About 30,000 to 40,000 feet up, a narrow band of fast wind flows west to east around each hemisphere. It forms above the sharpest boundary between cold polar air and warm tropical air. It meanders around the pole in a few huge waves, and those meanders decide whether your week is warm and dry or cold and stormy. Two short formulas explain most of it: the thermal wind for where the jet forms, and c = U − β/k² for how its waves move.
Pressure falls off faster with height in cold, dense air than in warm air. So at any altitude above a temperature contrast, pressure is lower on the cold side, and that difference grows the higher you go. Wind in balance with Earth's rotation follows the pressure difference, so the westerly wind grows with height too. It peaks near the top of the troposphere, right above where the contrast is sharpest. Drag across the cross-section to read the wind profile at any latitude.
How to read this: a slice through the air from the tropics (left) to the pole (right), ground at the bottom, with the jet blowing out of the screen toward you. Blue shading is colder air and brown is warmer. The gold lines ring the fastest wind and get brighter toward the glowing core. The strip on the right graphs wind speed against height at the green dashed line.
Try this: slide Front sharpness right and watch the gold rings tighten and the core speed up.
The wind keeps strengthening with height for as long as it stays colder toward the pole. Near the tropopause that stops. In the lower stratosphere the air over the tropics is colder than over the mid-latitudes, so the gradient reverses and the wind weakens again above it. The fastest wind sits at the turning point.
Real polar-front jet cores usually run 100 to 200 mph and occasionally exceed 200 mph, mostly in winter, when the pole-to-equator contrast is largest.
Idealized model: temperature falls 6.5 K/km to a tropopause at , with a tanh-shaped north–south contrast that weakens above and reverses by . The surface wind is set to zero, and the wind at each height is the running sum of thermal-wind increments from the ground up.
Spinning air keeps its total spin: its own spin plus the spin it gets from Earth, the Coriolis parameter f, which grows toward the pole. When the jet swings poleward, the air picks up planetary spin and has to give up some of its own, which curves it back toward the equator. A swing equatorward does the reverse. That restoring push makes the jet wave, and it always shoves the wave pattern westward. The wind carries it east. Which one wins depends on the wavelength.
How to read this: you're looking down on the North Pole. The orange loop is the jet, and the small pale arrows are the air racing along it. The gold dot rides one crest of the wave, so it shows where the pattern goes. Compare the gold dot with the arrows and with the green line to see whether the pattern drifts east, drifts west or stays put.
Try this: press Make it stationary. The gold dot stops while the arrows keep flowing through it.
How to read this: the gold curve shows how fast the wave pattern drifts for each number of waves around the globe. Above the middle line it drifts east, below it drifts west, and the green dashed line marks the count that stands still. The gold dot is your current setting.
At 45°N, β = 2 × 7.292×10⁻⁵ × cos 45° / 6.371×10⁶ m = 1.62×10⁻¹¹ m⁻¹ s⁻¹. The 45° latitude circle is around, so n waves each span 1/n of it. Carl-Gustaf Rossby derived this formula in 1939.
Upper-air maps usually show 4 to 6 long waves around the hemisphere. For typical jet winds that's close to the stationary wavelength, so these waves crawl or stall while the air races through them. Shorter waves riding on them move east with the wind and carry the day-to-day storms.
Idealized: a single jet at 45°N with no north–south wave structure, no friction and no mountains. Very long waves really do drift west at times, which forecasters call retrogression.
Where the jet bulges toward the pole it makes a ridge, and the air under it is usually warm and dry. Where it dips toward the equator it makes a trough that lets cold air spill south. Storms form and travel along the jet, especially just east of each trough. Drag the marker to put yourself anywhere on the map, then turn up the amplitude: a wavier pattern moves more slowly and holds the same weather in place for longer.
How to read this: the orange line is the jet blowing west to east across North America. Cold air sits above it (blue tint) and warm air below it (amber tint). Once the jet is wavy enough, a red L marks a surface low, where storms form, and H a surface high, usually calm and clear. The ring around "you" takes the color of your weather, and each sliver of the strip below is one hour at your marker as the pattern drifts past.
Try this: click Ahead of trough, then drag Waviness all the way right. The strip turns almost one solid color: a stalled pattern, called a block.
East of a trough the flow aloft comes from the southwest and spreads apart, so air rises underneath it. That's where surface lows deepen and rain and storms form. West of a trough the flow comes from the northwest, air sinks, and skies clear behind the cold front. Under a ridge, sinking air keeps things warm and dry.
When the waves grow large they tend to slow down. A pattern that stops moving is called a block, and it can hold heat, drought or repeated rain over the same place for a week or more.
Toy model: six waves on a jet centered at 45°N, drifting east at when nearly flat. The speed shrinks as the waviness grows by a made-up rule, 1 − (A/22°)², to mimic how amplified patterns slow down. Real blocks form through interactions between waves that this map doesn't simulate. The weather labels are rules of thumb from the jet's position and slope over your marker. The map is a lat–lon grid with no coastlines.
See W02 · Highs & lows for the surface systems and W04 · Global circulation for where the pole-to-equator contrast comes from.