1. Introduction

Turbulent mixing is integral to many regional and global-scale processes throughout the ocean, ranging from the ocean’s overturning circulation to the distribution of passive tracers. However, measurements resolving turbulence have been difficult to achieve since they rely on specialized ship-based instruments, and thus long temporal and large spatial scale patterns are hard to discern. Instruments capable of very high-resolution temperature measurements are starting to be deployed on long time scales (Moum et al. 2009), but these devices are not yet in wide use.

To fill this gap, finescale parameterizations are being applied to estimate the turbulent kinetic energy dissipation rate and diapycnal diffusivity from a variety of more common instruments such as CTDs, ADCPs, and Argo profiles (e.g., Gregg 1989; Polzin et al. 1995; Kunze et al. 2006; Huussen et al. 2012; Wu et al. 2011; Whalen et al. 2012). Strain- and shear-based finescale parameterizations aim to infer the centimeter-scale turbulent energy dissipation rate from what can be measured with relative ease with 2–15-m resolution using the scale-bridging physics of internal waves. These parameterizations rely on two major assumptions:

1. The observed variance in shear and/or strain on 10–100-m scales is caused mainly by internal waves.

2. The energy dissipation rate is primarily due to nonlinear interactions between internal waves that transfer energy from the finescale toward smaller-scale waves that subsequently break into turbulence.

Internal waves are associated with shear and strain variance observed on vertical scales of 10–100 m. Realizing that much of the finescale variance is caused by internal waves (assumption 1), and that energy transfers between these waves in the open ocean can be described using nonlinear interaction theory (assumption 2), has led previous authors to use a wave action approach (Henyey et al. 1986; Muller et al. 1986; Henyey and Pomphrey 1983). This allows for an expression of the downspectrum energy cascade in terms of the shear and/or strain spectra as explained in Polzin et al. (2014). Assuming that this internal wave energy cascade sets the energy dissipation rate (assumption 2) allows for calculation of the dissipation rate from the shear or strain spectra.

A key point is that finescale estimates of the dissipation rate are measuring distinctly different quantities than microstructure measurements. Both the length and time scales are larger for the finescale estimates than for the microstructure. It may be helpful to think of the microstructure profiles as a snapshot of the turbulence at an instant, while the finestructure profiles are estimates of the average dissipation rate expected over several wave periods. Therefore, comparing equivalent quantities requires averaging multiple microstructure profiles in time.

A set of parameterizations using finescale shear profiles has been tested in a variety of contexts, showing good agreement with microstructure in open-ocean conditions (Gregg 1989; Polzin et al. 1995; Winkel et al. 2002; Polzin et al. 2014). Implementation of shear parameterizations has revealed reasonable patterns of diapycnal mixing (Polzin et al. 1997; Kunze et al. 2006; Huussen et al. 2012). The shear-based parameterization is known not to be effective in regions where the underlying assumptions behind the parameterization do not apply (Polzin et al. 2014), such as on continental shelves (MacKinnon and Gregg 2003) and in strong geostrophic flow over rough topography (Waterman et al. 2014). Studies have also uncovered discrepancies in the presence of very large overturning internal waves (Klymak et al. 2008) and in submarine canyons (Kunze et al. 2002); however, Polzin et al. (2014) suggests that in these cases the disparity may be due to parameterization implementation choices rather than a violation of the underlying assumptions.

In contrast, the finescale strain-based version of this parameterization has only been tested to a limited extent. Observational work has uncovered a correlation between the finescale strain variance and the dissipation rate (Desaubies and Gregg 1981; Gregg and Kunze 1991; Wijesekera et al. 1993; Padman et al. 1990; Mauritzen et al. 2002), and it has been subsequently applied to yield reasonable large-scale patterns of diapycnal diffusivity (Kunze et al. 2006; Wu et al. 2011; Whalen et al. 2012). Individual profiles over seamounts have been compared in Hibiya et al. (2012). To the best of our knowledge, the only regional evaluations of the strain parameterization were conducted by Frants et al. (2013) and Waterman et al. (2014) using data from the Southern Ocean. The Frants et al. (2013) study found that applying the parameterization to CTD profiles characterizes the regional patterns better than using Thorpe scale overturns (Thorpe 1977) since the Thorpe scale method only includes overturns large enough to be fully resolved by the CTD. These estimates usually agreed with the microstructure within a factor of 2–3. Waterman et al. (2014) found that individual microstructure profiles agreed with finestructure within a factor of 2 more than 1500 m from the bottom, but discrepancies in a subset of the profiles caused the average finestructure to overestimate the microstructure within 1500 m of the seafloor. Possible explanations for this discrepancy are described in Waterman et al. (2014).

One motivation for an accurate parameterization is to assist estimates of the long-term mean dissipation rate and diffusivity both regionally and globally. This has the potential to aid our understanding of global processes such as the meridional overturning circulation, along with the heat and energy balance in the ocean. These estimates are hard to make but have been completed on a global scale using a catalog of microstructure measurements (Waterhouse et al. 2014) and in the Indian Ocean using a finescale parameterization (Huussen et al. 2012). The difficulty arises because in addition to the spatial variability of diapycnal mixing (e.g., Polzin et al. 1997; Kunze et al. 2006), there may also be temporal fluctuations on long enough scales to affect both microstructure and finescale estimates of the mean.

Origins of the temporal variability of diapycnal mixing on time scales longer than days include fluctuations of the two major internal wave energy sources, the winds and tides, as well as isolated events. Wind gusts in storms can perturb the mixed layer near the inertial frequency (D’Asaro et al. 1995), leading to downward-propagating near-inertial waves (e.g., Alford et al. 2012) and a seasonal cycle of these waves that tracks the seasonal cycle of storm activity (Alford and Whitmont 2007; Silverthorne and Toole 2009; Whalen et al. 2012). The spring–neap tidal cycle modulates the strength of flow over topography and subsequent internal waves that are generated, leading to a cycle in the dissipation rate (e.g., Klymak et al. 2006). Geostrophic flow, interacting with topography and generating lee waves (Bell 1975), can also be a source of episodic increases in the dissipation rate (Sheen et al. 2014). Sampling on scales much shorter than a month may misrepresent the mean close to generation sites of the internal tide with strong spring–neap cycles and less than a year could be problematic for areas with strong seasonal cycles. This is particularly true because of the lognormal nature of the mixing distribution; missing the large rare mixing events matters significantly (Gregg et al. 1993).

The following describes the first comparison between microstructure estimates and strain-based finescale estimates of the dissipation rate conducted for a wide variety of bathymetric features and forcing environments. This is accomplished by matching an updated global dataset of Argo float finestructure profiles previously discussed in Whalen et al. (2012) with nearby microstructure profiles. No temporal restrictions are utilized, implying that the profiles can be from different seasons or even decades. This is essentially assuming that both the microstructure (duration of <1 month) and finestructure (year-round) are estimates of the mean mixing at a particular location. The assumption is reasonable at the microstructure project locations since we find only a small (less than a factor of 2) or nondetectable seasonal cycle. When at least 10 estimates are averaged together, we find a good agreement between the two methods: 81% of the dissipation rate comparisons agree within a factor of 2, and 96% agree within a factor of 3.

2. Microstructure data

Instruments that measure microstructure typically include fast-sampling shear, temperature, and conductivity probes along with a CTD sampling at the standard rate. The microstructure measurements are used to calculate the dissipation rate ϵ (W kg⁻¹) and diffusivity K (m² s⁻¹) via the relation , where is the average adiabatically leveled buoyancy frequency (Bray and Fofonoff 1981), and a constant Γ = 0.2 is assumed.

These microstructure projects are a subset of the open-ocean dataset used in Waterhouse et al. (2014). Here, we additionally require that the profiles reach deeper than 300 m to enable a meaningful comparison with the finestructure estimates as explained in section 5. Projects are also only selected if they are near at least 10 Argo profiles and sample one particular area as opposed to a transect. The locations of every microstructure project are shown in Fig. 1a.

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(a) Locations of the microstructure projects plotted over regions with a variety of seafloor roughness as calculated from the Smith and Sandwell (1997) 14.1 ship-track dataset. (b) Power input into internal waves from the wind and tides as described in Waterhouse et al. (2014) for the microstructure locations. The global averages are shown on the right.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0167.1

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The projects include two from the Atlantic Ocean: the North Atlantic Tracer Release Experiment (NATRE) above an abyssal plane (Toole et al. 1994; Polzin and Ferrari 2004) and a dataset from 1997 including the Mid-Atlantic Ridge called the Brazil Basin Tracer Release Experiment (BBTRE97) (Ledwell et al. 2000; Polzin et al. 1997; St. Laurent et al. 2001). The projects in the Pacific Ocean are the Hawaii Ocean Mixing Experiment (HOME), which examines the Hawaiian Ridge as a site for the generation of internal waves (Lee et al. 2006; Rudnick et al. 2003; Klymak et al. 2006), and Larval Dispersal on the Deep East Pacific Rise (LADDER) (Thurnherr and St. Laurent 2011). A total of 25 equatorial microstructure profiles collected from 25 April to 3 May 2012 using a Rockland VMP-600 as part of the Mixing in the Equatorial Thermocline (MIXET) project were included (G. S. Carter, Principal Investigator, 2015, unpublished manuscript). The final project is the Southern Ocean Finestructure (SOFine) program in the Indian Ocean, which covers a section of the Antarctic Circumpolar Current (ACC) in the Kerguelen Plateau region (Waterman et al. 2013, 2014).

Specific microstructure profiles from each project were selected according to the criteria described in section 5. As shown in Fig. 1a, these locations cover a variety of latitudes and bottom topographies, including rough and smooth topography as defined by bottom height variance calculated over 30-km squares derived from the Smith and Sandwell (1997) 14.1 ship-track bottom bathymetry dataset. Waterhouse et al. (2014) demonstrate that these projects span the global range of power input levels into internal waves from the winds and tides, the presumed sources of turbulence. Figure 1b shows the average power input into internal waves for each project from the winds and tides within half a degree of a microstructure profile alongside the global mean calculated from a slab model (Alford et al. 2012; Simmons and Alford 2012). The details of the model configurations and key caveats are described in Waterhouse et al. (2014). In this selection of datasets, there is a bias toward areas of low power input; only HOME, SOFine, and BBTRE97 exceed the global average tidal input, and only SOFine exceeds the average global wind input.

3. Argo data

Every 10 days, a typical Argo float collects temperature, salinity, and pressure data between 2000 m and the surface before uploading the data via satellite for free access (www.argo.net/). Currently the float array consists of over 3500 floats scattered across every ocean. Their profiles have a variety of vertical resolutions due to communication constraints. For example, some profiles change from high resolution (5 m) to low resolution (40 m) with increasing depth. There are a growing number of Argo floats with Iridium communication that permit high vertical resolution (typically 2 m) throughout their 2000-m profile.

All Argo profiles with the program-designated quality rating of “A” (passed all real-time quality control tests) between January 2006 and March 2014 are selected for this analysis. They are additionally despiked and quality controlled to remove incomplete data and unrealistic values. After removal of the mixed layer and mode water (section 4), the profiles are cut into 200-m half-overlapping segments with constant resolution starting at the bottom of each profile. A length of 200 m was chosen to maximize the total number of usable segments in profiles that have variable resolution. Changing the segment length slightly (e.g., to 240 m) on average has a negligible effect on the resulting estimates (Whalen et al. 2012).

The vertical gradient of both salinity and temperature is sometimes very small (e.g., in the Weddell Sea). In this case, the actual finescale variance in salinity cannot be resolved, and the salinity profile appears to have steps as it shifts from one resolved value to another. These artifacts were removed by only selecting segments where both the maximum temperature range was greater than 0.2°C and the maximum salinity range was greater than 0.02 psu.

In the upper 250 m, the assumptions underlying the finescale parameterizations do not always apply (Kunze et al. 2006; Polzin et al. 2014). This is because near the surface of the ocean noninternal wave-driven variations in the buoyancy frequency, including mode water and surface processes, are particularly prevalent. The strainlike features that they produce are not internal waves and therefore are not subject to the energy cascade that we assume in the finescale parameterization. We will only consider estimates centered below 250 m to mitigate this issue.

4. Finestructure methods

The strain-based finescale estimates from Argo floats were calculated following Kunze et al. (2006), stemming from Polzin et al. (1995) and Gregg and Kunze (1991), in a nearly identical approach to Whalen et al. (2012). The dissipation rate is expressed in the strain version of the finescale parameterization in practical form as

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where and are the observed and Garrett–Munk (GM; Cairns and Williams 1976) strain variances, respectively; is the segment-averaged buoyancy frequency; describes the dependence on the ratio between shear and strain R_ω, which we set to 3; and is a latitudinal correction (Polzin et al. 1995; Gregg et al. 2003). The constants are N₀ = 5.24 × 10⁻³ rad s⁻¹ and ϵ₀ = 6.73 × 10⁻¹⁰ m² s⁻². This produces a single estimate of the dissipation rate for each 200-m segment. All the variable terms in Eq. (1), including , , and , are important in the parameterization. Each of these terms varies two to five orders of magnitude between 250 and 2000 m in the finescale estimates considered here.

The backbone of Eq. (1) is the buoyancy frequency, appearing in multiple terms including the strain

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where is the buoyancy frequency, and is a quadratic fit (Polzin et al. 1995; Huussen et al. 2012). The in the denominator is equivalent to a mean of . If the segment-scale and smaller variations in the buoyancy frequency are dominated by processes other than internal waves in violation of assumption 1 (see section 1), the dissipation rate will be incorrectly estimated (Polzin et al. 2014).

Segments partially spanning the mixed layer or mode water often have a high strain variance because of the large gradients between high- and low-stratification water, which are not predominantly caused by the presence of internal waves. Here, we remove the mixed layer using the variable temperature criterion (de Boyer Montegut et al. 2004), where the bottom of the mixed layer is defined as the depth where the increase in density from 10 m is equivalent to a decrease in temperature of 0.2°C using local conditions. The mode water is removed by applying the criterion a second time, using the bottom of the mixed layer analogously to the surface of the ocean. This method only removes the most prominent mode water and noninternal wave buoyancy frequency structure. Inclusion of these subtle mode-water profiles may lead to overestimation of diapycnal mixing in these areas.

Another violation of assumption 1, when the observed strain is not because of internal waves, occurs when large variations in the buoyancy frequency produce a strongly nonquadratic profile over a 200-m segment. This is occasionally noticeable immediately below the mixed layer. The quadratic fit in Eq. (2) does a poor job in these areas, so it noticeably increases the strain variance. We therefore choose to remove segments that vary in the squared buoyancy frequency by more than 5 × 10⁻⁴ s⁻² [slightly more conservative than in Whalen et al. (2012)]. Segments were also discarded if , since the strain signal at such low buoyancy frequencies is dominated by noise.

A finestructure feature that we are not correcting for is double-diffusive staircases generated by salt fingering or diffusive layering (Schmitt 1994). These processes have a narrowband finescale structure that falls within the scales we are considering here. While this is generally not observed to result in diapycnal mixing [for an exception see Polzin et al. (2014)], the resulting transport is of the opposite sign than what we expect from internal wave mixing and only results in a small amount of energy dissipation (Schmitt 2003). In these cases, the magnitude of the dissipation rate would be incorrectly assigned since the assumed physics (assumption 2) are not present.

To calculate the strain variance , each segment was detrended, windowed using a sin² 10% taper, and spatially Fourier transformed to generate the spectrum S_str for each segment. The spectrum is corrected for first differencing by dividing by the transfer function sinc² , where is the vertical resolution of the segment. Argo data are either point measurements or averages over a depth interval, which is currently not identified in the metadata. Here, we correct for this whenever we know the sampling scheme of the profile by dividing by the same transfer function a second time [a correction not included in Whalen et al. (2012)]. This slightly raises the variances in the bin-averaged cases. The average increase in the dissipation rate for the Atlantic Ocean is a factor of 1.02, with a range of 1 to 2.6.

The average strain spectra for segments centered between 250 and 500 m are shown in Fig. 2b for each experiment. The average spectra vary considerably in shape between each of the projects, ranging from LADDER, where the average spectrum closely follows the reference Garrett–Munk spectrum, to MIXET, where the spectrum is notably higher than the Garrett–Munk spectrum. The shape of the MIXET spectrum, and its contrast with a relatively whiter spectrum further from the equator, is consistent with previous observations of equatorial finescale strain in Johnson et al. (2002).

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(a) Locations of the microstructure profiles (colored) that are in close proximity to the Argo profiles (black) used to calculate finestructure for each project. (b) Averaged strain spectra (red) between 250 and 500 m along with the Garrett–Munk spectrum (black). Horizontal histograms of each average are in pink. Vertical dark gray lines are positioned at wavenumbers corresponding to the absolute integration limits of 100- and 10-m wavelengths. The light gray line corresponds to the 40-m wavelength, the smallest wavelength required for integration.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0167.1

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Each individual spectrum is integrated to obtain the strain variance:

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The integration range is delineated in Fig. 2 and begins at 100-m wavelength. This limit is chosen because we are constrained by the size of our segments (200 m), while it simultaneously avoids longer wavelength contamination by the background stratification (Kunze et al. 2006). The upper limit was set to the highest wavenumber possible between the wavelengths of 10 and 40 m while satisfying to avoid oversaturation of the spectrum that would thereby underestimate the variance (Gargett 1990).

Changing the integration limits slightly, for example, integrating to 120 m rather than 100 m using a 240-m-long segment, does not bias the average dissipation rate estimates (Whalen et al. 2012). However, in specific locations where the spectrum considerably deviates from the GM spectrum, the integration limits can impact the result. For example, the MIXET spectra are redder than GM, indicating that changing the integration range to include smaller wavelengths would decrease the resulting strain variance. In SOFine, Waterman et al. (2014) also found that changing the integration limits affected the finestructure estimates. The variance estimates were only averaged together after integration following Kunze et al. (2006) to avoid averaging spectra of different resolutions.

The energy cascade to smaller scales via wave–wave interaction is a function of the frequency limits of the local internal wave field and therefore is dependent on latitude. This is included by the term L(f, N) (Gregg et al. 2003; Polzin et al. 1995; Henyey 1991), with dependence on the local Coriolis frequency f, the Coriolis frequency at 30° f₃₀, and segment-averaged buoyancy frequency , given by

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The function h(R_ω) is given by

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where R_ω is the ratio between shear and strain, or aspect ratio of horizontal kinetic and potential energy of the internal wave field (Polzin et al. 1995; Kunze et al. 2006), which is variable with space and time throughout the ocean. Since we only have strain and no shear information, we are forced to assign a value for the Garrett–Munk wave field R_ω = 3 (Cairns and Williams 1976; Gregg 1989), reducing h(R_ω) to 1. This is also a reasonable choice for the upper 2000 m given previous global observations (Kunze et al. 2006, their Fig. 20). Comparisons between microstructure and finestructure presented later (section 6) show no scalar positive or negative bias that would be generated by choosing an incorrect constant for R_ω. However, using a constant R_ω rather than a variable one will cause some error (a factor of 2 if on average R_ω = 6) in the individual mean dissipation rate estimates. Maps of the average dissipation rate estimates achieved from applying this method to Argo float profiles are shown in Fig. 3.

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Average dissipation rate estimates derived by applying strain finestructure methods using Argo profiles between (a) 250 and 500, (b) 500 and 1000, and (c) 1000 and 2000 m. At least three estimates are required to show an average within a 1.5° square bin. These figures are updated from Whalen et al. (2012) and include data from 2006 to 2014.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0167.1

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5. Comparisons

b. Determining the number of samples

For the shear version of the finescale method, Gregg et al. (1993) determined for their particular dataset that at least 40 independent samples are required for an estimate of the mean possessing 95% confidence limits within a factor of 2. Here, we are using a different parameterization, applied to strain rather than shear, and in this case the underlying variable of strain is not normally distributed (Pinkel et al. 1991; Pinkel and Anderson 1992, 1997; Polzin 1996), as is true with the shear (Gregg et al. 1993). Instead we take an empirical approach to approximate the number of samples required to estimate the mean dissipation rate within reasonable bounds.

To empirically gauge the quantity of samples necessary to calculate a mean with reasonable confidence limits, means are calculated from subsamples of the dissipation rate estimates within a depth range. This is replicated 5000 times for each sample size ranging between 2 and 80, allowing for repetition. Using the distribution of the means obtained for each sample size, the ratio between the 5th and 95th percentile is calculated. Figure 4 shows this ratio of dissipation rates between 400 and 600 m for the NATRE dataset using Argo-derived finestructure estimates, profile-averaged microstructure estimates, and finescale estimates derived from the CTD on the microstructure profiler.

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The distribution of dissipation rate means between 400 and 600 m was calculated using 5000 means of 2–80 subsamples of estimates from the NATRE dataset. Shown here is the ratio between the 95th and 5th percentile of these distributions as a function of the number of samples contributing to the means for microstructure (blue) and finestructure using concurrent profiles from the CTD on the microstructure instrument (black) and Argo-derived finestructure (red).

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0167.1

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Figure 4 demonstrates that, as expected, there are larger variations in the mean for smaller sample sizes than for larger sample sizes. The Argo-derived finestructure and simultaneous CTD finestructure means are both determined within a factor of about 4 and the microstructure within a factor of 2 using 10 samples. We choose to average a minimum of 10 samples to get an estimate of the mean dissipation rate at a particular depth to allow for the inclusion of MIXET and LADDER. Ideally, 50 samples or more would be chosen to reduce the uncertainty to a factor of 2 for the estimated mean.

Only NATRE is shown because it is the only project with >50 dissipation rate values for both finestructure and microstructure. Also, the majority of the projects do not possess high enough quality temperature and salinity measurements from the CTD on the microstructure profiler to use the finescale parameterization (such as due to an unpumped CTD). We therefore take the NATRE dataset as an example and expect that data from different locations would converge to a mean at slightly different rates. For example, the finestructure from BBTRE97 and HOME converge to a factor of 5 for 10 samples. SOFine includes two very different mixing environments, both deep and shallow, and consequently converges slower, to a factor of 8 for 10 samples. In every case, averaging fewer than five finestructure samples is off by a factor of 10 or more, highlighting the importance of averaging to get an accurate estimate of the mean.

6. Results

a. Argo-derived finestructure and microstructure means

Profiles of both the microstructure and finestructure dissipation rates are presented in Fig. 5. Each profile is composed of 200 m half-overlapping bin averages between 250 and 2000 m. Since the vertical resolution of the Argo profiles frequently becomes too coarse at depth for the finescale parameterization to be applied, many comparisons do not reach 2000 m. Gaps in the finestructure estimates are because of a change in the resolution of Argo float profiles at that depth. The microstructure and Argo-derived finescale profiles agree very well; the majority of the points agree within the range of the 90% bootstrapped confidence intervals. The profiles of averaged finestructure estimates also tend to exhibit a trend similar to the microstructure profiles, for example, NATRE, BBRTRE97, HOME, SOFine, and MIXET all have higher dissipation rates near the top of the profile than closer to the bottom.

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Mean dissipation rate (W kg⁻¹) profiles from 200-m half-overlapping depth bins from microstructure (dots) and the Argo finestructure (bars) generally at most 60 km away from a microstructure profile. The 90% bootstrapped confidence intervals are represented by either bars or gray shading.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0167.1

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Figure 6 consolidates the comparisons between the microstructure and finestructure shown in Fig. 5 for the dissipation rate (Fig. 6a) and equivalent comparisons for the diffusivity (Fig. 6b). The dissipation rates calculated using these two methods span three orders of magnitude, agreeing within a factor of 2 for 81% and a factor of 3 for 96% of the comparisons. Overall, there is no noticeable positive or negative bias; the ratio between microstructure and finestructure dissipation rate is 1.03. However, individual projects sometimes exhibit a slight bias toward overestimation or underestimation. For example, 12 of 17 of the SOFine dissipation rate averages using microstructure are significantly larger than the finestructure estimates according to the 90% bootstrapped confidence intervals. The diffusivity agrees similarly well, within a factor 2 for 83% and a factor of 3 for 96% of the comparisons.

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Comparisons between Argo-derived finestructure estimates and microstructure measurements for (a) the dissipation rate and (b) the diffusivity K between 250 and 2000 m with bootstrapped 90% confidence intervals. The means are identical to the profiles shown in Fig. 5. Agreement within a factor of 2 or 3 is designated by the gray bands.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0167.1

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The agreement between the finestructure and microstructure in the equatorial MIXET project is particularly intriguing. The finescale strain parameterization has not been previously tested in the unique equatorial environment. Here, we find that 5 of 6 of the MIXET comparisons match within a factor of 3. This agreement adds confidence to the pattern of an elevated dissipation rate along the equator (Fig. 3) and the effectiveness of the form of latitudinal dependence in Eq. (4) (Gregg et al. 2003; Polzin et al. 1995; Henyey 1991). Adding further credence, Hennon et al. (2014) find evidence for a band of high internal wave activity along the equator at 1000 m. In their observations, the authors use Argo float time series data collected while the float was parked at depth between sampling vertically to generate the profiles used in our analysis. These internal waves may be dissipating through wave–wave interactions as described by the finescale parameterizations. However, caution should be used when interpreting this agreement since the assumptions underlying the finescale parameterization may be violated near the equator. Specifically, the red MIXET spectra (Fig. 2b) may be because of contamination from noninternal wave equatorial processes or a significant deviation from the Garrett–Munk reference internal wave field near the equator. Either possibility could cause spurious dissipation rate estimates in equatorial regions, despite the agreement we see at the MIXET location.

The dissipation rate distributions calculated from both the Argo-derived finestructure estimates and microstructure between 250 and 500 m are shown in Fig. 7 in the form of probability density functions (PDFs). Superimposed on each PDF are dots representing the mean in this depth range. The microstructure and finestructure have similar distributions at high values, while differing substantially at low values. Because of the non-Gaussian nature of the distributions, the large values are the most important for the averages, which is where the PDFs are in better agreement.

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PDFs of the dissipation rate (W kg⁻¹) derived from microstructure (blue) and Argo-derived finestructure (red) between 250 and 500 m, with means of each distribution (colored dots).

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0167.1

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There are no low dissipation values in the microstructure estimates since these calculations typically include a minimum dissipation value, for example, 10⁻¹¹ W kg⁻¹ in NATRE (Toole et al. 1994) or 3 × 10⁻⁹ to 3 × 10⁻⁸ W kg⁻¹ in HOME (Lee et al. 2006), depending on the noise floor of the measurements. As seen in Fig. 7, including this cutoff does not have a large effect on the averages. The steeper gradient in the lower values of the microstructure PDFs may also be because of the steepening of the PDFs at low values with added noise. Gregg et al. (1993) shows that adding random noise to finestructure data shifts the lower tail to the center of the distribution, thus facilitating better agreement. This is also true for the PDFs presented.

b. Seasonal variability of dissipation rates

These comparisons implicitly assume that both the microstructure and finestructure Argo estimates, sometimes months or even decades apart, are both measuring the mean dissipation rate at a particular location. If the dissipation rate varies intermittently, seasonally, or decadally, this may cause either measurement to be an inaccurate representation of the mean state. Here, we evaluate this assumption by investigating whether seasonal variability is significant at both the specific microstructure project locations and globally.

The seasonal cycle in the dissipation rate is presented in Fig. 8 using the Argo finestructure for NATRE, BBTRE97, HOME, and SOFine. Projects included have at least 10 estimates in each season for at least one depth range. The profiles of 200-m half-overlapping dissipation rate means in each season are depicted in Fig. 8a. The corresponding spectra between 250 and 1000 m are plotted in Fig. 8b. Both the profiles and spectra for NATRE (Northern Hemisphere) and BBTRE97 (Southern Hemisphere) show larger values in winter and spring than the summer and fall. The seasonal differences between the dissipation rate profiles are at most a factor of 2. The timing is consistent with the winter peak in near-inertial energy (Alford and Whitmont 2007; Silverthorne and Toole 2009; Whalen et al. 2012). We do not find a clear seasonal cycle in the dissipation rate for the HOME and SOFine datasets, which is either indicative of no seasonal cycle or an insufficient number of estimates to consider the seasonal cycle (see section 5b).

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(a) Profiles of the half-overlapping mean dissipation rates from Argo-derived finestructure estimates colored according to each season with 90% bootstrapped confidence intervals. Only averages with at least 10 estimates are shown, and only projects are shown that fit this requirement for every season. (b) The averaged spectra between 250 and 1000 m for each season. The Garrett–Munk spectrum is shown in black, and the vertical lines designate the wavenumbers corresponding to the upper wavelength integration limit (100 m) and the range for the lower limit (10–40 m).

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0167.1

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The shape of the spectra in Fig. 8b is consistent across all seasons for a particular project. For example, a winter BBTRE97 spectrum is more similar to BBTRE97 spectra of any season than any of the spectra at other locations. One exception to this is the high wavenumber portion of the SOFine spectrum, where the winter and fall spectra are bluer than during the remainder of the year.

In other regions the seasonal cycle is larger than observed in the NATRE, BBTRE97, HOME, and SOFine projects. Figure 9 shows the dissipation rate seasonal cycle from finestructure for 10° latitudinal bands between 50°S and 50°N and 250 and 1000 m, averaged over all longitudes. Each profile is constructed from half-overlapping means of at least 200 estimates, and a majority of the averages include over 1000 estimates. The largest variation is a factor of 2–5, occurring between March/April and September/October along the latitude band rich in storm tracks, 30°–40° in both hemispheres. The longitudinally averaged buoyancy frequency does not have an equivalent cycle (not shown). In contrast to the stormy 30°–40° latitudes, there is no significant seasonal cycle in the dissipation rate along 10°S–10°N and only a slight seasonal cycle in the other latitudinal ranges. All the microstructure projects considered here are at latitudes with low seasonal variability, outside of the 30°–40° range.

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The seasonal cycle in the Argo-derived finestructure dissipation rate for 10° latitudinal bands across the entirety of the (a) Northern Hemisphere and the (b) Southern Hemisphere. Each profile is constructed from half-overlapping means as in Fig. 8. The thin lines are 90% bootstrapped confidence intervals. At least 200 estimates are required for an average and over half of the averages include over 1000 estimates.

Citation: Journal of Physical Oceanography 45, 4; 10.1175/JPO-D-14-0167.1

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7. Discussion

We have presented comparisons between finescale estimates of the dissipation rate and diffusivity from Argo float profiles and microstructure measurements to 2000-m depth from six projects. The measurements were taken at different locations (separated up to 60 km) and various times (up to decades apart). The comparisons were made in diverse environments, from the equator to the ACC and from above abyssal plains to above ocean ridges. These locations represent a range of wind and tidal power input and topographic roughness. Of these Argo-derived strain finestructure estimates of the dissipation rate, 81% agreed within a factor of 2, and 96% agreed within a factor of 3. This good agreement over a wide range of open-ocean conditions strongly suggests that the underlying theory behind the finescale parameterization provides an accurate scaling; however, this finding is not definitive since the data presented here do not directly measure the nonlinear interactions on which this theory is based. Additionally, these results suggest that these methods can yield reasonable estimates of the mean dissipation rate.

Previous regional studies comparing strain finestructure to microstructure measurements have yielded consistent results. In the Southern Ocean near Drake Passage, finestructure from the ship CTD and microstructure dissipation rate estimates from nearly identical times typically agreed within a factor of 2–3, or at most a factor of 4 below 250 m (Frants et al. 2013), similar to what we find.

A second regional study, SOFine near the Kerguelen Plateau (Waterman et al. 2014), observed that the microstructure and finestructure agree within a factor of 2 far above the seafloor (>1500 m), but finestructure tended to overestimate the dissipation rate closer to the bottom. Waterman et al. (2014) attribute this near-bottom bias to non-GM conditions caused by lee waves generated in this region from geostrophic flow over rough topography, indicating that the finescale parameterization assumptions are not applicable close to the bottom in this environment. The majority of the Argo float profiles are not within this regime since they are >1500 m above the seafloor. Therefore, we find similar (within a factor of 3) agreement between the microstructure and Argo finestructure dissipation rates as Waterman et al. (2014) do far above the seafloor. In our case, the Argo finestructure slightly underestimates the dissipation rate.

In the SOFine region, temporal variability of internal waves may be the source of the uniformly smaller Argo finestructure dissipation rates via a mechanism similar to what is observed near Drake Passage (Sheen et al. 2014). Since lee-wave generation is a function of current speed over the topography (Bell 1975; Nikurashin and Ferrari 2010), and the fronts in Kerguelen Plateau region are particularly variable for the ACC (Sokolov and Rintoul 2009), we expect the internal wave generation and dissipation rate to vary as well. A mooring record at Kerguelen does show unusually high velocities during the time when the microstructure measurements were taken (S. Waterman 2014, personal communication). Therefore, temporal variability may be the source of the consistent underestimation (factor of 2–3) of the Argo finestructure compared to the microstructure for this project.

Considering six different projects in a variety of environments expands the previous Southern Ocean focus of regional comparisons between microstructure and strain-based finestructure estimates. Of the projects considered here, only BBTRE97, SOFine, and HOME have above average tidal power input, and only SOFine has above average wind power input. Environments that are still missing from this comparison include under storm tracks (30°–40°) and high latitudes (>50°). Expanding on the single station data from the MIXET project with more stations near the equator would also be useful. Evaluation of the strain–finescale parameterization as a means to detect a seasonal cycle would also be worthwhile. This is because the wind-induced near-inertial waves that are likely to contribute to the seasonal cycle have a high shear-to-strain ratio, which we are assuming to be constant. This would result in an underestimation of the seasonal cycle since the shear-to-strain ratio, and thus Eq. (5), would be biased low. A comparison between winter and summer microstructure and finestructure at a site with significant near-inertial wind forcing would be one way to discern if the strain parameterization is accurately representing the seasonal cycle.

Possible sources of the factor of 2–3 difference between the microstructure and finestructure averages include small violations of the parameterization’s assumptions, undersampling the temporal variability, and averaging too few samples to calculate the mean. We choose to use at least 10 samples to construct each average, which according the NATRE dataset implies that 90% of the time the finestructure mean estimates will be within a factor of 4 of each other and microstructure will estimate the mean within a factor of 2. This is similar to the spread of the comparisons presented here and may be the source of a significant portion of the factor of 2–3 difference we observe. This also suggests that it is necessary to average a number of estimates to achieve a reliable measure of the mean dissipation rate at a location.

The energy in the internal wave field fluctuates on seasonal and tidal time scales, yet in these comparisons we are assuming that the microstructure is measuring the mean dissipation rate in an area despite sampling over only a period of weeks. At a subset of locations (NATRE, BBTRE97, HOME, and SOFine) used in this study, this assumption is reasonable since the seasonal variability is small (at most factor of 2) or undetectable (Fig. 8). The remaining locations, LADDER and MIXET, are at low latitudes, which in the global averages do not show any seasonal cycle (Fig. 9). Tidal variability should not be significant in these comparisons since the microstructure projects considered here span either a full spring–neap cycle, are a great distance from internal tide generation sites, or are far above the topography. At these locations, seasonal and tidal variability will likely have a negligible effect on microstructure and finestructure estimates of the mean dissipation rate.

The mean dissipation rate and diffusivity are of wider interest for applications such as understanding the oceanic energy budget and overturning circulation. Estimating the mean dissipation rate is difficult because of the high levels of variability approximately lognormally distributed over multiple spatial and temporal scales. Here, we find evidence from finestructure estimates that seasonal variability is not very important for determining the global-mean dissipation rate in the upper ocean over most latitudes, assuming we are adequately resolving the seasonal cycle. This gives us confidence that measurements at these latitudes, such as microstructure projects, can on average accurately measure the mean dissipation rate as assumed in Waterhouse et al. (2014). However, at latitudinal ranges with strong winter storms (30°–40°), there is a significant cycle of a factor of 2–5 in the dissipation rate estimates from finestructure. At these latitudes, using data from only one season could lead to noticeable inaccuracies in the calculated mean. This is consistent with previous results that also find that spectra are typically invariable over time on the scales we consider here, except for some locations where the seasonal cycle is observed (Polzin and Lvov 2011). The relative steadiness of the dissipation rate on monthly to seasonal time scales over the majority of the ocean suggests that measurements of the dissipation rate in a region over these time scales are sufficient to produce reasonable average estimates of ocean mixing.

8. Conclusions

There are three main conclusions that arise from this study. First, the assumed fundamental physics behind the finescale parameterizations is consistent with our observations over a wide range of internal wave environments. Specifically, our findings are compatible with the notion that the majority of the turbulent energy dissipation in the open ocean is caused by internal waves transferring their energy to smaller scales through nonlinear interactions. Second, the mean dissipation rate is generally steady in the upper ocean over monthly to seasonal time scales. One notable exception to this is a significant seasonal cycle beneath storm tracks. Finally, the finescale strain parameterization is an effective tool for estimating the mean dissipation rate and diffusivity in the open ocean, provided that appropriate averaging is done over the internal wave field environment of interest.