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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
It is expressed as:
h
=
z
2
−
z
1
=
R
⋅
T
g
⋅
ln
[
P
1
P
2
]
{\displaystyle \ h=z_{2}-z_{1}={\frac {R\cdot T}{g}}\cdot \ln \left[{\frac {P_{1}}{P_{2}}}\right]}
where:
h
{\displaystyle \ h}
z
{\displaystyle \ z}
geopotential height [m]
R
{\displaystyle \ R}
gas constant for dry air
T
{\displaystyle \ T}
temperature in kelvins [K]
g
{\displaystyle \ g}
gravitational acceleration [m/s2 ]
P
{\displaystyle \ P}
pressure [Pa]In meteorology ,
P
1
{\displaystyle P_{1}}
P
2
{\displaystyle P_{2}}
isobaric surfaces and T is the average temperature of the layer between them. In altimetry with the International Standard Atmosphere the hypsometric equation is used to compute pressure at a given geopotential height in isothermal layers in the upper and lower stratosphere .
Derivation
The hydrostatic equation:
P
=
ρ
⋅
g
⋅
z
{\displaystyle \ P=\rho \cdot g\cdot z}
where
ρ
{\displaystyle \ \rho }
density [kg/m3 ], is used to generate the equation for hydrostatic equilibrium , written in differential form:
d
P
=
−
ρ
⋅
g
⋅
d
z
.
{\displaystyle dP=-\rho \cdot g\cdot dz.}
This is combined with the ideal gas law :
P
=
ρ
⋅
R
⋅
T
{\displaystyle \ P=\rho \cdot R\cdot T}
to eliminate
ρ
{\displaystyle \ \rho }
d
P
P
=
−
g
R
⋅
T
d
z
.
{\displaystyle {\frac {\mathrm {d} P}{P}}={\frac {-g}{R\cdot T}}\,\mathrm {d} z.}
This is integrated from
z
1
{\displaystyle \ z_{1}}
z
2
{\displaystyle \ z_{2}}
∫
p
(
z
1
)
p
(
z
2
)
d
P
P
=
∫
z
1
z
2
−
g
R
⋅
T
d
z
.
{\displaystyle \ \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 Ta, the average temperature between z1 and z2.
∫
p
(
z
1
)
p
(
z
2
)
d
P
P
=
−
g
R
⋅
T
a
∫
z
1
z
2
d
z
.
{\displaystyle \ \int _{p(z_{1})}^{p(z_{2})}{\frac {\mathrm {d} P}{P}}={\frac {-g}{R\cdot Ta}}\int _{z_{1}}^{z_{2}}\,\mathrm {d} z.}
Integration gives:
ln
(
P
(
z
2
)
P
(
z
1
)
)
=
−
g
R
⋅
T
a
(
z
2
−
z
1
)
{\displaystyle \ln \left({\frac {P(z_{2})}{P(z_{1})}}\right)={\frac {-g}{R\cdot Ta}}(z_{2}-z_{1})}
simplifying to:
ln
(
P
1
P
2
)
=
g
R
⋅
T
a
(
z
2
−
z
1
)
.
{\displaystyle \ln \left({\frac {P_{1}}{P_{2}}}\right)={\frac {g}{R\cdot Ta}}(z_{2}-z_{1}).}
Rearranging:
(
z
2
−
z
1
)
=
R
⋅
T
a
g
ln
(
P
1
P
2
)
{\displaystyle (z_{2}-z_{1})={\frac {R\cdot Ta}{g}}\ln \left({\frac {P_{1}}{P_{2}}}\right)}
or, eliminating the logarithm:
P
1
P
2
=
e
g
R
⋅
T
a
⋅
(
z
2
−
z
1
)
.
{\displaystyle {\frac {P_{1}}{P_{2}}}=e^{{g \over R\cdot Ta}\cdot (z_{2}-z_{1})}.}
References
![{\displaystyle \ h=z_{2}-z_{1}={\frac {R\cdot T}{g}}\cdot \ln \left[{\frac {P_{1}}{P_{2}}}\right]}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/c3a6cf963dfc2e86b65644e7041bd2391bf2e5e6)
