The China France Oceanography Satellite (CFOSAT) was launched in October, 2018 and records over the oceans the wind field as well as the ocean waves. We consider here the ocean wave data, which are given through the significant wave height (Hs). We analyse along-track fluctuations of $Hs$ by considering its fluctuations of the form $y_r = \Delta_r Hs = Hs(x+r)-Hs(x) $, with values of the spatial scale $r$ between 12.5 km and a global and large scale of 2000 km. For this we consider a cumulant approach: we estimate the cumulant generating function (of $\log y_r$) $\Psi(q) = \log <y_r^q>$. This function is considered in a log-stable framework, where its development is non-analytical of the form $\Psi(q)= C_1 q + C_{\alpha} q^{\alpha}$, where $C_1$ is the first cumulant ($C_1 = < \log y_r>$), $0< \alpha \leq 2$ is the non-analycity parameter and $C_{\alpha}$ a parameter. The analysis is done by partionning the global ocean into several oceans: Indian ocean, South and North Pacific, South and North Atlantic. The statistics of the three parameters are considered over the scale $r$ and for each ocean. This provides a global view of the significant wave height multi-scale fluctuations and is complementary to a previous analysis done using Fourier spectral analysis (Gao et al 2021).