Instructions for Using a Simple Model of Solar Irradiance

David R. Brooks, March 2006

The simple model of solar irradiance (insolation) presented on the Web page from which this document is linked provides a tool for predicting insolation and understanding, at least roughly, the effects of the atmosphere on insolation. It is important to understand that this is only an approximate model that provides some insight into the basic physical processes that control the amount of sunlight reaching Earth's surface under "clear sky" (no cloud) conditions. It is not intended as a replacement for actual measurements of insolation. Comparisons between the model and measured values should be viewed with caution, if for no other reason than the fact that the biggest deficiency of this model (and other models, too) is that it cannot take into account the primary effect of clouds on insolation. However, it is very useful as a learning tool that shows how location, time of year, and time of day affect insolation.

Insolation is defined as solar irradiance on a horizontal surface at Earth's surface. It is controlled primarily by the seasons and the weather. Maximum daily total insolation is greatest during the summer when the sun is highest in the sky. Even under clear sky conditions, the atmosphere reduces the amount of sunlight that reaches Earth's surface. Gas molecules (including water vapor) and aerosols (small particles suspended in the atmosphere) scatter sunlight, some of which is returned to space. Gases (including water vapor) and aerosols also absorb sunlight and some of this energy is re-radiated back to space. Of course, clouds have a major effect on insolation because they reflect a great deal of sunlight back to space.

The conditions that determine insolation are difficult to model with high accuracy even under clear skies. However, here's a simple model that accounts at least conceptually for the factors that reduce clear-sky insolation.

S = Socos(z)TrTgTwTa/R2

where So is the solar constant, z is the solar zenith angle ( z = 90 - elevation angle), and R is the Earth-distance in Astronomical Units (average distance is 1 AU).

Transmission factors are dimensionless values between 0 and 1 that account for reductions in transmission of sunlight through the atmosphere to Earth's surface. One paper (see references below) has given these factors for molecular scattering (also called "Rayleigh scattering," hence the "r" subscript), gas absorption and scattering, water vapor absorption, and aerosol absorption and scattering as:
TrTg = 1.021 - 0.084 [m (949 px10-6 + 0.051)]1/2
Tw = 1.0 - 0.077 (PW m)0.3
Ta = Am
where p is barometric pressure in millibars, PW is total column precipitable water vapor in units of cm of H2O, A is an aerosol transmission factor, and m is the relative air mass. (The relative air mass is 1 when the sun is directly overhead and varies approximately as 1/cos(z).) The values given here are typical values. For example, 1.42 cm of H2O is the value assigned to a so-called "standard atmosphere" that scientists use for modeling the behavior of the atmosphere. In very dry or high-elevation locations, PW can be as low as a few millimeters. In very wet locations, it can be as high as 6 or 7 cm. 1013 millibars is the standard atmospheric pressure at sea level. At any elevation above sea level, you need to use "station pressure" -- the actual barometric pressure. With only a few exceptions for research sites, weather reports always give pressure converted to sea level pressure. If you are at a higher elevation, you need to convert this value to your elevation. An approximate conversion is:

station pressure = (sea level pressure) - (elevation in meters)/9.2

That is, pressure decreases very roughly 1 millibar for each 10 meters increase in elevation.

You can change the typical values given here if you like, but you may get very strange results if you change them arbitrarily!

The solar constant -- the energy available at the average Earth-sun distance of 1 Astronomical Unit -- is about 1375 W/m2. (So is not really a "constant," and varies a little due to fluctuations in solar activity.) On a clear summer day in temperate latitudes, the maximum insolation near sea level is around 1100 W/m2 -- a little less in northern hemisphere summer and a little more in southern hemisphere summer. (Why? Because the sun is farther from Earth during northern hemisphere summer than it is during southern hemisphere summer.) If all this energy could be converted to electrical energy, it would power about 10 100-W lightbulbs. Unfortunately, we are a long way from being able to do that!

Also given in the above calculations is the maximum clear-sky insolation for the specified date, assuming atmospheric conditions such as barometric pressure do not change between the original time and the time of maximum insolation. To do this precisely, you have to know when "solar noon" occurs. This is the time at which the sun crosses your meridian. (Your meridian is the imaginary line running from north pole to south pole through your longitude.) Solar noon is, in general, never exactly the same as clock noon. It varies with time of year and the observer's longitude and can occur several minutes before or after clock noon. The calculations require the astronomically derived value known as the "equation of time." This gives the time by which actual solar noon occurs at the Greenwich Meridian relative to clock noon. Clock time is based on the apparent motion of a fictitious "mean sun" around Earth, periodically adjusted during leap years to keep seasons in sync with the sun over long periods of time.

The equations for the equation of time and the time of solar noon at your observing longitude are complicated. If you are interested, you can see all the calculations by viewing the source code for this Web page. These calculations haven't been checked exhaustively, but they appear to give values for the equation of time that agree to within less than 0.1 minute with the values given by NOAA's Surface Radiation Research Branch at their online solar position calculator.

"Local clock time" is based on 15 time zones, with the "0" time zone centered around the Greenwich Meridian, and ignores Daylight Savings Time and local anomalies in selecting a time zone. For example, on the East coast of the United States, UT is 5 hours later than Eastern Standard Time, but only 4 hours later than Eastern Daylight Savings Time. One of the output form fields shows the number of hours these calculations assume you have to add to your local clock time to get to Universal Time. Be sure you understand the relationship of this information to your actual local time! (The endless possibilities for confusion about this point is why scientific and astronomical calculations always use Universal Time rather than local clock time!)

These calculations include an approximate day length, to assist in calculating daily averaged insolation. This daylength calculation is based on the time at which the elevation of the center of the sun, based on astronomical calculations of the sun's geometric position relative to Earth, is 0. This is not the same as actual or perceived sunrise or sunset for two reasons. First, the sun is not a "point," but a disc with a size of about 0.5. Second, atmospheric refraction bends light traveling from the sun and makes it visible even when it is geometrically below the horizon. Thus, perceived sunrise, when the top of the sun just appears on the horizon, occurs earlier than the present calculation and perceived sunset, when the top of the sun just disappears below the horizon, occurs later. Nonetheless, this geometric calculation is a reasonable figure to use for calculating average daily insolation.

As a "zero order" approximation, clear-sky insolation varies as the cosine of the of the solar zenith angle, symmetrically around local solar noon, and reaches 0 at sunrise or sunset. This is not a very good approximation because insolation is due to both direct and scattered sunlight. Scattered sunlight does not follow a cosine curve at all and its relative contribution to total insolation varies during the day. Direct sunlight is attenuated as it passes through more atmosphere when it is near the horizon, after sunrise or before sunset. A considerably better approximation is to model insolation as a function of time as a somewhat "pinched" cosine curve:

insolation = (solar noon insolation)•cos(πt/D)x

where D is the daylength, t varies between -D/2 and +D/2, and x is an exponent greater than 1.

Some Comments and Precautions:
For the output fields, I have rounded off displayed calculated values to what I consider to be an appropriate number of digits to the right of the decimal point.

These calculations have been tested extensively but not exhaustively. In any event, common sense dictates a healthy skepticism about any results from online applications, especially when results do not appear to make sense. Values for latitudes poleward of the Arctic or Antarctic Circles may cause arithmetic errors leading to "values" of "NaN" appearing in their respective form fields when there is no sunlight.

If you find errors in the model calculations, please e-mail me.

References:
Duchon and O'Malley. Journal of Applied Meteorology, 38, pp 132-141, 1999.
Meeus, Jean, Astronomical Algorithms. Willmann-Bell, Inc., Richmond, VA, 1991.