Progress towards the determination of an empirical flux equation for asymmetry-induced transport

In previous work\footnote{D.~L. Eggleston and J.~M. Williams, Phys. Plasmas 15, 032305 (2008).} on asymmetry-induced transport, it was found useful to employ the hypothesis that the asymmetry frequency $\omega$ and the plasma rotation frequency $\omega_R$ always enter the physics in the combination $\omega - l\omega_R$, where $l$ is the azimuthal mode number of the asymmetry. Flux data points satisfying the condition $\omega - l\omega_R=0$ were shown to satisfy the equation $\Gamma_{sel} = - (B_0/B)^{1.33}D_0[\nabla n_0+f_0]$, where $B$ is the magnetic field, $\nabla n_0$ is the radial density gradient, and $B_0$, $D_0$, and $f_0$ are empirical constants. The general flux equation was then constrained to be of the form $\Gamma (\epsilon) = -(B_0/B)^{1.33}D(\epsilon)[\nabla n_0+f(\epsilon)] $, where $\epsilon=\omega -l\omega_R$ and $D(\epsilon)$ and $f (\epsilon)$ are unknown functions. We now examine data points adjacent to the $\epsilon=0$ points and compare them to a first order expansion of $\Gamma(\epsilon)$. We find that a plot of $d (\Gamma - \Gamma_{sel})/d\epsilon$ vs $r$ changes sign at about the same radius as $\nabla n_0 + f_0$, and show that this implies that $dD/d\epsilon(0)\not=0$. This, plus the requirement that $D(\epsilon=0)=D_0$, restricts the form of $D (\epsilon)$. In particular, it excludes a dependence on $\epsilon$ of the form found in resonant particle transport theory\footnote{D.~L. Eggleston and T.~M. O'Neil, Phys. Plasmas 6, 2699 (1999).}, i.e., $D(\epsilon)\propto\exp{(- C\epsilon^2)}$, with $C$ a parameter.