* Recasts topics in random fields by following a completely new way of handling both geometry and probability
 * Significant exposition of the work of others in the field
 * Presentation is clear and pedagogical
 * Excellent reference work as well as excellent work for self study

This monograph is devoted to a completely new approach to geometric problems arising in the study of random fields. The groundbreaking material in Part III, for which the background is carefully prepared in Parts I and II, is of both theoretical and practical importance, and striking in the way in which problems arising in geometry and probability are beautifully intertwined.

The three parts to the monograph are quite distinct. Part I presents a user-friendly yet comprehensive background to the general theory of Gaussian random fields, treating classical topics such as continuity and boundedness, entropy and majorizing measures, Borell and Slepian inequalities. Part II gives a quick review of geometry, both integral and Riemannian, to provide the reader with the material needed for Part III, and to give some new results and new proofs of known results along the way. Topics such as Crofton formulae, curvature measures for stratified manifolds, critical point theory, and tube formulae are covered. In fact, this is the only concise, self-contained treatment of all of the above topics, which are necessary for the study of random fields. The new approach in Part III is devoted to the geometry of excursion sets of random fields and the related Euler characteristic approach to extremal probabilities.

"Random Fields and Geometry" will be useful for probabilists and statisticians, and for theoretical and applied mathematicians who wish to learn about new relationships between geometry and probability. It will be helpful for graduate students in a classroom setting, or for self-study. Finally, this text will serve as a basic reference for all those interested in the companion volume of the applications of the theory. These applications, to appear in a forthcoming volume, will cover areas as widespread as brain imaging, physical oceanography, and astrophysics.

Random Fields and Geometry

A review of physical mechanisms of the rogue wave phenomenon is given. The data of marine observations as well as laboratory experiments are briefly discussed. They demonstrate that freak waves may appear in deep and shallow waters. Simple statistical analysis of the rogue wave probability based on the assumption of a Gaussian wave field is reproduced. In the context of water wave theories the probabilistic approach shows that numerical simulations of freak waves should be made for very long times on large spatial domains and large number of realizations. As linear models of freak waves the following mechanisms are considered: dispersion enhancement of transient wave groups, geometrical focusing in basins of variable depth, and wave-current interaction. Taking into account nonlinearity of the water waves, these mechanisms remain valid but should be modified. Also, the influence of the nonlinear modulational instability (Benjamin–Feir instability) on the rogue wave occurence is discussed. Specific numerical simulations were performed in the framework of classical nonlinear evolution equations: the nonlinear Schrodinger equation, the Davey–Stewartson system, the Korteweg–de Vries equation, the Kadomtsev–Petviashvili equation, the Zakharov equation, and the fully nonlinear potential equations. Their results show the main features of the physical mechanisms of rogue wave phenomenon.

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Physical Mechanisms of the Rogue Wave Phenomenon

Earlier models of random wave transformation are reviewed in the first section. Then the transformation of waves, including dissipation due to breaking and bottom friction, is described by an energy flux balance model. The wave height pdf of all waves (broken and unbroken) is shown by the field data to be well described by the Rayleigh distribution everywhere. The observed distributions of breaking and broken wave heights are fitted to simple analytical forms, and breaking wave dissipation is calculated by using a periodic bore formulation. The energy flux equation is integrated to yield local values of Hrms as a function of offshore wave conditions. Both analytical and numerical models are developed. In the last section the models are compared with results from random wave experiments in the laboratory and from an extensive set of field measurements.

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Transformation of wave height distribution

It is nearly three-quarters of a century since E. R. Watson (1904) and E. M. Wedderburn (1907) made the observations in Loch Ness which showed conclusively, and for the first time, that large bodies of water contain beneath their surface the wave motions which have now come to be known as internal waves. The observations and theory of these waves have developed much since those days, but the Loch is still very useful as a site in which to observe and examine phenomena which are also found in other bodies of water, particularly the ocean. In particular the Loch provides a large-scale natural ‘laboratory’ in which a variety of small-scale phenomena associated with turbulence in a stratified fluid may be studied. Observations have been made with a novel profiling instrument which measures the horizontal velocity of the water and its temperature, from which the density may be inferred. These observations serve to illustrate a variety of local conditions which occur in calm weather, as the Loch responds to the wind and during the passage of an internal surge. Analysis of the records is conducted in terms of an intermittency index (the fraction of fluid in which the density decreases with depth), the Richardson number and a length scale which characterizes the vertical scale of the regions which are found to be unstably stratified. Semi-empirical formulae for the eddy diffusion coefficient and the rate of dissipation of kinetic energy in the turbulent motion are examined to see whether they are consistent with observations. No universal value of the Richardson number is found, but this may be a consequence of the rather low values of Reynolds number found in the Loch thermocline.

Turbulence and mixing in a Scottish Loch

Oceanic rogue waves are surface gravity waves whose wave heights are much larger than expected for the sea state. The common operational definition requires them to be at least twice as large as the significant wave height. In most circumstances, the properties of rogue waves and their probability of occurrence appear to be consistent with second-order random-wave theory. There are exceptions, although it is unclear whether these represent measurement errors or statistical flukes, or are caused by physical mechanisms not covered by the model. A clear deviation from second-order theory occurs in numerical simulations and wave-tank experiments, in which a higher frequency of occurrence of rogue waves is found in long-crested waves owing to a nonlinear instability.

M. S. Longuet-Higgins

Papers

On the statistical distribution of the heights of sea waves