## 1. Introduction

Eddies are not explicitly resolved in standard-resolution, centennial-scale, global climate simulations; however, these simulations are sensitive to the representation of eddy mixing processes (e.g., Danabasoglu and Marshall 2007). This sensitivity has motivated efforts to estimate eddy mixing rates in the ocean that aim to reveal mixing processes leading to improved eddy parameterizations (e.g., Gille et al. 2012). Both float-based and tracer-based diagnostic approaches have been employed to estimate diffusivities in the tropical North Atlantic (e.g., Banyte et al. 2013), western boundary currents (e.g., Chen et al. 2014), the Southern Ocean (e.g., LaCasce et al. 2014; Tulloch et al. 2014), and the global surface (e.g., Abernathey and Marshall 2013).

*L*

_{mix}instead of the diffusivity itself:where Γ denotes the

*O*(1) mixing efficiency,

*u*

_{rms}is the eddy velocity magnitude, and

*κ*denotes the eddy diffusivity (e.g., Taylor 1915; Klocker and Abernathey 2014).

Bates et al. (2014) pointed out that ocean studies often assume that *L*_{mix} is of the same order of magnitude as the eddy size (e.g., Holloway 1986; Haine and Marshall 1998). This assumption is reasonable in an ocean without mean flows (Klocker and Abernathey 2014). In regions of the real ocean that contain mean flows, such as the Antarctica Circumpolar Current (ACC) and western boundary currents, eddy mixing in the cross-mean flow (cross stream) direction can be suppressed when eddies propagate relative to the mean flow. This concept dates back to Bretherton (1966) and Green (1970) and can be interpreted as follows: If eddies are stationary relative to the mean flow, eddies have sufficient time to stir and mix the same tracers, which also move with the mean flow; on the other hand, if the eddies propagate relative to the mean flow, eddies do not mix the same tracers, and mixing is suppressed (Klocker and Abernathey 2014). This idea has also recently been applied to studies about chaotic advection (Pratt et al. 2014).

**U**| denotes the mean flow magnitude. Note that although some previous studies (e.g., Klocker and Abernathey 2014) assume that the mean flow is zonal, this theory applies to the case with the mean flow in any arbitrary direction (Chen et al. 2014). Here,

*C*

_{w}denotes the phase speed along the mean flow direction; positive

*C*

_{w}corresponds to downstream propagation, and negative

*C*

_{w}corresponds to upstream propagation (Chen et al. 2014). The terms

*k*

_{eddy},

*L*, and

*γ*are the eddy wavenumber, eddy length scale, and the reciprocal of eddy decorrelation time, respectively. Surface eddy diffusivities from the F–N theory agree with those from altimetry (e.g., Ferrari and Nikurashin 2010; Klocker and Abernathey 2014; Bates et al. 2014). The F–N theory also captures the three-dimensional structure of float-based eddy diffusivities in the intense Kuroshio Extension jet area (Chen et al. 2014). On the other hand, the theory appears to break down in regions where large topographic gradients occur (e.g., Griesel et al. 2010; Naveira-Garabato et al. 2011; Chen et al. 2014). Bates et al. (2014) found that the F–N theory is also not effective in capturing the observed vertical structures of eddy diffusivities from the U.S./U.K. field program Diapycnal and Isopycnal Mixing Experiment in the Southern Ocean (DIMES; e.g., Ledwell et al. 2011) or from the North Atlantic Tracer Release Experiment (NATRE; e.g., Ledwell et al. 1998).

The breakdown of the F–N theory in some ocean scenarios is unsurprising, considering that it is built on a number of assumptions that are violated in the ocean, including a flat bottom, a spatially and temporally constant mean flow, and scale separation between the mean flow and eddies (e.g., Ferrari and Nikurashin 2010). One assumption upon which these theories, including the F–N theory, are based is that eddies only contain a single wave corresponding to the most unstable mode or that eddies are dominated by a single wave (e.g., Green 1970; Killworth 1997; Ferrari and Nikurashin 2010; Klocker et al. 2012a). In fact, the frequency–wavenumber spectra of sea surface height is a broad continuum with no robust peaks, indicating that the oceanic eddy field includes motions over a broad range of wavenumbers and frequencies (e.g., Wunsch 2010; Wortham 2013; Wortham and Wunsch 2014). Even a single Gaussian vortex can be decomposed into wave motions with a range of wavenumbers and frequencies (Chen et al. 2015). In the midlatitude ocean interior away from western boundary currents, the phase speeds for all the dominant wavenumbers are roughly the same; however, in the Gulf Stream, the Kuroshio Extension, and the ACC, the dominant waves are dispersive, and both eastward- and westward-propagating waves are nonnegligible [see Figs. 2–16 in Wortham (2013)].

Assuming that internal waves are small-amplitude Gaussian random processes, Holmes-Cerfon et al. (2011) developed a formula for one-particle horizontal diffusivity based on the internal wave spectra. Feddersen (2004) found that estimates of radiation stresses based on full spectra are different from those based on a peak frequency. Inspired by Holmes-Cerfon et al. (2011) and Feddersen (2004), and motivated by the discrepancy between the single-wavenumber assumption and the observed broadbanded oceanic spectra, here we formulate a multiwavenumber theory (MW theory) for eddy diffusivities in the mean flow (section 2). The theory can be derived from both float-based and tracer-based diffusivity formulas (sections 2a and 2b), and it links diffusivities with the frequency–wavenumber spectra of the Eulerian eddy velocity fields.

We choose the DIMES region (70°–45°S, 110°–20°W) as a test bed for the MW theory (Fig. 1). It covers both the region upstream of the Drake Passage, where the floats and tracers were released in the DIMES experiment (e.g., Ledwell et al. 2011; Watson et al. 2013; Sheen et al. 2013; LaCasce et al. 2014; Tulloch et al. 2014), and the region downstream of the Drake Passage (Scotia Sea), where eddy kinetic energy is larger and mixing is more intense than upstream of the Drake Passage. Eddy diffusivities are estimated at high spatial resolution from numerical floats deployed in a global eddying model. These float-based diffusivities are used not only to test the relevance of the single-wavenumber and multiwavenumber theories but also to help put the sparse mixing observations in the DIMES region into a larger spatial context.

This paper is organized as follows: Section 2 introduces the MW theory of eddy diffusivities and illustrates its consistency with the F–N theory in the single-wavenumber limit. Section 3 describes the configuration of the global eddying model with numerical floats, which we use to test the applicability of our theory. Section 4 presents eddy diffusivities in the DIMES area from the numerical floats in the eddying model. Section 5 provides the diffusivities diagnosed from both the F–N and MW theories and compares them with the float-based diffusivities. Sections 6 and 7 provide the discussion and summary, respectively.

## 2. The multiwavenumber theory from both float and tracer perspectives

Here, we derive the MW theory for the diffusivity tensor and cross-stream diffusivities from both the float and tracer perspectives. Its consistency with the F–N theory in the single-wavenumber scenario is demonstrated in appendix A.

### a. The multiwavenumber theory derived from Lagrangian eddy diffusivities

#### 1) Diffusivity tensor in autocorrelation form

*τ*goes to infinity and

**x**. Here,

*i*direction at time

*t*

_{0}+

*τ*for the float that passes position

**x**at time

*t*

_{0}. Residual velocities denote the float velocities subtracted from the local “time-mean” Eulerian velocity at the float position.

**U**is approximately a constant, and (ii) the eddy velocity magnitude is much smaller than the mean flow magnitude. Therefore, to leading order, floats move only with the mean flow. Consequently, the Lagrangian velocities

*i*direction.

*t*

_{0}to be zero, without loss of generality. Then substituting Eq. (4) into Eq. (3),We define the Eulerian eddy velocities at position

**x**+

**U**

*t*at time

*t*asthen we obtain from Eq. (5)Assuming that the eddy statistics are temporally stationary, we have

#### 2) Diffusivity tensor in spectral form

*ω*′ denotes frequency. Because the Fourier transform of a real variable is real at zero frequency,

*i*and

*j*differ. Following Randel and Held (1991), the one-dimensional cross spectrum of variables

*α*and

*β*at position

**x**,

**x**:Here, we use a rotated, coordinate system with wavenumber

*l*′ perpendicular to the mean flow and wavenumber

*k*′ aligned with and pointing in the mean flow direction.

**x**:

**U**| is the magnitude of the mean flow vector

**U**,

*k*′ is the wavenumber in the direction of

**U**, and

*l*′ is the wavenumber perpendicular to

**U**. Therefore,Substituting Eq. (15) into Eq. (13) leads to our multiwavenumber formula for the diffusivity tensor averaged over the patch, involving the three-dimensional spectrum of Eulerian eddy velocities at the patch:

**x**(see Fig. 7c):where

#### 3) Cross-stream diffusivities in spectral form

The full diffusivity tensor has received much attention in eddy parameterizations and tracer transport studies (e.g., Plumb and Mahlman 1987; Griffies 1998; Bachman and Fox-Kemper 2013). However, previous investigations of the role of the mean flow in mixing have focused primarily on cross-stream mixing (e.g., Ferrari and Nikurashin 2010; Griesel et al. 2014), which drives the eddy-induced meridional overturning circulation in the Southern Ocean (Tulloch et al. 2014). For consistency with recent work on cross-stream mixing, we next present cross-stream diffusivities in spectral form. Subsequent sections focus on testing the validity of the MW theory in representing cross-stream diffusivities. For brevity, in this paper we will not discuss the full diffusivity tensor.

*i*and

*j*components in Eq. (3) both represent the cross-stream direction, we obtain the float-based formula for cross-stream eddy diffusivities:where

*t*

_{0}+

*τ*for the float that passes position

**x**at time

*t*

_{0}.

*i*and

*j*direction in Eqs. (16) and (17). The cross-stream diffusivity averaged over the patch centered at

**x**iswhere

**x**iswhere

We find that the MW theory is a natural extension of the single-wavenumber theory to a more realistic regime with multiple wavenumbers. As shown in appendix A, in the limit of a single wavenumber, cross-stream diffusivities from the MW theory reduce to the diffusivities from the single-wavenumber F–N theory.

### b. The multiwavenumber theory derived from Eulerian eddy diffusivities

*C*satisfieswhere

**u**is the total velocity, and

*κ*

_{0}is the molecular or numerical diffusivity of the tracer. The Eulerian eddy diffusivity tensor

*u*

_{i}is the Eulerian velocity (e.g., Plumb and Mahlman 1987).

Consider the scenario from section 2a: eddies are of small amplitude compared to the mean flow and the system is spatially homogenous, with the spatial scale of the mean (e.g., mean flow and mean eddy flux) much larger than the eddy scale. Appendix B shows that *κ*_{0} is very small.

## 3. Testing the theory: A global eddying model with numerical floats

We use a global eddying model to test whether the MW theory from section 2 can capture diffusivity structures in realistic contexts. The Parallel Ocean Program (POP) simulation used in this study is the same as that described by Chen et al. (2014). Here, we briefly review the key model features. The domain is global and the grid has a nominal spatial resolution of *K*-profile parameterization (Large et al. 1994) is used to represent vertical mixing in the upper ocean, and biharmonic viscosity and diffusion parameterize horizontal mixing of momentum and tracers at subgrid scales.

As summarized by Abernathey et al. (2013), a number of methods exist for estimating eddy diffusivities. Here, we estimate Lagrangian eddy diffusivities from numerical floats (e.g., Davis 1987, 1991; Griesel et al. 2010). We deployed one million numerical floats at the beginning of the year 1994 uniformly over the entire globe at 23 vertical levels with a horizontal resolution of 0.25° in latitude and 2.5° in longitude. These floats were advected online by the three-dimensional Eulerian velocity fields for a full year. The float properties, including position, velocity, and density, were recorded at daily intervals.

Figure 1 shows our study domain, which extends from 70° to 45°S and from 110° to 20°W, roughly corresponding to the DIMES region. The barotropic streamlines, defined as *f* is the Coriolis parameter, *g* is gravity, and

The model agrees with observations and previous literature in the following two ways: First, the small change in horizontal structure with depth, shown in Fig. 2, is consistent with an equivalent barotropic flow field in the Southern Ocean, as has been identified and employed in previous studies (e.g., Killworth and Hughes 2002; Firing et al. 2011; Klocker et al. 2012b). The horizontal structure of eddy velocity magnitude is consistent with that observed in altimetry (e.g., Farneti et al. 2010): eddy velocity magnitudes are large along the path of the ACC and are also larger to the east of Drake Passage than to the west. Second, as reviewed in section 1, in the spectra of sea surface height from altimetry, both westward- and eastward-propagating signals are significant in the Southern Ocean and Kuroshio Extension patches, where the eastward mean flow is intense (Wortham 2013). Similar features exist in the POP model (Fig. 3). Note that the model resolution is high enough to be eddy permitting. In this area of the model, the domain average of the first baroclinic Rossby radius of deformation *R*_{d} is 12 km, which is twice as large as the POP grid spacing averaged over the study domain. Additionally, *R*_{d} is larger than the grid size at 88% of grid points in our study domain.

## 4. Eddy diffusivities in the DIMES region from numerical floats

### a. Methodology

Chen et al. (2014) calculated critical layer depths and float-based eddy diffusivities in the Kuroshio Extension. This study uses a similar approach to estimate diffusivities from floats. As in Griesel et al. (2010) and Chen et al. (2014), we diagnose cross-stream diffusivities using Eq. (18). Residual velocities

We use a clustering approach, described by Chen et al. (2014), to divide our domain into adaptive bins. These bins are irregularly distributed in space; however, the number of pseudotrajectories in each bin is roughly the same (Fig. 4). This statistical uniformity leads to more converged diffusivity estimates than the geographic binning approach (Koszalka and LaCasce 2010; Chen et al. 2014). We estimated diffusivities in these adaptive bins at 11 depth intervals in the upper 3000 m with layer thicknesses increasing from 45 to 500 m with depth. Using the Chen et al. (2014) technique to test for convergence, we found that diffusivity estimates in 95% of the adaptive bins were converged.

### b. Numerical results

The horizontal patterns of float-based cross-stream diffusivities, as shown in Fig. 5, are similar to those of the eddy velocity magnitudes shown in Fig. 2, with large values inside the ACC. This indicates that the spatial patterns of eddy mixing are probably mostly controlled by eddy kinetic energy. Similar to the eddy velocity magnitude (Fig. 2), the horizontal structures of eddy diffusivities change little with depth, consistent with the equivalent barotropic nature of the flow field in the Southern Ocean. Diffusivities in the ACC east of the Drake Passage are larger than those to the west. Negative float-based diffusivities exist at some spots, especially inside the ACC (Fig. 5). They correspond to upgradient eddy fluxes, which have been identified in many oceanic regions from both observations and modeling studies (e.g., Johnson et al. 1992; Morrow et al. 1992; Wilkin and Morrow 1994; Griesel et al. 2009; Chen et al. 2014).

The vertical structures of float-based diffusivities and mixing lengths from the POP model were examined by Griesel et al. (2014). They found that the mixing suppression in the jet core is not robust east of the Drake Passage, where the mean flow is composed of multiple thin jets. We repeated their analysis, and our findings support their conclusions (not shown).

LaCasce et al. (2014) and Tulloch et al. (2014) estimated meridional isopycnal diffusivities, using tracer and float observations collected during the DIMES experiment. Given the quasi-zonal orientation of the ACC west of the Drake Passage, their meridional diffusivities are expected to be approximately equivalent to cross-stream diffusivities. They extrapolated the vertical structures of the meridional diffusivities from the DIMES observations using a regional eddying model. However, their estimates do not resolve horizontal structures.

Figure 6 shows a comparison of our float-based diffusivities in a region west of the Drake Passage (61°–56°S, 110°–80°W) with those from a regional MITgcm simulation (Tulloch et al. 2014). Estimates from Tulloch et al. (2014) and our modeling results agree both in the order of magnitude of the diffusivities and the location of the deep maximum. Below 1000 m, the vertical structure of our diffusivity profile is similar to that of Tulloch et al. (2014), with both displaying peak values at around 2100 m; our estimate, however, is of a smaller magnitude (Fig. 6). At 500–1000 m, our float-based diffusivities decrease with depth, but theirs increase with depth. These diffusivity differences might be due to their model having higher horizontal and vertical resolution than ours. Also, their estimates are based on 12 patches of tracers, whereas ours are based on numerical floats deployed at high spatial resolution. Note, however, that the observational estimate from DIMES at 1500 m (the blue error bar in Fig. 6) encompasses our estimate, although our float-based diffusivity estimates are for the year 1994, roughly 15 yr prior to the initial DIMES float and tracer deployment (e.g., Ledwell et al. 2011).

## 5. Eddy diffusivities in the DIMES region from multiwavenumber theory: Estimations and comparisons

This section examines diffusivities and mixing lengths from the MW theory from section 2 and compares them with those from the F–N theory and from numerical floats.

### a. Methodology

To assess whether we can obtain improved mixing estimates by extending the single-wavenumber theory to the multiwavenumber scenario, we diagnose eddy diffusivities and mixing lengths from both the F–N and MW theories. In contrast to Tulloch et al. (2014), we consider the entire DIMES domain, both upstream and downstream of Drake Passage.

We use

The diffusivity **x** (red dot in Fig. 7c) and aligned with the mean flow direction at the position **x**. Then we extract Eulerian eddy velocities along the slice during the year 1994 to form the Hovmöller diagram [Figs. 7a(1) and 7b(1)]. The term

*k*

_{eddy}is chosen to be the centroid of the eddy kinetic energy spectrum. The eddy size and the reciprocal of the eddy decorrelation time scale are determined fromAs before,

*u*

_{rms}is the eddy velocity magnitude, and Γ denotes the mixing efficiency. We chose Γ to be 0.35, following Klocker and Abernathey (2014) and Chen et al. (2014). The phase speed

*C*

_{w}is determined from the Radon transform approach of Chen et al. (2014), and

**U**denotes the 2-yr (1994/95) mean flow vector. We can then determine

### b. Results

#### 1) Comparison of horizontal structures

Figure 8 shows cross-stream diffusivities from the two theories and numerical floats. Consistent with float-based results, both the MW and F–N theories represent large diffusivities occurring inside the ACC. However, both theories have large diffusivity estimates along the eastern coast of the South Pacific, which is not the case for the float-based estimates. Possible reasons for this overestimation are discussed in section 5b(3). Float-based diffusivities are negative at some spots, corresponding to upgradient eddy fluxes. However, neither the F–N nor MW theories can represent negative diffusivities, whose mechanism and parameterization are left for future work.

Cross-stream mixing lengths have complex horizontal patterns (Fig. 9). The MW theory reasonably represents the large mixing lengths inside the ACC below 1000 m, but the F–N theory does not (right panel of Fig. 9). Both the MW and F–N theories fail to capture effectively the large-scale structures of mixing lengths in the upper 1000 m (left panel of Fig. 9). In particular, the mixing length from the F–N theory is too large north of the ACC west of the Drake Passage because of the large eddy sizes *L*.

To quantify the skill of the two theories in representing diffusivities, we correlated the float-based estimates with those diagnosed using the MW and F–N theories (Fig. 10). Both theories capture some aspects of the horizontal structure of cross-stream diffusivities; the correlation coefficients between the float-based estimates and the MW theory estimates have roughly the same magnitude as those between floats and the F–N theory, with values of 0.4–0.6 (Fig. 10b). The similar skill of the two theories in capturing cross-stream diffusivity patterns is related to the fact that, in our study domain, the diffusivity patterns are mainly controlled by eddy kinetic energy. This indicates that obtaining accurate mixing lengths (e.g., Bates et al. 2014; Chen et al. 2014) may not be essential to obtaining the correct diffusivity structures in the DIMES region.

When we examine the horizontal structures of cross-stream mixing lengths below 500 m, we see that the correlation coefficients between the MW theory and floats are larger than those between the F–N theory and floats (Fig. 10a). The advantage of the MW theory is more noticeable with increasing depth. Below 2000 m, mixing lengths diagnosed from the F–N theory and float-based estimates decorrelate, whereas the correlation coefficient between the MW theory estimates and float-based estimates increases to 0.3–0.4 (Fig. 10a).

#### 2) Comparison of vertical structures

To assess how well the two theories capture the vertical structures of eddy mixing, we examined the correlation between float-based diffusivities/mixing lengths and their theoretical-based counterparts throughout the water column (Fig. 11). In the regions where correlation coefficients are not significantly positive, the theory is considered not to have skill in representing diffusivities. Larger positive correlation coefficients imply better skill in representing diffusivities. Table 1 quantitatively compares the skill of the two theories.

Percentage of the area that satisfies the criteria listed in the first column at 95% confidence level, in the entire domain, inside the ACC and outside the ACC. Here, we define corre(*α*, *β*) as the correlation between variables *α* and *β* in the entire water column over a 3.2° × 3.2° patch. Superscripts float, multi, and single, respectively, indicate estimates in the cross-stream direction from numerical floats, MW theory, and F–N theory. The terms ^{4} and 10^{5} m^{2} s^{−1}, approximately pass the northern and southern edges of the Drake Passage. They are chosen to be the boundaries of the ACC, the region inside the two black contours is defined to be inside the ACC, and the rest of the area in our study domain is defined to be outside the ACC. Reported uncertainties here are two standard errors, obtained from 10 realizations of a bootstrapping technique by randomly sampling the gridded values in the selected spatial domain.

The F–N theory has better skill representing diffusivities inside the ACC than outside the ACC [Figs. 11b(1) and 11b(2)]; it captures the vertical structures of the cross-stream diffusivities in 75% of the area inside of the ACC but in only 41% of the area outside of the ACC (Table 1). For vertical structures of cross-stream mixing lengths, the F–N theory provides skillful estimation in 55% of the area inside the ACC but only 19% of the area outside the ACC (Table 1).

The MW theory better represents the vertical structures of diffusivities and mixing lengths than the F–N theory both inside and outside the ACC (Fig. 11; Table 1). In 91% of the study domain, the MW theory represents vertical structures of diffusivities that are significantly correlated with float diffusivities. In contrast, the percentage is only 55% for the F–N theory. Table 1 reveals that the MW theory captures diffusivities outside the ACC as well as it does inside the ACC. In contrast, the F–N theory is better inside the ACC than outside. Nonetheless, the MW theory consistently outperforms the F–N theory. Overall, in roughly 90% of the area outside the ACC, the float-based estimates are better correlated with estimates from the MW theory, than with estimates from the F–N theory (Table 1). In contrast, the percentage is roughly 60% inside the ACC.

The MW theory better represents the mixing lengths to the east of the Drake Passage than to the west [Fig. 11a(2)]. The MW theory cannot represent negative diffusivities. The existence of negative diffusivities inside the ACC west of the Drake Passage (Figs. 8, 9) probably contributes to the small or negative correlation there. Note that the MW theory is based on the assumptions that the mean flow varies slowly spatially and that eddies have much smaller amplitude than the mean flow. These assumptions are poorly satisfied in the area north of the ACC west of the Drake Passage (not shown), leading to the poor representation of mixing lengths there [Fig. 11a(2)].

#### 3) Comparison of magnitudes

Though the MW and F–N theories capture the order of magnitude of eddy diffusivities and mixing lengths correctly, both theories overestimate the values (Fig. 12). In the upper 1000 m, the MW theory overestimates the domain-averaged diffusivities and mixing lengths more than the F–N theory; below 1000 m, the mismatch between the MW theory and floats is smaller than it is between the F–N theory and floats.

The magnitude of the diffusivities from the F–N theory depends on the choice of Γ, which denotes the mixing efficiency. The parameter Γ is generally chosen to be order one (e.g., Klocker and Abernathey 2014; Chen et al. 2014). We examined the diffusivities from the F–N theory using different choices of Γ. F–N theory’s overestimation of diffusivities between 85 and 3000 m is insensitive to Γ within the wide range from 0.1 to 10.

Since both the F–N and MW theories overestimate diffusivities, in particular, along the eastern coast of the South Pacific (Figs. 8, 12), the single-wavenumber assumption probably is not the leading-order explanation for the overestimation. When diagnosing the float-based diffusivities, in order to reduce the dispersion caused by the mean flow shear, as Griesel et al. (2010) did, we calculate residual velocities from Eq. (18) by subtracting the local mean flow rather than the spatially uniform mean flow. However, both the F–N and MW theories assume the mean flow to be constant and thus do not include a mechanism to reduce the dispersion due to mean flow shear. Therefore, the spatial homogeneity assumption inherent in these theories might contribute to the overestimation.

To test this hypothesis, we use Eq. (18) to recalculate the float-based cross-stream diffusivities *t*_{0}. In other words, the residual velocity is defined as the deviation of eddy velocity along the float trajectories from the mean flow at the float position at time *t*_{0}. Similar to the F–N and MW theories, the constant mean flow calculation does not include a mechanism to reduce the dispersion caused by the mean flow shear. As we expected, similar to the theory-based diffusivities, the domain-averaged

## 6. Discussion

The main goal of the MW theory was to modify the single-wavenumber theory to account for the multiwavenumber feature of eddies. The single-wavenumber theory and the MW theory are consistent in some respects. In the single-wavenumber limit, the MW theory reduces to the single-wavenumber mixing formula of Ferrari and Nikurashin (2010) (appendix A). In addition, both theories are based on an assumption of spatial homogeneity, which means that horizontal variations in the mean flow and eddy properties are negligible. Finally, both theories explicitly illustrate the effect of the mean flow **U** on mixing.

In spite of their consistency, the single-wavenumber and MW theories also have clear differences. The MW theory more clearly depicts the ocean, in that oceanic eddies are composed of a range of wavenumbers rather than a single dominant wavenumber. In addition, the MW theory provides formulas for the diffusivity tensor and cross-stream diffusivities [Eqs. (17) and (20)]; in contrast, the F–N theory only focuses on cross-stream diffusivities.

_{S}isThe advantage of Eq. (25) is that it is computationally more efficient than some other related frameworks (e.g., Chen 2013; Chen and Flierl 2015).

Our work regarding the MW theory has several implications. First, the fact that the MW theory outperforms the F–N theory in capturing the mixing length structures indicates that it is useful to consider the multiwavenumber regime in future development of mixing theories and eddy parameterizations. In addition, much effort has been devoted to characterizing and interpreting oceanic spectra in order to reveal the underlying processes of oceanic turbulence (e.g., Xu and Fu 2011). The MW theory links velocity spectra with mixing, underscoring the importance of spectra for mixing. Finally, critical layer theory suggests that elevated values of mixing lengths occur at the critical layer depth, where the wave phase speed matches the mean flow magnitude (e.g., Bretherton 1966; Green 1970). Critical layer depth has been estimated either by identifying the phase speed of the fastest growing mode from linear stability analysis (e.g., Smith and Marshall 2009) or by identifying the dominant phase speed of the eddy field through the Radon transform (Chen et al. 2014). Each wavenumber in the eddy field corresponds to a different critical layer depth, and this work indicates that it is important to consider the contribution of all the waves in the eddy field to mixing. Therefore, future studies would potentially benefit from estimating a critical layer depth specific to each wavenumber.

Though successful in many respects (as shown in Fig. 12), the MW theory, like the F–N theory, overestimates the domain-averaged cross-stream mixing lengths by roughly 5–10 km. As discussed in section 5b(3), the spatial homogeneity assumption inherent in these theories contributes to the overestimation. Further improvements to the mixing theory probably should take into account the spatial inhomogeneity (e.g., horizontal shear in the mean flow), which can be induced by topographic effects, the localized formation of coherent vortices, and so on. In addition, both the MW and F–N theories are built on the assumption that the magnitudes of eddies are smaller than that of the mean flow. This linear assumption does not hold in many oceanic regions, and taking this nonlinearity into account may lead to improvements of the mixing theory.

It is not trivial to take into account the spatial inhomogeneity in eddy parameterizations. Appendix B illustrates analytically that, in an inhomogeneous system, where the mean tracer gradient and eddy fluxes vary over short spatial scales, eddy mixing depends on both the local and nonlocal mean tracer gradients. However, the concept of the eddy diffusivity itself, on which the F–N and MW theories are built, is based on the assumption that eddy mixing and transport processes can be parameterized using the local mean tracer gradient.

While the shortcomings of the MW theory identified in this study indicate that nonlocal eddy parameterization schemes may ultimately prove valuable, the MW theory nonetheless merits further assessment, particularly since the assumption of spatial homogeneity is reasonable in regions such as the midlatitude ocean interior. In addition to cross-stream diffusivities, the MW theory also provides formulas for the diffusivity tensor. The relevance of the MW theory to mixing in other regions or other years has not yet been explored. Griesel et al. (2015, manuscript submitted to *Ocean Modell*.) diagnosed the diffusivities from the single-wavenumber theory using eddy parameters obtained from linear instability analysis and then compared them with those from numerical floats in the entire Southern Ocean. While they found evidence for enhanced Lagrangian integral time scales at the steering level depth, they concluded that a single frequency is not able to reproduce the oscillations in the velocity autocovariance. This suggests that it would be useful to assess whether the MW theory also better captures the vertical structure of mixing outside of the DIMES region.

To infer eddy diffusivities from the single-wavenumber and multiwavenumber theories, we need information about eddies (e.g., their spectra and dominant phase speed). Thus, these mixing theories are not directly applicable to eddy parameterization schemes, which are intended to link eddy diffusivities with large-scale, mean flow characteristics. However, the theories do underscore the value of including the critical-layer effect in eddy parameterization schemes, and the multiwavenumber theory shows that the critical-layer depth is wavenumber specific. The multiwavenumber theory also suggests that the link between the large-scale mean flow and eddy spectra may help formulate new eddy parameterization schemes. Previous studies linking mean flow with eddy properties exist (e.g., Venaille et al. 2011).

## 7. Summary

Though oceanic eddies contain motions spanning a wide range of wavenumbers (e.g., Wunsch 2010), previous theories of eddy mixing have often been based on the assumption that eddies are composed of a single or dominant wave (e.g., Green 1970; Killworth 1997; Ferrari and Nikurashin 2010; Klocker et al. 2012a). Motivated by this discrepancy, we formulated a MW theory of mixing, starting from the problem described by Klocker et al. (2012a). Our MW theory, which can be derived from both Lagrangian and Eulerian perspectives, is based only on the mean flow and eddy velocity spectra, which are unambiguous and straightforward to calculate.

We chose the DIMES region to compare mixing theories. Eddy diffusivities and mixing lengths in the cross-stream direction were estimated using numerical floats deployed in a global eddying model. Cross-stream diffusivities are large inside the ACC, where eddy amplitudes are large. Horizontal structures of cross-stream diffusivities and eddy kinetic energy vary little with depth, indicating the relevance of the equivalent barotropic nature in this region. These float-based mixing estimates at high spatial resolution can serve as a context for observational results from the DIMES experiment.

The float-based mixing estimates were then compared with those from both the F–N and MW theories. We found that the F–N and MW theories have similar skill in representing the horizontal structures of cross-stream eddy diffusivities, which are mainly controlled by eddy velocity magnitudes. Correlation analysis indicates that, compared to the F–N theory, the MW theory is better at capturing both the horizontal and vertical structures of cross-stream mixing lengths. Therefore, a so-called dominant wave is insufficient to capture the mixing length properties in the DIMES area and possibly in other ocean regions as well. Ideally the full range of waves in the ocean should be considered when developing new mixing theories or parameterization schemes.

## Acknowledgments

RC and JLM were supported by NSF Grant OCE0960914. JLM was also supported by the U.S. Department of Energy’s Office of Science Climate Modeling Programs via a Los Alamos National Laboratory subcontract. STG was supported by NASA NNXBAE446 and NSF OCE-1234473. We thank William R. Young and Laurence Armi for discussions, and Elena Yuleava for extracting the POP model output in the DIMES region. We also would like to thank Malte Jansen and an anonymous reviewer for their constructive comments about the manuscript. Raffaele Ferrari kindly provided us the diffusivity data from Fig. 10 in Tulloch et al. (2014) for the comparison shown in our Fig. 6. Computational resources for the model run were provided by NSF XSEDE Resource Grant TG-OCE100001. Model output is available at the National Center for Computational Sciences.

## APPENDIX A

### Consistency between the Single- and Multiwavenumber Theories in the Single-Wavenumber Scenario

Here, we illustrate that, in the single-wavenumber scenario, the cross-stream diffusivity from the MW theory in section 2 is consistent with the single-wavenumber formula for cross-stream diffusivities from Ferrari and Nikurashin (2010) (F–N theory).

#### a. Review of the F–N theory

*U*and buoyancy

*B*aswhere

*N*is the buoyancy frequency;

*f*is the Coriolis parameter. The eddy field satisfiesandwhere

*ψ*is eddy streamfunction,

*b*is surface eddy buoyancy,

*k*is the zonal wavenumber,

*l*is the meridional wavenumber,

*γ*is the linear damping rate, and

*r*(

*t*) is the white noise forcing with zero mean and autocorrelation function

*κ*is the wavenumber magnitude

*a*can be obtained from Eq. (A5); then the eddy variables

*ψ*and

*b*are solved.

*C*′ is the tracer concentration and Γ

_{c}is the mean tracer gradient. They obtain the solution to

*C*′ from Eq. (A6) by assuming

*C*′ has the same spatial structure as

*ψ*. Then the cross-stream Eulerian diffusivity can be obtained fromTheir final solution to

*C*

_{w}denotes the wave phase speedand EKE is the domain-averaged eddy kinetic energy

#### b. Cross-stream diffusivity from the multiwavenumber theory in the single-wavenumber limit

As shown in Eq. (19), to obtain cross-stream diffusivities from the MW theory, the key is to derive the frequency–wavenumber spectra of cross-stream eddy velocities. In the single-wavenumber scenario from Ferrari and Nikurashin (2010), we can obtain this spectrum from Eqs. (A4) and (A5).

*k*

^{2}, we obtain the spectrum for cross-stream (in this case, meridional) velocities:where

*r*(

*t*) is the white noise forcing with zero mean and autocorrelation

## APPENDIX B

### Derivation of the Multiwavenumber Theory from Tracers and Flux-Gradient Relation

Here, we show that the MW theory from section 2a can also be derived from an Eulerian diffusivity perspective using tracers. The mathematical symbols used below follow the convention of Eulerian diffusivities and apply only in this appendix.

*C*:andwhere

In the spatially inhomogeneous case, the accurate form [Eq. (B8)] differs from the standard form [Eq. (B9)]. First,

*κ*

_{0}in Eq. (B1) is small. Under the small eddy amplitude and small

*κ*

_{0}assumptions, Eq. (B3) is reduced toFor small enough excursions, the mean flow is roughly a constant, and thus we obtainwhere

*t*to be zero without the loss of generality. Then we obtain from Eq. (B14)where

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