Barometric formula

The barometric formula describes the vertical distribution of the (gas) particles in the atmosphere of the earth, so the dependency of the air pressure of the height. It is also described by a vertical pressure gradient, which can be described due to the high weather dynamics within the lower atmosphere only with approximations mathematically, however.

In its simplest form can be roughly assumed that the air pressure near sea level to a hPa (corresponding to 1 ‰ of the average air pressure) decreases eight meters increase in altitude.

• 3.1 Derivation 1 ( adiabatic atmosphere)
• 3.2 Derivation of 2
• 3.3 Typical temperature gradients

• 4.1 International height formula

• 5.1 Virtual Temperature
• 5.2 geopotential heights

• 6.1 Reduction to mean sea level 6.1.1 theory
• 6.1.2 practice
• 6.1.3 limits

Hydrostatic basic equation

The change of the pressure and density of the atmosphere with the amount of which is described by the hydrostatic equation. For the derivation, consider a rectangular volume element with the base and the infinitesimally small height, which contains the air density. From below acts on the base of only the force exerted by atmospheric pressure force. The acting from the top to the base force is made up of the weight of the air mass contained in the volume and the pressure exerted by the atmosphere pressure on the top power. The atmospheric pressure at this level by the amount different from the acting pressure on the underside; the force exerted by him is therefore.

In hydrostatic equilibrium, all air flows have come to rest. Thus, the equilibrium obtained and the volume element considered also remains at rest, the sum of all forces acting on it must be zero:

Delivers shortening and rearranging

According to the ideal gas law can be expressed as the density,

So that finally gives:

It is

• The average molar mass of the gas atmosphere ( 0.02896 kg mol -1),
• The acceleration of gravity ( 9.807 m s-2 ),
• The universal gas constant ( 8.314 J K-1 mol -1) and
• Is the absolute temperature.

The hydrostatic equation specifies the amount by which the atmospheric pressure changes when the height changes by a small amount. As the negative sign indicates negative when is positive; the pressure with increasing altitude so low. So take, for example, at mean air pressure at sea level ( = 1013 hPa) at a temperature of 288 K ( = 15 ° C), the pressure on one meter difference in height by 0.12 hPa respectively to 8.3 meters height difference at 1 hPa. The difference in height, which corresponds to a pressure difference of 1 hPa, is the barometric altitude level. At higher altitudes (smaller ) and at higher temperatures, the air pressure changed slowly, the barometric altitude level increases.

What is needed is usually explicit values ​​for pressure and density at given heights. It can be, if required, the pressure differences for large differences in height reading. The desired solution of the basic equation obtained by separation of variables

And subsequent integration between this height or the associated press:

Integration of the left-hand side yields. To integrate the right side of the altitude dependence of known and needs to be. The acceleration of gravity can be considered for not too great heights as a constant. The temperature varies in a complicated and hardly predictable manner with altitude. It must therefore be taken of the temperature development simplifying assumptions.

Isothermal atmosphere

The most frequently cited in introductory literature and in school teaching classical barometric height formula holds for the special case that the temperature, the atmosphere is thus equal isothermally at any height.

Derivation of the hydrostatic basic equation

The integration of the hydrostatic equation yields at constant reason:

By introducing the scale height is the height formula simplifies to

With each level increase the air pressure by a factor decreases. The scale height is thus a natural measure of the height of the atmosphere and the pressure curve in it. It is assumed here, in the model atmosphere at a temperature of 15 ° C to about 8.4 km.

For the density applies accordingly:

For an observer traveling downhill, the air pressure is constantly increasing, as a heavier and heavier column of air bearing down on him. The increase is exponential, since the air is compressible: for every meter height difference takes the weight of the weighing on a measuring surface air column to the weight of the adventitious on this route to the column volume. However, this weight depends on the density of the air and this in turn on the air pressure. The air pressure that is growing more rapidly, the higher it is already. Size changes always by an amount which is proportional to the size itself, so the change is happening exponentially.

Derivation from statistical mechanics

In a particle system, which is at the temperature in the thermal equilibrium (which thus in particular throughout the same temperature has ) and the particles can be taken continuously or discretely distributed energy levels, the probability that a particle is currently located at the energy level is given by the Boltzmann distribution

It is the Boltzmann constant, and a normalization factor ( the so-called partition ), which ensures that the sum of all the probabilities equals 1. The system consists of particles, as is the number of particles on the energy level on the average.

A gas particle of mass has potential energy because of its temperature and the average thermal energy, for a total energy in the gravitational field of the earth. Considering two equally sized volume elements on the heights or, the particles are at the level higher by the amount of energy. The probability of finding a particle in the higher volume element, therefore, is related to the probability of finding it in the lower volume element as

For a sufficiently large number of particles, the particle densities behave as the probabilities

And because of the ideal gas law follows for printing the same ratio

To give the molar mass and the gas constant, by multiplying the mass of particles or the Boltzmann constant to the Avogadro constant.

However, the equipartition theorem is provided here at the energy consideration. This requirement is generally met but only in a dense atmosphere, because only there the energies between the different degrees of freedom are replaced by collisions between the gas molecules. The reason that the equipartition theorem generally does not apply to the amount of energy is that the equipartition theorem can be applied directly only to potentials that enter quadratically in the Hamiltonian. Because the amount of energy only linearly enters into the Hamiltonian, one can not assume the equipartition theorem in very thin gases.

Atmosphere with a linear temperature profile

The strictly linear temperature gradient exists only in the idealized notion of a quiescent atmosphere without convection without compensating the temperature gradient by conduction. To make this more useful, the potential temperature has been introduced. Although the adiabatic gradient is a temperature gradient, the temperature is constant potential, i.e., an equilibrium. With the potential energy of a particle in the gravitational field (E = m · g · h) which has nothing to do. Is particularly evident with the number of degrees of freedom. Particles the same mass but different number of degrees of freedom have different temperature gradients.

As for the maintenance of the linear temperature profile, the heat conduction should play no role in the reality of permanent " heat transport " (heat conduction ) may by rapid circulation have only a minor impact. Because konvektionslos and circulation can not be both, the linear trend is always readily modified by heat transport of all kinds, the most famous is the condensation of water vapor, which leads to a lower temperature drop ( " moist - adiabatic" is a misleading term ).

Derivation 1 ( adiabatic atmosphere)

From the equation for the pressure change

And written with the help of logarithmic derivatives equation for the adiabatic change

Immediately follows the linear temperature decrease according

The average molar mass of the atmosphere gas M = 0.02896 kg mol-1, the gravity acceleration g = 9.807 m s -2, the universal gas constant R = 8.314 J K-1 mol -1, and the adiabatic exponent of ( dry ) air = 1.402 one obtains the temperature gradient

This is approximately the temperature gradient below. That is, however, essentially determined by the feuchtadiabatische expansion: the feuchtadiabatische adiabatic exponent is smaller than the adiabatic trockenadiabatische. In a pure water vapor atmosphere, the temperature gradient would be

Further limitations of the adiabatic approach: If the air is very cold, also changes in dry air of the adiabatic exponent. At very high altitudes (low density) and the mean free path is very large, so that the gas equations hardly still apply. What's more, that (no energy exchange with the environment ) is violated by the greenhouse effect and the adiabatic approach.

Derivation of 2

Generally, the temperature is not constant, but varies with height. The simplest approach to take account of such variability is based on a linear decrease with height. Therefore applies to the temperature

Where the ( positive to be taken ) the amount of the vertical atmospheric temperature gradient, which indicates by how many Kelvin decreases the air temperature per meter difference in height. The integral on the right side of the basic equation is thus

Because of

Is the solution of the integral

So that total from the integral over the fundamental equation

The barometric formula for linear temperature profile follows:

Or on account of:

For the density shall apply mutatis mutandis

Here, the exponent is decremented by 1, because the relationship between pressure and density is temperature dependent.

This extended barometric formula is the basis for the barometric height function of the standard atmosphere in aviation. First, the atmosphere is divided into sub-layers, each with linearly interpolated temperature profile. Then, beginning temperature and pressure is calculated with the lowermost layer on the upper limit of the respective sub- layer and used for the lower limit of the overlying layer. In this way, the model for the entire inductive atmosphere.

Typical temperature gradients

As measurements of the temperature profiles in the atmosphere show that assuming a linear temperature decrease on average is a good approximation, although in individual cases significant deviations occur, for example when temperature inversions. The main cause of the decrease in temperature with altitude, the warming of the lower layers of air of the heated surface of the sun, while the upper layers of air radiate heat to space. These trockenadiabatische or feuchtadiabatische temperature changes of individual ascending or sinking air parcels and additional modifications come through mixing processes between air masses of different origin. In warm air masses and at Aufgleitvorgängen the temperature gradient takes values ​​around 0.3 to 0.5 K per 100 m on, slumping in cold air usually around 0.6 to 0.8 K per 100 m, on average over all weather conditions 0.65 K per 100 m. In valleys common ground inversions can reduce the average temperature to 0.5 K per 100 m, in the winter months even at 0.4 K per 100 m.

The conditions described are limited to the troposphere. In the stratosphere, the temperature decreases more slowly, they usually even increases again, mainly due to the absorption of UV radiation in the ozone layer.

For a temperature gradient of 0.65 K per 100 m of the exponent takes the value of 5.255:

If the exponent is expressed by the Isentropenkoeffizienten, then:

This means 8.5 degrees of freedom.

From the temperature gradient and the mean heat capacity of air results in all weather conditions:

This value lies between the value of dry air { 1005 Ws / (kg K) } and water vapor { 2034 Ws / (kg K) }.

The following table shows the relationship between height and pressure (on average):

In this form, offers the height formula for the common case where the temperature and air pressure on one of the two heights are known, but not the currently existing temperature gradient.

The height levels

The barometric level, the vertical distance which must be traveled to achieve 1 hPa pressure change. Near the bottom of the barometric altitude level is about 8 meters, height 16 meters in 5 kilometers and 10 kilometers altitude 32 meters.

With the height formula, the following table is for the altitude and temperature dependence of the barometric stage:

As a rule of thumb for medium altitudes and temperatures considered " one hPa/30 ft". This rounding value drivers often use air for rough calculations.

International pressure formula

Substituting the reference altitude at sea level and takes for the atmosphere there a middle state how he is described by the International Standard Atmosphere (temperature 15 ° C = 288.15 K, 1013.25 hPa air pressure, temperature gradient 0.65 K per 100 m ), one obtains the International level formula for the troposphere (valid to 11 km altitude):

This formula allows the calculation of air pressure at a given height, without the temperature and temperature gradient are known. The accuracy in the case of actual use is limited, however, since the calculation instead of the actual atmospheric state a mean atmosphere is assumed.

General case

The solution of the hydrostatic basic equation is generally

Respectively

With still unattached integral.

Virtual temperature

The gas constant is a physical constant and can be pulled out of the integral. The average molar mass of the atmosphere gases, unless it is apart from the highly variable water vapor content, practically constant within the troposphere also and can also be taken outside the integral. The different scale heights of the different heavy atmospheric gases would indeed result in a resting atmosphere to a partial separation, so that heavier components would accumulate in the lower classes and lighter components in the higher layers; conditional by the weather patterns intensive mixing of the troposphere prevents this. The variable water vapor content, and also generalizes other minor changes of M (especially in the higher layers of the atmosphere ) can be taken into account by using the corresponding virtual temperature instead of the actual temperature. For M, therefore, the value for dry air are used at sea level.

Geopotential heights

The acceleration of gravity decreases with height, which must be considered when large height differences or high accuracy requirements. A variable gravitational acceleration in the integrand would significantly complicate the integration, however. This can be circumvented by using geopotential instead of geometric heights. Imagine this a test mass with a variable from sea level raised to the height. Because decreases with height, the thus obtained potential energy is smaller than if always had the sea level value. The geopotential height is the height, measured in geopotential meters, which is overcome by calculation, to supply the mass at an always constant acceleration due to gravity the same potential energy ( in other words. 's The same with divided gravity potential surfaces geopotential height are equipotential surfaces in a gravitational field ).

For the belonging to a geometric altitude geopotential height so it should apply:

From which it follows:

For the ratio of the gravitational acceleration in the amount of gravitational acceleration at sea level, since the gravitational field decreases quadratically with the distance from the center of the earth:

With the radius of the earth. integration of

Provides

Is to be set to the value of 6356 km. If necessary, must also be taken into account that the acceleration of gravity at sea level depends on the latitude.

In this way have the desired geometric heights in geopotential heights are converted before the invoice only once; in height formula can then be used instead of the variable gravitational acceleration is simply the constant sea level value. The difference between geometric and geopotential heights for not to high altitudes is low and often negligible:

With the acceleration of gravity at sea level, the geopotential heights and the virtual temperature and the overall height formula simplifies to

It remains to solve the integral over what only the temperature profile must be known. It can be determined in the real atmosphere, for example by radiosonde ascents. For simplified model atmospheres with constant or linearly varying temperature arise again height formulas of the initially treated type.

Applications

Reduction to mean sea level

Theory

The measured by a barometer air pressure depends on both the meteorological state of the atmosphere as well as the altitude of your location. If the details are different barometer in a larger area for meteorological purposes compared with each other ( for example, to determine the location of a low- pressure area or a front ), the influence of site heights from the measured data must be removed. For this purpose, the measured pressure values ​​to a common reference level, usually sea level equivalent. This conversion is done using a height formula. The converting is also known as reduction ( even if the numeric value is greater ), since the measurement value is freed of undesired parasitics. The result is the reduced air pressure at sea level.

Depending on the accuracy requirements of a suitable height formula must be used. For smaller claims, for example, from the height formula for constant temperature, a fixed conversion factor are derived, including a representative temperature has to be chosen:

For a site altitude of 500 m and using a mean annual temperature of 6 ° C results, for example, a reduction factor of 1.063, with which the measured values ​​are to be multiplied.

At slightly higher demands at least the actual air temperature must be considered. Their influence can be seen in the following example, in which a measured at 500 m altitude air pressure of 954.3 hPa with the height formula for linear temperature profile (a = 0.0065 K / m) is reduced through the adoption of various station temperatures at sea level:

Use of an incorrect temperature will therefore usually result in deviations from some hPa. If a higher accuracy is desired, current air temperatures are available and justify accuracy and calibration of the barometer used the effort, the reduction should be always using the current air temperature. The highlight formula there is the variant for linear temperature profile. But it can just as well the variant for constant temperature profile be used, provided the ruling halfway station altitude current temperature is used:

This variant is theoretically somewhat less accurate because it ignores the variability of temperature with height, while the linear variant, at least to some extent takes account of these. When the weather stations for commonly occurring heights and temperatures, however, the differences are quite negligible.

Recommended by the German Weather Service reduction formula corresponds to the variant with a constant temperature gradient. From the measured temperature at location altitude, the temperature is estimated to be half the height location using the standard temperature gradient. The humidity is taken into account through transition to the corresponding virtual temperature.

With

If no measured air humidity is available, the following approximation can also be estimated in accordance with, which is based on long-term average values ​​of temperature and humidity:

Practice

The collected here accuracy requirements for measured air pressure and height of the barometer will be unable to meet for an amateur meteorologists usually. In the barometers of amateur weather stations will be quite likely to systematic errors of at least one to two hPa. One such uncertainty corresponds to about the barometric stage an uncertainty of location height of ten to twenty meters. The ambition to determine the altitude of your location more accurately does not, most likely a more accurate result. In this light would be considered a consideration of the humidity and the necessity or superfluity.

If necessary, it is advisable not to use the real location height, but a fictitious amount which achieves the best match of the reduced air pressure with the details of a nearby reference barometer ( official weather station, airport, etc.). Such a calibration is a possible systematic error of the barometer can be compensated mostly. For this purpose, it is expedient first to use an approximate height to reduce or to compare the results of their own for an extended period of time ( especially at different temperatures ) with the reference information. If a systematic difference found can be calculated with the aid of a suitable height, the height difference formula, which shifts from the approximate location of the reduced height heights by the desired amount. The corrected in this manner is then used for future height reductions. The temperature is not considered in the reduction, the situation should be the basis of a representative temperature during calibration.

Simple Living barometers are usually set so that they directly show the reduced air pressure. Most often this is done by a screw on the rear panel, with which the bias of the compression spring can change doses. The calibration thus corresponds to a shift of the indicator scale. This is not strictly correct. As the height formulas show that reduction multiplied by a calibration factor must be done, and not by mere addition of a constant: The reduced air pressure changes by slightly more than one hPa when the air pressure changes at the site level to a hPa. The scale would have to, in addition to shift easily be stretched. The error thus caused is, however, less than the error caused by the fact that these devices ignore the temperature effect on the reduction. Since they have no input option for the Elevation, a calibration can be performed only by comparison with a reference barometer, which in turn simultaneously systematic zero-point error of the instrument are reduced. The calibration must be done for the location of the barometer (or any other place of comparable height). It's no use, " set right" to allow the unit to the dealer, if it is then suspended from a completely different place. If the barometer serves to drain from air pressure changes a short-term weather forecast, precise calibration is less important.

Confines

May generally has to be considered in the reduction of air pressure measurements that determined arithmetically added column of air for most locations in reality can not exist and therefore no " true" value for the " sea level pressure at the site " to give that precisely by sufficiently complicated arithmetic could be approximated. The reduction formulas are partly based only on conventions and serve, apart from specific scientific applications, mainly to make the measurements of the weather stations as far as possible among them. An example of the fictional character of the added air column: About flat terrain is not discharged to the cold air, moisture can form a clear night because of the heat radiation of the ground to cool the surface air noticeably (soil inversion). A weather station is located there register this reduced temperature and continue with the usual temperature gradient computation down. Befände to the terrain but at sea level, so that air would not be cooled because of the lack of ground now and now actually existing air column would have a significantly higher temperature than the calculated. The bill has thus assumed a too high density of the added column of air and results in a higher reduced air pressure than he would prevail at the same conditions, if the entire site would be at sea level.

Barometric altitude measurement

The altitude dependence of atmospheric pressure may also be used for height measurement. Barometric measurements are quick and relatively easy to perform, however, limited in their accuracy. A small amount for height determination is called a barometer or altimeter altimeter. The procedure is based on the purpose and accuracy requirements. Application, the procedures, among others, while hiking and with slightly higher accuracy demands in land surveying.