Solves for

Definitions

PyAQSC solves the following unknowns self-consistently. These quantities are managed by instances of aqsc.Equilibrium. On how to access these quantities, see aqsc.Equilibrium API.

• $X_n,Y_n,Z_n(\chi, \phi)$ - The coordinate transformation between the flux coordinate and a known Frenet frame: $$\textbf{x}(\psi, \chi, \phi) = \textbf{r}_0[l(\phi)]\\ +X(\psi, \chi, \phi) \hat{\boldsymbol{\kappa}}_0[l(\phi)]\\ +Y(\psi, \chi, \phi) \hat{\boldsymbol{\tau}}_0[l(\phi)]\\ +Z(\psi, \chi, \phi) \hat{\boldsymbol{b}}_0[l(\phi)].$$ By solving for $X_n, Y_n, Z_n$, pyAQSC constructs the flux coordinate self-consistently with other physical quantities.
• $B_{\psi n-2}, B_{\theta n}(\psi, \chi, \phi)$ - The magnetic field component. $B_\psi$ is always at $2$ orders lower than all other outputs.
• $p_{\perp n}, \Delta_n(\psi, \chi, \phi)$ - The perpendicular pressure and anisotropy: $$\nabla\cdot\Pi = \nabla\cdot(\Delta\textbf{bb} + p_\perp \mathbb{I})$$

When solving only the magnetic equations, $B_\theta, B_\psi$ and $Y$ will be treated as free parameters.

Expansion form

All of other quantities are expanded as: $$F(\psi, \chi, \phi)\\ =\sum_{n=0}^{n_{max}}\epsilon^nF_n(\chi,\phi) (\epsilon\equiv\sqrt{\psi})\\ =\sum_{n=0}^{n_{max}}\epsilon^n\sum_{m=0|1}^n e^{im} F_{n,m}(\phi) + e^{-im} F_{n,-m}(\phi)$$