Page 10 - Drift-wave stability in the field-reversed configuration

092518-10 Onofri et al. Phys. Plasmas 24, 092518 (2017) state, the densities of trapped (n ) and passing (n ) particles FPD2 aD2!" 1 r r
can be described by a system of two diffusion equations Dd2nT1⁄4nT nP; (17)
dr2 sT sP
P
w F T D 2w 1 k 2 k2 r rs #
1 k2 exp Dw
D w
T P nðrÞ1⁄4nwqexps
d2nP
D21⁄4 þþ; (18)
FPD2 aD2! r r n q exp s :
nT nP nP dr sT sP sk
þn
where the last term in Eq. (18) is present only in the SOL.
The system (17) and (18) describes the diffusion across
the magnetic field, the equilibration between trapped and
passing populations in velocity space, and the loss of pass-
ing particles in the SOL with characteristic time sk. The
times sT and sP are the characteristic times of transition
between the trapped and passing populations (sT) and
between the passing and trapped populations (s ). The P
transition times sT and sP are estimated as sT 1⁄4 FTsi and
s 1⁄4 F s , where s is the ion-ion collision time, F 1⁄4 PPiiT
F T D 2n D n
outside the separatrix, a 1⁄4 D2=D1; k 1⁄4 e n; n 1⁄4 rw rs , and
1
inside the
C; (26) rffiffiffiffiffi
(23) Here, a is the ratio of the diffusion coefficients inside and
n
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Similar expressions for the density scale length were found for the SOL in tokamaks.45,46 The integration constants nq, nw, and nn are determined by matching the internal and exter- nal solutions at the separatrix, imposing that the density pro- files and the fluxes are continuous functions
F2Ttanh2n nw ’ pffiffiffiffiffi pffiffiffiffiffi pffiffi
1 1=R and FP 1⁄4 1 FT are the fractions of velocity space for the trapped and passing populations, and R is the mirror ratio. The diffusion coefficient D can have different values inside the FRC and in the SOL, as should be expected from the observation that the turbulence in the SOL is two orders of magnitude higher than in the core.4
rffiffiffiffiffi si
We solve the system (17) and (18) using D 1⁄4 D
FRC and D 1⁄4 D2 in the SOL, and the internal and external solutions are matched to determine the boundary condi- tions at the separatrix, requiring that the densities and the fluxes are continuous functions. Using the condition of low collisionality, si sk, we can obtain simplified approximate expressions. The radial profiles for the densi- ties of trapped and passing particles inside the separatrix are
C r rs nTðrÞ1⁄4FTnsþDðrs rÞþnqexp D ; (19)
1q
C r rs nPðrÞ1⁄4FP nsþD ðrs rÞ nqexp D ; (20)
1q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Dw
the lengths Dw and Dn are the density scale lengths for the
“wide” population (mostly trapped ions) and “narrow” popu- lation (mostly passing ions),
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Dw ’ Dn ’
D2siFT; (24) pffiffiffiffiffiffiffiffiffi
D2sk: (25)
D1 si si
FT FP þ atanhn sk 3=2 pffiffi pffiffiffiffiffi
a þ a FP pffiffiffiffiffi pffiffiffiffiffi pffiffi
profile of trapped particles is much thicker than that of the passing particles, which are lost quickly in the parallel direction. This is illustrated in Fig. 16, where we show the typical radial profiles of passing and trapped particles. From the solutions of the diffusion equations, we can find a relationship between the density at the separatrix and the flux C,
FIG. 16. Illustration of typical radial density profiles of trapped and passing particles. The separatrix radius rs is shown by the thick vertical line.
nn ’
nq ’ pffiffiffi pffiffi
C; (27) (28)
aFP FPþa D1 pffiffiffi rffiffiffiffiffi
FT FT FP si FPþ a D1
C;
where tanhn 1⁄4 ð1 k2Þ=ð1 þ k2Þ. In the SOL, the density
where Dq 1⁄4
ticles (both trapped and passing) inside the separatrix, ns is the total density at the separatrix, and C is the total particle flux
C 1⁄4 D1dn=dr: (21)
Here, the particle flux C out of the separatrix of the FRC is assumed to be determined by the particle confinement in the FRC, i.e., C constitutes a number arising from the boundary condition at the separatrix. In the SOL, between the separa- trix radius rs and the wall radius rw, the densities of trapped and passing particles are
D1 si FT FP is the equilibration length of par-
" 1 rs r k2 r rs # nTðrÞ1⁄4nw 1 k2 exp D 1 k2 exp D
þ nn exp rs r ; Dn
(22)
ww