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<h2 class="hd hd-2 unit-title">What is convection?</h2>
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<h2>Convective heat transfer and radiative-convective equilibrium</h2>
<p>This section covers the vertical transport of heat in the Earth's atmosphere by convective motions. We will introduce convection as a means of improving upon the radiative equilibrium solution for the vertical structure of temperature in the Earth's atmosphere, and discuss the concept of radiative-convective equilibrium (RCE). We will use these concepts to begin to understand the role that convection plays in the Earth's climate, both through heat transfer and precipitation.</p>
<p><i>Instructor: Kerry Emanuel</i></p>
<h2>Module outline</h2>
<p>In this module, we will introduce the concept of atmospheric convection, and begin to understand the role it plays in the climate system. Specifically, the module includes the following parts:</p>
<ul>
<li>The importance of convective heat transfer in reconciling the flaws of the radiative equilibrium solution</li>
<li>The concept of <i>stability</i>, and what it means for a fluid to be convectively stable</li>
<li>Hydrostatic equilibrium</li>
<li>Developing a mathematical criterion for convective equilibrium using <b>buoyancy</b> and <b>entropy</b></li>
<li>A recalculation of the atmosphere's equilibrium state taking into account convection as well as radiation</li>
<li>Understanding how precipitation forms during atmospheric convection</li>
<li>Stability of the atmosphere to moist convection</li>
<li>Characterizing the properties of moist atmospheric convection</li>
<li>Introduction of a simple, two-layer model of radiative-convective equilibrium</li>
</ul>
<h2>What is Convective Heat Transfer?</h2>
<ol>
<li>The general scientific meaning of <i>Convection</i> is heat transfer that occurs due to movement of a mass of fluid from one place to another.</li>
<li>In meteorological contexts, including this course, <i>Convection</i> refers specifically to the <b>vertical</b> transfer of heat by rising warm air and sinking cool air.</li>
<li>In meteorology, the term <i>Advection</i> is used to refer to the <b>horizontal</b> transfer of heat by moving air. Example: an unusually warm day is caused by a strong, warm wind that blows from the south.</li>
<li><i>Moist Convection</i> indicates that the vertical motion is strongly influenced by a phase change of water, such as the condensation or freezing of water vapor to form liquid cloud droplets or solid ice crystals.</li>
</ol>
<h2>Background reading</h2>
<p>The material in this module relies heavily on the concept of <b>entropy</b>. Those without a strong thermodynamics background should read Chapter 8 of these <a href="https://www3.nd.edu/~powers/ame.20231/notes.pdf" target="_blank">lecture notes on thermodynamics</a> by Joseph M. Powers of the University of Notre Dame, or view Lectures 8 through 10 of these <a href="http://ocw.mit.edu/courses/chemistry/5-60-thermodynamics-kinetics-spring-2008/video-lectures/" target="_blank">MIT 5.60 video lectures</a> for background information on entropy.</p>
<p>Additionally, those students without strong thermodynamics backgrounds should view <a href="http://ocw.mit.edu/courses/chemistry/5-60-thermodynamics-kinetics-spring-2008/video-lectures/lecture-5-adiabatic-changes/" target="_blank">Lecture 5 of MIT OCW 5.60</a> to learn about <b>adiabatic processes</b>.</p>
<p>Some material in this section also makes use of the <b>Maxwell relations</b>, on which more information can be found in Section 11.1 of the <a href="https://www3.nd.edu/~powers/ame.20231/notes.pdf" target="_blank">Notre Dame thermodynamics lecture notes</a>.</p>
<h2>Lecture Slides</h2>
<p>Slides from these lectures can be found <a href="/assets/courseware/v1/7dedecee0085b448c810c0356a952068/asset-v1:MITx+12.340x+1T2020+type@asset+block/convective_heat_transfer_and_radiative-convective_equilibrium.pdf" target="[object Object]">here</a>.</p>
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Examples of convection
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<p>Which of the following is <b>not</b> an example of convection? </p>
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<text>Air cooled by evaporating rain moves downwards</text>
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<text>Air warmed by contact with the surface moves upwards</text>
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<text>Air is warmed by contact with a hot roof</text>
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<text>Water moves turbulently in a heated pot</text>
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An Unstable Equilibrium
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<p>Which of the following is an example of an <b>unstable</b> equilibrium? </p>
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<text>A ball stationary at the bottom of a bowl</text>
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<text>A ball rolling down a hill</text>
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<text>A buoy floating on the ocean</text>
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<text>A rollercoaster stationary at the highest point in its course</text>
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<h3 class="hd hd-3 problem-header" id="Characteristics_of_hydrostatic_Equilibrium-problem-title" aria-describedby="block-v1:MITx+12.340x+1T2020+type@problem+block@Characteristics_of_hydrostatic_Equilibrium-problem-progress" tabindex="-1">
Characteristics of hydrostatic Equilibrium
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<p>Which of the following does <b>not</b> accurately describe hydrostatic equilibrium? </p>
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<text>There is a balance between the pressure gradient and gravitational forces</text>
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<text>Pressure must decrease in the direction of the gravitational force</text>
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<text>Vertical acceleration can be neglected</text>
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<text>Pressure must increase in the direction of the gravitational force</text>
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Pressure distribution in an atmosphere at rest
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<p>Consider a number of planets each with atmospheres with compositions identical to that of Earth. Under which of the following conditions would pressure decrease most rapidly with height? </p>
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<text>High surface gravity, cold temperature</text>
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Adiabatic expansion and contraction
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<p>An air parcel moving horizontally encounters a mountain range and is forced to rise. The air parcel:</p>
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<text>Cools, because it expands and thus does work on its surroundings</text>
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<text>Warms, because it gets closer to the sun which is keeping it warm</text>
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Convective Temperature Profile
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<p>Which of the following is true of an atmosphere that is neutral to dry convection</p>
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<text>The temperature of the atmosphere is constant with height</text>
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<text>Air parcels displaced adiabatically experience no buoyancy force</text>
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<text>Density is constant with height</text>
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Dry vs moist convection
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<p>Dry radiative-convective equilibrium is an unrealistic model for the atmosphere because:</p>
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<text>The resultant temperature profile would be unstable to moist convection.</text>
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<text>Most of the convective energy transport in the atmosphere is latent heat rather than sensible heat.</text>
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<text>Not considering phase changes of water does not allow for the prediction of clouds or water vapor.</text>
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Precipitation Time Scales
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<p>Which of the following timescales is <b>not</b> important in setting the overall timescale of convective showers </p>
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<text>The timescale for water to condense on to atmospheric particles in super-saturated air</text>
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<text>The timescale of formation of rain drops (cm size) from cloud droplets (mm size)</text>
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<text>The timescale for an updraft beginning in the boundary layer to reach the tropopause</text>
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<text>The timescale for rain drops to fall from the upper troposphere to the surface</text>
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<p>An atmosphere stable to dry-convection may be unstable to moist convection because:</p>
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<text>Moist air is heavier than dry air</text>
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<text>There may not be enough cloud condensation nuclei in the atmosphere to support precipitation formation</text>
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<text>A saturated parcel of air cools more slowly when lifted because of the release of latent heat.</text>
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<text>The latent heat flux at the surface may be larger than the sensible heat flux.</text>
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<h2 class="hd hd-2 unit-title">Properties of moist convection</h2>
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Moist Convection
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<p>Which of the following is <b>not</b> a property of moist convection?</p>
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<text>In moist convection, the updrafts cover a much smaller area than the compensating downward motion. </text>
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<text>Moist convection has a profound effect on the distribution of water in the atmosphere.</text>
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<text>Moist convection is responsible for a large part of the vertical energy transport in the troposphere.</text>
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<h2 class="hd hd-2 unit-title">Two-layer radiative-convective model</h2>
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<h2>Two-layer radiative-convective model</h2>
<p>The <a href="https://courses.edx.org/courses/course-v1:MITx+12.340x_2+1T2016/courseware/Resources/Twolayer/" target="_blank">two layer-radiative-convective model</a> discussed in this video is available to use located under the 'Resources' tab, and is reproduced below. You are able to set the long-wave emissivity of each layer as well as the temperature difference required for convective neutrality. Play with this model by changing the parameters to get a feel for how the model behaves. If you make the threshold temperature differences very large, does convection have any impact on the model solution? Why, or why not? <br /> <br /> <br /> <iframe src="/asset-v1:MITx+12.340x+1T2020+type@asset+block/models_Twolayer_RC_2_Layer_simple.html" width="800" height="1000" frameborder="0" scrolling="no">
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<h2 class="hd hd-2 unit-title">Problem Set: Convection</h2>
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<h2>Convective Heat 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 concepts of hydrostatic equilibrium, static stability and buoyancy, and have a feeling for the importance of convection-- particularly moist convection-- in the global climate. This problem set contains three problems, each of which has multiple parts.</p><p>Problems may ask you to choose the correct answer between a list of alternatives, or may ask you to enter an answer in 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 class="hd hd-2 unit-title">Problem 1: Hydrostatic equilibrium</h2>
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<h2>Hydrostatic Equilibrium in a dry atmosphere</h2><p>In this problem, we will explore an example of hydrostatic equilibrium, where we assume the thermal structure of the atmosphere is set by dry convection. This is generally not how the temperature structure of the atmosphere is set in the vertical, but it will provide an example of the application of hydrostatic equilibrium to a non-isothermal atmosphere.</p>
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1a
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Derive an algebraic expression for pressure as a function of height for a hydrostatic, dry-adiabatic atmosphere, using the definition of a dry adiabat:
[mathjax] T(p) = T_0 \left(\frac{p}{p_0}\right)^{R/c_p}, [/mathjax]
and the hydrostatic relation:
[mathjax] \frac{1}{p}\frac{d p}{d z} = -\frac{g}{R T}. [/mathjax]
Terminology notes: Here, [mathjaxinline] p [/mathjaxinline] represents pressure [units: Pa], [mathjaxinline] p_0 [/mathjaxinline] represents surface pressure, [mathjaxinline]T[/mathjaxinline] represents temperature [units: K], [mathjaxinline]T_0[/mathjaxinline] represents surface air temperature ([mathjaxinline]\approx[/mathjaxinline] 290. K), [mathjaxinline]R[/mathjaxinline] is the ideal gas law constant for dry air ([mathjaxinline]\approx[/mathjaxinline]287 J/kg/K), [mathjaxinline]c_p[/mathjaxinline] is the heat capacity of dry air ([mathjaxinline]\approx[/mathjaxinline]1005 J/kg/K), [mathjaxinline]g[/mathjaxinline] is the gravitational constant ([mathjaxinline]\approx[/mathjaxinline]9.81 m/s[mathjaxinline]^2[/mathjaxinline]), and [mathjaxinline]z[/mathjaxinline] is the height above the surface [units: m].
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<text>[mathjaxinline]p(z) = p_0 \left( 1 -\frac{c_p T_0}{g z}\right)^{c_p/R}[/mathjaxinline]</text>
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<text>[mathjaxinline]p(z) = p_0 \left( 1 -\frac{g z}{c_p T_0}\right)^{c_p/R}[/mathjaxinline]</text>
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<text>[mathjaxinline]p(z) = p_0 \exp{\left(-\frac{gz}{R T_0}\right)}[/mathjaxinline]</text>
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<text>[mathjaxinline]p(z) = p_0 \exp{\left(-\frac{gz}{R (T_0-gz/c_p)}\right)}[/mathjaxinline]</text>
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1b
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Given the answer for 1A, does a dry-adiabatic atmosphere have a well-defined maximum height, where the pressure decays to zero?
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<text>No, because exponential decay never decays to exactly zero</text>
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<input type="radio" name="input_0d20c31c8967493db26d7755c125d1c0_2_1" id="input_0d20c31c8967493db26d7755c125d1c0_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="0d20c31c8967493db26d7755c125d1c0_2_1-choice_1-label" for="input_0d20c31c8967493db26d7755c125d1c0_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_0d20c31c8967493db26d7755c125d1c0_2_1">
<text>Yes, because when [mathjaxinline]z[/mathjaxinline] approaches [mathjaxinline]c_p T_0/g[/mathjaxinline], the argument of the exponential approaches negative infinity, and in this limit, pressure goes to zero </text>
</label>
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<input type="radio" name="input_0d20c31c8967493db26d7755c125d1c0_2_1" id="input_0d20c31c8967493db26d7755c125d1c0_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="0d20c31c8967493db26d7755c125d1c0_2_1-choice_2-label" for="input_0d20c31c8967493db26d7755c125d1c0_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_0d20c31c8967493db26d7755c125d1c0_2_1">
<text>Yes, because when [mathjaxinline]z[/mathjaxinline] equals [mathjaxinline]c_p T_0/g[/mathjaxinline], [mathjaxinline]\left(1-\frac{gz}{c_p T_0}\right)^{c_p/R}[/mathjaxinline] equals zero</text>
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1c
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<p>If you think that a dry-adiabatic atmosphere has a well-defined maximum height, enter an approximate value, in <b>kilometers</b>, corresponding to Earth-like parameter values (for gravity, surface temperature and pressure, and gas constants; see part 1A). If you think that there is no such maximum height for a dry-adiabatic atmosphere, enter the distance from the Earth to the Moon.</p>
<p>Enter your answer below as a number of kilometers. <b>Do not include units in your answer.</b></p>
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<div id="display_9ba73ca8d40647fbb9bb5d27bb71a5ee_2_1" class="equation">`{::}`</div>
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<h2 class="hd hd-2 unit-title">Problem 2: Composition and static stability</h2>
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<h2>Composition, Static Stability, and the Lake Nyos disaster</h2><p>Carbon dioxide can be deadly in ways other than its role in climate change. Lake Nyos, in Cameroon, lies above a pocket of magma that slowly leaks carbon dioxide into the water. Eventually, the water becomes supersaturated with carbon dioxide and violently degasses. In 1986, 1700 people were suffocated when Lake Nyos suddenly outgassed several thousand tons of carbon dioxide. In this problem, we discuss why the carbon dioxide leak was so deadly.</p>
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2a
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<p>Using the ideal gas law, calculate the density, in <b>kg/m<sup>3</sup></b>, of dry air at 300. K, and a pressure of 1000. hPa. You may assume, as in Problem 1, that the gas constant for dry air is [mathjaxinline]R[/mathjaxinline]= 287 J/kg/K.</p>
<p>Enter your answer below as a number of kg/m<sup>3</sup>. <b>Do not include units in your answer.</b></p>
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2b
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<p>What is the density of pure carbon dioxide at the same temperature and pressure? You may assume that the universal gas constant is [mathjaxinline]R_0 = 8.314 [/mathjaxinline] J/K/mol, and that the molar mass of carbon dioxide is given by [mathjaxinline]M_{{CO}_2} = 0.0440 [/mathjaxinline] kg/mol.</p>
<p>Enter your answer below as a number of kg/m<sup>3</sup>. <b>Do not include units in your answer.</b></p>
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2c
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Given the densities of air and pure carbon dioxide, which of the following pairs of words best fills in the blanks in this description of the Lake Nyos disaster:
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"Since carbon dioxide is __________ dense than air, a layer of carbon dioxide underneath air at the same temperature is statically __________ to dry convection. This allowed a surface layer with dangerously high concentrations to build up, without mixing with the overlying air, and suffocated many people."
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2d
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<p>Methane, in contrast to carbon dioxide, is less dense than dry air. If methane were emitted in a pulse from a lake, it would be unstable to convection, and most of it would rapidly rise away from the surface. Since methane is less dense than dry air, and carbon dioxide is more dense than dry air, a mixture of methane and carbon dioxide, with mixing fractions N and (1-N), thus has some critical value of N for which the mixture is just the same density as dry air. A mixture with more methane will be less dense than air, and will rise and dissipate; a mixture with less methane will be more dense than air, and will sink, posing a potential suffocation risk. What is the critical value of N, in decimal form?</p>
<p>You may assume that the universal gas constant is [mathjaxinline]R_0 = 8.314 [/mathjaxinline] J/K/mol, and that the molar mass of methane is given by [mathjaxinline]M_{{CH}_4} = 0.0160 [/mathjaxinline] kg/mol.</p>
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<h2 class="hd hd-2 unit-title">Problem 3: Lapse-rates</h2>
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<h2>Radiation, Convection, and Lapse Rates</h2><p>In this problem, we will use some idealized models of the atmosphere to explore why the radiative equilibrium solution is convectively unstable, and how moisture affects the static stability of the atmosphere. You may find the <a href="/courses/course-v1:MITx+12.340x+1T2020/courseware/Resources/Twolayer/">two layer-radiative-convective model</a> discussed in the videos helpful to answer some of these questions. This model is available under the resources tab, and is reproduced below.</p><br/><iframe src="/asset-v1:MITx+12.340x+1T2020+type@asset+block/models_Twolayer_RC_2_Layer_simple.html" width="800" height="800" frameborder="0" scrolling="no"/>
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3a
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<p>For fluids with very low compressibility (e.g. water), a convectively neutral temperature profile is almost isothermal. For the two-layer radiative-convective model described in the final video of this module, this would imply that [mathjaxinline]\Delta T_C = 0[/mathjaxinline]. Find the equilibrium temperature of the surface ([mathjaxinline]T_s[/mathjaxinline]) for this situation. Express your answer in terms of the emission temperature [mathjaxinline]T_e[/mathjaxinline] and the emissivities of the atmospheric layers [mathjaxinline]\epsilon_1 [/mathjaxinline] and [mathjaxinline]\epsilon_2[/mathjaxinline].</p>
<p>Hint: You can use the <a href="/courses/course-v1:MITx+12.340x+1T2020/courseware/Resources/Twolayer">2-layer radiative-convective equilibrium model</a> in the Resources section (and reproduced above) to check your answer.</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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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><span>[mathjaxinline]T_s [/mathjaxinline] =</span>
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</div><p>Question for discussion: does the atmosphere cause a greenhouse effect in this situation?</p>
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3b
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<p>For the 2-layer radiative-convective model described in the final video of this module, calculate the critical temperature difference between the upper and lower layers of the atmosphere, where the convective heat flux vanishes (the two layers will then be in radiative equilibrium). Express your answer in terms of the emission temperature T<sub>e</sub>.</p>
<p>Hint: You can use the <a href="/courses/course-v1:MITx+12.340x+1T2020/courseware/Resources/Twolayer">2-layer radiative-convective equilibrium model</a> in the Resources section (and reproduced above) to check your answer.</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>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><span>[mathjaxinline]\Delta T_C[/mathjaxinline] =</span>
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<div id="display_881bfba4097c4324b7d7ae9366b7266f_2_1" class="equation">`{::}`</div>
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3c
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<p>As the temperature increases, the moist-adiabatic lapse-rate decreases in magnitude. Since convecting atmospheres tend to lie close to convectively neutral, we would expect the lapse-rate of the atmosphere to decrease as the surface is warmed. Does this correspond to a positive or negative feedback on the surface temperature?</p>
<p>Hint: Use the <a href="/courses/course-v1:MITx+12.340x+1T2020/courseware/Resources/Twolayer">2-layer radiative-convective equilibrium model</a> in the Resources section (and reproduced above) to investigate how the surface temperature response to changes in the emissivity or solar constant and how this may be affected by changes to the convective threshold lapse-rate.</p>
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<input type="radio" name="input_c3f57da2f0bc4334adee909378ed39a8_2_1" id="input_c3f57da2f0bc4334adee909378ed39a8_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="c3f57da2f0bc4334adee909378ed39a8_2_1-choice_2-label" for="input_c3f57da2f0bc4334adee909378ed39a8_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_c3f57da2f0bc4334adee909378ed39a8_2_1"> No feedback on [mathjaxinline]T_s[/mathjaxinline]
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