Page 12 - Drift-wave stability in the field-reversed configuration

092518-12 Onofri et al.
tk 1⁄41: FT si
(44)
Phys. Plasmas 24, 092518 (2017) much faster for nearly all radii. The presence of closed field
lines in the FRC gives a much better confinement compared to the mirror trap. This difference in the density profiles in the FRC and in a mirror trap can be useful to infer the pres- ence of closed field lines and the geometry of the confining magnetic fields in experiments. In the mirror trap, the density profile is maximum on the cylindrical axis, while in the FRC, it has a maximum at the O-point, similar to the profiles observed in the C-2 experiment.5 This suggests that in C-2 discharges, the plasma is in an FRC configuration whenever experimental measurements show this tendency.
The simulations also show the effect of the closed field lines on the temperature confinement. The temperature in the FRC decreases much more slowly compared to the mirror trap, in which the temperature decreases quickly due to paral- lel conduction on open field lines. On the other hand, even though the confinement properties are distinctly different for the FRC and the mirror trap, we find a universal behavior for the parallel outflow speed when it is normalized to the local sound speed (note that the absolute value of the speed is greater in the FRC case). In the central region, where closed magnetic field lines exist in the FRC case, the velocities are different in the two cases, but they become equal to the local sound speed in both cases at the mirror throat. The normal- ized velocities are the same beyond the mirrors. Further uni- versality of the exhaust plasma speed as a function of temperature is also observed for simulations of FRCs. The parallel velocity is different in each of the examined cases, but when it is normalized to the local sound speed, all the dif- ferent axial velocity profiles collapse onto a universal curve.
We used the Q2D code to simulate the C-2 experiment, with realistic geometry and magnetic field, including neutral beams. We used the transport coefficients found in Ref. 11 with the Q1D code. In the experiment, the transports in the core and in the SOL are coupled,5 and the FRC lifetime is extended when the magnetic field in the mirror plugs is increased. Our simulations show that the mirror plugs have the effect of reducing the parallel particle flux and increasing the density gradient scale length Ln. This evolution is consis- tent with the experiment. However, Ln in simulations is shorter than what is observed in experiments (Fig. 11). We observe in simulations that the density scale length Ln in the SOL gets smaller and smaller as the fast parallel transport carries away particles, as seen in Fig. 8 and in Fig. 11. This phenomenon qualitatively resembles the experiment,4 but the evolved value of Ln is smaller than is observed in the experiment.
This difference in the density scale length between the simulations and the experiment is discussed in Sec. VI. One of the major reasons for this is the existence of additional effects that the present fluid description lacks, i.e., the kinetic mirror confinement of ions in the mirrors. The inclusion of the kinetic mirror effects improves the SOL confinement, as seen in Eqs. (36)–(40). One of the potential impacts of the improved parallel confinement which has not been treated here is the increase in Ln. In fact, we saw that when we increased the mirror ratio, we observed an improvement of the parallel confinement and an increase in Ln (Fig. 12 in Sec. IV), even though it was not the effect of kinetic mirror
We note that the particles inside the FRC migrate through the separatrix with the particle confinement time sFRC and enter the SOL, whose confinement time is given by Eq. (36). In the Conclusions section, we will comment on the potential feedback of the confinement by the mirrors on the transport properties of SOL and core, such as D1 and D2.
To simulate the effect of mirror confinement in the MHD code we require that the velocity of the fluid at the mirror is
C
where nM is the density at the mirror and SM is the plasma cross section at the mirror. This may be done by modifying the axial momentum equation in the mirror region with the addition of a viscous term that gives Vz ’ VM,
VM1⁄4M; (45) nM SM
dVz dP d2Vz
Mn dt 1⁄4 dzþ dz2 MðVz VMÞ;
(46) where is the viscosity and M is an artificial coefficient that
is added to the equation to obtain a velocity Vz ’ VM. VII. CONCLUSIONS
In the present paper, we have studied the particle trans- port properties of an FRC using the Q2D code. Q2D couples an MHD code with a Monte Carlo code for the beam compo- nent. The confinement properties of FRCs are affected by the coupling between the parallel transport in the SOL and the perpendicular transport in the core. With the Q2D code, it is possible to study self-consistently the effects of the parallel transport on the evolution of the density gradient and the core transport. This was not possible in previous studies of FRC transport that were done with a 1D code.11
To begin with, we have compared the simulations with the theories developed for the parallel outflow in the simple h-pinch configuration. The simulation shows that the plasma accelerates near the ends of the plasma column and the velocity at the ends is equal to the sound speed. The confine- ment time obtained from the simulation is in excellent agree- ment with the theory in Refs. 22 and 23. Not only do the individual values of our simulations agree with the theory quite well but also many simulation points systematically overlap on the theory curve, putting the parallel flow physics in the h-pinch firmly established.
After this benchmark, we have studied the particle paral- lel transport in a magnetic mirror and in an FRC. In both cases, the parallel flow becomes sonic at the mirror throats and supersonic beyond the mirrors. We find that the density evolution in the two configurations is substantially different. In the FRC, the plasma is confined in the core by closed field lines within the separatrix, while the plasma in the SOL is lost in the axial direction. This results in a steepening of the density profile in the FRC case because particles inside the separatrix are well confined. On the other hand, in the mirror trap, there are no closed field lines and the density decreases