Pressure-gradient force

The pressure-gradient force is the force which results when there is a difference in pressure across a surface. In general, a pressure is a force per unit area, across a surface. A difference in pressure across a surface then implies a difference in force, which can result in an acceleration according to Newton's second law, if there is no additional force to balance it. The resulting force is always directed from the region of higher-pressure to the region of lower-pressure. When a fluid is in an equilibrium state (i.e. there are no net forces, and no acceleration), the system is referred to as being in hydrostatic equilibrium. In the case of atmospheres, the pressure gradient force is balanced by the gravitational force, maintaining hydrostatic equilibrium. In Earth's atmosphere, for example, air pressure decreases at increasing altitudes above Earth's surface, thus providing a pressure gradient force which counteracts the force of gravity on the atmosphere.

Formalism

Consider a cubic parcel of fluid with a density $\rho$ , a height $dz$ , and a surface area $dA$ . The mass of the parcel can be expressed as, $m = \rho \cdot dA \cdot dz$ . Using Newton's second law, $F = m\cdot a$ , we can then examine a pressure difference $dP$ (assumed to be only in the $z$ -direction) to find the resulting force, $F = - dP \cdot dA = \rho \cdot dA \cdot dz \cdot a$ .

The acceleration resulting from the pressure gradient is then,

$a = \frac{-1}{\rho} \frac{dP}{dz}$ .

The effects of the pressure gradient are usually expressed in this way, in terms of an acceleration, instead of in terms of a force. We can express the acceleration more precisely, for a general pressure $P$ as,

$\vec{a} = \frac{-1}{\rho} \vec\nabla P$ .

The direction of the resulting force (acceleration) is thus in the opposite direction of the most rapid increase of pressure.

References

Roland B. Stull (2000) Meteorology for Scientists and Engineers, Second Edition, Ed. Brooks/Cole, ISBN 0-534-37214-7.

Meteorological data and variables

General	Adiabatic processes Advection Buoyancy Lapse rate Lightning Surface solar radiation Surface weather analysis Visibility Vorticity Wind Wind shear

Condensation	Cloud Cloud condensation nuclei (CCN) Fog Convective condensation level (CCL) Lifted condensation level (LCL) Precipitation Water vapor

Convection	Convective available potential energy (CAPE) Convective inhibition (CIN) Convective instability Convective momentum transport Convective temperature (T_c) Equilibrium level (EL) Free convective layer (FCL) Helicity K Index Level of free convection (LFC) Lifted index (LI) Maximum parcel level (MPL) Bulk Richardson number (BRN)

Temperature	Dew point (T_d) Dew point depression Dry-bulb temperature Equivalent temperature (T_e) Forest fire weather index Haines Index Heat index Humidex Humidity Relative humidity (RH) Mixing ratio Potential temperature (θ) Equivalent potential temperature (θ_e) Sea surface temperature (SST) Thermodynamic temperature Vapor pressure Virtual temperature Wet-bulb temperature Wet-bulb potential temperature Wind chill

Pressure	Atmospheric pressure Baroclinity Barotropicity Pressure gradient Pressure-gradient force (PGF)

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