Equation of Motion,  Cartesian Coordinate System

The Equation of Motion in rectangular coordinates, following our discussions of the last week or so, is:

 

        (1a, b, c)

 

 

We can make the following simplifications for the SYNOPTIC SCALE free atmosphere: 

  1. Frictional acceleration can be neglected above the boundary layer (above ~300 meters or so above the local ground level;

  2. In the free atmosphere, horizontally moving air parcels tend to be unaccelerated, except in restrictive circumstances, and only at certain times.    The result of this is that the left hand side of (1a, b) tends to be one order of   magnitude or more smaller than the magnitudes of the other terms.  Notethat this assumption is not true for strongly curved flow (discussed another      time) and in the upper troposphere.

  3. Vertical accelerations are small compared to horizontal accelerations and vertical velocities are small (generally, one to two orders of magnitude smaller) compared to horizontal accelerations and velocities.  At thesynoptic scale, vertical velocities and accelerations are one to two orders ofmagnitude smaller than the the other forces per unit mass that affect air   parcels.

Simplifications (1) and (2) transform (1a, b) to:

                                              (2a, b)

 

 

the component geostrophic wind equations, which state that the speed of the wind is directly proportional to the pressure gradient, and parallel to the isobars (and non-divergent, which you proved in Lab 2).

And simplification (3) transform (1c) to:

                                                   (2c)

 

the hydrostatic equation, which states that, with the simplifications accepted, the vertical pressure gradient acceleration is balanced by the acceleration of gravity.

The problem with Equations (2a,b) is that there is no direct way to measure density.  However, density is related to the vertical pressure gradient at the synoptic scale, as shown in (2c).  Thus, we can transform equations (2a,b) from the rectangular coordinate system to the x, y, p coordinate system by solving (2c) for density and inserting that into (2a, b).

                                                (3a, b)

 

 

where z is the geopotential height (and the derivatives measure the variation of geopotential height along the horizontal coordinate axes).