Smagorinsky

Anelastic thermodynamics equation (the first law of thermodynamics) with Smagorinsky turbulent heat flux, from Clark (1977), 'A Small-Scale Dynamic Model Using a Terrain-Following Coordinate Transformation'. Given the base-state air density rho_bar, the eddy heat diffusivity K_H, and the potential-temperature gradients, the component computes the three turbulent heat-flux components H1, H2, H3 = rho_bar*K_H*grad(theta) (Eq. 2.19), the heat-flux divergence term div_H = -div(H), and the residual of the anelastic thermodynamic equation rho_bar*dtheta/dt = -div(H) (Eq. 2.14) — which vanishes for an energy-conserving solution. Five purely-algebraic equations, no time integration. Migrated from AtmosphericDynamics.jl@816cc391 src/clark1977.jl:426 (`AnelasticThermodynamics` @component).

Smagorinsky (1963) subgrid-scale turbulence closure as used in Clark (1977), 'A Small-Scale Dynamic Model Using a Terrain-Following Coordinate Transformation', J. Comput. Phys. 24, 186-215. A diagnostic (purely algebraic) eddy-viscosity scheme: from the six independent components of the resolved-flow deformation tensor (D11, D22, D33, D12, D13, D23; Eq. 2.16) and the grid filter width Δ_grid it computes the total deformation magnitude |Def| (Eq. 2.18), the Smagorinsky eddy viscosity coefficient K_M = (k·Δ)²·|Def| (Eq. 2.17), and the turbulent heat diffusivity K_H, which Clark sets equal to the eddy viscosity (Eq. 2.19, unit turbulent Prandtl number). The model has 3 algebraic variables (no time integration), 3 equations, and 9 parameters (2 physical constants — the Smagorinsky constant k = 0.25 and a unit reference deformation rate — plus 7 inputs). All quantities are SI. Migrated from AtmosphericDynamics.jl@816cc391 src/clark1977.jl:214 (`SmagorinskyTurbulence` @component).