Annotated List of Vincent Larson's Publications

1999:  “The Relationship Between the Transilient Matrix and the Green’s Function for the Advection-Diffusion Equation.” V. E. Larson.  J. Atmos. Sci., 56, 2447-2453.  (See also slide 7 of the following presentation.)

The point of this paper is that if a single-column model contains information only about horizontal averages, it discards crucial information about  horizontal structure.  For instance, a single-column model may predict the average concentration of a pollutant at some altitude perfectly.  But a model that only predicts averages doesn't know whether the pollutant resides in an updraft or downdraft.  Hence the model doesn't know whether to transport the pollutant up or down at the next time step.  Therefore, the transport prediction degrades rapidly.  This problem was termed ``convective structure memory" by Roland Stull.  It can be quantified using Green's function theory.  This problem is part of the motivation for the papers below on probability density functions (PDFs).  PDFs do contain information beyond the horizontal averages.

2001: ``Systematic Biases in the Microphysics and Thermodynamics of Numerical Models that Ignore Subgrid Variability."  V. E. Larson, R. Wood, P. R. Field, J.-C. Golaz, T. H. Vonder Haar, W. R. Cotton.  J. Atmos. Sci., 58, 1117-1128.  (See also slides 8-12 of the following presentation, and this short conference paper.)

The paper above explores some systematic biases related to cloud processes. A systematic bias is a particular kind of error, namely one that always has the same sign.  A systematic bias is more harmful than a series of ordinary errors, because a systematic bias can never partially self-cancel when averaged over space or time.  

If a numerical model ignores variability on scales smaller than the grid box size, then a systematic bias can sometimes occur.  These biases are associated with convex functions.  One can find systematic biases by use of a theorem known as Jensen's inequality. 

2001: ``Small-scale and Mesoscale Variability of Scalars in Cloudy Boundary Layers: One-dimensional Probability Density Functions." V. E. Larson, R. Wood, P. R. Field, J.-C. Golaz, and T. H. Vonder Haar. J. Atmos. Sci.,  58, 1978--1994.  (See also the following short conference paper.)

The systematic biases mentioned above can be removed if a model can predict the appropriate probability density function (PDF).  A PDF is essentially a histogram.  It indicates the probability of finding a particular value of specific humidity, for instance, in a computational grid box.  This paper examines PDFs of clouds.  Some look quite complicated, with a long tail on the right or the left.  However, the level of complexity is manageable (we hope!).

2002: ``Small-Scale and Mesoscale Variability in Cloudy Boundary Layers: Joint Probability Density Functions."  V. E. Larson, J.-C. Golaz, W. R. Cotton.  J. Atmos. Sci., 59, 3519-3539.  (See also the following short conference paper.)

Whereas the prior paper discusses one-dimensional PDFs of cloud water and humidity, this paper discusses joint PDFs that include the vertical velocity.  Joint PDFs allow us to diagnose the buoyancy flux, which is the means by which convection generates turbulence.  Joint PDFs also allow us to diagnose fluxes of heat and moisture.  Therefore, joint PDFs can serve as the foundation of cloud and turbulence parameterizations in numerical models, as proposed and explored in the two following papers.  

2002: ``A PDF-Based Model for Boundary Layer Clouds.  Part I: Method and Model Description."  J.-C. Golaz, V. E. Larson, W. R. Cotton.  J. Atmos. Sci., 59, 3540-3551.  

2002: ``A PDF-Based Model for Boundary Layer Clouds.  Part II: Model Results."  J.-C. Golaz, V. E. Larson, W. R. Cotton.  J. Atmos. Sci., 59, 3552-3571.  

(See also slides 13-35 of the following presentation, and this short conference paper.)

Traditionally, cloud parameterization has been viewed as a multiplicity of tasks.  Such tasks include the prediction of heat flux, moisture flux, cloud fraction, and liquid water.  In contrast, the papers above adopt the alternative viewpoint that the goal of parameterization consists largely of a single task: the prediction of the joint PDF of vertical velocity, heat, and moisture.  Once the PDF is given, the fluxes, cloud fraction, and liquid water can be diagnosed.  

The above papers present a parameterization that can model both stratocumulus and cumulus clouds without case-specific adjustments.  This avoids the difficulty of having to construct a ``trigger function" that determines which cloud type should be modeled under which meteorological conditions.  

2005: ``Using Probability Density Functions to Derive Consistent Closure Relationships among Higher-Order Moments."  V. E. Larson and J.-C. Golaz.  Mon. Wea. Rev., 133, 1023-1042.  (See also slides 26-27 of the following presentation.)

The aforementioned papers show that if we choose an accurate PDF family, then we can solve for many of the unknowns in our one-dimensional cloud parameterization.  For some of these unknown terms, the present paper lists simple, analytic approximations.  All approximated formulas are based on the same PDF and hence are consistent with each other. 

A PDF may be constructed from a set of means, variances, and other moments of velocity, moisture, and temperature.  It is possible that a particular set of moments does not correspond to any real PDF in the family.  We call such a set of moments ``specifically unrealizable."  For instance, a set that includes asymmetric moments is specifically unrealizable with respect a PDF family of symmetric, bell-shaped curves.  This is because the bell shape family is too restrictive to include asymmetric moments.  We show that a broad class of moments is specifically realizable with respect to our PDF family.  That is, our PDF family is not restrictive. 

2007: ``From Cloud Overlap to PDF Overlap."  V. E. Larson.  Q. J. R. Meteorol. Soc., 133, 1877-1891.  (See also the following presentation.)

2005: ``Supplying Local Microphysics Parameterizations with Information about Subgrid Variability: Latin Hypercube Sampling."  V. E. Larson, J.-C. Golaz, H. Jiang, and W. R. Cotton.  J. Atmos. Sci., 62, 4010-4026. (See also slides 36-60 of the following presentation.)

One reason to predict the subgrid PDF is to drive microphysical parameterizations more accurately.  The most accurate way to do this is to integrate the relevant microphysical formulas over the PDF.  However, this may be analytically intractable or may require rewriting the microphysics code.  To avoid this, one may draw sample points from the PDF and input them into the microphysics code one at a time.  This is cheap and allows the use of existing microphysics codes, but it also introduces statistical noise due to imperfect sampling.  To reduce the noise, this paper proposes spreading out the sample points in a quasi-random fashion using "Latin hypercube sampling."

2007: ``A Single-Column Model Intercomparison of a Heavily Drizzling Stratocumulus-Topped Boundary Layer."  M. C. Wyant and Co-Authors.  J. Geophys. Res., 112, D24204, doi:10.1029/2007JD008536.

This paper compared the output from numerous single-column model that were all set up identically to simulate a cloud layer observed during the DYCOMS-II field experiment.  Part of the challenge was simulating drizzle.  In order to couple drizzle to the cloud fields, instead of drawing sample points from the PDF using the Latin hypercube method discussed above, we analytically integrated over the PDF.  This avoids the statistical noise inherent in Monte Carlo methods such a Latin hypercube sampling.   However, the analytic integration can be performed only if the drizzle equations are simple enough. 

2007: ``Elucidating model inadequacies in a cloud parameterization by use of an ensemble-based calibration framework."  J.-C. Golaz, V. E. Larson, J. A. Hansen, D. P. Schanen, and B. M. Griffin.  Mon. Wea. Rev., 135, 4077-4096. (See also the following oral presentation or slides, and this conference paper.)

It is often easy to see when an atmospheric model disagrees with data. It is usually much harder to locate the ultimate sources of model error.

It is particularly difficult to diagnose errors in a model's structure, that is, errors in the functional form of the model equations. One technique that may help is parameter estimation, that is, the optimization of model parameter values. Typically, parameter estimation is used solely to improve the fit between a model and observational data. In the process, however, parameter estimation may cover up structural model errors.

In a quite opposite application, parameter estimation may be used to uncover the ways in which a model is wrong. The basic idea is to separately optimize model parameters to two different data sets, and then identify parameter values that differ between the two optimizations. When no single value of a particular parameter fits both datasets, then there must exist a related structural error.

2004: ``Prognostic Equations for Cloud Fraction and Liquid Water, and Their Relation to Filtered Density Functions."  V. E. Larson.  J. Atmos. Sci., 61, 338-351.  (See also slides 29-32 of the following  presentation, and this short conference paper.)

This paper derives equations for cloud fraction and liquid water content.  Such equations are used in numerical models that cannot resolve all variability in wind, heat, and moisture because their grid boxes are too large (e.g., Tiedtke 1993).  The derivations show that closure of these equations requires information about the PDF of vertical velocity, heat, and moisture.  

Dry atmospheres in radiative-convective equilibrium: The goal of the two papers below is to move theory one step away from Rayleigh-Benard convection, which has proved so fruitful for understanding of buoyant fluids, and one step closer to atmospheric convection.  The problem considered here adds infrared radiation to the classical problem of fluid flow between two plates, the lower being heated and the upper being cooled.  When radiation is added, the stability properties do not change qualitatively as long as one substitutes a radiative Rayleigh number for the classical Rayleigh number.  However, when fluid motion occurs, the turbulent heat flux does change because the heat flux is strongly constrained by radiation.