is properly cited. Shelf Seas Baroclinic Energy Loss: Pycnocline Mixing and Bottom

) Abstract Observations of turbulent kinetic energy dissipation rate (  ) from a range of historical shelf seas data sets are viewed from the perspective of their forcing and dissipation mechanisms: barotropic to baroclinic tidal energy conversion, and pycnocline and bottom boundary layer (BBL) dissipation. The observations are placed in their geographical context using a high resolution numerical model (NEMO AMM60) in order to compute relevant maps of the forcing (conversion). We analyze, in total, 18 shear microstructure surveys undertaken over a 17 year period from 1996 to 2013 on the North West European shelf, consisting of 3,717 vertical profiles of shear microstructure: 2,013 from free falling profilers and 1,704 from underwater gliders. A robust positive relationship is found between model-derived barotropic to baroclinic conversion, and observed pycnocline integrated  .

In this article, we explore in more detail the influence of bed friction on pycnocline mixing (see Figure 1), speculated about briefly in previous studies Inall et al., 2000;MacKinnon & Gregg, 2003). While it has been shown that dissipation in the bottom boundary may not exert a direct control on pycnocline mixing (Rippeth, 2005), an indirect control is investigated here by considering separately the influence of internal friction and boundary friction on the internal wave energy flux divergence.
Currently, many shelf sea models include a turbulence closure vertical mixing scheme (e.g., Holt & Umlauf, 2008). However, when profiles of turbulent kinetic energy dissipation rate predicted using 1D versions of the closure schemes are compared with observations they have failed to correctly reproduce the dissipation rates within the pycnocline. Without the inclusion of artificial adjustments, including high levels of background diffusion (Simpson et al., 1996), the closure schemes are inadequate. Some improvements have been made in reproducing shelf seas mixing with high resolution numerical models, through the development and implementation of a hierarchy of second moment turbulence closure schemes (Holt & Umlauf, 2008). In order to simulate internal mixing, these schemes apply various different stability functions derived from the ratio of the local buoyancy frequency to local velocity shear, to relate prognostic turbulent length and time scale terms to the mean flow characteristics. However, without ad hoc enhancements these second moment turbulence closure schemes generally do not generate enough mixing across the pycnocline within seasonally stratified shelf seas (Holt & Umlauf, 2008;Rippeth, 2005). This has been taken to imply that the current models do not represent certain key processes which generate shear at the pycnocline; particularly internal waves or wind-driven inertial oscillations . Partly in order to resolve this issue, and also to improve model stability, an ad hoc and high level of background diffusion is INALL ET AL.  bottom boundary layer turbulence bbl  (which is also fueled directly from bottom friction acting on the barotropic tide). An idealized profile of shelf sea vertical density structure is shown demonstrating the pycnocline selection criteria described in the text. An idealized shelf sea velocity profile, ( ) U z is also shown. often applied throughout model domains (Jochum, 2009), justified on the dual grounds of tuning to observations and of numerical stability. This approach can lead to an improvement in reproducing observed levels of mixing in some areas, but often fails to do so within stratified regions (Luneva et al., 2019). Uniformly applied diffusive mixing also fails, by definition, to represent the significant temporal and spatial variability known to exist within ocean turbulence (Moum & Rippeth, 2009). Indeed, a recent study (Luneva et al., 2019), which examined a number of commonly used numerical mixing schemes adapted to overcome missing sub-grid scale dynamics in a 7 km resolution hydrodynamic model (the NEMO AMM7 model), demonstrated that many commonly used enhanced second order mixing schemes lead to an overly diffusive water column when implemented in the latest generation of shelf seas hydrodynamic models.
These model limitations have hampered our ability to make meaningful comparisons between modeled internal wavefield energetics and a shelf-wide observational database of turbulent kinetic energy dissipation estimates. However, recent enhancements of the NEMO shelf model to 1.8 km horizontal resolution (AMM60) have resulted in an increased ability to systematically permit internal tide generation and propagation (Guihou et al., 2018). AMM60 successfully simulated internal tides with realistic spectral energy at diurnal, inertial, semi-diurnal and quarter-diurnal bands, and tidally induced pycnocline displacements were diagnosed to vary with the spring neap cycle. A detailed study of vertical mixing and dissipative processes within AMM60 is yet to be undertaken, and is not the purpose here. Rather, our aim is to take the successes of AMM60 in reproducing internal tides (Guihou et al., 2018) and explore the relationship between model-based tidal conversion estimates and observations of turbulent kinetic energy dissipation within the pycnocline. Our exploration is primarily motivated by a small number of studies which noted the potentially significant role of lower boundary friction as the dominant energy sink from internal waves in shallow seas Inall et al., 2000;MacKinnon & Gregg, 2003). These previous observations suggest that the energy available for pycnocline mixing may be inextricably linked to energy loss in the lower boundary layer.
Our aim is achieved by collating and reanalyzing a large historical collection of microstructure observations of turbulent kinetic energy dissipation data (hereafter  ), all obtained on the relatively wide and flat North West European continental shelf ( Figure 2); an area of large tidal conversion (Baines, 1982;Egbert & Ray, 2001;Nycander, 2005), large amplitude internal tides, and enhanced pycnocline mixing (Inall et al., 2001(Inall et al., , 2011Palmer et al., 2008;Sharples et al., 2007;Sherwin, 1988). Historically, microstructure derived observations of  have often been targeted toward regions known to exhibit specific processes, such as internal waves (Moum et al., 2003), gravity currents (Kilcher & Nash, 2010) or indeed boundary layer processes (Rippeth et al., 2001;Simpson et al., 1996). This is due, at least in part, to an a priori expectation that observing the turbulence is key to understanding the dynamics of these processes, and how they are coupled to the larger scale mean flow. Microstructure surveys spanning larger horizontal scales are rare (Polzin et al., 1997;Vic et al., 2018Vic et al., , 2019, although the advent of microstructure equipped ocean gliders is beginning to address this limitation by extending both the duration and extent over which shear microstructure can be observed (Fer et al., 2014;Lucas et al., 2019;Palmer et al., 2015;Schultze et al., 2017). Due to the temporal and spatial restrictions related to microstructure observations, previous studies that investigate how microstructure derived turbulence varied across large spatial areas have consisted of a synthesis of previous observational campaigns, for example, (St. Laurent & Simmons, 2006;Waterhouse et al., 2014). The studies of St. Laurent and Simmons (2006) and Waterhouse et al. (2014) both investigated global patterns of turbulent mixing using the pre-existing databases of  measurements available at the time. Both of these studies focused on the deep ocean, purposefully excluding shelf sea regions. Here we draw together 18 shelf seas microstructure data sets, comprising in a total of more than 3,700 profiles INALL ET AL.   Table 1. Blue line shows 300 m depth contour. from the Northwest European shelf (Figure 2). This presents the first opportunity to investigate spatial patterns of turbulence across a wide shelf sea environment.
These 18 observational data sets from the Northwest European shelf are compared with two separate formulations of internal tide forcing computed using output from a 1.8 km resolution numerical model (Guihou et al., 2018). We view the resulting relationship through the lens of a commonly used pycnocline parameterization scheme in order to explore the relationship between internal and external (lower boundary) frictional energy losses. In all that follows, we stress the focus on spatial variability and energetic relationships averaged over time (tidal cycle) applicable to the stratified summer period.

Numerical Formulation of Baroclinic Forcing Terms
To place the 18  data sets within a common dynamical framework we use output from a three-dimensional hydrodynamic ocean simulation with a 1.8 km (  1 / 60 ) horizontal resolution, using the Nucleus for European Modeling of the Ocean (NEMO) framework based on v3.6. This NEMO configuration, AMM60, has 51 terrain following (s-sigma) vertical levels, is forced by ERA-Interim atmospheric forcing, TPXO7.2 tides and a North Atlantic NEMO configuration at the lateral boundaries. For details, see Guihou et al. (2018). AMM60 was the developmental precursor to FOAM AMM15, which in 2021 is the current UK Met Office operational model for the NW European Shelf Seas (Graham et al., 2018;Tonani et al., 2019).
Output from the AMM60 model (Guihou et al., 2018) is used to quantify the barotropic tidal forcing of the internal wavefield across the full region covering all microstructure surveys. Two linear forms of a barotropic to baroclinic tidal energy conversion term are implemented. The first approach mirrors that of Waterhouse et al. (2014), who reported a positive linear relationship between the tidal conversion from barotropic to baroclinic wave energy and observed turbulent kinetic energy dissipation derived from shear microstructure profiles in the deep ocean. The second approach we take is perhaps more suited to shelf seas, but we retain both approaches in part to allow an intercomparison between approaches and with the deep ocean results from Waterhouse et al. (2014).

Barotropic Form Drag
The first of the two conversion formulations used here is based on the macro-scale properties of total water depth, bathymetric slope, density stratification, and mean horizontal tidal currents. Following Green and Nycander (2013) we define a stress vector describing the tidal conversion as Where  0 is a reference density, u is the barotropic tidal velocity vector and c( , ) x y is the internal wave drag tensor (with units 1 ms ). We assume that  wave and u are parallel and can therefore replace c with a scalar coefficient formed according to the method of Zaron and Egbert (2006) and given by Where  is a scaling factor used to compensate for unresolved topography due to the horizontal resolution of the numerical model, h is the total water depth (positive) and  is the tidal frequency. The stratification terms are formed by assuming horizontally homogeneous stratification given by  0 ( ) ( / ) n n z n exp z l where n l is a vertical decay constant and 0 n is a background reference stratification. b n is then ( ) n z evaluated at the seabed   z h, and    0 [1 ( / )] / n n n l n exp h l h is the vertical average of ( ) n z . Further details of the application of these constituent parameters within this study can be found in Section 3.4 The dissipation of barotropic tidal energy per unit area as a result of the generation of internal waves over topography is then given by We use "P" here to describe the production of energy from tidal conversion, rather than "D" as in Green and Nycander (2013), in order to distinguish between the dissipative drag due to tidal conversion they describe, and pycnocline dissipation in this work, that is, observed turbulent kinetic energy dissipation.

Baines-Type Baroclinic Forcing
The second of the two conversion formulations used here is computed directly as a function of the vertical movement of isopycnal surfaces, under the influence of a barotropic tide over variable bathymetry. Following the philosophy of Baines (1982) and methodologies of Kang and Fringer (2012) and Fer et al. (2015), the barotropic to baroclinc conversion is given by: for  0 z at the surface and    0 z H at the bed, and where tilde variables represent time varying fields that are reconstructed from harmonic species, as follows. W is the barotropic vertical velocity induced by barotropic horizontal tidal flow over an uneven bathymetry: In this study, the density term   is reconstructed using simulated harmonics to ensure only harmonic contributions from the full hydrodynamic model are evaluated. The density term is inferred from harmonic vertical oscillations as follows. For a harmonic species (denoted with subscript  and of frequency   ) the vertical harmonic displacements are computed from the harmonic vertical velocity,  w (written in complex notation): Then, the corresponding density harmonic is given by These harmonic components are summed, in the usual way, to give   . Finally, in Equation 4, following (Guihou et al., 2018), the angle brackets ( . )   denote a Doodson filter (Doodson, 1921;IOC, 1985;Pugh, 1996) which is applied to hourly fields to remove the dominant diurnal and semi-diurnal tidal species.
Note that in this analysis, in contrast to the Zaron and Egbert (2006) formulation, the polarity of w P is determined by the phase relationship between density and vertical velocity fluctuations. Negative values of w P represent conversion from barotropic to baroclinic motions, that is, the barotropic vertical component of flow over a sloping bed being locally in phase with (or indeed generating) baroclinic vertical motion. Positive values of w P represent conversion from baroclinic motions to barotropic flow, for example, the damping of remotely generated internal waves by an out of phase, locally generated barotropic vertical component of flow over a local sloping bed.

Observational Surveys
The 18 shear microstructure and temperature data sets used in this study were collected over a 17 year period from 1996 to 2013, and span a large area of the North West European shelf, as illustrated in Figure 2. The surveys used four different instrument types, all equipped with airfoil velocity shear microstructure probes (Osborn & Crawford, 1977). Three of the types used were free falling: the FLY profiler (Dewey et al., 1987;Rippeth et al., 2003;Simpson et al., 1996), the MSS microstructure profiler (Prandke & Stips, 1998) and the VMP profiler . The third platform type was a Slocum glider fitted with a Rockland Scientific Instruments Microrider (OMG) (Fer et al., 2014;Palmer et al., 2015). All of the surveys were undertaken in summer during periods of well-developed seasonal stratification, as demonstrated in Figure 3. All data sets have a total duration exceeding that of a semi-diurnal tidal period, and a sampling resolution of at least six profiles per hour. All data sets were interpolated (if required) onto a 1 m vertical grid. The majority of these data sets have been the subject of previous publications as detailed in Table 1. Those denoted with the D340 prefix and the OMG JC88 data are presented here for the first time. The D340 data sets were processed following established techniques (Prandke & Stips, 1998)  Journal of Geophysical Research: Oceans was processed following . Although there are important subtleties in the three different processing methods, they all rely on the fundamental assumptions that the turbulence is isotropic and a relationship between microscale velocity shear   u z and dissipation  is given by 10.1029/2020JC016528 6 of 22 where  is the kinematic viscosity of seawater. In practice, the mean shear squared term defined in Equation 8 is calculated via integration of the shear power spectrum between two wave number bounds. Wave number spectra are derived from shear time series, making a frozen field assumption. The lower and upper wave number bounds are chosen to represent the portion of the shear spectrum that can be realistically resolved by the shear probes, typically between 2 cpm and 30-50 cpm (cycles per meter) (Rippeth et al., 2003).

Pycnocline Selection Criteria
Analysis of the microstructure data set is focused on the pycnocline, so a definition must be made for a vertical region of the water column that is energetically disconnected from upper and lower boundary layer turbulence. Reliable salinity data are not available for every data set so the assumption is made that temperature serves as a reliable proxy for density. This is supported by available salinity data, and the INALL ET AL.  Table 1 Microstructure Survey Location Metadata assumed lack of salinity control on the density structure, given both the geographical locations far from riverine inputs, and summertime conditions of all of the surveys. Hereafter, the terms pycnocline and thermocline are interchangeable. We therefore choose temperature criteria to define the pycnocline region from which  populations are drawn.
The vertical structure of conservative temperature (McDougall & Barker, 2011) during each survey is shown in Figure 3. During all of the survey periods, the water column was persistently stable with respect to temperature, exhibiting a clearly identifiable thermocline. For each vertical temperature profile within each survey ( ) T z , the upper and lower boundaries of the pycnocline are defined as upper z and lower z , where and min T and max T are the minimum and maximum temperatures from each profile. The upper and lower depth bounds that result from this criterion are shown as white solid lines in Figure 3.

Population Statistics and Survey Mean Values of Dissipation Rate
In order to compare microstructure derived turbulence metrics with the two forcing terms (computed for a given time and location), a single value of  that best represents a temporal average must be chosen. However, there exists an inherent difficulty in doing so given that survey-wide values of  are highly intermittent, and frequently span more than three orders of magnitude. In response to this challenge we follow a number of previous authors, as described in Lozovatsky et al. (2015), in choosing to view the pycnocline 1 m binned  values as statistical populations. Histograms of  values for which the pycnocline selection criterion described above are satisfied are shown in Figure 4. This method of selection provides large populations of  values, as a value for each 1 m depth bin within the pycnocline is available. In order to relate turbulence levels to baroclinic tidal energy conversion, expressed as a vertical integral, pycnocline integrated values of  are similarly vertically integrated, Also shown in Figure 5 are two representations of population averages; arithmetic means,  10( ) pyc log  and geometric means, ) 10( pyc log  . Viewing the pyc  populations in this way confirms that most of the  populations integrated over the pycnocline exhibit a strong tendency toward log-normality. Applying a one-sample Kolmogorov-Smirnov test, only four of the 18 data sets reject the null hypothesis that the logarithm of the data comes from a standard normal distribution, against the alternative that they do not come from such a distribution (at a 5% significance level), shown in Figure 5. This strong tendency toward log normality has been found in many previous studies of turbulent dissipation derived from both temperature microstructure (Gregg, 1980;Gregg et al., 1973;Washburn & Gibson, 1984) and shear microstructure (Belyaev et al., 1975;Crawford, 1982;Oakey, 1985;Osborn, 1978;Osborn & Lueck, 1985;Palmer et al., 2015;Thorpe et al., 2008). The pyc  populations all exhibit deviations at their upper and lower bounds, which a GEV model does a better job of capturing, also found by Lozovatsky et al. (2015). The presence of a small number of high pyc  values within each population is evident in the differences between the geometric and arithmetic means, with the former being significantly smaller in each case. Overall, this statistical analysis provides assurance of the data quality across the wide variety of data sets used, demonstrating also that all  distributions lie well above instrument detection limits of between ∼   values, we choose the geometric mean with which to compare to the macro scale barotropic to baroclinic conversion terms. For completeness, the same analysis was undertaken using the arithmetic mean, with the difference not changing the overall conclusions as detailed in Section 3.5.
Finally, in order to compare the observed mean dissipation rates with the tidal forcing terms (following, e.g., Waterhouse et al., 2014), an adjustment is made to account for the fraction of turbulent energy already converted to increased water column potential energy, where D  is the total energy dissipation rate within the pycnocline and ˆp yc  is the geometric mean of pyc  .
 is the proportion of energy that acts to change the potential energy of the water column through mixing,   (1 ) is the proportion that dissipates as heat, and is the proportion actually observed by the shear microstructure method. A canonical value of  0.2 (Osborn, 1980) is used. A sensitivity analysis is presented in Section 3.5, where upper and lower bounds on D  are computed using a range of values for    0.05 0.25, demonstrating a relatively small impact on the final comparison between conversion and dissipation.

Simulation Derived Barotropic to Baroclinic Energy Conversion
As described in Section 2, the two baroclinic forcing terms are computed using output from the high resolution AMM60 NEMO configuration covering the North West European Shelf and Atlantic margin (Guihou et al., 2018). The model output used for this study is centered around August 24, 2012. The reason for selecting one particular summertime period, rather than the actual times of each observational data set is rather prosaic: model output for the time period of the earliest observational data sets is not available. Nevertheless, since we are dealing with a system dominated by tidal currents and seasonal stratification, both of which are largely deterministic and repetitive, the use of a representative summer period, though a nonideal necessity, is deemed necessarily informative for our purposes since spatial distribution and range of forcing values is the focus. To demonstrate this, profile comparisons between the 1m depth binned survey averaged observed stratification and co-located 5-day mean modeled stratification centered on the 24th of August 2012 are shown in Figure 6 for each data set. Buoyancy frequency is computed from temperature profiles with a constant absolute salinity value of 35 for the observations, and both for this same constant salinity value and the actual modeled salinity for the model output. Broadly speaking, the modeled summer 2012 stratification matches that observed. There are of course differences between model and observed stratification, particularly severe at site D340BH, which is close to the poorly resolved topography of Barra Head. An investigation into the sensitivities of AMM60's ability to reproduce observed stratification is presented elsewhere (Luneva et al., 2019), and is not the focus here.
Values of the Baines type forcing, w P , are computed "offline" using tidal harmonics of density and velocity fields. Computing the barotropic form drag term, ZE P , requires the constituent terms describing the tidal velocity, the bathymetric gradient, the stratification and a scaling factor . The tidal velocity vector U , is the harmonically derived current amplitude of the M2 tidal constituent. The bathymetric gradient term is computed using the native horizontal resolution of the model bathymetry. In terms of stratification, we follow Green and Nycander (2013) in applying a vertical exponential profile, and determine the reference stratification 0 N , and decay length scale, Ln appropriate for this shelf seas application. These terms are derived by computing profiles of means are shown in red dashed and red solid vertical lines, respectively. A fitted normal and generalized extreme value distribution are shown in black and blue, respectively. Each panel is annotated with whether the null hypothesis that data comes from a standard normal distribution, against the alternative that it does not come from such a distribution, is accepted or rejected at a 5% significance level, using a one-sample Kolmogorov-Smirnov test.

Journal of Geophysical Research: Oceans
This value for  accounts for the ratio of the horizontal resolution of the AMM60 model (  1 / 60 ) to the  1 / 8 resolution of the model used in Green and Nycander (2013), where they apply   50. Histograms of ZE P and w P (Figure 7) show great similarity in shape, with the distribution of ZE P offset toward smaller values than w P by a factor of approximately two. The value for , a free parameter, is then tuned so that both populations align, giving a tuned value of   2 100 / (7.5 ). This re-tuning of ZE P is returned to in the discussion.
Maps of both w P and  _ ZE w P , referred to hereafter simply as ZE P , for regions within the model domain with total water depth shallower than 300m are shown alongside a data density binned scatter plot in Figure 8. Conversion values for later comparison with ˆp yc  of both w P and ZE P are then extracted, and arithmetically averaged within a radius of 5 km from the location of each of the observational  data sets.

Conversion Rates Versus Pycnocline-Averaged Dissipation Rate
The model derived tidal energy conversion terms, w P and ZE P , both demonstrate a positive and approximately linear relationship with the observationally derived dissipation term D  , in loglog space. A linear regression model is used to determine the gradient of the relationship between tidal conversion and pycnocline dissipation in loglog space. The resulting best fit lines in linear space relate to:   8 10 Wm , increasing as the energy entering the baroclinic wavefield increases further, following a power law of approximately one-third. This is an interesting result, and suggests that at higher conversion rates, the energy flux divergence of the internal wavefield due to energy dissipated within the pycnocline does not "keep-up" with increasing energy input into the internal wavefield.
Also noteworthy is the statistically significant result that pycnocline dissipation is higher than estimated conversion at low conversion rates (and, conversely, lower at high conversion rates as noted). This is consistent with the notion of an omnipresent internal wave "background" energy level, indicating the influence of other energy sources such as the wind or remotely generated internal waves. Locally, we may therefore expect that in low conversion regions, dissipation levels measured in the pycnocline will be greater than the local generation rate, because of wind or baroclinic energy radiating into the measurement zone from non-local sources.
Finally, with reference Figure 9, integrated over the full range of conversion space, that is from   5 1 10 to    3 2 9 10 Wm , pycnocline integrated  accounts for only ∼25% of conversion.

Pycnocline Versus Bottom Boundary Layer Dissipation
To examine in more detail the apparent one-third power law relationship between tidal energy conversion and pycnocline integrated dissipation, and the overall ∼75% dissipation deficit (Figure 9), we look first with-INALL ET AL.

10.1029/2020JC016528
14 of 22 in the baroclinic wave energy budget. The most obvious candidate mechanism is that of local wave energy dissipation within the bottom boundary layer. Since dissipation in the bottom boundary layer is known to exceed pycnocline dissipative energy losses in nonlinear internal waves (NLIWs), and more generally in internal tides Inall et al., 2000). Does the internal wave energy lost to bottom friction increasingly dominate the internal wave energy budget as wave amplitude, bc A increases? This is a reasonable question to ask, since bc A is expected to increase with increasing conversion rate (unless the wavefield is amplitude saturated), though no simple expression directly relates bc A to conversion.
Tidal conversion puts energy into baroclinic motions, in which turbulence may dissipate energy within a sheared pycnocline and within a turbulent bottom boundary layer through frictional bottom boundary drag acting on the near bed velocity. Full-depth turbulence observations are not available for many of the data sets. Even in data sets which do fully capture the bottom boundary layer, separate attribution of observed bottom boundary turbulence to barotropic tidal flow and to that generated by baroclinic motions is nontrivial, see for example the discussions in Inall et al. (2000) and Inall and Rippeth (2002). Barotropic and baroclinic tides, by their very nature, are phase locked with the same fundamental frequency, but their phase difference is spatially varying due to the large difference in wavelength of barotropic and baroclinic tides (factor of around 20, varying with stratification). This picture of spatial phase differences is complicated by time variation in barotropic forcing (e.g., spring/neap cycle) which may result in remotely forced baroclinic energy phase-shifted from the local barotropic signal , which in turn may result in more energetic baroclinic waves at a neap tide, rather than a spring tide, for example, Inall et al. (2000). A further complication may result from storm-forced variations in stratification which have been show to modify baroclinic wave energy flux across a wide shelf . For all of these reasons, and possibly others, it is not possible to look to the  observations or AMM60 estimations of  to determine the relative dissipative contributions of pycnocline and bottom friction as a function of local baroclinic wave amplitude, bc A . We can, however, turn to some commonly used parameterisations of pycnocline dissipation to examine this question.

Parameterized Models of Boundary and Internal Dissipation Rates
The simplest approach is to first consider the relative scaling of internal and boundary friction. The latter, as demonstrated for example in Simpson et al. (1996), is accurately approximated as Where  is near bed density and d C is a turbulent drag coefficient, usually taken to be   3 2.5 10 . Treating the water column initially as a two layer fluid with the upper and lower layers of thickness, U H and L H , which are later estimated from the vertical position of the maximum in the first mode vertical velocity structure, upper and lower layer baroclinic velocity amplitudes are related to bc A by Where bc A is the first mode internal wave amplitude, and U U and L U the upper and lower layer baroclinic velocities, and bc c the wave phase speed (which later is also determined from the internal wave eigenvalue problem). Energy dissipation in the bottom boundary layer therefore scales as  3 BBL bc A  . Internal wave shear, S, at the interface scales linearly as  bc S A , where with Z a finite measure of pycnocline thickness. Various empirically derived fine-scale parameterizations of pycnocline dissipation rate have been proposed in the literature. Here, we examine three commonly used versions, as discussed in Palmer, Polton, et al. (2013): denoted Gregg (Gregg, 1989); KWB (Kunze et al., 1990); and MG (MacKinnon & Gregg, 2003). The Gregg parameterization is defined as Where  G scales Gregg  to best match observed values, 0 N represents background levels of pycnocline N and angled brackets denote temporal averaging. GM S is the Garrett and Munk (1975)  internal wave shear spectrum, which as shown by Gregg (1989) can be estimated as a function of the local stratification, given by .
Where fr represents the fraction of the water that is thought to be gravitationally unstable. The z term is defined in Kunze et al. (1990) to be the region of the water column where  2 S N that is, . Finally, the MG parameterization is defined as where  MG is another free scaling parameter, and 0 S represents background levels of pycnocline S.
Reference is made here to the Gregg and KWB scalings for context, but are excluded from more detailed analysis for the following reasons. Both the Gregg and KWB scalings rely on resolving higher mode waves, while our analysis utilizes the first mode solutions only. The Gregg parameterization explicitly excludes the tidal contribution, the focus of this study, and the KWB scaling relies on an ability to resolve to a critical Richardson number, which is not possible with the methods here. Furthermore, for a given stratification, , it is evident that only the MG parameterization has a lower power law scaling for pycnocline dissipation compared to BBL dissipation, that is, MG has pycnocline dissipation linearly proportional to shear. KWB has cubic power dependence for pycnocline dissipation, same as BBL dissipation, while Gregg has a forth power dependence. On the basis of these notes, we would anticipate that only the MG parameterization will mirror the one-third power law behavior seen between observed dissipation and tidal conversion.
To make a direct comparison between BBL versus pycnocline dissipation and tidal conversion (rather than the baroclinic wave amplitude, as above), the MG mixing parameterization scheme is evaluated within an iterative approach to estimate the predicted values of pycnocline and bottom boundary layer turbulent dissipation rates for each of our survey locations. In order to compute the M2 tidally averaged layer-wise velocities, first mode internal wave eigenvalue solutions are computed to give the phase speed, bc c (solving the Taylor-Goldstein equation) using the same AMM60 output presented earlier used in computing the tidal conversion terms. Velocity shear is then computed as The stratification term is taken as the maximum value of buoyancy frequency, N, found within each of the modeled density profiles. For consistency (MacKinnon & Gregg, 2003) we apply regionally appropriate scaling constants This approach allows the first mode internal wave amplitude, bc A , to be determined iteratively for each data set as follows: The total internal wave dissipative energy loss may be expressed as a function of internal wave amplitude (  Figure 10. In essence, this method is used to reveal the partition, as a function of tidal conversion, between total TKE dissipation in the bottom boundary layer (given by a cubic dissipation law) and in the pycnocline as given by the MG parameterization. If pycnocline dissipation scales as the lower layer velocity (as in the MG parameterization), then one anticipates a one-third power law relationship between pycnocline dissipation and conversion. This is very nearly the case with a gradient of parameterized pycnocline dissipation to tidal conversion of 0.4. We acknowledge that the choice of scaling factor applied in the MG parameterization in Equation 16, may lead to some of the disparity between this and our observed D  . The absolute value of pycnocline integrated dissipation derived from this parameterization is much lower than observed, but an absolute comparison is not our focus. Rather we are interested in the one-third scaling with conversion which is invariant to choice of the scaling factor.

Discussion
The positive relationship between pycnocline integrated  and both tidal conversion estimates is perhaps unsurprising, though it is noteworthy as a general observation encompassing a large number of independent data sets covering a broad geographic range and a correspondingly wide range of dissipation and conversion values. That the relationship is not one-to-one does suggest that the concept of shelf seas internal wavefield being in some sense "saturated" (e.g., Sherwin, 1988;Thorpe & Liu, 2009) may be more nuanced, and strongly dependent on external interaction of the internal wavefield with a boundary. In that last statement, we interpret "saturated" to mean that the local rate of energy input into the internal wavefield equals the local rate of energy loss with minimal or no local growth in wave amplitude, noting also that the term "saturated" does not have a consistent definition in the literature.
If internal wave energy loss (dissipation) is occurring only in the stratified portion of the water column (e.g., as implicit in Thorpe & Liu, 2009), then increased energy input (i.e., conversion) would scale linearly with wave energy loss within the stratified portion of the fluid, which is not as observed here. The example given in Thorpe and Liu (2009) most closely related to internal waves in a shelf sea environment is the stratified and tidally swept Clyde Sea. Using an inviscid interpretation, they suggest the system is highly unstable (saturated, in some sense), and yet it has been demonstrated that internal wave energy loss there is dominated by friction in the bottom boundary . It is unclear how the interpretation of stability and saturation would change if the inviscid assumption were relaxed.
That we do see pycnocline dissipation increasing monotonically, but not proportionally with conversion is, however, consistent with the notion that the shelf seas pycnocline is maintained in a continual state of marginal stability (and by that we mean a bulk Richardson Number 1), by the BBL and/or wind/convection (Lincoln et al., 2016). Following this line of reasoning, even a small amount of additional energy given to the internal tide (i.e., greater conversion) gives rise to increased wave amplitude and therefore greater shear instability (and enhanced dissipation) internally and at the boundary, draining energy directly from the internal tide to mixing (change of water column potential energy) and to heat. The additional result here that pycnocline integrated  is higher than estimated conversion in locations of low conversion rates is also consistent with the notion of marginal stability: in regions of locally lower conversion, dissipation exceeds INALL ET AL. the energy locally entering the baroclinic wavefield because of the ubiquitous background baroclinic energy density from energy radiating into that region from non-local sources.
One of the two conversion estimates, ZE P , contains a free tuning parameter, . Using the deep ocean tuning (Green & Nycander, 2013), appropriately adjusted here for differing model resolution, results in ZE P values that are systematically smaller than w P (Figure 7). For application to the NW European shelf seas w P is used to provide a new tuning for ZE P , by adjusting  to force ZE P to match w P . This approach is justified on the grounds that ZE P contains a free parameter and w P does not, and there is no way to assess errors associated with the directly diagnosed w P . Tuned in this manner, ZE P provides a relatively simple method to calculate shelf seas tidal conversion using only knowledge of topography, barotropic tide and stratification, without recourse to a 3D, high resolution, baroclinic hydrodynamic numerical model.
Tuning  does not affect the power law relationship with pycnocline integrated  , and both conversion formulations exhibit similar power law relationships to pycnocline integrated  . Both exhibit a gradient on a loglog scale of  0.3 which in linear space relates to  1/3 Conversion  . This is an important result, though we are cautious in inferring anything about both conversion estimates having the same power law, since they are not completely independent (both using AMM60 velocity and stratification fields), and we acknowledge we are unable to assign error estimates to either of the conversion estimates.
The approximate one-third power law, shown to be robustly less than one, states that for every factor of 10 increase in barotropic energy conversion (perhaps near some steep topography, or region of strong barotropic tide), will result in only an approximate doubling in pycnocline dissipation (and hence associated vertical mixing and vertical heat/nutrient fluxes). This does not necessarily suggest a less-than-expected change in energy flux divergence in the baroclinic wavefield; just that we do not see a one-to-one relationship between change in energy input to the baroclinic tide (i.e., conversion), and a change in energy dissipated within the thermocline. It is also acknowledged that there is considerable scatter to the observations, and the observed power law fit could be between one-quarter and one-half. For example, observing that ZE P is proportional to  2 ( ) H a one half power law would be consistent, to first order, with dissipation varying with H, which is not unreasonable given that internal tide amplitude will scale as the dot product of barotropic tidal velocity and the local bathymetric slope. This reasoning, though, speaks only to the source of the energy conversion, it does not address wave energy flux divergence partitioning between pycnocline and other forms of dissipation, for example that occurring in the bottom boundary layer. It should also be re-stated that wind driven inertial energy has not been considered, and both conversion estimates are linear, that is, they do not account for supercritical flow over topography.
The approximate one-third power law relationship raises questions about possible mechanisms for dissipating the "excess" baroclinic energy conversion compared with pycnocline dissipation noted at higher conversion rates. BBL dissipation was selected and evaluated as the primary candidate mechanism. There are though (at least) three candidate processes, the second and third of which deserve some comment: 1. Local BBL dissipation. Supported by previous work (Inall et al., 2000;MacKinnon & Gregg, 2005;Rippeth, 2005) showing between  60 80% of IW energy to be dissipated in the BBL; 2. Remote internal wave breaking, or energy absorption into shelf seas fronts; 3. Nonlinear interaction with barotropic tides.
An explanation that invokes remote dissipation must counter the criticism that any point in the shelf seas will contain both locally and remotely generated internal waves, as we have illustrated with pycnocline integrated  exceeding conversion in low conversion locations. Remote dissipation hot spots, such as shoaling topography and fronts between stratified and well mixed water remains remain free of this criticism. Since BBL dissipation associated with NLIWs is known to exceed that in the pycnocline (Inall et al., 2001) and no measurement in our database were collected in fronts or over steep slopes, we have focused our attention on BBL dissipation.
There is a fundamental issue in trying to separately attribute dissipation in the BBL arising from internal tides and that arising directly from the barotropic tide. This issue is particularly acute when barotropic tidal velocity magnitude is similar to the internal tide induced velocity, as is the case on the NW European shelf. Barotropic and baroclinic tides are phase locked at any given location, their velocities above the boundary INALL ET AL.

10.1029/2020JC016528
18 of 22 will constructively or destructively interfere (or anything in between) in a consistent fashion at any given location (see discussion in Inall et al., 2000). The cubic dependence of BBL dissipation on near boundary velocity will therefore give rise to nonlinear, spatially varying interactions between barotropic and baroclinic signals, even on a flat seabed. The introduction of spatially varying topography further complicates the picture. The overall notion, therefore, is that dissipation in the BBL caused by barotropic and baroclinic motions is intrinsically inseparable. For example, high conversion rates are associated with large barotropic tidal velocities, and thus a cubic increase in BBL dissipation. This in turn might be viewed as creating a more viscous lower boundary over/through which the internal wave motions must propagate. This may consequently damp the internal tide/wavefield in a nonlinear fashion, thereby reducing wave amplitude, shear and pycnocline mixing. This line of reasoning, though somewhat speculative has, to the best of our knowledge received little attention in the literature and is mentioned in only a small number of studies (e.g., Inall et al., 2000;MacKinnon & Gregg, 2005). Bearing this caveat in mind, we have nonetheless considered the baroclinic motions in isolation of the barotropic tidal velocities, leaving analysis of their interaction for future study.
The simple scaling arguments of Section 3.6 suggest that the observed power law (Figure 9) is consistent with a linear dependence between pycnocline dissipation and baroclinic shear. It follows that an overall balance between conversion and dissipation is possible and consistent with this broad collection of 18 observational data sets of pycnocline dissipation. As already noted, the NW European shelf sea is often considered to be in a general state of marginal stability (with respect to a bulk Richardson Number -noting this to be a generalized statement, and stability thus defined will vary greatly in time and space). The success of the MG scheme in reproducing the observed power law dependence of pycnocline dissipation on tidal conversion is consistent with pyc  scaling with the product of N and S, in the sense that the shelf sea pycnocline sits at all times close to bulk Ri ∼ 1. Thus additional input in S will increase mixing, and any increase of N will result in greater baroclinic conversion and hence increased mixing (rather than stabilization of an unforced system).
As a final point of discussion, the one-third relationship reported here is different to the generalized relationship found by Waterhouse et al. (2014), who report  Conversion  . However, when Waterhouse et al. (2014) extract just internal tides (i.e., baroclinic conversion forcing) they find what appears to be a similar one-third power law (see left panel of Figure 4 in Waterhouse et al., 2014). This is a surprising observation. It is improbable that in the open ocean baroclinic tides dissipate largely through bottom friction, disproportionately increasing as a function of energy conversion into the baroclinic wavefield. A more likely interpretation is that the similarity in slope is a coincidence, and that the deep ocean sub-unity gradient reflects the widely accepted notion that the majority of deep ocean internal wave energy is dissipated at the ocean boundaries, including the marginal shelf seas, or lost to other processes, for example, acceleration of mean flow through wave-current interaction.

Conclusions
Whichever way one views the discussion above, we are left with two robust statements: (a) pycnocline dissipation is less than conversion at high local conversion rates and greater than for low local conversion rates; (b) the scaling of local pycnocline dissipation to local conversion rate follows an approximately onethird power law. Further, we suggest that these statements are consistent with an overall balance between conversion and dissipation only if one considers wave-induced dissipation within the BBL. At low conversion rates, local dissipation may exceed local conversion due to a remotely generated background baroclinic wave energy density, or a contribution from another source, that is, the wind. As conversion increases, there is a proportionate rise in the flux divergence of internal wave energy through increased BBL dissipation. However, as conversion increases there is not a proportionate rise in the flux divergence of internal wave energy through internal friction. Therefore, local diapycnal mixing does not increase linearly with tidal conversion, but rather with an approximately one-third power law. Such a simple algebraic relationship between conversion and dissipation, coupled with a straightforward method to compute conversion based only on topography, stratification and barotropic tide represents a new parameterization of diapycnal mixing in stratified shelf seas, applicable at least to the broad, tidally swept NW European Shelf.