Pressure-gradient force

In fluid mechanics, the pressure-gradient force is the force that 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 of motion, 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 altitudes above Earth's surface, thus providing a pressure-gradient force which counteracts the force of gravity on the atmosphere.

Magnus effect

The Magnus effect is an observable phenomenon that is commonly associated with a spinning object moving through a fluid. The path of the spinning object is deflected in a manner that is not present when the object is not spinning. The deflection can be explained by the difference in pressure of the fluid on opposite sides of the spinning object. The Magnus effect is dependent on the speed of rotation.

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, $m=\rho \,dA\,dz$ . Using Newton's second law, $F=ma$ , we can then examine a pressure difference $dP$ (assumed to be only in the $z$ -direction) to find the resulting force, $F=-dP\,dA=\rho a\,dA\,dz$ .

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.

v t e 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) Lifting condensation level (LCL) Precipitable water Precipitation Water vapor
Convection	Convective available potential energy (CAPE) Convective inhibition (CIN) Convective instability Convective momentum transport Conditional symmetric instability 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) Temperature anomaly Thermodynamic temperature Vapor pressure Virtual temperature Wet-bulb temperature Wet-bulb globe temperature Wet-bulb potential temperature Wind chill
Pressure	Atmospheric pressure Baroclinity Barotropicity Pressure gradient Pressure-gradient force (PGF)
Velocity	Maximum potential intensity

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