Wave Energy Coastal Waves Primer
Wave Energy Coastal Waves Primer
(R. Budd)
NIWA Internal Project Report: October 2004 NIWA Project: IRL05301 Wave Energy Coastal Waves Primer Murray Smith Craig Stevens Richard Gorman
Prepared for
FRST Wave Energy Project Group
NIWA Internal Report: October 2004 NIWA Project: IRL05301 National Institute of Water & Atmospheric Research Ltd 301 Evans Bay Parade, Greta Point, Wellington Private Bag 14901, Kilbirnie, Wellington, New Zealand Phone +64 4 386 0300, Fax +64 4 386 0574 www.niwa.co.nz
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Contents
1. Introduction 1
2. Linear theory Kinematics: 1 2.1 Wave classification 1 2.2 Velocity 2
3. Energy 4
4. Real waves 5
5. WaveSim 7
6. Wave Loading on Structures 8
7. Wave energy absorption 9
8. Measuring waves 10
9. Modeling Waves 10
10. Wave Resources 12
11. Bibliography/Links 15 11.1 Links 15 11.2 Literature 15
12. Brief Glossary: 17
1. Introduction
Any wave energy technology development requires a solid understanding of wave theory and practice. Here we summarise some important points relevant to the project.
2. Linear theory Kinematics:
2.1 Wave classification
It is convenient to classify waves as deep, intermediate (transitional) or shallow. We do this since deep and shallow waves exhibit quite different properties. This is shown for example a) in the way the water motion (and energy) is attenuated with depth b) the way the wave speed depends on wave period (or frequency).
Table 1: Wave classification, based on the ratio of depth (h) to wavelength (L), or kh where the wavenumber, k, is 2π/L:
Shallow h/L < 1/20 kh < π/10 Intermediate 1/20 < h/L < ½ π /10 < kh < π Deep ½ < h/L < ∞ π < kh < ∞
Table 2: Representative values of key wave parameters for typical coastal waves that we are likely to encounter, propagating in water depths, h. These are period, T, wavelength, L, and phase velocity, C:
h (m) T (s) L C h/L Classification Young waves, small 10 3 s 14 m 4.7 m/s .71 Deep fetch (harbour) Young waves, coastal 5 5 s 30 m 6.1 m/s .17 Intermediate Swell on coastal shore 3 12 s 64 m 5.3 m/s .05 Shallow Swell on open ocean 300 12 s 225 m 18.7 m/s 1.3 Deep
Note that ‘young’ waves generated in a shallow harbour can still be categorized as deep water waves, since their short wavelengths do not ‘feel’ the bottom.
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2.2 Velocity
We need to also distinguish between the phase velocity of the wave (i.e. the speed the wave form propagates), and the orbital velocity (i.e. the velocity that water particles or tracers travel). In deep water the orbital velocity ideally traces out circular motion, while in shallow water this becomes increasingly elliptical with a decreasing amount of vertical excursion. Ideally for small amplitude waves there is no net transport of water, only of energy. In reality, for finite amplitude waves, unclosed particle orbits do result in a small net migration (Stokes drift).
Figure 1: Wave orbital velocities are elliptical in shallow water, circular in deep water. (From Dean & Dalrymple, 1984)
The schematic in Figure 1 illustrates the shallow water elliptical motion in contrast to deep water circular motion. For deep water waves, the velocity (and pressure) is attenuated exponentially with depth. Thus most of the wave energy is confined to surface layers. In contrast for shallow water waves, the horizontal excursion is the same at all depths beneath the wave, but the total vertical excursion increases linearly from zero at the bed, to H (the wave height) at the surface.
One of the other important properties of water waves is that they are dispersive, i.e. the speed that they travel depends on their wavelength (or period), unlike non dispersive sound waves. This fact is often overlooked since we are accustomed to viewing shallow water waves breaking on the beach which, as we will see, appear to be non dispersive.
The relationship between frequency and wavelength is given by the general dispersion relationship:
ω2 = gk tanh (kh) ……. (1)
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where ω is the radian frequency (ie ω = 2πf = 2π/T), g is gravitational acceleration (9.81m/s 2), k is the wavenumber (2π/L), h is water depth, T the wave period, and L the wavelength. This transcendental equation is usually solved iteratively (e.g. Newton method) or using a lookup table.
The dispersion equation becomes simple for the two extremes of deep and shallow water:
Deep water: kh → ∞, tanh (kh) → 1, ω2 = gk i.e. frequency independent of depth
Shallow water: kh → 0, tanh(kh) → kh, ω2 = gk 2h.
Of immediate interest is the phase velocity, c (speed of crests) and group velocity, c g (speed of energy propagation).
Deep: c = ω/k = g/ ω cg = ∂ω /∂k = c/2
Shallow: c = √gh cg = ∂ω /∂k = c.
Waves slow down with decreasing depth, and in the ‘shallow’ water extreme, crests travel at the same speed regardless of frequency (or wavelength). This is what we observe on a surf beach, and contrasts to deep water where fast (long wavelength) waves are continually overtaking and moving through slower (shorter) ones. Note that the frequency is always invariant (although it can be Doppler shifted by currents); it is the phase velocity and wavelength that decrease in shallower water.
A practical consequence of the slowing phase velocity in shallow water is that wave crests refract to become parallel to the bathymetry. For a wave approaching the shore obliquely, those parts entering shallow water first are slowed, while deeper parts continue to catch up, thus swinging the wave crest around parallel to the depth contours and shoreline.
The distribution of wave height is often treated as a random Gaussian process. However this belies the fact that there is often an underlying group structure, which is evident when time space data is available. This structure is the ‘wave sets’ familiar to surfers. The practical implication of this is that the space time occurrence of wave breaking in deep water is not random, but structured, and in certain cases (narrow band wave spectra) is predictable. It is wave breaking that provides the strongest impact forces on structures. Wave Energy Coastal Waves Primer 3
3. Energy
The energy of water waves is equally divided between a) potential energy and b) kinetic energy. The potential energy relates to the raising of water from the trough to a
crest. The potential energy per unit area of a water surface raised by ς is given by E p = ½ ρ g ς 2. For a theoretical sinusoidal wave of amplitude H/2 averaged over a wave period this becomes Ep = 1/16 ρgH 2 .
Kinetic energy can be shown to have an equal value integrated over depth. However it can be seen in Figure 1 that the depth distribution of kinetic energy is quite different between deep and shallow water waves. The total average potential plus kinetic energy (after integrating over depth) per unit surface area is:
E = 1/8 ρgH 2
regardless of depth, and the power transported per unit crest length is:
2 P = 1/2 ρga Cg. More generally, when energy is distributed across a spectrum (S) of wave periods or frequencies (f) :
P = ρg ∫C g(f) S(f) df
(This may depend on the distances one is integrating over). The rate that energy is transported by the waves, the energy flux, F, is:
F = E C g = E (ω/k) ½ (1 + 2kh/sinh (2kh)).
On large scales energy will be lost to bottom friction, but if the depth changes rapidly with respect to this scale, we can treat E as constant. One consequence of the
conservation of energy is that as waves enter shallow water from depth h 1 to h 2,
E1Cg1 = E 2Cg2 .
Substituting for E, the wave height must increase to compensate for the decrease of group velocity:
H2 = H 1√(C g1/C g2).
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This shoaling equation tells us for example that a 10 s period wave approaching the shore from deep water will increase in amplitude by 11% by the time it reaches 5m depth, and 54% at 2m depth. Refraction is normally taken into account as well.
4. Real waves
In the real world, waves are not monochromatic sine waves (see Fig. 2 below), but due to the complex forcing mechanisms, occur as a continuous wave spectrum. In addition the spectrum of waves has a directional spread so a fully resolved measurement will also specify the direction from which a wave component is travelling.
Figure 2: Water surface velocities towards and away from the observer recorded using microwave radar from RV Tangaroa in 2004 showing the complex structure in time and space. Data slices are shown above and to the right of image panel.
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Figure 3: The wave example above comes from Wellington Harbour. The top panel shows a typical water elevation times series. Typically, it is not a simple sinusoid but shows a combination of frequencies. The bottom panel shows the resulting wave spectrum.
In order to define the wave parameters such as period and height, there are generally 2 approaches:
(1) from the timeseries e.g. the average time between zero crossings, Tz.
(2) from the spectrum. e.g. the dominant period is obtained from the peak of the spectrum, Tp. The moments of the spectrum can also be used to give the theoretical Tz. The significant wave height, Hs, is calculated from the area under the spectrum, m0. i.e. Hs =4√(m0).
The spectrum in the lower panel shows an approximate f 4 fall off in wave energy from the peak. In this case of ‘young’ waves, the peak frequency is 0.43 Hz (peak period 2.3s) and significant wave height, Hs = 0.30 m.
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Figure 4: Open ocean example, from the Bay of Plenty, is more typical than the earlier harbour spectrum in that it shows mixed wind sea and swell; the swell frequency is 0.08 Hz (period 12 sec) and the dominant sea frequency is 0.12 Hz (period ~8sec).
5. WaveSim
This Matlab tool was developed to display movies of real (and artificial) wave fields. The initial distribution took Waverider data from waveriders deployed at Mokohinau Island and Mangawhai.
Figure 5: WAVESIM3 still frame showing wave orbits and water surface elevation.
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6. Wave Loading on Structures
Forces on submerged wave affected structures are well described elsewhere (e.g. Grosenbaugh, 2002) with an emphasis on gas platforms and pipeline/cable scenarios. The Figure below summarizes the relevant forces on an ideal structure, including those
due to buoyancy, acceleration/inertia and drag (F b, F a, F d). These forces act in response to the local water motion (u) which is characterized as a function of time (t)
and space (x). Implicit in this is time variation where some forces (F b) are constant as
long as the structure is submerged, some are related to water velocity (F d) and some
are related to water acceleration (F a). The forces acting on the object (and to a lesser extent on the mooring line) are transferred along the mooring as both a force (and if it has any compressive strength a bending moment). To maintain equilibrium, there
must be a restorative force (F r) and bending moment (M r) acting at the base of the mooring.
Figure 6: Dynamics showing hydrodynamics and forces on a submerged body.
With respect to wave energy converter modelling we will need to determine
(1) the basic shape of the structure
(2) the flexibility of all mooring elements
(3) the coefficients of drag and added mass for all elements.
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Most of these properties can be derived from the literature, modelling of the combined response will need to be verified with laboratory/field validation.
7. Wave energy absorption
Considering wave energy from the perspective of simply absorbing it we can look to a number of successful floating breakwater approaches that can remove around 80% of energy that could be considered wind wave (e.g. Seymour and Hanes 1979).
Figure 7: Sketch of floating breakwater http://209.196.135.250/floating_breakwater.htm
The structural similarities between a floating breakwater and a NZ style mussel farm are clear.
Figure 8: Wave attenuation by a mussel farm. Energy Transmission Ratio ETR for the wave attenuation scaling analysis as a function of frequency (x axis) and initial wave height (given in legend). From Plew et al. (2005).
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8. Measuring waves
We can classify wave measurements in two categories: a) in situ and b) remote sensing, each with advantages and disadvantages.
Table 3: Wave measurement summary.
Device Sensor Advantage Disadvantages a) In-situ Wave rider buoy Accelerometer Suitable for longterm Expensive to (magnetometer) deployments purchase and maintain Pressure sensors Pressure Robust Limited to shallow (e.g. DOBIE) Inexpensive depths Suitable for longterm deployments Wave Staff Resistance or Accurate to high Limited to shallow Capacitance frequencies depths Less robust ADV Acoustic Also measures current Moderately expensive ($40k) ADP Acoustic 3 independent measures Moderately of wave height. expensive Measures current profile Robust at sea bed b) Remote sensing Radar Doppler effect Measures over a range Requires suitable of distances site Requires operator Satellite Electromagnetic Covers vast areas of Poor spatial backscatter or ocean resolution altimeter Very poor temporal resolution Not highly accurate
9. Modeling Waves
Wind generated waves can now be modeled well in open ocean deep water. The most common community developed model is WAM and is typically driven by meteorological analysis data. This is used in NIWA’s wave hindcasts (See below).
In shallower coastal water, the SWAN model is proving efficient except in particularly complicated situations e.g. variable bathymetry. Both of these models are spectral models in that they provide a statistical spectrum of wave energy. Wave growth is Wave Energy Coastal Waves Primer 10
modeled as a consequence of the imbalance between: energy advection and the ‘source terms’: wind input, dissipation due to breaking, dissipation by bottom friction, and non linear wave wave interactions.
Figure 9: New Zealand regional WAM wave model for a particular time. Windfields input from ECMWF reanalysis.
Figure 10: Mean Hs derived from NIWA wave hindcast averaging 20 years of the above results – see (Gorman et al 2003).
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To model an individual wave, various versions of a Boussinesq model are being developed. The current challenge is to incorporate breaking effects in shallow water, with the associated current generation.
For engineering applications, empirical relations are often used to predict wave heights and periods. Both wave height and period grow with fetch (distance over which the wind forcing is acting) and duration (over which wind forcing has been acting). The most common reference for this type of wave prediction is the US Army Corps Shore Protection Manual. Complications occur when: bathymetry is irregular, wind field is influenced by complex orography, or currents are strong.
10. Wave Resources
A wave hindcast (Gorman et al. 2003; Gorman 2003) provided a wave climate offshore of a number of selected sites from 1977 to 1997. The hindcast model was driven by European Centre for Medium Range Weather Forecasting re analysis windfields, and run on a 1.125x1.125 degree grid for the SW Pacific and Southern Ocean region. Directional spectra, saved at grid cells around the coast, have been interpolated to points on the 50m isobath and filtered to account for limited fetch to the coastline (Gorman et al. 2003). The hindcast generates 3 hourly estimates of significant wave height Hs (~2a) (Fig. 11), the frequency of the peak of the wave spectrum fp as well as a range of parameters relating to the energy flux. The Hs is the average of the highest 1/3 of the waves which is equivalent to the square root of the summed variance of the wave spectrum. It can be expected that the largest wave height within each 3 hr sample period will be about 2Hs.
Figure 11: Raw hindcast data for 4 coastal ocean sites spread around New Zealand. This represents around 58,000 data points per site.
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The analysis provided a suite of figures to demonstrate the resource availability these include
A site map showing the approximate location
A distribution of Hs occurrence statistics
A distribution of peak period occurrence statistics
A distribution of percentage occurrence of total omni directional wave energy flux occurrence statistics
Monthly averages of Hs and peak Hs. The 20 year peak Hs gives an indicator for survival modelling.
Event duration scatter diagram showing the number of events exceeding a particular Hs for a given period.
Figure 12: Monthly averages of Hs at four sites around New Zealand.
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Figure 13: Significant wave height occurrence statistics for a particular site.
Figure 14: Event duration occurrences for a site for three values of Hs (2, 3 & 4 m) where the symbols show the number of events exceeding Hs continuously for that duration.
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11. Bibliography/Links
11.1 Links
Banks Peninsula Wave Rider
NIWA http://www.niwa.co.nz/services/waves
Ecan http://www.ecan.govt.nz/Coast/Wave Buoy/wave buoy.html
WAM http://www.ecmwf.int/products/data/technical/wam/representations.html
Ocean Engineering University of New Hampshire: http://www.unh.edu/oe/
European Centre for Medium Range Weather Forecasts http://www.ecmwf.int/
SWAN wave model http://fluidmechanics.tudelft.nl/swan/default.htm
11.2 Literature Barnett P.S. and E.P.M. Brown (1987). The potential for wave generation off New Zealand. 8 th Australasian Conference on Coastal and Ocean Engineering, 87/11. IE Aust, 30 Nov – 4 Dec 1987, Launceston, Tasmania. Barstow, S., and R. Deo (1993). A wave energy resource climatology for the South Pacific. Proc. European Wave Energy Symposium, Edinburgh, Scotland, July '93. Brown, E.P.M. (1988). An estimate of New Zealand’s wave power resource for electricity generation. Proc. 1 st Symposium of the N.Z Ocean Wave Society, pp 17 27. Brown, E.P.M. (1990). Wave Power investigations in New Zealand. IPENZ Transactions, 1990, pp173 183. Brown, E.P.M. (1990). Comparison of Ocean Wave Power calculation methods. An estimate of New Zealand’s wave power resource for electricity generation. Proc. 2 nd Symposium of the N.Z Ocean Wave Society, pp 49 63 Dean, R.G. and R.A. Dalrymple 1984, Wave Mechanics for Engineers and Scientists, Englewood Cliffs: Prentice Hall, Inc. Falnes, J. 2002 Ocean Waves and Oscillating Systems, CUP. Gorman, R.M., Bryan, K.R. and Laing, A.K. (2003). Wave hindcast for the New Zealand region deep water wave climate. New Zealand Journal of Marine and Freshwater Research 37(3): 589 612.
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Gorman, R.M. and Laing, A.K. (2001). Bringing wave hindcasts to the New Zealand coast. Journal of Coastal Research Special Issue 34: 30 37. Gorman, R.M. (2003) The treatment of discontinuities in computing the nonlinear energy transfer for finite depth gravity wave spectra. Journal of Atmospheric and Oceanographic Technology, 20, 206 216. Grosenbaugh, M., S. Anderson, R. Trask, J. Gobat, W. Paul, B. Butman and R. Weller, “Design and performance of a horizontal mooring for upper ocean research,” J. Atmos. Oceanic Technol. , 19 , pp.1376 1389, 2002. Hornstra, M.W. (1983). Wave Power – a New Zealand study. M.E. Thesis, Dept of Mech. Eng., University of Auckland. 232pp. Komen, G.J., et al. , 1994. Dynamics and modelling of ocean waves. Cambridge, University Press Laing, A.K. 1993. Estimates of wave height data for New Zealand waters from numerical modelling. New Zealand Journal of Marine and Freshwater Research 27: 157 175. Pickrill, R.A., and J.S.Mitchell, (1979). Ocean wave characteristics around New Zealand. New Zealand journal of marine and freshwater research, 13(4):504 520. Plew. D.; Stevens. C.; Spigel, R.;Hartstein, D. 2005. Hydrodynamic implications of large offshore mussel farms. . IEEE Journal of Oceanic Engineering, Special Issue on Open Ocean Aquaculture Engineering 30:95 108. Reid, S.J. and B. Collen (1983). Analysis of wave and wind reports from ships in the Tasman Sea and New Zealand waters. N.Z Met Service, Misc. Pub 182. Seymour R.J. and D. M. Hanes, 1979 Performance analysis of tethered float breakwaters, J. Waterway, Port, Coastal & Ocean Eng., 105 , pp. 265 280. Smith, M.J.; Stevens, C.L.; Gorman, R.M.; McGregor, J.A.; Neilson, C.G. (2001) Wind wave development across a large shallow intertidal estuary: a case study of Manukau Harbour, New Zealand. NZ Journal of Freshwater and Marine Research, 35 : 985 1000. Stevens, C.L.; Hurd, C.L.; Smith, M.J. ( 2001). Water motion relative to subtidal kelp fronds , Limnology and Oceanography, 46(3) : 668 678. WAMDI group, 1988. The WAM model a third generation ocean wave prediction model. J. Phys. Oceanogr. 18 , pp. 1775 1810
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12. Brief Glossary:
ECMWF European Centre for Medium Range Weather Forecasts
Frequency (f) – number of crests (or troughs) passing a fixed point per second. The inverse of wave period.
Period (T) – time interval between two successive crests passing a fixed point
Phase velocity (c) speed that the waveform travels (L/T).
Significant wave height (Hs) – the most commonly used measure of wave height. It represents the height of approximately the highest third of waves. Hs = 4.0√(mo). Note that in a real sea in a three hour period you would expect to see a wave 2Hs in height.
Swell waves whose source region is very distant. As a result they have long periods (e.g. 8 12 sec)
Wavelength (L) – the distance between successive crests
Wind waves – generated by the local wind (e.g. 5 s period)
Young waves waves which are generated when either a) waves have not had time to develop to the point where the phase velocity approaches the wind speed or b) waves have not had the distance over which to develop to the point where the phase velocity approaches the wind speed
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