Editorial Type:
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Article Type:
Research Article

Irreversible Mixing Induced by Geostrophic Turbulence over the Global Ocean

Tongya Liu aState Key Laboratory of Satellite Ocean Environment Dynamics, Second Institute of Oceanography, Ministry of Natural Resources, Hangzhou, China
bSouthern Marine Science and Engineering Guangdong Laboratory (Zhuhai), Zhuhai, China

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https://orcid.org/0000-0001-6945-1393
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Yu-Kun Qian cState Key Laboratory of Tropical Oceanography, South China Sea Institute of Oceanology, Chinese Academy of Sciences, Guangzhou, China

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Xiaohui Liu aState Key Laboratory of Satellite Ocean Environment Dynamics, Second Institute of Oceanography, Ministry of Natural Resources, Hangzhou, China
bSouthern Marine Science and Engineering Guangdong Laboratory (Zhuhai), Zhuhai, China

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Shiqiu Peng cState Key Laboratory of Tropical Oceanography, South China Sea Institute of Oceanology, Chinese Academy of Sciences, Guangzhou, China

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Dake Chen aState Key Laboratory of Satellite Ocean Environment Dynamics, Second Institute of Oceanography, Ministry of Natural Resources, Hangzhou, China
bSouthern Marine Science and Engineering Guangdong Laboratory (Zhuhai), Zhuhai, China

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Online Publication:
04 Mar 2024
Print Publication:
01 Mar 2024
DOI:
https://doi.org/10.1175/JPO-D-23-0071.1
Page(s):
911–928
Free access

Abstract

Two recently proposed mixing diagnostics are employed to estimate the global surface irreversible mixing based on particle and tracer simulation driven by satellite-derived geostrophic velocities. These two novel diagnostics, similar to the traditional dispersion diffusivity and Nakamura’s effective diffusivity but defined in a localized and instantaneous sense, have the following advantages: 1) they reconcile the theoretical discrepancies between Eulerian-, particle-, and contour-based diffusivities and 2) they do not rely on the stationary and homogeneous assumptions of the turbulent ocean and are free from traditional average operators (e.g., Eulerian time–space or along-contour mean). Our results show that evident discrepancies among these three types of diffusivities do emerge when employing traditional estimates. However, these discrepancies could be significantly mitigated with the adoption of new diagnostic methods, implying that the three types of diffusivities can be effectively reconciled within a global framework. Moreover, finescale mixing structures and transient elevated mixing events due to geostrophic stirring can be clearly identified by the two new diagnostics, in contrast to previous estimates that are spatially and/or temporally smoothed. In particular, it is interesting to note that large values of the new diagnostics usually occur along narrow filaments/fronts associated with mesoscale eddies, and elevated mixing is observed to be located at the periphery of eddies. Our study presents a novel revisit of the global surface mixing induced by geostrophic eddies with an emphasis on irreversibility and provides new insights into previous questions regarding different mixing diagnostics in the community.

Significance Statement

Previous estimates of eddy mixing over the global ocean, using particle-based, tracer-based, and Eulerian-based diffusivities, have shown evident discrepancies. By using recently proposed novel mixing diagnostics, this study demonstrates that the three types of diffusivity estimates agree well with each other, indicating a practical unification of the three types of diffusivities. Also, since the new mixing diagnostics do not involve any traditional average operator, the local and instantaneous mixing maps over the global ocean are presented here, in contrast to previous spatial- or temporal-averaged ones. These new insights can address several unresolved issues in the mixing community.

© 2024 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Yu-Kun Qian, qianyukun@scsio.ac.cn

Abstract

Two recently proposed mixing diagnostics are employed to estimate the global surface irreversible mixing based on particle and tracer simulation driven by satellite-derived geostrophic velocities. These two novel diagnostics, similar to the traditional dispersion diffusivity and Nakamura’s effective diffusivity but defined in a localized and instantaneous sense, have the following advantages: 1) they reconcile the theoretical discrepancies between Eulerian-, particle-, and contour-based diffusivities and 2) they do not rely on the stationary and homogeneous assumptions of the turbulent ocean and are free from traditional average operators (e.g., Eulerian time–space or along-contour mean). Our results show that evident discrepancies among these three types of diffusivities do emerge when employing traditional estimates. However, these discrepancies could be significantly mitigated with the adoption of new diagnostic methods, implying that the three types of diffusivities can be effectively reconciled within a global framework. Moreover, finescale mixing structures and transient elevated mixing events due to geostrophic stirring can be clearly identified by the two new diagnostics, in contrast to previous estimates that are spatially and/or temporally smoothed. In particular, it is interesting to note that large values of the new diagnostics usually occur along narrow filaments/fronts associated with mesoscale eddies, and elevated mixing is observed to be located at the periphery of eddies. Our study presents a novel revisit of the global surface mixing induced by geostrophic eddies with an emphasis on irreversibility and provides new insights into previous questions regarding different mixing diagnostics in the community.

Significance Statement

Previous estimates of eddy mixing over the global ocean, using particle-based, tracer-based, and Eulerian-based diffusivities, have shown evident discrepancies. By using recently proposed novel mixing diagnostics, this study demonstrates that the three types of diffusivity estimates agree well with each other, indicating a practical unification of the three types of diffusivities. Also, since the new mixing diagnostics do not involve any traditional average operator, the local and instantaneous mixing maps over the global ocean are presented here, in contrast to previous spatial- or temporal-averaged ones. These new insights can address several unresolved issues in the mixing community.

© 2024 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Yu-Kun Qian, qianyukun@scsio.ac.cn
Keywords: Eddies; Mesoscale processes; Mixing

1. Introduction

Mesoscale eddies, primarily induced by the baroclinic instability of the large-scale density field, can stir, mix, and transport oceanic tracers on a global scale, which has a significant effect on the general ocean circulation and the climate-related issues (Lumpkin and Elipot 2010; Abernathey and Marshall 2013, hereinafter AM13; Busecke and Abernathey 2019; Liu et al. 2022). Proper estimates of the lateral eddy mixing (or diffusivity) would greatly benefit the accurate subgrid-scale parameterization and thus the performance of numerical models.

The essential question of estimating eddy mixing, as well as quantifying its spatial–temporal variations, is not how to measure but what to measure (Nakamura 2001). The existing mixing diagnostics, defining what to measure, typically fall in three categories: Eulerian, Lagrangian, and contour-based diffusivities (e.g., Marshall et al. 2006; Qian et al. 2019; Kamenkovich et al. 2021). One commonly used Eulerian diffusivity is the Osborn–Cox diffusivity (Osborn and Cox 1972; Nakamura 2001) defined as
KOC=κm|∇xyq(x,y)|2¯/|∇xyq¯(x,y)|2,
where q is a quasi-conservative tracer, κm the small-scale (molecular) diffusivity, ∇xy the horizontal gradient operator, and (   )¯ the temporal–spatial (Eulerian) average. It has been applied to diagnose diffusivity in the global atmosphere and oceans using gridded satellite or reanalysis data (e.g., Nakamura 2001; AM13). Lagrangian diffusivity, also known as particle dispersion diffusivity, is defined as the half growth rate of the mean squared displacement of Lagrangian particles (LaCasce 2008; Fox-Kemper et al. 2013):
KL(τ)=12ddτ⟨(D−⟨D⟩)2⟩L,
where D is the displacement of a particle during a time interval τ along a particular direction, and 〈 〉L is the ensemble average over many particles or realizations. Rich Lagrangian observations make this method widely used in regional (e.g., Bauer et al. 2002; Lumpkin et al. 2002; Rypina et al. 2012; Qian et al. 2013; LaCasce et al. 2014; Roach et al. 2016) and global estimates (e.g., Zhurbas et al. 2014; Roach et al. 2018). Effective diffusivity is one of the contour-based diffusivities that measures the rate of material transport across tracer contours (Nakamura 1996), and is defined as
Keff=κm⟨|∇xyq(x,y)|2⟩q∗/|∇XYq∗(X,Y)|2,
where q∗(X,Y) is an adiabatically rearranged (i.e., meridionally sorted and hence ∂q∗/∂X=0) state of tracer q so that it essentially becomes the lowest mixing-efficiency state among all possible ones (Fig. 1a). Here, ⟨  ⟩q∗ is the average along a contour of q with value q∗, and ∇XYq∗ is the gradient of the sorted state. Effective diffusivity would be large if tracer contours are extremely deformed by eddies (Fig. 1c), and reduced to κm in the sorted state if there is no stirring at all (Nakamura 2008). Marshall et al. (2006) introduced this diagnostic to the oceanic context and applied it to the estimate of meridional eddy transport over the Southern Ocean.
Fig. 1.
Fig. 1.

Schematic illustration of different turbulent mixing states: (a) meridional-sorted state, (b) Eulerian time-mean state, and (c) instantaneous eddying state. The plots in (a) and (b) represent the lowest mixing-efficiency states for Keff and KOC, respectively.

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0071.1

The aforementioned three types of diffusivities have been applied to measure eddy-induced mixing, either globally (e.g., AM13; Klocker and Abernathey 2014; Zhurbas et al. 2014; Groeskamp et al. 2020) or regionally (e.g., Lumpkin et al. 2002; Marshall et al. 2006; Abernathey et al. 2010; Lu and Speer 2010; Qian et al. 2013; Chen et al. 2014), offering different perspectives to understand their spatial and temporal variations. A thorough comparison of these different views then presents several problems that need to be resolved. First, mixing estimates from several frameworks reveal significant discrepancies in pattern and magnitude (e.g., Riha and Eden 2011; Klocker et al. 2012b). For example, the Lagrangian diffusivity by Sallée et al. (2008) reaches 104 m2 s−1 around the western boundary currents north of the Antarctic Circumpolar Current (ACC; their Fig. 3), which is one order of magnitude higher than the effective diffusivity presented by Marshall et al. (2006; their Fig. 5). In addition, several studies also indicated either a maximum or a minimum (e.g., Thompson 2008; Lu and Speer 2010) of diffusivity in the core of the ACC. A similar disparity also exists in the equatorial regions (cf. AM13; Zhurbas et al. 2014). These apparent discrepancies introduce large uncertainties in the eddy parameterization of noneddying climate models. Second, eddy mixing entangles adiabatic stirring and irreversible diffusion. Because of their distinct physical properties, it is best to isolate one from the other and parameterize them separately (Redi 1982; Gent and McWilliams 1990). While effective diffusivity is an accurate measurement of irreversible mixing1 (Nakamura 1996, 2008), Lagrangian diffusivity does not have an explicit linkage with irreversible mixing (i.e., κm does not explicitly present in its definition). Therefore, reversible undulation also leads to particle spreading and sometimes leads to a negative diffusivity in the presence of coherent flows (e.g., Griesel et al. 2010; Klocker et al. 2012b). Third, few diagnostics techniques thoroughly quantify the local and instantaneous estimates of mixing, primarily because of the use of various (temporal, spatial, ensemble, and along-contour) average operators on the basis of the assumptions required. For example, effective diffusivity gives only a contour-averaged result and cannot quantify the along-contour variation of mixing. Osborn–Cox diffusivity is a time-mean diagnostic suitable for stationary turbulence and thus may not be appropriate for dealing with the temporal variation of mixing. The difference in average operators between these mixing techniques also prevents a direct comparison.

Although these discrepancies may be a result of varied data sources and calculation procedures, the fundamental issue lies in the definition of diffusivity itself (i.e., what to measure). Efforts have been made to reconcile theoretical discrepancies between the different perspectives. Klocker et al. (2012b) presented that the Lagrangian diffusivity agrees with the effective diffusivity after the initial transient phase (i.e., the Lagrangian diffusivity reaches an asymptotic limit after tens of days). Wolfram and Ringler (2017) also tried to compute Nakamura’s effective diffusivity using Lagrangian particles. Their conceptual picture is clear, but their algorithm is a bit more complex than traditional dispersion calculations. Qian et al. (2019) proposed a novel view of particle displacement relative to tracer contours instead of fixed Eulerian positions. This new approach recovers the Lagrangian nature of both dispersion diffusivity and effective diffusivity, which leads to an exact unification of them. As a result, the new dispersion diffusivity, similar to effective diffusivity that directly links to the molecular diffusivity, is able to isolate reversible undulation and only sense irreversible mixing. Later, Qian et al. (2022) further extended this concept to introduce local instantaneous diffusivities without involving traditional average operators and reconciled Lagrangian and Eulerian diffusivities. The unification of the three types of diffusivities relies on the fact that the reversible mixing is isolated and only the irreversible part is quantified.

The present study aims to apply the three types of reconciled mixing tools to revisit the issue of ocean surface mixing by geostrophic eddies, in which previous discrepancies of different estimates can be addressed. Also, local and instant estimates can be demonstrated using the new diagnostics, which is not generally available in previous studies. The rest of the paper is organized as follows. Section 2 introduces the new local diffusivities and the data used for diffusivity estimates. Section 3 presents the comparison between different mixing estimates. Section 4 presents the conclusions and discussion.

2. Methods and data

a. Definitions of local mixing diagnostics

The theoretical developments in Lagrangian diffusivity are based on a contour-based coordinate identical to that of the effective diffusivity (Qian et al. 2019). Constructing the new coordinate is equivalent to sorting the tracer adiabatically in the meridional direction into a monotonic profile (Fig. 1a). Adiabaticity ensures that the area enclosed by a wavy tracer contour with value q∗ (herein defined as smaller than this value) equals to that south of the equivalent latitude Y(q∗) in the sorted space:
∬q<q∗dxdy=∬Y<Y(q∗)dXdY.
Therefore, one could obtain a one-to-one Y(q∗) relation and then a new X–Y space (Fig. 1a). Due to the presence of topography, the Y(q∗) relation is obtained numerically using the box-counting method (Nakamura 2008). Figures 1a and 1c, respectively, show the sorted tracer state and the original one schematically. The Lagrangian dispersion diffusivity is then introduced with a new definition of particle displacement D in the X–Y space (Qian et al. 2019):
D≡ΔY≈ΔqJ=Δq∂Y∂q∗.
Here, Δq is the variation of a tracer along a Lagrangian particle over a small time interval Δt, and J is the thickness between two tracer contours after the sorting (also known as Jacobian for coordinate transform from x–y space to X–Y space). This displacement D is different from the traditional one in x–y space: if the tracer evolution is fully adiabatic, its Lagrangian parcel will remain on the same tracer contour and does not move in the X–Y space (D ≡ 0). However, this parcel can still be displaced by adiabatic advection in the x–y space.
With this novel definition of displacement, a new local (single-particle) dispersion diffusivity:
K˜L≡12dD2dt,
is proposed by Qian et al. (2022). Note that K˜L is the decorrelated or asymptotic diffusivity here, so it does not depend on the time lag dt, in significant contrast to the traditional one. In addition, a local effective diffusivity:
K˜eff≡κm|∇xyq|2|∇XYq∗|2,
is also defined by dropping the along-contour mean operator ⟨  ⟩q∗ in the original definition Eq. (3). These two diagnostics are free from previous average operators, and thus could provide localized, instantaneous, irreversible estimates in a turbulent ocean that is far from being statistically stationary and homogeneous (Qian et al. 2022). After taking the along-contour mean ⟨  ⟩q∗, both methods recover Nakamura’s effective diffusivity exactly as ⟨K˜L⟩q∗=⟨K˜eff⟩q∗=Keff. Qian et al. (2022) also demonstrated that K˜L=K˜eff, indicating a local equivalence between particle-based and contour-based diffusivity when the particle’s displacement is defined in contour space.
If a Eulerian time mean ()¯ is applied to Eq. (7), and noting that q∗ evolves very slowly in response to small diffusion, then one has
K˜eff¯=κm|∇xyq|2/|∇XYq∗|2¯≈κm|∇xyq|2¯/|∇XYq∗|2¯≈κm|∇xyq|2¯/|∇XY(q¯)∗|2,
which is quite similar to the definition of KOC [Eq. (1)]. The difference lies in the denominators: KOC uses the time-mean tracer distribution as the lowest mixing-efficiency state (Fig. 1b), while K˜eff¯ uses the (time-mean) sorted tracer distribution (Fig. 1a). We will discuss this disparity in global estimates later.

b. Advection of tracers and particles using AVISO data

In this study, we use the satellite altimetry product distributed by Archiving, Validation and Interpretation of Satellite Oceanographic (AVISO) and Copernicus Marine Environment Monitoring Service (CMEMS) in the frame of the SSALTO/DUACS altimetry data processing (available at http://www.aviso.altimetry.fr/duacs/). The gridded product is generated based on the along-track measurements from several altimeter missions, and it provides several variables including sea level anomaly, absolute dynamic topography, and near-surface geostrophic velocities derived according to the geostrophic relation. Note that a higher-order vorticity balance is used to estimate the velocities in the equatorial region (between ±5°) where the geostrophy does not hold (Lagerloef et al. 1999). These daily variables are available on a 1/4° latitude–longitude grid. The period from January 1993 to December 2019 is selected for our analysis. Compared with the satellite data used in AM13, the grid resolution of the new version data is increased from 1/3° to 1/4°, and the time frequency is changed from 7 days to 1 day.

The precomputed geostrophic velocities are used to advect synthetic passive tracers and Lagrangian particles. Following AM13, the original AVISO velocity fields are linearly interpolated onto a 1/10° latitude–longitude grid. Note that this operation does not necessarily improve the resolution of the flow field as geostrophic currents fail to resolve small-scale/high-frequency processes, such as submesoscale flows, tides, and inertia–gravity waves (Liu and Abernathey 2023). Several potential reasons, including the meridional variation of Coriolis parameter and the algorithm for estimating velocities at the equator, render that the AVISO-derived velocities do not satisfy the nondivergent requirement (AM13). To conserve tracers in this two-dimensional flow field, a small correction is applied to the velocities to remove the divergence and to enforce the no-normal-flow boundary condition at the coastlines. For further details of the data correction, the reader is referred to the works of Marshall et al. (2006) and AM13. Figure 2 shows the original field, the corrected field, and their differences for eddy kinetic energy (EKE). It is found that the corrections are small in magnitude compared to the original velocity in the global ocean, except for the equatorial region. As a result, the stirring effect by eddies in the equatorial region could be somewhat underestimated.

Fig. 2.
Fig. 2.

(a) Original and (b) corrected distributions of EKE (m2 s−2), as well as (c) their differences. A logarithmic scale is used in (a) and (b), while (c) shows the absolute difference. Note that the areas possibly covered with sea ice in high latitudes are masked out during the selected period.

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0071.1

The Massachusetts Institute of Technology general circulation model (MITgcm; Marshall et al. 1997) is used to perform the tracer and particle advection in “offline” mode. For a tracer q in a two-dimensional flow, the “offline” model solves the advection–diffusion equation as
∂q∂t=−u⋅∇q+κm∇2q,
where u = (u, υ) is the velocity field loaded from the corrected AVISO data, and κm = 20 m2 s−1 is used in this study. Here the second-order moment Prather scheme (Prather 1986) suggested by Hill et al. (2012) is used for advection. This scheme uses extra coefficients to fit the subgrid-scale tracer variabilities, and thus is expected to have an effective resolution higher than that of the model grid. In AM13, three types of initial tracers, latitude (TrLAT) and climatological sea surface temperature (SST, TrSST) from World Ocean Atlas 2013, and streamfunction for the mean flow (TrPSI), are used. Here, only the first two tracers are used because it is not convenient to monotonously sort the contour of TrPSI with local maxima and minima. Also, local extrema could lead to an unbounded estimate of diffusivity (AM13) because of the zero-denominator problem. For TrSST, it peaks at the equatorial region and we sort it hemispherically to ensures the monotonicity of the tracer distribution in each hemisphere. We initialize the global tracer on the first day of a year, normalize it to values between 1 and 2, and run the model for one year from 1993 to 2019. Hence these processes are repeated for a total of 27 times. The integration time of one year is chosen with reference to the studies by AM13 and Busecke and Abernathey (2019). This ensures that tracer gradients will not be completely smeared out by diffusion over some regions, thus allowing all diffusivities to be well defined. Figure 3 shows the instantaneous concentrations for the two tracers on days 0, 100, and 365 in 2016 (randomly selected). Although the initial tracer patterns are quite different, the contours of the two tracers are basically aligned with the flow after an approximate 3-month spinup period (e.g., Shuckburgh et al. 2009).
Fig. 3.
Fig. 3.

Concentrations for the two tracers, (a),(c),(e) TrLAT and (b),(d),(f) TrSST, on days 0, 100, and 365 in 2016. The initial tracer concentrations are normalized to values between 1 and 2.

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0071.1

Besides the passive tracer, we deploy Lagrangian particles with a resolution of 0.1° over the global ocean on the first day of 2016. The kinematic equation dx/dt = u is solved to track all particles using the fourth-order Runge–Kutta scheme, where x = (x, y) is the particle position. Note that the same geostrophic flow field u is used to drive both the passive tracer and the Lagrangian particles. This eliminates the potential influence of different flows in resulting different diffusivity estimates. For more details on the particle tracking, readers are referred to the study of Liu and Abernathey (2023). Figure 4 shows sample trajectories of Lagrangian particles during the whole year of 2016, which clearly shows the particle motion under the geostrophic turbulence. Several particles are selected to highlight the difference in their meridional positions in the two coordinates. The meridional displacements in the geographic coordinates are mostly larger than those in the contour-based coordinates because the former is mainly caused by the reversible undulation of eddies, while the latter is caused only by irreversible motion (analog to the diapycnal motion in the context of vertical mixing). Note that particle tracking is conducted only for one year to demonstrate the traditional and new Lagrangian diffusivities.

Fig. 4.
Fig. 4.

(a) Sample trajectories of Lagrangian particles deployed uniformly over the global ocean. (b) Meridional positions in the geographical and equivalent latitudes for selected particles [large black dots with thick gray lines in (a)].

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0071.1

3. Results

a. Traditional mixing estimates

Prior to presenting the new mixing diagnostics [Eqs. (6) and (7)] introduced by Qian et al. (2022), we first review three traditional estimates using [Eqs. (1)–(3)]: the Osborn–Cox diffusivity KOC (Osborn and Cox 1972), Nakamura’s effective diffusivity Keff (Nakamura 1996), and the traditional Lagrangian diffusivity KL (e.g., LaCasce 2008). Note that traditional estimates of KOC, Keff, and KL employ the time average, along-contour average, and ensemble average, respectively.

Updated maps of KOC for two tracers are shown in Figs. 5a–d on a logarithmic scale. Although our estimated diffusivity is slightly stronger than that in AM13 because of using the daily data, the general pattern of KOC has been well reproduced. On the flanks of western boundary currents, such as the Kuroshio in the Pacific Ocean, the Gulf Stream in the Atlantic Ocean, and the Agulhas Current in the Indian Ocean, the values of KOC can exceed 104 m2 s−1. There are relatively low values (KOC < 103 m2 s−1) in the subpolar region corresponding to the weak EKE (Fig. 2a). The spatial patterns of KOC for two tracers (TrLAT and TrSST) are quite similar except for the tropics, which can be attributed to the fact that TrLAT has no initial local extrema but TrSST contains local maxima in the SST warm pool and cold tough regions. From the definition Eq. (1), it is clear that the vanishing of the mean tracer gradients at tropics for TrSST leads to a large area of diffusivity exceeding 104 m2 s−1. The successful duplication of the KOC pattern in AM13 verifies our numerical configurations and diagnostic calculations.

Fig. 5.
Fig. 5.

A comparison among three traditional diffusivity estimates. KOC (colors; m2 s−1) estimated from two tracer distributions: (a) TrLAT and (c) TrSST. (b),(d)The zonally averaged KOC (blue lines) is compared with Keff (red lines). The values are shown on a logarithmic scale. The dashed lines in (d) indicates the latitude band where contours of TrSST intersect the equator and hence a possibly degenerated Keff there. (e) Minor (or cross-stream) component of the traditional Lagrangian dispersion diffusivity tensor (m2 s−1) and (f) its zonal mean.

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0071.1

The effective diffusivity has been employed to estimate the irreversible mixing in the Southern Ocean (Marshall et al. 2006) and a sector of the east Pacific (AM13). These regions are mainly featured in substantial zonal flows without blocking effects of land. The zonally averaged KOC is compared with the effective diffusivity Keff over the global ocean in Figs. 5b and 5d. Unlike the good agreement between KOC and Keff in the east Pacific (see Fig. C2 in AM13), the global estimates show evident discrepancies in most latitudes, which is primarily due to the different definitions of the lowest mixing state in their denominators and the average operators. Since at most latitudes the sorted state has somewhat stronger meridional gradients, the profile of Keff is generally smaller than KOC (Figs. 5b,d). Note that in the case of TrSST, the tracer maximum is located at the equatorial region. As a consequence, the mixing at equator associated with maximum SST contour will be mapped to the northern boundary in the equivalent latitude space. To overcome this issue, we calculate Keff through sorting the tracer hemispherically to ensure monotonicity. However, the reliability of Keff might be low when tracer contours intersect the equator, as there are advective fluxes contributing the area changes enclosed by these contours (see the leveling off of Keff at the equator in Fig. 5d).

We also calculate the traditional Lagrangian dispersion diffusivity tensor using traditional pseudotracking methods (e.g., Swenson and Niiler 1996; Klocker et al. 2012b). That is, every single point in a 1° bin is taken as a “release” point and then calculate the statistics relative to this point. A practical time average over a time lag between 12 and 16 days is taken as the asymptotic or decorrelated estimate (e.g., Chen et al. 2014; LaCasce et al. 2014; Peng et al. 2015). Additionally, to minimize shear-induced dispersion, a principal-axes rotation is conducted to obtain the major and minor (or along stream and cross stream) components of the diffusivity tensor (Oh et al. 2000; Rypina et al. 2012). Figures 5e and 5f show the minor component of Lagrangian diffusivity, which is typically viewed as the asymptotic cross-stream component (e.g., Zhurbas et al. 2014). A large diffusivity is located in equatorial and subtropical regions, which is well consistent with the estimates using surface drifters reported in several studies (e.g., Lumpkin et al. 2002; Zhurbas and Oh 2004; Sallée et al. 2008; Zhurbas et al. 2014; Peng et al. 2015).

The aforementioned reproductions of the diffusivity maps clearly indicate that although the tracer and particle data are generated consistently by the same geostrophic flow, different methods still provide quite varied estimates. The reason leading to these discrepancies does not lie in different data types but in the different definitions of diffusivity. To tackle this problem, we will employ new mixing diagnostics.

b. New mixing diagnostics and reconciliation with previous ones

Here we present the global mixing diagnostics using two new diffusivities [Eqs. (6) and (7)], the local dispersion diffusivity K˜L and the local effective diffusivity K˜eff, both of which examine local, instantaneous, irreversible mixing. On the basis of theoretical arguments and idealized simulations, Qian et al. (2022) have proved that K˜L and K˜eff are equivalent, indicating a local equivalence between particle-based and contour-based diffusivities.

This can be observed from Fig. 6, which shows the instantaneous distributions of K˜L and K˜eff for TrLAT over the global ocean at four selected times (50th, 100th, 200th, and 300th day) in 2016 (a randomly selected year). Note that by definition, an along-contour average of Fig. 6 recovers exactly Nakamura’s standard effective diffusivity at each time step (i.e., ⟨K˜L⟩q∗=⟨K˜eff⟩q∗=Keff; Qian et al. 2022). The initial values are quite small (not shown) because the initial tracer concentration only has large-scale meridional gradients. As the tracer is stirred by geostrophic turbulence, the finescale filamentary structures emerge globally as revealed by the tracer distribution (see Fig. 3). On the 50th day, the general patterns of the two diffusivities show remarkable features, with the strongest local mixing at around five western boundary current regions. From the 100th to 300th day, the spatial patterns and magnitudes of the two diffusivities do not present significant variations with time, which means that the eddy activity has saturated and that there is a dynamical balance between filament generation by stirring and smoothing/homogenization by irreversible mixing. To measure how long the equilibrated K˜eff can be reached, an e-folding time scale is estimated through a least squares fit (e.g., Chiswell 2013) of the 27 yearly samples (figure not shown). The time scale over the Southern Ocean is about 2–3 months, which is consistent with that of Shuckburgh et al. (2009), and the global-averaged value is approximately 4 months.

Fig. 6.
Fig. 6.

Snapshots of (left) local Lagrangian diffusivity K˜L and (right) local effective diffusivity K˜eff for TrLAT (m2 s−1) over the global ocean on the 50th, 100th, 200th, and 300th days in 2016. The values are shown on a logarithmic scale. The red dashed lines in (b) are two meridional sections shown in Fig. 12.

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0071.1

The exceptional agreement between the particle-based K˜L and contour-based K˜eff over the global ocean at each time step relies on that both of them are measuring the same (irreversible) part of mixing. Physically, if a particle penetrates many contours locally, especially in the filamentary regions where contours are squeezed together in a compact form by stirring (Wagner et al. 2019), a large value of K˜L will generally be observed in that region. Then, all the particles passing through the same contour become the cross-contour flux that Nakamura’s effective diffusivity tries to measure. This is quite different from the traditional dispersion diffusivity that is close to effective diffusivity only in the channel-like region and only after an initial overshoot in autocorrelation (e.g., Klocker et al. 2012b). In the following analysis, we mainly focus on K˜eff because the gridded data are more accessible than the Lagrangian trajectory.

How about the relationship between K˜eff and KOC? As discussed in section 2a, the expression of K˜eff is very similar to that of KOC, both of which are the ratio of squared tracer gradients. One of their differences is that KOC can only be used to measure the climatological mixing after introducing the time-mean operator, while there is no such limitation for K˜eff. To compare with KOC, we calculate K˜eff for TrLAT and TrSST every day from 1993 to 2019 and take the time mean as the climatological K˜eff¯ in Fig. 7. Several prominent attributes of K˜eff¯ are comparable to those of KOC (Fig. 5 and AM13), including large values around the western boundary current regions (>104 m2 s−1), small values in the subpolar regions (<103 m2 s−1), and strong mixing in the tropics. The differences mainly lie in the spatial pattern along the main path of western boundary currents where KOC shows quite small values (<500 m2 s−1) in the jet core regions. This phenomenon has been noticed in previous studies (e.g., Ferrari and Nikurashin 2010; Klocker et al. 2012a; AM13; Groeskamp et al. 2020), and this mixing barrier reflects the suppression induced by strong currents. In addition, the meridional profile of K˜eff¯ is quite different from KOC even when both time and zonal means are taken (Figs. 7b,d).

Fig. 7.
Fig. 7.

Time means of local effective diffusivity K˜eff¯ (colors) estimated from two tracer distributions: (a) latitude and (c) SST. (b),(d)The zonally averaged K˜eff¯ (red lines) is compared with KOC (blue lines).

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0071.1

These differences arise mainly because the two diffusivities adopted quite different background states as the lowest mixing state, as already illustrated in Figs. 1a and 1b. Comparing Eq. (1) with Eq. (8), it is obvious that KOC and K˜eff¯ contain an identical numerator but different denominators. It is possible to reconcile their discrepancies by introducing the local effective diffusivity (normalized by κm) for the time-mean tracer distribution q¯ defined as (Qian et al. 2022):
K˜effq¯=|∇xyq¯|2|∇XY(q¯)∗|2,
where the denominator means sorting q¯ in the meridional direction. This is equivalent to taking q¯ as one kind of tracer distribution and evaluating its local effective diffusivity accordingly. Then, we could rectify K˜eff¯ by dividing K˜effq¯ so that the rectified diffusivity:
K˜eff′¯=K˜eff¯K˜effq¯≈κm|∇xyq(x,y)|2¯|∇XYq∗(Y)|2¯|∇XY(q¯)∗|2|∇xyq¯|2=KOC|∇XY(q¯)∗|2|∇XYq∗(Y)|2¯≈KOC
shares the same reference state as that of KOC. Figures 8a and 8c show the values of K˜effq¯ for the two tracers. The physical mean of K˜effq¯ is that for a time-mean q¯ shown in Fig. 1b, mixing is still enhanced relative to the sorted state in Fig. 1a. It is evident that K˜effq¯ is very close to 1 for most of the oceans, indicating the equivalence of K˜eff¯ to KOC in most of the regions. However, this climatological enhancement is significant over the regions of strong boundary currents and ACC (Figs. 8a,c), where semipermanent intense tracer fronts are located.
Fig. 8.
Fig. 8.

Values of (a),(c) normalized local effective diffusivity K˜effq¯ for the time-mean tracer distribution q¯ and (b),(d) base-10 logarithm of the factor of |∇XY(q¯)∗|2/|∇XYq∗(Y)|2¯ for two tracers.

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0071.1

Figure 9 shows the rectified local effective diffusivity K˜eff′¯, which is highly consistent (almost identical) with KOC (Fig. 5) in the spatial pattern and also the zonally averaged profile. Since the remaining factor |∇XY(q¯)∗|2/|∇XYq∗(Y)|2¯ in Eq. (11) is roughly of order unity (Figs. 8b,d) and thus negligible, the main difference between K˜eff and KOC is due to the different choices of the lowest mixing-efficiency state as reference. Also, there are many small-scale features after the rectification (Figs. 9a,c), which are mainly the imprints from the gradients of q¯. The local extrema of q¯, therefore, could lead to unbounded estimates of diffusivity (see equatorial regions in Fig. 9c for TrSST).

Fig. 9.
Fig. 9.

As in Fig. 7, but for rectified local effective diffusivity K˜eff′¯.

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0071.1

So far, we can conclude that the three types of diffusivities (particle-based K˜L, contour-based K˜eff, and Eulerian-based KOC) are eventually reconciled over the global scale. The primary difference between particle-based and contour-based diffusivities is that particle displacement should be measured relative to a wavy contour instead of a fixed Eulerian surface. The difference between contour-based and Eulerian-based diffusivities is that they adopt different reference states. Such differences can be reduced in the Southern Ocean or in the atmospheric context where there is no topography to interrupt the quasi-zonal tracer contours. That is, the sorted state is close to the time-mean state (e.g., Klocker et al. 2012b). However, it can cause quite large discrepancies when the climatological tracer gradients are too strong (need rectification, e.g., over the fontal regions) or too weak (zero-denominator problem, e.g., over the homogenized region), as compared to the sorted state.

c. Local and instantaneous estimates

Traditional diffusivities often involve averaging processes, such as ensemble pseudotrack average, along-contour average, or Eulerian time mean. These operations can eliminate local and instantaneous information that might be crucial for parameterization in numerical models. The new diagnostics, being free from these averaging processes, allow us to investigate the geographical distributions and temporal evolutions of mixing simultaneously, which is of interest here.

Figure 10 shows the instantaneous tracer distribution and its corresponding local effective diffusivity K˜eff in the Agulhas Current region. Snapshots every 10 days from 1 January to 11 February 2016 are considered. The sea level anomaly (SLA) field reflects the existence of a large number of mesoscale eddies in this region. At the initial time, K˜eff has the smallest value because the contours of TrLAT are aligned with latitudes. As the tracer contours are contorted, stretched, folded, and elongated in the turbulent flows, K˜eff becomes larger in an asymptotic sense, with an e-folding time scale of 2–3 months in this region. The most prominent feature is that large values of K˜eff usually occur along narrow filaments/fronts that are associated with mesoscale eddies. This means that fluid elements (e.g., temperature, salinity, or potential vorticity) are irreversibly transported across tracer contours in these filamentary structures with sharp gradients in density and large horizontal velocity (McWilliams 2021). At the center of coherent eddies, K˜eff is quite small due to the homogeneity of the trapped water masses. This spatial distribution of K˜eff successfully quantifies where a tracer contour is significantly stirred and where it is not (thus remains smooth), and thus solves the problem of quantifying the along-contour variation of Nakamura’s effective diffusivity [see the conceptual Fig. 1 in Shuckburgh et al. (2009)]. This new diagnostic clearly displays the finescale spatial structure and detailed temporal variation of the irreversible mixing intensity with the evolution of mesoscale processes (e.g., eddies, fronts, and filaments). Traditional diagnostics may not be able to capture this information because of the use of average operators.

Fig. 10.
Fig. 10.

(left) Instantaneous tracer distribution and (right) its corresponding local effective diffusivity (m2 s−1) in the Agulhas Current region every 10 days from 1 Jan to 11 Feb 2016. Black contours represent the SLA field.

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0071.1

Although K˜eff¯ can be reconciled with KOC by introducing a rectification with K˜effq¯, we need to further explain their differences in the western boundary current regions. Previous studies (e.g., Ferrari and Nikurashin 2010; Klocker et al. 2012a; AM13; Groeskamp et al. 2020) suggest that the mixing diffusivity can be suppressed by the propagation of eddies relative to the mean flow and show relatively weak mixing in the core of the western boundary currents where the mean flow is quite strong. However, the issue is that the stable mean flow can only be observed in the climate state after taking the long-term average. Figure 11 shows the differences between the climatological and instantaneous flow fields in the Kuroshio Extension and Agulhas Current regions. The instantaneous jet is much narrower and more flexuous than the mean flow, and is surrounded by rich eddies shedding from the jet. For KOC, the denominator |∇xyq¯(x,y)|2 provides large-scale tracer gradients in the main path of western boundary currents, turning these regions into mixing barriers. For an instant snapshot, active mesoscale eddies and filamentary processes might lead to the strong exchange of water masses across the climatological jet that does not permanently exist. This part of irreversible mixing cannot be represented by a climatological diffusivity KOC.

Fig. 11.
Fig. 11.

Comparison between climatological (red contours) and randomly selected instantaneous (green contours) absolute dynamic topography in the (a) Kuroshio Extension and (b) Agulhas Current regions.

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0071.1

Figure 12 shows the temporal evolution of K˜eff along two sections (180° and 330°E) with limited lands in 2016. It is clear that the spinup stage only takes tens of days. As shown in Fig. 10, the irreversible mixing is enhanced along the filamentary structures that can last from several weeks to several months. This mixing pattern has also been exemplified by Nakamura (2001), but the traditional effective diffusivity cannot illustrate the local information after taking the along-contour average. Unlike mesoscale eddies, which have propagation preferences in the zonal and meridional directions due to beta effects, the movement of these filamentary structures is mainly determined by the local flow field and does not show a distinct pattern. Considering their significant role in irreversible mixing, the propagation features of filamentary structures are worthy of further investigation.

Fig. 12.
Fig. 12.

Temporal evolution of local effective diffusivity K˜eff (m2 s−1) along (a) 180° and (b) 330°E in 2016. Two meridional sections are indicated in Fig. 6b.

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0071.1

4. Conclusions and discussion

It is a significant challenge for coarse-resolution climate models to accurately reproduce the current climate state, because these models are sensitive to the magnitude and structure of eddy diffusivities (Griffies et al. 2005; Danabasoglu and Marshall 2007; Marshall et al. 2017). However, different mixing diagnostics, typically falling into three categories as Eulerian, Lagrangian, and tracer-based (e.g., Marshall et al. 2006; Qian et al. 2019; Kamenkovich et al. 2021), define different aspects of eddy mixing. This gives rise to the diverging estimates of diffusivity and thus uncertainties of eddy mixing parameterization for ocean models. As such, it is important to understand why the three types of estimates differ (e.g., Klocker et al. 2012b) and how to reconcile them.

Following AM13, we used the surface geostrophic currents from satellite observations to advect passive tracers and synthetic Lagrangian particles and tried to revisit the eddy diffusivity in a global context instead of a channel-like domain (e.g., Riha and Eden 2011; Klocker et al. 2012b; AM13). Based on this consistent dataset, the three traditional estimates using the Osborn–Cox diffusivity KOC, the effective diffusivity Keff, and the Lagrangian diffusivity KL were first calculated. Results show that each of them is well consistent with the estimates in earlier studies (e.g., AM13; Klocker and Abernathey 2014; Zhurbas et al. 2014). However, large discrepancies among the three different methods were also clearly identified in spatial pattern and magnitude. This is then attributed to their fundamental definitions of what aspect of mixing is measured.

Two new mixing diagnostics recently proposed by Qian et al. (2022) were then employed to quantify surface mixing. One is local dispersion diffusivity K˜L with a new definition of particle displacement relative to a wavy tracer contour (Qian et al. 2019), and the other is local effective diffusivity K˜eff, in which no average operator is applied. These two mixing diagnostics are shown to be almost identical in magnitude and spatial pattern (Fig. 6), indicating a local equivalence between particle- and tracer-based diffusivities. Also, by definition, the along-contour average of both diffusivities recovers Nakamura’s (1996) effective diffusivity (Qian et al. 2022). In addition, the time mean of K˜eff, along with a rectification using the same lowest mixing-efficiency state, becomes almost identical to KOC (cf. Figs. 5 and 9). Therefore, the present study finally reconciles the three different types of diffusivities. Such an exact reconciliation depends on isolating reversible undulation and only measuring the irreversible part of eddy mixing.

The advantage of K˜eff (also K˜L) over traditional diagnostics is that it is able to quantify geographical distributions and temporal evolutions of irreversible mixing. Its application in the Agulhas Current region shows that large values of K˜eff usually occur along narrow filaments/fronts that are associated with mesoscale eddies. Also, elevated mixing is found to be located at the periphery of eddies rather than at their centers. In addition, the temporal evolution of irreversible mixing is shown, including a spinup of tens of days and subsequent statistical saturation when eddy stirring and homogenization are in dynamical balance.

Our revisit of surface mixing by geostrophic eddies with new mixing diagnostics has provided new insights into previous questions as follows:

  • Why do particle-based and contour-based diffusivities differ? Traditional particle dispersion diffusivity uses particle motion relative to the fixed Eulerian position. Thus, their motion is dominated by adiabatic advection. As a result, the shear dispersion makes a great contribution to particle dispersion, even in the cross-stream component (see Fig. 4b). Although removing the Eulerian mean velocity could reduce shear dispersion (e.g., Bauer et al. 2002), it becomes a mixed Eulerian–Lagrangian diagnostic (LaCasce 2008) and thus loses its exact connection with contour-based effective diffusivity. As long as particle’s displacement is defined relative to a tracer contour, it essentially represents a diascalar motion (e.g., Winters and D’Asaro 1996), conceptually identical to a diapycnal motion (Fig. 4b). As such, particle motion decorrelates immediately and allows a local and instantaneous estimate of dispersion diffusivity (Qian et al. 2019). This is not generally available using traditional method.

  • Is KOC equivalent to Keff? Although AM13 argued that they are equivalent after time and zonal means, this study showed that they are not (Fig. 5b). The reason lies in the definition of the lowest mixing-efficiency state (or background tracer gradient). Here, K˜effq¯ is introduced because it is an important factor that produces the discrepancies between them. It represents the mixing enhancement of the time-mean state (Fig. 1b) relative to the sorting state (Fig. 1a). Even if the climatological distribution of a tracer is close to the sorted state ( K˜effq¯ approaching unity), as in the reentrant domain (e.g., Klocker et al. 2012b; AM13), the Eulerian mean tracer gradient would generally be weaker than that in the sorted state, leading to somewhat larger estimates of KOC relative to Keff (Fig. 5b). This denominator problem has been recently addressed in the vertical mixing context (Arthur et al. 2017), but not well mentioned in the lateral mixing context.

  • Does the jet stream suppress or enhance mixing? Here, it is critical to select the reference relative to which mixing is being compared. Nakamura’s theory first defines a small-scale (molecular) diffusivity and a state of lowest mixing efficiency accurately (Fig. 1a). Any other kind of distribution is said to enhance mixing relative to this state. The suppression theory by Ferrari and Nikurashin (2010) first introduces an unsuppressed diffusivity (related to EKE), and then quantifies the extent to which mixing is reduced by a mean flow. In Nakamura’s context, those less “enhanced” places can be understood as regions of mixing suppression. Although semantic ambiguity is easy to understand, reconciling both theories still requires further investigation. In addition, estimates from traditional Lagrangian diffusivity usually support an enhancement over the jet stream, or at the very least, no clear suppression is identified (e.g., Sallée et al. 2008; Griesel et al. 2010; Riha and Eden 2011). One of the reasons is that the maximum dispersion diffusivity is used instead of the asymptotic one, and thus roughly scaled with EKE. However, we emphasize here that Lagrangian diffusivity can only be exactly equivalent to effective diffusivity when particle displacement is defined relative to tracer contours (Fig. 6). In this case, both dispersion diffusivity K˜L¯ (not shown) and contour-based diffusivity K˜eff¯ (Figs. 13a,b) demonstrate a less-enhanced feature (or suppression) of mixing over the core of ACC, except at the lee wakes of several topographic features such as Drake Passage, Kerguelen Plateau, Macquarie Ridge, and Eltanin Fracture Zone. Note that KOC has many small-scale features (Figs. 13c,d), and also indicate a near-perfect suppression of mixing at the locations of fronts. We argue that this is largely due to the use of ∇q¯ as the denominator in the definition. In this case, mixing tends to be overestimated where ∇q¯ vanishes and the suppression of mixing can be exaggerated where ∇q¯ is strong [see similar discussions of Arthur et al. (2017) in the context of vertical mixing].

Fig. 13.
Fig. 13.

Climatological distribution of K˜eff¯(m2 s−1) for (a) TrLat and (b) TrSST over the Southern Ocean. (c),(d) As in (a) and (b), but for KOC (m2 s−1). The black lines mark the Subantarctic Front, Polar Front, and Southern ACC Front from north to south, whereas the magenta lines mark the north and south boundaries of ACC (Park et al. 2019).

Citation: Journal of Physical Oceanography 54, 3; 10.1175/JPO-D-23-0071.1

The satellite data are used in this study because the derived geostrophic flow field is the only large-scale velocity observation that resolves mesoscale structures. Sea surface height is an integral property of the ocean’s dynamic state, and its temporal evolution reflects the interaction of various physical processes with different spatial–temporal scales. The variability of the sea surface height (or the SLA) contains a barotropic component (sea level fluctuations) and a baroclinic component induced by changes in temperature and salinity (e.g., Behnisch et al. 2013). However, this does not mean that the flow field can adequately capture submesoscale motions due to limitations in temporal and spatial resolution. The mixing rates presented here should be applicable within the framework of geostrophic turbulence over the surface mixed layer. The filamentary structures around mesoscale eddies can be attributed to the ageostrophic motions induced by geostrophic strain (Zhang et al. 2019). While recent studies (e.g., Balwada et al. 2018) have highlighted the significant role of submesoscale flows in irreversible tracer transport, addressing this issue falls beyond the scope of this study.

Also, scale-dependent (two-particle) Lagrangian mixing tools, such as relative dispersion and relative diffusivity, are not considered here. This is because the satellite altimetry data only capture the deformation scale. Any information below this scale, which holds significant relevance for relative dispersion, would yield spurious results due to the resolution limitation. After accounting for decorrelation, however, both single- and two-particle dispersion diffusivities should be essentially the same (e.g., LaCasce 2008). It is interesting to investigate relative dispersion using finer-resolution data, along with exploring it in contour-based coordinates. Nevertheless, we defer these endeavors to a future study.

Finally, this study only focuses on the diascalar component of the eddy diffusivity tensor and thus cannot address the anisotropy of mixing. For a practical application in Eulerian numerical models, the full diffusivity tensor should be investigated (e.g., Bachman et al. 2015; Kamenkovich et al. 2021; Uchida et al. 2023). However, diascalar diffusion is always changing its direction as contour evolves, and thus lose close connection with the tensor. Also, the linkage between eddy diffusivity tensor and irreversible mixing remains unclear, because in an adiabatic case with κm = 0, symmetric/divergent eddy fluxes can still arise through Eulerian temporal or spatial average. Besides, it is still an outstanding problem of tuning Eulerian eddy diffusivity in numerical models based on Lagrangian type of observations (Rühs et al. 2018). One potential solution to tackle all these problems could be to use a set of tracers as quasi-orthogonal coordinates, such as the 2D coordinates constructed by potential vorticity and potential temperature proposed in the atmospheric context (Hoskins 1991; McIntyre 1980; Nakamura 1995). This would allow the investigation of the mixing tensor in these multiple-contour coordinates, which will be pursued in a future study.

1

Here when small-scale or molecular diffusivity κm = 0, mixing is said to be reversible because fluid parcel does not lose their Lagrangian identity during stirring, in accordance with Nakamura’s theory. However, it is worth noting that in 2D nondivergent case even if κm = 0, tracer variance is still cascading towards smaller and smaller scales. This is essentially also irreversible in the statistical sense that the probability of restoring the tracer state back to its initial state is very unlikely. But here we keep the line with Nakamura and define irreversibility as the case when κm ≠ 0. Then stirring could greatly enhance the efficiency of irreversible mixing. Based on this definition, Lagrangian velocity decorrelation does not necessarily indicate irreversible mixing, as particles do not lose their Lagrangian identity when κm = 0.

Acknowledgments.

This work is jointly supported by the National Natural Science Foundation of China (42227901, 41931182, 41976023, 42376028, 42106008), Zhejiang Provincial Natural Science Foundation of China (LY24D060003), and the Open Project of the State Key Laboratory of Tropical Oceanography, South China Sea Institute of Oceanology, Chinese Academy of Sciences (LTO2107).

Data availability statement.

The satellite altimetry product used in this study is available at http://www.aviso.altimetry.fr/duacs/. The numerical ocean model is available at http://mitgcm.org/. The related algorithms for model configuration and data visualization can be found in a GitHub repository (https://github.com/liutongya/global_mixing).

REFERENCES

  • Abernathey, R., J. Marshall, M. Mazloff, and E. Shuckburgh, 2010: Enhancement of mesoscale eddy stirring at steering levels in the Southern Ocean. J. Phys. Oceanogr., 40, 170–184, https://doi.org/10.1175/2009JPO4201.1.

    • Search Google Scholar
    • Export Citation

  • Abernathey, R. P., and J. Marshall, 2013: Global surface eddy diffusivities derived from satellite altimetry. J. Geophys. Res. Oceans, 118, 901–916, https://doi.org/10.1002/jgrc.20066.

    • Search Google Scholar
    • Export Citation

  • Arthur, R. S., S. K. Venayagamoorthy, J. R. Koseff, and O. B. Fringer, 2017: How we compute N matters to estimates of mixing in stratified flows. J. Fluid Mech., 831, R2, https://doi.org/10.1017/jfm.2017.679.

    • Search Google Scholar
    • Export Citation

  • Bachman, S. D., B. Fox-Kemper, and F. O. Bryan, 2015: A tracer-based inversion method for diagnosing eddy-induced diffusivity and advection. Ocean Modell., 86, 1–14, https://doi.org/10.1016/j.ocemod.2014.11.006.

    • Search Google Scholar
    • Export Citation

  • Balwada, D., K. S. Smith, and R. Abernathey, 2018: Submesoscale vertical velocities enhance tracer subduction in an idealized Antarctic Circumpolar Current. Geophys. Res. Lett., 45, 9790–9802, https://doi.org/10.1029/2018GL079244.

    • Search Google Scholar
    • Export Citation

  • Bauer, S., M. S. Swenson, and A. Griffa, 2002: Eddy mean flow decomposition and eddy diffusivity estimates in the tropical Pacific Ocean: 2. Results. J. Geophys. Res., 107, 3154, https://doi.org/10.1029/2000JC000613.

    • Search Google Scholar
    • Export Citation

  • Behnisch, M., A. Macrander, O. Boebel, J.-O. Wolff, and J. Schröter, 2013: Barotropic and deep‐referenced baroclinic SSH variability derived from Pressure Inverted Echo Sounders (PIES) South of Africa. J. Geophys. Res. Oceans, 118, 3046–3058, https://doi.org/10.1002/jgrc.20195.

    • Search Google Scholar
    • Export Citation

  • Busecke, J. J. M., and R. P. Abernathey, 2019: Ocean mesoscale mixing linked to climate variability. Sci. Adv., 5, eaav5014, https://doi.org/10.1126/sciadv.aav5014.

    • Search Google Scholar
    • Export Citation

  • Chen, R., J. L. McClean, S. T. Gille, and A. Griesel, 2014: Isopycnal eddy diffusivities and critical layers in the Kuroshio Extension from an eddying ocean model. J. Phys. Oceanogr., 44, 2191–2211, https://doi.org/10.1175/JPO-D-13-0258.1.

    • Search Google Scholar
    • Export Citation

  • Chiswell, S. M., 2013: Lagrangian time scales and eddy diffusivity at 1000 m compared to the surface in the South Pacific and Indian Oceans. J. Phys. Oceanogr., 43, 2718–2732, https://doi.org/10.1175/JPO-D-13-044.1.

    • Search Google Scholar
    • Export Citation

  • Danabasoglu, G., and J. Marshall, 2007: Effects of vertical variations of thickness diffusivity in an ocean general circulation model. Ocean Modell., 18, 122–141, https://doi.org/10.1016/j.ocemod.2007.03.006.

    • Search Google Scholar
    • Export Citation

  • Ferrari, R., and M. Nikurashin, 2010: Suppression of eddy diffusivity across jets in the Southern Ocean. J. Phys. Oceanogr., 40, 1501–1519, https://doi.org/10.1175/2010JPO4278.1.

    • Search Google Scholar
    • Export Citation

  • Fox-Kemper, B., R. Lumpkin, and F. O. Bryan, 2013: Lateral transport in the ocean interior. Ocean Circulation and Climate, G. Siedler et al., Eds., International Geophysics Series, Vol. 103, Academic Press, 185–209.

  • Gent, P. R., and J. C. McWilliams, 1990: Isopycnal mixing in ocean circulation models. J. Phys. Oceanogr., 20, 150–155, https://doi.org/10.1175/1520-0485(1990)020<0150:IMIOCM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Griesel, A., S. T. Gille, J. Sprintall, J. L. McClean, J. H. LaCasce, and M. E. Maltrud, 2010: Isopycnal diffusivities in the Antarctic Circumpolar Current inferred from Lagrangian floats in an eddying model. J. Geophys. Res., 115, C06006, https://doi.org/10.1029/2009JC005821.

    • Search Google Scholar
    • Export Citation

  • Griffies, S. M., and Coauthors, 2005: Formulation of an ocean model for global climate simulations. Ocean Sci., 1, 45–79, https://doi.org/10.5194/os-1-45-2005.

    • Search Google Scholar
    • Export Citation

  • Groeskamp, S., J. H. LaCasce, T. J. McDougall, and M. Rogé, 2020: Full‐depth global estimates of ocean mesoscale eddy mixing from observations and theory. Geophys. Res. Lett., 47, e2020GL089425, https://doi.org/10.1029/2020GL089425.

    • Search Google Scholar
    • Export Citation

  • Hill, C., D. Ferreira, J.-M. Campin, J. Marshall, R. Abernathey, and N. Barrier, 2012: Controlling spurious diapycnal mixing in eddy-resolving height-coordinate ocean models – Insights from virtual deliberate tracer release experiments. Ocean Modell., 45–46, 14–26, https://doi.org/10.1016/j.ocemod.2011.12.001.

    • Search Google Scholar
    • Export Citation

  • Hoskins, B. J., 1991: Towards a PV-θ view of the general circulation. Tellus, 43A, 27–35, https://doi.org/10.1034/j.1600-0870.1991.t01-3-00005.x.

    • Search Google Scholar
    • Export Citation

  • Kamenkovich, I., P. Berloff, M. Haigh, L. Sun, and Y. Lu, 2021: Complexity of mesoscale eddy diffusivity in the ocean. Geophys. Res. Lett., 48, e2020GL091719, https://doi.org/10.1034/j.1600-0870.1991.t01-3-00005.x.

    • Search Google Scholar
    • Export Citation

  • Klocker, A., and R. Abernathey, 2014: Global patterns of mesoscale eddy properties and diffusivities. J. Phys. Oceanogr., 44, 1030–1046, https://doi.org/10.1175/JPO-D-13-0159.1.

    • Search Google Scholar
    • Export Citation

  • Klocker, A., R. Ferrari, and J. H. LaCasce, 2012a: Estimating suppression of eddy mixing by mean flows. J. Phys. Oceanogr., 42, 1566–1576, https://doi.org/10.1175/JPO-D-11-0205.1.

    • Search Google Scholar
    • Export Citation

  • Klocker, A., R. Ferrari, J. H. LaCasce, and S. T. Merrifield, 2012b: Reconciling float-based and tracer-based estimates of lateral diffusivities. J. Mar. Res., 70, 569–602.

    • Search Google Scholar
    • Export Citation

  • LaCasce, J. H., 2008: Statistics from Lagrangian observations. Prog. Oceanogr., 77 (1), 1–29, https://doi.org/10.1016/j.pocean.2008.02.002.

    • Search Google Scholar
    • Export Citation

  • LaCasce, J. H., R. Ferrari, J. Marshall, R. Tulloch, D. Balwada, and K. Speer, 2014: Float-derived isopycnal diffusivities in the DIMES experiment. J. Phys. Oceanogr., 44, 764–780, https://doi.org/10.1175/JPO-D-13-0175.1.

    • Search Google Scholar
    • Export Citation

  • Lagerloef, G. S. E., G. T. Mitchum, R. B. Lukas, and P. P. Niiler, 1999: Tropical Pacific near‐surface currents estimated from altimeter, wind, and drifter data. J. Geophys. Res., 104, 23 313–23 326, https://doi.org/10.1029/1999JC900197.

    • Search Google Scholar
    • Export Citation

  • Liu, T., and R. Abernathey, 2023: A global Lagrangian eddy dataset based on satellite altimetry. Earth Syst. Sci. Data, 15, 1765–1778, https://doi.org/10.5194/essd-15-1765-2023.

    • Search Google Scholar
    • Export Citation

  • Liu, T., H.-W. Ou, X. Liu, and D. Chen, 2022: On the role of eddy mixing in the subtropical ocean circulation. Front. Mar. Sci., 9, 832992, https://doi.org/10.3389/fmars.2022.832992.

    • Search Google Scholar
    • Export Citation

  • Lu, J., and K. Speer, 2010: Topography, jets, and eddy mixing in the Southern Ocean. J. Mar. Res., 68, 479–502.

  • Lumpkin, R., and S. Elipot, 2010: Surface drifter pair spreading in the North Atlantic. J. Geophys. Res., 115, C12017, https://doi.org/10.1029/2010JC006338.

    • Search Google Scholar
    • Export Citation

  • Lumpkin, R., A. M. Treguier, and K. Speer, 2002: Lagrangian eddy scales in the northern Atlantic Ocean. J. Phys. Oceanogr., 32, 2425–2440, https://doi.org/10.1175/1520-0485(2002)032%3C2425:LESITN%3E2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Marshall, J., A. Adcroft, C. Hill, L. Perelman, and C. Heisey, 1997: A finite‐volume, incompressible Navier Stokes model for studies of the ocean on parallel computers. J. Geophys. Res., 102, 5753–5766, https://doi.org/10.1029/96JC02775.

    • Search Google Scholar
    • Export Citation

  • Marshall, J., E. Shuckburgh, H. Jones, and C. Hill, 2006: Estimates and implications of surface eddy diffusivity in the Southern Ocean derived from tracer transport. J. Phys. Oceanogr., 36, 1806–1821, https://doi.org/10.1175/JPO2949.1.

    • Search Google Scholar
    • Export Citation

  • Marshall, J., J. R. Scott, A. Romanou, M. Kelley, and A. Leboissetier, 2017: The dependence of the ocean’s MOC on mesoscale eddy diffusivities: A model study. Ocean Modell., 111, 1–8, https://doi.org/10.1016/j.ocemod.2017.01.001.

    • Search Google Scholar
    • Export Citation

  • McIntyre, M. E., 1980: Towards a Lagrangian-mean description of stratospheric circulations and chemical transports. Philos. Trans. Roy. Soc., A296, 129–148, https://doi.org/10.1098/rsta.1980.0160.

    • Search Google Scholar
    • Export Citation

  • McWilliams, J. C., 2021: Oceanic frontogenesis. Annu. Rev. Mar. Sci., 13, 227–253, https://doi.org/10.1146/annurev-marine-032320-120725.

    • Search Google Scholar
    • Export Citation

  • Nakamura, N., 1995: Modified Lagrangian-mean diagnostics of the stratospheric polar vortices. Part I: Formulation and analysis of GFDL SKYHI GCM. J. Atmos. Sci., 52, 2096–2108, https://doi.org/10.1175/1520-0469(1995)052<2096:MLMDOT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Nakamura, N., 1996: Two-dimensional mixing, edge formation, and permeability diagnosed in an area coordinate. J. Atmos. Sci., 53, 1524–1537, https://doi.org/10.1175/1520-0469(1996)053<1524:TDMEFA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Nakamura, N., 2001: A new look at eddy diffusivity as a mixing diagnostic. J. Atmos. Sci., 58, 3685–3701, https://doi.org/10.1175/1520-0469(2001)058<3685:ANLAED>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Nakamura, N., 2008: Quantifying inhomogeneous, instantaneous, irreversible transport using passive tracer field as a coordinate. Transport and Mixing in Geophysical Flows, Springer, 137–164.

  • Oh, I. S., V. Zhurbas, and W. Park, 2000: Estimating horizontal diffusivity in the East Sea (Sea of Japan) and the Northwest Pacific from satellite‐tracked drifter data. J. Geophys. Res., 105, 6483–6492, https://doi.org/10.1029/2000JC900002.

    • Search Google Scholar
    • Export Citation

  • Osborn, T. R., and C. S. Cox, 1972: Oceanic fine structure. Geophys. Fluid Dyn., 3, 321–345, https://doi.org/10.1080/03091927208236085.

    • Search Google Scholar
    • Export Citation

  • Park, Y.-H., and Coauthors, 2019: Observations of the Antarctic Circumpolar Current over the Udintsev Fracture Zone, the narrowest choke point in the Southern Ocean. J. Geophys. Res. Oceans, 124, 4511–4528, https://doi.org/10.1029/2019JC015024.

    • Search Google Scholar
    • Export Citation

  • Peng, S., Y.-K. Qian, R. Lumpkin, P. Li, D. Wang, and Y. Du, 2015: Characteristics of the near-surface currents in the Indian Ocean as deduced from satellite-tracked surface drifters. Part II: Lagrangian statistics. J. Phys. Oceanogr., 45, 459–477, https://doi.org/10.1175/JPO-D-14-0049.1.

    • Search Google Scholar
    • Export Citation

  • Prather, M., 1986: Numerical advection by conservation of second-order moments. J. Geophys. Res., 91, 6671–6681, https://doi.org/10.1029/JD091iD06p06671.

    • Search Google Scholar
    • Export Citation

  • Qian, Y.-K., S. Peng, and Y. Li, 2013: Eulerian and Lagrangian statistics in the South China Sea as deduced from surface drifters. J. Phys. Oceanogr., 43, 726–743, https://doi.org/10.1175/JPO-D-12-0170.1.

    • Search Google Scholar
    • Export Citation

  • Qian, Y.-K., S. Peng, and C.-X. Liang, 2019: Reconciling Lagrangian diffusivity and effective diffusivity in contour-based coordinates. J. Phys. Oceanogr., 49, 1521–1539, https://doi.org/10.1175/JPO-D-18-0251.1.

    • Search Google Scholar
    • Export Citation

  • Qian, Y. K., S. Peng, X. Wen, and T. Liu, 2022: Quantifying local, instantaneous, irreversible mixing using Lagrangian particles and tracer contours. J. Phys. Oceanogr., 52, 741–757, https://doi.org/10.1175/JPO-D-21-0260.1.

    • Search Google Scholar
    • Export Citation

  • Redi, M. H., 1982: Oceanic isopycnal mixing by coordinate rotation. J. Phys. Oceanogr., 12, 1154–1158, https://doi.org/10.1175/1520-0485(1982)012<1154:OIMBCR>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Riha, S., and C. Eden, 2011: Lagrangian and Eulerian lateral diffusivities in zonal jets. Ocean Modell., 39, 114–124, https://doi.org/10.1016/j.ocemod.2011.02.002.

    • Search Google Scholar
    • Export Citation

  • Roach, C. J., D. Balwada, and K. Speer, 2016: Horizontal mixing in the Southern Ocean from Argo float trajectories. J. Geophys. Res. Oceans, 121, 5570–5586, https://doi.org/10.1002/2015JC011440.

    • Search Google Scholar
    • Export Citation

  • Roach, C. J., D. Balwada, and K. Speer, 2018: Global observations of horizontal mixing from Argo float and surface drifter trajectories. J. Geophys. Res. Oceans, 123, 4560–4575, https://doi.org/10.1029/2018JC013750.

    • Search Google Scholar
    • Export Citation

  • Rühs, S., V. Zhurbas, I. M. Koszalka, J. V. Durgadoo, and A. Biastoch, 2018: Eddy diffusivity estimates from Lagrangian trajectories simulated with ocean models and surface drifter data—A case study for the greater Agulhas system. J. Phys. Oceanogr., 48, 175–196, https://doi.org/10.1175/JPO-D-17-0048.1.

    • Search Google Scholar
    • Export Citation

  • Rypina, I. I., I. Kamenkovich, P. Berloff, and L. J. Pratt, 2012: Eddy-induced particle dispersion in the near-surface North Atlantic. J. Phys. Oceanogr., 42, 2206–2228, https://doi.org/10.1175/JPO-D-11-0191.1.

    • Search Google Scholar
    • Export Citation

  • Sallée, J. B., K. Speer, R. Morrow, and R. Lumpkin, 2008: An estimate of Lagrangian eddy statistics and diffusion in the mixed layer of the Southern Ocean. J. Mar. Res., 66, 441–463.

    • Search Google Scholar
    • Export Citation

  • Shuckburgh, E., H. Jones, J. Marshall, and C. Hill, 2009: Understanding the regional variability of eddy diffusivity in the Pacific sector of the Southern Ocean. J. Phys. Oceanogr., 39, 2011–2023, https://doi.org/10.1175/2009JPO4115.1.

    • Search Google Scholar
    • Export Citation

  • Swenson, M. S., and P. P. Niiler, 1996: Statistical analysis of the surface circulation of the California Current. J. Geophys. Res., 101, 22 631–22 645, https://doi.org/10.1029/96JC02008.

    • Search Google Scholar
    • Export Citation

  • Thompson, A. F., 2008: The atmospheric ocean: Eddies and jets in the Antarctic Circumpolar Current. Philos. Trans. Roy. Soc., A366, 4529–4541, https://doi.org/10.1098/rsta.2008.0196.

    • Search Google Scholar
    • Export Citation

  • Uchida, T., D. Balwada, Q. Jamet, W. K. Dewar, B. Deremble, T. Penduff, and J. Le Sommer, 2023: Cautionary tales from the mesoscale eddy transport tensor. Ocean Modell., 182, 102172, https://doi.org/10.1016/j.ocemod.2023.102172.

    • Search Google Scholar
    • Export Citation

  • Wagner, G. L., G. Flierl, R. Ferrari, G. Voet, G. S. Carter, M. H. Alford, and J. B. Girton, 2019: Squeeze dispersion and the effective diapycnal diffusivity of oceanic tracers. Geophys. Res. Lett., 46, 5378–5386, https://doi.org/10.1029/2019GL082458.

    • Search Google Scholar
    • Export Citation

  • Winters, K. B., and E. A. D’Asaro, 1996: Diascalar flux and the rate of fluid mixing. J. Fluid Mech., 317, 179–193, https://doi.org/10.1017/S0022112096000717.

    • Search Google Scholar
    • Export Citation

  • Wolfram, P. J., and T. D. Ringler, 2017: Computing eddy-driven effective diffusivity using Lagrangian particles. Ocean Modell., 118, 94–106, https://doi.org/10.1016/j.ocemod.2017.08.008.

    • Search Google Scholar
    • Export Citation

  • Zhang, Z., B. Qiu, P. Klein, and S. Travis, 2019: The influence of geostrophic strain on oceanic ageostrophic motion and surface chlorophyll. Nat. Commun., 10, 2838, https://doi.org/10.1038/s41467-019-10883-w.

    • Search Google Scholar
    • Export Citation

  • Zhurbas, V., and I. S. Oh, 2004: Drifter‐derived maps of lateral diffusivity in the Pacific and Atlantic Oceans in relation to surface circulation patterns. J. Geophys. Res., 109, C05015, https://doi.org/10.1029/2003JC002241.

    • Search Google Scholar
    • Export Citation

  • Zhurbas, V., D. Lyzhkov, and N. Kuzmina, 2014: Drifter-derived estimates of lateral eddy diffusivity in the world ocean with emphasis on the Indian Ocean and problems of parameterisation. Deep-Sea Res. I, 83, 1–11, https://doi.org/10.1016/j.dsr.2013.09.001.

    • Search Google Scholar
    • Export Citation
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  • Abernathey, R., J. Marshall, M. Mazloff, and E. Shuckburgh, 2010: Enhancement of mesoscale eddy stirring at steering levels in the Southern Ocean. J. Phys. Oceanogr., 40, 170–184, https://doi.org/10.1175/2009JPO4201.1.

    • Search Google Scholar
    • Export Citation

  • Abernathey, R. P., and J. Marshall, 2013: Global surface eddy diffusivities derived from satellite altimetry. J. Geophys. Res. Oceans, 118, 901–916, https://doi.org/10.1002/jgrc.20066.

    • Search Google Scholar
    • Export Citation

  • Arthur, R. S., S. K. Venayagamoorthy, J. R. Koseff, and O. B. Fringer, 2017: How we compute N matters to estimates of mixing in stratified flows. J. Fluid Mech., 831, R2, https://doi.org/10.1017/jfm.2017.679.

    • Search Google Scholar
    • Export Citation

  • Bachman, S. D., B. Fox-Kemper, and F. O. Bryan, 2015: A tracer-based inversion method for diagnosing eddy-induced diffusivity and advection. Ocean Modell., 86, 1–14, https://doi.org/10.1016/j.ocemod.2014.11.006.

    • Search Google Scholar
    • Export Citation

  • Balwada, D., K. S. Smith, and R. Abernathey, 2018: Submesoscale vertical velocities enhance tracer subduction in an idealized Antarctic Circumpolar Current. Geophys. Res. Lett., 45, 9790–9802, https://doi.org/10.1029/2018GL079244.

    • Search Google Scholar
    • Export Citation

  • Bauer, S., M. S. Swenson, and A. Griffa, 2002: Eddy mean flow decomposition and eddy diffusivity estimates in the tropical Pacific Ocean: 2. Results. J. Geophys. Res., 107, 3154, https://doi.org/10.1029/2000JC000613.

    • Search Google Scholar
    • Export Citation

  • Behnisch, M., A. Macrander, O. Boebel, J.-O. Wolff, and J. Schröter, 2013: Barotropic and deep‐referenced baroclinic SSH variability derived from Pressure Inverted Echo Sounders (PIES) South of Africa. J. Geophys. Res. Oceans, 118, 3046–3058, https://doi.org/10.1002/jgrc.20195.

    • Search Google Scholar
    • Export Citation

  • Busecke, J. J. M., and R. P. Abernathey, 2019: Ocean mesoscale mixing linked to climate variability. Sci. Adv., 5, eaav5014, https://doi.org/10.1126/sciadv.aav5014.

    • Search Google Scholar
    • Export Citation

  • Chen, R., J. L. McClean, S. T. Gille, and A. Griesel, 2014: Isopycnal eddy diffusivities and critical layers in the Kuroshio Extension from an eddying ocean model. J. Phys. Oceanogr., 44, 2191–2211, https://doi.org/10.1175/JPO-D-13-0258.1.

    • Search Google Scholar
    • Export Citation

  • Chiswell, S. M., 2013: Lagrangian time scales and eddy diffusivity at 1000 m compared to the surface in the South Pacific and Indian Oceans. J. Phys. Oceanogr., 43, 2718–2732, https://doi.org/10.1175/JPO-D-13-044.1.

    • Search Google Scholar
    • Export Citation

  • Danabasoglu, G., and J. Marshall, 2007: Effects of vertical variations of thickness diffusivity in an ocean general circulation model. Ocean Modell., 18, 122–141, https://doi.org/10.1016/j.ocemod.2007.03.006.

    • Search Google Scholar
    • Export Citation

  • Ferrari, R., and M. Nikurashin, 2010: Suppression of eddy diffusivity across jets in the Southern Ocean. J. Phys. Oceanogr., 40, 1501–1519, https://doi.org/10.1175/2010JPO4278.1.

    • Search Google Scholar
    • Export Citation

  • Fox-Kemper, B., R. Lumpkin, and F. O. Bryan, 2013: Lateral transport in the ocean interior. Ocean Circulation and Climate, G. Siedler et al., Eds., International Geophysics Series, Vol. 103, Academic Press, 185–209.

  • Gent, P. R., and J. C. McWilliams, 1990: Isopycnal mixing in ocean circulation models. J. Phys. Oceanogr., 20, 150–155, https://doi.org/10.1175/1520-0485(1990)020<0150:IMIOCM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Griesel, A., S. T. Gille, J. Sprintall, J. L. McClean, J. H. LaCasce, and M. E. Maltrud, 2010: Isopycnal diffusivities in the Antarctic Circumpolar Current inferred from Lagrangian floats in an eddying model. J. Geophys. Res., 115, C06006, https://doi.org/10.1029/2009JC005821.

    • Search Google Scholar
    • Export Citation

  • Griffies, S. M., and Coauthors, 2005: Formulation of an ocean model for global climate simulations. Ocean Sci., 1, 45–79, https://doi.org/10.5194/os-1-45-2005.

    • Search Google Scholar
    • Export Citation

  • Groeskamp, S., J. H. LaCasce, T. J. McDougall, and M. Rogé, 2020: Full‐depth global estimates of ocean mesoscale eddy mixing from observations and theory. Geophys. Res. Lett., 47, e2020GL089425, https://doi.org/10.1029/2020GL089425.

    • Search Google Scholar
    • Export Citation

  • Hill, C., D. Ferreira, J.-M. Campin, J. Marshall, R. Abernathey, and N. Barrier, 2012: Controlling spurious diapycnal mixing in eddy-resolving height-coordinate ocean models – Insights from virtual deliberate tracer release experiments. Ocean Modell., 45–46, 14–26, https://doi.org/10.1016/j.ocemod.2011.12.001.

    • Search Google Scholar
    • Export Citation

  • Hoskins, B. J., 1991: Towards a PV-θ view of the general circulation. Tellus, 43A, 27–35, https://doi.org/10.1034/j.1600-0870.1991.t01-3-00005.x.

    • Search Google Scholar
    • Export Citation

  • Kamenkovich, I., P. Berloff, M. Haigh, L. Sun, and Y. Lu, 2021: Complexity of mesoscale eddy diffusivity in the ocean. Geophys. Res. Lett., 48, e2020GL091719, https://doi.org/10.1034/j.1600-0870.1991.t01-3-00005.x.

    • Search Google Scholar
    • Export Citation

  • Klocker, A., and R. Abernathey, 2014: Global patterns of mesoscale eddy properties and diffusivities. J. Phys. Oceanogr., 44, 1030–1046, https://doi.org/10.1175/JPO-D-13-0159.1.

    • Search Google Scholar
    • Export Citation

  • Klocker, A., R. Ferrari, and J. H. LaCasce, 2012a: Estimating suppression of eddy mixing by mean flows. J. Phys. Oceanogr., 42, 1566–1576, https://doi.org/10.1175/JPO-D-11-0205.1.

    • Search Google Scholar
    • Export Citation

  • Klocker, A., R. Ferrari, J. H. LaCasce, and S. T. Merrifield, 2012b: Reconciling float-based and tracer-based estimates of lateral diffusivities. J. Mar. Res., 70, 569–602.

    • Search Google Scholar
    • Export Citation

  • LaCasce, J. H., 2008: Statistics from Lagrangian observations. Prog. Oceanogr., 77 (1), 1–29, https://doi.org/10.1016/j.pocean.2008.02.002.

    • Search Google Scholar
    • Export Citation

  • LaCasce, J. H., R. Ferrari, J. Marshall, R. Tulloch, D. Balwada, and K. Speer, 2014: Float-derived isopycnal diffusivities in the DIMES experiment. J. Phys. Oceanogr., 44, 764–780, https://doi.org/10.1175/JPO-D-13-0175.1.

    • Search Google Scholar
    • Export Citation

  • Lagerloef, G. S. E., G. T. Mitchum, R. B. Lukas, and P. P. Niiler, 1999: Tropical Pacific near‐surface currents estimated from altimeter, wind, and drifter data. J. Geophys. Res., 104, 23 313–23 326, https://doi.org/10.1029/1999JC900197.

    • Search Google Scholar
    • Export Citation

  • Liu, T., and R. Abernathey, 2023: A global Lagrangian eddy dataset based on satellite altimetry. Earth Syst. Sci. Data, 15, 1765–1778, https://doi.org/10.5194/essd-15-1765-2023.

    • Search Google Scholar
    • Export Citation

  • Liu, T., H.-W. Ou, X. Liu, and D. Chen, 2022: On the role of eddy mixing in the subtropical ocean circulation. Front. Mar. Sci., 9, 832992, https://doi.org/10.3389/fmars.2022.832992.

    • Search Google Scholar
    • Export Citation

  • Lu, J., and K. Speer, 2010: Topography, jets, and eddy mixing in the Southern Ocean. J. Mar. Res., 68, 479–502.

  • Lumpkin, R., and S. Elipot, 2010: Surface drifter pair spreading in the North Atlantic. J. Geophys. Res., 115, C12017, https://doi.org/10.1029/2010JC006338.

    • Search Google Scholar
    • Export Citation

  • Lumpkin, R., A. M. Treguier, and K. Speer, 2002: Lagrangian eddy scales in the northern Atlantic Ocean. J. Phys. Oceanogr., 32, 2425–2440, https://doi.org/10.1175/1520-0485(2002)032%3C2425:LESITN%3E2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Marshall, J., A. Adcroft, C. Hill, L. Perelman, and C. Heisey, 1997: A finite‐volume, incompressible Navier Stokes model for studies of the ocean on parallel computers. J. Geophys. Res., 102, 5753–5766, https://doi.org/10.1029/96JC02775.

    • Search Google Scholar
    • Export Citation

  • Marshall, J., E. Shuckburgh, H. Jones, and C. Hill, 2006: Estimates and implications of surface eddy diffusivity in the Southern Ocean derived from tracer transport. J. Phys. Oceanogr., 36, 1806–1821, https://doi.org/10.1175/JPO2949.1.

    • Search Google Scholar
    • Export Citation

  • Marshall, J., J. R. Scott, A. Romanou, M. Kelley, and A. Leboissetier, 2017: The dependence of the ocean’s MOC on mesoscale eddy diffusivities: A model study. Ocean Modell., 111, 1–8, https://doi.org/10.1016/j.ocemod.2017.01.001.

    • Search Google Scholar
    • Export Citation

  • McIntyre, M. E., 1980: Towards a Lagrangian-mean description of stratospheric circulations and chemical transports. Philos. Trans. Roy. Soc., A296, 129–148, https://doi.org/10.1098/rsta.1980.0160.

    • Search Google Scholar
    • Export Citation

  • McWilliams, J. C., 2021: Oceanic frontogenesis. Annu. Rev. Mar. Sci., 13, 227–253, https://doi.org/10.1146/annurev-marine-032320-120725.

    • Search Google Scholar
    • Export Citation

  • Nakamura, N., 1995: Modified Lagrangian-mean diagnostics of the stratospheric polar vortices. Part I: Formulation and analysis of GFDL SKYHI GCM. J. Atmos. Sci., 52, 2096–2108, https://doi.org/10.1175/1520-0469(1995)052<2096:MLMDOT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Nakamura, N., 1996: Two-dimensional mixing, edge formation, and permeability diagnosed in an area coordinate. J. Atmos. Sci., 53, 1524–1537, https://doi.org/10.1175/1520-0469(1996)053<1524:TDMEFA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Nakamura, N., 2001: A new look at eddy diffusivity as a mixing diagnostic. J. Atmos. Sci., 58, 3685–3701, https://doi.org/10.1175/1520-0469(2001)058<3685:ANLAED>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Nakamura, N., 2008: Quantifying inhomogeneous, instantaneous, irreversible transport using passive tracer field as a coordinate. Transport and Mixing in Geophysical Flows, Springer, 137–164.

  • Oh, I. S., V. Zhurbas, and W. Park, 2000: Estimating horizontal diffusivity in the East Sea (Sea of Japan) and the Northwest Pacific from satellite‐tracked drifter data. J. Geophys. Res., 105, 6483–6492, https://doi.org/10.1029/2000JC900002.

    • Search Google Scholar
    • Export Citation

  • Osborn, T. R., and C. S. Cox, 1972: Oceanic fine structure. Geophys. Fluid Dyn., 3, 321–345, https://doi.org/10.1080/03091927208236085.

    • Search Google Scholar
    • Export Citation

  • Park, Y.-H., and Coauthors, 2019: Observations of the Antarctic Circumpolar Current over the Udintsev Fracture Zone, the narrowest choke point in the Southern Ocean. J. Geophys. Res. Oceans, 124, 4511–4528, https://doi.org/10.1029/2019JC015024.

    • Search Google Scholar
    • Export Citation

  • Peng, S., Y.-K. Qian, R. Lumpkin, P. Li, D. Wang, and Y. Du, 2015: Characteristics of the near-surface currents in the Indian Ocean as deduced from satellite-tracked surface drifters. Part II: Lagrangian statistics. J. Phys. Oceanogr., 45, 459–477, https://doi.org/10.1175/JPO-D-14-0049.1.

    • Search Google Scholar
    • Export Citation

  • Prather, M., 1986: Numerical advection by conservation of second-order moments. J. Geophys. Res., 91, 6671–6681, https://doi.org/10.1029/JD091iD06p06671.

    • Search Google Scholar
    • Export Citation

  • Qian, Y.-K., S. Peng, and Y. Li, 2013: Eulerian and Lagrangian statistics in the South China Sea as deduced from surface drifters. J. Phys. Oceanogr., 43, 726–743, https://doi.org/10.1175/JPO-D-12-0170.1.

    • Search Google Scholar
    • Export Citation

  • Qian, Y.-K., S. Peng, and C.-X. Liang, 2019: Reconciling Lagrangian diffusivity and effective diffusivity in contour-based coordinates. J. Phys. Oceanogr., 49, 1521–1539, https://doi.org/10.1175/JPO-D-18-0251.1.

    • Search Google Scholar
    • Export Citation

  • Qian, Y. K., S. Peng, X. Wen, and T. Liu, 2022: Quantifying local, instantaneous, irreversible mixing using Lagrangian particles and tracer contours. J. Phys. Oceanogr., 52, 741–757, https://doi.org/10.1175/JPO-D-21-0260.1.

    • Search Google Scholar
    • Export Citation

  • Redi, M. H., 1982: Oceanic isopycnal mixing by coordinate rotation. J. Phys. Oceanogr., 12, 1154–1158, https://doi.org/10.1175/1520-0485(1982)012<1154:OIMBCR>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Riha, S., and C. Eden, 2011: Lagrangian and Eulerian lateral diffusivities in zonal jets. Ocean Modell., 39, 114–124, https://doi.org/10.1016/j.ocemod.2011.02.002.

    • Search Google Scholar
    • Export Citation

  • Roach, C. J., D. Balwada, and K. Speer, 2016: Horizontal mixing in the Southern Ocean from Argo float trajectories. J. Geophys. Res. Oceans, 121, 5570–5586, https://doi.org/10.1002/2015JC011440.

    • Search Google Scholar
    • Export Citation

  • Roach, C. J., D. Balwada, and K. Speer, 2018: Global observations of horizontal mixing from Argo float and surface drifter trajectories. J. Geophys. Res. Oceans, 123, 4560–4575, https://doi.org/10.1029/2018JC013750.

    • Search Google Scholar
    • Export Citation

  • Rühs, S., V. Zhurbas, I. M. Koszalka, J. V. Durgadoo, and A. Biastoch, 2018: Eddy diffusivity estimates from Lagrangian trajectories simulated with ocean models and surface drifter data—A case study for the greater Agulhas system. J. Phys. Oceanogr., 48, 175–196, https://doi.org/10.1175/JPO-D-17-0048.1.

    • Search Google Scholar
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  • Rypina, I. I., I. Kamenkovich, P. Berloff, and L. J. Pratt, 2012: Eddy-induced particle dispersion in the near-surface North Atlantic. J. Phys. Oceanogr., 42, 2206–2228, https://doi.org/10.1175/JPO-D-11-0191.1.

    • Search Google Scholar
    • Export Citation

  • Sallée, J. B., K. Speer, R. Morrow, and R. Lumpkin, 2008: An estimate of Lagrangian eddy statistics and diffusion in the mixed layer of the Southern Ocean. J. Mar. Res., 66, 441–463.

    • Search Google Scholar
    • Export Citation

  • Shuckburgh, E., H. Jones, J. Marshall, and C. Hill, 2009: Understanding the regional variability of eddy diffusivity in the Pacific sector of the Southern Ocean. J. Phys. Oceanogr., 39, 2011–2023, https://doi.org/10.1175/2009JPO4115.1.

    • Search Google Scholar
    • Export Citation

  • Swenson, M. S., and P. P. Niiler, 1996: Statistical analysis of the surface circulation of the California Current. J. Geophys. Res., 101, 22 631–22 645, https://doi.org/10.1029/96JC02008.

    • Search Google Scholar
    • Export Citation

  • Thompson, A. F., 2008: The atmospheric ocean: Eddies and jets in the Antarctic Circumpolar Current. Philos. Trans. Roy. Soc., A366, 4529–4541, https://doi.org/10.1098/rsta.2008.0196.

    • Search Google Scholar
    • Export Citation

  • Uchida, T., D. Balwada, Q. Jamet, W. K. Dewar, B. Deremble, T. Penduff, and J. Le Sommer, 2023: Cautionary tales from the mesoscale eddy transport tensor. Ocean Modell., 182, 102172, https://doi.org/10.1016/j.ocemod.2023.102172.

    • Search Google Scholar
    • Export Citation

  • Wagner, G. L., G. Flierl, R. Ferrari, G. Voet, G. S. Carter, M. H. Alford, and J. B. Girton, 2019: Squeeze dispersion and the effective diapycnal diffusivity of oceanic tracers. Geophys. Res. Lett., 46, 5378–5386, https://doi.org/10.1029/2019GL082458.

    • Search Google Scholar
    • Export Citation

  • Winters, K. B., and E. A. D’Asaro, 1996: Diascalar flux and the rate of fluid mixing. J. Fluid Mech., 317, 179–193, https://doi.org/10.1017/S0022112096000717.

    • Search Google Scholar
    • Export Citation

  • Wolfram, P. J., and T. D. Ringler, 2017: Computing eddy-driven effective diffusivity using Lagrangian particles. Ocean Modell., 118, 94–106, https://doi.org/10.1016/j.ocemod.2017.08.008.

    • Search Google Scholar
    • Export Citation

  • Zhang, Z., B. Qiu, P. Klein, and S. Travis, 2019: The influence of geostrophic strain on oceanic ageostrophic motion and surface chlorophyll. Nat. Commun., 10, 2838, https://doi.org/10.1038/s41467-019-10883-w.

    • Search Google Scholar
    • Export Citation

  • Zhurbas, V., and I. S. Oh, 2004: Drifter‐derived maps of lateral diffusivity in the Pacific and Atlantic Oceans in relation to surface circulation patterns. J. Geophys. Res., 109, C05015, https://doi.org/10.1029/2003JC002241.

    • Search Google Scholar
    • Export Citation

  • Zhurbas, V., D. Lyzhkov, and N. Kuzmina, 2014: Drifter-derived estimates of lateral eddy diffusivity in the world ocean with emphasis on the Indian Ocean and problems of parameterisation. Deep-Sea Res. I, 83, 1–11, https://doi.org/10.1016/j.dsr.2013.09.001.

    • Search Google Scholar
    • Export Citation
  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    Schematic illustration of different turbulent mixing states: (a) meridional-sorted state, (b) Eulerian time-mean state, and (c) instantaneous eddying state. The plots in (a) and (b) represent the lowest mixing-efficiency states for Keff and KOC, respectively.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    (a) Original and (b) corrected distributions of EKE (m2 s−2), as well as (c) their differences. A logarithmic scale is used in (a) and (b), while (c) shows the absolute difference. Note that the areas possibly covered with sea ice in high latitudes are masked out during the selected period.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    Concentrations for the two tracers, (a),(c),(e) TrLAT and (b),(d),(f) TrSST, on days 0, 100, and 365 in 2016. The initial tracer concentrations are normalized to values between 1 and 2.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    (a) Sample trajectories of Lagrangian particles deployed uniformly over the global ocean. (b) Meridional positions in the geographical and equivalent latitudes for selected particles [large black dots with thick gray lines in (a)].

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    A comparison among three traditional diffusivity estimates. KOC (colors; m2 s−1) estimated from two tracer distributions: (a) TrLAT and (c) TrSST. (b),(d)The zonally averaged KOC (blue lines) is compared with Keff (red lines). The values are shown on a logarithmic scale. The dashed lines in (d) indicates the latitude band where contours of TrSST intersect the equator and hence a possibly degenerated Keff there. (e) Minor (or cross-stream) component of the traditional Lagrangian dispersion diffusivity tensor (m2 s−1) and (f) its zonal mean.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    Snapshots of (left) local Lagrangian diffusivity K˜L and (right) local effective diffusivity K˜eff for TrLAT (m2 s−1) over the global ocean on the 50th, 100th, 200th, and 300th days in 2016. The values are shown on a logarithmic scale. The red dashed lines in (b) are two meridional sections shown in Fig. 12.

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    Time means of local effective diffusivity K˜eff¯ (colors) estimated from two tracer distributions: (a) latitude and (c) SST. (b),(d)The zonally averaged K˜eff¯ (red lines) is compared with KOC (blue lines).

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    Values of (a),(c) normalized local effective diffusivity K˜effq¯ for the time-mean tracer distribution q¯ and (b),(d) base-10 logarithm of the factor of |∇XY(q¯)∗|2/|∇XYq∗(Y)|2¯ for two tracers.

  • <sc>Fig</sc>. 9.
    View in gallery
    Fig. 9.

    As in Fig. 7, but for rectified local effective diffusivity K˜eff′¯.

  • <sc>Fig</sc>. 10.
    View in gallery
    Fig. 10.

    (left) Instantaneous tracer distribution and (right) its corresponding local effective diffusivity (m2 s−1) in the Agulhas Current region every 10 days from 1 Jan to 11 Feb 2016. Black contours represent the SLA field.

  • <sc>Fig</sc>. 11.
    View in gallery
    Fig. 11.

    Comparison between climatological (red contours) and randomly selected instantaneous (green contours) absolute dynamic topography in the (a) Kuroshio Extension and (b) Agulhas Current regions.

  • <sc>Fig</sc>. 12.
    View in gallery
    Fig. 12.

    Temporal evolution of local effective diffusivity K˜eff (m2 s−1) along (a) 180° and (b) 330°E in 2016. Two meridional sections are indicated in Fig. 6b.

  • <sc>Fig</sc>. 13.
    View in gallery
    Fig. 13.

    Climatological distribution of K˜eff¯(m2 s−1) for (a) TrLat and (b) TrSST over the Southern Ocean. (c),(d) As in (a) and (b), but for KOC (m2 s−1). The black lines mark the Subantarctic Front, Polar Front, and Southern ACC Front from north to south, whereas the magenta lines mark the north and south boundaries of ACC (Park et al. 2019).

Fig. 1.

Schematic illustration of different turbulent mixing states: (a) meridional-sorted state, (b) Eulerian time-mean state, and (c) instantaneous eddying state. The plots in (a) and (b) represent the lowest mixing-efficiency states for Keff and KOC, respectively.
Fig. 2.

(a) Original and (b) corrected distributions of EKE (m2 s−2), as well as (c) their differences. A logarithmic scale is used in (a) and (b), while (c) shows the absolute difference. Note that the areas possibly covered with sea ice in high latitudes are masked out during the selected period.
Fig. 3.

Concentrations for the two tracers, (a),(c),(e) TrLAT and (b),(d),(f) TrSST, on days 0, 100, and 365 in 2016. The initial tracer concentrations are normalized to values between 1 and 2.
Fig. 4.

(a) Sample trajectories of Lagrangian particles deployed uniformly over the global ocean. (b) Meridional positions in the geographical and equivalent latitudes for selected particles [large black dots with thick gray lines in (a)].
Fig. 5.

A comparison among three traditional diffusivity estimates. KOC (colors; m2 s−1) estimated from two tracer distributions: (a) TrLAT and (c) TrSST. (b),(d)The zonally averaged KOC (blue lines) is compared with Keff (red lines). The values are shown on a logarithmic scale. The dashed lines in (d) indicates the latitude band where contours of TrSST intersect the equator and hence a possibly degenerated Keff there. (e) Minor (or cross-stream) component of the traditional Lagrangian dispersion diffusivity tensor (m2 s−1) and (f) its zonal mean.
Fig. 6.

Snapshots of (left) local Lagrangian diffusivity K˜L and (right) local effective diffusivity K˜eff for TrLAT (m2 s−1) over the global ocean on the 50th, 100th, 200th, and 300th days in 2016. The values are shown on a logarithmic scale. The red dashed lines in (b) are two meridional sections shown in Fig. 12.
Fig. 7.

Time means of local effective diffusivity K˜eff¯ (colors) estimated from two tracer distributions: (a) latitude and (c) SST. (b),(d)The zonally averaged K˜eff¯ (red lines) is compared with KOC (blue lines).
Fig. 8.

Values of (a),(c) normalized local effective diffusivity K˜effq¯ for the time-mean tracer distribution q¯ and (b),(d) base-10 logarithm of the factor of |∇XY(q¯)∗|2/|∇XYq∗(Y)|2¯ for two tracers.
Fig. 9.

As in Fig. 7, but for rectified local effective diffusivity K˜eff′¯.
Fig. 10.

(left) Instantaneous tracer distribution and (right) its corresponding local effective diffusivity (m2 s−1) in the Agulhas Current region every 10 days from 1 Jan to 11 Feb 2016. Black contours represent the SLA field.
Fig. 11.

Comparison between climatological (red contours) and randomly selected instantaneous (green contours) absolute dynamic topography in the (a) Kuroshio Extension and (b) Agulhas Current regions.
Fig. 12.

Temporal evolution of local effective diffusivity K˜eff (m2 s−1) along (a) 180° and (b) 330°E in 2016. Two meridional sections are indicated in Fig. 6b.
Fig. 13.

Climatological distribution of K˜eff¯(m2 s−1) for (a) TrLat and (b) TrSST over the Southern Ocean. (c),(d) As in (a) and (b), but for KOC (m2 s−1). The black lines mark the Subantarctic Front, Polar Front, and Southern ACC Front from north to south, whereas the magenta lines mark the north and south boundaries of ACC (Park et al. 2019).
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