State-of-the-art stellarator optimization code

Tutorial: Boozer Transformation

This tutorial will walk the user through running BOOZ_XFORM on the free boundary equilibrium generated in the VMEC Free Boundary Tutorial. For this example the National Compact Stellarator Experiment (NCSX) configuration will be used. This machine is stellarator symmetric with a periodicity of three.

Edit the input text file.

The input file (in_booz.ncsx_c09r00_free) controls the execution of the BOOZ_XFORM code. The suffix of the input file will be appended to each of the output files as we will see after execution. Our input file looks something like

72 24
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 
32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 
60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 
88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99

Execute the code.

To execute the code simply pass the name of the in_booz file to the executable.

>~/bin_847/xbooz_xform in_booz.ncsx_c09r00_free
  0 <= mboz <=   71    -24 <= nboz <=   24
  nu_boz =   290 nv_boz =    98
             OUTBOARD (u=0)              JS          INBOARD (u=pi)
  v     |B|vmec    |B|booz    Error             |B|vmec    |B|booz    Error
  0    1.496E+00  1.496E+00  1.381E-14    2    1.526E+00  1.526E+00  3.157E-14
 pi    1.498E+00  1.498E+00  3.113E-15         1.524E+00  1.524E+00  3.336E-14
  0    1.485E+00  1.485E+00  1.496E-15    3    1.538E+00  1.538E+00  3.841E-14
 pi    1.488E+00  1.488E+00  1.194E-14         1.534E+00  1.534E+00  3.301E-14
  0    1.477E+00  1.477E+00  1.037E-14    4    1.545E+00  1.545E+00  3.434E-14
 pi    1.480E+00  1.480E+00  2.115E-14         1.540E+00  1.540E+00  3.850E-14
  0    1.471E+00  1.471E+00  4.982E-15    5    1.552E+00  1.552E+00  3.878E-14
 pi    1.474E+00  1.474E+00  2.877E-14         1.545E+00  1.545E+00  3.923E-14
  0    1.465E+00  1.465E+00  1.333E-14    6    1.557E+00  1.557E+00  2.966E-14
 pi    1.469E+00  1.469E+00  2.449E-14         1.549E+00  1.549E+00  3.239E-14
  0    1.461E+00  1.461E+00  1.186E-14    7    1.562E+00  1.562E+00  8.527E-16
 pi    1.464E+00  1.464E+00  3.306E-14         1.553E+00  1.553E+00  8.291E-15
  0    1.456E+00  1.456E+00  2.851E-14    8    1.567E+00  1.567E+00  1.601E-14
 pi    1.460E+00  1.460E+00  4.092E-14         1.557E+00  1.557E+00  4.435E-14
  0    1.452E+00  1.452E+00  2.507E-14    9    1.571E+00  1.571E+00  3.674E-14
 pi    1.455E+00  1.455E+00  5.432E-14         1.560E+00  1.560E+00  9.562E-14
  0    1.449E+00  1.449E+00  2.407E-14   10    1.576E+00  1.576E+00  8.555E-14
 pi    1.451E+00  1.451E+00  9.593E-14         1.564E+00  1.564E+00  1.779E-13
  0    1.445E+00  1.445E+00  2.996E-14   11    1.579E+00  1.579E+00  1.227E-13
 pi    1.448E+00  1.448E+00  1.235E-13         1.567E+00  1.567E+00  2.674E-13
  0    1.442E+00  1.442E+00  6.022E-14   12    1.583E+00  1.583E+00  1.811E-13
 pi    1.444E+00  1.444E+00  1.958E-13         1.570E+00  1.570E+00  3.872E-13
  0    1.439E+00  1.439E+00  9.276E-14   13    1.587E+00  1.587E+00  2.461E-13
 pi    1.440E+00  1.440E+00  2.517E-13         1.572E+00  1.572E+00  4.947E-13
  0    1.436E+00  1.436E+00  1.420E-13   14    1.590E+00  1.590E+00  3.259E-13
 pi    1.437E+00  1.437E+00  3.656E-13         1.575E+00  1.575E+00  6.623E-13
  0    1.433E+00  1.433E+00  1.777E-13   15    1.594E+00  1.594E+00  3.772E-13
 pi    1.434E+00  1.434E+00  4.911E-13         1.578E+00  1.578E+00  8.268E-13
  0    1.430E+00  1.430E+00  2.607E-13   16    1.597E+00  1.597E+00  4.400E-13
 pi    1.431E+00  1.431E+00  6.533E-13         1.580E+00  1.580E+00  1.001E-12
  0    1.428E+00  1.428E+00  3.350E-13   17    1.600E+00  1.600E+00  4.995E-13
 pi    1.428E+00  1.428E+00  8.321E-13         1.583E+00  1.583E+00  1.161E-12
  0    1.425E+00  1.425E+00  4.383E-13   18    1.603E+00  1.603E+00  4.998E-13
 pi    1.425E+00  1.425E+00  1.041E-12         1.586E+00  1.586E+00  1.327E-12
  0    1.423E+00  1.423E+00  5.640E-13   19    1.607E+00  1.607E+00  5.318E-13
 pi    1.422E+00  1.422E+00  1.262E-12         1.588E+00  1.588E+00  1.446E-12
  0    1.421E+00  1.421E+00  6.918E-13   20    1.610E+00  1.610E+00  5.167E-13
 pi    1.419E+00  1.419E+00  1.518E-12         1.590E+00  1.590E+00  1.541E-12
  0    1.418E+00  1.418E+00  8.262E-13   21    1.613E+00  1.613E+00  4.557E-13
 pi    1.416E+00  1.416E+00  1.777E-12         1.593E+00  1.593E+00  1.608E-12
  0    1.416E+00  1.416E+00  9.729E-13   22    1.616E+00  1.616E+00  3.790E-13
 pi    1.414E+00  1.414E+00  2.045E-12         1.595E+00  1.595E+00  1.639E-12
  0    1.414E+00  1.414E+00  1.134E-12   23    1.619E+00  1.619E+00  3.010E-13
 pi    1.411E+00  1.411E+00  2.282E-12         1.598E+00  1.598E+00  1.636E-12
  0    1.413E+00  1.413E+00  1.294E-12   24    1.622E+00  1.622E+00  1.962E-13
 pi    1.409E+00  1.409E+00  2.503E-12         1.600E+00  1.600E+00  1.669E-12
  0    1.411E+00  1.411E+00  1.449E-12   25    1.624E+00  1.624E+00  1.003E-13
 pi    1.406E+00  1.406E+00  2.698E-12         1.602E+00  1.602E+00  1.720E-12
  0    1.409E+00  1.409E+00  1.618E-12   26    1.627E+00  1.627E+00  3.739E-14
 pi    1.404E+00  1.404E+00  2.853E-12         1.605E+00  1.605E+00  1.860E-12
  0    1.407E+00  1.407E+00  1.808E-12   27    1.630E+00  1.630E+00  3.487E-14
 pi    1.402E+00  1.402E+00  2.933E-12         1.607E+00  1.607E+00  2.171E-12
  0    1.406E+00  1.406E+00  1.982E-12   28    1.633E+00  1.633E+00  1.001E-13
 pi    1.399E+00  1.399E+00  2.963E-12         1.609E+00  1.609E+00  2.719E-12
  0    1.404E+00  1.404E+00  2.186E-12   29    1.636E+00  1.636E+00  2.525E-13
 pi    1.397E+00  1.397E+00  2.864E-12         1.611E+00  1.611E+00  3.594E-12
  0    1.403E+00  1.403E+00  2.350E-12   30    1.638E+00  1.638E+00  5.204E-13
 pi    1.395E+00  1.395E+00  2.691E-12         1.614E+00  1.614E+00  4.938E-12
  0    1.401E+00  1.401E+00  2.463E-12   31    1.641E+00  1.641E+00  9.444E-13
 pi    1.393E+00  1.393E+00  2.399E-12         1.616E+00  1.616E+00  6.873E-12
  0    1.400E+00  1.400E+00  2.520E-12   32    1.644E+00  1.644E+00  1.579E-12
 pi    1.391E+00  1.391E+00  1.979E-12         1.618E+00  1.618E+00  9.529E-12
  0    1.398E+00  1.398E+00  2.425E-12   33    1.647E+00  1.647E+00  2.453E-12
 pi    1.389E+00  1.389E+00  1.461E-12         1.620E+00  1.620E+00  1.308E-11
  0    1.397E+00  1.397E+00  2.144E-12   34    1.649E+00  1.649E+00  3.603E-12
 pi    1.387E+00  1.387E+00  8.154E-13         1.623E+00  1.623E+00  1.770E-11
  0    1.396E+00  1.396E+00  1.674E-12   35    1.652E+00  1.652E+00  5.160E-12
 pi    1.385E+00  1.385E+00  4.392E-14         1.625E+00  1.625E+00  2.356E-11
  0    1.395E+00  1.395E+00  9.829E-13   36    1.655E+00  1.655E+00  7.180E-12
 pi    1.383E+00  1.383E+00  8.418E-13         1.627E+00  1.627E+00  3.094E-11
  0    1.394E+00  1.394E+00  8.732E-14   37    1.657E+00  1.657E+00  9.856E-12
 pi    1.381E+00  1.381E+00  1.811E-12         1.630E+00  1.630E+00  4.010E-11
  0    1.392E+00  1.392E+00  9.949E-13   38    1.660E+00  1.660E+00  1.320E-11
 pi    1.380E+00  1.380E+00  2.947E-12         1.632E+00  1.632E+00  5.116E-11
  0    1.391E+00  1.391E+00  2.167E-12   39    1.663E+00  1.663E+00  1.745E-11
 pi    1.378E+00  1.378E+00  4.172E-12         1.634E+00  1.634E+00  6.460E-11
  0    1.390E+00  1.390E+00  3.307E-12   40    1.665E+00  1.665E+00  2.263E-11
 pi    1.376E+00  1.376E+00  5.521E-12         1.636E+00  1.636E+00  8.067E-11
  0    1.389E+00  1.389E+00  4.258E-12   41    1.668E+00  1.668E+00  2.898E-11
 pi    1.375E+00  1.375E+00  6.940E-12         1.639E+00  1.639E+00  9.985E-11
  0    1.389E+00  1.389E+00  4.928E-12   42    1.670E+00  1.670E+00  3.639E-11
 pi    1.373E+00  1.373E+00  8.531E-12         1.641E+00  1.641E+00  1.221E-10
  0    1.388E+00  1.388E+00  5.274E-12   43    1.673E+00  1.673E+00  4.496E-11
 pi    1.372E+00  1.372E+00  1.024E-11         1.643E+00  1.643E+00  1.480E-10
  0    1.387E+00  1.387E+00  5.223E-12   44    1.676E+00  1.676E+00  5.458E-11
 pi    1.370E+00  1.370E+00  1.204E-11         1.645E+00  1.645E+00  1.777E-10
  0    1.386E+00  1.386E+00  4.936E-12   45    1.678E+00  1.678E+00  6.519E-11
 pi    1.369E+00  1.369E+00  1.396E-11         1.648E+00  1.648E+00  2.113E-10
  0    1.385E+00  1.385E+00  4.597E-12   46    1.681E+00  1.681E+00  7.658E-11
 pi    1.367E+00  1.367E+00  1.596E-11         1.650E+00  1.650E+00  2.488E-10
  0    1.385E+00  1.385E+00  4.528E-12   47    1.683E+00  1.683E+00  8.874E-11
 pi    1.366E+00  1.366E+00  1.808E-11         1.652E+00  1.652E+00  2.903E-10
  0    1.384E+00  1.384E+00  5.070E-12   48    1.686E+00  1.686E+00  1.015E-10
 pi    1.365E+00  1.365E+00  2.027E-11         1.655E+00  1.655E+00  3.357E-10
  0    1.383E+00  1.383E+00  6.593E-12   49    1.688E+00  1.688E+00  1.149E-10
 pi    1.364E+00  1.364E+00  2.256E-11         1.657E+00  1.657E+00  3.850E-10
  0    1.383E+00  1.383E+00  9.321E-12   50    1.691E+00  1.691E+00  1.291E-10
 pi    1.362E+00  1.362E+00  2.490E-11         1.660E+00  1.660E+00  4.381E-10
  0    1.382E+00  1.382E+00  1.331E-11   51    1.694E+00  1.694E+00  1.447E-10
 pi    1.361E+00  1.361E+00  2.725E-11         1.662E+00  1.662E+00  4.965E-10
  0    1.381E+00  1.381E+00  1.856E-11   52    1.696E+00  1.696E+00  1.612E-10
 pi    1.360E+00  1.360E+00  2.976E-11         1.664E+00  1.664E+00  5.577E-10
  0    1.381E+00  1.381E+00  2.470E-11   53    1.699E+00  1.699E+00  1.795E-10
 pi    1.359E+00  1.359E+00  3.238E-11         1.667E+00  1.667E+00  6.234E-10
  0    1.381E+00  1.381E+00  3.111E-11   54    1.701E+00  1.701E+00  1.994E-10
 pi    1.358E+00  1.358E+00  3.504E-11         1.669E+00  1.669E+00  6.921E-10
  0    1.380E+00  1.380E+00  3.712E-11   55    1.704E+00  1.704E+00  2.208E-10
 pi    1.357E+00  1.357E+00  3.769E-11         1.672E+00  1.672E+00  7.634E-10
  0    1.380E+00  1.380E+00  4.201E-11   56    1.706E+00  1.706E+00  2.434E-10
 pi    1.356E+00  1.356E+00  4.022E-11         1.674E+00  1.674E+00  8.363E-10
  0    1.379E+00  1.379E+00  4.510E-11   57    1.709E+00  1.709E+00  2.670E-10
 pi    1.355E+00  1.355E+00  4.255E-11         1.677E+00  1.677E+00  9.102E-10
  0    1.379E+00  1.379E+00  4.588E-11   58    1.712E+00  1.712E+00  2.910E-10
 pi    1.354E+00  1.354E+00  4.448E-11         1.679E+00  1.679E+00  9.838E-10
  0    1.379E+00  1.379E+00  4.407E-11   59    1.714E+00  1.714E+00  3.148E-10
 pi    1.353E+00  1.353E+00  4.577E-11         1.682E+00  1.682E+00  1.056E-09
  0    1.378E+00  1.378E+00  3.950E-11   60    1.717E+00  1.717E+00  3.379E-10
 pi    1.353E+00  1.353E+00  4.612E-11         1.684E+00  1.684E+00  1.126E-09
  0    1.378E+00  1.378E+00  3.224E-11   61    1.719E+00  1.719E+00  3.597E-10
 pi    1.352E+00  1.352E+00  4.520E-11         1.687E+00  1.687E+00  1.192E-09
  0    1.378E+00  1.378E+00  2.233E-11   62    1.722E+00  1.722E+00  3.799E-10
 pi    1.351E+00  1.351E+00  4.261E-11         1.689E+00  1.689E+00  1.252E-09
  0    1.378E+00  1.378E+00  9.926E-12   63    1.724E+00  1.724E+00  3.977E-10
 pi    1.350E+00  1.350E+00  3.793E-11         1.692E+00  1.692E+00  1.304E-09
  0    1.378E+00  1.378E+00  4.935E-12   64    1.727E+00  1.727E+00  4.128E-10
 pi    1.350E+00  1.350E+00  3.079E-11         1.695E+00  1.695E+00  1.346E-09
  0    1.378E+00  1.378E+00  2.214E-11   65    1.729E+00  1.729E+00  4.250E-10
 pi    1.349E+00  1.349E+00  2.098E-11         1.697E+00  1.697E+00  1.374E-09
  0    1.378E+00  1.378E+00  4.184E-11   66    1.732E+00  1.732E+00  4.341E-10
 pi    1.348E+00  1.348E+00  8.541E-12         1.700E+00  1.700E+00  1.387E-09
  0    1.378E+00  1.378E+00  6.408E-11   67    1.734E+00  1.734E+00  4.399E-10
 pi    1.347E+00  1.347E+00  5.965E-12         1.702E+00  1.702E+00  1.380E-09
  0    1.378E+00  1.378E+00  8.872E-11   68    1.737E+00  1.737E+00  4.425E-10
 pi    1.347E+00  1.347E+00  2.147E-11         1.705E+00  1.705E+00  1.350E-09
  0    1.378E+00  1.378E+00  1.151E-10   69    1.739E+00  1.739E+00  4.421E-10
 pi    1.346E+00  1.346E+00  3.599E-11         1.708E+00  1.708E+00  1.297E-09
  0    1.378E+00  1.378E+00  1.421E-10   70    1.742E+00  1.742E+00  4.394E-10
 pi    1.345E+00  1.345E+00  4.670E-11         1.710E+00  1.710E+00  1.219E-09
  0    1.378E+00  1.378E+00  1.672E-10   71    1.744E+00  1.744E+00  4.356E-10
 pi    1.345E+00  1.345E+00  4.983E-11         1.713E+00  1.713E+00  1.121E-09
  0    1.378E+00  1.378E+00  1.868E-10   72    1.747E+00  1.747E+00  4.328E-10
 pi    1.344E+00  1.344E+00  4.096E-11         1.716E+00  1.716E+00  1.008E-09
  0    1.378E+00  1.378E+00  1.960E-10   73    1.749E+00  1.749E+00  4.333E-10
 pi    1.343E+00  1.343E+00  1.548E-11         1.718E+00  1.718E+00  8.883E-10
  0    1.378E+00  1.378E+00  1.890E-10   74    1.752E+00  1.752E+00  4.408E-10
 pi    1.342E+00  1.342E+00  3.063E-11         1.721E+00  1.721E+00  7.790E-10
  0    1.378E+00  1.378E+00  1.597E-10   75    1.754E+00  1.754E+00  4.607E-10
 pi    1.342E+00  1.342E+00  1.002E-10         1.723E+00  1.723E+00  7.037E-10
  0    1.379E+00  1.379E+00  1.026E-10   76    1.756E+00  1.756E+00  4.987E-10
 pi    1.341E+00  1.341E+00  1.929E-10         1.726E+00  1.726E+00  6.859E-10
  0    1.379E+00  1.379E+00  1.350E-11   77    1.758E+00  1.758E+00  5.610E-10
 pi    1.340E+00  1.340E+00  3.044E-10         1.729E+00  1.729E+00  7.498E-10
  0    1.379E+00  1.379E+00  1.098E-10   78    1.761E+00  1.761E+00  6.551E-10
 pi    1.339E+00  1.339E+00  4.278E-10         1.731E+00  1.731E+00  9.293E-10
  0    1.379E+00  1.379E+00  2.675E-10   79    1.763E+00  1.763E+00  7.890E-10
 pi    1.338E+00  1.338E+00  5.527E-10         1.734E+00  1.734E+00  1.261E-09
  0    1.379E+00  1.379E+00  4.561E-10   80    1.765E+00  1.765E+00  9.718E-10
 pi    1.337E+00  1.337E+00  6.657E-10         1.737E+00  1.737E+00  1.787E-09
  0    1.380E+00  1.380E+00  6.670E-10   81    1.767E+00  1.767E+00  1.211E-09
 pi    1.336E+00  1.336E+00  7.522E-10         1.739E+00  1.739E+00  2.541E-09
  0    1.380E+00  1.380E+00  8.849E-10   82    1.769E+00  1.769E+00  1.517E-09
 pi    1.335E+00  1.335E+00  7.988E-10         1.742E+00  1.742E+00  3.564E-09
  0    1.380E+00  1.380E+00  1.088E-09   83    1.771E+00  1.771E+00  1.894E-09
 pi    1.334E+00  1.334E+00  7.934E-10         1.744E+00  1.744E+00  4.879E-09
  0    1.380E+00  1.380E+00  1.251E-09   84    1.773E+00  1.773E+00  2.348E-09
 pi    1.333E+00  1.333E+00  7.284E-10         1.747E+00  1.747E+00  6.515E-09
  0    1.380E+00  1.380E+00  1.351E-09   85    1.775E+00  1.775E+00  2.878E-09
 pi    1.331E+00  1.331E+00  5.984E-10         1.750E+00  1.750E+00  8.461E-09
  0    1.380E+00  1.380E+00  1.373E-09   86    1.777E+00  1.777E+00  3.481E-09
 pi    1.330E+00  1.330E+00  4.014E-10         1.752E+00  1.752E+00  1.072E-08
  0    1.380E+00  1.380E+00  1.307E-09   87    1.779E+00  1.779E+00  4.150E-09
 pi    1.328E+00  1.328E+00  1.336E-10         1.755E+00  1.755E+00  1.325E-08
  0    1.380E+00  1.380E+00  1.162E-09   88    1.781E+00  1.781E+00  4.872E-09
 pi    1.327E+00  1.327E+00  2.062E-10         1.757E+00  1.757E+00  1.602E-08
  0    1.380E+00  1.380E+00  9.490E-10   89    1.782E+00  1.782E+00  5.621E-09
 pi    1.325E+00  1.325E+00  6.261E-10         1.760E+00  1.760E+00  1.890E-08
  0    1.380E+00  1.380E+00  6.914E-10   90    1.784E+00  1.784E+00  6.365E-09
 pi    1.323E+00  1.323E+00  1.136E-09         1.762E+00  1.762E+00  2.178E-08
  0    1.380E+00  1.380E+00  4.127E-10   91    1.786E+00  1.786E+00  7.067E-09
 pi    1.322E+00  1.322E+00  1.749E-09         1.765E+00  1.765E+00  2.450E-08
  0    1.380E+00  1.380E+00  1.451E-10   92    1.787E+00  1.787E+00  7.672E-09
 pi    1.320E+00  1.320E+00  2.479E-09         1.767E+00  1.767E+00  2.682E-08
  0    1.380E+00  1.380E+00  7.666E-11   93    1.789E+00  1.789E+00  8.102E-09
 pi    1.318E+00  1.318E+00  3.336E-09         1.769E+00  1.769E+00  2.837E-08
  0    1.380E+00  1.380E+00  2.057E-10   94    1.790E+00  1.790E+00  8.254E-09
 pi    1.316E+00  1.316E+00  4.309E-09         1.772E+00  1.772E+00  2.868E-08
  0    1.380E+00  1.380E+00  2.038E-10   95    1.791E+00  1.791E+00  7.978E-09
 pi    1.314E+00  1.314E+00  5.352E-09         1.774E+00  1.774E+00  2.704E-08
  0    1.379E+00  1.379E+00  7.909E-11   96    1.793E+00  1.793E+00  7.026E-09
 pi    1.312E+00  1.312E+00  6.419E-09         1.777E+00  1.777E+00  2.232E-08
  0    1.379E+00  1.379E+00  6.602E-11   97    1.794E+00  1.794E+00  5.047E-09
 pi    1.309E+00  1.309E+00  7.557E-09         1.779E+00  1.779E+00  1.297E-08
  0    1.379E+00  1.379E+00  6.481E-11   98    1.795E+00  1.795E+00  1.240E-09
 pi    1.308E+00  1.308E+00  9.150E-09         1.781E+00  1.781E+00  4.472E-09
  0    1.379E+00  1.379E+00  2.040E-10   99    1.796E+00  1.796E+00  6.216E-09
 pi    1.306E+00  1.306E+00  1.174E-08         1.784E+00  1.784E+00  3.757E-08

Examine the output.

For this example three files were created (boozmn.ncsx_c09r00_free, RZ_BOOZER_VPLANE 1.txt, and RZ_VMEC_VPLANE 1.txt). The RZ files contain a set of points defining the VMEC and BOOZER boundaries respectively. The boozmn file contains the output from the BOOZ_XFORM code. Examination of the netCDF file shows it has the following format

 netcdf boozmn_ncsx_c09r00_free {
 dim_00038 = 38 ;
 radius = 99 ;
 comput_surfs = 98 ;
 mn_mode = 3504 ;
 mn_modes = 3504 ;
 pack_rad = 98 ;
 int nfp_b ;
 int ns_b ;
 double aspect_b ;
 double rmax_b ;
 double rmin_b ;
 double betaxis_b ;
 int mboz_b ;
 int nboz_b ;
 int mnboz_b ;
 char version(dim_00038) ;
 int lasym__logical__ ;
 double iota_b(radius) ;
 double pres_b(radius) ;
 double beta_b(radius) ;
 double phip_b(radius) ;
 double phi_b(radius) ;
 double bvco_b(radius) ;
 double buco_b(radius) ;
 int jlist(comput_surfs) ;
 int ixm_b(mn_mode) ;
 int ixn_b(mn_mode) ;
 double bmnc_b(pack_rad, mn_modes) ;
 double rmnc_b(pack_rad, mn_modes) ;
 double zmns_b(pack_rad, mn_modes) ;
 double pmns_b(pack_rad, mn_modes) ;
 double gmn_b(pack_rad, mn_modes) ;