Page 2 - Drift-wave stability in the field-reversed configuration

092518-2 Onofri et al.
Phys. Plasmas 24, 092518 (2017)
electrostatic potential in the expanding magnetic field region in the divertors. This electrostatic confinement makes it pos- sible to obtain high electron temperatures, as shown in GDT.34 In Sec. V, we study how the parallel outflow in an expanding magnetic field creates an electrostatic potential. We show how gas puff affects the formation of the electro- static potential and that it has a positive effect if it is injected close to the mirrors inside the confinement vessel. Using gas puff for fuelling can improve the confinement if the injection location is chosen correctly.
In the collisionless regime, the mirror confinement is kinetic, with distinct populations of trapped and passing par- ticles, which are not included in the isotropic MHD model. The parallel confinement in the SOL of FRCs is different from that of mirror machines due to the source of plasma coming from the core. These kinetic effects have an impact on the evolution of the density profile. In Sec. VI, we discuss how the kinetic mirror confinement modifies the parallel par- ticle transport, and we propose a model to incorporate these effects in the MHD code.
II. THE NUMERICAL MODEL
The Q2D transport code has been developed by coupling
The momentum equation contains the momentum trans- fer between fast ions and thermal ions FiNB and between fast ions and electrons FeNB
@u
Mn @tþu ru 1⁄4Jp B enfE rPþr ru
þ FiNB þ FeNB þ Su; (2)
where M is the ion mass, nf is the fast ion density, P is the pressure, Jp is the plasma current, is the viscosity, and Su is a momentum source due to charge exchange, ionization, and fast ions at the cutoff energy. The second term in the right- hand side is due to the difference between the electron den- sity and the thermal ion density.
The equation for the total thermal plasma energy w 1⁄4 P=ðc 1Þ þ Mnu2=2 is
2! @wþr cP þMnu u
@t c 1 2
1⁄4 u ðJp B enf EÞ þ kBr ðji?r?Ti þ jijjrjjTiÞ
þ kBr ðje?r?Te þ jejjrjjTeÞ þ gJp2 þ PiNB þ PeNB þ Sw: (3)
a 2D MHD code, LamyRidge, with a Monte Carlo code that calculates source terms due to fast ions produced by neu- tral beam injection. This is an extension of the Q1D transport code.11 The neutral beams make collisions with the plasma and form a population of fast ions through charge exchange and ionization, and they deposit energy and momentum on the plasma. The Monte Carlo code follows the 3D orbits of individual test particles (fast ions and neutrals) and is used to calculate fast ion current, momentum, and energy transferred to the plasma and the neutral density produced by charge exchange. The total particle number, momentum, and energy are conserved in the coupling between fast ions and thermal plasma. The fast ions have a slowing down distribution, and when their energy goes below a given threshold, they are considered part of the thermal plasma, they are removed from the fast ions followed by the Monte Carlo code, and they appear as source terms in the MHD equations. The code also solves fluid equations for the neutrals, which are cou- pled with the plasma and the fast particles through collisions, charge exchange, and ionization. The calculation of the source terms for density, momentum, and energy in the MHD and neutral equations includes the fast ions that charge exchange with neutrals producing fast neutrals and ions and the beam neutrals that charge exchange with thermal ions producing fast ions and warm neutrals.
The continuity equation solved by Q2D is
@n þ r nu 1⁄4 Sn; (1)
@t
where n is the thermal ion density, u is the fluid velocity, and Sn is a source term due to charge exchange, ionization, and fast ions that have an energy lower than the cutoff energy and are considered thermal plasmas.
resistivity, PiNB and PeNB are the energy transferred by fast ions to thermal ions and electrons due to collisions and charge exchange, and Sw is the energy source due to the cut- off energy, charge exchange, and ionization.
The code also solves an equation for the electron temperature

nekB @Te þue rTe þnekBTer ue c 1 @t
1⁄4 kBr ðje?r?Te þ jejjrjjTeÞ þgJ2þ kðT TÞþQe ; (4)
Here, T and T are the ion and electron temperatures, g is the 35 36ie
p eiBi e NB
where QeNB is the electron temperature change due to collisions with fast ions. The electron density is given by the sum of the thermal and fast ion densities, ne 1⁄4 n þ nf , and the electron velocity is ue 1⁄4 un=ne Jp=ene. The ion temperature is the difference between the total temperature T and the electron tem- perature, Ti 1⁄4 T Te, and T is obtained from the total pressure
T1⁄4 P nfTe: (5) nkB n
The magnetic field radial and axial components are derived from the poloidal flux
B 1⁄4 rW rh þ Bhu^h; (6) @W 1⁄4 rEh; (7)
@t
where W is the poloidal flux per radian, and the electric field
is given by Ohm’s law, including the Hall term
E1⁄4 u BþgJp þJp B rPe þFNB; (8) ene ene
e