Finite Difference of Variation of Geopotential Height 

With Respect to x, y and n Axes

 

Finite Difference – the discrete analog of the derivative

 

                               (1a)

 

                               (1b)

 

                              (1c)

 

In each case, location 2 is FURTHEST in the positive direction across the increment.  Along the x-axis, that would be furthest east.  Along the y-axis, that would be furthest north. 

 

The natural coordinate system is defined in such a manner so that that the s-axis is tangent to the wind (positive end downwind), with the n-axis at right angles.  The positive-n direction is 90 degrees counterclockwise to the positive-s axis.  See example below.

 

In natural coordinates, the geostrophic wind magnitude (in s-n-p coordinates) is given as

 

                                     (2)

 

Note that there is only component to the horizontal geostrophic wind in natural coordinates, as there is only one component of the horizontal wind in natural coordinates, by definition.

 

Note that the increment Æn will intersect the streamline locally at right angles at any location.  Thus, the positive-n direction will mostly NOT be aligned along a line of longitude.  It may also point south.