The
kinematic acceleration following the motion in natural coordinates (the left
hand side of the equation of horizontal motion in natural coordinates) can be
expanded into its constituent vectors.

(Equation
1)

This
says that the total acceleration is the sum of an
acceleration along a streamline and that normal to the streamline. In your reading in Martin, you
found out that the kinematic acceleration normal to the streamlines reduces to
a centrifugal acceleration, that only exists for
curved flow.

(Equation 2)

where r is the local radius of curvature of the streamlines and k
is the streamline curvature.
Substituting (2) into (1) gives

(Equation 3)

The
acceleration that acts along a streamline is entirely due to the pressure (or
height) gradient acceleration.
Now, segregating the accelerations that act along the s and n axes we get
the following:

t component

n component

(Equation
4a,b) Components of the Equation of Motion in Natural Coordinates

1. For geostrophic flow, which terms drop
out?

2. Manipulate the expressions above to
create an ̉Equation of MotionÓ for the state of zero flow (calm). This allows one to understand the basic
reason for air motion.

3. A problem develops, however, if the
contours are curved. Such a state
is not defined for the geostrophic wind, since by definition the geostrophic
wind is defined on the basis of unaccelerated flow
(straight streamlines with no along the streamline accelerations). However, a balanced flow does develop in
the case of curved contours/streamlines.
It is known as the Gradient Wind.

4. What is the relation between the
geostrophic and gradient wind? (Remember, in both, no accelerations along the streamlines occur).