google-deepmind / google-deepmind/torax

Add full "circuit equation" boundary condition on the psi equation

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Description

TORAX currently has the "loop voltage" boundary condition,

$$
\left. \frac{\partial \psi}{\partial t} \right\rvert_{\rho=\rho_B} = u(t)
$$

This allows an ad-hoc/approximate method for generating scenarios with prescribed inductive coil contributions (e.g. fully inductive scenarios, where $u(t) = 0$ and the ohmic current is hoped to be small).

For a better representation of solenoid-driven or solenoid-free scenarios, we should instead use the external circuit equation BC,

$$
u_{\rm inductive}(t) = \frac{d}{dt} (L_{\rm external} I_p) + V_{\rm loop}
$$

where $V_{\rm loop} = \left. \frac{\partial \psi}{\partial t} \right\rvert_{\rho=\rho_B} + 2\pi B_0 \mu(\rho_B) \frac{\partial \rho_B}{\partial t}$ (eq 51 and 74 from the ASTRA manual).

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