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<h2 class="hd hd-2 unit-title">Introduction to climate physics</h2>
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<h2>Climates of the past</h2>
<p>In this section we discuss the Earth's climate in the distant past. We have already seen a broad overview of Earth's climate history in the introduction section, but we now delve a little deeper into some of the major changes the Earth's climate has undergone, and some of the techniques scientists use to estimate past climates.</p>
<p><i>Instructor: David McGee</i></p>
<h2>Module outline</h2>
<p>In this module, we will discuss how the Earth's climate changes on a variety of different timescales. We will also be introduced to some of the myriad methods for estimating past climates using evidence from fossils, sediments and ice-cores. The module discusses different events and periods in Earth's history moving forward in chronological order:</p>
<ul>
<li>The <b>Faint young sun paradox</b>: Models of stellar evolution suggest that in Earth's early history the solar luminosity was much weaker than it is now. But the Earth maintained a relatively high temperature. How is this possible?</li>
<li>The <b>Snowball Earth hypothesis</b> states that at least once in Earth's history the globe was completely, or nearly completely covered in ice. How could we escape such a state?</li>
<li>The last 500 million years of Earth's history was characterized by large changes in temperature and extent of ice sheets. How can we infer such information?</li>
<li>The <b>Paleocene-Eocene Thermal Maximum</b> is a large spike in the Earth's surface temperature that is thought to have occured around 50 million years ago and lasted 200 thousand years. What could have caused such a catastrophic climate event?</li>
<li>Over the last two million years or so, variations in the Earth's climate are believed to be driven by slow changes in the Earth's orbit called the <b>Milankovitch cycles</b>. How do these tiny variations give us regular ice ages?</li>
<li>We have a relative abundance of records of the climate from the last two thousand years. How warm is the current period relative to the recent past?</li>
</ul>
<h2>Background reading</h2>
<p>Many techniques to estimate past climates make use of slight differences in the chemistry of different <it>isotopes</it> of the same chemical. Students who are not familiar with the concept of an isotope should read <a href="http://theconversation.com/explainer-what-is-an-isotope-10688" target="_blank">this short article</a> in <a href="http://theconversation.com" target="_blank">The Conversation</a> for a primer.</p>
<p>In this section we only touch on some of the techniques used to estimate past climates. For more information students may wish to read <a href="http://www.globalchange.umich.edu/globalchange1/current/lectures/kling/paleoclimate" target="_blank">this page</a> from the University of Michigan, which details some of the most common geochemical techniques for estimating past climates.</p>
<p>For some more detailed information on the current scientific understanding of the Earth's climate history section 6.3 and 6.4 of <a href="http://www.ipcc.ch/publications_and_data/ar4/wg1/en/ch6.html" target="_blank">Chapter 6</a> of the <a href="http://www.ipcc.ch/publications_and_data/ar4/wg1/en/contents.html" target="_blank">IPCC 4th Assessment report</a> contains a useful summary.</p>
<h2>Lecture Slides</h2>
<p>The slides for this lecture are available to download <a href="/assets/courseware/v1/d006eaf5364388b6d59a1380715d5fc0/asset-v1:MITx+12.340x+1T2020+type@asset+block/paleoclimate.compressed.pdf" target="[object Object]">here</a>.</p>
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<h2 class="hd hd-2 unit-title">The Faint Young Sun Paradox</h2>
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Why was the Earth so warm in the Archean?
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<p>Which of the following is a possible explanation for why the Earth was not completely frozen over in the Archean period (4-2.5 billion years ago) despite the solar flux being so low relative to today? </p>
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<text>The albedo was much higher.</text>
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<text>There was a much higher concentration of oxygen in the atmosphere.</text>
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<text>The Earth was in an orbit much closer to the sun.</text>
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<text>The concentration of methane may have been much higher than today.</text>
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<p>The ice-albedo feedback that is hypothesized to lead to the rapid global glaciations of the Snowball Earth can be modeled very simply using the one-layer radiative-equilibrium model with a temperature dependent albedo. An implementation of such an <a href="/courses/course-v1:MITx+12.340x+1T2020/courseware/Resources/ice_albedo/">ice-albedo feedback model</a> is reproduced below. Explore the parameter space of this model. Under what conditions does the surface freeze over? Are some solutions sensitive to the initial temperature?</p><iframe src="/asset-v1:MITx+12.340x+1T2020+type@asset+block/models_OneLayer_icealbedo_OneLayer.html" width="800" height="700" frameborder="0" scrolling="no"/>
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Climate and isotopes of oxygen
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<p>Which of the following would result in the lowest [mathjaxinline] \delta{}^{18} [/mathjaxinline]O measured in calcium carbonate shells of marine organisms?</p>
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<text>A high temperature period with little land ice.</text>
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<text>A low temperature period with a large icecap.</text>
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<text>A low temperature period with little polar land area on which to build ice sheets.</text>
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<text>A high temperature period with considerable high elevation land ice.</text>
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Natural and human impulses of atmospheric carbon dioxide
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<p>The current rate of emission of carbon dioxide by humans through fossil fuel use is around 7 Pg of Carbon per year. Suppose during the entire PETM 10 000 Pg of carbon was released into the atmosphere. Assuming this release took 200 thousand years*, what is the ratio of the current emission of carbon through the burning of fossil fuels to the average rate of release of carbon during the PETM?</p>
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<text>0.14</text>
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<text>1.4</text>
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<text>14</text>
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<text>140</text>
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<p>*NOTE: The estimate used here for the timescale over which Carbon was input into the atmosphere during the PETM is based on the duration of the entire event. Modern paleo-climatic studies have suggested the Carbon impulse may have been much shorter in duration - 10,000 years or even shorter. Using this shorter timescale would give a ratio 20 times smaller.</p>
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Obliquity and the seasonal cycle
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<p>At periods in the last million years, the Earth's obliquity was greater than it is today, being as large as 24.5 degrees. How would this affect the strength of the seasonal cycle of radiation in mid-latitudes in each hemisphere compared to today? </p>
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<text>The seasonal cycle becomes stronger in both hemispheres.</text>
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<text>The seasonal cycle becomes weaker in both hemispheres.</text>
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<text>One hemisphere experiences a stronger seasonal cycle, the other a weaker one. Which hemisphere has a strengthened seasonal cycle depends on the precession of the Earth's orbit.</text>
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<text>The seasonal cycle in unchanged - only precession affects the seasonal cycle of radiation at a given latitude.</text>
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The last 1500 years of climate variability
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<p>Which of the following is true of surface temperatures in the late 20th century as compared to the last 1500 years, according to proxy records of the Earth's climate? </p>
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<text>Many places on the globe are warmer than at any point in the past 1500 years, but some places have been warmer in the past.</text>
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<text>All points on the Earth are warmer than they have been for the past 1500 years.</text>
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<text>All points on the Earth were warmer than they are now in the medieval warm period.</text>
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<text>Most places were warmer in the medieval warm period that today, but a few isolated regions are warmer today than at any time in the last 1500 years.</text>
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<h2 class="hd hd-2 unit-title">Problem Set: Paleoclimate</h2>
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<h2>Problem Set: Paleoclimate</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 grasp of the use and utility of studying past climates, and some idea of how climate proxies are used to do this. You should also have a basic understanding of the Faint Young Sun paradox and the Snowball Earth hypothesis. This problem set contains three problems, each of which has multiple parts.</p>
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<h2>Oxygen isotopes and the climate</h2><p>In this problem we will seek to understand how changes in oxygen isotopes in the world's oceans can provide evidence for past climate change.</p><p>Oxygen atoms can exist as a number of different isotopes, each with a different atomic weight. For instance Oxygen-16, <sup>16</sup>O, is the most common isotope, with 8 protons and 8 neutrons, giving it an atomic weight of 16. A much less abundant isotope is <sup>18</sup>O, which has 2 extra neutrons compared to its more common counterpart. In processes such as evaporation and condensation, as well as chemical reactions, one isotope may be slightly favored energetically, and thus be more likely to be involved in the reaction. For instance, due to its larger molecular weight, <sup>18</sup>O does not evaporate as readily as <sup>16</sup>O. Hence, water evaporated from the ocean surface is slightly depleted in <sup>18</sup>O compared to the ocean. This property can be used to help understand past climates, as we shall see.
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1a
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<p>
The fraction of <sup>18</sup>O in a sample of water is commonly expressed in what is known as <i>delta notation</i>. The value of [mathjaxinline] \delta ^{18} [/mathjaxinline]O may be found from the following formula,
[mathjax]
\delta ^{18}O = 1000\left(\frac{\left[\frac{^{18}O}{^{16}O}\right]_{sample}}{\left[\frac{^{18}O}{^{16}O}\right]_{standard}} - 1\right)
[/mathjax]
Here [mathjaxinline]^{x}[/mathjaxinline]O is the molar concentration of the isotope [mathjaxinline]x[/mathjaxinline].The ratio of two isotopes in a sample are compared to a particular standard. For this particular case, the standard has an isotope ratio of 2.005e-3.
</p>
<p>
A sample of water contains 1 mole of [mathjaxinline] ^{16} [/mathjaxinline]O and 0.002054 moles of [mathjaxinline] ^{18} [/mathjaxinline]O. Find its value of [mathjaxinline] \delta ^{18} [/mathjaxinline]O.</p>
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<p>The figure above was shown in the second video of this section. It displays the relationship between the [mathjaxinline] \delta ^{18} [/mathjaxinline]O in certain marine organisms (Foraminifera) that use oxygen to create carbonate shells and the [mathjaxinline] \delta ^{18} [/mathjaxinline]O in the surrounding ocean as a function of temperature. The x-axis shows the [mathjaxinline] \delta ^{18} [/mathjaxinline]O in the carbonate shells minus the value of [mathjaxinline] \delta ^{18} [/mathjaxinline]O of the surrounding ocean water. The y-axis shows the temperature at which the carbonate shell was formed. </p>
<p>If the planet is ice free, one may assume the [mathjaxinline] \delta ^{18} [/mathjaxinline]O of the ocean remains somewhat fixed, and thus the [mathjaxinline] \delta ^{18} [/mathjaxinline]O in Foraminifera is a direct measure of the ocean temperature. </p>
<p>Imagine a world in which there are no polar land masses to support ice-caps. Under these conditions the amount of land ice remains small regardless of the temperature. In this ice-free world, the istopes of oxygen in the ocean are measured. For every one mole of <sup>16</sup>O in the ocean there are 0.002005 moles of [mathjaxinline] ^{18}[/mathjaxinline]O. The oxygen isotopes of a Foraminifera sample from the recent past are also measured. For every one mole of <sup>16</sup>O in the Foraminifera, there are 0.002000 moles of [mathjaxinline] ^{18} [/mathjaxinline]O.</p>
<p>Now suppose a much older fossilized Foraminifera from the same region of the ocean from a very cool period in the planet's history is found to have a isotopic ratio of <sup>18</sup>O to <sup>16</sup>O of 0.0020046. Use the calibration curve shown above to estimate how much cooler the ocean was during the time period when the older fossil was deposited.</p>
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1c
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As was discussed in the videos, the existence of ice caps complicates the interpretation of oxygen isotope records because of the fractionation of water by evaporation and precipitation before it becomes part of the ice cap. As the temperature increases or decreases we might expect the ice cap to grow or shrink. Assuming the ice cap does indeed respond to the temperature change in this way, for a given temperature change would you expect the resultant change in [mathjaxinline] \delta ^{18} [/mathjaxinline]O of Foraminifera to be larger or smaller with the existence of an ice cap?
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<h2 class="hd hd-2 unit-title">Problem 2 - Milankovitch cycles</h2>
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<h2>Milankovitch Cycles</h2><p>Long-term variations in the Earth's climate over the last million years are dominated by cyclical changes in the Earth's orbit around the sun named after Serbian astronomer Milutin Milankovic. In this problem we explore the effects on insolation of some of these cycles.
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2a
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Match the type of orbital variation to its name by dragging the boxes below.
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2b
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<p>In what way does increasing obliquity change the global- and annual-mean insolation reaching the Earth?</p>
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<text>Increases global- and annual-mean insolation</text>
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<input type="radio" name="input_9646a14b4aff486492d6e073acc9015d_2_1" id="input_9646a14b4aff486492d6e073acc9015d_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="9646a14b4aff486492d6e073acc9015d_2_1-choice_1-label" for="input_9646a14b4aff486492d6e073acc9015d_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_9646a14b4aff486492d6e073acc9015d_2_1">
<text>Decreases global- and annual-mean insolation</text>
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<text>Does not change the global- and annual-mean insolation</text>
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<text>It depends on the value of the eccentricity as to how increasing obliquity affects the global- and annual-mean insolation.</text>
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The insolation in the high Northern latitudes is seen as important for the climate over the entire globe during glacial and interglacial cycles. This is because the insolation at these latitudes influences the growth and decay of ice sheets. In the next few parts of the problem, we will calculate the daily average insolation for a given latitude and time of year, to understand how this is affected by obliquity.
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<p>
The solar zenith angle, [mathjaxinline]\theta[/mathjaxinline] for a given time and position on the Earth is given by,
[mathjax]
\cos\theta = \sin\phi\sin\delta + cos\phi\cos\delta\cos h,
[/mathjax]
where [mathjaxinline]\phi[/mathjaxinline] is the latitude, [mathjaxinline]\delta[/mathjaxinline] is the solar declination, and [mathjaxinline]h[/mathjaxinline] is the hour angle. The solar declination is the angle between the path of the sun and the celestial equator. At the summer solstice, which we will focus on, this value is simply the obliquity. The hour angle varies between [mathjaxinline]-\pi[/mathjaxinline] and [mathjaxinline]\pi[/mathjaxinline] throughout the day. Ignoring variations in the distance of the Earth to the sun (i.e. the eccentricity), the solar insolation [mathjaxinline]I(\phi,\delta,h)[/mathjaxinline] is then equal to
[mathjax]I(\phi,\delta,h) = S_0 \cos\theta, \,\,\,\, \cos\theta &gt; 0[/mathjax]
and zero otherwise. Here [mathjaxinline]S_0[/mathjaxinline] is the solar constant at the mean Earth-sun distance.
</p>
<p>
Calculate the value of [mathjaxinline]\cos h[/mathjaxinline] when sunset occurs. That is, find the largest value of [mathjaxinline]h[/mathjaxinline] for which [mathjaxinline]\cos\theta[/mathjaxinline] is equal to zero. Express your answer in terms of [mathjaxinline]\delta[/mathjaxinline] and [mathjaxinline]\phi[/mathjaxinline].
</p>
<p>Type "phi" to input the symbol [mathjaxinline]\phi[/mathjaxinline], and "delta" for the symbol [mathjaxinline]\delta[/mathjaxinline]. Explicitly indicate multiplication by the "*" symbol. Trigonometric functions can be input by typing the appropriate 3-letter abbreviation ("sin(phi)", "cos(phi)", "tan(phi)", "sec(phi)", "csc(phi)", or "cot(phi)")</p>
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<span>[mathjaxinline]\cos h [/mathjaxinline] =</span>
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2d
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Find the daily average insolation at a given latitude for a given solar declination by integrating the insolation between sunrise and sunset. This is equivalent to integrating between the values of [mathjaxinline][-h_0,h_0][/mathjaxinline], where [mathjaxinline]h_0[/mathjaxinline] is the hour of sunset calculated in part C. To find the mean insolation, one must divide by the total hour angle of a day, [mathjaxinline]2\pi[/mathjaxinline]. Thus, to calculate the average insolation for the day one must evaluate:
[mathjax]
I_{avg} = S_0\frac{1}{2\pi}\int_{-h_0}^{h_0} \sin\phi\sin\delta + \cos\phi\cos\delta\cos h \, dh.
[/mathjax]
</p>
<p> Express your answer in terms of the variables [mathjaxinline]S_0,h_0, \delta [/mathjaxinline], [mathjaxinline] \phi[/mathjaxinline] and the constant [mathjaxinline]\pi[/mathjaxinline]. Type "S_0" to input [mathjaxinline]S_0[/mathjaxinline], "h_0" to indicate [mathjaxinline]h_0[/mathjaxinline], "delta" to indicate [mathjaxinline]\delta[/mathjaxinline], "phi" to indicate [mathjaxinline]\phi[/mathjaxinline] and "pi" to indicate [mathjaxinline]\pi[/mathjaxinline]. Explicitly indicate multiplication with a "*" symbol.
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<p>Using the answers to the previous two parts of this problem, find the daily mean insolation at a latitude of 65 degrees North at the summer solstice for an obliquity similar to that of the present day, 23.5 degrees. You may assume a solar constant of 1368 W m[mathjaxinline]^{-2}[/mathjaxinline]. Remember to use radians for all angle measures, and express your answer in W m[mathjaxinline]^{-2}[/mathjaxinline].</p>
<p>NOTE: you may check your answers for the previous problem before you go on to ensure an incorrect answer for part C does not prevent you from getting credit for part D and E.</p>
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2f
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<p>Find the insolation at summer solstice for a low obliquity of 22 degrees.
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<h2 class="hd hd-2 unit-title">Problem 3: Multiple equilibria</h2>
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<h2>Multiple equilibria in the climate system</h2><p>The concept of multiple equilibria in the climate system refers to the possibility that there may be more than one steady state climate for a given set of boundary conditions. The state that the climate eventually reaches may depend on the historical evolution of the Earth-system, as well as the forcing agents. In this problem we investigate multiple climate equilibria in a very basic climate model that incorporates the ice-albedo feedback.</p><p>The model has a single layer atmosphere transparent to solar radiation and absorbing in the infra-red, as in the model discussed is the previous module. However, this model differs in that the albedo is set as a function of temperature. If the surface temperature is above freezing (273.15 K) the albedo is 0.3, if not, the surface is assumed to be ice covered and has a higher albedo of 0.6. Since there is the possibility of multiple equilibria in this model, one must also specify the initial surface temperature in order to distinguish between the possible steady states. Given these values, the model solves for a steady state climate that satisfies energy balance at the top of the atmosphere. </p><p>The <a href="/courses/course-v1:MITx+12.340x+1T2020/courseware/Resources/ice_albedo/">ice-albedo feedback model</a> is reproduced below.</p><iframe src="/asset-v1:MITx+12.340x+1T2020+type@asset+block/models_OneLayer_icealbedo_OneLayer.html" width="800" height="700" frameborder="0" scrolling="no"/>
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3a
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Play with the input parameters of the ice-albedo feedback model until you find a set of boundary conditions (i.e. solar constant and emissivity) that exhibit multiple equilibria (i.e. the model solution depends on the value you enter for the initial temperature). Enter the values for the parameters that give multiple climate steady states in the box below.
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Emissivity = <div id="inputtype_553698dcc2ec4e7e9eb6ea2d489191ef_2_1" class=" capa_inputtype textline">
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Solar constant (W/m^2) = <div id="inputtype_553698dcc2ec4e7e9eb6ea2d489191ef_2_2" class=" capa_inputtype textline">
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3b
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Use the ice-albedo model above to find solutions for the surface temperature for various solar constant values, with an atmospheric emissivity of 1.0. Plot these values on the graph below by entering the solutions into the input fields, one at a time, and clicking plot. Using your results, estimate the range of solar constants for which multiple equilibria in the model climate exist.
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Enter the lowest and highest values of the solar constant for which multiple solutions exist; be sure to enter both values before submitting your answer.
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3c
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In the ice-albedo feedback model above, the albedo of an ice surface is assumed to be 0.6. If instead we assumed an ice surface had a higher albedo - 0.7 - would the range over which multiple equilibria exist increase or decrease in size?
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<text>Increase</text>
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3d
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<p>In this problem we investigate hysteresis in our simple ice-albedo model. Run the model with the default parameters - S<sub>0</sub> = 1370 W m[mathjaxinline]^2[/mathjaxinline], [mathjaxinline]\epsilon [/mathjaxinline] = 1.0, T<sub>init</sub> = 288 K. Repeat the calculation for the following solar constants, keeping the emissivity at 1.0: 1200, 1000, 800, 1000, 1200 1370. Each time you run the model, use the equilibrium temperature from the previous run as your initial condition. Enter the surface temperature of the final simulation in the box below. Is it different from the results from the first simulation? The parameters are the same, so how can this be? You may like to plot your solutions using the interactive plotter in part B to visualize the hysteresis loop in this model.</p>
<p>NOTE: Ensure that you run the simulations with the parameters above in the same order as they are listed. In each case, use the equilibrium temperature for the previous simulation as the initial temperature for the next one.</p>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><span>T<sub>surf</sub> (K): </span>
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