Page 2 - Tomographic imaging system for measuring impurity line emission in a field-reversed configuration

022506-2 D. C. Barnes and L. C. Steinhauer
Phys. Plasmas 21, 022506 (2014)
a realistically steep scrape off layer (SOL) profile outside the separatrix. Additionally, sheared ion rotation is shown to affect the rotational modes. Taking all these calculations in sum, a new interpretation of rotational stability of FRC’s is provided.
The remainder of this paper is organized as follows. In Sec. II, the model equations are derived and reduced to a sin- gle ordinary differential equation (ODE) with appropriate boundary conditions, and the numerical solution methods are described. Section III applies this model to the equilibria of Ref. 7 and semi-quantitative agreement is shown. Section IV considers FRC’s with rigid ion rotation, while the subsequent section considers cases of more complicated sheared rota- tion. Physics conclusions are drawn in Sec. VI, where the pa- per is summarized.
II. MODEL EQUATIONS
We consider an equilibrium in which there is variation of all quantities with respect to cylindrical radius r alone. Electron mass is ignored and quasi-neutrality is assumed. The equilibrium equations, including multiple ion species and their solution, have been discussed earlier.16 In the pres- ent work, we use either one-dimensional (1D) solutions of the model of Ref. 16 with a singly charged, single ion spe- cies, or the rigid rotor model for both electrons and ions, including the centrifugal shift of the ion density maximum outward. This latter effect seems to give negligible correc- tions, as noted earlier in Ref. 7.
Equilibrium provides the quantities n; Te ; Ti ; Xe ; X; B, respectively, of equal electron and ion number density, elec- tron temperature, ion temperature, electron rotation rate, ion mass rotation rate, and magnetic field, respectively, as func- tions of r. Radial derivatives of these quantities of the required order are also computed from the equilibrium models.
Perturbation quantities are described by the perturbed mass flow momentum equation. Assuming for the moment that k is very small the relevant components are the radial (r) and azimuthal (h) components
!2n Mn ixuþi‘Xu 2Xv X rn
@ B B
1⁄4 @r Pþl0 r•PGV rþOðk2Þ;
Next, the assumption of incompressibility is used. Again, invoking the smallness of k, this implies the existence of a stream function U such that
u1⁄4U; v1⁄4iU0: (2) r‘
Taking the z component of the curl of Eq. (1), we find
i‘Mn ixuþi‘Xu 2Xv X2rn r n
þ1 d rMn ð ixvþi‘Xvþ2XuþrX0uÞ1⁄4QþOðk2Þ; r dr
where
(3)
Q1⁄4i‘ r•P$ GV r 1dr r•P$GV h (4) r r dr
$
is the gyro-viscous (GV) contribution.
Referring to Ref. 14, we see that the GV stress is of the
same form as that of Braginskii, but the form of the coeffi- cient g3 is modified. Accordingly, we write simply g for this and again denote equilibrium values by an over bar. The GV contribution consists of two terms. First, allowing for the possibility of sheared equilibrium rotation, there is a contri- bution from equilibrium flow and perturbed g which we write after a bit of algebra as
QX 1⁄4 2i‘ d gr2X0: (5) r2 dr
The GV coefficient of Eq. (5) is the perturbed GV coefficient. Since g is a function of n; T; B all of which are advected by the assumed incompressible perturbation flow we find
g 1⁄4 i g0 U; (6) rx^
where we have written here and subsequently the Doppler shifted mode frequency x^ 1⁄4 x ‘X.
The remainder of the GV contribution to Eq. (3) comes from the perturbed velocity with the equilibrium g. After a lengthy calculation, this contribution can be written as
QU 1⁄4 2 d g0U0 þ 2 ð‘2 1Þ g0 þ g00 : (7) r dr r3 r2
It is significant that only the radial derivative of the equilib- rium GV coefficient ( g0) enters the stability problem. This was found earlier in a different context with weak axial vari- ation and nearly incompressible motion assumed.17 Accordingly, it is essential to retain this profile effect when considering kinetic modifications for paraxial systems, as in our present context.
Substituting all these relations into Eq. (1) and simplify- ing, we come to a single ODE for U
x^ d AU0 þ FU 1⁄4 Oðk2Þ; (8) r dr
Mn ð ixv þ i‘Xv þ!2Xu þ rX0uÞ $
i‘ B B
1⁄4 r Pþl0 r•PGV hþOðk2Þ: (1)
In Eq. (1), u and v are the radial and azimuthal perturbed velocity components, M is the ion mass, x the mode fre- quency in the lab frame, equilibrium quantities are empha- sized by writing them with an over bar, while perturbation quantities are written without the over bar and the ’ indicates an ordinary radial derivative of any equilibrium quantity. It is also assumed that all perturbation quantities vary as eið‘hþkz xtÞ. Gyro-viscous stress (P$ GV) and finite k corrections will be discussed in detail subsequently, so are not written out explicitly in Eq. (1).