Equation of Horizontal Motion

(1)

x component

(2)

y component

The u-component of the geostrophic wind is obtained from (2)

(3)

divide both sides of equation (2) by f and substitute (3) into the result.

(4)

Equation (4) states that the difference between the real wind and the geostrophic wind is due to accelerations. Since the real wind can always be broken into a geostrophic and an ageostrophic component

u = u_{g}
+ u_{a }(5)

one can substitute (5) into (4) to get

(6)

Equation (6) simply states the obvious. The portion of the real wind that is ageostrophic is due to any acceleration that is present. If the wind is unacclerated, then there is no ageostrophic component to the real wind and, according to (5), the real wind is geostrophic. As we discussed in class, at the synoptic-scale, this only occurs approximately in the middle troposphere. Since the geostrophic wind is also non-divergent, both the assumptions of geostrophy and non-divergence work best there.