J. Havlíček, M. Komm, M. Peterka (IPP Prague)

L.C. Appel, I. Lupelli (CCFE)

J.-F. Artaud (CEA)

B. Faugeras (U. Nice)

Abstract

- FREEBIE inverse mode calculation for a given shape, p and .
- Reconstruction by EFIT from FREEBIE with available diagnostics (Rogowski coils, flux loops, TS)
- accuracy vs noise
- accuracy vs p', ff' degrees

- Rapid reconstruction by EQUINOX radial basis functions
- Possible enhancements -- synthetic dianostics
- MSE
- LCFS position (e.g. probes)

- FREEBIE in inverse mode for COMPASS as a shape designer, scenario designer)

*Optional*

*VDE simulation compared to experiments**METIS/FREEBIE comparison to experiments*

\label{sec:introduction}

We report here on validation and verification of tokamak equilibrium tools used for the COMPASS tokamak (Pánek 2006). We particularly focus on fundamental global plasma parameters and the shapes of magnetic flux surfaces, which are crucial in diagnostics interpretation and other analyses. EFIT++ (Appel 2006) is used for routine equilibrium reconstruction on COMPASS. FREEBIE (Artaud 2012) is a recent free-boundary equilibrium code; FREEBIE enables predictive equilibrium calculation consistent with the poloidal field (PF) components of the tokamak. In this study, FREEBIE is used in the so-called inverse mode, which predicts PF coils currents from a given plasma boundary and profiles. The third code employed in this study is VacTH (Faugeras 2014), which provides a fast reconstruction of the plasma boundary from magnetic measurements using a toroidal harmonics basis.

In order to verify and validate the aforementioned tools, we analyse EFIT++ and VacTH reconstructions of equilibria constructed with FREEBIE. Synthetic diagnostics (e.g., magnetic probes or flux loops) with optional artificial errors provide inputs for the reconstructions.

\label{sec:procedure}

Reliable MHD equilibrium reconstruction is very important for tokamak exploitation. Numerous diagnostics and subsequent analyses require as inputs equilibrium properties such as flux surface geometry, magnetic field, stored energy, internal inductance etc. We have set up a set of benchmarking tasks, which verify and validate equilibrium tools that are currently employed on COMPASS. The procedure is fundamentally following:

Equilibrium reconstruction of selected experimental cases using EFIT++.

Recalculate the equilibria using FREEBIE in inverse mode.

Optionally alter the equilibria in FREEBIE using e.g. experimental pressure profiles.

Reconstruct FREEBIE equilibria using EFIT++ and VacTH with various parameters and artificial input noise.

The first step employs a routine EFIT++ set-up for COMPASS with heuristically tuned parameters. In addition to the total plasma current \(I_\mathrm{p}\) and the currents in individual PF circuits, 16 partial Rogowski coils and 4 flux loops are employed in this reconstruction and \(p'\) and \(FF'\) are assumed to be linear functions of the poloidal flux \(\psi\).

In the second step, FREEBIE inputs \(I_\mathrm{p}\), \(p'\left( {\bar \psi } \right)\) and \(FF'\left( {\bar \psi } \right)\) profiles, the plasma boundary coordinates and an initial guess for the PF coils currents. Here, \(p\) is the plasma pressure, \(F = RB_\phi\) and \(\bar\psi\) is the normalized poloidal magnetic flux (\(\bar\psi = 0\) on the magnetic axis and \(\bar\psi = 1\) on the plasma boundary). \(p'\) comes either from the EFIT++ reconstruction or from Thomson scattering pressure profile \(p_\mathrm{TS} = 1.3 n_\mathrm{e} p_\mathrm{e}\). FREEBIE then seeks a solution to the Grad-Shafranov equation, including the PF coils currents, which minimizes the given plasma shape constraint. (This regime is called the inverse mode.)

It should be noted here that to set up a free-boundary equilibrium code, a rather complete machine description is necessary (in particular, the PF coils geometry and circuits, limiter, vessel and other passive PF elements and magnetic diagnostics configuration). We adopted the description that was already available for EFIT++ and transformed it to ITM CPO’s (Integrated Tokamak Modelling Consistent Physical Objects (Manduchi 2008)) structures, which are subsequently either used directly in FREEBIE or converted to VacTH specific input format.

FREEBIE can naturally output arbitrary synthetic diagnostics. We use here additional 24 poloidally and 24 radially oriented partial Rogowski coils (which are actually mounted on COMPASS) and an artificial set of 16 flux loops located at the same positions as the basic magnetic probes. Hereafter, the number of magnetic probes and flux loops are denoted \(n_\mathrm{mp}\) and \(n_\mathrm{fl}\). \(n_\mathrm{mp}=16\), \(n_\mathrm{fl}=4\) refers the basic set of magnetic measurements, \(n_\mathrm{mp}=64\) refers to a set of all presently mounted partial Rogowski coils on COMPASS and \(n_\mathrm{fl}=16\) implies artificial flux loops positioned at the same locations as the basic magnetic probes.

FREEBIE inputs can be modified in the optional third step. In the following analysis, we particularly use realistic pressure profiles, estimated by Thomson scattering diagnostics.

The final fourth step consists of reconstructing the equilibria form synthetic FREEBIE data using EFIT++ and VacTH. An artificial random noise is added to the calculated values of \(I_\mathrm{p}\), magnetic probes and flux loops. In particular, for a given noise level \(\epsilon\), \(\tilde X = \left( {1 + U\left( { - \epsilon, \epsilon} \right)^\mathrm{T}} \right)X\), where \(X\) is a row vector of the synthetic diagnostics data and \(U\left( { - \epsilon, \epsilon} \right)\) is a random vector of the same shape as \(X\) with a uniform distribution on \(\left( { - \epsilon, \epsilon} \right)\). The reconstructions are then compared to the original equilibrium, focusing on global parameters and geometry. Scans are performed over noise levels (\(\epsilon\)) and selected code parameters: \(p'\) and \(FF'\) polynomial degrees in EFIT++ (\(n_{p'}\), \(n_{FF'}\)) and the number of harmonics (\(n_P\), \(n_Q\)) in VacTH. The following quantities are used for the comparison.

\(R_{\mathrm ax}\), \(Z_{\mathrm ax}\) | \(R,Z\) coordinates of the magnetic axis |

\(R_{\mathrm in}\), \(R_{\mathrm out}\) | inner/outer \(R\) coordinate of LCFS at \(Z=Z_{\mathrm ax}\) |

\(Z_{\mathrm min}\), \(Z_{\mathrm max}\) | minimum/maximum \(Z\) coordinate of LCFS |

\(I_{\rm{p}}\) | plasma current |

\(\kappa = \frac{\left( {{Z_{{\rm{max}}}} - {Z_{{\rm{min}}}}} \right) } { \left( {{R_{{\rm{out}}}} - {R_{{\rm{in}}}}} \right) }\) | elongation |

\({l_{\rm{i}}} = {{\bar B_{\rm{p}}^2}} / {{B_{\rm{a}}^2}}\) | normalized internal inductance |

\({\beta _{\rm{p}}} = {{2{\mu _0}\bar p}} / {{B_{\rm{a}}^2}}\) | poloidal beta |

\( W = \int_0^V {\frac{3}{2}p{\rm{d}}V'}\) | stored plasma energy |

\( q_0\), \(q_{95}\) | safety factor at \(\bar \psi = 0,\ 0.95\) |

Here, \(\bar x = \int_0^V {x/V{\rm{d}}V'} \) is a volume average, \(V\) is the total plasma volume, \(B_{\mathrm a} = \mu _0 I_\mathrm{p} / l_{\mathrm a}\), \(l_{\mathrm a}\) is the poloidal LCFS (last closed flux surface) perimeter. We also define absolute differences \(\Delta x = {{x_0} - x} \) and relative differences \(\delta x = \left| {{x_0} - x} \right|/\left| {{x_0}} \right|\), where \(x\) is an arbitrary recontructed quantity and \(x_0\) its target value.

VacTH does not provide a full equilibrium but the plasma shape only as the target of VacTH is to provide such reconstructions in real time for a feedback control. The inputs (besides the machine description and code parameters) of VacTH are PF coil currents, \(I_{\rm{p}}\), magnetic probes and flux loops measurements and the initial axis and X-point coordinates (the coordinates can be fixed as code parameters).