October 9, Matt Landreman More accurate ion flow coefficients in the Pfirsch Schluter regime In these notes we repeat and slightly etend the calculation of Ref. [], computing the ion neoclassical flow in the Pfirsch Schluter (short mean free path) collisionality regime. he calculation relies on the solution of several generalized Spitzer problems. In the original work [], the Spitzer problems were solved using a Laguerre polynomial approimation. Here we derive the flow in terms of general integrals of Spitzer functions, so the coefficients may be evaluated as accurately as one wishes. he main reason for more accurate evaluation of these coefficients is for benchmarking of neoclassical codes in the limit of high collisionality. he final result of Ref. [] is that the parallel ion flow is where primes denote d / d, nv Ze p Icn p Ze k k k k ln (), () and the two dimensionless flow coefficients are k.8, k.7. () In eq () of Ref. [], more accurate values of the coefficients are given as k.77, k.5. () he main result of the present notes is that as the accuracy of the discretization is increased, (i.e. as more accurate Spitzer functions are used), the flow coefficients converge to k.97, k.96. (5) o a good approimation, the ratio of flu surface averages in () is, and so k k k.657. he calculation here (as in []) neglects ion electron collisions. Note that if ion electron collisions were included, the ion flow would change by ~ m / m. Preliminary steps We wish to solve the drift kinetic equation f C f d e i f v (6) for high collisionality: qr / where i / m is the thermal speed and i
ne ln (7) / / m is the ion ion collision frequency. he gradient in (6) is performed at fied / where /. Also, v d denotes the magnetic drift, and n f e (8) / i is the leading order Mawellian, with / i. We take C to be the ion ion collision operator, neglecting ion electron collisions. It will be convenient to epand the right hand side of (6) as mci p Ze 5 f Cf f Ze p. (9) We will need to annihilate the largest term, the collision operator, which is done by applying d..., d..., and d m / 5 /..., yielding moment equations. hese three moment equations are Icn p Ze d f b Ze p, () d f d f, () and 5 5 mci d f n m b m Ze. () It will be frequently useful in the following analysis to apply... to () to obtain d f. () Here denotes a flu surface average. j j We epand f f f f... where f. We begin the series with j since it turns out that f will have a term. Order kinetic equation: he leading order term in (6) is C f. () herefore p m 5 f V f (5) p for some functions p,, and. V Order constraints: Constraint () gives
V (6) so V A (7) for some flu function A. Constraint () tells us nothing at this order. Constraint () tells us p. (8) We may as well have kept all the flu surface averaged pressure in f, so Order kinetic equation: he terms in (6) are f C f p.. (9) Plugging in (5), 5 f A f Cf Notice that the first few Legendre polynomials are P, P, P. i Since Legendre polynomials are eigenfunctions of C, the solution of (9) must be p m 5 f V f p i A F f F f where p, F and F and V, and. () () () are new functions associated with the homogeneous solution, and are defined by the Spitzer problems CF f 5 f () C F f f. () Approimate solution of first Spitzer problem: Although we ultimate want to compute the Spitzer functions to arbitrarily high accuracy, to make contact with Ref. [], here we repeat the polynomial approimation used there.
It will turn out that we don t need the solution of () (for now), but we will need the solution of (). Notice that we can always add a constant to F, since C f. We consider an epansion in Laguerre polynomials of order /, keeping the first terms after the constant: / F a L k k k 5 a a 5 8. 8 Multiplying () by another Laguerre polynomial and integrating over velocity space, / 5 / / d Lj f d Lj C f aklk. (6) k From (6) of Ref. [], the right hand side of (6) is n / a / 5/6 a. (7) he left hand side of (6) is / 5 n / 5 d Lj f d L / j e (8) 5 n j,. hus, we have a linear system 5 / a / 5/6 a. (9) he solution is 5 a 6 () 5 a so 5 5 5 F 5 8 6 8 5 5 5 5 5 6 8 8 As noted above, we can shift F by a constant, so we may simplify it to 5 F () 8 making () into 5 i f f... () 8 Notice Ref. [] uses the collision time (5) ()
/ / mi m ne ne 8 ln ln so () becomes 5 i f f... (5) 8 in agreement with eq (5) (5) in Ref. []. Order constraints: Plugging () into () gives A d S In general, the integral and (). (6) do not vanish, so A must be. hus, 5 f f. (7) Plugging () into () then gives p, so again it is no loss in generality to take p, as we can absorb it into f. hus, we have m i 5 f V f F f f. (8) We net plug this result into (), finding i 5 5 mci d F f n m b (9) m Ze his result is basically just a statement of the incompressibility of the heat flu: q bq diamag for the classical q where q diamag is the diamagnetic heat flu. Defining a dimensionless coefficient 5 X d Fe () then 5 mci. () 6X Ze Consequently 5 mci A () 6X Ze for some A. Applying..., 5 mci A () 6X Ze so
5 mci. () 6X Ze For the polynomial approimation (), then 5 5 X de.6 8 (5) so () becomes.68 mci Ze in agreement with eq (5) in []. Net, we observe from (8) that the actual parallel flow is not V but rather Xn i nv d f nv 5X cni nv. X Ze where (6) (7) X d F e (8) is another dimensionless constant. Plugging (7) into (), we find nv n i Icn p Ze X Ze p. (9) hen 5 X cni Icn nv A p Ze X Ze Ze p (5) for some flu function A. Notice the total parallel flow (7) is Icn p Ze nv A Ze p. (5) Comparing this epression to (), we can define the parallel flow coefficient k in terms of A : A k Icn. (5) Ze For the polynomial approimation (), 5 X (5) 6 so the numerical coefficient in (5) is
in agreement with eq (5) in Ref. []. 5X. X (5) Order kinetic equation: he terms in (6) are mci p Ze 5 f C f f Ze p. (55) Epanding the first term, mci p Ze 5 Cf f Ze p m m Y V f f V f. (56) F 5 f f o evaluate several of the terms, we can recall. (57) It is apparent that the right hand side of (56) has Lj terms for j=,,. he solution is f YLYLYL f. (58) It turns out that only the j term is needed to compute the flow. hus, mci p Ze 5 f Ze p m m i. (59) Cf V f f i V Y f L j L i F f he terms in the right hand side of (59) have basic types of dependence, so there are basic Spitzer problems we must solve: CfFL L f, (6) CfFL L f, (6) CfFF L L ff, (6) for the functions F, F, and F F. he solution to (58) (59) then has the form
where Y d F d F d F (6) F F mcii p Ze 5 d V V Ze p, (6) d ci, (65) Ze i df 5 ci. 6 X Ze With a bit of manipulation, it can be seen that the p and A 5 X ci d n X Ze terms in (6) cancel, leaving (66) (67) Order constraint: At this order, the only constraint we need to consider is (). Due to the orthogonality of the Legendre polynomials, only the L component of f matters in this constraint, so we find de df df dfff. (68) We define the dimensionless coefficients so his constraint has the form 5 F X d e F X d e F X d e F dx dx df X 5 (69). (7) 5 5X X X5 5 6 6X X X 6X cni A. (7) 5X X5 5 Ze X X 6X
where and he solution is hus, from (5), At last, we obtain () where and A Q Q 5 5X X X5 5 cni Q 6 6X X X6X Ze Q (7) (7) 5X X5 5 cni. (7) X X 6X Ze ln A Q Q. (75) ln k Q Q. (76) Icn Ze k k 5X X 5 5 (77) X X 6X 5 5X X X 5 5. (78) 6 6X X X6X Summary and numerical solution he flow coefficient is given by () and (77) (78) where 5 X d e F X d e F X d e F X d e F X5 d e FF and the Spitzer problems we must solve are (79)
5 CF f f, (8) CF fl L f, (8) CF fl L f, (8) CFF fl L ff. (8) he Spitzer functions and integrals were computed using the matlab program m9 solvespitzerproblemsforpfirschschluterregimeflow.m, using the discretization techniques described in Ref. []. Here we do not numerically solve the finite collisionality problem (6), only the spatially independent problems (8) (8). Using 5 polynomials to reduce the 5 variation in the coefficients to < between successive numbers of polynomials, the converged values of the flow coefficients are given by (5), where all digits shown are converged. he plots below illustrate the convergence. he error shown in the bottom two plots is the difference between the value and the value for 5 grid points. -.5 -.6 -. -.7 -. k -.8 k -.6 -.9 -.8 - Number of grid points in v -. Number of grid points in v Error in k -5 Error in k -5 - Number of grid points in v - Number of grid points in v References [] Hazeltine, Phys. Fluids 7, 96 (97). [] Catto & Simakov, Phys. Plasmas 6, 5 (9).
[] Pusztai & Catto, Plasma Phys. Controlled Fusion 5, 756 (). [] Landreman & Ernst, J. Comp. Phys., ().