# Wave Climate and the Effect of Induced Currents over the Barrier Reef of the Cays of Alburquerque Island, Colombia

^{*}

## Abstract

**:**

## 1. Introduction

^{2}that is located between 12°08′–12°12′ N and 81°49′–81°54′ W, which is about 35 km SW of San Andrés Island in the San Andrés, Providence, and Santa Catalina Archipelago (SPSA), Colombia. This bank has a circular shape that encompasses a pre-reef terrace. The east–west diameter exceeds 8 km. The atoll has two cays (Cayo del Norte and Cayo del Sur or Pescadores), with a combined emerged area of 0.1 km

^{2}. Alburquerque is permanently occupied by military personnel, and it receives continuous visits from fishermen performing their fishing tasks.

^{2}[3]. This territory is also immersed in a legal dispute in which Nicaragua instituted proceedings against Colombia in 2001 before the International Court of Justice, which confirmed Colombia’s sovereignty over all of the islands in the archipelago, but drew maritime boundaries in favor of Nicaragua [4].

## 2. Materials and Methods

**I**of nonzero, and limited transformation, we obtain:

^{i}=

**Ve**

^{i}in Equations (1)–(3) are contravariant of the vector of the currents (

**V**) with the basic contravariant vector (

**e**

^{i}= ζ

^{i},), where ζ

^{i}= (ξ,χ), and i = 1, 2 (the contravariant components of the tensor metric

**g**

^{ik}=

**e**

^{i}

**e**

^{k}; and ${\Gamma}_{\mathrm{k}\mathrm{j}}^{\mathrm{i}}$, which are the Christoffel symbols of type II). Here, the chosen coordinate system is orthogonal; that is,

**g**

^{ik}= 0, i ≠ k; i, k = 1, 2 (the coordinates are Cartesian when

**g**

^{11}=

**g**

^{22}= 1).

^{i}(ξ,χ)) and the sea level (η (ξ,χ)). The other symbols are as follows: g: gravity; ρ: density of water; h, H: local and total depths, respectively (H = h + η); f: Coriolis parameter; T

^{1}and T

^{2}: the respective components of τ

_{sx}and τ

_{sy}in the Cartesian plane (x,y) of the wind stress in the curvilinear coordinates; γ

_{U}, γ

_{UV}, and γ

_{V}: expressions that parameterize the vertical structure of the flow [18].

^{i}was specifically introduced in Equations (2) and (3) to describe wave currents. These are radiative stress components that are produced by waves (wave-induced force per unit surface area (gradient of radiation stresses)) [19].

^{i}term in Equations (2) and (3) is related to the bottom friction effects. In the case of wave dynamics, these terms are presented as recommended in [20]. For the effect of currents, the quadratic law of friction applies. In the coupled form of these two effects, the formulas are linearized by employing an integral coefficient of the bottom friction (r). Thus, F

^{i}= rU

^{i}.

^{i}

_{m}(similar to α in Equation (1)), which do not have any physical sense (e.g., in the analogous comparison to Reynolds stresses), but are the products of mathematical operations. However, according to [21], these terms are tidal stresses that express the contribution of long tidal waves to the residual circulation:

**g**

_{ii}, produces the vortex equation in terms of the stream function:

^{i}) can then be presented in the following way:

_{k}), the impermeability conditions for the water flow were adjusted such that the flow that was perpendicular to the boundary was equal to zero. In terms of the stream function, this condition will be:

_{k}, in ∂Ω

_{k}, k = 1, …, N,

## 3. Results

#### 3.1. Analysis of the Wave Climate in Deep Waters

_{99%}of 2.48 m. The NE and E directions present probabilities of 13.21 and 8.7%, respectively, which indicates that these three directions contain 84.36% of the significant heights of the series. For the remaining directions, there is a 15.64% probability. The highest magnitude for Hs

_{99%}is 4.53 m in the N direction, with a 2.13% probability.

_{99%}is 10.6 s. The highest values of Tp

_{99%}belong to WNW (11.74 s) and SSE (11.56 s), with probabilities of 0.14 and 0.57%, respectively, which likely correspond to occasional bottom swells within the series (Table 3).

#### 3.2. Wave Propagation and Wave-Induced Currents

^{3}/s, with a perpendicular distance of 50–100 m.

^{−3}to 12 × 10

^{−3}m

^{2}/s (i.e., three times in energy terms), while the orbital speed at the bottom only increases from 0.8 to 1 m/s between the medium and extreme cases. The wavelength, in turn, does not show significant differences. Moreover, the wavelength’s impact on the steepness value is small compared to the increase in the wave height along the underwater transect.

## 4. Discussion

## 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**) The spatial location of the cays of Alburquerque Island alongside the general context on the nautical chart COL 007. (

**b**) Satellite image (taken from www.oceandots.com, accessed on 15 March 2021) and (

**c**) the bathymetry in the nautical chart COL 203 for the Bank of Alburquerque in the Colombian Caribbean.

**Figure 3.**Bathymetry (in m) and three domains of interest: Mesh 1 (magenta polygon line); Mesh 2 (green); Mesh 3 (blue).

**Figure 7.**Calculated time series of the significant height (

**top**) and peak periods (

**bottom**) from January 1979 to December 2010. The years 1988 and 1989 are missing (gaps in these figures).

**Figure 8.**Histogram showing that the frequency of the significant height (Hs) is mainly distributed in the height bands that range from 0 to 2 m.

**Figure 9.**The probabilities for the wave height, the Hs, and the direction show that the ENE, NE, and E directions present a probability of 84.36% of the significant heights of the series.

**Figure 10.**Significant wave-height field (in m) in Mesh 1 that corresponds to the propagation of the virtual buoy in deep water. Waves from the E. The barrier effect, moreover, is observed in the deformation of the wave fields.

**Figure 11.**Significant wave-height field (in m) in Mesh 2 that corresponds to the propagation of the virtual buoy in deep water. Waves are from the E-generating jagged rays that appear within the atoll.

**Figure 12.**Significant wave-height field (

**top**) showing jagged rays that appear within the atoll, which are not as noticeable in the wavelength field (

**bottom**) (in m) in Mesh 3, and which correspond to the propagation of the virtual buoy in deep water. Waves from the ENE, Case Hs

_{12}(Table 2 and Table 3).

**Figure 13.**Wave steepness (magnitude in colors) on the bottom relief (bathymetry contours in meters). Transect A and B indicates the position used for analysis. Waves from the ENE, Case Hs

_{12}.

**Figure 14.**Stream function of currents (isolines in m

^{3}/s) induced by waves corresponding to the E wave pattern and the Hs

_{12}swell. The maximum speed along the barrier reef is greater than 1 m/s (color scale in m/s). Arrows indicate the directions of the currents.

**Figure 15.**Stream function of currents (isolines in m

^{3}/s) induced by waves that correspond to the ENE wave pattern for Wave Hs

_{12}. The maximum speed along the barrier reef is greater than 1 m/s (color scale in m/s). Arrows indicate the directions of the currents.

**Figure 16.**Stream function of currents (isolines in m

^{3}/s) induced by waves that correspond to the ENE wave pattern; waves of the regime (color scale in m/s).

**Figure 17.**Profiles of depth, significant height (Hs), wave-energy-dissipation rate (Q), bottom orbital velocity (Uorb), length, and steepness for waves of the regime (solid line) and the Hs

_{12}(dashed line) along Transect A and B (in Figure 13).

Domain Number | 1 | 2 | 3 |
---|---|---|---|

X min (UTM), m | 400,770 | 405,958 | 408,046 |

X max (UTM), m | 413,500 | 412,043 | 410,565 |

Y min (UTM), m | 1,338,735 | 1,341,635 | 1,343,687 |

Y max (UTM), m | 1,350,185 | 1,349,260 | 1,345,650 |

X length, m | 12,730 | 6085 | 2519 |

Y length, m | 11,450 | 7625 | 1963 |

Resolution, m | 50 | 20 | 8 |

Grid points | 256 × 230 | 305 × 382 | 316 × 246 |

Direction | Dir. Probability | Hs_{50%} | Hs_{90%} | Hs_{99%} | Hs_{12} |
---|---|---|---|---|---|

N | 0.0213 | 1.4000 | 2.8700 | 4.5374 | 5.2143 |

NNE | 0.0448 | 1.4200 | 2.2900 | 3.4919 | 4.6730 |

NE | 0.1321 | 1.2400 | 1.8800 | 2.5868 | 3.1451 |

ENE | 0.6245 | 1.1200 | 1.8400 | 2.4800 | 2.9100 |

E | 0.0870 | 0.7100 | 1.3200 | 1.9120 | 2.2143 |

ESE | 0.0167 | 0.5600 | 0.9670 | 1.5728 | 2.0500 |

SE | 0.0081 | 0.6000 | 1.0500 | 1.5946 | 2.0342 |

SSE | 0.0057 | 0.6000 | 0.9900 | 1.7577 | 2.0264 |

S | 0.0073 | 0.6500 | 1.1660 | 2.0380 | 2.2734 |

SSW | 0.0260 | 0.9300 | 1.5900 | 2.1348 | 2.9715 |

SW | 0.0088 | 0.9500 | 1.6770 | 2.3482 | 2.5003 |

WSW | 0.0072 | 0.9700 | 2.2600 | 3.5935 | 4.1352 |

W | 0.0033 | 1.0250 | 2.5970 | 3.9160 | 4.4500 |

WNW | 0.0014 | 1.0300 | 2.1120 | 2.6481 | 2.6700 |

NW | 0.0012 | 0.8300 | 2.3570 | 2.7298 | 2.8400 |

NNW | 0.0045 | 1.2400 | 2.1820 | 3.6626 | 4.8162 |

Direction | Dir. Probability | Tp_{50%} | Tp_{90%} | Tp_{99%} | Tp_{12} |
---|---|---|---|---|---|

N | 0.0213 | 6.0800 | 7.7200 | 9.0500 | 9.7900 |

NNE | 0.0448 | 6.0800 | 7.7200 | 9.0500 | 9.7900 |

NE | 0.1321 | 7.1300 | 8.3600 | 9.7900 | 10.6000 |

ENE | 0.6245 | 7.7200 | 9.0500 | 10.6000 | 11.4800 |

E | 0.0870 | 7.1300 | 9.0500 | 9.7900 | 11.4800 |

ESE | 0.0167 | 6.5900 | 7.7200 | 9.6716 | 10.6000 |

SE | 0.0081 | 6.0800 | 7.7200 | 9.7900 | 20.0000 |

SSE | 0.0057 | 4.8000 | 7.1300 | 11.5646 | 20.0000 |

S | 0.0073 | 6.0800 | 7.7200 | 9.0500 | 11.1461 |

SSW | 0.0260 | 7.7200 | 9.0500 | 9.7900 | 10.6000 |

SW | 0.0088 | 5.1900 | 7.7200 | 9.0500 | 13.4500 |

WSW | 0.0072 | 5.1900 | 7.1300 | 9.0500 | 9.0500 |

W | 0.0033 | 5.6200 | 7.7200 | 9.0500 | 9.0500 |

WNW | 0.0014 | 5.1900 | 7.1300 | 11.7432 | 12.4200 |

NW | 0.0012 | 4.8000 | 7.7200 | 9.7900 | 9.7900 |

NNW | 0.0045 | 6.0800 | 7.7200 | 8.3600 | 9.7626 |

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**MDPI and ACS Style**

Lonin, S.; Andrade, C.A.; Monroy, J.
Wave Climate and the Effect of Induced Currents over the Barrier Reef of the Cays of Alburquerque Island, Colombia. *Sustainability* **2022**, *14*, 6069.
https://doi.org/10.3390/su14106069

**AMA Style**

Lonin S, Andrade CA, Monroy J.
Wave Climate and the Effect of Induced Currents over the Barrier Reef of the Cays of Alburquerque Island, Colombia. *Sustainability*. 2022; 14(10):6069.
https://doi.org/10.3390/su14106069

**Chicago/Turabian Style**

Lonin, Serguei, Carlos Alberto Andrade, and Julio Monroy.
2022. "Wave Climate and the Effect of Induced Currents over the Barrier Reef of the Cays of Alburquerque Island, Colombia" *Sustainability* 14, no. 10: 6069.
https://doi.org/10.3390/su14106069