A History of Surf Forecasting: Part I
Posted on June 20, 2007 @ 1:26 PM
Note that the SMB method is still being used nowadays, particularly by coastal engineers who want a quick idea of wave heights without running a complicated computer model. For example, the US Army Corps of Engineers publish a graph called a nomogram, from which one can quickly look up the height and period of the waves given the windspeed, fetch and duration (see diagram in box below).
The SMB method suffered from two major restrictions, both of which would be addressed in the decades following its invention. The first was that it was not based on a proper physical understanding of how waves are generated; and the second was that it used a single entity called the ‘significant wave’ to represent what is, in reality, an entire spectrum of wave heights, periods and directions.
By around the mid-1950s, work was already underway by J. Miles and O.M. Phillips to investigate how waves were really generated on the surface of the ocean, in an exact way that was not empirical or semi-empirical. They came up with the Miles-Phillips theory, which is still used nowadays. The theory describes how very small waves, called capillary waves, first begin to grow from an entirely flat sea. Then it describes how larger waves (called gravity waves) are subsequently formed from a sea already containing capillary waves. The capillary waves are generated by vertical perturbations in the surface wind, causing irregularities in the water surface. This then increases the surface roughness which, in turn, enables any further action of the wind to ‘grip’ the water surface, lifting it up even more.
The second mechanism is self-perpetuating – the rougher the surface, the more ‘grip’ and, therefore, the easier it is for a given wind to increase the height of the waves. The first mechanism causes the waves to grow linearly with time, but the second mechanism causes them to grow exponentially. Of course, eventually a point will be reached where a particular windspeed can’t lift up the surface of the sea any more – the force of gravity pulls the water back down again at the same rate as the wind lifts it up.
Apart from gravity itself, there are other physical mechanisms that work to reduce the height of the waves. One of these is the friction generated by the molecules under each wave interacting with other water molecules and with those on the seabed. Another mechanism is whitecapping, where the waves momentarily break in deep water, giving up a lot of energy to turbulence and sound.
Yet further mechanisms exist relating to the transfer of energy between the waves themselves. This is a curious concept, which involves a continual flux of energy from the shorter-period waves towards the longer ones. As the sea state grows, energy is diverted more and more away from the shorter-period waves and towards the longer ones, the short ones effectively being ‘gobbled up’ by the long ones, and the sea state becoming increasingly more dominated by long-period waves. This explains why, in a growing sea, the significant wave height not only gets bigger, but the significant period also gets longer.
(For those with a knowledge of physics, this ‘non-linear transfer’ can be explained by an input of energy at the highfrequency end of the spectrum, which gradually shifts through to the low-frequency end where it piles up, progressively skewing or ‘red-shifting’ the spectrum towards the low-frequency end).
So, after getting into the nitty-gritty of how waves are actually generated, scientists were now ready to write down, in mathematical terms, how wave energy on the sea surface grew and changed as the wind blew across that surface. Around the early 1960s, an equation was born which was to become the cornerstone behind all wave models developed thereafter. This was the Action Balance Equation, sometimes called the Radiative Transfer
Equation. It represented a major advance on earlier methods for three important reasons:
• It was a ‘dynamical’ equation, which meant it described the evolution of the sea state rather than predicting the sea in some ‘final’ state after being worked on by the wind.
• It was more closely related to the real underlying physics of wave growth and decay.
• It was a two-dimensional spectral equation, describing the wave energy evolution not just at a single frequency and direction, but over a whole range of both.
Here is a highly simplified schematic representation of the Action Balance Equation:
Each term in the equation has quite a complicated story behind it. I’ll briefly describe what each one means in simple terms. The box on the far left is the rate of change of wave energy with time, at a point on the ocean, over a number of different directions and frequencies. The ones on the right-hand side of the equals sign are the ingredients that go to make up the one on the left. The latter are known as source terms, some of which put
energy in to the waves, and some of which take it out.
• ‘Wind input’ is the energy transfer between the air and the water due to the wind blowing over the surface of the water (this is where all the formulae from the Miles-Phillips mechanism go).
• ‘Friction’ is the energy taken out of the water due to processes like molecular friction, whitecapping and wind blowing in the opposite direction to the waves.
• ‘Non-linear transfer’ is that transfer of energy between waves of different frequencies (the ‘gobbling-up’ of short waves by long waves).
By about 1971, a few models had been developed using this equation. These were termed ‘first-generation’ wave-forecasting models. Although they worked, it was obvious that more research was needed into the mathematics behind the source terms and their relative importance. There was also the fact that computing power was a big restricting factor in those days. For example, on the biggest supercomputers of the day, the models
could not be run with the ‘non-linear-transfer’ term included, otherwise the time taken to generate the simulation would far exceed real time. (Wouldn’t it be absurd, if tomorrow’s forecast were not available until next week?).
Ironically, it was soon to be found that the non-linear transfer was much more important than people first realised, and could not be ignored if the models were to work properly.
Continued on page 3...
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