# Large-scale impacts of the mesoscale environment on mixing from wind-driven internal waves

## Abstract

Oceanic mesoscale structures such as eddies and fronts can alter the propagation, breaking and subsequent turbulent mixing of wind-generated internal waves. However, it has been difficult to ascertain whether these processes affect the global-scale patterns, timing and magnitude of turbulent mixing, thereby powering the global oceanic overturning circulation and driving the transport of heat and dissolved gases. Here we present global evidence demonstrating that mesoscale features can significantly enhance turbulent mixing due to wind-generated internal waves. Using internal wave-driven mixing estimates calculated from Argo profiling floats between 30° and 45° N, we find that both the amplitude of the seasonal cycle of turbulent mixing and the response to increases in the wind energy flux are larger to a depth of at least 2,000 m in the presence of a strong and temporally uniform field of mesoscale eddy kinetic energy. Mixing is especially strong within energetic anticyclonic mesoscale features compared to cyclonic features, indicating that local modification of wind-driven internal waves is probably one mechanism contributing to the elevated mixing observed in energetic mesoscale environments.

## Main

The oceanic mesoscale is a diverse field of structures including eddies, fronts and filaments with length scales of order 10–100 km. The mesoscale is most active at the ocean’s surface, coincident with one of the major sources of internal wave energy: the wind. Storms and other events that create time-variable wind stress can induce a resonant response in the ocean mixed layer, which oscillates near the inertial frequency. Convergences and divergences of these inertial oscillations perturb the base of the mixed layer, transferring energy near the inertial frequency into internal waves that propagate into the stratified water below1. Eventually these waves break, turbulently mixing the water and dissipating their energy. The wind power driving these internal waves is comparable (within a factor of 2) to that going into internal tides; together they provide the bulk of the global power for turbulent mixing in the ocean interior2,3.

The spatial and temporal patterns of turbulent mixing due to near-inertial waves are influenced by both the geographical distribution and seasonal cycles of the winds, and by the processes that alter the propagation and breaking of these waves1. Near-inertial wave driven mixing has an impact on the location and timing of water mass transformation, sets global variations in stratification and facilitates the downward transport of heat and dissolved gases in the ocean2,4. Knowledge of the mechanisms that control the global patterns of near-inertial wave-driven mixing is therefore required for an accurate understanding of both basic oceanic physical processes and their subsequent representation in climate models.

Theory and modelling studies suggest that interactions between near-inertial waves and mesoscale vorticity5,6,7,8, strain rate5,9,10,11,12 and strongly sloping isopycnals6,13,14 are important factors that can set patterns of turbulent mixing in the global ocean. Interactions may occur both within the mixed layer during near-inertial wave generation and in the stratified water below. Consistent with these studies, observations have shown that estimates of turbulent mixing are globally correlated with mesoscale eddy kinetic energy15, and that internal wave energy is correlated with the strength of the local eddy field at a number of sites12. However, observations establishing a relationship between the behaviour of wind-driven near-inertial waves and the mesoscale field have been difficult to achieve due to the range of relevant scales, the episodic nature of near-inertial waves, and the diversity of the mesoscale environment. These difficulties have limited the majority of observations to specialized cases such as isolated individual eddies16,17,18,19.

Here we consider internal wave driven mixing over global spatial scales by inferring the diapycnal diffusivity K (hereafter referred to as the diffusivity) and dissipation rate $$\epsilon$$ (Fig. 1a,b), both measures of turbulent mixing. They are related by $$K = {\Gamma }\epsilon {\mathrm{/}}N^2$$, where N2 is the buoyancy frequency, and Γ = 0.2 is the assumed mixing efficiency20. This work primarily presents results in terms of diffusivity, as it is a natural way to examine patterns beyond the typically observed stratification dependence of turbulent dissipation rates. The diffusivity estimates are generated from Argo float density profiles using a fine-structure method described in previous work21, extended to include observations through July 2017 (see Methods). The fine-structure calculations of mixing are not direct, but apply the results of many theoretical and observational studies22,23,24,25,26 by using variations in vertical strain to infer the expected energy dissipated from the local internal wave field26. One source of uncertainty in this method is that other density variations (double diffusive layers, for example) can be mistaken for internal waves. Many of these additional density variations are persistent in time regardless of season or wind condition; this study therefore focuses on changes in diffusivity as a function of winds and seasons to minimize the influence of these density variations on our results.

Each mixing estimate is matched with concurrent characteristics of the mesoscale and wind fields using tools that are described in the Methods. Temporally averaged eddy kinetic energy is calculated from surface drifters (Fig. 1c), and the geostrophic vorticity is determined from satellite altimetry with 0.25° resolution. A slab model approximating the mixed layer response to winds without mesoscale currents is used to calculate the energy flux from winds into near-inertial oscillations, which is then averaged over a period of 30–50 days before each mixing estimate (Fig. 1d). The time span is within the range of previous observations27, equivalent to a near-inertial wave travelling 500 m vertically at a velocity of 10–17 m d−1.

Due to the strong seasonal cycle in mixing and large spatial variability in the magnitude of eddy kinetic energy (Fig. 1c), the latitudinal band between 30° and 45° N in the Northern Hemisphere Pacific and Atlantic oceans is chosen as the focus of this study. The Argo-derived mixing estimates show a seasonal cycle in both the zonal median diapycnal diffusivity and dissipation rate between 30° and 45° N, evident to the full depth of the 2,000 m profiles (Fig. 2a,b). The seasonal cycle is correlated with estimates of the near-inertial energy flux from winds into inertial oscillations (Fig. 2c,d), consistent with previous work indicating that the seasonal cycle of near-inertial kinetic energy28,29 and mixing15,21 is elevated beneath mid-latitude storm tracks. There is no significant difference between the magnitude of summer and winter eddy kinetic energy estimates from satellite altimetry, indicating that the basin-averaged mixing seasonal cycle is not likely to be caused by the modulating effect of the temporally variable eddy kinetic energy at mesoscales. However, variations that are regional or at scales smaller than can be resolved by satellite altimetry may contribute to the patterns of mixing observed in this study, including the seasonal cycle.

The seasonal cycle in mixing is amplified by the presence of an energetic mesoscale. In the summer months (July–September) the diffusivity is larger in regions of high (greater than the median) compared to low (less than the median) eddy kinetic energy (Fig. 3b,d). When the season changes to winter (January–March), storm activity increases and the diffusivity rises by a larger factor (2–3) in areas of high eddy kinetic energy throughout the 250–2,000 m profiles, as opposed to a smaller factor (1–2) in quiescent regions (Fig. 3c,e). The larger seasonal fluctuations in the presence of an energetic mesoscale field suggest that the lively internal wave field fostered by wintertime storms is interacting with mesoscale structures to enhance mixing.

We now consider the flux of near-inertial energy from the winds as a proxy for the strength of the locally forced internal wave field, as opposed to the previously discussed seasonal cycle that includes waves originating elsewhere. Evidence that the mixing is modulated by the near-inertial energy flux from the winds is seen in regions of both high and low eddy kinetic energy; the diffusivity between 250–500 m is correlated with the local wind flux (Fig. 4a). In regions of high eddy kinetic energy, the change in diffusivity in response to increases in wind energy flux is dramatic, climbing nearly an order of magnitude at a rate that is a factor of two larger than in regions with low eddy kinetic energy. The larger increase in diffusivity in response to changes in the winds over an energetic mesoscale is consistent with models30 that show intensification of near-inertial kinetic energy in western boundary currents and theories6,7,8,10,11,13 that near-inertial waves created in a rich mesoscale field are altered during their generation or propagation, increasing mixing in these regions.

Other studies have suggested two alternative processes that may enhance mixing in energetic mesoscale environments: mesoscale–internal tide interactions31,32 and the generation of internal waves directly by the mesoscale33,34,35. If either of those were globally important on long timescales, then the diffusivity would be elevated in an energetic mesoscale even when wind forcing is weak, since neither of these processes involve wind-driven near-inertial waves. We see no such relationship in the upper ocean (250–500 m, Fig. 4a), indicating that mesoscale–tide interactions and mesoscale-generated internal waves have little net impact on mixing at large spatial scales and long timescales in this depth range.

A kilometre deeper (1,250–1,500 m) the relationship changes; the diffusivity rises at a slower rate in response to increases in wind energy flux (Fig. 4b). Mixing from local wind-generated internal waves may be less important in this depth range compared with near-inertial waves travelling laterally from elsewhere36, resulting in a weaker relationship between local winds and mixing than occurs closer to the surface. There is also a larger separation between the diffusivity in high and low eddy kinetic energy regions for all values of wind energy flux (Fig. 4b) compared to shallower depths (Fig. 4a) indicating that interactions between the mesoscale and tidally generated internal waves, or internal waves produced by geostrophic flow over bottom topography37, might be more important in this depth range.

The observed larger-amplitude seasonal cycle of mixing in an energetic mesoscale field is probably due to processes in the ocean interior (such as internal wave–mesoscale interactions) as opposed to heightened storm activity above regions with an energetic mesoscale. Figure 4c shows the total dissipation rate normalized by the wind energy entering the internal wave field (see Methods) for all times when the wind energy flux is strong (above the median). The normalized dissipation rate is a measure of the fraction of wind energy dissipated within the depth range; however, this is only an approximation as large values can also indicate the presence of mixing due to other energy sources (such as tides or the mesoscale). These maps show a relatively large fraction of the local wind-energy input dissipating in regions of high eddy kinetic energy, indicating that mixing is more likely to be elevated due to interior ocean processes such as dissipation within the mesoscale eddy field rather than a strengthened source of wind energy. For example, the amplified seasonal cycle where the mesoscale is energetic (Fig. 3) is more likely to be caused by interactions between near-inertial internal waves and the mesoscale rather than a locally strong seasonal cycle in the winds. Alternatively, wind-generated near-inertial waves might be extracting energy from the mesoscale to form additional internal waves38,39, which would also appear as larger normalized dissipation rates in eddy-rich regions in Fig. 4c.

Only one mechanism that might explain the elevated mixing observed in energetic mesoscale environments can be investigated using global datasets: the interaction between the geostrophic vorticity field and near-inertial internal waves (or preceding inertial oscillations). Vorticity at mesoscales in theory6,7 and simulations8,40 can alter the dispersion relation of wind-forced internal waves, reducing the horizontal scales and concentrating the waves in anticyclonic vorticity, thereby dissipating their energy and causing turbulent mixing. Observations that support a specific case of this trapping process indicate that internal waves can propagate downwards until they reach a critical layer due to vorticity changes beneath an eddy core. This causes the vertical scales to rapidly shrink, triggering diapycnal mixing17,18,19. Observations also show that inertial oscillations can be modified by the geostrophic vorticity within the mixed layer41, potentially altering the internal waves that they generate. The influence of anticyclonic vorticity is explored next by focusing on the Kuroshio Extension (defined here as 150° E–170° W and 30°–45° N). This region is chosen due to relatively homogeneous high eddy kinetic energy, lack of substantial topographic features that could generate or scatter internal waves and sufficient data for statistical significance15.

Strong mesoscale anticyclonic vorticity (<−3 × 10−6 s−1) is associated with slightly larger diffusivities than cyclonic vorticity of the same magnitude in the Kuroshio Extension between 250–500 m depth (Fig. 5a,c). For very low values of wind energy flux the diffusivity is approximately equal in the presence of anticyclonic and cyclonic vorticity (Fig. 5c). With increasing wind energy flux, the diffusivity intensifies for strong vorticity of either sign; however, the rate is faster when the vorticity is anticyclonic compared to cyclonic. These observations are consistent with previous studies6,7,8,16,40 that find that anticyclonic vorticity is associated with mixing within or near vorticity anomalies. Our basin-wide observations therefore suggest that interactions between wind-driven near-inertial waves and mesoscale anticyclonic vorticity might be one of the mechanisms setting the location of turbulent mixing generated by these waves over large scales.

The influence of anticyclonic vorticity is not sufficient to fully explain the observed differences between high and low eddy kinetic energy environments (Figs. 3 and 4) because diffusivity is only slightly larger in regions of anticyclonic compared to cyclonic vorticity when the wind energy flux is strong (Fig. 5). Satellite altimetry may not fully resolve the relevant vorticity scales, causing an underestimation of the effects of vorticity. Alternatively, other mechanisms may also be important in the thermocline or mixed layer, irrespective of the sign of the vorticity, including additional internal waves generated by interactions between the mesoscale and near-inertial internal waves38,39, interactions between near-inertial waves and strongly sloping isopycnals with accompanying geostrophic shear6,13,14 or the horizontal mesoscale strain rate5,9,10,11,12,42,43. More work is needed to identify dominant mechanisms that govern the interactions between wind-driven internal waves and the mesoscale at large scales.

We have presented comprehensive observations showing that on basin scales the oceanic mesoscale plays a role in the mixing triggered by wind-driven internal waves. These results imply that the mesoscale assists in controlling the location, both in the horizontal and vertical, and timing of wind-driven internal wave mixing on large scales. Our observations also indicate that anticyclonic vorticity slightly enhances mixing arising from wind-driven internal waves, suggesting vorticity of this sign alters wave properties and propagation. Spatial and temporal patterns of mixing control the rate at which heat, greenhouse gasses, and freshwater are transported away from the surface, amplify or diminish associated air-sea coupled interactions, and set regional density-driven circulation patterns2. Therefore if the mesoscale is key in setting a share of these patterns as our results indicate, then the mesoscale’s involvement in internal wave driven mixing is crucial to parameterize in climate models and incorporate into our global view of ocean mixing.

## Methods

### Argo diffusivity and dissipation rate estimates

As direct measurements of diapycnal mixing are difficult to obtain, this study uses a method to infer mixing from the global Argo float dataset (www.argo.net/). Dissipation rate and diffusivity estimates are generated from 2–10 m vertical resolution temperature and salinity profiles between 2,000 m depth and the base of the mixed layer using Argo data from January 2006 to July 2017. A fine-structure parameterization that is built on theoretical and empirical studies that relate the energy dissipation rate to wave–wave interactions in the internal wave field22,23,24,25,26 is applied to these profiles to estimate the dissipation rate and diffusivity as described in previous work21. The method infers mixing by integrating the vertical wavenumber spectra of 200 m segments from each density profile to generate estimates of the turbulent energy dissipation rate and diapycnal diffusivity.

One source of uncertainty in this method is the assumption that all variations in the density profiles on 40–100 m vertical scales are internal waves, producing diffusivity estimates that are too large if non-internal wave density variability is present. These non-internal wave density variations may be more prevalent in regions of strong eddy kinetic energy, and frequently persistent across seasons (for example, double diffusion). However, if these density variations do not interact with near-inertial waves, then any spuriously elevated mixing estimates they may cause will be independent of the near-inertial internal wave state (that is, exist in both winter and summer). Therefore, this study focuses on the change in wind-driven mixing as a function of the seasons and wind state in environments with contrasting high and low eddy kinetic energy. The mixed layer and mode water are two other sources of non-internal wave density variability. Both the mixed layer and mode water are removed by identifying portions of the profiles with weak stratification21. However, mode water that has been partially destroyed, and is therefore more stratified, may not be detected; this may cause an overestimation of the magnitude of the seasonal cycle in the upper portion of the profile (250–500 m in the North Atlantic), where a seasonal cycle of partially destroyed mode water is observed44. Also, the ratio of internal wave shear to strain is also assumed to be constant in the version of the fine-scale method applied here, which could possibly influence the results in this study. For example, a constant shear to strain ratio could potentially lead to an of the change of diffusivity25 associated with the seasonal change in wind forcing because the ratio is expected to change seasonally as the fraction of near-inertial waves changes. Despite these caveats, the fine-structure method generally agrees well with direct measurements of the dissipation rate and diffusivity (within a factor of 2–3)21 and is therefore a useful method for studying large-scale mixing variability in the ocean.

The energy dissipation rate normalized by the near-inertial energy flux from the winds into internal waves $$\epsilon _f$$ is defined as

$$\epsilon _f = \frac{{\epsilon \rho L}}{{{\mit\Pi}}}$$

where $$\epsilon$$ is the dissipation rate from Argo profiles between 250–500 m depth, f is the Coriolis frequency, ρ is the density, L is the vertical length scale that $$\epsilon$$ is averaged over and Π (defined below) is the average near-inertial flux of energy from the winds into the internal wave field 30–50 days before the dissipation rate estimate. As the calculated dissipation rate is due to the dissipation of waves from all sources, including both tides and winds, $$\epsilon _f$$ may be greater than one in regions with significant internal tides.

### Slab model of near-inertial wind energy flux

A slab model following a number of authors45,46,47,48 was applied to reanalysis winds and observed mixed layer depths to generate estimates of the near-inertial flux from the winds into inertial oscillations. This model assumes that the inertial oscillations in the mixed layer do not interact with the background mesoscale velocity and vorticity fields, which studies have found can have an effect on the near-inertial energy flux and damping of the inertial oscillations5,49,50. As strong mesoscale environments may remove near-inertial energy from the mixed layer at a faster rate than the model assumes5,49, modification of the inertial currents in the mixed layer may contribute to the results presented in this study.

We briefly explain the model as it is used here. A full derivation of the slab model can be found in the work of other authors45,46,47,48. The velocity of the mixed layer (u and v) can be described by

$$\frac{{{\mathrm {d}}u}}{{{\mathrm {d}}t}} - fv = \frac{{\tau _x}}{H} - ru$$
(1)
$$\frac{{{\mathrm {d}}v}}{{{\mathrm {d}}t}} - fu = \frac{{\tau _y}}{H} - rv$$
(2)

where H is the mixed layer depth, τ the wind stress and r a chosen damping constant. The inverse of this damping constant 1/r ranges in recent observations between 1 and 5 d (refs 48,51), including in drifter observations that show decay timescales of approximately 1–2 d between 30° and 45° N51. Although the damping constant has a significant effect on the temporal variability of the near-inertial energy flux, it has been shown to not strongly influence the mean near-inertial flux compared to the wind forcing47,52. Therefore we do not expect the choice of damping constant to strongly affect our 20-day averaged near-internal flux estimates. However, a strong mesoscale can cause 1/r to be smaller in comparison with a weak mesoscale49, which may have some effect on the time-averaged near-inertial flux estimates when they are used here to compare high and low eddy kinetic energy environments. Here we choose 1/r = 2 d.

The slab model as derived by previous authors45,46,47,48 uses the complex variables

$$T = \frac{{\tau_{x} + {i}\tau_{y}}}{\rho },\,Z = u + iv\,{\mathrm{and}}\,\omega = r + if$$

to produce an energy relationship from equations (1) and (2)

$$\frac{{{\mathrm{d}}\left| {\frac{1}{2}Z_I} \right|^2}}{{{\mathrm{d}}{t}}} = - r\left| {Z}_{I} \right|^2 - {\mathrm{Re}}\left[ {\frac{{Z}_{I}}{{\omega ^ \ast H}}\frac{{{\mathrm {d}}T^\ast }}{{{\mathrm{d}}{t}}}} \right]$$
(3)

where ZI is the oscillating component of the full velocity Z = ZE + ZI, and ZE is the steady (Ekman) component. The asterisk (*) indicates the complex conjugate of a variable. Solutions of this system are in the form Z = Z0eiftrt. Equation (3) implies that the rate of change of the near-inertial energy (first term) is balanced by the dissipation of that energy in the mixed layer and energy loss through the mixed layer base (second term) and the flux of energy Π(H) to near-inertial oscillations (third term). If we assume that only wind at the inertial frequency is important, the near-inertial energy flux term reduces to

$${{\mit{\Pi}}}(H) = {\mathbf{u}} \cdot {\boldsymbol{{{\tau }}}}.$$

The near-inertial energy flux at each Argo float profile, leading up to the time of the profile, is calculated using the NASA Modern-Era Retrospective analysis for Research and Applications version 2 (MERRA-2) wind stress product generated from the Goddard Earth Observing System Model Version 5 (https://disc.gsfc.nasa.gov/). MERRA-2 provides hourly wind stress reanalysis on a grid of 0.5° latitude by 0.625° longitude, which we then subsample at 2 h intervals to save computation time. The value of H from the corresponding Argo profile was used unless the mixed layer depth was less than 20 m, in which case a minimum mixed layer depth of 20 m was applied. We select the mean near-inertial flux 30–50 d before each dissipation rate estimate as our measure of the near-inertial flux (Fig. 1d).

### Average eddy kinetic energy

The mean eddy kinetic energy product (provided by R. Lumpkin) used here is presented in Fig. 1c. The time-mean version of this dataset can be found at http://www.aoml.noaa.gov/phod/dac/53. The mean eddy kinetic energy product was calculated using the global surface drifter dataset as $$\left\langle {{\bf{U}}^{\prime 2}} \right\rangle$$, which is found by applying a 5 d low-pass filter to each 6 h drogued drifter observation (removing the tides and inertial oscillations), and separating the mean velocity from the deviation U = $${\bf{\bar U}}$$ + U′, and averaging the square of the deviation.

### Geostrophic velocity and vorticity

Geostrophic velocity (U, V) and vorticity (Vx − Uy) used in this study were calculated from the AVISO geostrophic velocity product created by Ssalto/Duacs (http://www.aviso.altimetry.fr/duacs/) at each time and location of an Argo-float derived diffusivity estimate. It was not possible to test the theory that the mesoscale strain rate is important for mesoscale/internal wave interactions as the resolution of the AVISO product was insufficient to make the required calculations. AVISO uses sea surface height anomalies from satellite altimetry to calculate geographic velocity daily on a 0.25° × 0.25° grid. The delayed version of the product was used for most of the Argo float locations unless it was unavailable, and then the near-real-time version was used. Eddy kinetic energy estimates (U2 + V2) from this AVISO data are confined to the larger scales that can be resolved from satellite data than the eddy kinetic energy estimates from drifters. Derivatives are calculated from the geostrophic velocities using a seven-point stencil, which improves vorticity calculations by incorporating more data points into the gradient calculations than the simpler three-point stencil method54.

### Code availability

The code used to generate the dissipation rate and diffusivity estimates is available from the corresponding author upon request.

### Data availability

All data used in this study is publicly available from the websites listed in Methods.

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## Acknowledgements

The authors are grateful for the support of NSF OCE-1259573 and for valuable comments from K. Polzin and E. Kunze.

## Author information

Authors

### Contributions

C.B.W conceived of the study, conducted the analysis, and wrote the manuscript. Both J.A.M and L.D.T. contributed to the analysis and writing of the manuscript.

### Corresponding author

Correspondence to C. B. Whalen.

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### Competing interests

The authors declare no competing interests.

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Whalen, C.B., MacKinnon, J.A. & Talley, L.D. Large-scale impacts of the mesoscale environment on mixing from wind-driven internal waves. Nature Geosci 11, 842–847 (2018). https://doi.org/10.1038/s41561-018-0213-6

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