Transmission of Sunlight through the atmosphere

Seasonal and Global Variations in Sunlight

I supervised a senior project by a physics major to investigate the direct beam transmission of sunlight through the atmosphere as a function of latitude and date for several typical atmospheric composition models. I developed a mathematical model of the transmission, absorption, and scattering of a direct beam of sunlight on a cloudless day. The scientific goal was to estimate the transmission of sunlight under varying conditions in order to determine the relative importance of various parameters. The project exemplified the Cal Poly learn by doing educational philosophy: using a simple but powerful "homemade" user-friendly mathematical model as it was being developed offered my student the opportunity to learn a variety of skills including code development, system analysis, data analysis, and graphical representation of results, in addition to an understanding of the physical phenomena that affect the availability of sunlight on the surface of a planet.

Sunlight is essential for life on Earth. Obviously, the entire food web is built on the photosynthesis, which uses the sun's energy to convert molecules like water and carbon dioxide into carbohydrates which are necessary for living organisms and for the organisms that feed on them. Perhaps it is less obvious that a waste product of photosynthesis is the oxygen in the atmosphere that is essential to plants and animals for respiration. Some of the oxygen in the upper atmosphere is converted to ozone by ultraviolet sunlight, which protects many organisms from the harmful effects of ultraviolet radiation. Perhaps least obvious is that the oceans absorb about 90% of the incident sunlight, contributing to a global warming that is essential to keeping the oceans from freezing, which would cut off the source of freshwater on land and destroy the environment for virtually all life on Earth.

Many atmospheric transmission models already exist, including LOWTRAN and HITRAN, which were developed in the 1970's by the Air Force for low resolution and high resolution transmission analyses. These codes are expensive and require expertise to operate. They are not necessarily structured to provide the information I need to solve problems that interest me. They also tend to devote enormous computational efforts to the enormous number (more than 100,000) of absorption lines in the infrared part of the spectrum, whereas I am only interested in the general trends in the infrared. More importantly, the underlying calculations are not transparent; and a student cannot readily modify the codes. These limitations made it more interesting to develop my own simple models.

My model used Mathcad because it is user-friendly and readily displays the results in graphs. The code looks a lot like equations in a textbook, so it is relatively compact and easy to understand. The graphs allow the user to quickly locate trends and more importantly, errors related to the models or the input parameters. Ultimately, it was necessary to display the data from multiple Mathcad runs and it turned out to be easy to save it in a Microsoft Excel spreadsheet and then plot it in simple bar charts there in order to provide easy comparisons between cases. It also turned out that the Mathcad cases required long run times like an hour because of the desire to separate the individual effects of absorption and scattering. The run times resulted from the fact that Mathcad is not a compiled program and that something like 10,000 to 100,000 integrals were evaluated for some of the cases. Several versions of the model were created to solve different related problems. Some problems can be solved very rapidly but are not as interesting as the ones investigated during this project.

There are four essential elements of a simple direct beam model:

1. The latitude and time of day of the observer on the Earth's surface.

2. The position of the Earth in its orbit around the sun.

3. The blackbody radiation of the sun at the radius of the Earth's nearly circular orbit.

4. The atmospheric losses due to absorption and scattering.

As the Earth rotates on its axis, the amount of sunlight incident on a horizontal surface (ignoring the atmosphere) varies. At local noon, the solar flux reaches a maximum. On the Equinoxes, the solar flux decreases as the local latitude increases away from the Equator. On the Equinoxes, the sun is above the horizon 12 hours at all latitudes because the axis of Earth is tilted at right angles to the direction of the Sun. The axis of the Earth is fixed in space as the Earth orbits the Sun, so that the North Pole is always pointing toward Polaris, the North Star. As the Earth orbits the Sun, its rotational axis points toward the Sun or away from the Sun. On the Northern Hemisphere Summer Solstice, the axis is tilted toward the Sun, exposing the Northern Hemisphere to more sunlight than the Southern Hemisphere or even the Equator.

There are two effects contributing to increased solar flux in the Spring and Summer: the apparent height of the Sun in the sky and the length of day that the Sun is above the horizon. Because the Earth's axis is tilted 23.45°, the Sun appears directly overhead at noon at the Tropic of Cancer (located at 23.45°N latitude) on the Summer Solstice. The zenith angle is defined as zero when the Sun is directly overhead. The peak solar flux (at local noon) decreases slightly just to the north because of the slow change of the cosine function near a zenith angle of zero. However the length of day continues to increase, so the average daily solar flux is greater to the north of the Tropic of Cancer despite the decrease in peak flux. Average flux is particularly interesting because it influences the average temperature and availability of sunlight for photosynthesis.

One version of the code calculated instantaneous flux at any time of day, including the peak at local noon. The code used for this project calculated average daily flux over a 24-hour period by calculating flux throughout the day at even intervals. This model was called SolarFluxBAS AF, for average flux (the BAS refers to blackbody spectrum with atmospheric absorption and scattering). The analysis with this model indicated that the average flux on a Summer Solstice tends to peak near a latitude of 35°, which is the location of San Luis Obispo County as well as many other major cities around the world.

In order to simplify the analysis, this project investigated two sets of cases in detail. We compared the average flux on the two solstices at one latitude, 35°. We also compared the average flux on the Equinox at two latitudes, 23.45° and 66.45°, which are the Tropic of Cancer and the Arctic Circle.

The entire solar spectrum had to be used in the calculations because absorption and scattering in the atmosphere vary with wavelength. Furthermore, it is interesting to investigate the effects in different spectral bands such as ultraviolet (UV), visible, infrared (IR), as well as the total spectrum, which was modeled from 0.3 to 3.0 microns, a band that includes nearly 95% of the energy incident on the top of the atmosphere. So all four of these cases were included in the detailed analyses of latitudes and dates.

Sunlight peaks around 0.5 microns because it is due to blackbody radiation of energy from the surface of the sun, whose temperature is approximately 5800K. This temperature is sufficient and necessary to radiate away the energy reaching the surface that was generated by thermonuclear reactions that occurred over a million years earlier. The initial form of this energy was predominately gamma rays from the transformation of matter into energy, but was degraded to mostly thermal energy as a result of numerous interactions with the matter in the sun over a period of more than a million years. An analysis of Planck's Law for blackbody radiation reveals that the solar flux varies as the fourth power of the temperature of the surface of the Sun. The surface of the Sun has been getting hotter for billions of years and will continue to warm for billions more. My model allows the solar surface temperature to be varied.

For simplicity, the Earth's orbit was treated as circular with a radius of 93 million miles. When the goal is to get the most accurate estimates or predictions, then the model needs to include as many significant factors as possible. In my case, it is desirable to ignore higher order effects since my goal was to understand the sensitivity of certain parameters to a few other parameters. It also makes the analysis simpler. Outside the Sun, the incident solar flux decreases as the square of the distance from the center of the Sun. Therefore my model depends on the square of the ratio of the radius of the Earth's orbit to the radius of the Sun.

People naturally think of clean dry air as transparent and imagine that sunlight passes through it undiminished and undisturbed. In reality, there are many constituents in the atmosphere that absorb and scatter sunlight. Clean dry air is approximately 78% nitrogen, 21% oxygen, and 1% argon. These gases absorb very little sunlight. Nitrogen and oxygen absorb very little except at very short ultraviolet wavelengths. The absorption of UV by oxygen molecules can dissociate the atoms, producing atomic oxygen at certain altitudes that can then combine with other oxygen molecules to form ozone (triatomic oxygen). Ozone has the beneficial property of absorbing harmful UV rays and its effect is included in my model. Most of the absorption of sunlight occurs in the infrared and is also mainly due to minor components that are triatomic molecules, namely water vapor and carbon dioxide. The absorption of sunlight warms the atmosphere. Eventually the energy is radiated away from the Earth by blackbody radiation of the atmosphere, which is peaked in the far infrared near ten microns.

Molecular scattering reduces the direct beam of sunlight transmitted through the atmosphere. Most of this is due to the nitrogen and oxygen molecules because they are the primary constituents of the atmosphere. Molecular scattering is known as Rayleigh scattering because the scattering particles are much smaller than the wavelength of light being scattered. Because Rayleigh scattering varies as the inverse fourth power of the wavelength, it is most intense in the ultraviolet and the short visible wavelengths and causes the sky to appear to be blue. The scattering loss of these wavelengths also causes sunsets to be orange. Molecular scattering has very little effect in the infrared band. Molecular scattering tends to send light in all directions. Some of the light is scattered into space and some of it is scattered toward the surface of the Earth. Light that is scattered into space cannot increase the temperature of the Earth. My model includes Rayleigh scattering and the wavelength dependence is slightly different from the fourth power because the optical properties of nitrogen and oxygen vary with wavelength.

Additional sunlight is scattered by aerosols even when the sky is cloudless. The two most important natural aerosols are dust and clusters of water molecules. Since these particles are larger than molecules, the wavelength dependence is not as strong as Rayleigh scattering. Dust can be due to eroded particles on the surface of the land or due to salt particles that enter the atmosphere when sea bubbles break. Water molecule clusters can be from the ocean or from freshwater sources. My model includes dust and water aerosols and uses the same concentration of water for absorption and for scattering. My scattering models vary with wavelength to the -0.75 for dust and -2 for water aerosol. My model assumes no absorption by dust.

In my model, the transmitted sunlight is the product of the transmission factors associated with each independent absorbing and scattering mechanism. The order of multiplication does not affect the result, so in my model the distribution of particles in the atmosphere does not matter. All absorbed and scattered light is treated as simply lost from the direct beam. My model also estimates absorption losses and scattering losses based on the light incident on the particles. The order is important in these calculations because any absorption or scatter of light decreases the light incident on subsequent particles. My model divides the atmosphere into ten layers so that the solar flux is fairly constant within each layer. Within each layer, we can calculate absorption losses first or scattering losses first without changing the estimate very much. This method is computationally very slow but faster alternatives were beyond the scope of this project.

As an extreme example, if molecules scatter 60% of the incident light at a particular wavelength and if ozone absorbs 60% at the same wavelength, then the transmitted energy is (100% - 60%) * (100% - 60%) = 16%. We know the total losses cannot be 60% + 60% = 120% of the incident light. The total energy lost from the direct beam is in fact 100% - 16% = 84%. If the scattering occurs first, it causes 60% of the losses and the absorption causes (100%-60%)*60% = 24% of the losses. Conversely, if the absorption occurs first, it causes 60% of the losses and the scattering causes (100%-60%)*60% = 24% of the losses. If on the other hand, the molecular scatterers and ozone absorbers have the same atmospheric distribution, we can guess that each effect causes a 42% loss of light.

In a real atmosphere, some particles are uniformly mixed at all altitudes and others are not. Molecular scattering by nitrogen and oxygen and absorption by carbon dioxide occur at all altitudes. Water and dust tend to be located at lower altitudes depending on where they are produced, circulation patterns, and dwell time in the atmosphere. Ozone is concentrated at a fairly high altitude where there is an abundance of both oxygen and UV light. These effects are only important when losses due to absorption and losses due to scattering occur at the same wavelength. The problem is most significant in the UV where absorption and scattering are both strong. But the total amount of UV is small and does not contribute much to the overall transmission of sunlight. Although my model assumes all molecules and aerosols are uniformly mixed at all altitudes, my approach could be upgraded to include non-uniform distributions of particles in the atmosphere by changing the composition of the ten layers in the model.

Figures

Latitude geometry

Solar blackbody spectrum after Iqbal’s out-of-print book on solar radiation

Four bar charts of transmitted, scattered, and absorbed sunlight

for four different atmospheres (CDA, CMA, DDA, DMA)

at 35N latitude on two solstices

for all wavelengths and for visible spectrum only

Same four bar charts for infrared spectrum only and for ultraviolet spectrum only

Four bar charts of transmitted, scattered, and absorbed sunlight

for four different atmospheres (CDA, CMA, DDA, DMA)

at two latitudes on an equinox

for all wavelengths and for visible spectrum only

Same four bar charts for infrared spectrum only and for ultraviolet spectrum only