Outputs¶

This page explains the output file formats for equilibrium solutions computed by DESC.

ASCII¶

The text output are ASCII files with the naming convention output.FILE_NAME. All of the necessary variables to fully define an equilibrium solution are output in the following order: grid parameters, fixed-boundary shape, pressure and rotational transform profiles, flux surface shapes, and the boundary function \(\lambda\). An example output file is included for reference at the end of this document. All integers are printed with a total width of 3 characters, and all floating point numbers are printed in exponential notation with a total width of 16 characters including 8 digits after the decimal points.

Grid Parameters¶

The first two lines of the output file specify some global parameters: next four lines contain the following information in order:

NFP (integer): number of field periods
Psi (float): total toroidal magnetic flux through the last closed flux surface, \(\psi_a\), in Webers

Fixed-Boundary Shape¶

The target shape of the plasma boundary is output for reference in the section of the output file with the heading Nbdry. This gives the number of boundary terms, followed by the coefficients. This is the fixed-boundary input that was used to compute the equilibrium, but the last closed flux surface generally does not match this desired shape exactly. The shape of the boundary surface is given as a double Fourier series of the form:

\[\begin{split}\begin{aligned} R^b(\theta,\phi) &= \sum_{n=-N}^{N} \sum_{m=-M}^{M} R^{b}_{mn} \mathcal{G}^{m}_{n}(\theta,\phi) \\ Z^b(\theta,\phi) &= \sum_{n=-N}^{N} \sum_{m=-M}^{M} Z^{b}_{mn} \mathcal{G}^{m}_{n}(\theta,\phi) \\ \label{eq:G} \mathcal{G}^{m}_{n}(\theta,\phi) &= \begin{cases} \cos(|m|\theta)\cos(|n|N_{FP}\phi) &\text{for }m\ge0, n\ge0 \\ \cos(|m|\theta)\sin(|n|N_{FP}\phi) &\text{for }m\ge0, n<0 \\ \sin(|m|\theta)\cos(|n|N_{FP}\phi) &\text{for }m<0, n\ge0 \\ \sin(|m|\theta)\sin(|n|N_{FP}\phi) &\text{for }m<0, n<0. \end{cases}\end{aligned}\end{split}\]

The Fourier coefficients \(R^{b}_{mn}\) and \(Z^{b}_{mn}\) are given by the output variables bR and bZ, respectively. The poloidal and toroidal mode numbers \(m\) and \(n\) that identify each coefficient are given by the variables m and n on the same line of the output file as bR and bZ. When stellarator symmetry is enforced, only the \(R^{b}_{mn}\) with \(mn > 0\) and the \(Z^{b}_{mn}\) with \(mn < 0\) are nonzero. Coefficients with \(mn = 0\) are nonzero for \(R^{b}_{mn}\) if one of the mode numbers is positive, and nonzero for \(Z^{b}_{mn}\) if one of the mode numbers is negative. The boundary surface given in the example is equivalent to (using Ptolemy’s identities):

\[\begin{split}\begin{aligned} R^b &= 10 - \cos\theta - 0.3 \cos(\theta-3\phi) \\ Z^b &= \sin\theta - 0.3 \sin(\theta-3\phi).\end{aligned}\end{split}\]

Pressure & Rotational Transform Profiles¶

The pressure and rotational transform profiles that were used to compute the equilibrium are also output for reference in the section of the output file with the heading Nprof, which also gives the number of profile coefficients. These are given as a power series in the flux surface label \(\rho \equiv \sqrt{\psi / \psi_a}\) as follows:

\[\begin{split}\begin{aligned} p(\rho) &= \sum_{l=0}^{2M} p_{l} \rho^{l} \\ \iota(\rho) &= \sum_{l=0}^{2M} \iota_{l} \rho^{l}.\end{aligned}\end{split}\]

The coefficients \(p_{l}\) and \(\iota_{l}\) are given by the output variables cP and cI, respectively. The radial order \(l\) that identifies each coefficient is given by the variable l on the same line of the output file as cP and cI. The profiles given in the example are:

\[\begin{split}\begin{aligned} p &= 3.4\times10^3 (1-\rho^2)^2 \\ \iota &= 0.5 + 1.5 \rho^2.\end{aligned}\end{split}\]

Flux Surface Shapes¶

The shapes of the flux surfaces are the solution to the equilibrium defined by the fixed-boundary and profile inputs. They are given by a Fourier-Zernike basis set with “fringe” indexing of the form:

\[\begin{split}\begin{aligned} R(\rho,\vartheta,\zeta) &= \sum_{n=-N}^{N} \sum_{m=-M}^{M} \sum_{l\in L} R_{lmn} \mathcal{Z}^{m}_{l}(\rho,\vartheta) \mathcal{F}^{n}(\zeta) \\ Z(\rho,\vartheta,\zeta) &= \sum_{n=-N}^{N} \sum_{m=-M}^{M} \sum_{l\in L} Z_{lmn} \mathcal{Z}^{m}_{l}(\rho,\vartheta) \mathcal{F}^{n}(\zeta)\end{aligned}\end{split}\]

where \(L = |m|, |m|+2, |m|+4, \ldots, 2 M\). \(\mathcal{F}^{n}(\zeta)\) is the toroidal Fourier series defined as

\[\begin{split}\mathcal{F}^{n}(\zeta) = \begin{cases} \cos(|n|N_{FP}\zeta) &\text{for }n\ge0 \\ \sin(|n|N_{FP}\zeta) &\text{for }n<0. \\ \end{cases}\end{split}\]

\(\mathcal{Z}^{m}_{l}(\rho,\vartheta)\) are the Zernike polynomials defined on the unit disc \(0\leq\rho\leq1\), \(\vartheta\in[0,2\pi)\) as

\[\begin{split}\mathcal{Z}^{m}_{l}(\rho,\vartheta) = \begin{cases} \mathcal{R}^{|m|}_{l}(\rho) \cos(|m|\vartheta) &\text{for }m\ge0 \\ \mathcal{R}^{|m|}_{l}(\rho) \sin(|m|\vartheta) &\text{for }m<0 \\ \end{cases}\end{split}\]

with the radial function

\[\mathcal{R}^{|m|}_{l}(\rho) = \sum^{(l-|m|)/2}_{s=0} \frac{(-1)^s(l-s)!}{s![\frac{1}{2}(l+|m|)-s]![\frac{1}{2}(l-|m|)-s]!} \rho^{l-2s}.\]

The Fourier-Zernike coefficients \(R_{mn}\) and \(Z_{mn}\) are given by the variables cR and cZ, respectively, in the section of the output file with the heading NRZ (which gives the total number of values). The indicies \(l\), \(m\), and \(n\) that identify each coefficient are given by the variables l, m, and n on the same line of the output file as cR and cZ. When stellarator symmetry is enforced, only the \(R_{mn}\) with \(mn > 0\) and the \(Z_{mn}\) with \(mn < 0\) are nonzero. Coefficients with \(mn = 0\) are nonzero for \(R_{mn}\) if one of the mode numbers is positive, and nonzero for \(Z_{mn}\) if one of the mode numbers is negative. Lines 45-46 of the example output file give the terms

\[\begin{split}\begin{aligned} R_{3,1,1} \mathcal{Z}^{1}_{3}(\rho,\vartheta) \mathcal{F}^{1}(\zeta) &= 5.26674681 \times 10^{-2} (3\rho^3-2\rho) \cos(\vartheta) \cos(3\zeta) \\ Z_{2,2,-1} \mathcal{Z}^{2}_{2}(\rho,\vartheta) \mathcal{F}^{-1}(\zeta) &= 5.01543691 \times 10^{-2} \rho^2 \cos(2\vartheta) \sin(3\zeta).\end{aligned}\end{split}\]

The magnetic field is computed in the straight field-line coordinate system \((\rho,\vartheta,\zeta)\) by

\[\mathbf{B} = B^\vartheta {\mathbf e}_{\vartheta}+ B^\zeta {\mathbf e}_{\zeta}= \frac{2\psi_a \rho}{2\pi \sqrt{g}} \left( \iota {\mathbf e}_{\vartheta}+ {\mathbf e}_{\zeta}\right).\]

The covariant basis vectors are defined as

\[\begin{split}{\mathbf e}_{\rho}= \begin{bmatrix} \partial_\rho R \\ 0 \\ \partial_\rho Z \end{bmatrix} \hspace{5mm} {\mathbf e}_{\vartheta}= \begin{bmatrix} \partial_\vartheta R \\ 0 \\ \partial_\vartheta Z \end{bmatrix} \hspace{5mm} {\mathbf e}_{\zeta}= \begin{bmatrix} \partial_\zeta R \\ R \\ \partial_\zeta Z \end{bmatrix}\end{split}\]

and the Jacobian of the coordinate system is \(\sqrt{g} = {\mathbf e}_{\rho}\cdot{\mathbf e}_{\vartheta}\times{\mathbf e}_{\zeta}\). The partial derivatives of \(R(\rho,\vartheta,\zeta)\) and \(Z(\rho,\vartheta,\zeta)\) are known analytically from the basis functions. The components of the magnetic field in the toroidal coordinate system \((R,\phi,Z)\) can be easily computed as \(B_i = \mathbf{B} \cdot \mathbf{e}_i\) with \({\mathbf e}_{R}= [1, 0, 0]^T\), \({\mathbf e}_{\phi}= [0, 1, 0]^T\), and \({\mathbf e}_{Z}= [0, 0, 1]^T\).

Boundary Function \(\lambda\)¶

The straight field-line angle \(\zeta\) is equivalent to the toroidal angle by definition: \(\zeta = \phi\). The function \(\lambda(\theta,\phi)\) relates the straight field-line angle \(\vartheta\) to the poloidal angle used to define the boundary surface \(\theta\) through the equation \(\vartheta = \theta + \lambda(\theta,\phi)\). It is used internally to enforce the boundary condition at the last closed flux surface, and is output for reference. The function is given as a doubles Fourier series of the form:

\[\begin{aligned} \lambda(\theta,\phi) &= \sum_{n=-N}^{N} \sum_{m=-M}^{M} \lambda_{mn} \mathcal{G}^{m}_{n}(\theta,\phi)\end{aligned}\]

where \(\mathcal{G}^{m}_{n}(\theta,\phi)\) was defined above for the boundary shape. The Fourier coefficients \(\lambda_{mn}\) are given by the variable cL in the section of the output file with the heading NL (which gives the number of \(\lambda\) coefficients). Their output format follows the same convention as the boundary coefficients bR and bZ. When stellarator symmetry is enforced, only the coefficients with \(mn < 0\) are nonzero. Coefficients with \(mn = 0\) are nonzero if one of the mode numbers is negative.

Example Output File¶

NFP =   3
Psi =   1.00000000E+00
Nbdry =  7
m:   0 n:   0 bR =   1.00000000E+01 bZ =   0.00000000E+00
m:   1 n:   0 bR =  -1.00000000E+00 bZ =   0.00000000E+00
m:  -1 n:   0 bR =   0.00000000E+00 bZ =   1.00000000E+00
m:   1 n:   1 bR =  -3.00000000E-01 bZ =   0.00000000E+00
m:  -1 n:  -1 bR =  -3.00000000E-01 bZ =   0.00000000E+00
m:  -1 n:   1 bR =   0.00000000E+00 bZ =  -3.00000000E-01
m:   1 n:  -1 bR =   0.00000000E+00 bZ =   3.00000000E-01
Nprof =   5
l:   0 cP =   3.40000000E+03 cI =   5.00000000E-01
l:   1 cP =   0.00000000E+00 cI =   0.00000000E+00
l:   2 cP =  -6.80000000E+03 cI =   1.50000000E+00
l:   3 cP =   0.00000000E+00 cI =   0.00000000E+00
l:   4 cP =   3.40000000E+03 cI =   0.00000000E+00
NRZ =   27
l:   0 m:   0 n:  -1 cR =   0.00000000E+00 cZ =  -2.90511418E-03
l:   0 m:   0 n:   0 cR =   9.98274712E+00 cZ =   0.00000000E+00
l:   0 m:   0 n:   1 cR =  -2.90180674E-03 cZ =   0.00000000E+00
l:   1 m:  -1 n:  -1 cR =   2.28896490E-01 cZ =   0.00000000E+00
l:   1 m:  -1 n:   0 cR =   0.00000000E+00 cZ =   9.48092222E-01
l:   1 m:  -1 n:   1 cR =   0.00000000E+00 cZ =  -2.27403979E-01
l:   2 m:   0 n:  -1 cR =   0.00000000E+00 cZ =  -2.41707137E-02
l:   2 m:   0 n:   0 cR =  -1.36531448E-01 cZ =   0.00000000E+00
l:   2 m:   0 n:   1 cR =  -2.41387024E-02 cZ =   0.00000000E+00
l:   1 m:   1 n:  -1 cR =   0.00000000E+00 cZ =   2.24346193E-01
l:   1 m:   1 n:   0 cR =   9.25944834E-01 cZ =   0.00000000E+00
l:   1 m:   1 n:   1 cR =   2.25843613E-01 cZ =   0.00000000E+00
l:   2 m:  -2 n:  -1 cR =   3.34519544E-02 cZ =   0.00000000E+00
l:   2 m:  -2 n:   0 cR =   0.00000000E+00 cZ =   1.58172393E-01
l:   2 m:  -2 n:   1 cR =   0.00000000E+00 cZ =  -5.03483447E-02
l:   3 m:  -1 n:  -1 cR =   4.81316537E-02 cZ =   0.00000000E+00
l:   3 m:  -1 n:   0 cR =   0.00000000E+00 cZ =   3.38024112E-02
l:   3 m:  -1 n:   1 cR =   0.00000000E+00 cZ =  -4.74860303E-02
l:   4 m:   0 n:  -1 cR =   0.00000000E+00 cZ =   2.08609498E-02
l:   4 m:   0 n:   0 cR =   1.33345992E-01 cZ =   0.00000000E+00
l:   4 m:   0 n:   1 cR =   2.07783052E-02 cZ =   0.00000000E+00
l:   3 m:   1 n:  -1 cR =   0.00000000E+00 cZ =   5.20291455E-02
l:   3 m:   1 n:   0 cR =   7.29416666E-02 cZ =   0.00000000E+00
l:   3 m:   1 n:   1 cR =   5.26674681E-02 cZ =   0.00000000E+00
l:   2 m:   2 n:  -1 cR =   0.00000000E+00 cZ =   5.01543691E-02
l:   2 m:   2 n:   0 cR =   1.56388795E-01 cZ =   0.00000000E+00
l:   2 m:   2 n:   1 cR =   3.32590868E-02 cZ =   0.00000000E+00
NL =  12
m:  -2 n:  -1 cL =  -0.00000000E+00
m:  -2 n:   0 cL =   9.55435813E-03
m:  -2 n:   1 cL =   2.53333116E-02
m:  -1 n:  -1 cL =  -0.00000000E+00
m:  -1 n:   0 cL =   9.91996517E-02
m:  -1 n:   1 cL =  -1.17417875E-02
m:   0 n:  -1 cL =   1.75103748E-04
m:   0 n:   0 cL =  -0.00000000E+00
m:   0 n:   1 cL =  -0.00000000E+00
m:   1 n:  -1 cL =   1.16506641E-02
m:   1 n:   0 cL =  -0.00000000E+00
m:   1 n:   1 cL =  -0.00000000E+00

HDF5¶

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