Northbound Transport of the Mediterranean Outflow and the Role of Time‐Dependent Chaotic Advection

The Mediterranean Sea releases approximately 1 Sv of water into the North Atlantic through the Gibraltar Straits, forming the saline Mediterranean Outflow Water (MOW). Its impact on large‐scale flow and specifically its northbound Lagrangian pathways are widely debated, yet a comprehensive overview of MOW pathways over recent decades is lacking. We calculate and analyze synthetic Lagrangian trajectories in 1980–2020 reanalysis velocity data. Sixteen percent of the MOW follow a direct northbound path to the sub‐polar gyre, reaching a 1,000 m depth crossing window at the southern tip of Rockall Ridge in about 10 years. Surprisingly, time‐dependent chaotic advection, not steady currents, drives over half of the northbound transport. Our results suggest a potential 15–20 years predictability in the direct northbound transport. Additionally, monthly variability appears more significant than inter‐annual variability in Lagrangian mixing and spreading the MOW.


Introduction
The middepth salinity and temperature fields of the North Atlantic Ocean (NA) contain a distinct high-salinity tongue originating from the Mediterranean Sea (Figure 1), a signature of the Mediterranean Outflow Water (MOW).The MOW exits the Straits of Gibraltar into the Gulf of Cadiz (GoC), entrains the (locally) fresher and cooler NA Central Water, and leaves the GoC upon crossing the Cape of Vicente at 500-1,500 m as a relatively salty plume of approximately 1 Sv and salinity between 36.3 and 37 psu (Baringer & Price, 1997;Iorga & Lozier, 1999;Sammartino et al., 2015).Beyond this point, the MOW begins spreading in the NA via various pathways (Bashmachnikov et al., 2015;Gasser et al., 2017;Iorga & Lozier, 1999;Sánchez-Leal et al., 2017).This significant input of salinity into the NA is thought to have an important effect on the strength and stability of the Atlantic Meridional Overturning Circulation in current and past climates by contributing to the salinity preconditioning of polar waters for formation of the NA Deep Water (NADW) (Calmanti et al., 2006;Chan & Motoi, 2003;Ferreira et al., 2018;Ivanovic et al., 2014;Rahmstorf, 1998;Reid, 1979;Rogerson et al., 2010;Voelker et al., 2006).The specific pathways taken by the MOW have been of interest and controversy at least since Reid (1978Reid ( , 1979)), who used tracer data from 1957 to 1971 to establish hydrographic evidence of a direct northbound pathway of MOW from the GoC past Porcupine Bank at 53°N into the sub-polar gyre (SPG) region via a middepth eastern boundary current.Since then, our understanding of these pathways has evolved by studies of hydrographic data, actual, and virtual drifters (Burkholder & Lozier, 2011;Iorga & Lozier, 1999;Jia et al., 2007;Lozier & Stewart, 2008;McCartney & Mauritzen, 2001;Sala et al., 2013).Several works found no evidence that an MOW core exists past Porcupine Bank (Iorga & Lozier, 1999;McCartney & Mauritzen, 2001); Iorga and Lozier (1999) used climatological mean fields from 1904to 1990, and McCartney and Mauritzen (2001) used hydrographic methods on data from the 1980's.In an attempt to reconcile the conflicting evidence, Lozier and Stewart (2008) suggested a temporal variability of flow fields, perhaps due to the NA Oscillation (NAO), that results in an east-west shift of the eastern limb of the SPG.This shift is measured by the SPG index (Hakkinen & Rhines, 2004;Lozier & Stewart, 2008), calculated from the principle component of the North Atlantic sea-surface height as described in detail in Text S2 of Supporting Information S1.
The time-dependent, 3D nature of oceanic flow over decadal timescales may result in significant chaotic advection of Lagrangian trajectories (Aref, 1984;Aref et al., 2017;Koshel & Prants, 2006;Ottino, 1990;Yang & Liu, 1994).The concept of chaos is asymptotic; in finite-time studies, oceanic chaotic advection refers to the divergence of trajectories and Lagrangian mixing (i.e., mixing due to advection and not due to diffusion) on a predetermined timescale (Hadjighasem et al., 2017;Haller, 2002;Haller & Poje, 1998).While a 3D steady incompressible flow is sufficient for chaotic advection (as opposed to a 2D incompressible steady flow), a special feature of time-dependent flows, both in 2D and 3D, is that streamlines do not equal material lines (Lagrangian trajectories).This allows what we denote "time-dependent chaotic advection": a transport mechanism that moves a tracer from point A to point B, located in two well-separated regions, and in a predefined timeframe, despite there being no streamlines from point A to point B at any snapshot in time, for example, the 2D oscillating doublegyre example (Aref, 1984;Yang & Liu, 1994).These studies illustrate a crucial effect of temporal variability on transport.Roughly, this mechanism is created in a flow with several instantaneous saddles (hyperbolic structures) that split nearby trajectories in different directions.If the splitting surfaces themselves are not stationary, a tracer can "jump" between cycles and reach places it could not reach in the predefined timescale (or, perhaps, ever).In incompressible 2D flows, it was shown that this mechanism is essential for recreating transport due to the integrability of steady trajectories: an average flow plus a reasonable isotropic diffusion will not imitate largescale time-dependent transport (Carlson et al., 2010).Furthermore, this mechanism allows a natural, significant, measurable enhancement of spreading and Lagrangian mixing without an increase in the kinetic energy of the flow.
In general, finding simple flow models such as steady, time-periodic, and various averaged flow models, that can imitate realistic spreading and Lagrangian mixing, can teach us about the relative importance of inner-and interannual variability, eddy kinetic energy, and small-scale dynamics, as well as the time-dependent chaotic advection mechanism.The dynamical origins underlying oceanic spreading and Lagrangian mixing of a localized source, in terms of areas covered and associated timescales, can be explored by a comparison between models of varying complexity.For example, a suitable simple model can assist in gaining better intuition from toy models that study the MOW impact on the Atlantic Ocean (see, e.g.Saporta-Katz et al. (2023), where such a toy model for the velocity field serves as an important component of the modeling).
In this work, we produce and analyze synthetic Lagrangian trajectories of the MOW in reanalysis velocity data from 1980 to 2020 to provide a comprehensive analysis of transport, spreading, and pathway statistics and associated timescales.To this aim, we compare virtual trajectories released at the GoC and advected by several types of oceanic models, with varying degrees of complexity.We examine the statistics obtained from the different simplified models of the full underlying velocity field, aiming to find simple models that provide MOW spreading that is similar to the one obtained with the full velocity data.Thus, we evaluate the effects of monthly, annual, and inter-annual variability, and the effect of time-dependent chaotic advection, on the northbound transport and overall spreading and Lagrangian mixing of the MOW in the NA.
We perform four distinct types of experiments, each corresponding to a different oceanographic model: 1. Full5day: 12 × 20 monthly releases in the full reanalysis 5-day data, from 1 January 1980 to 1 December 1999, denoted Full5day_#year_#month.2. FullMonthly: 12 × 20 monthly releases in the full reanalysis monthly-averaged data, from 1 January 1980 to 1 December 1999, denoted FullMonthly_#year_#month. 3. RepeatYear: 12 × 40 monthly releases in yearly-periodic velocity fields, from 1 January 1980 to 1 December 2019, denoted RepeatYear_#year_#month.For example, RepeatYear_n_m follows trajectories released in year n, month m, into a yearly-periodic velocity field that repeats year n for 20 times.4. RepeatMonth: 12 × 40 releases in steady velocity fields, from 1 January 1980 to 1 December 2019, that consist of the release month only, denoted RepeatMonth_#year_#month.For example, RepeatMonth_n_m follows trajectories released in year n, month m, into a steady velocity field that repeats year n, month m, for 12 × 20 times.
The first model is the most accurate and includes realistic inter-annual variability.The other models can be viewed as different versions of truncated series expansions of the first model of decreasing realism: FullMonthly is closest to Full5day; RepeatYear and RepeatMonth have the same spatial resolution and kinetic energy as in FullMonthly; RepeatYear corresponds to time-periodic velocity fields while RepeatMonth is steady.Since the RepeatMonth experiments repeat the velocity fields that, together, make up the FullMonthly velocity field, their Lagrangian trajectories are precisely segments of the (stationary) streamlines underlying the FullMonthly flow, namely these are the instantaneous streamlines of the FullMonthly flow.We use the streamlines calculated in the RepeatMonth experiments to evaluate the effect of time-dependent chaotic advection on the northbound transport of the FullMonthly experiments.To address the importance of inner-and inter-annual variability of flow patterns, as well as the ability of a simple flow model to recreate realistic spreading, we compare all four experiments.In Text S1 of Supporting Information S1, we also show results of the traditional climatological models.These exhibit limited transport and spreading; see Discussion section.
For each experiment, the particles are released once a month from the MOW section at the GoC at 7°W.This section is chosen since it is fully inside the GoC, westwards enough that it is beyond the major MOW sinking plume and eastwards enough that the velocity field is still clearly separated to an eastbound flow and a distinct westbound channel, see Figure 1.At this section, 7°W between 34°and 37°N, the salinity and depth threshold for defining the MOW is determined by the depth and salinity values at which the maximum salinity per depth is at its minimum, since any further rise in the salinity is attributed to the MOW (Figure 1c-1e).The MOW area is defined as the intersect between points with depth and salinity higher than the threshold values, and points with a westbound velocity.In Figure 1f, the overall MOW transport in Sverdrup according to this definition is shown per month for the FullMonthly and Full5day data sets, showing an average outflow of 1 Sv with a variability of up to 1.5 Sv.Every month, we release between 1,500 and 15,000 virtual particles, such that each particle carries 10 4 Sv of water, distributed randomly on the MOW section at 7°W as defined above.The virtual particles are tracked in their corresponding velocity field for 20 years in each of the four different oceanic models.

Spreading and Lagrangian Mixing of the Different Oceanic Models
We calculate the density of the final locations of MOW particles released over a single year and measured 20 years after the first release time (Figure 2), measuring the percentage of particles northwards of 53°N, entering the SPG region; westwards of 35°W, extending beyond the mid-Atlantic ridge; and deeper than 1,500 m, joining the NADW.In the Full5day and FullMonthly experiments, that are practically indistinguishable from each other, >30% of particles venture far enough to reach (at least) one of these regions.Specifically, an average of 13.5% and up to 20% of MOW particles reach the SPG region after approximately 15 years, at which point the entrance into and flushing out of the region reaches a balance, see plateau in Figure 2b.While differing in specifics, a similar qualitative picture is obtained for RepeatYear.The SODA velocity fields do not resolve convective processes, and their vertical velocity is calculated diagnostically (Delworth et al., 2012); nevertheless, we measure a significant amount of approximately 35% that sink under 1,500 m to join the NADW.
The steady RepeatMonth runs exhibit significant fluctuations in their northbound transport.While over 80% of the releases allow less than 5% of particles to cross 53°N, certain months exhibit a massive northbound transport.Specific RepeatMonth releases do exhibit statistics that are close to those of the full and RepeatYear runs, however they do not mix well (Figure 3), despite the steady velocity fields containing the same kinetic energy as the full runs.
To evaluate the degree of Lagrangian mixing, we use the symmetric Kullback-Leibler divergence (KLD) (Seghouane & Amari, 2007).The symmetric KLD is a measure of statistical distance between two distributions, with the attribute that it returns a distance of zero between two distributions that are identical up to permutations.This makes it a good measure for oceanic data, where features can retain their form while migrating with time.For two distributions P and Q, the symmetric KLD is defined as D sym KL (P,Q) = ∑ ⃗r∈Ω (P( ⃗r) Q( ⃗r) ) log( P( ⃗r) Q( ⃗r) ) .In Figures 3a-3c we perform an intercomparison between the different distributions using this tool.In Figures 3d  and 3e we compare all runs to a benchmark run, chosen as FullMonthly_1992, since its statistical spreading is closest to the average statistical spreading, see Figure 2. The RepeatMonth runs are clustered far away and exhibit a much more limited degree of Lagrangian mixing of the tracers.While Full5day and FullMonthly are again effectively indistinguishable, the RepeatYear experiments are clustered separately.Both the RepeatYear experiments and, even more so, the RepeatMonth experiments, have relatively large clusters of KLD mixing, reflecting the large inner-and inter-annual variability of the flow.
The small overlap between the clusters includes a single RepeatYear run that is close to the full data cluster (see circled run in Figure 3d).In our opinion, this overlap indicates that a yearly-periodic velocity field chosen with care could provide a good imitation of the degree of Lagrangian mixing and spreading of the full dynamics, despite its lack of inter-annual variability; see Discussion section.Generally, these results show that as the velocity field models become more realistic, the spreading patterns provide an increasingly better approximation of the full dynamics' KLD mixing; this statement also refers to the climatological experiments; see Figure S3 in Supporting Information S1.

The Northward Trajectories
The horizontal dynamics of the Lagrangian trajectories divide into three distinct groups (Figure 4): 1.An average of 61% of MOW trajectories are stationary, defined as contained in the box 25°-53°N, 5°-35°W throughout their 20 years of evolution; 2. 23% of MOW trajectories take a direct westbound path, taking on average 11.3 years to cross the mid-Atlantic ridge at 35°W; 3. 16% of trajectories follow a direct northbound path along the eastern boundary of the NA, taking on average 10.4 years to cross the 53°N mark.A negligible amount (<1%) of trajectories also exit the box from the south.Out of the northbound particles, most particles (55%) enter the northeastern box and stay there, whereas 34% continue into the subpolar gyre, reaching the northwestern box.Again, the FullMonthly and the Full5day trajectories are practically indistinguishable.
To study at which longitudes the MOW trajectories cross into the SPG region of the NA, we combine all Full-Monthly trajectories that cross the 53°N line, no matter through which pathway, and calculate the density plot on this latitude.A single window at 18°W and at a depth of approximately 1,000 m, situated at the southern border of the Rockall Ridge, provides a subsurface pathway for the vast majority of northbound trajectories, see Figure 5.It is a saddle-point of the velocity field, from which trajectories separate into two paths, one passing from the east, through the Rockall Trough, as seen in Burkholder and Lozier (2011); and one from the west of the ridge at 20°N.An average of 35% of trajectories choose the east over the west path.Per date of crossing, the percentage of eastbound trajectories has a positive, statistically significant (α < 0.001) correlation of R = 0.32 with the SPG index, as described in the introduction, calculated from the SODA data directly (Text S2 and S3 in Supporting Information S1); this fits the fact that the SPG index indicates the east-west shift of the eastern branch of the SPG.

Chaotic Advection and Correlations
We compare the percentage of trajectories that take a northbound path in the FullMonthly, RepeatYear, and RepeatMonth data sets, see Figure 4j.While there is a statistically significant (α < 0.001), albeit low, correlation of R = 0.26 between RepeatYear and RepeatMonth, the RepeatYear northbound trajectory statistics are systematically higher than those from the RepeatMonth data set, with 14% of RepeatYear trajectories and only 6% of RepeatMonth trajectories taking a direct northbound route, as defined in the previous section.An even stronger result emerges from the FullMonthly data, of which 16% of trajectories take a direct northbound route.Since the RepeatMonth runs follow the stationary streamlines of the underlying velocity fields that make up the Full-Monthly and the RepeatYear experiments, the difference in northbound trajectories between RepeatMonth and FullMonthly, and between RepeatMonth and RepeatYear, is attributed to time-dependent chaotic advection, as described in the introduction.Thus, we conclude that for most months, the main mechanism that transports particles from the GoC to the SPG in the FullMonthly experiments is time-dependent chaotic advection, and not a steady, direct northbound current, that would manifest in the RepeatMonth experiments.(Seghouane & Amari, 2007) distance of spreading distributions from FullMonthly_1992 (the year with statistics closest to their averages).The color scheme is the same as in Figure 2c.Panels (f and g) show the longitude-latitude density plots of the RepeatYear (f) and RepeatMonth (g) releases with the smallest statistics vector distances, as marked by black x's on (b-e).The black circle in (d) marks the RepeatYear experiment that has the smallest KLD distance from the origin.
We observe that the FullMonthly northbound trajectory statistics, out of which each datapoint contains data from 20 years ahead of the release date, exhibit a statistically significant (α < 0.001) high correlation with both the 8year-lagged and the 2-year-lagged RepeatYear data (respectively R = 0.27, R = 0.28); while a correlation with an 8-year lag is expected given the time it takes trajectories to cross the 53°N line, the 2-year-lag correlation is yet to be explained.These correlations may imply some degree of predictability, as marked on Figure 4j.
Finally, we note that the significant variability in the RepeatMonth and RepeatYear experiments (<5% to >30% northbound transport) indicate that some years are more prone to cross-gyre (specifically, northbound) transport than years.The origins of this variability are not well understood, and are left for future research; see Discussion section.

Discussion
We have created a thorough survey of the various Lagrangian pathways taken by the MOW over the past four decades, based on SODA3.4.2 reanalysis data.In the course of 20 years, the MOW mixes into the entire NA basin, from 10°N to 70°N and between the coasts (Figures 2 and 3), with a greater concentration around the Gibraltar  2).Panel (g) shows the percentage of northbound particles that either stayed in the northeast box after exiting the initial box (blue), moved on to the north-center box (purple), or continued to the northwest box (pink).Panels (h)/(i) show the average amount of years it took the particles released at a given date to cross, respectively, 35°W/53°N.(j) Similar to (f), the percentage of particles that take a direct northbound path for FullMonthly, RepeatMonth, and RepeatYear, out of all particles released at a given date.12-month moving averages (m.a.) are marked in bold.The black (red) prediction of the FullMonthly northbound percentage for 2000-2018 (2000-2012) is based on the 2-year (8-year) lag correlation between FullMonthly and RepeatYear; see Text S3 in Supporting Information S1.
Straits.After 15 years, the northbound influx and outflux rates reach a balance with an average of 13% of MOW particles situated in the SPG region, that is, beyond 53°N: this is the timeframe required to study MOW northbound transport.We identify a direct northbound path from the Gibraltar straits along the eastern section of the NA that leads beyond 53°N into the SPG region (Figure 4).Through this path, an average of 16% of MOW particles take a direct northbound path into the SPG region; in some release years, up to 35% take the direct northbound path.
The 53°N line contains a relatively narrow window at the southern tip of the Rockall Ridge, between 14 and 17°W and 800-1,200 m depth, through which the vast majority of northbound MOW particles cross into the SPG region (Figure 5).At this point they cross the ridge around either the east (35%) or the west (65%) of the ridge.There is a positive correlation between the SPG index and the percentage of particles that take the eastern path at a given date of crossing, supporting the idea that a high SPG index correlates with an eastward extension of the SPG's eastern boundary that blocks particles from taking the western path around the ridge.However, we have not yet found evidence that a high SPG index correlates with a decreased overall transport of the MOW into the SPG region.
Over half of the direct northbound transport in the full time-dependent experiments is a result of time-dependent chaotic advection, and not due to a steady northbound current, that would manifest in the instantaneous streamlines of the RepeatMonth experiments (Figure 4j).An observed statistically significant 2-year-lag correlation between the direct northbound transport in RepeatYear and FullMonthly indicates a possible predictability of the future northbound transport of the MOW.Exploration of the dynamical origins of the chaotic advection and the 2-year correlation is left to future studies, and may be related to the dynamical mechanism that causes some RepeatYear experiments to exhibit much more northbound transport than others.
Throughout all the diagnostics we consider, the FullMonthly and Full5day statistics are practically indistinguishable.We hypothesize that additional averaging, as done for example, in climatological oceanic models (Text S1 in Supporting Information S1), will begin to ruin the statistics at timescales significantly longer than the typical lifetime of mesoscale eddies, at around a few months.The yearly-periodic RepeatYear experiments differ from the FullMonthly experiments both in their statistics time series, and in their average degree of Lagrangian mixing (Figures 2c,2f,2i,and 3).In light of the presence of a RepeatYear run (indicated as the circled run in Figure 3d) that is close to the full runs' cluster, we suggest that it may be possible to construct a yearly-periodic flow model that can replicate the qualitative degree of MOW spreading and Lagrangian mixing in the full dynamics.On the other hand, the spreading of the MOW in all the steady RepeatMonth oceanic models (as well as in the steady climatological model, see Text S1 in Supporting Information S1) differs significantly from all the spreading observed in all of the FullMonthly cases, suggesting that some temporal variability is imperative, creating a natural enhancement of Lagrangian mixing and spreading without raising the kinetic energy of the flow.Numerous questions remain regarding the ability of a simple model to recreate realistic transport statistics; for example: Can one find a batch of yearly-periodic flow regimes that are statistically indistinguishable from the full flow experiments?Can one tailor a steady flow field (with a realistic amount of kinetic energy) to produce a realistic degree of Lagrangian mixing and spreading?;Lacking small spatial scales of the dynamics, can the climatological models produce a better Lagrangian mixing if injected with additional kinetic energy?;What is the effect of monthly variability; namely, what will be the transport properties of a yearly-averaged flow?These important questions are left for future studies.
Finally, we suspect that the mesoscale eddies do play an important role in the MOW spreading.Identifying the relative importance of mesoscale eddies and the kinetic energy content of the velocity fields in Lagrangian mixing and spreading the MOW would require using an eddy-resolving velocity data set.Additionally, it will be interesting to explore in this 3D context the possible compensation of the mesoscale eddies and/or of the velocity field time dependence by an isotropic diffusion term.

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Synthetic Lagrangian trajectories initialized at the Gibraltar Straits, based on SODA3.4.2 reanalysis velocity data from 1980 to 2020, are used to study the pathways, Lagrangian mixing, and spreading of the Mediterranean Outflow Water (MOW) • 16% of the MOW particles take a direct northbound route • Over half of the direct northbound transport results from time-dependent chaotic advection and not from a steady northbound current Supporting Information: Supporting Information may be found in the online version of this article.

Figure 1 .
Figure 1.Mediterranean Outflow Water (MOW) in the SODA3.4.2 reanalysis data.(a-e) Climatological annual data averaged 1980-2020: (a) Salinity at 1,000 m.(b) Salinity at 37°N latitude.(c) Maximum salinity per depth at 7°W; MOW is defined as the area with a westward velocity and salinity above 35.82psu at a depth below 330 m, see main text for details.(d) Salinity at 7°W.(e) Velocity at 7°W.(f) MOW transport in Sverdrup for 5-day and monthly SODA data.

Figure 2 .
Figure 2. Spreading of the Mediterranean Outflow Water (MOW) in the North Atlantic.(a, d, and g) Probability density plots of MOW particles released in FullMonthly data from January 1992 to December 1992 once a month, measured 20 years after initial release, at January 2012.(b, e, and h) The percentage of particles beyond 53°N, 35°W, and 1,500 m, respectively, per year, where each color signifies a different release year, for FullMonthly data.Note the difference in the y-axis between subplots.(c, f, and i) The overall percentage of particles crossing the benchmarks per release year, for all data types.The RepeatMonth experiments (blue) have a data point for every month repeated.

Figure 3 .
Figure 3. Lagrangian mixing and statistics comparison.(a-c) For each release year 1980-2000, the spreading statistics vector is defined as (% north spreading after 20 years, % west spreading after 20 years, % sinking spreading after 20 years).The pairwise Euclidean distance between every pair of spreading vectors is plotted.Panels (d and e) show the symmetric Kullback-Leibler divergence (KLD)(Seghouane & Amari, 2007) distance of spreading distributions from FullMonthly_1992 (the year with statistics closest to their averages).The color scheme is the same as in Figure2c.Panels (f and g) show the longitude-latitude density plots of the RepeatYear (f) and RepeatMonth (g) releases with the smallest statistics vector distances, as marked by black x's on (b-e).The black circle in (d) marks the RepeatYear experiment that has the smallest KLD distance from the origin.

Figure 4 .
Figure 4. Typical pathways of the Mediterranean Outflow Water.Panels (a-b)/(d-e) are typical examples of westbound/northbound trajectories: pathways that exit the release box through its western/northern boundary, respectively.Panels (c)/(f) show the percentage of particles that take the westbound/northbound path as a function of the release month (legend is the same as Figure2).Panel (g) shows the percentage of northbound particles that either stayed in the northeast box after exiting the initial box (blue), moved on to the north-center box (purple), or continued to the northwest box (pink).Panels (h)/(i) show the average amount of years it took the particles released at a given date to cross, respectively, 35°W/53°N.(j) Similar to (f), the percentage of particles that take a direct northbound path for FullMonthly, RepeatMonth, and RepeatYear, out of all particles released at a given date.12-month moving averages (m.a.) are marked in bold.The black (red) prediction of the FullMonthly northbound percentage for2000-2018 (2000-2012)  is based on the 2-year (8-year) lag correlation between FullMonthly and RepeatYear; see Text S3 in Supporting Information S1.

Figure 5 .
Figure 5. Panels (a and b) show two typical FullMonthly trajectories that cross 57°N, via either an eastern path (green) or a western path (red) around Rockall Ridge, both continuing into the sub-polar gyre (SPG).Panels (c and d) shows density sections at, respectively, 57°N and 53°N, of all trajectories from the FullMonthly releases that cross these sections.(e) Orange line-percentage of particles that cross from the east passage out of all particles that cross 57°N, versus the date at which they cross this section.Black line-its yearly moving average.The blue line is the yearly moving average of the normalized SPG index, see main text and Text S2 in Supporting Information S1.