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…
Budyko-Sellers 1-D EBM with ice-albedo feedback: dT/dt = [S(1-alpha(T))/4 - sigma T^4]/C
Budyko-Sellers energy-balance-model framework. Setup: the globally-averaged surface temperature T evolves under competition between…
Milankovitch orbital forcing: F(t) = sum A_k cos(omega_k t); spectrum on 3 frequencies
Milankovitch spectral-forcing framework. Setup: Earth's insolation at latitude phi is modulated on ~10 kyr timescales by three orbital…
Schwarzschild radiative-transfer: dI/d tau = I - B(T); linear 1st-order ODE in optical depth
Schwarzschild radiative-transfer framework. Setup: monochromatic plane-parallel radiative transfer in a grey absorbing-emitting atmosphere…
Theorem: Budyko warm-branch linearisation: F'(T) + 4 sigma T^3 = 0 (Planck feedback)
Theorem (Budyko warm-branch stability canonical): on the warm equilibrium branch the albedo is locally constant (alpha = alpha_o), so F(T)…
Theorem: distinct-frequency orthogonality integral_0^{2pi} cos(t) cos(2t) dt = 0
Theorem (Milankovitch spectral-orthogonality canonical): on the fundamental period [0, 2 pi], distinct-frequency cosines are…
Theorem: at tau = ln(2), I(tau) = (I_0 + B)/2 exactly (half-intensity optical depth)
Theorem (Schwarzschild half-intensity canonical): the formal solution I(tau) = I_0 exp(-tau) + B(1 - exp(-tau)) at tau = ln(2) gives I(ln…
Climate sensitivity (Charney 1979)
Charney report 1979: equilibrium climate sensitivity ECS = 1.5-4.5 C per CO2 doubling; bracket essentially unchanged through 40 yr of…
Milankovitch cycles (orbital forcing)
Milankovitch 1924: eccentricity (100k yr), obliquity (41k yr), precession (23k yr) drive ice-age cycles via N. hemisphere summer…
Planetary energy budget (Kiehl-Trenberth)
Kiehl-Trenberth 1997 / 2009 update: 341 W/m^2 incoming -> 102 reflected + 239 absorbed at TOA; 333 W/m^2 back-radiation from…
Water vapor feedback (positive)
Held-Soden 2006: warming -> more water-vapor (Clausius-Clapeyron 7%/K) -> stronger greenhouse -> more warming; ~doubles direct CO2 forcing;…
AMOC thermohaline circulation
Atlantic-Meridional-Overturning-Circulation: 17 Sv overturning of warm-saline surface water sinking in N Atlantic; weakening signal from…
Tipping points (Lenton 2008)
Lenton 2008 / 2019: 9 climate tipping elements (Greenland-IS, AMOC, Amazon, permafrost, etc.); cascading risk above 1.5-2 C; non-linear…
Manabe-Wetherald (1967)
S Manabe-R Wetherald 1967 1D radiative-convective model; modern GCM foundation; Manabe Nobel 2021 climate physics.
Hasselmann (1976)
K Hasselmann 1976 stochastic-climate-model; Brownian-motion-of-climate; Nobel 2021 + modern fingerprinting D&A.
Charney sensitivity (1979)
J Charney 1979 ECS=3.0±1.5°C; modern AR6 likely 2.5-4.0°C constrained by Sherwood 2020 multi-line-of-evidence.
IPCC AR6 (2021)
IPCC AR6 WG1 2021 'unequivocal' anthropogenic + 1.1°C since 1900; modern path to 1.5°C requires net-zero by 2050.
Budyko ice-albedo (1969)
M Budyko 1969 + Sellers 1969 ice-albedo-feedback; modern Snowball-Earth + tipping-point-research multiple-equilibria.
Bjerknes feedback (1969)
J Bjerknes 1969 ENSO Bjerknes-feedback; modern ENSO + Madden-Julian + Atlantic-Niño tropical-coupled-modes prediction.