3.1 Neutral transport model

The two largest sources of neutrals are gas injection and wall recombination. Relevant processes that act as source and sink terms in the neutral particle continuity equation are direct ionisation by electron-impact collisions, charge-exchange recombination with plasma ions and recombination with free electrons. Neutrals generated by charge-exchange recombination have a long mean free path and are directly lost to the wall.

We assume that the trajectory of neutral particles emanating from the wall are quickly randomised by hard-sphere ion–neutral collisions. Considering the ion–neutral collision frequency (Lieberman & Lichtenberg Reference Lieberman and Lichtenberg2005), effects from both ballistic motion and diffusion are likely important, but we justify this simple diffusion model a posteriori by how well the model fits the measurements (figure 4). Given the source and sink terms discussed above, the neutral continuity equation is

(3.1) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}N_{n}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D735}\boldsymbol{\cdot }\unicode[STIX]{x1D71E}=S_{\text{recomb}}-S_{\text{ion}}-S_{\text{CX}}, & \displaystyle\end{eqnarray}$$

(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D71E}=-D_{n}\unicode[STIX]{x1D735}N_{n}(r), & \displaystyle\end{eqnarray}$$

(3.3) $$\begin{eqnarray}\displaystyle & \displaystyle D_{n}=\frac{V_{th,n}^{2}}{N_{i}\unicode[STIX]{x1D70E}_{HS}V_{th,i}}, & \displaystyle\end{eqnarray}$$

with $D_{n}$ the diffusion coefficient of neutrals, $\unicode[STIX]{x1D70E}_{HS}$ the cross-section for hard-sphere collisions, $V_{th,x}$ the thermal velocity of the species $x$ , $S_{\text{recomb}}$ the recombination rate per unit of volume, $S_{\text{ion}}$ the ionisation rate per unit of volume and $S_{\text{CX}}$ is the rate background neutrals are lost to charge-exchange recombination with the ions. These rates can be estimated for typical plasma conditions (Janev et al. Reference Janev, Langer, Post and Evans1987; Sakabe & Izawa Reference Sakabe and Izawa1991; Huba Reference Huba2013) as

(3.4) $$\begin{eqnarray}\displaystyle & \displaystyle S_{\text{recomb}}\approx 10^{12}~\text{cm}^{-3}~\text{s}^{-1} & \displaystyle\end{eqnarray}$$

(3.5) $$\begin{eqnarray}\displaystyle & \displaystyle S_{\text{ion}}\approx 10^{15}~\text{cm}^{-3}~\text{s}^{-1} & \displaystyle\end{eqnarray}$$

(3.6) $$\begin{eqnarray}\displaystyle & \displaystyle S_{\text{exc}}\approx 10^{15}~\text{cm}^{-3}~\text{s}^{-1}. & \displaystyle\end{eqnarray}$$

These estimates allow us to neglect any volumetric source due to recombination. Note that the charge-exchange collision rate is high and shows the importance of the neutral drag in momentum loss. The ionisation and charge-exchange rates can be written as

(3.7) $$\begin{eqnarray}\displaystyle S_{\text{ion}}+S_{\text{CX}}=\langle \unicode[STIX]{x1D70E}v\rangle _{\text{ion}}N_{e}N_{n}+\langle \unicode[STIX]{x1D70E}v\rangle _{\text{CX}}N_{i}N_{n}\approx (\langle \unicode[STIX]{x1D70E}v\rangle _{\text{ion}}+\langle \unicode[STIX]{x1D70E}v\rangle _{\text{CX}})N_{e}N_{n}, & & \displaystyle\end{eqnarray}$$

where we have assumed quasi-neutrality in a single-ion species plasma. Assuming spherical symmetry, the neutrals follow a radial diffusion equation given by:

(3.8) $$\begin{eqnarray}\displaystyle r^{2}\left(\frac{\unicode[STIX]{x2202}^{2}N_{n}}{\unicode[STIX]{x2202}r^{2}}-\frac{1}{d^{2}}N_{n}\right)+2r\frac{\unicode[STIX]{x2202}N_{n}}{\unicode[STIX]{x2202}r}=0, & & \displaystyle\end{eqnarray}$$

with $d$ the depletion length (Cook et al. Reference Cook, King, Roberto, Stewart and Yokoyama1984; Yun, Taylor & Tynan Reference Yun, Taylor and Tynan2000; Benilov Reference Benilov2005, Reference Benilov1999),

(3.9) $$\begin{eqnarray}\displaystyle d=\sqrt{\frac{D_{n}}{(\langle \unicode[STIX]{x1D70E}v\rangle _{\text{ion}}+\langle \unicode[STIX]{x1D70E}v\rangle _{\text{CX}})N_{e}}}. & & \displaystyle\end{eqnarray}$$

Assuming $d$ to be constant in the plasma bulk and the density gradient to be zero in the centre, we have:

(3.10) $$\begin{eqnarray}\displaystyle N_{n}(r)=N_{n}(R_{\text{edge}})\times \frac{R_{\text{edge}}}{r}\frac{\sinh \left(\displaystyle \frac{r}{d}\right)}{\sinh \left(\displaystyle \frac{R_{\text{edge}}}{d}\right)}. & & \displaystyle\end{eqnarray}$$

3.2 Neutral particle local emissivity model

The emissivity of a spectral line, defined as the energy emitted by the decay from an excited state $i$ to a lower state $j$ per unit of volume, per steradian and per second is given by the relation:

(3.11) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D716}_{ij}=\frac{h\unicode[STIX]{x1D708}_{ij}}{4\unicode[STIX]{x03C0}}N_{i}A_{ij}, & & \displaystyle\end{eqnarray}$$

where $h$ is Plank’s constant, $\unicode[STIX]{x1D708}_{ij}$ is the frequency of the emitted light, $N_{i}$ is the density of the upper state and $A_{ij}$ is the Einstein coefficient. Several studies have used a collisional radiative model to measure the electron temperature and electron density of helium plasmas (MacWhirter Reference MacWhirter1965; Sasaki et al. Reference Sasaki, Takamura, Masuzaki, Watanabe, Kato and Kadota1995; Boivin et al. Reference Boivin, Loch, Ballance, Branscomb and Pindzola2007; Muñoz Burgos et al. Reference Muñoz Burgos, Schmitz, Loch and Ballance2012). Commonly used lines include 471.3, 492.2, 504.8, 667.8, 706.5 and 728.1 nm. Their ratio can be written as a function of $T_{e}$ and $N_{e}$ . Thanks to unmagnetisation, the electron transport is very efficient in the BRB plasma bulk. $N_{e}$ and $T_{e}$ are constant up to plasma edge where they decrease sharply. Examples of parameters profiles can be seen on Cooper et al. (Reference Cooper, Weisberg, Khalzov, Milhone, Flanagan, Peterson, Wahl and Forest2016) and Weisberg et al. (Reference Weisberg, Peterson, Milhone, Endrizzi, Cooper, Désangles, Khalzov, Siller and Forest2017). As a result, the He line emissivity ratios

(3.12) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D716}_{ij}}{\unicode[STIX]{x1D716}_{i^{\prime }j^{\prime }}}=\frac{\unicode[STIX]{x1D708}_{ij}N_{i}A_{ij}}{\unicode[STIX]{x1D708}_{i^{\prime }j^{\prime }}N_{i^{\prime }}A_{i^{\prime }j^{\prime }}}=f(T_{e},N_{e}) & & \displaystyle\end{eqnarray}$$

should also be constant. This assumption is verified by measurements shown in figure 3.

Figure 3.

(a) Emissivity profiles of each measured line. (b) The emissivity of the lines 447.2, 471.3 and 492.2 is plotted divided by the emissivity of the line 402.6 nm for an 7 eV plasma with a density of $7\times 10^{17}~\text{m}^{-3}$ (see run 3 in table 1).

Figure 4.

(a) Measurements of the radiance (cross) and fitting routine results in function of the impact factor. (b) Emissivity radial profile corresponding to the inversion of the computed radiant energy profile. Both are for the neutral helium line 471.3 nm, for run 2 and run 3.

Collisional radiative models are used to predict the quasi-steady state populations of the excited states $N_{i}$ based on lower state densities $N_{k}$ (which include the ground state and metastable states) and plasmas properties such as $n_{e}$ and $T_{e}$ . The total number density of neutral atoms ( $N_{n}$ ) at a particular location is the sum of the populations of all of the atomic states, but due to the assumption of coronal equilibration this sum is dominated by the sum of the lower states. Therefore, $N_{i}\propto \sum _{k}N_{k}\approx N_{n}$ . Equation (3.11) can be rewritten as:

(3.13) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D716}_{ij}\propto \frac{h\unicode[STIX]{x1D708}_{ij}A_{ij}}{4\unicode[STIX]{x03C0}}N_{n}. & & \displaystyle\end{eqnarray}$$

Combining equations (3.10) and (3.13), the emissivity in the bulk can be expressed as:

(3.14) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D716}_{ij}(r)\propto \frac{h\unicode[STIX]{x1D708}_{ij}\,A_{ij}}{4\unicode[STIX]{x03C0}}\,N_{n}(R_{\text{edge}})\times \frac{R_{\text{edge}}}{r}\frac{\sinh \left(\displaystyle \frac{r}{d}\right)}{\sinh \left(\displaystyle \frac{R_{\text{edge}}}{d}\right)}. & & \displaystyle\end{eqnarray}$$

The radiant flux from the plasma is collected by a collimating lens which fills a fibre that couples to the spectrometer. Some of the light may be reabsorbed by the plasma due to resonant excitation of the lower states of the observed transitions resulting in a certain amount of dimming of the emission lines. The effect of this reabsorption on the measured radiance is quantified by the optical depth of the plasma $\unicode[STIX]{x1D70F}_{ij}$ where the measured radiant flux $\unicode[STIX]{x1D6F7}^{\text{meas}}$ is reduced such that $\unicode[STIX]{x1D6F7}^{\text{meas}}=\unicode[STIX]{x1D6F7}^{\text{emitted}}\exp (-\unicode[STIX]{x1D70F}_{ij})$ . The optical depth and escape factor can be computed for each transition (MacWhirter Reference MacWhirter1965; Boivin, Kline & Scime Reference Boivin, Kline and Scime2001) and the plasma characteristics.

The obtained mean optical depth is of the order of $10^{-7}$ for all observed emission lines and all the light emitted by the plasma comes out of the experiment without being reabsorbed. We will then treat the plasma as optically thin. Measuring the radiance as a function of the impact factor (distance from the line of sight to the centre) we can determine the local emissivity of the plasma by comparing the set of radiance measurements to an Abel transform of our model emission profile as described § 4.1.

4.1 Measurement of the emissivity profiles from the radiant energy

Radiance profiles are measured as a function of the impact parameter by scanning the spectrometer’s collimator on the linear stage. The radiance is a line integrated measurement of the emissivity profile. These experimental measurements are compared to synthetic measurements calculated using a forward model for the emissivity. To compute the synthetic measurements, the edge of the plasma needs to be taken into account. The electron density is empirically modelled by the expression:

(4.1) $$\begin{eqnarray}\displaystyle N_{e}(r)=\frac{N_{e0}}{2}[1-\tanh (0.5(r-R_{\text{edge}}))], & & \displaystyle\end{eqnarray}$$

where $N_{e0}$ is the electron density measured by the interferometer and $R_{\text{edge}}$ is the plasma edge, approximately $138~\text{cm}$ . We combine equations (3.14) and (4.1) to include the plasma edge in the expression of the emissivity:

(4.2) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D716}_{ij}(r)=K[1-\tanh (0.5(r-R_{\text{edge}}))]\frac{\sinh \left(\displaystyle \frac{r}{d}\right)}{\left(\displaystyle \frac{r}{d}\right)}, & & \displaystyle\end{eqnarray}$$

where the amplitude $K$ and ionisation length $d$ are the fitting parameters of the model. Figure 4 shows the radiant energy measurements and the radiant energy profile computed using equation (4.2) on the top graph and the associated emissivity profile for the neutral helium line 471.3 nm on the bottom one.

4.2 Neutral density profile

We average the depletion length on the different emission lines in order to obtain one profile of depletion for each run. These results are presented in table 1. The associated emissivity profiles are renormalised to a neutral density profile using the measurements of the pressure at the edge. Neutral density profiles are plotted on figure 5. Different types of density profiles are observed, some with a small depletion in the centre like runs 1 and 2 and others with a bigger depletion of neutrals like runs 3, 4 and 5. The depletion lengths are also computed using (3.3) and (3.9) and tabulated collision rates (Janev et al. Reference Janev, Langer, Post and Evans1987; Sakabe & Izawa Reference Sakabe and Izawa1991). They are reported in table 1 as well. There is a good agreement between the inferred and the computed depletion lengths, with a maximum of 40 % of relative error between the two values for run 2 and the reproduction of the evolution of $d$ for the different runs.

Combining the neutral density profiles with the electron density measurements (table 1), one can compute the ionisation fraction profile (figure 5). The ten lanthanum-hexaboride cathodes and the high confinement time allow high ionisation fraction in the centre of the experiment, up to 80 %.

Figure 5.

(a) Inferred neutral density profiles for the different runs. (b) Inferred ionisation fraction profiles for the different runs.

4.3 Evaluation of the neutral drag and flow propagation

In the BRB geometry, the flow is stirred at the edge and propagates inward due to viscosity. The non-flowing neutrals exhibit a drag on the flowing ions via charge exchange. The equation of motion in the plasma can then be written as a classical stationary single fluid Navier–Stokes equation (4.3), where the fluid is the ion population, including a collision term between ions and neutrals which will be called neutral drag.

(4.3) $$\begin{eqnarray}\displaystyle Re[(\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{v}+\unicode[STIX]{x1D735}p]=\unicode[STIX]{x1D6FB}^{2}\boldsymbol{v}-\unicode[STIX]{x1D712}\boldsymbol{v}. & & \displaystyle\end{eqnarray}$$

In this equation, $Re$ is the kinetic Reynolds number, $\boldsymbol{v}$ is the velocity of the fluid, $p$ is the pressure and $\unicode[STIX]{x1D712}$ is the neutral drag which can be written as (Huba Reference Huba2013)

(4.4) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D712}=\frac{L^{2}}{\unicode[STIX]{x1D708}_{\text{visc}}\unicode[STIX]{x1D70F}_{in}}=38.2\frac{(n_{0}n_{i})_{18}Z^{3}L_{m}^{2}}{T_{i,eV}^{2}}. & & \displaystyle\end{eqnarray}$$

Here, $\unicode[STIX]{x1D70F}_{in}$ is the ion–neutral collision time and $\unicode[STIX]{x1D708}_{\text{visc}}$ is the Braginskii viscosity. From the neutral density profiles (figure 5), one can compute the neutral drag in function of the radial position using (4.4) (figure 6). It is interesting to notice that even though the neutral density is lower for run 5 than for runs 1 and 2, the neutral drag for these runs is comparable due to the differences in ion temperatures.

Figure 6.

Neutral drag (see (4.3) and (4.4)) computed using the neutral density profiles and ion temperature measurements.

A single fluid simulation code has been previously developed by Khalzov et al. (Reference Khalzov, Brown, Cooper, Weisberg and Forest2012), Khalzov, Cooper & Forest (Reference Khalzov, Cooper and Forest2013) to simulate the flow and the dynamo onset in the BRB vessel. The neutral drag has been implemented into this code to evaluate its influence on the flow. As velocity measurements had not been made for the runs of interest, we used Mach probe velocity measurements of another run to fix the boundary condition and to compare results of the simulation with measured flow. The simulation has been made with a boundary condition of $6.5~\text{km}~\text{s}^{-1}$ at 134 cm from the centre, with and without neutral drag, for the values of the neutral drag and Reynolds number of the different runs.

The velocity computed with the neutral drag decreases faster than the velocity computed with no neutral drag and better match the measured velocity profiles. This agreement is verified for every obtained value of the drag. It demonstrates that the neutral drag plays an important role in the ion dynamics. The difference in velocity profiles without any drag is due to the specific Reynolds number of each run. This difference is small compared to the effects of the neutral drag. A low neutral drag significantly helps to viscously couple the edge torque across the plasma.

Khalzov et al. (Reference Khalzov, Brown, Cooper, Weisberg and Forest2012, Reference Khalzov, Cooper and Forest2013) have also studied the impact of the geometry of the flow drive on the onset of an MHD instability in BRB. They have shown that an optimised flow can produce a growing MHD mode for a moderate $Re=150$ and Rm ranging between two critical values. Using their drive geometry, one can study the impact of neutral drag on the MHD stability. Without neutral drag, the flow resulting from the applied torques should produce an instability as predicted by Khalzov et al. (Reference Khalzov, Brown, Cooper, Weisberg and Forest2012, Reference Khalzov, Cooper and Forest2013). With a uniform neutral density profile, the resulting charge-exchange drag slows the flow enough to stabilise this MHD instability. Accounting for the hollow neutral profile due to the neutral dynamics, the flow is slightly less damped illustrating the importance of quantifying neutral dynamics in these plasmas.

Figure 7.

Typical measured velocity profile and simulated velocity profile of the plasma for run 3 with and without the neutral drag taking into account in the simulations.