<?xml version="1.0" encoding="UTF-8" standalone="yes"?><oembed><version><![CDATA[1.0]]></version><provider_name><![CDATA[Azimuth]]></provider_name><provider_url><![CDATA[https://johncarlosbaez.wordpress.com]]></provider_url><author_name><![CDATA[John Baez]]></author_name><author_url><![CDATA[https://johncarlosbaez.wordpress.com/author/johncarlosbaez/]]></author_url><title><![CDATA[Mathematics of the Environment (Part&nbsp;4)]]></title><type><![CDATA[link]]></type><html><![CDATA[<p>We&#8217;ve <a href="https://johncarlosbaez.wordpress.com/2012/10/13/mathematics-of-the-environment-part-3/">been looking</a> at some very simple models of the Earth&#8217;s climate.  Pretty soon I want to show you one that illustrates the <b>ice albedo effect</b>.  This effect says that when it&#8217;s colder, there&#8217;s more ice and snow, so the Earth gets lighter in color, so it reflects more sunlight and tends to get even colder.  In other words, it&#8217;s a <b><a href="http://en.wikipedia.org/wiki/Positive_feedback">positive feedback</a></b> mechanism: a reaction that strengthens the process that caused the reaction. </p>
<p>Since a higher temperature leads to a higher radiation and therefore to cooling, and a lower temperature leads to a lower radiation, according to the Planck distribution, there is always also a <b><a href="http://en.wikipedia.org/wiki/Negative_feedback">negative feedback</a></b> present in the climate system of the earth. This is dubbed the <b>Planck feedback</b>, and this is what ultimately protects the Earth against getting arbitrarily hot or cold.</p>
<p>However, the ice albedo effect may be important for the &#8216;ice ages&#8217; or more properly &#8216;glacial cycles&#8217; that we&#8217;ve been having for the last few ten million years&#8230; and also for much earlier, much colder <a href="http://en.wikipedia.org/wiki/Snowball_Earth">Snowball Earth</a> events.  In reverse, melting ice now tends to make the Earth darker and even warmer.  So, this is an interesting topic for many reasons&#8230; including the math, which we&#8217;ll get to later.</p>
<p>Now, obviously the dinosaurs did not keep records of the temperature, so how we estimate temperatures on the ancient Earth is an important question, which deserves a long discussion&#8212;but not today!  Today I&#8217;ll be fairly sketchy about that.  I just want you to get a feel for the overall story, and some open questions.</p>
<h3> The Earth&#8217;s temperature since the last glacial period </h3>
<p>First, here&#8217;s a graph of Greenland temperatures over the last 18,000 years:</p>
<div align="center"><a href="http://math.ucr.edu/home/baez/temperature/greenland.18kyr.gif"><img width="450" src="https://i0.wp.com/math.ucr.edu/home/baez/temperature/greenland.18kyr.gif" /></a></div>
<p>(As usual, click to enlarge and/or get more information.)   This chart is based on ice cores, taken from:</p>
<p>&bull; Richard B. Alley, <em>The Two-Mile Time Machine: Ice Cores, Abrupt Climate Change, and our Future</em>, Princeton U. Press, Princeton, 2002.</p>
<p>This is a good book for learning how people reconstruct the<br />
history of temperatures in Greenland from looking at a two-mile-long ice core drilled out of the glaciers there.</p>
<p>As you can see, first Greenland was very cold, and then it warmed up at the end of the last &#8216;ice age&#8217;, or glacial period.  But there are lot of other things to see in this graph.  For example, there was a severe cold spell between 12.9 and 11.5 thousand years ago: the <a href="http://en.wikipedia.org/wiki/Younger_Dryas"><b>Younger Dryas</b></a> event.</p>
<div align="center"><a href="http://en.wikipedia.org/wiki/Dryas_octopetala"><img width="350" src="https://i1.wp.com/upload.wikimedia.org/wikipedia/commons/5/5b/Mountainavens2.jpg" /></a></div>
<p>I love that name!   It comes from the tough little Arctic flower<br />
<em><a href="http://en.wikipedia.org/wiki/Dryas_octopetala">Dryas octopetala</a></em>, whose plentiful pollen in certain ice samples gave evidence that this time period was chilly.    Was there an Older Dryas?  Yes: before the Younger Dryas there was a warm spell called the <a href="http://en.wikipedia.org/wiki/Aller%C3%B8d_Oscillation">Aller&oslash;d</a>, and before that a cold period called the <a href="http://en.wikipedia.org/wiki/Older_Dryas">Older Dryas</a>.</p>
<p>The Younger Dryas lasted about 1400 years.  Temperatures dropped dramatically in Europe: about 7 &deg;C in only 20 years!   In Greenland, it was 15 &deg;C colder during the Younger Dryas than today.  In England, the average annual temperature was -5 &deg;C, so glaciers started forming.  We can see evidence of this event from <a href="http://math.ucr.edu/home/baez/temperature/temperature.html#greenland.20kyr">oxygen isotope records</a> and <a href="http://www.agu.org/revgeophys/mayews01/node6.html">many other things</a>.  </p>
<p>Why the sudden chill?  One popular theory is that the melting of the ice sheet on North America lowered the salinity of North Atlantic waters.  This in turn blocked a current called the<br />
<a href="http://www.azimuthproject.org/azimuth/show/AMOC">Atlantic meridional overturning circulation</a>, or AMOC for short, which normally brings warm water up the coast of Europe.   Proponents of this theory argue that this current is what makes London much warmer than, say, Winnipeg in Canada or Irkutsk in Russia.  Turn it off and&mdash;<em>wham!</em>&mdash;you&#8217;ll get glaciers forming in England.</p>
<p>Anyway, whatever caused it, the Younger Dryas ended as suddenly at it began, with temperatures jumping 7 &deg;C.   Since then, the Earth continued warming up until about 6 thousand years ago&#8212;the <a href="http://en.wikipedia.org/wiki/Holocene_climatic_optimum">mid-Holocene thermal maximum</a>.   The earth was about 1&deg; or 2&deg; Celsius warmer than today.  Since then, it&#8217;s basically been cooling off&#8212;not counting various smaller variations, like the global warming we&#8217;re experiencing in this century.</p>
<p>However, these smaller variations are very interesting!  From 6000 to 2500 years ago things cooled down, with the coolest<br />
stretch occurring between 4000 and 2500 years ago: the <a HREF="http://en.wikipedia.org/wiki/Iron_Age_Cold_Epoch"><b>Iron Age Cold Epoch</b></a>.  </p>
<p>Then things warmed up for a while, and then they cooled down<br />
from 500 to 1000 AD.  Yes, the so-called &quot;Dark Ages&quot; were also chilly!  </p>
<p>After this came the <a href="http://en.wikipedia.org/wiki/Medieval_Warm_Period"><b>Medieval Warm Period</b></a>, a period from about 1000 to 1300 AD:  </p>
<div align="center"><a href="http://www.globalwarmingart.com/wiki/File:2000_Year_Temperature_Comparison_png"><img width="450" src="https://i2.wp.com/upload.wikimedia.org/wikipedia/commons/c/c1/2000_Year_Temperature_Comparison.png" /></a></div>
<p>From 1450 AD to 1890 there was a period of cooling, often called the <a href="http://en.wikipedia.org/wiki/Little_Ice_Age"><b>Little Ice Age</b></a>.  This killed off the Icelandic colonies in Greenland, as described in this gripping book:</p>
<p>&bull; Jane Smiley, <em>The Greenlanders</em>, Ballantine Books, New York, 1996. </p>
<p>However, the term &quot;Little Ice Age&quot; exaggerates the importance of a small blip in the grand scheme of things.  It was nowhere near as big as the Younger Dryas: temperatures may have dropped a measly 0.2&deg; Celsius from the Medieval optimum, and it may have happened only in Europe&#8212;though this was a subject of debate when I last checked.</p>
<p>Since then, things have been warming up:</p>
<div align="center"><a href="http://www.globalwarmingart.com/wiki/File:1000_Year_Temperature_Comparison_png"><img width="450" src="https://i2.wp.com/upload.wikimedia.org/wikipedia/commons/b/bb/1000_Year_Temperature_Comparison.png" /></a></div>
<p>The subject has big political implications, and is thus subject to enormous <a href="http://en.wikipedia.org/wiki/Temperature_record_of_the_past_1000_years">controversy</a>. But, I think it&#8217;s quite safe to say that we&#8217;ve been seeing a rapid temperature rise since 1900, with the Northern Hemisphere average temperature rising roughly 1 &deg;C since then.  Each of the last 11 years, from 2001 to 2011, was one of the <a href="http://en.wikipedia.org/wiki/Instrumental_temperature_record#Warmest_years">12 warmest years since 1901</a>.  (The other one was 1998.)</p>
<div align="center"><a href="http://en.wikipedia.org/w/index.php?title=File:Instrumental_Temperature_Record_%28NASA%29.svg&amp;page=1"><img src="https://i0.wp.com/upload.wikimedia.org/wikipedia/commons/thumb/7/75/Instrumental_Temperature_Record_%28NASA%29.svg/450px-Instrumental_Temperature_Record_%28NASA%29.svg.png" /></a></div>
<p>All these recent variations in the Earth&#8217;s climate are very much worth trying to understand.   but now let&#8217;s back off to longer time periods!  We don&#8217;t have many Earth-like planets whose climate we can study in detail&#8212;at least not yet, since they&#8217;re too far away.  But we do have one planet, the Earth, that&#8217;s gone through many changes.  The climate since the end of the last ice age is just a tiny sliver of a long and exciting story!</p>
<h3> The Earth&#8217;s long-term climate history </h3>
<p>Here&#8217;s a nice old chart showing estimates of the Earth&#8217;s average temperature in the last 150 years, the last 16,000 years, the last 150,000 years and the last million years:</p>
<div align="center"><a href="http://math.ucr.edu/home/baez/temperature/1Myr.jpg"><img width="450" src="https://i1.wp.com/math.ucr.edu/home/baez/temperature/1Myr.jpg" /><br />
</a></div>
<p>Here &#8220;ka&#8221; or &#8220;kilo-annum&#8221; means a thousand years.  These temperatures are estimated by various methods; I got this chart from:</p>
<p>&bull; Barry Saltzman, <i>Dynamical Paleoclimatology: Generalized Theory of Global Climate Change</i>, Academic Press, New York, 2002, fig. 3-4. </p>
<p>As we keep zooming in towards the present we keep seeing more detail:</p>
<p>&bull; Over the last million years there have been about ten glacial periods&#8212;though trying to count them is a bit like trying to count &#8216;very deep valleys&#8217; in a hilly landscape!</p>
<p>&bull; From 150 to 120 thousand years ago it warmed up rather rapidly. From 120 thousand years ago to 16 thousand years ago it cooled down&#8212;that was the last glacial period. Then it warmed up rather rapidly again.</p>
<p>&bull; Over the last 10 thousand years temperatures have been unusually constant.</p>
<p>&bull; Over the last 150 years it&#8217;s been warming up slightly. </p>
<p>If we go back further, say to 5 million years, we see that temperatures have been colder but also more erratic during this period:</p>
<div align="center"><a href="http://www.globalwarmingart.com/wiki/File:Five_Myr_Climate_Change_Rev_png"><img width="450" src="https://i2.wp.com/math.ucr.edu/home/baez/temperature/5Myr.png" /></a></div>
<p>This figure is based on this paper:</p>
<p>&bull; L. E. Lisiecki and M. E. Raymo, A Pliocene-Pleistocene stack of 57 globally distributed benthic &delta;<sup>18</sup>O records, <i><a href="http://www.agu.org/pubs/crossref/2005/2004PA001071.shtml">Paleoceanography</a></i> <b>20</b> (2005), PA1003. </p>
<p>Lisieki and Raymo combined measurements of oxygen isotopes in the shells of tiny sea creatures called <a href="http://en.wikipedia.org/wiki/Foraminifera">foraminifera</a> from 57 globally distributed deep sea sediment cores.  But beware: they constructed this record by first applying a computer aided process to align the data in each sediment core.  Then the resulting stacked record was tuned to make the positions of peaks and valleys match the known <a href="http://en.wikipedia.org/wiki/Milankovitch_cycles">Milankovitch cycles</a> in the Earth&#8217;s orbit.  The temperature scale was chosen to match <a href="http://www.ncdc.noaa.gov/paleo/icecore/antarctica/vostok/vostok.html">Vostok ice core data</a>.  So, there are a lot of theoretical assumptions built into this graph.</p>
<p>Going back 65 million years, we see how unusual the current glacial cycles are:</p>
<div align="center"><a href="http://math.ucr.edu/home/baez/temperature/65Myr.png"><img width="450" src="https://i1.wp.com/math.ucr.edu/home/baez/temperature/65Myr.png" /><br />
</a></div>
<p>Click to make this graph bigger; it&#8217;s from:</p>
<p>&bull; Robert Rohde, <a href="http://en.wikipedia.org/wiki/Image:65_Myr_Climate_Change.png">65 million years of climate change</a>, at <a href="http://www.globalwarmingart.com/">Global Warming Art</a>.</p>
<p>This graph shows the Earth&#8217;s temperature since the extinction of the dinosaurs about 65 million years ago&mdash;the end of the <a href="http://en.wikipedia.org/wiki/Mesozoic">Mesozoic</a> and beginning of the <a href="http://en.wikipedia.org/wiki/Cenozoic">Cenozoic</a>.  At first the Earth warmed up, reaching its warmest  50 million years ago: the &quot;Eocene Optimum&quot;. The spike before that labelled &quot;PETM&quot; is a fascinating event called the <a href="http://en.wikipedia.org/wiki/Paleocene-Eocene_Thermal_Maximum"> Paleocene-Eocene Thermal Maximum</a>.   At the end of the <a href="http://en.wikipedia.org/wiki/Eocene">Eocene</a> the Earth cooled rapidly and the Antarctic acquired year-round ice. After a warming spell near the end of the  <a href="http://en.wikipedia.org/wiki/Eocene">Oligocene</a>, further cooling and an increasingly jittery climate led ultimately to the current age of rapid glacial cycles.  </p>
<p>Why is the Earth&#8217;s climate so jittery nowadays?  That&#8217;s a fascinating puzzle, which I&#8217;d like to discuss in the weeks to come.  </p>
<p>Why did the Earth suddenly cool at the end of the Eocene 34 million years ago?  One theory relies on the fact that this is when Antarctica first became separated from Australia and South America.  After the <a href="http://a.ennyu.com/pdf/Exon_JoiJour_00.pdf">Tasmanian Gateway</a> between Australia and Antarctica opened, the only thing that kept water from swirling endlessly around Antarctica, getting colder and colder, was the connection between this continent and South America.  South America seems to have separated from Antarctica around the end of the Eocene. </p>
<p>In the early Eocene, Antarctica was fringed with a warm temperate to sub-tropical rainforest.  But as the Eocene progressed it became colder, and by the start of the Oligocene it had deciduous forests and vast stretches of tundra.  Eventually it became almost completely covered with ice.</p>
<p>Thanks to the ice albedo effect, an icy Antarctic tends to keep the Earth cooler.  But is that the only or even the <i>main</i> explanation of the overall cooling trend over the last 30 million years?  Scientists argue about this.</p>
<p>Going back further:</p>
<div align="center"><a href="http://math.ucr.edu/home/baez/temperature/4600Myr.jpg"><img width="450" src="https://i0.wp.com/math.ucr.edu/home/baez/temperature/4600Myr.jpg" /></a></div>
<p>Here &quot;Ma&quot; or &quot;mega-annum&quot; means &quot;million years&quot;.   This chart was drawn from many sources; I got it from: </p>
<p>&bull; Barry Saltzman, <em>Dynamical Paleoclimatology: Generalized Theory of Global Climate Change</em>, Academic Press, New York, 2002, fig. 1-3. </p>
<p>Among other things on this chart, you can sort of see hints of the <a href="http://en.wikipedia.org/wiki/Snowball_Earth">Snowball Earth</a> events that may have happened early in the Earth&#8217;s history.  These are thought to have occurred during the <a href="http://en.wikipedia.org/wiki/Cryogenian">Cryogenian period</a> 850 to 635 million years ago, and also during the <a href="http://en.wikipedia.org/wiki/Huronian_glaciation">Huronian glaciation</a> 2400 to 2100 million years ago.  In both these events a large portion of the Earth was frozen&#8212;much more, it seems, than in the recent glacial periods!  Ice albedo feedback plays a big role in theories of these events&#8230; though also, of course, there must be some explanation of why they <i>ended</i>.</p>
<p>As you can see, there&#8217;s a lot of things a really universal climate model might seek to explain.  We don&#8217;t <i>necessarily</i> need to understand the whole Earth&#8217;s history to model it well <i>now</i>, but thinking about other eras is a good way to check our understanding of the present-day Earth.</p>
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