Dirichlet algebra

The hypsometric equation, also known as the thickness equation, relates an atmospheric pressure ratio to the equivalent thickness of an atmospheric layer under the assumptions of constant temperature and gravity. It is derived from the hydrostatic equation and the ideal gas law.

Equation

The hypsometric equation is expressed as:

$\ h=z_{2}-z_{1}={\frac {R\cdot {\bar {T}}}{g}}\cdot \ln \left({\frac {p_{1}}{p_{2}}}\right)$

where:

$\ h$ = thickness of the layer [m]

$\ z$ = geometric height [m]

$\ R$ = specific gas constant for dry air

$\ {\bar {T}}$ = mean temperature in Kelvin [K]

$\ g$ = gravitational acceleration [m/s²]

$\ p$ = pressure [Pa]

In meteorology, $p_{1}$ and $p_{2}$ are isobaric surfaces. In altimetry with the International Standard Atmosphere the hypsometric equation is used to compute pressure at a given height in isothermal layers in the upper and lower stratosphere.

Derivation

The hydrostatic equation:

$\ p=\rho \cdot g\cdot z$

where $\ \rho$ is the density [kg/m³], is used to generate the equation for hydrostatic equilibrium, written in differential form:

$dp=-\rho \cdot g\cdot dz.$

This is combined with the ideal gas law:

$\ p=\rho \cdot R\cdot T$

to eliminate $\ \rho$:

${\frac {\mathrm {d} p}{p}}={\frac {-g}{R\cdot T}}\,\mathrm {d} z.$

This is integrated from $\ z_{1}$ to $\ z_{2}$:

$\ \int _{p(z_{1})}^{p(z_{2})}{\frac {\mathrm {d} p}{p}}=\int _{z_{1}}^{z_{2}}{\frac {-g}{R\cdot T}}\,\mathrm {d} z.$

R and g are constant with z, so they can be brought outside the integral. If temperature varies linearly with z (as it is assumed to do in the International Standard Atmosphere), it can also be brought outside the integral when replaced with ${\bar {T}}$, the average temperature between $z_{1}$ and $z_{2}$.

$\ \int _{p(z_{1})}^{p(z_{2})}{\frac {\mathrm {d} p}{p}}={\frac {-g}{R\cdot {\bar {T}}}}\int _{z_{1}}^{z_{2}}\,\mathrm {d} z.$

Integration gives:

$\ln \left({\frac {p(z_{2})}{p(z_{1})}}\right)={\frac {-g}{R\cdot {\bar {T}}}}(z_{2}-z_{1})$

simplifying to:

$\ln \left({\frac {p_{1}}{p_{2}}}\right)={\frac {g}{R\cdot {\bar {T}}}}(z_{2}-z_{1}).$

Rearranging:

$(z_{2}-z_{1})={\frac {R\cdot {\bar {T}}}{g}}\ln \left({\frac {p_{1}}{p_{2}}}\right)$

or, eliminating the ln:

${\frac {p_{1}}{p_{2}}}=e^{{g \over R\cdot {\bar {T}}}\cdot (z_{2}-z_{1})}.$

References

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