M.-J Messias

C Andrieh

L M E H Mery

H Mercier

Tracing the North Atlantic Deep Water through the Romanche and Chain fracture zones with chloro#uoromethanes

Chloro#uoromethanes (CFMs) F-11 and F-12 were measured during August 1991 and November 1992 in the Romanche and Chain Fracture Zones in the equatorial Atlantic. The CFM distributions showed the two familiar signatures of the more recently ventilated North Atlantic Deep Water (NADW) seen in the Deep Western Boundary Current (DWBC). The upper maximum is centered around 1600 m at the level of the Upper North Atlantic Deep water (UNADW) and the deeper maximum around 3800 m at level of the Lower North Atlantic Deep Water (LNADW). These observations suggest a bifurcation at the western boundary, some of the NADW spreading eastward with the LNADW entering the Romanche and the Chain Fracture Zones. The upper core ( "34.70 kg m\) was observed eastward as far as 53W. The deep CFM maximum ( "45.87 kg m\), associated with an oxygen maximum, decreased dramatically at the sills of the Romanche Fracture Zone: east of the sills, the shape of the CFM pro"les re#ects mixing and deepening of isopycnals. Mean apparent water &&ages'' computed from the F-11/F-12 ratio are estimated. Near the bottom, no enrichment in CFMs is detected at the entrance of the fracture zones in the cold water mass originating from the Antarctic Bottom Water #ow.

Introduction

Deep waters (colder than 43C potential temperature) represent the major part of the world ocean water masses. Formed at high latitudes, the cold and dense North Atlantic Deep Water (NADW) and Antarctic Bottom Water (AABW) #ow through the Atlantic Ocean as deep western boundary currents. The deep Atlantic ocean is divided into two basins, west and east by the Mid-Atlantic Ridge (MAR). The contribution of the MAR fracture zones, which allow the passage of the deep and bottom waters from the western to the eastern basins, is a key issue in understanding the ventilation and circulation of the deep ocean (Mantyla and Reid, 1983). Evidence of the impact of the fracture zones on the circulation between the two basins has been shown in the Charlie-Gibbs Fracture Zone (westward #ux of over#ow &2.4 Sv: 1Sv"1;10 m s\) near 533N [START_REF] Saunders | The #ux of over#ow water through the Charlie-Gibbs Fracture Zone[END_REF] and in the Vema Fracture Zone at mid-latitude around 113N (geostrophic transport colder than 2.03C &2 Sv at the Vema exit, [START_REF] Speer | Tracing Lower North Atlantic deep water across the equator[END_REF]. The present paper focuses on the Romanche and the Chain Fracture Zones (referred to as RFZ and CFZ in the following), which are both possible pathways for the LNADW (Lower North Atlantic Deep Water) and the AABW across the MAR at the equator (Fig. 1). Some authors [START_REF] Speer | Tracing Lower North Atlantic deep water across the equator[END_REF]Friedrichs and Hall, 1993) have examined the splitting of the Deep Western Boundary Current (DWBC) eastward near the equator and con"rmed the role of these equatorial fracture zones in guiding the circulation of deep water. [START_REF] Mercier | Flow pathways of bottom water through the Romanche and chain fracture zones[END_REF] have described the pathways followed by the AABW through the equatorial fracture zones. The transport of bottom water through these fracture zones has been indirectly estimated to be comprised between 2.6 to 5.1 Sv for water colder than 2.073C by [START_REF] Schlitzer | Renewal rates of east Atlantic deep water estimated by inversion of C data[END_REF], and to be &2 Sv for water below 4000 m depth by [START_REF] Warren | Deep circulation in the eastern South Atlantic Ocean[END_REF]. Recent direct velocity measurements [START_REF] Mercier | Transport of bottom water in the Romanche fracture zone and the chain fracture zone[END_REF] estimated an eastward bottom water transport of 1.2 Sv.

Based on the hydrographic background of the region and previous tracer studies [START_REF] Metclaf | The sill depth of the Mid-Atlantic Ridge in the equatorial region[END_REF][START_REF] Metclaf | The sill depth of the Mid-Atlantic Ridge in the equatorial region[END_REF][START_REF] Weiss | Atmospheric chloro#uoromethanes in the deep equatorial Atlantic[END_REF][START_REF] Schlitzer | Renewal rates of east Atlantic deep water estimated by inversion of C data[END_REF], we expected to "nd recently ventilated deep water with detectable anthropogenic chloro-#uoromethanes (CFMs or freons) in these passages. Released to the atmosphere since the early 1930s, CFMs have tagged the surface ocean and entered the ocean interior in the areas of deep water formation, providing a very useful tool for the investigation of the circulation of newly formed water masses (Bullister, 1989). Ten years ago, during the Transient Tracers in the Ocean (TTO) program, the Upper North Atlantic Deep Water (UNADW) was traced using its core property of CFM maximum from the Labrador Sea along the western boundary [START_REF] Weiss | Atmospheric chloro#uoromethanes in the deep equatorial Atlantic[END_REF]. The signal has reached the equator, where it is found in the DWBC at a depth of about 1600 m (density anomaly relative to 1500 m, "34.63 kg m\). There, this water mass bifurcates, part extending east along the equator and part continuing south in the Brazil Basin DWBC. Fine and Molinari (1988), [START_REF] Molinari | The deep western boundary current in the tropical North Atlantic ocean[END_REF], and [START_REF] Rhein | The Atlantic deep western boundary current: water masses and transports near the equator[END_REF] have observed two cores of high CFMs in the DWBC between 26.53N and 133N, 14.53N and the equator, and downstream of 53S, respectively: the shallow UNADW core mentioned previously and a deeper core at about 3000}4000 m corresponding to the LNADW. Along the equator, CFM concentrations have also been measured near 1400}2250 and 4000 dbar (Doney and Bullister, 1992;AndrieH et al., 1998). Using CFM concentrations, the northward spreading of AABW has been observed as far as 53Sin the Brazil basin (AndrieH et al., 1998;[START_REF] Rhein | The Atlantic deep western boundary current: water masses and transports near the equator[END_REF].

The present paper is structured as follows. Section 2 is a presentation of both the Romanche cruises and the experimental methods. In Section 3, we describe and analyze the characteristics of the deep and bottom water masses at the entrance of the RFZ and CFZ as well as the CFM spatial distributions along these fracture zones. In Section 4 and 5, we describe the CFM maxima and the transformations of the water masses as they #ow eastwards. In Section 6, we then compute the CFM ratio and estimate the &&apparent ages'' of the deep water masses tagged by the CFM extrema as well as the &&apparent dilution'' factors. Finally, Section 7 gives a summary and some concluding remarks.

Data

The Romanche 1 and 2 cruises were part of the Deep Basin Experiment (DBE) of the World Ocean Circulation Experiment (WOCE). They consisted of bathymetric and hydrographic surveys of the Romanche and the Chain Fracture Zones (0345S and 193W, 232S and 153W, respectively) from the N.O. ¸1Atalante. Maps of the area with the CFM station positions are shown in Fig. 2. Romanche 1 (August 10 to September 8, 1991) was an exploration of the hydrography and the bathymetry of the two fracture zones. A description of the bathymetry of the Romanche and Chain Fracture Zones has been published by [START_REF] Mercier | Flow pathways of bottom water through the Romanche and chain fracture zones[END_REF]. We summarize brie#y the main structure of the bathymetry: the RFZ (about 800 km long) and CFZ (about 300 km long) are deep openings across the MAR that have average heights of about 3000 m (with summits from 3400 m to less than 2000 m). The width of the CFZ is narrower than that of the RFZ (5}20 km versus 10}40 km). The RFZ and CFZ main sills are found at depths of 4350 and 4050 m, respectively. In the eastern part of the MAR, an alternating pattern of rises and valleys, referred to as the communication region by [START_REF] Mercier | Flow pathways of bottom water through the Romanche and chain fracture zones[END_REF], connects the CFZ to the RFZ. During the Romanche 2 cruise (November 11 to 30, 1992), the hydrographic coverage was better resolved in the sill region and extended eastward.

At each station, 16}32 samples were taken from an 8 l sampling-bottle rosette equipped with a Neil Brown conductivity-temperature-depth-oxygen pro"ler (CTDO

). The tabulated data, the calibration of the CTDO measurements and the validation of the bottle data are published in data reports (Mercier et al., 1992;[START_REF] Rhein | The Atlantic deep western boundary current: water masses and transports near the equator[END_REF]. The error in the pressure was estimated to be $3 dbar. The errors in the oxygen, salinity and temperature are, respectively, on the order of $0.003 ml/l, $0.003 and $0.0033C and those for nutrients (measured only during the cruise Romanche I with an Auto-Analyser II Technicon) of $0.1%.

The chloro#uoromethanes F-11 (CCl F) and F-12 (CCl F ) were measured by electron capture detection gas chromatography coupled to a purge-and-trap extraction technique similar to that of Bullister and Weiss (1988). Samples were taken in glass syringes. On the Romanche 1 cruise, contamination problems precluded the use of the F-12 data for stations 4 to 39. The F-11 and F-12 data are reported on the SIO 1986 calibration scale. The atmospheric concentrations were estimated, with a precision of $1% from daily measurements, to be 500.5 ppt (part per trillion) for F-12 and 268.4 ppt for F-11 during Romanche 1 and 515.5 ppt of F-12 and 275.3 ppt for F-11 during Romanche 2. The near surface equatorial Atlantic is subject to an intense seasonal upwelling that brings the water originally downwelled in the subtropical gyres to the surface, and we found a CFM saturation in surface water of about 88% during Romanche 1 (upwelling intensi"ed in August) and 92% during Romanche 2 (November). All the CFM measurements were corrected for the sample blank (bottle, glass syringes, storage) and the analytical system blank. The blanks have been estimated from the CFM concentrations measured in assumed CFM-free deep water masses (i.e. Circumpolar Deep Waters). Because we observed an evolution of the blank values during the cruises, the blanks were estimated at each station. During Romanche 1, blanks for F-11 ranged from 0.008 to 0.021 pmol kg\ (mean: 0.015$0.004 pmol kg\). Blanks for F-12 ranged from 0.011 to 0.019 pmol kg\ for station 41 to 71 (mean: 0.015$0.003 pmol kg\). During the Romanche 2 cruise, blanks for F-11 were similar, with values from 0.008 to 0.019 (mean: 0.014$0.004). For F-12, the blanks had lower values, from 0.005 to 0.013 (mean: 0.009$0.003). In order to check the blank level of all hydrographic bottles, the entire rosette was tripped at a single depth (3000 dbar) in CFC-free water for the Romanche 2 cruise. The averages of the measurements (around 0.010$0.005 pmol kg\ for F-11 and 0.007$0.003 pmol kg\ for F-12 at the end of the cruise) on these samples were close to the blanks estimated by station. The standard deviation of the blanks (including the accuracy of the analytical system as well as the e!ects of contamination) determines the detection limit, which is around 0.005 pmol kg\ for F-11 and 0.003 pmol kg\ for F-12 for both cruises. Samples from deep levels were sometimes quite close to the detection limit, but the signi"cant number of similarly measured values shows that they represent a real signal.

Observations

Water mass properties and CFM vertical proxles

The sources and the deep water mass characteristics in the equatorial Atlantic have been recently investigated by several authors [START_REF] Weiss | Atmospheric chloro#uoromethanes in the deep equatorial Atlantic[END_REF]Fine and Molinari, 1988;[START_REF] Pickart | Determining the strength of the deep western boundary current using Chlorofuoromethane ratio[END_REF][START_REF] Speer | Tracing Lower North Atlantic deep water across the equator[END_REF][START_REF] Molinari | The deep western boundary current in the tropical North Atlantic ocean[END_REF]Friedrichs and Hall, 1993;[START_REF] Smethie | Tracing the thermohaline circulation in the western North Atlantic using chloro#uorocarbons[END_REF][START_REF] Rhein | The Atlantic deep western boundary current: water masses and transports near the equator[END_REF]AndrieH et al., 1998). In particular, hydrography of the LNADW and the AABW along the RFZ and the CFZ during the cruise Romanche 1 has been presented by [START_REF] Mercier | Hydrography of the Romanche and chain fracture zones[END_REF]. At the fracture entrances, the CFM distribution (Fig. 3) was found to be structured as in the DWBC (Fine and Molinari, 1988;[START_REF] Molinari | The deep western boundary current in the tropical North Atlantic ocean[END_REF]. In the NADW, two familiar CFM maxima observed at 1600 and 3800 m appear as evidence of eastward feeding from the DWBC towards the fracture entrances of the recently ventilated components of the warm, saline, oxygen rich NADW. The fresh, low oxygen and high nutrient water masses of circumpolar origin [START_REF] Reid | On the total geostrophic circulation of the South Atlantic Ocean: #ow patterns, tracers, and transports[END_REF]) are found at 1000 and 3000 m depth and at the bottom below 4000 m, sandwiching the NADW. Circumpolar Deep Water (CDW) results from the mixing of deep-waters from the Indian, Paci"c and Atlantic Oceans along a path around Antarctica. Due to the relatively long time since these deep waters were in contact with the atmosphere, these waters are CFM-free at the entrance to the RFZ and CFZ. Below we detail the main hydrographic characteristics of the deep-water masses in accordance with the CFM signals (Figs. 3 and4).

The F-11 distribution clearly identi"es the main components of the NADW. The upper F-11 maximum ('0.04 pmol kg\) coincides with the salinity maximum (34.98 at "34.66 kg m\ of the UNADW that WuK st (1935) attributed to the Mediterranean Sea Water in#uence. It enhances the core of the most recently ventilated NADW originating from the southern Labrador Sea [START_REF] Pickart | Water mass components of the North Atlantic deep western boundary current[END_REF]. The F-11 core extends from 1300 to 1900 m depths and is centered around 1600}1700 m ( "34.63 kg m\and "3.83C, all temperatures referred to in the following are potential temperatures). Between 1700 and 1800 m, it is associated with phosphate (1.1 mol kg\ at "36.94 kg m\) and silicate (20 mol kg\ at "36.89 kg m\) minima, as previously observed in the equatorial Atlantic [START_REF] Weiss | Atmospheric chloro#uoromethanes in the deep equatorial Atlantic[END_REF][START_REF] Reid | On the total geostrophic circulation of the South Atlantic Ocean: #ow patterns, tracers, and transports[END_REF]Doney and Bullister, 1992). The top of the UNADW is marked here by a slight maximum in temperature of 4.423C around 1200 m (1150}1300 m, "27.6 kg m\). Above, the "rst free-CFM level corresponds to the upper part of the Circumpolar Deep Water (UCDW), centered around 1000 m ( "27.3 kg m\), in agreement with [START_REF] Reid | On the total geostrophic circulation of the South Atlantic Ocean: #ow patterns, tracers, and transports[END_REF] description of CDW at the equator. The UCDW is characterized by a minimum in temperature around 4.363C and low oxygen values (162 mol kg\). Here, it lies below the Antarctic Intermediate Water (AAIW), which is recognized by its salinity minimum (34.5) and silicate maximum (33.8 mol kg\) at approximately 700}800 m. In the North Atlantic [START_REF] Smethie | Tracing the thermohaline circulation in the western North Atlantic using chloro#uorocarbons[END_REF]) and downstream at 83N [START_REF] Molinari | The deep western boundary current in the tropical North Atlantic ocean[END_REF], the upper CFM core has been observed centered at 4.53C. This suggests that the CFM maximum is truncated in the upper part of the UNADW by the UCDW at the equator as suggested by [START_REF] Rhein | The Atlantic deep western boundary current: water masses and transports near the equator[END_REF]. The oxygen maximum between 1700 and 2400 m depths de"nes the Middle North Atlantic Deep Water (MNADW) according to WuK st (1935). It is centered at about 2000 m ( " 34.7 kg m\, "36.95 kg m\) and is not associated with the CFM core. This oxygen maximum is thought to be the classical Labrador Sea Water (LSW; [START_REF] Pickart | Water mass components of the North Atlantic deep western boundary current[END_REF], which has been associated with a relative minimum of F-11 in the DWBC [START_REF] Molinari | The deep western boundary current in the tropical North Atlantic ocean[END_REF]. The upper part of the oxygen maximum overlaps the high salinity layer and the CFM signal, which decreases to undetectable levels around 2500 m. Below, the second CFM-free level ( "45.83 kg m\) is associated with a minimum in oxygen and includes components of both NADW and CDW. The CFM free NADW component (2500}2900 m) corresponds to Gibbs Fracture Zone Water (GFZW), which is a mixture of Iceland Scotland Over#ow Water (ISOW) and North East Atlantic water (NEAW) [START_REF] Smethie | Tracing the thermohaline circulation in the western North Atlantic using chloro#uorocarbons[END_REF][START_REF] Reid | On the total geostrophic circulation of the North Atlantic Ocean: #ow patterns, tracers, and transports[END_REF][START_REF] Rhein | The Atlantic deep western boundary current: water masses and transports near the equator[END_REF]. The ISOW is fed by Norwegian Deep Water not recently renewed. Hence, the GFZW contributes little to the CFM signal when it joins the DWBC in the western Atlantic through the Gibbs Fracture Zone [START_REF] Smethie | Tracing the thermohaline circulation in the western North Atlantic using chloro#uorocarbons[END_REF]. Moreover, [START_REF] Molinari | The deep western boundary current in the tropical North Atlantic ocean[END_REF], [START_REF] Rhein | The Atlantic deep western boundary current: water masses and transports near the equator[END_REF]AndrieH et al. (1998) suggest weak ventilation of the GFZW in the DWBC, which accentuates the CFM minimum. The Circumpolar Deep Water found beneath also has oxygen and CFM minima, but it is marked by a slight maximum in nitrate (1.39 mol/l), phosphate (1.39 mol/l) and silicate (39.8 mol/l). In the DWBC, [START_REF] Molinari | The deep western boundary current in the tropical North Atlantic ocean[END_REF] attribute lower salinity near 3500 m to a water mass of circumpolar origin. The characteristics of this southern water mass are more marked in the range of 3000}3500 m (Fig. 4) at the CFZ (1.223S) than at the RFZ (0.273S) entrance and then decrease north-eastwards. The lower F-11 core, centered at 3800 m ( "45.87 kg m\) tags the highly oxygenated (259 mol kg\) and salty densest layer of the NADW (LNADW according to WuK st, 1935) originating from the Denmark Strait Over#ow (DSOW). This F-11 maximum is lower (0.02 pmol kg\) than the maximum observed in the UNADW. The feeding of the DSOW used to be attributed principally to the upper Arctic Intermediate Water [START_REF] Swift | The contribution of the Denmark Strait over#ow to the deep North Atlantic[END_REF][START_REF] Smethie | Tracing the thermohaline circulation in the western North Atlantic using chloro#uorocarbons[END_REF] formed in the Iceland and Greenland Seas by winter convection. This water mass would exit the Nordic Seas through the Denmark Strait 1}2 years after its formation, loaded with high CFM concentrations [START_REF] Smethie | The Tritium: Krypton-85 age of Denmark strait over#ow water and Gibbs fracture zone water just south of Denmark Strait[END_REF][START_REF] Smethie | Tracing the thermohaline circulation in the western North Atlantic using chloro#uorocarbons[END_REF]. Recently, based on new data sets, on tracer conservation and considerations on heat #uxes, Mauritzen (1996) has presented an alternative to this classical scheme. After strong buoyancy losses during their northward journey in the Norwegian sea, Atlantic waters are carried into the Arctic Ocean. They are then modi"ed and #ow back southward through the Fram Strait, in the East Greenland Current, with considerable speed, and almost no mixing. The LNADW at the entrance of the fracture zones is associated with minima in silicate and phosphate. The temperature range of the lower F-11 core was found between 2.23C and around 1.6}1.53C. The historic delineation for AABW, 1.93C (McCartney and Curry, 1993), corresponds in our data to the lower CFM maximum, certainly due to upstream vertical mixing between the AABW and the LNADW [START_REF] Mercier | Hydrography of the Romanche and chain fracture zones[END_REF]. The transition between the LNADW and AABW corresponds to important monotonic gradients. Below 4700 m, the pro"les are homogeneous in temperature (0.75}0.623C), salinity (34.76}34.75), oxygen (231.6}230.4 mol kg\) and density (45.997( (46.001).

These characteristics associated with a CFM free level, a silicate (100 mol kg\) and a phosphate maximum (1.96 mol kg\) identify the Lower Circumpolar Deep Water (LCDW with '45.92), which is less dense than the other component of AABW, i.e. the oxygen-CFM-rich Weddell Sea Deep Water (46.006(

). The Weddell component has been observed at the bottom of the Brazil basin at around 53S}4330S (AndrieH et al., 1998;[START_REF] Rhein | The Atlantic deep western boundary current: water masses and transports near the equator[END_REF]. AndrieH et al. (1998) suggested a pathway of these two components of the AABW at the exit of the Brazil basin: the WSDW component would spread on the western side, while the older, less dense component would instead lie on the eastern side. The characteristics of the bottom water at the entrance of the RFZ seem to indicate that some of the CFM free last component could have turned northeast to feed the RFZ. 2 sections along the RFZ (Figs. 5 and9) are compared. The F-11 pro"les below 500 m depth for the station reoccupied during Romanche 2 are presented in Fig. 10. The concentrations measured during the Romanche 2 cruise always appear equal to or higher than the concentrations measured during Romanche 1. The mean increase of the LNADW F-11 maximum is around 0.01 pmol kg\. At the level of the UNADW, the F-11 increase between the two cruises appears to be larger, with concentration di!erences reaching 0.05 pmol kg\.

The F-11 distributions along the RFZ and CFZ show the upper and lower F-11 NADW cores extending eastwards in a CFM-free deep ocean (Figs. 5, 6 and9). The lower core, constrained by the topography, forms a plume, which decreases eastward in the two fractures, suggesting a transport from west to east. The LNADW F-11 values are stronger in the RFZ (193W}0.453S) than the CFZ (153W}1.53S), which is farther southeast from the source. This observation can also be associated with the large-scale meridional gradient observed in other tracer distributions upstream of the fracture zones at 253W [START_REF] Tsuchiya | Water-mass distributions in the western South Atlantic; a section from South Georgia Island (54S) northward across the equator[END_REF]. Fig. 8, drawn from the Romanche 2 measurements, is the meridional section (33S}53N) through both the RFZ (153W) and the CFZ (13350W): the F-11 LNADW signal presents a southward decrease from the RFZ, where levels as high as 0.03 pmol kg\ are observed (whereas in the CFZ the levels are less than 0.02 pmol kg\). These meridional F-11 gradients seem principally to re#ect the gradients at the entrances of the fracture zones. Along the RFZ valley (Fig. 5), the LNADW F-11 signal during the Romanche 1 cruise appears a!ected at the "rst sill of 4750 m depth (sill W) decreasing below 0.02 pmol kg\. After the main sill (4350 m), the signal is not detectable in the Romanche 1 data. The F-11 distribution across the RFZ can be visualized in three cross channel sections (Fig. 7) in the RFZ, at 163W, 153W and 143W, respectively. The F-11 lower core seems to be slightly intensi"ed at the RFZ northern wall although the resolution is too coarse to ascertain that feature. Note that this feature is observed right from 163W before the RFZ south wall vanishes and where it should not be linked to these numerous gaps. In the CFZ (Fig. 6), the F-11 signal decreases below the detection limit at station 48 before the main sill. In the deep valleys of the communication region (not shown), some faint scattered F-11 signals (around 0.01 pmol kg\) were measured at stations 46, 54, 44 and 57 around 4200, 3800, 4750 and 4400 m depths, respectively, suggesting an in#uence of recently ventilated LNADW. According to [START_REF] Mercier | Flow pathways of bottom water through the Romanche and chain fracture zones[END_REF], [START_REF] Mercier | Flow pathways of bottom water through the Romanche and chain fracture zones[END_REF] showed that both basins are fed by bottom water following two distinct pathways after a branching, occurring at about 13312W. In Fig. 8, the bottom F-11 signal visible around 23N (Sta. 3) corresponds to the northern exit feeding the Sierra Leone Basin, whereas the F-11 signal within the axis of the RFZ feeds the Guinea Basin. The section along the RFZ channel (Fig. 9) shows that the F-11 maximum is a!ected while crossing the sills as in the Romanche 1 observations, with values decreasing by 0.02 pmol kg\ between Stations 4, 5 and 14. The F-11 maximum located between 3600 and 4300 m (2.43C/1.53C) within the RFZ deepens abruptly towards the eastern basin after passing over the main sill. Using the High-Resolution Pro"ler, [START_REF] Polzin | Intense mixing of Antarctic bottom water in the equatorial Atlantic Ocean[END_REF] showed that the bottom water #ow accelerates downstream of the RFZ main sill, becomes unstable, and feeds intense turbulence, which implies vertical mixing between CFM-free and CFM loaded waters and could explain the observed pertubated tracer signal of the out#ow. In the Guinea abyssal plain, the F-11 maximum was found between 4500 m and the bottom within the densest water (1.453/1.653C). The F-11 concentration weakens eastward and seems to decrease below the detection limit at about 113W from the Romanche data set. More recently in early 1995, evidence of the eastward invasion of the anthropogenic transient tracer in the Guinea Basin was detected at 93W (Messias and MeH mery, 1998). Along the pathway toward the Sierra Leone Basin, we followed the F-11 maximum up to 23N. Stations 18}20 along 123W (not shown) through the northern exit present concentrations around 0.015 pmol kg\, which are also at greater depths (4700, 4870 and 5088 m) than within the RFZ. This observation suggests that intense mixing also occurs on the pathway toward the Sierra Leone Basin, where [START_REF] Polzin | Intense mixing of Antarctic bottom water in the equatorial Atlantic Ocean[END_REF] found evidence for a boundary current.

The UNADW F-11 core was present at all the stations during both Romanche surveys (up to 53W). Both data sets (Figs. 5, 6 and 8) con"rm the southern position (relative to the equator) of the UNADW maxima for both summer and winter seasons. The meridional section (33S}53N) crossing the RFZ and the CFZ locates this maximum (values higher than 0.06 pmol kg\) between 035S (1500 m) and 33S (1600 m). The same indication of a southern shift of the CFM maximum is noted by Doney and Bullister (1992) at 283W and by [START_REF] Rhein | The Atlantic deep western boundary current: water masses and transports near the equator[END_REF]

The shallow CFM maximum

Fig. 11 shows F-11, oxygen and salinity versus between 950 and 3050 m depth. The oxygen maximum is centered at 2000 m ( "34.7 kg m\), whereas the CFM and salinity maxima are shallower, between 1600 and 1700 m ( "34.63 kg m\). The di!erence in depth of the CFM and oxygen maxima was previously observed along the DWBC to the north [START_REF] Pickart | Water mass components of the North Atlantic deep western boundary current[END_REF] and in the equatorial Atlantic (Weiss et al., 1985;[START_REF] Molinari | The deep western boundary current in the tropical North Atlantic ocean[END_REF][START_REF] Rhein | The Atlantic deep western boundary current: water masses and transports near the equator[END_REF]. Several explanations have been given in the literature. At 403N, [START_REF] Pickart | Water mass components of the North Atlantic deep western boundary current[END_REF] observed that the oxygen content of the UNADW is reduced by consumption and vertical mixing with low oxygen water lying above. [START_REF] Rhein | The Atlantic deep western boundary current: water masses and transports near the equator[END_REF] add that this water mass, located around 600}700 m in the northern Atlantic, has a higher oxygen consumption than deeper water masses. [START_REF] Rhein | The Atlantic deep western boundary current: water masses and transports near the equator[END_REF] also suggest that convection to 500 m depth, which occurs for the UNADW [START_REF] Talley | Distribution and circulation of Labrador Sea Water[END_REF][START_REF] Pickart | Water mass components of the North Atlantic deep western boundary current[END_REF], would produce higher CFM concentrations compared to convection to 2000 m depth, which occurs for the classical LSW (Gascard and Clarke, 1983). Note that incomplete convection during the 1960s and 1970s could have also produced low CFM LSW [START_REF] Wallace | Anthropogenic chloro#uoromethanes in newly formed Labrador Sea water[END_REF].

The CFM and salinity distributions show features that are strikingly correlated (Fig. 11). During the Romanche 1 cruise, we observed a variability of the depth of the CFM maxima, which were found between 1400 and 1900 m (Figs. 12a/d). The southern (37,38,41) and western (4) stations present regular, well-de"ned maxima in CFM and salinity at the same depth near 1700 m (see for example Fig. 12a andb, station 41). In contrast, to the northeast of the studied region (stations 70, 58), the F-11 maximum (near 1550 m) seems truncated from below (see for example Fig. 12a andb, station 70). Simultaneously, the corresponding salinity pro"les are diminished and pertubed. This suggests the in#uence of a water mass with lower CFM and salinity in the northeastern region of the RFZ at the density of "34.65. Station 19 shows the intermediate situation. We observe an increase both in the salinity and the F-11 concentrations between Romanche 1 and Romanche 2 cruises (Fig. 11). This increase is clearly seen by comparing pro"les (Fig. 12a}d) of the stations done at the same position during the two cruises (0.012 in salinity and 0.03 pmol kg\ in F-11 relative to station 70 of Romanche 1). In the Romanche 2 data, the upper CFM signal in the northeast of the region is split into two maxima separated by slight minima at 1600 m depth. The high salinity of the upper CFM core can result from double di!usive mixing with the overlying water of Mediterranean origin [START_REF] Sarmiento | Three-dimensional ocean models for predicting the distribution of CO between the ocean and atmosphere in the changing carbon cycle: a global analysis[END_REF]. BoK ning and Schott (1993) show a map of the salinity tongue at the 1500 m depth #owing parallel to the equator (their Fig. 1). The salinity gradient observed in this map in the N-E zone of the Romanche 1 area could be responsible for the variability of the Romanche salinity pro"les. Another explanation suggested by the particular feature of the upper transient tracer signal could be as follows. The lateral distribution of CFMs shows zones of alternating high and low concentration extrema in the most northeastern stations. This distribution of the high F-11 lenses seems to indicate that the upper CFM signal is not monotonically transported eastward along the equator. One possible mode of transport is by non-stationary currents or jets (horizontal mixing). A vertical pro"le of velocities measured on 17 January 1989 in the equatorial Atlantic (0300N, 30322W) reveals westward #ows from 1400 to 1600 m and eastward #ows between 1600 and 2200 m [START_REF] Ponte | Equatorial deep jets in the Atlantic Ocean[END_REF]. SOFAR #oat data at nominal depths of 1800 m [START_REF] Richardson | Deep cross-equatorial #ow in the Atlantic measured with SOFAR #oats[END_REF] show the presence of reversing equatorial currents with timescales of the order of 18 months. [START_REF] Richardson | Deep cross-equatorial #ow in the Atlantic measured with SOFAR #oats[END_REF] found an eastward equatorial current followed a year later by a reversing westward equatorial current. They suggest that this could have advected the observed F-11 tongue to the east along the equator. The high variability of the F-11 pro"les that we found can result from the e!ects of these reversing #ows in the north of the main CFM core. The model calculations of BoK ning and Schott (1993) yielded seasonally reversing eastward and westward transport with only a small mean annual eastward transport. BoK ning and Schott (1993) add that the latter was su$cient to maintain the salinity tongue. The temporary westward current may perhaps diminish the spreading of the transient tracers and in particular CFMs (see Section 6).

The deep CFM maximum

Two elements can explain the dramatic decrease of the F-11 signal in the RFZ: the transient behavior of the CFM and the dilution by lateral and vertical mixing. Direct current measurements in the fracture zone have shown permanent and important eastward velocities [START_REF] Polzin | Intense mixing of Antarctic bottom water in the equatorial Atlantic Ocean[END_REF]Mercier and Speer, 1997). Due to the very short transit time for #uid parcels within the RFZ, transient e!ects should not be important in establishing the zonal gradient. As seen in Section 3.1, the deep CFC maximum used to trace the LNADW is associated with an oxygen maximum at the entrances of both fracture zones (initial density between "45.85 and 45.95). Comparison of the oxygen and the F-11 signal along the RFZ and the CFZ shows that the two tracer maxima have a similar evolution (Fig. 13a andb), decreasing through the sills. At the eastern exit of the RFZ, the oxygen and F-11 pro"les below 3500 m are relatively homogenous (station 58, Fig. 13b). In Fig. 14, we observe a simple linear trend between F-11 and oxygen characteristics for deep waters. After the sills, the F-11 } oxygen correlation is no longer evident due to the low F-11 concentrations (Fig. 14b). In the eastern basin, the lower CFM signal is weaker in oxygen and also colder, deeper, less salty and richer in nutrients than at the entrance of the channel, giving evidence of the AABW in#uence. The coldest and purest bottom water used to trace the AABW, shown in Fig. 15, has undetectable values in F-11 within the RFZ. The anticorrelated hydrographic characteristic changes in the LNADW compared to the AABW supports the idea that vertical mixing is the primary mechanism for the modi"cation of both water masses through the RFZ sills, in agreement with the "ndings of [START_REF] Polzin | Intense mixing of Antarctic bottom water in the equatorial Atlantic Ocean[END_REF]Ferron et al. (1998), who inferred highly e$cient vertical mixing and averaged diapycnal di!usion coe$cients of 150 cm s\ with maximum values as large as 0.1 m s\ downstream of the Romanche Fracture Zone main sill.

Apparent ages and dilution factors

Based on the continuous increase from 1931 until the mid-1970s of the F-11 : F-12 ratio in the atmosphere, we can estimate the date at which a water parcel became isolated from its source region by its F-11 : F-12 concentration ratio (Fig. 16). The apparent age of the water sample is then de"ned as the time di!erence between the year corresponding to the original atmospheric ratio of the water sample and the date of the sampling. We follow the method of [START_REF] Weiss | Atmospheric chloro#uoromethanes in the deep equatorial Atlantic[END_REF]: the original atmospheric ratio is computed from the sample F-11 and F-12 concentrations using solubility functions [START_REF] Warner | Solubilities of chloro#uorocarbons 11 and 12 in water and seawater[END_REF]. The estimated precision and accuracy of the CFM solubility functions are &0.7 and 1.5%, respectively. The source water is assumed to be in approximate solubility equilibrium with the atmosphere. We have computed the apparent ages for the UNADW and the LNADW CFM cores using the northern hemisphere 1931}1990 input function as provided to us by Weiss (personal communication, 1992). The method assumes that mixing occurs with CFM-free water so that the F-11 : F-12 ratio is not altered by mixing after the water parcel becomes isolated from the source region. In reality, the residence time of the water mass in its formation area or mixing with water loaded in CFM tends to lower the F-11 : F-12 ratio and leads to an overestimation of the apparent age [START_REF] Pickart | Determining the strength of the deep western boundary current using Chlorofuoromethane ratio[END_REF]. The two CFM tongues observed in the Romanche area are surrounded by CFM-free waters. But, mixing with water loaded with CFMs could have occurred upstream, notably in the DWBC (with adjacent water accumulating CFMs, see [START_REF] Pickart | Determining the strength of the deep western boundary current using Chlorofuoromethane ratio[END_REF]. In the present study, no correction of the F-11 : F-12 ratio relative to these processes was made and the computed age estimates are an upper bound of the true age. A dilution factor is derived by dividing the CFM source surface water concentration at the time of formation inferred from the apparent age by the observed CFM concentration. The dilution factor critically depends on the water mass source saturation. Equilibrium between atmosphere and ocean is not always achieved in deep water formation regimes, a!ecting the CFM concentration of newly formed deep water [START_REF] Rhein | The deep western boundary current: tracers and velocities[END_REF]. [START_REF] Smethie | Tracing the thermohaline circulation in the western North Atlantic using chloro#uorocarbons[END_REF] in equilibrium [START_REF] Warner | Solubilities of chloro#uorocarbons 11 and 12 in water and seawater[END_REF] with reconstructed atmospheric concentrations in the Northern Hemisphere.

Saturations of only 60% have been reported, for example, in the case of the newly formed LSW [START_REF] Wallace | Anthropogenic chloro#uoromethanes in newly formed Labrador Sea water[END_REF]. Therefore, the observed concentrations give a lower value of dilution [START_REF] Weiss | Atmospheric chloro#uoromethanes in the deep equatorial Atlantic[END_REF][START_REF] Pickart | Determining the strength of the deep western boundary current using Chlorofuoromethane ratio[END_REF]. A limitation in the Romanche study is the very low CFM concentrations that lead to large errors in the F-11 : F-12 ratio (ranging from 10% for F-11"0.06 pmol kg\ to 100% for values below 0.015 pmol kg\), which leads to even larger uncertainties for the derived parameters.

In Fig. 17, we report the apparent age versus the F-11 : F-12 ratio calculated for the Romanche 1 and Romanche 2 stations. For the UNADW, the results are relatively homogenous. The calculated atmospheric F-11 : F-12 ratio of the UNADW (average for each station) ranges from 1.65 to 2. It is characteristic of surface waters from 1966 to 1972 and implies a mean travel time from the source region to the RFZ of around 25$4 years. Mean dilution factors for Romanche 1 and Romanche 2 are 27$10 and 36$12, respectively, for the transient tracer compounds in this layer. The values of the apparent age obtained are consistent with values previously inferred from CFMs along the DWBC in the literature, which increase in the downstream direction (Table 1a). [START_REF] Smethie | Tracing the thermohaline circulation in the western North Atlantic using chloro#uorocarbons[END_REF] gave an apparent age for the UNADW of &11 years at 443N and &18 years at 333N. Further south, the values reach &20 years at 7. 33N (AndrieH et al., 1997) and &23 years at the equator [START_REF] Weiss | Atmospheric chloro#uoromethanes in the deep equatorial Atlantic[END_REF]. The F-11 value of the upper core was found to be diluted by a factor of 5 at the equator in 1983 [START_REF] Weiss | Atmospheric chloro#uoromethanes in the deep equatorial Atlantic[END_REF]. AndrieH et al. (1998) estimated dilution factors ranging from 8$1a t 7330N within the DWBC to around 50$30 at 335W in the equatorial eastern Atlantic. [START_REF] Rhein | The Atlantic deep western boundary current: water masses and transports near the equator[END_REF] reported a dilution factor of 29 before the water reaches the equator. The high dilution factor in the UNADW along the equator may re#ect the e!ects of time-dependent reversing equatorial currents or jets [START_REF] Richardson | Deep cross-equatorial #ow in the Atlantic measured with SOFAR #oats[END_REF] discussed above (Section 4). These #ows advect the CFM tongue eastward into the surrounding CFM free water, but also return westward, increasing the mixing of the transient tracer compounds in this layer. In the present work, the LNADW CFM core appears signi"cantly older but less dilute than the UNADW core. The F-11/F-12 ratios measured in the LNADW core range from 0.6 to 1.5, giving an average apparent age of &36$5 years and a dilution of 10$5. These values are consistent with previous CFC-based estimates in the DWBC further north (Table 1b) [START_REF] Pickart | Determining the strength of the deep western boundary current using Chlorofuoromethane ratio[END_REF] have shown how mixing can reduce the F11 : F12 ratio in the core of the DWBC. Assuming that CFM-free mixing occurs in the current and that the CFM ratio is not altered during deep water formation and over#ow, the predicted core speed from the F11/F12 ratio, was 2 cm s\.M o d e l i n gt h e DWBC by simple advective}di!usive boundary current models coupled to a convection model of the Iceland}Scotland over#ow, [START_REF] Pickart | Determining the strength of the deep western boundary current using Chlorofuoromethane ratio[END_REF] obtained core speeds of 5}10 cm s\ for the DWBC from matching model solutions to CFM data collected along the western boundary. These values are of the same order of magnitude as DWBC velocities observed with current meters, generally 5}7cms\ south of the Grand Banks (Watts, 1991). Using the ideas of [START_REF] Pickart | Determining the strength of the deep western boundary current using Chlorofuoromethane ratio[END_REF], [START_REF] Rhein | The deep western boundary current: tracers and velocities[END_REF] obtained by modeling transient tracer a mean velocity of 4.8 cm s\ for the DWBC south of the Faroe Bank to 103S. as well as with a dilution factor of about 16 (8}32) calculated by Doney and Jenkins (1994) at 203N using TTO Tritium-He. Compared to the UNADW core, the low LNADW dilution factor (&10 found within the RFZ before the sills) suggests a more steady #ow guided by the topography that limits mixing with CFM-free water. Dilution factors increase to around 30$20 beyond the sills. In Fig. 18, we have reported the F-11 and F-12 concentrations, the apparent age and the dilution factor along the RFZ for the LNADW and the UNADW. We use only stations along the RFZ axis with well-de"ned CFM maxima. The dramatic decrease of the LNADW CFM concentration along the fracture axis is not linked to the age of the water, which stays constant. In contrast, the dilution factor increases from the west to the east. Hence, the decrease of the CFM concentration appears to be mainly the result of mixing by dilution with CFM-free water rather than the result of the transient behavior of the CFMs. However, the very low concentration of CFMs in the LNADW implies an error close to 100% in the ratio F-11 : F-12 and raises concerns about the possibility of estimating the transient behavior.

The apparent mean current speed has been calculated from the F-11 : F-12 ratio apparent age and the distance from the source. The travel time of about 25 years from the Labrador Sea source and 35 years from the Nordic Seas to the Romanche area indicates mean apparent velocities of 1.4}1.3 cm s\ for the UNADW and 1.2 cm s\ for the LNADW. These values fall in the range of previous estimates for the NADW in the DWBC or along the equator based on CFMs (Table 1) and Tritium-He estimates (on the order of 0.75}1.5 cm s\, Doney and Jenkins, 1994) but are signi"cantly smaller than those measured using direct methods. The interpretation of the tracer-based velocity estimates has been much discussed in recent years [START_REF] Pickart | Determining the strength of the deep western boundary current using Chlorofuoromethane ratio[END_REF], McCartney, 1993;Smethie Jr., 1993, Doney andJenkins, 1994). Mean spreading rates estimated with transient tracers include the e!ects of mixing and recirculation with the alongstream advection along the travel path and lead to a lower limit. CFM modeling experiments [START_REF] Pickart | Determining the strength of the deep western boundary current using Chlorofuoromethane ratio[END_REF][START_REF] Rhein | The deep western boundary current: tracers and velocities[END_REF] also indicate that the earlier assumptions of CFM core mixing with CFM-loaded water could cause the estimated speed to be underestimated (Table 1).

Summary and concluding remarks

Chloro#uoromethanes have been measured in the deep water in the Romanche and Chain Fracture Zones in the equatorial Atlantic during two surveys, in August 1991 and November 1992. These two openings in the Mid-Atlantic Ridge allow deep water exchanges between the western and eastern basins. The identi"cation of the eastward spreading of two NADW CFM signals (at depths of 1600 and 3800 m) originating from the DWBC con"rms the earlier interpretation of [START_REF] Weiss | Atmospheric chloro#uoromethanes in the deep equatorial Atlantic[END_REF] and Doney and Bullister (1992) regarding a splitting at the DWBC near the equator. One of the most interesting results of this work is the evidence of the lower CFM signal (LNADW) penetrating the Romanche (193W}0.453S) and the Chain (153W}1.53S) Fracture Zones.

The upper CFM maximum (UNADW) appears south of the Equator and was followed eastward as far as 53W}1.453N. The equatorial UNADW F-11 core was more recently intersected at 335W in January 1993 (AndrieH et al., 1998) and 23Wi n January 1995 (Messias and MeH mery, 1998). In particular, on the meridional section along 23W, south of the main equatorial UNADW core, scattered CFC lenses were visible until 123S, giving some evidence for a southward #ow of the UNADW from the equatorial region along the eastern boundary (Messias and MeH mery, 1998).

The concentration of the lower CFM maximum found in the fracture zone is in agreement with the suggestion that the splitting of the LNADW at the DWBC explains the north}south CFM gradient observed at the equator (between 53N and 53S) as pointed out by [START_REF] Rhein | The Atlantic deep western boundary current: water masses and transports near the equator[END_REF]AndrieH et al. (1998). The lower CFM core evolution within the fracture zones appears to be closely related to the topography. The LNADW CFM signal is stronger in the RFZ than in the CFZ and was followed down to the eastern basin. The characteristics of the CFM signals change as they pass the sills of the RFZ, showing clear qualitative transformation of the water mass: the CFM signal decreases and sinks to greater depths in the eastern basins. The observations suggest that a mixture of CFM-loaded LNADW, lateral water and AABW extends down to the bottom, east into the Guinea Basin and northeast into the Sierra Leone Basin. These mixing process have also been seen in the CTDO and nutrient data by [START_REF] Mercier | Flow pathways of bottom water through the Romanche and chain fracture zones[END_REF] and [START_REF] Mercier | Hydrography of the Romanche and chain fracture zones[END_REF]. The CFM data reveal that there is an active ventilation of the eastern basin by waters carrying anthropogenic CFM derived from the Romanche fracture zone. In the deep Angola Basin at 193S, F-11 and F-12 concentrations were undetectable; however, traces of anthropogenic CCl were found in the bottommost water on the eastern side of the Mid-Atlantic Ridge [START_REF] Wallace | Carbon tetrachloride and chloro#uorocarbons in the South Atlantic Ocean, 193S[END_REF]. CCl is a transient tracer with a longer water-dating scale than F-11 and F-12, since it was introduced to the ocean at the beginning of the 20th century. [START_REF] Wallace | Carbon tetrachloride and chloro#uorocarbons in the South Atlantic Ocean, 193S[END_REF] suggested the possibility of ventilation of the Angola Basin from the RFZ and through the Walvis Ridge. The agreement with earlier CFM results in the deep equatorial Atlantic ocean encourages the study of the evolution of this transient-tracer signature in the eastern basin.

The attempt to date the UNADW and LNADW CFM core from the F-11/F-12 ratio gives constant mean ages of 25$4 years and 36$5 years, respectively, along the Romanche fracture zone. The UNADW appears to be &younger' but more dilute (20}40) than the LNADW (5}15), perhaps a!ected by reversing equatorial currents. The dilution factor of the LNADW CFM core increases through the sills from around 10 to 30. These estimates can be biased because of upstream mixing e!ects and because of the very low CFM concentrations measured here. The interpretation of these results should therefore be used with caution. A comparison with a numerical model that includes an explicit treatment of circulation and mixing could yield further insights into the apparent age estimates computed here.

Fig. 1 .

 1 Fig. 1. Bathymetry of the Equatorial Atlantic showing the location of the Romanche and Chain Fracture Zones in the Mid-Atlantic-Ridge. Adapted from GEBCO (1978).

Fig. 2 .

 2 Fig. 2. Bathymetry of the Romanche and Chain Fracture Zones and CFM station locations (dots) occupied during (a) the Romanche 1 cruise and (b) the Romanche 2 cruise. Black areas indicate bottom depths less than 4000 m.

Fig. 3 .

 3 Fig. 3. Vertical pro"les below 500 dbar of salinity, oxygen ( mol kg\), potential temperature (3C) from CTDO data and F-11 (pmol kg\) from bottle samples (stars), for August 1991 (Romanche 1) from (a) station 4 (0327S}19318W) located at the entrance of the RFZ and (b) station 38 (1322S}14359W) located at the entrance of the CFZ. In (a) the core layers of several water masses referred to in the text are identi"ed. UNADW, MNADW and LNADW are Upper, Middle, and Lower North Atlantic Deep Water, according to classi"cation WuK st (1935).

Fig. 4 .

 4 Fig. 4. Vertical pro"les below 500 dbar of nitrate ( mol kg\), silicate ( mol kg\) and phosphate ( mol kg\) for August 1991 (Romanche 1) from (a) station 4 located at the entrance of the RFZ and (b) station 38 located at the entrance of the CFZ. In (b) the core layers of Circumpolar Deep Water (CDW) masses referred to in the text are identi"ed. UCPW and LCDW are Upper and Lower Circumpolar Deep Water.

Fig. 5 .

 5 Fig. 5. Deep water section of F-11 (pmol kg\) versus pressure along the RFZ for August 1991 (Romanche 1). This F-11 section and those following (except Fig. 7) have been hand contoured. Dots show positions of water samples. The smallest F-11 contour intervals are close to the precision of the measurements. See Fig. 2 for station positions.

  Fig. 6. Deep water section of F-11 (pmol kg\) versus pressure along the CFZ for August 1991 (Romanche 1), as in Fig. 5.

Fig. 7 .

 7 Fig. 7. Meridional deep-water sections of F-11 (pmol kg\) versus pressure across the RFZ realized at (a) 163W, (b) 153W and (c) 143W during August 1991 (Romanche 1), as in Fig. 5.

Fig. 8 .

 8 Fig. 8. Meridional deep water section of F-11 (pmol kg\) versus pressure across the RFZ and CFZ near 143W for November 1992 (Romanche 2), as in Fig. 5.

Fig. 9 .

 9 Fig. 9. Deep water section of F-11 (pmol kg\) versus pressure along the RFZ axis from the sill region into the Guinea basin for November 1992 (Romanche 2), as in Fig. 5.

Fig. 10 .

 10 Fig. 10. Comparison of vertical F-11 (pmol kg\)p r o "les below 500 dbar measured at the same position in August 1991 (Romanche 1 } solid line) and in

  and AndrieH et al. (1998) at 353W. Tsuchiya et al. (1994) also present a similar shift of the UNADW salinity maximum at 253W. AndrieH et al. (1998) attribute this shift to the eastward bifurcation induced by the rise, around 2000 m depth, of the Parnaiba ridge near 33S at the western boundary. Two other interesting features of the UNADW found in both Romanche surveys are the presence of lenses of high F-11 concentration and a constriction of the tongue near 13}143W (Fig. 5 and 9).

Fig. 11

 11 Fig. 11. F-11 (pmol kg\), salinity and oxygen ( mol kg\) versus (34.2}34.8) for (a) Romanche 1 stations and (b) Romanche 2 stations.

Fig. 12 .

 12 Fig. 12. Pro"les of F-11 (pmol kg\) and salinity versus (34.55}34.75) for (a, b) Romanche 1 stations 70, 19 and 41 and (c, d) Romanche 2 stations 20, 14 and 9 made at the same position.

Fig. 13 .

 13 Fig. 13. Evolution of the (a) F-11 (pmol kg\) and (b) oxygen (ml l\) deep maxima versus (45.8}46) along the RFZ from Romanche 1 cruise.

Fig. 14

 14 Fig. 14. F-11 (pmol kg\) versus oxygen (ml l\) for water samples below 3000 dbar in the RFZ for (a) Romanche 1 stations and (b) Romanche 2 stations. Stars represent stations located at the exit of the sills.

Fig. 15

 15 Fig. 15. F-11 (pmol kg\) versus potential temperature (3C) for bottom water samples ( '45.9) for (a) Romanche 1 stations and (b) Romanche 2 stations. Station numbers are plotted at the F-11/potential temperature points.

Fig. 16

 16 Fig. 16. F-11 to F-12 concentration ratio in the assumed DSOW source surface waters. The curve represents values for water of !0.53C and 34.8 (for upper AAIW;[START_REF] Smethie | Tracing the thermohaline circulation in the western North Atlantic using chloro#uorocarbons[END_REF] in equilibrium[START_REF] Warner | Solubilities of chloro#uorocarbons 11 and 12 in water and seawater[END_REF] with reconstructed atmospheric concentrations in the Northern Hemisphere.

Fig. 17

 17 Fig. 17. F-11 : F-12 ratio versus apparent age calculated for the deep CFM maxima from Romanche 1 (only for the UNADW: on the Romanche 1 cruise, contamination problems precluded the use of the F-12 data on station 4}39) and Romanche 2 (UNADW and LNADW). The symbol codes are (#) for the UNADW CFM maximum and ( * ) for the LNADW CFM maximum.

  (a) the upper CFM maximum and (b) the lower CFM maximum, obtained from the Romanche cruises and from previous cruises upstream of the RFZ along the and Jenkins (1994) calculated a dilution factor of about 16 (8}32) at 203N for the LNADW using TTO Tritium-3He data in a simple boundary current/abyssal ventilation model for the western basin of the North Atlantic.

Fig. 18 .

 18 Fig. 18. F-12 and F-11-concentrations (pmol kg\), apparent ages and dilution factors calculated from the F-11 : F-12 ratio versus longitude along the RFZ axis for the upper CFM maximum (a, c, e) and the lower CFM maximum (b, d, f ).

Table 1

 1 Comparison of the properties for

Acknowledgements

The analytical system was built at the Laboratoire de GeH ochimie Isotopique (LGI): we wish to acknowledge all the technical sta! and in particular Philippe Jean-Baptiste for his assistance and helpful advice. We would also like to acknowledge the captains and the crews of the N.O. ¸'Atalante for outstanding cooperation and the scienti"c party of the cruise for splendid assistance. Our thanks go to the technical sta! of the Laboratoire de Physique des OceH ans (LPO). We thank Kurt Polzin for extremely valuable comments on the manuscript. This work was supported by l'Institut National des Sciences de l'Univers (INSU) and l'Institut Franc7 ais de Recherche pour l'Exploitation de la Mer (IFREMER) through le Programme National d'Etude de la Dynamique du Climat (PNEDC), as well as a grant received from le Centre d'Energie NucleH aire (CEN), Saclay.

AndrieH , C., Ternon, J.-F., Messias, M.-J., MeH mery, L., Bourle`s, B., 1998. Chloro#uoromethane distributions in the deep equatorial Atlantic during January}February 1993. Deep-Sea Research I 45, 903}929. BoK ning, C.W., Schott, F.A., 1993. Deep currents and the eastward salinity tongue in the Equatorial Atlantic: results from an eddy } resolving, primitive equation model. Journal of Geophysical Research 98, 6991}6999. Bullister, J.L., 1989. Chloro#uorocarbons as time-dependent tracers in the ocean. Oceanography 2, 12}17. Doney, S.C., Bullister, J.L. 1992. A chloro#uorocarbon section in the eastern North Atlantic. Deep-Sea Research 39, 1857}1883. Doney, S.C., Jenkins, W.J., 1994. Ventilation of the deep western boundary current and Abyssal Western North Atlantic: estimates from Tritium and He distributions. Journal of Physical Oceanography, 24, 638}659. Ferron, B., Mercier, H., Speer, K.G., Gargett, A., Polzin, K.L., 1998. Mixing in the Romanche Fracture Zone. Journal of Physical Oceanography 28, 1919}1945. Fine R.A., Molinari, R.L., 1988. A continuous deep western boundary current between Abaco (26.53N) and Barbados (133N). Deep-Sea Research 35, 1441}1450. Friedrichs, M.A.M., Hall, M.M., 1993. Deep circulation in the tropical North Atlantic. Journal of Marine Research 51, 697}736. Gascard, J.-C., Clarke, R.A., 1983. The formation of Labrador Sea Water. Part II: mesoscale and smaller-scale processes. Journal of Physical Oceanography 13, 1779}1797. Heezen, B.C., Bunce, E.T., Hersey, J.B., Tharp, M., 1964. Chain and Romanche Fracture Zones. Deep-Sea Research 11, 11}33. Lazier, J.R.N., 1988. Temperature and salinity changes in the deep Labrador Sea, 1962}1986. Deep-Sea Research 35, 1247}1253. Mantyla, A.W., Reid, J.L., 1983. Abyssal characteristics of the World Ocean waters. Deep Sea Research 30, 805}833. Mauritzen, C.A., 1996. Production of dense over#ow waters feeding the North Atlantic across the Greenland-Scotland Ridge. Part 1: evidence for a revised circulation scheme. Deep Sea Research I 43(6) 769}806. McCartney, M.S., 1993. Crossing of the equator by the deep western boundary current in the western Atlantic Ocean. Journal of Physical Oceanography 23, 1953}1974. McCartney, M.S., Bennett, S.L., Woodgate-Jones, M.E., 1991. Eastward #ow through the Mid-Atlantic Ridge at 113N and its in#uence on the abyss of the eastern basin. Journal of Physical Oceanography 21, 1089}1121. Mercier, H., Billant, A., Branellec, P., Morin, P., Messias, M.J., MeH mery, L., Thomas, C., Honnorez, J., 1992. Campagne Romanche 1-DonneH es CTDO , Chimie et BathymeH trie. Rapport Interne Laboratoire de Physique des Oce & ans 92-02. Mercier, H., Billant, A., Branellec, P., AndrieH , C., Messias, M.J., Gouriou, Y., Laguadec, C., 1995. Campagne Romanche 2-DonneH es de la sonde CTDO , Mesures de SaliniteH ,d 'Oxyge`ne Dissous et des Chloro-#uoromeH thanes, CourantomeH trie Acoustique Doppler. Rapport Interne Laboratoire de Physique des Oce & ans 95-02. Mercier, H., Morin, P., 1997. Hydrography of the Romanche and the chain fractures zones. Journal of Geophysical Research 102, 10,373}10,389.


D Boichard

S Fritz

P Croiseau

V Ducrocq

T Tribout

M Barbat

B C D Cuyabano

Methods to Estimate Erosion Factors of Genomic Breeding Values of Candidates due to Long-Distance Linkage Disequilibrium

Keywords: Genomic evaluation, inflation, erosion of genomic values, validation methods

Most validation studies of genomic evaluation observe inflation, i.e. regression coefficients of the later phenotypes on early predictions smaller than one. This pattern does not reflect a bias in the evaluation model, it rather reflects long distance associations between markers and quantitative trait loci (QTLs). Due to linkage disequilibrium (LD), SNP effects estimated from a reference data capture non-zero contributions from distant QTLs located not only in the same, but also in the other chromosomes, and we show that some across-chromosome LD does exist in different French dairy cattle breeds. This LD results from limited effective population size and, more importantly, from the relationship within the reference population. Long distance associations are partly broken and rebuilt at random at each generation. Therefore, corresponding SNP effects are partly lost in the next generations and we shall refer to this effect loss as erosion. This erosion can be predicted by different methods based on the following equations applied to simulated QTLs. If the breeding values are Pq with P the QTL genotypes and q their effects, the expected contribution of QTL j to the estimated SNP effect i is ci M' Pj qj, where M is the matrix of SNP genotypes and ci is line i (corresponding to SNP i) of C = (M'M +  I) -1 . Two methods based on simulations are proposed to estimate the erosion factor . In Method 1, the direct genomic value (DGV) of the progeny based on SNP effects estimated in this new simulated generation are regressed on the DGV of the same progeny based on SNP effects estimated in the reference population. In Method 2 all the QTL contributions to SNP effects are regressed based on SNP-QTL recombination rates and summed to predict the breeding value at the next generation. The regression coefficient of the DGV based on eroded contributions on the raw DGV is also an estimate of erosion. An illustration is given with the French Normande female reference population in 2021. Method 1 is simpler to implement on a routine basis, and yields good estimates of erosion over one generation. Erosion is also dependent on the distance between the young candidates and their reference population and formulae are proposed to apply erosion. We recommend accounting for erosion in genetic evaluations to provide unbiased predictions for the young candidates. Accordingly, erosion has been accounted for in the French Single Step bovine evaluation since March 2022.

Introduction

In genomic evaluation, single nucleotide polymorphism (SNP) effects are estimated in a reference population and applied to selection candidates. This method is extensively used to select candidates at an early stage of their life or not yet with phenotypic information. The standard interpretation is that SNPs are in close LD with causal mutations (or QTL) and therefore, good proxies for these QTL. Implicitly, this assumes that estimated SNP effects reflect those of the neighbouring causal mutations. Under this assumption, SNP effects observed in the reference sample should be very similar in the next generation as short-distance LD erodes slowly due to recombination. It is, however, well known that genomic evaluation efficiency is highly dependent on the close relationship of the candidates to the reference sample [START_REF] Habier | The impact of genetic relationship information on genome-assisted breeding values[END_REF][START_REF] Habier | Genomic BLUP decoded: A look into the black box of genomic prediction[END_REF][START_REF] Legarra | Performance of genomic selection in mice[END_REF][START_REF] Pszczola | Reliability of direct genomic values for animals with different relationships within and to the reference population[END_REF]. Many studies have shown the limited gain in accuracy in multibreed evaluation [START_REF] Erbe | Improving accuracy of genomic predictions within and between dairy cattle breeds with imputed high-density single nucleotide polymorphism panels[END_REF][START_REF] Hozé | Efficiency of multi-breed genomic selection for dairy cattle breeds with different sizes of reference population[END_REF], illustrating that distant reference data are not informative. Other studies have shown a decrease in accuracy over generations when the reference population is not updated (Soneson et al, 2009;[START_REF] Solberg | Persistence of accuracy of genome-wide breeding values over generations when including a polygenic effect[END_REF]. Moreover, it has been observed that the absence of parents in the reference population directly influences the prediction accuracy of the selection candidates. All these results suggest that SNP effects erode as the distance between candidates and reference sample increases.

Validation studies of genomic evaluations are generally based on the regression of later performances on the early predictions. These studies frequently observe an inflation pattern, i.e., the regression coefficient is systematically lower than 1, meaning that later performances of the best candidates are below those initially predicted (and later performances of the worst candidates, if any, are above those initially predicted).

One plausible interpretation is the existence of long-range LD, even across different chromosomes. Consequently, many markers may capture partial effects of supposedly unlinked QTL. Although long-distance LD is notably lower than short-distance LD, the number of long-distant variants is considerably higher and their combined effects can account for a substantial proportion of the genetic variance in a genomic prediction.

In the first part of this study, we demonstrate that markers do capture part of the effects of distant QTL due to the long-distance LD. Because this long-range LD gradually decays over generations, it is imperative to account for the erosion of marker effects to predict the genomic values for the candidates. In the second part, we propose two methods to estimate the specific erosion factor of a reference population and suggest how to use it in practice to adjust DGV.

Materials & Methods

Evidence for linkage disequilibrium across chromosomes

LD across chromosomes was assessed using data from the 2021 female reference populations of six French dairy cattle breeds (Holstein, Montbéliarde, Normande, Abondance, Tarentaise, Vosgienne). The reference populations exhibited varying sizes, ranging from 2617 to 362,363 animals. It is worth noting that Vosgienne, Tarentaise, and Abondance are local mountain breeds, whereas Montbéliarde and Normande are national populations, comprising 18% and 7% of the French dairy herd, respectively. On the other hand, the international Holstein breed accounts for 70% of the French dairy cattle population. In our analysis, we selected one every 20 SNP of the Illumina EuroGMD BeadChip on the 29 autosomes, resulting in a sample of ~3 million r 2 values for each breed (vs ~1.2 billion in total for all SNPs).

Table 1 presents various LD statistics across chromosomes in the female reference populations of six French dairy cattle breeds. While the average r 2 values appear to be small, suggesting limited across-chromosome disequilibrium, it is important to note that the focus here is on the parameter r, as the impact of a QTL on a SNP effect is directly proportional to the correlation between them. These correlation values decreased when the size of the breed (reference population, number of females in the breed, or effective size) increased. The proportion of SNP pairs with |r| exceeding 5% fluctuated notably, ranging from 1.5% to 33%, depending on the breed. Notably, as the reference population size increased, this proportion also tended to decrease. Nevertheless, it is worth highlighting that even with a small percentage, we still observe nonnull correlations between 600 to several thousand SNP with a QTL located on a different chromosome (assuming that these r distributions between SNP and QTL are the same as between SNP).

Impact of long distance LD on genomic predictions

To investigate the impact of long-distance LD on SNP effects, we focused on the Normande population, which consisted of 69,220 cows (N) with genotypes and phenotypes. In this study, we considered the first five chromosomes comprising 13,608 SNP. Two hundred additive causal mutations (nq=200) were randomly sampled among SNPs with a minor allele frequency (MAF) higher than 0.02. The additive effects of these mutations were independently drawn from a normal distribution, assuming a heritability of 0.3. Two scenarios were tested: (1) the SNP-BLUP model accounted for the ns=13,408 SNPs excluding the QTL; (2) in addition to these SNPs, the SNP-BLUP model also accounted for an additional residual polygenic effect explaining 20% of the genetic variance. Note that in a previous study [START_REF] Boichard | Longdistance associations generate erosion of genomic breeding values of candidates for selection[END_REF], we have shown with a similar approach that erosion was minimal when the causal variants were included in the analysis, and this scenario is not replicated here. This also agrees with de los Campos et al ( 2015) who showed that missing heritability does not exist when causal variants are in the model.

The strategy used to compute erosion relied on the determination of the contribution of each QTL to each SNP, as follows. Omitting fixed effects, the SNP-BLUP equations can be written as

[ M'M +  I ] 𝒔 ̂ = M' y
with M the (N x ns) matrix of cantered and scaled genotypes, s the vector of SNP effects, y the vector of phenotypes adjusted for the fixed effects, and  = e 2 / s 2 with e 2 and s 2 the residual and the SNP variances, respectively. According to the simulation, the phenotype can be written as y = Pq + e, i.e., the sum of nq QTL effects and an error term, with P the (N x nq) matrix of genotypes at the QTL level and q the vector of true QTL effects. Therefore, the equations can be rewritten as

𝒔 ̂ = [ M'M +  I ] -1 M'(Pq + e) [1] Let us denote C = [ M'M +  I ] -1 the inverse of the coefficients matrix. If ci is line i (corresponding to SNP i) of C, the contribution of QTL j to SNP effect i is fij = ci M' Pj qj.
[2]

There were nq x ns = 200 x 13,408 = 2,681,600 such contributions, distributed in the 4 following categories based on the distance (d) between the QTL and the SNP: (1) d < 5 Mb;

(2) 5 < d < 20 Mb; (3) d > 20 Mb with both the QTL and the SNP located on the same chromosome; (4) the QTL and the SNP are located on different chromosomes. Within each of these categories, we computed a partial DGV for each cow within the reference population. Summary statistics were then calculated over 30 replicates quantifying their relative contributions to the total DGV and their correlations.

The same strategy can also be applied in a model including a residual polygenic effect, denoted as u. The equations corresponding to s and u are as follows:

[ 𝒁 ′ 𝒁 + 𝜅𝑨 -𝟏 𝒁 ′ 𝑴 𝑴 ′ 𝒁 𝑴 ′ 𝑴 + 𝜆 𝑰 ] [ 𝒖 ŝ ̂] = [ 𝒁 ′ 𝒚 𝑴 ′ 𝒚 ] [3]
with Z being the incidence matrix linking the records of y to u and  = e 2 / u 2 the corresponding variance ratio.

The u equations can be absorbed into s equations, resulting in the following formula:

Let us denote C* the inverse of the coefficient matrix after absorption

𝐂 * = [ 𝐌 ′ (𝐈 -𝐙(𝐙 ′ 𝐙 + κ𝐀 -𝟏 ) -𝟏 𝐙 ′ ) 𝐌 + λ 𝐈 ] -1
and M* the adjusted genotype matrix after absorption

𝑴 * ′ = 𝑴′ (𝑰 -𝒁(𝒁 ′ 𝒁 + 𝜅𝑨 -𝟏 ) -𝟏 𝒁′)
Then the contribution of QTL j to each SNP effect i is:

fij = c*i M*' Pj qj [4]
where c*i is line i of C*

Results and Discussion

In the model without a polygenic effect (scenario 1), the partial DGV derived from contributions of the QTL close to markers explained approximatively three-quarters of the DGV variance. Notably, more distant markers (d>20 Mb) and markers located on other chromosomes together accounted for ~13% of the total DGV variance. Markers located on other chromosomes explained more variance than markers at more than 20 Mb on the same chromosome. This result can be attributed to the larger number of marker-QTL pairs when markers are located on different chromosomes. The likely underlying reason of this pattern lies in the strong shrinkage of the effects of markers situated close to the QTL due to the influence of the prior information. Indeed, all markers receive the same prior variance, and the parameter  had a relatively high value compared to the diagonal elements of the matrix M'M. Consequently, the estimated effects of markers in proximity to the QTL experience substantial shrinkage and are much smaller, even altogether, than the true QTL effect. As a result, the unexplained part of the true QTL effect becomes available for distant markers, potentially leading to their contribution to total DGV. This effect would be probably maximized when the number of true QTL is much lower than the number of SNPs (q 2 >> s 2 ), and when the reference population is relatively small (diagonal (M'M) does not dominate ). It is important to note that when the size of the reference population is very large, the influence of prior information decreases, and this observed pattern is likely to gradually decrease. The inclusion of a polygenic effect in the model (scenario 2) resulted in a higher proportion of variance being explained by distant markers and by markers located on different chromosomes. At first glance this result may seem counterintuitive as one could

[𝑴 ′ (𝑰 -𝒁(𝒁 ′ 𝒁 + 𝜅𝑨 -𝟏 ) -𝟏 𝒁 ′ ) 𝑴 + 𝜆 𝑰] 𝒔 = 𝐌 ′ (𝐈 -𝐙(𝐙 ′ 𝐙 + κ𝐀 -𝟏 ) -𝟏 𝐙′) (𝐏𝐪 + 𝐞)
expect that the polygenic effect would help account for these long-distance effects since it captures the genetic relationships between individuals. Distant markers and markers on another chromosome altogether explained around 17% of the DGV variance whereas the share due to close markers (d<5Mb) decreased to 69%. As in scenario 1, a possible interpretation is the shrinkage of estimated SNP effects. Indeed, in the presence of a polygenic effect with variance u 2 , the total variance due to SNPs is reduced to g 2 -u 2 and results in an increased variance ratio  (e 2 / s 2 ). It can then be concluded that inclusion of a polygenic effect into the model should primarily be motivated by the need to account for the genetic variance not captured by the SNPs, rather than as a means to reduce inflation. Tables 3a and 3b present the correlations between the partial DGVs derived from different categories of QTL-SNP distances in scenarios 1 and 2, respectively. Both scenarios presented low to moderate positive correlations, illustrating that distant QTL contribute to the effects of many markers. Inclusion of a polygenic effect in the model (scenario 2, table 3b) increases these correlations showing that long distance effects are reinforced.

Methods to estimate erosion factor of SNP effects

The concept of erosion of the genomic breeding values has two distinct components: (a) a component that is characteristic of the reference population itself, and (b) a component that is specific to each candidate and its genetic distance from the reference population. The extent of long-distance LD in the reference population is influenced by the effective population size (Ne). Notably, when Ne is small, a non-zero LD baseline persists. More importantly, the level of long-distance LD is also strongly dependent on the genetic relatedness within the reference population, which can be different from the relatedness in the overall population. A higher average relationship between individuals within the reference population results in more longdistance LD. It can be argued that, on average, the long-distance LD appears relatively stable across generations, but this stability does not hold for individual pair of markers. At a given generation, existing LD is halved in the subsequent generation due to recombination, but new LD can emerge from different marker pairs as a consequence of the random processes associated with the sampling of parents and genetic drift, making average LD stable.

The theoretical derivation of the erosion factor  requires additional investigation. Nevertheless, practical and efficient solutions can be obtained through simulation. In this paper, we present two simulation-based approaches which offer practical and effective means to address the erosion phenomenon and to estimate .

well with practical observations. Therefore, breeding schemes with accelerated generations without updating reference data tend to accumulate more erosion than initially anticipated. These schemes may be less appealing due to the rapid erosion effect on genomic values. Furthermore, as shown by [START_REF] Dekkers | Predicting the accuracy of genomic predictions[END_REF], erosion also affects reliability, but with a coefficient equal to  2 instead of . The loss in genomic accuracy is therefore very fast. Theoretical accuracies calculated based on the inverse of the coefficient matrix tend to overestimate the reliabilities for candidates and must be adjusted accordingly. In the French evaluation system, reliabilities are computed by combining effective record contributions (ERC) associated to polygenic information and genomic information, the latter (and the latter only) being eroded in candidates.

Conclusions

The practical implications of erosion in genetic evaluations and breeding programs are important. When considering the overall prediction of selection candidates, contributions from short-distance LD tend to remain relatively stable because they are only mildly eroded by recombination [START_REF] Dekkers | Predicting the accuracy of genomic predictions[END_REF]. However, contributions from long-distance LD are halved at each generation. The extent of erosion varies with the relative weight of short and long-distance LD, but it should never be disregarded.

Further investigations are needed to theoretically determine the erosion factor . Nonetheless it is clear this factor depends on baseline LD in the population, i.e., effective population size (Ne) and genome length (L), as well as on the structure of the reference population. Additionally, it is likely influenced by the genetic architecture of the traits (such as the number of QTL and magnitude of their QTL effects) and the model used (SNP-BLUP/GBLUP vs Bayesian models, the latter being likely less affected by erosion).

It is also relevant to explore the impact of the structure of the reference population, such as its heterogeneity in terms of time span, selection, and relationship. For instance, the impact on erosion of old generation data in the reference population would be worth investigating. While the proposed erosion methods consider the smallest distance between the candidate and the reference population, alternative approaches using the barycentre of the reference population warrant investigation.

However, one can anticipate that:

(1) Inflation factors, frequently observed between 0.8 and 0.9, give the magnitude of the erosion phenomenon;

(2) Models including causal variants tend to be more persistent and less subject to erosion, as demonstrated by [START_REF] Boichard | Longdistance associations generate erosion of genomic breeding values of candidates for selection[END_REF];

(3) Models that incorporate a residual polygenic component may appear to have less inflated predictions for candidates because they combine two estimates of the MS term: the genomic estimate, which is inflated, and a polygenic estimate, which is equal to zero (i.e., 100% deflated). However, the polygenic effect does not capture long-distance LD effects and, therefore, does not improve predictions in terms of persistence. We believe that accounting for erosion is a more rigorous and accurate approach, even if it requires post-processing. This methodology has been implemented in the French Single Step bovine evaluation since March 2022.

Table 1 .

 1 Statistics of |r| and r 2 values across the 29 chromosomes in female reference populations of six French dairy cattle breeds (selection of one every 20 SNP within chromosome).

	Breeds	# cows	Mean	%	Mean(r 2 )
			(|r|) *	|r| >
				0.05
	Vosgienne	2617	0.0420	33	0.0029
	Tarentaise	3788	0.0225	18	0.0015
	Abon-	7115	0.0268	15	0.0012
	dance
	Normande	69,220	0.0206	7	0.00073
	Montbe-	185,053	0.0173	4	0.00053
	liarde
	Holstein	362,363	0.0148	1.5	0.00038
	*statistics based on 2,812,741 to 3,231,800 SNP
	pairs per breed

  Table 2 presents the relative contribution of the 4 categories based on QTL-SNP distance to the total DGV variance.

Table 2 .

 2

	-Relative contributions (%) of each of the
	4 classes of QTL-SNP pairs defined according to
	their distance (d). The contribution of a class is the
	percentage of DGV variance explained by each
	partial DGV in the reference population, over 30
	replicates.
	Classes of partial	Scenario 1	Scenario 2
	DGV defined	Model without	Model with
	according to QTL-	polygenic	polygenic
	SNP distance (d)	effect	effect
	1: d < 5 Mb	73.5	68.9
	2: 5 Mb < d < 20 Mb	13.7	14.3
	3: d > 20 Mb	4.9	5.2
	4: QTL and SNP
	located on different	8.0	11.5
	chromosomes

Table 3a .

 3a Average correlations between partial DGV in Scenario 1, without polygenic effect.

	Results over 30 replicates
	Partial DGV class	<5 Mb 5-20 Mb >20 Mb
	5-20 Mb	0.30
	>20 Mb	0.17	0.15
	Other Chromosomes	0.11	0.05	0.12
	Table 3b. Average correlations between partial
	DGV in Scenario 2, with polygenic effect. Results
	over 30 replicates
	Partial DGV class	<5 Mb 5-20 Mb >20 Mb
	5-20 Mb	0.54
	>20 Mb	0.26	0.29
	Other Chromosomes	0.25	0.22	0.27

Method 1: by simulating a new generation

The real reference population Gr of the breed for a given trait is considered with its SNP genotypes M. As previously, nq QTL are simulated in this reference population by sampling SNP which are thereafter excluded from the analysis. Expectations of SNP effects are estimated by 𝐬 𝐫 ̂= (𝐌′M + 𝐈) -𝟏 𝐌′ 𝐏 𝐪, assuming the same previous notations.

A new generation, Gn, is then simulated, by sampling parents (at random or following a predefined design) in the reference population and performing matings. The expected DGV of this new generation is obtained from the genotypes Mn and from the SNP effects estimated in the reference population: DGVr = Mn 𝐬 𝐫 ̂ From these two sets of DGV, an estimate of the erosion  between the two generations is obtained through a regression analysis, where:

Method 2: by regressing contributions of QTL to marker effects.

As above, the real reference population Gr of the breed for a given trait is considered with its genotypes M and nq QTL are simulated in this reference population. All contributions fij of the QTL j to the effect of SNP i are computed as shown in equation [2]:

DGVr in the reference population is the sum of all contributions:

with 1q being a vector of 1 of size q.

Note that the DGV can also be obtained as DGVr = M 𝐬 𝐫 ̂, as in Method 1. Then, all fij are regressed according to the genetic map, with coefficients (1-rij) varying from 1 to 0.5, rij being the recombination rate between the loci i and j: hij = rij fij.

[6]

New eroded DGV (DGVe) are the sum of all regressed contributions

DGVe = M h 1q

An estimate of √𝜌 is obtained through a regression analysis, where:

Comparison of both methods

Both methods are based on QTL simulation, and their results are influenced by assumptions, particularly regarding the number of QTLs. However, as far as the number of QTLs is smaller than the number of markers and longdistance LD is present, we can anticipate that erosion exists.

Method 1 is relatively straightforward to implement, as it involves simulating one additional generation and estimating expected SNP effects in both the reference population and in the new generation using standard software. In contrast, Method 2 requires a specific program to compute all the contributions and erode them. Nevertheless, it provides an explicit biological basis to understand and interpret erosion across generations.

Method 1 generates progeny from pairs of parents, leading to erosion on both sire-progeny and dam-progeny pathways. Method 2, on the other hand, simulates erosion through recombination at only one meiosis, resulting in the erosion factor estimated by method 2 being the square root of that estimated by method 1. This scale difference should be considered when interpreting results.

Additionally, method 2 considers the whole reference population whereas method 1 generates a new generation based on assumptions about the number and the choice of parents sampled. Therefore, results between the two methods can exhibit slight variations.

Numerical example

In our numerical example, we applied both method 1 and method 2 to the same 2021 Normande female reference population. As before, we focused on only 5 chromosomes and simulated 200 QTLs. In method 1, a new generation was created by sampling 1,000 sires and 50,000 dams. Each sire had 50 progeny, while each dam had one progeny, resulting in a new generation of 50,000 animals. The results obtained with both methods are presented in Table 4. Recombination rates were based on ARS-UCD1.2 bovine genome assembly, assuming 1 cM for 1 Mb. The number of replicates was set to 30. Variability across replicates was small (at least with such a large reference population) and this number of replicates was sufficient to obtain reliable estimates of .

Discussion on how to apply erosion in practice

Here, we consider that, by definition, individuals in the reference population are assumed to possess non-eroded SNP effects. It is worth noting that this assumption may warrant discussion due to potential heterogeneity within the reference population. The estimated SNP effect represents an expectation and may not align with individual situations. However, this point goes beyond the scope of this initial study.

Erosion primarily concerns selection candidates, i.e. genotyped individuals without phenotype data and, therefore, out of the reference population. When their parents, referred to as s and d, are part of the reference population, we assume that the parent's average (PA) DGV, denoted as PA = 0.5 (DGVs + DGVd), remains unaffected. Indeed, their DGVs are based on performances; and this is especially the case for sires with progeny evaluation, and therefore with very reliable DGVs. Erosion influences the deviation from PA, i.e., the predicted Mendelian sampling term. We propose applying the following formula:

When the parents of a candidate are not in the reference population, erosion applies at each generation between the reference population and the candidate. Following a similar approach as described by [START_REF] Dekkers | Predicting the accuracy of genomic predictions[END_REF], the number of generations between the candidate and its closest relatives within the reference population is determined on both the sire and dam pathways and k is their sum. Erosion is applied following equation [9]:

When the parents themselves are candidates, i.e., genotyped and not in the reference population, erosion also applies to them. This erosion affects the PA of their progeny in the following way:

This formula should be applied recursively, processing parents before progeny. This recursive approach highlights that the DGV of candidates born from very young parents experience significant erosion, which aligns