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| Fig. 1: Diagram showing the physical attributes of a water wave. (Source: Wikimedia Commons) |
California is a national and global leader in harnessing renewable energy resources. With its extensive coastline, taking advantage of the ocean as a source of renewable energy is a attractive possibility. Indeed, many federal and industry energy research entities have put much effort into assessing the feasibility of extracting energy from the ocean. While much of this research has been grounded in physical reality, attempts to drum up excitement in the technology have seeded many extraordinary claims that the ocean is host to infinite or unlimited energy up for the taking. Unfortunately, these claims are not grounded in reality.
To estimate the average power of waves incident on the California coastline, we can approximate all waves as simple, uniform waves (technically called plane progressive waves). Such waves have several defining features, including a constant wavelength, period, height, and direction of motion, as shown in Fig. 1. There are two types of energy stored in a wave: kinetic and potential, or the energy stored due to the wave's bulk motion and height, respectively. The total average energy density stored for a unit patch of ocean surface (e.g. 1 m2) is
| E | = | 1 2 |
ρgh2 | , |
where ρ = 1000 kg m-3 is the density of water, g = 9.8 m sec-2 is the acceleration due to gravity, and A is the amplitude of the wave, or half the total height from trough to crest (see Fig. 1). [1] However, wave height is most commonly reported using the significant wave height, Hs, which is about 16/5A. This gives an average energy density per unit area of ocean surface of
| E | ≈ | 25 512 |
ρgHs2 | . |
Instead of using energy, it is often most useful to compare energy resources using their power output, or energy per unit time (often reported in Watts). To calculate the power density of ocean waves per unit length of coastline (e.g. 1 m), we can multiply the energy density, E, with the wave's bulk (or "group") velocity, v. In general, a wave's speed depends on both its wavelength, λ, and period, T. However, in the case of plane water waves, the wavelength and period are not independent. The wavelength can be written in terms of the period as
| λ | = | gT 2 2π |
. |
For plane water waves, the bulk velocity is half of the "phase" velocity, vp=λ/T. Thus, the bulk velocity is
| v | = | λ 2T |
= | gT 4π |
. |
Putting all of these equations together we find the wave power density per unit length of coastline is
| P | = | Ev | ≈ | 25 2048π |
ρTg2Hs2. |
Studies of historical wave measurements along the U.S. West Coast measure an average value for Hs as 2.1 m and 1.3 m for Northern and Southern California, respectively. [2] The average period of waves is about 10 seconds. Substituting these values, we find the average wave power density per meter of coastline is about 17 kW/m and 6 kW/m in Northern and Southern California, respectively. Excluding inlets, tidal areas, and other coastal areas where waves are significantly disturbed, California has about 1,350 km of coastline. [3] Dividing the coastline evenly between the North and South and multiplying these two values with the power, we find if we were able to harness the waves over the entire California coast, they would provide at most about 15 GW of power. As our model is quite simple, it does not include more detailed effects such as mass transport and turbulence, but it is in good agreement with more detailed studies, which report power availability closer to 20 GW. [4] Using the more optimistic value of 20 GW and assuming this power output is relatively constant throughout the day/night cycle and is averaged over the seasons — off the West Coast, wave energy is generally higher in the summer — then the total energy available in a year is about 180 TWh. To compare, for the year 2019, California's energy usage was 2282 TWh. [5] If we assume we could potentially harness 10% of wave energy resources, which would be an incredibly ambitious undertaking, at best wave energy could only support about 0.7% of California's energy needs.
To place this result into context, we can compare the power output of ocean waves to that of sunshine over the entire area of the Mojave Desert. Both sources of energy are large-scale, non-residential (unlike, e.g. residential rooftop solar panels), and would need significant organized infrastructure deployed as a public utilities project or privatized enterprise. Thus, these two energy sources, while quite different in nature, are of comparable scales.
The sun provides about 340 W of power per meter squared at the top of the Earths atmosphere. [6] The atmosphere absorbs and reflects about 50% of the sunlight's energy reaching the Earth. This includes factors such as average cloud coverage, thus using this percentage is rather conservative for the Mojave Desert, where cloud coverage is minimal. The Mojave Desert is about 100,000 km2. [7] Multiplying these numbers, we find the theoretical total power from sunlight over the entire Mojave Desert is about 17 TW. Of course, the sun only shines for a portion of the day; using 5 hours a day when the sun provides the most energy, this gives us just over 30,000 TWh per year. This is about 170 times more power than theoretically available in ocean waves off the California coast. Harnessing 7% of this solar energy could supply the entirety of California's energy needs, assuming complete conversion of fossil fuel usage, such as natural gas heating and gasoline-powered vehicles, to electricity.
As waves provide only a tiny fraction of what is available through solar power, in order to be a cost effective energy source, energy extraction from the ocean and delivery to end users must be much cheaper than solar. However, this is not the case. An early foray in testing wave technology in 2008-2009 by Pacific Gas and Electric Company (PG&E), the WaveConnect project, faced considerable difficulties, finding ocean wave energy conversion technology would cost about 50-100% more than wind or solar photovoltaic resources in USD/kWh. [8] Thus, significantly more research and development is necessary to make wave energy conversion technology efficient enough to be competitive with conventional wind and solar.
Of course, there are many other variables to consider when evaluating the potential of ocean wave energy. A key point is how close a location is to the ocean. It may be more cost effective and reliable for a Northern California coastal community with modest power usage to make use of the nearby ocean resource than to pay for electricity from a faraway solar farm. Further, implementing diverse sources of energy can enhance the resilience of the electrical grid and prevent mass power shortages or blackouts if a certain resource is adversely affected. On the other hand, as with all renewable energy resources including conventional wind and solar photovoltaic systems, the environmental impact of ocean energy infrastructure is a significant concern. Implementing wave turbines, anchored buoys, and seabed cables would disturb the ocean floor, a potentially sensitive ecosystem. Indeed, one of the leading causes for PG&E to abandon the WaveConnect project beyond cost was difficulty in assessing environmental impact. [8]
Ocean wave energy can only provide for a small portion of California's power consumption. The technology may eventually be useful for coastal communities, particularly remote areas where wave energy would be the leading local renewable energy resource. Currently, wave energy conversion is infeasible, even for relatively small scale facilities. However, more research and development is underway. In early 2022, the U.S. Department of Energy's Water Power Technologies Office awarded eight projects a total of $25 million to support further research, development, and demonstration of their ocean energy technologies at the open-water wave energy test facility, PacWave South, seven miles off the Oregon Coast. [9] Only time will tell whether development of these and related technologies will prove fruitful in the effort to make California's energy sector carbon-free.
© Theo Schutt. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] J. N. Newman, Marine Hydrodynamics (MIT Press, 2018), pp. 271-277.
[2] Z. Wang et al., "Characteristics and Variability of the Nearshore Wave Resource on the U.S. West Coast," Energy 203, 117818 (2020).
[3] J. C. Beaver, "U.S. International Borders: Brief Facts," Congressional Research Service, RS212719, 9 Nov 2006.
[4] L. Kilcher, M. Fogarty, and M. Lawson, "Marine Energy in the United States: An Overview of Opportunities," U.S. National Renewable Energy Laboratory, "NREL/TP-5700-78773, February 2021.
[5] "State Energy Consumption Estimates: 1960 Through 2020," U.S. Energy Information Administration, DOE/EIA-0214(2020), June 2022, pp. 80-81.
[6] J. T. Kiehl and K. E. Trenberth, "Earth's Annual Global Mean Energy Budget," Bull. Am. Meteorol. Soc. 78, 197 (1997).
[7] L. R. Walker and F. H. Landau, A Natural History of the Mojave Desert (University of Arizona Press, 2018), p. 44.
[8] B. P. Dooher et al., "PG&E WaveConnect Program: Final Report," Pacific Gas and Electric Company, December 2011.
[9] S. Patel, "DOE Picks First Marine Energy Projects for PacWave Test Site in Oregon," Power Magazine, 27 Jan 22.
