Covariant form of magnetic field in $(\psi , \theta , \zeta )$ coordinate system

In the above, we have given the covariant form of the magnetic field in $(\psi , \theta , \phi )$ coordinates (i.e., Eq. (177)). Next, we derive the corresponding form in $(\psi , \theta , \zeta )$ coordinate. In order to do this, we need to express the $\nabla \phi$ basis vector in terms of $\nabla \psi$, $\nabla \theta$, and $\nabla \zeta$ basis vectors. Using the definition of the generalized toroidal angle, we obtain

$$g \nabla \phi = g \nabla (\zeta + q \delta)$$

$$= g \nabla \zeta + g q \nabla \delta + g \delta \nabla q$$

$$= g \nabla \zeta + g q \left( \frac{\partial \delta}{\partial \psi}... ...artial \delta}{\partial \theta} \nabla \theta \right) + g \delta q' \nabla \psi$$

$$= \left( g q \frac{\partial \delta}{\partial \psi} + g \delta q' \r... ...si + g q \frac{\partial \delta}{\partial \theta} \nabla \theta + g \nabla \zeta$$

$$= g \frac{\partial (q \delta)}{\partial \psi} \nabla \psi + g q \frac{\partial \delta}{\partial \theta} \nabla \theta + g \nabla \zeta .$$ (265)

Using Eq. (265), the covariant form of the magnetic field, Eq. (177), is written as

$$\mathbf{B}= \left( \Psi' \frac{\mathcal{J}}{R^2} \nabla \psi \cdo... ...cal{J}}{R^2} \vert \nabla \psi \vert^2 \right) \nabla \theta + g \nabla \zeta .$$ (266)

The expression (266) can be further simplified by using the equation (240) to eliminate $\partial \delta / \partial \theta$, which gives

$$\mathbf{B} = \left( \Psi' \frac{\mathcal{J}}{R^2} \nabla \psi \cdot \nabla \th... ...bla \psi \vert^2 \right) \frac{\mathcal{J}}{R^2} \nabla \theta + g \nabla \zeta$$

$$= \left( \Psi' \frac{\mathcal{J}}{R^2} \nabla \psi \cdot \nabla \th... ...}{\mathcal{J}} \right) \frac{\mathcal{J}}{R^2} \nabla \theta + g \nabla \zeta .$$ (267)

Using $B_0^2 = (\vert \nabla \Psi \vert^2 + g^2) / R^2$, the above equation is written as

$$\mathbf{B} = \left( \Psi' \frac{\mathcal{J}}{R^2} \nabla \psi \cdot \nabla \th... ...^2}{\mathcal{J}} \right) \frac{\mathcal{J}}{R^2} \nabla \theta + g \nabla \zeta$$

$$= \left( \Psi' \frac{\mathcal{J}}{R^2} \nabla \psi \cdot \nabla \th... ...- \frac{B_0^2}{\Psi'} \mathcal{J}- g q \right) \nabla \theta + g \nabla \zeta .$$ (268)

Equation (268) is the covariant form of the magnetic field in $(\psi , \theta , \zeta )$ coordinate system. For the particular choice of the radial coordinate $\psi = - \Psi$ and the Jacobian $\mathcal{J}= \alpha (\psi) / B_0^2$, Eq. (268) reduces to

$$\mathbf{B}= \left( - \frac{\mathcal{J}}{R^2} \nabla \psi \cdot \n... ...}{\partial \psi} \right) \nabla \psi + I (\psi) \nabla \theta + g \nabla \zeta,$$ (269)

with $I (\psi) = \alpha (\psi) - g q$. The magnetic field expression in Eq. (269) frequently appears in tokamak literature.