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 $g\simeq 0.11\pm 0.03$.