Stirring of water by mesoscale currents (“eddies”) leads to large‐scale transport of many important oceanic properties (“tracers”). These eddy‐induced transports can be related to the large‐scale tracer gradients, using the concept of turbulent diffusion. The concept is widely used to describe these transports in the real ocean and to represent them in climate models. This study focuses on the inherent complexity of the corresponding coefficient tensor (“*K*‐tensor”) and its components, defined here in all its spatio‐temporal complexity. Results demonstrate that this comprehensive *K*‐tensor is space‐, time‐, direction‐ and tracer‐dependent. Using numerical simulations with both idealized and comprehensive models of the Atlantic circulation, we show that these properties lead to upgradient eddy fluxes and the potential importance of all tensor components. The uncovered complexity of the eddy transports calls for reconsideration of how they are estimated in practice, included in the general circulation models and theoretically interpreted.

Mesoscale eddies, loosely defined as ocean currents on the spatial scales of tens to hundreds of kilometers, are ubiquitous in the World Ocean. Relentless stirring of water by these eddies leads to large‐scale transport and redistribution of such important oceanic properties as heat, salinity, and anthropogenic carbon. The efficiency of this process has been conventionally described by turbulent (“eddy”) diffusion. Our study focuses on the inherent complexity of the corresponding transport tensor, defined here in all its complexity, without any space and/or time averaging. Results from this study demonstrate that this transport tensor varies with location and time. Using numerical simulations with both simplified and realistic models of the North Atlantic circulation, we show that these properties lead to eddy fluxes that act to sharpen tracer fronts, rather than smooth them. We also show that all components of the comprehensive tensor are potentially important for tracer distributions, and, therefore, cannot be generally neglected. Our results further demonstrate that the comprehensive diffusivity tensor depends on the tracer that it is derived from. The uncovered complexity of the eddy transports calls for reconsideration of how they are estimated in practice, included in the general circulation models and theoretically interpreted.

Both idealized and realistic numerical simulations uncover new properties of the transport tensor used to represent mesoscale tracer transports

The full spatio‐temporal complexity reveals strong time‐ and tracer‐dependence, as well as the systematic presence of negative diffusivities

The complexity of the eddy transports calls for reconsideration of how they are represented in models and estimated from observations

Mesoscale eddies, loosely defined as ocean currents on the spatial scales of tens to hundreds of kilometers, are ubiquitous in the World Ocean (Chelton et al., 2007). Relentless stirring of water by these eddies leads to large‐scale transport and redistribution of many dynamically and climatically important oceanic properties (“tracers”), including heat, salinity, and anthropogenic carbon (McWilliams, 2008). As a result, mesoscale eddies play a key role in determining the current and future states of the World Ocean and the Earth Climate, as manifested by strong sensitivity of ocean and climate simulations to the magnitude and distribution of eddy transports (Gnanadesikan et al., 2013; McWilliams, 2008; Wiebe & Weaver, 1999). At the same time, vast majority of ocean components in modern climate models either completely miss the eddies or only partially resolve them (Adcroft et al., 2019; Delworth et al., 2012; Williams et al., 2015). The eddy‐induced transports in these models need to be additionally expressed (“parameterized”) in terms of known large‐scale properties. This task requires a thorough study of eddy transport properties and their significance for tracer distributions. Below, we report on several new and important properties of the eddy transport using the framework of turbulent eddy diffusion, which is defined next.

By analogy between turbulent transport and molecular diffusion, the corresponding turbulent flux *c* can be written as a linear function of the large‐scale tracer gradient (Prandtl, 1925; Taylor, 1921; Vallis, 2017).*K* will be referred to as “*K*‐tensor” (also called “transport tensor” in literature) and the angle brackets denote the large‐scale component of a field. This flux‐gradient relation, with some common simplifications, has been traditionally used in numerical models to parameterize turbulent fluxes due to the important unresolved part of the flow. The divergence of the eddy flux enters the full tracer equation in these models, along with advection by the large‐scale flow

Equation 2 can also be written for the large‐scale tracer tendency *c*.

Because of the joint effect of planetary rotation and ocean stratification, the stirring of water by mesoscale eddies is primarily along neutral density surfaces (Iselin, 1939; Mcdougall, 1987; McDougall et al., 2014), which can be approximated by isopycnal (constant density) surfaces in the interior ocean. This is particularly convenient in this study, which uses isopycnal numerical models. Therefore, the focus here is on the lateral material transport. The general *K*‐tensor in a two‐dimensional (2D) flow can be written as a 2 × 2 matrix

The seeming simplicity of the flux‐gradient relation (Equation 1) hides the incredible complexity of the *K*‐tensor. Only in purely homogeneous, stationary and isotropic turbulence are the off‐diagonal tensor zero (*K*_{xy}* = K*_{yx} = 0) and the diagonal tensor elements are equal to each other (*K*_{xx}* = K*_{yy}). In realistic oceanic flows, all *K*‐tensor elements are generally nonzero, distinct (i.e., the eddy‐induced mixing is anisotropic) and vary in space and time (i.e., the mixing is inhomogeneous and nonstationary). Observation‐ and model‐based estimates of the simplified eddy diffusivity exhibit strong dependence on depth, geographical location (Abernathey & Marshall, 2013; Cole et al., 2015; Canuto et al., 2019; Griesel et al., 2010; Groeskamp et al., 2020; Lumpkin et al., 2002; Marshall et al., 2006), and time (Busecke & Abernathey, 2019; Haigh et al., 2020). These estimates usually involve some spatio‐temporal averaging and can be based on either drifter (“particle”) trajectories or tracer distributions. Both particle‐based statistics (Griesel et al., 2010; Kamenkovich et al., 2009, 2015; McClean et al., 2002; O'Dwyer et al., 2000; Rypina et al., 2012; Sallee et al., 2008) and tracer‐based estimates (Abernathey et al., 2013; Bachman et al., 2017, 2020; Eden, 2007; Haigh et al., 2020) also exhibit significant anisotropy. This anisotropy is important in the typical oceanic case of strong eddies embedded in relatively weak large‐scale circulation (Kamenkovich et al., 2015).

The diffusion approach (Equation 1) is built on an inherent assumption that the *K***‐**tensor is unique for any given turbulent flow. However, some model estimates report significant sensitivity of a simplified *K*‐tensor to the tracer field (Abernathey et al., 2013; Bachman et al., 2020, 2015; Eden & Greatbatch, 2009; Haigh et al., 2020). This sensitivity complicates interpretation of the *K*‐tensor because even the exact solution of Equation 1 for one particular pair of tracers will lead to biases in *F* for another set.

The other serious complication is that *F* contains some large nondivergent (“rotational”) component (Haigh et al., 2020; Jayne & Marotzke, 2002; Marshall & Shutts, 1981) that does not affect tracer distribution, because only divergence of the eddy fluxes matters, but influences *K* in the flux‐gradient relation (Equation 1). The rotational flux can be tracer‐dependent (Bachman et al., 2015) and can lead to negative diffusivities (Marshall & Shutts, 1981). The separation of *F* into rotational and divergent components via the Helmholtz decomposition is, unfortunately, not unique and depends on the boundary conditions (Fox‐Kemper et al., 2003; Jayne & Marotzke, 2002; Maddison et al., 2015; Roberts & Marshall, 2000), which are usually known for the total *F* but not for its rotational and divergent components, separately. The need to remove the rotational component thus leads to another source of ambiguity in estimating the *K*‐tensor.

This study describes the complexity of the eddy‐induced transports, using the *K*‐tensor framework in its entire spatio‐temporal complexity, assuming neither temporal nor zonal averaging. The analysis explores tensor’s dependence on tracer, importance of all its components, opposite signs of its diffusivities, and significant spatial inhomogeneity and temporal variability that cannot be removed by commonly applied spatio‐temporal averaging. This complexity strongly suggests the need to expand the traditional flux‐gradient relation of Equation 1 to include new functional and/or stochastic terms.

Two types of simulations are used in this study to guarantee the robustness of conclusions. The first type is the idealized quasi geostrophic (QG) double‐gyre flow. This flow contains all the essential elements of the mid‐latitude North Atlantic or North Pacific: large‐scale subpolar and subtropical gyres, separated by a coherent meandering jet, representing eastward extensions of the Gulf Stream and Kuroshio currents, and an ambient eddy field. The model is formulated in a square‐box, flat‐bottom ocean basin, which is a classical idealization that facilitates the analysis and numerical simulations (Haigh et al., 2020). The numerics employ the CABARET scheme (Karabasov et al., 2009) on a uniform Cartesian grid with 1025 by 1025 grid points and the grid spacing

The second model is a comprehensive, general circulation model (GCM) of the entire Atlantic, used in the “offline” regime, which means that tracers are simulated using previously computed daily physical fields, thus, making the model computationally very efficient (Kamenkovich et al., 2017). The physical variables used in offline models are calculated in a separate “online” simulation with the hybrid coordinate ocean model (HYCOM) (Bleck, 2002; Chassignet et al., 2003), which uses isopycnal coordinates in the open ocean and below the mixed layer. HYCOM's coordinate system dynamically transitions to other coordinate types (sigma‐ and *z*‐coordinates) to provide optimal resolution in the surface‐mixed layer, in high‐latitude unstratified regions, and near coasts. The online simulation has a global domain with 1/12° spatial resolution; the horizontal grid is rectilinear south of 47°N followed by an Arctic bipolar patch. The vertical grid has 41 layers.

Both model solutions are initialized with 2D tracer configurations which initially are vertically uniform but have different horizontal profiles (see supporting information). The QG model is integrated for 180 days, while the GCM is used for several overlapping segments, 110 days each.

The definition of the large‐scale circulation and large‐scale tracer field is not unique, and the resulting *K*‐tensor depends significantly on it. The mesoscale is not clearly separated from the large‐scale in ocean models and observations (McWilliams, 2008), and an unambiguous definition of the eddies is missing. The large scales are often defined as long‐term time mean (Vallis, 2017), although the utility of this definition is far from clear for transient tracers. Thus, a fundamental uncertainty in defining the eddies leads to uncertainty in defining the eddy diffusivity. This study defines mesoscale using spatial filtering, which is relevant to the task of spatial resolution of eddies in numerical models (Stanley et al., 2020). For example, the QG analysis in this study employs the low‐pass spatial filtering <…> intended to remove scales shorter than 112.5 km (Rossby deformation radius is 40 km), while the GCM analysis uses a square filter width of approximately 2° longitude. Large‐scale GCM velocities are also averaged over 5 years for the most efficient separation of the large‐scale circulation (Kamenkovich et al., 2017).

The flux‐gradient relation can be solved exactly for any pair of independent tracers. In the QG simulations, we use 6 tracers that are initially linear (constant gradient) and 6 nonlinear tracers (15 independent pairs in each set). The linear tracers are of the form

The rotational component is removed from each tracer flux, using the Helmholtz decomposition (Lau & Wallace, 1979):

In the above equations, *K*‐tensor is derived from

The *K*‐tensor can be decomposed into the symmetric and antisymmetric components with distinct physical interpretations (Griffies, 1998; Plumb & Mahlman, 1987):

The symmetric *θ* (Rypina et al., 2012; Kamenkovich et al., 2015).

The angle *θ* defines the direction of the maximal tracer diffusivity, and the first eigenvalue

When the eddy‐induced stirring (*K*‐tensor) is isotropic and homogeneous, these two components of the full tensor correspond to the divergent (zero curl and nonzero divergence) and rotational (zero divergence and nonzero curl) components, ^{−9} s^{−1} (tracer is unitless), and the curl of these components is 6.5 × 10^{3} s^{−1} and 6.7 × 10^{3} s^{−1}, respectively. Because the rotational component is exactly zero in the full

An intriguing new feature of the comprehensive *θ*) and squeezed in the direction normal to that (direction of negative diffusivity), leading to transient filamentation of the tracer field. Moreover, the polarity, which is ubiquitous in both QG and GCM solutions, is a robust feature of the instantaneous flow and is observed regardless of whether and how the rotational component of *F* is removed.

All components of the comprehensive tensor have significant time dependence, with the standard deviations comparable with and exceeding the corresponding time‐mean values (Figure 3). The uncovered time dependence has important implications not only for transient tracer behavior, but also for time‐mean tracer structure. The latter point can be illustrated by the time‐average eddy flux *K*‐tensor defined from the time‐mean eddy fluxes and tracer gradients. The above relation implies that (i)

Due to the nonstationary nature of *θ* both change in time. Although the polarity is reduced in *K*‐tensor and the sign of the along‐flow diffusivity. Another possibility is that negative eigenvalues are associated with nondivergent, rotational component of

Another unexpected property of the diffusion tensor

The rotational component can be naturally suspected of being the cause of the above nonuniqueness of the *K*‐tensor. Nevertheless, our results demonstrate that the nonuniqueness (as measured by *F* and *F*. (Figures 4a and 4b). We conclude that the presence of the rotational component cannot be the main cause of nonuniqueness.

Using the exact solution for *K*‐tensor implies that the parameterized eddy flux divergence *K*‐tensor estimates, according to our analysis. Since an exact match between *K* is practically impossible, it is important to estimate what properties of the *K*‐tensor are most important for tracer distribution. This study describes several examples of such properties.

The *K*‐tensor depends on the flow decomposition (definition of the large‐scale <…>), which is loosely defined in most cases. This study defines mesoscale based on spatial scales, which is more directly relevant to the issue of its parameterization in numerical models. The spatial filter characteristics cannot, however, be easily derived from model resolution alone, since it is unclear to what extent different dynamical scales are actually resolved. The *K*‐tensor is also nonstationary, and a meaningful definition of *K*‐tensor polarity reflects the actual properties of eddy fluxes, negative diffusivity in numerical simulation should be implemented with caution, in order to avoid singularities. Observation‐based estimates, on the other hand, present additional challenges. Given the discovered complexity, obtaining accurate estimates of *K* from drifter and float trajectories (Lagrangian observations) appears highly problematic, because these asymptotic and spatially nonlocal methods will not be able to accurately capture the spatial and temporal variability of the *K*‐tensor, as well as its negative eigenvalues and its advective component.

Finally, the comprehensive *K*‐tensor is a function of the tracer field, formally violating assumptions of the classical, tracer‐independent flux‐gradient relation. A practical approach to this problem is to use multitracer ensemble‐averaged estimates of

Negative diffusivities and tracer dependence clearly illustrate the fact that the stirring driven by mesoscale currents is dramatically more complex than the molecular diffusion, which is used to motivate the flux‐gradient representation. An alternative solution is to expand the traditional flux‐gradient representation of the eddy flux by adding new, nondiffusive and/or nongradient terms. Such expansion can involve terms that explicitly depend on either the tracer concentration or its curvature, as well as purely stochastic components. Another alternative is to represent *K*‐tensor as a stochastic process, as it has been done in the past for other transport parameters (Berloff & McWilliams, 2003; Grooms, 2016). Finally, approximating eddy flux divergence, instead of eddy fluxes themselves, will help to avoid ambiguity associated with the presence of the rotational component.

Since the most important properties and aspects of *K*‐tensor suggest that the flux‐gradient relationship is not suitable for representing eddy‐induced fluxes in terms of large‐scale properties and alternative types of parameterizations may be needed.

I. Kamenkovich and Y. Lu were supported by NOAA grant NA16OAR4310165 and NSF grant 1849990. The authors declare no conflicts or competing interests. P. Berloff gratefully acknowledges funding by NERC, UK Grants *NE*∕*R*011567∕1 and *NE*∕*T*002220∕1, and by Leverhulme Trust, UK Grant *RPG*−2019−024, and by the Moscow Center for Fundamental and Applied Mathematics (supported by the Agreement 075‐15−2019−1624 with the Ministry of Education and Science of the Russian Federation). I. Kamenkovich would like to thank Zulema Garraffo for the continuous help with the offline HYCOM model; P. Berloff, M. Haigh, and L. Sun would like to thank James McWilliams for fruitful discussions on the topics of this study.

Model data used to produce figures in this study are available from