Comment on "Pfirsch-Tasso versus standard approaches in the plasma stability theory including the resistive wall effects" [Phys. Plasmas 24, 112513 (2017)]

TL;DR

This paper defends the rigor of the Pfirsch-Tasso resistive wall mode theorems by providing detailed derivations that address previous criticisms, reaffirming their validity in plasma stability theory.

Contribution

It offers comprehensive derivations confirming the completeness and correctness of the original theorems, countering claims of limitations and omissions in prior critiques.

Findings

The proofs of the resistive wall mode theorems are rigorous and complete.

The operator $ abla\times\nabla\times$ is self-adjoint, ensuring certain terms vanish.

The additional term in Ohm's law claimed previously is shown to be zero.

Abstract

In the commended paper it is claimed that the proves of the "Resitive-Wall-Mode theorem" by Pfirsch and Tasso [Nucl. Fusion \textbf{11}, 259 (1971)] and extensions of that theorem for time dependent wall resistivity and equilibrium plasma flow are not detailed and that there are limitations restricting their applicability. In response, we provide here pertinent detailed derivations showing that the proves of the above mentioned theorems are rigorous and complete, unlike the considerations in the commended paper which ignore the self adjointness of the operator $\nabla\times\nabla\times$ and the fact that the force operator in the linearized ideal MHD momentum equation remains self adjoint in the presence of equilibrium flows. As a matter of fact it is proved here that, because of the self adjointness of the operator $\nabla\times\nabla\times$, a claimed in the commended paper additional term in Ohm's law vanishes identically.