(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of 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). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. 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[ 18519, 539]*) (*NotebookOutlinePosition[ 19389, 567]*) (* CellTagsIndexPosition[ 19345, 563]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Spectral Model for Thermal Convection with Free-Slip, Constant Temperature Boundaries\ \>", "Title", TextAlignment->Center], Cell[TextData[{ "Solves the equations for the spectral representation of thermal convection \ in a layer. Certain simplifying symmetry restrictions are made on the \ convection.\nPlots the results.\n\n", StyleBox[ "This note book should be executed before the subsequent notebooks are \ used.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1], Background->GrayLevel[1]] }], "Text"], Cell[CellGroupData[{ Cell["Constants", "Section"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ StyleBox[\( (*The\ critical\ Rayleigh\ number*) \), "Commentary"], "\n", \(Rc = 27*Pi^4/4\)}]], "Input"], Cell[BoxData[ \(\(27\ \[Pi]\^4\)\/4\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Subroutines", "Section"], Cell[BoxData[ RowBox[{ StyleBox[\( (*Dirac\ delta\ function*) \), "Commentary"], "\n", "\t", \(id[m_, n_] := If[m == n, 1, 0]\)}]], "Input"], Cell[BoxData[ RowBox[{"\t", StyleBox[\( (*nonlinear\ interaction\ coefficient\ for\ buoyancy*) \), "Commentary"], "\n", "\t\t", \(ct[mi_, ni_, mj_, nj_, mk_, nk_] := \n\t Module[\n\t\t{norm}, \n\t\t\t\tnorm = If[mi == 0, .125, .25]; \n\t norm*\((\n\t\t mj*nk*\n\t\t \((id[mj + mk, mi] + id[mj - mk, mi] + id[mk - mj, mi])\)\n* \((id[nj + nk, ni] + id[nj - nk, ni] - id[nk - nj, ni]) \)\n + mk*nj*\n\t\t \((\(-id[mj + mk, mi]\) + id[mj - mk, mi] + id[mk - mj, mi]) \)\n\ *\(( id[nj + nk, ni] - id[nj - nk, ni] + id[nk - nj, ni])\)\n \t\t)\)\n\t]\)}]], "Input"], Cell[BoxData[ RowBox[{"\t", StyleBox[ \( (*nonlinear\ interaction\ coefficient\ for\ stream\ \(functio n\)*) \), "Commentary"], \(cp[mi_, ni_, mj_, nj_, mk_, nk_] := Module[{}, \n \t\t .25* \((\n\tmj*nk*\n \((id[mj + mk, mi] - id[mj - mk, mi] + id[mk - mj, mi])\)\n* \((id[nj + nk, ni] + id[nj - nk, ni] - id[nk - nj, ni]) \)\n - mk*nj*\n \((id[mj + mk, mi] + id[mj - mk, mi] - id[mk - mj, mi])\)\n* \((id[nj + nk, ni] - id[nj - nk, ni] + id[nk - nj, ni])\)\n) \)\n]\), "\n"}]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\t", StyleBox[\( (*individual\ spectral\ adection\ term\ for\ buoyancy*) \), "Commentary"], "\n", \(advt[i_, j_, k_] := \(p[j]\)[t]*\(t[k]\)[t]*\n\t ct[\n\t\t\ttmodes[\([i, 1]\)], tmodes[\([i, 2]\)], \n\t\t\t pmodes[\([j, 1]\)], pmodes[\([j, 2]\)], \n\t\t\t tmodes[\([k, 1]\)], tmodes[\([k, 2]\)]\n\t\t]\)}]], "Input"], Cell[BoxData[ \(General::"spell1" \( : \ \) "Possible spelling error: new symbol name \"\!\(pmodes\)\" is similar \ to existing symbol \"\!\(tmodes\)\"."\)], "Message"] }, Open ]], Cell[BoxData[ RowBox[{ StyleBox[ \( (*individual\ spectral\ adection\ term\ for\ \(vorticit y\)*) \), "Commentary"], \(advom[i_, j_, k_] := \(-\(p[j]\)[t]\)*\(p[k]\)[t]*\n\t\t \((a^2*pmodes[\([k, 1]\)]^2 + b^2*pmodes[\([k, 2]\)]^2)\)*\n\t cp[\n\t\t\tpmodes[\([i, 1]\)], pmodes[\([i, 2]\)], \n\t\t\t pmodes[\([j, 1]\)], pmodes[\([j, 2]\)], \n\t\t\t pmodes[\([k, 1]\)], pmodes[\([k, 2]\)]\n\t\t]\)}]], "Input"], Cell[BoxData[ RowBox[{ StyleBox[ \( (*buoyancy\ production\ term\ , \ \((the\ one\ with\ the\ Rayleigh\ number)\)*) \), "Commentary"], \(ray[i_, j_] := \(p[j]\)[t]*a*pmodes[\([j, 1]\)]*\n\t\t id[tmodes[\([i, 1]\)], pmodes[\([j, 1]\)]]\n\t\t* id[tmodes[\([i, 2]\)], pmodes[\([j, 2]\)]]\)}]], "Input"], Cell[BoxData[ RowBox[{ StyleBox[\( (*baroclinic\ production\ of\ vorticity*) \), "Commentary"], \(baro[i_, j_] := \(t[j]\)[t]*a*tmodes[\([j, 1]\)]*\n\t\t id[pmodes[\([i, 1]\)], tmodes[\([j, 1]\)]]\n\t\t* id[pmodes[\([i, 2]\)], tmodes[\([j, 2]\)]]\)}]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ StyleBox[\( (*sum\ of\ buoyancy\ advection\ terms*) \), "Commentary"], "\n", \(advectt[i_] := Sum[\n\t\tSum[advt[i, j, k], \n\t\t\t{j, 1, nump}], \n \t\t\t{k, 1, numt}]\)}]], "Input"], Cell[BoxData[ \(General::"spell1" \( : \ \) "Possible spelling error: new symbol name \"\!\(numt\)\" is similar to \ existing symbol \"\!\(nump\)\"."\)], "Message"] }, Open ]], Cell[BoxData[ RowBox[{ StyleBox[\( (*sum\ of\ \(vorticit y\)\ advection\ terms*) \), "Commentary"], "\n", \(advectom[i_] := Sum[\n\t\tSum[advom[i, j, k], \n\t\t\t{j, 1, nump}], \n \t\t\t{k, 1, nump}]\)}]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ StyleBox[\( (*sum\ of\ buoyancy\ production\ terms*) \), "Commentary"], "\n", \(rayt[i_] := Sum[\n\t\tray[i, j], \n\t\t\t{j, 1, nump}]\)}]], "Input"], Cell[BoxData[ \(General::"spell1" \( : \ \) "Possible spelling error: new symbol name \"\!\(rayt\)\" is similar to \ existing symbol \"\!\(ray\)\"."\)], "Message"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ StyleBox[\( (*sum\ of\ baroclinc\ \(generatio n\)\ terms*) \), "Commentary"], "\n", \(barop[i_] := Sum[\n\tbaro[i, j], \n\t\t\t{j, 1, numt}]\)}]], "Input"], Cell[BoxData[ \(General::"spell1" \( : \ \) "Possible spelling error: new symbol name \"\!\(barop\)\" is similar to \ existing symbol \"\!\(baro\)\"."\)], "Message"] }, Open ]], Cell[BoxData[ RowBox[{ StyleBox[ \( (*defines\ some\ global\ variables\ based\ on\ set\ of\ modes\ tm\ and\ pm*) \), "Commentary"], "\n", \(setup[tm_, pm_] := Module[{}, \n\t\ttmodes = tm; \n\t\tpmodes = pm; \n\t\t\t numt = \(Dimensions[tmodes]\)[\([1]\)]; \n\t\t nump = \(Dimensions[pmodes]\)[\([1]\)]; \n\t\t]\)}]], "Input"], Cell[BoxData[ \(whatis[q_, t_, sol_] := q[t] /. sol[\([1, 1]\)]\)], "Input"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Programs ", StyleBox[ "distinction between programs and subroutines is somewhat arbitrary", "Commentary"] }], "Section"], Cell[BoxData[ RowBox[{ StyleBox[ \( (*A\ model\ is\ a\ list\ of\ things, \ see\ below ... *) \), "Commentary"], "\n", \(makemodel[tm_, pm_] := Module[{}, \n\t\tsetup[tm, pm]; \n \t{makeode, makevar, tmodes, pmodes}\n\t]\)}]], "Input"], Cell[BoxData[ RowBox[{ StyleBox[ RowBox[{"(*", RowBox[{ StyleBox["Makes", "Commentary"], StyleBox[" ", "Commentary"], StyleBox["a", "Commentary"], StyleBox[" ", "Commentary"], StyleBox["list", "Commentary"], StyleBox[" ", "Commentary"], StyleBox["of", "Commentary"], StyleBox[" ", "Commentary"], StyleBox[\(ODE'\), "Commentary"], "s", " ", "compatible", " ", "with", " ", "NDSolve"}], "*)"}], "Commentary"], "\n", \(makeode := \n\t\t\t Join[Table[\n\t\t \(t[i]'\)[t]\n\t\t\t\t + \((a^2*tmodes[\([i, 1]\)]^2 + \n\t\t\t\t b^2*tmodes[\([i, 2]\)]^2)\) \(t[i]\)[t]\n \t\t\t\t\t\t == \n\t\t\t\(-a\)*b*advectt[i]\n\t\t + rayt[i]\n \t\t, \n\t\t\t\t{i, 1, numt}], \n\t\t\t Table[\n\t\t \(p[i]'\)[t]\n\t\t\t\t + P \((a^2*pmodes[\([i, 1]\)]^2 + \n\t\t\t\t b^2*pmodes[\([i, 2]\)]^2)\) \(p[i]\)[t]\n \t\t\t\t\t\t == \n\t\t\t \((\(-a\)*b*advectom[i]\n\t\t + P*R*barop[i])\)\n\t\t/ \((a^2*pmodes[\([i, 1]\)]^2 + \n\t\t\t\t b^2*pmodes[\([i, 2]\)]^2)\)\n\t\t, \n\t\t\t\t{i, 1, nump}] \n\t\t]\), "\n"}]], "Input"], Cell[BoxData[ RowBox[{ StyleBox[ \( (*Makes\ a\ list\ of\ initialization\ equations\ \(c ompatible\)\ with\ NDSolve*) \), "Commentary"], \(makeinit[themodel_, m_, n_, amp_] := Module[{}, \n\t\tsetup[themodel[\([3]\)], themodel[\([4]\)]]; \n\t\t Join[\n\t\t\t\t Table[\n\t\t \(t[i]\)[0] == amp*id[tmodes[\([i, 1]\)], m]*id[tmodes[\([i, 2]\)], n], \n \t\t\t\t\t{i, 1, numt}], \n\t\t\t Table[\n\t\t\(p[i]\)[0] == 0, \n\t\t\t\t{i, 1, nump}]\n\t]\n\t\t] \), "\n", "\t"}]], "Input"], Cell[BoxData[ RowBox[{ StyleBox[ \( (*Makes\ a\ list\ of\ the\ variables\ that\ will\ be\ solved\ for\ in\ NDSolve*) \), "Commentary"], "\n", \(makevar := \n\t\t Join[\n\t\t\t\tTable[\n\t\tt[i], \n\t\t\t\t\t{i, 1, numt}], \n\t\t\t Table[\n\t\tp[i], \n\t\t\t\t{i, 1, nump}]\n\t]\), "\n", "\t"}]], "Input"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ StyleBox[ RowBox[{"(*", RowBox[{ StyleBox["integrates", "Commentary"], StyleBox[" ", "Commentary"], StyleBox["a", "Commentary"], StyleBox[" ", "Commentary"], StyleBox["model", "Commentary"], " ", "forward", " ", "in", " ", "time", StyleBox[" ", "Commentary"], StyleBox["using", "Commentary"], StyleBox[\(\ \ \), "Commentary"], StyleBox["NDSolve", "Commentary"]}], "*)"}], "Commentary"], \(runmodel[themodel_, init_, parms_, tstop_] := \n\t Module[{}, \n\t\tsetup[themodel[\([3]\)], themodel[\([4]\)]]; \n \t\t{NDSolve[Join[themodel[\([1]\)] /. parms, init], themodel[\([2]\)], {t, 0, tstop}, MaxSteps -> 10000], \n\t\t\t parms, themodel[\([3]\)], themodel[\([4]\)], maketfun, makepfun}\n \t\t]\), "\n", "\t"}]], "Input"], Cell[BoxData[ \(General::"spell1" \( : \ \) "Possible spelling error: new symbol name \"\!\(makepfun\)\" is similar \ to existing symbol \"\!\(maketfun\)\"."\)], "Message"] }, Open ]], Cell[BoxData[ RowBox[{ StyleBox[ \( (*makes\ a\ function\ b \((x, z)\)\ that\ can\ be\ plotted*) \), "Commentary"], "\n", \(maketfun := \n\t 1 - z*b/Pi + \n\t\t Sum[\n\t\t\t \(t[i]\)[t]*Cos[a*tmodes[\([i, 1]\)]*x]\n\t\t\t\t\t\t\t* Sin[b*tmodes[\([i, 2]\)]*z], \n\t\t\t\t\t\t\t{i, 1, numt}\n \t\t\t\t\t\t]\)}]], "Input"], Cell[BoxData[ RowBox[{ StyleBox[ RowBox[{"(*", RowBox[{"makes", " ", "a", " ", "function", " ", StyleBox[ RowBox[{ StyleBox["p", "Commentary"], "si"}]], \((x, z)\), " ", "that", " ", "can", " ", "bi", " ", "plotted"}], " ", "*)"}], "Commentary"], "\n", \(makepfun := \n\t\t Sum[\n\t\t\t \(p[i]\)[t]*Sin[a*tmodes[\([i, 1]\)]*x]\n\t\t\t\t\t\t\t* Sin[b*tmodes[\([i, 2]\)]*z], \n\t\t\t\t\t\t\t{i, 1, nump}\n \t\t\t\t\t\t]\)}]], "Input"] }, Closed]], Cell[CellGroupData[{ Cell["Plotting Programs", "Section"], Cell["\<\ What do these do? They are best explained by seeing them in action \ in subsequent notebooks.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(modeplot[tlist_, plist_, mmax_, nmax_] := \n\t Module[{backpoints, rpoint}, \n\t\tbackpoints = {}; \n\t\t rpoint[q_] := Text[StyleForm["\<\[Cross]\>", FontSize -> 24], q]; \n Do[\n\t\tDo[\n\t\t\tbackpoints = Append[backpoints, {i, j}]\n \t\t\t, {i, 0, mmax, 1}], \n\t\t{j, 1, nmax, 1}]; \n\t\t theplot = ListPlot[{{0, 0}}, PlotRange -> {{\(-1\), mmax}, {0, nmax}}, AspectRatio -> nmax/\((mmax + 1)\), AxesOrigin -> {\(-1\), 0}, \n \tEpilog -> {\n \t\t\t\t\t{RGBColor[1, 0, 0], Map[Point, backpoints]}, \n \t\t\t\t\t\t{RGBColor[ .7, .7, 1], AbsolutePointSize[9], Map[Point, tlist]}, \n \t\t{RGBColor[0, 0, 0], Map[rpoint, plist]}\n\t\t\t\t}]; \n \t\t]\)], "Input"], Cell[BoxData[ \(General::"spell1" \( : \ \) "Possible spelling error: new symbol name \"\!\(plist\)\" is similar to \ existing symbol \"\!\(tlist\)\"."\)], "Message"], Cell[BoxData[ \(General::"spell1" \( : \ \) "Possible spelling error: new symbol name \"\!\(nmax\)\" is similar to \ existing symbol \"\!\(mmax\)\"."\)], "Message"] }, Open ]], Cell[BoxData[ \(specplot[sol_, time_, what_, size_, color_, mmax_, nmax_] := Module[{hack, splot, alltheblobs}, \n\t\t splot[q_] := {AbsolutePointSize[size*Sqrt[Abs[q[\([3]\)]]]], Point[{q[\([1]\)], q[\([2]\)]}]}; \n\t\t If[what == "\", \n\t\t\t \(hack = Table[{sol[\([4, i]\)], whatis[p[i], time, sol]} // Flatten, {i, 1, nump}]; \)\n, \n\t\t\t\t \(hack = Table[{sol[\([3, i]\)], whatis[t[i], time, sol]} // Flatten, {i, 1, numt}]; \)\n]; \n\talltheblobs = Map[splot, hack]; \n\t\t\t theplot = ListPlot[{{0, 0}}, PlotRange -> {{\(-1\), mmax}, {0, nmax}}, AspectRatio -> nmax/\((mmax + 1)\), AxesOrigin -> {\(-1\), 0}, \n \tEpilog -> {\n\t\t\t\t\t{color, alltheblobs}\n\t\t\t\t}]; \n\t] \)], "Input"], Cell[BoxData[ \(BlueGrayRed[z_] := RGBColor[z, 2*z*\((1 - z)\), 1 - z]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Gray[z_] := RGBColor[ .3 + .6*z, .3 + .6*z, .3 + .6*z]\)], "Input"], Cell[BoxData[ \(General::"spell1" \( : \ \) "Possible spelling error: new symbol name \"\!\(Gray\)\" is similar to \ existing symbol \"\!\(ray\)\"."\)], "Message"] }, Open ]], Cell[BoxData[ \(\(MyTable = Gray; \)\)], "Input"], Cell[BoxData[ \(conp[sol_, time_] := Module[{thefunc, theratio}, \n\t thefunc = \(\(sol[\([6]\)] /. sol[\([1, 1]\)]\) /. t -> time\) /. sol[\([2]\)] // N; \n\t\ttheratio = a/b /. sol[\([2]\)]; \n\t\t theplot = ContourPlot[thefunc, \n \t\t{x, 0, Pi/a /. sol[\([2]\)]}, {z, 0, Pi/b /. sol[\([2]\)]}, ColorFunction -> MyTable, ContourLines -> True, AspectRatio -> theratio, Frame -> False\n\t]; \n\t\t]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(cont[sol_, time_] := Module[{thefunc, theratio}, \n\t thefunc = \(\(sol[\([5]\)] /. sol[\([1, 1]\)]\) /. t -> time\) /. sol[\([2]\)] // N; \n\t\ttheratio = a/b /. sol[\([2]\)]; \n\t\t theplot = ContourPlot[thefunc, \n \t\t{x, 0, Pi/a /. sol[\([2]\)]}, {z, 0, Pi/b /. sol[\([2]\)]}, ColorFunction -> MyTable, ContourLines -> True, AspectRatio -> theratio, Frame -> False, \n\t\t Contours -> { .1, .2, .3, .4, .5, .6, .7, .8, .9}]; \n \t\t]\)], "Input"], Cell[BoxData[ \(General::"spell1" \( : \ \) "Possible spelling error: new symbol name \"\!\(cont\)\" is similar to \ existing symbol \"\!\(conp\)\"."\)], "Message"] }, Open ]], Cell[BoxData[ \(hist[sol_, time1_, time2_, color_] := \n\t Module[{thefunc}, \n\t\tthefunc = \(t[1]\)[t] /. sol[\([1, 1]\)]; \n\t\t theplot = Plot[\n\tthefunc, \n\t\t{t, time1, time2}, PlotRange -> {\(- .5\), .5}, PlotStyle -> {{color}}]; \n\t\t] \)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(butterfly[sol_, time1_, time2_, color_] := \n\t Module[{tfunc, t2func}, \n\t\ttfunc = \(t[1]\)[t] /. sol[\([1, 1]\)]; \n \t\tt2func = \(t[2]\)[t] /. sol[\([1, 1]\)]; \n\t theplot = ParametricPlot[\n\t{tfunc, t2func}, \n\t\t{t, time1, time2}, PlotPoints -> \((time2 - time1)\)*100, PlotStyle -> {{color}}, PlotRange -> {{\(- .5\), .5}, {\(- .7\), .1}}]; \n\t\t]\)], "Input"], Cell[BoxData[ \(General::"spell1" \( : \ \) "Possible spelling error: new symbol name \"\!\(t2func\)\" is similar \ to existing symbol \"\!\(tfunc\)\"."\)], "Message"] }, Closed]] }, Closed]] }, Open ]] }, FrontEndVersion->"Macintosh 3.0", ScreenRectangle->{{0, 1152}, {0, 850}}, WindowSize->{618, 678}, WindowMargins->{{96, Automatic}, {Automatic, 44}}, StyleDefinitions -> "ok.nb", MacintoshSystemPageSetup->"\<\ 00<0001804P000000]P2:?oQon82n@960dL5:0?l0080001804P000000]P2:001 0000I00000400`<300000BL?00400@00000000000000060801T1T000000@0000 00000000004000000000000000000000\>" ] (*********************************************************************** Cached data follows. 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