Earth
Ice and albedo feedback
Watch a planet’s ice caps in the light of its sun: each latitude band absorbs sunlight according to its albedo, radiates to space and shares heat with its neighbours. Today’s sun puts the ice edge near 66 degrees; dim it slowly and the edge creeps towards the equator until, at nine tenths of today’s sunshine, the bright ice reflects so much that the whole planet freezes over in one jump. Brighten the sun again and the snowball stays frozen far past where it froze - it thaws only at more than six fifths. Make ice and water equally dark and the feedback is gone: the equilibrium is then the closed-form P2 solution in every band.
What happens
Sixty latitude bands in x = sin(latitude) over one hemisphere. Each absorbs Q s(x)(1 − alpha(T)) with Q = 340 W/m² times the sunshine factor and North’s s = 1 − 0.482 P2(x), radiates A + B T with A = 203.3 W/m² and B = 2.09 W/m²·K, and exchanges heat by diffusion d/dx (D (1 − x²) dT/dx) with D = 0.40. The albedo turns from 0.30 for water to 0.62 for ice over a tanh range around −10 °C. Implicit steps of 0.05 year in diffusion and radiation, explicit in the albedo, a heat capacity of a 100 m mixed layer. With a constant albedo the equilibrium is the P2 solution; with the feedback the ice line jumps and the climate has two branches.
Try this
Read today’s ice edge; dim the sun or press the sweep and watch the snowball come; brighten it again and see how far the frozen branch holds; make the albedos equal and compare with the P2 solution.
The model and its limits
A zonally averaged, diffusive energy balance with a fixed outgoing law, no seasons, ocean dynamics or clouds; the thresholds belong to this idealized planet, not to the Earth’s history.
Equations used in the module
Equations C·dT/dt = Q·s(x)·(1 − α(T)) − (A + B·T) + d/dx(D·(1 − x²)·dT/dx) s(x) = 1 − 0.482·P2(x), α = α_i + (α_w − α_i)·(1 + tanh((T − T_f)/ΔT))/2 T(x) = (Q·(1 − α) − A)/B − 0.482·Q·(1 − α)·P2(x)/(B + 6D) x_s = sin φ_ice: two stable branches, a snowball below Q_c
