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<h2 class="hd hd-2 unit-title">Introduction to atmospheric circulation</h2>
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<h2>The general circulation of the atmosphere</h2>
<p>While the last two sections have covered the vertical structure of Earth's atmosphere and the physical mechanisms that govern it, in this section we will study the horizontal structure and motion of the atmosphere. The module will provide an overview of the most important features of the three-dimensional circulation of the atmosphere and its effects on climate.</p>
<p><i>Instructor: Kerry Emanuel</i></p>
<h2>Module outline</h2>
<p>In this module, we will describe the <i>general circulation</i> of the atmosphere and its underlying physical theory. Specifically, the module covers the following topics:</p>
<ul>
<li>The seasonal variability of temperature and wind patterns at the Earth's surface, and their relationship with incoming solar radiation</li>
<li>Mathematical concepts of <b>global energy balance</b>, and the roles played by atmospheric and oceanic circulations in its maintenance</li>
<li>Description of an exact solution for the motions of the atmosphere of an idealized planet</li>
<li>Derivation of the equations of <b>geostrophic balance</b> and <b>thermal wind</b></li>
<li>Description of the <b>Hadley circulation</b> and its effects on climate</li>
<li>Other important tropical circulations, including the <b>Walker circulation</b> and <b>monsoons</b></li>
<li>The concept of <b>baroclinic instability</b></li>
<li>Characterization of baroclinic eddies and their effects on climate</li>
<li>Known and unknown effects of global warming on the general circulation</li>
</ul>
<h2>Background reading</h2>
<p>In this section, we will derive some important equations of atmospheric motion using the meteorological coordinate system in which <i>x</i> and <i>y</i> correspond to longitude and latitude, respectively.</p>
<h2>Supplemental reading</h2>
<p>The general circulation of the atmosphere is a very broad topic about which entire semesters' worth of material could be taught. Since we are only interested in characterizing the atmospheric circulation for the purposes of understanding climate change, there are several parts of this week's material that will not go into great detail. Students who wish to learn more about these topics are encouraged to read the following resources:</p>
<ul>
<li>The Walker circulation and El Niño: <a href="https://www.climate.gov/enso" target="_blank">NOAA's El Niño page</a></li>
<li>Hadley cell: See notes from <a href="http://ocw.mit.edu/courses/earth-atmospheric-and-planetary-sciences/12-810-dynamics-of-the-atmosphere-spring-2008/lecture-notes/chapter_7.pdf" target="_blank">MIT 12.803</a> and <a href="http://paoc.mit.edu/labguide/circ_exp_slow.html" target="_blank">Weather in a Tank lab experiment</a>.</li>
<li>Baroclinic instability: See notes from <a href="http://ocw.mit.edu/courses/earth-atmospheric-and-planetary-sciences/12-803-quasi-balanced-circulations-in-oceans-and-atmospheres-fall-2009/lecture-notes/MIT12_803F09_lec19.pdf" target="_blank">MIT 12.803</a> and <a href="http://paoc.mit.edu/labguide/circ_exp_fast.html" target="_blank">Weather in a Tank lab experiment</a>.</li>
<li>Effects of climate change on the general circulation: Read sections <a href="http://www.climatechange2013.org/images/report/WG1AR5_Chapter11_FINAL.pdf" target="_blank">11.3.2.4</a> and <a href="http://www.climatechange2013.org/images/report/WG1AR5_Chapter12_FINAL.pdf" target="_blank">12.4.4</a> of the <a href="http://www.ipcc.ch/report/ar5/wg1/" target="_blank">IPCC AR5 report</a>.</li>
</ul>
<h2>Lecture Slides</h2>
<p>The slides for this weeks lecture can be downloaded in two parts, <a href="/assets/courseware/v1/26c7127fc0b561f99043af7b1f994b89/asset-v1:MITx+12.340x+1T2020+type@asset+block/atmospheric_circulation_a.pdf" target="[object Object]">here</a> and <a href="/assets/courseware/v1/3b307c0c031ea7a173a8ffdeba3a7b71/asset-v1:MITx+12.340x+1T2020+type@asset+block/atmospheric_circulation_b.pdf" target="[object Object]">here</a>.</p>
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<h2 class="hd hd-2 unit-title">Phenomenology of the atmospheric circulation</h2>
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Energy balance and transport
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<p>If the ocean and atmosphere stopped transporting energy, what would be the likely effect on the Earth's temperatures? </p>
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<text>The poles would cool and the tropics would warm.</text>
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<text>The poles would warm and the tropics would cool.</text>
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<text>The globe would warm roughly uniformly.</text>
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<text>The globe would cool roughly uniformly.</text>
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Hydrostatic equilibrium review
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<p>The hydrostatic relation is given by,
[mathjax]
\frac{\partial{} p}{ \partial z} = -\rho g,
[/mathjax]
where [mathjaxinline]p[/mathjaxinline] is the pressure and [mathjaxinline]z[/mathjaxinline] is height. The ideal gas law may be written,
[mathjax]
p = \rho R T,
[/mathjax]
where [mathjaxinline]\rho[/mathjaxinline] is the density, [mathjaxinline]T[/mathjaxinline] is the temperature, and [mathjaxinline]R[/mathjaxinline] is the gas constant for air. For an atmosphere in which temperature does not vary in height, and surface pressure [mathjaxinline]p_0[/mathjaxinline], what is the correct form of the distribution of pressure with height?
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<text>[mathjaxinline]p = p_0\exp(-\frac{RTz}{g})[/mathjaxinline]</text>
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<text>[mathjaxinline]p = p_0(\frac{gz}{RT})^{R/c_p}[/mathjaxinline]</text>
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Coriolis force
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<p>The equations of motion for a parcel of fluid on a rotating sphere may be approximately written
[mathjax]
\frac{du}{dt} = fv - \alpha\frac{\partial p}{\partial x},
[/mathjax]
[mathjax]
\frac{dv}{dt} = -fu - \alpha\frac{\partial p}{\partial y},
[/mathjax]
where [mathjaxinline]f = 2\Omega\sin\theta[/mathjaxinline] is the Coriolis parameter. What is the direction of the Coriolis force on a parcel of air in the Northern Hemisphere moving at 3 m/s toward the Northeast?
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<text>towards the Northeast.</text>
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<text>towards the Southeast.</text>
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<text>towards the East.</text>
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Geostrophically balanced wind
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<p>The above figure shows a hypothetical pressure distribution in the mid-troposphere in the <it>Northern Hemisphere</it>. The high pressure center is marked "H", while the low pressure center is marked "L". In what direction would the geostrophically balanced wind be at point "X" in the above figure? In meteorological notation, an "Easterly" wind blows from the East, while an "Eastward" wind blows toward the East. You may assume that up is North.</p>
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Thermal wind
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<p>Due to the large heat-capacity of an ocean surface, land areas typically respond much more quickly than oceanic areas to diurnal or seasonal variations in solar heating. over Southern Asia, the warming of the continent in the Spring and summer is so rapid that the temperature gradient in the upper atmosphere typically reverses, so that the warmest air is over Northern India, and the temperature <it>decreases</it> as one moves toward the equator. </p>
<p>Assuming the winds near the surface are weak, what direction are the upper atmospheric winds in this region?</p>
<p>Remember, in meteorological notation An "Easterly" wind blows from the East, while an "Eastward" wind blows toward the East.</p>
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Unstable equilibria
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<p>Which of the following is an example of an unstable equilibrium?</p>
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<text>A cup of water heated in the microwave to 120 degrees Celsius</text>
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<text>A can of gasoline sitting next to a box of matches</text>
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Eddy heat transport
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<p>The above figure shows the distribution of meridional velocity (v) and atmospheric energy content (E) as a function of longitude at a particular latitude on Earth. In what direction is the eddy heat transport implied by the longitudinal distributions of meridional velocity and energy depicted above?</p>
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<text>Northward</text>
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<text>There is no implied heat transport</text>
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Effects of a changing circulation
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<p>Which of the following would be most likely to occur as a result of an increase in the width of the Hadley cell which many models predict will occur as a result of future climate change?</p>
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<text>Stronger tropical cyclones.</text>
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<text>A poleward expansion of the subtropical dry-zones in which many of the world's deserts exist.</text>
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<text>More storminess at mid-latitudes.</text>
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<text>More rain in the Inter-tropical convergence zone.</text>
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<h2 class="hd hd-2 unit-title">Problem Set: Atmospheric circulation</h2>
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<h2>Atmospheric circulation</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 characteristics of the circulation, and how they differ between the tropics and extra-tropics. You should understand some of the reasons why a simple radiative-convective equilibrium climate cannot exist on the Earth, and hence why we have weather. 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>
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<h2 class="hd hd-2 unit-title">Problem 1: Hadley Cell</h2>
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<h2>Problem 1: Hadley Cell</h2><p>In this problem we look in more detail at the Hadley circulation. We consider the hypothetical radiative-convective equilibrium solution described in the lecture videos, show that it cannot exist, and outline an angular momentum conserving conserving solution that better describes the observed circulation.</p>
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1a
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<p>The angular momentum [mathjaxinline]M[/mathjaxinline] of a parcel on the surface of the Earth at latitude [mathjaxinline]\phi[/mathjaxinline] may be expressed,
[mathjax]
M = a\cos\phi\left(u+\Omega a\cos\phi\right).
[/mathjax]
Here [mathjaxinline]a[/mathjaxinline] is the radius of the Earth, [mathjaxinline]\Omega[/mathjaxinline] is the angular velocity of the Earth's rotation, and [mathjaxinline]u[/mathjaxinline] is the velocity of the parcel in the zonal direction. </p>
<p>The angular momentum of a parcel at rest (i.e. when [mathjaxinline]u=0[/mathjaxinline]) increases as one moves toward the equator. If the circulation of the atmosphere is zonally symmetric, angular momentum is conserved. Under these conditions the maximum angular momentum achievable by any parcel is that of a stationary parcel at rest at the equator. The maximum velocity of such a parcel at any given latitude is the velocity it achieves if it conserves its angular momentum starting from rest at the equator. Write an expression for this maximum velocity, which we will denote [mathjaxinline]u_M[/mathjaxinline], as a function of the latitude [mathjaxinline]\phi[/mathjaxinline]. </p>
<p>Type "Omega" to input [mathjaxinline]\Omega[/mathjaxinline], and "phi" to input [mathjaxinline]\phi[/mathjaxinline], and explicitly indicate any mathematical operations with the appropriate symbol (e.g. '+', '-', '*', '/', '^'). Trigonometric functions of [mathjaxinline]\phi[/mathjaxinline] can be entered as "sin(phi)", "cos(phi)", "tan(phi)", "csc(phi)", "sec(phi)", "cot(phi)", and if you wish to put in a power of a trig function, you should enter it as, e.g., "cos(phi)^2", not "cos^2(phi)" (the latter will look right inline, but will throw a syntax error).</p>
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<span>[mathjaxinline]u_M [/mathjaxinline] =</span>
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<div id="display_d3841be0b7404990a946679fd2009189_2_1" class="equation">`{::}`</div>
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1b
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<p>We now consider the radiative-convective equilibrium solution described in the lecture videos. In this solution, the atmospheric winds are in <i>thermal wind balance</i> with the temperature gradient. But is there enough angular momentum on the Earth to sustain such winds?</p>
<p>Suppose the radiative-convective equilibrium solution for the atmospheric temperature is isothermal in the vertical (not realistic!), and has a latitudinal dependence of the form,
[mathjax]
\ln\left(\frac{T}{T_0}\right) = 1 - \frac{1}{3}\Delta_H\left(3\sin^2(\phi) - 1\right)
[/mathjax]
where [mathjaxinline]T[/mathjaxinline] is the temperature, with global mean [mathjaxinline]T_0[/mathjaxinline], and [mathjaxinline]\Delta_H[/mathjaxinline] is the radiative-convective equilibrium temperature difference between equator and pole.</p>
<p>The thermal wind equation may be written,
[mathjax]
2\Omega\sin\phi\frac{\partial u}{\partial z} = -\frac{g}{a}\frac{\partial \ln(T)}{\partial \phi}.
[/mathjax]
Using the expression for [mathjaxinline]\ln(T)[/mathjaxinline], find an expression for the zonal velocity at the tropopause, where [mathjaxinline]z=H[/mathjaxinline], that is in thermal wind balance with the radiative-convective equilibrium solution for the temperature. We denote this velocity [mathjaxinline]u_T[/mathjaxinline]. You may assume that the winds at the surface are negligible.</p>
<p>Type "Omega" to input [mathjaxinline]\Omega[/mathjaxinline], "Delta_H" to input [mathjaxinline]\Delta_H[/mathjaxinline] and "phi" to input [mathjaxinline]\phi[/mathjaxinline]. To indicate the natural logarithm of a variable [mathjaxinline]x[/mathjaxinline], type "ln(x)". Explicitly indicate any mathematical operations with the appropriate symbol (e.g. '+', '-', '*', '/', '^').</p>
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<span>[mathjaxinline]u_T [/mathjaxinline] =</span>
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<div id="display_04dc0490d9564504bf0a473e0619caf3_2_1" class="equation">`{::}`</div>
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1c
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<p>The thermal wind equation is not valid at the equator, as at this point the Coriolis force itself approaches zero, and geostrophic balance no longer holds. Nevertheless, we can ask the question: What happens in the limit as one approaches the equator?</p>
<p>How does [mathjaxinline]u_T[/mathjaxinline] compare to [mathjaxinline]u_M[/mathjaxinline] as one approaches the equator? Please assume [mathjaxinline]\Omega[/mathjaxinline] and [mathjaxinline]\Delta_H[/mathjaxinline] are positive-definite parameters.</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_5d7c7d3e6f2f4a43886f077452e34c8f_2_1">
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<div class="field">
<input type="radio" name="input_5d7c7d3e6f2f4a43886f077452e34c8f_2_1" id="input_5d7c7d3e6f2f4a43886f077452e34c8f_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="5d7c7d3e6f2f4a43886f077452e34c8f_2_1-choice_0-label" for="input_5d7c7d3e6f2f4a43886f077452e34c8f_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_5d7c7d3e6f2f4a43886f077452e34c8f_2_1"> [mathjaxinline]u_T \lt u_M[/mathjaxinline]
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<input type="radio" name="input_5d7c7d3e6f2f4a43886f077452e34c8f_2_1" id="input_5d7c7d3e6f2f4a43886f077452e34c8f_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="5d7c7d3e6f2f4a43886f077452e34c8f_2_1-choice_1-label" for="input_5d7c7d3e6f2f4a43886f077452e34c8f_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_5d7c7d3e6f2f4a43886f077452e34c8f_2_1"> [mathjaxinline]u_T = u_M[/mathjaxinline]
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<input type="radio" name="input_5d7c7d3e6f2f4a43886f077452e34c8f_2_1" id="input_5d7c7d3e6f2f4a43886f077452e34c8f_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="5d7c7d3e6f2f4a43886f077452e34c8f_2_1-choice_2-label" for="input_5d7c7d3e6f2f4a43886f077452e34c8f_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_5d7c7d3e6f2f4a43886f077452e34c8f_2_1"> [mathjaxinline]u_T \gt u_M[/mathjaxinline]
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<input type="radio" name="input_5d7c7d3e6f2f4a43886f077452e34c8f_2_1" id="input_5d7c7d3e6f2f4a43886f077452e34c8f_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="5d7c7d3e6f2f4a43886f077452e34c8f_2_1-choice_3-label" for="input_5d7c7d3e6f2f4a43886f077452e34c8f_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_5d7c7d3e6f2f4a43886f077452e34c8f_2_1"> depends on the ratio of the parameters [mathjaxinline]H[/mathjaxinline] and [mathjaxinline]a[/mathjaxinline]
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<p>Think about why this indicates that the radiative-convective equilibrium solution is unattainable given the amount of angular momentum available.</p>
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1d
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<p>The previous results indicate that the radiative-convective equilibrium solution cannot be valid near the equator. In this region, one may construct a solution to the equations in which a Hadley-cell like circulation exists and conserves angular momentum in its upper branch. The upper level zonal velocity according to this solution is thus the velocity [mathjaxinline]u_M[/mathjaxinline] described in part A. What is the limit of the angular momentum conserving velocity as one approaches the pole?</p>
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<input type="radio" name="input_4d945f866194482ead596b691c5cb0e1_2_1" id="input_4d945f866194482ead596b691c5cb0e1_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="4d945f866194482ead596b691c5cb0e1_2_1-choice_1-label" for="input_4d945f866194482ead596b691c5cb0e1_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_4d945f866194482ead596b691c5cb0e1_2_1"> [mathjaxinline]\infty[/mathjaxinline] for any [mathjaxinline]\Omega \gt 0[/mathjaxinline]
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<input type="radio" name="input_4d945f866194482ead596b691c5cb0e1_2_1" id="input_4d945f866194482ead596b691c5cb0e1_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="4d945f866194482ead596b691c5cb0e1_2_1-choice_2-label" for="input_4d945f866194482ead596b691c5cb0e1_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_4d945f866194482ead596b691c5cb0e1_2_1"> [mathjaxinline]\Omega a[/mathjaxinline]
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<input type="radio" name="input_4d945f866194482ead596b691c5cb0e1_2_1" id="input_4d945f866194482ead596b691c5cb0e1_2_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="4d945f866194482ead596b691c5cb0e1_2_1-choice_3-label" for="input_4d945f866194482ead596b691c5cb0e1_2_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_4d945f866194482ead596b691c5cb0e1_2_1"> [mathjaxinline]\frac{\pi}{2} \Omega a[/mathjaxinline]
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<p>Discussion questions: Could a hemispheric Hadley cell that brings air from the equator to polar regions exist? What would be the value of the zonal wind velocity for a parcel brought from the equator to 60 degrees latitude conserving its angular momentum? Is this a realistic value for Earth's atmosphere?</p>
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<h2 class="hd hd-2 unit-title">Problem 2: Geostrophic balance and thermal wind</h2>
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<h2>Problem 2: Geostrophic balance and thermal wind</h2><p>In this problem we consider geostrophic and thermal wind balance, and their application to the analysis of weather maps.</p>
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2a
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<p>The weather map above shows the surface pressure (black lines) over North America, as analyzed for a numerical weather prediction model on February 3, 2014. Based on geostrophic balance, what is the direction of the wind near the surface at the location of the red "X" on the figure?</p>
<p>Remember the meteorological conventions for discussing the directions of the wind.</p>
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2b
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<p>Again using geostrophic balance, estimate the <b>magnitude</b> (absolute value) of the geostrophic wind near the surface at the point marked with a red "X" in the figure from part A.</p>
<p>The location of the "X" is close to the border of the Canadian state of Manitoba and the Northwest territories. This is at 60. degrees North. You may assume the density of the air near the surface is 1.2 kg/m<sup>3</sup>, and that the surface pressure at the location of the "X" varies 12 hPa over a distance of 500. km.</p>
<p>Remember to use SI units for all variables, and that the angular rotation velocity of the Earth is [mathjaxinline]7.29\times10^{-5}[/mathjaxinline] s[mathjaxinline]^{-1}[/mathjaxinline].</p>
</span>
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2c
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<p>The blue and red contours plotted on the figure above represent the "thickness" of the 1000 hPa - 500 hPa pressure interval-- that is, the vertical distance [mathjaxinline]\Delta z[/mathjaxinline] between the 1000 hPa level and the 500 hPa level. On the figure it is expressed in decameters, where 1 decameter = 10 meters.</p>
<p>Use the hydrostatic relation,
[mathjax]
\frac{d p}{d z} = -\rho g
[/mathjax]
and the ideal gas law,
[mathjax]
p = \rho R T
[/mathjax]
to express this height difference in terms of the temperature, [mathjaxinline]T[/mathjaxinline], gas constant [mathjaxinline]R[/mathjaxinline] and gravity [mathjaxinline]g[/mathjaxinline]. You may assume the temperature to be uniform in the vertical.</p>
<p>Explicitly indicate any mathematical operations with the appropriate symbol (e.g. '+', '-', '*', '/', '^'), and indicate the natural logarithm of a quantity [mathjaxinline] x [/mathjaxinline] with "ln(x)".</p>
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<span>[mathjaxinline]\Delta z [/mathjaxinline] =</span>
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<h2 class="hd hd-2 unit-title">Problem 3: Heat transport by eddies</h2>
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<h2>Problem 3: Heat transport by eddies</h2><p>In this problem we consider the transport of energy in the mid-latitudes. We will consider the importance of eddies in this region of the atmosphere, and discuss the concept of eddy transports.</p>
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3a
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<p>The figure below shows the zonal- and annual-mean mass <i>streamfunction</i> for the atmosphere as estimated by what is known as a 'reanalysis'. A 'reanalysis' is constructed thorough the use of a general circulation model in combination with observations to produce a best estimate of the atmospheric circulation. The streamfunction is defined so that the mean circulation follows its contours, and the difference in value between adjacent contours indicates the mass flux between them. On this figure, positive values of the streamfunction correspond to circulation in the clockwise sense.</p>
<p>Mark the position of the center of the Northern Hemisphere Hadley circulation on the figure. This is the position at which there is no mean meridional or vertical motion within the Hadley Cell.</p>
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3b
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<p>The secondary cells shown in the figure exist in the mid-latitudes and are called Ferrel cells. Notice that these circulations point in the opposite direction to the Hadley cells.</p>
<p>Use the figure above to estimate the maximum value of the annual-mean mass flux, in kg/s, of the Southern Hemisphere Ferrel cell. This is the amount of mass that the Ferrel cell moves meridionally through the mid-latitudes in each direction.</p>
<p>Enter your answer below as a number of kg/s. <b>Do not include units in your answer.</b></p>
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3c
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<p>The energy transport by mean overturning cells such as the Hadley and Ferrel cells can be thought of as their total mass flux multiplied by a characteristic difference in the energy content of air moving equatorward compared to air moving poleward,
[mathjax]
F_E = \overline{M}\cdot\overline{\Delta E},
[/mathjax]
where [mathjaxinline]\overline{M}[/mathjaxinline] is the mass flux of the Ferrel cell, and [mathjaxinline]\overline{\Delta E}[/mathjaxinline] is the zonal mean energy difference between the upper and lower branches. Here the energy content of the atmosphere may be written,
[mathjax]
E = c_p T + gz + L_v q,
[/mathjax]
where [mathjaxinline]c_p[/mathjaxinline] is the specific heat capacity of air at constant pressure, [mathjaxinline]T[/mathjaxinline] is the temperature, [mathjaxinline]g[/mathjaxinline] is the gravitational acceleration, [mathjaxinline]z[/mathjaxinline] is height in the vertical, [mathjaxinline]L_v[/mathjaxinline] is the latent heat of vaporization and [mathjaxinline]q[/mathjaxinline] is specific humidity. We have neglected the small contribution of kinetic energy to the total energy in this calculation.
</p>
<p>Assume the temperature in the upper branch of the Ferrel cell is 30.0 K colder than the lower branch, while the mean altitude of the upper branch is 8.00 km and the mean altitude of the lower branch is 1.00 km. Finally, assume the lower branch has a mean specific humidity of 0.010 kg/kg, while the upper branch has negligible water vapor. Based on your answer to part B, what is the <b>absolute value</b> of the mean energy transport at mid-latitudes by the Ferrel Cell?</p>
<p>You may assume:
[mathjax]
c_p = 1004 \mbox{ J/kg/K} \\
g = 9.81 \mbox{ m/s}^2 \\
L_v = 2.50 \times 10^6 \mbox{ J/kg}
[/mathjax]
</p>
<p>Express your answer in Petawatts (1 PW = [mathjaxinline]10^{15}[/mathjaxinline] W).</p>
<p>Enter your answer below as a number of PW. <b>Do not include units in your answer.</b></p>
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3d
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<p>In which direction is the energy transport by the Ferrel cell?</p>
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3e
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<p>The Ferrel cell represents energy transport by the zonal-mean circulation, but eddies - motions that vary in longitude - can also transport energy.</p>
<p>Consider a flow in which the meridional velocity [mathjaxinline]v[/mathjaxinline] has a sinuisoidal dependence on longitude [mathjaxinline]\theta[/mathjaxinline],
[mathjax]
v = v_0\sin(5\theta).
[/mathjax]
Here, [mathjaxinline]v_0[/mathjaxinline] is a typical surface wind speed at mid-latitudes which we set to 10.0 m/s. Now, suppose the temperature [mathjaxinline]T[/mathjaxinline] also had a sinusoidal dependence on longitude,
[mathjax]
T = \overline{T} + \Delta T\sin(5\theta),
[/mathjax]
where we set [mathjaxinline]\Delta T[/mathjaxinline] to a typical horizontal temperature fluctuation at mid-latitudes of 10.0 degrees Celsius.
</p>
<p>The energy flux by such a circulation may be estimated by the integral,
[mathjax]
F_E^{eddy}=a H\int_0^{2\pi} \rho v\Delta E d\theta,
[/mathjax]
where [mathjaxinline]a[/mathjaxinline] is the radius of the Earth (6368 km), [mathjaxinline]\rho[/mathjaxinline] is the density of air (which we assume to be 1.00 kg/m[mathjaxinline]^3[/mathjaxinline]), and [mathjaxinline]H[/mathjaxinline] is the depth of the circulation, which we take as 5.00 km. The quantity [mathjaxinline]\Delta E[/mathjaxinline] is the deviation of the energy content of the atmosphere from the zonal mean. Estimate the heat transport by this circulation assuming the moisture content of the atmosphere does not vary with longitude.</p>
<p>Express your answer in Petawatts (1 PW = [mathjaxinline]10^{15}[/mathjaxinline] W).</p>
<p>Enter your answer below as a number of PW. <b>Do not include units in your answer.</b></p>
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3f
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<div><p>The value found in the above problem is an upper bound, as we assumed the temperature and velocity were completely in phase. Calculate the energy transport in the case where the temperature is not in phase with the meridional velocity, and instead has a cosine dependence on longitude,
[mathjax]
T = \overline{T} + \Delta T \cos(5\theta)
[/mathjax]
</p><p>Express your answer in Petawatts (1 PW = [mathjaxinline]10^{15}[/mathjaxinline] W).</p><p>Enter your answer below as a number of PW. <b>Do not include units in your answer.</b></p><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_aa150f3adf4f42ba9a767aff6e8c3d2a_2_1" class="text-input-dynamath capa_inputtype textline">
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