Remarks on the Scaling of Kurtosis with Squared Skewness

Recent analysis of density fluctuations in TORPEX\footnote{B. Labit \emph{et al.}, Universal statistical properties of drift-interchange turbulence in TORPEX plasmas, Phys.\ Rev.\ Lett. \textbf{98}, 255002 (2007).} support the relationship $K = aS^2 + b$ between the skewness $S$ and (excess) kurtosis $K$, where $a \approx 1.5$ and $b \approx -0.2$. (A realizability constraint is $K \ge S^2 - 2$.) Remarkably, essentially the same result has been shown to hold for a global dataset of fluctuations of sea-surface temperature,\footnote{P. Sura and P. D. Sardeshmukh, A global view of non-Gaussian SST variability, J. Phys.\ Oceanogr.(2007), in press.} and a simple theoretical (nonlinear Langevin) model has been proposed\footnotemark[3] that leads to $a = 3/2$ and $b = 0$. This is obviously suggestive, but it is a challenge to justify the Langevin model in detail for magnetized plasma turbulence. Previous results on higher-order statistics,\footnote{J. A. Krommes, Non-Gaussian statistics, classical field theory, and realizable Langevin models, Phys.\ Rev.\ E \textbf{53}, 4865 (1996).} dimensionally compatible with $K \sim S^2$, are reviewed and an attempt is made to derive $a$ and $b$ for a model involving coupled modes and linear waves. The extent to which the values of $a$ and $b$ are sensitive discriminants for details of the underlying turbulence is discussed.