EQUATIONS OF MOTION IN X, Y, Z COORDINATES

In the free atmosphere, F~0. The order of magnitude of other terms are given above.

How to estimate the order of magnitudes ?

Look at a vertical cross-section of a cyclone.

Estimate u~v~10 m/s

Vertical scale H~10 km

Horizontal scale L~1000 km

The simplified equations of motion

Eliminates all terms 2 order of magnitudes smaller than the largest term and use the definition of the Coriolis parameter:

we get:

The third equation is the hydrostatic equation.

Geostrophic wind approximation

Definition: The PGF balances the CF and the net acceleration is zero:

Equations for geostrophic wind:

Example:

For simplicity, let us use f = 10-4 s-1 and r = 1 kg m-3. If the distance between the isobars is 200 km, then Dp = -4 mb = -400 Pa, and Dy = 200 km = 200000m.

Substitute the values into the geostrophic equation, we get:

ug = 20 m s-1 and vg = 0 m s-1

Properties of geostrophic wind

1. If we stand with our backs to the wind, the low pressure will be to our left (Buys-Ballot's Law).

2. Geostrophic wind speed is inversely proportional to spacing between isobars.

3. For the same pressure gradient (spacing between isobars), geostrophic wind speed is inversely proportional to f.

At lower latitudes, f is small and ug is large.

At higher latitudes, f is large and ug is small.

4. At equator, f = 0, and the geostrophic wind relation breaks down, i.e., the wind cannot be geostrophic at equator.

Adjustment to geostrophic wind

Given the following pressure pattern and no air motion at the beginning

At pt A, there is no motion and no CF. The PGF will push the air parcel northward.

At points B and C, the air parcel gains speed, and the CF will turn it toward east.

At points D and E, the direction of the air parcel is due east, the CF can balance the PGF and achieve the geostrophic wind.

NOTE: This is an extremely simplified explanation. In reality, if the pressure pattern does not change, the air parcel starting at point A will go through an oscillation and never achieve a balanced state. Please see homework 1.

Geostrophic departure

In the free atmosphere, the observed winds (u, v) are close to the geostrophic wind (ug, vg) but usually u does not equal ug and v does not equal vg.

Definition of ageostrophic wind:

ua = u - ug

va = v - vg

Consider only the u component

u > ug , ua > 0, supergeostrophic

CF > PGF

Under the same environment:

If the wind is supergeostrophic, the air parcel will turn toward the RHS (right hand) side of the air movement direction (south in this example).

Similarly,

u < ug , ua < 0, subgeostrophic

CF < PGF

Under the same environment:

The air parcel will turn toward the LHS (left hand side) of the air movement direction (north in this example).