## 1. Introduction

[2] Arctic climate is influenced in strong and complex ways by mixed-phase Arctic clouds. We cite two examples here. First, mixed-phase Arctic clouds influence radiative transfer and are often observed to persist for long times [*Pinto*, 1998; *Prenni et al.*, 2007]. Several modeling studies suggest that this longevity is possible only if ice nuclei concentrations are limited in order to prevent ice concentrations from increasing and depleting liquid water [*Harrington et al.*, 1999; *Prenni et al.*, 2007]. Second, many Arctic cloud layers are thin enough to be partly transparent to longwave radiation. Because of this, some researchers have hypothesized that if Arctic clouds experience an increase in droplet number concentration, these clouds will emit more longwave radiation and hence cause a relative warming of the surface [*Garrett et al.*, 2002; *Garrett and Zhao*, 2006].

[3] Given this complexity, it is perhaps unsurprising that climate and regional simulations differ markedly from each other in their estimates of Arctic clouds [*Walsh et al.*, 2002; *Kattsov and Källén*, 2005; *Rinke et al.*, 2006; *Prenni et al.*, 2007]. For instance, *Kattsov and Källén* [2005] mention the “dramatic scatter between the total cloud amounts . [which] approaches 60% in winter” that is simulated by the climate simulations they examine. Furthermore, even if a model correctly predicts the presence of cloud, it may not necessarily predict the correct phase of water. The regional models studied by *Prenni et al.* [2007] severely underpredict liquid water in wintertime Arctic clouds, probably due to excessive ice nuclei concentrations in the simulations. These uncertainties in simulations of clouds lead to uncertainties in other components of the simulated Arctic climate.

[4] A key difficulty in improving regional and climate models is the difficulty of parameterizing small-scale spatial variability in hydrometeors. Regional and climate models have superkilometer horizontal grid spacings, but such large grid volumes contain considerable variability. This variability ought to be taken into account when driving microphysics. Therefore, developing accurate formulas for aerosol and microphysics is necessary, but not sufficient. Also needed is an accurate parameterization of subgrid variability that is implemented in a regional or climate “host” model. Given the resolved fields predicted by the host model, the parameterization would need to predict the relevant aspects of subgrid variability and feed them into a microphysics scheme [*Golaz et al.*, 2002; V. E. Larson and B. M. Griffin, Analytic upscaling of local microphysics parameterizations, part I: Theory, submitted to *Quarterly Journal of the Royal Meteorological Society*, 2011]. B. M. Griffin and V. E. Larson (Analytic upscaling of local microphysics parameterizations, part II: Simulations, submitted to *Quarterly Journal of the Royal Meteorological Society*, 2011) simulated a drizzling stratocumulus cloud and found that accounting for subgrid variability in a microphysics scheme led to enhanced autoconversion of cloud droplets to drizzle and accretion of cloud droplets onto drizzle drops. This, in turn, led to a 75% increase in drizzle mixing ratio near the ocean surface.

[5] A particularly difficult aspect is parameterizing correlations among hydrometeor species. This is useful, e.g., for estimating the rates of collection of droplets by snow particles. Although parameterizations of subgrid distributions of moisture-related variables have been developed [e.g., *Tompkins*, 2002; *Morrison and Gettelman*, 2008], typically these distributions are univariate. Therefore, they do not contain information on the covariability of the hydrometeors. The information on covariability is needed, for instance, to compute the collection rate, which depends on whether snow falls preferentially through parts of the cloud that contain greater or lesser cloud water mixing ratio.

[6] One possible approach to predicting correlations among hydrometeors is to develop a prognostic or diagnostic equation for each of these correlations based on fundamental physical equations. While this approach is perhaps the most satisfying from a theoretical point of view, it suffers two drawbacks. First, it is computationally expensive. If the number of hydrometeors is *n*, then the number of correlations among those hydrometeors is *n*(*n* − 1)/2. That is, as *n* grows large, the number of correlations becomes proportional to *n*^{2}, which is very large. For instance, if a microphysics scheme predicts both the number concentration and mixing ratio of cloud water, rain, cloud ice, and snow, then *n* = 8 and *n*(*n* − 1)/2 = 28. Hence there is an incentive to limit the cost of computing each correlation. Second, if each correlation is individually predicted using a physically based estimate, then the correlations so estimated may be inconsistent with each other. To take an extreme example, if variate *X*_{1} and variate *X*_{2} are perfectly correlated with each other, then correlations of a third variate *X*_{3} with *X*_{1} and *X*_{2} must be identical. However, simple physical estimates, which inevitably contain errors, will not ensure such a result.

[7] This paper presents a method that mitigates these two problems. It starts with an established mathematical framework that guarantees the internal consistency of the correlations. Within the strictures of this framework, it diagnoses values of each correlation using an inexpensive formula based on guidance from rigorous bounds on the correlations. The formula contains an adjustable parameter that must be empirically fit to observations or, in the case of this study, model-generated data.

[8] The new method is tested noninteractively using output from a large-eddy simulation (LES) model. The LES model does not serve as a host model for the correlation parameterization; rather, in this study, the LES model produces turbulent fluxes and other moments that serve as input to the new method for the purpose of noninteractive tests. The LES model also produces correlations that serve as validation data. We perform LESs of three mixed-phase Arctic clouds. The first case is based upon the Indirect and Semi-Direct Aerosol Campaign (ISDAC) field experiment; the second is based upon the Mixed-Phase Arctic Cloud Experiment (M-PACE) field experiment, period B; and the third is based on M-PACE period A. These three cloud cases differ in their surface fluxes and microphysical characteristics. Correlations computed from 3D snapshots of the LES are compared to estimates provided by the new method. In this preliminary effort, we have not implemented the correlation estimates interactively in a large-scale host model.

[9] Section 2 describes the LES model that we use. Section 3 describes the three Arctic cloud cases that we will simulate. Section 4 presents correlation matrices as computed by LES. Section 5 defines the parameterization problem that we address. Section 6 discusses rigorous lower and upper bounds on the correlations. These are used later to guide the parameterization of these correlations. Section 7 presents the spherical parameterization and the cSigma method of parameterizing its coefficients. Section 8 compares the spherical parameterization with a prognostic approach. Finally, Section 9 discusses the results and concludes.