Hydrostatic Balance

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The Hydrostatic Balance is when in the absence of atmospheric motions the gravity force must be exactly balanced by the vertical component of the pressure gradient force.

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{dp}{dz} = -\rho g

This condition of hydrostatic balance provides an excellent approximation for the vertical dependence of the pressure field in the real atmosphere. Only for intense small-scale systems such as squall lines and tornadoes is it necessary to consider departures from hydrostatic balance.

Integrating from a height z to the top of the atmosphere we find that:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): p(z) = \int_z^\infty {\rho g dz}

so that the pressure at any point is simply equal to the weight of the unit cross section column of air overlying the point. Thus, mean sea level pressure p(0) = 1013.25 hPa is simply the average weight per square meter of the total atmospheric column.

Note that Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): d\Phi = g dz and that Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \alpha = \frac{RT}{p}

So we can also express the hydrostatic equation as:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): g dz = d\Phi = -\frac{RT}{p} dp = -RT d \ln p

To continue with integration, see Hypsometric Equation