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<h2 class="hd hd-2 unit-title">Forcings and feedbacks in the climate system</h2>
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<h2>Welcome to Part III 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 explored the specifics of the physics and chemistry of individual components of the climate system. Now, in the final part of the course, we will synthesize all of this information with the goal of understanding the climate system as a whole, and the ways in which it is modeled and analyzed by scientists.</p>
<h2>The general circulation of the atmosphere</h2>
<p>To this point in the course, we have studied individual components of the climate system, and the physical and chemical ways in which they affect the Earth's planetary climate. In this section, we will consider the interactions of some of these individual components, and the ways in which they provide <b>forcings</b> and <b>feedbacks</b> to the climate system.</p>
<p><i>Instructor: Kerry Emanuel</i></p>
<h2>Module outline</h2>
<p>In this module, we will address theoretical bases and observational evidence of several of Earth's climate forcings and feedbacks. Specifically, the module covers the following topics:</p>
<ul>
<li>An overview of the concept of climate forcings and feedbacks</li>
<li>A mathematical treatment of forcings, feedbacks, and climate sensitivity</li>
<li>Orbital forcing</li>
<li>Solar forcing</li>
<li>Greenhouse gas forcings</li>
<li>Aerosol forcings</li>
<li>Change over time in radiative forcings, and a breakdown of the components of total anthropogenic forcing, and their uncertainties</li>
<li>Overview of some known climate feedbacks</li>
<li>Feedbacks as observed numerical climate models</li>
</ul>
<h2>Supplemental reading</h2>
<p>A very detailed examination of the radiative forcings present in the Earth's climate system can be found in <a href="http://www.climatechange2013.org/images/report/WG1AR5_Chapter08_FINAL.pdf" target="_blank">Chapter 8</a> of the <a href="http://www.ipcc.ch/report/ar5/wg1/" target="_blank">IPCC AR5 report</a>. See <a href="http://www.climatechange2013.org/images/report/WG1AR5_Chapter09_FINAL.pdf" target="_blank">Section 9.7</a> of that report for additional discussion of climate sensitivity and feedbacks.</p>
<h2>Exit survey</h2>
<p>After you have completed this week's material, please take the <b>important exit survey</b>, which is accessible in the last section of Part III: The global climate system and climate modeling.</p>
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<h2 class="hd hd-2 unit-title">Overview of Climate Forcings and Feedbacks</h2>
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The Planck Feedback
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<p>The increase in emitted longwave radiation by Earth with temperature, due to the dependence of blackbody emission on temperature, is a key stabilizing mechanism for the climate. We refer to the derivative of [mathjaxinline] \sigma T_e^4 [/mathjaxinline] with respect to [mathjaxinline] T_e [/mathjaxinline] as the "Planck Feedback." For Earth, what is the magnitude of the Planck feedback, in W/(m<sup>2</sup>&#8226;K)?</p>
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Ice Age forcing
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<p>Why is it useful to look at variations in summer insolation at 65<sup>o</sup>N due to orbital variations as a forcing that might drive ice ages?</p>
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<text>Low summer insolation implies higher winter insolation, which helps melt ice</text>
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<text>High summer insolation effectively melts ice, leading to deglaciation</text>
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<text>A rapid change from high to low summer insolation is associated with a rapid change in ice volume</text>
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The Solar Constant and Radiative Forcing
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<p>If the Earth's albedo is 0.3, how large of a radiative forcing would occur for a 1 W/m<sup>2</sup> increase in the solar constant?</p>
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CO2 Radiative Forcing at the LGM
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<p>The radiative forcing by carbon dioxide is approximately logarithmic in the ratio of final to initial concentrations. Specifically,
[mathjax]
RF = 5.37 \ln{(C/C_0)},
[/mathjax]
where the radiative forcing is expressed in W/m<sup>2</sup>, [mathjaxinline]C[/mathjaxinline] is the final concentration, and [mathjaxinline]C_0[/mathjaxinline] is the is initial carbon dioxide concentration. What was the radiative forcing of carbon dioxide at the last glacial maximum, with [mathjaxinline]C=190[/mathjaxinline]ppm, relative to pre-industrial carbon dioxide concentrations?
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Black carbon anthropogenic forcing
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<p>Why does black carbon have a different sign of anthropogenic radiative forcing from that of other aerosols?</p>
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<text>Black carbon absorbs sunlight more strongly than it scatters it to space, so increases in anthropogenic emissions of black carbon have decreased Earth's albedo</text>
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<text>Like other aerosols, black carbon scatters sunlight more strongly than it absorbs, but decreases in black carbon emissions over the historical period have decreased Earth's albedo</text>
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<text>Unlike other aerosols, black carbon causes clouds to darken and reflect less sunlight</text>
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<text>The vertical distribution of black carbon in the atmosphere is highly concentrated towards the ground, and the additional scattered light is more likely to be absorbed by the surface than the direct solar beam</text>
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<h3 class="hd hd-3 problem-header" id="Uncertainty_in_anthropogenic_forcing-problem-title" aria-describedby="block-v1:MITx+12.340x+1T2020+type@problem+block@Uncertainty_in_anthropogenic_forcing-problem-progress" tabindex="-1">
Uncertainty in anthropogenic forcing
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<p>Which forcing agent contributes the largest uncertainty to the estimate of total anthropogenic radiative forcing?</p>
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<text>CO<sub>2</sub></text>
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<text>CH<sub>4</sub> and N<sub>2</sub>O combined</text>
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<text>changes in surface albedo</text>
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<text>aerosols</text>
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<text>O<sub>3</sub> (stratospheric and tropospheric)</text>
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<text>halocarbons</text>
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Stabilizing Feedbacks
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<p>Which of the following is the most strongly negative feedback in the CMIP 3 and CMIP 5 models?</p>
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<text>water vapor</text>
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<text>clouds</text>
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<text>surface albedo</text>
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<h2 class="hd hd-2 unit-title">Problem Set: Forcings and feedbacks</h2>
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<h2>Forcings and feedbacks</h2><p>This problem set is aimed at elaborating on the discussion of radiative forcing in this week's lectures. This problem set only contains one multi-part problem. This will be the last problem set for the course, and it is due only a few days before the start of the final, so we suggest that you take this relatively light week to review content across the other weeks of the course in preparation for the final.</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: Aerosol Forcing Revisited</h2>
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<p>In this problem we will revisit the problem of aerosol forcing of the climate system, which was originally discussed in Problem 1 of Problem Set 7, on Clouds, Aerosols, and climate.</p><p>In that problem, we noted that we can think of sulfate aerosols as having a huge scattering cross-section, of a few thousand square meters per kilogram of aerosol! We'll start by comparing this to the case of cloud droplets, then discuss why sulfate aerosols are such efficient scatterers, and then we'll go on to consider how strongly the volcanic input of aerosols can force the climate system.</p>
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1A
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<span><p>Why can such a small amount of aerosol (or cloud water) have such a huge effect on the climate system? Consider expressions for the mass and surface area of aerosol particles in an atmospheric column, per unit ground area:</p>
[mathjax]
M = \frac{4}{3} \rho \pi r^3 N \\
A = 4 \pi r^2 N.
[/mathjax]
<p>Here, we apply the following definitions: [mathjaxinline]M[/mathjaxinline] is the mass of aerosol per unit ground area in the atmospheric column, in kg/m<sup>2</sup>, [mathjaxinline]\rho[/mathjaxinline] is the density of the individual aerosol particles, in kg/m<sup>3</sup>, [mathjaxinline]r[/mathjaxinline] is the radius of the individual aerosol particles in m, [mathjaxinline]N[/mathjaxinline] is the number density of aerosol particles per unit ground area 1/m<sup>2</sup>, and [mathjaxinline]A[/mathjaxinline] is the surface area of aerosol particles per unit ground area in an atmospheric column, which is dimensionless (it has units of m<sup>2</sup>/m<sup>2</sup>, which cancel).</p>
<p>Use the above to express [mathjaxinline]A[/mathjaxinline] in terms of [mathjaxinline]M[/mathjaxinline], but not [mathjaxinline]N[/mathjaxinline].</p>
<p>Type your answer in the box below. Enter the mass per unit area [mathjaxinline]M[/mathjaxinline] as "M", the density [mathjaxinline]\rho[/mathjaxinline] "rho", and the aerosol radius [mathjaxinline]r[/mathjaxinline] "r". Explicitly indicate any mathematical operations with the appropriate symbol (e.g. '+', '-', '*', '/', '^').</p>
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<span>[mathjaxinline] A [/mathjaxinline]=</span>
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1B
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<p>Based on the answer for part A (feel free to check the solution before moving on), how much surface area per unit ground area would the cloud droplets in a column have, given an average of 40.0 g/m<sup>2</sup> of cloud water, and an average cloud droplet radius of 10.0 microns? Be careful with the units for density of water.</p>
<p>Enter your answer below (do not include units in your answer).</p>
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1C
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<p>Reflection of light by a mixture of substances, like droplets of water suspended in air, or air bubbles trapped in ice, occurs primarily at the interfaces between the two phases, due to the discontinuity of the index of refraction between different substances. More surface area between the two phases means that light is more readily scattered by the mixture. A glass full of crushed ice looks white, but you can see through a glass full of large ice cubes.</p>
<p>If the same amount of cloud water from problem 1B were instead transferred into raindrops (you can and should look up the average size of raindrops), how would the amount of light scattered by raindrops compare to the amount of light scattered by cloud droplets?</p>
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1D
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<p>As discussed somewhat in a previous problem set, sulfate aerosols from large volcanic eruptions can have a large impact on the climate in part because they are long-lived -- they make it into the stratosphere where they are not rained out, and take a long time to disappear. But why is such a small mass of aerosol so effective at scattering sunlight? The answer for this is that volcanic aerosols are small, on the order of 1 micron in radius, so they have a huge amount of surface area per unit mass!</p>
<p>In 1991, Mt. Pinatubo in the Philippines erupted, and it is estimated that about 30. teragrams (1 teragram=10<sup>12</sup> grams) of sulfate aerosol wound up in the stratosphere as a consequence. Assuming that the resulting aerosols had a density of 1.63 g/cm<sup>3</sup>, that the aerosol particles had a radius of about 0.50 microns, and that the area of the Earth is 510 million km<sup>2</sup>, what would have been the surface area of sulfate aerosol per unit ground area, averaged over the whole Earth?</p>
<p>Enter your answer below (do not include units).</p>
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1E
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<span><p>Now, we want to come up with an estimate for how much this aerosol loading affected the solar radiation absorbed by the Earth. The total surface area of aerosols is not actually the best metric to use for this; it is better to use the one-sided shadow area of the droplets to calculate how much light is scattered: [mathjaxinline]\pi r^2[/mathjaxinline] rather than [mathjaxinline]4 \pi r^2[/mathjaxinline]. This allows us to appoximate the aerosol optical depth, [mathjaxinline]\tau[/mathjaxinline], as:</p>
[mathjax]
\tau = \frac{3 M}{4 \rho r}
[/mathjax]
<p>Along with the answer to D, this gives an estimate for the aerosol optical depth from the Pinatubo eruption. For small optical thicknesses (much less than 1), we can approximate the direct radiative forcing from aerosols as:
[mathjax]
RF_{Aer} = -b\tau\frac{S_0(1-\alpha_0)}{4},
[/mathjax]
where [mathjaxinline]b[/mathjaxinline] is the fraction of the light that is back-scattered, or reflected from the planet, after interacting with an aerosol particle, [mathjaxinline]S_0[/mathjaxinline] is the solar constant, and [mathjaxinline]\alpha_0[/mathjaxinline] is the normal albedo of the Earth (before adding extra aerosol). We will assume [mathjaxinline]S_0=1370[/mathjaxinline] W/m<sup>2</sup> and [mathjaxinline]\alpha_0=0.30[/mathjaxinline].</p>
<p>The final piece of the puzzle for calculating the radiative forcing from aerosols is then that we need to know what fraction of light, [mathjaxinline]b[/mathjaxinline], is scattered backwards. This is actually a quite interesting aspect of atmospheric radiative transfer. In PS7, we assumed [mathjaxinline]b=1[/mathjaxinline], but for most aerosols, this is a poor assumption. Aerosols and cloud droplets are not like little solid metal spheres that directly reflect light backwards. They are instead transparent, and most of the light they interact with is scattered forward - scattered in the sense that the direction of the light is changed, but usually the deflection from the original direction is small. It turns out that for cloud droplets, the backscattered fraction of light is only about 7%: [mathjaxinline]b\approx0.07 [/mathjaxinline]!</p>
<p>Assume that sulfate aerosols have [mathjaxinline]b=0.25[/mathjaxinline], and calculate the radiative forcing, in W/m<sup>2</sup>, due to the Pinatubo aerosols. The sign convention is such that a negative sign means additional solar reflection; a positive sign means additional absorption.
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<p>Enter your answer below as a number of W/m<sup>2</sup>. Do not include units in your answer.</p>
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