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Determination of the First Critical Gradient in Plasma Energy Functional Profiles

The first critical gradient is identified using canonical profiles theory for function μ based on Euler equation solution for the free plasma energy. Boundary conditions and dimensionless current density are considered to determine the constants and the solution. The first dimensionless critical gradients for temperature and density are computed as ΩTc = -2/3.Ric'/ic and Ωnc = -1/3.Ric'/ic.

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Determination of the First Critical Gradient in Plasma Energy Functional Profiles

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  1. DIII-D, #89943, t=1.7 sec

  2. 1. The first critical gradient We find the first critical gradient by the canonical profiles theory. The canonical profile for the function  = 1/q(denoted below as с) can be found by the solution of the Euler equation for the free plasma energy functional [1] 2Gc2/ + (/2) / ((1/ V) (VGc)) = Cc/V.(1) Here:indexSmeans the plasma boundary, iс = 1/V/(G Vс)is the dimensionless current density, Vis the plasma volume,V= V/, G = R2<(grad )2/r2>is the metric coefficient.

  3. The solution of Eq. (1) and the constants Сandare determined by the following four boundary conditions: c(0) = 0 ~ 1, c(0) = 0,c(max) = S, X [ic/(2Gc)]S = G(a)1/2S/0 (2) The first dimensionless critical gradientsfor the temperature and density are following: Tc = R/LTc -RTc/Tc = - 2/3 Ric/ic, nc -Rnc/n = - 1/3Ric/ic. (3)

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