  In this paper, we investigate and develop a new approach to the numerical
analysis and characterization of random fluctuations with heavy-tailed
probability distribution function (PDF), such as turbulent heat flow and solar
flare fluctuations. We identify the heavy-tailed random fluctuations based on
the scaling properties of the tail exponent of the PDF, power-law growth of
$q$th order correlation function and the self-similar properties of the contour
lines in two-dimensional random fields. Moreover, this work leads to a
substitution for fractional Edwards-Wilkinson (EW) equation that works in
presence of $\mu$-stable L\'evy noise. Our proposed model explains the
configuration dynamics of the systems with heavy-tailed correlated random
fluctuations. We also present an alternative solution to the fractional EW
equation in the presence of $\mu$-stable L\'evy noise in the steady-state,
which is implemented numerically, using the $\mu$-stable fractional L\'evy
motion. Based on the analysis of the self-similar properties of contour loops,
we numerically show that the scaling properties of contour loop ensembles can
qualitatively and quantitatively distinguish non-Gaussian random fields from
Gaussian random fluctuations.
