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<h2 class="hd hd-2 unit-title">Introduction to radiative heat transfer</h2>
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<h2>Welcome to Part II of the course</h2>
<p>In Part I, you got an overview of the components of the Earth's climate system, and the science behind them. In Part II, we'll take a more in-depth look and learn in a bit more detail about the physics of climate. We'll begin with an overview of the physics of heat transfer by radiation.</p>
<h2>Radiative heat transfer</h2>
<p>In this section we will give an introduction to basic concepts of heat transfer by radiation, including the scattering, absorption and emission of radiation by gases and particles in the atmosphere. The material covered here will give you an appreciation for why some gases are strong greenhouse gases and others are not, as well as why clouds may have warming or cooling effects on the climate system depending on their properties.</p>
<p><i>Instructor: Kerry Emanuel</i></p>
<h2>Module outline</h2>
<p>In this module we will introduce some basic concepts of <i>radiative heat transfer</i>. Specifically, the module covers the following topics:</p>
<ul>
<li>Some definitions of terms used in radiative transfer theory such as <b>intensity</b>, <b>flux density</b> and <b>total flux</b></li>
<li>The concept of <b>Black-body radiation</b> including Planck's law, Wein's displacement law and the Stefan-Boltzmann equation</li>
<li><b>Absorption</b> and <b>emission spectra</b> is gases, and the quantum mechanical transitions that lead to these spectra</li>
<li>A description of the most important gases in the atmosphere and their effect on <b>Earth's emission spectrum</b></li>
<li>Scattering and absorption of radiation by clouds and aerosols</li>
<li><b>Radiative equilibrium</b> and its application to the vertical structure of the Earth's atmosphere</li>
</ul>
<h2>Background reading</h2>
<p>Some of the material in this module is based on the fundamentals of the quantum mechanical structure of atoms. While this will not be addressed in detail, students unfamiliar with these concepts can view these <a href="http://ocw.mit.edu/courses/chemistry/5-61-physical-chemistry-fall-2007/lecture-notes/lecture12to15.pdf" target="_blank">MIT 5.61 lecture notes</a> for some more rigorous background.</p>
<h2>Supplemental reading</h2>
<p>Radiative transfer is a complex topic, and we only scratch the surface in this week of lectures. While it will not be required for understanding the material in this course, interested students may wish to obtain some more detailed information on some of the topics we touch on. To get more in-depth notes on radiative transfer and its effects on planetary climate, view the <a href="http://ocw.mit.edu/courses/earth-atmospheric-and-planetary-sciences/12-815-atmospheric-radiation-fall-2008/lecture-notes/" target="_blank">lecture notes from MIT 12.815</a>.</p>
<h2>Lecture Slides</h2>
<p>Slides from these lectures can be found <a href="/assets/courseware/v1/db210484832084b851a9f0675c5505c4/asset-v1:MITx+12.340x+1T2020+type@asset+block/radiative_heat_transfer.pdf" target="[object Object]">here</a>.</p>
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<h2 class="hd hd-2 unit-title">Radiative transfer basics</h2>
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Outgoing long-wave radiation
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<p>The amount of outgoing long-wave radiation emitted by the Earth system is maximum in the subtropics, slightly lower in the deep tropics, and lower still in the polar regions. The reasons for this are: </p>
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<text>The amount of radiation emitted <i>increases</i> with temperature, but the subtropical deserts tend to have very high albedo and thus a high amount of outgoing longwave radiation. </text>
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<text>The amount of radiation emitted <i>decreases</i> with temperature, but the greenhouse effect of low polar clouds allows the polar regions to emit more radiation.</text>
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<text>The amount of radiation emitted <i>increases</i> with temperature, but the greenhouse effect of high tropical clouds reduces the outgoing long-wave radiation in the deep tropics. </text>
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<text>The amount of radiation emitted is independent of temperature, but the outgoing long-wave radiation pattern reflects the distribution of clouds in the atmosphere. </text>
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<h3 class="hd hd-3 problem-header" id="Important_atmospheric_absorbers-problem-title" aria-describedby="block-v1:MITx+12.340x+1T2020+type@problem+block@Important_atmospheric_absorbers-problem-progress" tabindex="-1">
Important atmospheric absorbers
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<p>Why are trace gases such as water vapor and carbon dioxide so important for the radiative balance of the atmosphere?</p>
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<text>The major components of the atmosphere (Nitrogen, Oxygen, Argon) are not able to absorb radiation in the long-wave spectrum due to their inability to form electric dipoles. Trace gases that are able to form electric dipoles can have a large influence on the overall radiation balance. </text>
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<text>The major components of the atmosphere are not emitted by fossil fuel use. </text>
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<text>Carbon dioxide is heavier than air, and thus is able to trap heat near the Earth's surface, while other gases allow heat to escape to the upper atmosphere, and eventually to space. </text>
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<text>Lorentzian and Doppler broadening are much more efficient at widening the absorption bands of Carbon dioxide and water vapor than Oxygen and Nitrogen, and thus carbon dioxide and water vapor are more important to the radiative balance of the Earth. </text>
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Effective emission temperature
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<p>Suppose there are a number of planets in the same orbit around a star such that the solar constant for each planet is equal to 2000 W m<sup>-2</sup>. Which of the following planets would have the highest effective emission temperature?</p>
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<text>Planet Xorba, with an albedo of 0.6 and a pure carbon dioxide atmosphere. </text>
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<text>Planet Yanov, with an albedo of 0.3 and an atmosphere with an equal mixture of Nitrogen and Oxygen. </text>
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<text>Planet Zykal, with an albedo of 0.45 and an atmosphere with an equal mixture of carbon dioxide and water vapor. </text>
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Why is the sky blue?
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<p>Why is the sky blue? </p>
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<text>Because it is cold - blue is a cool color. </text>
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<text>Ozone absorbs the much of the ultraviolet portion of the spectrum, making the sky appear blue to our eyes. </text>
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<text> Reflections from the ocean make the atmosphere appear blue. On Mars the sky is pink. </text>
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<text> Molecules in the atmosphere scatter the shortest wavelengths much more efficiently than longer wavelengths. These shorter wavelengths are on the blue end of the spectrum, making the sky appear blue to our eyes.</text>
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<h2 class="hd hd-2 unit-title">Problem Set: Radiation</h2>
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<h2>Radiative Transfer</h2>
<p>This problem set is aimed at testing your understanding of the material presented in the videos of this sequence. By now you should have a basic understanding of the components of the radiative balance of the atmosphere, and the gases and particles that influence this balance. This problem set contains one problem, which has multiple parts.</p>
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<h2 class="hd hd-2 unit-title">Problem: Multi-layer models of the atmosphere</h2>
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<h2>Multi-layer models of the atmosphere</h2><p>The following problems refer to a two-layer model of the global climate, extending the single layer model discussed in class. A schematic of this model is presented in Problem 1A below.</p>
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1a
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A schematic of a two layer model of the Earth-atmosphere energy balance is shown in the image below.
The atmosphere is assumed to be transparent to solar radiation, and the total solar radiation absorbed by the surface is written in terms of the emission temperature defined,
[mathjax] \sigma T_e^4 = \frac{S_0(1-\alpha)}{4} [/mathjax]
There are two atmospheric layers, with temperatures T<sub>1</sub> and T<sub>2</sub> respectively, while the surface temperature is denoted T<sub>s</sub>.
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<p>Assuming the surface and each atmospheric layer have long-wave emissivities of 1, fill in the values of the long-wave fluxes represented by each of the red arrows by dragging and dropping the appropriate values into the boxes.</p>
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1b
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<p>Using the diagram from the previous problem, construct equations for the energy balance of each atmospheric layer of the model. Solve these equations to find the surface temperature in terms of the emission temperature T<sub>e</sub>.</p>
<p>Type your solution in the box below. Use "T_e" to indicate T<sub>e</sub>, and explicitly indicate any mathematical operations with the appropriate symbol (e.g. '+', '-', '*', '/', '^').</p>
<p><i>Hint:</i> You can use the <a href="/courses/course-v1:MITx+12.340x+1T2020/courseware/Resources/Nlayer/">multi-layer radiative-equilibrium model</a> in the Resources section to check your answer.</p>
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<span>T<sub>s</sub> =</span>
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<div id="display_f614a1efbbb34c1fa41422a17adff212_2_1" class="equation">`{::}`</div>
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1c
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<p>In Problem 1B, you derived a formula for the surface temperature in terms of the emission temperature (T<sub>e</sub>) for a two-layer model. Now, do the same for a model with three atmospheric layers. Based on your answer and the answer to the previous question, and the solution to the one-layer model given in class, deduce a formula for the radiative equilibrium temperature for a surface beneath N layers of atmosphere, each with an emissivity of 1. Remember, you can check your answer against the multi-layer radiative model.</p>
<p>Type your solution in the box below. Use "T_e" to indicate T<sub>e</sub>, and "N" for the number of atmospheric layers. Explicitly indicate any mathematical operations with the appropriate symbol (e.g. '+', '-', '*', '/', '^').</p>
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<div id="display_33a79cdcb5f84410adb1b1b48eed1dd3_2_1" class="equation">`{::}`</div>
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1d
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<p>Suppose that the second layer of atmosphere now has an emissivity smaller than 1, which we denote [mathjaxinline] \epsilon [/mathjaxinline]. Fill in the value of the long-wave fluxes as in problem 1A.</p>
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Drag and drop the appropriate values into the boxes.
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1e
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<p>Consider the limit in which [mathjaxinline]\epsilon[/mathjaxinline] approaches zero. Under these conditions, the second layer of atmosphere has almost no effect on the solution, and, based on energy balance, one can deduce the temperature of layer 1. Find this temperature in terms of the emission temperature assuming [mathjaxinline]\epsilon[/mathjaxinline] is small enough that its absorption of radiation is insignificant.</p>
<p>Type your solution in the box below. Use "T_e" to indicate T<sub>e</sub>, and explicitly indicate any mathematical operations with the appropriate symbol (e.g. '+', '-', '*', '/', '^').</p>
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<span>T<sub>1</sub> =</span>
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<div id="display_ac204e225cb24f4db18232036cfd955c_2_1" class="equation">`{::}`</div>
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<h3 class="hd hd-3 problem-header" id="970422086083490ab5b0a3e15d610eaa-problem-title" aria-describedby="block-v1:MITx+12.340x+1T2020+type@problem+block@970422086083490ab5b0a3e15d610eaa-problem-progress" tabindex="-1">
1f
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<p>Derive the energy balance equation for layer 2 of the model illustrated in problem 1D. Again, assuming [mathjaxinline]\epsilon[/mathjaxinline] approaches zero and has no effect on the other layers, deduce the temperature of layer 2 in terms of the emission temperature. This temperature is called the <b>radiative skin temperature</b>. It is the temperature of the very outer reaches of an atmosphere in radiative equilibrium.</p>
<p>Type your solution in the box below. Use "T_e" to indicate T<sub>e</sub>, and explicitly indicate any mathematical operations with the appropriate symbol (e.g. '+', '-', '*', '/', '^').</p>
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<span>T<sub>2</sub> =</span>
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<div id="display_970422086083490ab5b0a3e15d610eaa_2_1" class="equation">`{::}`</div>
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