the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Turbulent kinetic energy dissipation rate and associated fluxes in the western tropical Atlantic estimated from ocean glider observations
Gillian M. Damerell
Philip J. Leadbitter
Karen J. Heywood
Rob A. Hall
Ocean gliders enable us to collect the highresolution microstructure observations necessary to calculate the dissipation rate of turbulent kinetic energy, ε, on timescales of weeks to months: far longer than is normally possible using traditional shipbased platforms. Slocum gliders have previously been used to this end; here, we report the first detailed estimates of ε calculated using the Batchelor spectrum method on observations collected by a FP07 fast thermistor mounted on a Seaglider. We use these same fast thermistor observations to calculate ε following the Thorpe scale method and find very good agreement between the two methods. The Thorpe scale method yields larger values of ε, but the average difference, which is less than an order of magnitude, is smaller than reported elsewhere. The spatiotemporal distribution of ε is comparable for both methods. Maximum values of ε (10^{−7} W kg^{−1}) are observed in the surface mixed layer; values of approximately 10^{−9} W kg^{−1} are observed between approximately 200 and 500 m depth. These two layers are separated by a 100 m thick layer of low ε (10^{−10} W kg^{−1}), which is colocated with a highsalinity layer of Subtropical Underwater and a peak in the strength of stratification. We calculate the turbulent heat and salt fluxes associated with the observed turbulence. Between 200 and 500 m, ε induces downward fluxes of both properties that, if typical of the annual average, would have a very small influence on the heat and salt content of the overlying salinitymaximum layer. We compare these turbulent fluxes with two estimates of doublediffusive fluxes that occur in regions susceptible to salt fingers, such as the western tropical Atlantic. We find that the doublediffusive fluxes of both heat and salt are larger than the corresponding turbulent fluxes.
Turbulence in the ocean and the mixing of different water masses that it induces are of fundamental importance to ocean dynamics. Over relatively small scales, turbulent mixing often controls the distribution of key water mass properties and tracers; over the world ocean, the sum of these smallscale processes is responsible for the closure of the thermohaline circulation and for the primary production that relies on the upward flux of nutrients to the euphotic zone.
Estimating the dissipation rate of turbulent kinetic energy, ε, from highresolution observations of shear and temperature (e.g. Lueck et al., 2002; Oakey, 1982; Ruddick et al., 2000) has, historically, required considerable ship time, plus specialist instruments and expertise, so there are relatively few of such estimates. Methods such as Thorpe scaling (Thorpe, 1977) and finescale parameterisation (Polzin et al., 2014; Whalen et al., 2015) have been developed to enable ε to be estimated from standard conductivity–temperature–depth (CTD) and acoustic Doppler current profile (ADCP) observations of temperature, salinity and current velocity – hereafter referred to as finescale methods and observations. Although finescale methods do not require specialist instruments (e.g. Fer et al., 2010b; Whalen et al., 2012, 2015), they are dependent on more assumptions; their results tend not to be valid over as wide a range of conditions as those derived from highresolution, microstructurescale observations (Polzin et al., 2014; Whalen, 2021) – hereafter referred to as microstructure methods and observations. Thus, despite the widespread use of finescale methods, the potential remains for discrepancies between estimates of ε based on finescale and microstructure observations (Howatt et al., 2021).
Given the proliferation in the use of buoyancydriven ocean gliders over the last decade, there is growing interest in using them to collect microstructure observations. Because of a glider's smooth flight through the water column, it resembles the freefalling, loosely tethered profilers traditionally used to collect microstructure observations. A growing body of literature makes use of microstructure observations collected by gliders and sets out the best ways of processing such data sets (e.g. Fer et al., 2010b; Peterson and Fer, 2014; Palmer et al., 2015; Schultze et al., 2017; Scheifele et al., 2018; Scott et al., 2021). Up until now, the majority of studies have used microstructure shear observations collected by Slocum gliders (Palmer et al., 2015). Some authors have used microstructure temperature observations to estimate ε (e.g. Peterson and Fer, 2014; Scheifele et al., 2018), while Rainville et al. (2017) briefly discuss the microstructure system developed for use on Seagliders, another commonly used glider platform, and present estimates of the rate of dissipation of temperature variance, χ. However, observational studies of turbulence using autonomous platforms other than Slocum gliders are known to be lacking (FrajkaWilliams et al., 2021). Here, we report in detail the first microstructurebased estimates of ε calculated from Seaglider microstructure temperature observations, and we compare the results with estimates of ε calculated by applying the Thorpe scale method, commonly applied to finescale temperature and salinity observations, to the same microstructure observations.
The western tropical Atlantic (Fig. 1) is known for the persistent presence of the saltfingering regime of double diffusion (Schmitt et al., 1987; Rollo et al., 2022). For salt fingering to occur, warm, saline water must overlie cooler, fresher water: the water column is therefore stably stratified by temperature but unstably stratified by salinity. Such conditions are maintained in the western tropical Atlantic by the presence of Subtropical Underwater (SUW) at the base of the mixed layer, a warm, highsalinity water mass common to tropical regions (Schmitt et al., 1987; Fer et al., 2010a). Beneath SUW, temperature and salinity both decrease with depth. In a saltfingering regime, the slow molecular diffusion of salt relative to the fast diffusion of heat leads to the development of salt fingers: narrow, smallscale filaments of alternately upwelling warming water and downwelling cooling water. Over time, doublediffusive convection and salt fingers promote the formation of thermohaline staircases: temperature and salinity profiles characterised by a series of homogeneous mixed layers separated by sharp, narrow gradient layers. Such staircases have previously been observed in the western tropical Atlantic (Schmitt et al., 1987; Rollo et al., 2022). Importantly for studies of ocean mixing, doublediffusive convection enables the vertical transport of heat and salt by a mechanism other than the mechanical, turbulent mixing captured by ε.
Here, we use microstructure temperature observations collected by a Seaglider to estimate ε using the Batchelor spectrum method (Sect. 2.2 herein; Batchelor, 1959) and using the Thorpe scale method (Sect. 2.3 herein; Thorpe, 1977), and we compare the results (Sect. 3.1). From these estimates of ε, we derive turbulent fluxes of heat and salt through an observed layer of elevated ε (Sect. 3.2) and compare these with heat and salt fluxes driven by salt fingers and doublediffusive mixing (Sect. 3.3). We discuss the results in Sect. 4.
2.1 Glider observations
As part of the EUREC4A field campaign (Stevens et al., 2021), Seaglider 620 was deployed at 14.2^{∘} N, 57.3^{∘} W, approximately 200 km northeast of Barbados (Fig. 1) on 23 January 2020. It completed 131 dives to 750 m before being recovered on 5 February 2020. The glider carried an unpumped conductivity–temperature (CT) sail measuring in situ conductivity and temperature, and a microstructure system. Given the shape of the Seaglider's hull, it is not possible to mount an allinone microstructure payload, such as the Rockland Scientific International (RSI) MicroRider that is used on Slocum gliders (e.g. Fer et al., 2014; Schultze et al., 2017; Scheifele et al., 2018). Instead, a reconfigured payload is used, consisting of a pair of RSI MicroPod sensor modules mounted on either side of the CT sail; a dedicated pressure housing containing the system's data logger is mounted inside the Seaglider's aft fairing (Creed et al., 2015). The system draws its power from the Seaglider and can be turned on and off on a divebydive basis.
During the EUREC4A campaign, the glider was equipped with one MicroPod carrying a shear probe and one MicroPod carrying an FP07 fastresponse temperature sensor. There was a fault with the shear probe on this deployment, and the observations could not be used. The fast thermistor sensor samples at 512 Hz and has a sensitivity of better than 0.1 mK (Sommer et al., 2013); thermal inertia of the sensor is such that its effective resolution is estimated to be 10 ms (i.e. 100 Hz; Sommer et al., 2013). Microstructure temperature observations are better suited than shear observations to estimating ε in lowdissipation environments (Scheifele et al., 2018) and have the added advantage of being less readily contaminated by platform vibration (FrajkaWilliams et al., 2021); here we focus on the temperaturebased estimates of ε. The glider's hydrodynamic flight model, which is used to estimate alongpath speed, is tuned following FrajkaWilliams et al. (2011), and the thermal lag of the standard CT sail is corrected following Garau et al. (2011).
2.2 Estimating ε using the Batchelor spectrum method
We estimate ε from the glider's fast thermistor temperature microstructure observations using the Batchelor spectrum method; we hereafter refer to these estimates as ε_{μ}. For this, we use the MATLAB toolbox produced by Benjamin Scheifele and Jeffrey Carpenter (https://github.com/bscheife/turbulence_temperature, last access: 17 January 2023) and recently used by Howatt et al. (2021). The method is described in detail by Scheifele et al. (2018); much of the underlying theory, as well as a similar methodology, is described by Peterson and Fer (2014), so we here give only an outline.
We divide the temperature time series from the fast thermistor into halfoverlapping segments of 32 s length. Within each 32 s segment, we further divide the measurements into fifteen 4 s, halfoverlapping subsegments. From each subsegment, we calculate a temperature power spectrum, Δ_{4}. We then average these 15 Δ_{4} to produce 1 power spectrum, Δ_{32}, that is representative of the original 32 s segment. Values of Δ_{32} at high frequencies, where the thermal inertia of the fast thermistor is such that its temporal response is inadequate, are corrected using the transfer function of Sommer et al. (2013). We convert each Δ_{32} from frequency space to wavenumber space (Fig. 2) using the glider's alongpath speed averaged over the same 32 s and assuming the validity of Taylor's frozen turbulence hypothesis (Scheifele et al., 2018).
We transform each Δ_{32} into a temperaturegradient spectrum, Ψ=(2πk)^{2}Δ_{32}, which should resemble the Batchelor spectrum, Ψ_{B} (Batchelor, 1959), the theoretical spectrum that describes temperaturegradient spectra and which is commonly used when calculating ε_{μ} (e.g. Oakey, 1982; Ruddick et al., 2000; Peterson and Fer, 2014; Scheifele et al., 2018). The Batchelor spectrum is a function of k_{B}, the Batchelor wavenumber, and of χ, the rate of destruction of temperature variance (Osborn and Cox, 1972). A comprehensive mathematical treatment of the use of Ψ_{B} when estimating ε is given by Peterson and Fer (2014). Here, we require k_{B} in order to calculate ε_{μ} (W kg^{−1}) according to
where ν is the kinematic viscosity of seawater, and D_{T} = 1.44 × 10^{−7} m^{2} s^{−1} is the molecular diffusion coefficient of temperature.
We calculate χ according to
where χ_{obs} is that part of χ derived by integrating Ψ, and χ_{l} and χ_{u} are correction terms derived from Ψ_{B}. The factor of 6 comes from assuming isotropic turbulence. The wavenumbers k_{l} and k_{u} are, respectively, the lower and upper wavenumber limits of the range over which Ψ is considered reliable; the criteria for choosing k_{l} and k_{u} are explained fully by Scheifele et al. (2018). Given an estimate of χ, the maximum likelihood estimation procedure of Ruddick et al. (2000) is used to find the value of k_{B} corresponding to the Ψ_{B} that is the best fit to Ψ between k_{l} and k_{u}. On the first iteration, χ_{l} and χ_{u} are set to zero, and hence χ=χ_{obs}. On subsequent iterations, the previous value of χ and the previous bestfit value of k_{B} are used to estimate Ψ_{B} and hence χ_{l} and χ_{u}, and the estimate of k_{B} is further refined.
An observed spectrum that deviates from the shape of the relevant theoretical spectrum should not be used to estimate ε_{μ}. To discriminate between acceptably and unacceptably shaped spectra, we fit secondorder polynomials of the form
to both the observed and the theoretical spectra, following the method of Scott et al. (2021), where k is wavenumber, and a, b and c are the polynomial coefficients to be determined. Prior to fitting, we normalise each spectrum by dividing by its corresponding estimate of ε_{μ}; this enables the same criteria to be used when assessing goodness of fit over spectra that otherwise span many orders of magnitude. We also multiply spectra by k in order to preserve variance. P is defined over the same range of wavenumbers over which the observed spectrum is integrated when estimating χ.
We accept a spectrum if

the value of a fitted to Ψ is positive (note that a fitted to Ψ_{B} is always positive)

the ratio of the a values fitted to Ψ_{B} and Ψ (${a}_{{\mathrm{\Psi}}_{\mathrm{B}}}/{a}_{\mathrm{\Psi}}$) is less than 2.
In addition, following Scheifele et al. (2018), we remove an estimate of ε_{μ} if
 3.
fewer than six points are included in the spectra fit;
 4.
the quantity $U/({\mathit{\epsilon}}_{\mathit{\mu}}/N{)}^{\mathrm{1}/\mathrm{2}}$ is less than 5, where U is the glider's speed in the direction of travel, and $({\mathit{\epsilon}}_{\mathit{\mu}}/N{)}^{\mathrm{1}/\mathrm{2}}$ is an estimate of the turbulent flow velocities (Fer et al., 2014), indicating that Taylor's frozen turbulence hypothesis is invalid;
 5.
the sum of the correction terms χ_{u} and χ_{l} is greater than the observed term χ_{obs} (Eq. 2).
Finally, following Peterson and Fer (2014), we remove an estimate of ε_{μ} if
 6.
the mean absolute deviation, which quantifies the goodness of fit between Ψ_{obs} and Ψ_{B}, is greater than $\mathrm{2}(\mathrm{2}/d{)}^{\mathrm{1}/\mathrm{2}}$, where d is the degrees of freedom, calculated as 1.9 multiplied by the number of subsegments within each 32 s segment, i.e. 1.9 × 15;
 7.
the estimate of ε_{μ} is greater than 2 × 10^{−7} W kg^{−1} (even having corrected estimates of Δ_{32} at high frequencies, the effective resolution of the fast thermistor, 100 Hz, is such that values of ε_{μ} greater than this cannot be reliably estimated).
Examples of accepted and rejected spectra are presented in Fig. 2. After quality control, 84 % of ε_{μ} estimates remained. Qualitycontrolled estimates of ε_{μ} were binned, profile by profile, into 25 m vertical bins; we use the geometric mean in preference to the arithmetic (i.e. ordinarily used) mean, as it is better for representing the average of observations that span many orders of magnitude and which are not normally distributed.
2.3 Thorpe scale estimates
We apply the Thorpe scale method (Thorpe, 1977) to the fast thermistor microstructure potential temperature observations to derive a second, independent estimate of the turbulent kinetic energy dissipation rate, hereafter referred to as ε_{T}. Given that the effective resolution of the fast thermistor is estimated to be 100 Hz, we apply a lowpass, 12thorder Butterworth filter with a cutoff frequency of 100 Hz to remove the highestfrequency variability. This prevents instrumental noise erroneously manifesting as small density overturns (Mater et al., 2015; Ijichi and Hibiya, 2018). Temperature observations were then binned into 10 ms bins using the arithmetic mean, giving an effective vertical resolution of 3 ± 0.5 mm.
Each temperature profile is then reordered in depth so that temperature always decreases with depth (i.e. is stable with respect to temperature). From the reordered profile we calculate the vertical Thorpe displacement, Δz: the difference between an observation's original depth and its reordered depth. We identify an overturn as a vertical segment in which the cumulative sum of Δz is nonzero and which is bounded above and below by segments in which the cumulative sum of Δz is zero. Following Ijichi and Hibiya (2018) and Howatt et al. (2021), we combine all overturns that are smaller than 2 m and are within 1 m of an adjacent overturn until the region is larger than 2 m. Overturns smaller than 2 m that are farther than 1 m from an adjacent overturn are not aggregated. This assumes that a succession of small overturns is the signal of a once larger, single overturn that has recently decayed into a series of smaller overturns (Smyth et al., 2001). The Thorpe scale, L_{T} (Fig. 3), is then the root mean square of Δz over an overturn:
where angular brackets indicate the mean. Larger values of L_{T} correspond to larger mixing rates; the largest values of L_{T} (approximately 1 m) are found principally between 200 and 500 m, although much lower values (approximately 0.01 m and lower) are also found within this depth range (Fig. 3). Similarly high values of L_{T} are also found in the bottom 200 m sampled towards the beginning of the deployment. Above 200 m, values of L_{T} are much lower, being generally below 0.01 m (Fig. 3).
Finally, we calculate ε_{T} (W kg^{−1}) from L_{T} by relating the Ozmidov scale, ${L}_{O}=({\mathit{\epsilon}}_{T}/{N}^{\mathrm{3}}{)}^{\mathrm{1}/\mathrm{2}}$ (Ozmidov, 1965), to L_{T} by the empirical relation L_{O}=0.8L_{T} (Dillon, 1982), which yields
where N is the background buoyancy frequency calculated using the Seaglider's finescale temperature and salinity observations, binned into 5 m vertical bins, then smoothed in the vertical using a Gaussianwindowed running mean over nine bins (i.e. 45 m).
Our estimates of ε_{T} are derived from potential temperature rather than potential density; we must therefore assume that temperature is the dominant control on density. In regions where this is not the case – i.e. in regions where salinity is the dominant control on density – temperature perturbations may not correspond to the density perturbations that the Thorpe scale method takes to be indicative of turbulent overturns. To identify regions where salinity is the dominant control on density, we use the density ratio, R_{ρ},
where α is the thermal expansion coefficient, Θ_{z} is the vertical temperature gradient, β is the haline contraction coefficient, and S_{z} is the vertical salinity gradient. We calculate Θ_{z} and S_{z} from finescale glider observations binned into 5 m vertical bins using the arithmetic mean and find R_{ρ} at the depth of each overturn. Where $\mathrm{1}<{R}_{\mathit{\rho}}<\mathrm{1}$, salinity is the dominant control on density, and we discard any overturns and associated value of ε_{T}. In total, 2388 overturns are discarded by the R_{ρ} quality control criterion, 4.03 % of the total. In the majority of the water column, temperature is the dominant control on density, and so temperature observations may be reliably used to estimate ε. Estimates of ε_{T} that are discarded correspond principally to large values of L_{T} (> 1 m) in middepth regions (i.e. between 200 and 600 m). This is the part of the water column in which the majority of the thermohaline staircases are found (Rollo et al., 2022).
Finally, we discard all values of ε_{T} shallower than 75 m because a temperature inversion in the mixed layer is erroneously identified as an overturn. Remaining estimates of ε_{T} are binned into 25 m vertical bins using the geometric mean, as for ε_{μ}.
3.1 Estimates of ε
The water masses observed are typical of the region (e.g. Schmitt et al., 1987). A warm (>26 ^{∘}C) surface mixed layer of intermediate salinity overlies SUW, a salinitymaximum (>37.6 g kg^{−1}) layer located in the upper thermocline (Fig. 4). Beneath SUW, temperature and salinity steadily decrease with depth into the Antarctic Intermediate Water layer that lies beneath (Fig. 4). Two maxima in buoyancy frequency are observed: an upper maximum at the base of the surface isohaline layer and a lower maximum at the base of the surface isothermal layer (Fig. 4).
There is generally good agreement between ε_{μ} and ε_{T}. Histograms of the two distributions are very similar, and, when scattered one against the other, the points are clustered around the onetoone line (Fig. 6a and b). The principal difference is at depths between 400 and 600 m in the first 2 d of the deployment, when ε_{μ} (> 10^{−9} W kg^{−1}) is noticeably higher than ε_{T} (< 10^{−10} W kg^{−1}; Fig. 5). This is a region with a clear pattern in the differences between ε_{μ} and ε_{T}: in many other regions, the differences are fairly randomly distributed (Fig. 5c).
Of the two estimates of ε, the higher of the two is ε_{T}, the geometric mean of which is 3.42 × 10^{−9} W kg^{−1}; the geometric mean of ε_{μ} is 3.05 × 10^{−9} W kg^{−1}. We note that, on average, the difference between the two estimates is small and much less than an order of magnitude. The more variable of the two is ε_{μ}, the geometric standard deviation factor (GSDF) of which is 5.58. The GSDF of ε_{T} is 4.43. (Note that GSDF is multiplicative, not additive, and is therefore dimensionless. The range is from the geometric mean$/$GSDF to geometric mean × GSDF.) Averages and GSDFs are calculated from bins only where estimates of both ε_{μ} and ε_{T} are available.
The highest values of ε are found in approximately the top 50 m of the water column, in the surface mixed layer above the upper boundary of SUW (> 10^{−8}; Fig. 5; ε_{μ} only). The upper boundary of SUW corresponds to the shallowest band of high buoyancy frequency (Fig. 4); a peak in the strength of the stratification might be expected to arrest the downward penetration of surface mixing. But these highest values are infrequently observed: a large proportion of ε_{μ} estimates in the mixed layer are greater than 2 × 10^{−7}, beyond the range for which the FP07 fast thermistor observations and the Batchelor spectrum method yield meaningful results (Sect. 2.2 herein; Peterson and Fer, 2014). Consequently, the remaining values are all below this threshold; indeed, considerable variability in ε_{μ} is observed in the mixed layer – this is reflected in the relatively high GSDF for ε_{μ} in this region (Fig. 5) – and many remaining estimates are low (< 10^{−10}). The mean ε_{μ} in the upper water column is therefore likely to be biased towards these low values. In the remainder of the water column, ε_{μ} and ε_{T} are predominantly below $\mathrm{2}\times {\mathrm{10}}^{\mathrm{7}}$ (Fig. 5); hence the comparison between the two is meaningful.
Between approximately 100 and 200 m, a thin layer with moderate values of ε lies within SUW in the upper thermocline (Figs. 4 and 5). Here, values of both ε_{μ} and ε_{T} are commonly between 10^{−9.5} and 10^{−9} W kg^{−1}. Below this lowε SUW layer, between approximately 200 and 500 m, is a relatively thick layer with higher values of ε (10^{−9} < ε < 10^{−8} W kg^{−1}; Fig. 5). Values of ε_{T} in this layer are generally higher than values of ε_{μ} (Fig. 5), which would explain why the distribution of ε_{T} is slightly skewed to higher values than the distribution of ε_{μ} (Fig. 6a). A few values of ε_{T} are in excess of 10^{−7.5} W kg^{−1} (Fig. 5). The thickness of this higherε_{μ} and higherε_{T} layer increases by 50 to 100 m over the course of the deployment. Below 700 m, both ε_{μ} and ε_{T} are less than 10^{−10} W kg^{−1} between 28 January and 4 February; the differences between the two estimates also tend to be lower within this spatiotemporal range (Fig. 5c). This is in contrast to higher values of ε_{μ} and ε_{T} (> 10^{−10} W kg^{−1}) within the 700 to 800 m depth range at the beginning and end of the deployment.
3.2 Heat and salt fluxes
We calculate diapycnal diffusivity, κ_{ρ}, from ε using the Osborn relation:
where Γ is mixing efficiency, which is here taken to be 0.2 (Osborn, 1980). We use ε_{μ} in preference to ε_{T} because the former has better coverage in the mixed layer (Fig. 5).
Below approximately 250 m, the distribution of κ_{ρ} resembles that of ε_{μ} due to the relatively low variability in N^{2} at these depths (Fig. 4c). Above 200 m, N^{2} increases substantially in the pycnocline, and thus κ_{ρ} decreases. Beneath the surface mixed layer, κ_{ρ} is highest between 400 and 500 m ($<{\mathrm{10}}^{\mathrm{4.5}}$ m^{2} s^{−1}; Fig. 7a), with low values predominating in the core of the highsalinity SUW (< 10^{−6} m^{2} s^{−1}; Fig. 7a).
Vertical turbulent heat and salt fluxes, Q_{h} (W m^{−2}) and Q_{S} (kg m^{−2} s^{−1}), respectively, can be calculated from κ_{ρ}:
where ρ is density, C_{p} is the specific heat capacity of seawater, Θ_{z} is the vertical gradient of conservative temperature, and S_{z} is the vertical gradient of absolute salinity.
Beneath the surface mixed layer, both Q_{h} and Q_{S} are predominantly negative (i.e. downward) because temperature and salinity decrease with depth (Fig. 4a and b). The most prominent feature of the distributions of both is the broad region of negative (i.e. downward) turbulent heat and salt transport between approximately 200 and 500 m (Fig. 7b and c). This corresponds to the elevated values of κ_{ρ} (> 10^{−5} m^{2} s^{−1}) found within the same depth range (Fig. 7a). Within the surface mixed layer, notwithstanding the limited coverage of the observations, Q_{h} is positive in the top 50 m and negative between 50 and 100 m; Q_{S} is positive throughout the surface mixed layer (Fig. 7b and c). This is due to weak thermal and haline inversions near the surface; the depths of the temperature and salinity maxima are indicated by the black lines in Fig. 7b and c, respectively.
We now focus on the 200 to 500 m depth range, the region of highest κ_{ρ} and, consequently, the region in which Q_{h} and Q_{S} are most pronounced. Over the period of the observations, the arithmetic mean Q_{h} between 200 and 500 m was −1.40 W m^{−2}. This and all subsequent flux estimates are summarised in Table 1. The arithmetic mean Q_{S} between 200 and 500 m was −5.84 × 10^{−8} kg m^{−2} s^{−1}. This is a relatively lowturbulence region; the attendant turbulent fluxes are correspondingly relatively small and likely have little influence on the region's hydrography. For instance, integrated over a year, Q_{h} results in an annual turbulent heat flux of −4.43 × 10^{7} J m^{−2}, which would reduce the temperature of the overlying SUW layer (assumed to be 100 m thick) by just 0.11 ^{∘}C. Similarly integrated over a year, Q_{S} results in an annual turbulent salt flux of −1.84 kg m^{−2}, which would reduce the salinity of the overlying SUW layer by just 0.02 g kg^{−1}.
3.3 Salt fingers and associated fluxes
Diffusivity as estimated using the Osborn relation, κ_{ρ}, (Eq. 7) and the associated fluxes presented above are derived from ε and so are applicable to transports of heat and salt that are driven by turbulent, mechanical mixing (turbulent regime). However, these εbased estimates do not account for the fluxes driven by the salt fingers (i.e. doublediffusive mixing; saltfinger regime) that are characteristic of the thermohaline staircases prominent in the western tropical Atlantic (Schmitt et al., 1987; Rollo et al., 2022); Seaglider 620 was deployed at the edge of the region identified by Schmitt et al. (1987) as being the location of strong staircase structures. Such structures are prominent in the temperature observations (Fig. 8). In the turbulent regime, thermal diffusivity, κ_{Θ}, and haline diffusivity, κ_{S}, are the same. But in the saltfinger regime, κ_{S} may be approximately twice κ_{Θ}; hence they shall hereafter be considered separately. Moreover, in lowturbulence regions, such as that of the present study, salt fingers give rise to fluxes of heat and salt that can be larger than those driven by mechanical turbulence (Schmitt, 1988).
Those regions of the water column that are susceptible to salt fingers may be identified using the Turner angle, Tu: salt fingers can occur where 45^{∘} < Tu < 90^{∘}. The following equations are applied only where this condition is met. Tu is calculated from temperature and salinity binned into 5 m vertical bins; all subsequent diffusivities are calculated from variables binned into 5 m vertical bins in order to match Tu.
We compare two different estimates of κ_{Θ} and κ_{S} in the saltfinger regime. Firstly, and given that theoretical flux laws can overestimate κ_{S} in the real ocean (e.g. Taylor and Veronis, 1996; Kelley et al., 2003; Radko, 2005), we follow van der Boog et al. (2021) in using the empirical relations of Radko and Smith (2012) to calculate κ_{Θ} and κ_{S} from the density ratio, R_{ρ} (Eq. 6), and the molecular diffusivity of heat, K, estimated from temperature and salinity at each grid point:
From κ_{S}, again following van der Boog et al. (2021) and Radko and Smith (2012), we then calculate κ_{Θ}:
where $\mathit{\gamma}=\mathrm{2.709}{e}^{\mathrm{2.513}{R}_{\mathit{\rho}}}+\mathrm{0.5128}$ is the density flux ratio in the presence of salt fingers. We refer to these saltfinger diffusivities as κ_{Θ} and κ_{S} from R_{ρ} (Fig. 9a and b).
Secondly, we use the Osborn and Cox (1972) relation to estimate κ_{Θ} and κ_{S} using χ (Eq. 2; see also St Laurent and Schmitt, 1999; Schmitt et al., 2005; Ijichi and Hibiya, 2018), which we calculated as an intermediary step in the calculation of ε_{μ} (Sect. 2.2):
The relationship between κ_{Θ} and κ_{S} when calculated from χ remains as in Eq. (11) (Schmitt et al., 2005; van der Boog et al., 2021). We refer to these second saltfinger diffusivities as Osborn–Cox κ_{Θ} and κ_{S}; we note that the Osborn–Cox relation can include a contribution of mechanical mixing to χ and hence on κ_{Θ}.
There is approximately an orderofmagnitude difference between κ_{Θ} and κ_{S} from R_{ρ}, the latter being the greater (Figs. 6c and 9a and b). Neither distribution exhibits the same pronounced structure as κ_{ρ} (Fig. 7a): whereas κ_{ρ} resembles the distribution of ε_{μ}, from which it was derived (Fig. 5a) – i.e. with elevated values between 200 and 500 m, and lower values below and, in particular, immediately above – values of κ_{Θ} and κ_{S} from R_{ρ} are relatively constant in depth and time. Of the two, κ_{S} is the larger, being generally greater than 10^{−4.5} m^{2} s^{−1}; κ_{Θ} is generally between 10^{−6} and 10^{−5} m^{2} s^{−1}, although some values higher than this are present (Figs. 6c and 9a and b). Estimates of both κ_{Θ} and κ_{S} from R_{ρ} are higher than estimates of κ_{ρ} at the same depth (Figs. 7a and 9a and b), indicating that the vertical mixing of properties in the saltfinger regime is higher than in the turbulent regime.
We then substitute κ_{Θ} from R_{ρ} (Fig. 9a) into Eq. (8) in place of κ_{ρ} (Fig. 7a). Averaged (arithmetic mean) between 200 and 500 m, κ_{Θ} from R_{ρ} gives rise to a heat flux of −1.71 W m^{−2} in the saltfinger regime (Table 1), an annual temperature reduction of 0.13 ^{∘}C in the SUW layer. This flux is larger than the value reported above for the turbulent regime (Sect. 3.2). Similarly, we substitute κ_{S} from R_{ρ} (Fig. 9b) into Eq. (9) in place of κ_{ρ}. Averaged (arithmetic mean) between 200 and 500 m, κ_{S} from R_{ρ} gives rise to a salt flux of −1.83 × 10^{−7} kg m^{−2} s^{−1} in the saltfinger regime (Table 1), or an annual reduction of 0.06 g kg^{−1} in the salinity of the SUW layer. This flux is over 3 times larger than the corresponding salt flux in the turbulent regime. Note that, for the calculation of the arithmetic mean heat and salt fluxes in the saltfinger regime, we set both κ_{T} and κ_{S} from R_{ρ} to zero outside of the saltfinger regimes, i.e. where Tu < 45^{∘} and Tu > 90^{∘}, and include these zeros in our averages. Because we first filter out regions of the water column that are not susceptible to salt fingering, gaps in the record therefore indicate an absence of the process to be averaged (i.e. saltfingerdriven fluxes) rather than an absence of data.
The distributions of κ_{Θ} and κ_{S} from the Osborn–Cox relation resemble the distribution of ε, which itself resembles the distribution of χ that was used to calculate both diffusivities (not shown): the highest diffusivities are found between 200 and 500 m, with lower values being found above and below. In both cases, the highest diffusivities are between 10^{−6} and 10^{−5} m^{2} s^{−1}; within this range, κ_{S} is greater (Figs. 6d and 9c and d). Above and below this depth range, κ_{Θ} decreases to values below $\mathrm{10}{\times}^{\mathrm{6.5}}$ m^{2} s^{−1} (Fig. 9c), whereas κ_{S} decreases only to values of approximately 10^{−5.5} m^{2} s^{−1} (Fig. 9c).
We then substitute κ_{Θ} from the Osborn–Cox relation (Fig. 9c) into Eq. (8) in place of κ_{ρ}. Averaged (arithmetic mean) between 200 and 500 m, κ_{Θ} from the Osborn–Cox relation gives rise to a heat flux of −1.49 W m^{−2} in the saltfinger regime (Table 1), an annual temperature reduction of 0.11 ^{∘}C in the SUW layer. This heat flux is very similar to that reported for the turbulent regime and less than that predicted by the empirical relation of Radko and Smith (2012) reported above (i.e. κ_{Θ} and κ_{S} from R_{ρ}). And κ_{S}, averaged (arithmetic mean) between 200 and 500 m, gives rise to a salt flux of −9.40 × 10^{−8} kg m^{−2} s^{−1} (Table 1), an annual salinity reduction of 0.03 g kg^{−1} in the SUW layer. This flux is approximately 1.6 times that reported for the turbulent regime, but half that predicted by the empirical relation of Radko and Smith (2012). As before, we set both κ_{T} and κ_{S} from the Osborn–Cox relation to zero outside of the saltfinger regimes.
4.1 Comparing ε_{μ} and ε_{T}
Our finding that estimates of ε_{T} are higher than estimates of ε_{μ} is in agreement with the findings of Howatt et al. (2021). They report that, while the Thorpe scale method can well represent qualitative aspects of the spatiotemporal distribution of ε, the method can, when applied to relatively lowresolution, finescale observations, artificially inflate ε_{T} by an order of magnitude relative to estimates from, for instance, the Batchelor spectrum method. This is for the simple reason that finescale observations lack the resolution to capture smallscale overturns; the resultant distribution of L_{T}, and hence of ε, is consequently biased towards high values.
This is supported by our observations: the differences between ε_{μ} and ε_{T} reported here are much lower, being, on average, far less than an order of magnitude (Sect. 3, Figs. 5 and 6). Further, we directly compare L_{T} calculated using the FP07 microstructure observations (512 Hz, 100 Hz when accounting for the sensor's response time; Fig. 3) to L_{T} calculated using the Seaglider's ordinary finescale temperature and salinity observations (0.2 Hz). The distribution of L_{T} derived from microstructure observations has a pronounced peak at very low values (> 0.25 m; Fig. 10). In contrast, the finescale glider observations are unable to resolve values of L_{T} at this scale (Fig. 10). The better agreement between ε_{μ} and ε_{T} reported here than by Howatt et al. (2021) therefore appears to be explained by the higher vertical resolution of our observations. Consequently, and in agreement with Howatt et al. (2021), we suggest that, in a lowturbulence environment, the Thorpe scale method is unable to produce reliable quantitative estimates of ε unless it is applied to microstructure observations. In a highturbulence environment where overturns are large, finescale observations may be better able to accurately resolve the distribution of L_{T}, and their relatively low resolution may not introduce such a systematic bias.
A potential problem with using the Thorpe scale method on Seaglider observations is the nonvertical profile that the Seaglider collects: unlike, for instance, a shipdeployed CTD or a vertical microstructure profiler, a Seaglider follows a slanted trajectory, typically covering a horizontal distance of 4 km over a 1000 m deep dive–climb cycle. Any resultant sampling of horizontal gradients, particularly in the presence of internal waves, could artificially inflate estimates of ε_{T} due to false overturn detection (Thorpe, 2012). However, this is a concern only when the internal wave slope exceeds the slope of the Seaglider's trajectory. Following Howatt et al. (2021), we calculate that the mean (± 1 standard deviation) of SG620's trajectory slope (from the horizontal) during the Eurec4a deployment was 0.70 ± 0.09, i.e. greater than the upper limit of the slope of internal waves (approximately 0.3; Thorpe, 1978; Howatt et al., 2021). Consequently, false overturn detection due to the Seaglider's sloping trajectory is unlikely to lead to overestimation of ε_{T} in this data set.
4.2 Observed dissipation in spatial and temporal context
Values of ε in the western tropical Atlantic are broadly consistent with other openocean regions away from shallow topography. For instance, Sheen et al. (2013) and Naveira Garabato et al. (2016) report background ε values of approximately 10^{−10} to 10^{−9} W kg^{−1} downstream of the Drake Passage, with values in excess of 10^{−8} W kg^{−1} in the upper 1000 m and in the vicinity of rough topography. George et al. (2021) report ε values between 10^{−10} and 10^{−8} W kg^{−1} in the upper layers of the southwestern Bay of Bengal. And Peterson and Fer (2014) report missionmean values of between 10^{−8} and 10^{−7} W kg^{−1} from observations collected near the Faroe Islands in the northern Atlantic.
Limited estimates of ε are available for the western tropical Atlantic. Two profiles from the region, collected by a microstructure turbulence profiler, were presented by FernándezCastro et al. (2014). Similarly to our profiles, they find surface ε values between approximately 10^{−7} and 10^{−6} W kg^{−1} and values between approximately 10^{−9} and 10^{−8} W kg^{−1} between 50 and 300 m. Below 50 m, their estimates of κ_{ρ} are similar to ours, although they report nearsurface values several orders of magnitude larger than ours (e.g. > 10^{−3} m^{2} s^{−1}). Our methods do not yield reliable estimates of high turbulence in the mixed layer, however, so a full comparison is not possible. Two sets of observations taken several years apart and separated by up to hundreds of kilometres cannot be readily compared; moreover, their profiles do not extend deeper than 300 m. Interannual variability and geographic differences could be pronounced. Nevertheless, there is no evidence in their deepest observations of an increase in ε that might indicate the presence of the elevatedε layer that we observe beneath SUW (Figs. 4 and 5) and which, to our knowledge, has not previously been described.
4.3 Diffusivity and its influence on hydrography
The distribution of κ_{Θ} and κ_{S} from the Osborn–Cox relation (i.e. from χ) resembles that of κ_{ρ} – itself derived from ε_{μ} – far more closely than do either κ_{Θ} or κ_{S} from R_{ρ} (Figs. 7a and 9). Notably, neither κ_{Θ} nor κ_{S} from R_{ρ} seems to be particularly influenced by features of the water column that might be expected to influence vertical diffusivity, such as stratification (Fig. 4c) or temperature and salinity gradients (not shown). Given that the Osborn–Cox relation explicitly relates diffusivity to a mixing variable (i.e. χ), we suspect that it is more accurate than the empirical relation of Radko and Smith (2012), i.e. κ_{Θ} and κ_{S} from R_{ρ}.
As calculated from R_{ρ} (Radko and Smith, 2012), heat flux due to salt fingers is 1.22 times greater than that due to turbulent mixing; as calculated from χ (Osborn and Cox, 1972), the heat flux due to salt fingers is 1.06 times that due to turbulent mixing. And as calculated from R_{ρ}, the salt flux due to salt fingers is 3.31 times greater than that due to turbulent mixing; as calculated from χ, the salt flux due to salt fingers is 1.6 times that due to turbulent mixing. In all instances, heat and salt fluxes due to salt fingers (i.e. doublediffusive convection) are higher than turbulent fluxes. The empirical relations of Radko and Smith (2012) predict a relatively large difference, whereas the Osborn–Cox relation predicts a difference that is relatively small – indeed, for heat, the turbulent and saltfinger fluxes are almost the same. The choice of equation used to calculate diffusivities in the saltfinger regime makes a considerable difference to the result.
The dissipation rates and diffusivities presented above appear to have a limited influence on the hydrography of the study region. If the turbulent fluxes estimated above are representative of annual average conditions, heat and salt from the SUW layer penetrate downwards into the ocean interior relatively slowly; over the 12 d of the Eurec4a Seaglider deployment, the hydrographic influence of the fluxes would be negligible. As calculated from R_{ρ}, the doublediffusive heat flux due to salt fingers is 1.2 times greater than that due to turbulent mixing; as calculated from χ the doublediffusive heat flux due to salt fingers is the same as that due to turbulent mixing. Salt fingers potentially have a greater influence on hydrography than turbulent mixing, but the overall effect would still be nearly inconsequential.
We demonstrate that microstructure temperature observations collected by a Seaglidermounted FP07 fast thermistor may be used to estimate the dissipation rate of turbulent kinetic energy, ε. We estimate ε using the Batchelor spectrum method (ε_{μ}) and the Thorpe scale method (ε_{T}). The results from the two methods agree well, although ε_{T} is on average higher than ε_{μ}. This is in agreement with previous studies, although the difference reported here, which is less than an order of magnitude, is below that reported elsewhere. This improved agreement is due to ε_{T} being calculated using the same highresolution observations as were used to calculate ε_{μ}: other studies (e.g. Howatt et al., 2021) have compared ε_{μ} to ε_{T} calculated using lowerresolution, finescale temperature observations that are unable to resolve overturns below a certain size, thus biassing estimates of ε towards high values.
We identify a layer of elevated ε values between 200 and 500 m that lies immediately below Subtropical Underwater, a highsalinity subsurface water mass that is colocated with a maximum in stratification. We estimate that, over the period of the deployment, this elevatedε layer is responsible for a mean heat flux of −1.40 W m^{−2} and a mean salt flux of −5.84 × 10^{−8} kg m^{−2} s^{−1}. Given the prevalence of double diffusion and salt fingers in the region, we estimate thermal and haline diffusivities in the saltfinger regime and the resultant heat and salt fluxes using both the empirical relation of Radko and Smith (2012), which depends on the density ratio, R_{ρ}, and the relation of Osborn and Cox (1972), which depends on the rate of destruction of temperature variance, χ. Fluxes estimated using the relation of Radko and Smith (2012) are higher than those estimated using the relation of Osborn and Cox (1972). Nevertheless, turbulent and saltfinger fluxes are small, and their influence on hydrography is likely limited.
The MATLAB toolbox of Scheifele et al. (2018) to calculate turbulent kinetic energy dissipation rate following the Batchelor spectrum method is available at https://github.com/bscheife/turbulence_temperature.
Standard hydrographic observations from SG620 (Rollo, 2021) are available from the British Oceanographic Data Centre at https://doi.org/10.5285/c596cdd7c709461ae0536c86abc0c127. Processed turbulent kinetic energy dissipation rate estimates (Rollo et al., 2023), from both Batchelor and Thorpe scale methods, are available from the British Oceanographic Data Centre at https://doi.org/10.5285/f173b9c1bb500b75e0536c86abc02a4a.
PMFS and GMD calculated turbulent kinetic energy dissipation rate following the Batchelor spectrum method. PJL calculated turbulent kinetic energy dissipation rate following the Thorpe scale method. All authors contributed to the analysis. PMFS wrote the paper with assistance from GMD and PJL, and with comments and feedback from KJH and RAH.
The contact author has declared that none of the authors has any competing interests.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
We thank Beth Siddle, Callum Rollo, and the crew and scientists of the RV Meteor (cruise M161) for assistance in the deployment and recovery of the glider. We thank Gareth Lee and Marcos CobasGarcia for the preparation of the glider and the UEA glider group for piloting. We are indebted to two anonymous reviewers, whose feedback has greatly improved this paper.
This research has been supported by the European Research Council (grant no. 741120) and UK Research and Innovation (grant no. NE/N012070/1).
This paper was edited by Katsuro Katsumata and reviewed by two anonymous referees.
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