Algebraic-System

Anelastic momentum equations from Clark (1977), 'A Small-Scale Dynamic Model Using a Terrain-Following Coordinate Transformation', Journal of Computational Physics 24, 186-215. A diagnostic (purely algebraic) component that evaluates the residuals of the three nonlinear anelastic momentum equations (Eqs. 2.1-2.3) in Cartesian coordinates. Given the velocity components (u, v, w), their material derivatives (du/dt, dv/dt, dw/dt), the perturbation-pressure gradients (∂p'/∂x, ∂p'/∂y, ∂p'/∂z), the base-state and perturbation densities (ρ̄, ρ'), the Reynolds-stress divergences ((∇·τ)₁, (∇·τ)₂, (∇·τ)₃), the Coriolis parameter f and the Rayleigh-friction time scale τ_R, it returns the three momentum-equation residuals mom_u, mom_v, mom_w. Each residual is the sum of the inertial term ρ̄·(dV/dt), the Coriolis term ρ̄·V·f (horizontal components only), the perturbation-pressure gradient, the negative Reynolds-stress divergence, the buoyancy term ρ'·g (vertical component only) and the negative Rayleigh-friction sink ρ̄·V/τ_R; the residual vanishes when the momentum balance is exactly satisfied. The model has 3 algebraic variables (no time integration), 3 equations and 17 parameters (1 physical constant + 16 inputs). All quantities are SI. Migrated from AtmosphericDynamics.jl@816cc391 src/clark1977.jl:282 (`AnelasticMomentum` @component).

Complete 3D anelastic atmospheric dynamics system in terrain-following coordinates from Clark (1977), 'A Small-Scale Dynamic Model Using a Terrain-Following Coordinate Transformation', Journal of Computational Physics 24, 186-215. The full nonlinear Clark (1977) model assembled as a hierarchical composition of eight subsystems: the isentropic base state (Eqs. 2.6-2.9), Witch-of-Agnesi mountain topography (Eq. 7.1), the terrain-following coordinate transformation (Eqs. 2.20-2.27), the Smagorinsky turbulence closure (Eqs. 2.15-2.19), the anelastic momentum residuals (Eqs. 2.1-2.3), anelastic mass continuity (Eq. 2.4), anelastic thermodynamics with turbulent heat flux (Eq. 2.14) and the diagnostic pressure equation (Section 4). Migrated from AtmosphericDynamics.jl@816cc391 src/clark1977.jl:582 (`Clark1977AnelasticSystem` @component) — a pure subsystem composition (`System(Equation[], t; systems=[...])`, no coupling equations): the eight subsystems are inlined verbatim under `subsystems` (ESS spec §4.7) and remain dynamically independent. Purely algebraic — every subsystem is a diagnostic (observed-only) system with no time derivative. SI units throughout.

Parameterized atmospheric-boundary-layer surface fluxes following Holtslag & Boville (1993), 'Local Versus Nonlocal Boundary-Layer Diffusion in a Global Climate Model', J. Climate 6, 1825-1842. A diagnostic (purely algebraic) bulk-aerodynamic scheme: from the lowest-model-level wind (u₁, v₁), the surface and lowest-level potential and virtual potential temperatures (θ₀/θ₁, θᵥ₀/θᵥ₁), and specific humidities (q₀, q₁), it computes the kinematic surface fluxes of momentum (wu₀, wv₀; Eqs. 2.1-2.2), heat (wθ₀; Eq. 2.3) and moisture (wq₀; Eq. 2.4). The stability-dependent exchange coefficients Cₘ and Cₕ (Eqs. 2.5-2.6) are the neutral coefficient Cₙ (Eq. 2.7, a logarithmic function of the roughness ratio) modulated by the stability functions fₘ and fₕ, which switch on the sign of the surface bulk Richardson number Ri₀ (Eq. 2.8): the unstable branch uses the Louis et al. (1982) forms (Eqs. 2.9-2.10) and the stable branch the Holtslag & Beljaars (1989) form (Eq. 2.11). The model has 11 algebraic variables (no time integration), 11 equations, and 15 parameters (4 physical constants + 11 inputs). All quantities are SI. Migrated from AtmosphericDynamics.jl@816cc391 src/holtslag_boville_1993.jl:35 (`HoltslagBovilleSurfaceFlux` @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).