Effects of Smooth Divergence-Free Flows on Tracer Gradients and Spectra: Eulerian Prognosis Description

Abstract

Abstract Ocean eddies play an important role in the transport of heat, salt, nutrients, or pollutants. During a finite-time advection, the gradients of these tracers can increase or decrease, depending on a growth rate and the angle between flow gradients and initial tracer gradients. The growth rate is directly related to finite-time Lyapunov exponents. Numerous studies on mixing and/or tracer downscaling methods rely on satellite altimeter-derived ocean velocities. Filtering most oceanic small-scale eddies, the resulting smooth Eulerian velocities are often stationary during the characteristic time of tracer gradient growth. While smooth, these velocity fields are still locally misaligned, and thus uncorrelated, to many coarse-scale tracer observations amendable to downscaling [e.g., sea surface temperature (SST), sea surface salinity (SSS)]. Using finite-time advections, the averaged squared norm of tracer gradients can then only increase, with local growth rate independent of the initial coarse-scale tracer distribution. The key mixing processes are then only governed by locally uniform shears and foldings around stationary convective cells. To predict the tracer deformations and the evolution of their second-order statistics, an efficient proxy is proposed. Applied to a single velocity snapshot, this proxy extends the Okubo–Weiss criterion. For the Lagrangian-advection-based downscaling methods, it further successfully predicts the evolution of tracer spectral energy density after a finite time, and the optimal time to stop the downscaling operation. A practical estimation can then be proposed to define an effective parameterization of the horizontal eddy diffusivity. Significance Statement An analytical formalism is adopted to derive new exact and approximate relations that express the clustering of tracers transported by upper-ocean flows. This formalism bridges previous Eulerian and Lagrangian approaches. Accordingly, for slow and smooth upper-ocean flows, a rapid prognosis estimate can solely be performed using single-time velocity field observations. Well suited to satellite-altimeter measurements, it will help rapidly identify and monitor mixing regions occurring in the vicinity of ocean eddy boundaries.