**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:

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

- 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.

- 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).