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The formula for estimating the wind speed, given a map of height contours, is:
In the above equation, v is the wind speed, g is the acceleration due to gravity (9.8 m/s2), f, called the coriolis parameter, is equal to twice the local rotation rate of the earth (approximately 0.0001/s), and delta Z is the change in height over a given distance (delta X). The change in height should be measured in the direction that the height changes most rapidly. With a calculator and some conversions of units, you could use this formula to compute wind speeds in geostrophic balance.
But why bother, when there's a simpler way. Notice that g and f are constants. If we pick a particular distance for delta X, and never change it, then delta X is a constant, too. So if we multiply all those constants together, we would get a formula which looks like
v = constant * delta z
which, if we're lucky, we can compute in our heads.
For a distance (delta X), I'm going to pick the height of Kansas. That's three degrees of latitude, or 333 km. Now I'll multiply everything out:
So if I have delta Z in meters, I can multiply by 0.3 and get the wind speed in meters per second.
But, I usually have height contours plotted in decameters, and I want the wind speed in knots! So...
The geostrophic wind speed (in knots) is therefore equal to the change in height across a distance equal to the height of Kansas, times 6.
How do you use this formula, now that we've got it? Suppose you want to know the wind speed at a particular spot, and all you have are the height contours. Take your pencil or something, stick the end of it against the northern border of Kansas, and place your thumbnail on the pencil over the southern border of Kansas. There, you've marked off a distance equal to the height of Kansas. Now, without moving your thumb on the pencil, lay the pencil across the height contours at whatever point you want the wind speed. How much (in decameters) does the height change from where the end of your pencil is and where your thumb is? Take that number, multiply by six, and you've got the wind speed.
Change in height across pencil end to thumbnail is 558 - 540 = 18.
18 * 6 = 108 knots.
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