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Notebook[{

Cell[CellGroupData[{
Cell[BoxData[
    \(\(\( (*\ Preliminary\ Definitions*) \)\(\[IndentingNewLine]\)\(<< 
      Calculus`VectorAnalysis`\[IndentingNewLine]
    SetCoordinates[Cylindrical[R, theta, Z]]\[IndentingNewLine]
    ClearAll[m, kz, lambda]\[IndentingNewLine]
    B = B0 {0, BesselJ[1, lambda\ R], 
          BesselJ[0, lambda\ R]}\[IndentingNewLine]
    Curl[B] - lambda\ B // FullSimplify\[IndentingNewLine]
    Div[B]\[IndentingNewLine]
    k = {0, m/R, \ kz}\[IndentingNewLine]
    k . B\[IndentingNewLine]
    F[R_] = R\ \(\((k . B)\)^2/k . k\)\ /B0^2 // Simplify\[IndentingNewLine]
    G[R_] = \((\((m^2 - 1)\)/R^2\  + \ kz^2)\) 
            F[R] + \ \((2/R)\) \(kz^2/\((k . k)\)^2\)/
              B0^2\ \((kz^2\ B[\([3]\)]^2\  - \((m/R)\)^2\ B[\([2]\)]^2)\)\  // 
        Simplify\[IndentingNewLine]\[IndentingNewLine]\[IndentingNewLine]\
\[IndentingNewLine]\[IndentingNewLine]\[IndentingNewLine]\[IndentingNewLine]
    \)\)\)], "Input"],

Cell[BoxData[
    \(Cylindrical[R, theta, Z]\)], "Output"],

Cell[BoxData[
    \({0, B0\ BesselJ[1, lambda\ R], B0\ BesselJ[0, lambda\ R]}\)], "Output"],

Cell[BoxData[
    \({0, 0, 0}\)], "Output"],

Cell[BoxData[
    \(0\)], "Output"],

Cell[BoxData[
    \({0, m\/R, kz}\)], "Output"],

Cell[BoxData[
    \(B0\ kz\ BesselJ[0, 
          lambda\ R] + \(B0\ m\ BesselJ[1, lambda\ R]\)\/R\)], "Output"],

Cell[BoxData[
    \(\(R\ \((kz\ R\ BesselJ[0, lambda\ R] + m\ BesselJ[1, lambda\ R])\)\^2\)\
\/\(m\^2 + kz\^2\ R\^2\)\)], "Output"],

Cell[BoxData[
    \(\(\(1\/\((m\^2 + kz\^2\ R\^2)\)\^2\)\((R\ \((2\ kz\^4\ R\^2\ BesselJ[0, \
lambda\ R]\^2 - 
            2\ kz\^2\ m\^2\ BesselJ[1, lambda\ R]\^2 + \((kz\^2 + \(\(-1\) + \
m\^2\)\/R\^2)\)\ \((m\^2 + 
                  kz\^2\ R\^2)\)\ \((kz\ R\ BesselJ[0, lambda\ R] + m\ \
BesselJ[1, lambda\ R])\)\^2)\))\)\)\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\( (*\ 
      Determine\ MHD\ Stability\ Boundaries\ \
*) \)\(\[IndentingNewLine]\)\(\(ClearAll[kappa1];\)\[IndentingNewLine]
    \(kappa1[x_] = \(\(-BesselJ[1, x]\)/x\)/
          BesselJ[0, x];\)\[IndentingNewLine]
    \(atable = 
        Table[{kappa1[5.2 - \((i - 1)\)/100], 5.20 - \((i - 1)\)/100}, {i, 
            301}];\)\[IndentingNewLine]
    \(a1 = Interpolation[atable];\)\[IndentingNewLine]
    \(ClearAll[lambda];\)\[IndentingNewLine]
    btable = Table[{0, 0}, {i, 161}]; 
    Do[kzr = \(-0.7\) + \((i - 1)\)/100; 
      btable[\([i]\)] = {kzr, \(\(FindRoot[
                BesselJ[1, Sqrt[lambda^2 - kzr^2\ lambda^2]] + 
                    kzr\ lambda\ \ Sqrt[
                          lambda^2 - kzr^2\ lambda^2]/\((lambda - 
                            kzr\ lambda)\) 
                      BesselJ[0, Sqrt[lambda^2 - kzr^2\ lambda^2]] \[Equal] 
                  0, {\ lambda, 
                  3 + \((kzr - 0.2)\)^2\ 8\  + 
                    If[kzr < 0, \((kzr +  .2)\)^3\ 20, 
                      0]}]\)[\([1]\)]\)[\([2]\)]}, {i, 
        161}];\[IndentingNewLine]
    \(a2 = Interpolation[btable];\)\[IndentingNewLine]
    kappaminus = \(\(FindRoot[
            a1[x] - a2[x] \[Equal] 
              0, {x, \(- .3\)}]\)[\([1]\)]\)[\([2]\)]\[IndentingNewLine]
    kappaplus = \(\(FindRoot[
            a1[x] - a2[x] \[Equal] 
              0, {x,  .3}]\)[\([1]\)]\)[\([2]\)]\)\)\)], "Input"],

Cell[BoxData[
    \(\(-0.23650606600691793`\)\)], "Output"],

Cell[BoxData[
    \(0.27236682023963227`\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\( (*Main\ program\ for\ calculating\ radial\ displacement\ ksi \((x)\
\)\ inside\ and\ outside\ singular\ surface \((s)\) . \ Set\ parameters\ m, \ 
      kz, \ and\ lambda\ here\ *) \)\(\[IndentingNewLine]\)\(\(m = 1; 
    lambda = 3.6; kz = 0.4  lambda; ClearAll[ksi];\)\(\[IndentingNewLine]\)
    \(MHDStable = 
        If[kappaminus < kz/lambda && kappaplus > kz/lambda && 
            lambda > a1[kz/lambda], 0, 1];\)\(\[IndentingNewLine]\)
    \(Print[
      If[MHDStable \[Equal] 
          0, "\<MHD Unstable\>", "\<MHD Stable\>"]]\)\(\[IndentingNewLine]\)
    \(rsep1 = \(\(FindRoot[
            F[R] \[Equal] 0, {R, 
              0.8}]\)[\([1]\)]\)[\([2]\)]\)\(\[IndentingNewLine]\)\( (*\ 
      Note\ that\ for\ kz < \(-0.5\)\ lambda\ a\ second\ Singular\ surface\ \
appears*) \)\(\[IndentingNewLine]\)
    \(rsep2a = \(\(FindRoot[
              F[R] \[Equal] 0, {R, 
                0.3}]\)[\([1]\)]\)[\([2]\)];\)\(\[IndentingNewLine]\)
    \(rsep2 = 
        If[rsep2a < 0.1, 0.001, 
          If[rsep2a > 1, 0.001, rsep2a]];\)\(\[IndentingNewLine]\)
    \(rsep = If[rsep1 < 1 && rsep1 > 0.1, rsep1, 1];\)\(\[IndentingNewLine]\)
    \(rsep = 
        If[rsep == 1 && rsep2 > 0.1, rsep2, rsep1];\)\(\[IndentingNewLine]\)
    \(rsep1 = rsep;\)\(\[IndentingNewLine]\)
    \(rsep2 = If[Abs[rsep2 - rsep1] < 0.1, 0, rsep2]\)\(\[IndentingNewLine]\)
    \(Print[
      If[rsep1 < 1 && rsep >  .1, 
        StringJoin["\<Singular Surface at R=\>", 
          ToString[If[rsep2 > 0.1, rsep2 "\< and \>", "\< \>"]], 
          ToString[
            rsep1]], "\<No Singular Surface inside plasma\>"]]\)\(\
\[IndentingNewLine]\)
    \(epsilon = 
        0.0005;\)\(\[IndentingNewLine]\)\( (*Calculate\ inner\ ksi\ as\ ksix\ \
and\ outer\ ksi\ as\ ksix1\ *) \)\(\[IndentingNewLine]\)
    \(rmin = 
      If[rsep2 > 0.1 && rsep1 == 1, 0.01, 
        rsep2 + epsilon]\)\(\[IndentingNewLine]\)
    \(rmax = 
      If[rsep2 > 0.1 && rsep1 == 1, 1, 
        rsep1 - epsilon]\)\(\[IndentingNewLine]\)
    \(sol = 
      NDSolve[{F[R] \(ksi'\)[R] \[Equal] 
            z[R], \(z'\)[R] - G[R] ksi[R] \[Equal] 0, 
          ksi[rmin] \[Equal] rmin, z[rmin] \[Equal] 0}, {ksi, z}, {R, rmin, 
          rmax}]\)\(\[IndentingNewLine]\)
    \(ksix[
          x_] = \(Evaluate[
            ksi[x] /. sol]\)[\([1]\)];\)\(\[IndentingNewLine]\)
    \(sol1 = 
      NDSolve[{F[R] \(ksi'\)[R] \[Equal] 
            z[R], \(z'\)[R] - G[R] ksi[R] \[Equal] 0, ksi[1] \[Equal] 0, 
          z[1] \[Equal] 1}, {ksi, z}, {R, 
          If[rsep < 1, rsep + epsilon, 0.001], 1}]\)\(\[IndentingNewLine]\)
    \(ksix1[
          x_] = \(Evaluate[
            ksi[x] /. 
              sol1]\)[\([1]\)];\)\(\[IndentingNewLine]\)\( (*Normalization\ \
of\ Inner\ and\ Outer\ solutions, \ 
      so\ that\ delta\ B\ radial\ is\ continuopus\ across\ Singular\ Surface\
*) \)\(\[IndentingNewLine]\)
    \(y0 = 
        1/\((\((k . B)\)/B0 /. R \[Rule] \ rsep1 - epsilon)\)\ ksix[
            If[rsep1 > 1, 1, rsep1 - epsilon]];\)\(\[IndentingNewLine]\)
    \(y1 = 
        1/\((\((k . B)\)/B0 /. R \[Rule] \ rsep1 + epsilon)\)\ ksix1[
            If[rsep1 > 1, 0.01, 
              rsep1 + epsilon]];\)\(\[IndentingNewLine]\)\( (*\ 
      Optional\ delta\ W\ calculation\ for\ MHD\ Stability\ *) \)\(\
\[IndentingNewLine]\)
    \( (*deltaW1 = \ \ \(If[rsep1 > 1, 
              NIntegrate[\ 
                  F[x] \((\(ksix1'\)[x])\)^2 + \ \ G[x]\ ksix1[x]^2, {x, 
                    0.02, 1}]/
                y1^2, \ \ NIntegrate[\ 
                  F[x] \((\(ksix1'\)[x])\)^2 + \ \ G[x]\ ksix1[x]^2, {x, 
                    rsep + \ 20  epsilon, 1}]/
                y1^2]\[IndentingNewLine]deltaW2 = \(If[rsep1 > 1, 
                0, \ \ NIntegrate[\ 
                    F[x] \((\(ksix'\)[x])\)^2 + \ \ G[x]\ ksix[x]^2, {x, 0.1, 
                      rsep - 20  epsilon}]/
                  y0^2]\[IndentingNewLine]\[IndentingNewLine]deltaW = \((Pi/
                  2)\) \((deltaW1 + deltaW2)\)\)\)*) \)\)\)\)], "Input"],

Cell[BoxData[
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Cell[BoxData[
    \(0.8418181581881599`\)], "Output"],

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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            TagBox[\(InterpolatingFunction[{{0.0015`, 0.8413181581881599`}}, 
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            TagBox[\(InterpolatingFunction[{{0.0015`, 0.8413181581881599`}}, 
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              Editable->False]}]}], "}"}], "}"}]], "Output"],

Cell[BoxData[
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      RowBox[{"{", 
        RowBox[{
          RowBox[{"ksi", "\[Rule]", 
            
            TagBox[\(InterpolatingFunction[{{0.8423181581881598`, 1.`}}, 
                "<>"]\),
              False,
              Editable->False]}], ",", 
          RowBox[{"z", "\[Rule]", 
            
            TagBox[\(InterpolatingFunction[{{0.8423181581881598`, 1.`}}, 
                "<>"]\),
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              Editable->False]}]}], "}"}], "}"}]], "Output"]
}, Open  ]],

Cell[BoxData[""], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\( (*Plot\ bradial\ and\ do\ Delta\ Prime\ calculation . \ 
          Do\ only\ if\ there\ is\ a\ singular\ surface\ inside\ plasma\ \
\((rsep1 < 
            1)\)\ and\ the\ plasma\ is\ MHD\ Stable\ *) \)\(\
\[IndentingNewLine]\)\(\(Rmin = 
        If[rsep1 > 1, 0.5, rsep1 - 0.05];\)\[IndentingNewLine]
    Rmin = rsep - 0.05\[IndentingNewLine]
    \(y1 = 
        1/\((\((k . B)\)/B0 /. R \[Rule] \ rsep1 + epsilon)\)\ ksix1[
            If[rsep1 > 1, 0.01, rsep1 + epsilon]];\)\[IndentingNewLine]
    \(gg1 = 
        Plot[\ \ \((k . B)\)/B0\ ksix1[R]/y1, {R, 
            If[rsep1 > 1, 0.01, rsep1 + epsilon], 1}, PlotRange \[Rule] All, 
          DisplayFunction \[Rule] Identity];\)\[IndentingNewLine]
    \(y0 = 
        1/\((\((k . B)\)/B0 /. R \[Rule] \ rsep1 - epsilon)\)\ ksix[
            If[rsep1 > 1, 1, rsep1 - epsilon]];\)\[IndentingNewLine]
    \(gg2 = 
        Plot[\ \ \((k . B)\)/B0\ ksix[R]/y0, {R, Rmin, 
            If[rsep1 > 1, 1, rsep1 - epsilon]}, PlotRange \[Rule] All, 
          DisplayFunction \[Rule] Identity];\)\[IndentingNewLine]
    \(Brsep = \((\((k . B)\)/B0\ ksix[R]/y0 /. 
            R \[Rule] rsep1 - epsilon)\);\)\[IndentingNewLine]
    \(DeltaPrime = \(\(\(-\((\((\((\((k . B)\)/B0\ ksix[R]/y0 /. 
                            R \[Rule] 
                              rsep1 - epsilon)\) - \((\((k . B)\)/B0\ ksix[R]/
                                y0 /. R \[Rule] 
                              rsep1 - 
                                1.1  epsilon)\))\) - \
\[IndentingNewLine]\((\((\((k . B)\)/B0\ ksix1[R]/y1 /. 
                            R \[Rule] 
                              rsep1 + 1.1\ epsilon)\) - \((\((k . B)\)/
                                B0\ ksix1[R]/y1 /. 
                            R \[Rule] rsep1 + epsilon)\))\))\)\)/
              epsilon\)/ .1\)/Brsep;\)\[IndentingNewLine]
    Show[gg1, gg2, Frame \[Rule] True, 
      DisplayFunction \[Rule] $DisplayFunction, 
      PlotLabel \[Rule] 
        StringJoin["\<\[CapitalDelta]'=\>", 
          ToString[DeltaPrime]]]\[IndentingNewLine]
    Print[
      If[DeltaPrime > 
          0, "\<Computational Resistive MHD Unstable\>", "\<Computational \
Resistive MHD Stable\>"]]\[IndentingNewLine]
    \(lambdamin = a2[kz/lambda];\)\[IndentingNewLine]
    Print[
      If[lambda > 
          lambdamin, "\<Theoretical Resistive MHD Unstable\>", "\<Theoretical \
Resistive MHD Stable\>"]]\)\)\)], "Input"],

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