Page 3 - Gyrokinetic particle simulation of a field reversed configuration

052307-3 Gupta et al.
Phys. Plasmas 23, 052307 (2016)
The first complement is that the spatial connection between inner and outer field lines leads to rapid equilibration along field lines of certain physical quantities. Second, the length of the overall configuration must change in order to be in axial force balance. The last two-dimensional effect included is axial loss in the open field region using a simple parallel transport model. Even though Q1D formulation misses actual geometrical effects, such as magnetic field curvature, changes to FRC shape, different axial length, X-points, and detailed end losses, Q1D can provide quick parametric stud- ies, which are important for the transport analysis.
Here, ne and ni are the electron and ion densities, Te and
Ti are the electron and ion temperature, ue and ui are the elec-
tron and ion radial velocity, and Xe and Xi are the electron
and ion mass flow angular rotation frequency. In case of
multiple ion species, dynamics for each ion species are gov-
erned by Equations (1)–(4). In Equations (1)–(6), the terms
with superscript i/o such as ni=o ; pi=o ; wi=o , and wi=o are the iii e
source terms due to two-dimensional effect of axial equili- bration along the field lines. All the terms with coefficient c L represent the effect of axial length changes in order to satisfy axial force balance. The source terms Si, S Te , and S Ti are the source/sink terms due to ionization and charge exchange pro- cess, pellet fueling, and open field line losses. The terms Snb; Jnb; Finb; Fenb; Pinb, and Penb are the source/sink terms in density, fast particle current, momentum, and energy depos- ited on background plasma by neutral beam ions and calcu- lated using a Monte-Carlo (MC) code described later in this section. Here, J is the total current calculated from Ampere’s law and Jo is the Ohkawa current,32 and are given as
1 @B 1@w J1⁄4 l @r; B1⁄4r@r;
X0 Zi
Jo1⁄4ePr ZiniXi 1 Z ; (7)
i eff
i Z i2 n i Zeff 1⁄4PZn :
iii
Here, Ohkawa current ðJoÞ originates due to partial cancela- tion of current by electrons in the presence of multiple ion species with different charge (Zi). In either case of a single ion or multiple ions with identical charge, Jo goes to zero. As we have eliminated the electron density and their velocities, these quantities are recovered using quasi-neutrality condition as
X
Zini þnf; XX Jrnb
i
The one-dimensional transport equations solved in Q1D Ion continuity equation:
are:
Ion radial momentum equation:

AMn @ui X2r 1⁄4ZenðX XÞrB @knT iii@ti iiie @rBii
Zini @k nT 1@r @ui: ne@rBee r@ri@r
@niþ1@rnuþcn1⁄4ni=oþSþS : @tr@rii Li i i nb
(1)
(2)
Ion azimuthal momentum equation:

AMn @r2Xi þu @r2Xi
iii @t i@r
1⁄4 Zenðu uÞrBþ1@ r3 @Xi iiie r@ri@r
þr2 X ðX XÞþ Fi þZini Fe þpi=o:
ij i j nb ne nb i Ion energy equation:
j61⁄4i
(3)
3k n @Tiþu@Ti þk nT 1@ruiþc 2Bi@ti@r Biir@rL
Ziniui þ e ; (8) J 1⁄4 er ZiniXi erneXe þ Jhnb:
ne 1⁄4 neue 1⁄4
i
i
i=o kB @ @Ti @Xi 2 1⁄4wiþXrvi þ ir
r @r @r @r þkB vijðTj TiÞþ1gJ2 þPi
Here, nf is the fast ions density, and Jrnb and Jhnb are the radial and azimuthal components of the neutral beam fast particle current. The closed field FRC region is coupled to the surrounding open field region using a simple one-point parallel transport model, where it is assumed that particle and ion energy transports are convective, while the electron energy transport is conductive. With these assumptions, the particle and energy sink rate for sink terms in Equations (1), (4), and (5) are given as
j61⁄4i 2p nb þ kBvieðTe TiÞ þ STi:
(4)
(5)
(6)
Electron energy equation:
3 @T @T 1@ru kneþueþknTeþc

2 B e @t e @r B e e r @r L 1⁄4wi=o þkB @ rv @Te þ1gJ2 þPe
aCs aCs jT5=2 rN1⁄4;rTi1⁄4 andrTe1⁄4e;(9)
e r@r e@r 2p nb þkBvieðTi TeÞ þ STe Prad:
Magnetic flux equation:
L L L2
@wþu@w1⁄4 rgðJ J JÞ rFe: @t e@r nb o ene nb
where a 1⁄4 0:3 and j is the electron parallel thermal conduc- tivity coefficient multiplier.
Q1D implements two different transport models for re- sistivity, electron, and ion perpendicular thermal conductiv- ities. The two models are: multiple times classical and Bohm