﻿ Useful Relationships Derived from the Primitive Equations

Hypsometric Relation

Heating the atmosphere causes it to expand (special application of the gas law, explained below) obtained by substitution of the hydrostatic equation (4) into the equation of state.(1).  This relation provides the basis of explaining many, many things that synoptic meteorologists see on weather maps and charts.

Thickness of layer between two pressure surfaces is directly related to the mean temperature of the layer.

Also, if we consider the thickness of a layer that is often of importance to synoptic meteorologists, the layer approximately between the ground (1000 mb) and about 6 km (500 mb), the Hypsometric Relation is

where k = R/g ln 2 and ∆Z is the thickness of the layer between 1000 mb and 500 mb and Tv is the Virtual Temperature, which we will assume here is very close to the actual temperature.

 Applications:          Tropopause is higher over the Equator than at the Poles (generally, the tropopause corresponds in winter to the 300 mb surface)          Since the surface pressure is nearly 1000 mb, deep cold air masses are associated with areas of low heights (troughs) in the middle and upper troposphere (and vice versa)          Since the surface pressure is nearly 1000 mb, the 1000-500 mb thickness pattern can be used as a first guess approximation of the 500 mb height pattern          Since fronts are defined as the surface expressions of boundaries between deep air masses with significant temperature differences, as a first guess they can be found on the warm air side of the packing of, for example, 500 mb height (crudest), 1000-500 mb thickness (crude) fields          The relation of surface winds to thickness contours allows one to assess temperature advection, and to make a better estimate of both the location and type of surface fronts (Note: In reality, finding surface fronts requires a careful analysis of actual temperature fields, instead of the layer mean temperatures inferred from thickness or height maps. Here is the NCEP analysis; note that the general position of and type of fronts for the eastern two thirds of the US was well "guessed", but the complications associated with the actual wind and temperatures in southwest TX made our "first guess" poor there)