Velocity Perturbation

The flux-freezing constraint of marginally-stable ideal-MHD takes the form

$${\bf E} + {\bf V}\times {\bf B} = 0.$$ (3.55)

[See Equation (2.379).] Taking the curl of the previous equation, combining with Maxwell's equations, and linearizing, we obtain

$$\frac{\partial\delta {\bf B}}{\partial t} = (\delta{\bf B}\cdot\n... ...\cdot\nabla)\,\delta{\bf V}_\perp - (\delta {\bf V}_\perp\cdot\nabla)\,{\bf B},$$ (3.56)

where

$$\delta {\bf V}_\perp= \nabla\phi\times {\bf e}_z,$$ (3.57)

and use has been made of Equation (3.50), as well as the facts that $\nabla\cdot{\bf B}=\nabla\cdot\delta {\bf B} = \nabla\cdot{\bf V}=\nabla\cdot\delta {\bf V}_\perp=0$. The radial component of the previous equation yields

$$\phi =-\left(\frac{\omega-{\mit\Omega}}{F}\right)\delta\psi,$$ (3.58)

where use has been made of Equations (3.9), (3.19), (3.32), (3.52), and (3.53). Equations (3.54) and (3.58) imply that

$$\delta v= 0.$$ (3.59)

Hence, we deduce that the perturbed plasma flow associated with a tearing mode in a low-$\beta$, large aspect-ratio, tokamak plasma is divergence-free [see Equations (3.51) and (3.59)], and is specified by Equations (3.57) and (3.58).