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 q_a there are unique natural equilibrium profile shapes for the current density j(r) (and consequently q(r)) and the electron temperature T_e(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/q_a) (1/q_a, empirically); (2) 〈T_e〉/T_e0 = F₂(l/q_a); and (3) a unique scaling law for the central electron temperature where q(r₁) = 1, 〈T_e〉 is the volume average electron temperature, and F₃(q_a) = q_a/q₀. Since Ohm's law relates j(r) to T_e(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), I_p, V_L, B_T, etc. The paper presents a rather complete and detailed analysis of this self-consistency of the measured values of T_e(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/q_a is an acceptable solution of q(r₁) = 1 for all q_a dependent profiles. (2) A comparison between experiment and the present analytic results yields [〈T_e〉/T_e0]_EXP [〈T_e〉/T_e0]_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 q_a independent profiles, F₃(q_a) = q_a/q₀ = constant, and, consequently, for all q_a dependent profiles, F₃(q_a) = q_a/q₀ ∼ q_a when r₁/a 1/q_a, and, consequently, (4) Coppi's and Ohkawa's forms of x_e(r) yield while the INTOR value of x_e(r) yields. (5) For r₁/a 1/q_a, 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 ΔT_e/T_e ∼ 1/q_a, in agreement with the experimental observations. (6) For q_a dependent models, universality of profiles exists in suitably reduced co-ordinates when r₁/a 1/q_a.