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 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.



Formalism


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


The acceleration resulting from the pressure gradient is then,


a=−1ρdPdz{displaystyle a={frac {-1}{rho }}{frac {dP}{dz}}}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{displaystyle P}P as,


a→=−1ρ∇→P{displaystyle {vec {a}}={frac {-1}{rho }}{vec {nabla }}P}{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, .mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
    ISBN 0-534-37214-7.








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