Estimating Dissipation Rates Associated With Double Diffusion
Abstract
Double diffusion refers to a variety of turbulent processes in which potential energy is released into kinetic energy, made possible in the ocean by the difference in molecular diffusivities between salinity and temperature. Here, we present a new method for estimating the kinetic energy dissipation rates forced by doublediffusive convection using temperature and salinity data alone. The method estimates the upgradient diapycnal buoyancy flux associated with double diffusion, which is hypothesized to balance the dissipation rate. To calculate the temperature and salinity gradients on small scales we apply a canonical scaling for compensated thermohaline variance (or ‘spice’) on submeasurement scales with a fixed buoyancy gradient. Our predicted dissipation rates compare favorably with microstructure measurements collected in the Chukchi Sea. Fine et al. (2018), https://doi.org/10.1175/jpod180028.1, showed that dissipation rates provide good estimates for heat fluxes in this region. Finally, we show the method maintains predictive skill when applied to a subsampling of the Conductivity, Temperature, Depth (CTD) data.
Key Points

We develop a method for estimating dissipation rates due to doublediffusive convection from temperature and salinity data using an energetic argument

The method reproduces the patterns and magnitude of dissipation rates and heat fluxes from microstructure measurements across an arctic eddy

The technique performs well even when applied to lower resolution Conductivity, Temperature, Depth (CTD) data
Plain Language Summary
Understanding the transport of heat in the Arctic Ocean is vital for predicting the fate of seaice in the decades to come. Smallscale turbulence is an important driver of heat transport and one of the major forms of this turbulence is known as ‘doublediffusive convection’. This is where gradients in temperature and salinity drive turbulence. It is only possible because salinity diffuses significantly slower than temperature, hence the name ‘doublediffusive’. The most direct measurements of ocean mixing require sampling velocity or temperature gradients on scales <1 mm, socalled microstructure measurements. Here we present a new method for estimating the energy dissipated by doublediffusive convection using temperature and salinity measurements on larger scales (100s–1000s of metres). We apply the method to a highresolution survey of temperature and salinity through a subsurface Arctic eddy and compare the results with simultaneous microstructure measurements. The new technique can reproduce up to 72% of the observed dissipation rates to within the measurement uncertainty. This suggests the method could be used to estimate the dissipation and so heat fluxes associated with doublediffusive convection in regions without microstructure measurements. The method also performs well when applied to data subsampled on larger scales.
1 Introduction
Recent decades have seen a decline in Arctic seaice thickness and extent (E. Carmack et al., 2015). Seaice melting is influenced by vertical ocean heat transport (Timmermans & Marshall, 2020). For example, a doubling of the Beaufort Gyre halocline heat content has influenced seaice extent (Shimada et al., 2006; Timmermans et al., 2018). Heat exchange between the surface layer and subsurface water masses depends on smallscale turbulent processes and identifying turbulent regions helps to quantify this heat flux (Fine et al., 2018). The most direct measurements of turbulent mixing require highresolution measurements (microstructure) which are costly and difficult to obtain across large spatial scales. A natural solution is to parameterize turbulence based on properties measured on the finescale (e.g., Polzin et al., 2014; Whalen et al., 2015). However, common shear and strain based parameterisations for internal wave breaking do not apply well to areas of doublediffusive convection (Gregg, 1989).
Buoyancy gradients in the upper Arctic Ocean are dominated by salinity, which enables heat to be trapped subsurface. Opposing vertical gradients in temperature (T) and salinity (S) and a relatively weak internal wave field make the Arctic favorable for doublediffusive convection (e.g., Bebieva & Timmermans, 2016), where it can play a dominant role in upwards transport of the trapped heat (E. C. Carmack et al., 2012). Doublediffusive convection refers to convective processes that rely on the difference between the molecular diffusivities of heat () and salt () (Schmitt, 1994). A variety of flow structures may develop due to double diffusion. For example, vertical gradients in temperature and salinity can force convective motions, leading to steplike temperature and salinity profiles in a thermohaline ‘staircase’ (e.g., Timmermans et al., 2008). Other results of double diffusion include thermohaline intrusions (Ruddick & Kerr, 2003; Ruddick & Richards, 2003) and salt fingers (Kunze, 2003). Traditionally these processes are analyzed and parameterized separately, however, recently Middleton and Taylor (2020) presented an energetic framework that describes the dynamics of all cases where potential energy is released into kinetic energy via double diffusion (our definition of doublediffusive convection).
Several studies have shown a close connection between largescale circulation and doublediffusive fluxes. Shibley et al. (2017) showed that the prevalence of thermohaline staircases, and by inference the doublediffusive fluxes, are modified by the largescale circulation patterns within the Arctic basins. Individual layers associated with thermohaline staircases can extend for several hundred kilometers with significant variations in T/S properties along this length (Bebieva & Timmermans, 2019). This suggests a possible connection between lateral fluxes of heat and salt and vertical fluxes associated with doublediffusive convection.
Estimating doublediffusive heat fluxes generally requires either microstructure measurements or parameterizations of specific doublediffusive processes. For example, Shibley et al. (2017) use a 4/3 flux law to estimate heat fluxes across diffusive interfaces in thermohaline staircases, and Inoue et al. (2007) combine two parameterisations for different doublediffusive regimes, distinguished using the density ratio . Fine et al. (2018) showed that the turbulent dissipation rate can be used to infer the vertical heat flux in doublediffusive conditions. Here, we propose a new statistical technique to estimate turbulent dissipation rates associated with general doublediffusive processes from a Conductivity, Temperature, Depth (CTD) section. We rely on the notion that lateral stirring of T/S along isopycnals (surfaces of constant density) gives rise to large T/S gradients that are ‘compensated’ in terms of their contribution to density (Rudnick & Ferrari, 1999). This compensated T/S (or ‘spice’) variance is preferentially fluxed to small scales.
The central concept behind our method is that large spice gradients generated by alongisopycnal stirring drive doublediffusive motions through an upgradient buoyancy flux and that the resulting smallscale turbulence fluxes energy further down scale until it is ultimately dissipated. We use a synthetic model spectrum to extrapolate alongisopycnal spice variance measured from a CTD section down to the scale of the largest turbulent overturns and use this result along with other assumptions to estimate the upgradient buoyancy flux and the turbulent dissipation rate through the framework described in Middleton and Taylor (2020).
The standard model for sheardriven ocean mixing is that energy is input to small scales via shear forced by a larger structure (e.g., internal waves and mean flows). Some portion of this energy gets transferred to microscales and ultimately dissipated at a rate ϵ, and some portion is lost to potential energy via the turbulent buoyancy flux w′b′; used to irreversibly mix the density field. This division is quantified using a parameter Γ, the flux coefficient (Gregg et al., 2018), that has been evaluated from observations, experiments and simulations (e.g., Holleman et al., 2016; Howland et al., 2020; Jackson & Rehmann, 2003). The complication is that the exchange of potential energy and kinetic energy via w′b′ is reversible and in fact dominated by adiabatic internal waves. This was resolved by Winters et al. (1995) who divided the potential energy into a ‘Background Potential Energy’ (BPE) and an ‘Available Potential Energy’ (APE) and considered their budgets. They found that APE is transformed into BPE via a diapycnal (across surfaces of constant density) buoyancy flux ϕ_{d}, which they associated with ‘irreversible mixing’ of density as ϕ_{d} is always positive in a singlecomponent fluid. However, in doublediffusive convection, variations in density (i.e., APE) are created by the action of diffusion, so the standard model of sheardriven irreversible mixing of density does not apply.
Extending the Winters et al. (1995) framework to include double diffusion, Middleton and Taylor (2020) showed that the ‘mixing’ of density may be reversed; in a doublediffusive fluid, BPE (as defined in Winters et al., 1995) can be converted into APE via an upgradient diapycnal buoyancy flux, i.e., ϕ_{d} < 0. In a statistically steady state, we might then expect the generation of APE to be converted into kinetic energy through the vertical turbulent buoyancy flux, i.e., 〈w′b′〉 ≃ −〈ϕ_{d}〉, averaging across an appropriate volume. Further, if doublediffusive convection is the only source of turbulence, then the input of kinetic energy comes entirely from potential energy, so the average rate of dissipation of kinetic energy, 〈ϵ〉, should balance the vertical turbulent buoyancy flux 〈w′b′〉, which gives the balance 〈ϵ〉 ≃ 〈w′b′〉 ≃ −〈ϕ_{d}〉. The form for ϕ_{d} can be expressed in terms of the local gradients in temperature and salinity and their respective molecular diffusivities, which provides a pathway for predicting the dissipation rate associated with double diffusion from T and S alone.
The diapycnal buoyancy flux, ϕ_{d}, as formulated by Middleton and Taylor (2020), involves the molecular diffusive fluxes of heat and salt. Therefore, diagnosing ϕ_{d} requires threedimensional gradients of T and S at small scales. While this is possible in direct numerical simulations, it is not practical using observational data. Here, we follow the theory of Haynes and Anglade (1997) and Smith and Ferrari (2009) and assume that spice is stirred along isopycnals by quasigeostrophic motions. The possibility of doublediffusive instabilities developing in response to alongisopycnal stirring of spice was raised in Garrett (1982) and Smith and Ferrari (2009). However, we believe this letter is the first attempt to use this principle to estimate the dissipation rate associated with doublediffusive processes.
To demonstrate our method and test the underlying assumptions, we use a CTD section through a subsurface warmcore eddy reported in Fine et al. (2018) and compare our estimated dissipation rates with those inferred from microstructure measurements. Fine et al. (2018) surveyed an intrahalocline warmcore eddy on the Chuchki Slope in September 2015, collecting shear and thermal microstructure data as well as finescale T and S across multiple eddy transects. Eddies such as this are thought to be an important source of heat into Canada Basin (Watanabe, 2011), where the implications for seaice are not well understood. This was confirmed by Fine et al. (2018), who showed that the regions of enhanced heat flux at the top of the eddy, inferred from thermal microstructure data, could lead to a significant upwards heat flux if extrapolated over the eddy's lifetime. Fine et al. (2018) found that the top of the eddy was also associated with enhanced dissipation rates, which were used to estimate the heat flux.
The method we introduce in this letter uses physical reasoning to link alongisopycnal stirring of spice with doublediffusive convection and ultimately energy dissipation. The primary assumption that we make is that doublediffusive convection is the sole driver of turbulence i.e., shear production is negligible. This is analogous to the assumption made in finescale internal wave parameterisations that shear production from internal waves is the sole driver of turbulence (e.g. Polzin et al., 2014). When compared with the data from Fine et al. (2018), the dissipation rate inferred using our method closely matches the dissipation rate inferred from a microstructure probe except in regions with relatively low Richardson number where Fine et al. (2018) hypothesized that sheardriven turbulence was active. This shows that our method can be used in conjunction with more direct microstructure measurements to identify the relative importance of double diffusion. When microstructure measurements are not available, our method provides a means to estimate the turbulent dissipation rate in regions that are dominated by double diffusion from a high resolution CTD section.
This letter is organized as follows. In the first section we introduce the observations from Fine et al. (2018). In the second section we describe the method for estimating turbulent dissipation rates associated with doublediffusive convection. In the third section we compare our results to the observations. We find a good match across most of the section, including the top of the eddy, where dissipation rates can be linked to the vertical heat flux. We also find a good match across ‘doublystable’ regions without opposing background T/S gradients. The estimated doublediffusive dissipation rate is significantly smaller than the microstructure dissipation rate only in regions where the gradient Richardson number is low and the buoyancy Reynolds number Re_{b} > 20, indicating active sheardriven turbulence. Finally, we test how the method performs on a coarse subsampling of the data, suggesting applicability to data with lower spatial resolution.
2 Observations
In this section we briefly discuss the observations from the Arctic Ocean that we use to test the new method for estimating the doublediffusive dissipation rate. This data will also be used to test some assumptions made in the next section. Fine et al. (2018) described data collected from an anomalously warm anticyclonic eddy on the Chukchi slope in September 2015. The temperature and salinity from one of the sections are shown in Figure 1. Isopycnals are shown in gray in panel b and the 3°C isotherm which outlines the eddy is shown in black. In most places density variations are controlled by salinity, although temperature plays a role within the eddy. Concurrent microstructure measurements of the kinetic energy dissipation rate, ϵ, and thermal variance dissipation, χ_{T}, were used to identify regions of high dissipation along the top, sides and bottom of the eddy. Practical issues regarding the processing of microstructure data in the presence of thermohaline layering and instrument noise floors are discussed in Fine et al. (2018) and not included here. The heat flux was also inferred from the microstructure profiles and used to estimate the lifetime of the eddy. The lateral heat flux along the eddy flanks (∼ 2,000 Wm^{−2}) was found to have the largest impact on the eddy evolution, giving an estimated eddy lifetime of ∼1 year, although the heat flux along the top of the eddy (5 Wm^{−2}) was also significant. The dissipation rate, ϵ, was used to estimate the heat flux using the Osborn (1980) model, where a good agreement was found with the estimates from χ_{T} using the Osborn and Cox (1972) model.
The data from Fine et al. (2018) provides a useful testing grounds for our method since it contains a variety of regimes. Different processes were identified by Fine et al. (2018) based on visual identification of coherent features (layers and intrusions), Richardson number (see Figure 1) and the relative alignment of temperature and salinity gradients. The density ratio R_{ρ} (plotted in Figure 1, smoothed with a median filter) indicates a ‘Diffusive Convection’favorable (DC) region along the top of the eddy where staircases were observed (0 < R_{ρ} < 1), a ‘SaltFingering’favorable (SF) region in the middle and at the base of the eddy (R_{ρ} > 1) and a ‘DoublyStable’ region within and below the eddy (R_{ρ} < 0) where sheardriven mixing was thought to occur based on the enhanced Richardson number below the eddy (panel d). Traditionally, different types of doublediffusive convection are parameterized separately (e.g., Inoue et al., 2007). However, our method does not depend on the type of double diffusion based on measured T/S gradients, instead assuming that the opposing gradients necessary for double diffusion are created by submeasurement scale spice variance. The T/S gradients can take any alignment relative to one another, subject to the condition for an upgradient diapycnal buoyancy flux (Φ_{d}) derived in Middleton and Taylor (2020).
3 Method
In this section we start by presenting a new form for the diapycnal buoyancy flux, ϕ_{d}, in terms of spice gradients before describing our method for downscaling the observed spice gradients (i.e., inferring their size at submeasurement scales) in order to estimate ϕ_{d} and the dissipation rate, ϵ.
Our method can be summarized as follows. First, assuming doublediffusive convection is the driver of turbulence, we take 〈ϵ〉 = −〈ϕ_{d}〉, where 〈⋅〉 is an implicit average over the volume between observations. The diapycnal flux ϕ_{d} is then reexpressed in terms of spice gradients and buoyancy gradients. We assume that the buoyancy and spice gradients are anticorrelated, justified with reference to the observations, which gives an approximate upper bound for −〈ϕ_{d}〉. Hypothesizing that the submeasurement scales are dominated by quasigeostrophic (QG) stirring along isopycnals, we assume the buoyancy gradients are constant and the spice variance is modeled with the QG wavenumber scaling for passive tracers i.e., in stretched coordinates. This scaling is taken to hold down to the critical scale at which ϕ_{d} drives convective overturns. The amplitude of the synthetic spice power spectrum is estimated using a twopoint correlation calculated locally from the observations. The power spectrum is then used to estimate the mean spice gradient magnitude down to the overturning scale, which is then used to estimate ϕ_{d} and ϵ.
3.1 Diapycnal Buoyancy Flux
In the singlecomponent limit where the two diffusivities are equal, the second term disappears, so we can consider it as a doublediffusive contribution. Using molecular values for the diffusivities, κ_{T} ≫ κ_{S} so κ_{T} + κ_{S} ≈ κ_{T} ≈ κ_{T} − κ_{S} and hence the two prefactors are of similar size, aside from the factor of g.
In order to estimate the dissipation rate due to double diffusion, we will assume that the dissipation is entirely forced by an upgradient buoyancy flux as discussed in the introduction. Middleton and Taylor (2020) showed that BPE is converted into APE when the average of ϕ_{d} over a given volume (Φ_{d}) is negative. Here, we attempt to estimate where angle brackets denote an average over a volume bounded by neighboring T/S measurement points. We then assume that all of the potential energy released by the upgradient buoyancy flux is converted into kinetic energy and ultimately dissipated and set .
When Equation 2 is evaluated directly with buoyancy and spice gradients calculated using the available observations, the corresponding dissipation rate is very small ∼O(10^{−12}) W/kg. This shows, unsurprisingly, that molecular diffusion is weak at the resolution of the observations (∼300 m in the horizontal and ∼0.5 m in the vertical). The central hypothesis of our method is that spice variance is transferred downscale by alongisopycnal stirring until it reaches a scale where the upgradient buoyancy flux drives doublediffusive motions. Since the hypothesized stirring is alongisopycnal, we take the buoyancy gradient on submeasurement scales to be constant. Specifically, this allows us to assume first that 〈∇b‖∇s_{p}〉 ≈ 〈∇b〉〈∇s_{p}〉, and second that 〈∇b〉 = 〈∇b〉, so the buoyancy gradient magnitude can be estimated by the value on the measurement scale. However, we have not assumed that the spice gradient is constant on submeasurement scales, so 〈∇s_{p}〉 ≠ 〈∇s_{p}〉 in general and we must estimate 〈∇s_{p}〉 by other means. In the next section we describe a method for estimating the magnitude of the spice gradient on submeasurement scales.
3.2 DownScaled Spice Gradients
To estimate the submeasurementscale spice gradients, we use a statistical method based on the assumption that spice is stirred along isopycnals by quasigeostrophic (QG) motions. The Rossby number associated with the eddy shown in Figure 1 is about 0.2 which is not inconsistent with the QG limit (Fine et al., 2018). Based on theory from Charney (1971), Haynes and Anglade (1997) and Smith and Ferrari (2009) showed that QG flow generates filaments of passive tracers (such as spice) with a horizontal to vertical aspect ratio of N/f where N is the buoyancy frequency and f is the Coriolis parameter. The data appear to be generally consistent with an N/f aspect ratio. Lines with a slope matching the local f/N slope are indicated in panel a of Figure 1. Figure 2b shows a joint PDF of spice gradients in stretched coordinates. The peak in the distribution lies close to the y = x line, corresponding to isotropy in the stretched coordinate system.
The alongisopycnal spice variance spectrum from the Fine et al. (2018) data (Figure 2c) shows a scaling, consistent with other observations of passive tracers at these scales (Callies & Ferrari, 2013; Klymak et al., 2015). Our modeled spectrum at submeasurement scales is indicated with a dashed line. MacKinnon et al. (2016) shows observational evidence for a k^{−1} scaling of temperature in salinitycontrolled mixed layers for two separate observational datasets, however the transition to this scaling occurs at horizontal wavelengths ∼300 m which is smaller than the available observations. In Section 4.1 we discuss the sensitivity of our results to the model spice power spectrum.
To estimate the spice gradients that contribute to ϕ_{d} we extrapolate the spice variance spectrum down to a scale corresponding to the convective overturns driven by double diffusion. Our underlying assumption is that the APE generated by the upgradient buoyancy flux at this scale drives convective motions and that the resulting turbulence fluxes kinetic energy further downscale until it is dissipated at the same rate. Hence, although spice gradients will persist down to the haline Batchelor scale, we estimate the spice gradients at the scale of the largest 3d turbulent motions which we take to be the Ozmidov scale . This choice is consistent with the scaling for the layer thickness in diffusive convection proposed by Fernando (1989). Other choices are discussed in Zhou et al. (2016) who suggest that provides a better match with observations and lab experiments. However, since and Re_{b} is small for the Fine et al. (2018) data set (see Figure 3g) the two lengthscales yield similar dissipation rate estimates and we use L_{Oz} for simplicity.
Evaluating the spice gradient magnitude ∇s_{p} in Equation 2, we get the form for the average dissipation rate 〈ϵ〉. In summary, the method can be written as a series of steps
4 Results
4.1 Comparison With Microstructure
The result of our method as applied to the data from Fine et al. (2018) are shown in Figure 3a. The match between the estimated and measured dissipation rates is generally very good. Both the pattern and magnitude of the dissipation are captured by the method, even in areas of doubly stable stratification, not traditionally thought of as active areas of doublediffusive convection. In the centre of the eddy, where the stratification is very weak, the method predicts very low levels of dissipation (<10^{−10} m^{2}s^{−3}), however the noise floor for the shear probe observations of ϵ is about 10^{−10} m^{2}s^{−3} so we cannot compare these small values. Comparing the horizontal averages of dissipation rate on the right we can also see the close match between observations and our method. Figure 3d shows the horizontal average of the heat flux inferred from the dissipation rate using the Osborn (1980) method using a mixing efficiency Γ = 1, following Fine et al. (2018) and St. Laurent and Schmitt (1999). Following the method described in Fine et al. (2018) we exclude values where ‘interfaces’ were identified using a threshold ). The agreement between the Osborn (1980) method for microstructure and our prediction implies that we can also estimate heat fluxes along the top and sides of the eddy. There is a discrepancy beneath the eddy that is discussed below, which suggests that double diffusion is not the dominate source of turbulence in this region.
Microstructure measurements of ϵ have an uncertainty of a factor of around 2 (Fine et al., 2018). Histograms of the ratio of the estimated and observed dissipation rates are shown in Figure 3e and 3f (points with ϵ_{pred} below the instrument noise floor of 10^{−10} m^{2}s^{−3} are excluded). The comparison in panel e shows a lognormal distribution around an average bias of 0.89 with a tail to the distribution for small values. The mean bias corresponds to an average observed dissipation of 9.7 × 10^{−10} m^{2}s^{−3} and average predicted dissipation of 8.5 × 10^{−10}m^{2}s^{−3}. Excluding points with buoyancy Reynolds numbers (a threshold value for doublediffusion used by Inoue et al. (2007) amongst others) removes the tail such that 72% of the estimated dissipation rates are within a factor of 3 of the observed values, although note that very little data here has Re_{b} > 20 (as shown in Figure 3g). Differences between our prediction and observations (at low Re_{b}) are similar to those occurring due to natural variability (Moum et al., 1995). The enhanced Re_{b} may be the result of shear as the Richardson number is smaller above and below the eddy (Figure 1), however it is still unclear when shear will damp or enhance doublediffusive convection (Padman, 1994; Shibley & Timmermans, 2019). The enhanced Re_{b} beneath the eddy explains the discrepancy between the horizontally averaged prediction and observation in Figure 3c.
Panel f divides the data between areas usually identified as ‘doublediffusive convection favorable’ (0 < R_{ρ} < 1), ‘salt fingering favorable’ (R_{ρ} > 1) and ‘doublystable’ (R_{ρ} < 0). This panel shows that the method works well for ‘DC’ and ‘DS’ regimes but potentially overestimates the dissipation rate in ‘SF’ regimes. It is possible that saltfingers are present, which may change the appropriate overturning length scale. Another explanation is that our assumption of anticorrelating buoyancy and spice gradients breaks down for R_{ρ} > 1. However, there is too little ‘SF’data here to fully analyze these possibilities and the method still works relatively well in ‘SF’ regions. Alternatively, if we were to assume a persistent scaling down to the Ozmidov scale, the patterns of the estimated dissipation rate would be qualitatively similar, but the magnitude of the dissipation rate would decrease by about a factor of 3 (see panel e). The model spice spectrum represents the largest sensitivity in the method and further observations of the m scales could help test and improve the method.
4.2 Method Sensitivity
The T/S section from Fine et al. (2018) has a high resolution (Δx ∼ 200–1000 m in the horizontal and Δz = 0.25 m in the vertical). A natural question is whether the method would work when applied to a lower resolution section of T/S. To test this, we subsample the original data at lower horizontal resolution. Figure 4 shows three different resolution comparisons in the three columns. In each comparison we calculate the dissipation rate using subsampled temperature and salinity sections, and we compare the result with the observed doublediffusive dissipation rate subsampled at the same resolution. Note that the data in Figure 4 are calculated based on horizontally gridded T/S data, whereas the method in Figure 3a is applied to ungridded data (which gives a better agreement with observations). Using lower resolution T/S data somewhat increases the estimated dissipation rate, although the method performs well even at significantly reduced resolution. This suggests that our method could be applied to lower resolution CTD sections which are much more widely available.
Equation 11 more clearly shows the dependence of 〈ϵ〉 on the rate of generation of spice variance via A, the buoyancy frequency N and the term which gives a measure of the curvature of isopycnals. If we set the term constant, we underestimate the dissipation rate along the top of the eddy, where the isopycnals heave, suggesting that the largescale variations in density may have a direct effect on the small scale turbulence.
If we vary the spice spectrum amplitude A (Equation 7), by a scale factor a, although the pattern of the dissipation rate remains similar to Figure 3a, the magnitude changes by a factor of about a^{1/3} (by considering Equation 11). Therefore, an error in A by a factor of 10 would result in an estimate of ϵ that is incorrect by a factor of ∼2.15. This relatively low sensitivity to A is the reason for the lack of sensitivity to the scale at which the spectrum transitions to in our model. This also explains why the method maintains skill at low resolution (Figure 4) despite the disagreement between the assumed spectrum and the observed spectrum (Figure 2). This effect may be corrected for by using a more complex model spectrum, however that is left as future work. The size of the eddy is also important here. The slope is hypothesized to be the result of the eddy stirring, so for larger eddies we may expect the model spectrum to hold at larger scales.
Although we calculate α and β at each grid point using the fully nonlinear equation of state, we assume a linear equation of state on submeasurement scales in our method. If we use a single constant value for α and β across the whole section, the estimated dissipation rate would change by up to a factor of 1.8, so the effects of a nonlinear equation of state also represent an uncertainty in our method that may explain some of the discrepancies in Figure 4.
5 Conclusion
We have introduced a new method for estimating the dissipation rates associated with double diffusive turbulence from a temperature and salinity section. We used an energetic argument and a statistical approach which applies to general doublediffusive processes. For points with low buoyancy Reynolds number (Re_{b} < 20) in the data from Fine et al. (2018), our method closely matches observations (72% of estimated dissipation rates are within the observational uncertainty). Fine et al. (2018) used observations of dissipation rate to infer heat fluxes that matched those calculated using thermal microstructure observations. The close match in dissipation rates between our method and observations suggests our method may be used to infer heat fluxes in other locations where double diffusion is active. Finally, we showed that our method can be applied (with additional uncertainties) to T/S data with coarse spatial resolution, opening the door to applying the method to more widelyavailable lower resolution CTD sections.
Acknowledgments
The authors thank Barry Ruddick and an anonymous reviewer for their valuable input on the development of this manuscript. This work was supported by the Natural Environment Research Council (grant number NE/L002507/1).
Open Research
Data Availability Statement
Shipboard and microstructure data are available for download (http://www.rvdata.us/catalog/SKQ201511S and https://microstructure.ucsd.edu, respectively). Code for applying our method to CTD data can be found at https://doi.org/10.5281/zenodo.4665777.