Internal tide energy flux over a ridge measured by a co-located ocean glider and moored ADCP

Internal tide energy flux is an important diagnostic for the study of energy pathways in the ocean, from large-scale input by the surface tide, to small-scale dissipation by turbulent mixing. Accurate calculation of energy flux requires repeated full-depth measurements of both potential density (ρ) and horizontal current velocity (u) over at least a tidal cycle and over several weeks to resolve the internal spring-neap cycle. Typically, these observations are made using full-depth oceanographic moorings that are vulnerable to being ‘fished-out’ by commercial trawlers when deployed on continental shelves and slopes. 5 Here we test an alternative approach to minimise these risks, with u measured by a low-frequency ADCP moored near the seabed and ρ measured by an autonomous ocean glider holding station by the ADCP. The method is used to measure the M2 internal tide radiating from the Wyville Thompson Ridge in the North Atlantic. The observed energy flux (4.2±0.2 kW m−1) compares favourably with historic observations and a previous numerical model study. Error in the energy flux calculation due to imperfect co-location of the glider and ADCP is estimated by sub-sampling 10 potential density in an idealised internal tide field along pseudorandomly distributed glider paths. The error is considered acceptable (<10%) if all the glider data is contained within a ‘watch circle’ with a diameter smaller than 1/8 the mode-1 horizontal wavelength of the internal tide. Energy flux is biased low because the glider samples density with a broad range of phase shifts, resulting in underestimation of vertical isopycnal displacement and available potential energy. The negative bias increases with increasing watch circle diameter. If watch circle diameter is larger than 1/8 the mode-1 horizontal wavelength, 15 the negative bias is more than 3% and all energy fluxes within the 95% confidence limits are underestimates. Over the Wyville Thompson Ridge, where the M2 mode-1 horizontal wavelength is ≈100 km and all the glider dives are within a 5 km diameter watch circle, the observed energy flux is estimated to have a negative bias of only 0.5% and an error less than 4% at the 99% confidence limit. With typical glider performance, we expect energy flux error due to imperfect co-location to be <10% in most mid-latitude shelf slope regions. 20 Copyright statement. The works published in this journal are distributed under the Creative Commons Attribution 4.0 License. This license does not affect the Crown copyright work, which is re-usable under the Open Government Licence (OGL). The Creative Commons Attribution 4.0 License and the OGL are interoperable and do not conflict with, reduce or limit each other.


Introduction
Internal tides are a ubiquitous hydrodynamic feature over continental shelves and slopes as they are commonly generated at the shelf break by across-slope tidal flows (Baines, 1982;Pingree et al., 1986;Sharples et al., 2007). However, direct measurement of internal tides can be a challenge in these regions due to intense commercial fishing activity leading to an increased risk of oceanographic mooring loss (Sharples et al., 2013). Calculation of internal tide energy flux, a key diagnostic 5 for the understanding of baroclinic energy pathways, requires repeated full-depth measurements of both potential density (ρ) and horizontal current velocity (u) over at least a tidal cycle (Nash et al., 2005). If an objective is to resolve the internal springneap cycle or observe the effect of seasonal changes in stratification on the internal tide field, repeated full-depth measurements over several weeks or months are required. Typically, these measurements are made using a full-depth oceanographic mooring incorporating an acoustic Doppler current profiler (ADCP) and a string of conductivity-temperature loggers (e.g., Hopkins 10 et al., 2014), or a profiling mooring with a CTD and acoustic current meter (e.g., Zhao et al., 2012). On continental shelves and slopes, these full-depth moorings are vulnerable to being 'fished-out' by demersal and pelagic trawling activity. Hall et al. (2017b) describe a novel alternative approach to minimise these risks, with u measured by a low-frequency ADCP moored near the seabed and ρ measured by an autonomous ocean glider holding station by the ADCP as a 'virtual mooring'.
Commercial fishing activity on continental shelves and their adjacent slopes is often intense because these regions are highly 15 biologically productive. However, steps can be taken to reduce the risk of ADCP loss, including deploying deeper than 600 m, keeping mooring lines short, or using trawl-resistant frames. Being relatively small, gliders are unlikely to be fished-out, and the risk can be further reduced by real-time evasive action in response to vessel proximity guided by the maritime Automatic Identification System (AIS). However, this alternate approach was not comprehensively tested by Hall et al. (2017b) because of glider navigation and telemetry problems. In this study we test the method using a co-located glider and ADCP dataset from the 20 Wyville Thompson Ridge in the North Atlantic, a topographic feature that previous observations and numerical model studies suggest is an energetic internal tide generator (Sherwin, 1991;Hall et al., 2011). We also estimate the error in the energy flux calculation due to imperfect co-location of the glider and ADCP and find that, with typical glider performance, it is acceptable in most mid-latitude shelf slope regions.
Ocean gliders have previously been used to observe internal waves and internal tides Rainville et al., 25 2013; Boettger et al., 2015;Hall et al., 2017a), including the calculation of energy fluxes using current velocity measurements from gliders equipped with ADCPs . However, ADCPs are not routinely integrated with commercially available glider platforms (Seaglider, Slocum, and SeaExplorer), in part due to their higher power requirement. Synergy with moored ADCP data allows accurate calculation of internal tide energetics without the endurance limitations and data analysis complexities of an ADCP-equipped glider (e.g., Todd et al., 2017). 30 In Section 2 the temporal resolution constraints of glider measurements are explained and the observations used in this study described. The calculation of internal tide energy flux from co-located glider and moored ADCP data is fully described in Section 3. Observations of the internal tide radiating from the Wyville Thompson Ridge are presented in Section 4 and compared with historic observations and a previous numerical model study. In Section 5 the error in the energy flux calculation due to imperfect co-location is estimated. Key results are summarised and discussed in Section 6.

Observations
High temporal resolution is crucial for internal tide observations; Nash et al. (2005) suggest a minimum of four evenlydistributed independent profiles of u and ρ are required per tidal cycle for an unbiased calculation of energy flux. Over conti-5 nental shelves and upper shelf slopes, this temporal resolution is achievable using gliders. Typical glider vertical velocities are 15-20 cm s −1 , so a complete dive cycle to 1000 m can take as little as three hours. This yields eight profiles (four dives) per semidiurnal (≈12-hour) tidal cycle, but near the surface and seabed the descending and ascending profiles converge in time so the number of independent samples is halved to four. For diurnal (≈24-hour) internal tides, 16 profiles per cycle are possible.
In shallower water the temporal resolution of glider measurements increases further; 40-minute full-depth dives are achievable The observations used to test the method were collected from the northern flank of the Wyville Thompson Ridge (WTR) in the North Atlantic (Fig. 1a). A Kongsberg Seaglider (SG613; Eriksen et al., 2001) was deployed from NRV Alliance between 2nd and 5th June 2017 during the fourth Marine Autonomous Systems in Support of Marine Observations mission 5 (MASSMO4). The glider was navigated from the deeper waters of the Faroe-Shetland Channel (FSC) to the WTR and held station for 40 hours by a short oceanographic mooring, deployed five days previously from MRV Scotia (Fig. 1b). The mooring was sited close to the 800 m isobath and instrumented with an upwards-looking 75 kHz RDI Long Ranger ADCP at approximately 722 m and an Aanderaa Seaguard acoustic current meter at 784 m, yielding observations of horizontal current velocity over 78% of the water column. When on-station by the ADCP mooring, the glider made repeated 2-hour dives to 700 m or 10 the seabed, whichever was shallower. This yielded approximately 12 profiles (six independent samples near the surface and seabed; 12 independent samples at mid-depth) per semidiurnal tidal cycle.
Glider location at the surface, before and after each dive, was given by GPS position. Subsurface sample locations were approximated by linearly interpolating surface latitude and longitude onto sample time. When on-station, the glider stayed within 2.5 km of the mooring and the mean horizontal distance between temporally coincident glider and ADCP measurements 15 was 1.3 km. This spatial scattering of the glider data is small compared to the semidiurnal mode-1 horizontal wavelength over the WTR (≈100 km, calculated from the observed buoyancy frequency profile) and so the glider data are initially considered a fixed-point timeseries with no spatial-temporal aliasing.
As the glider was on-station for only 40 hours, the co-located timeseries is not long enough to resolve the internal springneap cycle. As a result, M 2 harmonic fits to the glider and mooring data (Section 3) are contaminated with S 2 variability. To 20 acknowledge this, we refer to the estimated M 2 component of the co-located timeseries as D 2 following Alford et al. (2011).
The comparative numerical model (Section 4.1) only includes the M 2 tidal constituent so we refer to model diagnostics as M 2 .

Data processing
The glider was equipped with a standard Sea-Bird Electronics conductivity-temperature (CT) sail sampling at 0.2 Hz and the data processed using the UEA Seaglider Toolbox (https://bitbucket.org/bastienqueste/uea-seaglider-toolbox) following Queste 25 (2014). Conductivity data were corrected for thermal hysteresis following Garau et al. (2011) and the Seaglider flight model regressed using a method adapted from Frajka-Williams et al. (2011). As the CT sail was unpumped, salinity samples were flagged when the glider's speed was less than 10 cm s −1 or it was within 8 m of apogee 1 . Temperature-salinity profiles from descents and ascents were independently averaged (median value) in 5-m depth bins, typically with 4-5 samples per bin. Sample time was averaged into the same bins to allow accurate temporal analysis at all depths. Absolute salinity (S A ), conservative 30 temperature (Θ), and potential density (ρ) in each bin were calculated using the TEOS-10 equation of state (IOC et al., 2010).
1 Apogee is the phase of the dive between descent and ascent, when the glider pitches upwards and increases its buoyancy. Flow through the conductivity cell is unpredictable during this phase and so salinity spikes are common. The 75 kHz ADCP was configured in narrowband mode with 10-m bins and 24 pings per 20-minute ensemble. The ADCP data were processed using Marine Scotland Science's standard protocols, including correction for magnetic declination and quality assurance based on error velocity, vertical velocity, and percentage good ping thresholds. The ADCP data were then linearly upsampled onto the same ∆5-m depth levels as the glider data. The acoustic current meter was configured with a 10minute sampling interval and linearly downsampled onto the same 20-minute sampling interval as the ADCP. Good velocity 5 data were recovered for all depths levels between 85 m and 705 m, as well as 780-785 m. In addition to the ADCP/current meter measurements, horizontal velocity was inferred from GPS position and the Seaglider flight model using a dive-average current method (DAC; Eriksen et al., 2001;Frajka-Williams et al., 2011). DAC was only calculated for dives deeper than 500 m so that values were representative of the majority of the water column. All velocities were transformed into along-slope and across-slope components. We take the northern flank of the WTR to be orientated exactly northwest-southeast so along-slope 10 (u) is positive southeast and across-slope (v) is positive northeast (down-slope).
The full 3-day glider timeseries of conservative temperature and absolute salinity is shown in Figure 2a.  Mode-1 horizontal velocity, calculated from the observed buoyancy frequency profile, reverses at approximately 505 m, slightly above the pycnocline.

Internal tide energy flux
Following Kunze et al. (2002) and Nash et al. (2005), internal tide energy flux is calculated F = u bc p . The method requires repeated full-depth measurements of ρ and u over at least a tidal cycle in order to determine pressure perturbation (p ) and 20 baroclinic velocity (u bc ), respectively.

Pressure perturbation
For the 40-hour window when the glider was on-station by the ADCP mooring, potential density anomaly is calculated by subtracting the window-mean density profile from measured potential density, (1) 25 Before subtraction, ρ(z) is smoothed with a 50-m gaussian tapered running mean (σ = 10 m) to yield a suitable background density profile. Vertical isopycnal displacement is then calculated To separate D 2 internal tide variability from other physical processes, M 2 tidal period (T = 12.42 hours) harmonics are fit to ξ on each ∆5-m depth level following Emery and Thomson (2001). This analysis is only applied to depth levels between temporal resolution due to the glider going into apogee above 700 m. To obtain a full-depth timeseries, the D 2 component of ξ is linearly extrapolated assuming ξ = 0 at the surface (z = 0) and bottom (z = −H, where H is water depth). Buoyancy frequency squared, N 2 = −g/ρ 0 (∂ρ/∂z) is also linearly extrapolated, assuming N 2 = 10 −6 s −2 at the surface and bottom.
Pressure perturbation is then calculated by integrating the hydrostatic equation from the surface, where p surf is pressure perturbation at the surface due to the internal tide, determined by applying the baroclinicity condition for pressure,

Baroclinic velocity
For the same 40-hour window, horizontal velocity perturbation is calculated where u(z) is the window-mean horizontal velocity profile. There are three spatial gaps in the timeseries: above 85 m, between the ADCP and current meter (705-780 m including blanking distance), and from the current meter to the seabed (785-800 m).

5
To obtain a full-depth timeseries, u is linearly interpolated between the ADCP and current meter, and extrapolated to the surface and the bottom using a nearest neighbour method. Baroclinic velocity is then calculated where u bt is barotropic velocity, assumed here to equal the depth-mean velocity, calculated 10 The D 2 components of u bc and u bt are extracted using the same harmonic analysis method applied to ξ.

Internal tide energetics
Profiles of internal tide energy flux, available potential energy (APE), and horizontal kinetic energy (HKE) are calculated 20 where · denotes an average (mean) over an integer number of M 2 cycles and ρ 0 = 1028 kg m −3 is a reference density.

Results
Maximum D 2 vertical isopycnal displacement is 42 m and occurs at 565 m (Fig. 4a), within the main pycnocline. This is comparable with historic observations of a semidiurnal internal tide over the northern flank of the WTR. Sherwin (1991)  analysed CTD data from a 17-hour repeat station (30 minutes between casts) that was 6.7 km east of the mooring (Fig. 1b) and determined maximum D 2 vertical isopycnal displacement to be 37 m at 580 m, again within the pycnocline. Here, almost all APE is contained within the pycnocline (Fig. 4c), because maximum ξ occurs at a similar depth to maximum N 2 (4.9 × 10 −5 s −2 at 525 m). D 2 baroclinic velocity is maximum (≈20 cm s −1 ) near-bottom (Fig. 4a), as is HKE (Fig. 4c). Depthintegrated HKE and APE are 5.5 kJ m −2 and 1.7 kJ m −2 , respectively.

5
Both the across-and along-slope components of D 2 internal tide energy flux are maximum (≈12.7 kW m −2 ) near-bottom and go to zero at the depth of maximum N 2 (Fig. 4d), characteristic of a low-mode internal tide with a pycnocline in the lower half of the water column. Depth-integrated energy flux magnitude is 4.2 kW m −1 , directed almost due east (7 • counterclockwise from east). In comparison, Sherwin (1991) estimated the D 2 mode-1 internal tide energy flux at the nearby CTD repeat station to be 4.7 kW m −1 , but was unable to diagnose the direction.

Model comparison
In Figure 5 the observations are compared with the regional tide model described by Hall et al. (2011). The model is a configuration of the Princeton Ocean Model (POM; Blumberg and Mellor, 1987) for the FSC and WTR region, initiated with typical late-summer stratification, and forced at the boundaries with M 2 barotropic velocities (see Hall et al., 2011, for full details). Maximum N 2 in the model is slightly higher than observed (Table 1) ( Fig. 5b). The northern flank of the WTR is an area of energetic internal tide generation, up to 4 W m −2 , and radiates an internal tide into the southern FSC. Modelled internal tide energy fluxes are spatially variable, but >5 kW m −1 at some locations (Fig. 5a). The mooring was located east of the most energetic generation and up-slope of the largest energy fluxes.
For direct comparison, the model output is interpolated onto the exact location of the mooring (Table 1). The modelled M 2 internal tide energy flux is 6-7% larger than the observed D 2 energy flux, but within 10 • of its direction. Maximum modelled 5 vertical isopycnal displacement is 41 m (slightly smaller than observed), but is compensated by the higher maximum N 2 and results in modelled APE being 30% larger than observed; modelled HKE is 10% smaller than observed.

Surface tidal ellipses
As well as measuring potential density by the ADCP mooring, the glider is used to infer a second estimate of barotropic velocity.
Harmonic analysis is used to extract the D 2 component of DAC velocity (all dives deeper than 500 m) and compared to the 10 D 2 component of u bt from the ADCP/current meter. Barotropic velocity is highest in the across-slope direction (maximum 15 cm s −1 , Fig. 3a) and there is a very close match between the DAC and ADCP estimates (r.m.s. difference is 0.8 cm s −1 ).
In the along-slope direction, where barotropic velocity is lower (maximum 0.5 cm s −1 , Fig. 3b), the DAC estimate lags the ADCP estimate by 35 minutes, but their amplitudes closely match (r.m.s. difference is 1.2 cm s −1 ). The resulting surface tidal ellipses have similar semi-major axis lengths and phases (Table 1), but the DAC estimate is less eccentric (more circular) and 15 rotated 3 • counter-clockwise (Fig. 5c). Compared with M 2 surface ellipses from the regional tide model described by Hall et al. (2011), both observational estimates are less eccentric and have shorter semi-major axes (Table 1; Fig. 5c). However, the inclination of observed and modelled ellipses are comparable, with their semi-major axes orientated across-slope. This is the orientation required to generate an energetic internal tide at the WTR.
5 Glider sampling error 20 The separation of spatial and temporal variability is a common problem when interpreting glider data due to their slow speed (Rudnick and Cole, 2011) and imperfect positioning. In this context, the inability of the glider to perfectly hold station by the ADCP mooring leads to error in the calculation of internal tide energy flux (Section 3) due to mis-sampling of the spatially and temporally varying density field. An understanding of this error is important for both mission planning and interpretation of results. Other missions along the European continental slope (e.g., Hall et al., 2017a) have shown that a glider operating 25 as a virtual mooring by repeatedly diving to 1000 m around a fixed station can maintain a 'watch circle' with a diameter of approximately 5 km, i.e., all dives start and end within 2.5 km of the target location. The ability to do this is dependent on environmental conditions, particularly tidal and slope currents, but the lower limit is effectively set by the glide angle; a steep, 45 • glide angle will result in around 2 km horizontal travel over a complete dive cycle to 1000 m.
The size of the energy flux error is related to the length-scale of the sampling cloud (d, the diameter of the watch circle) and 30 the horizontal wavelength of internal tide being measured (λ). If d λ we can consider the glider data a fixed-point timeseries with no spatial-temporal aliasing, and so the error will be small. As d increases the glider will increasingly sample density at the wrong phase of the internal tide and so the error will increase because the measured pressure perturbation (p Glider ) will deviate from the pressure perturbation at the ADCP (p ADCP ), located at the centre of the watch circle. If d λ the glider will sample density at random phases of the internal tide and so p Glider and p ADCP will be uncorrelated.
Here we use a Monte Carlo approach to estimate the energy flux error. Potential density in an idealised internal tide field is sub-sampled along pseudorandomly distributed glider paths contained within watch circles of varying diameters. The 'true' 5 depth-integrated energy flux at the ADCP, F true = 0 −H u bc p ADCP dz, is then compared with the 'observed' depth-integrated energy flux, F obs = 0 −H u bc p Glider dz. In both equations u bc is baroclinic velocity at the ADCP. An idealised M 2 multimode internal tide field is created for a 1000-m deep water column with uniform stratification (Appendix A). The mode-1 horizontal wavelength (λ) is 80 km and mode-1 vertical isopycnal displacement is 50 m, typical of mid-latitude shelf slope regions. Glider sampling is modelled as a group of twelve 1000-m dives, over 37 hours (≈3 M 2 cycles), within a watch circle 10 of diameter d. Each dive is 2 hours 50 minutes long, with 15 minutes at the surface between dives. Horizontal distance travelled during each dive cycle is between 1.5 km and 4 km (typical of real glider missions), but there is no surface drift. The glider's path during each dive is determined by randomly selecting a start position within the watch circle then randomly selecting an end position 1.5-4 km away, but still within the watch circle. The start position of the following dive is the same as the end position. Potential density is linearly interpolated onto this pseudorandom glider path and the resulting density 'observations' analysed using the method described in Section 3.1 to yield p Glider .
Nine cases are investigated, with d ranging from λ/32 (2.5 km) to λ/4 (20 km), and for each case 5000 different twelve-dive scenarios are simulated. A different random set of baroclinic mode phases is used for each scenario. Example pseudorandomly distributed glider paths for four cases are shown in Figure 6. Energy flux relative error is defined F err = (F obs − F true )/F true , 5 so positive error indicates an overestimation and negative error indicates an underestimation. Similarly, APE relative error is and APE obs is 'observed' depth-integrated APE (calculated from ξ Glider ).  Figure 8. Histograms of (a) energy flux relative error and (b) APE relative error due to mis-sampling density in an idealised M2 multimode internal tide field for the four watch circle diameter cases shown in Figure 6. Positive error indicates an overestimation, negative error indicates an underestimation. Distribution of (a) energy flux relative error and (b) APE relative error against watch circle diameter, including the 99% and 95% confidence limits and the bias (median value). The four cases in panels (a) and (b) are indicated with black and grey triangles. The red triangle is the case most appropriate for the observations. The white triangles are four additional cases.

Single-scenario example
A single-scenario for the d = λ/4 case is shown in Figure 7 to highlight the impact of mis-sampling density on observed energy flux and APE. This is an extreme example, with all the glider dives 6-10 km from the ADCP (Fig. 7d), and features near-bottom internal tide intensification similar to observed on the northern flank of the WTR. In this example, the error in measured density is maximum in the lower half of the water column (where ξ ADCP is up to 80 m; Fig. 7a,b); the resulting 5 ξ Glider underestimates ξ ADCP by up to 20 m and leads by up to 40 minutes. Observed energy flux and APE underestimate true energy flux and APE over the majority of the water column (Fig. 7c) and, depth-integrated, underestimate by 772 W m −1 (F err = −0.09) and 615 J m −2 (APE err = −0.2), respectively.

Energy flux error
Histograms of F err (0.005-wide bins) for four watch circle diameter cases are shown in Figure 8a. The peaked distribution for the d = 1/32λ case broadens with increasing watch circle diameter as well as becoming biased towards negative error.
The negative bias results from two related mechanisms. Firstly, the amplitude of ξ Glider (and therefore p Glider ) is typically underestimated for large watch circles because the glider samples density with a broad range of phase shifts, causing spectral 5 smearing and poor harmonic fits to ξ. Secondly, maximum energy flux occurs when p and u bc are exactly in-phase so any error in the phase of p Glider , positive or negative, will also result in a negative bias.
F err distributions for all nine watch circle diameter cases are shown in Figure 8c, including the 99% and 95% confidence limits and the bias (median value). As watch circle diameter increases, the width of the confidence intervals increases and the bias becomes progressively more negative. For the d = 1/32λ case, F err is ±0.04 at the 99% limit and the bias is near zero 10 (−0.002). For the d = 1/4λ case at the other extreme, F err is 0 to −0.31 at the 99% limit and the bias is −0.1.

APE error
Histograms of APE err for four watch circle diameter cases are shown in Figure 8b. Compared with F err , the distributions are broader and with a more negative bias for small watch circles. The broader distribution is explained by the error in ξ Glider being squared in Equation 9. The negative bias is explained by the first mechanism described in Section 5.2. APE err distributions 15 for all nine watch circle diameter cases are shown in Figure 8d. Similar to F err , the width of the confidence intervals increases and the bias becomes progressively more negative as watch circle diameter increases. For d = 1/32λ, APE err is ±0.08 at the 99% limit and the bias is only −0.005. For d = 1/4λ, APE err is 0.02 to −0.33 at the 99% limit and the bias is −0.08. Unlike F err , the bias converges towards a constant value for very large watch circles.
6 Summary and discussion 20 A novel approach to measuring internal tide energy flux using a co-located ocean glider and moored ADCP is tested using a dataset collected from the WTR in the North Atlantic. Gliders cannot perfectly hold station, even when operating as a virtual mooring, so error in the energy flux calculation due to imperfect co-location of the glider and ADCP is estimated by subsampling potential density in an idealised internal tide field along pseudorandomly distributed glider paths. If we consider the maximum acceptable energy flux error to be 0.1 (10%), all the glider data must be contained within a watch circle with a 25 diameter smaller than 1/8 the mode-1 horizontal wavelength of the internal tide. Energy flux is biased low and the negative bias increases with increasing watch circle diameter. If watch circle diameter is larger than 1/8 the mode-1 horizontal wavelength, the negative bias is more than −0.03 (3%) and all realisations within the 95% confidence interval are underestimates. When on-station over the WTR, the glider stayed within 2.5 km of the mooring so watch circle diameter, d = 5 km. The local D 2 mode-1 horizontal wavelength, λ ≈ 100 km so the d/λ = 0.05 case (Table 2) is the most appropriate for the observations 30 presented here. The observed energy flux is estimated to have a negative bias of only −0.004 (0.4%) and an error of less than   ±0.03 (3%) at the 95% confidence limit. This estimate does not include the effect of internal tide advection by the barotropic tide (Stephenson et al., 2016), which can lead to an additional negative bias if barotropic velocity amplitude is of a similar size to baroclinic phase speed. Over the WTR, D 2 mode-1 phase speed is ≈2.2 m s −1 and barotropic velocity amplitude is <0.2 m s −1 so we expect this effect to be negligible for our observations.
At mid-latitudes, D 2 mode-1 horizontal wavelength for a 1000-m deep water column is typically in the range 40-160 km.

5
The results presented here suggest energy flux error due to imperfect co-location can be reduced to an acceptable level (10%) if the glider maintains a 5-km to 20-km diameter watch circle. In the absence of strong tidal and slope currents, a well-trimmed glider diving to 1000 m with a relatively steep glide angle can usually maintain a watch circle with a diameter of 5 km or less, so energy flux error will typically be <10%. Where horizontal wavelengths are shorter, for example at lower latitudes or in shallower and less stratified water columns, a smaller watch circle will be required to maintain an acceptable level of 10 error. In shallower water, smaller watch circles are generally achievable because horizontal travel over a complete dive cycle scales with dive depth. Diurnal internal tides have longer horizontal wavelengths so larger watch circles are acceptable. For mission planning, the mode-1 horizontal wavelength of a tidal frequency ω can be estimated λ = 2πc 1 / ω 2 − f 2 , where f is the inertial frequency and c 1 = N H/π is an approximation of mode-1 eigenspeed. If the assumption of uniform stratification is not appropriate, c 1 can be calculated by solving the boundary value problem for a given N (z) (Gill, 1982). Table 2 can then 15 be used to estimate the energy flux bias and error that can be expected for a given value of d/λ.
Including the above estimate of error due to imperfect co-location, the observed depth-integrated D 2 internal tide energy flux over the northern flank of the WTR is 4.2±0.2 kW m −1 . This is considerably larger than previous internal tide observations Ridge (up to 33 kW m −1 ; Lee et al., 2006) and Luzon Strait (up to 41 kW m −1 ; Alford et al., 2011). More comparable to the WTR is the Mendocino Escarpment, where a ridge is orientated perpendicular to the continental slope and the observed energy flux is 7 kW m −1 (Althaus et al., 2003).
The 40-hour co-located timeseries presented here is not long enough to resolve the internal spring-neap cycle. Peak neap tide occurred on yearday 153, one day before the majority of the co-located timeseries. Assuming the internal tide is generated 5 locally at the WTR, the surface and internal spring-neap cycles will be in phase. The observed D 2 energy flux is therefore representative of neap internal tide and so an underestimate of the true M 2 internal tide. This may somewhat explain the slight underestimate compared to the M 2 -only regional tide model. Interestingly, the CTD timeseries used by Sherwin (1991) was recorded two days after peak spring tide so is more representative of spring internal tide. The fact that two observational estimates of D 2 vertical isopycnal displacement, 6.7 km apart and at different phases of the internal spring-neap cycle, are so 10 similar, implies that there are compensating spatial gradients in internal tide magnitude. The regional tide model shows the possible extent of these gradients and suggests that accurate siting of moorings is crucial for repeated, long-term observations.
For future experiments, spatial gaps in the timeseries can be minimised with conductivity-temperature loggers and additional current meters on the mooring line. We have also shown that glider-inferred DAC can provide an accurate estimate of tidal current velocity that could be used to constrain barotropic velocity in the absence of full-depth data coverage by ADCPs and 15 current meters. However, the major limitation of the dataset presented here is the short length of the co-located timeseries.
Future glider missions will hold station by an ADCP mooring for several weeks to resolve the internal spring-neap cycle.
Calculating D 2 internal tide energetics in a 36-hour moving window will yield a time-varying energy flux that can be related to seasonal changes in stratification, advection by mesoscale eddies, spatial and temporal patterns in internal tide-driven turbulent mixing, and the resulting biogeochemical response.

20
Code and data availability. The Seaglider data were processed using the UEA Seaglider Toolbox (https://bitbucket.org/bastienqueste/ueaseaglider-toolbox) and are available from the UEA Glider Group. The ADCP and acoustic current meter data are available from Marine Scotland Science. Data analysis code is available on request from the corresponding author.

Appendix A: Idealised internal tide field
An idealised M 2 multi-mode internal tide field is created for a 1000-m deep water column with uniform stratification (N 2 = 25 6.1 × 10 −6 s −2 ). Horizontal current velocity, u = (u, v), and vertical isopycnal displacement, ξ, are defined by summing the first ten baroclinic modes, u(x, y, z, t) = 10 n=1 u n sin(k n x − ωt − φ n )A n (z), v(x, y, z, t) = 10 n=1 u n f ω cos(k n x − ωt − φ n )A n (z), and (A2) ξ(x, y, z, t) = 10 n=1 u n sin(k n x − ωt − φ n )B n (z) 1 ω where u n and φ n are the velocity amplitude and the phase of the n-th baroclinic mode, respectively, ω = 1.41 × 10 −4 s −1 is 5 the M 2 frequency, and f = 1.26 × 10 −4 s −1 is the inertial frequency at 60 • N. A n (z) and B n (z) are the vertical structures of horizontal current velocity and vertical isopycnal displacement for each baroclinic mode, and are equivalent to cos(nπz/H) and sin(nπz/H), respectively, where n is mode number. Horizontal wavenumber, k n = ω 2 − f 2 /c n , where c n = N H/nπ is an approximation of mode eigenspeed (Gill, 1982). Velocity amplitude decays with mode number, u n = u 1 e −0.5(n−1) , where u 1 is the mode-1 velocity amplitude. This decay rate results in a well-defined internal tide beam if velocity phase is 10 approximately equal for each baroclinic mode. However, a different random set of baroclinic mode phases (φ n ) is used for each scenario simulated so internal tide beams are only apparent in a subset of scenarios. u 1 = 0.28 m s −1 yields a mode-1 vertical isopycnal displacement amplitude of 50 m, but energy flux error and APE error are not sensitive to absolute amplitude.
The time-varying potential density field is then 15 where ρ(z) is a background density profile with a vertical gradient equivalent to N 2 . Barotropic velocity (u bt ) and residual flow (u) are both zero so u bc = u.
Competing interests. The authors have no competing interests.
Acknowledgements. SG613 is owned and maintained by the UEA Marine Support Facility. The glider and ADCP mooring were deployed as part of the fourth Marine Autonomous Systems in Support of Marine Observations mission (MASSMO4; funded primarily by the Defence

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Science and Technology Laboratory) and the Marine Scotland Science Offshore Monitoring Programme. The cooperation of the captain and crew of NRV Alliance (CMRE) and MRV Scotia (Marine Scotland) are gratefully acknowledged. The glider data were processed by Gillian Damerell, the ADCP data were processed by Helen Smith and Barbara Berx, and the acoustic current meter data were processed by Jennifer Hindson and Helen Smith. Assistance with glider piloting was provided by the UEA Glider Group. Helpful comments on the manuscript were provided by two reviewers.