The European Center for Medium-Range Weather Forecasts (ECMWF) has a long history of reanalysis in climate monitoring applications. The state-of-the-art product of ERA5 was released in April 2019, replacing the widely used ERA-Interim reanalysis. ERA5 is the fifth generation reanalysis product from the ECMWF for global climate and weather over the past 4 to 7 decades. Besides, ERA5 is highly regarded in the Copernicus Climate Change Service (C3S), which provides a precise and consistent record for a large number of basic climate variations for the C3S Climate Data Store (CDS). See the detail descriptions in Hersbach et al. (2020).

In this study, the hourly and monthly ocean-wave products are used from 1979 to 2020 from CDS, interpolated to a regular grid with a 0.5° × 0.5° spatial resolution. The key parameters include the significant wave height, mean wave period, and mean wave direction, as derived from the wave spectrum. These parameters were classified into three types: wind-sea wave components, swell components, and ensemble waves of both.

Wave power P_w (W/m) is defined as the wave energy flux per unit of wave-crest length (Dean and Dalrymple, 1991) and wave energy transport at wave group velocity (c_g ). Thus, wave power in the wave propagation direction can be written as P w = ρ g ∫ 0 ∞ ∫ 0 2 π S ( f , θ ) c g ( f , d ) d f d θ , where S(f,θ) is the directional spectrum corresponding to the wave frequency (f) and direction (θ). It is simplified as P _w =E _w ·c _g , with E w = 1 8 ρ g H s 2 (J/m²) being the averaged wave energy density per unit horizontal area (including the kinetic and potential energy), where H_s is the significant wave height; g=9.80 (m/s²) is the acceleration of gravity; and the ρ=1025 (kg/m³) is the averaged density of the seawater. Furthermore, the wave power (P_w ) can be determined from T_e and H_s in deep waters as follows:

where T_e is the wave energy period. The determination of this parameter was controversial in previous studies. Reguero et al. (2015); Reguero et al. (2019) assumed that T _e =α·T ₀₁=0.538T ₀₁ (s) for the JONSWAP spectrum. While Fairley et al. (2017) and Rusu and Rusu (2021) used the mean wave periods for wave power assessment. As introduced in ECMWF documentation, the mean period (T _m−1) is also known as the energy period (ECMWF, 2020) and can be used in Eq.(1) directly.

Due to the “shoaling” effects, as waves move from the open ocean into shallow water, their crests become steeper, increase in height, and shorten in wavelength. The difference in averaged wave power between deep and shallow water is caused by their group velocities (Izadparast and Niedzwecki, 2011). According to the dispersion relationship in shallow water (ω ²=gk ² d , where ω is the angular frequency; k is wave number; and the d is the water depth), the group and phase velocities are both determined solely from water depth ( c g = g d ), and wave power in shallow waters can be expressed as:

Thus, wave power varies with the square root of water depth and is independent of wave period. Generally, average wavelengths near shore are less than 100 m, and Eq. (2) should be applied when water depths are less than 5 m based on the shallow-water limit ( d < 1 20 λ ) of linear wave theory. Since the grid resolution of ERA5 data is 0.5° (about 55 km in ground distance) and a minimum water depth is 5 m, the impacts of shallow water conditions on wave power estimation are trivial and can be neglected in this study. Additionally, a parametrization scheme of subgrid bathymetry was implemented in ERA5 data to correct the wave propagation and wave energy flux (ECMWF, 2020). Ozkan and Mayo (2019) indicated that the simplified equation potentially underestimates available wave power in coastal Florida compared with the spectral wave power equation. However, only three wave parameters mentioned above were selected in the ERA5 hourly product, but bring a huge amount of data. The computational burdens limit the application of spectral methods in wave power assessment. Altimeter measurements can only provide H_s in the assimilation process of wave modeling (Hersbach et al., 2020), potentially supporting more reliable wave height products. Therefore, Eq. (1) was used to estimate the wave power in this study.

Furthermore, the mean wave direction (oceanographical convention) for wave power analysis is introduced. The mean wave direction was decomposed into zonal and meridional components based on significant wave height (θ _Hs =[H _s ·sinθ ₀,H _s ·cosθ ₀] ) and wave power (θ _WP =[P _w ·sinθ ₀,P _w ·cosθ ₀] ). Then the climate direction of H_s and P_w could be estimated.

Various factors that affect wave power can be used, for example, since surface waves are extensions of past wind forcing, predicted changes in wind patterns can inform future predictions of surface waves. Particularly, the interannual or decadal variations in GWP reflect the long-term climate variability (e.g., Vieira et al., 2020; Reguero et al., 2019), which can be spontaneously connected with the climate teleconnection patterns.

According to geostrophic relationships, pressure gradients determine wind fields (Pedlosky, 1987). Therefore, the climate indexes related to pressure (or wind) are good candidates for analyzing potential relationships with wave power. Chen (2014) has made systematic works on characterizing the spatiotemporal patterns of significant atmospheric oscillations over global oceans. Thus, the following indexes were selected for analysis: The Pacific Decadal Oscillation (PDO; index for Pacific climate variability based on SST anomalies with time scales usually greater than 10 years), the North Atlantic Oscillation (NAO; index based on surface sea-level pressure difference between the subtropical (Azores) high and the subpolar low pressure), the Southern Oscillation (SOI; a bimodal variation index based on sea-level barometric pressure differences between observation stations at Darwin, Australia and Tahiti), the Arctic Oscillation (AO; a climate pattern index characterized by winds circulating counterclockwise around the Arctic at a latitude around 55°N, with a positive phase when colder air masses are confined in polar regions, and a negative phase when southward penetration occurs), and the Southern Annual Mode (SAM; i.e., the Antarctic Oscillation; dependent on atmospheric pressures at the Antarctic and at about 40°S-50°S). Each of these indexes is related to the variability in atmospheric circulation and, therefore, is linked to surface waves via wind forcing. The SAM index was acquired from National Center for Atmospheric Research (NCAR; Marshall, 2003), while all other climate index data were downloaded from the National Oceanic and Atmospheric Administration (NOAA).

In this section the wave power from total wave, swell, and wind-sea are presented and computed at the global level based on monthly ERA5 data. Figure 1 shows the averaged H_s , T_e , and P_w fields, calculated from the total wave fields over the 42-year time interval (1979~2020). The H_s and P_w fields exhibited similar patterns, with higher values concentrated in the latitude bands from 40° to 60° (interior of the prevailing westerlies zones) in both hemispheres, and smaller values mainly appearing in tropical and nearshore areas. The distributions and magnitudes were similar to those obtained by Rusu and Rusu (2021). T_e exhibited a pattern that was different from H_s and P_w , with values in the Southern Hemisphere being significantly larger than those in the Northern Hemisphere, and the Pacific and Indian Oceans having larger values than the Atlantic Ocean. The eastern portions of ocean basins generally had larger values than their respective western portions, and some jet-shaped patterns can be seen in Figure 1B , with values exceeding 11 s. Secondly, the maximum value of P_w beyond 100 kW/m was located in the Southern Ocean, and P_w transport occurred in three main directions, as indicated by the arrows in Figure 1C . The pattern in wave power showed deflection southward (northward) upon approaching the Antarctic continent (the lower latitude ocean), and along the path eastward there was a sharp decrease to the east of the Drake Passage (See the schematic arrows in the figure). The tracks leading to the Atlantic Ocean exhibited a lower P_w field than the Indian and Pacific Oceans in the southern hemisphere. As a result, the strongest wave power field (approximately 60 kW/m) of the Atlantic Ocean was in the westerly zone of the Northern Hemisphere. Additionally, there were dominant trends showing ocean waves traveling and transporting energy from high latitude to equatorial regions and towards shores. Meanwhile, P_w values were reduced sharply during the propagating processes, and were only one-third of their original magnitudes after leaving the westerlies (see Figure 1C ).

Studies on global wave power (GWP) have rarely compared swell and wind-sea waves. However, wave parameters integrated from the entire wave spectrum might only provide a limited description of the wavefield (Qian et al., 2019). The swell fields ( Figures 2A–C ) exhibited the same patterns as the total wave fields ( Figures 1A–C ), but had slightly larger wave periods and lower significant wave heights (H_s ) and wave power (P_w ). Wind-sea wave fields exhibit the weakest magnitude of wave parameters (see Figures 2D–F ). There was a region with remarkably low P_w values along the equator in the wind-sea field (the white belt zone in Figure 2F ), which corresponded to a calm belt near the equator. Furthermore, the transport directions of P_w in the wind-sea field were different from those in the swell field. In particular, all the H_s and P_w in the Southern Ocean moved poleward across the westerlies and approached Antarctica. By comparing the averaged fields of the total waves, swell, and wind-sea, it was found that significant wave energy potential existed in the two zones with prevailing westerlies, and the swell components contributed most of the total wave power.

The period of surface waves was generally less than twenty seconds. However, they were forced by the seasonal or long-term winds (or wind stress curl) and, as such, carry the atmosphere’s imprint. Analyzing temporal variability is helpful for evaluating P_w and accurately predicting its trends. Thus, the time-series of averaged global wave power (GWP) were calculated at yearly and monthly resolutions using the ERA5 monthly products and at daily and hourly resolutions using the ERA5 hourly products. Note that valid data are used for GWP statistics (e.g., some locations at certain times of the year may experience ice coverage, and only grid with ice-free periods longer than half a year are valid). The results over the 42-year period are shown in Figure 3 , and the GWPs of the total wave, swell, and wind-sea are marked by the black, red, and blue lines, respectively.

These results also support that swell components dominate the total GWP, including the trends in magnitudes and variation. One unanticipated finding was that annual GWP ( Figure 3A ) did not exhibit a steady increase, instead there was a sharp increase after 1991 accompanied by a (quasi) decadal fluctuation. As suggested by Hersbach et al. (2020), the altimeter assimilation of wave information began in 1991, and validation of matched buoy results has also shown much smaller errors (scatter index less than 16%) in the ERA5 products since then. Similar “jump” phenomenon can be found in the results of Reguero et al. (2019); and Muhammed Naseef and Sanil Kumar (2019); which may indicate the intrinsic dependence of contemporary wave models on altimeter assimilation, and potential underestimate wave power or wave fields prior to the appearance of altimeter observations. Validation results with in-situ observations (e.g., Muhammed Naseef and Sanil Kumar, 2019; Wang and Wang, 2021) give us more confidence in the wave model products in the altimeter era. The grey error bars represent the standard deviations, and the longer-term averaged GWPs tended to have lower standard deviations. Interestingly, the variabilities in the total GWP at the inter-annual ( Figure 3A ) and seasonal ( Figure 3B ) scales were remarkable, while there was barely any variability at daily and hourly scales ( Figures 3C, D ). Moreover, the values of the averaged GWP from the monthly product were slightly smaller than from the hourly product. The former fluctuated over the range from 4 to 5 GW/m, while the latter was steady at 5 GW/m. This may have been because the short-term variability in winds resulted in short-term high P_w values in the hourly product, while the process of merging monthly products smoothed out these short-term anomalies.

By focusing on the fluctuating trends in total GWP ( Figure 3A ), the decadal oscillation phases in the series appear to be separated by years 1991, 2003, and 2014 (marked by the green dotted lines in Figure 3A ). Because long-term variability was our primary concern, the distribution of averaged P_w was further compared within the four phases to understand the cause of GWP variation, and their time-averaged fields were shown in Figures 4A–D . There were significant differences in the intensities of P_w among the Southern Ocean (SO), Northern Pacific, and Atlantic oceans (NP and NA), but the mean directions of wave powers were generally similar. Furthermore, Figure 4E shows the positive discrepancy between the averages of years 1992-2003 ( Figure 4B ) and 1979-1991 ( Figure 4A ), with a maximum difference that was more than 10 (5) kW/m in the SO (NP and NA). In contrast, a negative discrepancy field ( Figure 4F ) was obtained when Figure 4C was subtracted from Figure 4B . The wave power reduction in Figure 4F was significantly lower than the increases in wave power in Figure 4E . In short, there was a net increment of GWP during 1979 ~ 2013. Notably, the reduction in wave power in the SO was smaller than those in the Northern Hemisphere basins, which may imply that SO had a larger role in the overall enhancement of total GWP (in Figure 3A ) after 1991. Meanwhile, another major positive difference field was produced (not shown) by comparing Figures 4D, C . Furthermore, when looking differences between the four phases, the averaged rates of change in GWP were 12.89%, -3.97%, and 4.05%, respectively. Therefore, the sudden increases, general fluctuations, and decadal oscillations in GWP were quantified in our results. Particularly, accurate simulation of wave fields in the Southern Ocean is the core to improving wave models in the future.

Additionally, Figure 3B shows evidence of seasonal variation. The maximum GWP values were near 4.7 GW/m in March and July (which likely corresponded to times of high intensity wave action in both hemispheres), while the minimum value was slightly less than 4.1 GW/m in November. Moreover, a slight GWP peak in the wind-sea appeared from June to October. Therefore, the seasonal wave power fields (seasons here correspond to seasons in the Northern Hemisphere) are calculated, as shown in Figure 5 . The strongest wave power field occurred in winter (DJF), with extreme values of more than 100 kW/m in the NP and NA ( Figure 5A ), while this season (i.e., summer in the Southern Hemisphere) had the weakest wave power field in the SO with extreme values remaining below 60 kW/m. In contrast, the weakest wave power field in the Northern Hemisphere (almost below 10 kW/m) occurred in summer ( Figure 5C ). Notably, the Northern Indian Ocean had the highest P_w field (exceeding 50 kW/m) in the year due to the incoming wave power from the Southern Ocean Swell. The P_w distributions exhibited similar patterns in spring and autumn (i.e., two transition seasons between winter and summer), and there were only slight differences in P_w between the two westerlies.

The stability of wave resources is an essential factor for reliable energy harvesting. Thus, the coefficient of variation (standard deviation divided by the mean, C V = σ x x ¯ × 100 ) is calculated within the four seasonal fields. The lowest variabilities in P_w were observed in winter ( Figure 6A ) and summer ( Figure 6C ), with CVs lower than 25%, while the highest CV values were mainly concentrated in nearshore areas. However, there was significant variability in P_w in spring ( Figure 6B ) and autumn ( Figure 6D ). Specifically, their CVs were higher than 35% in the Northern Hemisphere, while a small variability (lower than 10%) was exhibited in the tropical oceans but the magnitudes of P_w in these oceans were also lower than 10 kW/m (see Figures 5B, D ). These results show that the Southern Hemisphere contains more stable wave resources which make it an ideal energy harvesting field.

The hourly variation in GWP was rarely trivial, as suggested by Figure 3D . However, it is well understood that the air-temperature differences between day and night can significantly change the wind field which will influence ocean waves. Therefore, to examine the difference between day and night periods, the distributions of wave power during the day (i.e., the local time between 6 am to 6 pm) and night were calculated separately. Figures 7A, B show the day and night wave power distributions. As expected, their patterns were very similar (i.e., the energetic wave power was mainly concentrated in the westerly zones). However, the difference between the day and night distribution produced a novel result, revealing some large-scale “bubble” or wave-like patterns, as shown in Figure 7C , and significant differences were mainly concentrated in higher latitudes. Since this result was obtained by averaging hourly climate data, these spatial scales were too large to be attributed to local noise. Their morphological features were intuitively reminiscent of the widespread eddies or Rossby waves which form in the ocean and atmosphere. These striped patterns also revealed a significant trend of westward intensification or propagation from latitudes ~ ± 40° equatorward, with more divergence patterns in higher latitudes. These patterns may be related to wind stress, and exhibited similar patterns in previous studies (e.g., Chelton et al., 2004). Due to the lack of further evidence and because its dynamic mechanisms were outside of our scope of this study, we supposed this may be a mirror effect imposed on the ocean by atmospheric forcing in the sea-air coupled model of ERA5 products.

The ultimate goal of studying the spatiotemporal distribution and variability of GWP is to scientifically identify the best strategy for the development and harvesting of ocean wave resources. Therefore, the probability distribution of P_w per 0.5° cell over the 42-year data series with a high temporal resolution is calculated in Figure 8 . The probability is also considered as the evaluation of the potential working time of the wave energy converters during which a specific P_w threshold would be met. The low-efficiency region where the probability was less than 0.3 (i.e., the effective working time would be less than 2600 hours in a year) is defined. In contrast, the high-efficiency regions had probabilities greater than 0.8 (i.e., the effective working time is higher than 7,000 hours in a year). Moreover, to identify optimal locations for power generation, water depth must be considered. In nearshore areas, wave energy is dissipated as waves interact with the seabed. However, deploying wave power plants and connecting it to a shore-based power station is often impractical and uneconomical in deeper water conditions. Generally, optimal depths are between 40 and 1000 m, which had large wave periods and amplitudes (Scruggs and Jacob, 2009). Therefore, the 1000 m water depth is marked on the maps with red contours.

As is shown in Figure 8A , the high-efficiency regions covered almost the entire the ocean basins when the criterion of P_w ≥ 5 kW/m was used, except in nearshore areas and oceanic western boundary zones. In particular, China’s most promising wave resources were apparent in the South China Sea where the probability was near 0.55, indicating that the working time would be approximately 4800 hours/year. Furthermore, the highly-effective regions were significantly reduced in the Northern Hemisphere when a threshold of 10 kW/m was used ( Figure 8B ), while there were only a few differences in the Southern Hemisphere. Notably, only the westerlies of the Southern Hemisphere were identified as high-efficiency regions when the criterion was increased to 20 kW/m ( Figure 8C ). The nearshore areas close to the Southern Ocean are geographically predisposed to obtaining more wave energy. According to the optimized criteria for water depth, effective working times, and wave power, regions with high potential for wave energy harvesting are identified (white rectangles in Figure 8C ). These included the southern coast of Africa, the western and southern coasts of Australia, the nearshore area of New Zealand, and the southern coast of South America. These regions were all close to the westerlies of the Southern Ocean, benefitting from their proximity to the significant wind fields therein ( Figure 8D ). Therefore, these regions are optimal for the efficient harvest of wave and wind energies simultaneously.