A hurricane is a heat engine running on warm ocean water, but its shape comes from math: rotation turns inflowing air into a spiral, conservation spins it up, a cube law makes it destructive, and chaos makes its path hard to predict.
Low pressure at the center pulls air inward from every side. On a rotating planet, moving air gets deflected sideways: to the right in the Northern Hemisphere, to the left in the Southern. That's the Coriolis effect. Each parcel misses the center and curls into a spiral. Slide the latitude through the equator and watch the spin reverse, then vanish.
How to read this: you're looking down on a storm. Each streak is a puff of air being pulled toward the low-pressure center in the middle. Watch which way the streaks curl.
Try this: drag Latitude from 20° N to 20° S and watch the curl reverse. Then set Surface friction to 0: the air circles without ever reaching the center. Release new air starts a fresh batch.
Ω is Earth's spin rate. The deflection depends on sin(latitude): strongest at the poles, zero at the equator, and opposite in sign across it.
Pressure pulls it toward the low. Coriolis pushes it sideways, at right angles to its motion. Friction with the sea surface slows it. Without friction, air would circle the low forever. Friction is what lets it spiral in.
A figure skater spins faster by pulling in their arms. Air spiraling toward the eye does the same, because angular momentum is conserved. Drag the parcel inward and watch its speed climb.
How to read this: a storm seen from above. The amber dot is one puff of air. Move it closer to the eye and watch it whip around faster. The motion is sped up for viewing, and the distance rings are squeezed together toward the edge.
A parcel that starts almost still 311 mi out has a small spin, but dividing by a small r makes it huge. Without friction it would pass 440 mph near the eye. Real storms lose most of that to the sea surface and top out near 180 mph.
As r shrinks, the speed needed to keep that angular momentum grows without limit. Air can't get any closer, so it turns upward in the eyewall. That leaves a calm, clear eye in the middle, often 12–37 mi across.
The rain bands curl in as rough logarithmic spirals, r = a·ebθ, the same family as the golden spiral in 001 · Golden ratio.
The Saffir-Simpson scale sorts hurricanes by sustained wind speed, but damage doesn't grow in step with it. The force of wind on a wall grows with the square of speed, and the energy it delivers grows with the cube. Slide from Category 1 to Category 5.
How to read this: each curve is measured against the weakest hurricane wind (74 mph). The labels v² and v³ mean "grows like the speed squared" and "grows like the speed cubed". The red curve (energy delivered) climbs much faster than the wind itself.
Try this: slide from 100 mph to 150 mph and compare the amber and red readouts.
Faster air hits harder (one factor of v), more air arrives every second (another), and each bit of air carries more energy (a third). Going from 100 to 150 mph is 1.5× the speed, but 2.25× the force and about 3.4× the power.
The category measures wind only. Storm surge, rainfall and flooding cause most hurricane deaths, and they depend on size, forward speed and coastline shape, none of which the category captures. A slow, large Category 1 can be deadlier than a compact Category 4.
Forecasters run a model many times, each with its starting conditions nudged slightly. This collection of runs is called an ensemble. The tracks agree at first, then fan out. The forecast cone is drawn so the storm's center stays inside it about two-thirds of the time. Below it, the Lorenz system shows why the fan-out can't be avoided.
How to read this: each thin green line is one computer forecast of the storm's path. The gold line is their average and the shaded gold area is the cone. Labels show hours from now.
Try this: push Starting uncertainty up, let go, and watch the cone widen. The Runs and uncertainty sliders take effect when you release them; Run new forecast rolls a fresh set.
The cone shows where the center might go, not how big the storm is. Impacts often reach far outside it. It widens with time because small errors in today's winds grow as the forecast runs.
How to read this: this isn't a map. Each dot is the whole state of a toy weather model moving through time. At first the amber dot hides behind the green one. When they visibly separate, the forecast has failed.
Try this: pick 10⁻¹² and see how little later the split happens.
Edward Lorenz built this tiny weather model in 1963. Two runs that differ by a millionth still split apart: the gap grows exponentially, doubling over and over. Try making the starting gap a thousand times smaller. It only buys a fixed amount of extra time, which is why forecasts stay reliable for days, not months.