Planetary-scale energy balance, atmospheric radiative transfer, and dynamical response to forcings. Zero-dimensional energy-balance model: absorbed solar radiation πR²·S·(1−α) equals emitted longwave radiation 4πR²·σT⁴ at equilibrium,…
climate-physics
Stefan-Boltzmann energy balance: T_eq = [S(1−α)/(4σ)]^(1/4)
Zero-dimensional energy-balance model (Sellers 1969; Budyko 1969): a spherical planet of radius R intercepts solar irradiance S across its…
One-layer grey-atmosphere greenhouse: T_s = 2^(1/4)·T_e
Manabe-Strickler 1964 one-layer grey-atmosphere model: an isothermal atmosphere that is fully transparent to incoming shortwave radiation…
Planck feedback λ_P = 4σT_e³ ≈ 3.2 W/(m²·K)
The Planck feedback (sometimes 'blackbody feedback') is the intrinsic thermodynamic temperature-dependence of outgoing longwave radiation…
T_eq closed form: √2·S^(1/4)(1−α)^(1/4)/(2σ^(1/4)) (exact)
Closed-form evaluation of the zero-dimensional radiative-balance equilibrium. Setting absorbed solar πR²·S(1−α) = emitted longwave…
One-layer: T_s⁴ − 2T_e⁴ ≡ 0 (exact polynomial identity)
Closed-form polynomial identity of the one-layer grey-atmosphere model. Setting T_s = 2^(1/4)·T_e and expanding T_s⁴ − 2T_e⁴ gives exactly…
λ_P = 4σT_e³ (exact symbolic derivative)
Closed-form symbolic derivative of the Stefan-Boltzmann law. OLR(T_e) = σT_e⁴; dOLR/dT_e = 4σT_e³ — a cubic monomial. Sympy symbolic…