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Turbulent heat fluxes describe the exchange of energy between the snow surface and the near-surface atmospheric boundary layer. Friction at the surface-atmosphere interface creates dissipation of momentum through both direct (molecular) viscous drag and turbulent eddy motions. Direct viscous effects are confined to within a few centimeters of the surface and are generally minor; the energy fluxes in (3.1) are primarily associated with turbulent eddies. If the surface is rough and the wind is strong enough to create turbulence, sensible and latent energy at the surface, QH and Qe, are created as a by-product of the dissipation of momentum and kinetic energy in the atmospheric flow over the snow or ice.

The equations used to estimate turbulent energy fluxes are a complex mixture of theory and empiricism. The simplest approach is to assume a well-mixed atmospheric boundary layer, with vertical fluxes of sensible and latent energy proportional to the wind speed. This can be parameterized through

where pa and cpa are the density and specific heat capacity of the air, Ls/v is the latent energy of sublimation or vaporization, ua is the wind speed, and CH and CE are dimensionless bulk exchange parameters for heat and moisture. The atmospheric variables 9 and q refer to the potential temperature and specific humidity of the air. Surface values of potential temperature and humidity in (3.12), sometimes referred to as "skin" values, are those within ca. 1 mm of the surface. These are taken to be the surface temperature of the snow or ice and the saturation specific humidity at this temperature.

Snow and ice surfaces are characterized by a stable boundary layer, with cold air near the surface and strong vertical gradients in temperature, humidity, and wind speed. Stability adjustments can be built into the aerodynamic coefficients of (3.12) or the bulk aerodynamic formulas can be modified to include more realistic assumptions about boundary-layer profiles. The latter approach introduces a different model for closure, variously known as Prandtl theory, flux-gradient theory, the profile method, scalar transfer theory, or the eddy-diffusivity model of turbulent fluxes. This treatment is conceptually different from the bulk aerodynamic approach of the slab mixed layer, although the two methods converge mathematically in some applications.

Based on assumptions about the vertical gradients of the thermodynamic variables in the near-surface boundary layer, Prandtl theory essentially parameterizes turbulent fluxes as a form of bulk diffusion, with eddy diffusivities KH and KE. There are direct parallels between these parameters and the idea of eddy viscosity that is used in oceano-graphic modeling. Eddy viscosity is a construct, unrelated to molecular viscosity. Similarly, the eddy diffusivities for heat and moisture resemble thermal and hydraulic dif-fusivities, but they are not material properties. To some degree it is reasonable to think of turbulent transfers as a form of diffusion: momentum, heat, and moisture are transferred from high to low concentrations, creating a more homogeneous lower boundary layer. The analogy with true diffusion has limitations. Turbulent exchange is not always effective at mixing the lower boundary layer, and the efficacy of mixing depends on surface roughness properties, wind strength, wind shear, and lower boundary layer stability. In principle, these influences can be incorporated in the eddy diffusivities, KH and KE.

According to classical Prandtl theory, wind speed increases with height above the surface following

Parameter k is von Karman's constant, which has an empirically determined value of 0.4, and u. is a characteristic velocity. This can be integrated to give the well-known log relationship for boundary layer winds, where z0 is an integration constant and is known as the surface roughness. Mathematically, surface roughness length is defined as the height at which the wind speed goes to zero. Physically, the roughness length relates to the degree of mechanical coupling of the snow surface and the boundary-layer airflow. Rougher surfaces impose a greater viscous perturbation and are more likely to lead to turbulent eddies, given sufficient airflow. Aerodynamic roughness values are less than the geometric roughness elements, but z0 is generally proportional to geometrical asperities and undulations in the surface. Published values of z0 range from 0.1 mm to a few centimeters over snow and ice, based on wind profile measurements in neutral stability conditions. Sub-millimeter measurements refer primarily to grain-scale roughness elements. Values of 1-10 mm are typical of melting surfaces, which can develop sun cups and relatively large-scale surface undulations.

Working from the velocity profile in (3.14) and with similar assumptions for the profiles of potential temperature and specific humidity,

qh pacpakh paf?^

Here, z0H and z0E are the roughness length scales for sensible and latent heat fluxes. These are distinct from the roughness length in turbulent momentum exchange. By analogy, however, they can be taken as the heights at which qa and qv are equal to the surface (skin) values. Implicit in (3.15) is the assumption that the eddy dif-fusivities for momentum, sensible heat, and latent heat transport are equal. This expression also assumes neutral stability in the lower boundary layer. Equation (3.15) can be adjusted to parameterize the effects of atmospheric stability, which will amplify or limit the extent of turbulent energy exchange.

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