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 HYDROSTATIC EQUATION

METEOROLOGIST JEFF HABY

The hydrostatic equation is one of the most important and most basic equations in meteorology. Understanding the equation makes it easier to physically interpret analysis and thickness charts. The equation is:

dP/dz = - density*gravity

Written in English, this is the change in pressure with the change in height is equal to the average density of the air times the gravitational constant. The negative sign is due to the fact that pressure decreases with height (usually when graphing, the values on the y-axis increase with height, this is the opposite with pressure in the atmosphere). The term that varies on the right hand side of the equation is density. Density of air is a function of temperature and moisture content. Increasing water vapor content and/or increasing the temperature causes the density to decrease. In very cold air, the air is very dense. Therefore, the dP/dz is large in cold air (the change in pressure with height is large). This means that pressure decreases rapidly in cold air and the thickness of a cold air mass is small (the atmosphere is thinner in the vertical when it is cold). In warm air this is the opposite case, warm air expands and takes up a larger volume. The thickness of warm air and the depth of the atmosphere is greater in a warm air mass. Since warm air has a low density, the change in pressure with height in warm air is smaller than in cold air. Example: The 500-millibar level is closer to the surface in arctic regions than tropical regions because the change in pressure with height is greater in arctic regions than tropical regions.

Upper air analysis charts show isohypses (lines of equal geopotential height). In association with a trough, heights will be lower. This is because rising air cools, becomes more dense, and thus lowers heights. Heights will also lower aloft when a cold air mass advects equatorward. In association with a ridge, heights will be higher. This is because sinking air warms dry adiabatically, becomes less dense and expands, thus raising heights. Heights will also rise aloft when a warm air mass advects poleward. An important analysis chart is the 1000 to 500 millibar thicknesses. Thickness is a function of the average temperature and average moisture content of the air. Lower thicknesses are associated with cold air while high thicknesses with warm air. The thickness chart can be used to assess cold air advection, warm air advection, frontal boundaries, the position of the jet stream, the intensity of troughs and ridges, and the thermal gradient in the troposphere.