gradually warmed while the free tropospheric sounding remains unchanged, causing the boundary layer to deepen, the surface relative humidity to decrease, and surface latent heat fluxes to increase. Diurnally averaged insolation is used.

For entrainment closures in which entrainment rate is related to a large-eddy convective velocity scale *w**, the MLM predicts an increasingly prominent layer of negative buoyancy fluxes below cloud base as the sea surface temperature warms. Decoupling of the mixed layer can be inferred when the MLM-predicted negative buoyancy fluxes become too large for the internal circulations to sustain. The authors show that decoupling is mainly driven by an increasing ratio of the surface latent heat flux to the net radiative cooling in the cloud, and derive a decoupling criterion based on this ratio. Other effects such as drizzle, the vertical distribution of radiative cooling in the cloud, and sensible heat fluxes, also affect decoupling but are shown to be less important in typical subtropical boundary layers. A comparison of MLM results with a companion numerical study with a cloud-resolving model shows that the decoupling process is similar and the same decoupling criterion applies. A regional analysis of decoupling using Lagrangian trajectories based on summertime northeast Pacific climatology predicts decoupling throughout the subtropical stratocumulus region except in coastal zones where the boundary layer is under 750 m deep.

A “flux-partitioning” entrainment closure, in which the entrainment rate is chosen to maintain a specified ratio of some measure of negative subcloud buoyancy fluxes to positive buoyancy fluxes within the cloud and near the surface, was also considered. By construction, such an MLM never predicts its own breakdown by decoupling. The changed entrainment closure had only a minor influence on the boundary layer evolution and entrainment rate. Instead, the crucial impact of the entrainment closure is on predicting when and where the mixed-layer assumption will break down due to decoupling.

## Abstract

Decoupling during the “Lagrangian” evolution of a cloud-topped boundary layer advected equatorward by the trade winds in an idealized eastern subtropical ocean is studied using a mixed-layer model (MLM). The sea surface temperature is gradually warmed while the free tropospheric sounding remains unchanged, causing the boundary layer to deepen, the surface relative humidity to decrease, and surface latent heat fluxes to increase. Diurnally averaged insolation is used.

For entrainment closures in which entrainment rate is related to a large-eddy convective velocity scale *w**, the MLM predicts an increasingly prominent layer of negative buoyancy fluxes below cloud base as the sea surface temperature warms. Decoupling of the mixed layer can be inferred when the MLM-predicted negative buoyancy fluxes become too large for the internal circulations to sustain. The authors show that decoupling is mainly driven by an increasing ratio of the surface latent heat flux to the net radiative cooling in the cloud, and derive a decoupling criterion based on this ratio. Other effects such as drizzle, the vertical distribution of radiative cooling in the cloud, and sensible heat fluxes, also affect decoupling but are shown to be less important in typical subtropical boundary layers. A comparison of MLM results with a companion numerical study with a cloud-resolving model shows that the decoupling process is similar and the same decoupling criterion applies. A regional analysis of decoupling using Lagrangian trajectories based on summertime northeast Pacific climatology predicts decoupling throughout the subtropical stratocumulus region except in coastal zones where the boundary layer is under 750 m deep.

A “flux-partitioning” entrainment closure, in which the entrainment rate is chosen to maintain a specified ratio of some measure of negative subcloud buoyancy fluxes to positive buoyancy fluxes within the cloud and near the surface, was also considered. By construction, such an MLM never predicts its own breakdown by decoupling. The changed entrainment closure had only a minor influence on the boundary layer evolution and entrainment rate. Instead, the crucial impact of the entrainment closure is on predicting when and where the mixed-layer assumption will break down due to decoupling.

## Abstract

Mesoscale variability in entrainment across the inversion capping the cloud-topped atmospheric boundary layer (CTBL) has been proposed as an explanation for mesoscale variability in cloud thickness. The relevance of this mechanism, called mesoscale entrainment instability, or MEI, to some typical atmospheric boundary layers is investigated. The results indicate that MEI is of relevance only if the potential temperature jump across the inversion is small, ∼1–2 K, and the stable layer virtual potential temperature above is very strongly stratified, ∼40 K km^{−1}. Thus, MEI does not appear to be a viable explanation for mesoscale cellular convection in most CTBLs.

Two parameters are also investigated whose effect on the growth rate can be substantial. They are the rate of horizontal turbulent diffusion and the effect of variations in solar heating due to cloud thickness fluctuations. Decreases in the horizontal turbulent transport rate do not greatly affect the growth rate but can substantially decrease the wavelength of the instability. The solar heating effect can as much as double the growth rate but probably not enough to make the instability significant.

## Abstract

Mesoscale variability in entrainment across the inversion capping the cloud-topped atmospheric boundary layer (CTBL) has been proposed as an explanation for mesoscale variability in cloud thickness. The relevance of this mechanism, called mesoscale entrainment instability, or MEI, to some typical atmospheric boundary layers is investigated. The results indicate that MEI is of relevance only if the potential temperature jump across the inversion is small, ∼1–2 K, and the stable layer virtual potential temperature above is very strongly stratified, ∼40 K km^{−1}. Thus, MEI does not appear to be a viable explanation for mesoscale cellular convection in most CTBLs.

Two parameters are also investigated whose effect on the growth rate can be substantial. They are the rate of horizontal turbulent diffusion and the effect of variations in solar heating due to cloud thickness fluctuations. Decreases in the horizontal turbulent transport rate do not greatly affect the growth rate but can substantially decrease the wavelength of the instability. The solar heating effect can as much as double the growth rate but probably not enough to make the instability significant.