Sources and transfer terms
External sources are
\(S_n =\) Source of plasma ions
\(S_p =\) Source of pressure, related to energy source \(S_E = \frac{3}{2}S_p\)
In the simulations carried out so far, these source functions are both constant between midplane and X-point, and zero from X-point to target.
Transfer channels
There are several transfer channels and sinks for particles, energy and momentum due to rates of recombination, ionisation, charge exchange, electron-neutral excitation, and elastic collisions with units of m\(^{-3}\)s\(^{-1}\):
where \(n\) is the plasma density; \(n_n\) is the neutral gas density; \(\sigma_{cx}\) is the cross-section for charge exchange; \(\sigma_{rc}\) is the cross-section for recombination; and \(\sigma_{iz}\) is the cross-section for ionisation. Each of these processes’ cross-section depends on the local density and temperatures, and so changes in time and space as the simulation evolves.
\(S =\) Net recombination i.e neutral source (plasma particle sink). Calculated as Recombination - Ionisation:
\[\begin{aligned} S &=& \mathcal{R}_{rc} - \mathcal{R}_{iz}\end{aligned}\]\(R =\) Cooling of the plasma due to radiation, and plasma heating due to 3-body recombination at temperatures less than 5.25eV.
\[\begin{split}\begin{aligned} R &=& \left(1.09 T_e - 13.6\textrm{eV}\right)\mathcal{R}_{rc} \qquad \mbox{\textrm{(Recombination)}}\\ &+& E_{iz}\mathcal{R}_{iz} \qquad \mbox{\textrm{(Ionisation)}} \\ &+& \left(1\textrm{eV}\right)\mathcal{R}_{ex} \qquad \mbox{\textrm{(Excitation)}} \\ &+& R_{z,imp} \qquad \mbox{\textrm{(Impurity radiation)}} \end{aligned}\end{split}\]The factor of 1.09 in the recombination term, together with factor of \(3/2\) in \(E\) below, is so that recombination becomes a net heat source for the plasma at \(13.6 / 2.59 = 5.25\)eV. \(E_{iz}\) is the average energy required to ionise an atom, including energy lost through excitation.
If excitation is not included (
excitation = false) then following Togo et al., \(E_{iz}\) is chosen to be 30eV. If excitation is included, then \(E_{iz}\) should be set to \(13.6\)eV.\(E =\) Transfer of energy to neutrals.
\[\begin{split}\begin{aligned} E &=& \frac{3}{2} T_e \mathcal{R}_{rc} \qquad \mbox{\textrm{(Recombination)}} \\ &-& \frac{3}{2} T_n \mathcal{R}_{iz} \qquad \mbox{\textrm{(Ionisation)}} \\ &+& \frac{3}{2}\left(T_e - T_n\right)\mathcal{R}_{cx} \qquad \mbox{\textrm{(Charge exchange)**}} \\ &+& \frac{3}{2}\left(T_e - T_n\right)\mathcal{R}_{el} \qquad \mbox{\textrm{(Elastic collisions)**}} \end{aligned}\end{split}\](**) Note that if the neutral temperature is not evolved, then \(T_n = T_e\) is used to calculate the diffusion coefficient \(D_n\). In that case, \(T_n\) is set to zero here, otherwise it would cancel and leave no CX energy loss term.
\(F =\) Friction, a loss of momentum from the ions, due to charge exchange and recombination. The momentum of the neutrals is not currently modelled, so instead any momentum lost from the ions is assumed to be transmitted to the walls of the machine.
\[\begin{split}\begin{aligned} F &=& m_iV_{||}\mathcal{R}_{rc} \qquad \mbox{\textrm{(Recombination)}} \\ &-& m_iV_{||n}\mathcal{R}_{iz} \qquad \mbox{\textrm{(Ionisation)}} \\ &+& m_i\left(V_{||} - V_{||n}\right)\mathcal{R}_{cx} \qquad \mbox{\textrm{(Charge exchange)}} \\ &+& m_i\left(V_{||} - V_{||n}\right)\mathcal{R}_{el} \qquad \mbox{\textrm{(Elastic collisions)}}\end{aligned}\end{split}\]
All transfer channels are integrated over the cell volume using Simpson’s rule:
where \(J\) is the Jacobian of the coordinate system, corresponding to the cross-section area of the flux tube, and subscripts \(L\), \(C\) and \(R\) refer to values at the left, centre and right of the cell respectively.
Recycling
The flux of ions (and neutrals) to the target plate is recycled and
re-injected into the simulation. The fraction of the flux which is
re-injected is controlled by frecycle:
frecycle = 0.95 # Recycling fraction
The remaining particle flux (5% in this case) is assumed to be lost from the system. Note that if there are any external particle sources, then this fraction must be less than 1, or the number of particles in the simulation will never reach steady state.
Of the flux which is recycled, a fraction fredistribute is
redistributed along the length of the domain, whilst the remainder is
recycled at the target plate
fredistribute = 0.8 # Fraction of recycled neutrals redistributed evenly along length
The weighting which determines how this is redistributed is set using
redist_weight:
redist_weight = h(y - pi) # Weighting for redistribution
which is normalised in the code so that the integral is always 1. In these expressions \(y\) is uniform in cell index, going from \(0\) to \(2\pi\) between the boundaries. The above example therefore redistributes the neutrals evenly (in cell index) from half-way along the domain to the end.
When neutrals are injected, some assumptions are needed about their energy and momentum
When redistributed, neutrals are assumed to arrive with no net parallel momentum (so nothing is added to \(NV_n\)), and they are assumed to have the Franck-Condon energy (3.5eV currently)
When recycled from the target plate, neutrals are assumed to have a parallel momentum away from the target, with a thermal speed corresponding to the Franck-Condon energy, and is also added to the pressure equation. NOTE: This maybe should be one or the other, but not both…