Trang VU, Laurent LEFEVRE and Bernhard MASCHKE
Abstract— A port-Hamiltonian model is derived for tokamak plasma current proﬁle dynamics. Magnetohydrodynamics and energy balance equations are ﬁrst written in port-Hamiltonian formalism using Stokes-Dirac structures in 3D forms and a speciﬁc interconnection structure for the magneto-hydrodynamical coupling. A balance preserving reduction scheme is applied to the 3D models, assuming classical cylindrical approximations and neglecting the diamagnetic effects. .
I. I NTRODUCTION A plasma is a gas in which an important fraction of the particles is ionized, so that the electrons and ions are separately free. A tokamak is a facility (whose main ideas came in the 1950’s from a group of Russian scientists including Andrei Sakharov and Igor Tamm) constructed with the shape of a torus (or dough-nut) in which a plasma is magnetically conﬁned and heated in order to produce nuclear fusion reactions. The magnetic conﬁnement of the plasma particles in the vacuum vessel torus is obtained through the combination of toroidal and poloidal ﬁelds produced by external coils (see ﬁg.I.1) with the additional ﬁeld produced by the electrical current ﬂowing along the plasma ring. This plasma current is generally ﬁrstly generated by induction (the plasma ring can then be considered as the secondary loop of a transformer whose primary loop is the ohmic ﬁeld coil (see ﬁg.I.1). It allows to heat up the plasma, which behaves as a resistive conductor. However, ohmic heating and current drive do not allow to reach the adequate plasma temperature and duration required for future fusion reactors. Indeed the plasma resistivity decreases with temperature and technology limits the ohmic ﬁeld coil current. Non inductive heating and current drive methods were thus developed to take it over, namely high power microwave or fast neutral beams injection. A complete review of tokamak’s