On Correlation between Wind and Wave Storms

Abstract

The paper is focused on the formulation of an adequate criterion for associating wave storm events to the generating wind storm ones, and on the study of correlation between their characteristic parameters. In this context, the sea storm definition commonly used for storm identification from significant wave height data is adapted for wind storm, by processing wind speed data. A sensitivity analysis is proposed as function of the storm thresholds aiming at identifying optimal combination of wind speed and significant wave height thresholds allowing the association of relatively large number of events ensuring high correlation between wind and wave storm parameters. The analysis is carried out using as input data wind speeds and significant wave heights from four meteorological (buoys and anemometers) stations of the National Data Buoy Center moored off the East Coast of the United States. Results reveal that an optimal threshold combination is achieved assuming both wind speed and significant wave height threshold as 1.5 time their respective averages.

1. Introduction

The study of offshore climate plays a key role for any activity developed in offshore sea environment, including navigation, oil and gas, and offshore wind and wave farms. In this context the research community focuses on a variety of topics that ranges from the forecasting of environmental data [1,2,3,4,5] to the investigation of climate changes [6,7,8,9], energy resource assessment [10,11,12,13,14], simulation methods [15,16], and design and operation of offshore systems [17,18,19]. With regard to offshore structures and energy systems, the knowledge of climatic events occurring and their impact on any device is paramount both at design and at operational stages. Usually, one refers to environmental states within a time duration compatible with the stationarity assumption, described by two sea state parameters—which could be the significant wave height and the spectral peak period—and a wind state with given average wind speed and turbulence intensity. Among all the possible states, the most representative ones are selected from in-situ scatter diagram at site. However, it could be interesting to investigate the system response under nonstationary conditions, identified as sequences of environmental states exceeding critical thresholds and defined as storms. In order to do so, an in-depth study of such events and their characteristics is needed. The separation of wind events from long wind speed data sets is a crucial step for conducting both statistical and wind-excited structural response analyses. The approaches employed for this kind of extrapolation differ from each other if the phenomenon is investigated from a meteorological or engineering perspective. In the first case, wind events identification is performed by analyzing related meteorological parameters [20,21,22,23,24,25,26,27]. In the second approach, wind peaks are extrapolated from long wind speed time series with a systematic selection method without any consideration of climatic aspects, and are used for long-term statistical estimations and design value calculation; among these selection criteria are: Gumbel’s method [28] in which the largest values per fixed time period (e.g., one year) is selected; its extension with the selection of the r-largest value over a given time duration [29]; the method of independent storms [30,31] that uses a period of wind speeds below a selected (low) threshold to separate storms; and the Peak-over-threshold (POT) method that extracts maxima from sample data series to produce a series of extreme values above a chosen, adequately high, threshold [32,33].

In this paper, a criterion for associating a wave storm to its generating wind storm is formulated, starting with results achieved through wind and wave data analysis. The last is based on the storm concept intended as sequence of stationary wind or wave states exceeding pre-established critical thresholds. Specifically, it is aimed at identifying the wind and wave thresholds such that it is possible to associate a quite large number of events ensuring a high correlation between the storms’ parameters. Trends between the wind and wave storm parameters are investigated as well. The paper is organized as follows: section 2 describes the data used and the site, section 3 introduces the storms identification and association criterion, section 4 shows data analysis on the basis of the proposed criterion, and section 5 illustrates some concluding remarks. Results of this analysis are useful for survivability, reliability, and operability assessments of combined wind and wave systems, which take advantage of the complementarity of these resources. In this regard, the understanding of how nonstationary wind and wave events follow each other, their shift in terms of beginning time, time lag between peaks, and correlation over common time window where both persist over certain ranges of values, represent key elements in the development of combined technologies. Further, they can provide preliminary information on how to couple wind and wave states over longer time interval than those commonly considered in response assessments of offshore wind systems, for which wind and waves represent the main load factors.

2. Wind and Wave Data and Study Area

Data used as input of the analysis proposed in this work are those provided by the National Data Buoy Center (NDBC) (https://www.ndbc.noaa.gov/ (accessed on 8 December 2021)). The NDBC has deployed buoys around the USA coastline and Great Lakes since 1970s. From that date, the NDBC has deployed at least eight directional buoys and over a hundred wave measurement stations for operational and experimental use, covering areas from the western Atlantic to the Pacific Ocean around Hawaii, and from the Bering Sea to the South Pacific. Stations have been subject to modifications throughout the years, due to instrumental improvements and changing data storage systems as well. The NDBC time series contain wave data in terms of significant wave height, dominant (peak spectral) and average (mean) periods, mean wave direction at the peak frequency, spectral wave density data, the four Fourier directional parameters for spectral wave direction data, and meteorological data as wind direction, wind speed, wind gust, air pressure at sea level, air temperature, and sea surface temperature. In this work only significant wave height and wind speed data are considered. Regarding the wind speed data, it is worth to mention that anemometer heights on NDBC buoys vary according to buoy type. Anemometers on 3 m discus and 6 m Navy Oceanographic and Meteorological Automatic Device (NOMAD), buoys are located approximately 5 m above the waterline. Anemometers on 10 and 12 m buoys are located 10 m above the waterline. Anemometer heights at Coastal-Marine Automated Network (C-MAN) stations vary widely, depending on site structure and elevation above sea level. The NDBC adjusts wind speeds to conform to the universally accepted reference standard of 10 m. NDBC also adjusts wind speeds to 20 m, a height closer to that of typical ship anemometers. These standardized wind speeds are stored in a different file than the one which can be downloaded in the historical data page, where only unadjusted data are included. The latter are those used in this paper for the coupled analysis of wind and wave storm events. Four buoy stations were selected off the East Coast of the United States: 44004, 44008, 44014, and 41025. The time duration covered by each time series data and station characteristics are summarized in

Table 1. NDBC 44004, 44008, 44014, and 41025 buoy stations: time covering, geographic location, anemometer height, and water depth.

Buoy	Time	Latitude	Longitude	Anemometer Height [m]	Water Depth [m]
44004	28 March 2003–31 December 2020	38.484 N	70.433 W	5	3182.1
44008	18 August 1982–31 December 2020	40.498 N	69.251 W	4.1	68.9
44014	01 October 1990–31 December 2020	36.609 N	74.842 W	3.2	47
41025	28 March 2003–31 December 2020	35.010 N	75.454 W	3.8	48.8

.

This study area was selected because of the great interest faced by companies in proposing new offshore wind farm projects and it is already involved in pilot projects. Among them, the Coastal Virginia Offshore Wind (CVOW) located about 43 km (27 mi) off the coast of Virginia Beach, Virginia, US, whose initial phase, was a 12 MW pilot project constructed in 2020, consisting of two 6-megawatt offshore wind turbines [34,35]. Another example is Sunrise Wind [36,37], a 880 MW offshore wind farm to be developed in New York state, about 48km east of Montauk Point on Long Island.

The results of the analysis carried out in this work could be useful in the context of reliability and operability assessment of existing offshore wind farms, for designing and planning new wind farms, and for the integration of systems with wave energy converters and the design of combined technologies.

3. Methodology: Wind and Wave Storm Identification and Association Criteria

In this section the method used for identifying storm events and the association criteria of a wave event to its preceding wind one will be described. Note that in the context of this work the term “storm” refers to a nonstationary either wind or wave event, which is characterized by growth, peak, and decay stages (see Figure 1). In this regard, for what concerns the wind storm extrapolation, it is performed regardless to Beaufort’s scale [38,39,40]. Specifically, the sea storm definition adopted in long-term analysis via Storm Models [41,42,43] is adjusted for extrapolation of wind storms from wind speed data time series. Following the above-mentioned approach, a sea storm is a sequence of sea state during which the significant wave height H_s exceeds a given threshold named critical and does not fall below it for a time interval greater than twelve hours. This definition takes into account that a calm period may occur for a maximum duration of twelve hours and the storms characterized by this condition may have more than one peak. In this paper, the above definition is slightly reviewed removing the calm condition and considering only the part of the storm characterized by the sequence of sea states above the critical threshold h_crit. Thus, when H_s falls below the critical threshold it is considered the end of the storm. The adaptation of this definition to wind storm events simply consists in repeating the sea storm definition by replacing the significant wave height H_s and the critical threshold h_crit with the average wind speed u and critical wind speed u_crit, respectively. The critical thresholds of both average wind speed and significant wave height are fixed as equal or multiple times that of their averages (calculated from the whole time series) at site. The criterion adopted in this paper for storm identification and association is described in the flow chart represented in Figure 2. It involves as first steps, the calculation of average values of wind speed and significant wave height at site and the related critical thresholds u_crit, h_crit. Subsequently, the identification of sequences of wind storm events according to the definition above is performed. Then, for each wind event the related generated sea storm, if any, is extrapolated from H_s data. Specifically, considering the time interval (t_{in wind}, t_{fin wind}) (being t_{in wind}, t_{fin wind} the starting time and final time instants of the wind storm, respectively) covering the whole wind storm evolution (growth, peak and decay) if an excessed of at least one value of H_s over its critical threshold is detected it is assumed the wind storm generates a wave storm. Otherwise, no wave storm is generated and the considered wind storm is excluded from the correlation analysis. Furthermore, only events characterized by a duration of more that twelve hours are considered as relevant ones. Another important aspect is related to the time spacing between two successive storm events (either wind or wave). In this regard, it is assumed that only events with a minimum time separation of 48 h are relevant, in agreement with previous studies [31,44].

4. Data Analysis

This section proposes data analysis of wind and wave storms based on the criterion presented in the previous section. Specifically, wind and wave storms are extrapolated from time series of average wind speed and significant wave height, respectively, according to the proposed criterion. Then, for each sample wind and wave storm, correlation between relevant storm parameters is investigated. This analysis is essentially divided in two main parts: the first part deals with the proposition of a sensitivity analysis to the couple of thresholds (h_crit, u_crit) aiming to identify their most adequate combination; the second shows the correlated storm parameters trends when the optimal couple (h_crit, u_crit) is considered in storms identification and association.

4.1. Variability of Storm Number and Correlation Parameters Assuming Different Thresholds

In this section, the sensitivity of wind and wave events to their critical thresholds (h_crit, u_crit) is investigated for our storm sample. In this regard, both critical wind speed and significant wave height thresholds are correlated to the average value of related variables (see

Table 2. NDBC 44004, 44008, 44014, and 41025 average wind speed and significant wave height.

Buoy	Hs [m]	U [m/s]
44004	2.05	7.33
44008	1.71	6.39
44014	1.44	6.16
41025	1.51	7.35

). In particular, the following ratios between critical thresholds and average values of both wind speed and significant wave height are considered: 1, 1.5, 2, 2.5. The effects of all the possible combinations of these thresholds are examined. For each couple of thresholds (h_crit, u_crit) the analysis proceeds as follows:

• Wind storms are identified as a “sequence of wind states with average wind speed exceeding the critical threshold u_crit”. Those having a too short duration (less than 12 h) and of those not satisfying the condition of established minimum time separation (48 h intended as time distance between peaks) are removed. Note that, when two wind storm peaks have a time separation less than 48 h, the one characterized by the highest value of maximum wind speed is kept in the sample and the other one is eliminated.
• The storm sample prepared following the procedure described above is divided in two subsamples: one including the wind storms during whose time window none of the significant wave height values exceed the critical threshold h_crit, and the other constituted by all wind storms during which at least one significant wave height value is over h_crit. The last subsample is assumed to be made up of those wind storms which generate a wave storm.
• For each wind storm in the second subsample the corresponding complete storm event is extrapolated from the significant wave height time series. Specifically, starting from the time instant at which the maximum H_s value is detected in the wind storm time window, H_s data are processed backward and forward the preceding and the following time instants at which H_s is below h_crit. Note that, the last two time instants define the time window over which the wave storm event evolves and the related duration.
• Finally, some parameters are calculated to characterize the events. They are: maximum wind speed u_max in the wind storm, wind storm duration D_wind, maximum significant wave height H_smax in the wave storm, wave storm duration D_wave, time distance Δt u_max − H_{s max} between u_max, and H_{s max}. Note that, Δt u_max − H_{s max} is calculated as t (H_smax) − t(u_max), thus if H_{s max} occurs after u_max it is greater than 0, negative if u_max occurs after H_{s max}, and equal to 0 if they occur simultaneously.

At this stage of the analysis, for each couple (h_crit, u_crit) the following quantities are investigated:

• Total number of wind storms N_wind including both generating and non-generating wave storms (Figure 3, Figure 4, Figure 5 and Figure 6a);
• Number of wind storms generating a wave storm (Figure 3, Figure 4, Figure 5 and Figure 6a);
• Correlation between u_max and H_smax for those wind storms generating a wave storm (Figure 3, Figure 4, Figure 5 and Figure 6b);
• Correlation between D_wind and D_wave for those wind storms generating a wave storm (Figure 3, Figure 4, Figure 5 and Figure 6c);
• Number of events in which H_{s max} occurs before u_max (Δt u_max − H_{s max} < 0) (Figure 3, Figure 4, Figure 5 and Figure 6d).
• Number of events characterized by time distance Δt u_max − H_{s max} greater than one day in absolute value (Figure 3, Figure 4, Figure 5 and Figure 6e).

A comparison among these Figures shows that, for the investigations at points 1, 4, and 5 general considerations can be developed, while for points 2 and 3 the trend is different from one location to another. In general, from Figure 3, Figure 4, Figure 5 and Figure 6a it is seen that the events number decreases as the threshold increases, as expected. For a fixed ratio $u_{crit}/\overline{u}$ < 2, the number of wind events generating a wave storm increases as the as the ratio $h_{crit}/{\overline{H}}_s$ decreases, while for both ratios $u_{crit}/\overline{u}$ , $h_{crit}/{\overline{H}}_s$  $\ge 2$ it is possible to associate a wave storm to each identified wind storm. Certainly, by fixing high ratios $u_{crit}/\overline{u}$,$h_{crit}/{\overline{H}}_s$ the storm sample is small and it does not include the less intense and moderate wind and wave storms, but only extreme events. For a more adequate and numerous sample, one should select lower ratios of both $u_{crit}/\overline{u}$ and $h_{crit}/{\overline{H}}_s$.An appropriate thresholds combination could be created by fixing the wind ratio $u_{crit}/\overline{u}$ equal to 1.5 and the ratio of the significant wave height $h_{crit}/{\overline{H}}_s$ in the range (1–2), or alternatively, by taking both $u_{crit}/\overline{u}$ and $h_{crit}/{\overline{H}}_s$ equal to 1 or 1.5. The assumption of the last option above leads to a good correlation of maximum wind speed u_max and significant wave height H_smax, which in such situation is always between 0.6 and 0.7 (see Figure 3, Figure 4, Figure 5 and Figure 6b). In this regard, it is important to say that the correlation of H_smax and u_max does not exhibit a regular trend and varies from one location to another. However, for wind ratios $u_{crit}/\overline{u}$ < 2 it keeps itself always above 0.5 (see Figure 3, Figure 4, Figure 5 and Figure 6b). The correlation between wind and wave storm duration is completely irregular and does not show any particular trend: sometimes it is very high and sometimes it can be very low. Then, focusing on the time distance Δt u_max −H_{s max} between u_max and H_smax, from Figure 3, Figure 4, Figure 5 and Figure 6d it could be noticed that the peak of wind speed could proceed or follow that of significant wave height. Another insight into Figure 3, Figure 4, Figure 5 and Figure 6d is that the number of events characterized by Δt u_max − H_{s max} (H_s peak occurring before u peak) decreases with increasing $u_{crit}/\overline{u}$ and $h_{crit}/{\overline{H}}_s$, and for $u_{crit}/\overline{u}$ $\ge 2$ does not depend upon the significant wave height ratio $h_{crit}/{\overline{H}}_s$. Lastly, observing Figure 3, Figure 4, Figure 5 and Figure 6e it is seen that Δt u_max − H_{s max} can be greater than one day and the number of events characterized by such distance between peaks decreases when both $u_{crit}/\overline{u}$ and $h_{crit}/{\overline{H}}_s$ increase up to 2, while for higher values it tends to zero. This result is quite intuitive because with such high ratios the sample includes only extreme events in which it is expected wind speed and significant wave height peaks to be very close to each other (see Figure 1). In light of the results analyzed in this section, a suggestable combination of wind and wave ratios $u_{crit}/\overline{u}$ and $h_{crit}/{\overline{H}}_s$ could be achieved by assuming both of them to be equal to 1.5. In this regard, it is worth mentioning that the value of 1.5 is the one generally used for sea storm identification. Furthermore, any combination of $u_{crit}/\overline{u}$ and $h_{crit}/{\overline{H}}_s$ in the range (1–1.5) is adequate for reliable analysis.

4.2. Trend and Correlation between Wind and Wave Storm Parameters Associated with Selected Thresholds

Based on the analysis proposed in previous section, the trends of relevant storm parameters are investigated, taking as wind and wave ratios $u_{crit}/\overline{u}$ and $h_{crit}/{\overline{H}}_s$ with a value of 1.5. Starting from our wind and wave storm sample, for each event, the following parameters are determined:

• Maximum wind speed u_max;
• Wind storm duration D_wind;
• Maximum significant wave height H_smax;
• Wave storm duration D_wave;
• Time distance Δt_{u max − Hs max} between wind storm and wave storm peaks (H_smax, u_max);
• Standard deviation of wind speed σ_wind;
• Standard deviation of significant wave height σ_wave;
• Correlation coefficient ρ(u(t), H_s(t)) between time histories of wind speed and significant wave height over a time window including the whole evolution of both wind and wave storms. In this regard it is important to specify that this time window could include a time interval in which either the wind speeds or the significant wave heights are below their critical thresholds.

The estimations listed above are used to draw graphics to elucidate some interesting trends. Specifically, Figure 7, Figure 8, Figure 9 and Figure 10a–d, show wind storm duration D_wind versus maximum wind speed u_max, wave storm duration D_wave versus maximum significant wave height H_smax, wind speed standard deviation σ_wind versus maximum wind speed u_max, and significant wave height standard deviation σ_wave versus maximum significant wave height H_smax. Data summarized in these figures essentially confirm some well known trends. For instance, both wind and wave storm durations increase with the storm intensity and standard deviations of both wind speed and significant wave height increase as well. Figure 7, Figure 8, Figure 9 and Figure 10e–i show the relationship between wind and wave storm parameters. In detail, in Figure 7, Figure 8, Figure 9 and Figure 10e is represented H_smax versus u_max, in Figure 7, Figure 8, Figure 9 and Figure 10f D_wave versus D_wind, in Figure 7, Figure 8, Figure 9 and Figure 10g σ_wave versus σ_wind, in Figure 7, Figure 8, Figure 9 and Figure 10h Δt u_max − H_{s max} versus u_max, and in Figure 7, Figure 8, Figure 9 and Figure 10i ρ(u(t), H_s(t)) versus u_max. Looking at this last group of figures, some general considerations can be developed:

• The wave storm peak H_smax increases with the wind storm peak u_max;
• The wave storm duration D_wave increases with the wind storm duration D_wind;
• Considering a given storm event the significant wave height standard deviation σ_wave is always less than the wind speed σ_wind standard deviation. This result is interesting in the context of offshore renewables energies because it indicates which energy source is more regular and justifies the interest in combined systems to the aim of smoothing the power output.
• The time distance Δt u_max − H_{s max} may be greater than one day, in most of the storms shorter than 10 h in absolute value;
• The correlation ρ(u(t), H_s(t)) increases with the wind storm intensity u_max.

5. Conclusions

In this paper, a criterion is proposed to identify and associate wind and wave storm events, based on a sensitivity analysis as a function of wind and wave critical threshold. An optimal combination of such thresholds is suggested as a result of the proposed analysis. It consists of assuming both wind and wave critical thresholds as 1.5 times the average of wind speed and significant wave height, respectively. Thanks to this assumption, it is possible to associate a large number of correlated wind and wave storm events. Furthermore, the relationship between wind and wave storm parameters are investigated with these thresholds, elucidating that, as the wind speed increases the significant wave height increases as well. However, the peak of the significant wave height does not always occur after that of the wind speed. In general, the structure of a wind storm is very similar to that of a wave storm. In addition, the wind speed is characterized by a higher variability with respect to the significant wave height. All the results mentioned above could be useful both for sample extrapolation in the context of a statistical analysis, offshore energy resource assessment, and wind and wave condition combination for response analysis of offshore structures.