1. Introduction
Mixing in the ocean occurs primarily along isopycnals because of turbulent large-scale stirring, for example, by baroclinic eddies at the mesoscale, and secondarily across isopycnals, for example, because of smaller-scale phenomena including breaking internal waves, boundary mixing, developed baroclinic instability, and double-diffusive convection (McDougall 1984). Surface ventilation at outcropped isopycnal layers provides an opportunity for the transport of heat, freshwater, and biogeochemical constituents across the atmosphere–ocean interface and into the deep ocean. Isopycnal mixing, quantified by diffusivity, modulates the rate at which these constituents are subducted into the deep ocean (Siegenthaler 1983; Dutay et al. 2002; Gnanadesikan et al. 2004). Quantifying the magnitude and structure of isopycnal diffusivity is fundamental to understanding the role of mesoscale eddies in setting the global ocean climate.
Two approaches are presently available for quantifying isopycnal diffusivity: a tracer-based Eulerian approach and a particle-based Lagrangian approach. Each approach comes with its own strengths and weaknesses. Estimation of isopycnal diffusivity within an Eulerian framework, particularly for models with nonisopycnal vertical coordinates, is still maturing. The two general tracer approaches include single tracer-based approaches accounting for mixing across bounded tracer regions (Nakamura 2001) and eddy flux gradient–based approaches with multiple tracers via an overdetermined matrix system (Bachman and Fox-Kemper 2013; Bachman et al. 2015). Analysis can be challenging because results are dependent on numerically diffusive tracer advection schemes and may include nondivergent eddy fluxes (Marshall and Shutts 1981; Plumb and Mahlman 1987; Lee et al. 1997; Bratseth 1998; Marshall et al. 2006; Ferrari and Nikurashin 2010; Klocker et al. 2012b; Bachman and Fox-Kemper 2013; Abernathey et al. 2013). The particle-based approach is more mature, dating back to Taylor (1921), and offers a perspective of the ocean flow not readily available from Eulerian-based dynamical cores (Davis 1987; Bennett 1987; Veneziani et al. 2005; LaCasce and Ohlmann 2003; LaCasce 2008; Keating et al. 2011; Abernathey et al. 2013; Griesel et al. 2014; Zhurbas et al. 2014). Since the particle-based approach is not intimately tied to any of the dynamical core prognostic equations, care must be taken in interpreting the results. Consistency between the Nakamura (2001) tracer- and particle-based estimates of diffusivity strongly suggests both tracer- and particle-based approaches are viable routes toward quantifying isopycnal mixing in the global ocean system (Klocker et al. 2012b; Abernathey et al. 2013).
We quantify isopycnal diffusivity using a Lagrangian framework where mixing is estimated directly from particle statistics tagging fluid motion. Isopycnal diffusivity can be readily computed provided particle motion is constrained to isopycnal surfaces. The connection between Eulerian diffusivity and Lagrangian particle statistics was provided by Taylor (1921, 1935), resulting in the classic observation that diffusivity is proportional to the time rate of change of particle dispersion if the particle motion is decorrelated and the turbulent motion is homogeneous and isotropic. Davis (1987) generalized Taylor diffusivity to inhomogeneous flows and developed the elaborated flux versus gradient law for particles, indicating wider utility of particle statistics. These results hinge on a key caveat. The underlying assumption of isotropic turbulence is violated for oceanic flows because the velocity probability density function is not strictly Gaussian (Bracco et al. 2000). However, the consistency between tracer- and particle-based estimates of diffusivity indicates that this is not a fatal flaw of the particle-based approach (Klocker et al. 2012b; Abernathey et al. 2013).
Single-particle statistics produce absolute diffusivities biased by the large-scale background mean flow. A better alternative, double-particle statistics, computes diffusivity from the relative dispersion between particles, mitigating the mean flow bias (Batchelor 1952; Bennett 1987). Particle statistics have enjoyed great applicability in a variety of oceanic contexts and an excellent review of single- and double- particle statistics, including derivations and applications, is presented by LaCasce (2008). One’s choice of method depends on the desired application because single- and double-particle methods have been shown to yield consistent diffusivity estimates, provided there are enough samples and asymptotic diffusivities are obtained (Klocker et al. 2012b; Abernathey et al. 2013).
Cluster statistics, a generalization of double-particle statistics, are best for our application because they readily allow computation of the diffusivity tensor, help isolate the effect of background advection that can contaminate single-particle statistics, and better incorporate Lagrangian particle information to enhance the accuracy of computed diffusivity. Fewer studies have used particle clusters relative to single- and double-particle statistics and typically clusters are used to compute divergence or vorticity (LaCasce 2008). However, an improved method to compute relative diffusivity using clusters is provided by Aris (1956) via the use of statistical moments (covariances), yielding a convenient way to estimate the diffusivity tensors for scalars (Fischer et al. 1979; Kamenkovich et al. 2009; Holleman et al. 2013; Holleman and Stacey 2013; Wolfram and Fringer 2013). The mathematical connection between double-particle statistics and cluster covariances is demonstrated in the LaCasce (2008) review. This underused cluster approach has the benefit of being particularly robust in computing the diffusivity arising from mesoscale baroclinic eddies, as typified in an idealized, wind-forced, double-gyre circulation to be explored below.
Isopycnal diffusivity occurs primarily because of straining by mesoscale eddies initiated from baroclinic instability. The dominant length scale controlling eddy size is proportional to the Rossby radius of deformation (RRD) corresponding to the first baroclinic mode, approximated by 
The discovery of important spatial and temporal scales driving mixing is vital to modeling efforts. Knowledge of dominant scales associated with mesoscale eddies is useful to better understand the resolution requirements for future Coupled Model Intercomparison Projects, for example, CMIP6, particularly at weakly eddying resolutions. Models with insufficient resolution to resolve these dynamically important scales will correspondingly fail to properly reproduce mesoscale mixing. Determination of relevant regimes where simplified scaling arguments exist can suggest local resolution requirements and improvements for mesoscale eddy closure schemes. Our diagnosis of resolution dependence on mesoscale eddy diffusivity can additionally be used to evaluate the cost of underresolving mesoscale eddies by suggesting the amount of mesoscale mixing that is neglected by the coarse resolution used in global climate simulations.
The rest of the paper is organized as follows: Section 2 describes the Simulating Ocean Mesoscale Activity (SOMA) configuration used to represent an idealized, eddying, midlatitude, double-gyre system. An implementation outline for the Lagrangian in Situ Global High-Performance Particle Tracking (LIGHT) analysis member we added to the Model for Prediction across Scales Ocean (MPAS-O) model is presented in section 2c. Analysis techniques related to computing diffusivity from particle trajectories and Lagrangian scaling are discussed in section 3 and illustrate the use of Lagrangian analysis techniques to better understand the role of resolution, sub-RRD velocity scales, and the suitability of scaling estimates for isopycnal diffusivity. Results from the application of these methods to the SOMA simulation are presented in section 4. A discussion of diffusivity characteristics, dominant mixing length scales, and mixing model scaling follows in section 5, with concluding remarks in section 6.
2. The Simulating Ocean Mesoscale Activity experimental design
a. The Simulating Ocean Mesoscale Activity configuration
The SOMA configuration is designed to investigate equilibrium mesoscale activity in a setting similar to how ocean climate models are deployed. It simulates an eddying, midlatitude ocean basin with latitudes ranging from 21.5° to 48.5°N and longitudes ranging from 16.5°W to 16.5°E. In contrast to previous idealized double-gyre studies (Holland and Lin 1975; Figueroa and Olson 1994; Poje and Haller 1999; Berloff et al. 2002; Straub and Nadiga 2014), this basin is circular instead of rectangular and features more realistic curved coastlines with a 150-km-wide, 100-m-deep continental shelf. The simulation setup is detailed in appendix A.
The basin is forced by an asymmetric zonal wind stress driving a double-gyre circulation with a strengthened subtropical gyre with respect to a subpolar gyre. The gyres form northward- and southward-traveling western boundary currents that coalesce and separate at midlatitude to form an unsteady, baroclinic jet. Momentum and mass transport within the jet enhance local mixing because of the interaction of large and small baroclinic eddies. The vast majority of the available potential energy used to grow baroclinic eddies is generated from upwelling and downwelling along the western boundary. We expect the maximum baroclinic eddy size to be related to the RRD associated with the first baroclinic mode, which is approximately 30 km with a corresponding phase speed of 2.5 m s−1.
b. Model configuration
Simulations are performed using the unstructured MPAS-O (Ringler et al. 2013). MPAS-O is a global, multiscale ocean model that simulates spatial and temporal scales ranging from coastal dynamics to basinwide circulations. Grids are based on variable-resolution spherical Voronoi tessellations (Ju et al. 2011) generated using a scalable mesh-generation algorithm (Jacobsen et al. 2013). The grids used here are primarily composed of hexagonal cells. Mean grid spacings of 32, 16, 8, and 4 km are used, which correspond to approximately 1, ½, ¼, and ⅛ times the RRD of the first baroclinic mode. We use 40 vertical z-star levels with 60% of the levels residing in the upper 500 m. Recent analysis of the z-star vertical coordinate reveals that its implementation in MPAS-O is minimally diffusive (Petersen et al. 2015). Additional model configuration details are provided in appendix B.
The suite of simulations is composed of 30 ensemble members for each of the four different grid resolutions and velocity-filtered cases. A single 10-yr baseline run is performed for each experiment. During the course of each simulation, restart files are created at 2-month intervals beginning on 1 January of year 5. Each restart serves as an initial condition for an ensemble member, which is then integrated for 30 days. As discussed immediately below in section 2c, each ensemble member is considered an independent realization from which we estimate diffusivity based on particle dispersion calculated from particle positions archived every 2 days.
c. Lagrangian, in Situ, Global, High-Performance Particle Tracking
In the past, particle tracking has been performed with offline solvers because of inexpensive storage and the relative difficultly of performing online parallelized particle tracking. However, as supercomputing architectures evolve, high-performance in situ methods will be necessary for particle-tracking techniques to be viable in terms of accuracy and performance because the relative cost of memory and storage continues to grow more rapidly than the cost of computation. We overcome these issues by implementing LIGHT in MPAS-O’s core analysis diagnostics.
LIGHT is an online, extensible, particle-tracking analysis diagnostic that uses MPAS-O’s native Eulerian velocities and state information to compute Lagrangian trajectories and properties associated with each particle’s path along these trajectories. Computations are performed with the in situ Eulerian dynamical core (dycore) variables. LIGHT’s computational cost is minimal because it uses the same high-performance computing communication pattern as the host dynamical core. For example, particles are advected on subdomains corresponding to the Eulerian domain decomposition. Particle lists are formed for particles advected off a given processor’s domain. Processor to processor communication of particle lists is used to facilitate this data transfer for efficiency. Particle properties are readily extensible via modification of an xml file, allowing code flexibility and adaptation.
LIGHT is executed simultaneously with the ocean dynamics and uses the same grid decomposition as the MPAS-O dynamic core. Particles “live” on the same processor as the cell they currently reside in and share the same time step as the native ocean core, thereby preventing spurious errors due to insufficient time resolution (Qin et al. 2014). The communication pattern to integrate (1) is similar to that used by the MPAS-O dynamic core partial differential equation solver; that is, we pass particle information and “ownership” between processors as particles are advected from one processor’s domain into another.
Particles cannot leave the domain for Courant–Friedrichs–Levey advection conditions less than or equal to unity because 1) no-slip boundary conditions are enforced when the Eulerian velocity is interpolated to cell vertices, and 2) Wachspress interpolation is bounds preserving. When target isopycnal surfaces do not exist because of outcropping or intersection with bathymetry, particles are placed at the center of the top or bottom z-star layer, respectively.
For each ensemble member, we estimate isopycnal diffusivity along five potential density surfaces ρ, shown in Table 1, corresponding to approximate depths of 100, 250, 400, 600, and 1000 m. Each ensemble member is “seeded” with 303 665 Lagrangian particles divided equally between the five density surfaces and spread uniformly across each surface, resulting in particles spaced at approximately 16 km or one-half the principal RRD. In total, the results presented below are derived from 55 million particle trajectories over 30 realizations of six resolution and filter cases, each extending for a duration of 30 days.
Mean and standard deviations of depth for layer-averaged potential density surfaces at 4-km resolution.
3. Analysis methods
a. Computing isopycnal diffusivity using Lagrangian particle clusters
Let a particle be described by a set of positions x along an individual fluid path corresponding to the motion computed from (1) and an initial condition specified by the particle’s spatial location a at time ts, that is, x(a, ts | t) (Bennett 2006). The set of realizations used in ensembles corresponds to sets sharing the same initial locations in space at the start of each realization over different times, that is, for a single particle 
Identical seeding of particles and their associated clusters are used with Ne = 30 different initial conditions to produce an ensemble of diffusivity estimates. Particles are grouped into clusters corresponding to the set of particles that are within a radius of 100 km of locations spaced throughout the domain. Clusters typically consist of Nc ≥ O(100) particles, which is comparable to the number of particles used in other studies (Koszalka and LaCasce 2010; Koszalka et al. 2011) at approximately the 100-km scale (Poulain 2001; Sætre 1999; Jakobsen et al. 2003; Koszalka et al. 2011). This choice balances competing goals of spatial fidelity and statistical robustness, physically ensuring that the quantified mixing is of appropriate scale to the mesoscale eddies that are set by the 30-km RRD scale within the turbulent jet. We now describe cluster statistics, which are used to compute the isopycnal diffusivity.
1) Cluster moments
Statistics corresponding to each cluster can be used to compute bulk fluid properties, including diffusivity (Aris 1956). In the following particle statistics equations, let p denote an index over the Nc particles in a realization cluster such that 
2) Cluster diffusivity
Equations (6) and (7) are applicable provided particle motions are decorrelated, that is, for t > Td. Additionally, t must be small enough that particle motions are not adversely affected by the boundary (Artale et al. 1997) or by flow and particle statistics inhomogeneity that impact spatial fidelity of the computed diffusivity. In some cases, these conditions may be incompatible, rendering computation of classical diffusivity intractable.
We discretize time derivatives in (6) and (7) with a discrete definition of the derivative starting at τs over a 2-day interval. Diffusivities computed with (6) and (7) are reasonable, provided results are not strongly sensitive to τs and short times thereafter. In principal, τs should strictly occur after the decorrelation time Td. However, in real world systems, Td may be difficult to ascertain because of inhomogeneity. The selection of τs is then a balance of establishing that the diffusivity magnitude and shape are convergent in time and also ensuring that τs is not so large that it oversmooths the underlying diffusivity. Assuming τs is appropriate, diffusivity is computed as a function of cluster location, cluster size, and realization.
3) Ensemble diffusivity for clusters
We can gauge statistical confidence in our estimate of diffusivity because we have 30 realizations for every experiment, where each realization estimates the spatial structure of κC across each target density surface. The estimated standard deviation over 30 turbulent realization members is used in conjunction with the Student’s t test to compute the mean confidence interval error 
b. Computing the decorrelation and integral time scales
Integral time scales obtained with both methods are comparable. We present results using the direct cluster method because they only depend on the Lagrangian particle positions, do not require a complicated nonlinear fit, can be used for direct comparisons with computed cluster diffusivities, and are largely consistent with the analytical-fit method. However, they overestimate the integral time scale in the near-surface layer within the western boundary current near the separation of the subpolar and subtropical gyres relative to the analytical-fit method.
The integral time scale is not a direct proxy for the decorrelation time. However, it indicates the appropriate order of magnitude time scale for decorrelation. The assumed decorrelation time τs is assessed by compromising between minimizing the autocorrelation and preserving spatial fidelity of the diffusivity calculation because computed cluster diffusivities are themselves Lagrangian and therefore are advected throughout the fluid with increasing time, reducing spatial fidelity of the diffusivity estimate. This assessment resulted in the inferred practical basin time scale of τs = 10 days used in the diffusivity calculation that is compatible with estimates of the eddy decorrelation time scale in the global ocean of 4 to 8 days (Klocker and Abernathey 2014). Furthermore, this result is not too surprising because the integral time scale and mixing lengths are reduced when the autocorrelation is oscillatory (Klocker et al. 2012a; Klocker and Abernathey 2014), as is the case for the SOMA flow.
c. Diffusivity and characteristic scales of mixing
All quantities in (18) must be approximated. In principal, for isotropic turbulence, α ≡ 1. Although there are errors in the computation of κ, u′, and TL, we expect α to be an O(1) constant assuming mixing is describable in terms of the diffusivity theory that is adapted from isotropic turbulence. Thus, we can assess the error of the calculations and suitability of the applied theory based on the range of values for α estimated from (18). The applicability of computed mixing quantities is inversely proportional to the complexity of spatial structure in α. Additionally, α also indicates the relative agreement between the mixing model that arises from single-particle statistics and the computed diffusivity that is derived directly from double-particle statistics via covariances. The term α is a mixing efficiency.
d. Resolution and sub-Rossby radius of deformation velocity scale dependence
Results for diffusivity will vary with resolution because the baroclinic eddies are resolved to varying degrees. Consequently, we evaluate diffusivity at 4-, 8-, 16-, and 32-km scales, with varying resolution of the largest RRD of approximately 30 km. This approach evaluates the diffusivity dependence on the range of eddy scales simulated by the model and demonstrates whether the computed spatial diffusivity is well resolved. However, this approach is unable to ascertain the contributions made to mixing by spatial velocity scales finer than RRD.
To carry out this assessment, we apply low-pass filters to the velocity field in order to remove sub-RRD scales of motion. Low-pass spatial velocity filtering is used to compare the relative effect of sub-RRD scale motion on the computed isopycnal diffusivity (Grooms et al. 2013). In this study, filtering is performed at the 2Δx and 8Δx levels with the notation 4 → 32′ km indicating filtering of the 4-km velocity field with an 8Δx filter to produce a velocity field with an effective resolution of 32′ km. These filtered velocities are used to advect the particles in the absence of spatially fine velocity scales. Derivations of the unstructured, generalized filters used are provided in appendix C. Comparison of isopycnal diffusivity for filtered and unfiltered cases at similar effective resolutions are used to assess the contribution of sub-RRD velocity scales to isopycnal diffusivity.
4. Results
a. Hydrodynamics of the Simulating Mesoscale Ocean Activity simulations
The ensemble-mean paths of particles are designated in Fig. 1 by black lines terminating in arrows corresponding to mean particle motion on the potential density surface ρ = 1026.9 kg m−3 for the 32-km grid. The mean paths originate at the “seed” location of each cluster 
Particle ensemble-mean paths and covariance on the potential density surface ρ = 1026.9 kg m−3 for the 32-km grid over 60 days with 2-day interval.
Citation: Journal of Physical Oceanography 45, 8; 10.1175/JPO-D-14-0260.1
Shaded regions surrounding the mean paths in Fig. 1 designate the spread of particles with respect to the ensemble mean and are the union of covariance ellipses. Covariance ellipses are computed to enclose a volume of 0.118 underneath the covariance distribution at a particular location in time. At any location along the mean particle path, the width of the shading indicates the relative spatial extent of particles in the ensemble. The rate at which the width of the shading increases when moving along the mean particle path indicates the rate of particle dispersion, which is the diffusivity. Strong diffusivity occurs when there is rapid growth in the shaded region along the mean particle path. The diffusivity is greatest in the jet interior between 15° and 5°W at a latitude of 32.5°N and decreases as the jet extends to the east.
Mixing characteristics are quantified by first examining integral time scales and the characteristic eddy speed, followed by scalar diffusivity dependence on spatial location, resolution, buoyancy surface, and low-pass filtering of sub-RRD velocity scales. Spatial dependence of the anisotropic tensor diffusivity is explored, followed by a discussion of dominant physical mixing length scales and mixing length scaling.
b. Dominant scales
Mixing is describable in terms of a time scale over which mixing occurs, that is, the Lagrangian time scale TL, a characteristic eddy speed u′, and a Lagrangian length scale LL. These are the simplest component scales of mixing utilized in a simple mixing model such as κ = αu′2TL.
The integral time scale is on the order of 10 days as shown in Fig. 2, which details the horizontal and vertical structure of TL at a resolution of 4 km. The fastest time scale occurs in the asymmetric turbulent jet with a characteristic time of less than 10 days. The Lagrangian time scale mildly increases with depth corresponding to the attenuation of the baroclinic eddies. The computed autocorrelations in the western boundary current near the separation region use the direct cluster method and are unreasonable, particularly compared to analytical-fit results (not shown) that have integral time scales of less than 5 days. Therefore, estimates for TL and derived α on the shelf are suspect. However, the dynamics of the shelf region is not driven by eddies arising from baroclinic instability. Consequently, more accurate estimation of shelf diffusivities is not pursued.
Lagrangian time scale TL on the 4-km grid for buoyancy surfaces of (a) ρ = 1025.6 kg m−3, (b) ρ = 1027.4 kg m−3, (c) ρ = 1027.7 kg m−3, and (d) ρ = 1028.1 kg m−3. White contours designate TL values of 5, 10, 15, and 20 days.
Citation: Journal of Physical Oceanography 45, 8; 10.1175/JPO-D-14-0260.1
Based on the large-scale structures in Fig. 2, we infer that a reasonable estimate of the characteristic decorrelation time scale for the basin is 10 days. Use of this time scale as τs provides a reasonable balance between establishing that mixing is likely within the diffusive regime for mesoscale eddies while ensuring that the computed result is not artificially smoothed over longer time scales where the cluster can sample multiple disparate mixing regimes within the basin.
The characteristic eddy speed u′ (Fig. 3) is derived from particle positions via the root-mean-square of (14) and approaches a maximum scale of over 0.75 m s−1 in the region of greatest turbulent activity within the jet interior of the surface layer near 32.5°N, 10°W. Strong attenuation occurs with depth for a reduction to approximately 0.25 m s−1 for the potential density ρ = 1028.1 kg m−3. Characteristic eddy velocities in excess of 0.50 m s−1 extend to a depth of approximately 300 m. These results imply that the diffusivity should be at a maximum within the jet and subsequently be attenuated with depth.
Characteristic eddy speed u′ (m s−1) on the 4-km grid for buoyancy surfaces of (a) ρ = 1025.6 kg m−3, (b) ρ = 1026.9 kg m−3, (c) ρ = 1027.4 kg m−3, and (d) ρ = 1028.1 kg m−3. White contours designate u′ values of 0.25, 0.50, and 0.75 (m s−1).
Citation: Journal of Physical Oceanography 45, 8; 10.1175/JPO-D-14-0260.1
When combined, the Lagrangian time scale TL and characteristic eddy speed u′ yield a Lagrangian length scale LL = u′TL (not shown). The length scale is O(10–100) km and is greatest in the region of the turbulent jet. This scaling supports our choice of forming clusters from particles within a 100-km radius to ensure that the computed diffusivity is responsive to these length scales. Furthermore, this choice is consistent with bin sizes of (2° × 1°) used in the studies of the Nordic Seas (Poulain 2001; Sætre 1999; Jakobsen et al. 2003; Koszalka et al. 2011), which is of a similar spatial extent as SOMA with smaller RRD.
c. Scalar diffusivity
1) Resolution dependence
Scalar diffusivity is shown in Fig. 4 for the 32-, 16-, 8-, and 4-km simulations (Figs. 4a,b,c,d) along the 1026.9 kg m−3 density surface. All simulations show that scalar diffusivity is largest within the western boundary currents and meandering jet regions. Note, however, that Berloff et al. (2002) also found that particles tended to stay within their respective gyres, with strong intergyre mixing but weak cross-jet mixing. Within the western boundary currents and the meandering jet regions, the 4- and 8-km simulations are, for practical purposes, identical. Even the 16-km simulation shows a strong quantitative resemblance to the 4-km simulations within these regions. Only when the resolution is decreased to 32 km does the spatial structure of scalar diffusivity begin to change substantially in the western half of the domain. Both the magnitude and spatial extent of the jet, in terms of its width spanning approximately 27.5° to 37.5°N as well as the extent of enhanced mixing along the western shelf, are substantially enhanced at 4-km resolution relative to the coarsest resolution of 32 km in Fig. 4a. Diffusivity’s strongest dependence on resolution is found in the eastern half of the domain with κ decreasing steadily from approximately 104 m2 s−1 in the 4-km simulation to less than 103 m2 s−1 in the 32-km simulation. The east and west exhibit different diffusivity sensitivity to resolution, suggesting that different dynamics may mediate mixing in the east and west. This topic is revisited in the discussion.
Scalar diffusivity magnitude log10κC (m2 s−1) computed using Lagrangian clusters for the ρ = 1026.9 kg m−3 buoyancy surface at resolutions of (a) 32, (b) 16, (c) 8, and (d) 4 km. White contours designate the order of magnitude changes in diffusivity corresponding to integer log10κC.
Citation: Journal of Physical Oceanography 45, 8; 10.1175/JPO-D-14-0260.1
2) Vertical structure
Scalar diffusivity decreases with increasing potential density, as shown in Fig. 5 for the 4-km simulation. Initially, the 104 m2 s−1 contour extends through nearly the entire domain on the 1025.6 kg m−3 potential density surface in Fig. 5a. However, the spatial extent of the 104 m2 s−1 contour contracts steadily to approximately 2.5°E, 0°, and 2.5°W as density increases to 1027.4, 1027.7, and 1028.1 kg m−3, respectively. Its meridional width ranges from approximately 10°, 7.5°, and 5° over the same vertical extents. In the eastern half of the domain, the dependency of scalar diffusivity on vertical position is even more abrupt. In the east (near 32.5°N, 10°E), scalar diffusivity decreases from approximately 1 × 104 to 5× 103 m2 s−1, a factor of 2, over a vertical distance of approximately 300 m. In contrast, in the west (near 32.5°N, 7.5°W), diffusivity decreases from 3 × 104 to 2 × 104 m2 s−1, a factor of 1.5, over this same vertical extent. The fastest vertical attenuation of diffusivity occurs in the east, even though the magnitude of diffusivity is greatest in the west.
Scalar diffusivity magnitude log10κC (m2 s−1) computed using Lagrangian clusters on the 4-km grid for buoyancy surfaces of (a) ρ = 1025.6 kg m−3, (b) ρ = 1027.4 kg m−3, (c) ρ = 1027.7 kg m−3, and (d) ρ = 1028.1 kg m−3. White contours designate order of magnitude changes in diffusivity corresponding to integer log10κC.
Citation: Journal of Physical Oceanography 45, 8; 10.1175/JPO-D-14-0260.1
3) Dependence on sub-Rossby radius of deformation velocity scales
Diffusivity in Fig. 6 is computed on the ρ = 1026.9 kg m−3 potential buoyancy surface. Diffusivity in the top row (Figs. 6a,b) is computed using unfiltered velocity. Diffusivity in the bottom row (Fig. 6c) uses low-pass filtered velocity to a cutoff scale of 8Δx, removing small spatial scales in the velocity. The first columns (Figs. 6a,c) are simulations on the 4-km grid, and the second (Fig. 6b) is on the 32-km grid. Lack of substantial difference between the 4 (Fig. 6a) and 4 → 32′ km (Fig. 6c) cases indicates that the bulk of mixing is primarily occurring at scales larger than 32 km and suggests that results obtained on the 4-km grid are numerically well converged, at least with respect to mixing initiated by mesoscale eddies associated with the first RRD. Comparison of Figs. 6a and 6c indicate that velocities at sub-RRD scales are relatively unimportant in terms of their direct contribution to total scalar diffusivity. However, model resolution is critically important in terms of accurately simulating mixing, as demonstrated by a comparison of the 4 → 32′-km (Fig. 6c) and 32-km cases (Fig. 6b). Diffusivity on the 32-km grid is spatially underdeveloped in the baroclinic jet and is reduced in the easternmost region of the domain relative to the resolved, but filtered, 4-km case. Diffusivity in the eastern region of the domain is dependent upon resolution, as the magnitude of diffusivity is an order of magnitude smaller for the 32-km resolution relative to the 4- and 32′-km resolutions. Comparisons of the cases (i) 4 km, (ii) 4 → 32′ km, and (iii) 32 km in Fig. 6 demonstrate that the spatial extent and overall magnitude of the diffusivity is diminished with respect to the reduced resolution of the Eulerian velocity field. However, filtering the Lagrangian velocity at 8Δx scales does not qualitatively change the diffusivity. This demonstrates that motion due to eddies at scales smaller than the RRD is not responsible for the magnitude and structure of the computed diffusivity.
As in Fig. 4, but at effective filtered resolutions of (a) 4, (b) 32, and (c) 4 → 32′ km. The 32′-km simulation was performed on the 4-km grid with low-pass filtering applied to reach a cutoff of the effective filtered resolution, 32′ km.
Citation: Journal of Physical Oceanography 45, 8; 10.1175/JPO-D-14-0260.1
d. Tensor diffusivity
The scalar diffusivity, although useful to diagnose the magnitude of mixing, cannot quantify the anisotropy or dominant direction of mixing. The tensor representation in (9), in contrast, provides a more complete description of the mixing occurring in the basin. The ellipses in Fig. 7 represent the diffusivity tensor. The orientation of the ellipses denotes the principal axes of the tensor, while the lengths of the major and minor axes denote the magnitude of the diffusivity. Consistent with scalar diffusivity, the diffusivity tensor is greatest for near-surface potential density ρ = 1025.6 kg m−3 and is attenuated for increasing density surfaces and depths.
Diffusivity tensor ellipses κC (105 m2 s−1) computed using Lagrangian clusters for the 4-km grid at buoyancy surfaces of (a) ρ = 1025.6 kg m−3, (b) ρ = 1026.9 kg m−3, (c) ρ = 1027.4 kg m−3, and (d) ρ = 1028.1 kg m−3. The basin shoreline is designated by the thicker black line. The distance of the minor axis shown in the inset is equivalent to κC = 105 m2 s−1.
Citation: Journal of Physical Oceanography 45, 8; 10.1175/JPO-D-14-0260.1
Diffusivity is mostly anisotropic in the jet and boundary currents. The ratio of the major to minor axes is approximately 2 to 3 within the meandering, eastward jet and greater than 3 within the western boundary currents. The diffusivity tensor is primarily aligned with the mean flow within the jet and boundary currents and is more isotropic far away from the jet or boundary. Diffusivity is most anisotropic in boundary currents, and while important regions of anisotropic mixing are clearly present, the large portion of the interior domain is well characterized by the scalar diffusivity.
5. Discussion
a. Magnitude and spatial structure of diffusivity
Mean scalar diffusivity within each buoyancy surface is presented in Table 2 to provide a gross assessment of the eddy-driven mixing as a function of model resolution and filter scale. However, this metric fails to properly account for differences in spatial patterns of diffusivity. Hence, we use the coefficient of determination r2 shown in Table 3 to quantify deviations in shape by comparing each estimate of diffusivity to the converged 4-km simulation, which is taken to be the true diffusivity. We expect diffusivities with similar shapes to have r2 ≈ 1 because r2 measures the linearity between cases. Taken together, Tables 2 and 3 quantify the aggregate diffusivity dependence on resolution and filter scales.
Mean κC (103 m2 s−1) on each buoyancy surface. Filtered case 8′ km corresponds to use of the 2Δx filter on the 4-km grid and 32′ km to the 8Δx filter on the 4-km grid. The Prandtl case corresponds to u′2TL obtained from the 4-km grid.
Coefficient of determination r2 for diffusivities compared to 4-km simulations. The coefficient r2 for 4 km is unity because it is the base case and is perfectly correlated with itself. The disagreement between each case and the 4-km base case is 1 − r2, a measurement of the fraction variance that differs between compared cases. Filtered case 8′ km corresponds to use of the 2Δx filter on the 4-km grid, and 32′ km to the 8Δx filter on the 4-km grid. The Prandtl case corresponds to comparison with u′2TL obtained from the 4-km grid.
The mean diffusivity decreases by a factor of approximately 2 as the resolution is decreased from 4 to 32 km, with only approximately two-thirds the variance described by the 32-km case relative to the 4-km case. In contrast, there is only a 2% reduction in mean diffusivity for the 4 → 32′-km effective filtered resolution relative to the 4-km case. Furthermore, filtering from 4 to 32′ km results in a better estimate of diffusivity as compared with the 8-km case, which has a smaller mean κC by about 7%.
In general, we observe a mean reduction of approximately 0.1 in the coefficient of determination for each factor of 2 decrease in resolution. Likewise, there is a reduction in mean diffusivity to 90%, 80%, and 50% for the 8-, 16-, and 32-km cases, respectively, relative to the mean diffusivity of the 4-km case. In contrast, mean diffusivity and spatial structure are almost entirely preserved for filtered cases based at the 4-km resolution, with a mean κC reduction of 0.5% and 2% for the 2Δx and 8Δx filtered cases.
The spatial structure of diffusivity is well resolved with nearly indistinguishable spatial fidelity between the 4- and 8-km cases, as shown in Figs. 4c and 4d where the 8-km mean diffusivity is 90% of the 4-km diffusivity. Furthermore, approximately 90% of the spatial variance is represented by the 8-km resolution relative to the 4-km reference diffusivity. Diffusivity magnitude and structure are nearly indistinguishable between the 4-km and 4 → 8′-km cases, with a maximum difference in diffusivity of 0.5%. Qualitatively, the extent of the baroclinic jet shows excellent agreement with elongated regions of enhanced mixing spanning meridionally along the western shelf. The primary discrepancy between the 4- and 8-km cases is the region of underestimated diffusivity in the eastern part of the domain, a trend observable in the progression of enhanced resolution from Figs. 4a to 4d.
b. Western and eastern mixing dynamics
The diffusivity in the eastern region appears to be more sensitive to resolution than the diffusivity in the western region, indicating that relevant time and space scales may differ between the two regions. To explore these differences, we choose a representative location in the west (32.5°N, 7.5°W) and east (32.5°N, 10.0°E) for comparison. For the east and west regions, Fig. 8 shows the vertical profiles of (i) the first two eigenvectors for the horizontal velocity, (ii) Brunt–Väisälä frequency, (iii) diffusivity κC, (iv) characteristic eddy velocity scale u′, (v) inferred Lagrangian time scale TL = κC/u′2, and (vi) inferred characteristic mixing length scale Lmix = κC/u′. The striking difference between the east and west regions is the faster attenuation of diffusivity in the east relative to the west. Scalar diffusivity decreases by a factor of 2 between 150 and 350 m in the east but only by a factor of 1.5 in the west over the same vertical distance. While the Brunt–Väisälä frequency shows structure in the top 250 m in the east that is not present in the west, the first two eigenvectors are virtually identical between the two regions. The characteristic velocity scale in the west is more rapidly attenuated than in the east between depths of 150 and 350 m by 0.2 versus 0.1 m s−1. The western characteristic velocity at a depth of 150 m is 3 times larger, whereas at a depth of 1200 m it is only approximately 2 times larger.
Vertical profiles on the 4-km grid where red denotes values in the west (32.5°N, 7.5°W) and blue values in the east (32.5°N, 10.0°E) for (a) the first and second baroclinic horizontal velocity eigenvectors R1 (solid lines) and R2 (dashed lines); (b) buoyancy frequency N; (c) diffusivity κC; (d) the characteristic eddy speed u′; (e) the Lagrangian time scale TL = κ/u′2; and (f) the Lagrangian length scale LL = κC/u′. All values presented are horizontally averaged in circles with a 2.5° radius.
Citation: Journal of Physical Oceanography 45, 8; 10.1175/JPO-D-14-0260.1
The characteristic Lagrangian time scale TL and length scale LL = κC/u′ are derived directly from these measured quantities. The eastern Lagrangian time scale is nearly 3 times as large as for the west near the surface. However, at a depth of 1200 m the time scales are the same. The depth-averaged mixing length scale in the west is nearly twice the first RRD of 30 km, whereas in the east it is the same scale as the first RRD. This implies that mixing dynamics in the east may begin to depend on higher RRD modes relative to the west where mixing is at a scale larger than the first RRD, corresponding to the baroclinic jet that appears to be largely driven by the first RRD and baroclinic eigenvector. However, the eastern region is more complex and appears to require resolution of sub-RRD scales. This suggests that the submesoscale may contribute to mixing and failure to resolve these dynamics may inhibit the prediction of diffusivity at coarse scales, for example, loss of over an order of magnitude in accuracy for an eddy-resolving 4-km resolution relative to an eddy-permitting resolution of 32 km.
c. Cluster versus single-particle diffusivity and mixing model scaling
Diffusivity in baroclinic jets can be readily estimated with cluster statistics or scaling arguments if (18) is reasonably close to unity. Estimates for the integral time scale TL (Fig. 2), the characteristic eddy speed u′ (Fig. 3), and the scalar diffusivity κC (Fig. 5) subsequently yield estimates for α = κC/(u′2/TL) in (18). We use plots of log10α (Fig. 9) to highlight the applicability of (18) because the region between white contours of −0.5 and 0.5 indicate reasonable applicability for the scaling, namely, 0.32 ≤ α ≤ 3.2.
Spatial distribution of mixing model scaling coefficient log10α on 4-km grid for buoyancy surfaces of (a) ρ = 1025.6 kg m−3, (b) ρ = 1027.4 kg m−3, (c) ρ = 1027.7 kg m−3, and (d) ρ = 1028.1 kg m−3. White contours designate log10α values of −1.0, −0.5, 0.0, 0.5, and 1.0 corresponding to α of 0.10, 0.32, 1.0, 3.2, and 10.0.
Citation: Journal of Physical Oceanography 45, 8; 10.1175/JPO-D-14-0260.1
Diffusivity is largely overestimated by the u′2TL scaling with α = 1. Results for α in the western boundary current are unfortunately inconclusive because the estimate for TL is not accurate. However, α = 0.6 for large parts of the domain and ranges primarily from α = 0.4 to α = 0.8, as shown in Fig. 10. These results are consistent with Γ = 0.35 for the Klocker and Abernathey (2014) equation, Kmix = ΓurmsLmix, which describes mean flow suppression of mixing by eddies, where Kmix is the suppressed diffusivity, urms = u′, Lmix is a mixing length scale that is suppressed relative to the mixing length L for isotropic turbulence, and Γ ≤ 0.35. For equivalence with (18) note that L ≈ u′TL. Because the scalar diffusivity is the trace of the tensor, it is approximately half the unidirectional diffusivity Kmix, implying for our flow that α is actually reduced by a factor of 2. Direct estimates for αtracer from tracer studies in the global ocean, where Ktracer = αtracerurmsLtracer, suggest reasonable values are αtracer = 0.15 (Klocker and Abernathey 2014) and αtracer = 0.16 (Wunsch 1999). These values are close to our results obtained after converting κC to a unidirectional diffusivity by dividing by two in isotropic regions of the domain, for example, α ranging from 0.2 to 0.4. Our computed mixing efficiency α is reasonable within the context of prior mixing efficiency estimates. Additional analysis, left to a future study, is needed to determine the physical mechanisms responsible for SOMA’s reduced mixing efficiency.
Histogram of mixing model scaling coefficient α on 4-km grid.
Citation: Journal of Physical Oceanography 45, 8; 10.1175/JPO-D-14-0260.1
Although the value and structure of α provides insight into mixing dynamics, the overall spatial structure of diffusivity may be most easily represented by the coarse model as shown in Table 3 because 
The magnitude of diffusivity may be parameterized in terms of well-resolved (4 km) Lagrangian quantities such as the characteristic eddy speed, an integral time scale, and a mixing efficiency. However, diffusivity can be more challenging to compute with particle statistics and a mixing model instead of cluster diffusivity via covariances. This is because the cluster method does not explicitly require computation of the integral time scale and only requires that particle motions be decorrelated at the instant in time at which the rate of dispersion is calculated. The scaling method, in contrast, is fraught with difficulty because the autocorrelation oscillates, the integral time scale must be computed over long times to mitigate this effect, and the mixing efficiency must also be assessed.
6. Conclusions
We have demonstrated that Lagrangian particle tracking is a powerful analysis technique to estimate the horizontal and vertical structure of isopycnal diffusivity. A Lagrangian, in situ, Global, High-Performance Particle Tracking (LIGHT) approach provides a complementary description of dynamics provided by Eulerian ocean modeling because it faithfully represents the physics simulated by the dynamical core. LIGHT utilizes the same grid decomposition, time step, and similar communication patterns as the host MPAS-O dynamic core partial differential equation (PDE) solver. By leveraging the high-performance computing algorithms of the host dynamical core, we can use a large number of particles to accurately estimate scalar and tensor diffusivity over a spectrum of model resolutions and filtering scales.
Computations of diffusivity indicate that eddying simulations can be successful in accurately computing diffusivity provided there is enough horizontal resolution to appropriately resolve the baroclinic Rossby radius. For example, a resolution of one-fourth the Rossby radius is sufficient to compute diffusivity in the baroclinic jet. While the scales between one-fourth RRD and RRD are shown to be insignificant with respect to their direct contribution to mixing, these scales are required in order to energize motions at and above the characteristic mixing length scale of O(100) km. Consequently, eddying global ocean models will require finer resolution than the RRD in order to appropriately simulate mixing arising from mesoscale eddies. Simple scaling models are reasonable away from the basin shelf and in regions of strong mixing such as the baroclinic jet. However, at the shelf the arguments are less applicable. The difficulty with these arguments is, of course, obtaining the correct eddy velocity speed u′, integral time scale TL, and mixing efficiency α factors that are more challenging to compute than the diffusivity from cluster covariances.
Earth system simulations for the study of climate and climate change are now beginning to include “weakly eddying” ocean model components (e.g., Delworth et al. 2012). Since the 32-km simulation presented above is a counterpart to these weakly eddying ocean model configurations, we can attempt to infer the relative strengths and weakness of using weakly eddying configurations in place of traditional parameterizations. The isopycnal diffusivity produced by the 32-km simulation shown in Fig. 4 shows a richness in horizontal structure that should be emergent in weakly eddying global ocean simulations. While the 32-km simulation broadly captures the same horizontal structure of isopycnal mixing as found in the resolved, 4-km simulation, the magnitude is typically too low by a factor of 2 (Table 2). This suggests that augmenting weakly eddying simulations with parameterized isopycnal mixing will likely be required in order to correctly represent ventilation time scales of deep-ocean waters.
Calculations of isopycnal diffusivity, such as those presented above, should be useful in further developing parameterizations of Redi diffusion (Redi 1982) for ocean configurations that do not permit eddies. Broadly, our results support the notion of constructing prognostic eddy closures of the form suggested by Eden and Greatbatch (2008) and Marshall and Adcroft (2010). For example, the closure proposed by Eden and Greatbatch (2008) includes an additional prognostic equation for eddy kinetic energy and a diagnostic equation for characteristic length scale. These quantities are combined to estimate eddy diffusivity. Our results indicate that the isopycnal diffusivity can be well approximated with knowledge of the local eddy kinetic energy and either an integral time scale or characteristic length scale. Such an approximation allows for one nondimensional parameter, which we call α, as shown in Fig. 9, that physically parameterizes the mixing efficiency. Within the context of typical nondimensional parameters, α exhibits relatively small variations, indicating that prognostic closures have at least the opportunity to be robust.
State-of-the-art eddy parameterizations account for the vertical structure of isopycnal diffusivity; for example, following the results of Ferreira et al. (2005), the scheme proposed by Danabasoglu and Marshall (2007) assumes the vertical structure of κ to be self-similar to stratification N2. Our results largely support this form for the vertical structure of κ. Both N and u′ have a vertical structure that is qualitatively similar to the first horizontal eigenvector R1. The Prandtl model would suggest that κ goes as u′2, which is equivalent to a vertical form of 
Looking to the future, we believe that Lagrangian particle tracking must be implemented through a dynamically coupled, in situ, high-performance solver for exascale high-performance computing because computation is becoming cheap relative to storage. Our implementation of LIGHT in MPAS-O addresses these anticipated needs in support of next generation global climate models. As a result, we foresee no road blocks that would prevent LIGHT from exhibiting the same scaling efficiency as the PDE solver at exascale. Incorporating diffusivity analysis via LIGHT in eddying global climate simulations will likely lead to new understanding of the global ocean system.
Acknowledgments
This research was supported by the Office of Science, Office of Biological and Environmental Research of the U.S. Department of Energy Regional, and Global Climate Modeling Program (RGCM) and used computational resources provided by the Los Alamos National Laboratory Institutional Computing facility. We thank Drs. Scott Reckinger and Sergey Danilov for helpful discussions during early preparation of this work and two anonymous reviewers for comments leading to an improved manuscript.
APPENDIX A
SOMA Double-Gyre Simulation Configuration
a. Geometry
The ocean basin is defined on the surface of the sphere. Let a point, xc = (xc, yc, zc), on the surface of the sphere of radius a be located at (λc, θc) = (35°N, 0°W). Let all distances be measured along great arcs, in that, d = aarccos(x⋅xc), where x = (x, y, z) is any point on the surface of the sphere. Distances measured in the meridional direction relative to xc are expressed as Δy = a(θ − θc).
b. Basin specification
c. Initial conditions
d. Rossby radius of deformation
e. Forcing
APPENDIX B
MPAS-O Model Configuration Details
a. Vertical mixing and bottom drag
We use the Richardson number–based parameterization of Pacanowski and Philander (1981) as our vertical mixing model. Background values of νυ = 1.0 × 10−4 m2 s−1 and κυ = 1.0 × 10−5 m2 s−1 are used for viscosity and diffusivity away from the mixed layer and convective mixing. The coefficient for Richardson number vertical mixing function is that suggested by Pacanowski and Philander (1981), namely, cRi = 0.005 m2 s−1. Vertical mixing values of νυ = κυ = 1.0 m2 s−1 are used to remove gravitationally unstable profiles.
Bottom drag is required to extract energy supplied by wind forcing. We use a bottom drag parameterization of D = −cd|uB|uB, where uB is the velocity in the bottom layer and cd = 1.0 × 10−3.
b. Transport schemes
Horizontal tracer advection uses a third-order accurate tracer flux divergence reconstruction with an upwind bias of β = 0.25, following Skamarock and Gassmann (2011). Vertical advection uses a flux-corrected third-order accurate transport scheme. Tracer transport is monotone through use of the flux-corrected transport scheme of Zalesak (1979).
c. Other simulation parameters
Table B1 highlights simulation parameters that vary depending upon grid resolution. All other simulation parameters, shown in Table B2, are the same across resolutions. The bottom depths (m) of the initial z-star levels used are 4.6, 9.7, 15.4, 21.7, 28.6, 36.4, 45.0, 54.5, 65.0, 76.7, 89.6, 104.0, 119.8, 137.4, 156.9, 178.5, 202.5, 229.0, 258.3, 290.8, 326.7, 366.5, 410.6, 459.4, 513.3, 573.0, 639.1, 712.1, 793.0, 882.4, 981.2, 1090.6, 1211.4, 1345.1, 1492.8, 1656.1, 1836.6, 2036.1, 2256.5, and 2500.0.
Grid-dependent parameters for the SOMA simulation cases. The term Δt is the simulation time step, νh is the biharmonic turbulence closure viscosity, and n is the number barotropic subcycles used in the split explicit time-stepping method.
Parameters for the SOMA simulations.
APPENDIX C
Sharpened 2mΔx Filters
Filtering is performed using a sharpened Laplacian filtering procedure. Let L1 be a single Laplacian smoothing operation. By construction, a single application of L1 completely removes 2Δx oscillations in the velocity field. For example, on an equally spaced 1D grid the perfect 2Δx filter has the form 
Magnitude of filter transfer function attenuation |G| for low-pass filters 
Citation: Journal of Physical Oceanography 45, 8; 10.1175/JPO-D-14-0260.1
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