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Christopher S. Bretherton

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Christopher S. Bretherton

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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.

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Christopher S. Bretherton

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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 Rs, 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 Nc 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 λcRslog(μ−1). If the cloud spacing λ < λc, 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 > λc.

Numerical experiments with dry stability Nd = Nc corroborate the conclusions from the cloud field equations when Nc 2/N c0 2 is less than ten. As Nc 2 increases, the numerically determined λc. becomes approximately 1.8Rs ≈ 1.8Nd. There is a second threshold spacing λt ≈ 1.6Ndc 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 Nc 2/N c0 2 is less than 10, all initial conditions lead to steady uniformly spaced fields of identical clouds. If Nc 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, Rs increases rapidly to Nd. The clouds are spaced much closer than λt 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.

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Christopher S. Bretherton

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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.

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Christopher S. Bretherton

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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.

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Christopher S. Bretherton

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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 Rs. A simple approximate formula for Rs 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 Rs and the strength of convection.

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Peter Caldwell and Christopher S. Bretherton

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This paper describes a series of 6-day large eddy simulations of a deep, sometimes drizzling stratocumulus-topped boundary layer based on forcings from the East Pacific Investigation of Climate (EPIC) 2001 field campaign. The base simulation was found to reproduce the observed mean boundary layer properties quite well. The diurnal cycle of liquid water path was also well captured, although good agreement appears to result partially from compensating errors in the diurnal cycles of cloud base and cloud top due to overentrainment around midday. At all times of the day, entrainment is found to be proportional to the vertically integrated buoyancy flux. Model stratification matches observations well; turbulence profiles suggest that the boundary layer is always at least somewhat decoupled. Model drizzle appears to be too sensitive to liquid water path and subcloud evaporation appears to be too weak. Removing the diurnal cycle of subsidence had little effect on simulated liquid water path. Simulations with changed droplet concentration and drizzle susceptibility showed large liquid water path differences at night, but differences were quite small at midday. Droplet concentration also had a significant impact on entrainment, primarily through droplet sedimentation feedback rather than through drizzle processes.

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Gad Levy and Christopher S. Bretherton

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Zhiming Kuang and Christopher S. Bretherton

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In this paper, an idealized, high-resolution simulation of a gradually forced transition from shallow, nonprecipitating to deep, precipitating cumulus convection is described; how the cloud and transport statistics evolve as the convection deepens is explored; and the collected statistics are used to evaluate assumptions in current cumulus schemes. The statistical analysis methodologies that are used do not require tracing the history of individual clouds or air parcels; instead they rely on probing the ensemble characteristics of cumulus convection in the large model dataset. They appear to be an attractive way for analyzing outputs from cloud-resolving numerical experiments. Throughout the simulation, it is found that 1) the initial thermodynamic properties of the updrafts at the cloud base have rather tight distributions; 2) contrary to the assumption made in many cumulus schemes, nearly undiluted air parcels are too infrequent to be relevant to any stage of the simulated convection; and 3) a simple model with a spectrum of entraining plumes appears to reproduce most features of the cloudy updrafts, but significantly overpredicts the mass flux as the updrafts approach their levels of zero buoyancy. A buoyancy-sorting model was suggested as a potential remedy. The organized circulations of cold pools seem to create clouds with larger-sized bases and may correspondingly contribute to their smaller lateral entrainment rates. Our results do not support a mass-flux closure based solely on convective available potential energy (CAPE), and are in general agreement with a convective inhibition (CIN)-based closure. The general similarity in the ensemble characteristics of shallow and deep convection and the continuous evolution of the thermodynamic structure during the transition provide justification for developing a single unified cumulus parameterization that encompasses both shallow and deep convection.

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Gad Levy and Christopher S. Bretherton

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The governing vorticity and divergence equations in the surface layer are derived and the roles of the different terms and feedback mechanisms are investigated in semigeostrophic and nongeostrophic cold-frontal systems. A planetary boundary layer model is used to perform sensitivity tests to determine that in a cold front the ageostrophic feedback mechanism as defined by Orlanski and Ross tends to act as a positive feedback mechanism, enhancing vorticity and convergence growth. Therefore, it cannot explain the phase shift between convergence and vorticity as simulated by Orlanski and Ross. An alternative plausible, though tentative, explanation in terms of a gravity wave is offered. It is shown that when the geostrophic deformation increases, nonlinear terms in the divergence equation may become important and further destabilize the system.

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