Two forces on moving air
Air has weight, so it feels pressure from all sides. If pressure is higher on one side, the air is pushed to the lower side. This push is the pressure gradient force. It always points from high to low pressure, straight across the isobars. Isobars are lines joining places with the same pressure.
The more crowded the isobars, the steeper the pressure change, and the stronger the push.
Earth spins, so moving air seems to curve. This is the Coriolis effect. In the northern half it turns air to the right. In the southern half it turns air to the left. It is zero at the equator and biggest at the poles. It also grows with the wind speed. It can only turn the air; it never speeds it up.
Geostrophic wind: the two forces balance
Air starts to move across the isobars. As it speeds up, the Coriolis force grows and turns it. This goes on until the Coriolis force is exactly equal and opposite to the pressure force. Then there is no net force and the air moves in a straight line along the isobars. This is the geostrophic wind.
The speed is v = (1 / ρf) × (Δp / d). Here ρ is air density, f is the Coriolis number (about 0.0001 per second in mid-latitudes), and Δp / d is the pressure change per metre across the isobars. Closer isobars mean a larger Δp / d and a faster wind.
This works well above about 1 km, where the ground does not rub the air. It does not work near the equator, because f is almost zero there.
Gradient wind: curved isobars
Real isobars are often circles round a high or a low. To go round a curve, air needs a net force towards the centre. This is the gradient wind.
- Around a low: the pressure force points in, the Coriolis force points out. The pressure force is bigger. The wind circles anticlockwise in the north (clockwise in the south). It is a bit slower than the geostrophic wind.
- Around a high: the Coriolis force points in and is bigger than the pressure force pointing out. The wind circles clockwise in the north (anticlockwise in the south). It is a bit faster than the geostrophic wind for the same isobar spacing.
Near the ground: friction and Buys Ballot's law
Near the ground, friction slows the wind. A slower wind feels less Coriolis force, so the pressure force wins a little and the wind drifts across the isobars towards low pressure, by about 10 to 30 degrees. This is why air spirals into a low and rises, bringing cloud and rain.
Buys Ballot's law: in the northern half, stand with your back to the wind. Low pressure is on your left. In the southern half it is on your right.
Try it: a turntable and a pencil
Put a paper plate on a record player or a slowly spinning stool. Spin it anticlockwise (like the northern half of Earth seen from above). Hold a ruler still and draw a "straight" line from the centre outwards. When you stop, the line is curved. The line curves because the plate moved under your pencil, just as Earth turns under moving air. Now guess: which way would the line curve if you spun the plate clockwise? Try it and check.
Key formulas and definitions
- Pressure gradient = Δp / d (Pa per metre)
- Geostrophic wind: v = Δp / (ρ f d)
- Coriolis number: f = 2Ω sin φ (Ω = 7.3 × 10⁻⁵ s⁻¹, φ = latitude)
- Balance: pressure force = Coriolis force
Worked examples
1. The pressure drops 4 hPa over 100 km. Find the geostrophic wind. Use ρ = 1.2 kg/m³ and f = 1 × 10⁻⁴ s⁻¹.
Δp = 4 hPa = 400 Pa; d = 100 km = 100 000 m. Gradient = 400 / 100 000 = 0.004 Pa/m. v = 0.004 / (1.2 × 0.0001) = 0.004 / 0.00012 ≈ 33 m/s.
2. If the isobars are twice as far apart (same pressure drop), what happens to the wind?
Δp / d becomes half. So v becomes half. Wide isobars mean a weaker wind.
3. You stand in the northern half with the wind on your back. Where is the low pressure?
On your left (Buys Ballot's law).
4. Find f at 30° N. (Ω = 7.3 × 10⁻⁵ s⁻¹)
f = 2Ω sin 30° = 2 × 7.3 × 10⁻⁵ × 0.5 = 7.3 × 10⁻⁵ s⁻¹.
Common mistakes
- Thinking wind blows straight from high to low pressure. High up it blows along the isobars.
- Thinking the Coriolis force makes the wind faster. It only turns it.
- Using the geostrophic formula at the equator. There f is zero, so it fails.
- Mixing up directions: around a low it is anticlockwise in the north, but clockwise in the south.