Clark-1977

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).

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).

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.

Diagnostic pressure equation for the anelastic atmosphere in terrain-following coordinates, from Clark (1977) "A Small-Scale Dynamic Model Using a Terrain-Following Coordinate Transformation", Section 4. Observed-only algebraic component: from the pressure-gradient-force divergences, the pressure-perturbation Laplacian terms, and a momentum-divergence source it computes the pressure-gradient-force divergence div_PF (with the terrain-following Jacobian factor G^(1/2) scaling the vertical term), the pressure Laplacian laplacian_p, and the diagnostic pressure equation residual pressure_residual (Eq. 4.2), which vanishes when the anelastic mass-continuity constraint is satisfied. Migrated from AtmosphericDynamics.jl@816cc391 src/clark1977.jl:499 (`DiagnosticPressure` @component); the upstream @constant g is declared as a fixed-default parameter.

Isentropic base-state atmosphere for the Clark (1977) terrain-following mountain-wave model, "A Small-Scale Dynamic Model Using a Terrain-Following Coordinate Transformation", Eqs. 2.6-2.9. The base state assumes a constant potential temperature Θ — the deep, isentropic reference atmosphere of Ogura and Phillips (1962) adopted in Clark (1977) Eq. 2.6 — and derives the height-dependent thermodynamic profiles that satisfy hydrostatic balance: from the reference potential temperature Θ, the surface pressure P_0, and the height above the surface z, the component computes the isentropic scale height H_s = c_p*Θ/g, the base-state temperature T_bar = Θ*(1 - z/H_s) (Eq. 2.7), the base-state pressure p_bar = P_0*(1 - z/H_s)^(c_p/R_d) (Eq. 2.8), and the base-state density ρ_bar = P_0/(R_d*Θ)*(1 - z/H_s)^(c_p/R_d - 1) (Eq. 2.9). Temperature falls linearly with height while pressure and density fall as powers of the same (1 - z/H_s) factor, all three vanishing at the top of the isentropic atmosphere z = H_s. Four purely-algebraic equations, no time integration. Migrated from AtmosphericDynamics.jl@816cc391 src/clark1977.jl:49 (`IsentropicBaseState` @component); the five upstream @constants — g, R_d, c_p and the derived exponents inv_kappa = c_p/R_d and inv_kappa_m1 = c_p/R_d - 1 — are declared as fixed-default parameters.

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).

Terrain-following coordinate transformation from Clark (1977) "A Small-Scale Dynamic Model Using a Terrain-Following Coordinate Transformation", Eqs. 2.20-2.23 and 2.27. Observed-only algebraic component: from the terrain surface height z_s, the domain height H, the transformed vertical coordinate z_bar, and the two terrain slopes dz_s_dx and dz_s_dy it computes the transformation Jacobian G^(1/2) = 1 - z_s/H (G_sqrt), the two metric coefficients G^(1/2)G^13 and G^(1/2)G^23 that carry the terrain slope into the transformed equations (G_sqrt_G13, G_sqrt_G23), the physical height z_phys recovered from z_bar (the inverse of the mapping z_bar = H(z - z_s)/(H - z_s)), and the diagnostic Jacobian factor ω_diag used to recover the contravariant vertical velocity ω (Eq. 2.27). The transformation maps the irregular lower boundary z = z_s(x,y) to the flat computational surface z_bar = 0 while preserving the rigid lid z = H at z_bar = H. Migrated from AtmosphericDynamics.jl@816cc391 src/clark1977.jl:157 (`TerrainFollowingTransform` @component).

"Witch of Agnesi" bell-shaped mountain topography from Clark (1977) "A Small-Scale Dynamic Model Using a Terrain-Following Coordinate Transformation", Equation 7.1 — the canonical idealized terrain for mountain-wave simulations. Algebraic component: from the mountain half-width a, peak height h_mtn, and horizontal position x relative to the mountain center it computes the terrain surface height z_s = a^2 h_mtn / (a^2 + x^2) (Eq. 7.1) and the terrain slope dz_s_dx = d(z_s)/dx. Migrated from AtmosphericDynamics.jl@816cc391 src/clark1977.jl:107 (`WitchOfAgnesi` @component); z_s and dz_s_dx are upstream `@variables` bound by algebraic equations and migrate as state-typed variables.