4 Atmosphere models for aircraft altimetry
The elements presented in this section are mainly taken from the International Civil Aviation Organization (ICAO) Standard Atmosphere Manual, and the revised atmosphere model ATMORev? used in the Eurocontrol Base of Aircraft Data (BADA) performance model.
4.1 The International Standard Atmosphere (ISA)
4.1.1 The Hydrostatic Equation
where
4.1.2 The Ideal Gaz Law
In this equation,
4.1.3 The Geopotential Altitude
In Equation 4.1, the acceleration
To simplify the expression of the atmospheric model equations, we introduce a new quantity, the geopotential altitude
4.1.4 Characteristics of the Standard Atmosphere at Mean Sea Level
The main characteristics of the ISA atmosphere at mean sea level are shown in Table 4.1.
4.1.5 Temperature as a Function of Geopotential Altitude
The temperature is a piecewise linear function of the geopotential altitude.
The different atmospheric layers, with the values of the temperature gradient up to the altitude of 80 km, are described in Table 4.2. Note that Equation 4.4 can be used for negative altitudes, with the parameters of the layer
N |
Altitude géopotentielle | Limite inf. | Gradient | Nom |
0 | 0 | 288.15 | -6.5 | troposphère |
1 | 11 | 216.65 | 0 | stratosphère |
2 | 20 | 216.65 | +1.0 | stratosphère |
3 | 32 | 228.65 | +2.8 | stratosphère |
4 | 47 | 270.65 | 0 | stratosphère |
5 | 51 | 270.65 | -2.8 | mésosphère |
6 | 71 | 214.65 | -2.0 | mésosphère |
Commercial aviation is concerned by the first two layers, in the troposphere and the beginning of the stratosphere. The troposphere and the stratosphere are separated by an isobaric surface, the tropopause, at
4.1.6 Atmospheric pressure
Equation 4.2 gives us an expression for the air density
Taking into account the expression of Equation 4.4 in the altitude layer (numbered
This leads to two possible expressions of the pressure as a function of the geopotential altitude, depending on the value of the temperature gradient:
4.1.7 Geopotential Altitude as a Function of Atmospheric Pressure
Conversely, starting from Equation 4.5, we can easily express the geopotential altitude
4.1.8 Air Density as a Function of Geopotential Altitude
The density of air is simply expressed from Equation 4.2.
It is expressed as a function of geopotential altitude by replacing pressure and temperature by their expressions from Equation 4.6 and Equation 4.4. We then find Equation 4.9, where
4.1.9 Speed of Sound
The speed of sound in air is given by the following equation, where
4.2 Non-ISA atmospheres for Altimetry
In general, the real atmosphere does not satisfy the assumptions of the International Standard Atmosphere (ISA). An atmosphere can be non-ISA in many different ways. First of all, it is not always composed of dry air only. Also, the temperature and pressure conditions at sea level and/or the temperature gradient may be different from those defined by the standard atmosphere.
For the measurement of aircraft altitude, however, relatively simple assumptions about the atmosphere are made, using the notion of pressure altitude, which we will detail in the rest of this section.
4.2.1 Concept of Pressure Altitude
Let
The pressure altitude is denoted
4.2.2 Hydrostatic Equilibrium, Law of Perfect Gases, Humidity
For altimetry purposes, the non-ISA atmosphere modeled in this section is assumed to be at hydrostatic equilibrium, and to follow the law of perfect gases. The humidity of the air is not taken into account, as in the ISA model. Equation 4.1 and Equation 4.2 which were made for the ISA model remain valid, with the same specific constant
4.2.3 Assumptions on Temperature Gradient
The following assumptions are made about the temperature profile, with respect to the altimetry requirements:
the layers of atmosphere are defined in pressure altitude
, and not in geopotential altitude 1. For aviation purposes, we only consider the two lowest layers : the troposphere, and the stratosphere, separated by the tropopause located at the pressure altitude given by Equation 4.11:the temperature depends linearly on pressure altitude (i.e. the altitude that would be observed if the atmosphere were ISA), with the following gradient:
In the following, we will denote
4.2.4 Temperature and “ISA Temperature”
Let
According to the assumptions made in this section, in each layer of the atmosphere, the temperature is a linear function of the pressure altitude, with an identical temperature gradient for
Taking the isobar
4.2.5 Reference Altitudes and Levels in Altimetry
4.2.5.1 The isobar it hPa (StdRef)
By definition of pressure altitude, the isobar StdRef is at pressure altitude
4.2.5.2 Mean Sea Level (MSL).
By definition, mean sea level is at geopotential altitude
4.2.6 Relation between Geopotential and Pressure Altitudes
Combining Equation 4.1 and Equation 4.2, we see that a variation in pressure
The same variation of pressure in an ISA atmosphere would correspond to a variation of pressure altitude
Dividing the expression Equation 4.17 by Equation 4.18, we obtain the relation between geopotential altitude variation and pressure altitude variation, given by the following Equation 4.19.
The relation Equation 4.20 between
where
4.2.7 Pressure as a Function of Pressure Altitude
Equation 4.6 can be transposed directly to the non-ISA case, replacing
4.2.8 Pressure Altitude as a Function of Pressure
with
4.2.9 Air Density as a Function of Pressure Altitude
4.2.10 Speed of Sound
The speed of sound in the non-ISA atmosphere is given by
List of Acronyms
- BADA
- Base of Aircraft Data
- ICAO
- International Civil Aviation Organization
Note that the pressure altitude and geopotential altitude are the same, when the atmosphere is standard↩︎