Page 11 - Drift-wave stability in the field-reversed configuration

092518-11
Onofri et al. FT
Phys. Plasmas 24, 092518 (2017) In Fig. 17, we plot sSOL=si as a function of the mirror ratio R
for a 1⁄4 1 and n 1 (no particles are lost radially). The con- finement time in the SOL increases with the mirror ratio, and for high R, it can be approximated as
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !
(37)
rffiffiffiffiffiffiffiffi !
sSOL’si 1 1 : (38) 2aR
In the limit n 1, sSOL becomes
pffiffiffiffiffi 2 FTtanh n
rffiffiffiffiffi
ns ’ pffiffiffi
FP þ atanhn
si D1
C: (29)
pffiffi
We can calculate the fluxes of trapped and passing particles that cross the separatrix
CT C
CP C
pffiffi
aFT tanhn
s’s1 1 1r1 1 r1 1:
’ pffiffiffiffiffi pffiffi FPþ atanhn
s n In the case of no radial losses (n 1), sSOL becomes
;
pffiffiffiffiffi pffiffi
FP þ aFPtanhn
(30) : (31)
SOL
i
2aRtanhn r w n coshn s
’ pffiffiffiffiffi pffiffi FPþ atanhn
The flux of particles that are lost at the wall radius rw is pffiffi
CSOL 1⁄4 D2
dn
dr r1⁄4rw
’ FT C
1 coshn
atanhn pffiffiffiffiffi pffiffi
: (32) The case where no particles are lost radially can be obtained
for n 1. In this case, 1=coshn’0 and tanhn’1, and from Eq. (32), it is found that CSOL ’0. In the limit n 1 (large diffusion coefficient D2 in the SOL, when fluctuations in the SOL are much greater than in the core,4), CSOL is
(33)
which is equal to the flux of trapped particles CT across the separatrix [Eq. (30)] in the same limit. This shows that when D2 D1, most of the trapped particles are lost radially. The SOL confinement time due to the mirrors can be estimated as
FP þ atanhn
sSOL ’
and for large mirror ratio, R 1,
siF2T Dw ; (39) 1þ Dq rs
FC CSOL’ T ;
rw rs
pffiffiffiffiffiffiffiffiffi rffiffiffiffiffi!rffiffiffiffiffi
1þ Dq rw rs
s’s1 siD11D2; (40) SOL i rw rs 2R D0
where D0 1⁄4 rs2=si.
After the particles diffuse across the separatrix into the
SOL, they move in the parallel direction towards the mirrors, and during this time, some trapped particles are converted into passing particles, and they are lost through the mirrors. Under the assumption that no particles are lost in the radial direction in the FRC region, the flux of particles that are lost through the mirrors, CM, is given by the flux of passing par- ticles that cross the separatrix, CP, plus the flux of trapped particles, CT, that are converted into passing particles during the time they take to reach the mirrors,
sSOL 1⁄4 CS
;
(34)
NSOL
where Ssep 1⁄4 2prsL is the separatrix surface and NSOL is the number of particles in the SOL. From the solution of the dif- fusion equations, we find
sep
ðrw 2pffiffi tk
FT atanhn NSOL 1⁄4 nrdr2pL ’ si pffiffiffiffiffi pffiffi
rs Fpþ atanhn 1 1 1
rw n coshn rs n 1 C2pL; (35) 2pffiffi
CM 1⁄4 CP þ fMCT; (41) FT si
where fM 1⁄4 1 CSOL=CT and tk is defined as
ð dz
tk 1⁄4 ; (42)
Vk
and Vk is the fluid parallel velocity. Inserting the expressions given by Eqs. (30)–(32) into Eq. (41), the flux at the mirrors CM is
sSOL ’ si pffiffiffiffiffi Fp þ
pffiffi rw atanhnrs

rs
1 : (36)
FT atanhn 1
1
n
1 coshn
1
n
pffiffiffiffiffi pffiffi
FP þ aFPtanhn
CM ’ pffiffiffiffiffi pffiffi C FP þ atanhn
pffiffi tk aFTtanhn 1
þ pffiffiffiffiffi pffiffi 1 C: (43) FTsi FP þ atanhn coshn
FIG. 17. Ratio of the confinement time to the collisional time sSOL=si as a function of the mirror ratio R.
The second term in the last parenthesis is due to the particles escaping radially through the SOL. In the limit n 1, the term in parenthesis is zero, and no trapped particles are lost through the mirrors. In a steady state with no radial losses, we have CM 1⁄4 C, and from Eq. (43), we get