Numerical assessment of a novel technique for the reconstruction of 3D magnetic fields in tokamaks

Abstract

In this paper, we aim at numerical validating a novel technique for 3D reconstruction of magnetic fields in tokamaks from a discrete set of magnetic measurements. The identification is based on the solution of a classical inverse problem, after modeling the direct problem by using a well-tailored expansion for the plasma and the other field sources. Suitable boundary conditions are defined on two virtual nested surfaces, one inside the plasma, the other surrounding the plasma and including the magnetic sensors. Some a-priori information on the problem, as the expected major periodicity of 3D effects and the known poloidal field sources, are also taken into account for the field evaluation. An influence matrix is calculated, to get the measurement values due to each basis function and then a classical pseudo-inverse procedure identifies the coefficients in the expansion. The reconstructed magnetic field is then available to trace the plasma boundary with conventional methods. In this way, accurate 3D plasma shape identifications are available, at affordable computational cost. A significant set of magnetic configurations is adopted as test-bed for validating the technique, allowing in particular to separately evaluate the efficiency in catching the intrinsically 2D effects and the fully 3-D ones.

Keywords

Magnetic fields reconstruction plasma boundary identification tokamaks diagnostics

1. Introduction

The problem of the identification of the plasma boundary in tokamaks is fundamental for the plasma shape control, which in turn is a basic crucial point to achieve the nuclear fusion. Different approaches have been proposed in the literature [1, 2, 3, 4], but still room is available for developing robust and numerically reliable techniques, when 3D perturbations are considered. In this paper we numerically validate a technique based on a specific formulation for the direct problem, which is very well-suited for the typical tokamak geometries. The reconstruction algorithm is applied, for its validation, to some significant plasma configurations.

2. Field representation and identification

The problem of the plasma boundary 3-D identification from a discrete set of magnetic measurements, taken in the region outside the plasma but well inside the Tokamak structure, can be seen as the reconstruction, with prescribed accuracy, of the magnetic field map in a proper hollow region fully containing the plasma boundary itself. The magnetic field within such region should be represented with a high accuracy, but with a relatively small number of free parameters, in order to reasonably match the realistic number of measurements available for a real tokamak. This obviously requires an optimal combination of a smart formulation of the problem, a well-suited expansion for the field, as well as the correct inclusion of all available a-priori information. According to such goals, we define two suitable surfaces, a Virtual Internal Wall (VIW) well inside the plasma region and along the plasma centroid, and a Virtual External Wall (VEW) well outside the plasma region but well inside the external active structures in the tokamak. A schematic draw of such surfaces is given in Fig. 1. For the reconstruction of the field map within the defined interest region it is helpful to consider the field as the combination of three different contributions:

${\rm{\bf B}}={\rm{\bf B}}_{\textit{int}}+{\rm{\bf B}}_{\textit{ext}}+{\rm{\bf B% }}_{\textit{source}}$ (1)

Figure 1.

A poloidal section of the plane of the plasma chamber, with first wall (FW) and the axi-symmetric virtual internal (VIW) and external (VEW) walls.

where B ${}_{\textit{int}}$ represent the contribution to the field of the boundary conditions on the internal VIW (and zero on the VEW), B ${}_{\textit{ext}}$ represent the contribution to the field of the boundary conditions on the external VEW (and zero on the VIW), B ${}_{\textit{source}}$ represent the contribution to the field of all the sources external to the plasma, considered known with sufficient accuracy either from measurements or simulations. Due to the peculiarities of the field in a Tokamak structure, and considering that the interest region is current free (that constrains the divergence free magnetic field to be additionally curl free), we assume:

${\rm{\bf B}}-{\rm{\bf B}}_{\textit{source}}=\nabla\psi\times\nabla\varphi+f_{0% }\nabla\varphi-\nabla\Omega$ (2)

where $\varphi$ is the toroidal angle in a $\left({r,\varphi,z}\right)$ coordinate system, $f_{0}=\textit{const}$ , the functions $\psi\left({r,z}\right)$ and $\Omega\left({r,\varphi,z}\right)$ which are split (according to the above assumptions) as $\psi=\psi_{\textit{int}}+\psi_{\textit{ext}}$ , $\Omega=\Omega_{\textit{int}}+\Omega_{\textit{ext}}$ are subject to:

$\displaystyle-\frac{\partial}{\partial r}\ \left(\frac{1}{\mu_{0}r}\frac{% \partial\psi}{\partial r}\right)-\frac{\partial}{\partial z}\left(\frac{1}{\mu% _{0}r}\frac{\partial\psi}{\partial z}\right)=0,\left.\psi\right|_{\textit{VIW}% }=\psi_{i}\left({r,z}\right),\left.\psi\right|_{\textit{VEW}}=\psi_{e}\left({r% ,z}\right)$ (3a) $\displaystyle\nabla^{2}\Omega=0,\left.\Omega\right|_{\textit{VIW}}=\Omega_{i}% \left({r,z}\right);\left.\Omega\right|_{\textit{VEW}}=\Omega_{e}\left({r,z}\right)$ (3b)

It can be noted how the assumption of Eq.fon the structure of the field perturbation is reflected intrinsically in an axisymmetric poloidal component, given by the term $\nabla\psi\times\nabla\varphi$ , an axisymmetric toroidal component $f_{0}\nabla\varphi$ , and a fully 3-D term $-\nabla\Omega$ .

According with the field structure as assumed in Eq. (2), the unknown functions $\psi_{k}$ and $\Omega_{k},k\in\left\{{\text{int,ext}}\right\}$ , are expanded on the virtual boundaries as:

$\displaystyle\psi_{k}\left({r,z}\right)=\sum\limits_{m=1}^{M_{\psi k}}{c_{km}% \alpha_{\psi km}\left({r,z}\right)}$ (4a) $\displaystyle\Omega_{k}\left({r,\varphi,z}\right)=\sum\limits_{m=0}^{M_{\Omega k% }}{\sum\limits_{n\in S}{\left({a_{km}\cos n\varphi+b_{km}\sin n\varphi}\right)% }}\alpha_{\Omega\textit{kmn}}\left({r,z}\right)$ (4b)

Figure 2.

Examples of shape functions $\alpha_{\psi\textit{km}},\alpha_{\Omega\textit{kmn}}$ based on Fourier expansion. For instance, the shape function in the upper right corner verifying Eq. (3b), has zero value on the VIW and cos (3 $\vartheta$ ) on the VEW, where $\vartheta$ is the poloidal angle measured from the center of the circular cross section of the VIW.

where $S$ is a proper set of integers representing the expected toroidal harmonics.

The choice of the basis functions $\alpha_{\psi km},\alpha_{\Omega\textit{kmn}}$ of expansions in Eq. (4a) plays a key role. In this paper we consider the Fourier expansions in the poloidal plane (Fig. 2). However, different possible choices can alternatively be proposed.

The recurrent numerical solution of the problem Eq. (3a),(3b) for each term of boundary conditions as in Eq. (4) allows to define an “influence matrix” whose terms $g_{ij}$ represent the field at the location $i$ -th of the magnetic sensors due to the $j$ -th component in the expansion Eq. (4) . On these basis the identification is formulated as an inverse problem in the form:

$\displaystyle\left[{{\begin{array}[]{*{20}c}{g_{11}}\hfill&\cdots\hfill&{g_{1F% }}\hfill\\ \vdots\hfill&\ddots\hfill&\vdots\hfill\\ {g_{S1}}\hfill&\ldots\hfill&{g_{SF}}\hfill\\ \end{array}}}\right]\left[{{\begin{array}[]{*{20}c}{x_{1}}\hfill\\ \vdots\hfill\\ {x_{F}}\hfill\\ \end{array}}}\right]=\left[{{\begin{array}[]{*{20}c}{m_{1}}\hfill\\ \vdots\hfill\\ {m_{S}}\hfill\\ \end{array}}}\right]$ (5)

where ${\rm{\bf G}}=\left[{g_{ij}}\right]$ is the influence matrix, ${\rm{\bf M}}=\left[{m_{i}}\right]$ is the column vector of the measurements set, and the unknown ${\rm{\bf X}}=\left[{x_{i}}\right]$ vector represents the amplitudes of the respective coefficients $a_{km},b_{km},c_{km}$ for the corresponding terms in expansions Eq. (4a), (4b). System Eq. (5) is an over-determined rectangular one, that can be solved with classical least square methods, e.g., pseudo-inversion via singular value decomposition (SVD) with suitable regularization procedures.

3. Test cases and numerical assessment

In this section we aim to validate the formulation by considering a significant set of different magnetic configurations referred to the typical parameters of the Divertor Test Tokamak (DTT) facility [5] summarized in Table 1.

Table 1
Main parameters of the reference plasma in the DTT facility

Parameter	Value
Major radius R (m)	2.15
Minor radius a (m)	0.70
Plasma current $I_{p}$ (MA)	6.0
Toroidal field $B_{T}$ (T)	6.0
Plasma volume $V_{p}$ (m ${}^{3}$ )	33.0

With reference to such parameters, we define a set of test cases well suited to validate the formulation also in the presence of 3D field perturbations in a tokamak. Such cases are: i) axisymmetric single null equilibrium; ii) magnetic field of a circular kinked wire (not coaxial with the torus); iii) a set of sources with periodic (sinusoidal) distribution along circular wires coaxial with the torus; iv) 3D ripple field generated by 18 toroidal field coils.

Figure 3.

Location of the triaxial magnetic field sensors: a) 294 (882 measurements) for cases i–iii; b) 104 (312 measurements) for case iv.

For cases i–iii we assume that 882 local measurements are available from a set of triaxial magnetic field sensors placed at 49 poloidal locations inside the vessel at six evenly spaced toroidal angles (Fig. 3a). For case iv there are 312 measurements, available from two sets of triaxial magnetic field sensors placed at 49 poloidal locations at two toroidal angles (0 and 10 deg) complemented by 6 additional triaxial sensors located at a single poloidal location evenly spaced between $\varphi=$ 0 and $\varphi=$ 10 deg (Fig. 3b).

3.1 Axisymmetric single null equilibrium

Figure 4.

Magnetic field map in the poloidal section for a single null axisymmetric configuration: (a) actual map; (b) comparison between actual and reconstructed plasma boundary.

As a general consistency proof, we consider the case of a 2D axisymmetric single null magnetic configuration, referring to the DTT parameters [5]. The nominal magnetic field map in the poloidal section is reported in Fig. 4a. The expansions Eq. (4) are limited to $M_{\psi\textit{int}}=M_{\psi\textit{ext}}=40$ , $M_{\Omega\textit{int}}=M_{\Omega\textit{ext}}=40$ , and $n=1$ . Thus the number of degrees of freedom is $M_{\psi\textit{int}}+M_{\psi\textit{ext}}+2M_{\Omega\textit{int}}+2M_{\Omega% \textit{ext}}+1=$ 241. For the identification, we use SVD, truncating the singular values below 0.1% of the first one, thus considering 140 out of the 241 singular values. The global error of the reconstructed field map yields $\varepsilon_{\textit{est}}=$ 0.08% and $\varepsilon_{\textit{val}}=$ 0.08%, where $\varepsilon_{\textit{est}}$ is the relative fitting error for the 294 signals used for reconstruction and $\varepsilon_{\textit{val}}$ the relative error for the remaining 588 measurements. The plasma boundary reconstruction using the method proposed in [4, 6] is shown in Fig. 4b. In Fig. 5 we show the comparison of the actual and reconstructed sensor signals. It has to be underlined how the reconstruction of this 2-D example has been obtained with the full 3D expansion of Eq. (4), where the coefficients of the $\Omega$ expansion result correctly suppressed.

Figure 5.

Single null axisymmetric configuration: (a) comparison of the actual and reconstructed field values at the sensors position; (b) absolute reconstruction error The comparison on the toroidal component of the magnetic field has been omitted because its actual magnitude is zero and the absolute reconstruction error is orders of magnitude lower than that obtained on the vertical and radial components.

3.2 Kinked current wire

In this case, which is fully 3D in the original coordinate system, we consider a kinked circular filamentary current of 1 MA (obtained by a circular wire located at $R=$ 2.15 m, $Z=$ 0, after a 2 mm shift along the x-axis and a 0.5 deg tilt around the x-axis), as sketched in Fig. 6.

Figure 6.

Kinked filamentary current of test case ii ( $\Delta x=$ 2 mm, $\Delta\vartheta=$ 0.5 deg).

The expansions Eq. (4) are limited to $M_{\psi\textit{int}}=M_{\psi\textit{ext}}=40$ , $M_{\Omega\textit{int}}=M_{\Omega\textit{ext}}=$ 40, and $n=1$ . We use SVD, truncating the singular values below 0.1% of the first one, thus considering 140 out of the 241 singular values, obtaining $\varepsilon_{\textit{est}}=$ 0.08% and $\varepsilon_{\textit{val}}=$ 0.09%. In Fig. 7 we show the reconstructed maps of $\psi$ and $\Omega$ . In Fig. 8 we report the coefficients in expansion Eq. (4). As expected, the amplitudes of 2D and 3D terms are now comparable.

Figure 7.

Kinked current wire: reconstructed maps of $\psi(r,z)$ and $\Omega(r,\varphi=0,z)$ .

Figure 8.

Kinked current wire: expansion coefficients for $\psi$ and the $n=$ 1 component of $\Omega$ .

3.3 Cosinusoidal fictitious magnetic charge distribution on a ring

A significant test case is conceived by considering a cosinusoidal fictitious magnetic charge distribution on a ring, whose periodicity is an $n=1$ toroidal mode [7]. The expansions Eq. (4) are limited to $M_{\psi\textit{int}}=M_{\psi\textit{ext}}=$ 40, $M_{\Omega\textit{int}}=M_{\Omega\textit{ext}}=$ 40, and $n=1$ . Also here we use SVD, truncating the singular values below 0.1% of the first one, thus considering 140 out of the 241 singular values, obtaining $\varepsilon_{\textit{est}}=$ 0.05% and $\varepsilon_{\textit{val}}=$ 0.05%. Figure 9 shows the reconstructed $\psi$ and $\Omega$ maps.

Figure 9.

Cosinusoidal fictitious magnetic charge distribution on a ring: reconstructed maps of $\psi(r,z)$ and $\Omega(r,\varphi=0,z)$ .

As expected, this kind of source generates only an $n=1$ periodic perturbation on the flux density field, so suppressing the coefficients $c_{km}$ related to the poloidal flux per unit radian $\psi$ . Since the source is proportional to the cosine of the toroidal angle, the values obtained for the coefficients $b_{km}$ are orders of magnitude smaller than those of $a_{km}$ .

3.4 TF coils ripple field

Finally, we consider the 3D ripple generated by the periodic structure of the toroidal field coils in a tokamak. In this case, there is only the virtual external wall VEW. With reference to the DTT parameters summarized in Table 1 [5], the expansions Eq. (4) are limited to $M_{\psi\textit{ext}}=$ 14, with $M_{\Omega\textit{ext}}=$ 40 for the $n=$ 18 perturbation. The choice of such a set of identification parameters is related to the knowledge of information regarding the geometry of the field sources and their 20 deg periodicity along the torus. The truncation of the singular values below 0.1% of the first one, thus considering 100 out of the 203 singular values, yields values well below 0.01% for both $\varepsilon_{\textit{est}}$ and $\varepsilon_{\textit{val}}$ .

4. Conclusions

We have successfully assessed a new technique aimed to the 3D tokamak plasmas identification, which is based on the proper formulation of a classical SVD inversion after a well-suited definition of the direct problem. This technique extends to the 3D case the formulations usually adopted for the 2D axisymmetric equilibria [8, 9, 10, 11, 12]. In particular, the peculiar decomposition of the 3-D problem into the axisymmetric and non axisymmetric parts is tested with specifically designed test cases. Our analysis shows how such procedure is able to deal with a significant class of 3D perturbations, thanks to its flexibility. With a number of field sensors which is realistic for existing tokamaks the technique shows clearly its ability to reconstruct the actual field within an accuracy of 1%.

Footnotes

Acknowledgments

The authors wish to thank Profs. G. Miano and L. Verolino for providing a set of Green functions including the analytical solution for the test case described in Section 3.3.

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Numerical assessment of a novel technique for the reconstruction of 3D magnetic fields in tokamaks

Abstract

Keywords

1. Introduction

2. Field representation and identification

Table 1 Main parameters of the reference plasma in the DTT facility

4. Conclusions

Footnotes

Acknowledgments

References

Table 1
Main parameters of the reference plasma in the DTT facility