Page 8 - Drift-wave stability in the field-reversed configuration

092518-8
Onofri et al.
Phys. Plasmas 24, 092518 (2017)
FIG. 12. Radial density profiles at t 1⁄4 0.6 ms in the C-2 experiment for shot 36 691 (black line) and in two simulations with mirror plug fields of 2 T (red line) and 4 T (green line).
dPek þPe? Pek dBþend/ 1⁄4 0; (13) dl B dl dl
where Pek and Pe? are the parallel and perpendicular elec- tron pressures and l is a coordinate parallel to the field lines. In Q2D, the electron temperature is calculated by Eq. (5), which assumes an isotropic temperature. In the experimental conditions of C-2 and C-2 U, the electron collision time is of the order of few ls; therefore, the electron temperature is isotropic. In these conditions, the temperature is equilibrated between the parallel and perpendicular components and also uniform along field lines because of the high parallel con- ductivity. Under such conditions, a hydromagnetic code can be applied to the study of the electrostatic potential along field lines in the expander. The theoretical arguments for the creation of an electrostatic barrier in the expander divertor can be summarized as follows:29 From Eq. (13), we see that the electron pressure balance in the plasma outflow with isotropic and uniform electron temperatures gives an elec- trostatic potential that can be related to the plasma density profile
e/ 1⁄4 TelnðnM=nÞ; (14)
where nM is the plasma density at the mirror throat.42
The density is determined by the outflow physics exam- ined in Sec. IV. In the simplified case of weak ionization in the divertor, the density can be estimated from the conserva- tion of total plasma flux, nVkS const, where S is the cross section of a flux tube. Using the conservation of magnetic
Cold ions, Ti Te, create a favorable condition for the formation of a steeper electrostatic potential. In this case, the outflow velocity is determined mostly by the electron pres- sure, and substantial plasma acceleration can occur, making the density decrease faster and forming a higher and steeper electrostatic potential profile according to Eq. (14).42
The above simplistic considerations only indicate major effects of the electrostatic potential formation in the expanding magnetic field. In the real experimental condi- tions, there are several effects reducing the formation of the electrostatic potential, such as the accumulation of neu- trals in the divertor chamber, which slow down the plasma flow and increase the plasma density through ionization. In the case of very low electron collisions, the temperature isotropy cannot be maintained, which also leads to a decrease in the electrostatic potential compared to the ide- alized case. However, these negative effects start affecting the potential formation in the expander at the far end of the divertor, where plasma density and collisionality go down substantially. Closer to the mirror, where K 20, simple basic effects always dominate.43
Q2D simulations of the plasma outflow from the main confinement vessel into the divertor help better understand these phenomena and are useful for future FRC experiments, where two expanders are located beyond the mirrors to improve the electron parallel confinement. We did two simu- lations of mirror configurations with two different ion tem- peratures, Ti 1⁄4 1000 eV (relevant for C-2 experiments) and Ti1⁄4100eV (relevant for GDT experiments) and electron temperature Te 1⁄4 250 eV. We keep the density and tempera- ture constant in the middle of the confinement vessel, and we look at the potential in the expander when the system is in a steady state. The simulations use Neumann boundary condi- tions for the electron temperature at the ends of the divertors, and the electron temperature is approximately uniform on field lines. The MHD model cannot calculate correctly the electron heat conduction in a regime with a long mean free path, which would need a kinetic treatment. However, as long as the electron temperature is approximately uniform, the results are close to what is found in kinetic calcula- tions,43 and the MHD model can be used for the calculation of the electrostatic potential along the field lines, before the sheath.
As discussed above, if the velocity was uniform, we would expect an electrostatic potential e/ 1⁄4 TelnK. However, the acceleration of the plasma flow in the expander makes the potential profile steeper, and a higher potential is reached at lower expansion. This effect is already present in the case of hot ions (Ti 1⁄4 1000 eV), but it is stronger in the case of cold ions (Ti 1⁄4 100 eV), where the velocity at the mirror is lower and the flow accelerates more in the expander. Figure 13 shows the potential / on the axis as a function of magnetic expansion K for the two simulations, which agree well with the theoretical curves obtained from Eq. (16). The simulations show that in the case of lower ion temperature, which produces a higher acceleration of the plasma flow in the divertors, the electrostatic potential pro- file becomes steeper.
flux, this can be written as
nVjj=B const;
(15)
(16)
In the case of hot ions, Ti Te, there is little plasma acceler- ation beyond the mirror because the maximum ion kinetic energy is limited by their initial thermal energy and the acceleration caused by electron pressure is negligible. Then, from Eq. (16), when V VM, the electrostatic potential can be estimated as e/ ’ TelnK, where K is the magnetic expan- sion defined as K 1⁄4 BM=B (BM is the magnetic field at the mirror throat).
and Eq. (14) becomes
e/1⁄4Teln BV :

BM V M