warming up your workspace

Ice, albedo, and Snowball Earth, when a planet has two climates at once

Roughly 700 million years ago, geologists find evidence of glaciers at the equator. The leading explanation is Snowball Earth: the entire planet, oceans included, froze over. What makes this more than a historical curiosity is the mechanism, because it reveals something unsettling about climate in general. The same feedback that could freeze the whole planet is active right now, and it means that for a single fixed amount of sunlight, Earth does not have one stable temperature. It has two. Which one you get depends on where you started, and flipping between them is far easier in one direction than the other. Build a small energy-balance model and this stops being abstract.

The one idea

Start from the fact that a planet's temperature is set by the balance of absorbed sunlight against emitted heat. Now add one feedback: the albedo, the fraction of sunlight reflected, is not fixed. Warm ocean and land are dark and absorb sunlight; ice and snow are bright and reflect it. So if the planet cools enough to grow ice, it reflects more sunlight, which cools it further, which grows more ice. That is a runaway loop with a floor, once everything is frozen, there is no more ocean to freeze. The consequence is that the balance equation can have more than one solution: a warm state where dark ocean keeps things warm, and a frozen state where bright ice keeps things cold, both perfectly stable under identical sunlight.

Build it

Make albedo depend on temperature, dark when warm, bright when frozen, and relax the temperature toward the point where absorbed sunlight equals emitted heat. The emitted-heat side uses the empirical linear form climatologists fit to real data, which is what lets the feedback produce two states.

A, B = 210.0, 2.1          # outgoing heat = A + B*T  (W/m^2, T in Celsius)

def albedo(T):
    if T > 0:   return 0.30           # warm: dark ocean
    if T < -10: return 0.62           # frozen: bright ice
    return 0.30 + (0.62-0.30)*(-T/10) # partial ice in between

def net_flux(T, S):
    return (S/4)*(1-albedo(T)) - (A + B*T)   # absorbed minus emitted

def equilibrium(S, T_start):
    T = T_start
    for _ in range(2000):
        T += 0.02 * net_flux(T, S)    # nudge toward balance
    return T

Three details that matter:

  • The albedo jump between dark and bright is the whole engine. Remove it, make albedo constant, and the planet has exactly one temperature for a given sun. The feedback is what creates the second state.
  • The outgoing heat is modeled as A + B*T, a straight line, not the pure fourth-power radiation law. This linear form absorbs the greenhouse effect and other feedbacks into two empirical numbers, and it is the standard choice for exactly this kind of stability analysis.
  • "Relaxing" toward equilibrium mimics real thermal inertia: the planet does not jump to its steady temperature, it drifts there as oceans warm or cool. Where it drifts to depends on where it starts, which is the entire point.

Proof: two climates, and a trap

Ask where the planet settles under today's sunlight, once from a warm start and once from a frozen start:

from a warm start:   13.4 C
from a frozen start: -38.4 C   <- two stable climates, same sun

Same star, same equation, two answers. A warm Earth stays warm; a frozen Earth stays frozen. Neither is "wrong". Now watch the trap spring. Slowly dim the sun, let the climate follow, then brighten it back:

sun 100.0% ->   13.4 C  (warm)
sun  90.0% ->    2.1 C  (warm)
sun  88.0% ->  -45.8 C  (FROZEN)     <- collapse
...
sun 100.0% ->  -38.4 C  (FROZEN)     <- back to full sun, still frozen
sun 106.0% ->  -34.7 C  (FROZEN)     <- even brighter, still frozen

At 88 percent sunlight the warm state simply ceases to exist and the planet crashes into the frozen one. But turning the sun back up does not undo it. At 100 percent, the same brightness that was comfortably warm before, the planet stays locked in ice, because now the bright ice is reflecting away the sunlight that would melt it. You have to overshoot far past the original brightness to escape. This asymmetry, easy to fall in, hard to climb out, is called hysteresis, and it is why Snowball Earth was so hard to end (it took volcanic carbon dioxide building up over millions of years).

Where this shows up

This is a tipping point, and it is the clearest possible example of why climate scientists worry about them. A system with feedback like this does not respond to a nudge proportionally; it can be stable, stable, stable, then flip to a completely different state that resists going back. Real climate has several such feedbacks, ice sheets, permafrost carbon, ocean circulation, each with its own hysteresis. The tiny model you built is the conceptual core of why "we can just reverse it later" is a dangerous assumption.

If you want to build latitude-resolved versions where the ice line itself becomes unstable, and couple this to a carbon cycle, that is what the climate-modeling track on IWTLP constructs, one feedback at a time.