State-of-the-art stellarator optimization code
The NESCOIL (NEumann Solver for fields produced by externals COILs) (P. Merkel 1987 //Nucl. Fusion// **27** 867) code calculates a surface current on the exterior surface of two toroidally closed surfaces such that the normal field on the interior surface is minimized.
We seek a current potential on a toroidal surface (D) which results in the existance of an enclosed flux surface (R).
The field produced by current potential can be written as
\[\vec{B}\left(\vec{x}\right) = \frac{\mu_0}{4\pi}\int\frac{\vec{j}\times\left(\vec{x}-\vec{x}'\right)}{\left|\vec{x}-\vec{x}'\right|^3}d^3x'\]where we define the current density by a current potential \(\vec{j} = \hat{n}\times\nabla\Phi\). Which can be separated into secular and non-secular parts
\[\Phi\left(u,v\right) = \sum_{m=0}^M\sum_{n=-N}^N \Phi_{mn} sin\left(2\pi mu+2\pi nv\right) - \frac{I_{pol}}{N_p}v - I_{tor}u\]Here \(I_{pol}\) is the total poloidal current per field period of the equilbrium and \(I_{tor}\) is the total toroidal current. Note that the argument to sine is not the same as that of VMEC. The plasma surface and current potential surface also use this formulation and not the VMEC one.
The two scalar parts of the potential represent magnetic fields arrising
from a toroidal field (\(\frac{I_{pol}}{N_p}v\)) and a vertical field
(\(I_{tor}u\)). While these two quantities are free parameters, the
BNORM code normalizes to the poloidal current per field period, therefore
\(\frac{I_{pol}}{N_p}=1\) is assumed. However, the user is free to modify
CUP and CUT to meet their needs.
NESCOIL is distributed as part of the STELLOPT package of codes through Git.
The NSCOIL code takes an ‘nescin.ext’ file as input and optionally a similarly named ‘bnorm.ext’ file (as produced by the BNORM code). If no ‘bnorm.ext’ file is found then it is assumed that the normal plasma field is zero (vacuum condition). The ‘nescin’ file has the following format
------ Spatial dimensions ----
nu, nv, nu1, nv1, npol, ntor, lasym_bn
256 256 256 256 64 10 F
------ Fourier Dimensions ----
mf, nf, md, nd (max in surf and bnorm files)
24 15 24 22
------ Plasma information from VMEC ----
np iota_edge phip_edge curpol
4 0.0000000000000000 0.32524904170259739 -16.089222854438592
------ Current Controls ----
cut cup ibex(=1,use fixed background coils)
0.0000000000000000 1.0000000000000000 0
------ SVD controls -----
mstrt, mstep, mkeep, mdspw, curwt, trgwt
0 0 0 4 0.0000000000000000 0.0000000000000000
------ Output controls -----
w_psurf w_csurf w_bnuv w_jsurf w_xerr w_svd
0 0 0 0 0 0
------ Plasma Surface ----
Number of fourier modes in table
10
Table of fourier coefficients
m,n,crc,czs,cls,crs,czc,clc
0 0 4.000000000000E+00 0.000000000000E+00 0.000000000000E+00 0.000000000000E+00 0.000000000000E+00 0.000000000000E+00
1 0 1.000000000000E+00 1.000000000000E+00 -1.925732563734E-01 0.000000000000E+00 0.000000000000E+00 0.000000000000E+00
2 0 0.000000000000E+00 0.000000000000E+00 -5.090906726166E-02 0.000000000000E+00 0.000000000000E+00 0.000000000000E+00
3 0 0.000000000000E+00 0.000000000000E+00 6.458084248335E-03 0.000000000000E+00 0.000000000000E+00 0.000000000000E+00
4 0 0.000000000000E+00 0.000000000000E+00 -5.915484436642E-03 0.000000000000E+00 0.000000000000E+00 0.000000000000E+00
5 0 0.000000000000E+00 0.000000000000E+00 7.030074268844E-04 0.000000000000E+00 0.000000000000E+00 0.000000000000E+00
6 0 0.000000000000E+00 0.000000000000E+00 -1.024498213757E-03 0.000000000000E+00 0.000000000000E+00 0.000000000000E+00
7 0 0.000000000000E+00 0.000000000000E+00 -1.091616780958E-04 0.000000000000E+00 0.000000000000E+00 0.000000000000E+00
8 0 0.000000000000E+00 0.000000000000E+00 6.174851363263E-06 0.000000000000E+00 0.000000000000E+00 0.000000000000E+00
9 0 0.000000000000E+00 0.000000000000E+00 -1.438747872521E-06 0.000000000000E+00 0.000000000000E+00 0.000000000000E+00
------ Current Surface: Coil-Plasma separation = 1.000000000000E+00 -----
Number of fourier modes in table
2
Table of fourier coefficients
m,n,crc2,czs2,crs2,czc2
0 0 4.000000000000E+00 0.000000000000E+00 0.000000000000E+00 0.000000000000E+00
1 0 2.000000000000E+00 2.000000000000E+00 0.000000000000E+00 0.000000000000E+00
The following table explains each parameter as defined by the preceding line:
| Input Parameter Name | Description |
|---|---|
| nu | Number of poloidal grid points on current surface |
| nv | Number of toroidal grid points on current surface |
| nu1 | Number of poloidal grid points on plasma surface |
| nv1 | Number of toroidal grid points on plasma surface |
| npol | Number of poloidal Fourier modes in potential solution |
| ntor | Number of toroidal Fourier modes in potential solution |
| mf | Number of poloidal Fourier modes for plasma surface |
| nf | Number of toroidal Fourier modes for plasma surface |
| md | Number of poloidal Fourier modes for current potential surface (and B normal) |
| nd | Number of toroidal Fourier modes for current potential surface (and B normal) |
| np | Number of field periods |
| iota_edge | Equilibrium rotational transform at edge |
| phip_edge | Equilibrium toroidal flux derivative at edge |
| curpol | Equilibrium total poloidal current in Amps per field period |
| cut | Normalized toroidal current |
| cup | Normalized poloidal current |
| ibex | Include external 1/R field ibex=1 |
| mstrt | Method + svdscan start if > 1, >=0 Berr, <=0 Least square |
| mstep | Method + svdscan stepsize <=0 least square, =0 use F04ABE, no svd |
| mkeep | svd/scan control 0 svdscan, else keep nkeep wgts |
| mdspw | 2 + exponent of dsur multiplying bfn, ben |
| curwt | Weight for surface current minimization (only in LSQ branch) |
| trgwt | Not yet implemented |
| For the Output control values -2 means just option 2, +2 means 1 and 2 | |
| w_psurf | Write plasma surface info (1: R/Z, 2: X/Y/Z, 3: NX/NY/NZ, 4: dXdu/dYdu/dXdv/dYdv ) |
| w_csurf | Write coil surface info (1: R/Z, 2: X/Y/Z, 3:NX/NY/NZ, 4: jx/jy/jz) |
| w_bnuv | Write Bnorm field info (2: BN_EXT) |
| w_jsurf | Write J surface current info (1: Potential, 2: Current) |
| w_xerr | Write X error (displacement) info |
| w_svd | Write SVD solution info |
The user should run the BNORM code to generate a ‘nescin’ file with offset winding surface.
To invoke NESCOIL
> xnescoil nescin.example
A text file is output with the name 'nescout.ext' where 'ext' in the same extension as the input file which generated the output. In the file, the various input parameters are output in tables along with the Fourier harmonics of the surface potential. Setting the w_psurf, w_csurf, w_bnuv, w_jsurf, w_xerr, and w_svd flags can modify the parameters which are output to this file. This file is essentially a self-documenting text file.
To calculate the potential one simply need evaluate:
\[\Phi\left(u,v\right) = \sum_{m=0}^M\sum_{n=-N}^N \Phi_{mn} sin\left(2\pi mu+2\pi nv\right) - \frac{I_{pol}}{N_p}v - I_{tor}u\]where \(u\) and \(v\) are defined on the unit circle from 0 to 1. When cutting coils one simply should calculate contours of constant potential. And use the \(u,v\) paris to evaluate R,phi, and Z. Note that \(v\) is define over one field period.
NCSX NESCOIL Example Solovev NESCOIL Example