This study is dedicated to the tidal dynamics in the Sylt-Rømø Bight with a focus on the non-linear processes. The FESOM-C model was used as the numerical tool, which works with triangular, rectangular or mixed grids and is equipped with a wetting/drying option. As the model's success at resolving currents largely depends on the quality of the bathymetric data, we have created a new bathymetric map for an area based on recent studies of Lister Deep, Lister Ley, Højer Deep and Rømø Deep. This new bathymetric product made it feasible to work with high-resolution grids (up to 2 m in the wetting/drying zone). As a result, we were able to study the tidal energy transformation and the role of higher harmonics in the domain in detail. For the first time, the tidal ellipses, maximum tidally induced velocities, energy fluxes and residual circulation maps were constructed and analysed for the entire bight. Additionally, tidal asymmetry maps were introduced and constructed. The full analysis was performed on two grids with different structures and showed a convergence of the results as well as fulfilment of the energy balance. A great deal of attention has been paid to the selection of open-boundary conditions, model validation against tide gauges and recent in situ current data. The tidal residual circulation and asymmetric tidal cycles largely define the circulation pattern, transport and accumulation of sediment, and the distribution of bedforms in the bight; therefore, the results presented in the article are necessary and useful benchmarks for further studies in the area, including baroclinic and sediment dynamics investigations.

The Sylt-Rømø Bight (SRB) is one of the largest tidal catchments in the Wadden Sea, which stretches from the Dutch island of Texel to Skallingen, a peninsula in Denmark. The SRB is characterized by the barrier islands Sylt (Germany) and Rømø (Denmark), which are separated by a tidal inlet called Lister Deep. Two artificial causeways, the Hindenburg Damm (1927) and the Rømøvej (1948), connect the islands Sylt and Rømø to the mainland and create a semi-enclosed back-barrier environment. Water exchange with the North Sea takes place through the 2.8 km wide Lister Deep. The main channels draining the tidal back-barrier environment are called Lister Ley, Højer Deep and Rømø Deep (Fig. 1).

The bight is characterized by large intertidal areas occupying about 40 %
of the entire bight. The domain of interest has an average water depth of

The bathymetry of the considered domain. The red lines indicate acoustic Doppler current profiler (ADCP) transects, the blue rhombuses show positions of the tide gauges. The hatching represents the zone with the recent multibeam echo sounder (MBES) data.

The tides play a major role in the local bight dynamics. Estimates of
maximum horizontal tidal velocities in Lister Deep vary from 1.2 to 2 m s

There is a pronounced asymmetry in the tidal water level and current velocities behaviour, caused by complex morphological features and by the general shallowness of the area (e.g. Austen, 1994; Becherer et al., 2011; Lumborg and Windelin, 2003; Nortier, 2004; Ruiz-Villarreal et al., 2005). It is known that the Lister Deep can be characterized as an ebb-dominated area, i.e. the velocities during ebb are larger compared to flood velocities in the mean and maximum senses (e.g. Hayes, 1980; Oost et al., 2017; Fig. 1). The analysis of bedforms based on seismic profiles revealed that the area around Lister Deep is represented by a complex spatial pattern of the flood- and ebb-dominated subaqueous dunes (Boldreel et al., 2010).

The tidal residual circulation and asymmetric tidal cycles largely define
the transport and accumulation of sediment and the distribution of bedforms
in the bight (e.g. Boldreel et al., 2010; Burchard et al., 2008; Hayes,
1980; Nortier, 2004; Postma, 1967). Note that the tide entering the Wadden
Sea leaves

The article is organized as follows. The “Model setup” section contains information about the numerical ocean solution, the grids, the open-boundary conditions and the bathymetric data we used. The next section contains information about the observational data we used to verify the simulations. The Results section contains model validation results and detailed information about tidally induced barotropic dynamics in the area, with a focus on the non-linear dynamics. The Discussion section contains information about possible sediment dynamics outputs based on our results, and about grid performance. The last sections summarize the article and provide the supplemental data.

FESOM-C is a coastal branch of the global Finite volumE Sea ice–Ocean Model (FESOM2; Danilov et al., 2017). FESOM-C has cell-vertex finite volume discretization and works on any configuration of triangular, quadrangular or hybrid meshes (Androsov et al., 2019; Danilov and Androsov, 2015). It has split barotropic and baroclinic modes and a terrain-following vertical coordinate, and it is equipped with 3rd-order upwind horizontal advection schemes, implicit 3rd-order vertical advection schemes, implicit vertical viscosity, biharmonic horizontal viscosity augmented to the Smagorinsky viscosity, and the General Ocean Turbulence Model (GOTM; Umlauf and Burchard, 2005) for the vertical mixing. Wetting and drying of intertidal flats have been included because this is a crucial point for the reconstruction of the non-linear dynamics in the shallow zone.

All results except intercomparison of the different tidal solutions are
obtained based on multi-layer barotropic simulations. In all simulations
only tidal forcing was turned on. The

The spin-up duration was determined by the total energy stationary case and was about 3 months. For the analysis, we considered two last lunar (synodic) months (59 d). We simulated the tidal dynamics in 2018, which is expressed in Doodsen correction of the prescribed amplitudes and phases at the open boundary; therefore, we were able to compare simulated and observed velocities second to second.

Unless otherwise indicated, the figures visualize depth-averaged behaviour.

We have created two grids for the area of interest; all simulations were performed on these two grids. One grid is curvilinear and the other is unstructured and contains mostly arbitrary quads with a few triangles.

The curvilinear grid contains 119 305 nodes; the resolution varies from 14 to 261 m. The finest resolution is in Lister Deep near the south-western boundary and in the eastern area of the internal part of the domain. The curvilinear grid was generated by the elliptic method (Thompson, 1982).

The unstructured grid contains 208 345 nodes (10 398 triangles; 201 141 quads); its resolution varies from 2 m in the wetting/drying zones to 304 m in the deepest area of the external (seaward) part of the considered domain (Fig. 1). The size of grid cells is determined by the information about the bathymetry, the bathymetry gradient and the zones of particular interest (Lister Deep; main draining channels). The unstructured grid was built using the mesh generation software package of the Surface Water Modeling System (SMS version 12.3, AQUAVEO).

Both grids have nearly the same open boundary position. The nodal areas at both grids are presented in Fig. A1 in Appendix A.

We relied on four sources for the open-boundary conditions of the tidal elevation: TPXO 8.1 and 9 (TPXO database, 2019; Egbert et al., 2002); the output of NEMO (Nucleus for European Modelling of the Ocean; Gurvan et al., 2017) simulations for the north-west European Shelf (Copernicus Marine Database, 2019; Tonani et al., 2019); and FES2014 (Finite Element Solution 2014; AVISO database, 2019; Carrere et al., 2016).

TPXO 8.1 and 9 are fully global models of ocean tides which best fit, in a
least-squares sense, the Laplace tidal equations and altimetry data. They
provide information about 13 harmonic constituents (or 15 with TPXO 9). TPXO
8.1 and 9 atlases are combinations of the

The European north-west shelf model results include information about hourly instantaneous sea level. The sources are full baroclinic simulations based on version 3.6 of NEMO with data assimilation (vertical profiles of temperature, salinity and satellite sea-level anomaly). The model is forced by lateral boundary conditions from the UK Met Office North Atlantic ocean forecast model and, at the Baltic boundary, by the Copernicus Marine Environment Monitoring Service (CMEMS) Baltic forecast product. We obtained information about the tidal constituents by performing an FFT (fast Fourier transform) analysis of the elevation signal at our open boundary based on data for 1 year (2017).

FES2014 is a global finite-element hydrodynamic solution with assimilated
altimeter data and a grid resolution of

Bathymetric data for a given area were generated based on two sources: the
AufMod database (Valerius et al., 2013), with a 50 m resolution for the whole
area, and newly obtained data for the inlet and the main tidal channels,
with a grid resolution of

The new bathymetric data in Lister Deep tidal inlet and the main tidal
channels were obtained during winter of 2017–2018 using the hull-mounted ELAC
SeaBeam 1180 multibeam system on board the RV

The summary of the five cruises on board RV

The observed velocities represented the base for validation of the model and
testing of the different tidal open-boundary conditions. The data set is
composed of observed profiles of the water currents gathered on five cruises
of the RV

Wind data were automatically measured by an anemometer mounted on the RV

The tide gauge data for the area are represented by three stations: ListTG
(55.017

Model validation was organized into the following two steps: (1) selection of the best open-boundary conditions using ADCP data and (2) validation of the best open boundary solution against existing tide gauge data, in particular using ListTG, VidåTG and HavnebyTG data. For step (1), the measurements were done during different tidal periods (spring, neap, ebb and flood) in the area of Lister Deep and the main inlets (Fig. 1), which are characterized by the largest depths (up to 35 m) in the domain and by the highest as well as by complex tidally induced velocities. We performed the frequency analysis using the MATLAB package T-TIDE (Pawlowicz et al., 2002) and identified the errors in amplitude and phase for the main tidal constituents (with maximum amplitudes), including higher harmonics.

The intercomparison of the observed and simulated velocities based
on different open-boundary conditions in the area of Lister Deep. The table
presents root mean square deviation (RMSD, m s

Table 2 represents the root mean square deviation (RMSD) and correlation coefficients of the observed velocities (ADCP data) and the modelled velocities obtained for each open boundary solution. The comparison is based on depth-averaged velocities since the measurements were performed in the deep part of the domain and our task here was to check the performance of the different tidal forcing. In Table 2, results for all the open boundary solutions we used are shown only for the first grid since the difference between the results on different grids is < 0.01 for correlation coefficients and < 0.01 m for the RMSD. However, we would like to note that the unstructured grid provides slightly better results despite its somewhat coarser resolution (the unstructured grid has a larger number of cells, mainly due to detailed representation of the wetting/drying zone). Based on additional experiments (Kuznetsov et al., 2019), we think the reason is that the unstructured grid reflects the bathymetric gradients.

We used different bottom-friction coefficients (

The observed and modelled depth-averaged velocities on 29 May 2018. The coloured arrows indicate the current magnitude (m s

Table 2 shows that the TPXO 9 solution fits the ADCP data best, second best
are the TPXO 8.1 and NEMO solutions, and the FES 2014 solution follows.
For all solutions except FES, the correlation coefficients are higher during
spring tides as well as in the deepest part of the domain; this is true in
particular for the measurements performed on 29 and 30 May and despite
the quite strong winds often ranging from 10 to 15 m s

Despite the quite good results of TPXO 8.1 (Table 2), the unstructured grid yields a number of vortex structures near the open boundary that cannot be removed by the sponge layer, which dumps advection and diffusion near the open boundary. It is known that grids that employ an arbitrary normal at the open boundary are subject to the quality of the open boundary signal (Danilov and Androsov, 2015). For TPXO 8.1, the behaviour of the phase is not realistic near the solid boundary: the phase goes through zero simultaneously near the western and eastern solid boundaries.

Simulated and observed amplitudes (cm) and phases (

Once we determined the best open-boundary conditions solution – TPXO 9,
based on ADCP data – we moved on to the second verification stage. For this,
we switched to a 3-D simulation with 10 vertical layers, which are crowding
near the sea bed. The optimal roughness height was 0.001 m; this value
agreed with the one estimated from observations in a similar region (Werner et
al., 2003) and, in terms of the mean, with a

Figure 3a shows the probability, in the unstructured grid case, of each node
being wet with tidal forcing alone. This figure is based on simulation
results for two lunar (synodic) months (29.5 d

An extensive intertidal subarea situated in the western and southern parts of the considered domain as well as in the Königshafen embayment can be seen, and the Jordsand creates a secondary bight with the Rømø Deep main channel (Figs. 1, 3a).

Figure 3b shows the maximum velocities at each grid point within a lunar
(synodic) month. Thus, Fig. 3b shows the highest possible tidally induced
velocities in the domain. Figure 3b exhibits, as expected, a correlation
with the depth (Fig. 1) but there are many peculiarities which emphasize
the large role of non-linear processes in the domain. The maximum velocities
can be found at the opening of Lister Deep and near the edge of Sylt during
spring ebb and are

The analysis of the energy budget and energy flux distribution provides an
important insight into the evolution of energy in the modelled region. The
energy balance for the vertically averaged equations for the barotropic case
has the following form (e.g. Androsov et al., 2002):

The energy budget for the depth-averaged solution with open-boundary conditions from the TPXO 9 database for summary tide, in watts (W): in blue (dashed line) is the energy change in time, in red (dash-dotted line) is the flow through the open boundaries, in green (dotted line) is the bottom friction and in cyan (solid line) is the imbalance.

The first term on the right-hand side of Eq. (2) is the total flux of energy across
the open boundary, and the second term is the rate of energy dissipation due
to the bottom friction. Figure 4 shows the energy balance for the whole area
based on the unstructured grid for the summary tide for one M

The maximum change in system energy and maximum fluxes through the open boundaries take place during ebb tide; the ebb phase duration is, on average, 0.85 of the flood phase. These general conclusions mask very patchy dynamics in the area, which are considered in detail in the next sections.

The flux of tidal energy (summary tide):

Figure 5 demonstrates the energy fluxes in the area. The tidal energy flux,
represented by the sum of the potential and kinetic energy fluxes, is
estimated using the following definition (Crawford, 1984; Kowalik and
Proshutinsky, 1993):

Residual circulation of the summary tide for

The residual circulation in the area is characterized by the large number of
vortex structures with different rotation directions in the area of Lister
Deep and the main channels (Figs. 1, 6a). Note that the residual circulation
in the external part of the domain is defined by kinetic energy fluxes,
because here the role of non-linearity in the continuity equation is minor
(the water depth compared to the tidal amplitude is relatively large). These
vortexes organize a complex residual circulation pattern; the strongest
circulation (up to 0.45 m s

The resulting movement in the SRB represents a superposition of the tidal
waves reflected off the region's solid boundaries. As a result of this
interference, a standing wave occurs, containing only one component of the
velocity for the main tidal harmonics, which is to say currents are close
to reverse. The Coriolis effect does not lead to significant cross-currents.
Figure 7a and b show the ellipses of the wave M

The ellipses induced by the M

Axes of the

The major higher harmonics in the area are M

The ebb-flood dominance asymmetry maps. Left panels: ratio of

The domain configuration and the foregoing results of the tidal energy
transformation and evolution are signals of a pronounced tidal asymmetry in
the tidal water level and in the current velocity behaviour in the area.
There are several reasons for the tidal asymmetry – in particular, the
presence of the non-linear advection and bottom friction terms together with
complex topography, resulting in a non-trivial wave interaction in the
system (e.g. Friedrichs and Aubrey, 1988). The key geometric features
directly impacting tidal distortion are the bathymetry relative to the tidal
amplitude, the bottleneck width and its variation during the tidal
cycle, as well as the area occupied by the intertidal zone and its distance to the
main tidal inlets. For data about tidal asymmetry to inform sediment
dynamics analysis, we concentrate on the ebb and flood durations as well as
the mean and maximum velocity relations. Ebb is defined as a period when
the water level at the current location is decreasing, and flood is defined as the
period when it is increasing. Figure 9a (left panel) represents the ratio
between the maximum velocities during spring ebb and flood. Figure 9b (left
panel) represents the ratio of mean velocities during ebb and flood periods.
Figure 9c (left panel) represents the ratio of mean ebb and flood durations.
The analysis reflects near-bottom velocities; however, the pattern of ratios
is nearly the same for the depth-averaged solution. We note generally that
the near-bottom solution provides a more pronounced ebb or flood dominance.
The features represented in Fig. 9 (left panel) show the

Figure 9 (right panel) comprises 57 floods and ebbs. (We considered 28.5 d, but the first and last flood and ebb of the lunar (synodic) month – 29.5ḋ – were removed to make sure that we were considering the beginning of the flood and ebb.) Then we calculated how often the mean and maximum velocities of the ebb are larger during the following flood (Fig. 9a, b right panels). We also calculated how often the ebb duration was smaller than that of the following flood (Fig. 9c, right panel). In other words, Fig. 9a (right panel) shows for example the frequency of the maximum velocity during ebb being larger than that of the following flood; a value of 57 means that across all 57 cycles, maximum velocities during ebb are larger than during flood. We should note that flood or ebb dominance is caused not only by non-linear effects but also due to the diurnal inequality of tide. From all these pictures, we have removed subareas which did not take part at least one flood–ebb cycle.

Figure 9 was generated from simulations on the curvilinear grid. But an important result is that the difference between the ratios (asymmetry indicators) on either grid touches only a few details, but not a general pattern. The maximum disagreement occurred during comparison of the ratios of velocities, going up to 0.2 for ratios of mean and maximum velocities. However, we would like to stress that the position of the ebb and flood dominance areas are the same with only a small difference at the edge of the zones.

The patterns presented in the left and right panels show some common features, and this is to be expected. For example, if the ebb velocities are larger than the flood velocities in nearly every ebb–flood cycle, then the averaged value of the velocity ratios will be large, and the corresponding figure in the right panel will show a high frequency of the ebb-dominance event. Figure 9 (right panel) shows that there are a lot of zones that may behave differently during different periods of a lunar (synodic) month. (The frequency is between 0 and 57.) Note also that Fig. 9a and b represent largely different patterns than Fig. 9c. For, example, if the mean and maximum velocities in the internal part of the domain are larger during flood, it does not mean that the flood will be shorter. There can be no mass conservation for the ebb–flood cycle for the particular area subunit. Note that the ebb in the whole domain under consideration is typically shorter than the flood, except in the south-western part of the internal area (Fig. 9c, left panel).

Figure 9 demonstrates that it is not correct to regard ebb or flood dominance based only on velocity characteristics or typical flood or ebb durations when these characteristics can yield opposite answers. Additionally, these characteristics are not necessarily the same within one lunar (synodic) month (29.5 d).

To explain the complexity of the presented pattern, we have split the domain under consideration into four parts designated as zones 1 through 4 (Fig. 9a, left panel).

Zone 1 represents an area where we observe a progressive Kelvin wave (Figs. 5, 9). The flood-dominated (orange-coloured) zone, in terms of maximum
velocities (Fig. 9a, left panel), matches the area where we have a
comparatively small residual circulation (Fig. 6). Gravity waves traverse
the shallow zone non-dispersively at a speed defined by

In zone 2, we have an extensive intertidal zone and maximum amplitudes of
the semidiurnal tides compared to other zones; also, this zone lies far from
the Lister Deep area. In this zone, the role of higher harmonics is greatest
(Fig. 8), revealing the major role of non-linear friction effects and
non-linearity in changes of the water-layer thickness: here, the ratio of
tidal amplitude to depth is greatest (increasing from Lister Deep toward the
southern part of the zone). This also can be seen in Fig. 7, which shows
M

Snapshots of direction relative to the

In zone 3 we have very patchy dynamics in terms of velocities during flood and ebb (there are a lot of subzones characterized by the flood dominance and ebb dominance); however, it is typical for the whole area that ebb is shorter than flood (flood duration is between 1.1 and 1.3 of ebb duration). Lister Deep can be characterized generally by higher mean and maximum velocities during ebb, but some side channels can be characterized by higher velocities during flood. As the tide turns at low water, strong currents are still flowing seaward out of the main ebb channel. As the water level rises, the flood currents seek the path of least resistance around the margin of the delta. This creates horizontal segregation of flood and ebb currents in the tidal channels for a time. This is apparent in Fig. 10, depicting the transitional moment from ebb to flood.

In this zone, we emphasize subareas A and B as prime examples of the subdomains where mean current ratio and maximum current ratio (Fig. 9a, b) are not synchronized with the durations ratio (Fig. 9c). For example, Fig. 9a and b demonstrate that the velocities at B are larger during flood in terms of the mean and maximum; but flood duration is longer than ebb. This signals that the water parcels are travelling different pathways during flood and ebb.

Zone 4 represents a small semi-enclosed bight with a flood-dominated main
channel and an ebb-dominated area around, as well as in the inner part of the bight
in terms of velocity. However, we would like to stress that flood dominance, though
typical, is not a constant feature. For several cycles within a lunar
(synodic) month (29.5 d), the mean and maximum velocities will be larger
during ebb. For the whole zone, ebb is shorter than flood for almost a lunar
(synodic) month. The reason the pattern is opposite to that of zone 3 lies
in the small volume of intertidal storage and large variation in the “main
channel” width. In this zone, frictional drag is insignificantly greater at
low water than at high water. Also, the role of M

The most interesting question for future study is how the given asymmetry and residual circulation pattern line up with the bedform peculiarities. Therefore, the next step will be an intercomparison between measurements in the frame of the planned multibeam echosounder surveys and results of the current paper. The prediction of the bedform peculiarities in the area requires a coupled sediment module as well as wind and wave forcing. However, as soon as we have shown (Table 2, Fig. 2) that tides can explain a large part (more than 80 %) of the current velocities in the area of Lister Deep, some prognoses can be made at the current stage. The results suggest the presence of subaqueous dunes with stable characteristics in areas where mean and maximum current velocities are permanently higher or lower during ebb and flood than during flood and ebb. With this definition (see Fig. 9), our results agree with those presented in Boldreel et al. (2010) and Mielck et al. (2012). In those studies, conducted in the working area of Lister Deep and adjacent to Königshafen, dunes of various sizes, escarpments and other erosional features were analysed based on hydro-acoustic data (seismic profiles, side-scan sonar and the RoxAnn seafloor classification system) to determine dune characteristics and orientations (and hence flood or ebb dominance) of the particular areas. Boldreel et al. (2010) state that the flood-dominated dunes are larger than ebb-dominated dunes in the area of the bottleneck. Our study reveals that, in this zone, the ebb is generally shorter than the flood (Fig. 9c), which may explain this observation. The map of the residual circulation (Fig. 6) suggests probable directions of subaqueous dune migration where bottom currents are strong enough.

Each presented grid has advantages and disadvantages. The curvilinear grid offered minimum numerical viscosity but was not very flexible when it came to choosing grid cell size as compared to an unstructured grid (see also, Figs. A1, A2; Danilov and Androsov, 2015; Androsov et al., 2019). This is especially crucial in the zones of large bathymetric gradients and in the wetting/drying zones. The unstructured grid is more dissipative and more sensitive to the quality of the open boundary solutions. The reproduction of the non-linear effects on the grids of different structures represents an additional scientific question, which should be disentangled in the future.

This study is dedicated to tidally induced dynamics in the SRB, with a focus
on the non-linear component. The newly obtained high-quality bathymetric
data supported the use of high-resolution grids (up to 2 m in the
wetting/drying zone) and elaboration of the details of tidal energy
transformation in the domain. The FESOM-C model was used as the numerical
tool. In preparation, different open-boundary conditions for the summary
tide (major diurnal and semidiurnal as well as M

All experiments were performed on two grids with different structures and resolution details. Energy balance fulfilment was shown for both grids. The generated maps generally showed the same pattern on both grids, which allowed us to conclude that further increasing the resolution will not lead to pronounced changes in the results. The obtained results are a necessary and useful benchmark for further studies in the area, including for work on baroclinic and sediment dynamics. It would be fruitful to pursue further research about how the obtained maps reflect bedform peculiarities in the area in order to predict bedform characteristics in the shallow zones, which are hard-to-reach places for the ship surveys.

The observed profiles of the water currents gathered on five cruises of the RV

VF designed the experiments, carried them out, prepared the unstructured grid and new bathymetry product, and suggested asymmetry analysis. AA constructed the curvilinear grid, encouraged VF to investigate some aspects of the non-linear dynamics, wrote the part about tidal ellipses and helped to visualize the results. LS carried out and provided the multi-beam data, participated in the construction of the new bathymetry product, and consulted VF a lot during the verification stage. IK and HCH discussed with VF all experiments and results providing valuable comments and remarks. FA carried out and processed ADCP data, prepared the reference list, and, together with HCH and LS, significantly improved the article readability. HCH provided the information about bedform peculiarities in the area. KHW supervised the research. All authors contributed to the final article, provided critical feedback and helped to shape the research.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Developments in the science and history of tides (OS/ACP/HGSS/NPG/SE inter-journal SI)”. It is not associated with a conference.

Authors are indebted to Hans Burchard for the useful remarks and fruitful discussion and to Natalja Rakowsky and Sven Harig for the technical support. Also we are grateful to the three anonymous reviewers for the valuable comments and suggested improvements to the article.

The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association.

This paper was edited by Mattias Green and reviewed by three anonymous referees.