Waves, turbulent motions and mixing - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 90

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Waves, turbulent motions and mixing

In this chapter we shall look at waves and turbulence, two forms of motion
which are of particular importance in the shelf seas because of their roles in
bringing about the mixing which re-distributes properties such as heat, salt,
momentum and substances dissolved or suspended in the water. There is a
marked contrast in character between the two: waves are generally highly
ordered motions which are amenable to precise mathematical description, while
turbulence is chaotic in nature and it can usually only be represented in terms of
its statistical properties. Both waves and turbulence are large scientific topics in
their own right and are the subject of more than a few specialised textbooks.
Here, we shall focus on those aspects of surface waves, internal waves and
turbulence theory which are necessary to the understanding of processes in shelf
seas, and we shall leave the more specialised aspects for the interested student
to pursue from the further reading list.

4.1 Surface waves

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We have already developed the theory of long waves in Chapter 3 from the basic
equations and shown how such waves can help us to understand tidal motions in
shelf seas. The more general theory of surface wave motions, in which there is no
restriction on wavelength, is more involved and a full treatment is beyond the scope
of this topic. Here we shall simply present the assumptions and the principal results
of the theory of infinitesimal waves and give a physical description of the motion
involved. As well as being useful in themselves, the results of surface wave theory
introduce us to many of the concepts relevant to the understanding of the more
complicated motions involved in internal waves.

4.1.1 The first order velocity potential

The starting point is the recognition that the waves are irrotational, which means
that, if we were able to freeze (i.e. make solid) a small water particle anywhere in the
wave field, we would find that it was not rotating. In fluid mechanics, this essential