Every weather map is drawn in pressure. Air starts moving because pressure differs from place to place. Earth's rotation then turns it sideways until it flows along the isobars, and friction near the ground lets some of it leak into the lows. That leak lifts air into cloud, which is why lows bring rain and highs bring clear skies.
The map below holds a high and a low, drawn as isobars every 4 hPa. Parcels of air start at rest and move under the forces you switch on. Pressure force alone sends them straight downhill into the low. Add Earth's rotation and they turn until they run along the isobars. Add surface friction and they cross the isobars at an angle, out of the high and into the low. Drag the H and the L.
How to read this: a weather map seen from above. The lines join places of equal pressure. Each small dot is a bit of air. Follow the big amber dot: the colored arrows above sit on it and show the forces pushing it.
Try this: click Pressure only, then + Coriolis, then + friction, and watch how the dots' paths change.
The three terms are the pressure force, the Coriolis force (with f = 2Ω sin φ) and a simple linear friction. Here ρ = 1.2 kg/m³, φ = 45°, k = 5×10⁻⁵ s⁻¹, and each system is a Gaussian bump in pressure. This is an idealized model of air near the surface on a flat map.
Once the pressure force and the Coriolis force cancel, the wind is geostrophic. Its speed follows from the spacing of the isobars, the pressure step between them, the air density and the latitude. Near the ground friction slows the wind and turns it toward the low. Around curved isobars a third term, the centripetal acceleration, changes the answer.
How to read this: the dot is a bit of air between two pressure lines. Each arrow is a push on it, and longer means stronger; the numbers in mm/s² say how hard each one pushes. When the pushes cancel, the wind (amber) blows steadily along the lines.
Try this: drag Isobar spacing to the left (closer lines) and watch the amber wind arrow grow.
Coriolis force grows with sin φ. At 20° it is about half its value at 45°, so the same isobar spacing needs about twice the wind before rotation can balance it. That's why tropical weather maps show weak pressure gradients, and why forecasters there often draw streamlines instead of isobars.
At the equator f is zero and no geostrophic balance exists. The Rossby number Ro = U/(fL) measures this: below about 0.1 the balance holds well, and near 1 or above it fails. Strong small vortices like W01 · Tornadoes are balanced by the curve of their own motion instead.
A mid-latitude low usually forms on the boundary between cold polar air and warm subtropical air. The Norwegian cyclone model, developed by the Bergen school around 1920, describes four stages: a stationary front, a wave on it, a mature low with a warm front and a cold front, and finally an occlusion. The map is a top view. Below it, a cross-section just south of the low shows warm air riding up over cold air.
How to read this: top, a weather map of one storm from above, with its low (L) in the middle. Fronts are the edges between cold and warm air, and the little triangles and bumps point the way each front is moving. Bottom, a side-on slice through the storm: warm air (dark red) slides up over cold air (blue), and the rising air makes the pale clouds.
Try this: press Play life cycle, or drag Stage from 0 to 3.
Margules' formula gives the slope at which a sloping boundary between two air masses stays in balance. Here f is set for 45°, T̄ = 280 K, and Δv is the jump in along-front wind. Real fronts are only a few kilometers deep and hundreds of kilometers wide, so the cross-section is stretched vertically.
Near the ground, friction makes air spiral into a low, as in Lab 1. That air piles up in the middle and has nowhere to go but up. Rising air cools, its water vapor condenses, and cloud and rain follow. A low can only deepen if more air leaves at the top, through divergence high up near the jet stream, than flows in at the bottom. A high works the other way round: air converges aloft, sinks, warms, dries out and spreads at the ground.
The vertical motion is only a few centimeters per second, too slow to measure directly. Forecasters diagnose it from the large-scale fields with the quasi-geostrophic omega equation. In plain terms, air tends to rise where warm air is being carried in and where cyclonic spin is being carried in more strongly aloft than below, as happens ahead of an upper-level trough. This page states that result without deriving it. W06 · Weather models covers how computers solve the full equations.
The classic picture, in which the cold front catches up with the warm front, is a simplification. Many real lows wrap their fronts around the center in other ways.