The swell that dates its own storm

The packet that moves at another speed showed that a packet of waves in a dispersive medium goes at the group speed and that this is not the speed of its crests. The whistle that arrives sorted took that into the ionosphere: a lightning stroke’s crackle is stretched over thousands of kilometres of plasma into a descending tone, with its high notes first. The sea does the same thing, the other way up. A storm is a sudden broadband source, a churning of the surface containing many wave periods. Long waves outrun short ones, and by the time the waves reach a far coast they arrive sorted, with the longest first, in a stretched-out sequence over a week. Sailors have long noticed long-period swell arriving ahead of a storm. What was not obvious was how much information is in the arrival.

Walter Munk and his colleagues set out in the early 1960s to read the sorting of swell in the Pacific. They fixed a pressure sensor on the seabed in shallow water off San Clemente Island, and three others in different places. The records were analysed over months in 1959, 1960 and 1963, using the spectral methods then newly available from computers. A low, slow rising tone appeared on their charts again and again: the peak frequency climbing steadily over days. Each was a storm, and with no information at all about the storm except this tone, they found out where it had been.

A ruler in the dispersion relation #

Waves in deep water obey the relation that the speed that depends on the length derived, $\omega^2 = gk$. The phase speed is $g/\omega$, and the group speed is half that:

$$c_g = \frac{g}{2\omega} = \frac{gT}{4\pi}.$$

This has no depth in it, and no amplitude in it (in the linear limit), and no wind. A swell’s energy travels at a speed set solely by its period, and the only other quantity involved is the acceleration of gravity. In SI units the group speed is about 0.78 times the period, and hence for a 15 s swell it is about 11.7 m/s, about 42 km/h, roughly what a person cycles at.

A storm a long way off lasts about a day and is a few hundred kilometres across. At a distance of thousands of kilometres, those dimensions are small compared with how the waves have spread out in time. So the storm looks like a single event, a point source in space and time. Each period is then found at each distance at a definite time:

$$t = t_0 + \frac{4\pi D f}{g}.$$

Equivalently, at a fixed place, the frequency in the record at time $t$ is

$$f(t) = \frac{g\,(t - t_0)}{4\pi D}.$$

This is a straight line in frequency and time, starting from zero at the moment of the storm, with a slope that depends on one thing.

The ridges can overlap in time without confusion, because water waves of low steepness pass through one another with little interaction, a property of linear waves that when two waves meet relies on. So a single instrument over a month records a family of lines at different slopes, and each slope is a distance. It is a little like a passive sonar, where there is nothing to send and everything to hear, and the difference is that it needs only one instrument and no knowledge of what it is listening to, since dispersion is doing the rangefinding.

What the sea actually does at one place #

It is one thing to draw lines, and it is another to see what the actual surface does. So here a surface record has been simulated by adding hundreds of independent wave components, each arriving at the group speed time for its period, with random phase.

The early waves are low and long. They may be barely visible, and an arrival at twenty seconds may be a few centimetres high on the open sea. It is why their first sign is often in a seismometer on land or a pressure gauge rather than to the eye. They are noticed best at a beach, as the waves that always arrive square to the beach showed, because shoaling amplifies them there and the long waves feel the bottom early. Later arrivals are shorter and more energetic, at the frequency where the storm had the most energy, followed by a decline. Surfers wait for the first long-period arrival, as it is often the strongest for its size and the cleanest. Modern surf forecasts are a kind of commercial version of Munk’s chart.

A point of interest is that the waves in the record appear in groups. They come in sets, of a kind any surfer can count. No group structure has been built in. What is there is a sum of nearby frequencies with random phases, which the phases that turn a glow into pulses dealt with in a different context, and the beat between close frequencies does the rest. A narrow range of frequencies produces beats lasting many waves. So sets are a straightforward outcome of swell being narrowband after a long trip. Longer trips produce narrower bands, and hence longer, more distinct sets.

Reading a storm off a graph #

The arithmetic for extracting a distance is short. Measure the peak frequency at a series of times, with whatever errors a record provides. Fit a straight line.

So the slope gives $D$ and the intercept gives $t_0$. That is a remarkable result from a single record at a fixed place, given that the storm itself was never seen. What cannot be read directly is the bearing. A ridge says how far away a storm was and when, but not in which direction. Munk’s group had a triangle of instruments that gave a rough direction, and they combined these with the weather maps of the time to identify which storm in which ocean might have been the source. A great circle at about the right distance was compared with reports of storms. Some of their sources lay south of New Zealand and some south of Australia and even in the Indian Ocean. Swell from those sources had crossed the entire Pacific from its southern edge to reach California.

That result led to a still more ambitious experiment, the Waves Across the Pacific project of 1963. Munk and collaborators, including Frank Snodgrass, ran six stations on a line from New Zealand to Alaska, including a tiny manned platform (the research vessel FLIP), and followed individual groups of swell along a great circle. It was the first such tracking across an ocean and it remains the classic one. They found that swell of periods above about 13 s lost surprisingly little energy across that distance, which ran contrary to what was previously assumed about wave decay in a real ocean. So a storm near Antarctica really can send energy to Alaska, and it arrives. This changed oceanographers’ understanding of how far waves travel.

These days a satellite altimeter can see large swell from orbit and synthetic aperture radar can map the wave field, and a good many such events have been followed. In 2009 Fabrice Ardhuin and colleagues reported tracking from space swell over 10,000 km or more, at rates of decay that could be measured. These rates have been shown to depend on the steepness of the swell, with low swells losing very little.

How far it can go #

A large ocean storm’s swell faces the geometry of a sphere, and it does something that a flat surface cannot do.

Energy leaving a point on a sphere spreads over a circle whose circumference grows until the circle is 10,000 km away, then shrinks. On a smooth water-covered sphere, swell from a storm should therefore refocus at the point opposite. In an actual ocean, with continents in the way, the point opposite to most storms in the Southern Ocean lies in the Northern Hemisphere on land. An exact antipodal focus has never been shown in the sea. But the same geometry does apply to the Earth as a whole for seismic surface waves, where a large earthquake’s waves do focus at the antipode and go round several times, as the tone the whole Earth rings at dealt with. There is no similar drama in the sea, where the coasts stop it, but the geometric spreading is less bad than a flat calculation would imply when a swell is on a very long path.

A single scale for loss has also been put on the same chart, though a real value depends on the period and steepness. Reasonable e-folding distances for long, low swell in open ocean are thousands of kilometres, perhaps 15,000 km for swell of the longest period and much less for steep waves. Those are mostly due to friction in the air at the surface, in a way that was mysterious in Munk’s time and is still not settled, but which the satellite observations appear to link to viscous effects in the air at the wave’s surface.

A forecast that went to war #

The physics of swell’s speed was already in Lamb’s hydrodynamics by 1895, and the first people to need it urgently were planning an invasion. In 1943 the United States Navy approached Harald Sverdrup, then director of the Scripps Institution, and his student Walter Munk, who had noticed in a training exercise that amphibious landings in the Carolinas were being stopped by a swell that nobody had predicted. Landing craft could be used only in low surf, and at that time the only forecast available for an invasion beach was the guess of an experienced sailor. Sverdrup and Munk developed a method, relating waves to wind speed, duration and distance from the fetch, and then a dispersive propagation to the beach, and trained officers to use it. The forecasts were used for the landings in North Africa, in the Pacific and in Normandy. On the Normandy decision, a forecast of a brief gap in poor conditions for 6 June 1944 was among the things on which the invasion was launched, and the surf that day was at the limit of what was workable. It is one of the very few cases in which an equation in the mechanics of fluids had a place in such a decision. Wave forecasting as a science dates from that work, and the fetch law was the starting point for all later models.

Twenty years later, the same Walter Munk wrote the papers described above. His reason for going back to the subject was the opposite of war: he wanted to know how the energy spread over the world’s oceans, and whether there was an upper limit to how far it could go.

The tone in the ground #

There is an unexpected place where the swell can be heard at great distance from any coast. Every seismometer on Earth, in the middle of a continent, records a continuous background of very small motion at periods of about 5 to 20 seconds, with no earthquake in sight. It is called the microseism. It was known early in the twentieth century, and it was noticed to be stronger in winter and correlated with storms at sea. No one could at first explain its strongest band, which lies around half of the typical swell period. Waves on the surface should not reach the sea bed in deep water, as their motion decays exponentially with depth, and a pressure at the bottom ought to be absent.

Michael Longuet-Higgins explained it in 1950. When two opposing wave trains of the same period overlap, the first-order pressure at depth vanishes, but a second-order term does not. It pulses at twice the frequency and it does not decay with depth at all, as it is a change in the average height of the water column as a whole. So regions where swell meets its own reflection from a coast or where a storm’s waves meet an opposing swell act as a forcing on the deep sea floor, at double the frequency. This drives seismic waves, which travel through the Earth to any instrument. A record taken in Kansas contains the sea state off Newfoundland, and seismologists’ records from the early twentieth century have been used to reconstruct wave climates from before any buoys existed. A related effect, at the swell’s own period, comes from waves feeling the shallow bottom at a shelf. And lower still is the Earth’s hum, around 5 mHz, which the tone the whole Earth rings at touched upon, for which ocean waves of very long period seem to be responsible.

This noise is not merely a nuisance. Because it is produced everywhere and at all times, it provides a seismic source that can be correlated between pairs of stations, to measure the Earth’s structure between them with no earthquakes, in the way that the echo hidden in random noise described. So the swell that was produced by a storm in the Southern Ocean has helped to make maps of the crust beneath California. It seems unlikely that Munk, tracking his ridges, would have predicted that use of the same waves.

What the ridge cannot be trusted for #

The ridge assumes that the source is a point in space and in time. A real storm occupies about a thousand kilometres and lasts a day or more, and its wind field moves. Near to the storm, those are not negligible and the ridge is blurred and slightly curved. At very long ranges the ridge becomes sharp, since dispersion has outrun the storm’s own extent. So the method gets better with distance, which is an unusual property for a range-finding technique. For a nearby storm, within a few hundred kilometres, it has little use.

The deep-water dispersion relation is also only a first approximation. Long swell feels ocean currents, and a current bends the path of the waves. It also changes the arrival time. In the Pacific, the effect is modest, but Munk’s own later work showed that a chain of islands blocks and scatters swell, and that seamounts cast shadows. So a storm’s waves do not arrive by a single path. A record may contain two or more ridges from the same storm, with slightly different slopes, from rays that went around an island on different sides. The islands of the Pacific also dissipate a good share of what passes through. In their 1963 study, these effects were among the most striking in the data. All of this is lost in the simple figure, as are wave–wave interaction effects, which are weak for low-steepness swell but do move energy to lower frequencies, slightly shifting the ridge.

There is also the simulation itself. The record above is a sum of Gaussian arrivals in time, using a chosen spectrum and a chosen width, and the reader should not take its exact look as what is measured. A real pressure record contains shorter-period noise, tides and a local wind sea on top of the swell, and the extraction of a ridge requires some work with spectra. Most of the difficulty in the original work lay in that.

Still open: where the energy goes on a very long trip #

Swell’s slow decay has become an important unknown in the coupled ocean–atmosphere system. Long swell may hand energy upward to the wind, driving a thin “wave-driven wind” in calm conditions, an effect that has been seen in very light winds over swell. Where the energy lost by a swell finally goes, whether into the air’s turbulence or the ocean’s upper layer or a thin viscous sub-layer, matters for how climate models represent the air–sea boundary, and it matters also for a future in which wave energy might be harvested or storm climates shift. Satellite instruments such as the CFOSAT mission and the SWOT altimeter, launched in the last decade, can now follow individual swell fields in far greater detail than Munk’s stations could. It is now possible to measure, over the open ocean, whether swell at a given steepness loses energy as predicted, and whether what is lost returns to the air. Those records are now being assembled, and the answer to how far a swell will go is still being refined some sixty years after the question was first asked.

Deep-water swell travels at $gT/4\pi$, so a distant storm’s arrivals at a point draw a straight line of frequency against time, with a slope of $g/4\pi D$ and an intercept at the moment of the storm: a fit recovers 10,600 km against a true 10,500 km and a birth time to about a tenth of a day from a single instrument. A coast keeps an accurate record of events it never saw. The habit worth carrying is to treat a dispersive medium as a recording, since anything that sorts itself in transit has written down how far it came.