Invariant Measure and Turbulent Pinch in Tokamaks

It is shown that electron transport due to a generic low-frequency electrostatic turbulence in tokamak geometry results in the relaxation to a peaked, self-sustained plasma density profile $n_0\left(r\right)$, rather than to a diffusion-induced flat distribution. The relaxed density profile depends on the magnetic geometry and the distribution of turbulence. The associated inward pinch velocity ${\mathbf{V}}_r=D\mathbf{\nabla }\ln n_0$ results from the competition of the turbulent diffusion of trapped electrons over the poloidal magnetic flux coordinate $\psi$ and the collisional relaxation toward a Maxwellian distribution function.