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<h2 class="hd hd-2 unit-title">Introduction to climate physics</h2>
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<h2>A little climate physics</h2>
<p>This section will provide a broad overview of the physics of climate. Specifically, we will cover the basics of atmospheric climate physics, including its role in the Earth's energy balance. The material covered here will prepare you to learn about paleoclimate and the instrumental record of Earth's climate in the next two sections. It will also serve as background for the remainder of the course.</p>
<p><i>Instructor: Kerry Emanuel</i></p>
<h2>Module outline</h2>
<p>In this module, we will introduce the concept of planetary energy balance, and begin to understand the properties of the Earth-system that may affect it. Specifically, the module includes the following parts:</p>
<ul>
<li>Basic facts about the atmosphere, including its composition and vertical structure</li>
<li>The introduction of <i>planetary energy balance</i>, and how the Earth's temperature is related to solar radiation</li>
<li>The <b>greenhouse effect</b></li>
<li>Developing a simple analytical model for planetary energy balance</li>
<li>Details of the observed <b>radiative budget</b> of the Earth</li>
<li>Observations of seasonal and other variability in the climate system</li>
</ul>
<h2>Supplemental reading</h2>
<p>This module will address the global climatology of surface temperature, as well as the current composition of the atmosphere. To learn about observed changes in surface temperature and atmospheric composition, read <a href="http://www.climatechange2013.org/images/report/WG1AR5_Chapter02_FINAL.pdf" target="_blank">Chapter 2.2 and Chapter 2.4</a> of the <a href="http://www.ipcc.ch/report/ar5/wg1" target="_blank">IPCC AR5 Report</a>.</p>
<h2>Lecture Slides</h2>
<p>Slides for these lectures are available for download <a href="/assets/courseware/v1/29bb607b677c454ac8eba07b434b10a9/asset-v1:MITx+12.340x+1T2020+type@asset+block/a_little_climate_physics.pdf" target="[object Object]">here</a>.</p>
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<h2 class="hd hd-2 unit-title">Basics of the climate system</h2>
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Atmospheric composition
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<p>Together, Nitrogen (N<sub>2</sub>) and Oxygen (O<sub>2</sub>) account for what proportion of the atmosphere? </p>
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<text>About 10 %.</text>
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<text>Close to half.</text>
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<text>approximately 78 %.</text>
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<text>More than 95 %.</text>
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<h2 class="hd hd-2 unit-title">Planetary energy balance</h2>
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Which terms are being &#39;balanced&#39;?
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<p>In equilibrium, Earth's energy balance can be expressed as:</p>
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<text>A balance between the radiative fluxes entering and leaving the Earth system.</text>
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<text>A balance between the downward and upward radiative fluxes at the surface.</text>
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<text>A balance between the radiative fluxes entering and leaving the atmosphere.</text>
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<text>A balance between heating by sunlight and cooling by the wind.</text>
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Causes of the greenhouse effect
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<p>The greenhouse effect is a result of:</p>
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<text>The atmosphere absorbing radiation emitted from the sun.</text>
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<text>The Earth absorbing radiation from the sun.</text>
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<text>The atmosphere absorbing radiation emitted from the surface.</text>
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<text>The Earth absorbing solar radiation.</text>
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<p>An implementation of the simple <a href="/courses/course-v1:MITx+12.340x+1T2020/jump_to_id/N-layer_radiation_model" target="_blank">one-layer model</a> of the global climate described in this video may be found under the resources tab, and is reproduced below. This version allows for multiple atmospheric layers, and allows each atmospheric layer to absorb different fractions of the long-wave radiation incident upon it. We define an emissivity [mathjaxinline]\epsilon[/mathjaxinline], ranging between 0 and 1, that characterizes both the absorption and emission of a layer. A layer of atmosphere with temperature [mathjaxinline]T[/mathjaxinline] absorbs a fraction [mathjaxinline]\epsilon[/mathjaxinline] of the radiation incident upon it and emits [mathjaxinline]\epsilon\sigma{} T^4[/mathjaxinline], both upwards and downwards.</p>
<p>Explore the ways in which the surface temperature changes as you add more layers, or change the emissivity of the layers. When the upper most layer has an emissivity of one, what is its temperature? Does this depend on how many layers are below it? What happens to the surface temperature when the emissivity of the atmosphere is set close to zero?</p>
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Components of surface radiative balance
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<p>The Earth's surface emits considerably more long-wave radiation than the amount of short-wave radiation it absorbs from the sun. The primary reason for this is:</p>
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<text>The surface also loses energy via convection.</text>
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<text>The Earth is warming up.</text>
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<text>The surface also absorbs long-wave radiation emitted downwards by the atmosphere.</text>
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<text>The Earth is cooling.</text>
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<p>Note: in this course 'solar radiation' and 'short-wave radiation' are used interchangably. Similarly, 'terrestiral radiation' and 'long-wave radiation' both refer to radiation emitted by the Earth system.</p>
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<h2 class="hd hd-2 unit-title">Problem Set: Climate Physics</h2>
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<h2>Problem Set: Climate physics</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 the Earth's energy balance and the greenhouse effect, and have some sense for how the Earth's radiation balance varies across he globe. We encourage you to experiment with the <a href="/courses/course-v1:MITx+12.340x+1T2020/jump_to_id/N-layer_radiation_model" target="_blank"> multi-layer radiative model</a> given in the resources section before attempting the problem set.</p>
<p>Problems may ask you to choose the correct answer between a list of alternatives, or may ask you to enter an answer directly. Be sure to follow the instructions for each problem carefully to ensure you receive full credit for your responses.</p>
<p>In general you will be allowed two attempts at any multiple choice questions, and three attempts for symbolic or numerical responses, but this will be clearly indicated to you on each problem. Many problems have parts that are dependent on previous answers, and you are encouraged to check that these are correct before you proceed.</p>
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<h2>Radiative equilibrium in a "leaky greenhouse" atmosphere</h2>
<p>Consider the radiative equilibrium temperature of a one-layer atmosphere which is transparent for short-wave radiation, but absorbing for long-wave radiation. This is similar to the model discussed in the lectures, but now we will add a slight twist: The atmosphere will no longer be completely absorbing in the long-wave part of the spectrum. We define an emissivity [mathjaxinline]\epsilon[/mathjaxinline], between 0 and 1, that characterizes both the absorption and emission of the layer. So, the atmosphere absorbs a fraction [mathjaxinline]\epsilon[/mathjaxinline] of upwelling radiation from the surface and emits [mathjaxinline] \epsilon \sigma T_a^4 [/mathjaxinline] both upwards and downwards.</p>
<p>You can investigate the properties of such a model using the <a href="https://courses.edx.org/courses/course-v1:MITx+12.340x_2+1T2016/courseware/Resources/Nlayer/" target="_blank">N-layer radiative model</a> available under the Resources tab, and reproduced below.</p>
<p><iframe src="/asset-v1:MITx+12.340x+1T2020+type@asset+block/models_Nlayer_N_layer.html" width="800" height="600" frameborder="0"></iframe></p>
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1a
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<p>Complete the schematic of the leaky greenhouse model by dragging and dropping the appropriate values of the radiative fluxes into the empty boxes. Here [mathjaxinline]T_s[/mathjaxinline] is the surface temperature, [mathjaxinline]T_a[/mathjaxinline] is the temperature of the atmosphere, [mathjaxinline]\epsilon[/mathjaxinline] is the atmospheric emissivity, and [mathjaxinline]\sigma=5.67\times10^{-8}[/mathjaxinline] W m<sup>-2</sup> K<sup>-4</sup> is the Stefan-Boltzmann constant. We have expressed the net incoming short-wave radiation using the definition of the effective emission temperature [mathjaxinline]T_e[/mathjaxinline],
[mathjax]
\sigma T_e^4 = \frac{S_0(1-\alpha_p)}{4},
[/mathjax]
where [mathjaxinline]S_0[/mathjaxinline] is the solar constant, and [mathjaxinline]\alpha_p[/mathjaxinline] is the planetary albedo.
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1b
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<p>By considering equations for top-of-atmosphere and surface energy balance, derive an expression for the radiative-equilibrium surface temperature with a one-layer "leaky greenhouse" atmosphere. </p>
<p>Feel free to check your answer using the multi-layer radiative equilibrium model provided in the resources section, and reproduced above. In particular you should verify that the answer makes sense for the atmospheric emissivity approaching the limiting cases of 0 and 1 that were explored in the lecture.</p>
<p>Enter an expression into the box below in terms of the variables [mathjaxinline]\epsilon[/mathjaxinline] and [mathjaxinline]T_e[/mathjaxinline].</p>
<p>NOTE: In order to enter [mathjaxinline]T_e[/mathjaxinline], type 'T_e', and in order to enter [mathjaxinline]\epsilon[/mathjaxinline] type 'epsilon'. Multiplication must be indicated explicitly using the '*' character.</p>
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<span>[mathjaxinline]T_s [/mathjaxinline] =</span>
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<div id="display_6bef4076b57c415599ac7a6d91338484_2_1" class="equation">`{::}`</div>
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1c
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<p>In the limit of vanishingly small atmospheric emissivity (i.e. [mathjaxinline] \epsilon \rightarrow 0[/mathjaxinline]), what is the temperature of the atmosphere?</p>
<p>Hint: Solve for the atmospheric temperature and attempt to substitute [mathjaxinline] \epsilon = 0 [/mathjaxinline] into your solution.</p>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_a7e1b6afc82048f3b4e18dc45471fab3_2_1">
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<input type="radio" name="input_a7e1b6afc82048f3b4e18dc45471fab3_2_1" id="input_a7e1b6afc82048f3b4e18dc45471fab3_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="a7e1b6afc82048f3b4e18dc45471fab3_2_1-choice_0-label" for="input_a7e1b6afc82048f3b4e18dc45471fab3_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_a7e1b6afc82048f3b4e18dc45471fab3_2_1"> [mathjaxinline]T_e[/mathjaxinline]
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<input type="radio" name="input_a7e1b6afc82048f3b4e18dc45471fab3_2_1" id="input_a7e1b6afc82048f3b4e18dc45471fab3_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="a7e1b6afc82048f3b4e18dc45471fab3_2_1-choice_1-label" for="input_a7e1b6afc82048f3b4e18dc45471fab3_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_a7e1b6afc82048f3b4e18dc45471fab3_2_1"> [mathjaxinline]T_e/2^{1/4}[/mathjaxinline]
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<input type="radio" name="input_a7e1b6afc82048f3b4e18dc45471fab3_2_1" id="input_a7e1b6afc82048f3b4e18dc45471fab3_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="a7e1b6afc82048f3b4e18dc45471fab3_2_1-choice_2-label" for="input_a7e1b6afc82048f3b4e18dc45471fab3_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_a7e1b6afc82048f3b4e18dc45471fab3_2_1"> [mathjaxinline]2^{1/4}T_e[/mathjaxinline]
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<h2 class="hd hd-2 unit-title">Problem 2: Equator-to-pole temperature difference</h2>
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<h2>Radiative-equilibrium equator-to-pole temperature difference</h2>
<p>We think of the differential supply of solar radiation between the tropics and high-latitudes as a major driver of atmospheric motions (the winds). Here, we will explore a simple model for what the atmospheric temperature difference between the equator and poles would be in the absence of atmospheric energy transport.</p>
<p>The annual mean incident solar radiation [mathjaxinline]I[/mathjaxinline] as a function of latitude [mathjaxinline]\phi[/mathjaxinline] is given approximately by, [mathjax] I(\phi) = 342 \left(1 - 0.48 *(1/2)*\left(3 \sin^2\phi -1 \right) \right) [/mathjax]</p>
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2a
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<p>Which of the following gives the correct expression for the effective emission temperature (the temperature at which the top of the atmosphere radiates to space) as a function of latitude? Here, [mathjaxinline]\alpha(\phi)[/mathjaxinline] is the albedo as a function of latitude, while [mathjaxinline]\alpha_p[/mathjaxinline] is the average albedo of the Earth.</p>
<p>Hint: For calculations of radiative equilibrium, there is no energy transport by the atmosphere, and thus one can consider the temperatures at each latitude to be independent of the temperature at any other latitude.</p>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_704f6620f9b8479d82512fade1676b5b_2_1">
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<input type="radio" name="input_704f6620f9b8479d82512fade1676b5b_2_1" id="input_704f6620f9b8479d82512fade1676b5b_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="704f6620f9b8479d82512fade1676b5b_2_1-choice_0-label" for="input_704f6620f9b8479d82512fade1676b5b_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_704f6620f9b8479d82512fade1676b5b_2_1"> [mathjaxinline]T_e(\phi) = \left[\frac{(1-\alpha_p)I(\phi)}{\sigma}\right]^{1/4}[/mathjaxinline]
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<input type="radio" name="input_704f6620f9b8479d82512fade1676b5b_2_1" id="input_704f6620f9b8479d82512fade1676b5b_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="704f6620f9b8479d82512fade1676b5b_2_1-choice_1-label" for="input_704f6620f9b8479d82512fade1676b5b_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_704f6620f9b8479d82512fade1676b5b_2_1"> [mathjaxinline]T_e(\phi) = \left[\frac{(1-\alpha_p)I(\phi)}{4\sigma}\right]^{1/4}[/mathjaxinline]
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<input type="radio" name="input_704f6620f9b8479d82512fade1676b5b_2_1" id="input_704f6620f9b8479d82512fade1676b5b_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="704f6620f9b8479d82512fade1676b5b_2_1-choice_2-label" for="input_704f6620f9b8479d82512fade1676b5b_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_704f6620f9b8479d82512fade1676b5b_2_1"> [mathjaxinline]T_e(\phi) = 2^{1/4}\left[\frac{(1-\alpha(\phi))I(\phi)}{\sigma}\right]^{1/4}[/mathjaxinline]
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<input type="radio" name="input_704f6620f9b8479d82512fade1676b5b_2_1" id="input_704f6620f9b8479d82512fade1676b5b_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="704f6620f9b8479d82512fade1676b5b_2_1-choice_3-label" for="input_704f6620f9b8479d82512fade1676b5b_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_704f6620f9b8479d82512fade1676b5b_2_1"> [mathjaxinline]T_e(\phi) = \left[\frac{(1-\alpha(\phi))I(\phi)}{\sigma}\right]^{1/4}[/mathjaxinline]
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2b
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<div><span><p>From the model presented in class, in which the atmosphere is transparent to short-wave radiation, but opaque to long-wave radiation, we found that [mathjaxinline]T_a = T_e[/mathjaxinline]. Based on this, and the expression for [mathjaxinline]I(\phi)[/mathjaxinline] given above, calculate an approximate value for the <b>atmospheric temperature difference</b> between the equator and pole in radiative equilibrium, [mathjaxinline] \Delta T_a = T_a(0^\circ{}) - T_a(90^\circ{})[/mathjaxinline]. You may also assume that the albedo for all latitudes is 0.3; i.e. [mathjaxinline]\alpha(\phi) = \alpha_p = 0.3[/mathjaxinline].</p>
Enter your answer below in units of Kelvin (K).
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[mathjax] \Delta T_a = [/mathjax]
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2c
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Water vapor is the most important greenhouse gas, and the amount of water vapor increases with temperature. Using the "leaky greenhouse" model from problem 1, we could model this by introducing a latitude-dependent atmospheric emissivity, [mathjaxinline]\epsilon(\phi)[/mathjaxinline]. If [mathjaxinline]\epsilon(0^\circ{}) &gt; \epsilon(90^\circ{}) [/mathjaxinline], how would that affect the radiative equilibrium atmospheric temperature difference between the equator and the poles, [mathjaxinline]\Delta T_a[/mathjaxinline]?
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<input type="radio" name="input_0c8e8461ad1e46808f7e61483be27cdf_2_1" id="input_0c8e8461ad1e46808f7e61483be27cdf_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="0c8e8461ad1e46808f7e61483be27cdf_2_1-choice_0-label" for="input_0c8e8461ad1e46808f7e61483be27cdf_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_0c8e8461ad1e46808f7e61483be27cdf_2_1">
<text>[mathjaxinline]\Delta T_a[/mathjaxinline] would be <b>increased</b> by the introduction of a latitudinally dependent emissivity.</text>
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<text>[mathjaxinline]\Delta T_a[/mathjaxinline] would be <b>decreased</b> by the introduction of a latitudinally dependent emissivity.</text>
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<text>There would be <b>no change</b> in [mathjaxinline]\Delta T_a[/mathjaxinline] as a result of introducing a latitudinally dependent emissivity.</text>
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<text>There is insufficient information to determine what would happen to [mathjaxinline]\Delta T_a[/mathjaxinline] if a latitudinally dependent emissivity was introduced.</text>
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<h2 class="hd hd-2 unit-title">Problem 3: Transmitted radiation</h2>
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<h2>Radiation transmitted by the atmosphere</h2><p>The spectral properties of the atmosphere are extremely important for understanding climate. In this problem we consider in more detail the figure showing atmospheric transmission of radiation that was shown in the lecture slides and is reproduced below.</p>
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3a
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<p>CO<sub>2</sub> is an important greenhouse gas predominantly due to strong absorption of radiation of wavelengths around 15 [mathjaxinline]\mu[/mathjaxinline]m. In this area of the spectrum, almost all radiation emitted from the surface is absorbed by the atmosphere.</p>
<p>This absorption feature can be seen in the panel marked "Carbon dioxide" in the plot below, while its effect on spectral intensity can be seen in the colored panel (also in the figure below). Click on the region of the plot of spectral intensity affected by the CO<sub>2</sub> absorption feature in question.</p>
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3b
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<p>Rayleigh scattering is important for the visible part of the spectrum, and it is what makes the sky blue. Based on the figure shown above, is Rayleigh scattering important for the long-wave radiation emitted by the Earth?</p>
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3c
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<p>We often idealize the Earth's atmosphere as being nearly transparent to solar radiation. But as the figure above shows, that assumption is only 70-75% correct! Which of the following gases plays the most important role in absorption of solar radiation?</p>
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<text>CO<sub>2</sub></text>
</label>
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<text>H<sub>2</sub>O</text>
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<text>N<sub>2</sub>O</text>
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<text>CH<sub>4</sub></text>
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<div class="solution-span">
<span id="solution_1fd299402e7d4d6590b43cecace8e82f_solution_1"/>
</div></div>
<div class="action">
<input type="hidden" name="problem_id" value="3c" />
<div class="submit-attempt-container">
<button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_1fd299402e7d4d6590b43cecace8e82f" >
<span class="submit-label">Submit</span>
</button>
<div class="submission-feedback" id="submission_feedback_1fd299402e7d4d6590b43cecace8e82f">
<span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span>
</div>
</div>
<div class="problem-action-buttons-wrapper">
</div>
</div>
<div class="notification warning notification-gentle-alert
is-hidden"
tabindex="-1">
<span class="icon fa fa-exclamation-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="1fd299402e7d4d6590b43cecace8e82f-problem-title">
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification warning notification-save
is-hidden"
tabindex="-1">
<span class="icon fa fa-save" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="1fd299402e7d4d6590b43cecace8e82f-problem-title">None
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
<div class="notification general notification-show-answer
is-hidden"
tabindex="-1">
<span class="icon fa fa-info-circle" aria-hidden="true"></span>
<span class="notification-message" aria-describedby="1fd299402e7d4d6590b43cecace8e82f-problem-title">Answers are displayed within the problem
</span>
<div class="notification-btn-wrapper">
<button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button>
</div>
</div>
</div>
"
data-graded="True">
<p class="loading-spinner">
<i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i>
<span class="sr">Loading…</span>
</p>
</div>
</div>
</div>
</div>
</div>
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