Page 3 - Drift-wave stability in the field-reversed configuration

092518-3 Onofri et al.
Phys. Plasmas 24, 092518 (2017)
where the last term is due to collisions between fast ions and electrons. The toroidal component of the magnetic field is given by
@Bh 1⁄4 1⁄2r E h: (9) @t
The plasma current density Jp is defined as
Jp 1⁄4J Jf; (10)
where Jf is the fast ion current density and J is the total cur- rent density
J1⁄4l1r B: (11) 0
Similar equations for density, momentum, and energy are solved for the neutral fluid, which is coupled to the plasma through ionization and charge exchange collisions. The code can work with realistic wall geometry, with an arbitrarily shaped, axisymmetric resistive wall, using a uniform r, z mesh, and it incorporates boundary conditions that include external coils and conductors. The magnetic field is the result of imposed external currents and internal plasma currents. The eddy currents induced in the surrounding metallic walls are also calculated.
III. BENCHMARK OF THE PARALLEL TRANSPORT
Plasma confinement in linear systems with open mag- netic field lines, such as h pinches and mirror machines, is strongly determined by parallel transport along field lines and end losses. Several theories have been proposed to explain the end losses observed in experiments and numeri- cal simulations. However, considerable discrepancies existed between early theories and experiments. Most experi- ments15,16 and simulations17 show that the confinement time is not as sensitive to b as many theories predict. The typical theoretical prediction for the confinement time of a plasma column with sharp boundaries as b ! 1 is ^s / ð1 bÞ 1=2 (Refs. 24 and 25) (^s 1⁄4 2Css=L is the normalized confine- ment time, Cs is the sound speed at z 1⁄4 0, and L is the length of the plasma column). On the contrary, the theory proposed in Refs. 22 and 23 does not predict a divergent confinement time for b ! 1 and is in good agreement with previous simu- lations and experiments. This theory takes into account the curvature of magnetic field lines due to the propagation of an area wave, and they find that the plasma reaches sonic speed at the ends of the column. This theory is used to benchmark the parallel transport in the Lamy Ridge code. We used the Lamy Ridge code to simulate a system similar to the config- uration studied in Refs. 22 and 23 but in cylindrical geome- try. The initial condition is a plasma column confined by a magnetic field with straight field lines in the axial direction (z direction). The plasma has a uniform density for radius r < R0, and the magnetic field is uniform for r < R0 and for r > R0. The size of the computational domain is 0.7 m in the radial direction and 2.8 m in the axial direction, and the spa- tial grid is 150 300. The radius of the plasma column is R0 1⁄4 0:18 m, and the density is n0 1⁄4 1019 m 3. Outside the
plasma column, a density floor ne 1⁄4 1016 m 3 is used, and the temperature is uniform everywhere T0 1⁄4 103 eV. The ini- tial internal and external magnetic fields (B0i and B0e) are determined by the plasma b 1⁄4 2l0P0=B20e and the pressure balance condition,
P0 þ B20i 1⁄4 B20e ; (12) 2l0 2l0
where P0 is the pressure of the plasma column. In order to compare the simulations to the results described in Refs. 22 and 23, the pressure is derived from an adiabatic equation with adiabatic index c1⁄42. We use the Neumann boundary conditions for the velocity, and the plasma that touches the ends at z6L=2 is lost. The simulation is done without neutral beams and without neutrals. We see that the plasma acceler- ates near the ends of the plasma column, and as expected, the axial velocity at the ends is equal to the local sound speed (Fig. 1). Figure 2 shows the density at t 1⁄4 6 ls for a case with b 1⁄4 0:8. The plasma column narrows at the ends, but the magnetic field lines remain open, allowing the plasma to escape, and the mass decays exponentially, for all values of b. These findings agree well with the results obtained in Refs. 22 and 23.
We calculated the plasma confinement time for different B2
values of b 1⁄4 P0= 0e , and the results are shown in Fig. 3, 2l0
together with the predictions of previous simulations, experi- ments, and different theories. Figure 3 shows the normalized confinement time ^s 1⁄4 2Css=L as a function of b. The theo- ries from Refs. 18–21 predict a divergent confinement time for b ! 1, while experiments15,16 and previous simula- tions22,23 are in better agreement with the theory in Refs. 22 and 23. Our results are in very good agreement (nearly on top of each other) with the theory in Refs. 22 and 23 through- out the b range. These results may be regarded as a good benchmark of the present code.
IV. PARALLEL CONFINEMENT IN AN FRC AND IN A MIRROR TRAP
With benchmarking the parallel transport calculated by the Lamy Ridge code accomplished, we go on to study the confinement of a plasma in a Field Reversed Configuration (FRC) and in a magnetic mirror machine. For the FRC case, the initial condition is an equilibrium obtained from the
FIG. 1. Axial velocity on the cylindrical axis for a simulation with b 1⁄4 1. The velocity is normalized to the local sound speed.