1. Introduction
Baroclinic instability produces mesoscale eddies (Smith and Marshall 2009, and references therein) that are ubiquitous in the global ocean (Chelton et al. 2011; Petersen et al. 2013). The eddies occur at the scale of the baroclinic Rossby radius of deformation (RRD), which is typically a few tens of kilometers at high latitudes (Chelton et al. 1998). These eddies have associated time scales ranging from days, corresponding to Lagrangian time scales (Balwada et al. 2016), to a few weeks, corresponding to eddy velocity decorrelation time scales (Klocker and Marshall 2014), to potentially hundreds of days for the eddy life cycle (Petersen et al. 2013). Both the space and time scales of mesoscale eddies are very short relative to ocean basin and ocean climate scales, suggesting that a separation of scales might exist. Temporal-scale separation between resolved and unresolved scales, for example, the mean and eddy flow, would imply that Reynolds averaging is reasonable (Bachman et al. 2015) and validate assumptions used to develop mixing suppression via weakly nonlinear theory (Ferrari and Nikurashin 2010; Klocker and Abernathey 2014). A clean scale separation, if it exists, would inform climate modeling efforts because mesoscale eddies strongly contribute to transport processes within the global climate system by enhancing mixing via stirring along isopycnals. For example, mixing by mesoscale eddies helps drive carbon sequestration into the deep ocean (Gnanadesikan et al. 2015), overturning circulations in the Southern Ocean (Naveira Garabato et al. 2011), and is likely a leading-order process for basin-scale ocean circulation and transport (Marshall and Speer 2012). Overall, an accurate parameterization of lateral mixing is necessary to understand the general circulation of the ocean (Fox-Kemper et al. 2013, and references therein).
The role of mesoscale eddies in the climate is complex and exhibits both advective- or diffusive-like transport characteristics (Garrett 2006). For example, mixing averaged to long time scales may result in advective-like behavior due to the along-tracer contour component of diffusivity, that is, an eddy-induced velocity (Gent and McWilliams 1990). Similarly, gradients in diffusivity may also produce advective-like fluxes (Garrett 2006). In contrast, at sufficiently large spatial and temporal scales, mixing may be parameterized by a downgradient Redi (1982) diffusivity that approximately describes the irreversible mixing corresponding to the covariance of eddy and scalar fluctuations. The time-averaged regime is typically quantified in mixing studies and presumes notions of scale separation and decorrelation (Corrsin 1975; Papanicolaou and Pironeau 1981; LaCasce 2008; Klocker et al. 2012b; Chen et al. 2015).
A decomposition of the Redi (1982) diffusivity term into slow and fast processes, to the authors’ knowledge, does not exist. Part of the challenge is that mixing is a time integral of cascading processes whereby stirring stretches material lines that, in turn, produce strong tracer gradients that are ultimately diffused across tracer contours. Mixing arising from the residual cross terms of a mean and eddy diffusivity decomposition (e.g., geostrophic turbulence by nonlinear eddies affected by the mean flow), may not be well understood because of the challenge of untangling complexity originating from the temporal integration, for example, because of chaos (Aref 1984; Pierrehumbert 1991; del Castillo-Negrete and Morrison 1993; Ngan and Shepherd 1997; Prants 2014). Strong progress in understanding interactions between eddies and the mean flow, especially with respect to mixing suppression, has been made (Bates et al. 2014, and references therein). However, the processes producing mixing suppression via eddy and mean flow interactions are still not fully understood, although work on contributing processes is occurring (e.g., understanding shear contributions to suppression) (Srinivasan and Young 2014). A deeper understanding of the effect of the role of residual, eddy, and mean flow interactions on diffusivity is needed to fundamentally understand mixing due to mesoscale eddies. Physical contributions to diffusivity via a diffusivity decomposition could quantify the relative mechanisms and roles of mixing in mesoscale eddying regimes. Ultimately, this knowledge will be needed to conduct accurate climate simulations at global and decadal scales that cannot directly resolve the mesoscale and instead rely upon parameterization.
In this paper, we hypothesize that mixing for an idealized baroclinic jet is due to the combination of interactions between mesoscale eddies and the background mean flow. Hence, we quantify interactions between mesoscale eddies and a sheared mean flow for an idealized baroclinic jet, producing a decomposition of mixing into mean shear, wave, and turbulent components via assessment of diffusivity for temporally decomposed Eulerian velocities. Short temporal scales produce initial filamentation via eddy motions as quantified using a high-pass filter. Long temporal scales are quantified using a low-pass filter, for example, and are associated with large-scale straining of filaments by time-mean shear. The resultant residual diffusivity quantifies turbulent contributions to mixing for interactions between the slow and fast Eulerian velocity time scales. The goal of the paper is to better understand mixing by quantifying the importance of the residual diffusivity following the removal of mean and eddy mixing from the full diffusivity, with evaluation within the context of mixing suppression and critical layer (MSCL) theory (e.g., Ferrari and Nikurashin 2010).
To this end we begin by reviewing the literature related to mean flow effects on diffusivity, that is, diffusivity suppression, in section 2. We design an experiment to study mixing in an idealized circumpolar current (ICC) in section 3a via application of Lagrangian in Situ High-Performance Global Particle Tracking (LIGHT), as detailed in section 3b. Lagrangian trajectories derived from integration of the full and decomposed eddy and mean Eulerian velocity fields are used to assess diffusivities and compute an associated residual diffusivity, as outlined in section 4. Results are presented for the ICC flow in section 5, highlighting the hydrodynamics and isopycnal mixing arising from the background flow, the eddies, and the residual interaction of the two. These results are used to analyze MSCL theory using the residual diffusivity decomposition in section 6, followed by concluding remarks in section 7.
2. Diffusivity suppression: MSCL theory
The mixing suppression prescribed in (1) is due to the absolute phase speed for nonlinear eddies cw, the mean flow 
3. The idealized circumpolar current experimental design
a. Model configuration
Previous idealized Southern Ocean flows (Abernathey et al. 2011; Stewart and Thompson 2013; Abernathey et al. 2013; Saenz et al. 2015) motivate the ICC configuration. To quantify the role of mean flow effects on meridional isopycnal diffusivity, the ICC is configured such that baroclinic eddies evolve under the influence of a strong zonal-mean flow (Fig. 1). Additionally, the ICC is also used to understand the role of eddy–mean flow interactions on the momentum budget (Ringler et al. 2017).
Time-averaged and instantaneous isometric view of the zonally periodic ICC flow. Eulerian instantaneous (green) and time-mean (gold) kinetic energy (m2 s−2) are shown for 500 < x < 1000 km. The Eulerian time-mean zonal velocity (m s−1 red, white, blue) is shown for 0 < x < 500 km.
Citation: Journal of Physical Oceanography 47, 8; 10.1175/JPO-D-16-0101.1
The ICC is simulated using the unstructured grid Model for Prediction across Scales Ocean (MPAS-O; Ringler et al. 2013). MPAS-O is the ocean model component of the U.S. Department of Energy’s Accelerated Climate Model for Energy (ACME) and uses variable-resolution spherical Voronoi tessellations (Ju et al. 2011) that are capable of simulating ocean motions across a range of spatial and temporal scales.
The ICC simulation is conducted on a planar zonally periodic domain that is 1000 km wide with a meridional extent of 2000 km. The shelf break is 500 km from the southern boundary with a tanh half-width of 100 km. The on-shelf water depth is 500 m, and the deep water depth is 2500 m. Regular hexagons with cell centers separated by 5 km are used to tessellate the plane in a 200 zonal by 460 meridional cell pattern. The vertical resolution varies from approximately 0.6 m at the surface to 90 m at depth using 100 vertical layers; approximately half the vertical layers are in the upper 250 m with a z-star style arbitrary Lagrangian–Eulerian coordinate (Petersen et al. 2015). The baroclinic time step is 3 min with 28 split-explicit, barotropic subcycles per baroclinic time step.
A westerly wind stress of 0.2 N m−2 and easterly wind stress of −0.05 N m−2 are applied using sin2 profiles with 1600-km wavelengths centered at y = 1400 and 400 km, respectively, following the zonal Southern Ocean winds approximation of Large and Yeager (2009) as used by Stewart and Thompson (2013). A linear equation of state with linear thermal expansion coefficient of α = 0.255 kg m−3 °C−1 is employed. Interior temperature restoring within sponge layers at the north and south boundaries are used to drive an overturning circulation. The northern and southern wall interior restoring is similar to Saenz et al. (2015), but with ze = 1 km and an exponential decay e-folding length scale of Le = 80 km. The surface temperature is restored to the tanh profile of Saenz et al. (2015), but with Tb = 1.0 °C and a piston velocity of 1.93 × 10−5 m s−1. A β-plane approximation is used with f0 = −1 × 10−4 s−1 and β = 1 × 10−11 m−1 s−1 (Abernathey et al. 2011; Stewart and Thompson 2013). Additional configuration details are summarized in Ringler et al. (2017).
General MPAS-O details are given in Ringler et al. (2013) and Petersen et al. (2015). Grid-scale enstrophy is removed from the simulation by rotational and divergent hyperviscosities of 3.9 × 108 and 3.9 × 109 m4 s−1, respectively (Ringler et al. 2017). Bottom drag is parameterized by quadratic bottom drag with cd = 3 × 10−3. The surface mixed layer is parameterized via the K-profile parameterization (Large et al. 1994) via CVMix (Griffies et al. 2015) with a background viscosity of 10−4 m2 s−1 and diffusion of 5 × 10−6 m2 s−1.
The model has a constant salinity of 34 psu and uses the initial temperature profile of Saenz et al. (2015). The simulation is started from rest and is integrated under its steady forcing for 100 years at 20-km resolution, interpolated to 5-km resolution, and then integrated for another 35 years. Across the last 10 years of simulation, particles are released from their initialized state every month to develop 120 ensemble members from 1.2 × 108 particle trajectories with daily output.
b. Lagrangian in Situ Global High-Performance Particle Tracking
Lagrangian particle tracks are simulated using MPAS-O’s online LIGHT analysis member (Wolfram et al. 2015). This approach computes particle trajectories during simulation run time, ensuring that Lagrangian particles are advected with the same spatial and temporal fidelity as the Eulerian model. Particle trajectories are purely diagnostic and are integrated directly from a particular Eulerian velocity field without parameterization of diffusion. This online particle tracking approach is preferable to offline post hoc analysis because Lagrangian trajectories obtained from undersampled velocity output are less accurate and accuracy is not recoverable, even by addition of stochastic particle diffusion (Qin et al. 2014). The incremental cost for the simulation is an increase in runtime of approximately 15% for a particle time step of 3 min with daily output. The relatively small incremental cost for use of LIGHT is possible because its input, output, and computation are all fully parallel via the message passing interface (MPI), similar to the host MPAS-O dynamical core.
Particles are advected along isopycnal surfaces shown via the time-averaged contours in Fig. 2. At the beginning of each simulated month, 1 × 106 particles are seeded in a 200 × 460 × 11 pattern by placing one particle at each cell center across 11 potential density surfaces. The potential density surfaces (kg m−3) are shown in Fig. 2 with values ranging from 1028.5 to 1030.0 in increments of 0.15. Isopycnally constrained advection ensures, by construction, that diagnosed mixing is representative of quasi-adiabatic processes along interior surfaces of potential density. Advection on outcropping layers is performed using the velocity on the nearest potential density surface resulting in some adiabatic mixing with the surface layer. Horizontally, particles remain in the domain for Courant–Friedrichs–Levey numbers less than one by construction because Wachspress interpolation from cell vertices is employed (Gillette et al. 2012).
Time-mean potential density (kg m−3) in coordinate space with isopycnal particle surfaces used in this study designated by bold white lines. The thickest white line designates the 1029.7 kg m−3 potential density surface.
Citation: Journal of Physical Oceanography 47, 8; 10.1175/JPO-D-16-0101.1
4. Analysis methods
a. Particle diagnostics
Particle diagnostics such as the mean velocity, eddy velocity, decorrelation time, and integral time scale are computed as in Wolfram et al. (2015) based on the statistics of ensembles of particle clusters. Quantities are plotted in buoyancy space because a potential density coordinate is a natural frame of reference to analyze isopycnal mixing. Furthermore, this choice is consistent with the diagnostic frame of reference because isopycnally constrained trajectories from LIGHT are used to compute Lagrangian diagnostics.
The size of the cluster should correspond to the largest scales of mixing, for example, several multiples of the Rossby radius of deformation, to ensure statistical accuracy by including as many particles as possible, provide a reasonable time to decorrelation, and not impair the spatial fidelity of the computed spatial diffusivity field (Wolfram et al. 2015). A cluster radius of 100 km is sufficient to this end. Particles on outcropped potential density surfaces at the start of each realization are excluded from the calculation of cluster diffusivities to minimize the influence of the surface layer and, instead, emphasize the eddy-driven mixing along isopycnals.
Velocity autocorrelation R for each isopycnal layer in the ICC core (y = 1500 km) using a 10-km grid.
Citation: Journal of Physical Oceanography 47, 8; 10.1175/JPO-D-16-0101.1
b. Comparison with MSCL theory
For purposes of comparison with diagnostics in buoyancy space the potential density at the critical layer is computed using the inverse mapping between the time-averaged layer z coordinate and the time-averaged potential density. However, even independent of this mapping, the location of the critical layer based on the phase speed is an approximation that employs assumptions of linearity. In reality, the phase speed is much more complex and is formed from a relationship between β, 
c. Decomposing the Eulerian velocity into high-pass and low-pass components
Online time filtering is employed to decompose the Eulerian velocity field into high- and low-pass components, corresponding to a decomposition of the velocity field into eddy and mean components, respectively. Typically, offline decompositions employ a running mean. However, use of a running mean is not possible for online computations. Consequently, a recursive filter is employed to decompose the velocity into its mean and eddy components. Derivation of the single-pole, recursive, low-pass exponential impulse filter employed in this study is presented in appendix A. A filter time constant of τ = 90 days, which is used to filter out time scales occurring at or faster than the decorrelation time scale of O(25) days, is chosen as the time scale for separating the low-frequency, background flow from the high-frequency eddying flow. Success of the filtering approach is indicated by the full and high-pass filtered flow having similar structure and magnitudes of urms (not shown), while the high-pass filtered flow also exhibits a zonal flow 
d. Diffusivity decomposition from the Eulerian velocity decomposition
This decomposition approach allows direct comparison to MSCL theory and is distinct from the generalized Lagrangian mean (GLM) theory (Andrews and McIntyre 1978; Craik 1985; Bühler 2014, chapter 10) because the Eulerian velocity field is first filtered and then used to produce Lagrangian trajectories, whereas in GLM the unfiltered Eulerian velocity field is used to produce Lagrangian trajectories, that is, xFULL, which are then decomposed into their slow- and fast-varying components.
Thus, interactions of the mean flow with eddy variability can be quantified by computing diffusivity for six cases: 1) a full diffusivity κFULL computed from Lagrangian particle trajectories advected by the full Eulerian velocity field in (11); 2) a low-pass or mean flow diffusivity κLOW for particle trajectories advected by the low-pass filtered Eulerian velocity field in (13); 3) a high-pass or eddy diffusivity κHIGH for particle trajectories advected by the high-pass filtered Eulerian velocity field in (14); 4) a residual diffusivity κDIFFU based on decomposition via (16); 5) the dispersion-based diffusivity corresponding to residual particle positions given by (15), κDIFFX; and 6) an estimate for the unsuppressed diffusivity case UNSPR, κUNSPR (see below). Note that a pure shear flow does not have a true diffusivity associated with it because particle motion is never decorrelated. However, for the purposes of our decomposition, we estimate its potential contribution as a “diffusivity” even though κLOW does not necessarily represent a physical diffusivity.
In a model that resolves the mean flow but not the eddies, the diffusivity derived from the resolved flow κFULL should be implemented, which is approximately composed of the sum of the eddy κHIGH and residual κDIFFU diffusivities. Because eddies are not represented in the model, the effect of eddies and the residual, κHIGH and κDIFFU, respectively, must be parameterized to compute the full κFULL.
The κDIFFU and κDIFFX estimates of mixing account for the asymptotic cases of homogeneous turbulence (e.g., high-pass filtered velocity) as well as mean meridional shear flow mixing (e.g., low-pass filtered velocity) because κDIFFU = 0 and xDIFFX = 0 for either uFULL = uHIGH or uFULL = uLOW. As explained in appendix B, the difference κDIFFX − κDIFFU provides physical insights into diffusivity suppression.
e. Estimation of unsuppressed diffusivity UNSPR
To place the components of the decomposed diffusivity in context, we estimate a range of unsuppressed diffusivities (case UNSPR) using a lower bound, which is constrained to the peak diffusivity observed near the critical layer location for the FULL diffusivity, and an upper bound Rhines scale estimate (Klocker and Abernathey 2014; Klocker et al. 2016).
5. Results
a. Hydrodynamics of the ICC simulations
An overview of the zonally periodic ICC flow is shown in isometric view in Fig. 1. The plot is split into two sections where isosurfaces for the mean zonal velocity are plotted for 0 < x < 500 km. The resultant eastward-propagating ICC that approximates an Antarctic Circumpolar Current can be seen in red, approximately centered at y = 1500 km, and the westward-propagating slope front current (SFC) that approximates an Antarctic Slope Front is over the shelf break at y = 500 km.
Instantaneous and mean kinetic energy are plotted for 500 < x < 1000 km. The instantaneous kinetic energy, shown in green, reveals the presence of strong eddies, and the mean kinetic energy, shown in transparent gold, is largely coherent with the mean flow. As shown, these coherent structures are of a scale of O(100) km. The mean kinetic energy and mean flow are fairly zonally homogeneous. Additional details related to the ICC’s hydrodynamics are discussed by Ringler et al. (2017).
1) Lagrangian trajectories
Lagrangian pathlines and mean fluid speed over 7 days are shown in Fig. 4 for a single realization for a single buoyancy surface. This view of the flow demonstrates the strong eddy and mean eastward flow in the ICC as well as the smaller eddy and westward mean flow over the shelf break. Smaller eddies have turnover times on the order of a week, which correspond to the pathlength shown in the figure (e.g., in the ICC at approximately x = 500 km and y = 1700 km). Lagrangian paths in the ICC typically encounter large eddies but are not typically entrained within a coherent eddy, as evident by only a few Lagrangian trajectories advecting westward in the ICC. Instead, the majority of trajectories propagate toward the east with their paths curved by the superimposed eddy and mean flow. In contrast, eddies near the shelf break and on the shelf are much smaller than in the deep part of the channel. These trajectories lack strong, coherent zonal advection in the eastward direction. Particles within the SFC are advected toward the west, and particles on the shelf are advected primarily by eddies with an indistinct zonal propagation in regions away from the SFC. Although useful to qualitatively describe the flow, the unprocessed particle paths are unable to directly quantify mixing within ICC. To this end, the Lagrangian mean and eddy flows are quantified using cluster statistics developed from clustering of individual Lagrangian trajectories in space and over realizations to compute spatial structure (Wolfram et al. 2015) for the FULL case.
Lagrangian particle 7-day pathlines on potential density surface 1029.7 kg m−3 overlaid on 7-day averaged current speed (m s−1). Lines designating pathlines are progressively thinner for previous particle positions. Particles are subsampled at an interval of approximately 50 km. Particles on this potential density surface outcrop onshore of the shelf break at approximately y ≤ 200 km.
Citation: Journal of Physical Oceanography 47, 8; 10.1175/JPO-D-16-0101.1
2) Zonal-mean and eddy velocity
The zonal-mean flow in Fig. 5 is characterized by a strong 0.25 m s−1 zonal eastward-flowing transport within the ICC (e.g., at y = 1500 km) with a smaller westward mean flow of 0.05 m s−1 over the shelf break within the SFC (e.g., at y = 500 km). Within the ICC the flow is coherent throughout the water column with the strongest velocity at the surface. The SFC, however, is strongest at depth at the shelf break at y = 500 km with very little mean flow on the shelf.
Zonal-mean velocity (m s−1) in buoyancy coordinates computed with Lagrangian particle clusters using the FULL velocity via (11).
Citation: Journal of Physical Oceanography 47, 8; 10.1175/JPO-D-16-0101.1
Like the zonal-mean velocity, the eddy velocity is greatest within the core of the ICC, as shown in Fig. 6. The spatial structure of the eddy velocity is similar to the zonal-mean flow in Fig. 5 with surface intensification in the core of the ICC at y = 1500 km with a fairly coherent vertical structure. The eddy velocity, however, attenuates more rapidly in the vertical than does the zonal-mean flow. The eddy velocity is smallest directly over the shelf break at y = 500 km and low eddy velocities occur for 0 ≤ y ≤ 1000 km. This result is consistent with the Lagrangian trajectories presented in Fig. 4, where Lagrangian trajectories are dominated by large eddies and large mean flows within the ICC from 1000 ≤ y ≤ 2000 km and by smaller eddies otherwise.
As in Fig. 5, but for eddy velocity (m s−1).
Citation: Journal of Physical Oceanography 47, 8; 10.1175/JPO-D-16-0101.1
3) Dominant scales
The Rossby radius of deformation, shown in Fig. 7, is largely governed by the mean stratification and water depth. It ranges from approximately 20 km in the ICC to less than 5 km on the shelf break and onshore. Its associated linear Rossby wave speed scaling 
Meridional distribution of the absolute phase speed of nonlinear eddies cw (m s−1) and Rossby radius of deformation LD (km).
Citation: Journal of Physical Oceanography 47, 8; 10.1175/JPO-D-16-0101.1
The meridional flow is approximately decorrelated after 20 days (Fig. 3), and we compute meridional diffusivities at t = 25 days. Spatial structure and magnitude for the diffusivity and its decomposed components are not particularly sensitive to this choice, for example, diffusivities at 13, 25, and 50 days are comparable in terms of magnitude and structure (not shown). However, spatial fidelity is lost at longer assessment times because particles diverge from their initial cluster locations so that a shorter time scale is preferable (Wolfram et al. 2015). The integral time scale is biased by the presence of a negative lobe in the autocorrelation (Abernathey et al. 2013) that produces a peak value at approximately 2 to 10 days. The integral time scale ranges from hours to several days (not shown), and the decorrelation time is consequently much larger than the integral time scale. Furthermore, the decorrelation scale of several weeks is also qualitatively consistent with Fig. 4, which shows particles traversing several fractions of a complete orbit, for example, near x = 600 km and y = 1700 km, over a period of 7 days.
b. Full diffusivity 
Figure 8 shows the spatial structure of the complete diffusivity κFULL that is computed based on Lagrangian particles advected with the full Eulerian velocity field uFULL. Diffusivity values are plotted on a log10 scale with contours at 0.3 intervals corresponding to doubling of diffusivity magnitude. All spatial plots for diffusivity are presented with identical color maps and contour interval.
Zonally and temporally averaged meridional section of meridional diffusivity κyy (m2 s−1) corresponding to particle advection using the full velocity FULL. An estimate of the critical layer location is computed using the full velocity and is designated by a thick white line in each panel. Diffusivity values above 1000 m2 s−1 are contoured with numbered contours indicating doubled values.
Citation: Journal of Physical Oceanography 47, 8; 10.1175/JPO-D-16-0101.1
Mixing by the FULL flow in Fig. 8 is greatest within the ICC, reaching values of 6000 m2 s−1, and is weakest at the shelf break and onshore with values O(100) m2 s−1, an observation that is qualitatively corroborated by the scale of eddies represented by pathlines for the realization in Fig. 4. The diffusivity maximum occurs on a subsurface potential density surface corresponding to approximately the 1029.4 kg m−3 surface. The spatial location of the diffusivity maximum is in the vicinity of the estimated buoyancy surface for the critical layer and near the region of maximum wind stress in the ICC. The role of the eddies and the mean flow in contributing to the mixing within this region can subsequently be explored by considering diffusivities derived from particle trajectories from the low- and high-pass filtered Eulerian velocities, that is, (13) and (14).
c. Diffusivity 
Mixing by the LOW flow, as approximated by the low-pass Eulerian velocity filter, results in the κLOW shown in Fig. 9a. Values are one to two orders of magnitude smaller than for κFULL and are largely potential density independent for each meridional location. This result is to be expected because mixing by the LOW flow is largely due to the low-pass filter extracting the mean flow, which is predominantly characterized by meridionally varying zonal shear (e.g., Fig. 5) that does not enhance cross-stream mixing, for example, κyy, as it does zonal along-stream mixing κxx via classic shear dispersion (Taylor 1953; Fischer et al. 1979; Oh et al. 2000; Griesel et al. 2010, 2014), but this mixing enhancement is not considered here.
Zonally and temporally averaged meridional sections of meridional diffusivity κyy (m2 s−1) corresponding to particle advection using (a) the low-pass filtered velocity LOW, (b) the high-pass filtered velocity HIGH, (c) the residual diffusivity arising due to nonlinearity interactions between the mean and eddy flow via velocity DIFFU, and (d) the residual diffusivity arising due to nonlinearity in particle positions DIFFX. An estimate of the critical layer location is computed using the full velocity and is designated by a thick white line in each panel. Diffusivity values above 1000 m2 s−1 are contoured with numbered contours indicating doubled values.
Citation: Journal of Physical Oceanography 47, 8; 10.1175/JPO-D-16-0101.1
d. Diffusivity 
Figure 9b quantifies κHIGH, based on trajectories using the high-pass Eulerian filtered velocity field that is analogous to the standard Eulerian eddy velocity computed by removal of the mean. Diffusivities are highest at the surface, corresponding to the greatest eddy kinetic energy with attenuation toward depth. The slope of the diffusivity contours in buoyancy space at the ICC is aligned with the gradient of the nonlinear wave phase speed with values increasing toward the north. It is modulated by a mixing length that is dependent upon the Rossby radius of deformation LD, which decreases toward the shelf and is the dominant spatial scale for the eddies. Similar to κFULL in Fig. 8, diffusivity is small over the shelf break. Characteristic κHIGH values are more than a factor of 5 smaller than for κFULL diagnosed using the FULL velocity, which is a surprising finding in light of other studies (Klocker et al. 2012b; Abernathey and Marshall 2013) that found eddy-produced diffusivities, for example, κHIGH, to be larger. The relevance and consequence of this interesting result is explained in the discussion section.
e. Residual diffusivity 
The residual diffusivity resulting from the difference between the full and combined mean and eddy flows is presented in Fig. 9c. Comparison of the residual κDIFFU in Fig. 9c to the FULL velocity in Fig. 8 suggests that κDIFFU ≈ κFULL. Note that the operation in (16) is applied prior to zonal averaging and consequently values within the figures may not strictly satisfy (16). This diffusivity measures the mixing due to the turbulent combination of eddies and the background mean. The maximum diffusivity occurs near the critical layer location within the ICC at y = 1500 km, and the diffusivity is of similar magnitude to that for the FULL flow. Diffusivity for outcropping portions of the layer in the southernmost portion of layers is very small.
f. Dispersion-based diffusivity 
The dispersion-based κDIFFX is obtained by computing the diffusivity associated with the residual particle positions in (15) and is shown in Fig. 9d. It is a factor of 3 times larger in the core of the ICC at the surface relative to the critical layer and the MSCL-predicted maximum diffusivity at depth is absent. It is larger than κFULL and is intensified in the vicinity of the critical layer by 25% relative to κDIFFU.
6. Discussion
a. Characteristics of decomposed diffusivities
1) Magnitude
The subsurface maximum diffusivity predicted by MSCL theory is evident for κFULL and residual κDIFFU in Figs. 8 and 9c. However, the maximum at the critical layer depth does not occur for κLOW, κHIGH, or the dispersion-based κDIFFX, that is, Figs. 9a, 9b, and 9d, respectively. Furthermore, the magnitude of mixing for κLOW and κHIGH in Figs. 9a and 9b is much smaller than for κFULL in Fig. 8, for example, more than 4 times smaller near the critical layer. The implication is that the mean flow effect on diffusivity is particularly important for mixing as κDIFFX is much larger than κFULL and κDIFFU and is qualitatively very similar in magnitude to κFULL, although the spatial extent of the maximum diffusivity is reduced.
2) Spatial structure
The diffusivity for κLOW in Fig. 9a is largely vertically homogeneous across potential density surfaces on account of the depth coherency of the mean flow shown in Fig. 5 within interior layers of the fluid. In contrast, κHIGH in Fig. 9b is primarily depth attenuated, corresponding to decreased eddy velocities at depth. Diffusivity for outcropping layers is largely driven by eddy activity with minimal contributions by the mean flow, as demonstrated by similar magnitudes and structure for the southernmost portion of layers in Figs. 8, 9a, and 9b. The residual κDIFFU in Fig. 9c displays the subsurface maximum as predicted by MSCL theory because it arises due to a suppressed mixing length for κFULL. In contrast, the dispersion-based κDIFFX in Fig. 9d shows a monotonic decrease from surface-intensified diffusivity corresponding to the vertical eddy kinetic energy maximum. Additionally, the spatial structure of κDIFFX is reminiscent of the eddy and mean zonal flow, for example, compare Fig. 9d to Figs. 5 and 6.
3) Mean flow effects on diffusivity
The κDIFFU mixing within the ICC contributes to approximately 80% of κFULL (Fig. 9c). Furthermore, the location of the maximum diffusivity κDIFFU is more closely aligned with the estimated location of the critical layer, shown by the thick white line, than for κFULL in Fig. 8. These observations indicate that mixing within the ICC is strongly governed by HIGH and LOW flow interactions with a subsurface maxima obtained via minimized suppression for the FULL and DIFFU cases. Additionally, the strong dispersion-based κDIFFX in Fig. 9d, particularly at the surface, indicates that the mean flow effect on diffusivity is to constrain the total amount of mixing that could occur because κFULL < κDIFFX, suggesting that diffusivity suppression serves to both constrain the mixing near the ocean surface and minimally reduce mixing for subsurface isopycnals near the critical layer, as predicted by MSCL theory.
b. Mixing dynamics of an inhomogeneous eddying flow
The ICC hydrodynamics within the ICC region at y = 1500 km can be conceptualized as a flow composed of quasi-independent eddies and a background mean flow. To illustrate the role of the turbulent eddy and mean flow in enhancing mixing, consider the clusters of particles that have evolved 25 days from their initial condition of 100-km clusters distributed at y = 500, 1000, and 1500 km, shown in Fig. 10. For the case of the FULL flow in Fig. 10a, with particle positions xFULL computed via (11), eddies serve to produce the initial filamentation of the clusters, and then the background mean flow further serves to elongate these filaments, resulting in advection from one eddy to the next. The implication is that mixing driven by stretching and straining by the combined eddy and mean flow is large, even for the critical layer where 
Particle dispersion for the first realization at 25 days following particle releases on the 1029.7 kg m−3 potential density surface. Initial particle positions are within radii of 100 km and are centered at (x, y) = (500, 500), (500, 1000), and (500, 1500) km corresponding to the dark purple, the medium blue, and light green. Particle positions are computed from (a) the FULL velocity, (b) the low-pass filtered LOW velocity, (c) the high-pass HIGH filtered velocity, and (d) the position resulting in DIFFX mixing obtained from the residual particle position xDIFFX via (15) where particles are translated back to their initial positions for plotting.
Citation: Journal of Physical Oceanography 47, 8; 10.1175/JPO-D-16-0101.1
Particle dispersion is smallest for the case computed with only the LOW flow via (13) in Fig. 10b, xLOW, because the background flow with its meridional shear is not strong enough, in and of itself, to efficiently spread the particle cluster. The LOW case can be thought of as producing sheared mixing that occurs due to the mean flow. If the background flow were constant, no relative dispersion would occur because each point is advected precisely the same amount from one time to another. In this extreme case no real mixing occurs outside that caused by background diffusion because there is no filamentation to amplify background gradients to instigate downgradient mixing. However, for the HIGH case computed by (14) in Fig. 10c, xHIGH, the initial filamentation produced by the eddies is localized and not dispersed past the mixing length scale for the FULL flow. Mixing consequently occurs much more slowly than for the FULL case in Fig. 10a.
In MSCL theory the mean flow and eddy phase speed reduces the maximum diffusivity, for example, the denominator in (1). Consequently, κDIFFX, computed with (15), is designed to circumvent mixing suppression by reversing the effect of the individual contributions of the LOW and HIGH transport on the FULL. Conceptually, xDIFFX will estimate motion occurring due to residual motion following removal of the filtered eddy and mean trajectories. This residual motion strongly enhances dispersion through the combined action of eddies producing initial filamentation and further stretching and straining of these filaments via the background mean flow as demonstrated in the FULL flow of Fig. 10a and even more so for the DIFFX case Fig. 10d for xDIFFX. Dispersion-based κDIFFX computed from particles in Fig. 9d is even larger than κDIFFU computed with (16) in Fig. 9c, and κDIFFU arises from a decomposition due to differences in particle dispersion, whereas κDIFFX is derived from a decomposition of particle positions that are used to derive dispersion and diffusivity. The implication is that mixing driven by stretching and straining is larger than mixing due to the combined eddy and mean flow, even for the critical layer where 
c. Vertical structure of decomposed diffusivities in the baroclinic jet
Vertical profiles in y–z space are used to better understand the diffusivity decomposition and evaluate the applicability of scalings for parameterization. The profiles in Fig. 11 are for meridional diffusivities that are averaged over the range 1400 < y < 1600 km and then projected from buoyancy space back to depths via mean buoyancy layer depths, for example, as shown in Fig. 2. The approximate location of the critical layer is also shown for reference. A scaling estimate for the unsuppressed diffusivity is included in Fig. 11a, where κDIFFX is observed to approximate κUNSPR near the critical layer and away from the free surface. Unsuppressed diffusivity computed using 
Vertical profiles of meridional isopycnal diffusivity κyy (m2 s−1) for (a) the LOW (shear), HIGH (linear wave), DIFFU (nonlinear turbulence), FULL, DIFFX (estimated unsuppressed), and UNSPR (unsuppressed scaling) cases centered on the ICC with (b) fits to the FULL diffusivity using (1) via (2) and (3) plotted with thick and thin dotted black lines, respectively. The theoretical estimate from (24) with depth variable Lmix via (22) is also plotted against the HIGH case with the red dotted line.
Citation: Journal of Physical Oceanography 47, 8; 10.1175/JPO-D-16-0101.1
Hatching denotes the z positions inside the ventilation-defined surface layer. As discussed in more depth in Ringler et al. (2017), the ventilation-defined surface layer is the set of buoyancy coordinates that ever reach the ocean surface. We note that the κDIFFU, κFULL, and κDIFFX diffusivities all show a local minimum at the base of the ventilation-defined surface. As described in Ringler et al. (2017), the base of the ventilation-defined surface layer corresponds to an unusual location within the ICC where mesoscale eddies are fluxing Ertel potential vorticity up the mean potential vorticity gradient.
In particular in Fig. 11a, note that the diffusivity κLOW corresponds to mixing due to shear and is very small because there is no turbulent motion to enhance mixing. In contrast, κHIGH, which corresponds to mixing by temporal high-frequency variability of the Eulerian velocity field, for example, eddies, is surprisingly much smaller than κFULL and requires further analysis.
1) Eddy diffusivity departure from previous studies
The κHIGH effectively quantifies the mixing occurring due to the eddy velocity resulting from the removal of the Eulerian time-mean velocity from the full velocity. However, previous researchers (Griesel et al. 2010, 2014; Klocker et al. 2012b; Abernathey and Marshall 2013) noted that the eddy-produced diffusivity was larger than the κFULL that corresponds to mixing suppression by the mean flow. Our results, for example, Figs. 8 and 9b, are at odds with these previous findings.
There are two primary reasons for this departure from the previous literature. First, eddy results in Klocker et al. (2012b) and Abernathey and Marshall (2013) are not generated by removal of the mean flow from a full velocity field. Instead, the results are produced using an AVISO-generated eddy velocity field where the full flow is constructed by addition of an independent mean flow, which subsequently results in mixing suppression in accordance with MSCL theory. This approach is the inverse of this study where the eddy flow is constructed by the removal of the mean from the full flow. Second, results in Griesel et al. (2010, 2014) are obtained performing the eddy decomposition in a similar fashion to this study via construction by removal of the mean from a full flow, albeit offline. They found that the full flow had the largest along-stream dispersion resulting in unconvergent along-stream diffusivities (Griesel et al. 2010, 2014) due to residual shear dispersion in the binning method (Bauer et al. 1998; Oh et al. 2000; Griesel et al. 2010; Koszalka et al. 2011). To the best of the authors’ knowledge they did not report results for the cross-stream meridional diffusivity for the case of the eddy-only flow, as considered here. Hence, a direct comparison to their work is not possible.
Our approach is distinct from both these types of analyses because it decomposes the full Eulerian velocity to obtain the Eulerian eddy velocity and does not consider the along-stream diffusivity. Fundamentally, the mixing that we are computing is derived from the combined action of both the mean and eddy components of the flow field, working together, to produce mixing. The addition of a velocity field, for example, to remove the mean flow, results in Doppler shifting of the eddies, thereby altering the nonlinear parameter r. This is predictable, as demonstrated in the next subsection by the high accuracy of the curve fit for the HIGH pass diffusivity. This approach demonstrates the cause of the significant reduction of the full diffusivity to the eddy diffusivity quantified via the HIGH case.
2) Shifted eddy phase speeds for the HIGH case
Mixing consequently transitions from arising due to a geostrophic turbulence regime dominated by nonlinear eddies to a regime characterized by linear wavelike behavior (Klocker and Abernathey 2014; Klocker et al. 2016) under high-pass filtering depending upon the relative magnitudes of the eddy and mean flow velocities. Consequently, (14) does not correspond to motion for an unsuppressed velocity field but instead corresponds to motion for a wave-dominated regime if 
Fitted model of ΓurmsLmix to high-pass filtered diffusivity with correlation coefficient of 0.985 and p = 1.4 × 10−6.
Citation: Journal of Physical Oceanography 47, 8; 10.1175/JPO-D-16-0101.1
3) Physical interpretation of diffusivity decomposition
The decomposed diffusivity can now be described in terms of its physical significance. The κLOW corresponds to mixing occurring due to the mean flow, which in this case is dominated by shear as indicated by Fig. 10b. As demonstrated in the previous section, κHIGH corresponds to mixing due to reduced nonlinearity of the eddies, for example, as would be expected for linear Rossby waves (Klocker and Abernathey 2014; Klocker et al. 2016). The term κDIFFU quantifies the turbulent diffusivity produced from decomposition of the FULL (suppressed), LOW (shear), and HIGH (linear wave) diffusivities via (16), assuming that 
d. Quantifying eddy and mean flow interaction reduction to turbulent mixing
e. Diffusivity parameterization
Although urms and its vertical structure is relatively easy to approximate or compute in a resolved simulation, the structure of Lmix and TL must be parameterized in a more nuanced way. This is evident by examination of two other fits using MSCL theory in (1). One uses vertically varying urms and an assumed vertically constant Lmix via (2), as designated by the black dashed line. Another fit uses the vertical structure of 
7. Conclusions
Mixing quantified by meridional diffusivity for the ICC is highly dependent on both the turbulent eddy flow, the mean flow, and their combined mixing as quantified using a residual diffusivity that separates diffusivity associated with the high- and low-pass filtered Eulerian velocity fields from the full diffusivity. Initial filamentation by eddy scales is stretched to larger-scale structures by mean flow straining. Mixing is due to combined stirring by the full velocity. Decomposition of diffusivity into eddy and mean constituents demonstrates that turbulent eddy and mean flow interactions ultimately account for 80% of the diffusivity. Interactions between the mean and eddy flow both enhance and suppress diffusivity, leading to a maximum at depth, as predicted by MSCL theory.
A simple thought experiment may help aid insight into important interactions. Let the mean flow velocity be a constant in the zonal direction and the eddy component be standing eddies, for example, a mean flow perturbed frozen-field case (Lumpkin et al. 2002). The mean motion is simply xMEAN = uMEANt + x0 and the eddy motion orbits within an eddy length scale Leddy of the particle initial position with mixing occurring due to perturbations transporting particles from eddy to eddy. Dispersion for the eddy case is small and dependent upon the background diffusivity providing the perturbation. In contrast, dispersion for the mean case is quadratic in time corresponding to ballistic motion in the zonal direction, zero otherwise. However, dispersion is not maximized under either case but is instead increased when transport occurs under the combined velocity as particles may rapidly advect from one eddy to the next, allowing spreading even in directions normal to the mean flow, assuming the eddies are not perfectly aligned in the zonal direction.
In the idealized simulation the removal of the mean affects the phase speed of eddies and shifts turbulence and its associated mixing from that dominated by nonlinearity to linearity as quantified via r (Klocker et al. 2016). This is directly accounted for in the diffusivity decomposition. To the best of the authors’ knowledge, a technique to recover the unsuppressed eddy velocity field from uFULL that does not alter r via modification of the eddy absolute phase speed, that is, convert (20) to (21), is an open question. Better estimates for the unsuppressed diffusivity would be available given this capability.
The estimated strength of the interaction of the combined eddy and mean to the full diffusivity demonstrates that residual eddy and mean flow interactions dominate mixing with a turbulent contribution κDIFFU accounting for 80% of the diffusivity. Dispersion plots of a realization (Fig. 10a) demonstrate that initial filamentation by eddy scales is stretched to larger-scale structures by mean flow straining. Ultimately though, mixing due to stirring by the full velocity is constrained by restoring due to linear waves arising from Doppler shifting due to the velocity field decomposition (appendix B). At present, parameterizations by MSCL reasonably capture the effects of mixing within the vicinity of the critical layer but further progress is clearly needed, particularly to better parameterize diffusivity at depth (e.g., Fig. 11b).
The implication of coupled eddy and mean flow processes, for example, via nonhomogeneous turbulence, being responsible for the mixing in baroclinic jets is potentially profound for mixing parameterizations. First, mean flow suppression formulations demonstrating diffusivity amplification at depth appear necessary, as suggested by MSCL theory. But parameterizations based on linear baroclinic growth rates are likely unable to capture the enhancement of diffusivity because the time integral “memory” required to capture the turbulent interactions of eddies with the mean flow is lacking (Sinha and Abernathey 2016). The diffusivity based on the linear baroclinic growth rate (Bretherton 1966; Green 1970; Bates et al. 2014; Griesel et al. 2015) can be represented using (1) and (3) following appropriate choices for b and TL, which can be computed from our data via the nonlinear fit shown in Fig. 11b. Following the fit, the diffusivity is 20% too large near the critical layer but nearly a factor of 4 too small near the channel bottom, reflecting the inability of the theory to appropriately account for the vertical structure of the diffusivity. Third, linear mixing length theories ultimately may not be able account for nonlinear baroclinic growth rates unless Kmax is nonlinear beyond the structured imposed by depth variable urms, with depth variable Lmix, TL, and/or mixing efficiency Γ, that is, in (2) and (3) eddy size and mixing memory arising from interactions with the mean flow appear to be important. For example, this is why Lmix is typically scaled directly from the Rossby radius of deformation but requires additional information for its estimation, for example, evaluation via r as in (22).
Therefore, we ultimately conclude that a better understanding of the vertically variable feedback between the eddy and mean flows is necessary to build better parameterizations for mixing. Progress toward this endeavor is challenging and has been made here. However, a full understanding of parameterization will likely require more advanced decompositions potentially using both Eulerian spatial and temporal filtering. Lagrangian decomposition of xFULL via generalized Lagrangian mean theory (Andrews and McIntyre 1978; Craik 1985; Bühler 2014, chapter 10) may also be fruitful (B. Fox-Kemper 2017, personal communication). The limited applicability of linear parameterizations based on linear baroclinic growth rates or standard MSCL theory implies that parameterizations may require large-eddy simulation approaches to model nonlinear subgrid-scale interactions (Eden and Greatbatch 2008; Marshall and Adcroft 2010), as also noted by Sinha and Abernathey (2016). In general, the vertical structure of diffusivity does not appear to be simply self-similar to the stratification N2 or the first Rossby radius of deformation eigenvector (Bates et al. 2014; Wolfram et al. 2015). Instead, it requires consideration of mixing suppression over depth to represent the vertical diffusivity maxima in the ICC; additional improvements will likely also require some parameterization of the residual diffusivity κDIFFU that modifies the vertical structure of the diffusivity, particularly below the critical layer. These conclusions should be interpreted within the context of our diagnostic’s capabilities. Particle methods typically only track fluid motions. Mixing is a product of the time integral instantaneous straining and stretching, background fluid diffusivity, and the time history of a particular concentration field and its gradients as it is strained over time by a particular Eulerian velocity field. Simplistic particle statistics that lack representation of the concentration gradient cannot therefore directly quantify the time integral because they assume concentration invariance, as demonstrated via the connection made between particle dispersion and effective diffusivity in Klocker et al. (2012b). The concentration invariance assumption may be violated for certain spatiotemporal scales. The shortcoming of diffusivity metrics based on particle dispersion is that reversible and irreversible mixing are indistinguishable.
However, the particle statistics do loosely quantify the maximum mixing that may occur due to straining. Classical techniques (Taylor 1921, 1935) require a decorrelation time scale because the latent assumption, particularly for homogeneous turbulence, is that the straining eventually results in decorrelation that is representable by a diffusion operator or Redi (1982) diffusivity. In principle, this implies that diffusivity metrics relying upon particle statistics provide upper bound estimates for mixing. For example, no mixing occurs for a highly strained fluid if the passive tracer is already fully mixed. It is also possible that the conclusions derived in this paper, namely, that mixing is highly dependent upon mixing enhancement due to turbulent eddy and mean flow interactions, are exaggerated because the computed diffusivity may also be exaggerated relative to transport of a passive tracer in the real ocean. However, consistency between particle, tracer, and effective diffusivity methods has been demonstrated in certain cases (Klocker et al. 2012b; Abernathey et al. 2013), suggesting that the results in this paper are at a minimum a reasonable provisional estimate.
Further diffusivity decompositions, perhaps with Eulerian spatial and/or bandpass temporal filters or via use of generalized Lagrangian mean theory for the decomposition of Lagrangian trajectories (B. Fox-Kemper 2017, personal communication; Andrews and McIntyre 1978; Craik 1985; Bühler 2014, chapter 10), are likely needed to better explain the components of turbulent diffusivity and provide improved estimates for the unsuppressed diffusivity, ultimately working toward a diffusivity closure. Investigations with this focus are likely necessary to provide the key insights needed to develop robust parameterizations of diffusivity arising from mesoscale eddies.
Acknowledgments
This research was supported by the Office of Science, Office of Biological and Environmental Research of the U.S. Department of Energy Accelerated Climate Model for Energy (ACME) and used computational resources provided by the Los Alamos National Laboratory Institutional Computing facility. Code developments and simulations relied heavily on the work of the MPAS dynamical core development team at LANL and NCAR and in particular the contributions from the MPAS-Ocean development team at LANL. We thank Matt Rocklin for his support of the Dask (Rocklin 2015; Dask Development Team 2016) and distributed Python packages that we use to perform off-node parallel calculations, Stephan Hoyer and Joe Hamman for help with use and development of the Xarray Python package (Hoyer and Hamman 2017), Andrew Stewart for discussions of idealized Southern Ocean physics, Matt Rayson for sharing particle pathline plotting code, Luke van Roekel for contributions to the design of ICC, Mat Maltrud for discussions of mixing theory, Juan Saenz for discussions related to design of idealized Southern Oceans and transport along isopycnal surfaces, Andy Hogg and Ru Chen for their helpful comments on preliminary results leading to this work, Andreas Klocker for sharing preprints of his work, and reviewers Andreas Klocker, Dhruv Balwada, and Baylor Fox-Kemper whose comments greatly improved the quality of this manuscript.
APPENDIX A
Online Temporal Filtering
APPENDIX B
Quantifying Reduction of Nonlinear Turbulent Mixing
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