The neutral fluid is generally important in determining the overall plasma proper- ties in any plasma device, with neutral collisions affecting plasma transport prop- erties, the neutral particles acting as the source for ion production, and neutral particles otherwise being heavily involved in plasma chemistry (in particular neg- ative ion production and plasma recombination in hydrogen). At low powers and ionisation fractions, the neutral profile is not strongly influenced by the plasma discharge. It is therefore common to calculate neutral flow properties entirely in- dependently from the plasma discharge[61]. In mid-to-high power plasma devices, such as MAGPIE, the ionisation fraction can approach unity, plasma pressure can far exceed neutral pressure, and the neutral fluid can be strongly heated by the discharge[165]. This results in the neutral dynamics being directly coupled to the discharge properties, and is especially important in the temporal evolution of the discharge. In order to accurately calculate the overall plasma discharge properties,
we must therefore account for the dynamics of the neutral species.
3.6.1
Neutral Species Profiles and Transport
As discussed in Section 3.1.2, the profile of the molecular hydrogen density is determined from a particle conserving ideal gas equation of state, based on an experimentally measured fill pressure profile. This approach is particularly insen- sitive to the temporal dynamics of molecular hydrogen, due to the assumption that the pressure and density each respond instantaneously to changing temper- atures, the fact that neutral flows and pumping are not considered beyond the measured pressure gradient, and that body forces acting on the neutral fluid due to ion-neutral momentum transfer are not considered, thus bypassing an explicit treatment of molecular hydrogen transport entirely. Furthermore, atomic hydro- gen density and transport is calculated under a dilute drift diffusion assumption, which can break down at the high dissociation fractions observed in the simula- tion and ignores bulk flows. Fortunately the assumptions involved in the above treatment of the neutral species remain suitable for treating the plasma steady state (with the addition of terms to account for neutral depletion as described in the following section), when calibrated empirically by experimental observations. The lack of an explicit treatment of neutral transport, and the associated lim- itations on temporal sensitivity for the neutral profiles is currently one of the primary factors which hinders the model’s applicability and ab initio predictive capability when simulating operating conditions where experimental data is un- available. While it is currently beyond the scope of this thesis, it would be feasible to include a relatively complete and self-consistent treatment of neutral transport. This would involve the inclusion of some form of multicomponent diffusion (likely an approximation of Maxwell-Stefan diffusion[166]) accounting for nondilute dif- fusive transport between atomic and molecular hydrogen, as well as the inclusion
of a mixture averaged Navier-Stokes equation incorporating body forces acting on the neutral fluid due to ion-neutral and electron-neutral collisions in regions of ion and electron pressure gradients and rotation. This would go a long way towards calculating realistic neutral dynamics, but it may not be sufficient on its own if turbulence plays a significant role in neutral flows.
Turbulence may be expected to play a role in both mass and heat transport in a device such as MAGPIE[167], especially in the radial direction where strong azimuthal plasma flows and the irregular internal geometry of the target chamber are likely to promote turbulent flow in the neutral fluid, and it may need to be taken into account if the previously described treatment were to be found insufficient in describing the observed behaviour of neutrals and their effect on the discharge.
3.6.2
Neutral Depletion
It is a well known observation that low pressure helicon discharges can often reach an upper density limit where increasing the applied power no longer leads to an increase in plasma density[165, 168–171], and this is also observed in MAGPIE (see Section 2.4.3). While it may be considered a trivial expectation that the high efficiency helicon coupling mechanism may lead to a density limit where the plasma is fully ionised and there are no longer any remaining neutrals to fuel the plasma and allow for a further increase in plasma density, the density limit is found to occur well below the original background density of the neutrals[165,
169]. This indicates that direct ionisation is not the primary mechanism which removes neutrals from the core of the plasma volume and this effect is termed ‘neutral depletion’. We must therefore consider additional mechanisms by which a depletion of the neutral density may occur.
are two proposed mechanisms by which it is commonly thought to occur[165]. The first description proceeds as an indirect result of ionisation. Neutral particles in the core of the plasma are initially ionised which allows the electrostatic fields to accelerate the ion out of the plasma volume at speeds approaching the Bohm velocity (depending on how close to the chamber walls the ion is), before being neutralised at the chamber wall. Due to the difference between the Bohm velocity (typically ∼ 20000 ms−1 in hydrogen) and the neutral thermal speed (typically
∼3000 ms−1), we would then expect a flux imbalance to create a neutral density
gradient, causing neutral depletion. Under standard conditions, MAGPIE oper- ates with strong magnetic fields which greatly reduces radial ion transport. As such, we would not expect this mechanism to be important in MAGPIE. Fur- thermore, electrostatic transport of ions is included in this model and it is found that this is insufficient to observe neutral depletion. We note that this finding does not preclude this mechanism from being important under different operating conditions and other devices. In particular it may be important in unmagnetised discharges with strong plasma potentials.
The second description involves the displacement of neutrals due to direct mo- mentum transfer in ion-neutral and electron-neutral collisions. When the plasma pressure becomes comparable to the neutral pressure, gradients in the plasma pres- sure will begin to apply strong body forces on the neutral fluid. As the plasma pressure is centrally peaked due to magnetic confinement, this leads to a radial outflow of neutrals from the plasma volume, resulting in neutral depletion in the core. In addition to the effects of static pressure gradients, we would also expect the azimuthal drifts of the ions to contribute to the collisional momentum transfer, with these collisions resulting in radial outflow of neutrals since they are not con- fined to rotational paths by the magnetic fields like the ions are. This mechanism is suitable for explaining the observed neutral depletion in MAGPIE, but to include it self-consistently in the model would require the improved treatment of neutral
transport described in the previous section. As such we consider approximate, empirically informed treatments of neutral depletion for each molecular hydrogen and atomic hydrogen. The treatment of neutral depletion is therefore subject to the same limitations as our treatment of neutral dynamics more generally, that is it has limited temporal sensitivity and ab initio predictive capability.
For molecular hydrogen, we assume that the molecular hydrogen pressure is displaced by the plasma pressure by the following relation
pH2 =
pH2,back
pH2,back+Cndpp
(3.69)
where pH2,back is the initial background fill pressure of molecular hydrogen as
calculated from the equation of state, and Cnd is a coupling coefficient which
describes how efficiently the plasma pressure,pp displaces the molecular hydrogen.
For atomic hydrogen, whose density is calculated from a particle balance equa- tions, we can include the effect of ion-neutral and electron-neutral collisions some- what more consistently. We assume that the outward neutral flux will be directly proportional to the plasma pressure gradient and include this as an additional term in the flux of atomic hydrogen
Γnd =
−Cnd∇pp
mHνm,H
(3.70)
with Cnd also describing how efficiently momentum is transferred to atomic hy-
drogen by ion-neutral and electron-neutral collisions. The coupling coefficient is found empirically by comparison with experimental observations. We find that under standard operating conditions a value of 0.2 is sufficient to approximately match observations. In Section 4.2.2 we examine the sensitivity of the model to our assumed value of Cnd.