Coupling scheme
The derivation and implementation of the (kinetic) neutral model for JOREK is described in S. Q. Korving et al. PoP 2023.
For the neutral (ncs) and impurity (ics) coupling scheme, sources and sinks for particles are projected for the density, (parallel) momentum, and energy equations. However, as the JOREK form of the equations is different than the typical Euler form of the equations, the source terms need to be corrected. The sources are corrected as follows:
\[\begin{aligned} \left( \frac{\partial \rho}{\partial t}\right) &= S_\rho \\ \rho \left( \frac{\partial v}{\partial t}\right) &= \vec{S}_v = \vec{S}_{\rho v}-\vec{v}S_\rho \\ \left( \frac{\partial \rho T}{\partial t}\right) &= S_T = (\gamma -1)\left( S_E - \vec{v}\cdot \vec{S}_{\rho v} + \frac{1}{2} v^2 S_\rho \right), \\ \end{aligned}\]Where $S_E$, $\vec{S}_{\rho v}$ and $S_\rho$ are the physical source terms.
The particles are two-way coupled to the MHD equations, the schematics can be seen in the picture on the right:
Overview of source terms
| Reaction | $S_\rho$ | $\vec{S}_{\rho v}$ | $\vec{S}_{v}$ | $S_{E,i}$ | $S_{Ti}$ | $S_{E,e}$ | $S_{Te}$ |
|---|---|---|---|---|---|---|---|
| n Ion | $m_i \Gamma_{ion}$ | $m_i \Gamma_{ion} \vec{v}_n$ | $m_i \Gamma_{ion} (\vec{v}_n - \vec{v}_n)$ | $\Gamma_{ion} ( \frac{m_i}{m_n}\frac{T_n}{\gamma-1} + \frac{1}{2} m_i v_n^2)$ | $\Gamma_{ion} ( T_n + \frac{1}{2} m_i (\vec{v}_n-\vec{v}_i)^2)$ | $-\Gamma_{ion} \|E_{Bind}\|$ | $-\Gamma_{ion} \|E_{Bind}\|$ |
| n rec | $-m_i \Gamma_{rec}$ | $-m_i \Gamma_{rec} \vec{v}_i$ | - | $-\Gamma_{rec}( \frac{T_i}{\gamma-1}+\frac{1}{2} m_i v_i^2 )$ | $-\Gamma_{rec} T_i$ | $-\Gamma_{rec} \frac{T_e}{\gamma-1}$ | $-(L_{PRB}-S_{rec}\|E_{bind}\|)n_e n_i$ |
| n CX | - | $w_p (v_n^{old} - (N_{rng} \sqrt{\frac{T_i}{m_i}}+v_{drift}))$ | idem | $\frac{1}{2} w_p ((v_n^{old})^2-(v_n^{new})^2)$ | $S_E^{CX} - v \cdot S_v^{CX}$ | ||
| Imp ion/rec | - | - | - | - | - | $-w_p \|E_{bind}\|$ | $-(\gamma-1)w_p \|E_{bind}\|$ |
| Imp rad | - | - | - | - | - | $-n_e w_p (PLT + PRB -S_{rec}\|E_{bind}\|)\delta t$ | $-(\gamma-1)n_e w_p (PLT + PRB -S_{rec}\|E_{bind}\|)\delta t$ |
| Imp coll | - | $w_p m_{imp}(v_{old}-v_{new})$ | idem | $\frac{1}{2} w_p m_{imp}(v_{old}^2-v_{new}^2)$ | $(\gamma - 1)\frac{1}{2} w_p m_{imp}(v_{old}^2-v_{new}^2)$ |
Note 1, the terms including $ T + \frac{1}{2} m_i (\vec{v}_a-\vec{v}_b)^2 $ is a fluid picture, where the macroscopic drift velocity can be slpit from the random microscopic velocity resulting in the temperature. For the particle picture, it is just the kinetic energy.
Note 2, there is a difference between a radiation rate, and cooling rate. The energy sink for the plasma is the cooling rate. Not all radiation is couling the plasma, e.g. the dielectric cascade in i-e recombination