Let’s pick up with the geostrophic wind scale — Page 159, Lesson 135

Let’s pick up with the geostrophic wind scale. I want to walk you through how we actually determine the speed of the geostrophic wind from a weather chart. Earlier, you learned that there is a relationship between how closely the isobars are spaced, the pressure gradient force, and the wind speed. The geostrophic wind scale is the tool that lets us put a number on that relationship. Here is how you use it: you measure the distance perpendicular between two adjacent isobars — that is, the shortest distance across the pressure gradient. Then you take that measured distance and place it on the geostrophic wind scale, reading from left to right. The scale directly gives you the wind speed in knots. Now, there is a very important limitation. The geostrophic wind only blows above the friction layer. Within the friction layer — the layer of air closest to the Earth's surface — surface friction slows the wind down. When the wind speed is reduced, the Coriolis force also reduces because Coriolis force depends on wind speed. That means the two forces — pressure gradient force and Coriolis force — are no longer in balance. So the geostrophic wind concept only applies above that friction layer. How high is the friction layer? It varies depending on the nature of the surface — rough terrain versus smooth ocean — and the time of day. But generally, we consider the geostrophic wind to be found between 2000 and 3000 feet above the surface. Now, let's look at the effect of latitude. For a geostrophic wind, the pressure gradient force equals the Coriolis force. So if you keep the same pressure gradient force — which means the same spacing between isobars — and you move to a higher latitude, the Coriolis force would need to stay the same for balance. But the Coriolis force depends on the sine of the latitude. As latitude increases, sine of latitude increases. For the Coriolis force to remain constant with a larger sine of latitude, the wind speed must decrease. That is the key relationship. Let me show you with the numbers from the geostrophic wind scale. A certain isobar spacing at 40° north gives a wind speed of 25 knots. The exact same isobar spacing at 70° north gives a wind speed of only 15 knots. So the same pressure gradient produces a slower wind at higher latitude. The formula that describes this is: V equals PGF divided by 2 times Omega times rho times sine theta. Let me unpack that. V is the geostrophic wind speed. PGF is the pressure gradient force. Omega is the angular velocity of the Earth's rotation. Rho is the air density. And sine theta — where theta is the latitude — accounts for the latitude effect. You can see that as sine theta increases in the denominator, the wind speed V decreases, assuming everything else stays the same. So the geostrophic wind scale must be corrected for latitude. The scale you use is designed for latitudes between 40° and 70°. And here is an important boundary condition: within 5 degrees of the Equator, the Coriolis force is close to zero. Within 15 degrees of the Equator, the Coriolis force is very small. That means the geostrophic formula is no longer valid near the Equator — the balance between pressure gradient force and Coriolis force simply does not hold there.