  We formulate the problem of confined L\'evy flight on a comb. The comb
represents a sawtooth-like potential field $V(x)$, with the asymmetric teeth
favoring net transport in a preferred direction. The shape effect is modeled as
a power-law dependence $V(x) \propto |\Delta x|^n$ within the sawtooth period,
followed by an abrupt drop-off to zero, after which the initial power-law
dependence is reset. It is found that the L\'evy flights will be confined in
the sense of generalized central limit theorem if (i) the spacing between the
teeth is sufficiently broad, and (ii) $n > 4-\mu$, where $\mu$ is the fractal
dimension of the flights. In particular, for the Cauchy flights ($\mu = 1$),
$n>3$. The study is motivated by recent observations of
localization-delocalization of transport avalanches in banded flows in the Tore
Supra tokamak and is intended to devise a theory basis to explain the observed
phenomenology.
