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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
The hypsometric equation is expressed as:[ 1]
h
=
z
2
−
z
1
=
R
⋅
T
¯
g
⋅
ln
(
p
1
p
2
)
{\displaystyle \ h=z_{2}-z_{1}={\frac {R\cdot {\bar {T}}}{g}}\cdot \ln \left({\frac {p_{1}}{p_{2}}}\right)}
where:
h
{\displaystyle \ h}
z
{\displaystyle \ z}
R
{\displaystyle \ R}
gas constant for dry air
T
¯
{\displaystyle \ {\bar {T}}}
temperature in Kelvin [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. 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
=
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
T
¯
{\displaystyle {\bar {T}}}
z
1
{\displaystyle z_{1}}
z
2
{\displaystyle z_{2}}
∫
p
(
z
1
)
p
(
z
2
)
d
p
p
=
−
g
R
⋅
T
¯
∫
z
1
z
2
d
z
.
{\displaystyle \ \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
(
p
(
z
2
)
p
(
z
1
)
)
=
−
g
R
⋅
T
¯
(
z
2
−
z
1
)
{\displaystyle \ln \left({\frac {p(z_{2})}{p(z_{1})}}\right)={\frac {-g}{R\cdot {\bar {T}}}}(z_{2}-z_{1})}
simplifying to:
ln
(
p
1
p
2
)
=
g
R
⋅
T
¯
(
z
2
−
z
1
)
.
{\displaystyle \ln \left({\frac {p_{1}}{p_{2}}}\right)={\frac {g}{R\cdot {\bar {T}}}}(z_{2}-z_{1}).}
Rearranging:
(
z
2
−
z
1
)
=
R
⋅
T
¯
g
ln
(
p
1
p
2
)
{\displaystyle (z_{2}-z_{1})={\frac {R\cdot {\bar {T}}}{g}}\ln \left({\frac {p_{1}}{p_{2}}}\right)}
or, eliminating the ln:
p
1
p
2
=
e
g
R
⋅
T
¯
⋅
(
z
2
−
z
1
)
.
{\displaystyle {\frac {p_{1}}{p_{2}}}=e^{{g \over R\cdot {\bar {T}}}\cdot (z_{2}-z_{1})}.}
References
