Ferromagnetic effects in the theory of slow and fast resistive wall modes in tokamaks

The traditional theory of the resistive wall modes (RWMs) in the toroidal fusion systems was developed assuming the magnetic permeability μ of the wall the same as the vacuum one μ 0 . Here, we analyze the dynamics of unstable RWMs at the presence of a ferromagnetic wall with μ ̂ ≡ μ / μ 0 ≤ 4 . This choice with μ ̂ = c o n s t corresponds to the saturated state of ferritic materials in a strong magnetic field, as it should be in a tokamak reactor. The study is based on the cylindrical dispersion relation valid for arbitrary s / d w , where s is the skin depth and d w is the wall thickness. This equation is solved numerically, and the solutions are compared with analytical asymptotes obtained for slow ( s ≫ d w ) and fast ( s ≪ d w ) RWMs. Within the model, only very slow RWMs are found insensitive to variations of μ ̂ , while slightly above the no-wall stability limit the growth rate of the modes increases with larger μ ̂ . It is shown that at s < d w this increase is roughly given by a factor of μ ̂ compared to a similar case with μ ̂ = 1 . The dependence of the transition from slow to fast RWMs on μ ̂ is discussed, and the accuracy of the available analytical relations is evaluated.