Describing geophysical turbulence with a Schrödinger–Coriolis equation in velocity space

Abstract

In this paper, we examine the predictions of the scale-relativity approach for a turbulent fluid in rotation. We first show that the time derivative of the governing Navier–Stokes equation in the usual x-space can be transformed into a Schrödinger-like equation in velocity space with an external vectorial field to account for the rotation, together with a local velocity harmonic oscillator (VHO) potential in the v-space. The coefficients of this VHO are given by the second order x-derivatives of the pressure. We can then give formulas for the velocity and acceleration probability distribution functions (PDF). Using a simple model of anisotropic harmonic oscillator, we compare our predictions with relevant data from both direct numerical simulations (DNS) and oceanic drifter velocity measurements. We find a good agreement of the predicted acceleration PDF with that observed from drifters and some possible support in DNS for the existence of gaps in the local velocity PDF, expected in the presence of a Coriolis force.