An Analytical APproach to the Turbulence—Relaxed State in Tokamak Plasmas

An Analytical Approach to the Turbulence-Relaxed State in Tokamak Plasmas

Starting from the continuity, temperature, and motion equations of the trapped electron fluid in general tokamak magnetic field with positive or reversed shear and the definition of Lagrangian invariant, dL/dt≡(\partial_t+ u•▽)L=0, where u is convective velocity, the trapped electron dynamics is considered in the following two assumptions: (i) the turbulence is low frequency electrostatic, and (ii) L is a functional only of the density n, temperature T, and magnetic field B, and the effect of perturbation potential φ is included in the convective velocity u, i.e., u is a functional of n, T, B, and φ. The Lagrangian invariant hidden in the trapped electron dynamics is strictly found: L=ln(n/B)^c₁(T/B^2/3)^c₂], where c₁ and c₂ are dimensionless changeable parameters and c₁∝c₂. From this Lagrangian invariant the turbulent particle and electron thermal transport scaling laws are derived: 〈n>_ψq(ψ)=const. and 〈T^3/2>_ψq(ψ)=const., which, in the limit of large aspect ratio, reduce to n(r)q(r)=const. and T^3/2(r)q(r)=const., respectively. The latter two scaling laws are compared with existent experimental results, being in good agreement.