Scale height

For planetary atmospheres, scale height is the increase in altitude for which the atmospheric pressure decreases by a factor of e. The scale height remains constant for a particular temperature. It can be calculated by

{{cite web

|title= Glossary of Meteorology - scale height

|url=http://glossary.ametsoc.org/wiki/Scale_height

|publisher= American Meteorological Society (AMS)

}}

{{cite web

|title= Pressure Scale Height

|url=http://scienceworld.wolfram.com/physics/PressureScaleHeight.html

|publisher= Wolfram Research

}}

H = \frac{k_\text{B}T}{mg},

or equivalently,

H = \frac{RT}{Mg},

where

: k_B = Boltzmann constant = {{physconst|k|round=3}}

: R = molar gas constant = 8.31446 J⋅K⁻¹⋅mol⁻¹

: T = mean atmospheric temperature in kelvins = 250 K{{cite web

|title= Daniel J. Jacob: "Introduction to Atmospheric Chemistry", Princeton University Press, 1999

|url= http://acmg.seas.harvard.edu/people/faculty/djj/book/bookchap2.html

|access-date= 2013-04-18

|archive-date= 2013-04-10

|archive-url= https://web.archive.org/web/20130410233524/http://acmg.seas.harvard.edu/people/faculty/djj/book/bookchap2.html

|url-status= dead

}} for Earth

: m = mean mass of a molecule

: M = mean molar mass of atmospheric particles = 0.029 kg/mol for Earth

: g = acceleration due to gravity at the current location

The pressure (force per unit area) at a given altitude is a result of the weight of the overlying atmosphere. If at a height of z the atmosphere has density ρ and pressure P, then moving upwards an infinitesimally small height dz will decrease the pressure by amount dP, equal to the weight of a layer of atmosphere of thickness dz.

Thus:

$$\backslash frac\{dP\}\{dz\}\; =\; -g\backslash rho,$$

where g is the acceleration due to gravity. For small dz it is possible to assume g to be constant; the minus sign indicates that as the height increases the pressure decreases. Therefore, using the equation of state for an ideal gas of mean molecular mass M at temperature T, the density can be expressed as

$$\backslash rho\; =\; \backslash frac\{MP\}\{RT\}.$$

Combining these equations gives

$$\backslash frac\{dP\}\{P\}\; =\; \backslash frac\{-dz\}\{\{k\_\backslash text\{B\}T\}/\{mg\}\},$$

which can then be incorporated with the equation for H given above to give

$$\backslash frac\{dP\}\{P\}\; =\; -\; \backslash frac\{dz\}\{H\},$$

which will not change unless the temperature does. Integrating the above and assuming P₀ is the pressure at height z = 0 (pressure at sea level), the pressure at height z can be written as

$$P\; =\; P\_0\backslash exp\backslash left(-\backslash frac\{z\}\{H\}\backslash right).$$

This translates as the pressure decreasing exponentially with height.{{cite web | title = Example: The scale height of the Earth's atmosphere | url = http://iapetus.phy.umist.ac.uk/Teaching/SolarSystem/WorkedExample4.pdf | archive-url = https://web.archive.org/web/20110716205659/http://iapetus.phy.umist.ac.uk/Teaching/SolarSystem/WorkedExample4.pdf | url-status = dead | archive-date = 2011-07-16 }}

In Earth's atmosphere, the pressure at sea level P₀ averages about {{val|1.01|e=5|u=Pa}}, the mean molecular mass of dry air is {{val|28.964|ul=Da}}, and hence m = {{val|28.964|u=Da}} × {{val|1.660|e=-27|u=kg/Da}} = {{val|4.808|e=-26|u=kg}}. As a function of temperature, the scale height of Earth's atmosphere is therefore H/T = k_B/mg = {{val|1.381|e=-23|u=J.K-1}} / ({{val|4.808|e=-26|u=kg}} × {{val|9.81|u=m.s-2}}) = {{val|29.28|u=m/K}}. This yields the following scale heights for representative air temperatures:

: T = 290 K, H = 8500 m,

: T = 273 K, H = 8000 m,

: T = 260 K, H = 7610 m,

: T = 210 K, H = 6000 m.

These figures should be compared with the temperature and density of Earth's atmosphere plotted at NRLMSISE-00, which shows the air density dropping from 1200 g/m³ at sea level to 0.125 g/m³ at 70 km, a factor of 9600, indicating an average scale height of 70 / ln(9600) = 7.64 km, consistent with the indicated average air temperature over that range of close to 260 K.

Note:

• Density is related to pressure by the ideal gas laws. Therefore, density will also decrease exponentially with height from a sea-level value of ρ₀ roughly equal to {{val|1.2|u=kg.m-3}}.
• At an altitude over 100 km, the atmosphere is no longer well-mixed, and each chemical species has its own scale height.
• Here temperature and gravitational acceleration were assumed to be constant, but both may vary over large distances.

Approximate atmospheric scale heights for selected Solar System bodies:

{{unsourced section|date=December 2024}}

File:Scale Height Force Balance.jpg

For a disk of gas around a condensed central object, such as, for example, a protostar, one can derive a disk scale height which is somewhat analogous to the planetary scale height. We start with a disc of gas that has a mass small relative to the central object. We assume that the disc is in hydrostatic equilibrium with the z component of gravity from the star, where the gravity component is pointing to the midplane of the disk:

\frac{dP}{dz} = -\frac{GM_*\rho z}{(r^2 + z^2)^{3/2}},

where

: G = Newtonian constant of gravitation ≈ {{physconst|G|round=3}}

: r = the radial cylindrical coordinate for the distance from the center of the star or centrally condensed object

: z = the height/altitude cylindrical coordinate for the distance from the disk midplane (or center of the star)

: M_* = the mass of the star/centrally condensed object

: P = the pressure of the gas in the disk

: $\backslash rho$ = the gas mass density in the disk

In the thin disk approximation, $z\; \backslash ll\; r$, and the hydrostatic equilibrium, the equation is

$$\backslash frac\{dP\}\{dz\}\; \backslash approx\; -\backslash frac\{GM\_*\backslash rho\; z\}\{r^3\}.$$

To determine the gas pressure, one can use the ideal gas law:

$$P\; =\; \backslash frac\{\backslash rho\; k\_\backslash text\{B\}\; T\}\{\backslash bar\{m\}\}$$

with

: T = the gas temperature in the disk, where the temperature is a function of r, but independent of z

: $\backslash bar\{m\}$ = the mean molecular mass of the gas

Using the ideal gas law and the hydrostatic equilibrium equation, gives

$$\backslash frac\{d\backslash rho\}\{dz\}\; \backslash approx\; -\backslash frac\{GM\_*\; \backslash bar\{m\}\backslash rho\; z\}\{k\_\backslash text\{B\}\; Tr^3\},$$

which has the solution

$$\backslash rho\; =\; \backslash rho\_0\; \backslash exp\backslash left(-\backslash left(\backslash frac\{z\}\{h\_\backslash text\{D\}\}\backslash right)^2\; \backslash right),$$

where $\backslash rho\_0$ is the gas mass density at the midplane of the disk at a distance r from the center of the star, and $h\_\backslash text\{D\}$ is the disk scale height with

h_\text{D} = \sqrt{\frac{2k_\text{B}Tr^3}{GM_* \bar{m}}} \approx

0.0306 \sqrt{\frac{(T/100~\text{K})(r/1~\text{au})^3}{(M_* / M_\odot)(\bar{m}/2~\text{Da})}}~\text{au},

with $M\_\backslash odot$ the solar mass, $\backslash text\{au\}$ the astronomical unit, and $\backslash text\{Da\}$ the dalton.

As an illustrative approximation, if we ignore the radial variation in the temperature $T$, we see that $h\_\backslash text\{D\}\; \backslash propto\; r^\{3/2\}$ and that the disk increases in altitude as one moves radially away from the central object.

Due to the assumption that the gas temperature T in the disk is independent of z, $h\_\backslash text\{D\}$ is sometimes known{{cn|date=December 2024}} as the isothermal disk scale height.

{{unsourced section|date=December 2024}}

A magnetic field in a thin gas disk around a central object can change the scale height of the disk.{{cite journal |last1=Lovelace |first1=R. V. E. |last2=Mehanian |first2=C. |last3=Mobarry |first3=C. M. |last4=Sulkanen |first4=M. E. |title=Theory of Axisymmetric Magnetohydrodynamic Flows: Disks |journal=Astrophysical Journal Supplement |date=September 1986 |volume=62 |page=1 |doi=10.1086/191132 |bibcode=1986ApJS...62....1L |url=https://ui.adsabs.harvard.edu/link_gateway/1986ApJS...62....1L/ADS_PDF |access-date=26 January 2022 |doi-access=free }}{{cite journal |last1=Campbell |first1=C. G. |last2=Heptinstall |first2=P. M. |title=Disc structure around strongly magnetic accretors: a full disc solution with turbulent diffusivity |journal=Monthly Notices of the Royal Astronomical Society |date=August 1998 |volume=299 |issue=1 |page=31 |doi=10.1046/j.1365-8711.1998.01576.x |bibcode=1998MNRAS.299...31C |doi-access=free }}{{cite journal |last1=Liffman |first1=Kurt |last2=Bardou |first2=Anne |title=A magnetic scaleheight: the effect of toroidal magnetic fields on the thickness of accretion discs |journal=Monthly Notices of the Royal Astronomical Society |date=October 1999 |volume=309 |issue=2 |page=443 |doi=10.1046/j.1365-8711.1999.02852.x |bibcode=1999MNRAS.309..443L |doi-access=free }} For example, if a non-perfectly conducting disk is rotating through a poloidal magnetic field (i.e., the initial magnetic field is perpendicular to the plane of the disk), then a toroidal (i.e., parallel to the disk plane) magnetic field will be produced within the disk, which will pinch and compress the disk. In this case, the gas density of the disk is

\rho(r, z) = \rho_0(r) \exp\left(-\left(\frac{z}{h_\text{D}}\right)^2\right) -

\rho_\text{cut}(r) \left[1 - \exp\left(-\left(\frac{z}{h_\text{D}}\right)^2\right)\right],

where the cut-off density $\backslash rho\_\backslash text\{cut\}$ has the form

\rho_\text{cut}(r) = (\mu_0 \sigma_\text{D} r)^2 \frac{B_z^2}{\mu_0}

\left(\frac{\Omega_*}{\Omega_\text{K}} - 1\right)^2,

where

: $\backslash mu\_0$ is the permeability of free space

: $\backslash sigma\_\backslash text\{D\}$ is the electrical conductivity of the disk

: $B\_z$ is the magnetic flux density of the poloidal field in the $z$ direction

: $\backslash Omega\_*$ is the rotational angular velocity of the central object (if the poloidal magnetic field is independent of the central object, then $\backslash Omega\_*$ can be set to zero)

: $\backslash Omega\_\backslash text\{K\}$ is the keplerian angular velocity of the disk at a distance $r$ from the central object.

These formulae give the maximum height of the magnetized disk as

H_\text{B} = h_\text{D} \sqrt{\ln\left(1 + \rho_0/\rho_\text{cut}\right)},

while the e-folding magnetic scale height is

h_\text{B} = h_\text{D} \sqrt{\ln\left(1 + \frac{1 - 1/e}{1/e + \rho_\text{cut}/\rho_0} \right)}.

• Time constant

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Category:Atmospheric dynamics

Category:Vertical position