Relationship between the vertical velocity
in x,y,z and that in x,y,p
The vertical velocity in x,y,p coordinates is the rate at which a vertically-moving air parcel experiences a change in pressure following its motion. Recall the Lagrangian derivative of pressure is:
The pressure change due to vertical motion alone is
Substituting the hydrostatic approximation into the right hand side of (2) gives the following definition of the vertical velocity in x,y,p coordinates:
Equation (3) says that the vertical velocity in the two coordinate systems differ by a factor proportional to densityXgravity. For a typical surface density of 1.0 X 10-3 g/cm-3 and gravity of 9.8 X 102 cm s-1, at the synoptic scale we get vertical velocities on the order of -1 microbars per second and 10 centimeters per second (roughly).