Natural (s, n, z) Coordinate
s: along the wind direction
n: perpendicular to the wind direction, positive to the left
The Natural coordinate can be obtained by
rotating the Cartesian Coordinate until the x direction is along
the wind direction, then:
x is in the s direction
y is in the n direction
and
where V is the wind speed.
This coordinate is convenient because we
can use the observed wind speed and do not have to decompose the
wind into the x and y components.
NOTE:
In reporting the wind, we give speed in
knots and direction in 10s of degrees. The direction of wind
is 0 degree for northerly wind, 90 for easterly wind, 180 for
southerly wind, and 270 for westerly wind. For southerly wind
at 10 knots, we will report it as 1018.
The mathematical direction q
and the wind reporting direction dd can be related as:
q=270°-dd
To convert wind speed V and direction dd
to u and v, we use the following equations:
Wind reporting direction Mathematical direction
Starting from the equations of motion in
the Cartesian Coordinate
After rotating the coordinate until the
x direction is along the wind direction, the equations of motion
in the Natural coordinate are:
These equations can be obtained by changing
x to s, y to n, u to V, and v to zero from the equations of motion
in the Cartesian Coordinate. Even though we set v=0 in this coordinate
transformation, we cannot set 
to zero.
now is the acceleration in the n (perpendicular
to the wind) direction, and it causes the wind direction to change.
, therefore, is equal to the centrifugal
force needed to change the wind direction and
R is positive for cyclonic (counterclockwise)
rotation and negative for anticyclonic (clockwise) rotation.
If R > 0, the acceleration is in the positive n direction and
it causes the wind to turn left (counterclockwise). If R <
0, the acceleration is in the negative n direction and it causes
the wind the turn right (clockwise).
In s, n, p coordinate, the equations of
motion can be similarly written as
Geostrophic wind in Natural coordinate
Use the Buys-Ballot's Law, the direction
of s will be along the geostrophic wind direction and n will be
toward the low pressure.
On the surface map
Set (no acceleration)
The geostrophic wind is
On an upper air maps
Summary
Horizontal equations of motion
x, y, z coordinate x, y, p coordinate
s, n, z coordinate s, n, p coordinate
Geostrophic wind relation
x, y, z coordinate x, y, p coordinate
s, n, z coordinate s, n, p coordinate
Derivation of 
