Wang-2020

Smoothed Heaviside surface boundary condition for coupling surface and subsurface water flow, from Wang et al. (2020) (Advances in Water Resources 137, 103499), Eqs. 3 and 5. The component switches a single surface node continuously between a ponded (saturated) boundary and an unsaturated boundary as the pressure head h crosses zero, replacing a hard step with the smooth approximations eta_omega(h) = (1/pi)*atan(h/omega) + 1/2 (smoothed Heaviside, Eq. 5a) and delta_omega(h) = (1/pi)*omega/(omega^2 + h^2) (smoothed Dirac delta, Eq. 5b; delta_omega = d eta_omega/dh). The head obeys the boundary condition d/dt[eta_omega(h)*h] = P - I_infil (Eq. 3), stated as the explicit ODE D(h) = (P - I_infil)/(delta_omega*h + eta_omega). h is the single integrated state; eta_omega and delta_omega are algebraic (observed). Migrated from EnvironmentalTransport.jl@a48559c4 src/surface_runoff.jl:232 (`HeavisideBoundaryCondition` @component). Because the head equation conserves eta_omega(h)*h - (P - I_infil)*t, every ODE-integration test below is checked against the exact conservation law, not merely against a prior solver run.

Full 1D Saint-Venant PDE system for surface overland flow: mass conservation ∂h̃/∂t = −∂q/∂l + (P − I) and momentum conservation ∂q/∂t = −∂F/∂l + g·h̃·(S₀ − Sᶠ), where the momentum flux F = q²/h̃ + g·h̃²/2 is an auxiliary (observed) PDE field. Migrated from EnvironmentalTransport.jl@a48559c4 src/surface_runoff.jl:117 (SaintVenantPDE builder). This is a DISCRETIZATION-AGNOSTIC leaf (esm-spec §9.7.10): the spatial fluxes are written with the rewrite-target spatial derivative op `D(·, wrt: l)` and the component imports NO discretization of its own, so it can be composed under different schemes. Its inline tests and examples inject a discretization (form C) — here the sibling adapter `saint_venant_upwind1_inflow.esm` (first-order upwind, per-variable inflow BC) — for that run only; the persisted leaf keeps its spatial `D` intact. The 1D spatial variable l is the along-slope coordinate; the sibling SurfaceRunoff component (surface_runoff_component.esm) is the single-node ODE reduction of this same PDE system with the spatial-divergence terms exposed as input parameters.

Single-node surface-runoff model based on the Saint-Venant overland-flow equations, from Wang et al. (2020), Eq. 1. The component tracks two ODE states at one surface computing node — the ponded flow depth h_tilde and the surface runoff flux per unit width q — plus one algebraic (observed) state, the Manning friction slope S_f. Mass conservation (Eq. 1a) reads dh_tilde/dt = -dqdl + (P - I_infil): the flow depth rises with the net of precipitation P over infiltration I_infil and falls with the spatial divergence of runoff flux dqdl. Momentum conservation (Eq. 1b) reads dq/dt = -dFdl + g*max(h_tilde,h_tilde_0)*(S_0 - S_f): the runoff flux is driven by the imbalance between the surface slope S_0 and the friction slope S_f, minus the spatial divergence of momentum flux dFdl. The friction slope closes the system via Manning's equation S_f = (n_mann*q)^2 / h_tilde^(10/3), written here in non-dimensionalized form so the fractional powers act on dimensionless bases. Because this is a single-node reduction of a spatial PDE, the two spatial-divergence terms dqdl and dFdl are exposed as input parameters to be supplied by a discretization scheme or a coupled model; with both at zero the component reduces to the local mass/momentum response. Migrated from EnvironmentalTransport.jl@a48559c4 src/surface_runoff.jl:35 (`SurfaceRunoff` @component).