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:
Simplifications (1) and (2) transform (1a, b) to:
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:
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).
where z is the geopotential height (and the derivatives measure the variation of geopotential height along the horizontal coordinate axes).