GWH Oct 18, 1999: This version of collisions.txt includes slight clarifications of the notation. In the following, v_te=sqrt(T_e/m_e), and v_t2e=sqrt(2*T_e/m_e), and v_t=sqrt(T_i/m_i), v_t2=sqrt(2*T_i/m_i). Date: Wed, 7 Apr 1999 12:56:43 -0400 From: Bill Dorland To: hammett@pppl.gov Subject: Collisionality in GS2 Reply-to: bdorland@ipr.umd.edu Cc: mtk@pauling.ph.utexas.edu Greg, You sent us a memo back in September about the collision operator in GS2. I'm sorry we didn't respond sooner. We have finally worked through it, and have resolved the questions. First, on the collisionality. Starting from either the Formulary or Hinton and Hazeltine's Eq. 5.2, one gets the expression from your memo, i.e., C_ei(f_e) = (nu^ei_0/2) d/dzeta [ (1-zeta**2) df_e/dzeta]. This is different from the expression the Kotschenreuther, Rewoldt, Tang paper, by a factor of 2. The KRT paper is apparently wrong, but GS2 is correct. For my own records, I'll show the work here, starting from HH: C(f) = nu_ei L f 3 sqrt(pi) 4 sqrt(2 pi) n_i Z_i**2 e**4 ln(Lambda) 1 d df = -------------------------------------------------- - -- (1-z**2) -- 4 (v/v_t2e)**3 3 sqrt(m_e) T_e**1.5 2 dz dz = nu d/dz (1-z**2) df/dz. This defines what I mean by nu. Change T_e into T_kev units: T_e**1.5 = 6.4 x 10**-14 T_kev**1.5. The time normalizing factor in GS2 is omega_*, evaluated with the minor radius "a" instead of a scale length: omega_* = cT/eB k_theta / a Thus, the collisionality in the code should be nu_tilde = nu / omega_* 612 n_19 Z_eff ln(Lambda) = ------------------------------- . (v/v_t2e)**3 T_kev**1.5 omega_* In the units that Mike used (which include the sqrt(2) in the definition of v_t2,=sqrt(2*T_i/m_i), i.e., v_tMK=v_t2), omega_* is omega_* = cT/eB (k_theta / a) = (mc/eB) (T/m) (k_theta/a) = v_t2**2 / Omega (k_theta / a) 1/2 = (k_theta rho_MK) (1/2) (v_tMK / a) = "ky" / 2 (v_tMK/a). Here, ky is an input parameter which is defined to be k_theta rho_MK. Using the formulary (which doesn't include the factor of sqrt(2) in the thermal velocity definition): omega_* = (ky/2) [sqrt(2) 9.79 x 10**5 sqrt(1000 T_kev)/sqrt(2)/(100 a_m)] = 1.55 x 10**5 ky (sqrt(T_keV) / a) where the second factor of sqrt(2) comes from the fact that it is assumed that deuterium is the main species at this point. Thus, nu_tilde = 3.95 x 10**-3 n_19 Z_eff ln(Lambda) / ky / Te(keV)**1.5 / Ti(kev)**0.5 * a / (v/v_t2e)**3 ------------------------------------------------------------------------------ The documentation for GS2 (see comments in s1.in on the GS2 site) says that the electron collisionality parameter vnewk should be vnewk = 0.00394 * coulog * ne_19 / Te(keV)**1.5 / Ti(kev)**0.5 * a where the minor radius is supposed to be in meters. There are a couple of other steps, in which the remaining factors of 1/ky, Z_eff, (from the "parameters" namelist, not self-consistently from the different species) and 1/(v/v_t2e)**3 are included; the end result is that the correct collisionality is used. The factor of two that is evidently missing from the KRT paper is in the code. This probably came about from a transcription of the HH Eqs. (5.2 ff), where the factor of 1/2 in Eq. (5.5) was accidentally left out. I still need to double-check to make sure that the collisionality is normalized properly when one uses v_t/a time units instead of omega_* units (i.e., for nonlinear runs). It looks right, but I will double-check. With respect to momentum conserving terms, Mike and Peter put these in some time in the last year in preparation for nonlinear runs with the RH effect. They can be seen in collisions.f90. Thanks for checking this closely! Talk to you later, --Bill