Analysis of the dispersion relation for resistive wall modes in tokamaks with account of the skin effect

The accuracy of the dispersion relations derived earlier analytically for the resistive wall modes (RWMs) in tokamaks is evaluated. Existing models use the expansions in the ratio of the wall thickness dw to the skin depth s. This parameter is small in the standard theory of "slow" RWMs, but it is large in the recently developed approach for the "fast" RWMs. Here, a dispersion relation applicable not only to these extreme cases, but also to the intermediate range is derived in the single-mode cylindrical approximation without restrictions on s/dw. The derived equation is solved numerically, the result is compared with the predictions of the earlier analytical models with either s ≫ dw or s ≪ dw. The applicability limits of the asymptotic expressions for the RWM growth rate are clarified. In particular, the limits are specified in which their error does not exceed 10%. It is shown that these expressions always underestimate the growth rate. The study confirms the necessity of the skin effect incorporation in the description of RWM dynamics.