Atmospheric Air Mass Models

Introduction

The air mass coefficient defines the path length of sunlight through the atmosphere (e.g. the column depth), and is a key input for estimating solar extinction and the irradiance intensity on the Earth’s surface.  The challenge in modeling air mass and solar extinction is an atmosphere that is highly variable, exhibiting unique behavior at different altitudes.  Atmospheric models seek to overcome some of the errors in the interpolative models of air mass.  Specifically, atmospheric models:

  • Define a non-homogenous atmosphere with temperature and pressure by altitude;
  • Estimate the column depth of dry air using real-time or forecast data;
  • Estimate the column depth of water vapor and the impact of water droplets; and
  • Estimate the column depth and concentrations of dust and other aerosols.   

The practical goal is to use the best measurement data and atmospheric models to  understand air mass, solar irradiance, and the performance of solar technology. Readers who are not interested in the math or code supporting atmospheric models are encouraged to skip the following and go direct to Terrestrial Irradiance here.

Geometry of a Non-variable Atmosphere

Airmass_geometryA radially symmetric, uniform  atmosphere is one where:

  • the earth and its atmosphere are assumed to be round,
  • air density, \normalsize \rho, as a function of height is constant, and
  • there is no refraction or bending of sunlight.

Under these restrictive conditions, Shoenberg (1929) has used basic geometry to define the path length of sunlight, \normalsize s, through the atmosphere at a zenith angle of \normalsize \theta_{Z}:

(1)   \begin{equation*} \normalsize s = \sqrt{R^2_{E}\cos^2\theta_{Z} + 2R_{E}y_{atm} + y^2_{atm}} - R_{E}\cos\theta_{Z} \end{equation*}

where \normalsize R_{E} is the radius of the Earth.  The derivation of the path length is shown in Neal Clarke’s diagram.1.  If air density is constant and the fluid atmosphere is at rest, then the atmospheric height is defined under hydrostatic conditions as:

(2)   \begin{equation*} \normalsize y_{atm} = \frac{kT_{0}}{mg} \end{equation*}

where \normalsize k is Boltzmann’s constant (1.380 6488×10−23),  \normalsize T_{0} is the sea-level temperature, \normalsize m is the molecular mass of air (28.9644×1.6605×10−27), and \normalsize g is the acceleration due to gravity (9.80665 m/s2).

Air Mass and the Variable Density Atmosphere

In the real atmosphere, air density varies with temperature and pressure as a function of altitude.  The column depth or path length through the dry atmosphere now depends on air density data \normalsize \rho at elevation \normalsize R:

(3)   \begin{equation*} \normalsize N(\theta_{Z}) = \int^{R_{mes}}_{R_{0}} \rho ~ dR \end{equation*}

More specifically, \normalsize R is the distance from the center of the Earth for the observer at altitude \normalsize 0 and for the top of atmosphere, \normalsize R_{med}, which is defined by the mesospause boundary layer (not the true top of the atmosphere as explained below).  Meanwhile, a fluid atmosphere will refract sunlight according to Snell’s law.  The  column depth or path length of sunlight becomes:

(4)   \begin{equation*} \normalsize N(\theta_{Z}) = \int^{R_{mes}}_{R_{0}} \frac{\rho}{\sqrt{1 - \frac{R_{0}^2 ~ \mu_{0}^2 ~ \sin^2{\theta_{Z}}}{R^2 ~ \mu^2}}} ~ dR \end{equation*}

where \normalsize \mu is the air’s index of refraction at altitude \normalsize R, and \normalsize \theta_{Z} is the apparent zenith angle.  The equation is the column depth of the atmosphere with a correction factor in the denominator for the the refraction of light.  To get the air mass, \normalsize N(\theta_{Z}) must be normalized by the column depth \normalsize N(\theta_{0}, as shown previously).

The index of refraction, \normalsize \mu, is also defined by Garfinkel (1967) as a function of the air density, \normalsize \rho, which can be rewritten as:2

(5)   \begin{equation*} \normalsize \mu^2 = \frac{3 + 4 c\rho}{3 - 2 c \rho} \end{equation*}

where \normalsize c is a wavelength-varying constant depending on the chemical composition of the air.  Air’s index of refraction at a given wavelength appears in the CRC Handbook of Chemistry and Physics as:

(6)   \begin{equation*} \normalsize c = \left(2875.66 + \frac{13.42}{L^2 10^8} + \frac{0.3777}{L^4 10^-10}\right)10^-7 * I\left(\frac{273.15K}{101325Pa}\right) \end{equation*}

where \normalsize L is the wavelength in Angstroms and \normalsize I is the ideal gas constant.  Since \normalsize c has units of volume or mass, \normalsize I needs to be expressed in energy/mass/K).

The Standard Atmosphere

Plot138cThe atmospheric air mass equation now relies on air density as a function of altitude.  Air density is estimated using a temperature profile combined with the ideal gas law (which expresses density in terms of pressure and temperature), along with the assumption of hydrostatic stability (which also defines density in terms of air pressure).

The problem of the temperature profile is determined by solving equations for the fluid dynamics of the atmosphere, using gridded reanalysis data and appropriate boundary conditions.  This is the fundamental challenge of numerical weather prediction for which there are standard models (WRF, MM5, etc.) and gridded data sets for historic or forecast time frames.  

Clearly, advanced physics models of the atmosphere are the preferred approach to defining temperature, pressure and air density profiles by altitude.  In practice, these models are computationally intensive and demanding to implement, and both issues  detract from the immediate focus of explaining how air mass impacts solar irradiance.  For this purpose, the Standard Atmosphere is appropriate and can still be used to obtain alternative measures of air mass.  

Applying the Standard Atmosphere

The temperature and pressure profiles for the Standard Atmosphere are displayed above.  Keep in mind that the plotted values are average values only and profile temperature for North America.  In reality, the location of the different temperature layers are high variable and the profiles can assume a wide range of shapes for different locations and times of year.

A temperature profile can be estimated by interpolating between observed or modeled values.  Meyer (2000) provides a model of air mass optical adjustments used in astronomy by assuming the atmosphere is broken down into several layers within which the temperature changes  linearly with altitude.3.   In any given layer \normalsize n, the temperature is

(7)   \begin{equation*} \normalsize T_{Q} = T_{n} + \beta_{n} (Q - Q_{n}) \end{equation*}

where \normalsize beta_{n} is the slope of the temperature rise with altitude within the layer n, \normalsize T_{n} is the temperature at the bottom of that layer, Q is the height (defined below), and \normalsize Q_{n} is the height of  the bottom of the layer.  This is the methodology used in the U.S. Standard Atmosphere, 1972, wherein a Standard Atmosphere appropriate for the continental United States is defined.  A calculator for the Standard Atmosphere coded in R appears at the end of this post.

In practice, it makes sense to adjust the standard atmosphere for any location and season.  The bottom eight temperature layers, corresponding to the troposphere, stratosphere, and mesosphere, extend 100 km above mean sea level.  The ability to generalize the standard atmosphere so it is appropriate for other locations requires a change in the values for the lowest two layers (the troposphere and lower stratosphere).  

Lower Atmosphere Adjustments

Let the boundary between the two layers (the tropopause; \normalsize Q_{1}) vary as a function of time of year and latitude.  \normalsize \beta_{0}, the temperature falloff in the first layer, also varies by location and time.   The bottom of the first layer (\normalsize Q_{0}) is set to the height \normalsize Q of the observer’s location.  The boundary  condition temperature at bottom of first layer (\normalsize T_{0}) is set to the actual local temperature at the observer.  The condition \normalsize T_{1} equals the temperature at the top of the first layer, i.e., as determined from the  above expression for \normalsize T with \normalsize n = 0 and \normalsize Q = Q_{1}.  Finally, \normalsize \beta_{1} is set by forcing the temperature at the top of the second layer to match \normalsize T_{2} (fixed at some value as reported by gridded data sets, such as MERRA data supplied by NASA).  

Although the temperatures in the second layer and higher (altitudes > 20 km) do vary with season and latitude, the variations are less severe than in the troposphere and lower stratosphere.  Fortunately, the atmosphere has already greatly thinned out before the stratosphere and the higher layers will contribute  little to the solar airmass and solar extinction results.  Regardless, the model defines the effective top of the atmosphere at the mesopause boundary.

Model Calibration

The two values that change with time and latitude, and which define the lower two layers (along with \normalsize T_{0}) are \normalsize Q_{1} and \normalsize \beta_{0}.  NASA MERRA data provides values of the temperature distribution within the troposphere as a function of location and pressure layer by hour.  A polynomial fit of the data will provide a solid in-sample function definition for \normalsize \beta_{0}.  Another polynomial fit will define a function for \normalsize R_{1}, the altitude of the troposphere, which is related to \normalsize Q_{1}.

Air Mass and the Air Density Profile

Values for the \normalsize \beta‘s, \normalsize n‘s, \normalsize T_{n}‘s and \normalsize Q_{n}‘s are known using the calibration method above for the bottom two layers of the atmosphere.  The simplicity of the time-varying temperature  model leads to easily solvable equations, which result in a closed-form expression for the air density:

(8)   \begin{equation*} \normalsize \frac{\rho}{\rho_{n}} = \begin{cases} \left(\frac{T_{n}}{T_{n} + \beta_{n}(Q - Q_{n})}\right)^{(1+g_{E}/(R\beta_{n}))} & \beta_{n}\neq 0\\ exp\left(\frac{g_{E}}{R T_{n}}(Q_{n}-Q)\right) & \beta_{n} = 0 \end{cases} \end{equation*}

where \normalsize n refers to the bottom of the temperature layer \normalsize (0 <= n < 8) and \normalsize g_{E} is the standard value for the acceleration of gravity at the Earth’s surface (9.80665 m/s2).

In order to use equation 8 in equation 4, Meyers defines \normalsize R in terms of \normalsize Q.  The Standard Atmosphere defines the geopotential altitude, \normalsize h, as

(9)   \begin{equation*} \normalsize h = \int^{R_{mes}}_{0} \frac{g}{g_{E}} dR \end{equation*}

where \normalsize g is the gravity at some \normalsize R .  The Standard Atmosphere then defines the eight lowest temperature layers in terms of the geopotential altitude.  The alternative definition removes the problem that the gravity field varies with height, thus simplifying the calculation.  Meyers then defines the quantity \normalsize Q by  

(10)   \begin{equation*} \normalsize Q = h - \frac{R_{E}^2}{R_{msl}} \end{equation*}

where again \normalsize R_{E} is “standard” radius of the Earth, \normalsize g is the gravitational constant, and \normalsize R_{msl} is the actual radius of the Earth for some given latitude at mean sea level (e.g. msl).    In practice, \normalsize R_{msl} varies since the Earth is shaped as an oblate spheroid, meaning \normalsize R is larger at the equator when compared to the polar axis).  Note that \normalsize Q is always negative.

Combining definitions for \normalsize h and \normalsize Q facilitates the definition of \normalsize R in terms of \normalsize Q:

(11)   \begin{equation*} \normalsize R = - \frac{R_{E}^2}{Q} \end{equation*}

Hence, \normalsize Q can be defined for any altitude a by

(12)   \begin{equation*} \normalsize Q = - \frac{R_{E}^2}{R_{msl} + a} \end{equation*}

\normalsize R_{msl} is given by the analytic geometry of the oblate spheroid and the standard definition of geographic latitude:

(13)   \begin{equation*} \normalsize R_{msl}^2 = \frac{y^4 + (x^4 - y^4) \cos^2{\phi}}{y^2 + (x^2 - y^2) \cos^2{\phi}} \end{equation*}

where x is the major axis of the spheroid, y is the minor axis and \normalsize phi is the observer’s latitude.  Plugging equation (11) into (4) yields

(14)   \begin{equation*} \normalsize N_{Z} = \int^{Q_{mes}}_{Q_{0}} \frac{\rho R_{E}^2}{R^2 \sqrt{1 - \frac{Q^2 \mu^2_{0} \sin^2{z}}{Q^2_{0}\mu^2_{0}}}} dQ \end{equation*}

Plugging equations 5, 6 and 8 into equation 14 and integrating defines the path length of the atmosphere and the numerator for the air mass coefficient.  Setting \normalsize Z=0 and using the same equations and integration process defines the denominator of the air mass coefficient.

Conclusion

Atmospheric models define an alternative and preferred approach for estimating air mass when compared to interpolative models.  Air mass, in turn, is essential to understanding solar extinction and irradiance intensity on the Earth’s surface.  The air mass model for astronomy proposed by Meyers relies on adjustments to the Standard Atmosphere model that serve to capture how the air filters, bends and dims sunlight.  Use of the Standard Atmosphere model is convenient from an implementation perspective, and represents a stepping stone for the use of physics models like WRF and MM5 that better describe the behavior of the fluid atmosphere.

B.Horn

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Show 3 footnotes

  1. Neal Clarke, Air Mass (Astronomy), Wikipedia, The Free Encyclopedia.
  2. Garfinkel, B. (1967). Astronomical Refraction in a Polytropic AtmosphereAstronomical Journal 72: 235-254.
  3. Reed D. Meyer, Air Mass, http://reed.gigacorp.net/airmass.c