Global Circulation
Sunlight falls hardest on the equator and glances off the poles. The air moves to even that out, and a spinning planet turns the moving air into trade winds and jets.
The tropics take in more sunlight than they give back to space, and the poles give back more than they take in. Air rises over the warm equator, travels poleward near the top of the atmosphere, sinks in the subtropics and returns along the surface. That overturning is the Hadley cell, and it carries the surplus heat away from the equator. As the poleward air moves it comes closer to the axis the planet turns about, so it gains eastward speed, the way a spinning skater quickens on pulling the arms in. That is the subtropical jet. The air coming back along the surface moves the other way and so blows from the east: the trade winds. How far the cell can reach is set by how fast the planet spins. Spin it faster and the eastward gain is steeper, the air runs out of room sooner, and the cell is narrower.
Pole to Pole, Surface to 15 km
The Wind, by Latitude
What Sets the Width
a planet radius · φ latitude · Ω spin rate · u eastward wind
A ring of air carries the angular momentum M above. Left alone it keeps it. Air that leaves the equator with no eastward wind of its own and keeps its M arrives at latitude φ blowing east at Ω a sin²φ / cos φ. On Earth that is 14 m/s at 10°, 134 m/s at 30° and 697 m/s at 60°. The gold dashes on the wind plot are that wind. It rises so steeply that the cell cannot reach far: long before the air gets near the pole the wind it would need is impossible, and the cell stops.
Held and Hou’s estimate. Setting the heat the cell must carry against that limit gives a cell edge near √(5 Δ g H / 3 Ω²a²), which for Earth’s spin and a 50 °C equator-to-pole difference is 24.9° (Held & Hou 1980). This model puts its jet at 25.7°. The observed Hadley cell ends between about 25° and 31°, with the subtropical jet above its poleward edge.
The jet is the measure used here. In this theory the jet sits where the cell ends, so its latitude is the cell’s width. An earlier version of this page measured the width from a contour of the overturning instead. That was dropped: at strong forcing the contour crosses at a different latitude on a finer grid, while the jet does not move.
Nothing reaches the limit. The jet here is 46% of the angular-momentum wind at Earth’s spin and 80% at half of it. Friction bleeds momentum away on the journey, so the air arrives slower than a free ring would. The readout tracks that share as you move the sliders.
Why the Reading Is Open
Three things, classified separately. The program running the panel is an attractlet model under § 7.4, as on the Lorenz page. The circulation in the model is a second thing. The atmosphere is a third.
Name the boundary. Take X to be the moving air, with the uneven heating supplied from outside X.
The case for attractlet. The test on the Weather page asks whether a structure’s own activity releases the energy that drives it, or only carries energy delivered to it. This circulation only carries. Press Flatten the heating to take the equator-to-pole difference away: the trade winds are gone within 10 days, the jet falls from 45 to 26 m/s by day 20 and to 5.6 by day 160, and the overturning is down to an eighth of its strength by day 160 and a twenty-eighth by day 320. That is the fifth test of § 7.4, a configuration that “relaxes to an inert or passive state when uncoupled.”
What is left open. The overturning is the air’s own doing. Sunlight does not arrange a cell; it sets up a temperature difference, and the air builds the cell out of it. That is the same shape as the convection cells, and the same shape as the coronal loop: the machinery inside the boundary, the gradient outside it. Whether a configuration that builds its own conversion machinery out of a supplied gradient can satisfy Recursion Lock is the question raised on the sunspot cycle page and not yet settled. This page does not settle it either.
The Shape of the Basin
One state, from five different starts. At Earth’s spin and a 50 °C difference, the model was started from still air, from a scrambled state with random winds up to 30 m/s and a scrambled temperature field, from the spun-up state of a planet turning four times as fast, from the spun-up state of a 60 °C difference, and from the running state with every wind wiped to zero. All five end with the jet at 45.0 to 45.1 m/s at 25.7°, trade winds of 3.18 to 3.20 m/s, and the same overturning to within 2%.
It takes about two months. From still air the jet reaches nine tenths of its final strength in 57 days. Press Wipe the winds, which leaves the temperature pattern standing but stops every wind dead, and it is back in 55 days. The recovery is violent: with the winds gone the temperature pattern is far out of balance, and for a few days the overturning runs about 230 times stronger than it does at rest before settling back.
Spin, and the cell draws in. Holding the temperature difference at 50 °C and turning the spin up from half Earth’s to four times it, the jet moves in from 36.7° to 14.7° and weakens from 82.7 to 11.8 m/s, the trades fall from 4.95 to 1.00 m/s, and the overturning falls by a factor of 32. Every step of the slider moves all four the same way.
Heat, and the cell strengthens far more than it widens. Holding the spin at Earth’s and raising the difference from 25 to 60 °C, the jet goes from 22.2 to 54.5 m/s and the overturning more than triples, while the jet moves out only from 22.0° to 25.7°. Held and Hou have the edge moving from 17.6° to 27.3° over that range. The direction is right and the size is not, and the next panel says so.
Where This Model and the Air Part Company
The cell is too weak. Trade winds of 3.2 m/s here against roughly 6 m/s over the real trade wind belt. That is not a bug so much as the known result: a dry cell with no eddies and no latent heat moves far less air than the real Hadley cell does, which Held and Hou pointed out about their own solution.
It answers to spin more weakly than the theory says. Over the range of the slider Held and Hou’s estimate falls from 49.9° to 6.2°, a factor of eight; the jet here moves from 36.7° to 14.7°, a factor of two and a half. The two agree almost exactly at Earth’s spin, 24.9° against 25.7°, and part company at the ends. The theory is for a nearly frictionless cell; this model has enough friction to put a floor under how narrow the cell can get and a ceiling on how wide.
It answers to the heating contrast more weakly still. The same estimate has the edge moving out as the square root of the temperature difference, 17.6° to 27.3° across the slider. The jet here moves 22.0° to 25.7°, which on this grid is a single point of latitude. The direction is right, the size is not, and the reason has not been run down. Read the slider as changing how strong the cell is, which it does faithfully, rather than how wide.
There are no westerly winds in the middle latitudes. On the real Earth the surface wind turns from easterly to westerly near 30°, and those westerlies are driven by the travelling highs and lows that this model averages away. Here the surface wind is easterly at nearly every latitude. The red curve on the wind plot should not be read as the real surface wind outside the tropics.
What holds up. Angular momentum is conserved to three parts in 10¹⁰ over 400 days once the surface drag is switched off, so the winds are not being fed or starved by the arithmetic. Every corner of both sliders was run against a grid of twice the resolution in each direction: the jet and the trade winds agree within 9% and the jet’s latitude within 1.7°, with the ordinary setting agreeing to 3%. The overturning strength is the loosest of the four and is quoted here only as a rough size. The two hemispheres come out identical to three decimal places at every setting tested, and at every setting the run is steady to within 0.02 m/s over its last 40 days.
Held, I. M., & Hou, A. Y. (1980). Nonlinear axially symmetric circulations in a nearly
inviscid atmosphere. Journal of the Atmospheric Sciences, 37, 515.
Lindzen, R. S., & Hou,
A. Y. (1988). Hadley circulations for zonally averaged heating centered off the equator. Journal of
the Atmospheric Sciences, 45, 2416.
Schneider, T. (2006). The general circulation of the
atmosphere. Annual Review of Earth and Planetary Sciences, 34, 655.