Equation of motion in x, y, p coordinates

The x and y components of the horizontal pressure gradient force in x, y, z coordinates are:

In x, y, p coordinates, they are

where f is the geopotential defined as:

.

The variable g is the local gravity, which varies with altitudes and from location to location. Geopotential height Z is defined as:

The geopotential height is used in all the aerological reports. For simplicity, z, instead of Z will be used to represent the geopotential height and g will be understood as a constant of 9.8 in subsequent presentations.

Derivation of the pressure gradient force terms in x, y, p coordinate

and the hydrostatic equation

is used to obtain

Similarly

Pressure gradient force in p coordinate:

Equations of motion in z coordinates

Equations of motion in p coordinates

Geostrophic wind in x, y, p coordinate

Comparing these equations with the geostrophic wind equation in x, y, z coordinate

the relationship between and the geostrophic wind in x, y, p coordinate is similar to the relationship between p and the geostrophic wind in x, y, z coordinate.

Example:

On a 500 mb weather map below, what is the geostrophic wind speed and direction if the spacing between the contour lines is 200 km?

For simplicity, let us use f = . Dz = -60 m, and Dy = 200 km = 200000m.

Substitute the values into the geostrophic wind equation, we get:

ug = 29.4 and vg = 0

NOTE: If =1 , the relationship between f and the geostrophic wind in x, y, p coordinate will be identical to the relationship between p and the geostrophic wind in x, y, z coordinate.