Hydrostatic Law

up:: 034 MOC Atmospheric Physics

Given a differential element of area $dA$ and height $dz$,¹ it will feel a pressure $P$ above it, and a pressure $P+dP$ below it.

Thus, the resultant force it feels is

$$(P+dP-P)dA=dPdA$$

The acceleration it feels is $-g$, and its mass is $\rho dAdz$.

Equalling both, we have

$$\begin{array}{rl}dPdA& =-\rho gdAdz\end{array}$$

Therefore,

$$dP=-\rho gdz$$

This equation motivates the definition of a Geopotential

$$\Phi :={\int }_0^{z}gdz$$

and a Geopotential Height

$$Z:=\frac{\Phi (z)}{g_0}$$

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References

• WALLACE, John Michael; HOBBS, Peter Victor. Atmospheric science: an introductory survey. 2nd ed ed. Amsterdam Paris: Academic press, 2006.

Footnotes

1. And given a gravitational acceleration $g$ and constant density $\rho$. ↩