Ocean neutral transport: sub-Riemannian geometry and hypoelliptic diffusion

TL;DR

This paper models ocean neutral transport using sub-Riemannian geometry and hypoelliptic diffusion, revealing how water parcels move along neutral planes in a complex, three-dimensional manner with implications for understanding ocean mixing.

Contribution

It introduces a novel geometric and stochastic framework for modeling ocean neutral transport, connecting contact geometry with hypoelliptic diffusion processes.

Findings

Neutral transport is globally three-dimensional despite local two-dimensionality.

The hypoelliptic diffusion model captures the non-confinement of trajectories to neutral surfaces.

Monte Carlo simulations estimate the timescale of vertical ocean transport under neutrality constraints.

Abstract

Transport and mixing of tracers in the ocean is thought to be preferentially along neutral planes defined by the potential temperature and salinity fields. This gives rise to a conceptual model of ocean transport in which water parcel trajectories are everywhere neutral, that is, tangent to the neutral planes. Because the distribution of neutral planes is not integrable, neutral transport, while locally two dimensional, is globally three dimensional. We describe this form of transport, building on its connection with contact and sub-Riemannian geometry. We discuss a Lie-bracket interpretation of local dianeutral transport, the quantitative meaning of helicity and the implications of the accessibility theorem. We compute sub-Riemnanian geodesics for climatological neutral planes and put forward the use of the associated Carnot--Carath\'eodory distance as a diagnostic of the strong anisotropy of neutral transport. We propose a stochastic toy model of neutral transport which represents motion along neutral planes by a Brownian motion. The corresponding diffusion process is degenerate and not (strongly) elliptic. The non-integrability of the neutral planes however ensures that the diffusion is hypoelliptic. As a result, trajectories are not confined to surfaces but visit the entire three-dimensional ocean. The short-time behaviour is qualitatively different from that obtained with a non-degenerate highly anisotropic diffusion. We examine both short- and long-time behaviours using Monte Carlo simulations. The simulations provide an estimate for the time scale of ocean vertical transport implied by the constraint of neutrality.