# Spreading of Lagrangian Particles in the Black Sea: A Comparison between Drifters and a High-Resolution Ocean Model

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials

#### 2.1.1. Drifters

_{0}~3.5–5 km, which is much smaller than R

_{d}(internal Rossby radius) ranging from 10 to 30 km for the BS [2,28,29]. The initial drifter pair tracks were considered every 5 days along the trajectories. This time interval is larger than the integral time scale estimated in the BS (T

_{L}~3 days) (see [2,23] for details).

#### 2.1.2. Satellite Altimetry

#### 2.1.3. Models

_{0}~3.5–5 km as the real drifter pairs displayed above by adapting the [24] approach.

#### 2.2. Methods

#### 2.2.1. The Turbulent Relative Dispersion Statistics

^{(1)}and r

^{(2)}are the Lagrangian arc length of the particle trajectories forming the pair, $\mathrm{t}$ is the time and D

_{0}is the initial pair separation distance.

_{0}.

^{(1)}and V

^{(2)}are the Lagrangian velocities of two drifters separated with a predefined distance.

#### 2.2.2. The Turbulent Absolute Dispersion Statistics

## 3. Results

#### 3.1. Circulation of the BS

^{2}s

^{−2}) were observed in winter in the branches of the RC located along the Anatolia coast, the Crimea peninsula, and in the western corner of the BS (Figure 4a and Figure 6a), in agreement with previous studies [2,16].

#### 3.2. Dispersion

#### 3.2.1. Relative Dispersion

#### Surface Relative Dispersion

#### Intermediate and Deep Relative Dispersion

#### 3.2.2. Absolute Dispersion

#### Surface Absolute Dispersion

^{5/4}(Figure 13c and Figure 14e,g) and t

^{5/3}(Figure 13d and Figure 14f,h), for the hyperbolic and elliptic regimes, respectively. In fact, a quasi-plateau was found, reflecting the dominance of the hyperbolic and elliptic regimes in any cases discussed above. The occurrence of the hyperbolic regime over a relatively cold season (winter/spring and January) is related to the dominance of hyperbolic regions, which were detected by the Q* parameter; more than 70% of hyperbolic areas were detected in this scenario (see the insets of Figure 4d and Figure 6d). The (5/3) law was only observed in relatively warm seasons (summer/fall, August), essentially due to the fact that particles were launched into the elliptic regions for a persistent meandering flow (blue regions, Figure 5d and Figure 7d).

#### Intermediate and Deep Absolute Dispersion

^{2}and random-walk~t) were also detected at depth.

## 4. Discussion

_{d}, from model results (Figure 2a). The presence of this regime can be related to the effect of coherent structures on the pair separations at small spatio-temporal scales rather as the ability of the model to solve the exponential regime for small scales. This is because this regime is related to the role of the coherent structures at scales less than R

_{d}. According to these results, it is clear that the model is able to resolve the complex turbulent variability and could also help to explore the non-local regime along the water column (Figure 11 and Figure 12).

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) General surface circulation scheme (red arrows) and the bathymetry (colors) of the BS and (

**b**) mean surface circulation in the BS (1999–2009) in bins of 0.25° × 0.25° by merging drifters and altimetry datasets. Acronyms: RC: Rim Current, WG: Western Gyre, EG: Eastern Gyre, SevE: Sevastopol Eddy, BosE: Bosphorus Eddy, SakE: Sakarya Eddy, SE: Sinop Eddy, KizE: Kizilirmak Eddy, BE: Batumi Eddy, CauE: Caucasus Eddy and CrE: Crimea Eddy.

**Figure 2.**The well-known turbulent (

**a**) relative and (

**b**) absolute dispersion regimes, were t and T are the times, T

_{L}is the integral time scale, R

_{d}is the internal Rossby radius and δ is the separation distance between the pairs. The time parameter T allows estimating the theoretical exponential fit of Ku.

**Figure 3.**The surface circulation in January and August of 2002 and 2006, respectively, and in bins of 0.25° × 0.25° by merging the drifters and altimetry datasets (

**a**–

**d**) and modeled virtual particle tracks (

**e**–

**h**).

**Figure 4.**Circulation of BS superimposed with the Kinetic Energy (KE) (left panels) and the normalized Okubo–Weiss parameter Q* (right panels) obtained from the model in January 2002 at 15 m (

**a**,

**d**), 150 m (

**b**,

**e**), 750 m (

**c**,

**f**) depths. The insets show the percentage of areas that have Q* > 0 (hyperbolic grid cells) and Q* < 0 (elliptic grid cells).

**Figure 5.**Circulation of BS superimposed with the Kinetic Energy (KE) (left panels) and the normalized Okubo–Weiss parameter Q* (right panels) obtained from the model in August 2002 at 15 m (

**a**,

**d**), 150 m (

**b**,

**e**), 750 m (

**c**,

**f**) depths. The insets show the percentage of areas that have Q* > 0 (hyperbolic grid cells) and Q* < 0 (elliptic grid cells).

**Figure 6.**Circulation of BS superimposed with the Kinetic Energy (KE) (left panels) and the normalized Okubo–Weiss parameter Q* (right panels) obtained from the model in January 2006 at 15 m (

**a**,

**d**), 150 m (

**b**,

**e**), 750 m (

**c**,

**f**) depths. The insets show the percentage of areas that have Q* > 0 (hyperbolic grid cells) and Q* < 0 (elliptic grid cells).

**Figure 7.**Circulation of BS superimposed with the Kinetic Energy (KE) (left panels) and the normalized Okubo–Weiss parameter Q* (right panels) obtained from the model in August 2006 at 15 m (

**a**,

**d**), 150 m (

**b**,

**e**), 750 m (

**c**,

**f**) depths. The insets show the percentage of areas that have Q* > 0 (hyperbolic grid cells) and Q* < 0 (elliptic grid cells).

**Figure 8.**Drifter pairs (red color) superimposed with the initial drifter positions (black dots) and their trajectories (yellow lines) for initial separation distances between 3.5 km and 5 km. These were used to calculate the surface relative dispersion (at 15 m depth) in winter/spring (

**a**) and January of 2002 (

**b**) and 2006 (

**c**), as well as summer/fall (

**d**), August 2002 (

**e**) and 2006 (

**f**).

**Figure 9.**Surface relative diffusivity (left panels), time scale-dependent pair separation rate as function of pair separation distances (right panels) and the fourth moment of pair dispersion distances (Kurtosis) as a function of time in the insets. The theoretical exponential, Richardson and diffusive regimes are shown with the gray/dashed gray, thick dashed black, and thin black lines, respectively. The red and blue curves in the insets show the theoretical exponential regimes. The dashed thin black lines indicate the 90% confidence intervals from bootstrap resampling.

**Figure 10.**Model derived pairs as the pairs shown in Figure 8 for depths of 150 m (left panels) and 750 m (right panels) in January 2002 (

**a**,

**e**), August 2002 (

**b**,

**f**), January 2006 (

**c**,

**g**), and August 2006 (

**d**,

**h**).

**Figure 11.**Relative diffusivity and the fourth moment of pair dispersion distances (Kurtosis) in the insets (left panels), time scale-dependent pair separation rate as function of pair separation distances (right panels), at 150 m (

**a**,

**c**) and 750 m (

**b**,

**d**) in 2002. The theoretical exponential, Richardson and diffusive regimes are shown with the gray/dashed gray, thick dashed black, and thin black lines, respectively. The red and blue curves in the insets show the theoretical exponential regimes. The dashed thin black lines indicate the 90% confidence intervals from bootstrap resampling.

**Figure 12.**Relative diffusivity and the fourth moment of pair dispersion distances (Kurtosis) in the insets (left panels), time scale-dependent pair separation rate as function of pair separation distances (right panels), at 150 m (

**a**,

**c**) and 750 (

**b**,

**d**) in 2006. The theoretical exponential, Richardson and diffusive regimes are shown with the gray/dashed gray, thick dashed black, and thin black lines, respectively. The red and blue curves in the insets show the theoretical exponential regimes. The dashed thin black lines indicate the 90% confidence intervals from bootstrap resampling.

**Figure 13.**Absolute dispersion calculated from real drifters as a function of time in winter/spring (

**a**) and summer/fall (

**b**). (

**c**) The absolute dispersion normalized by t

^{5/4}and (

**d**) by (t

^{5/3}). In the insets, we show the individual drifters (black dots) for separation distance 0–100 km. The theoretical turbulent absolute dispersion regimes: ballistic (thick black line), random-walk (thick gray line), the elliptic (dashed red line), and the hyperbolic (red full line) laws. The horizontal full red line shows the presence of plateau for the occurrence of the hyperbolic regime, while the dashed horizontal red line investigates the elliptic regime. The dashed black lines indicate the 90% confidence intervals from bootstrap resampling.

**Figure 14.**Absolute dispersion calculated by model as a function of time in the surface layer in January 2002 (

**a**,

**e**), August 2002 (

**b**,

**f**), January 2006 (

**c**,

**g**) and August 2006 (

**d**,

**h**). (

**c**) Absolute dispersion normalized by t

^{5/4}and (

**d**) by (t

^{5/3}). Theoretical turbulent absolute dispersion regimes are shown: ballistic (thick black line), random-walk (thick gray line), the elliptic (dashed red line), and the hyperbolic (red full line) laws. The horizontal full red line shows the presence of plateau for the occurrence of the hyperbolic regime, while the dashed horizontal red line investigates the elliptic regime. The dashed black lines indicate the 90% confidence intervals from bootstrap resampling.

**Figure 15.**Absolute dispersion calculated by model as a function of time at 150 m depth in January 2002 (

**a**,

**e**), August 2002 (

**b**,

**f**), January 2006 (

**c**,

**g**) and August 2006 (

**d**,

**h**). (

**c**) Absolute dispersion normalized by t

^{5/4}and (

**d**) by (t

^{5/3}). Theoretical turbulent absolute dispersion regimes are shown: ballistic (thick black line), random-walk (thick gray line), the elliptic (dashed red line), and the hyperbolic (red full line) laws. The horizontal full red line shows the presence of plateau for the occurrence of the hyperbolic regime, while the dashed horizontal red line investigates the elliptic regime. The dashed black lines indicate the 90% confidence intervals from bootstrap resampling.

**Figure 16.**Absolute dispersion calculated by model as a function of time at 750 m depth in January 2002 (

**a**,

**e**), August 2002 (

**b**,

**f**), January 2006 (

**c**,

**g**) and August 2006 (

**d**,

**h**). (

**c**) Absolute dispersion normalized by t

^{5/4}and (

**d**) by (t

^{5/3}). Theoretical turbulent absolute dispersion regimes are shown: ballistic (thick black line), random-walk (thick gray line), the elliptic (dashed red line), and the hyperbolic (red full line) laws. The horizontal full red line shows the presence of plateau for the occurrence of the hyperbolic regime, while the dashed horizontal red line investigates the elliptic regime. The dashed black lines indicate the 90% confidence intervals from bootstrap resampling.

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Bouzaiene, M.; Menna, M.; Elhmaidi, D.; Dilmahamod, A.F.; Poulain, P.-M.
Spreading of Lagrangian Particles in the Black Sea: A Comparison between Drifters and a High-Resolution Ocean Model. *Remote Sens.* **2021**, *13*, 2603.
https://doi.org/10.3390/rs13132603

**AMA Style**

Bouzaiene M, Menna M, Elhmaidi D, Dilmahamod AF, Poulain P-M.
Spreading of Lagrangian Particles in the Black Sea: A Comparison between Drifters and a High-Resolution Ocean Model. *Remote Sensing*. 2021; 13(13):2603.
https://doi.org/10.3390/rs13132603

**Chicago/Turabian Style**

Bouzaiene, Maher, Milena Menna, Dalila Elhmaidi, Ahmad Fehmi Dilmahamod, and Pierre-Marie Poulain.
2021. "Spreading of Lagrangian Particles in the Black Sea: A Comparison between Drifters and a High-Resolution Ocean Model" *Remote Sensing* 13, no. 13: 2603.
https://doi.org/10.3390/rs13132603