Equation of Horizontal Motion in Natural Coordinates

 

 

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).