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This study considers the isopycnal eddy transport of mass and passive tracers in eddy-resolving double-gyre quasigeostrophic oceanic circulation. Here we focus on advective transport, whereas a companion paper focuses on eddy-induced diffusive tracer transport. To work towards parameterising eddy tracer transport we quantify the eddy tracer flux using a transport tensor with eddies defined using a spatial filter, which leads to results distinct from those obtained via a temporal Reynolds eddy decomposition. The advection tensor is the antisymmetric part of the transport tensor, and is so named since the associated tracer transport can be expressed as advection of the large-scale tracer field by a rotational eddy-induced velocity (EIV) u ∗ c with streamfunction A . The EIV u ∗ c is fastest ( ∼ 1 m s−1) where eddy activity is strongest, e.g., in the upper layer, near the eastward jet and western boundary current. Our results suggest that a stochastic closure for the eddy transport would be most suitable since A exhibits a probabilistic distribution when conditioned on, for example, the large-scale relative vorticity. Consistent with closures in ocean circulation models, we quantify eddy mass (isopycnal layer thickness) fluxes as eddy-induced advection by the thickness EIV u ∗ h . The divergent part of u ∗ h – the only part relevant for mass transport in the quasigeostrophic limit – tends to be oriented down the thickness gradient suggesting it quantifies some baroclinic eddy effects similar to those parameterised by the Gent & McWilliams (GM90) EIV. Although u ∗ h has some qualitative similarities to u ∗ c , our results suggest that eddy-induced tracer advection is driven by more than just the thickness-determined EIV and, in turn, more than just the GM90 EIV.

Oceanic mesoscale currents (‘eddies’) can have significant effects on the distributions of passive tracers. The associated inhomogeneous and anisotropic eddy fluxes are traditionally parametrised using a transport tensor (K-tensor), which contains both diffusive and advective components. In this study, we analyse the eddy transport tensor in a quasigeostrophic double-gyre flow. First, the flow and passive tracer fields are decomposed into large- and small-scale (eddy) components by spatial filtering, and the resulting eddy forcing includes an eddy tracer flux representing advection by eddies and non-advective terms. Second, we use the flux-gradient relation between the eddy fluxes and the large-scale tracer gradient to estimate the associated K-tensors in their entire structural, spatial and temporal complexity, without making any additional assumptions or simplifications. The divergent components of the eddy tracer fluxes are extracted via the Helmholtz decomposition, which yields a divergent tensor. The remaining rotational flux does not affect the tracer evolution, but dominates the total tracer flux, affecting both its magnitude and spatial structure. However, in terms of estimating the eddy forcing, the transport tensor prevails over its divergent counterpart because of the significant numerical errors induced by the Helmholtz decomposition. Our analyses demonstrate that, in general, the K-tensor for the eddy forcing is not unique, that is, it is tracer-dependent. Our study raises serious questions on how to interpret and use various estimates of K-tensors obtained from either observations or eddy-resolving model solutions.

This study provides an interpretation of isopycnal eddy transport for mass and passive tracers in double-gyre eddy-resolving oceanic circulation. This paper focuses on a transport/diffusion tensor representation of the eddy tracer flux, and a companion paper will focus on advective eddy-induced tracer and mass transports. We use a spatial filter to separate the large and small scales, which leads to results distinct from those obtained via a temporal Reynolds eddy decomposition. To work towards a parameterisation, we relate the eddy tracer flux to the large-scale tracer gradient via the transport tensor K . The symmetric part of K is the diffusion tensor, S , which parameterises diffusive fluxes and whose mixing properties are determined by the signs of its eigenvalues. The eigenvalues of S are robustly of opposite sign (polar) and thus quantify filamentation of the tracer via both up- and down-gradient fluxes. Given the prevalence of polar eigenvalues – which are also obtained for Reynolds eddy fluxes – representing their associated effects should be a target of future eddy tracer transport closures. Given the inherent inhomogeneity and anisotropy of the eddy-induced transport, we argue that a full transport tensor is better suited to this task than scalar coefficients or diagonal tensors. The diffusion axis, which represents the direction of preferential mixing, tends to align with the large-scale velocity vector and contours of large-scale relative vorticity and layer thickness. Strong shears can inhibit this alignment. We show that the large-scale velocity gradient matrix may be suitable for parameterising the transport tensor, in particular at depth. Furthermore, since entries of K and S exhibit probabilistic distributions when conditioned on certain large-scale flow features, we suggest that a stochastic closure for the eddy transport would be most suitable.

This study provides estimates of the mean eddy-induced diffusivities of passive tracers in a three-layer, double-gyre quasigeostrophic (QG) simulation. A key aspect of this study is the use of a spatial filter to separate the flow and tracer fields into small-scale and large-scale components, and we compare results with those obtained using Reynolds temporal averaging. The eddy tracer flux is related to a rank-2 diffusivity tensor via the flux-gradient relation, which is solved for a pair of tracers with misaligned large-scale gradients. We concentrate on the symmetric part of the resulting diffusivity tensor which represents irreversible mixing processes. The eigenvalues of the symmetric tensor exhibit complicated behaviour, but a particularly dominant and robust feature is the positive/negative eigenvalue pairs, which physically represent filamentation of the tracer concentration. The large off-diagonal diffusivity tensor component is the primary contributor to the eigenvalue polarity, and since this is such a prevalent feature we argue that the (horizontal) eddy-induced diffusivity should always be treated as a full 2 × 2 tensor. Diffusivity magnitudes are largest in the upper layer and in the eastward jet region, where the eddying flow is strongest. After removing the rotational part of the eddy tracer flux, typical mean diffusivities (eigenvalues) in the upper-layer are on the order of 10 3 m2 s−1 in the jet region and 10 2 m2 s−1 elsewhere. We also confirm that the time-mean of the diffusivity calculated from instantaneous fluxes is not the same as the diffusivity associated with the time-mean fluxes.