Ocean neutral transport: sub-Riemannian geometry and hypoelliptic diffusion

TL;DR

This work reframes ocean parcel transport as motion constrained to a nonintegrable distribution of neutral planes defined by a dianeutral 1-form $\bm{\eta}$ (dual to the normal $\bm{n}$), giving rise to a contact and sub-Riemannian geometric description. It develops a stochastic toy model of neutral transport as Brownian motion on this sub-Riemannian manifold, yielding a degenerate but hypoelliptic diffusion whose short-time dynamics are distinctly anisotropic and whose long-time behavior explores the full three-dimensional ocean. The paper derives the CC distance, analyzes geodesics, and interprets helicity $\mathcal{H}=\bm{\eta}\wedge d\bm{\eta}$ as a quantitative measure of non-integrability and dianeutral transport via Lie brackets. Numerical experiments with WOCE-derived neutral fields show two-scale dispersion: rapid, horizontal diffusion along neutral planes versus slow vertical (dianeutral) diffusion, with mean first passage times to the mixed layer on the order of hundreds to thousands of years, implying a substantial but finite vertical transport timescale under neutrality constraints. Overall, the framework integrates ocean transport physics with rigorous sub-Riemannian geometry to quantify anisotropy and to provide diagnostic tools (CC distance, geodesics) for understanding long-term tracer spread under neutral constraints.

Abstract

The motion of water parcels in the ocean is thought to be preferentially along neutral planes defined by climatological potential temperature and salinity fields. This gives rise to a conceptual model of ocean transport in which 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--Carathéodory 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.

Figures (10)

• Figure 1: Distribution of neutral planes defined by a field of normal $\bm{n}$. A neutral path, with $\bm{x}$ the initial position and $\bm{\varphi}_t(\bm{x})$ the position at time $t$, is shown as the blue curve.
• Figure 2: Lift of a closed path in the $(x^1,x^2)$-plane to an open neutral trajectory.
• Figure 3: The distribution $\Delta$ of neutral planes can be characterised by two vector fields $\bm{v}_1$ and $\bm{v}_2$.
• Figure 4: Dianeutral transport, i.e. transport across the neutral plane (indicated by the blue shading), results from the successive transport along the vectors $\bm{v}_1$, $\bm{v}_2$$-\bm{v}_1$ and $-\bm{v}_2$ for a short time $t \ll 1$. A parcel path is shown as the black line. Its endpoint, with position $t^2 [\bm{v}_1,\bm{v}_2] + O(t^3)$, has dianeutral component $t^2 \bm{\eta}([\bm{v}_1,\bm{v}_2]) + O(t^3)$.
• Figure 5: Helicity $\mathcal{H}$ at $z=-1300$ m computed from WOCE data.