{"id":6597,"date":"2018-07-29T09:21:51","date_gmt":"2018-07-29T07:21:51","guid":{"rendered":"http:\/\/www.randform.org\/blog\/?p=6597"},"modified":"2023-04-24T11:30:44","modified_gmt":"2023-04-24T09:30:44","slug":"simple-greenhouse-gas-models","status":"publish","type":"post","link":"https:\/\/www.randform.org\/blog\/?p=6597","title":{"rendered":"simple greenhouse gas models"},"content":{"rendered":"<p><script src=\"http:\/\/www.randform.de\/blog\/plotly-latest.min170117.js\"><\/script><\/p>\n<div id=\"0f2e4c3f-2996-4c32-b8ca-15494b99be79\" style=\"width:600px;height:450px;\"><\/div>\n<p>    <script>\n    PLOT = document.getElementById('0f2e4c3f-2996-4c32-b8ca-15494b99be79');\n    Plotly.plot(PLOT, 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1.000)\",\"family\":\"sans-serif\"},\"bordercolor\":\"rgba(0, 0, 0, 1.000)\",\"x\":1.0},\"paper_bgcolor\":\"rgba(255, 255, 255, 1.000)\",\"height\":400,\"margin\":{\"r\":0,\"l\":0,\"b\":0,\"t\":20}});\n    <\/script><\/p>\n<p>Rotatable with your mouse after you click on &#8220;<a href=\"http:\/\/www.randform.org\/blog\/?p=6597#more-6597\">Read the rest of this entry<\/a>&#8220;.<\/p>\n<p>Most of randform readers might have heard that the socalled <a href=\"https:\/\/en.wikipedia.org\/wiki\/Greenhouse_effect\">greenhouse effect<\/a> is one of the main causes of global warming.<\/p>\n<p>The effect is not easy to understand. There are two posts which give a nice intro to the greenhouse effect on Azimuth. One is by <a href=\"https:\/\/johncarlosbaez.wordpress.com\/2011\/07\/02\/a-quantum-of-warmth\/#comment-87677\">Tim van Beek<\/a> and one is by <a href=\"https:\/\/johncarlosbaez.wordpress.com\/2012\/10\/13\/mathematics-of-the-environment-part-3\/\">John Baez<\/a>.<br \/>\nThe greenhouse effect can also be understood in a slightly more quantitative way by looking at an <a href=\"https:\/\/en.wikipedia.org\/wiki\/Idealized_greenhouse_model\">idealized greenhouse model.<\/a> <\/p>\n<p>In the above diagram I now enhanced this  <a href=\"https:\/\/en.wikipedia.org\/wiki\/Idealized_greenhouse_model\">idealized greenhouse model<\/a> (as of Jan 2017) in order to get an idea about the hypothetical size of the effect of an absorption of non-infrared sunlight and it&#8217;s reradiation as infrared light, i.e. the possibly effect size of a certain type of <a href=\"https:\/\/en.wikipedia.org\/wiki\/Fluorescence\">fluorescence.<\/a> <\/p>\n<p>I sort of felt forced to do this, because at the time of writing (February 2017) the current climate models did not take the absorption of UV and near infrared light in methane (here a possible candidate for that above mentioned hypothetical greenhouse gas) into account and I wanted to get an insight into how important such an omission might be. The simple model here is far from any realistic scenario &#8211; in particular no specific absorption lines but just the feature of absorption and reradiation is looked at.<br \/>\n&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8211;<br \/>\ninsert April 24, 2023<br \/>\nMy first email asking about the inclusion of shortwave (UV and near infrared) radiation into the radiative forcing of methane to Gunnar Myrhe was on Sept. 10, 2015. The last answer I got was in January 2017, when I pointed out in a new email that there are problems with the evaluation of solar radiation, as explained in the randform post <a href=\"http:\/\/www.randform.org\/blog\/?p=6524\"> Information about solar irradiance measurements sought<\/a> (see also <a href=\"https:\/\/www.randform.org\/blog\/?p=7312\">What is going on at the Sun?).<\/a> The answer was basically that he is busy with meetings etc. <\/p>\n<p>Meanwhile I had found out, by searching the internet, that the problems had been somewhat adressed.<\/p>\n<p>That is in: <a href=\"https:\/\/agupubs.onlinelibrary.wiley.com\/doi\/epdf\/10.1002\/2016GL071930\">Radiative forcing of carbon dioxide, methane, and nitrous oxide: A significant revision of the methane radiative forcing<\/a> by M. Etminan1 , G. Myhre2 , E. J. Highwood 1 , and K. P. Shine it was found that<\/p>\n<blockquote><p>Methane\u2019s RF is particularly impacted because of the inclusion of the shortwave forcing; the 1750\u20132011 RF is about 25% higher (increasing from<br \/>\n0.48Wm\u22122 to 0.61Wm\u22122) compared to the value in the Intergovernmental Panel on Climate Change (IPCC) 2013 assessment; the 100year global warming potential is 14% higher than the IPCC value.<\/p><\/blockquote>\n<p>The new value came from the inclusion of absorption spectra with shorter wavelengths.<\/p>\n<p>The new value was inserted into climate simulations in:<br \/>\n<a href=\"https:\/\/agupubs.onlinelibrary.wiley.com\/doi\/epdf\/10.1029\/2018GL079826\">Understanding Rapid Adjustments to Diverse Forcing Agents<\/a><br \/>\nAccordingly the new IPCC report writes in chapter 7: <a href=\"https:\/\/www.ipcc.ch\/report\/ar6\/wg1\/chapter\/chapter-7\/\">7.3.2.2 Methane:<\/a> <\/p>\n<blockquote><p>The SARF for methane (CH4) has been substantially increased due to updates to spectroscopic data and inclusion of shortwave absorption (Etminan et al., 2016). Adjustments have been calculated in nine climate models by Smith et al. (2018b). Since CH4 is found to absorb in the shortwave near infrared, only adjustments from those models including this absorption are taken into account. For these models the adjustments act to reduce the ERF because the shortwave absorption leads to tropospheric heating and reductions in upper tropospheric cloud amounts. The adjustment is \u201314% \u00b1 15%, which counteracts much of the increase in SARF identified by Etminan et al. (2016). Modak et al. (2018) also found negative forcing adjustments from a methane perturbation including shortwave absorption in the NCAR CAM5 model, in agreement with the above assessment. The uncertainty in the shortwave component leads to a higher radiative modelling uncertainty (14%) than for CO2 (Etminan et al., 2016). When combined with the uncertainty in the adjustment, this gives an overall uncertainty of \u00b120%. There is high confidence in the spectroscopic revision but only medium confidence in the adjustment modification.<\/p><\/blockquote>\n<p>A very new evaluation in <a href=\"https:\/\/agupubs.onlinelibrary.wiley.com\/doi\/full\/10.1029\/2022GL098270\">Methane&#8217;s Solar Radiative Forcing<\/a> finds:<\/p>\n<blockquote><p>Including the impact of SW absorption on stratospheric temperature increases tropopause SARF by 0.039 W m\u22122 (or 7%) compared to the LW-only SARF.<\/p><\/blockquote>\n<p>The article <a href=\"https:\/\/agupubs.onlinelibrary.wiley.com\/doi\/epdf\/10.1029\/2018GL079826\">Understanding Rapid Adjustments to Diverse Forcing Agents<\/a> however also evaluated what a 2% increase of the solar constant parameter shows in the climate models and found (see Fig. 1) that the radiative forcing increases as if one would double the CO2 parameter or triple the CH4 parameter.<\/p>\n<p>&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8211;<\/p>\n<p>The above diagramm shows the earth temperature in Kelvin as a function of two parameters, as given by this enhanced model. The two parameters can be seen as being (somewhat) proportional to densities of a hypothetical greenhouse gas, which would display this type of fluorescence. That is the parameter x is seen as (somewhat) proportional to the density of that hypothetical greenhouse gas within the atmossphere, while y is (somewhat) proportional to the density near the surface of the earth. Why I wrote &#8220;somewhat&#8221; in brackets is explained below. <\/p>\n<p>The middle of the &#8220;plate&#8221; is at x=0, y=0 (please hover over the diagram) which is the &#8220;realistic&#8221; case of the idealized greenhouse model, i.e. the case where infrared absoptivity is 0.78 and the reflectivity of the earth is 0.3. The main point of this visualization is that linearily increasing x and y in the same way leads to an increase of the temperature. Or in other words, although raising x by a certain amount leads to cooling this effect is easily trumped by raising y by the same amount. <\/p>\n<p>As far as I learned from discussions with climate scientists the omission of non-infrared radiation in the climate models was mostly motivated by the fact that an abpsorption of non-infrared is mostly happening in the upper atmossphere (because methane is quickly rising (but there are also <a href=\"https:\/\/forum.azimuthproject.org\/discussion\/comment\/14864\/#Comment_14864\">circulations<\/a>)) and thus leading rather to a global cooling effect than a global warming effect and so it in particular doesn&#8217;t contribute to global warming. The enhanced simple model here thus confirms that if absorption is taking place in the upper athmossphere then this leads to cooling. The enhanced model however also displays that the <strong>contribution of methane that has not risen, i.e. methane that is close to the earth surface, is to warm <\/strong> upon absorption of non-infrared light and that the <strong>effect of warming is much stronger than the cooling effect in the upper athmosphere.<\/strong> Unfortunately I can&#8217;t say how much stronger for a given amount of methane, since for assessing this one would need to know more about the actual densities (see also discussion below and the comment about <a href=\"https:\/\/forum.azimuthproject.org\/discussion\/comment\/14864\/#Comment_14864\">circulations<\/a>). Nonetheless this is a <strong>quite<\/strong> disquieting observation. <\/p>\n<p>I had actually exchanged a couple of emails with <a href=\"https:\/\/www.ipcc.ch\/pdf\/assessment-report\/ar5\/wg1\/WG1AR5_Chapter08_FINAL.pdf\">Gunnar Myrhe, the lead author of this corresponding chapter in the IPCC report<\/a>, who confirmed that non-infrared light absorption in methane hasn&#8217;t sofar been taken into account, but that some people intended to work on the near-infrared absorption. He didn&#8217;t know about the UV absorption that I had found <a href=\"http:\/\/joseba.mpch-mainz.mpg.de\/spectral_atlas_data\/cross_sections_plots\/Alkanes+alkyl%20radicals\/Alkanes\/CH4_VUV_110-165nm_lin.jpg\">e.g. here<\/a> (unfortunately my email to Keller-Rudek and Moortgat from 2015 whether there is more data for methane especially in the range 170nm-750nm stayed unanswered) and thanked for pointing it out to him. He appeared to be very busy and as drowning in (a lot of administrative) work, so that I fear that those absorption lines still might not have been looked at. That is also why I decided to publish this now. I sent a copy of this post to Gunnar Myrhe, Zong-Liang Yang and John Baez in June 2017, where I pointed out that:<\/p>\n<blockquote><p>I have strong concerns that the estimations of the global warming potential of methane need to be better assessed and that the new value might eventually be very different then the current one. <\/p><\/blockquote>\n<p>&#8211; but I got no answer.<\/p>\n<p><!--more--><br \/>\n<\/br><\/p>\n<p>The Wikipedia entry on the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Idealized_greenhouse_model\">Idealized greenhouse model<\/a> is based on <a href=\"http:\/\/www.geo.utexas.edu\/courses\/387H\/Lectures\/chap2.pdf\">course notes<\/a> of the <a href=\"http:\/\/www.geo.utexas.edu\/courses\/387H\/default.htm\">course 387H: Physical Climatology<\/a> by instructor: <a href=\"http:\/\/www.jsg.utexas.edu\/researcher\/zong-liang_yang\/\">Zong-Liang Yang<\/a> where he used this simple model for motivating more complicated models with many layers:<\/p>\n<p><a href=\"http:\/\/www.randform.org\/blog\/wp-content\/2017\/02\/IdealizedGreenhouseEmissivity78.png\"><img src=\"http:\/\/www.randform.org\/blog\/wp-content\/2017\/02\/IdealizedGreenhouseEmissivity78.png\" alt=\"\" title=\"IdealizedGreenhouseEmissivity78\" width=\"450\" height=\"350\" class=\"aligncenter size-full wp-image-6606\" \/><\/a><br \/>\n<span style=\"font-size:80%\">Solution of idealized greenhouse model with emissivity by author <a href=\"https:\/\/en.wikipedia.org\/wiki\/Idealized_greenhouse_model#\/media\/File:IdealizedGreenhouseEmissivity00.png\">Incredio<\/a>, licence <a href=\"http:\/\/creativecommons.org\/licenses\/by-sa\/3.0\">CC BY-SA 3.0<\/a><\/span><\/p>\n<p>As said, I now enhanced this simple model in a certain way in order to get some insight into the temperature sensitivity of absorption of non-infrared light and it&#8217;s conversion into infrared light.  I currently don&#8217;t have access to a commercial computer algebra system and I sofar haven&#8217;t got along with the Sage syntax, so in particular solving spherical Navier-Stokes equations as done in <a href=\"https:\/\/en.wikipedia.org\/wiki\/General_circulation_model\">GCM&#8217;s<\/a> is quite out of reach. So I tried to use this enhanced model with <a href=\"https:\/\/en.wikipedia.org\/wiki\/Julia_(programming_language)\">Julia<\/a>. The code is below.<br \/>\nThe enhanced model is depicted in the following image:<br \/>\n<a href=\"http:\/\/www.randform.org\/blog\/wp-content\/2017\/02\/Greenhouse.png\"><img src=\"http:\/\/www.randform.org\/blog\/wp-content\/2017\/02\/Greenhouse.png\" alt=\"\" title=\"Greenhouse\" width=\"450\" height=\"370\" class=\"aligncenter size-full wp-image-6598\" \/><\/a><\/p>\n<p>The notation is as in the Wikipedia  article (see first image above), with a few alterations. That is $latex S=\\frac{1}{4}S_0 = 341 W\/m^2$ is here one fourth of the total incoming solar radiation (the factor one fourth is because the area of a sphere (i.e. here the earth) is four times the area of its circular shadow, this is e.g. <a href=\"http:\/\/www.randform.org\/blog\/?p=12\">motivated here<\/a>) and $latex \\alpha_p$ is set here $latex \\alpha_p = \\rho_s$ where I chose $latex \\rho$ as in &#8220;reflected&#8221;. I kept the notation for the subscripts as they were already used for the temperatures $latex T$ in Wikipedia, so the subscripts are $latex s$ as in &#8220;surface&#8221; and $latex a$ as in &#8220;atmosphere&#8221;. The symbol $latex \\epsilon_{IR}$  denotes the absorptivity\/emissitivity of infrared light in the atmossphere (in the Wikipedia entry just $latex \\epsilon$), likewise $latex \\epsilon_{UV}$ denotes the absorptivity\/emissitivity of ultraviolet and other noninfrared light, which is here now assumed to be reradiated as infrared light within the atmossphere. <\/p>\n<p>As there seems no &#8220;simplify&#8221; in Julia, I had to shuffle the algebraic expressions by hand, which is of course error-prone, but I hope there are no mistakes. Below the code and intermediate steps.<\/p>\n<p>Anyways if you look at the code then you see that $latex \\epsilon_{UV}$ and $latex \\rho_s$ are dependent on the variables delta1 and delta2. In the model they describe &#8220;small deviations&#8221; from some standard values. delta1 describes the deviation from the UV absorptivity and delta2 the deviation from the reflectivity of the earth. The idea behind is that if there is some greenhouse gas which absorbs noninfrared and reradiates this as infrared then as delta1 increases the noninfrared absorptivity of the atmosshpere, this is as if there would be &#8220;more of that absorbing&#8221; greenhouse in the atmossphere. So in the beginning I wrote the word &#8220;somewhat&#8221; in brackets, because I don&#8217;t know the exact relations between absorptivity and density of a greenhouse gas, apart from this I don&#8217;t know much about actual densities (see comment about circulation and <a href=\"http:\/\/www.randform.org\/blog\/?p=5965\">this post<\/a>). Likewise delta2 could describe a &#8220;more of that greenhouse gas&#8221; at the surface of the earth. In the diagram delta1 is x and delta2 is y.<\/p>\n<p>#code is GPL by Nadja Kutz<br \/>\nS= 341.5<br \/>\ndeltaAt = 0.0<br \/>\ndeltaSur = 0.0<br \/>\nepsuv = 0.0+deltaAt<br \/>\nepsir = 0.78<br \/>\n#epsir=0.78 is corresponding to usual CO2forcing<br \/>\nrhos = 0.3-deltaSur<br \/>\nsigma = 0.00000005670367<br \/>\nTs= (1\/((1-0.5*epsir)*sigma)*(((1-rhos)*(1-epsuv) -0.5*(-epsuv*(1-rhos)-rhos + (1-epsuv)^2*rhos))*S))^0.25<br \/>\nTa=(1\/((epsuv + epsuv*(1-epsuv)*rhos +  epsir)*sigma)*((1-(1-epsuv)^2*rhos)*S-(1-epsir)*sigma*Ts^4))^0.25<br \/>\nprintln(&#8220;deltaAt=&#8221;,deltaAt,&#8221;  deltaSur=&#8221;,deltaSur,&#8221;  Ts=&#8221;,Ts,&#8221;  Ta=&#8221;,Ta)<\/p>\n<p>#calculation see image Greenhouse.svg<br \/>\n#Term 1<br \/>\n-(1-(1-epsuv)^2*rhos)*S + (epsuv + epsuv*(1-epsuv)*rhos +  epsir)*sigma*Ta^4 + (1-epsir)*sigma*Ts^4<br \/>\n#Term 2<br \/>\n(1-rhos)(1-epsuv)*S + (epsuv + epsuv*(1-epsuv)*rhos + epsir)*sigma*Ta^4 &#8211; sigma*Ts^4<\/p>\n<p>#Term1 + Term2 !=0<br \/>\n-epsuv (1-rhos)*S -rhos*S + (1-epsuv)^2*rhos*S + 2* (epsuv + epsuv*(1-epsuv)*rhos +  epsir)*sigma*Ta^4 + (-epsir)*sigma*Ts^4<\/p>\n<p>#Solve Term1 + Term2 !=0 for Ta<br \/>\nTa^4= -1\/(2*(epsuv + epsuv*(1-epsuv)*rhos +  epsir)*sigma)*((-epsuv*(1-rhos)-rhos + (1-epsuv)^2*rhos)*S + (-epsir)*sigma*Ts^4)<\/p>\n<p>#Into Term 2<br \/>\n(1-rhos)(1-epsuv)*S -0.5*((-epsuv*(1-rhos)-rhos + (1-epsuv)^2*rhos)*S + (-epsir)*sigma*Ts^4)- sigma*Ts^4<\/p>\n<p>#Simplify<br \/>\n((1-rhos)(1-epsuv) -0.5*(-epsuv*(1-rhos)-rhos + (1-epsuv)^2*rhos))*S + (0.5*epsir-1)*sigma*Ts^4<\/p>\n<p>#Solve for Ts<br \/>\nTs= (1\/((1-0.5*epsir)*sigma)*(((1-rhos)*(1-epsuv) -0.5*(-epsuv*(1-rhos)-rhos + (1-epsuv)^2*rhos))*S))^0.25<\/p>\n<p>The plot of the function (with some help from Tim) can be got from this code:<\/p>\n<p>function Surfacetemp(deltaAt,deltaSur)<br \/>\nS= 341.5<br \/>\nepsuv = 0.0+deltaAt<br \/>\nepsir = 0.78<br \/>\n#epsir=0.78 wird als CO2forcing angenommen<br \/>\nrhos = 0.3-deltaSur<br \/>\nsigma = 0.00000005670367<br \/>\n(1\/((1-0.5*epsir)*sigma)*(((1-rhos)*(1-epsuv) -0.5*(-epsuv*(1-rhos)-rhos + (1-epsuv)^2*rhos))*S))^0.25<br \/>\n#Ta=(1\/((epsuv + epsuv*(1-epsuv)*rhos +  epsir)*sigma)*((1-(1-epsuv)^2*rhos)*S-(1-epsir)*sigma*Ts^4))^0.25<br \/>\nend<br \/>\nusing Plots<\/p>\n<p>plotly()<\/p>\n<p>x = y = linspace(-0.1, 0.1, 20)<br \/>\nplot(x,y,Surfacetemp,st=:surface)<\/p>\n<p>thanks to Tim for helping me deciphering the Julia documentation<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Rotatable with your mouse after you click on &#8220;Read the rest of this entry&#8220;. Most of randform readers might have heard that the socalled greenhouse effect is one of the main causes of global warming. The effect is not easy to understand. There are two posts which give a nice intro to the greenhouse effect [&hellip;]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[3,25,15,26,7,6,14],"tags":[],"_links":{"self":[{"href":"https:\/\/www.randform.org\/blog\/index.php?rest_route=\/wp\/v2\/posts\/6597"}],"collection":[{"href":"https:\/\/www.randform.org\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.randform.org\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.randform.org\/blog\/index.php?rest_route=\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.randform.org\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=6597"}],"version-history":[{"count":77,"href":"https:\/\/www.randform.org\/blog\/index.php?rest_route=\/wp\/v2\/posts\/6597\/revisions"}],"predecessor-version":[{"id":7660,"href":"https:\/\/www.randform.org\/blog\/index.php?rest_route=\/wp\/v2\/posts\/6597\/revisions\/7660"}],"wp:attachment":[{"href":"https:\/\/www.randform.org\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6597"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.randform.org\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6597"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.randform.org\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6597"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}