1 Introduction

The North Sea is located on the continental shelf of northwestern Europe. It connects to the Atlantic Ocean in the north, through the English Channel in the southwest, to the Baltic Sea in the east and ends at the latitude of the Orkney/Shetland Islands in the north (Schrum 2001). The German Bight is a coastal region of global importance located in the southeastern part of the North Sea. It is a key link between the English Channel and Europe’s major ports. The maritime industries, including shipping, fishing, aggregate extraction and energy production, play a significant role due to the variety of stakeholders in the area, including residents, fishermen, tourism operators and researchers, making it a well-studied and documented site (Otto et al. 1990; Huthnance 1991; Becker et al. 1992).

The shallow water shelf of the German Bight is subject to semi-diurnal tidal waves (M2), resulting in a cyclonic residual circulation influenced by non-linear tidal interaction and prevailing westerly winds (Stanev and Ricker 2020). Ocean waves influence the circulation through a variety of processes: turbulence to breaking and non-breaking waves, momentum transfer from breaking waves to currents in deep and shallow water, wave interaction with planetary and local vorticity, and Langmuir turbulence. Some studies have found that including these wind-wave-driven effects in a circulation model improves the simulated data (Alari et al. 2016; Staneva et al. 2021b).

Lagrangian modeling calculates trajectories by solving integral curves via equations of motion expressed in a Lagrangian format (Bennett 2006). It has widespread use in transportation analysis, and Hainbucher et al. (1987) were among the first to apply a Lagrangian model to the North Sea to estimate long-term pollution transport pathways. Schönfeld (1995) simulated the dispersion of artificial radionuclides released into the English Channel at Cap de La Hague. Callies et al. (2011) performed a time-reversed Lagrangian simulation to assess the degree to which the outcomes of such transport simulations are dependent on the selection of a specific hydrodynamic model.

Model-simulated Lagrangian trajectories have been the focus of research over the past years. This Lagrangian approach is useful for the study of flow-driven transport problems in the marine environment (Stanev et al. 2019). As well as for applications such as the transport of fish eggs and larvae (Mariani et al. 2010) or marine litter (Gutow et al. 2018; Ricker et al. 2021; Meyerjürgens et al. 2023), among others. In this context, it is necessary to take into account the underlying physical mechanisms governing oceanic mixing and diffusion. Relative dispersion plays a significant role in Lagrangian transport processes and in the resolution of subgrid effects. It defines the gradual mixing of passive tracers in a turbulent flow field over time, which is influenced by a variety of physical processes such as advection, diffusion, and turbulence (Spydell et al. 2021). In oceanic and atmospheric sciences, Lagrangian methods are becoming more prevalent for simulating fluid particle motion. However, their accuracy depends on the inclusion of relative dispersion to account for subgrid effects, including small-scale turbulence and mixing, that are not explicitly represented in the models (Corrado et al. 2017). Modeling subgrid effects accurately is critical for simulating large-scale phenomena such as the transport of pollutants, nutrients, and plankton in the ocean (Ricker et al. 2021). Therefore, researching relative dispersion in Lagrangian simulations is necessary for enhancing comprehension of transportation processes and fluid dynamics at small scales, ultimately resulting in the improved prediction of large-scale phenomena in the ocean and atmosphere (Ricker and Stanev 2020).

In the study of turbulent flows, the relative diffusivity is also an important statistical metric used to quantify the stirring of a flow field. It is defined as the rate of change of relative dispersion. Additionally, it is used to resolve sub-grid effects that are not explicitly represented in models. The incorporation of relative diffusivity in Lagrangian simulations is, therefore, necessary for accurately representing transport processes and improving our understanding of the dynamics of the fluid motion in the ocean (LaCasce 2008). By analyzing the relative diffusivity, we can gain insights into the physical mechanisms that contribute to ocean mixing, which is crucial for accurately modeling ocean circulation.

Fig. 1
figure 1

Bathymetry of the GCOAST model system. The red rectangle delineates the study area (the German Bight)

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Our research serves as a sensitivity analysis, focusing on various parameters. We utilize a hydrodynamic model enhanced by the integration of wind-wave-driven mixing parameters, including sea state-dependent momentum flux, energy flux, and wave-induced mixing. The output from this model is employed to drive the Lagrangian model, OpenDrift, enabling the examination of their combined effects on particle dynamics. Previous publications have not thoroughly investigated these dynamics, particularly their effects on relative dispersion and diffusivity. Moreover, this research underscores the significance of diffusion, an essential element that earlier models have not emphasized for accurately simulating particle trajectories. Furthermore, the study considers the influence of cluster size and how groupings of particles behave under different environmental conditions. By integrating these dynamics, our approach not only extends existing research but also provides new insights into the mechanisms influencing particle transport.

This study enhances our comprehension of oceanic transport processes by examining the role of a two-way ocean-wave model on the relative dispersion and diffusivity. We use a fully coupled ocean-wave model and simulations from a Lagrangian particle-drift model conducted from October to December 2018 and January 2019. Section 2 presents the methods, including a detailed description of the coupled model components as well as the Lagrangian model setup, and experiments. In Section 3, the results are evaluated using surface drifter observations and in situ measurements. Section 4 provides a discussion and conclusions based on the results, underscoring the implications of our findings for improving the accuracy and applicability of marine transport models.

2 Methods

The study utilizes GCOAST (Geesthacht Coupled cOAstal model SysTem), which is a flexible and comprehensive model system that incorporates key elements of regional and coastal models. GCOAST consists of different models, each developed for a specific compartment of the Earth system, including atmosphere-ocean-wave interactions, dynamics and fluxes in the land-sea transition zone, and coupling of the marine hydrosphere and biosphere (Fig. 1). The selection of different GCOAST model components depends on the specific scientific question under investigation. For the present study, we utilize the NEMO-WAM two-way coupled hydrodynamic and wave circulation models (Ho-Hagemann et al. 2020; Staneva et al. 2021a; Grayek et al. 2023), as well as the OpenDrift model (Dagestad et al. 2018) for Lagrangian simulations (See Section 2.4).

2.1 Wave model

The WAM model (Group 1988) is a third-generation wave model that combines the basic transport equations describing the evolution of a two-dimensional ocean wave spectrum. WAM is based upon the spectral description of the wave conditions in frequency and directional space at each of the active model grid points of a specific model area. The energy balance equation, complemented with a suitable description of the relevant physical processes, is used to follow each wave spectral component’s evolution. WAM estimates the two-dimensional wave and variance spectrum through the integration of the transport equation in spherical coordinates. The driving force for the wave model is the one-hourly ERA5-U10-wind fields provided by the atmospheric model system IFS Cycle 41 of the ECMWF (European Centre for Medium-Range Weather Forecasts) with a spatial resolution of \(0.25 * 0.25\) degrees (Hersbach et al. 2020).

The wave model boundary information used at the open boundaries is taken from the regional wave model EWAM for Europe, which consist of nine runs twice a day in the operational wave forecast routine at the DWD (German Weather Service).

2.2 Ocean model

The NEMO (Nucleus for European Modelling of the Ocean) framework is an advanced tool used in ocean and climate sciences research and forecasting (Madec 2016). It employs the 3D primitive equations, utilizing both hydrostatic and Boussinesq approximations (Ho-Hagemann et al. 2020). The vertical grid follows the NEMO s-z-hybrid grid structure with 50 levels and tangential stretching below 200 m. The model has a minimum water depth of 8 m and a maximum depth of 6300 m. Horizontal resolution is approximately 2 nm, providing instantaneous hourly surface velocity fields. Atmospheric pressure and tidal potential (e.g. (Egbert and Erofeeva 2002)) are included in the model forcing. Lateral open boundary and initial condition fields (temperature, salinity, velocities and sea level) are derived from the MetOffice Forecasting Ocean Assimilation Model (FOAM) AMM7 (7 km horizontal resolution (O’dea et al. 2012; Lewis et al. 2019; Staneva et al. 2021a) used by the Copernicus Marine Environment and Monitoring Service (CMEMS) as an operational service.

2.3 NEMO-WAM coupling

NEMO has been enhanced to incorporate various wave effects (See Fig. 2 in Alari et al. (2016)). These effects include the Stokes-Coriolis forcing (Hasselmann 1970), sea state-dependent momentum flux (Janssen 1989; Staneva et al. 2021a), sea state-dependent energy flux (Craig and Banner 1994; Breivik et al. 2015; Staneva et al. 2017), and wave-induced mixing (Staneva et al. 2021a). The motion of fluid particles in waves does not follow a completely closed orbit due to the different velocities of wave crests and troughs. This results in a difference between the average Lagrangian flow velocity of a fluid parcel and the Eulerian flow velocity, known as the Stokes drift (Stokes 1847). Similar to wind-induced currents, the Stokes drift is influenced by the Earth’s rotation, leading to a phenomenon called the Stokes-Coriolis force (Hasselmann 1970), which causes the ocean currents to veer. The momentum flux between the wave model WAM and the ocean model NEMO is determined by the wave-modified drag coefficient, which affects the air-side stress and the ocean-side stress. The ocean-side stress depends on the balance between wave growth and dissipation, as described by Staneva et al. (2017). In NEMO, the wave-induced turbulent kinetic energy flux (TKE) at the sea surface is dependent on the wave energy factor \(\alpha \) (Craig and Banner 1994) and is kept constant throughout the simulations. Overall, these modifications and considerations in NEMO account for the complex interactions between waves and ocean dynamics, providing a more comprehensive representation of the physical processes involved.

2.4 Lagrangian model

OpenDrift (Dagestad et al. 2018) is a freely available open-source offline Lagrangian model, which contains several modules for the advection of, e.g. oil spills (Jones et al. 2016), larvae or passive tracers (Christensen et al. 2018).

For this study, the OceanDrift module is used, which is a buoyant passive particle trajectory model based on OpenDrift. It is a generic module for particles that move in three dimensions with the possibility to add subgrid diffusion.

Fig. 2
figure 2

Initial positions and trajectory of one element for the different clusters: Cluster 1 (C1) (red circle),Cluster 2 (C2) (blue circle) as well as the FINO-3 research platform station (black circle)

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The diffusion is determined by the grid spacing of the ocean model (Schönfeld 1995) according to the 4/3 law of oceanic diffusion (Stommel 1949):

$$\begin{aligned} D(A_H) = c_1 * dl^{4/3} \end{aligned}$$
(1)

where D(\(A_H\)) is the horizontal diffusion coefficient and dl is the grid spacing. According to diffusion experiments conducted in the North Sea on length scales between 1 and 5 nm (Okubo 1971; Weidemann 1984), a horizontal diffusion coefficient of \(D(A_H) = 5.8\) \(m^2/s\) was determined for a grid spacing of \(dl= 3.5\) km and a coefficient of \(c_1 = 1.1 * 10^{-4}\) m/s. This equation determines the degree of spreading of the particles due to natural turbulence and mixing processes in the ocean. The OceanDrift module in OpenDrift examines two sources of diffusion: from current and/or wind. The diffusion process is introduced into the system as random deviations to simulate the unpredictable movement of elements in response to fluctuating environmental conditions. Both meridian and zonal velocities are scaled with a normal distribution with a mean of zero and a variance of one. Empirical formulas from Callies et al. (2011) and Stommel (1949) are the basis of its formulation, where \(u'\) represents the stochastic component of the velocity:

$$\begin{aligned} u' = \sqrt{\frac{2D}{dt }}*R \end{aligned}$$
(2)

D represents again the diffusion coefficient as formulated by Schönfeld (1995), while R is the random component. This diffusion formulation was previously used by De Dominicis et al. (2012). The trajectories are calculated off-line, i.e. after GCOAST has been integrated and the velocity fields have been stored. This allows many more trajectories to be calculated than would be possible online (i.e. while GCOAST is running), as it reduces the computational load on the GCOAST model, allowing it to run more efficiently.

The actual windage of the drifters is unknown, therefore, estimates for these values are necessary. In Meyerjürgens et al. (2020), the wind drift factor was theoretically determined to be 0.27 \(\%\). However, in this study, the wind drift factor was increased to 0.5\(\%\). Increasing the wind drift factor in the Lagrangian model was necessary to improve the accuracy of the model’s simulations. A higher wind drift factor accounts for additional wind effects on particle movement in the ocean, resulting in more realistic trajectories.

The advection scheme is a 2nd-order Runge-Kutta scheme, and diffusion is in some experiments added to analyze the impact it has on relative dispersion and relative diffusivity. If a particle comes ashore, it is no longer advected and is considered beached, meaning these particles are no longer taken into account.

2.5 Drifter data

The drifters’ data originated from the RV Heincke cruise in the German Bight in October 2018 (Meyerjürgens et al. 2019, 2020). The data set includes 14 surface drifters that were equipped with satellite tracking systems and deployed in two clusters. Cluster 1 included eight drifters, resulting in 28 possible pairs. Cluster 2 included six drifters, resulting in 15 possible pairs (43 pairs in total). The primary objective of the study was to gather current data from the upper 0.5 m of the water column. To attain sufficient precision in analyzing submesoscale processes, the drifters’ position data was transmitted via Globalstar satellite telemetry at 10-minute intervals. The drifters’ positions were tracked with around 2.5 m accuracy. It is important to note that the drifters experienced a direct wind slip, referred to as Uslip, of 0.27 %. This slip would lead to wind-induced velocities ranging from 0.0027 to 0.027 m/s, corresponding to wind speeds within the range of 1 to 10 m/s.

2.6 Relative dispersion and diffusivity

Particle dispersion can be studied in two distinct frameworks: absolute dispersion, which considers the trajectory of a single particle as a distance to a fixed point in space (Taylor 1922), and relative dispersion, which considers the trajectory of two particles as a distance between them. Richardson’s 1926 pioneering work investigating the relative motion of particles within isotropic turbulence established the foundations of relative dispersion. Relative dispersion is then calculated as the average of the square of the separation distance of two particles at different times:

$$\begin{aligned} D^2(t) = \frac{1}{N} \sum _{i,j} [x_i (t) - x_j (t)]^2 + [y_i (t) -y_j (t)]^2 \end{aligned}$$
(3)

The coordinates of each pair in the cluster of N model-run pairs are denoted by i and j and represented by \(x_{i,j}(t)\) and \(y_{i,j}(t)\) respectively in the Universal Transverse Mercator (UTM) projection at time t. The rate of change of the relative dispersion defines the mixing properties of the flow field as a function of the length scales. It is a function of the relative separation between particles and of the relative diffusivity (K):

$$\begin{aligned} K = \frac{1d}{4dt} [D^2 (t)] \end{aligned}$$
(4)

The separation distances of the elements are determined by the interaction between the kinetic energy in the flow field and the time t.

Richardson (1926) presented a study of pair dispersion in the turbulent flow field of the atmospheric boundary layer. The Richardson regime characterizes a phase in which the mean squared separation distance, \(D^2(t)\), between particle pairs expands cubically with time, suggesting \(D^2(t) \backsim t^3\), which is mainly driven by processes with spatial scales similar to the particle separation. Consequently, this relationship indicates that the separation distance, D(t), grows according to a \(t^{3/2}\) power law. In addition, Richardson’s analysis extends to the diffusion coefficient, K, which he theorized to be proportional to \(D^{4/3}\). This proportional relationship highlights a dynamic where diffusivity increases with separation distance, emphasizing the influence of larger and more energetic turbulent scales as particles drift apart. The \(D^{4/3}\) growth rate thus not only reflects an increase in particle dispersion velocity at larger separations but also implies a scale-dependent diffusivity that is essential in environments dominated by high-scale turbulence. The ballistic regime is another regime that has a squared growth pattern and is related to local dispersion effects. Diffusive dispersion regimes are characterized by a linear growth of relative dispersion in time.

A regime where dispersion is driven by processes larger than the model-run separation distances shows the nonlocal exponential growth of dispersion in time:

$$\begin{aligned} D^2 (t) = D^2 (0) e^{\frac{t}{\tau }} \end{aligned}$$
(5)

where \(\tau \) is the unfolding time that is related to the strain rate (Koszalka et al. 2009; Badin et al. 2011; Dräger-Dietel et al. 2018).

Table 1 Summary of experiments with the initial position based on the drifters position from Meyerjürgens et al. (2019)
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2.7 Model experiments

In this study, two clusters were implemented to analyze the effect of cluster size. Cluster 1 (C1) with 8 elements and cluster 2 (C2) with 6 elements, with an element referring to a passive virtual particle that represents a small mass of oil, plastic, or other material that can drift at the ocean surface (Fig. 2). The initial positions of the drifters were employed as starting positions for the virtual particles in our simulations. The experiments are divided into two parts: those that account for wind-wave-driven mixing parameters and those that do not. Diffusion is either activated or not (Table 1).

Fig. 3
figure 3

Time series of zonal u (left) and meridional v (right) velocities at 4 m depth, from January 7th to January 13th, 2016. The red line represents the data measured by the ADCP (FINO-3 station), the black one represents experiment 1, including wind-wave coupling, and the blue one is experiment 3, without the wind-wave coupling

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2.8 Model validation

The velocity simulations are assessed against Acoustic Doppler Current Profiler (ADCP) data (Fig. 3), for the period of January-February 2016, obtained from the FINO-3 station. The validation was conducted over a three-month period, from January 1, 2016 to March 1, 2016. However, the resulting plots only encompass seven days.

The analysis was based on three-hourly averaged ADCP data, which was used to facilitate a point-to-point comparison. This was necessary since the model output provides three-hourly means, while the ADCP provides 10-minute instantaneous data. For the zonal component of the velocity (u), both experiments agree well with the ADCP. However, the model underestimates the peaks measured by the ADCP.

Nevertheless, it is observed that the velocity obtained from experiment 1 agrees better with the data measured by the ADCP than the one from the stand-alone model simulations. The meridional component of the velocity (v) of experiment 1 is the one that most closely matches the data measured by the ADCP.

When all the wave-wind coupling parameters are taken into account, the peaks measured by the ADCP are better represented. The meridional component of the experiment without the coupling parameters is sometimes very low or even above that measured by the ADCP. In general, it is observed that the meridional component of the velocity tends to have a greater magnitude than the zonal component.

A variety of statistical parameters were calculated to assess the model’s skill. Given the limited temporal resolution of the model output, which may not capture all high-frequency variabilities observed in the data, the correlation coefficients for the zonal and meridional components are 0.867 and 0.885, respectively. Additionally, the biases for the zonal and meridional components are 0.008 m/s and -0.003 m/s, respectively, suggesting that the model predictions are closely aligned with the observations, with minimal deviation. Moreover, the root mean square deviation (RMSD) for the zonal and meridional components is 0.095 m/s and 0.091 m/s, respectively. These RMSD values demonstrate the accuracy of the model simulations, indicating a reliable model performance in capturing the essential dynamics of the observed data. These parameters were calculated for experiment 1.

3 Results

3.1 Experiments including wind-wave coupling parameters

Figure 4a presents the relative dispersion for C1 for experiment 1. Here we observe two regimes. The Richardson regime describes a dispersion following \(D^2 (t) \backsim t^3\). It describes processes driven with spatial scales similar to the model separation. The second regime shows an exponential growth. This appears when processes larger than the separation distance of the model drive the dispersion: \(D^2(t)= D^2(0)e^{\frac{t}{\tau }}\). The initial exponential regime indicates a rapid increase in relative dispersion. The exponential growth implies that dispersion processes are dominating over time, likely due to the interaction with the waves and the resulting rapid movement of particles. The transition from the exponential regime to the Richardson regime indicates a change in dispersion behavior. The Richardson regime is characterized by slower growth of relative dispersion. In this regime, the dispersion is driven by mechanisms with larger spatial scales and longer timescales, indicating a shift from localized turbulent mixing to more organized and coherent flow patterns.

Fig. 4
figure 4

Relative dispersion for experiment 1 (ALL-on) and experiment 2 (ALL-off). For C1 and C2. The blue line displays the average over all elements, while thin gray lines indicate the relative dispersion of the individual elements. The theoretical fits are shown as a black dashed line, and the black line (Drifter) indicates the average over all pairs from Meyerjürgens et al. (2019)

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Experiment 1 for C1 (Fig. 4a) is the one with the highest relative dispersion (\(10^4 km^2\)). The magnitude of relative dispersion indicates the level of spreading or dispersal achieved, reflecting the efficiency of wave effects, wind stress, Stokes drift, and wave-induced turbulence in driving particle motion and mixing.

The incorporation of diffusion into the module results in a more uniform and continuous curve that includes both the \(e^{\frac{t}{\tau }}\) and \(t^3\). This is in contrast to the absence of the ballistic regime \((D^2(t)\backsim t^2)\), which is not present due to the rapid rise of the curve. Although the particles start at the same position as the drifters, the experiments show that diffusion acts faster in the model system, causing the particles to disperse more quickly than the drifters in the ocean.

For C2 (Fig. 4b), the relative dispersion is reduced compared to C1. The exponential and ballistic regimes are observed, while the Richardson regime is absent. This could be due to the size of the cluster. In the experiments with wind-wave-driven mixing, the absence of diffusion resulted in a lack of distinct regimes within the two clusters (Fig. 4c and d). The interplay between wind and waves, which is often crucial for dispersal dynamics, did not result in observable dispersal patterns within the studied clusters. The lack of regimes indicates that the complex mixing processes facilitated by wind and waves, which would normally result in identifiable dispersion behaviors, are not effectively captured in the experimental conditions without diffusion.

Figures 5a and b present the relative diffusivity for C1 and C2. The diffusivity demonstrates a growth rate proportional to \(D ^{\frac{4}{3}} \), which is consistent with the theoretical prediction of Richardson (1926), suggesting the occurrence of a local dispersion regime for both small and large scales. Compared to the drifters, the model shows a higher relative diffusivity. When accounting for diffusion in OpenDrift, the relative diffusivity becomes a smoother curve with a small peak towards the end. In contrast, in the absence of diffusion (Fig. 5c and d), the element separation D is significantly reduced, as is the relative diffusivity.

3.2 Experiments without the wave coupling parameters

Figures 6a and b demonstrate that when the wind-wave-driven mixing parameters are not included, the amplitude of the dispersion is smaller. Similar to the results obtained when the wind-wave-driven mixing parameters were considered, two regimes, namely the ballistic regime and exponential growth, can be observed. However, their magnitudes are smaller. On the other hand, for experiments without diffusion (Fig. 6c and d), the amplitude of the relative dispersion is negligible, less than \(10^0 km^2\).This indicates that no regime is observed, and only towards the end a few peaks can be seen. The small relative dispersion also results in a reduction of the relative diffusivity and element separation D.

The relative diffusivity is almost as high as when the wind-wave driven mixing parameters are included (Fig. 7a and b). However, the separation D between particles is significantly reduced. When considering the wind-wave-driven mixing parameters, we can observe that the separation of the elements D reaches a maximum of \(10^2\) km (Fig. 5a), while without them, D only reaches approximately \(10^1\) km (Fig 7a).

Fig. 5
figure 5

Relative diffusivity for experiment 1 (ALL-on) and experiment 2 (ALL-off) for C1 and C2

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Fig. 6
figure 6

Relative dispersion for experiment 3 (NO-on) and experiment 4 (NO-off). For C1 and C2. The blue line displays the average over all elements, while thin gray lines indicate the relative dispersion of the individual elements. The black line (Drifter) indicates the average over all pairs from Meyerjürgens et al. (2019)

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Fig. 7
figure 7

Relative diffusivity for experiment 3 (NO-on) and experiment 4 (NO-off) for C1 and C2

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4 Discussion and conclusions

The effect of wave coupling on particle dispersion has been investigated in this study using a Lagrangian model and a fully coupled ocean wave model, and has been compared to dispersion obtained by surface drifters deployed in the North Sea. The results show that the inclusion of wave coupling parameters, such as sea state-dependent momentum flux, sea state-dependent energy flux, and wave-induced mixing, has a significant impact on the magnitude of the relative dispersion. When the wind-wave-driven parameters are included, the relative dispersion shows two distinct regimes - the Richardson regime and exponential growth - with a peak that reaches up to \(10^4 km^2\). On the other hand, when the wind-wave-driven mixing parameters are not included, the relative dispersion is smaller, reaching a peak of \(10^3 km^2\), and in some cases, no regime can be identified. These findings suggest that the wind-wave-driven parameters play a crucial role in the dispersion of particles in the ocean, and their inclusion should not be ignored in particle tracking studies.

Furthermore, the reduction in separation distance when wave coupling parameters are not considered implies that particles will likely disperse over smaller distances, and the influence of local physical processes such as turbulence may have a greater impact on the dispersal patterns. Moreover, the results also indicate that the effects of the wind-wave-driven mixing parameters have a significant impact on the dispersal of particles and the separation between them. Their inclusion leads to a larger separation between particles, which results in a higher relative diffusivity. On the other hand, excluding these parameters leads to a smaller separation between particles, which reduces the relative diffusivity.

Experiments 2 and 4 show a reduced dispersion (Fig. 6a and b) due to the absence of wind-wave-induced mixing, a process that significantly affects particle dispersion in the ocean. The impact of wind-wave-driven parameters on particle separation is observed to be greater over longer distances (Fig. 4a). As particles move through ocean currents, their dispersion patterns become more noticeable over larger distances. Wave-induced mixing occurs due to the interaction of waves and turbulence, resulting in the mixing of water masses and dispersion of particles (Staneva et al. 2021a). The relevance of including Stokes drift in modeling simulations is discussed by De Dominicis et al. (2016). It improves the accuracy of drifter trajectories or, in our case, particle trajectories. Stokes drift is needed to reproduce the transport of particles that are not only driven by surface ocean currents. It accounts for the additional displacement caused by wave-induced motion. Neglecting these processes could lead to an underestimation of particle dispersion in the ocean. Therefore, it is crucial to account for wave-induced mixing in particle dispersal modeling.

It is also relevant to consider the minimum depth of the NEMO model. The minimum depth of the model is 8 m, which may result in limitations in accurately representing coastal processes in the wave model, particularly in shallow water regions. It is therefore necessary to place these limitations within the context of the study’s scope and objectives. The limitations of the wave model in fully capturing shallow water phenomena such as shoaling and diffraction can result in discrepancies in the simulation of particle dispersion near the coast. These processes have been demonstrated to have a significant impact on wave dynamics, which in turn affects coastal currents (Staneva et al. 2017). It is, however, important to note that the primary focus of our study is to investigate large-scale phenomena rather than fine-scale coastal processes. Consequently, while the model’s representation of the coastal zone may not be entirely exact, its performance remains adequate for the intended purpose of investigating larger-scale and rather offshore dynamics. Future research may benefit from the use of alternative modeling approaches or the incorporation of parameterizations to more accurately account for shallow water processes, thereby improving the accuracy of coastal simulations.

When comparing our results to those of Meyerjürgens et al. (2020), we observed that their relative dispersion figure displays three distinct regimes: exponential, ballistic, and Richardson. In each of these regimes, the relative dispersion demonstrates different growth behavior over time. Our implemented model accurately captures the exponential regime, indicating an initial rapid spread or separation of particles. The Richardson regime is also represented in the model’s framework if the model accounts for the effects of wind-wave-driven mixing and diffusion (experiment 1). However, it does not simulate the ballistic regime, which is characterized by a quadratic growth of the dispersion. This suggests that the model may not fully capture the dynamics of particle trajectories during the ballistic phase. Therefore, it may be necessary to refine the model by introducing adjustments, to account for the ballistic dispersion behavior observed in the empirical data. In the absence of the wind-wave-driven mixing parameters (experiments 3 and 4), the Richardson regime is absent while the ballistic regime is present. This absence suggests a potential limitation in the model’s ability to accurately replicate dispersion influenced by turbulent processes. Therefore, it is necessary to include these processes in the simulations to obtain accurate results.

The inclusion of diffusion in the Lagrangian model is another key aspect. None of the regimes described in the method section could be observed without diffusion (Fig. 4c and d and Fig. 6c and d), indicating that particles do not disperse as much. Even when considering only wind-wave-induced mixing (experiment 3), the results were inadequate, showing that diffusion is necessary for an accurate representation of the system. However, experiment 2, which only used diffusion, showed lower magnitudes in some regimes, indicating that both wind-driven mixing and diffusion are necessary. In the OpenDrift framework, diffusion is modeled using a random component, which is scaled by a normal distribution with a mean of zero and a variance of one. This approach is based on research by Stommel (1949) and Callies et al. (2011), and represents a common approach (Döös et al. 2011; LaCasce 2008; De Dominicis et al. 2016) to capturing the stochastic nature within the model. The introduction of diffusion is inherently challenging due to the complex dynamics of oceanic processes. However, the omission of this component results in significantly less accurate representations of particle dispersion, as observed in experiments 3 and 4 (Fig. 6). Nevertheless, it is acknowledged that this method may not always align perfectly with physical observations. Our findings indicate that this approach tends to accelerate dispersion rates. Although this approach is effective in capturing a broad range of particle dynamics, it may not fully reflect the subtleties of natural diffusion processes observed in the ocean. A comparison of this approach with other studies reveals that there are different ways to introduce diffusion into the model. One such approach is the Smagorinsky model (Meneveau et al. 1996). The Smagorinsky model is a turbulence closure scheme employed in large eddy simulations to approximate the effects of subgrid-scale turbulence by calculating a dynamic turbulent viscosity based on the local rate of strain (Smagorinsky et al. 1965). While enhancing the realism and stability of numerical simulations, it can also lead to excessive diffusion if not carefully calibrated, particularly in less turbulent regions of the flow. Future iterations of the OpenDrift framework may benefit from the implementation of alternative parametrizations of diffusion within the model. This could facilitate a more comprehensive and precise simulation of oceanic diffusion processes, which are inherently complex and dynamic.

The study by Rühs et al. (2018) emphasizes the role of diffusion in characterizing ocean processes and aims to contribute to the understanding of diffusivity. The authors highlight the importance of Lagrangian analysis and diffusivity estimation in understanding the dispersion of oceanic particles. Additionally, the study suggests the need for further research to address methodological choices and improve the tuning of diffusivity coefficients based on Lagrangian estimates. The De Dominicis et al. (2012) diffusion parameterization is similar to the one we used and it is consistent with our idea of the relevance of including diffusion as well as the role of diffusion parameterization. However, there are cases where running the model without diffusion can be advantageous, especially when studying the impact of ocean currents or wind drift. In such situations, particle dispersion becomes minimal or even negligible, providing a valuable opportunity to assess the accuracy of the model.

In our study, we observed that the particle dispersion behavior varies with the cluster size and the parameters considered. In particular, the Richardson regime, which indicates large-scale turbulence and is characterized by a cubic increase in particle separation with time, was detected in larger clusters (C1). This agrees with the theoretical predictions of Richardson (1926). In contrast, this regime was not evident in smaller clusters (C2) due to their reduced size, which limits their capacity to adequately represent larger turbulent scales. It is possible that smaller clusters are unable to completely capture the full spectrum of turbulence, which could result in dispersion measurements that underestimate the actual variability observed with larger clusters. Our results suggest that the detectability of the Richardson regime is closely related to cluster size. This underscores the need to account for cluster size in the setup of experiments in order to effectively capture the diverse behavior of dispersion regimes.

Figures 4 and 6 illustrate the relative dispersion of our model in comparison to established theoretical regimes. It is important to note that not all three regimes -exponential, ballistic, and Richardson- are shown simultaneously in each experiment. Specifically, in experiment 1 for C1 (Fig. 4a), we observe transitions from the exponential to the Richardson regime, while in experiment 1 for C2 (Fig. 4b), transitions occur from the exponential to the ballistic regime. Although the model generally captures the trends of these regimes, there are still some discrepancies that require further examination. Notably, the model aligns well with the theoretical curves in the exponential regime. However, there are noticeable discrepancies during the transition to the Richardson regime in experiment 1-C1 and to the ballistic regime in experiment 1-C2. These discrepancies may be attributed to several factors inherent in modeling complex systems, including the resolution of the model and the parameterization of certain processes. Additionally, the size of the clusters plays a significant role in influencing the dispersion behavior. This complexity and variability in real-world data may also explain why we do not observe all three dispersion regimes concurrently in a single experiment. For experiment 3 (Fig. 6) only two regimes were observed (exponential and ballistic), due to the aforementioned factors.

When considering the complex hydrodynamic systems under investigation, particularly those in the German Bight, regional-specific influences may account for the observed discrepancies. The interplay of physical processes in this coastal zone is intricate, with local wind stress and wave-induced turbulence exerting a significant impact on water movement and, consequently, particle dispersion.

The circulation flow in the German Bight, influenced by factors such as wind stress and wave-induced turbulence (Staneva et al. 2016), contributes to the prevalence of the exponential regime as well as the ballistic regime. The observed agreement with Richardson’s findings further confirms the presence of this regime in the dynamics of the German Bight. Furthermore, our analysis indicates the emergence of the Richardson regime at larger separation distances, which is associated with the influence of larger-scale processes that contribute to enhanced particle spreading over longer distances. The specific characteristics of the German Bight, including the semi-diurnal tidal waves and the interaction between tidal processes and prevailing westerly winds, likely play a significant role in the development of the Richardson regime, which was also concluded by Meyerjürgens et al. (2019).

Our study builds upon the work of Ohlmann et al. (2012) and Dugstad et al. (2019), which simulated surface current dispersion in the coastal ocean using a Lagrangian approach. We employ a fully coupled NEMO-WAM model with feedback between the upper ocean and waves. The paper by Döös et al. (2011) suggests that incorporating a more realistic representation of the 3D nature of the velocity field into model simulations can improve dispersion. Our simulations, which take diffusion into account and use a Lagrangian particle-drift model, produce accurate results that support this finding. Comparisons between 2D and Lagrangian 3D trajectories reveal that relative dispersion is stronger when trajectories are advected with a 3D velocity field.

In conclusion, this study demonstrates the significance of incorporating wind-wave-driven mixing parameters and diffusion mechanisms into particle tracking studies and provides insight into their major impact on the accurate simulation and interpretation of particle dispersion in the ocean. These considerations not only improve the precision of modeling but also provide valuable knowledge that contributes to a more refined understanding of the complex processes that govern particle behavior.