Electromagnetic thin-wall model for simulations of plasma wall-touching kink and vertical modes

Abstract

The understanding of plasma disruptions in tokamaks and predictions of their effects require realistic simulations of electric current excitation in three-dimensional vessel structures by the plasma touching the walls. As discovered at JET in 1996 (Litunovski JET Internal Report contract no. JQ5/11961, 1995; Noll et al., Proceedings of the 19th Symposium on Fusion Technology, Lisbon (ed. C. Varandas & F. Serra), vol. 1, 1996, p. 751. Elsevier) the wall-touching kink modes are frequently excited during vertical displacement events and cause large sideways forces on the vacuum vessel which are difficult to withstand in large tokamaks. In disruptions, the sharing of electric current between the plasma and the wall plays an important role in plasma dynamics and determines the amplitude and localization of the sideways force (Riccardo et al., Nucl. Fusion, vol. 40, 2000, p. 1805; Riccardo & Walker, Plasma Phys. Control. Fusion, vol. 42, 2000, p. 29; Zakharov, Phys. Plasmas, vol. 15, 2008, 062507; Riccardo et al., Nucl. Fusion, vol. 49, 2009, 055012; Bachmann et al., Fusion Engng Des., vol. 86, 2011, pp. 1915–1919). This paper describes a flat triangle representation of the electric circuits of a thin conducting wall of arbitrary three-dimensional geometry. Implemented into the shell simulation code (SHL) and the source sink current code (SSC), this model is suitable for modelling the electric currents excited in the wall inductively and through current sharing with the plasma.

The JOREK non-linear extended MHD code and applications to large-scale instabilities and their control in magnetically confined fusion plasmas

“…Via the derivation shown in references [93][94][95], plasma currents flowing directly into conducting structures or out of them (current sharing between plasma and wall), can be treated 35 Note, that the natural boundary condition, however, leads to a less sparse matrix structure for boundary degrees of freedom. Local interactions between neighbouring grid nodes are replaced by global interactions on the boundary.…”

Simulation of the electromagnetic wall response during Vertical Displacement Events (VDE) in ITER tokamak

“…According to Helmholtz decomposition theorem, our surface current density d w j in the conducting shell (a vector field twice continuously differentiable in 3D) can be split into two components: an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field [9]…”

“…Therefore, we have used a weak formulation -a finite element method formulation -for our problem. Thus, φ S was obtained by minimizing the functional W S [9,12,13,14,15]…”

“…For the divergence-free part of the surface current i, the energy principle looks like [9,13,14] ∂W S /∂ φ S = 0, ∂W I /∂ I n = 0, ∂W I /∂ φ S = 0, leads to a linear systems of equations with Hermitian symmetric-positive definite matrices which can be solved using the Cholesky decomposition.…”

“…Understanding that in disruptions the sharing of electric current between the plasma and the wall plays an important role in plasma dynamics [1,2,7,8], we have developed a wall model that covers both eddy currents, excited inductively, and source/sink currents due to current sharing between the plasma and the wall. We have adopted a triangle representation of the plasma facing wall surface [9] (simplicity and analytical formulas for magnetic field B and magnetic vector potential A of a uniform current in a single triangle). We have considered the wall in its thin wall approximation (reasonable for thin stainless steel structures of the vacuum vessel of about 1-3 cm thick with conductivity σ = 1.3810 6 Ω −1 m −1 ).…”