Definition of the field-line-following coordinates $(\psi , \theta , \alpha )$

Noting that $\nabla q \times \nabla \psi = 0$, the magnetic field in Eq. (250) can be further written

$$\mathbf{B}= - \Psi' \nabla (\zeta - q \theta) \times \nabla \psi .$$ (275)

Define a new coordinate $\alpha$ by

$$\alpha \equiv \zeta - q \theta,$$ (276)

to replace $\zeta$, then the magnetic field in Eq. (275) is written

$$\mathbf{B}= \Psi' \nabla \psi \times \nabla \alpha,$$ (277)

which is called the Clebsch form of the magnetic field. Equation (277) implies that

$$\mathbf{B} \cdot \nabla \alpha = 0,$$ (278)

and

$$\mathbf{B} \cdot \nabla \psi = 0,$$ (279)

which indicates that both $\alpha$ and $\psi$ are constant along a magnetic field line. Taking scalar product of Eq. (277) with $\nabla \theta$, we obtain

$$\mathbf{B} \cdot \nabla \theta = - \frac{\Psi'}{\mathcal{J}},$$ (280)

which is nonzero, i.e., only $\theta$ among $(\psi , \theta , \alpha )$ is changing along a magnetic field line. (Here $\mathcal{J}= (\nabla \psi \times \nabla \theta \cdot \nabla \alpha)^{- 1}$ is the Jacobian of the coordinate system $(\psi , \theta , \alpha )$, which happens to be equal to the Jacobian of $(\psi , \theta , \zeta )$ coordinates.) Due to this fact, $(\psi , \theta , \alpha )$ coordinates are usually called ``field-line-following coordinates'' or ``field-aligned coordinates'' [1,4].

Using Eq. (277), the magnetic differential operator $\mathbf{B} \cdot \nabla$ in the new coordinate system $(\psi , \theta , \alpha )$ is written

$$\mathbf{B} \cdot \nabla f = - \Psi' \frac{1}{\mathcal{J}} \frac{\partial}{\partial \theta} f,$$ (281)

which is just a partial derivative over $\theta$ and this is not surprising since only $\theta$ is changing along a magnetic field line.

It is widely accepted that turbulence in tokamak plasmas usually has $k_{\parallel} \ll k_{\perp}$, where $k_{\parallel}$ and $k_{\perp}$ are the parallel and perpendicular wavenumbers, respectively. Due to this elongated structure along the parallel direction, less grids can be used in the parallel direction than in the perpendicular direction in turbulence simulation. In this case, the field-aligned coordinates $(\psi , \theta , \alpha )$ provide suitable coordinates to be used, where less grids can be used for $\theta$ coordinate in simulations and even some $\partial / \partial \theta$ derivatives can be neglected (high-n approximation), which simplifies the equations that need to be solved.

Using Eq. (244), the definition of $\alpha$ in Eq. (276) can be written as

$$\alpha = \phi - \int_0^{\theta} \frac{\mathbf{B} \cdot \nabla \phi}{\mathbf{B} \cdot \nabla \theta} d \theta$$ (282)

$$= \phi - \int_0^{\theta} \hat{q} d \theta,$$ (283)

where $\hat{q} =\mathbf{B} \cdot \nabla \phi /\mathbf{B} \cdot \nabla \theta$ is the local safety factor. For later use, define $\overline {\delta }$ by $\overline{\delta} = \int_0^{\theta} \hat{q} d \theta$.