# Swell (ocean)

Breaking swell waves at Hermosa Beach, California

A swell, in the context of an ocean, sea or lake, is a series of mechanical waves that propagate along the interface between water and air and thus are often referred to as surface gravity waves. These surface gravity waves are not wind waves, which are generated by the immediate local wind, but instead are generated by distant weather systems, where wind blows for a duration of time over a fetch of water. More generally, a swell consists of wind-generated waves that are not—or are hardly—affected by the local wind at that time. Swell waves often have a long wavelength, but this varies due to the size, strength, and duration of the weather system responsible for the swell and the size of the water body. Swell wavelength also varies from event to event. Occasionally, swells that are longer than 700m occur as a result of the most severe storms. Swells have a narrower range of frequencies and directions than locally generated wind waves, because swell waves have dispersed from their generation area, have dissipated and therefore lost an amount of randomness, taking on a more defined shape and direction. Swell direction is the direction from which the swell is coming. It is measured in degrees (as on a compass), and often referred to in general directions, such as an NNW or SW swell.

## Formation

Large breakers observed on a beach may result from distant weather systems over a certain distance of the uninterrupted ocean. Five factors influence the formation of wind waves[1] which will go on to become ocean swell:

• Wind speed or strength relative to wave speed – the wind must be moving faster than the wave crest (in the direction in which the wave crest travels) for net energy transfer from air to water; stronger prolonged winds create larger waves
• The uninterrupted distance of open water over which the wind blows without significant change in direction (called the fetch)
• Width of area affected by fetch
• Wind duration – the time over which the wind has blown over a given area
• Water depth

All of these factors work together to determine the size of wind waves:

Effects of deepwater wave on movement of water particles (Stokes drift).

A fully developed sea has the maximum wave size theoretically possible for a wind of a specific strength, duration, and fetch. Further exposure to that specific wind could only cause a loss of energy due to the breaking of wave tops and formation of "whitecaps". Waves in a given area typically have a range of heights. For weather reporting and for scientific analysis of wind wave statistics, their characteristic height over a period of time is usually expressed as significant wave height. This figure represents an average height of the highest one-third of the waves in a given time period (usually chosen somewhere in the range from 20 minutes to twelve hours), or in a specific wave or storm system. The significant wave height is also the value a "trained observer" (e.g. from a ship's crew) would estimate from visual observation of a sea state. Given the variability of wave height, the largest individual waves are likely to be somewhat less than twice the reported significant wave height for a particular day or storm.[2]

The phases of an ocean surface wave: 1. Wave Crest, where the water masses of the surface layer are moving horizontally in the same direction as the propagating wavefront. 2. Falling wave. 3. Trough, where the water masses of the surface layer are moving horizontally in the opposite direction of the wavefront direction. 4. Rising wave.

### Sources of wind-wave generation

Cross sea of shallow-water swell waves near the Whales Lighthouse (Phare des Baleines), Île de Ré

Wind waves are generated by many kinds of disturbances such as seismic events, gravity, and crossing wind. The generation of wind waves is initiated by the disturbances of the crosswind field on the surface of the water. Two major mechanisms of surface wave formation by winds (the Miles-Phillips mechanism) and other sources (e.g. earthquakes) of wave formation can explain the generation of wind waves.

However, if one sets a flat water surface (Beaufort Scale 0) and abrupt crosswind flows on the surface of the water, then the generation of surface wind waves can be explained by two mechanisms, which are initiated by normal pressure fluctuations of turbulent winds and parallel wind shear flows.

### Surface wave generation by winds

The wave formation mechanism

From "wind fluctuations": Wind wave formation is started by a random distribution of normal pressure acting on the water from the wind. By this mechanism, proposed by O.M. Phillips in 1957, the water surface is initially at rest, and the generation of the wave is initiated by turbulent wind flows and then by fluctuations of the wind, normal pressure acting on the water surface. Due to this pressure fluctuation arise normal and tangential stresses that generate wave behavior on the water surface.

The assumptions of this mechanism are as follows:

1. The water is originally at rest;
2. The water is inviscid;
3. The water is irrotational;
4. The normal pressure to the water surface from the turbulent wind is randomly distributed; and
5. Correlations between air and water motions are neglected.[3]

From "wind shear forces": In 1957, John W. Miles suggested a surface wave generation mechanism that is initiated by turbulent wind shear flows, ${\displaystyle Ua(y)}$, based on the inviscid Orr-Sommerfeld equation. He found that the energy transfer from wind to water surface as a wave speed, ${\displaystyle c}$, is proportional to the curvature of the velocity profile of wind, ${\displaystyle Ua''(y)}$, at the point where the mean wind speed is equal to the wave speed (${\displaystyle Ua=c}$, where ${\displaystyle Ua}$ is the mean turbulent wind speed). Since the wind profile, ${\displaystyle Ua(y)}$, is logarithmic to the water surface, the curvature, ${\displaystyle Ua''(y)}$, has a negative sign at point ${\displaystyle Ua=c}$. This relation shows the wind flow transferring its kinetic energy to the water surface at their interface, and thence arises wave speed, ${\displaystyle c}$. The growth-rate can be determined by the curvature of the winds (${\displaystyle (d^{2}Ua)/(dz^{2})}$) at the steering height (${\displaystyle Ua(z=z_{h})=c}$) for a given wind speed, ${\displaystyle Ua}$.

The assumptions of this mechanism are:

1. 2-dimensional, parallel shear flow, ${\displaystyle Ua(y)}$.
2. Incompressible, inviscid water/wind.
3. Irrotational water.
4. Small slope of the displacement of the surface.[4]

Generally, these wave formation mechanisms occur together on the ocean surface, giving rise to wind waves that eventually grow into fully developed waves.[5] If one supposes a very flat sea surface (Beaufort number, 0), and sudden wind flow blows steadily across it, the physical wave generation process would be like this:

1. Turbulent wind flows form random pressure fluctuations at the sea surface. Small waves with a few centimeters order of wavelengths are generated by the pressure fluctuations (Phillips mechanism).[3]
2. The cross wind keeps acting on the initially fluctuated sea surface. Then the waves become larger, and as they do so, the pressure differences increase, and the resulting shear instability expedites wave growth exponentially (Miles mechanism).[3]
3. The interaction among the waves on the surface generates longer waves (Hasselmann et al., 1973)[6] and this interaction transfers energy from the shorter waves generated by the Miles mechanism to those that have slightly lower frequencies than at the peak wave magnitudes. Ultimately, the wave speed becomes higher than that of the cross wind (Pierson & Moskowitz).[7]
Conditions necessary for a fully developed sea at given wind speeds, and the parameters of the resulting waves
Wind conditions Wave size
Wind speed in one direction Fetch Wind duration Average height Average wavelength Average period and speed
19 km/h (12 mph) 19 km (12 mi) 2 h 0.27 m (0.89 ft) 8.5 m (28 ft) 3.0 s, 2.8 m/s (9.3 ft/s)
37 km/h (23 mph) 139 km (86 mi) 10 h 1.5 m (4.9 ft) 33.8 m (111 ft) 5.7 s, 5.9 m/s (19.5 ft/s)
56 km/h (35 mph) 518 km (322 mi) 23 h 4.1 m (13 ft) 76.5 m (251 ft) 8.6 s, 8.9 m/s (29.2 ft/s)
74 km/h (46 mph) 1,313 km (816 mi) 42 h 8.5 m (28 ft) 136 m (446 ft) 11.4 s, 11.9 m/s (39.1 ft/s)
92 km/h (57 mph) 2,627 km (1,632 mi) 69 h 14.8 m (49 ft) 212.2 m (696 ft) 14.3 s, 14.8 m/s (48.7 ft/s)
• (Note: Most of the wave speeds calculated from the wavelength divided by the period are proportional to the square root of the length. Thus, except for the shortest wavelength, the waves follow the deep water theory described in the next section. The 8.5 m long wave must be either in shallow water or between deep and shallow.)

## Dissipation

The dissipation of swell energy is much stronger for short waves, which is why swells from distant storms are only long waves. The dissipation of waves with periods larger than 13 seconds is very weak but still significant at the scale of the Pacific Ocean.[8] These long swells lose half of their energy over a distance that varies from over 20,000 km (half the distance round the globe) to just over 2,000 km. This variation was found to be a systematic function of the swell steepness: the ratio of the swell height to the wavelength. The reason for this behavior is still unclear, but it is possible that this dissipation is due to the friction at the air-sea interface.

## Swell dispersion and wave groups

Swells are often created by storms thousands of nautical miles away from the beach where they break, and the propagation of the longest swells is only limited by shorelines. For example, swells generated in the Indian Ocean have been recorded in California after more than half a round-the-world trip.[9] This distance allows the waves comprising the swells to be better sorted and free of chop as they travel toward the coast. Waves generated by storm winds have the same speed and will group together and travel with each other, while others moving at even a fraction of a meter per second slower will lag behind, ultimately arriving many hours later due to the distance covered. The time of propagation from the source t is proportional to the distance X divided by the wave period T. In deep water it is ${\displaystyle t=4\pi X/(gT)}$ where g is the acceleration of gravity. For a storm located 10,000 km away, swells with a period T=15 s will arrive 10 days after the storm, followed by 14 s swells another 17 hours later, and so forth.

This disperse arrivals of swells, long periods first with a reduction in the peak wave period over time, can be used to tell the distance at which swells were generated.

Whereas the sea state in the storm has a frequency spectrum with more or less the same shape (i.e. a well defined peak with dominant frequencies within plus or minus 7% of the peak), the swell spectra are more and more narrow, sometimes as 2% or less, as waves disperse further and further away. The result is that wave groups (called sets by surfers) can have a large number of waves. From about seven waves per group in the storm, this rises to 20 and more in swells from very distant storms.

## Coastal impacts

Just like for all water waves, the energy flux is proportional to the significant wave height squared times the group velocity. In deep water, this group velocity is proportional to the wave period. Hence swells with longer periods can transfer more energy than shorter wind waves. Also, the amplitude of infragravity waves increases dramatically with the wave period (approximately the square of the period), which results in higher run-up.

As swell waves typically have long wavelengths (and thus a deeper wave base), they begin the refraction process (see water waves) at greater distances offshore (in deeper water) than locally generated waves.[10]

Since swell-generated waves are mixed with normal sea waves, they can be difficult to detect with the naked eye (particularly away from the shore) if they are not significantly larger than the normal waves. From a signal analysis point of view, swells can be thought of as a fairly regular (though not continual) wave signal existing in the midst of strong noise (i.e., normal waves and chop).

Swells were used by Micronesian navigators to maintain course when no other clues were available, such as on foggy nights.[11]

## References

1. ^ Young, I. R. (1999). Wind generated ocean waves. Elsevier. ISBN 0-08-043317-0. p. 83.
2. ^ Weisse, Ralf; von Storch, Hans (2009). Marine climate change: Ocean waves, storms and surges in the perspective of climate change. Springer. p. 51. ISBN 978-3-540-25316-7.
3. ^ a b c Phillips, O. M. (1957), "On the generation of waves by turbulent wind", Journal of Fluid Mechanics 2 (5): 417–445, Bibcode1957JFM.....2..417P, doi:10.1017/S0022112057000233
4. ^ Miles, J. W. (1957), "On the generation of surface waves by shear flows", Journal of Fluid Mechanics 3 (2): 185–204, Bibcode1957JFM.....3..185M, doi:10.1017/S0022112057000567
5. ^
6. ^ Hasselmann K., T.P. Barnett, E. Bouws, H. Carlson, D.E. Cartwright, K. Enke, J.A. Ewing, H. Gienapp, D.E. Hasselmann, P. Kruseman, A. Meerburg, P. Mller, D.J. Olbers, K. Richter, W. Sell, and H. Walden. Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP)' Ergnzungsheft zur Deutschen Hydrographischen Zeitschrift Reihe, A(8) (Nr. 12), p.95, 1973.
7. ^ Pierson, Willard J., Jr. and Moskowitz, Lionel A. Proposed Spectral Form for Fully Developed Wind Seas Based on the Similarity Theory of S. A. Kitaigorodskii, Journal of Geophysical Research, Vol. 69, p.5181-5190, 1964.
8. ^ Observation of swell dissipation across oceans, F. Ardhuin, Collard, F., and B. Chapron, 2009: Geophys. Res. Lett. 36, L06607, doi:10.1029/2008GL037030
9. ^ Directional recording of swell from distant storms, W. H. Munk, G. R. Miller, F. E. Snodgrass, and N. F. Barber, 1963: Phil. Trans. Roy. Soc. London A 255, 505
10. ^
11. ^ "Home". www.penn.museum.