Radial differential operator

In solving the MHD eigenmode equations in toroidal geometry, we also need the radial differential operator $\nabla \psi \cdot \nabla$. Next, we derive the form of the operator in $(\psi , \theta , \zeta )$ coordinates. Using

$$\nabla f = \frac{\partial f}{\partial \psi} \nabla \psi + \frac{\... ...rtial \theta} \nabla \theta + \frac{\partial f}{\partial \zeta} \nabla \zeta,$$

the radial differential operator is written as

$$\nabla \psi \cdot \nabla f = \vert \nabla \psi \vert^2 \frac{\partial f}{\partial \psi} + (\na... ...al \theta} + (\nabla \zeta \cdot \nabla \psi) \frac{\partial f}{\partial \zeta}$$

$$= \vert \nabla \psi \vert^2 \frac{\partial f}{\partial \psi} + (\na... ...q \delta (\psi, \theta)] \cdot \nabla \psi \} \frac{\partial f}{\partial \zeta}$$

$$= \vert \nabla \psi \vert^2 \frac{\partial f}{\partial \psi} + (\na... ...\theta} - \nabla [q \delta] \cdot \nabla \psi \frac{\partial f}{\partial \zeta}$$

$$= \vert \nabla \psi \vert^2 \frac{\partial f}{\partial \psi} + (\na... ...a \delta + \delta \nabla q] \cdot \nabla \psi \frac{\partial f}{\partial \zeta}$$

$$= \vert \nabla \psi \vert^2 \frac{\partial f}{\partial \psi} + (\na... ...elta q' \nabla \psi \right] \cdot \nabla \psi \frac{\partial f}{\partial \zeta}$$

$$= \vert \nabla \psi \vert^2 \frac{\partial f}{\partial \psi} + (\na... ...eta} \nabla \theta \cdot \nabla \psi \right] \frac{\partial f}{\partial \zeta},$$ (273)

where $\partial (q \delta) / \partial \psi$ and $q \partial \delta / \partial \theta$ are given respectively by Eqs. (246) and (240). Using the above formula, $\nabla \psi \cdot \nabla \zeta$ is written as

$$\nabla \psi \cdot \nabla \zeta = - \left[ \frac{\partial (q \delt... ...rac{\partial \delta}{\partial \theta} \nabla \theta \cdot \nabla \psi \right] .$$ (274)

This formula is used in GTAW code.