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Transport and mixing properties of the ocean's circulation is crucial to dynamical analyses, and often have to be carried out with limited observed information. Finite-time coherent sets are regions of the ocean that minimally mix (in the presence of small diffusion) with the rest of the ocean domain over the finite period of time considered. In the purely advective setting (in the zero diffusion limit) this is equivalent to identifying regions whose boundary interfaces remain small throughout their finite-time evolution. Finite-time coherent sets thus provide a skeleton of distinct regions around which more turbulent flow occurs. Well known manifestations of finite-time coherent sets in geophysical systems include rotational objects like ocean eddies, ocean gyres, and atmospheric vortices. In real-world settings, often observational data is scattered and sparse, which makes the difficult problem of coherent set identification and tracking challenging. I will describe mesh-based numerical methods [3] to efficiently approximate the recently defined dynamic Laplace operator [1,2], and rapidly and reliably extract finite-time coherent sets from models or scattered, possibly sparse, and possibly incomplete observed data. From these results we can infer new chemical and physical ocean connectivities at global and intra-basin scales (at the surface and at depth), track series of eddies, and determine new oceanic barriers.[1] G. Froyland. Dynamic isoperimetry and the geometry of Lagrangian coherent structures. Nonlinearity, 28:3587-3622, 2015[2] G. Froyland and E. Kwok. A dynamic Laplacian for identifying Lagrangian coherent structures on weighted Riemannian manifolds. Journal of Nonlinear Science, 30:1889–1971, 2020.[3] Gary Froyland and Oliver Junge. Robust FEM-based extraction of finite-time coherent sets using scattered, sparse, and incomplete trajectories. SIAM J. Applied Dynamical Systems, 17:1891–1924, 2018.

Mesoscale eddies stir along the neutral plane, and the resulting neutral diffusion is a fundamental aspect of subgrid-scale tracer transport in ocean models. Calculating neutral diffusion traditionally involves calculating neutral slopes and three-dimensional tracer gradients. The calculation of the neutral slope traditionally occurs by computing the ratio of the horizontal to vertical locally referenced potential density derivative. However, this approach is problematic in regions of weak vertical stratification, prompting the use of a variety of ad hoc regularization methods that can lead to rather nonphysical dependencies for the resulting neutral tracer gradients.Here we introduce VENM; a search algorithm that requires no ad hoc regularization and significantly improves the numerical accuracy of calculating neutral slopes, neutral tracer gradients, and associated neutral diffusive fluxes. We compare and contrast VENM against a more traditional method, using an independent objective neutrality condition combined with estimates of spurious diffusion, heat transport, and water mass transformation rates. VENM is more accurate, both physically and numerically, and should form the basis for future efforts involving neutral diffusion calculations from observations and possibly numerical model simulations.

This paper uses a suite of Earth system models which simulate the distribution of He isotopes and radiocarbon to examine two paradoxes in Earth science, each of which results from an inconsistency between theoretically motivated global energy balances and direct observations. The helium–heat paradox refers to the fact that helium emissions to the deep ocean are far lower than would be expected given the rate of geothermal heating, since both are thought to be the result of radioactive decay in Earth's interior. The isopycnal mixing paradox comes from the fact that many theoretical parameterizations of the isopycnal mixing coefficient ARedi that link it to baroclinic instability project it to be small (of order a few hundred m2 s−1) in the ocean interior away from boundary currents. However, direct observations using tracers and floats (largely in the upper ocean) suggest that values of this coefficient are an order of magnitude higher. Helium isotopes equilibrate rapidly with the atmosphere and thus exhibit large gradients along isopycnals while radiocarbon equilibrates slowly and thus exhibits smaller gradients along isopycnals. Thus it might be thought that resolving the isopycnal mixing paradox in favor of the higher observational estimates of ARedi might also solve the helium paradox, by increasing the transport of mantle helium to the surface more than it would radiocarbon. In this paper we show that this is not the case. In a suite of models with different spatially constant and spatially varying values of ARedi the distribution of radiocarbon and helium isotopes is sensitive to the value of ARedi. However, away from strong helium sources in the southeastern Pacific, the relationship between the two is not sensitive, indicating that large-scale advection is the limiting process for removing helium and radiocarbon from the deep ocean. The helium isotopes, in turn, suggest a higher value of ARedi below the thermocline than is seen in theoretical parameterizations based on baroclinic growth rates. We argue that a key part of resolving the isopycnal mixing paradox is to abandon the idea that ARedi has a direct relationship to local baroclinic instability and to the so-called "thickness" mixing coefficient AGM.