Governing equations

PyAQSC expands its governing equations in the flux coordinate $$(\epsilon\equiv\sqrt{\psi}, \chi\equiv\theta-\frac{N}{M}\phi, \phi),$$ where $\psi$ is the toroidal flux normalized by $2\pi$ ($\psi = \Psi/2\pi$. The definition of $\psi$ is identical to that in DESC.), $\theta$ is the poloidal angle, and $\phi$ is the toroidal angle. The coordinate transformation $\textbf{r}(\psi, \chi, \phi)$ will be self-consistently solved with other physical quantities.

The governing equations solved by pyAQSC are:

• The Jacobian equation $$J^{-1} = (\nabla\psi\times\nabla\theta\cdot\nabla\phi) = \frac{B^2}{B_\alpha^2}$$
• The co/contravariant equation $$\textbf{B}=B_\theta\nabla\chi +(B_\alpha-\bar{\iota}B_\theta)\nabla\phi +B_\psi\nabla\psi\\ =\nabla\psi\times\nabla\chi +\bar{\iota}\nabla\phi\times\nabla\psi.$$
• The force balance equation $$\textbf{j}\times\textbf{B} = \nabla\cdot\Pi = \nabla\cdot(\Delta\textbf{bb} + p_\perp \mathbb{I}).$$ The first two equations, the "magnetic equations", encodes QS and nested flux surfaceconditions. They can be solved independenctly with pyAQSC to yield a globally QS magnetic field with no force balance.

The last equation enforces force balance. Solving all three equations together yields a globally QS equilibrium with anisotropic pressure.

For more details, see Ref.1-3.