Jean-Christophe Golaz

A PDF-Based Model for Boundary Layer Clouds. Part I: Method and Model Description

A new cloudy boundary layer single-column model is presented. It is designed to be flexible enough to represent a variety of cloudiness regimes—such as cumulus, stratocumulus, and clear regimes—without the need for case-specific adjustments. The methodology behind the model is the so-called assumed probability density function (PDF) method. The parameterization differs from higher-order closure or mass-flux schemes in that it achieves closure by the use of a relatively sophisticated joint PDF of vertical velocity, temperature, and moisture. A family of PDFs is chosen that is flexible enough to represent various cloudiness regimes. A double Gaussian family proposed by previous works is used. Predictive equations for grid box means and a number of higherorder turbulent moments are advanced in time. These moments are in turn used to select a particular member from the family of PDFs, for each time step and grid box. Once a PDF member has been selected, the scheme integrates over the PDF to close higher-order moments, buoyancy terms, and diagnose cloud fraction and liquid water. Since all the diagnosed moments for a given grid box and time step are derived from the same unique joint PDF, they are guaranteed to be consistent with one another. A companion paper presents simulations produced by the single-column model.

Systematic Biases in the Microphysics and Thermodynamics of Numerical Models That Ignore Subgrid-Scale Variability

A grid box in a numerical model that ignores subgrid variability has biases in certain microphysical and thermodynamic quantities relative to the values that would be obtained if subgrid-scale variability were taken into account. The biases are important because they are systematic and hence have cumulative effects. Several types of biases are discussed in this paper. Namely, numerical models that employ convex autoconversion formulas underpredict (or, more precisely, never overpredict) autoconversion rates, and numerical models that use convex functions to diagnose specific liquid water content and temperature underpredict these latter quantities. One may call these biases the ‘‘grid box average autoconversion bias,’’ ‘‘grid box average liquid water content bias,’’ and ‘‘grid box average temperature bias,’’ respectively, because the biases arise when grid box average values are substituted into formulas valid at a point, not over an extended volume. The existence of these biases can be derived from Jensen’s inequality. To assess the magnitude of the biases, the authors analyze observations of boundary layer clouds. Often the biases are small, but the observations demonstrate that the biases can be large in important cases. In addition, the authors prove that the average liquid water content and temperature of an isolated, partly cloudy, constant-pressure volume of air cannot increase, even temporarily. The proof assumes that liquid water content can be written as a convex function of conserved variables with equal diffusivities. The temperature decrease is due to evaporative cooling as cloudy and clear air mix. More generally, the authors prove that if an isolated volume of fluid contains conserved scalars with equal diffusivities, then the average of any convex, twice-differentiable function of the conserved scalars cannot increase.

Using Probability Density Functions to Derive Consistent Closure Relationships among Higher-Order Moments

Parameterizations of turbulence often predict several lower-order moments and make closure assumptions for higher-order moments. In principle, the low- and high-order moments share the same probability density function (PDF). One closure assumption, then, is the shape of this family of PDFs. When the higher-order moments involve both velocity and thermodynamic scalars, often the PDF shape has been assumed to be a double or triple delta function. This is equivalent to assuming a mass-flux model with no subplume variability. However, PDF families other than delta functions can be assumed. This is because the assumed PDF methodology is fairly general. This paper proposes closures for several third- and fourth-order moments. To derive the closures, the moments are assumed to be consistent with a particular PDF family, namely, a mixture of two trivariate Gaussians. (This PDF is also called a double Gaussian or binormal PDF by some authors.) Separately from the PDF assumption, the paper also proposes a simplified relationship between scalar and velocity skewnesses. This PDF family and skewness relationship are simple enough to yield simple, analytic closure formulas relating the moments. If certain conditions hold, this set of moments is specifically realizable. By this it is meant that the set of moments corresponds to a real Gaussian-mixture PDF, one that is normalized and nonnegative everywhere. This paper compares the new closure formulas with both large eddy simulations (LESs) and closures based on double and triple delta PDFs. This paper does not implement the closures in a single-column model and test them interactively. Rather, the comparisons are diagnostic; that is, low-order moments are extracted from the LES and treated as givens that are input into the closures. This isolates errors in the closures from errors in a single-column model. The test cases are three atmospheric boundary layers: a trade wind cumulus layer, a stratocumulus layer, and a clear convective case. The new closures have shortcomings, but nevertheless are superior to the double or triple delta closures in most of the cases tested.

Small-Scale and Mesoscale Variability of Scalars in Cloudy Boundary Layers: One-Dimensional Probability Density Functions

A key to parameterization of subgrid-scale processes is the probability density function (PDF) of conserved scalars. If the appropriate PDF is known, then grid box average cloud fraction, liquid water content, temperature, and autoconversion can be diagnosed. Despite the fundamental role of PDFs in parameterization, there have been few observational studies of conserved-scalar PDFs in clouds. The present work analyzes PDFs from boundary layers containing stratocumulus, cumulus, and cumulus-rising-into-stratocumulus clouds. Using observational aircraft data, the authors test eight different parameterizations of PDFs, including double delta function, gamma function, Gaussian, and double Gaussian shapes. The Gaussian parameterization, which depends on two parameters, fits most observed PDFs well but fails for large-scale PDFs of cumulus legs. In contrast, three-parameter parameterizations appear to be sufficiently general to model PDFs from a variety of cloudy boundary layers. If a numerical model ignores subgrid variability, the model has biases in diagnoses of grid box average liquid water content, temperature, and Kessler autoconversion, relative to the values it would obtain if subgrid variability were taken into account. The magnitude of such biases is assessed using observational data. The biases can be largely eliminated by three-parameter PDF parameterizations. Prior authors have suggested that boundary layer PDFs from short segments are approximately Gaussian. The present authors find that the hypothesis that PDFs of total specific water content are Gaussian can almost always be rejected for segments as small as 1 km.

A PDF-Based Model for Boundary Layer Clouds. Part II: Model Results

A new single-column model for the cloudy boundary layer, described in a companion paper, is tested for a variety of regimes. To represent the subgrid-scale variability, the model uses a joint probability density function (PDF) of vertical velocity, temperature, and moisture content. Results from four different cases are presented and contrasted with large eddy simulations (LES). The cases include a clear convective layer based on the Wangara experiment, a trade wind cumulus layer from the Barbados Oceanographic and Meteorological Experiment (BOMEX), a case of cumulus clouds over land, and a nocturnal marine stratocumulus boundary layer. Results from the Wangara experiment show that the model is capable of realistically predicting the diurnal growth of a dry convective layer. Compared to the LES, the layer produced is slightly less well mixed and entrainment is somewhat slower. The cloud cover in the cloudy cases varied widely, ranging from a few percent cloud cover to nearly overcast. In each of the cloudy cases, the parameterization predicted cloud fractions that agree reasonably well with the LES. Typically, cloud fraction values tended to be somewhat smaller in the parameterization, and cloud bases and tops were slightly underestimated. Liquid water content was generally within 40% of the LES-predicted values for a range of values spanning almost two orders of magnitude. This was accomplished without the use of any case-specific adjustments.

Supplying Local Microphysics Parameterizations with Information about Subgrid Variability: Latin Hypercube Sampling

One problem in computing cloud microphysical processes in coarse-resolution numerical models is that many microphysical processes are nonlinear and small in scale. Consequently, there are inaccuracies if microphysics parameterizations are forced with grid box averages of model fields, such as liquid water content. Rather, the model needs to determine information about subgrid variability and input it into the microphysics parameterization. One possible solution is to assume the shape of the family of probability density functions (PDFs) associated with a grid box and sample it using the Monte Carlo method. In this method, the microphysics subroutine is called repeatedly, once with each sample point. In this way, the Monte Carlo method acts as an interface between the host model’s dynamics and the microphysical parameterization. This avoids the need to rewrite the microphysics subroutines. A difficulty with the Monte Carlo method is that it introduces into the simulation statistical noise or variance, associated with the finite sample size. If the family of PDFs is tractable, one can sample solely from cloud, thereby improving estimates of in-cloud processes. If one wishes to mitigate the noise further, one needs a method for reduction of variance. One such method is Latin hypercube sampling, which reduces noise by spreading out the sample points in a quasi-random fashion. This paper formulates a sampling interface based on the Latin hypercube method. The associated family of PDFs is assumed to be a joint normal/lognormal (i.e., Gaussian/lognormal) mixture. This method of variance reduction has a couple of advantages. First, the method is general: the same interface can be used with a wide variety of microphysical parameterizations for various processes. Second, the method is flexible: one can arbitrarily specify the number of hydrometeor categories and the number of calls to the microphysics parameterization per grid box per time step. This paper performs a preliminary test of Latin hypercube sampling. As a prototypical microphysical formula, this paper uses the Kessler autoconversion formula. The PDFs that are sampled are extracted diagnostically from large-eddy simulations (LES). Both stratocumulus and cumulus boundary layer cases are tested. In this diagnostic test, the Latin hypercube can produce somewhat less noisy time-averaged estimates of Kessler autoconversion than a traditional Monte Carlo estimate, with no additional calls to the microphysics parameterization. However, the instantaneous estimates are no less noisy. This paper leaves unanswered the question of whether the Latin hypercube method will work well in a prognostic, interactive cloud model, but this question will be addressed in a future manuscript.

A large-eddy simulation study of cumulus clouds over land and sensitivity to soil moisture

A series of large-eddy simulations (LES) of non-precipitating cumulus clouds over land was performed. These simulations were idealized from observed conditions at the Southern Great Plains ARM site on 21 June 1997 and were intended to investigate the effect of initial soil moisture on the structure of the cloudy boundary layer. The surface fluxes were either dominated by latent heat or sensible heat flux, with the transition between one regime and the other occurring over a very narrow soil moisture range. The effect on clouds was mixed. Cloud fraction was nearly identical throughout all experiments. Simulations with dominant sensible heat fluxes led to more turbulent boundary layers and higher cloud bases. Simulations dominated by latent heat flux tended to have fewer but stronger updrafts in the cloud layer.

The Liquid Water Oscillation in Modeling Boundary Layer Cumuli with Third-Order Turbulence Closure Models

A hierarchy of third-order turbulence closure models are used to simulate boundary layer cumuli in this study. An unrealistically strong liquid water oscillation (LWO) is found in the fully prognostic model, which predicts all third moments. The LWO propagates from cloud base to cloud top with a speed of 1 m s21. The period of the oscillation is about 1000 s. Liquid water buoyancy (LWB) terms in the third-moment equations contribute to the LWO. The LWO mainly affects the vertical profiles of cloud fraction, mean liquid water mixing ratio, and the fluxes of liquid water potential temperature and total water, but has less impact on the vertical profiles of other second and third moments. In order to minimize the LWO, a moderately large diffusion coefficient and a large turbulent dissipation at its originating level are needed. However, this approach distorts the vertical distributions of cloud fraction and liquid water mixing ratio. A better approach is to parameterize LWB more reasonably. A minimally prognostic model, which diagnoses all third moments except for the vertical velocity, is shown to produce better results, compared to a fully prognostic model.

Evaluation of the Warm Rain Formation Process in Global Models with Satellite Observations

AbstractThis study examines the warm rain formation process over the global ocean in global climate models. Methodologies developed to analyze CloudSat and Moderate Resolution Imaging Spectroradiometer (MODIS) satellite observations are employed to investigate the cloud-to-precipitation process of warm clouds and are applied to the model results to examine how the models represent the process for warm stratiform clouds. Despite a limitation of the present study that compares the statistics for stratiform clouds in climate models with those from satellite observations, including both stratiform and (shallow) convective clouds, the statistics constructed with the methodologies are compared between the models and satellite observations to expose their similarities and differences. A problem common to some models is that they tend to produce rain at a faster rate than is observed. These model characteristics are further examined in the context of cloud microphysics parameterizations using a simplified one-dim...

Turbulence and Vertical Fluxes in the Stable Atmospheric Boundary Layer. Part II: A Novel Mixing-Length Model

This is the second part of a study about turbulence and vertical fluxes in the stable atmospheric boundary layer. Based on a suite of large-eddy simulations in Part I where the effects of stability on the turbulent structures and kinetic energy are investigated, first-order parameterization schemes are assessed and tested in the Geophysical Fluid Dynamics Laboratory (GFDL)’s single-column model. The applicability of the gradient-flux hypothesis is first examined and it is found that stable conditions are favorable for that hypothesis. However, the concept of introducing a stability correction function fm as a multiplicative factor into the mixing length used under neutral conditions lN is shown to be problematic because fm computed a priori from large-eddy simulations tends not to be a universal function of stability. With this observation, a novel mixing-length model is proposed, which conforms to large-eddy simulation results much better under stable conditions and converges to the classic model under neutral conditions. Test cases imposing steady as well as unsteadyforcingsaredevelopedtoevaluatetheperformanceofthenewmodel.Itisfoundthatthenewmodel exhibitsrobustperformanceas thestabilitystrengthis changed,while othermodels aresensitiveto changesin stability. For cases with unsteady forcings, which are very rarely simulated or tested, the results of the singlecolumn model and large-eddy simulations are also closer when the new model is used, compared to the other models. However, unsteady cases are much more challenging for the turbulence closure formulations than cases with steady surface forcing.