Progress in Oceanography Ocean Shelf Exchange, NW European Shelf Seas: measurements, estimates and comparisons

Transports across the continental shelf edge enhance shelf-sea production, remove atmospheric carbon and imply an active boundary to ocean circulation. We estimate relatively large overall transport across three contrasted sectors of north-west European shelf edge: the Celtic Sea south-west of Britain, the Malin-Hebrides shelf west of Scotland, the West Shetland shelf north of Scotland. The estimates derive from measurements in the project FASTNEt (Fluxes across sloping topography of the North East Atlantic): drifters, moored current meters, effective “diffusivity” from drifter dispersion and salinity surveys, other estimates of velocity variance contributing to exchange. Process contributions include transport by along-slope flow, internal waves and their Stokes drift, tidal pumping, eddies, Ekman transports in the wind-driven surface layer and bottom boundary layer. Overall exchange across the shelf edge is estimated as several m2s-1: if extrapolated globally even 1 m2s-1 is large compared with oceanic transports and potentially important to shelf-sea and adjacent oceanic budgets.  In our context, most exchange is in tides, and other motion with periods ~one day or less, and so effective only for water properties that evolve on such short time-scales.  Nevertheless, cross-slope fluxes, and exchange by low-frequency motion (periods > two days), are large by global standards and also very variable.  Deployment-mean fluxes nearest the shelf break were in the range 0.3 – 4 m2s-1; mean exchanges from low-frequency motion were 0.8 – 3 m2s-1.  Deeper longer-term moorings and drifters crossing 500 m depth gave much larger fluxes and exchanges up to 20 m2s-1.  These transports’ significance depends on distinctive properties of the water, or its contents, and on internal shelf-sea circulation affecting further transport.  For the NW European shelf, transports across the shelf edge enable its disproportionately strong CO2 “pump”. The complex context, and small scales of numerous processes enabling cross-slope Powered by Editorial Manager® and ProduXion Manager® from Aries Systems Corporation transports, imply a need for models.  Measurements remain limited in extent and duration, but widely varied contexts, particular conditions, events, processes and behaviours are now available to support model validation, especially around the northwest European continental shelf edge.  Variability still renders observations insufficient for stable estimates of transports and exchanges, especially if partitioned by sector and season; indeed, there may be significant inter-annual differences.  Validated fineresolution models give the best prospect of spatial and temporal coverage and of estimating present-day and potential future shelf-sea sensitivities to the adjacent ocean. Suggested Reviewers: Susan Allen The University of British Columbia sallen@eoas.ubc.ca Expert on canyon effects Kenneth Brink Emeritus Research Scholar, Woods Hole Oceanographic Institution kbrink@whoi.edu Expert on ocean-shelf exchange Sandro Carniel Istituto di scienze marine Consiglio Nazionale delle Ricerche sandro.carniel@ismar.cnr.it Studies of ocean-shelf exchange Xavier Durrieu de Madron Universite de Perpignan demadron@univ-perp.fr Studies of down-slope transport Bernard Le Cann Universite de Brest: Universite de Bretagne Occidentale blecann@univ-brest.fr Studies of flow at the continental shelf edge Kon-Kee Liu National Central University kkliu@cc.ncu.edu.tw Overview of ocean margin science

Reviewer 1 "Consider Table 1. These are statistics from 11-day deployments of current meters. Look at "ST1 bottom" (just to pick one line). The standard deviation is 38 (the units should be included in the table! I am guessing cm/sec, but everything else seems to be mks!). The 95% confidence bounds are about +/-0.5. Given that 95% confidence for a mean should be about +/-2*std(u)/sqrt(N), this says that there are thousands of degrees of freedom in an 11 day time series. Maybe there is a typo, and it should be standard deviation is 3.8. That still says that there are more than 100 degrees of freedom in an 11 day time series. I have seen lots of error bars on means of ocean time series, and I have never seen anything like this. There is something really wrong here." Response. We thank Reviewer 1 for this comment which has caused us to look again at the estimation and application of degrees of freedom. A coding mistake, now corrected, has been found in the application of degrees of freedom to 95% confidence bounds. The particular case of ST1 appears to relate to the next comment of the reviewer.
"See line 465: The correct independence time scale to use is the integral (out to large lags) of the autocorrelation function. Instead, the time scale is given here as an e-folding of that function. I am guessing that the reason this was not done conventionally is that the authors found themselves taking the autocorrelation of what is very nearly a pure (tidal) sinusoid. This (of course) does not have a well-defined integral, and so the authors come out with their astounding result (e folding of a 12.42 hour sinusoid giving T_e maybe around 2 hours?) as a way to get some sort of answer. (The autocorrelation of a sinusoid is itself sinusoidal.) Doing this correctly is not a trivial business, but one approach might be to use a least squares fit to remove the tides, and then compute the time scale correctly since what is left ought to be a more broad-band time series with a convergent integral. (The tides contribute little information: only 2 degrees of freedom -amplitude and phase-for each component)." Response. The reviewer is correct that a tidal sinusoid is indeed dominant in the total flux analysed; this is especially so at ST1 for which the residual after removing tides appears to be mainly high-frequency "noise". We did in fact use a least-squares fit (by six constituents) to remove most of the tides before calculating Te. However, the resulting value of Te for the ST1 residual series is only 0.18 hours, much less than at most other moorings. Although use of Te is widely advocated, the experience here is that its value can be "accidental", i.e. sensitive to particularities of the time series. Accordingly, we have now changed to a more "robust" estimate of the time scale as the integral of the autocorrelation function of ures (the residual after removing the fit by six tidal constituents) as suggested by the reviewer. As pointed out by the reviewer, the autocorrelation of a sinusoid is sinusoidal; any residual periodic motion would therefore have negative autocorrelation for portions of the integral and so reduce the estimated time scale. To avoid this, we "conservatively" integrate |autocorrelation| to avoid over-estimating degrees of freedom. Then upper and lower confidence bounds (95% meaning 2½% to 97½%) have been calculated using a Student's tdistribution: CI = mean(x) +/-Tscore*SE where x is the time series of depth integrated flux/exchange, standard error SE = std(x) / sqrt (df Various contrasting locations have been studied. Around European margins, down-slope particle 140 fluxes were emphasised in the north-west Mediterranean (Guarracino et al. 2006; ECOMARGE -141 Monaco et al. 1990) and in the Bay of Biscay (e.g. ECOMARGE - Heussner et al. 1999). MORENA 142 (off Portugal) emphasised mainly summer upwelling (enhanced off capes), along slope flow (more 143 prominent in winter) and hydrography (Fiuza et al. 1998; Stevens et al. 2000). ARCANE, SEFOS 144 and INTERAFOS measured general and mesoscale Lagrangian circulations over the Bay of Biscay 145 abyssal plain and slopes (Serpette et al. 2006). OMEX studied physics and biogeochemical fluxes 146 over Goban Spur (south-west of Britain) and off north-west Spain (Pingree et al. 1999 Off eastern North America, an early study was at the Scotian shelf edge (Smith 1978). "Shelf Edge 155 Exchange Processes" (SEEP-I and -II) studied the Middle Atlantic Bight (Walsh et al. 1988;Biscaye 156 et al. 1994). Other specific studies concern along-shelf convergence to infer off-shelf export near Rossby number, so relaxing the geostrophic constraint (also moderated by strong stratification) and 191 facilitating cross-isobath flow. Canyons cut across any along-slope tidal flow component, often 192 generating baroclinic tides and internal waves; they can also focus internal waves. Allen et al. (2009) 193 give a review of canyon effects. 194 195 1

.5 Motivation for this study 196
Despite the previous studies reviewed above, we still lack knowledge of seasonal and inter-annual 197 variability in behaviours of different exchange mechanisms. Measurements have been especially 198 difficult in winter (when wind-forced mechanisms may be at their strongest). Yet seasonality in 199 physical exchange is vital to meaningful estimates of biogeochemical fluxes. There also remains a 200 challenge to integrate individual processes for regional-scale estimates of transports across the shelf 201 edge as: (a) individual process contributions based on measurements may not be simply additive; (b) 202 the small-scale physical processes enabling transport across steep slopes may not be resolved or 203 parameterised in regional numerical models. Improved understanding of exchange requires numerical 204 modelling to provide evidence with a density and coverage beyond the scope of observations alone. 205 However, we need measurement data of sufficient variety to test such models' representation of 206 numerous known significantly-contributing processes (as listed above). "Variety" implies different 207 seasons and contrasting shelf-edge sectors. Accordingly, project FASTNEt (Fluxes across sloping 208 topography of the North East Atlantic) around the NW European shelf edge included aims: 209 (i) To determine through measurements the seasonality of physical gradients and exchange across 210 the shelf edge; 211 (ii) To quantify key exchange mechanisms and obtain new data to test and improve fine-resolution 212 models of the shelf edge, by carrying out process studies in contrasting shelf-edge sectors. 213 The second aim entails fine spatial resolution of transports, at scales comparable to or less than model 214 resolution, to understand the very local scale (10's km or less) variability in exchange, in varied shelf-215 edge sectors having different combinations of exchange processes. 216 217 Here we attempt to synthesise estimates of transports and contributions thereto, on the basis of the Estimates of some process contributions thereto are presented in section 4. A discussion section (5) 225 includes some global comparisons and significance for shelf-sea budgets and cycling. Conclusions 226 (section 6) include The shelf seas bordering the NE Atlantic are broad and irregular, from ~50 km wide around Ireland to 233 as much as 400 km in the Celtic Sea ( Figure 1). Depths are typically between 100 and 150m. The 234 shelf slope is steep (super-critical to internal tides) in the south, becoming less steep (sub-critical) 235 north-west of Scotland. The Celtic Sea margin is irregular. The definition of "along-slope" varies 236 locally owing to the many canyons which may focus internal waves instigating sediment movement very variable; the proportions of Atlantic-origin to coastal-source water on the Hebrides shelf around 262 57°N 7°W vary from more than 62%:38% to less than 6%:94% as winds vary from strong westerly to 263 strong easterly for sustained periods . Salinity here and on the Malin shelf is 264 strongly affected by these ratios of Atlantic-and coastal-source waters (and hence by wind stress and 265 direction; Jones et al. 2018); there is no particular periodicity but salinity variability is greater in 266 winter. Prevailing winds are often strong, generating large waves, turbulence and consequent mixing, 267 to which strong tidal currents also contribute (see below).

269
This is an eastern ocean boundary; there is no strong (wind-driven) western boundary current, nor 270 large associated eddies. However, along the upper continental slope there is a current, which is 271 usually poleward. The North Atlantic Current meets the continental slope off the Celtic Sea and in 272 Rockall Trough; 1 -2 Sv of its transport is converted to barotropic transport over the slope. The slope 273 current transport increases significantly from Rockall Trough to the Faroe-Shetland Channel (Zhou 274 and Nost 2013). This slope current is believed to be the result of increasing poleward density at the 275 sloping ocean margin (Huthnance 1986 seasonal September/October -March/April variation (Pingree et al. 1999). The slope current here is 282 perhaps weaker than further north owing to non-meridional alignment and indentations in the Celtic 283 Sea slope; around Goban Spur it may sometimes overshoot off-shelf rather than follow the depth 284 contours (Pingree et al. 1999).

286
Slope current transport is determined by a balance between the meridional oceanic density gradient, 287 wind stress and bottom friction over the slope (Huthnance et al. 2020). Where the slope current is 288 poleward, the bottom frictional boundary layer is expected to give off-shelf Ekman transport. 289 Measurements over the Hebrides slope west of Scotland showed slope-current speeds of order 0. At any one location we define flux as the net transport through depth (− < < 0) in a layer: 373 where ̅ is the profile of current averaged over time . Moreover 375 where ′ = − ̅ is the local instantaneous departure from the time-mean flow and |..| denotes 377 magnitude (also averaged over time ; magnitude avoids cancellation of flows reversing in time or 378 space that comprise exchange and, being linear unlike standard deviation, avoids giving extra 379 "weight" to large values).

381
For a layer above or below a moving interface at = −ℎ( ), flux in the layer is taken as 382 where the z-integration is over the ranges (−ℎ, 0) and (− , −ℎ) respectively for the layers. This is 384 relevant for (e.g.) biogeochemical interests which may want transport estimates in a 2-layer 385 framework, e.g. to account for nutrient imports supporting production along with carbon export in a To support FASTNEt, an Atlantic margin model "AMM60" was developed with resolution 1 nautical 391 mile = 1/60 degree (~ 1.8 km) and 51 hybrid s-sigma terrain-following layers (Guihou et al. 2018). It 392 is based on version 3.6 of the NEMO ocean model (Madec et al. 2016). AMM60 spans from Spain to 393 Norway, including all this sector of the Atlantic margin and the North Sea ( Figure 1). Lateral 394 viscosity is 50 m 2 s -1 , lateral diffusion is Smagorinsky (1963) with factor kH = 0.7, vertical mixing is 395 GLS Canuto A (k-ε) formulation with background viscosity 10 -7 m 2 s -1 .

397
Surface forcing is by the ERA-interim atmospheric reanalysis (Dee et al. 2011). Lateral oceanic 398 forcing for AMM60 is from a NEMO-based Northern North Atlantic model "NNA" using the same 399 atmospheric and tidal forcing (Holt et al. 2014). Freshwater input from rivers is also included from a 400 synthesized dataset (see Guihou et al. 2018 In this section each observational method is considered in turn. Precise methodology for the data-type 418 is given, which then allows flux, exchange and effective diffusivity to be evaluated. Directly 419 following each precise methodology, results are presented and discussed without reference to other 420 results. Later, in Section 3.6 and the discussion of Sections 4 and 5, the variously derived estimates of 421 fluxes, exchange and effective diffusivity are synthesised. 422 423 3.1 Fluxes and exchange from moorings 424 Fluxes and exchanges were calculated from moorings deployed in the Celtic Sea, on the Malin and 425 Hebrides shelves and on the West Shetland slope (the eastern side of the Faroe-Shetland Channel) 426 ( Figure 1). Mooring terminology, duration, location, depth range sampled and sampling interval are 427 given in Table S1.

429
At each location, the across-slope component of flow, , was defined based on the orientation of the 430 local bathymetric slope and the observed deployment-mean current direction at mid-depths over the 431 slope (presumed best-geostrophically-constrained along the slope; details are provided in the 432 respective sections below, Appendix B and At all moorings, the barotropic tidal and tidal residual exchanges are each less than the exchange by 452 the total current. However, according to whether tidal currents are dominant or not, usually either the 453 tidal or the tidal residual exchange is close to the total; their sum is always greater than the total, i.e. 454 contributions to exchange are not simply additive.

456
We report standard deviations (± in respective tables) for each of the deployment mean total across-457 shelf flux and exchange calculations and also 95% confidence intervals (meaning 2½% to 97½%; in 458 parentheses in respective tables) based on Student's t-distribution. The numbers of degrees of freedom 459 ( ) were calculated as 460 where is the autocorrelation of the depth integrated flux/exchange (x) calculated using (tidal 462 signal removed), ∆ is the sampling interval and the total number of data points. 463 464 3.1.1 Celtic Sea 465 Flux and exchange were calculated for five sites (ST1-ST5 within 30 km of the shelf break; Table S1, 466 Figure 1c) over an 11-day period in June 2012 (21 M2 tidal periods) when all the moorings were in the 467 water simultaneously. Along-and across-slope direction was determined from the long-term mooring 468 LT1 on the slope (Table S1, Figure 1c), see Appendix B and respectively for all these moorings, including LT1. On the shelf (ST2-ST5) the baroclinic currents at 476 each site were dominated by a mode-1 semi-diurnal internal tidal wave structure, with opposing 477 surface and bottom layer current directions ( Figure S1). Maximum baroclinic current velocities  On the shelf (ST2-ST5), deployment-mean flux directions ( Figure 1c) and off-shelf components were 489 very varied (Table 1), 1-4 m 2 s -1 in the space of 30-40 km. Full-depth fluxes calculated from the 490 filtered ( ) current time-series closely match the total; low-frequency (< 1/48 cycles h -1 ) processes 491 were responsible for most of this transport. ST4 alone had an on-shelf flux, in the bottom layer.

493
Mooring ST1 on the slope was dominated by a 0.1 m s -1 equatorward slope current for an along-slope 494 flux of 57 m 2 s -1 (Figure 1c and Table S4). Across-shelf flux (Table 1) was attributable to low-495 frequency processes, as for the on-shelf sites but much larger.

497
Across-shelf transports varied substantially within the 11 days common deployment at all sites ST1-498 ST5 ( Figure S1). The 297 days at mooring LT1 (Table S1, Figure 1c) in 1503 m depth show seasonal variability in 508 cross-shelf fluxes. Time-series of flux and exchange averaged over four semi-diurnal periods are 509 shown in Figure 2; monthly across-and along-slope fluxes are shown in Figure 3.   Most exchange on the shelf at ST3 to ST5, and at ST1 on the slope, was driven by the barotropic tide 528 (Table 2). Close to the shelf break, at ST2 and ST3, total across-shelf exchange was smaller although 529 exchange associated with residual currents ( ) was comparable with ST4 and ST5. Within the 530 internal tide generation zone (ST1 and ST2) the exchange had a large tidal residual component 531 comprising baroclinic and non-tidal currents (Error! Reference source not found.2). ST1 on the 532 slope had a ~100 m thick bottom boundary layer where maximum across-shelf exchange took place 533 ( Figure S1; discussed in section 5.3). Spring-tide exchanges on the shelf (ST3 to ST5) exceeded those 534 at neaps, but were smaller than at neaps on the upper slope (ST1 and ST2).

632
At all mooring locations except NWS-C, long-term mean fluxes were primarily along the channel to 633 the north-east (Figure 1f). At NWS-C near the north-western side of the Channel, long-term mean 634 flux was southward. Thus cross-shelf mean fluxes (Table 5) were much smaller than respective mean 635 along-slope fluxes ≥ 4.6 (NWS-F), 24.8 (NWS-E,H), 43.2 m 2 s -1 (NWS-D,G; Table S7). Differences 636 between mean fluxes based on total and filtered currents are insignificant (Table 5 shows only the 637 former). Fluxes were strongest on mid-slope, decreasing to the shelf edge ( Figure 1f). Cross-shelf 638 flux was generally onto the shelf, although some sustained off-shelf fluxes occurred in winter 2014/15 639 (Figure 6a, c).

641
Much variability on time-scales of two days or more is indicated by the standard deviations in Table  642 5.
[The confidence intervals are narrowed by the many degrees of freedom in 6-12 month records]. 643 Monthly-mean fluxes retain a substantial fraction of this variability (e.g. at NWS-E; Figure 6a, c). 644 Multi-year deployments showed larger mean fluxes along the slope in winter than in summer ( Table  645 6), in accord with the seasonal cycle of slope current transport in the Faroe-Shetland Channel in Berx 646 et al. (2013). The mainly-winter deployments suggested (with exceptions) less across-slope flux and 647 more cross-slope exchange than mainly-summer deployments (Table 6).  Table 6. Mean along-slope fluxes, across-slope fluxes and across-slope exchanges ± standard errors 650 based on over all mooring deployments separated into those mainly in winter (November-651 March) and those mainly in summer, Faroe-Shetland Channel. Numbers of seasons in parentheses.

715
Flux and exchange estimates, from drifter crossings of depth contours, are shown in Table 8 for the 716 Celtic, Malin and West-Shetland deployments. The estimates are based on daily 24-hour-average 717 positions (derived from 3-hourly recorded positions) so as to remove most of the tidal displacements. 718 We do not expect bias from the choice of depth contours; some depth contour is crossed in any 719 interval ; the chosen contours are of interest (as "conventional" depth choices) but should be 720 representative. However, the fact of analysing only occasions of contour crossing may introduce bias 721 (contours are probably more likely to be crossed when flow is faster). Such bias can be estimated by 722 comparing mean distance travelled on contour-crossing occasions with the overall mean distance 723 travelled in (three or) 24 hours. This was carried out for the daily positions and the "bias" (factor) for "June" 2.5 ± 8.6 5.7 ± 1.4 1.1 ± 6.2 5.7 ± 1.8 "July" 6.9 ±16.2 9.8 ± 4.2 -1.1 ± 2.3 3.3 ± 0.4 -1.2 ± 1.9 3.5 ± 0.4 "August" 7.5 ± 12.9 8.5 ± 2.3 1.7 ± 2.3 3.4 ± 0.6 4.2 ± 2.5 3.5 ± 0.8 "September" -9.4 ± 12.7 9.0 ± 3.7 6.7 ± 4.4 3.
7.1 ± 0.4 3.3 ± 1.6 2.1 ± 0.5 2.8 ± 0.2 0.20 ± 0.04 "July" 14.3 ± 3.9 6.3 ± 1.2 13.3 ± 2.4 4.7 ± 0.6 3.6 ± 2.2 4.8 ± 0.7 "August" 4.4 ± 4.4 8.8 ± 1.4 "September" 1.9 ± 4.7 6.9 ± 1.7 "October" -8.5 ± 11.0 6.7 ± 2.7 "November"  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 27 -8.0 ± 21.1 46.5 ± 3.7 20.1 ± 9.8 18.3 ± 2.6 3.9 ± 1.8 1.8 ± 0.6 June 11.2 ± 14.7 24.5 ± 3.3 -0.75 ± 2.7 2.3 ± 1.0 0.03 ± 1.9 2.0 ± 0.6 July 1.17 1.41 1.11 Bias factor with open-ocean eddies; some drifters over the slope were apparently caught in short-lived eddies that 744 often exist over the Biscay slope (in association with the irregular topography) and appear to favour 745 ocean-shelf exchange there (Porter et al. 2016a). The strongest drifter-derived cross-contour currents 746 ( ̅ and | ′ |) from the three FASTNEt areas (Tables S9-11) were from the Celtic Sea deployment, 747 supporting the suggestion that the irregular Biscay topography favours ocean-shelf exchange. allowing for the "bias" are typically 2-4 m 2 s -1 transport and 2-7 m 2 s -1 exchange, significantly less but 770 still large compared with exchange estimates from FASTNEt moorings' (sections 3.1, 3.6) and 771   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 28 compared with exchange estimates elsewhere (sections 1.3, 5.6). These large estimates may be biased 772 by drifter drogues at depths 15-70 m in the upper, typically faster-moving, part of the water column. winter. There is more variance in summer off NW Ireland; Malin-Hebrides along-shelf variability is 779 larger. Then exchange ½ ℎ| ′ | (section 2.3) is estimated from these values as ~ 0.5 to 1 m 2 s -1 (range 780 given by standard error) or more in summer off NW Ireland. As the estimates are based on weekly 781 maps of altimetry interpolated to a regular grid, they can fully represent only low-frequency, large-782 scale motion and are probably under-estimates (albeit aliasing may add a contribution from higher-783 frequency motion). Moreover, the ageostrophic component of variability is not considered, 784 reinforcing the view that the altimetry variance represents a lower bound for the velocity variance. 785 786 3.4 Effective diffusivity from drifters 787 Relative dispersion of deployed drifter clusters (Table S8) enables calculations of effective diffusivity. 788 Dispersion along co-ordinate is defined as 2 = < ( − 0 ) 2 > where <. . > denotes an average 789 over all drifter pairs (labelled ); is the -separation between two drifters with initial -separation 790 0 . Dispersion along co-ordinate is defined similarly. The -diffusivity is estimated after 791 Batchelor (1952) as = ½ ( 2 )/ and -diffusivity similarly. 90% confidence intervals (5th 792 and 95th percentiles) were defined as (1 ± 1.65(2/ ) 1/2 , where is the ( -or -) diffusivity and 793 is the number of drifters, as in e.g. LaCasce and Bower (2000). A synthesis of the results from the 794 three FASTNEt drifter releases is shown in Figure 7. Across-slope dispersion roughly followed a 2 curve for the first 10 days. Once away from the slope, 804 the across-slope dispersion was roughly linear in (as Fickian diffusion). For the first 60 days overall 805 (13 drifters, mostly over the abyssal plain), drifters' relative dispersion indicated effective diffusivity 806 of about 700 m 2 s -1 along and across the slope. Nevertheless, the dispersion character was anisotropic: 807 along-slope dispersion was dominated by turbulence and shear; across-slope separation was driven 808 mostly by diffusive-like processes (Porter et al. 2016a). The 2014 Faroe-Shetland Channel drifters initially moved north-eastwards (Figure 7a). Two entered 822 the North Sea between Orkney and Shetland, two moved to deeper water in the Faroe-Shetland 823 Channel and the others dispersed across the slope in 0°-1°W. Meridional and zonal relative 824 dispersion (squared separations) appear to evolve similarly but there is a clear magnitude difference 825 between along-and across-slope components (Figure 7d). Both show a near-exponential increase (e-826 folding time ~ 10 hours) for the first three days, after which the relative growth rate slows (e-folding 827 time increases). During days 8 to 30, both relative dispersion components increase approximately as 828 2 (across-slope relative dispersion with more fractional fluctuation). Then along-slope effective 829 diffusivity increases ~linearly to O(3700 m 2 s -1 ); across-slope diffusivity fluctuates with a mean value 830 ~140 m 2 s -1 over days 8 to 30 but arguably a linear increase to O(280 m 2 s -1 ). This exponential and then 831 transition to 2 increase, with increasing time and separation scales, appears to prevail in the North 832 Atlantic ( To estimate diffusivity from salinity gradients, we consider a steady-state salinity ( ) budget. In 837 Appendix C for an along-shelf sector "box" of length from "south" (subscript ) to "north" have uncertainties: formal (standard) errors in | / | (Table 9) are relatively small; the necessary 871 use of "global" values on the right-hand side of (8) and possible non-coincidence of / | = 0 872 with the centre of slope-current transport suggest possible errors ±50% in the Table 9 values for . 873 874 Effective diffusivity has equivalence with exchange. Constituent transport by unresolved small- via is most relevant. In practice (in Table 9) is taken as ∫ | | where is the autocorrelation 885 of the depth integrated low-pass filtered current, consistent with the section 3.1 equation (4) estimates 886 of degrees of freedom in time series. Equivalent exchange values are then derived as ½ℎ( / ) 1/2 887 and shown in Table 9. There is further uncertainty in that time-series of order one year have larger 888 values 2 (in parentheses in Table 9) suggesting longer-term correlations that are not represented 889 in the FASTNEt records of order two weeks. The larger values in Table 9 reduce the Malin and  (Table S3). SSB transects 896 were entirely on the shelf side of (did not include) the salinity maximum or "core" of the along-slope 897 flow; in (8) we took / | = 0 and / | as estimated by a least-squares fit of a quadratic to 898 ( ) along the whole length of each transect. Depth ℎ was taken as the greatest depth in the transect 899 (variously 151 to 202 m); as previously, = 10 6 m 3 s -1 was taken as the along-slope transport. Celtic   (outlined "Historical CTD sections" in Figure 1a); / was estimated for each and an overall value 921 for / was estimated. Separate summer and winter calculations were made (Jones 2016). 922 Effective diffusivities (Table 10) follow from (8) using assumed values of the slope-current transport 923 which was supposed centred with a salinity maximum at 500 m depth. Equivalent exchange values 924 were derived as for the Malin shelf using slope current transport and as shown in Table 10.

925
Salinity gradients may be underestimated: they come from salinity values in many sections, tending to 926 cause scatter in regression against cross-slope location. Hence and derived exchange ½ℎ( / ) 1/2 927 in turn may be over-estimated. Here ℎ = 500 m has been used both onshelf and offshelf. At typical 928 shelf-edge depths 150 m this results in an under-estimate of by a factor 3.3 but an over-estimate of 929 exchange by a factor 1. Comparing summer and winter, Tables 9 and 10 suggest increased effective diffusivity in winter, 939 when cross-slope salinity gradients were smaller (Table 9). For the Hebrides shelf (Table 10), cross-940 slope salinity gradients were similar despite a stronger winter slope current tending to intensify them. waters. Clearly different deeper-water dynamics do not apply to exchange across the shelf break. 956 957

Malin and Hebrides shelves 958
On the Malin shelf, July 2013 moorings showed persistent downslope transport in the canyon at 500 959 m depth: more than 5 m 2 s -1 estimated in the bottom 300 m; stronger episodes correlated with stronger 960 along-shelf flow. This canyon location recorded the largest flux (Table 3), off the shelf. In locations 961 LA to the west-south-west and SD to the north, fluxes at depth tended to be on-shelf. Excepting the 962 canyon and the shallowest location SG on the shelf, fluxes were O(1 m 2 s -1 ) associated with low-963 frequency variability (periods > 48 h), as for the Celtic Sea. Exchanges in 500 m and deeper were as 964 33 large as 15 m 2 s -1 if tides are included, but low-frequency exchanges (periods > 48 h) were O(5 m 2 s -1 ) 965 or less (Table 4). On-shelf exchanges were O(7 m 2 s -1 ) including tides, O(1 m 2 s -1 ) due to low 966 frequencies. Drifters' transverse "diffusivities" increased from less than 100 m 2 s -1 to a maximum 967 300-350 m 2 s -1 ; they correspond to exchange estimates ½ℎ( / ) 1/2 up to 3 m 2 s -1 if depth ℎ = 140 m, 968 = 42 hours (Table 9). Effective diffusivities from cross-slope salinity gradients in winter glider 969 sections average to a corresponding exchange 5.7 m 2 s -1 .

971
On the Hebrides shelf in 1995-96, typical filtered cross-shelf fluxes were ± 3 m 2 s -1 at both moorings 972 (Table 3). In 140 m, there was a baroclinic element; on-shelf flow (0.025 to 0.075 m s -1 ) formed in an 973 intermediate layer at/just below the seasonal thermocline from early summer to the end of autumn. 974 Off-shelf flow above 50 m and below 100 m was stronger during winter months. The overall depth 975 integral was on to the shelf and small on long-term average (Table 3)  Fluxes vary markedly in space on scales of 10-30 km, as shown by the contemporaneous (albeit short-1001 duration) moorings, and in time on scales from days to seasons, as shown by the (albeit few) longer-1002 duration moorings. Exchanges are more consistent; they have a large tidal contribution whilst longer-1003 period contributions generally decrease from deeper water to the shelf break. Exchange estimates 1004 derived from Tables 2, 4, 7 (moorings) and In Section 3, fluxes, exchange and effective diffusivity were presented without regard to process 1019 attribution. Process mechanisms are considered in this Section, including: lenses and internal waves 1020 (4.1); Ekman transport from wind (4.2) and at the bottom (4.3); boundary/slope currents and 1021 associated diversions, meanders and eddies (4.4); tides (4.5). 1022 1023

1185
The resulting downslope transport was integrated within the bottom Ekman layer, defined (in the 1186 absence of turbulence measurements) as the layer adjacent to the seabed within which the (filtered) The downstream impact of the canyon on slope current stability, and consequently the Ekman drain, is 1203 apparent as the anomalously large upslope transport (max. = 22.86 m 2 s -1 ) on days 192-193.

1205
Time-mean Ekman (downslope) transport, calculated over the period of each mooring deployment, is 1206 given in Table 14.
[Beyond the scope of this study is an evaluation of the correct slope heading 1207 within the canyon, which due to the curvature of the canyon makes such a determination challenging. 1208 Future work could investigate the impact of curvature in the flow induced by the underlying 1209 bathymetry and how this impacts on estimates of cross-slope transport as the flow may not follow 1210 isobaths as closely as along straight slopes.] Bottom Ekman transport was largest within the canyon 1211 (SB) where a persistent anticlockwise veering in current direction was observed (Figure 9b). Although 1212 these records are shorter than was hoped for (the long term moorings were lost to trawling), they are 1213 long enough to provide a clear comparison of Ekman transport at sub-inertial timescales from the 1214   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 40 three locations located upstream, downstream and within the canyon. They show the influence of the 1215 canyon (rather than dynamical influence of longer timescale processes) where the largest time-mean 1216 value occurred. The largest bottom Ekman layer transport at a given moment, however, occurred 1217 downstream of the canyon at SC on day 192 due to an instability in the along-slope current ( Figure 9) Values ± standard deviations with 95% confidence intervals in parentheses (see section 3.1).

1224
Estimates of down-slope bottom Ekman transport a little further north off the Hebrides shelf were 1.6 1225 The dye showed weak tendencies to cool (~ 0.015 °C day -1 ), increase density (~ 0.002 kg m -3 day -1 ) 1243 and move down in the water column by 10 ± 4.7 m day -1 (95% confidence limits). These tendencies 1244 appear to be broadly consistent with diapycnal flow needed to supply the divergent bottom Ekman 1245 layer from above; down-slope Ekman transport is expected to increase beyond the shelf break towards 1246 the maximum in the poleward slope current at O(500 m or greater) water depth. However, the 1247 ultimate source of the water supplying this Ekman "drain" remains unclear; a convergence of cross-1248 shelf transport is probable but whether via onshore transport from the ocean or offshore transport from 1249 nearer-shore is not yet determined and, in a local study such as this, the role of along-slope 1250 convergence cannot be dismissed.

1252
This section 4.3 clearly shows down-slope flow near the bottom, albeit of unresolved origin. Along-1253 slope flow that is weaker on the shelf than over the slope implies divergence of the bottom Ekman 1254 layer. This could be supplied by local along-slope convergence, but a more extensive, regional down-1255 slope flow must be supplied principally from cross-slope convergence and downwelling of overlying 1256 water (e.g. as shelf-ward flow, balancing along-slope density and pressure gradients, encounters the 1257 slope). The relative contributions of oceanic and shelf water are not known. Regarding slope current alignment and meanders (ii and iii), alignment of flow at individual moorings 1268 is problematic (Appendix B): the along-slope component tends to be relatively strong, so that any 1269 mis-alignment has a large effect on cross-slope estimates; moreover, effective direction of meandering 1270 depth contours is ill-defined. Hence drifters may give better estimates of effective cross-slope 1271 transport and exchange resulting from slope-current meanders (section 3.2 and Table 8 of the dye patch, as it was tracked northwards, was similar to that in the first dye experiment (section 1296 4.3.1).

1298
As the slope current passed northwards, the relatively shallow canyons here appeared to have a 1299 minimal destabilising effect. Based on the separation of iospycnals, the dyed portion of the water 1300 column (a density band θ = 27.20-27.24 kg/m 3 ) stretched vertically by ~22% as the slope current 1301 failed to follow depth contours and bottom depth increased over the canyons. There is evidence that 1302 stretching occurred throughout the upper water column and increased with depth, although estimates 1303 for undyed portions of the water column are not demonstrably Lagrangian. This suggests cyclonic 1304 relative vorticity over the canyon and a possible overall displacement off-shelf. The degree of 1305 downstream recovery was unclear, but overall destabilisation was small. Such behaviour broadly 1306 follows model runs by Klinck (1996). Theoretically the effect would be very different for a flow in 1307 the opposite direction, in view of the propagation direction of trapped waves (an equatorward flow 1308 here could stall propagation and build the amplitude of a canyon perturbation). Canyon mooring SB 1309 43 showed sinking (downslope flow) in a much thicker lower layer than the common "Ekman Drain" 1310 (sections 3.1.3, 4.3). At the downstream location SC, a distinct two-day on-shore flow accompanied 1311 intensification in poleward, along-slope flow commencing on day 192. Results from the dye release 1312 experiment suggest that the slope current, where re-joining the slope, experiences perturbations in the 1313 steady flow. These measurements do not enable a transport estimate.

1315
The topography near 55.5°N apparently also causes along-slope Atlantic water to flow onto the Malin 1316 shelf (c.f. section 3.2) albeit the precise mechanism is not clear. In lifetime < 4 weeks] that probably prevail in shallow, frictional shelf seas. The section 3.3 exchange 1332 estimates from altimetry, ~ 1 m 2 s -1 , account for meanders and eddies at somewhat smaller scales, but 1333 still low-pass filtered according to satellite footprint and pass frequency. As the first baroclinic 1334 Rossby radius at the shelf break is O(20 km) and greatly reduced on the shelf (LaCasce and 1335 Groeskamp, 2020), the eddy field and its associated exchange contribution is most probably under-1336 represented. Counter to this, altimetric estimates (based on geostrophy) include other flows with 1337 surface elevation expression. The influence of the large scale is therefore somewhat conflated with 1338 mesoscale driven flux. However, the influence of the former is not likely to offset the variance 1339 associated with unsampled eddy flux, and so the altimetry-based estimates are likely a lower bound. 1340 In addition, some eddy transports might not show a surface signal, though this is less likely if we 1341 focus on the shallow shelf break with limited scope for stratification to decouple the flow from the 1342 surface slope. For these reasons, we have no firm estimate for eddy contributions to exchange. Exchanges on short time scales of order one day or less were often large relative to exchanges from 1349 motion on longer time scales (Tables 2, 4, 7) except for the deepest Faroe-Shetland Channel 1350 moorings. However, short-term exchange may be ineffective, if it is only movement to-and-fro 1351 without constituent transfer to a different water "parcel" or time to change character. Shear dispersion 1352 may nevertheless cause longer-term transfer from shorter-period oscillatory motion; shear generates 1353 vertical gradients from horizontal gradients and so facilitates vertical diffusion (probably turbulent).

1354
To model long-term 137 Cs transport on the north-west European continental shelf, Prandle (1984) 1355 44 assumed effective " -" and " -" diffusivities | |, | | to represent shear dispersion by largely 1356 barotropic tidal currents, being the predominant M2 tidal current amplitude; he found a best fit to 1357 observed distributions for ~10 3 sec. On this basis, and using Table 2 Tables 2, 4  1361 and 7, take into account tides, surges and inertial currents. However, the caveat remains: short-term 1362 displacements may not transfer constituents effectively. by smaller-scale processes may be less sensitive to slope direction definition, because their lesser 1391 constraint by geostrophy reduces the relative "contamination" by alongslope currents.

1393
Drifters may partly avoid these difficulties of localised moorings and uncertainty in slope direction 1394 and representativeness; drifter crossings of depth contours are clear. However, their tracks and hence 1395 spatial coverage are not controlled, and indeed are sensitive to conditions during deployment and 1396 subsequent interaction with ocean "weather". Response to the 2012 Celtic Sea storm is an example, 1397 with initial off-shelf crossings into deeper Biscay waters. Deployment location may also introduce 1398 biases, e.g. deployment in less than 500 m implies initial crossings of the 500 m contour can only be 1399 to deeper water (negative ̅). Malin drifters were deployed in more than 600 m but the few crossing 1400 the 150 m contour in July 2013 were still within ~ 8 km of each other; their velocities were similar but 1401 45 not widely representative. The 3-hourly interval for drifter locations is also limiting; drifter-based 1402 estimates of exchange ½h|u'| only partly include tidal excursions. This is demonstrated by a much 1403 smaller increase in exchange estimates from 3-hourly drifter positions (relative to daily positions) than 1404 is shown in estimates from moorings' total currents compared with low-frequency currents.

1406
Across-slope transport as internal tides' Stokes Drift may be estimated from moorings with useful 1407 accuracy. However, this contribution to transport is a lowest-order difference between Eulerian and 1408 Lagrangian approaches. Lagrangian transport onshore is necessarily zero at the coast, so any on-or 1409 off-shore Stokes drift tends to be offset by an opposing Eulerian transport (or by along-shore 1410 divergence or convergence). Transport estimates depend on appropriate interpretation and distinction 1411 between Stokes Drift and its bolus velocity component (section 4.1).

1413
Partition between "flux" and "exchange" depends on (subjective) choices of domain and averaging 1414 period. Consider for example a one-month record and variations from week to week. If "flux" is 1415 calculated as the average over the month, then weekly departures from that average (as well as within-1416 week variations) will be accounted as "exchange". If "flux" is calculated as the average each week, 1417 then only within-week variations will be accounted as "exchange". A similar transfer from 1418 "exchange" to "flux" applies if the spatial domain is sub-divided. Hence a small domain or short-1419 period averaging favours a time-varying flux and small exchange. A large domain or long-period 1420 averaging favours a relatively small or slowly-varying flux but large exchange (with more scope for 1421 deviations from the period average, for example). Here, for flux, we have typically averaged over 1422 about one month (necessarily less for moorings of shorter duration). Exchange at the long-term Celtic 1423 mooring LT1 highlights the effect, being much larger for record-length averaging ( The distinctive assumption (limitation), in using a salinity budget to infer effective diffusivity and 1436 hence exchange, is that the salinity distribution and large-scale transports controlling it are in a steady 1437 state. The inference process then involves (at least) two uncertain factors. (i) Extensive sections are 1438 used to estimate the salinity gradient and hence effective diffusivity ; the gradient cannot then be 1439 identified with a particular location and depth ℎ. (ii) Equivalence of − / and ′ ′ ̅̅̅̅̅ is used in 1440 section 3.5 to estimate exchange as ½ℎ( / ) 1/2 ; there is uncertainty in the appropriate to use as 1441 the time-scale associated with the "eddy" motions presumed to be responsible for the effective 1442 diffusivity . In respect of (i), the estimation of presumes a salinity maximum which some of the 1443 sections (particularly in SSB) do not include. However, in the wider context we know that the 1444 maximum exists: salinity values are greater in the slope current than in the adjacent deep ocean and 1445 decrease onto the shelf. More uncertainty, most evident in the historical sections (Table 10) The relatively direct estimates of transports (fluxes plus exchanges), from currents at moorings and 1510 drifters crossing depth contours, give shelf-break values in the range 1-5 m 2 s -1 excluding exchanges 1511 from tidal currents. The observations are localised (moorings) or spatially indiscriminate (drifters) so 1512 that representativeness and resolution by along-shelf sector are problematic. Limited duration also 1513 prevents resolution by season from observations alone. There are suggestions of enhanced cross-1514 slope dispersion off-shelf compared with on-shelf, more on the Malin shelf in summer than in winter 1515 (from altimetry) but otherwise more in winter than summer (from salinity sections).

1517
Overall along about 5000 km of shelf edge from Biscay to north of Shetland (following the 200 m 1518 depth contour closely) fluxes plus exchanges O(2 m 2 s -1 ) amount to O(10 Sv). This exceeds the 2.5 Sv 1519 modelled in Huthnance et al. (2009). However, the discussion there suggests processes adding to 1520 more than 2 m 2 s -1 exchange or more than 10 Sv overall in 5000 km. The time-scale to "renew" water on the shelf is proportional to the relevant shelf-sea volume and 1532 inversely proportional to the exchange (rate) with the adjacent ocean. Budgeting of transports for a 1533 shelf-sector "box" is formalised in Appendix C. Consider along-shelf inflow at the "southern" end, 1534 outflow at the "northern" end and exchange across the ocean side. Including shorter-scale processes 1535 (in the sense of section 5.4) but excluding tides, indicative exchange is O(2 m 2 s -1 ). For dimensions 1536   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  At each Malin shelf mooring site, more than one ADCP was deployed to ensure full water column 1958 coverage. Adjustments were therefore necessary to correct for compass misalignment. "Along-slope" 1959 was chosen (as for the Celtic Sea) as the direction of the mid-depth deployment-mean current, 1960 assumed to be near-geostrophic and therefore parallel with the isobaths. There was about 8° 1961 difference between mean current directions at sites LA and LB upstream of the canyon. Directions at 1962 sites (SD, SC) downstream of the canyon agree more closely. Signs and magnitudes of across-shelf 1963 fluxes are very sensitive to the defined "along-shelf" direction and to any ADCP compass errors, 1964 (much) more than 100% per 10° except at SF and SG. This is a consequence of strong along-slope 1965 fluxes at LA, LB, SB, SC, SD.

1967
At Hebrides 1995-1996 (SES) moorings S140, S400, across-slope flux estimates are very sensitive to 1968 the definition of "along-shelf" direction and to compass errors, again much more than 100% per 10°.

1969
Moreover, identifying the "along-shelf" direction for the SES ADCPs was problematic. The mean 1970 direction of flow was assessed for each deployment; estimates varied between -5° and 51° clockwise 1971 from North; one outlier was 168° at S140. It was concluded that S400 had a different compass bias 1972 for each deployment, corroborated by independent current meter data during deployments 1 and 5. 1973 S140 directions were confirmed by independent current meter data. To ameliorate the S400 compass 1974 errors, "corrections" were applied so that for each deployment the mean direction in the mid-depth 1975 range 100-250 m was N10°E; this chosen "along-shelf" direction is between the mean current-meter 1976 direction at S400 (N8.2°E) and the mean direction (N12.1°E) from ADCP and current meter 1977 deployments at S140. The realignment of S400 data should provide more realistic cross-shelf 1978 estimates, possibly at the expense of real seasonal variability in flux estimates (this procedure 1979 minimises net "cross-slope" flow through much of the depth range).

1981
For the Faroe-Shetland Channel, using the same approach as for the Celtic Sea, an along-slope 1982 direction N 38° E was adopted.

Declaration of interests
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.