How WebGCM works
Part 2: A spinning planet

What a turning planet does to everything that moves across it.

Part 1 ended with a row of columns of air, the warmed ones pushing their air straight outward at the top and the neighbors sending it back along the ground. On the real planet the outflow would not have gone straight. It would have turned. The Earth spins, and everything that moves across it appears to curve, which is why the wind on a weather map blows along the lines of equal pressure rather than across them. This part is about that turn and nothing else: no grid yet, that is Part 3's business, just a parcel of air on a turning planet, and in the last figure a push from the pressure.

The turntable

The turntable shows what turning does. The puck slides in a straight line at a steady speed. Only when you ride the turntable does the path bend, always to the right on this turntable, which turns counterclockwise as the northern hemisphere does seen from above the pole, and the faster the turntable spins, the more it bends.

The force in three dimensions

That apparent push is the Coriolis force, and on a sphere it does not always lie along the ground. For each kilogram of air it is −2Ω × v, where v is the velocity and Ω is the planet's spin, an arrow pointing north along the axis whose length is the rate of turning, 7.29 × 10⁻⁵ radians per second, one turn every 23 hours 56 minutes, measured against the stars. The × is a cross product, which makes the force 2Ω times the speed times the sine of the angle between the velocity and the axis, at right angles to both, so it vanishes for motion along the axis. The figure sets a parcel sliding over the globe with no friction and no pressure differences, so nothing pushes it along the ground but the Coriolis force: pick its latitude, which way it heads and how fast. The globe is drawn turning with the planet, like the turntable's second view, so the globe and its lines of latitude and longitude hold still and only the parcel moves. Beneath it the same parcel is seen twice more from above the north pole, standing still on the left and turning with the planet on the right; those two disks, and the last slider, which changes how fast the planet spins, make sense once the parcel's circles do, and the end of the section after next comes back to both.

At the equator the axis is parallel to the ground, pointing north, so a force at right angles to it lies in the plane of up and east; at right angles to a velocity along the ground as well, it can only point up or down: straight up for a parcel heading east, straight down for one heading west, and nothing at all for one heading due north or south, which moves parallel to the axis. None of it pushes sideways. A parcel launched east or west along the equator runs all the way around it without ever being turned. Launched a degree off due east, it is steered back by the turning on either side of the equator, and at the figure's default 100 m/s it never strays more than 0.3° from it; launched a degree off due west, it is thrown away from the equator, as far as 38°, so only the eastward run holds. Hurricanes, which need that turning to spin up, almost never form within about 5° of the equator. At the pole the axis is the local vertical, so the force lies flat along the ground at its full strength, 2Ω times the speed: 5.3 m/s an hour for a 10 m/s wind. Slide the latitude to the equator with the heading east, or to the pole, to see both cases.

In between it does some of each. The part along the ground is f times the speed, where f = 2Ω sin φ at latitude φ is the Coriolis parameter, twice the rate Ω sin φ at which the ground turns about the local vertical, so this piece of the force is the turntable again. It points at right angles to the velocity, to its right in the northern hemisphere and to its left in the southern. The part along the vertical is 2Ω cos φ times the eastward part of the velocity, upward for a parcel heading east and downward for one heading west. At 45° the two are the same size, 3.7 m/s an hour, or 1.03 × 10⁻³ m/s², for a 10 m/s east wind, but they meet very different competition. Along the ground the Coriolis push is as large as the pressure pushes that drive the wind, so it shapes every wind. Along the vertical it is up against gravity, 9.81 m/s², and makes the air lighter by about one part in 9,500. That is the Eötvös effect, first noticed in gravity measured on ships, which came out lower when they sailed east than when they sailed west; it is real, and small enough that models like WebGCM drop it, along with the small sideways push that rising and sinking air would feel. Keeping only the part along the ground, as hydrostatic models generally do, is called the traditional approximation.

In between, the green arrow is the part along the ground, the pink the part along the vertical, the thin lines show how the force splits into the two, and the force itself always lies in the pale disk through the parcel, at right angles to the axis; drag the globe to see the pink part from the side, since at 45° it points nearly at you. The readout gives both parts.

Inertial circles

Because the push along the ground is always at right angles to the velocity, it changes the parcel's direction but never its speed. With nothing else acting, the parcel goes around in a circle, an inertial circle, whose radius is the speed divided by f and whose period is 2π/f: half the time the planet takes to turn once, 11.97 hours, divided by the sine of the latitude, so 16.9 hours at 45°, 12 at the pole, and no circle at all at the equator. The globe's speeds are unrealistic on purpose. A 10 m/s wind at 45° circles 97 km in radius, a dot on a globe this size, so the slider runs from there up to 300 m/s, and the default 100 m/s starts on a curve 970 km in radius. Set the globe back to 45° north, heading east, and watch it for a few days: the loops creep westward, because the turning is tighter on the poleward side of each loop than on the equatorward side, so no loop quite closes. That creep has a name, beta drift, after the Rossby parameter β, which measures how fast f changes with latitude. A related effect steers hurricanes slowly westward and toward the pole. At 100 m/s it is exaggerated along with everything else: the loops are lopsided, reaching down to 23° north, where the turning is gentler, so each takes 22 hours rather than 16.9, and they march west by 14° of longitude a day. Launched east from 45° north faster than about 136 m/s, the parcel no longer loops at all: it swings across the equator into the southern hemisphere, as far south as it started north, and back. Closer to the equator it takes less: at the default 100 m/s, any eastward launch within about 38° of the equator swings across. A path that crosses into the southern hemisphere is drawn faint there, seen through the planet from above the north pole.

Beneath the globe the same parcel is seen from above the north pole, standing still on the left and turning with the planet on the right: the turntable's two views again. Launched at the pole, the pair looks like the turntable's, a straight line on the left and a circle on the right, with one difference. The real planet bulges slightly at the equator, by just enough that gravity and the outward fling of the spin together pull straight down into the ground everywhere, and the figure's gravity is set up the same way; seen standing still, that leaves a small pull toward the axis, so the parcel runs out along the line, back through the pole and out the other side once a day instead of sliding off. Launched east along the equator, it rides the rim in both, once around in 20 hours on the left, carried by the planet, and once in 4.6 days on the right. In between, the path seen standing still circles the pole once a day with the loop riding on it as a wobble, because the ground carries the parcel around at 330 m/s at 45° while the launch speed is 100, so only at the pole does the standing-still view show a straight line.

The last slider sets the planet's spin. The Coriolis parameter f is proportional to Ω, so at twice Earth's spin the inertial period at 45° is half as long, 8.5 hours, and the circle half the size, 485 km at 100 m/s. The loops, less lopsided now, reach down only to 35° and take 9.3 hours each, and the planet's day is half as long too, so the disk on the left turns twice as fast. At half Earth's spin the period and the radius double, to 33.8 hours and 1,939 km, but at 100 m/s the parcel no longer loops: it swings across the equator, along the same path Earth gives a parcel at 200 m/s, traced at half the pace. At zero there is no Coriolis force at all. The parcel runs straight, along a great circle, once around the planet every 4.6 days at 100 m/s, and the two disks show the same thing, because the one on the left no longer turns. Turn the planet backward and Ω points south, down the axis. Because f changes sign with Ω, the parcel turns to its left in the northern hemisphere, as it would in the southern hemisphere on Earth: launched east from 45° north, it loops on the poleward side, up to 61°, the loops creep east instead of west, and the disk on the left turns clockwise.

Adding a push

The next figure zooms in on a patch around one latitude, seen from above, with the parcel moving at a realistic 10 m/s. With the pressure even it shows the same circle at its true size, creeping west about 7 km a day at 45°. Now add a pressure difference. The parcel is pushed toward low pressure by the pressure gradient force of Part 1, the Coriolis force turns it to the right, and it ends up looping around one line of equal pressure, never straying more than a loop's radius from it, while the loops as a whole drift sideways at a steady speed: along the lines, not across them, with the low pressure on the left in the northern hemisphere. Smooth out the loops and that drift is all that remains. The figure can show this too: switch it to twelve parcels, started together and each shoved at the same speed relative to the drift but in a different direction. Every one of them loops, but their loops cancel in the average, which glides along the line at the drifting speed, wobbling only slightly because the parcels feel slightly different Coriolis forces at their different latitudes. That drift is the geostrophic wind, and the lines it follows are the isobars, the lines of equal pressure a weather map draws. It is also what the outflow in Part 1's last figure would have done with a second dimension to turn into: it would have swung sideways instead of heading straight for the neighbors.

Now switch the figure from an even slope to a circle. Released from rest beside a low, the parcel heads for the center, but the Coriolis force turns it before it gets there, and it ends up circling the low counterclockwise, swinging in and out a little, rather than falling in. That is why the air around a storm spins instead of filling it. In the northern hemisphere a low is circled counterclockwise and a high clockwise, the other way around south of the equator, and in both the air stays close to its circle of equal pressure. Around a low the balanced speed is a little less than the geostrophic wind, because part of the pressure push is spent bending the path into a circle; the version of the balance that accounts for that is called the gradient wind. The twelve parcels work here too: started together beside the low, each moving with the balanced wind plus a small shove in its own direction, they loop on their own while their average circles the low smoothly at the balanced speed. Try the same beside a high and it holds only while the high is weak. A strong high cannot keep a balanced circle close to its center at all. There the Coriolis force has to hold the air against the outward push of the pressure and also bend it around a tight circle, and past a certain steepness of pressure it cannot do both, whatever the speed. That is one reason highs are broad and gentle while lows can be intense.

That is what a turning planet does to the air. Pushed once, a parcel circles; pushed steadily, by a difference in pressure, it loops its way along the lines of equal pressure, and the drift that remains when the loops are smoothed out is a wind in which the Coriolis force holds the air against the push. Most of the wind in the parts to come is some version of that balance, bent by the shape of the pressure and by friction. Part 3 builds the grid that will hold these winds, and one of its pieces is there to hand the turn a wind the grid does not store.