Algebraic

Fire spread from a single ignition point assuming an elliptical fire shape — calculates eccentricity, spread rate in arbitrary direction, backing fire spread, fire length and width, flanking distance, and ellipse geometric parameters (semi-major axis, focus offset, semi-minor axis). Migrated from WildlandFire.jl@126976f2508ceacca6c486b21ae6da5b168c6f46:src/fire_spread_direction.jl:174 (`EllipticalFireSpread` @component). Algebraic-only: R_H, Z, γ, elapsed_time are inputs (parameters); e, R_γ, R_B, D_H, D_B, L, W, D_F, f, g, h are algebraic outputs.

Fire perimeter spread rate and fireline intensity — calculates spread rate normal to the elliptical fire perimeter and Byram's fireline intensity. Migrated from WildlandFire.jl@126976f2508ceacca6c486b21ae6da5b168c6f46:src/fire_spread_direction.jl:264 (`FirePerimeterSpread` @component). Algebraic-only: R_H, f, g, h, γ, H_A are inputs (parameters); cos_θ, sin_θ, θ, R_ψ, ψ, I_B are algebraic outputs.

McTigue (1987) finite-sphere model for volcanic surface deformation. Extends the Mogi (1958) point-source model by incorporating higher-order correction terms that account for the finite radius of a spherical deformation source. The correction depends on ε = a/d and improves accuracy for larger ε: within 5% of FE models for ε < 0.59, vs ε < 0.37 for Mogi. Three purely-algebraic equations: R = sqrt(x²+d²) (source-to-surface distance), Ur = (Mogi_Ur) × C (Eq. 3, radial displacement), Uz = (Mogi_Uz) × C (Eq. 4, vertical displacement), where C = 1 + (a/d)³ × [(1+ν)/(2(−7+5ν)) + 15d²(−2+ν)/(4R²(−7+5ν))] is the McTigue finite-sphere correction factor. Migrated from Geodynamics.jl@d680af419676cc576d60b16af0d3dd40c0b23eea src/mogi_mctigue.jl:107 (`McTigueModel` @component).

Mogi (1958) point-source model for volcanic surface deformation. Computes horizontal-radial (Ur) and vertical (Uz) surface displacements due to a pressurized spherical cavity in an elastic half-space, using the small-sphere (point-source) approximation (ε = a/d ≪ 1). Three purely-algebraic equations: R = sqrt(x²+d²) (source-to-surface distance), Ur = a³ΔP(1−ν)x / (G·R³) (Eq. 1, radial displacement), Uz = a³ΔP(1−ν)d / (G·R³) (Eq. 2, vertical displacement). Migrated from Geodynamics.jl@d680af419676cc576d60b16af0d3dd40c0b23eea src/mogi_mctigue.jl:48 (`MogiModel` @component).

Wildfire plume-rise model based on Sofiev et al. (2012). Computes plume top height H_p (m) as a function of atmospheric boundary-layer height H_abl (m), free-troposphere Brunt-Väisälä frequency N_ft (s⁻¹), and fire radiative power P_fr (W) via one empirical algebraic equation: H_p = α·H_abl + β·(P_fr/P_f0)^γ · exp(−δ·N_ft²/N_0²). The five empirical constants (α=0.24, β=170 m, γ=0.35, δ=0.6, N_0=√(2.5×10⁻⁴) s⁻¹) are from Sofiev et al. (2012). In a coupled model H_abl and N_ft are supplied by GEOSFP reanalysis; the additional variable lev_p (plume-top pressure level index) is computed from H_p by the GEOSFP coupler via a Newton-iteration height-to-level mapping. Migrated from EnvironmentalTransport.jl@a48559c4114eb9f06668ecf9f06235aad20f63f0 src/plume_rise/sofiev_2012.jl:14.

Stage (1973) static regression model for height to crown base (HCB) of lodgepole pine. Predicts HCB from tree height, stand density (CCF), elevation, relative tree size (DBH/RMSQD), and habitat type. Single explicit-algebraic equation: HCB = one_foot*(b0 + b1*HT_ft + b2*ln(CCF) + b3*EL_100ft + b4*DBH_in/RMSQD_in + HAB). All regression coefficients are in feet-space; result converted to meters. Migrated from Vegetation.jl@010c3774dc5646d81806c6119c8a029a70f6c9a0:src/stage_prognosis.jl:257.