Density and Temperature Perturbations

According to the equations of marginally-stable ideal-MHD, (2.375)–(2.380), the electron number density, the electron temperature, and the ion temperature all satisfy equations of the form

$$\nabla_\parallel A\propto {\bf B}\cdot\nabla A = 0.$$ (3.39)

Linearization of the previous equation yields

$$\delta {\bf B}\cdot\nabla A + {\bf B}\cdot\nabla \delta A = 0,$$ (3.40)

where $A(r)$ denotes an equilibrium quantity. It follows from Equations (3.8), (3.19), and (3.32) that

$$\delta A = - \frac{m}{r}\,\frac{A'}{F}\,\delta\psi.$$ (3.41)

More explicitly, we conclude that the perturbations in the electron number density, the electron temperature, and the ion temperature that are associated with a tearing mode in a low-$\beta$, large aspect-ratio, tokamak plasma take the respective forms

$$\delta n_e = - \frac{m}{r}\,\frac{n_e'}{F}\,\delta\psi,$$ (3.42)

$$\delta T_e = - \frac{m}{r}\,\frac{T_e'}{F}\,\delta\psi,$$ (3.43)

$$\delta T_i = - \frac{m}{r}\,\frac{T_i'}{F}\,\delta\psi.$$ (3.44)