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Work notes on infragravity

Jerome Aucan

Theory

Surf beat

Infragravity motions or surf beat are low frequency motions, with period between 30 and 300 seconds, associated to shorter, higher frequency surface gravity waves. First described as surf beat (Munk, 1949), infragravity motions were later shown to have a significant importance for nearshore processes. Further investigation showed that infragravity could be separated in two categories, a bound or forced part, and a free part. In addition, the time variation of the wave induced setup, can also be part of these infragravity motions.

Bound infragravity

Open ocean surface gravity waves are usually irregular in size, and travel in packets of alternatively high and low waves. In both deep and shallow water, but seaward of the surf zone, it can be observed that the mean sea level is depressed under the groups of high waves. This can be explained by using the concept of radiation stress (Longuet-Higgins and Stewart, 1962). Upon breaking the fate of this bound infragravity wave is uncertain. It is commonly written that it is released in the form of a freely propagating wave. When approaching shore, resonance occur when the group speed of the swell train comes closer to the phase speed of the bound wave.
Bound infragravity is proportional at a given depth to the square of the swell energy (Elgar et al, 1992), and for a given swell the bound infragravity motions are proportional to $ h^{-5}$ . Forced waves motions depend only on the local forcing $ E(f,\theta)$, and the water depth (Herbers et al. JPO 1994 Part. I)

Free Infragravity

Free infragravity waves are of two main forms : The edge waves, obliquely incident, and refractively trapped on the shore or on the shelf, and the shore normal leaky wave. Although generation mechanisms are unclear, edge waves are of significant importance in surf zone dynamics. For a given depth, the amount of free wave energy is approximately lineraly proportional to the swell energy (Herbers et al. JPO 1995 Part. II), in contrast to the quadratric dependence of forced energy on swell energy.
For a given swell , the predicted relation between leaky surfave gravity waves and depth is $ h^{-1/2}$. Observation of relations closer to $ h^{-1}$ are consistent with the refractive trapping of directionally broad free wave field.

Alternative model : Time varying setup

Rarely mentioned in the litterature about infragravity, is the effect of a time varying setup. Within the surf zone, between the breaking point and the shoreline, one can observe an shoreward increasing mean sea level, refered to as wave induced setup. As this setup is a local function of the wave height, the setup is modulated by the groups of incident short waves, so it is experiencing the same temporal variations as the bound waves, but now with the same sign, ie a group of high waves corresponds to a higher mean sea level (Symonds et al., 1982).This would also predict a weaker than quadratic dependence on swell energy, consistent with observations. In the surf zone, bound infragravity is supposedly released, edge waves dominate the infragravity budget and decrease in amplitude seaward. The wave induced setup, only present between the breaking zone and the shore also decreases seaward.

Observations, and conclusions in previous studies

Elgar et al. 1992

Barbers Point 8m and Duck 8 and 13 m (bispectrum). Overall, the amplification of IG between 8 and 13 m is much smaller than if it was only bound IG (ie. proportional to $ h^{-5}$). but during energetic swell the amplification is also much greater than if it was only shoaling leaky or edge waves ($ h^{-1}$ or $ h^{-1/2}$). Conclusion : bound wave is a smaller contribution to the total IG than the free wave, but it can be significant during energetic swells.

Okihiro et al. 1992

Imperial Beach Ca. 8 -13 meters and Pt Conception 183 m. IG is larger for swells than seas for a given total energy. Ratio bound/free increases with total IG increasing and decreasing depth. Pt. Conception, $ <.1$ of IG is bound, even during high swells. At all sites, free IG are more energetic than bound. In 183 m, total IG is small, so the free IG is refractively trapped and does not reach deep water. Strong dependence of bound wave on directional spreading.

Herbers, Elgar, Guza 1994 JPO Part I. Forced waves

Field study at Duck with 24 P-sensors, 13 m depth. Forced waves can be extracted with bispectral analysis. Statistical uncertainties in the bispectral estimates is probably the dominant source of errors, when non linearities are weak. Results are better for total energy than for individual frequncy bins.

Herbers, Elgar, Guza 1995 JPO Part II. Free waves.

(similar to Herbers et al. 1995 JGR). Very good correlation between swell and IG. Forced wave is proportional to $ E_{swell}^2$ while free wave is approximately proportional to $ E_{swell}$ (Duck, 8 and 13m) or a little more ( $ E_{swell}^{1.4}$ for the Barbers Point data).

Hawaii Data

Infragravity motions are more tractable to observation when they are associated to a narrow banded swell in frequency and direction ( Holman, 1981). The north and west facing shores of the hawaiian Islands are subject to those high energy narrow banded swell, while sheltered from locally generated wind seas. The hawaiian islands also show a narrow steep shelf that would successfully segregate refractively trapped edge waves to a narrow band near the shore. Waimea, on the north shore of O'ahu is a highly reflective pocket beach inside a bay, and it displays an intermitent breaking zone at the entrance of the bay. It is subject to seasonal, low frequency, narrow banded swells from the NW. Kailua Bay, in contrast, is a large reef bottom bay, subject to frequent, high frequency locally generated windswells.

Waimea Bay

Two bottom mounted pressure sensors were deployed for 3 weeks near Waimea Bay during the winter 2001-2002. During this period, the island experienced succesive low frequency energetic swell events. A directional waverider buoy was also deployed in 200 meters of water, 3 miles offshore of the bay. The pressure sensors were deployed in 10 and 17 meters of water. They simultaneously recorded bottom pressure with a sampling rate of 1 Hz during 30 minutes bursts every 4 hours. The deep pressure sensor was located outside the intermitent breaking zone, while the shallow one was located just inside of this intermitent breaking zone, but still outside of the beach breakers. The time series were detrended. Two spectrum analysis were performed : The first one for the swell/sea band (frequencies between 0.025 Hz and .6 Hz), and a second one for the infragravity band (frequencies between .004 Hz and .04 Hz). For each 30 minutes segment, the time serie of sea surface elevation at infragravity frequencies was reconstructed. For each of those segment, the envelope of the swell was calculated using the peak frequency of the swell. Because the samples were only 30 minutes long, a bispectral analysis similar to Herbers (1994) will not be statistically relevant. We can still have a idea of the amount of forced infragravity by calculating the correlation coefficient between the swell enveloppe and the IG elevation time serie.
On the other hand, a directional waverider buoy offshore provided us with full directional spectrum, which can be used to compute the predicted forced wave at that location. A numerical wave model can be used to propagate the spectrum to shore and then measure a theoritical amount of forced IG at all points of the model grid.

Kailua Bay

During 3 weeks in the winter 2000-2001, 4 pressure sensors (depth between 2.5 and 30 m) and 2 waverider buoys (120 m and 30 m) were deployed. Data analysis was similar to Waimea Bay.

Tidal Gauges data

In addition to our temporary deployment of pressure sensors, 5 permanent pressure sensors located around O'ahu, used primarily for the detection of tsunamis, can provide infragravity energy levels in harbors

Directional wave measurements

Datawell waverider buoys have been deployed by the department of Oceanography. The Mokapu buoy, on the windward side is operational since Aug. 9 2000, and the Waimea buoy is operational since Dec 2001. For earlier experiment, data from NOAA Discus buoy can be available. A predicted forced infragravity wave can be derived from the directional spectrum measured by those buoys.

Additional data

Data from previous studies can also be obtained for Barbers Point, O'ahu and Kahului, Maui.

Results

early results from the Waimea Bay data

Total infragravity

Figure 1 shows the correlation between the total infragravity energy and the swell energy. For both the deep and the shallow location, the amount of total infragravity is strongly correlated to the amount of incoming swell energy.

Figure 1: Correlation between swell and IG energy

\includegraphics[height=4in]{infra_versus_swell.eps}

This rules out any remote sources for infragravity and supports the hypothesis that infragravity motions are linked to the local wave field. The shoreward amplification of infragravity energy between the two sensors averages 73 percent, but varies a lot during the time of the experiment (std =44 percent). The ratio of IG energy between inside and outside is between .9 and 3 , averaging 1.75. The relation of infragravity to swell energy is $ E_{swell}^{1.2}$ outside, and $ E_{swell}^{1.14}$ inside the Bay

Bound infragravity at the offshore pressure sensor

The bound infragravity motions is phase-locked to the incoming short waves packets. The mean sea-level is depressed during packets of high waves and is higher during packets of low waves. Figure 2 shows the time series of low frequency elevation and the corresponding swell envelope at the outside pressure sensor during the most energetic swell.
Figure 2: surface elevation at infragravity frequencies and swell envelope
\includegraphics[height=4in]{envellope.eps}

For each 30 minutes segment, the correlation between the time serie of elevation at the infragravity frequencies and the enveloppe of the incoming swell is calculated. Significant negative correlation occurs at the outside pressure sensors during the most energetic events, while correlation is quite low during less energetic events (figure 3).

Figure 3: Wave height and correlation between enveloppe and IG surface elevation
\includegraphics[height=4in]{outside_correlation.eps}

evolution of IG between the 2 pressure sensors

At the shallow pressure sensor, correlation between the swell envelope and the infragravity motion is very low. This indicates that just outside the breaking zone, and during the energetic events, the bound infragravity makes a significant contribution to the total infragravity, while after breaking or during low to intermediate event, little bound infragravity can be observed. There is only a small amplification on average of total IG between offshore and inshore ($ O(1)$). For a bound wave amplification should be $ 15$, for a leaky it should be $ 1$ and for an edge wave $ 1.7$. The amplification is very variable during the time of the experiment. Nonetheless, IG energy at the 2 pressure sensors is very well correlated, supporting the theory of the free wave being generated locally by the forced wave. More careful examination of the amplification ratio might help decide if the bound wave is dissipated between the 2 sensors and/or if the edge wave turning point is also between the 2 sensors. Those results will of course vary with time as the breaking line outside is intermitent and varies in space with the incoming swell energy.

Some results from Kailua Bay

Figure 4: Swell/Sea height and IG wave height in Kailua Bay
\includegraphics[height=4in]{ig_swell.eps}
At the most shallow pressure sensor, the amount of infragravity is almost $ O(1)$ of the swell energy (figure(4)), and the correlation with swell is very good. This is consistent with IG motion on very dissipative beaches, where a significant amount of the surface elevation energy is at the infragravity frequencies. At the other pressure sensors, the correlation, the correlation between swell and IG decreases (fig. (4)), and is much smaller than for the Waimea Bay experiment. The wave climate and shoreline configuration are very different at those 2 sites.

Further analysis

Herbers et most others base some of their conclusions about forced versus free IG on the $ h^{-5}$ relation of the forced wave on the water depth. In area where the shelf is very steep, and where wave properties can vary over distances of a few wavelength, this relation may not hold. More analysis and/or modeling is needed to answer this question. A better knowledge of the evolution of the bound wave when the short wave directional spectrum is rapidly modified upon shoaling can only help the investigation of the generation of the free wave, still to to be properly explained.

Discussion

This experience is reproducing observations of infragravity motions obtained in very different nearshore environments (gently sloping beaches, wide shelves) :The ratio of infragravity energy to swell energy is increasing shoreward; The contribution of the forced infragravity to the total energy, seaward of the surf zone is increasing with increasing total energy. Beside the presence of edge waves, the little contribution of the forced wave to the total energy at the inside sensor might be explained by the opposite effect of the setup. The bound wave is associated with the short waves group stucture. These waves are dissipated through the surf zone, so the bound wave might be dissipated along with the short waves. The wave induced setup increases shoreward, and is higher during the groups of high waves. Between two groups of high waves, the forcing for the wave-induced setup is smaller, so the previously high setup could decrease a send a progressive seaward wave. This would be coherent with the first observations of surf beat by Munk (1949) where he suggests that the surf beat is an seaward propagating wave radiated by the surf zone, or in other word the periodic relaxation of the wave induced setup, which occurs with the same periodicity as the incoming short waves groups, and can then be measured as an infragravity signal.

Bibliography

1
Herbers, T. H. C., S. Elgar, R. T. Guza, Infragravity-Frequency (0.005-0.05 Hz) Motions on the shelf. Part I : Forced Waves, J. Phys. Ocean., 24, 917-927, 1994.

2
Herbers, T. H. C., S. Elgar, R. T. Guza, Infragravity-Frequency (0.005-0.05 Hz) Motions on the shelf. Part II : Free Waves, J. Phys. Ocean., 25, 1063-1079, 1995.

3
Herbers, T. H. C., S. Elgar, R. T. Guza, Generation and propagation of infragravity waves, J. Geophys. Res., 100, 24863-24872, 1995.

4
Longuet-Higgins, M. S. and R. W. Stewart, Radiation stress and mass transport in surface gravity waves with application to surf beats, J. Fluid Mech., 13, 481-504, 1962.

5
Longuet-Higgins, M. S. and R. W. Stewart, Radiation stress in water waves : A physical discussion with applications, Deep Sea Res., 11, 529-562, 1964.

6
Munk, W. H., Surf beats, EOS Trans. AGU,30, 849-854, 1949.

7
Okihiro, M., R. T. Guza, and R. J. Seymour, Excitation of seiche observed in a small harbor J. Geophys. Res., 98, 18201-18211, 1993.

8
Symonds, G., D. A. Huntley, and A. J. Bowen, Two-dimensional surf beat : Long wave generation by a time varying breakpoint J. Geophys. Res., 87, 492-498, 1982.

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Work notes on infragravity

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Jerome Aucan 2004-05-18