Sensitive dependence of trajectories on tracer seeding positions-coherent structures in German Bight surface drift simulations

Backward drift simulations can aid the interpretation of in situ monitoring data. Some trajectories, however, are sensitive to even small changes of the tracer release position. A corresponding spread of backward simulations implies convergence in the forward passage of time. Such uncertainty about the probed water body’s origin complicates the interpretation of measurements. This study examines surface drift simulations in the German Bight (North Sea). Lines across which drift behaviour changes non-smoothly are obtained as ridges in the fields of the finite-time Lyapunov exponent (FTLE), a parameter 5 used in dynamical systems theory to identify Lagrangian coherent structures (LCS). Results are shown to closely resemble those obtained considering a) two-particle relative dispersion and b) the average divergence of Eulerian velocities that tracers experience. Structures observed in simulated sea surface temperature and salinity further corroborate the FTLE results.


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In the German Bight area exists a comprehensive monitoring network, including the Marine Environmental Monitoring Network in the North Sea (MARNET), the Coastal Observing System for the North and Arctic Seas (COSYNA) and other stations.
Details on the type of data being collected can be found in Baschek et al. (2017). Stanev et al. (2016) discuss issues related to modelling and data assimilation with spatiotemporal optimal interpolation. Multivariate statistical methods could also be used for optimizing the design of observational arrays (e.g. Chen et al., 2016;Kim and Hwang, 2020). However, data analysis based 15 on a merely statistical description of spatial connectivity falls short of what can be achieved if hydrodynamic current fields from either models or remote sensing are available. This applies all the more, when it comes to the interpretation of data from a whole array of in situ monitoring stations.
Backward trajectories of Lagrangian tracers seeded at monitoring stations provide valuable insight into the background of water bodies that are probed (e.g. d 'Ovidio et al., 2015). They help distinguish between temporal and spatial variability, 20 i.e. local changes and advection from somewhere else. Because of considerable uncertainties, however, following just single particle trajectories is likely to be misleading. Trajectories accumulate deficiencies of the underlying hydrodynamic fields, including the effects of unresolved sub-grid scale hydrodynamic structures. Initially moderate deviations may possibly transfer a trajectory to another submesoscale circulation structure. Backtracking water bodies from hypothetical monitoring stations in the vicinity of Helgoland, Callies et al. (2011, their Fig. 3) provide an example of how quasi-chaotic mixing may transform 25 initially regular into quite contorted structures. Also in nature drifters released pairwise may separate quite fast (e.g. Callies et al., 2019;Meyerjürgens et al., 2020), which sets a limit to the reliability of simulations that can be achieved in the best case.
For these reasons, Lucas et al. (2016) for instance, studying the variation of bacterial community composition at station Helgoland Roads in the German Bight (North Sea), considered the behaviour of a whole bundle of backward trajectories, seeded within an extended region around the observational site. Uncertainties due to sub-grid scale eddies unresolved in the model 30 were dealt with by a random walk component superimposed to each individual trajectory. This blanket approach implicitly deals also with the problem the present study focusses on: A possibly high sensitivity of backward trajectories (either simulated or observed) to where exactly they are seeded. A statistical measure for such particle spreading is relative dispersion, the mean square particle distance as function of time. LaCasce (2008) reviews how this parameter relates to the energy spectrum of a turbulent flow. Relative dispersion is called non-local if particle separation is dominated by eddies much larger than particle 35 separation. In this case, characterized by a steep energy spectrum, particle separation is expected to grow exponentially. The very high sensitivity to initial particle positions implies what in dynamical systems theory is called chaotic advection.
Dynamical systems theory aims at a description of the kinematics of turbulent mixing. The approach is based on flow maps that describe particle advection over some time interval, according to Haller (2015) "thereby mimicking experimental flow visualization by tracers". This technique has widely been applied for analysing the microstructure of chaotic mixing processes 40 in two dimensions (e.g. Pierrehumbert and Yang, 1993), describing how chaotic advection may transform initially small disks of fluid into complex filamentary structures. Trying to improve the sometimes vague definitions of such structures, Haller and Yuan (2000) introduced the framework of Lagrangian coherent structures (LCS). Their method seeks to identify material lines that function as only weekly permeable barriers for water body transport, attracting or repelling neighboured trajectories. Peacock and Haller (2013) provide a nice overview of the topic.

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In case of flows with arbitrary time dependence, identification of LCSs can still be difficult. Hadjighasem et al. (2017) compare twelve candidate approaches that could be used. Among those, calculation of finite-time Lyapunov exponents (FTLE) is one of the most common methods. It is closely related to the finite-scale Lyapunov exponent (FSLE), originally introduced by Aurell et al. (1996Aurell et al. ( , 1997 and used in experiments for diagnosing scale dependent separation rates between drifter pairs (LaCasce and Ohlmann, 2003;Sansón et al., 2017). Karrasch and Haller (2013), however, report some limitations for FSLE in 50 LCS detection suggesting that an approach based on FTLE distributions may be more reliable. The FTLE fields are independent of an observer's reference frame (Haller, 2015), representing the rate at which neighbouring tracers diverge according to the largest eigenvalue of the so-called Cauchy-Green strain tensor. Ridges in the FTLE field are indicators of LCSs. Building on work by Haller (2001), Shadden et al. (2005) even define LCS in terms of these ridges, assuming that those approximately act as transport barriers. In to dimensions the LCSs are material lines transported with the flow.

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Haller (2011) discusses examples in which substantial material flows crossing a FTLE ridge occur. It may also happen that a LCS does no produce a FTLE ridge or that a LCS suggested by FTLE does not exist. Therefore Haller (2011) developed a more sophisticated variational theory that also involves the eigenvectors of the Cauchy-Green strain tensor. Farazmand and 2 https://doi.org/10.5194/os-2020-83 Preprint. Discussion started: 31 August 2020 c Author(s) 2020. CC BY 4.0 License. Haller (2012) present a corresponding numerical algorithm for two-dimensional application, based on the specification of strainlines along which exponential stretching occurs (to be distinguished from simple shear). Recently Tian et al. (2019) 60 applied a variational method to identify the outer bounds of the Kuroshio current system. Wiggins (2005) makes reservations that, as contrasted with many engineering applications, the presence and interaction of very different scales in geophysical flows can restrict the possibility of simulating detailed particle drift paths. The present study will therefore adhere to the simple conventional FTLE analysis. German Bight residual currents change with changing atmospheric winds (Schrum, 1997;Callies et al., 2017a) so that a description of exchange processes in a quasi-persistent hydrodynamic space-time structure like gyres or jets (Wiggins, 2005) is not the topic here. Analysing surface transports simulated by the operational hydrodynamic model BSHcmod, FTLE fields will be compared with statistical measures like single-particle absolute and two-particle relative dispersion, but also with the Lagrangian divergence (the average divergence that tracers experience along their trajectories (Huntley et al., 2015)). It turns out that all these parameters deliver very consistent results.
The paper is organized as follows: Section 2 first describes how Lagrangian drift simulations were performed based on 70 pre-calculated hydrodynamic surface current fields. It follows a short compilation of the definitions of the FTLE, Lagrangian divergence and statistical measures of dispersion. Section 3 then reports three prototypical situations, evaluated also with regard to fields of sea surface temperature and salinity. Example trajectories illustrate the relevance of FTLE ridges as material separatrices. Three snapshots from a video available in the supplement illustrate the temporal variability of LCSs. A general discussion and a short summary conclude the paper.

Study area
The North Sea is a semi-enclosed shelf sea that connects to the north-eastern Atlantic at its northern boundary and through the English Channel at its southwest (Sündermann and Pohlmann, 2011). Strong tidal forcing occurs as a co-oscillation triggered by Atlantic tidal waves. This study focusses on the German Bight, the shallow south-eastern part of the North Sea with water 80 depths of mainly 20-40 m, adjoining the Dutch, the German and the Danish coasts (Becker et al., 1992). In the German Bight, a mean cyclonic North Sea circulation corresponds with residual currents from the southwest to the north. Superimposed to this mean circulation, a strong weather driven variability occurs on short time scales (Schrum, 1997;Callies et al., 2017a). A fresh water plume emerging from the Elbe River and, to a minor extent, also the Weser river (see Fig. 1) can be observed as a permanent feature. Transient eddies and meanders depend on bottom topography, baroclinic instabilities and wind effects. The 85 most important topographic feature is the old Elbe Glacial Valley, opening from today's Elbe estuary towards the northwest (west of Helgoland) into the open North Sea. Frontal structures depend on season but vary also on a short term basis (Budéus, 1989;Schrum, 1997). In the warm season, strong stratification occurs at water depths greater than approximately 30 m, mainly in the Elbe Glacial Valley. A baroclinic tidal mixing front (James, 1984;Holt and Umlauf, 2008) separates this region from the well-mixed more shallow coastal water, where stratification is prevented by strong tidal mixing (Krause et al., 1986).

Hydrodynamic fields
Offline drift simulations were based on surface currents taken from archived BSHcmod model output. Fields of surface temperature and salinity were taken from the same data base. BSHcmod is run operationally by the Federal Maritime and Hydrographic Agency (BSH). The model covers North Sea and Baltic Sea and is two-way nested with approximately 900 m resolution in the German Bight area and approximately 5 km in the open North Sea (Dick et al., 2001). In the vertical, a dy-95 namical coordinate is used (Dick et al., 2008). The model's atmospheric forcing on an hourly basis is provided by the regional model COSMO-EU (Consortium for Small-Scale Modelling; Schulz and Schättler (2014)), run by the German Meteorological Service (Deutscher Wetterdienst -DWD). For an inclusion of wind stress, the parametrization by Smith and Banke (1975) is used. Stokes drift remains disregarded in archived operational model output.
In the process of archiving, BSHcmod hydrodynamic fields with originally higher vertical resolution were re-gridded. Con-100 serving transport rates, this was done in such a way that the stored surface currents used in this study approximately represent the uppermost 5 metres of the water column.

Lagrangian drift simulations
Drift simulations were performed using the Lagrangian transport program PELETS-2D (Callies et al., 2011), based on BSHcmod model output archived on a 15 min basis. Originally, the PELETS toolbox developed at Helmholtz-Zentrum Geesthacht 105 was designed for its use with hydrodynamic currents on unstructured triangular grids. Current fields provided on a regular grid (like those from BSHcmod) must be preprocessed, splitting each rectangular grid cell into two triangles. This transformation of grid topology does not affect the information content of hydrodynamic fields.
All simulations in this study were produced using the fourth-order Cash Karp method (Press et al., 1992) that belongs to the Runge Kutta family of solvers. It should be mentioned, however, that a simple Euler forward scheme used in other PELETS 110 applications (e.g. Callies et al., 2011Callies et al., , 2017bCallies et al., , 2019 gave very similar results. The maximum time step is set to 15 min. Velocities are updated earlier if a tracer particle moves to another triangular grid cell.

Finite-time Lyapunov exponents (FTLE) as indicators of Lagrangian coherent structures (LCS)
Definition of the FTLE is based on a consideration of Lagrangian flow motions. A flow map Φ relates particle locations x 0 , where particles were seeded at time t 0 , to their destinations x at later time t = t 0 + τ : Taking the spatial gradient ∇Φ τ t0 = ∂x(t 0 + τ ; t 0 , x 0 )/∂x 0 , one obtains the following Cauchy-Green strain or deformation tensor (e.g. Shadden et al., 2005;Haller, 2015): Lyapunov exponent is based on its largest eigenvalue λ max : The absolute value of integration time τ is used because integration of particle drift can be conducted either forward or backward in time. The geometric interpretation of the FTLE refers to the maximum separation rate of neighbouring particles.
Maximum separation among particles started on a small circle around location x 0 occurs for those particles that end up along 125 the largest principal axis of an ellipse that evolved from the initially circular structure (see Haller, 2015, his Fig. 4).
For the computation of FTLE fields, a regular Cartesian grid of tracers was released. Initial locations with 1 km resolution covered the German Bight area east of 6.5 (2)) was performed involving trajectories seeded at neighbouring locations of the regular FTLE grid.
If at least one of the trajectories needed for FTLE calculation reached the coastline, the FTLE value was treated as missing.
Corresponding gaps in the FTLE fields depend on prevailing atmospheric forcing. As BSHcmod covers the whole North Sea, no such problem occurs for particles that cross the open boundaries of the FTLE grid.

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An incompressible two-dimensional flow field preserves the area of a Lagrangian patch during arbitrary deformations. In the present study this is not the case as the two-dimensional surface currents being used were extracted from 3D hydrodynamic fields, allowing for vertical exchange of water masses. Huntley et al. (2015) developed a concept that splits FTLE values into contributions that come from area-preserving stretching on the one hand and dilation on the other. With the area of a deformed elliptical Lagrangian patch being proportional to the product of the two eigenvalues λ i of the Cauchy-Green strain tensor, 140 Huntley et al. define a dilation rate ∆ in a two-dimensional flow field as: and demonstrate its potential for supporting the interpretation of satellite based observations of surface chlorophyll a patches.
In the present study, FDLD values were calculated at all locations with valid FTLE values. Eulerian divergences needed for the evaluation of Eq. (5) were computed based on a discretization using auxiliary points at a 250 m distance. Velocities at these auxiliary locations were obtained by linear interpolation in the respective grid triangle.

Absolute and relative dispersion
Absolute and relative dispersion are statistical measures for analysing Lagrangian data. Generally, absolute dispersion is defined as the second moment of the single particle displacement PDF, i.e. the variance of particle displacements relative to their starting position, which must not be confused with cloud variance (LaCasce, 2008). Ensemble averaging could be performed with respect to either different locations or different realizations at some fixed location. Here, following Haller and Yuan 155 (2000), the simpler density of absolute dispersion is considered, describing just a single particle's squared displacement from its release point: By contrast, relative dispersion describes the mean square separation of particle pairs with nearby initial release points. Relative dispersion at each node of the FTLE grid will be calculated combining information from four particle pairs, where δx i denotes the distance vector between neighbouring nodes. For a comparison with FTLE and FDLD fields, the logarithm of absolute and relative dispersion is a reasonable choice. Exponential growth of pair separations indicates the presence of Lagrangian chaos dynamical systems theory deals with (Wiggins, 2005).

Examples
The following examples are intended to illustrate the occurrence of Lagrangian structures in German Bight surface currents.
None of these structures are persistent, occurrence and specific details depend on the past evolution of environmental conditions. if they were zero). All locations that gave rise to trajectories hitting the coast were disregarded.
At the time the plot refers to, the most prominent feature of the FTLE field is a south-north running ridge that separates the region of interest more or less into two halves. Further west, a less pronounced parallel second ridge occurs which, however,     Fig. S1).

Third example
The third example, referring to 29 February 2016 (11:00), provides an analysis in terms of statistical dispersion measures. A measure that directly concentrates on small scale changes in drift behaviour is two-particle relative dispersion (Fig. 3b).
Maps of absolute and relative dispersion are in very good agreement, relative dispersion highlighting sharp transitions in the 215 graph of absolute dispersion. The two plots include the same example trajectories. Two test trajectories near the horizontal divide south of MARNET station 4 illustrate a stepwise change of advection speed, giving rise to the enhanced level of absolute dispersion for the test station located more to the south (green). Note that a pure change of drift direction, maintaining advection speed, would have affected relative but not absolute dispersion. Three additional magenta trajectories, seeded at MARNET stations 1, 2 and 6, were included to just visualize spatial variability of transports.

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Finally, it is to be noted that the relative dispersion graph in Fig. 3b closely resembles the backward FTLE field (Fig. S2).
FTLE ridges subdivide the area of interest in the same way as relative dispersion does, differences can hardly be distinguished.  13 https://doi.org/10.5194/os-2020-83 Preprint. Discussion started: 31 August 2020 c Author(s) 2020. CC BY 4.0 License. Fig. 4a, referring to the situation in Fig. 1, shows a south-north oriented zone of relatively cool water. This belt is made up by a couple of patches that bear a striking structural resemblance to patches of positive divergence in Fig. 1b. These patches and 230 the overall belt are delimited by the FTLE ridges shown in Fig. 1a. In the temperature field these lines of convergence (Fig. 1b) appear as being relatively warm. Fig. 1b suggests that some features of the temperature distribution in Fig. 4a can indeed be explained in terms of up-and downwelling.
Similar effects occur on 26 March 2018 (Fig. 4b, corresponding with Fig. 2a). The sharp west-east oriented ridges in Fig. 2a reappear in Fig. 4b as lines of relatively warm water (e.g. near MARNET station 6 or between MARNET stations 1 and 3).

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On the other hand, three tongues of relatively cool water extend westward from the coast into the areas between the lines of converging surface currents. Note that the eye-catching pronounced westward transition towards generally higher temperatures in the open sea (a transition broadly corresponding with increasing water depth towards the old Elbe Glacial Valley) does actually not always coincide with the main FTLE ridge neighbouring MARNET station 4. In particular to the north of this station, the FTLE ridge produces a line of relatively warm water that is clearly separate and shifted eastward (Fig. 4b).
240 Fig. 4c, showing the temperature field for 29 February 2016, corresponds with dispersion rates in Fig. 3. In this case, sharp transitions in the temperature field correspond with lines of large relative backward dispersion (Fig. 3b) or backward FTLE ridges (Fig. S2). Fig. 4, some related structures can be identified also in salinity fields (see Fig. S3). See Krause et al. (1986) and Budéus (1989) for a report on observations regarding the roles of temperature and salinity in different 245 kinds of German Bight frontal structures. Long FTLE ridges aligned in a meridional direction (Fig. 5a) evolve into a more cellular structure (Fig. 5c).

Time evolution of coherent structures
The FTLE field in Fig. 5b is much less compartmentalized than the fields in Figs. 2a and 3b, for instance. Instead, it contains more filamentary ridges that sometimes come very close. To illustrate the relevance of such narrow filaments, Fig. 5b combines a simulated backward trajectory starting at MARNET station 4 with another two trajectories (red and green) initialized slightly further east. Between the three seeding positions, FTLE ridges indicate enhanced backward particle separation (i.e. conver-255 gence in forward mode). Accordingly, the three trajectories end points are clearly much more separated from each other than tracers were at the outset. All three trajectories clearly reflect a reversal of the residual circulation that occurred during 22-24 November, when a pronounced cyclonic circulation changed to an anticyclonic circulation 2 . However, the more the observation position is shifted to the east, the more any hypothetical measurements would reflect conditions the probed water parcel experienced further south. 15 https://doi.org/10.5194/os-2020-83 Preprint. Discussion started: 31 August 2020 c Author(s) 2020. CC BY 4.0 License.

Discussion
Taking a monitoring perspective, this study focussed on an analysis of attracting LCSs, technically identified as repelling LCSs in backward simulations. LCSs help delineate regions in situ observations are possibly representative for. A structure like the one shown in Fig. 1a, for instance, provides a warning that in the vicinity of the central south-north oriented FTLE ridge, even measurements at neighbouring locations might see water bodies with very different backgrounds. In Fig. 2a

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In this study, FTLE fields were analysed on a grid with 1 km resolution, nearly matching resolution of the underlying hydrodynamic current fields. Generally, defining FTLE fields on a finer grid to look at structures smaller than the resolution of the Eulerian hydrodynamic model would have been possible (see Huhn et al., 2012, for instance). Generated by chaotic advection with exponential material stretching rates, small scale structures arise from tracer simulations over distances much exceeding numerical grid resolution (Huhn et al., 2012). Generally, Harrison and Glatzmaier (2010) found locations of major 290 LCSs to be fairly robust to spatial resolution.
According to Lekien et al. (2005), the relevance of FTLE ridges may be classified with regard to their length rather than the size of FTLE values. Here, LCSs often turned out to have considerable length and to be connected, sometimes forming a whole network of closed subregions. Throughout the study, all FTLE values were calculated based on trajectories integrated Vigo estuary in Spain. Experiments reducing integration time to just 25 hours, revealed that key FTLE ridges tended to become less sharp but to not change their locations (not shown). This finding agrees with expectations (e.g. Peng and Dabiri, 2009;Shadden et al., 2009). According to Peng and Dabiri (2009), in practice integration time should be chosen such that it makes LCSs well resolved and clearly visible. As in Huhn et al. (2012), the tidal signal did not dominate the choice of integration time. The example trajectories shown, illustrate how changing residual currents, driven by wind forcing, play a major role for 300 particle separation. This is very different in a Norwegian fjord, for instance, with topographically constrained currents driven mainly by tides (Orre et al., 2006). Branicki and Malek-Madani (2010)  sistent with an accumulation of drifting material near tidal mixing fronts (Simpson and Pingree, 1978;Thiel et al., 2011).  Lehahn et al. (2007) found satellite observations of chlorophyll filaments in the northeast Atlantic to well agree even with simulated geostrophic transports, contracting at and stretching along material lines. Referring to Lapeyre and Klein (2006), Lehahn et al. argue that an ageostrophic secondary circulation injecting nutrients from deeper layers may trigger further chlorophyll production.

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Combining SeaWiFS ocean-colour data with altimetry-derived surface currents in the Brazil-Malvinas confluence zone, d 'Ovidio et al. (2010) found that stirring by mesoscale currents can play an important role in structuring phytoplankton communities and even create what they call fluid dynamical niches, sharply delimited by LCSs. Hernández-Carrasco et al. (2018) study this topic at the submesoscale, using currents observed with High-Frequency Radar (HFR) in coastal waters. According to Scales et al. (2018) attracting LCSs can also be targeted by fisheries, lead by lines of drifting foam or debris. However, 325 Abraham and Bowen (2002), employing the FTLE for estimating a stirring rate from surface velocity data in the East Australian Current region, emphasize that a model beyond a simple passive tracer concept would be needed to better understand chlorophyll distribution.
Relatively stable FTLE ridges connected to the island of Helgoland, for instance, could also be relevant for sedimentation processes. However, again an analysis of ideal passive tracer trajectories is likely to be too simplistic for studying such effects.

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Movements of inertial tracers can substantially differ from those of fluid parcels. Therefore the idea of LCSs has been generalized to include dynamics of inertial particles (Sapsis and Haller, 2009;Sudharsan et al., 2016;Günther and Theisel, 2017).
This theoretical concept has successfully been applied on the scale of ocean eddies (Beron-Vera et al., 2015) but also on the very small scale of jellyfish feeding (Peng and Dabiri, 2009;Sapsis et al., 2011).
In this study, drift simulations were not validated against data. However, it was shown that to some extent the LCSs iden-335 tified in model output manifested themselves also in simulated fields of surface temperature (Fig. 4) and salinity (Fig. S3) as intrinsic tracers. A relationship between frontal structures and FTLE ridges confirms the relevance of LCSs for surface current transports. Becker et al. (1992) summarize different types of fronts (river plume, thermal and upwelling fronts) that occur in the German Bight. Schrum (1997) showed how the spatial extent of thermohaline stratified areas, a precondition for the occurrence of tidal mixing fronts, depends on wind forcing possibly inducing differential advection. In a recent paper, Chegini 340 et al. (2020) provided a more detailed analysis of different processes that affect stratification and destratification, including freshwater buoyancy input. Location of the Elbe River plume again depends on the wind driven residual circulation. Against this backdrop, it can be assumed that atmospheric forcing is also a key driver for the generation, movement and extinction of German Bight LCSs.
Although some patterns in the temperature (and salinity) field seem clearly related to ridges in the FTLE fields, it must 345 nevertheless be noted that there is no one to one relationship. An example for this provides the rather smooth FTLE field in Fig. 2b. The corresponding temperature field (Fig. S4a) shows small-scale structures with less clear counterparts in the FTLE field. According to Fig. S4b, the Lagrangian divergence FDLD reproduces structures seen in the temperature field, but FDLD values are clearly smaller than those in Fig. 2b. Combining pure flow dynamics with a simple representation of the dynamics of temperature itself might be necessary for an explanation of these structures in the temperature field (Abraham and Bowen,350 2002). Note that large divergences in coastal regions are likely to be artefacts because of water depths in tidal waters falling below the depth of the assumed 5 m surface layer (remember the same type of discrepancies also between Figs. 1b and 1c).
FTLE barriers may move, disappear or newly arise under changing environmental conditions. Numerical models are valuable tools for making observers aware of this fact. However, hydrodynamic models can never provide a perfect surrogate nature.
In a comparative study, Hufnagl et al. (2017) found considerable discrepancies between the results from a large number of 355 different North Sea tracer simulations essentially based on vertical mean currents. For surface drift simulations, additional simulation errors may arise from the necessity to specify the extent to which near surface currents are exposed to wave related Stokes drift or a direct wind drag. In field studies, corresponding parameters may be tuned empirically (e.g. Callies et al., 2017b). Altogether, simulated FTLE distributions will always be imperfect. Guo et al. (2016) propose concepts to extend the conventional analysis of deterministic FTLE fields and ridges to uncertain flow conditions. However, even in case of inaccurate 360 simulations, the simulated FTLE would at least warn about key sensitivities of model output. If an observation is taken close to a simulated FTLE ridge, a simulated backward trajectory for this location must be used with due care.
This study did not address repelling LCSs in prediction mode. However, it is obvious that the above difficulties also occur when forward simulations are employed for search and rescue (Breivik et al., 2013), for instance. A forward FTLE field could possibly warn users against particularly sensitive dependences on the assumed location of numerical drift simulations.

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In tracer experiments, substantial model data discrepancies could result from just a slight misspecification of initial locations or a moderate displacement of simulated LCSs relative to reality.

Conclusions
The analysis of backward surface tracer simulations in the German Bight revealed the intermittent presence of linear structures (LCSs) across which the past history of water bodies substantially changes. Such sensitive dependences, represented by ridges 370 in the fields of either backward FTLE or backward relative dispersion are potential sources of uncertainty in the interpretation of in situ observational data.
In the presence of repelling LCSs, large differences between observed and simulated tracer trajectories do not necessarily reflect poor model performance. If the location of a simulated LCS does not fully agree with reality, a tracer release point may come to lie on different sides of the separatrix in the model and in nature. In this case, a naive comparison of emerging 375 trajectories could much exaggerate inconsistencies. The same arguments pertain to a comparison of different drift models.
Conventional evaluations based on drift paths might be supplemented with a comparison of simulated FTLE fields that highlight spatial variability of prediction uncertainty.
Examples illustrated the variability of LCSs in the German Bight. For a more comprehensive picture it would be useful to establish a link between the recent history of atmospheric forcing, tidal movements and the main characteristics of the 380 backward FTLE fields to be expected. Due to sometimes complex filamentary structures, a decomposition of FTLE fields in terms of a mean field plus the sum of a number of weighted anomaly fields (empirical orthogonal function analysis) seems not very promising. Classification of FTLE fields into a limited number of categories might be useful. This problem is left to future research.
Code and data availability. The hydrodynamic data analysed in this paper were obtained from the repository of the Federal Maritime and Author contributions. The author performed all analyses and prepared the manuscript.