Inverse energy cascade in ocean macroscopic turbulence: Kolmogorov self-similarity in surface drifter observations and Richardson-Obhukov constant

Abstract: We combine two point velocity and position data from surface drifter observations in the Benguela upwelling region off the coast of Namibia. The compensated third order longitudinal velocity structure function $\left\langle{\Delta u_{\ell}^{\rm 3}}\right\rangle/s$ shows a positive plateau for inertial separations $s$ roughly between $9~\rm{km}$ and $120~\rm{km}$ revealing an inverse energy cascade with energy transfer rate $\varepsilon\simeq 1.2 \pm 0.1 \cdot 10^{-7} m^3/s^2$. Deviations from Gaussianity of the corresponding probability distribution $P(\Delta u_{\ell} |s)$ of two-point velocity increments $\Delta u_{\ell}$ for given pair separation $s$ show up in the n$^{th}$ antisymetric structure functions $S_{-}^{(n)}(r)=\int u^n(P(u)-P(-u)d u$, which scale in agreement with Kolmogorov's prediction, $S_{-}^{(n)}(r)\sim r^{(n/3)}$, for $n=2,4,6$. The combination of $\varepsilon$ with Richardson dispersion $\left\langle s^2(t)\right\rangle=g\varepsilon t^3$, where $\left\langle s^2(t)\right\rangle$ is mean squared pair separation at time $ t$, reveals a Richardson-Obhukov constant of $g\simeq 0.11\pm 0.03$.