Earth Energy Balance and CO2 Warming Calculator

The models are deliberately separate. They share solar irradiance, and the ocean and ice models share CO2 forcing. Their temperature predictions are not added together.

Global radiation balance. With solar irradiance $S_0$, planetary albedo $\alpha$ and effective infrared emissivity $\epsilon$:

Use absolute kelvin inside the fourth-power law; convert the displayed temperatures to °C. The schematic's reflected and outgoing fluxes describe this baseline equilibrium. NASA explains the geometric factor and the measured global budget. The selected emissivity is an illustrative effective graybody parameter.

CO2 forcing and two-box response. Myhre et al. (1998), Table 3, and IPCC TAR Table 6.2 give $F=5.35\ln(C/C_0)$ W/m². Held et al. (2010), attached manuscript p.5, Eqs. (1)–(3), motivate

Both initial anomalies are zero. Default coefficients are the published CMIP5 example repeated by Fredriksen & Rypdal (2017), §3a: $C_u=7.3$, $C_d=106$ W yr m⁻² K⁻¹, $\lambda=1.13$, $k=0.73$ W m⁻² K⁻¹. They are not a calibration to Held's CM2.1 experiment.

The exact decay rates solve $r^2-pr+q=0$, where $p=(\lambda+k)/C_u+k/C_d$ and $q=\lambda k/(C_uC_d)$. The equilibrium anomaly is $F/\lambda$. Surface response is a weighted sum of two exponential step responses; the deep response follows from the same eigenmodes. A linear forcing ramp is evaluated by analytically integrating each exponential response, then subtracting a delayed ramp at the hold time. The rates and full curves do not read the playing time. Only the selected values and schematic change per frame. The top-of-atmosphere imbalance is $N=F(t)-\lambda T_u$, while deep-ocean uptake is $H=k(T_u-T_d)$.

One-dimensional Budyko ice-albedo model. This is the mean-transport member of the Budyko–Sellers family, following Walsh & McGehee, attached preprint pp.3–6; it does not implement Sellers' spatial diffusion operator. Let $x=\sin|\varphi|$, $0\le x\le1$, with $s(x)=1.241-0.723x^2$. Set albedo to 0.32 equatorward of ice line $\eta$ and 0.62 poleward; the edge uses their mean. Meridional transport is $K(\bar T-T)$ with $K=3.04$ W m⁻² K⁻¹. Its global integral is zero.

With $B=1.9$ W m⁻² K⁻¹ and $A_e=A-F$,

The solar-weighted albedo is integrated exactly using $\int_0^\eta s(x)dx=1.241\eta-0.241\eta^3$. Solve the averaged edge temperature equal to the critical ice-line temperature. A projected slow ice-line rule, $\dot\eta\propto T(\eta)-T_c$, constrained to [0,1], defines branch stability and endpoint nucleation. The turning point is located by bounded minimization, and 34 bisections resolve the stable partial-ice root. The hysteresis chart sweeps effective $A$ quasi-statically, not time or observed CO2.

• Annual global averages; spherical Earth and constant mean solar irradiance.
• The zero-dimensional emissivity remains fixed. CO2 forcing is not also added to its graybody temperature, avoiding a second greenhouse adjustment.
• The ocean model has constant coefficients, heat-uptake efficacy one, and zero initial anomalies relative to the entered reference concentration. An abrupt experiment changes forcing at time zero. A ramp changes forcing linearly until its entered duration, then holds it constant.
• With a logarithmic forcing law, a linear forcing ramp corresponds to exponential concentration evolution. The 70-year doubling ramp closely approximates 1% per year; arbitrary target and duration settings are not the standardized transient climate response experiment.
• The ice model is symmetric between hemispheres and assumes instantaneous thermal equilibration at each slowly varying ice-line position. Its fixed two-region albedos and linear OLR are independent of the global graybody controls.
• Ice-line boundaries use the mean-edge-albedo convention of the constrained ice-line dynamics. These idealized nucleation and escape thresholds differ from a fully resolved ice-sheet calculation.
• All heat capacities are per global planetary area. A temperature anomaly in K has the same numerical magnitude as a temperature change in °C.
• Reservoir fill in the schematic shows the magnitude of each anomaly as a fraction of the target equilibrium anomaly. Warm colors indicate positive forcing, blue indicates cooling; the exchange arrow follows the sign of ocean heat flow.

This is a conceptual climate sensitivity and energy balance tool for education and parameter studies, not a weather forecast or a regional impact model. The default CO2 doubling is illustrative, not a current concentration observation.

The historical Myhre logarithmic approximation is intended here for target/reference ratios between 0.5 and 4; a note appears outside that range. It omits other gases, aerosols, rapid tropospheric adjustments and modern effective-radiative-forcing refinements. Held's CM2.1 model diagnosed a different doubling forcing (3.48 W/m² in the manuscript p.10), so its GCM warming numbers are not direct checks on a calculation using 5.35 ln(2).

Ocean results are perturbations from an equilibrium baseline. They do not include the present observed ocean disequilibrium, changing circulation, varying feedbacks or the carbon cycle. The reported equilibrium anomaly assumes the target forcing is maintained indefinitely.

The one-dimensional model omits seasons, geography, ice thickness and time-dependent ice sheets. Its temperature profile has a deliberate jump at the two-albedo interface. The warm and cold paths are idealized quasi-static histories; changing the history selector does not reconstruct a user's previous inputs. Large excursions of the OLR intercept demonstrate hysteresis mathematically and do not establish real-world snowball or ice-loss thresholds. The ocean animation and ice equilibria are not dynamically coupled.