SELF-CONSISTENCY OF THE PRINCIPLE OF PROFILE CONSISTENCY RESULTS FOR SAWTOOTHING TOKAMAK DISCHARGES

The principle of profile consistency states that for fixed limiter safety factor qa there are unique natural equilibrium profile shapes for the current density j(r) (and consequently q(r)) and the electron temperature Te(r) for any tokamak plasma, independent of the shapes of the heating power deposition profiles. The mathematical statements of the three basic consequences of this principle for sawtoothing discharges (i.e. discharges with q(0) ≤ 1) are: (1) r1/a = F1(l/qa) (1/qa, empirically); (2) 〈Te〉/Te0 = F2(l/qa); and (3) a unique scaling law for the central electron temperature where q(r1) = 1, 〈Te〉 is the volume average electron temperature, and F3(qa) = qa/q0. Since Ohm's law relates j(r) to Te(r), the principle of profile consistency dictates that this unique set of functions F1, F2 and F3 remain the same for all sawtoothing discharges in any tokamak, regardless of its size (i.e. a and R), Ip, VL, BT, etc. The paper presents a rather complete and detailed analysis of this self-consistency of the measured values of Te(r), F1, F2 and F3 for sawtoothing TFTR discharges. In particular, the analytical predictions of the profiles of Coppi's Gaussian, exponential, modified exponential and trapezoidal models, as well as the model profiles of Kadomtsev and Campbell et al. are compared with TFTR and TFR data. Some of the principal results are: (1) The empirical profile consistency relation r1/a = 1/qa is an acceptable solution of q(r1) = 1 for all qa dependent profiles. (2) A comparison between experiment and the present analytic results yields [〈Te〉/Te0]EXP [〈Te〉/Te0]AN + 0.05 for the profiles of Coppi's Gaussian model, and for the profiles of the models of Kadomtsev and Campbell et al. (3) For all qa independent profiles, F3(qa) = qa/q0 = constant, and, consequently, for all qa dependent profiles, F3(qa) = qa/q0 ∼ qa when r1/a 1/qa, and, consequently, (4) Coppi's and Ohkawa's forms of xe(r) yield while the INTOR value of xe(r) yields. (5) For r1/a 1/qa, the profiles of Coppi's Gaussian model and those of the models of Kadomtsev and Campbell et al. all predict that the normalized sawtooth amplitude ΔTe/Te ∼ 1/qa, in agreement with the experimental observations. (6) For qa dependent models, universality of profiles exists in suitably reduced co-ordinates when r1/a 1/qa. © 1990 IOP Publishing Ltd.