# Analysis of the Variability of Wave Energy Due to Climate Changes on the Example of the Black Sea

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Data and Methods of Analysis

## 3. Fluctuations of the Average Annual Wave Heights and Periods in Connection with the Changes of Climatic Indices

^{−1}; the longest period is 6.4 s, with the shortest period at 3 s, and the average annual wave period for the considered 51 years is 4.1 s. It can be assumed that the change of the average annual wave period has a low-frequency periodicity: a sharp decrease was observed beginning in the 1960s and then a visible increase, starting approximately in 2005.

^{−1}. Additionally, changes of the average annual period in the frequency range 0.05–0.13 year

^{−1}are non-stationary. For example, the characteristic frequency scale of 0.07 year

^{−1}increases with time to 0.1 year

^{−1}(Figure 3a).

^{−1}has an increasing trend to 0.08 year

^{−1}. For frequency scales in the range 0.1–0.16 year

^{−1}, beginning in 1985, the frequency scale 0.1 year

^{−1}increases to 0.16 year

^{−1}; after 1985, there are no trends and the frequency 0.16 year

^{−1}is stable (Figure 3b).

^{−1}, for NAO—0.022 and 0.06 year

^{−1}, for PDO—0.025 year

^{−1}, for AO—0.026 and 0.06 year

^{−1}, and for EA-WR—0.035 and 0.1 year

^{−1}. Wavelet analysis shows that, for all climatic indices, there are different trends of frequencies changes in the range 0.09–0.13 and 0.2–0.3 year

^{−1}.

^{–1}. Wavelet diagram of the changes of the average period are similar to wavelet diagrams of the changes seen in PDO and AMO. This similarity indicates the occurrence of an identical long-term (decadal and multi-decadal) periodicity of changes of average annual wave heights, periods, and corresponding climatic indices.

^{−1}and wavelet frequency scales of 0.05, 0.07–0.08, and 0.2–0.25 year

^{−1}for average annual wave heights (Figure 5a). As seen in Figure 5a, the figure appears as a ridge parallel to the wavelet frequencies axis at spavlet′s frequency of 0.016 year

^{−1}. As seen in Figure 5, the ridge on the spavlet diagram at spavlet′s frequency equal to the double wavelet′s frequency is not significant because it was produced by the doubling of wavelets coefficients frequencies at the creation of its modules. The average annual period of waves is modulated by the same low frequency but only on a wavelet frequency of 0.03 year

^{−1}(Figure 5b). On the basis of the structure of the spavlet, one can approximate functions for the process of periodicity of changes of average annual wave heights in the form:

_{H}(t) = A

_{1}cos(0.016t) + B

_{1}cos(0.016t)cos(0.05t) + C

_{1}cos(0.016t)cos(0.075t) + D

_{1}cos(0.016t)cos(0.22t),

_{T}(t) = A

_{2}cos(0.016t) + B

_{2}cos(0.016t)cos(0.03t),

^{−1}. The fluctuations of average annual waves and the NAO index have the similar spavlet structures: the low-frequency modulation at a frequency of 0.016 year

^{−1}of the main wavelet frequency scales 0.075 year

^{−1}(Figure 5a and Figure 6a). In general, the spavlet structure of the fluctuations of the average annual wave period is similar to the structure of the PDO index: the low-frequency modulation by the frequency of 0.016 year

^{−1}of the main wavelet frequency scales 0.03 year

^{−1}(Figure 5b and Figure 6b).

## 4. Fluctuations of the Power of Wave Energy Flux in the Black Sea

_{s}is the significant wave height, T

_{e}= 0.9T

_{p}, T

_{p}is the peak period of wave spectrum.

^{3}and Formula (3) can be simplified to

_{s}

^{2}T

_{e}kW/m.

^{−1}(Figure 9).

_{E}(t) = A

_{3}cos(0.016t) + B

_{3}cos(0.016t)cos(0.03t) + C

_{3}cos(0.016t)cos(0.075t),

^{−1}is also present in the spavlets of the average annual heights, periods, and of the power flux of wave energy; it is also present in spavlets of all considered climatic indices as the fundamental frequency and as the modulating frequency. It is possible that the fluctuations of this period reflect the trends of global climate change.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Distribution of observation points for period 1960–2011 along the Black Sea (

**a**), the total number of observations per year (

**b**), and the dependence of the average annual wave height on the number of observations (

**c**).

**Figure 3.**Wavelet transforms of the average annual wave periods (

**a**) and heights (

**b**). For wavelet analysis, the mean values of the time series were deleted.

**Figure 4.**Wavelet transforms of NAO (

**a**), AMO (

**b**), AO (

**c**), PDO (

**d**) and EA/WR (

**e**). For wavelet analysis, the mean values of the time series were deleted.

**Figure 7.**Coefficients of wavelet-correlation functions between climatic indices and average annual wave periods (

**a**) and heights (

**b**) at zero time lags.

**Figure 10.**Coefficients of wavelet-correlation functions between the climatic indices and the power flux of wave energy at zero time lags.

Parameter | VOS | Altimetry | Buoys Data |
---|---|---|---|

Wave height | 0.5 m | 0.4 m | 0.2 m |

Wave period | 1 s | – | ≥1 s |

Wind speed | 1 m/s | 1.5 m/s | 1 m/s |

Direction | 10° | 17–20° | 10° |

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Saprykina, Y.; Kuznetsov, S. Analysis of the Variability of Wave Energy Due to Climate Changes on the Example of the Black Sea. *Energies* **2018**, *11*, 2020.
https://doi.org/10.3390/en11082020

**AMA Style**

Saprykina Y, Kuznetsov S. Analysis of the Variability of Wave Energy Due to Climate Changes on the Example of the Black Sea. *Energies*. 2018; 11(8):2020.
https://doi.org/10.3390/en11082020

**Chicago/Turabian Style**

Saprykina, Yana, and Sergey Kuznetsov. 2018. "Analysis of the Variability of Wave Energy Due to Climate Changes on the Example of the Black Sea" *Energies* 11, no. 8: 2020.
https://doi.org/10.3390/en11082020