Form of operator $(B \times \nabla \psi / B^2) \cdot \nabla$ in $(\psi , \theta , \zeta )$ coordinates

In solving the MHD eigenmode equations in toroidal geometries, besides the $\mathbf{B} \cdot \nabla$ operator, we will also encounter another surface operator $(\mathbf{B} \times \nabla \psi / B^2) \cdot \nabla$. Next, we derive the form of the this operator in $(\psi , \theta , \zeta )$ coordinate system. Using the covariant form of the equilibrium magnetic field [Eq. (268)], we obtain

$$\frac{\mathbf{B} \times \nabla \psi}{B^2} = \frac{1}{B^2} \left( ... ...bla \theta \times \nabla \psi + \frac{g}{B^2} \nabla \zeta \times \nabla \psi .$$ (270)

Using this, the $(\mathbf{B} \times \nabla \psi / B^2) \cdot \nabla$ operator is written as

$$\frac{\mathbf{B} \times \nabla \psi}{B^2} \cdot \nabla = \frac{1}{B^2} \left( \frac{B^2}{\Psi'} \mathcal{J}+ g q \right) \... ...rtial \zeta} + \frac{g}{B^2} \mathcal{J}^{- 1} \frac{\partial}{\partial \theta}$$ (271)

$$= \left( \frac{1}{\Psi'} + g \frac{\mathcal{J}^{- 1}}{B^2} q \right... ...tial \zeta} + g \frac{\mathcal{J}^{- 1}}{B^2} \frac{\partial}{\partial \theta},$$ (272)

which is the form of the operator in $(\psi , \theta , \zeta )$ coordinate system.

Examining Eq. (272), we find that if the Jacobian $\mathcal{J}$ is chosen to be of the form $\mathcal{J}= \alpha (\psi) / B^2$, where $\alpha$ is some magnetic surface function, then the coefficients before the two partial derivatives will be independent of $\theta$ and $\zeta$. It is obvious that the independence of the coefficients on $\theta$ and $\zeta$ will be advantageous to some applications. The coordinate system $(\psi , \theta , \zeta )$ with the particular choice of $\mathcal{J}= \alpha (\psi) / B^2$ is called the Boozer coordinates, named after A.H. Boozer, who first proposed this choice of the Jacobian. The usefulness of the new toroidal angle $\zeta$ is highlighted in Boozer's choice of the Jacobian, which makes both $\mathbf{B} \cdot \nabla$ and $(\mathbf{B} \times \nabla \psi / B^2) \cdot \nabla$ be a constant-coefficient differential operator. For other choices of the Jacobian, only the $\mathbf{B} \cdot \nabla$ operator is a constant-coefficient differential operator.