(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 83332, 2309]*) (*NotebookOutlinePosition[ 97920, 2843]*) (* CellTagsIndexPosition[ 97876, 2839]*) (*WindowFrame->Normal*) Notebook[{ Cell["Energy Balance Model of the Earth", "Title"], Cell[TextData[{ "CEGEOS/CHEG 4/598 ", StyleBox[" Global Climate modeling", FontSlant->"Italic"], " \nLecture #4, 01/21/04" }], "Subsubsection"], Cell[TextData[{ "This notebook has been written in ", StyleBox["Mathematica ", FontSlant->"Italic"], "by \n\n", StyleBox["Mark J. McCready\nProfessor and Chair of Chemical Engineering\n\ University of Notre Dame\nNotre Dame IN 46556\nUSA", FontSize->14], "\n\n", ButtonBox["Mark.J.McCready.1@nd.edu", ButtonData:>{ URL[ "mailto:mccready.1@nd.edu"], None}, ButtonStyle->"Hyperlink"], "\n", ButtonBox["http://www.nd.edu/~mjm/", ButtonData:>{ URL[ "http://www.nd.edu/~mjm/"], None}, ButtonStyle->"Hyperlink"], "\n\n\nIt is copyrighted to the extent allowed by whatever laws pertain to \ the World Wide Web and the Internet.\n\nI would hope that as a professional \ courtesy, that if you use it, that this notice remain visible to other users. \ \nThere is no charge for copying and dissemination \n\nVersion: 01/21/04\n\ More recent versions of this notebook might be available at the web site:\n", ButtonBox["http://www.nd.edu/~mjm/energy_balance_lect_4.nb\n\n", ButtonData:>{ URL[ "http://www.nd.edu/~mjm/energy_balance_lect_4.nb"], None}, ButtonStyle->"Hyperlink"], "The Excel spreadsheet for the one-dimensional energy balance calculations, \ in case you don't want to use this notebook for calculations is", Cell[BoxData[ FormBox[ ButtonBox[\(\(\ \)\(\(\(http\)\(:\)\) // \(www . nd . edu/\(\(~\)\(mjm\)\)\)/energy_balance \(\( _calc . xls\)\(\ \)\)\)\), ButtonData:>{ URL[ "http://www.nd.edu/~mjm/energy_balance_calc.xls "], None}, ButtonStyle->"Hyperlink"], TraditionalForm]]] }], "Text"], Cell[TextData[{ "Reference: ", StyleBox["K. McGuffie and A. Henderson-Sellers, A Climate Modeling Primer, \ 1997.", FontFamily->"Times New Roman"], StyleBox["\n\n", FontFamily->"Times New Roman", FontSize->14, FontWeight->"Plain"], StyleBox["F. P. Incropera and D. P. DeWitt, Fundamentals of Heat and Mass \ Transfer, 1995.", FontFamily->"Times New Roman"] }], "Subsubsection"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematical m", FontSlant->"Italic"], "odels in science and Engineering" }], "Subtitle"], Cell[CellGroupData[{ Cell["What is a model?", "Section"], Cell[TextData[{ "Generally if we are thinking of a model, we mean a set of mathematical \ equations that can ", StyleBox["predict", FontSlant->"Italic"], " behavior, Simply interpolating within existing data could also be \ modeling, but this is less useful.\n\nTo be able to ", StyleBox["predict", FontSlant->"Italic"], ", a model usually needs to have some finite degree of match with the \ physical laws, conservation of mass, conservation of energy, conservation of \ momentum, applicable rate processes.\n\nPredictions for a \"cold winter\" are \ often based on correlations of data from previous events. Note that these \ are often wrong and we don't know why. Another desirable property of a model \ is that we should be able to estimate the expected ", StyleBox["accuracy", FontSlant->"Italic"], "!" }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Why do we need models?", "Section"], Cell[TextData[{ "Engineers use models for design of new devices, structures and processes. \ For example if we want to build a steel structure, say a \"truss\", we need \ to know how big to make the beams, the best angles and the best answer for \ \"how many\" or which size. The word \"best\" is perhaps being overused. \ Why, because by \"best\" we mean \"optimal\" and to figure this out we need \ to define the criteria for optimal and then do calculations or experiments \ until we find it. Of course calculations sound, at least, easier, quicker \ and cheaper. (If we have a good model, if it can be solved sufficiently \ quickly ...)\n\nEngineers also use models to help understand experimental \ data that could arise from a study of fundamental questions, (e.g., rates and \ mechanisms of chemical reactions, equilibrium solubilities of complex \ mixtures, molecular transport across a membrane) or developmental questions \ (rates of drug uptakes and lifetimes in people). Even if the entire \ experimental space (vary all pertinent variables) could be explored, \ experiments typically give the answer at only one point, so you would need to \ piece together the behavior between the points. Further, if you don't ", StyleBox["understand ", FontSlant->"Italic"], "why certain behavior is occurring, your uncertainty could be lead to \ problems. If you have a model, based on good fundamentals, often different \ qualitative trends can be identified that improve understanding. " }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["How do Earth Scientists use models?", "Section"], Cell["\<\ How will the earth temperature change in the future ? Difficult to \ get from an experiment!!! Even regarding the past: What causes the ice ages? How much do changes in \ solar output change the earth temperature, what is the effect of \ anthropogenic CO2, what is the origin of El Nino and how does it affect \ weather, how come the ocean currents are as they are and what could change \ them, if the ocean currents are altered, what would be the effect on the \ climate? \ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "What will it take to model the ", StyleBox["temperature", FontSlant->"Italic"], " of the earth?" }], "Section"], Cell[TextData[{ "It is relatively easy to make a case that if we want to predict or \ interpret the temperature of the earth, we need to keep track of how much \ energy the earth receives from the sun. If we know some basic principles of \ heat transfer, we would realize the heat would also be leaving earth. The \ mechanism of heat transfer for each of these processes is called \ \"radiation\", which is transfer of energy in the form of electromagnetic \ radiation.\n\nTo formally (and accurately) keep track of energy, we do what \ is called an", StyleBox[" energy balance", FontSlant->"Italic"], " that relies on the principle of ", StyleBox["conservation of energy -- ", FontSlant->"Italic"], " which is also known as the first law of thermodynamics. " }], "Text"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Energy balance for the earth", "Subtitle"], Cell[CellGroupData[{ Cell["Background and Derivation ", "Section"], Cell[CellGroupData[{ Cell["In terms of words, the energy balance for the earth is", "Subsection"], Cell["\<\ Rate of accumulation of energy in the earth = Rate at which energy \ is absorbed by the earth - rate at which energy is radiated by the \ earth.\ \>", "Subsubsection"], Cell[TextData[{ "We might just mention a few ", StyleBox["things that arise from thinking too hard", FontSlant->"Italic"], ". The accumulation of energy will occur in only in the atmosphere, the \ top part of the earth and some portion of the ocean. We have some difficulty \ in quantifying exactly how much heat would be held by this region. There is \ also the issue of internal earth heat that could leak into the surface \ region, The earth is hot inside and there is radioactive decay. However, \ this source of heat is much less than solar radiation. " }], "Text"], Cell[TextData[{ "If we want to write some symbols for the energy balance\n\n", Cell[BoxData[ \(TraditionalForm\`d\ [total\ energy\ in\ this\ top\ layer\ of\ the\ \ earth]\/dt\)]], " = Rate of energy that the earth absorbs - rate at which earth is \ radiated by the earth" }], "Subsubsection"], Cell[TextData[{ "Can the two terms on the right side be equal? If so, the earth's energy \ is in what is called a \"", StyleBox["steady state", FontWeight->"Bold"], "\" that is not changing with time. This does not mean there is no flow of \ energy to and from the earth, or that there are no dynamics of energy \ transfer within the energy containing region of the earth. \n\nWe certainly \ hope that earth is at a steady state, or very close. Of course the data and \ modeling that demonstrate evidence of change in the temperature of the earth \ suggest that the earth is not quite at a steady state. We will ask below ", StyleBox["how perilously perched are we ", FontVariations->{"Underline"->True}], "?\n\nIf the two terms are to be steady and equal, we should find out what \ affects their behavior." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ For a given flux[energy/area/time] from the sun what affects \ absorbance?\ \>", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Show[ Graphics[{RGBColor[0.002, .8, 0.01], PointSize[ .2], Table[Point[{i, 2}], {i, 1, 1}]}]];\)\)], "Input", CellLabel->"In[105]:="], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.47619 0.0147151 0.147151 [ [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .002 .8 .01 r .2 w .5 .30902 Mdot % End of Graphics MathPictureEnd \ \>"], "Graphics", CellLabel->"From In[105]:=", ImageSize->{288, 177.938}, ImageMargins->{{0, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHgo8F>o003oHklQ Hkl00?mS_b5S_`00of>o8F>o003oHklQHkl00?mS_b5S_`00of>o8F>o003oHklQHkl00?mS_b5S_`00 of>o8F>o003oHklQHkl00?mS_b5S_`00of>o8F>o003oHklQHkl00?mS_b5S_`00of>o8F>o003oHklQ Hkl00?mS_b5S_`00of>o8F>o003oHklQHkl00?mS_b5S_`00of>o8F>o003oHklQHkl00?mS_b5S_`00 of>o8F>o003oHklQHkl00?mS_b5S_`00of>o8F>o003oHklQHkl00?mS_b5S_`00of>o8F>o003oHklQ Hkl00?mS_b5S_`00of>o8F>o003oHklQHkl00?mS_b5S_`00of>o8F>o003oHklQHkl00?mS_b5S_`00 of>o8F>o003oHklQHkl00?mS_b5S_`00of>o8F>o003oHklQHkl00?mS_b5S_`00of>o8F>o003oHklQ Hkl00?mS_b5S_`00of>o8F>o003oHklQHkl00?mS_b5S_`00of>o8F>o003oHklQHkl00?mS_b5S_`00 of>o8F>o003oHklQHkl00?mS_b5S_`00of>o8F>o003oHklQHkl00?mS_b5S_`00RV>o2`o0026 HklC0b27Hkl008AS_aL388ES_`00PV>o6`o0020HklO0b21Hkl007mS_b43881S_`00OF>o9@

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Of course if the disk were shiny silver we would reflect \ almost all of the light giving even less absorbance. Interestingly, the radiation of the earth, or any object, also depends on the \ color in the same way as the apparent absorbance. A dark color radiates more \ strongly than a light color. The effect of \"color\" also depends on the temperature of emission, or the \ wavelength of the incident light.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["General equation for radiation", "Subsection"], Cell[TextData[{ "The", StyleBox[" Stefan-Boltzman", FontSlant->"Italic"], " equation governs radiation:\n\n", Cell[BoxData[ \(TraditionalForm\`E\_b\)], FontSize->14], StyleBox["= \[Epsilon] \[Sigma] ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`T\^4\)], FontSize->14], ", \[Sigma]=5.670 ", Cell[BoxData[ \(TraditionalForm\`10\^\(-8\)\)]], "W/", Cell[BoxData[ \(TraditionalForm\`m\^2\)]], Cell[BoxData[ \(TraditionalForm\`K\^4\)]], ", the Stefan-Boltzman constant, the \"emissivity\", \[Epsilon], depends \ on the nature of the surface and its color.\n\n\[Epsilon] could be <.1 for a \ shiny metal. Approaches 1 for dark, dull surfaces. The energy flow/area is \ the energy flux, ", Cell[BoxData[ \(TraditionalForm\`E\_b\)], FontSize->14], "." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Different possible steady state earth temperatures", "Subsection"], Cell["\<\ So what if the earth is initially covered with ice, will the \ absorbance be the same if the earth is initially ice free? No, so we could envision a value for absorbance for an icy earth -- which \ gives one absorbance. However, if the earth had no ice, we would get a different absorbance. This \ is a much higher value. These two different absorbances would be different and would have different \ temperatures associated with them! We will have to check this with numbers, but the possibility is intriguing! \ \ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Fill in the accumulation term of the Energy Balance", "Subsection"], Cell[TextData[{ "Rate of change of total energy\n\nd", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"[", RowBox[{"\[Rho]", " ", FormBox[\(\(\(\(C\_p\) T\ \ V\)\(]\)\)\(\ \)\), "TraditionalForm"]}]}], "dt"], TraditionalForm]]], ", where \[Rho] is the density, [mass/volume] and ", Cell[BoxData[ \(TraditionalForm\`\(\(C\_p\ [energy/mass\ - \ K]\)\(\ \)\)\)]], "is the \"heat capacity\" of the material we are doing the energy balance \ on. We could make some arguments that what we really need to think about \ holding the heat is the top layer of the oceans. Also, we can take \[Rho], V \ and ", Cell[BoxData[ \(TraditionalForm\`C\_\(\(P\)\(\[IndentingNewLine]\) \)\)]], " as constants. \n\nThus our left side becomes:\n\n", Cell[BoxData[ FormBox[ RowBox[{"\[Rho]", " ", FormBox[\(\(C\_p\)\(\ \)\(V\)\(\ \)\), "TraditionalForm"]}], TraditionalForm]]], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"d", " ", FormBox[\(\(T\)\(\ \ \)\), "TraditionalForm"]}], "dt"], TraditionalForm]]], "\n\n=\n\nrate of energy flux to the earth * area - rate of radiation \ out." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Characteristic time for temperature change", "Subsection"], Cell[TextData[{ "A very useful engineering tool is called dimensional reasoning. The \ origin of this is thought comes form us not being able to compare apples to \ oranges, to know how much is a lot, how far it is to Chicago or if this is \ far ( or long), or a good way to judge the competence of people you meet and \ have to deal with. For all of these you make comparisons based on specific \ \"dimensions\", length, mass, time, \"intelligence\", etc.\n\nIf we are \ worried how precarious the earth temperature is. How can we get the simplest \ estimate?\n\nFind the rate at which the temperature is expected to be \ changing. \n \nRate at which energy is being absorbed by the earth [=] \ energy/time, Q = 342 W/", Cell[BoxData[ \(TraditionalForm\`m\^2\)]], "* ", Cell[BoxData[ \(5.0845070422535217`*^8\)]], Cell[BoxData[ \(TraditionalForm\`km\^2\)]], "\n\nCapacitance of the ocean upper layer ", Cell[BoxData[ FormBox[ RowBox[{"\[Rho]", " ", FormBox[\(\(C\_p\)\(\ \)\(V\)\(\ \)\), "TraditionalForm"]}], TraditionalForm]]], "[=] energy/K = 1.05 ", Cell[BoxData[ \(TraditionalForm\`10\^23\)]], " J/K.\n\nThus the ratio, ", Cell[BoxData[ FormBox[ RowBox[{"\[Rho]", " ", FormBox[\(\(C\_p\)\(\ \)\(V\)\(\ \)\), "TraditionalForm"]}], TraditionalForm]]], "/Q has the dimensions of time/K, which seems to tell us the time it will \ take to make the temperature change. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(areaofearth\)\(\ \)\(=\)\(N[ 4\ Pi\ \((12756/2\ \ km)\)^2\ ]\)\(\ \)\)\)], "Input", CellLabel->"In[11]:="], Cell[BoxData[ \(TraditionalForm\`5.111859325225255`*^8\ km\^2\)], "Output", CellLabel->"Out[11]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(heatin = 342\ \(J/s\)/m^2\ \ areaofearth\ \ \ \((1000\ m/km)\)^2\)], "Input", CellLabel->"In[12]:="], Cell[BoxData[ \(TraditionalForm\`\(1.748255889227037`*^17\ J\)\/s\)], "Output", CellLabel->"Out[12]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"capacitance", " ", "=", " ", RowBox[{"1.05", FormBox[\(10\^23\), "TraditionalForm"], \(J/K\)}]}]], "Input", CellLabel->"In[11]:="], Cell[BoxData[ \(\(1.0499999999999999`*^23\ J\)\/K\)], "Output", CellLabel->"Out[11]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(timetoheat = capacitance/heatin\)], "Input", CellLabel->"In[12]:="], Cell[BoxData[ \(\(603829.5183943236`\ s\)\/K\)], "Output", CellLabel->"Out[12]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(timetoheat\ 1\ \(hr/3600\)/s\)], "Input", CellLabel->"In[13]:="], Cell[BoxData[ \(\(167.730421776201`\ hr\)\/K\)], "Output", CellLabel->"Out[13]="] }, Open ]], Cell["\<\ So this is fast enough to worry about. Heating from the Sun is \ significant. Now if we could use the entire ocean (the mixing time is 1500 \ years), rate of change would be considerably slower. Of course there is \ radiation out (fortunately.) \ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ You might be asking, did we make something up, or does it have some \ validity?\ \>", "Subsubsection"], Cell[TextData[{ "The rest of the energy balance\n\n\n", Cell[BoxData[ FormBox[ RowBox[{"\[Rho]", " ", FormBox[\(\(C\_p\)\(\ \)\(V\)\(\ \)\), "TraditionalForm"]}], TraditionalForm]]], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"d", " ", FormBox[\(\(T\)\(\ \ \)\), "TraditionalForm"]}], "dt"], TraditionalForm]]], " = (1- \[Alpha]) ", Cell[BoxData[ \(TraditionalForm\`S\/4\ A\)]], "- \[Epsilon] ", Cell[BoxData[ \(TraditionalForm\`\[Tau]\_a\)]], " \[Sigma] ", Cell[BoxData[ \(TraditionalForm\`\(T\^4\) A\)]], ",\n\nwhere S = solar flux in Watts/", Cell[BoxData[ \(TraditionalForm\`m\^2\)]], ", A, is the surface area of the earth, ", Cell[BoxData[ \(TraditionalForm\`\[Tau]\_a\)]], "is the infrared transmissivity of the atmosphere, \[Alpha] is the albedo \ which tells how much of the incident radiation is reflected back by the \ surface (a key issue depending upon how much ice is present).\n\nLet's write \ the equation in ", StyleBox["Mathematica,", FontSlant->"Italic"] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"eq1", "=", RowBox[{\(\[Rho]\ \(C\_p\) V\ D[T[t], t]\), " ", "-", " ", RowBox[{\((1 - \[Alpha])\), FormBox[\(S\/4\ A\), "TraditionalForm"]}], "+", RowBox[{"\[Epsilon]", FormBox[\(\[Tau]\_a\), "TraditionalForm"], "\[Sigma]", FormBox[\(\(T[t]\^4\) A\), "TraditionalForm"], " "}]}]}]], "Input", CellLabel->"In[13]:="], Cell[BoxData[ FormBox[ RowBox[{\(A\ \[Epsilon]\ \[Sigma]\ \[Tau]\_a\ \(T(t)\)\^4\), "-", \(1\/4\ A\ S\ \((1 - \[Alpha])\)\), "+", RowBox[{"V", " ", "\[Rho]", " ", \(C\_p\), " ", RowBox[{ SuperscriptBox["T", "\[Prime]", MultilineFunction->None], "(", "t", ")"}]}]}], TraditionalForm]], "Output", CellLabel->"Out[13]="] }, Open ]], Cell["If we divide through to get just the temperature derivative,", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(eq2 = Expand[\(\(eq1/\[Rho]\)/\ C\_p\)/V]\)], "Input", CellLabel->"In[14]:="], Cell[BoxData[ FormBox[ RowBox[{\(\(A\ \[Epsilon]\ \[Sigma]\ \[Tau]\_a\ \(T(t)\)\^4\)\/\(V\ \ \[Rho]\ C\_p\)\), "+", RowBox[{ SuperscriptBox["T", "\[Prime]", MultilineFunction->None], "(", "t", ")"}], "+", \(\(A\ S\ \[Alpha]\)\/\(4\ V\ \[Rho]\ C\_p\)\), "-", \(\(A\ S\)\/\(4\ V\ \[Rho]\ C\_p\)\)}], TraditionalForm]], "Output", CellLabel->"Out[14]="] }, Open ]], Cell["The term of the incident radiation is", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(term1 = eq2[\([1]\)]\)], "Input", CellLabel->"In[20]:="], Cell[BoxData[ \(TraditionalForm\`\(-\(\(A\ S\)\/\(4\ V\ \[Rho]\ C\_p\)\)\)\)], "Output",\ CellLabel->"Out[20]="] }, Open ]], Cell["\<\ If we look at just this first term, its inverse is exactly the \ quantity we calculated above. Dimension reasoning can tell us useful \ info!!\ \>", "Text"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Steady state energy balance, no spatial variation.", "Section"], Cell["\<\ If we assume that a steady state exists, so we drop the time \ derivative, we can calculate a steady state temperature for the earth because \ T appears explicitly. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(sseq1 = \ eq2 /. \(T'\)[t] -> 0\)], "Input", CellLabel->"In[21]:="], Cell[BoxData[ \(TraditionalForm\`\(A\ \[Epsilon]\ \[Sigma]\ \[Tau]\_a\ \ \(T(t)\)\^4\)\/\(V\ \[Rho]\ C\_p\) + \(A\ S\ \[Alpha]\)\/\(4\ V\ \[Rho]\ C\_p\ \) - \(A\ S\)\/\(4\ V\ \[Rho]\ C\_p\)\)], "Output", CellLabel->"Out[21]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ssans1 = Solve[sseq1 == 0, \ T[t]]\)], "Input", CellLabel->"In[22]:="], Cell[BoxData[ \(TraditionalForm\`{{T( t) \[Rule] \(-\(\@\(S - S\ \[Alpha]\)\%4\/\(\@2\ \@\[Epsilon]\%4\ \ \@\[Sigma]\%4\ \@\(\[Tau]\_a\)\%4\)\)\)}, {T( t) \[Rule] \(-\(\(\[ImaginaryI]\ \@\(S - S\ \ \[Alpha]\)\%4\)\/\(\@2\ \@\[Epsilon]\%4\ \@\[Sigma]\%4\ \ \@\(\[Tau]\_a\)\%4\)\)\)}, {T( t) \[Rule] \(\[ImaginaryI]\ \@\(S - S\ \[Alpha]\)\%4\)\/\(\@2\ \@\ \[Epsilon]\%4\ \@\[Sigma]\%4\ \@\(\[Tau]\_a\)\%4\)}, {T( t) \[Rule] \@\(S - S\ \[Alpha]\)\%4\/\(\@2\ \@\[Epsilon]\%4\ \@\ \[Sigma]\%4\ \@\(\[Tau]\_a\)\%4\)}}\)], "Output", CellLabel->"Out[22]="] }, Open ]], Cell[TextData[{ "If we put in some numbers we have:\n \n S = 1370 W/", Cell[BoxData[ \(TraditionalForm\`m\^2\)]], "\n A = ", Cell[BoxData[ \(5.0845070422535217`*^8\ km\^2\)]], ", but it cancels\n \[Sigma]=5.670 ", Cell[BoxData[ \(TraditionalForm\`10\^\(-8\)\)]], "W/", Cell[BoxData[ \(TraditionalForm\`m\^2\)]], Cell[BoxData[ \(TraditionalForm\`K\^4\)]], "\n ", Cell[BoxData[ RowBox[{"\[Epsilon]", FormBox[\(\[Tau]\_a\), "TraditionalForm"]}]]], " = .62, or \[Epsilon] = .98 and ", Cell[BoxData[ \(TraditionalForm\`\[Tau]\_a\)]], "=.626\n \[Alpha] = .3" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ssans1\)], "Input", CellLabel->"In[19]:="], Cell[BoxData[ \({{T[ t] \[Rule] \(-\(\((S - S\ \[Alpha])\)\^\(1/4\)\/\(\@2\ \[Epsilon]\ \^\(1/4\)\ \[Sigma]\^\(1/4\)\ \[Tau]\_a\%\(1/4\)\)\)\)}, {T[ t] \[Rule] \(-\(\(\[ImaginaryI]\ \((S - S\ \[Alpha])\)\^\(1/4\)\)\ \/\(\@2\ \[Epsilon]\^\(1/4\)\ \[Sigma]\^\(1/4\)\ \[Tau]\_a\%\(1/4\)\)\)\)}, \ {T[t] \[Rule] \(\[ImaginaryI]\ \((S - S\ \[Alpha])\)\^\(1/4\)\)\/\(\@2\ \ \[Epsilon]\^\(1/4\)\ \[Sigma]\^\(1/4\)\ \[Tau]\_a\%\(1/4\)\)}, {T[ t] \[Rule] \((S - S\ \[Alpha])\)\^\(1/4\)\/\(\@2\ \ \[Epsilon]\^\(1/4\)\ \[Sigma]\^\(1/4\)\ \[Tau]\_a\%\(1/4\)\)}}\)], "Output", CellLabel->"Out[19]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"PowerExpand", "[", RowBox[{"ssans1", "/.", RowBox[{"{", RowBox[{\(S -> 1370\ \(\(J/s\)/m\)/m\), ",", " ", RowBox[{"\[Sigma]", "->", RowBox[{"5.670", FormBox[\(10\^\(-8\)\), "TraditionalForm"], \(\(\(\(J/s\)/m\)/m\)/K\^4\)}]}], ",", " ", \(\[Epsilon] -> .98\), ",", RowBox[{ FormBox[\(\[Tau]\_a\), "TraditionalForm"], "->", ".626"}], ",", \(\[Alpha] -> .3\)}], "}"}]}], "]"}]], "Input", CellLabel->"In[23]:="], Cell[BoxData[ \(TraditionalForm\`{{T(t) \[Rule] \(-288.13352143528607`\)\ K}, {T( t) \[Rule] \(-288.13352143528607`\)\ \[ImaginaryI]\ K}, {T( t) \[Rule] 288.13352143528607`\ \[ImaginaryI]\ K}, {T(t) \[Rule] 288.13352143528607`\ K}}\)], "Output", CellLabel->"Out[23]="] }, Open ]], Cell["\<\ Without sweating any Math, we like the last of these best, an earth \ temperature of 288 K = about 59 F, -- exactly what we claim!!\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(288.1*9/5 - 459.4\)], "Input", CellLabel->"In[24]:="], Cell[BoxData[ \(TraditionalForm\`59.180000000000064`\)], "Output", CellLabel->"Out[24]="] }, Open ]], Cell[TextData[{ "If however there is no absorbance of IR radiation by the atmosphere, the \ earth would radiate very efficiently at its surface temperature. To model \ this we take ", Cell[BoxData[ FormBox[ FormBox[\(\[Tau]\_a\), "TraditionalForm"], TraditionalForm]]], " = 1. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"PowerExpand", "[", RowBox[{"ssans1", "/.", RowBox[{"{", RowBox[{\(S -> 1370\ \(\(J/s\)/m\)/m\), ",", " ", RowBox[{"\[Sigma]", "->", RowBox[{"5.670", FormBox[\(10\^\(-8\)\), "TraditionalForm"], \(\(\(\(J/s\)/m\)/m\)/K\^4\)}]}], ",", " ", \(\[Epsilon] -> .98\), ",", RowBox[{ FormBox[\(\[Tau]\_a\), "TraditionalForm"], "->", "1"}], ",", \(\[Alpha] -> .3\)}], "}"}]}], "]"}]], "Input", CellLabel->"In[25]:="], Cell[BoxData[ \(TraditionalForm\`{{T(t) \[Rule] \(-256.29336920838574`\)\ K}, {T( t) \[Rule] \(-256.29336920838574`\)\ \[ImaginaryI]\ K}, {T( t) \[Rule] 256.29336920838574`\ \[ImaginaryI]\ K}, {T(t) \[Rule] 256.29336920838574`\ K}}\)], "Output", CellLabel->"Out[25]="] }, Open ]], Cell["\<\ We see that the earth average temperature is below the freezing \ point of water. So that the effect of atmospheric absorption is very \ substantial! The \"albedo\" could also vary. If this is adjusted to .6, you get a very \ cold earth. To get this high reflectivity, it would take an ice coverered \ surface and maybe a lot of clouds. So a frozen earth steady-state is possible! (with this simple model) \ \>", \ "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"PowerExpand", "[", RowBox[{"ssans1", "/.", RowBox[{"{", RowBox[{\(S -> 1370\ \(\(J/s\)/m\)/m\), ",", " ", RowBox[{"\[Sigma]", "->", RowBox[{"5.670", FormBox[\(10\^\(-8\)\), "TraditionalForm"], \(\(\(\(J/s\)/m\)/m\)/K\^4\)}]}], ",", " ", \(\[Epsilon] -> .98\), ",", RowBox[{ FormBox[\(\[Tau]\_a\), "TraditionalForm"], "->", "1"}], ",", \(\[Alpha] -> .6\)}], "}"}]}], "]"}]], "Input", CellLabel->"In[26]:="], Cell[BoxData[ \(TraditionalForm\`{{T(t) \[Rule] \(-222.8321538719781`\)\ K}, {T( t) \[Rule] \(-222.8321538719781`\)\ \[ImaginaryI]\ K}, {T( t) \[Rule] 222.8321538719781`\ \[ImaginaryI]\ K}, {T(t) \[Rule] 222.8321538719781`\ K}}\)], "Output", CellLabel->"Out[26]="] }, Open ]], Cell["\<\ A maybe more interesting calculation is the effect of solar flux. \ If the flux is 3 percent larger, the temperature goes up 2 degrees. \ \>", \ "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"PowerExpand", "[", RowBox[{"ssans1", "/.", RowBox[{"{", RowBox[{\(S -> 1.03*1370\ \(\(J/s\)/m\)/m\), ",", " ", RowBox[{"\[Sigma]", "->", RowBox[{"5.670", FormBox[\(10\^\(-8\)\), "TraditionalForm"], \(\(\(\(J/s\)/m\)/m\)/K\^4\)}]}], ",", " ", \(\[Epsilon] -> .98\), ",", RowBox[{ FormBox[\(\[Tau]\_a\), "TraditionalForm"], "->", ".626"}], ",", \(\[Alpha] -> .3\)}], "}"}]}], "]"}]], "Input", CellLabel->"In[27]:="], Cell[BoxData[ \(TraditionalForm\`{{T(t) \[Rule] \(-290.27062844534254`\)\ K}, {T( t) \[Rule] \(-290.27062844534254`\)\ \[ImaginaryI]\ K}, {T( t) \[Rule] 290.27062844534254`\ \[ImaginaryI]\ K}, {T(t) \[Rule] 290.27062844534254`\ K}}\)], "Output", CellLabel->"Out[27]="] }, Open ]], Cell["\<\ However, this would make more water vapor and thus increase the \ albedo, say to .4\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"PowerExpand", "[", RowBox[{"ssans1", "/.", RowBox[{"{", RowBox[{\(S -> 1.03*1370\ \(\(J/s\)/m\)/m\), ",", " ", RowBox[{"\[Sigma]", "->", RowBox[{"5.670", FormBox[\(10\^\(-8\)\), "TraditionalForm"], \(\(\(\(J/s\)/m\)/m\)/K\^4\)}]}], ",", " ", \(\[Epsilon] -> .98\), ",", RowBox[{ FormBox[\(\[Tau]\_a\), "TraditionalForm"], "->", ".626"}], ",", \(\[Alpha] -> .35\)}], "}"}]}], "]"}]], "Input", CellLabel->"In[28]:="], Cell[BoxData[ \(TraditionalForm\`{{T(t) \[Rule] \(-284.9422979155387`\)\ K}, {T( t) \[Rule] \(-284.9422979155387`\)\ \[ImaginaryI]\ K}, {T( t) \[Rule] 284.9422979155387`\ \[ImaginaryI]\ K}, {T(t) \[Rule] 284.9422979155387`\ K}}\)], "Output", CellLabel->"Out[28]="] }, Open ]], Cell[TextData[{ "But the temperature would go down. But wait, the IR absorbance would go \ back up, say ", Cell[BoxData[ \(TraditionalForm\`\[Tau]\_a\)]], "=0.55. Now it is much warmer! " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"PowerExpand", "[", RowBox[{"ssans1", "/.", RowBox[{"{", RowBox[{\(S -> 1.03*1370\ \(\(J/s\)/m\)/m\), ",", " ", RowBox[{"\[Sigma]", "->", RowBox[{"5.670", FormBox[\(10\^\(-8\)\), "TraditionalForm"], \(\(\(\(J/s\)/m\)/m\)/K\^4\)}]}], ",", " ", \(\[Epsilon] -> .98\), ",", RowBox[{ FormBox[\(\[Tau]\_a\), "TraditionalForm"], "->", ".55"}], ",", \(\[Alpha] -> .35\)}], "}"}]}], "]"}]], "Input", CellLabel->"In[29]:="], Cell[BoxData[ \(TraditionalForm\`{{T(t) \[Rule] \(-294.31326272081753`\)\ K}, {T( t) \[Rule] \(-294.31326272081753`\)\ \[ImaginaryI]\ K}, {T( t) \[Rule] 294.31326272081753`\ \[ImaginaryI]\ K}, {T(t) \[Rule] 294.31326272081753`\ K}}\)], "Output", CellLabel->"Out[29]="] }, Open ]], Cell["We have probably worn this one out!", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["One dimensional energy balance", "Section"], Cell["\<\ We see that with some \"tweaking\" we can get the average \ temperature of the earth. With only a little more work, we can do better. We can get the change in \ temperature with latitude. Knowing the change with latitude has benefits such as determining how much \ melting of polar ice, and recession of ice covered regions, can greatly \ change the water levels. Cooling of the tropics could change the frequency \ and severity of storms. It also possible the cooling of the polls will have \ a lessor effect as might warming of the topics. Any change in the mid \ latitudes could cause problems. What we will get is a first order prediction of the effect of solar output \ and greenhouse effect with latitude. \ \>", "Text"], Cell[CellGroupData[{ Cell["Formulation of 1-D energy balance", "Subsubsection"], Cell[TextData[{ "If we break the earth in to \"i\" regions of different latitude we could \ write an energy balance for each region,\n\n", Cell[BoxData[ \(TraditionalForm\`S\_i\)]], " (1- ", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_i\)]], ") = R \[UpArrow](", Cell[BoxData[ \(TraditionalForm\`T\_i\)]], ") + F (", Cell[BoxData[ \(TraditionalForm\`T\_i\)]], ") ,\n\nwhere R \[UpArrow](", Cell[BoxData[ \(TraditionalForm\`T\_i\)]], "), is short hand for energy leaving a given region by radiation and F (", Cell[BoxData[ \(TraditionalForm\`T\_i\)]], ") is the loss of energy from a region to its cooler neighboring regions by \ ocean currents and wind. ", Cell[BoxData[ \(TraditionalForm\`S\_i\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_i\)]], "are the flux and albedo for a given latitude.\n\nThe solar fluxes, ", Cell[BoxData[ \(TraditionalForm\`S\_i\)]], ", will be largest at the equator and smallest at the poles. The function \ will not be a perfect sine function because of the tilt of the earth. We \ will specify these.\n\nThe albedos will be 0.3 for latitudes with no ice and \ 0.6 at latitudes with ice cover. This a very big difference and thus there \ will be some level of error associated with it., If we get better \ resolution, we might expect the answer to change. We can pick a specific \ temperature necessary for ice conditions.\n\nWe can model the radiation with \ a linear function because the range of possible temperatures is small (is \ this a big error?) . \nR \[UpArrow](", Cell[BoxData[ \(TraditionalForm\`T\_i\)]], ") = A +B ", Cell[BoxData[ \(TraditionalForm\`T\_i\)]], " \nPresumably we can make a sensible argument for values of A and B.\n\n\ The heat lost to cooler latitudes can also me modeled with a linear function.\ \nF (", Cell[BoxData[ \(TraditionalForm\`T\_i\)]], ") = K(", Cell[BoxData[ \(TraditionalForm\`T\_i\)]], "- ", Cell[BoxData[ \(TraditionalForm\`T\&\[LongDash]\)]], ")\nAgain we ", StyleBox["hope", FontSlant->"Italic"], " that we can obtain a reasonable value for K. " }], "Text"], Cell["Here is the equation with the model terms substituted in", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"oneDSSeq", "=", RowBox[{ RowBox[{\(S\_i\), RowBox[{"(", RowBox[{"1", "-", FormBox[\(\[Alpha]\_i\), "TraditionalForm"]}], ")"}]}], " ", "==", " ", RowBox[{"A", "+", RowBox[{"B", FormBox[\(T\_i\), "TraditionalForm"]}], " ", "+", " ", RowBox[{"K", RowBox[{"(", RowBox[{ FormBox[\(T\_i\), "TraditionalForm"], "-", FormBox[\(T\&\[LongDash]\), "TraditionalForm"]}], ")"}]}]}]}]}]], "Input", CellLabel->"In[338]:="], Cell[BoxData[ \(TraditionalForm\`S\_i\ \((1 - \[Alpha]\_i)\) \[Equal] 3.87`\ \((T\_i - 9.04975391729917`)\) + 2.17`\ T\_i + 204\)], "Output", CellLabel->"Out[338]="] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[" This is readily solved.", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[oneDSSeq, T\_i]\)], "Input", CellLabel->"In[339]:="], Cell[BoxData[ \(TraditionalForm\`{{T\_i \[Rule] \(-0.16556291390728478`\)\ \ \((\(\(168.9774523400522`\)\(\[InvisibleSpace]\)\) - 1.`\ S\_i\ \((\(\(1.`\)\(\[InvisibleSpace]\)\) - 1.`\ \[Alpha]\_i)\))\)}}\)], "Output", CellLabel->"Out[339]="] }, Open ]], Cell[TextData[{ "To evaluate this equation, we need the value of the average temperature, \ ", Cell[BoxData[ FormBox[ FormBox[\(T\&\[LongDash]\), "TraditionalForm"], TraditionalForm]]], ". At the start of problem, we don't know this. We need the ", Cell[BoxData[ \(TraditionalForm\`T\_i\)]], "'s, appropriately weighted, to calculate ", Cell[BoxData[ FormBox[ FormBox[\(T\&\[LongDash]\), "TraditionalForm"], TraditionalForm]]], ". Thus the solution will have to be iterative. \n\nTo accomplish this, \ we need to start with initial guesses for the ", Cell[BoxData[ \(TraditionalForm\`T\_i\)]], "'s. Calculate a ", Cell[BoxData[ FormBox[ FormBox[\(T\&\[LongDash]\), "TraditionalForm"], TraditionalForm]]], ". Then check to see if the ", Cell[BoxData[ \(TraditionalForm\`T\_i\)]], "'s have changed. Note also that and ", Cell[BoxData[ \(TraditionalForm\`\(\(\[Alpha]\_i\)\(\ \)\)\)]], "could change along the way.\n\nIf all goes well, this calculation will \ converge. If not, you would need to use a more elaborate iteration scheme. 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For the given values of all of the constants, what solar \ fraction value is necessary to completely glaciate the earth? 2. Vary the value of K in a range close to the given value. How does the \ mean temperature vary? 3. Now use very small values of K. What is the behavior. Physically, what \ does this correspond to? 4. Examine the effect of the critical ice temperature. For the earth, \ regions of ice coverage start occurring in the range of 0C down to as low as \ -13C. 5. The albedo of ice covered regions varies from 0.5 to 0.8. Investigate \ this. 6. Investigate the effect of B on the climate. What physically does this \ correspond to? 7. \"Break the model\" -- not the computer! That is, adjust something to \ give a nonsensible answer or a nonconvergent answer. 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