# Pressure gradient (atmospheric)

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Editor-In-Chief: C. Michael Gibson, M.S., M.D. [1]

## Overview

In atmospheric sciences (meteorology, climatology and related fields), the pressure gradient (typically of air, more generally of any fluid) is a physical quantity that describes in which direction and at what rate the pressure changes the most rapidly around a particular location. The pressure gradient is a dimensional quantity expressed in units of pressure per unit length. The SI unit is pascal per metre (Pa/m).

## Mathematical description

Assuming that the pressure p is an intensive quantity, i.e., a single-valued, continuous and differentiable function of three-dimensional space (often called a scalar field), i.e., that

$p=p(x,y,z)$

where x, y and z are the coordinates of the location of interest, then the pressure gradient is the vector quantity defined as

$\nabla p = \begin{pmatrix} {\frac{\partial p}{\partial x}}, {\frac{\partial p}{\partial y}}, {\frac{\partial p}{\partial z}} \end{pmatrix}$

## Physical interpretation

Strictly speaking, the concept of pressure gradient is a local characterization of the air (more generally of the fluid under investigation). The pressure gradient is defined only at those spatial scales at which pressure (more generally fluid dynamics) itself is defined.

Within planetary atmospheres (including the Earth's), the pressure gradient is a vector pointing roughly downwards, because the pressure changes most rapidly vertically, increasing downwards. The value of the strength (or norm) of the pressure gradient in the troposphere is typically of the order 9 Pa/m (or 90 hPa/km).

The pressure gradient often has a small but critical horizontal component, which is largely responsible for the wind circulation. The horizontal pressure gradient is a 2-dimensional vector resulting from the projection of the pressure gradient onto a local horizontal plane.

Near the Earth's surface, this horizontal pressure gradient is typically pointing towards high pressure air masses (anticyclones), its particular orientation at any one time and place depends strongly on the weather situation. At mid-latitudes, the typical horizontal pressure gradient may take on values of the order of 10-2 Pa/m (or 10 Pa/km), although rather higher values occur within meteorological fronts.

## Weather and climate relevance

Differences in air pressure between different locations are critical in weather forecasting and climate. As indicated above, the pressure gradient constitutes one of the main forces acting on the air to make it move as wind. Note that the pressure gradient force points from high towards low pressure zones, it is thus oriented in the opposite direction from the pressure gradient itself.

Sound waves and shock waves are events that can induce very large pressure gradients, but these are often transitory disturbances.

## References

• Edward N. Lorenz (1967) The nature and theory of the general circulation of atmosphere, World Meteorological Organization, Publication No. 218, Geneva, Switzerland.
• Robert G. Fleagle and Joost A. Businger (1980) An Introduction to Atmospheric Physics, Second Edition, Academic Press, International Geophysics Series, Volume 25, ISBN 0-12-260355-9.
• John M. Wallace and Peter V. Hobbs (2006) Atmospheric Science: An Introductory Survey, Second Edition, Academic Press, International Geophysics Series, ISBN 0-12-732951-X.