Elements of Physical Oceanography: A derivative of the Encyclopedia of Ocean Sciences

By Academic Press

Elements of Physical Oceanography is a derivative of the Encyclopedia of Ocean Sciences, Second Edition and serves as an important reference on current physical oceanography knowledge and expertise in one convenient and accessible source. Its selection of articles—all written by experts in their field—focuses on ocean physics, air-sea transfers, waves, mixing, ice, and the processes of transfer of properties such as heat, salinity, momentum and dissolved gases, within and into the ocean. Elements of Physical Oceanography serves as an ideal reference for topical research.

• References related articles in physical oceanography to facilitate further research
• Richly illustrated with figures and tables that aid in understanding key concepts
• Includes an introductory overview and then explores each topic in detail, making it useful to experts and graduate-level researchers
• Topical arrangement makes it the perfect desk reference

• Language: English
• Publisher: Academic Press
• Release date: Aug 26, 2009
• ISBN: 9780123757210

Book preview

Level

Elements of Physical Oceanography: Introduction

Physical Oceanography is one of the several different, if not entirely distinct, sciences of the ocean. It is concerned with kinematics and dynamics, fluxes and stress, waves, tides, flows and mixing. These factors all impinge in fundamental ways on (and gain understanding from) the other companion sciences, marine biology, geochemistry and geology, the ocean flows advecting and diffusing the dissolved solutes and particulate matter - living or inanimate - in the water, and largely controlling their distributions and movements. This volume is a selection of articles from the Encyclopedia of Ocean Sciences about ocean physics, air–sea transfers, waves, mixing, ice (a topic of rapidly growing interest) or, more generally, about the processes of transfer of properties such as heat, salinity, momentum and dissolved gases, within and into the ocean. It provides a general source of reference to the present state of knowledge of some aspects of the subject.

The articles are arranged in eight sections in an order and with connections that might be found useful in teaching courses on ocean physics at undergraduate or postgraduate level. The volume does not, however, follow the conventional order of textbooks or courses in physical oceanography which commonly begin with a discussion of the properties of seawater and their effect on density (a discussion which is difficult to make attractive to students whose excitement and interest are more easily aroused by dynamical processes which are visible and might be used to recreational advantage, such as waves — surfing and sailing being popular sports) before going on to set up relationships to describe the behaviour of fluids. Nor does the volume include a very important component of a physical oceanography course - that of ocean currents and circulation, the form they take in different oceans and seas and how they are driven - a subject that has its own separate volume in this series: Ocean Currents.

The order of sections initially follows a progress downwards from the sea surface. The first section is about surface waves and other changes in the level of the sea surface. Included are articles on the ‘rogue’ waves that cause damage to ships, and tsunamis that have resulted in severe damage and loss of life in recent years. The tides are perhaps the oldest marine subjects of scientific observation, enquiry and speculation, beginning with the Ancient Greeks (Cartwright, 1999). Articles on tides and tidal energy are included in this section as well as the climate-related topic of sea level change.

This is followed by a section in which are described the exchanges of heat, momentum and gases between the atmosphere and the ocean, topics of particular current importance because of the part played by the ocean in the composition of the atmosphere and in climate change. The presence of surface films may restrict the exchange of gases, whilst bubbles created by breaking waves enhance the transfer.

Much of the mixing and turbulent dissipation of kinetic energy within the ocean takes place in its boundary layers. Three different boundary layers are identified as sub-topics in the third section, the upper ocean, the under-ice, and the benthic boundary layers. (It appears logical to include the benthic boundary layer with its companions although the order of descent through the ocean is interrupted.) Mixing in the three differs. The upper ocean boundary layer is strongly affected by the wind, and the consequent waves and Langmuir circulation, as well as by the buoyancy changes resulting from solar radiation, and heat and freshwater fluxes. The mixing in the benthic boundary layer is driven largely by the stress on the seabed, very little by geothermal heat (or buoyancy) flux. Like the others, the under-ice boundary layer is also driven by stress, but also by buoyancy resulting from freezing, salt rejection or ice melting.

Internal waves and tides are the subject of the short fourth section and, because in breaking they contribute to mixing, they provide an introduction to the subject addressed in the following section, processes of diapycnal mixing, including turbulent mixing, convection, double diffusive convection and diffusion. There is presently keen debate about how the deep ocean is mixed, and consequently this is a subject of active and developing investigation. It is central to the study of ocean physics, relating the sources of energy to the mixing and circulation of the oceans (Wunsch and Ferrari, 2004).

Articles on the vitally important subjects of horizontal dispersion and the lateral transport of heat around the World are included in the sixth section. (The oceans transport about as much heat from the equatorial to the arctic regions as does the atmosphere!) The major interannual oscillations of the Pacific and Atlantic Oceans, ENSO and NAO, the former characterized by sea-surface temperature fluctuations and the latter by differences in sea level pressure are, included in this section, as are two articles-on water masses and neutral surfaces-that relate indirectly to dispersion and to the consequent density of seawater.

The final two sections are (returning to the sea surface) about ice and polynyas, and processes in the coastal and shelf seas.

Because of space limitations, it has been necessary to be selective and to omit some important topics that would be included in a thoroughly comprehensive account (or perhaps in a broad-ranging taught course). Most notably these are an account of some of the instruments used in measurement, such as current meters, floats, CTD and turbulence sensors. These are described in articles in the full Encyclopedia and in the companion special topic volume on ‘Measurement Techniques, Platforms and Sensors.’ Some related topics, e.g., models of ocean circulation, are found in the full Encyclopedia. It is regretted that there are also no articles about the development of sedimentary waves, ripples and dunes or about the important relationship between turbulence and particles of sediment or algae.

The author of each article is an expert in his or her field. They are all distinguished researchers who have given time to write concisely and lucidly about their subjects, and the Editors are indebted to them all for the time given and the care taken in preparing these accounts.

The articles in this volume would not have been produced without the considerable help of the several members of the Encyclopedia’s Editorial Advisory Board listed below. Each provided advice and suggestions about the content and authorship of particular subject areas covered in the Encyclopedia. In addition to thanking the authors of the articles in this volume, the Editors wish to thank the members of the Editorial Board for the time they gave to identify and encourage authors, to read and comment on (and sometimes to suggest improvements to) the written articles, and to make this venture possible.

Editorial Advisory Board Members who helped in the production of this volume

Garry Bass, Ken Brink, Robert Duce, John Gould, Ann Gargett, Chris Garrett, Peter Liss, Nick McCave, Dennis McGillicuddy, Ken Melville, Jim Moum, Colin Summerhayes, Stewart Turner, Bob Weller and James Yoder.

Steve A. Thorpe

Editor

REFERENCES

Cartwright DE. Tides: A Scientific History. Cambridge: Cambridge University Press; 1999 p. 292.

Wunsch C, Ferrari R. Vertical mixing, energy and the general circulation of the oceans. Annual Review of Fluid Mechanics. 2004;36:281–314.

Surface Gravity and Capillary Waves

W.K. Melville    Scripps Institution of Oceanography, University of California, San Diego, La Jolla, USA

Introduction

Ocean surface waves are the most common oceanographic phenomena that are known to the casual observer. They can at once be the source of inspiration and primal fear. It is remarkable that the complex, random wave field of a storm-lashed sea can be studied and modeled using well-developed theoretical concepts. Many of these concepts are based on linear or weakly nonlinear approximations to the full nonlinear dynamics of ocean waves. Early contributors to these theories included such luminaries as Cauchy, Poisson, Stokes, Lagrange, Airy, Kelvin and Rayleigh. Many of the current challenges in the study of ocean surface waves are related to nonlinear processes which are not yet well understood. These include dynamical coupling between the atmosphere and the ocean, wave–wave interactions, and wave breaking.

For the purposes of this article, surface waves are considered to extend from low frequency swell from distant storms at periods of 10 s or more and wavelengths of hundreds of meters, to capillary waves with wavelengths of millimeters and frequencies of O(10) Hz. In between are wind waves with lengths of O(1–100) m and periods of O(1–10) s. Figure 1 shows a spectrum of surface waves measured from the Research Platform FLIP off the coast of Oregon. The spectrum, Φ, shows the distribution of energy in the wave field as a function of frequency. The wind wave peak at approximately 0.13 Hz is well separated from the swell peak at approximately 0.06 Hz.

Figure 1 (A) Surface displacement spectrum measured with an electromechanical wave gauge from the Research Platform FLIP in 8 m s —1 winds off the coast of Oregon. Note the wind-wave peak at 0.13 Hz, the swell at 0.06 Hz and the heave and pitch and roll of FLIP at 0.04 and 0.02 Hz respectively. (B) An extension of (A) with logarithmic spectral scale, note that from the wind sea peak to approximately 1 Hz the spectrum has a slope like f —4 , common in wind-wave spectra. (Reproduced with permission from Felizardo FC and Melville 1995. Correlations between ambient noise and the ocean surface wave field. Journal of Physical Oceanography 25: 513–532.)

Ocean surface waves play an important role in air–sea interaction. Momentum from the wind goes into both surface waves and currents. Ultimately the waves are dissipated either by viscosity or breaking, giving up their momentum to currents. Surface waves affect upper-ocean mixing through both wave breaking and their role in the generation of Langmuir circulations. This breaking and mixing influences the temperature of the ocean surface and thus the thermodynamics of air–sea interaction. Surface waves impose significant structural loads on ships and other structures. Remote sensing of the ocean surface, from local to global scales, depends on the surface wave field.

Basic Formulations

The dynamics and kinematics of surface waves are described by solutions of the Navier-Stokes equations for an incompressible viscous fluid, with appropriate boundary and initial conditions. Surface waves of the scale described here are usually generated by the wind, so the complete problem would include the dynamics of both the water and the air above. However, the density of the air is approximately 800 times smaller than that of the water, so many aspects of surface wave kinematics and dynamics may be considered without invoking dynamical coupling with the air above.

The influence of viscosity is represented by the Reynolds number of the flow, Re = UL/μ, where U is a characteristic velocity, L a characteristic length scale, and v = μ/P is the kinematic viscosity, where μ is the viscosity and ρ the density of the fluid. The Reynolds number is the ratio of inertial forces to viscous forces in the fluid and if Re>>1, the effects of viscosity are often confined to thin boundary layers, with the interior of the fluid remaining essentially inviscid (v = 0). (This assumes a homogeneous fluid. In contrast, internal waves in a continuously stratified fluid are rotational since they introduce baroclinic generation of vorticity in the interior of the fluid). Denoting the fluid velocity by u = (u, v, w, the flow is said to be irrotational. From Kelvin’s circulation theorem, the irrotational flow of an incompressible (∇.u = 0) inviscid fluid will remain irrotational as the flow evolves. The essential features of surface waves may be considered in the context of incompressible irrotational flows.

For an irrotational flow, u = ∇ϕ where the scalar ϕ is a velocity potential. Then, by virtue of incompressibility, ϕ satisfies Laplace’s equation

   [1]

We denote the surface by z =(x, y,t), where (x, y) are the horizontal coordinates and t is time. The kinematic condition at the impermeable bottom at z = −h, is one of no flow through the boundary:

   [2]

There are two boundary conditions at z = η:

   [3]

   [4]

The first is a kinematic condition which is equivalent to imposing the condition that elements of fluid at the surface remain at the surface. The second is a dynamical condition, a Bernoulli equation, Which is equivalent to stating that the pressure p_ at z = η_, an infinitesimal distance beneath the surface, is just a constant atmospheric pressure, pa, plus a contribution from surface tension. The effect of gravity is to impose a restoring force tending to bring the surface back to z = 0. The effect of surface tension is to reduce the curvature of the surface.

Although this formulation of surface waves is considerably simplified already, there are profound difficulties in predicting the evolution of surface waves based on these equations. Although Laplace’s equation is linear, the surface boundary conditions are nonlinear and apply on a surface whose specification is a part of the solution. Our ability to accurately predict the evolution of nonlinear waves is limited and largely dependent on numerical techniques. The usual approach is to linearize the boundary conditions about z= 0.

Linear Waves

Simple harmonic surface waves are characterized by an amplitude a, half the distance between the crests and the troughs, and a wavenumber vector k with |k|= k=2π/λ, where λ is the wavelength. The surface displacement, (unless otherwise stated, the real part of complex expressions is taken)

   [5]

where σ = 2π/T is the radian frequency and T is the wave period. Then ak is a measure of the slope of the waves, and if ak <<1, the surface boundary conditions can be linearized about z = 0.

Following linearization, the boundary conditions become

   [6]

   [7]

where the linearized Laplace pressure is

   [8]

where Γ is the surface tension coefficient.

Substituting for η and satisfying Laplace’s equation and the boundary conditions at z= 0 and -h gives

   [9]

Where

   [10]

and

   [11]

are known as dispersion relations, and for linear waves provide a fundamental description of the wave kinematics. The phase speed,

   [12]

is the speed at which lines of constant phase (e.g., wave crests) move.

For waves propagating in the x-direction, the velocity field is

   [13]

   [14]

   [15]

and the pressure

   [16]

The velocity decays with depth away from the surface, and, to leading order, elements of fluid execute elliptical orbits as the waves propagate.

For shallow water, kh <<1,

   [17]

is independent of the wavenumber. Such waves are said to be non-dispersive. Waves propagating towards shore eventually attain this condition, and, as the depth tends to zero, nonlinear effects become important as ak increases.

For very deep water, kh>> 1,

   [18]

so that the water particles execute circular motions that decay exponentially with depth. The horizontal motion is in phase with the surface displacement, and the phase speed of the waves

   [19]

These deep-water waves are dispersive; that is, the phase speed is a function of the wavenumber as shown in the wavelength λ = 1.7cm and the phase speed is a minimum at c gravity is the dominant restoring force, the wavelength is greater than 1.7 cm, and the phase speed increases as the wavelength increases.

Figure 2 The phase speed of surface gravity-capillary waves as a function of wavelength λ. A minimum phase speed of 23cm s —1 occurs for λ= 0.017 m. Shorter waves approach pure capillary waves, whereas longer waves become pure gravity waves. Note that there are both capillary and gravity waves for a given phase speed. This is the basis of the generation of parasitic capillary waves on the forward face of steep gravity waves.

The Group Velocity

Using the superposition principle over a continuum of wavenumbers a general disturbance (in two spatial dimensions) can be represented by

   [20]

where, as above, only the real part of the integral is taken. Assuming the disturbance is confined to wavenumbers in the neighbourhood of ko, and expanding σ(k) about ko gives

   [21]

whence

   [22]

where

   [23]

is the group velocity. Eqn [22] demonstrates that the modulation of the pure harmonic wave propagates at the group velocity. This implies that an isolated packet of waves centered around the wavenumber ko will propagate at the speed cg, so that an observer wishing to follow waves of the same length must travel at the group velocity. Since the energy density is proportional to a.

For deep-water gravity waves,

   [24]

so the wave group travels at half the phase speed, with waves appearing at the rear of a group propagating forward and disappearing at the front of the group.

For deep-water capillary waves,

   [25]

so waves appear at the front of the group and disappear at the rear of the group as it propagates.

For shallow water gravity waves, kh<<1, cg = c.

Second Order Quantities

The energy density (per horizontal surface area) of surface waves is

   [26]

being the sum of the kinetic and potential energies. In the case of gravity waves, the potential energy results from the displacement of the surface about its equilibrium horizontal position. For capillary waves, the potential energy arises from the stretching of the surface against the restoring force of surface tension.

The mean momentum density M is given by

   [27]

where the unit vector e = k/k.

To leading order, linear gravity waves transfer energy without transporting mass; however, there is a second order mass transport associated with surface waves. In a Lagrangian description of the flow it can be shown that for irrotational inviscid wave motion the mean horizontal Lagrangian velocity (Stokes drift) of a particle of fluid originally at z = zo is

   [28]

which reduces to (ak)²ce²kzoe when kh>>1. This second order velocity arises from the fact that the orbits of the particles of fluid are not closed. Integrating eqn [28] over the depth it can be shown that this mean Lagrangian velocity accounts for the wave momentum M in the Eulerian description. The Stokes drift is important for representing scalar transport near the ocean surface, but this transport is likely to be significantly enhanced by the intermittent larger velocities associated with wave breaking.

Longer waves, or swell, from distant storms can travel great distances. An extreme example is the propagation of swell along great circle routes from storms in the Southern Ocean to the coast of California. For waves to travel so far, the effects of dissipation must be small. In deep water, where the wave motions have decayed away to negligible levels at depth, the contributions to the dissipation come from the thin surface boundary layer and the rate of strain of the irrotational motions in the bulk of the fluid. It can be shown that the integral is dominated by the latter contributions, and the timescale for the decay of the wave energy is just

   [29]

or σ/8πvk² wave periods. This gives negligible dissipation for long-period swell in deep water over scales of the ocean basins. More realistic models of wave dissipation must take into account breaking and near surface turbulence which is sometimes parameterized as a ‘super viscosity’ or eddy viscosity, several orders of magnitude greater than the molecular value. When waves propagate into shallow water, the dominant dissipation may occur in the bottom boundary layer.

Eqn [27] shows that dissipation of wave energy is concomitant with a reduction in wave momentum, but since momentum is conserved, the reduction of wave momentum is accompanied by a transfer of momentum from waves to currents. That is, net dissipative processes in the wave field lead to the generation of currents.

Waves on Currents: Action Conservation

Waves propagating in varying currents may exchange energy with the current, thus modifying the waves. Perhaps the most dramatic examples of this effect come when waves propagating against a current become larger and steeper. Examples occur off the east coast of South Africa as waves from the Southern Ocean meet the Aghulas Current; as North Atlantic storms meet the northward flowing Gulf Stream, or at the mouths of estuaries as shoreward propagating waves meet the ebb tide.

For currents U = (U, V) that only change slowly on the scale of the wavelength, and a surface displacement of the form

   [30]

where a . and the x- and yThe frequency seen by an observer moving with the current U is

   [31]

which is equal to the intrinsic frequency σ Thus

   [32]

which is just the Doppler relationship.

We also have,

   [33]

which can be interpreted as the conservation of wave crests, where k is the spatial density of crests and ω the wave flux.

The velocity of a wave packet along rays is

   [34]

which is simply the vector sum of the local current and the group velocity in a fluid at rest. Furthermore, refraction is governed by

   [35]

where the first term on the right represents refraction due to the current and the second is due to gradients in the waveguide, such as changes in the depth. It is this latter term which results in waves, propagating from deep water towards a beach, refracting so that they propagate normal to shore.

For steady currents, the absolute frequency is constant along rays but the intrinsic frequency may vary, and the dynamics lead to a remarkable and quite general result for linear waves. If E the wave action, is conserved:

   [36]

In other words, the variations in the intrinsic frequency σ and the energy density E, are such as to conserve the quotient.

This theory permits the prediction of the change of wave properties as they propagate into varying currents and water depths. For example, in the case of waves approaching an increasing counter current, the waves will move to shorter wavelengths (higher k), larger amplitudes, and hence greater slopes, ak. As the speed of the adverse current approaches the group velocity, the waves will be ‘blocked’ and be unable to propagate further. In this simplest theory, a singularity occurs with the wave slope becoming infinite, but higher order effects lead to reflection of the waves and the same blocking effect. This theory also forms the basis of models of long-wave-short-wave interaction that are important for wind-wave generation and the interpretation of remote sensing measurements of the ocean surface, including the remote sensing of long nonlinear internal waves.

Nonlinear Effects

The nonlinearity of surface waves is represented by the wave slope, ak. For typical gravity waves at the ocean surface the average slope may be O(10—2-10—1); small, but not negligibly so. Nonlinear effects may be weak and can be described as a perturbation to the linear wave theory, using the slope as an expansion parameter. This approach, pioneered by Stokes in the mid-nineteenth century, showed that for uniform approach deep-water gravity waves,

   [37]

and

   [38]

Weakly nonlinear gravity waves have a phase speed greater than linear waves of the same wavelength. The effect of the higher harmonics on the shape of the waves leads to a vertical asymmetry with sharper crests and flatter troughs.

The largest such uniform wave train has a slope of ak = 0.446 a phase speed of 1.11c, and a discontinuity in slope at the crest containing an included angle of 120°. This limiting form has sometimes been used as the basis for the models of wave breaking; however, uniform wave trains are unstable to side-band instabilities at significantly lower slopes, and it is unlikely that this limiting form is ever achieved in the ocean.

With the assumption of both weak nonlinearity and weak dispersion (or small bandwidth, δk/κo<<1), it may be shown that if

   [39]

where σ0 = σ(κ0) and Re means that the real part is taken, then the complex wave envelope A(x, y, t) satisfies a nonlinear Schrӧodinger equation or one of its variants. Solutions of the nonlinear Schrӧdinger equation for initial conditions that decay sufficiently rapidly in space evolve into a series of envelope solitons and a dispersive tail. Solitons propagate as waves of permanent form and survive interactions with other solitons with just a change of phase. Attempts have been made to describe ocean surface waves as fields of interacting envelope solitons; however, instabilities of the two-dimensional soliton solutions, and the effects of higher-order nonlinearities, random phase and amplitude fluctuations in real wave fields give pause to the applicability of these idealized theoretical results.

Resonant Interactions

Modeling the generation, propagation, interaction, and dissipation of wind-generated surface waves is of great importance for a variety of scientific, commercial and social reasons. A rigorous theoretical foundation for all components of this problem does not yet exist, but there is a rational theory for weakly nonlinear wave–wave interactions.

For linear waves freely propagating away from a storm, the spectral content at any later time is explicitly defined by the initial storm conditions. For a nonlinear wave field, wave–wave interactions can lead to the generation of wavenumbers different from those comprising the initial disturbance. For surface gravity waves, these nonlinear effects lead to the generation of waves of lower and higher wave-number with time. The timescale for this evolution in a random homogeneous wave field is of the order of (ak)⁴ times a characteristic wave period; slow, but significant over the life of a storm.

The foundation of weakly nonlinear interactions between surface waves is the resonant interaction between waves satisfying the linear dispersion relationship. It is a simple consequence of quadratic nonlinearity that pairs of interacting waves lead to the generation of waves having sum and difference frequencies relative to the original waves. Thus

   [40]

If in addition, σi(i = 1,2,3) satisfies the dispersion relationship, then the interaction is resonant. In the case of surface waves, the nonlinearities arise from the surface boundary conditions, and resonant triads are possible for gravity capillary waves, and gravity waves in water of intermediate depth.

For deep-water gravity waves, cubic nonlinearity is required before resonance occurs between a quartet of wave components:

   [41]

These quartet interactions comprise the basis of nonlinear wave–wave interactions in operational models of surface gravity waves. Exact resonance is not required, since even with detuning significant energy transfer can occur across the spectrum. The formal basis of these theories may be cast as problems of multiple spatial and temporal scales, and higher-order interactions should be considered as these scales increase, and the wave slope increases.

Parasitic Capillary Waves

The longer gravity waves are the dominant waves at the ocean surface, but recent developments in air–sea interaction and remote sensing, have placed increasing importance on the shorter gravity-capillary waves. Measurements of gravity-capillary waves at sea are very difficult to make and much of the detailed knowledge is based on laboratory experiments and theoretical models.

Laboratory measurements suggest that the initial generation of waves at the sea surface occurs in the gravity-capillary wave range, initially at wavelengths of O(1) cm. As the waves grow and the fetch increases, the dominant waves, those at the peak of the spectrum, move into the gravity-wave range. A simple estimate of the effects of surface tension based on the surface tension parameter Σ using the gravity wavenumber k would suggest that they are unimportant, but as the wave slope increases and the curvature at the crest increases, the contribution of the Laplace pressure near the crest increases. A consequence is that so-called parasitic capillary waves may be generated on the forward face of the gravity wave (Figure 3).

Figure 3 (A)-(D) Evolution of a gravity wave towards breaking in the laboratory. Note the generation of parasitic capillary waves on the forward face of the crest. (Reproduced with permission from Duncan JH et al. (1994) The formation of a spilling breaker. Physics of Fluids 6: S2.)

The source of these parasitic waves can be represented as a perturbation to the underlying gravity wave caused by the localized Laplace pressure component at the crest. This is analogous to the ‘fish-line’ problem of Rayleigh, who showed that due to the differences in the group velocities, capillary waves are found ahead of, and gravity waves behind, a localized source in a stream. In this context the capillary waves are considered to be steady relative to the crest. The possibility of the direct resonant generation of capillary waves by perturbations moving at or near the phase speed of longer gravity waves is implied by the form of the dispersion curve in Figure 2. Free surfaces of large curvature, as in parasitic capillary waves, are not irrotational and so the effects of viscosity in transporting vorticity and dissipating energy must be accounted for. Theoretical and numerical studies show that the viscous dissipation of the longer gravity waves is enhanced by one to two orders of magnitude by the presence of parasitic capillary waves. These studies also show that the observed high wavenumber cut-off in the surface wave spectrum that has been observed at wavelengths of approximately O(10—3-10—2) m can be explained by the properties of the spectrum of parasitic capillary waves bound to short steep gravity waves.

Wave Breaking

Although weak resonant and near-resonant interactions of weakly nonlinear waves occur over slow timescales, breaking is a fast process, lasting for times comparable to the wave period. However, the turbulence and mixing due to breaking may last for a considerable time after the event. Breaking, which is a transient, two-phase, turbulent, free-surface flow, is the least understood of the surface wave processes. The energy and momentum lost from the wave field in breaking are available to generate turbulence and surface currents, respectively. The air entrained by breaking may, through the associated buoyancy force on the bubbles, be dynamically significant over times comparable to the wave period as the larger bubbles rise and escape through the surface. The sound generated with the breakup of the air into bubbles is perhaps the dominant source of high frequency sound in the ocean, and may be used diagnostically to characterize certain aspects of air–sea interaction. Figure 4 shows examples of breaking waves in a North Atlantic storm.

Figure 4 Waves in a storm in the North Atlantic in December 1993 in which winds were gusting up to 50–60 knots and wave heights of 12–15 m were reported. Breaking waves are (A) large, (B) intermediate and (C) small scale. (Photographs by E. Terrill and W.K. Melville; reproduced with permission from Melville, (1996).)

Since direct measurements of breaking in the field are so difficult, much of our understanding of breaking comes from laboratory experiments and simple modeling. For example, laboratory experiments and similarity arguments suggest that the rate of energy loss per unit length of the breaking crest of a wave of phase speed c is proportional to ρg-1c⁵ with a proportionality factor that depends on the wave slope, and perhaps other parameters. Attempts are underway to combine such simple modeling along with field measurements of the statistics of breaking fronts to give an estimate of the distribution of dissipation across the wave spectrum. Recent developments in the measurement and modeling of breaking using optical, acoustical microwave and numerical techniques hold the promise of significant progress in the next decade.

See also

Breaking Waves and Near-Surface Turbulence. Heat and Momentum Fluxes at the Sea Surface. Internal Waves. Langmuir Circulation and Instability. Surface Films. Wave Energy. Wave Generation by Wind. Whitecaps and Foam.

Further Reading

Komen GJ, Cavaleri L, Donelan M, et al. Dynamics and Modelling of Ocean Waves. Cambridge: Cambridge University Press; 1994.

Lamb H. Hydrodynamics. New York: Dover Publications; 1945.

LeBlond PH, Mysak LA. Waves in the Ocean. Amsterdam: Elsevier; 1978.

Lighthill J. Waves in Fluids. Cambridge: Cambridge University Press; 1978.

Mei CC. The Applied Dynamics of Ocean Surface Waves. New York: John Wiley; 1983.

Melville WK. The role of wave breaking in air–sea interaction. Annual Review of Fluid Mechanics. 1996;28:279–321.

Phillips OM. The Dynamics of the Upper Ocean. Cambridge: Cambridge University Press; 1977.

Whitham GB. Linear and Nonlinear Waves. New York: John Wiley; 1974.

Yuen HC, Lake BM. Instability of waves on deep water. Annual Review of Fluid Mechanics. 1980;12:303–334.

Wave Generation by Wind

J.A.T. Bye    The University of Melbourne, Melbourne, VIC, Australia

A.V. Babanin    Swinburne University of Technology, Melbourne, VIC, Australia

Introduction

The prime focus in this article is on ocean waves (which have always captured the scientific imagination), although results from wind-wave tank studies are also introduced wherever appropriate. Growth mechanisms fall naturally into three phases: (a) the onset of waves on a calm sea surface, (b) mature growth in the confused sea state under moderate winds, and (c) sea-spray-dominated wave environments under very high wind speeds. Of these three phases, (b) has the greatest general importance, and numerous practical formulas have been developed over the years to represent its properties. Figure 1 illustrates the sea state which occurs at the top end of phase (b) in a strong gale (wind speed c. 25 ms-1, Beaufort force 9).

Figure 1 The sea state during a strong gale.

An important consideration is that wave generation by wind involves three main physical processes: (1) direct input from the wind, (2) nonlinear transfer between wavenumbers, and (3) wave dissipation. This article is specifically dedicated to (1); however, we briefly review (2) and (3) below.

Nonlinear interactions within the wave system can only be neglected for infinitesimal waves. To a first approximation, the wind wave can be regarded as almost sinusoidal with negligible steepness (i.e., linear), but its very weak mean nonlinearity (i.e., finite steepness and deviation of its shape from the sinusoid) is generally believed to control the evolution of the wave field. Theoretical models of the air–sea boundary layer indicate that the input of momentum from the wind is centered in the short gravity waves. The wind pumps energy mostly into short (high-frequency) and slowly moving waves of the wave field which then transfer this energy across the continuous spectrum of waves of all scales mainly toward longer (lower-frequency) components, which may be traveling at speeds close to the wind speed, thus allowing them to grow into the dominant waves of frequencies close to the peak frequency of the wave (energy) spectrum. The transfer of energy toward shorter (higher-frequency) waves where it is dissipated occurs at a much less significant rate.

Wave breaking is the major player in the third important mechanism, which drives wave evolution – wave energy dissipation. The Southern Ocean has the greatest potential for wave growth due to the never ceasing progression of intense storm systems over vast expanses of sea surface, unimpeded by land masses. Yet, wave models (http://www.knmi.nl/waveatlas/) indicate that the significant wave height (the average crest-to-trough height of the one-third highest waves) rarely goes beyond 10 m. The process, which controls the wave growth, is the dissipation by wave breaking, and to a lesser extent radiation of wave energy away from the storm centers, and into the adjacent seas.

Theories of Wave Growth

Phase (a): The Onset of Waves

We consider firstly the initial generation of waves over a flat water surface, independently of the simultaneous generation of a surface drift current. The key theoretical result is that the initial wavelength which can be excited on the air-water interface is a wave of wavelength 17 mm, which is the capillary gravity wave of minimum phase speed 230 mms–1, controlled by gravity and surface tension. The classical Kelvin-Helmholtz analysis completed in 1871, which relies on random natural disturbances present on the water surface, shows that this wave can only be excited by a velocity shear across the sea surface exceeding 6.5 ms–1.

Observations, however, show that waves are generated at much lower wind speeds, of order 1–2 ms–1. In order to resolve this dilemma, another mechanism was proposed by Phillips in 1957. It takes into account the turbulent structure of wind flow. Turbulent pressure pulsations in the air create infinitesimal hollows and ridges in the water surface, which, once the pressure pulsation is removed, may start propagating as free waves (similarly to the waves from a thrown stone). If the phase speed of such free waves is the same as the advection speed of the pressure pulsations by the wind, a resonant coupling can occur which will then lead these waves to grow beyond the infinitesimal stage. The first wave to be generated as the wind speed increases is likely to be the wave of minimum phase speed, propagating at an angle to the wind direction. Laboratory observations indicate that at slightly higher wind speeds, wave growth results from a shear flow instability mechanism. These two processes acting in the open ocean give rise to cat’s paws, which are groups of capillary-gravity wavelets (ripples) generated by wind gusts.

These results are applicable for clean water surfaces. In the presence of surfactants (surface-active agents), which lower the surface tension, ripple growth is inhibited, and at a sufficiently high surfactant concentration it may be totally suppressed. Phytoplankton are a major source of surfactants that produce surface films, and hence slicks, which are regions of relatively smooth sea surface.

Phase (b): Mature Growth

Once the finite-height waves exist, other and much more efficient processes take over the air–sea interaction.

Jeffreys in 1924 and 1925 pioneered the analytical research of the wind input to the existing waves by employing effects of the wave-induced pressure pulsations in the air. Potential theory predicts such pressure fluctuations to be in antiphase with the waves, which results in zero average momentum/energy flux. Jeffreys hypothesised a wind-sheltering effect due to presence of the waves which causes a shift of the induced pressure maximum toward the windward wave face and brings about positive flux from the wind to the waves.

The original theory of Jeffreys was based on an assumed phenomenon of the air-flow separation over wave crests. Experiments conducted between 1930 and 1950 with wind blown over solid waves found such an effect to be small and the theory fell into a long disrepute. Jeffreys’ sheltering ideas are now coming back, with both experimental and theoretical evidence lending support to his qualitative conclusions.

The period from 1957 until the beginning of the new century was dominated by the Miles theory (MT) of wave generation. This linear and quasi-laminar theory, originally suggested by Miles, was later modified by Janssen to allow for feedback changes of the airflow due to growing wind-wave seas. MT regards the air turbulence to be important only in forming the mean boundary-layer wind profile. In such a profile, a critical height exists where the wind speed equals the phase speed of the waves (Figure 2). Wave-induced air motion at this height leads to water-slope-coherent air-pressure perturbations at the water surface and hence to energy transfer to the waves.

Figure 2 Mean streamlines in the turbulent flow over waves according to the MT, in a frame of reference moving with the wave. The critical layer occurs at the height (Z ) where the wave speed (C ) equals the wind speed ( U(Z )). Reproduced from Phillips OM (1966) The Dynamics of the Upper Ocean, figure 4.3. Cambridge, UK: Cambridge University Press, with permission from Cambridge University Press.

MT however fails to comprehensively describe known features of the air–sea interaction. For example, for adverse winds the critical height does not exist and therefore no wind-wave energy transfer is expected, but attenuation of waves by such winds is observed. Therefore, a number of nonlinear and fully turbulent alternatives have been developed over the past 40 years.

One of the most consistent fully turbulent approaches is the two-layer theory first suggested by Townsend, and advanced by Belcher and Hunt (TBH). TBH revives the sheltering idea in a new form: by considering perturbations of the turbulent shear stresses, which are asymmetric along the wave profile. While still in need of experimental verification, particularly for realistic non-monochromatic three-dimensional wave fields, this theory has been extensively and successfully utilized in phase-resolvent numerical simulations of the air–sea interaction by Makin and Kudryavtsev. TBH and similar theories attract serious attention because the nature of the air–sea interface is often nonlinear and always fully turbulent.

Air–sea interaction is also superimposed by a variety of physical phenomena, which alter the wave growth. Wave breaking appears to cause air-flow separation, which brings the ideas of Jeffreys back in their original form; and gustiness and non-stationarity of the wind, the presence of swell and wave groups, nonlinearity of wave shapes, modulation of surface roughness by the longer waves have all been found to cause either a reduction or an enhancement of the wind-wave input.

These processes of active wave generation give rise to the windsea in which a simple measure of the sea state, relevant to wave growth, is the wave age (c/u*) where c is the wave speed of the dominant waves, and u* is the friction velocity in the air (the square root of the wind stress divided by the air density). The age of the windsea increases with fetch (the distance from the coast over which the wind is blowing), and the windsea becomes ‘fully developed’, that is, the energy flux from the wind and the dissipation flux are in balance, at a wave age of about 35. Empirical relations for the properties of the fully developed sea in terms of the wind speed (U) at 10 m (approximately the height of the bridge on large ships) given by Toba are: Hs=0.30 U²/g and Ts=8.6 U/g in which Ts (=2πc/g) is the significant wave period and g is the acceleration of gravity. As the fetch increases, Hs and Ts both increase toward their fully developed values, and the wave spectrum spreads to lower frequencies. Older seas of wave age greater than 35 can also exist after the wind has moderated.

The observations of the velocity structure in the atmospheric boundary layer by Hristov, Miller, and Friehe have shown directly the existence of the MT critical layer mechanism for fast-moving waves of wave age about 30. It is not yet known whether it operates for younger wave age, where a quasi-laminar theory may not be appropriate.

Phase (c): Very High Wind-Speed Wave Environments

The processes discussed in the previous two subsections are all grounded in two-layer fluid dynamics in which there exists a sharp interface between the two fluids. In recent times, it has been realized that this model is inadequate, especially at very high wind speeds. The link between the two phases is the breaking wave. In moderate winds (less than about 25 ms–1) the sea state is characterized by whitecapping due to the production of foam in a roller on the wave crests, and also foam streaks on the sea surface (Figure 1), whereas at very high wind speeds (greater than about 30 ms–1, Beaufort force 12) the air is filled with foam.

This transition arises from the structure of the breaking waves. In moderate winds, the roller remains attached to the parent wave and dissipates by the formation of foam streaks down its forward face, the trailing face of the wave remaining almost foam free. In this situation the airflow separates over the troughs and reattaches at the crests of the wave, producing Jeffreys-like phase shifts between the pressure and the underlying wave surface which enhance the energy flux to the wave. At very high wind speeds, on the other hand, the foam detaches from the wave crests, and is jetted forward into the air where it disperses vertically and horizontally before returning to the water surface. This process implies a return of momentum to the atmosphere, and hence the sea surface drag coefficient (which is an overall measure of the efficiency of momentum transfer from the atmosphere to the ocean both to waves and turbulence), which has been rising in phase (b), becomes ‘capped’ and possibly even reduces in phase (c). The all-pervasive presence of spray in extreme winds has prompted the anecdotal statement that ‘‘in hurricane conditions the air is too thick to breathe and too thin to swim in.’’

In summary, at very high wind speeds, the airflow effectively streams over the wave elements, which are reduced to acting as sources of spray. The spray then stabilizes the wind profile, and caps the sea surface drag coefficient, and interestingly, this feedback most likely allows the hurricanes to exist in the first place. This analysis has been greatly stimulated by the dropwindsonde observations of Powell, Vickery, and Reinhold in which wind profiles in hurricanes were measured for the first time, and also subsequently by experiments in high-wind-speed wind-wave tanks.

Experiments and Observations

Direct Measurements of Wave Growth Rates

The wind-to-wave energy input, which for each wave component is proportional to the time average of the product of the sea surface slope and sea surface atmospheric pressure, is the only source function, responsible for wave development, which can so far be measured directly, although this is an extremely difficult experimental task, and only a handful of attempts have been undertaken. The principal theoretical difficulty is that the sea surface atmospheric pressure must be estimated by extrapolating downward from the measurement level.

The pressure pulsations of interest are of the order of 10–5 —10–4 of the mean atmospheric pressure and therefore require very sensitive probes. The surface-coherent oscillations are superposed, at the same frequencies, by random turbulent fluctuations, which are tens and hundreds of times greater in magnitude. This implies that a sophisticated data analysis is required to separate the signal buried in the noise. The wave-induced pressure decays rapidly away from the wavy surface and thus, particularly for short wave scales, it has to be sensed very close to the surface, below the wave crests of dominant waves. At the same time, the air-pressure probes have to stay dry. The last requirement leads either to measurements being conducted above the crests, which limits the estimates to the amplification of the dominant waves only, or to the use of a wave-following technique. The latter has a limited capability beyond the laboratory conditions and involves further complications due to multiple corrections needed to recover the signal contaminated by air motion in the tubes connecting the pressure probes with pressure transducers.

The first field experiment of the kind, conducted by Snyder and others in 1981, resulted in a parameterization of wind input across the wave spectrum, which has been frequently used until now. Most of these measurements, however, were taken by stationary wave probes above the wave crests, and the winds involved were very light, mostly around 4 ms–1. Waves at such winds are known not to break, and this fact implies an air–sea energy balance, very different from that at moderate and strong winds. Therefore, extrapolation of these results into normal wave conditions has to be exercised with great caution.

Another field experiment was conducted by Hsiao and Shemdin in 1983. It used a wave-following technology and thus was able to obtain a spectral set of measurements somewhat beyond the dominant wave scales. This study produced a parameterization in which the growth rates were very low. Its drawback comes from the fact, that in the majority of circumstances the measured waves were ‘quite old’, half of the records being above the limit for the fully developed windsea. For such waves, the growth rates are expected to be very small if not zero, and given the measurement and analysis errors, the interpretation of the low growth values becomes quite uncertain.

On the other hand, a set of precision wave-following measurements conducted by Donelan in 1999 in a wind-wave tank where the waves were very young (c/u*≈1), produced a growth rate 2.5 times that of Hsiao and Shemdin’s, and also demonstrated a very significant wave attenuation rate by the adverse wind.

The differences between these two data sets stimulated the latest campaign undertaken by Donelan and others in 2006. The Lake George experiment in Australia employed precision laboratory instruments in a field site. The site was chosen such that it provided a variety of wind-wave conditions, including very strongly wind-forced and very steep waves normally unavailable for measuring in the open ocean. The results revealed some new properties of the air–sea interaction, in which wave growth rates merged with previous results at moderate winds, but deviated significantly in strong wind conditions with continually breaking steep waves, in which full flow separation, that is, detachment of the streamlines of the airflow at the wave crest and reattachment well up the windward face of the preceding wave, occurred leading to a reduction of the wind input. This reduction means that as the winds become stronger the wind-to-wave input will keep growing, but the growth rates will be reduced compared to simple extrapolations to extreme conditions of the input measured at moderate winds. This behavior, which is consistent with that in very high wind speeds in the open ocean, did not appear to be associated with spray production.

It is worth mentioning that one of the key properties of the wind input – its directional distribution – has never been measured. It was assumed to be a cosine function by Plant, but no data on the wind input directional distribution are available. Such measurements cannot be made adequately in a wind-wave tank, and are a formidable task in the field where a spatial array of wave-following pressure probes would have to be operated. Directional wave input distribution, nevertheless, is an integral part of any wave forecast model and therefore this problem remains a major challenge for the experimentalists.

Reverse Momentum Transfer

Nonlinear interactions transfer energy to the longer, faster-propagating waves, which after leaving the region of generation are known as swell. The swell may travel at a speed greater than the local wind speed, and even propagate in the opposite direction to the wind, leading to the possibility of reverse momentum transfer from the waves to the wind.

The direct effect of waves propagating faster than the wind has been measured in a wind-wave tank by Donelan; however, when the results were applied to swell propagating in the ocean, the damping effect was found to be much too large. This is well known to surfers, who rely on the arrival of swells from distant storms: their propagation across the Pacific Ocean (over a distance of c. 10 000 km) was measured in a classical campaign conducted by Snodgrass and others in 1963.

Reverse momentum transfer has also been observed in wind profiles. In Lake Ontario, while a swell was running against a very light wind, the wind speed increased downward (toward the sea surface) due to the propagation of the swell, rather than the normal decrease. This is a clear example of reverse momentum transfer arising from the presence of a wave train of nonlocal origin. Reverse momentum transfer, however, is a ubiquitous process in windseas in which part of the wind input is returned to the atmosphere by the dissipation process, especially the injection of spray.

Numerical Modeling of the Wind Input

Over the past few decades, numerical modeling of ocean waves has developed into a largely independent field of study. Two different kinds of models have been used to study the wind input. Historically, spectral models based on known physics were the first. Their progress is described in great detail in the book by Komen and others. Given the uncertainties of such predictions due to simultaneous action of the multiple wave dynamics processes, the capacity of such models to scrutinize the wind input function is limited. For example, very high quality synoptic analyses of weather systems are necessary in order to discriminate between the various coupling mechanisms for wave growth by comparing observational wave data from wave buoys with the predictions of coupled wind-wave simulations. In these models the formulation of the wave energy dissipation is based on tuning the total energy balance. Csanady, in a lucid textbook on air–sea interaction, notes that in a fetch-limited windsea only about 6% of the momentum transferred from the wind to the water supports the downwind growth of the dominant waves, the remainder being accounted for locally by the dissipation stress, that is, the rate of loss of momentum from the wave field to the ocean.

The phase-resolvent models are another kind of numerical simulations of air–sea interaction, which reproduce wind input and wave evolution in physical rather than wavenumber space. Such models solve the basic fully nonlinear equations of fluid mechanics explicitly and recent advances in numerical techniques allow us to reproduce the water surface, airflow, and wave motion with potentially absolute precision and unlimited temporal and spatial resolution. Use of such models to forecast waves globally is obviously not feasible, but they now constitute a very effective tool for dedicated studies of wind–wave interaction. The interested reader is referred to recent research by Makin and Kudryavtsev, and by Chalikov and Sheinin.

Conclusions

It is clear from this article that there are still many tasks ahead to fully understand wave generation by wind. The nonlocal aspects of wave generation by wind are a particularly challenging topic. Contemporary interest lies with our climate system. The interface between the atmosphere and the ocean is vital in this regard. This holistic view calls urgently for further study, especially of extreme events in which major momentum transfers occur, affecting the land through the initiation of hurricanes, and the sea through mixing below the wave boundary layer into the deep ocean.

See also

Breaking Waves and Near-Surface Turbulence. Surface Gravity and Capillary Waves.Tsunami. Wind- and Buoyancy-Forced Upper Ocean.

Further Reading

Belcher SE, Hunt JCR. Turbulent flow over hills and waves. Annual Review of Fluid Mechanics. 1998;30:507–538.

Bye JAT, Jenkins AD. Drag coefficient reduction at very high wind speeds. Journal of Geophysical Research. 2006;111: C03024 (doi:10.1029/2005JC003114).

Chalikov D, Sheinin D. Modeling extreme waves based on equations of potential flow with a free surface. Journal of Computational Physics. 2005;210:247–273.

Csanady GT. In: Air–Sea Interaction Laws and Mechanisms. Cambridge, UK: Cambridge University Press; 2001:239.

Donelan MA. Wind-induced growth and attenuation of laboratory waves. In: Sajjadi SG, Thomas NH, Hunt JCR, eds. Wind-Over-Wave Couplings:Perspectives and Prospects. Oxford, UK: Clarendon; 1999:183–194.

Donelan MA, Babanin AV, Young IR, Banner ML. Wave follower measurements of the wind input spectral function. Part 2: Parameterization of the wind input Journal of Physical Oceanography. 2006;36:1672–1688.

Hristov T, Friehe C, Miller S. Dynamical coupling of wind and ocean waves through wave-induced air flow. Nature. 2003;422:55–58.

Jones ISF, Toba Y, eds. Wind Stress over the Ocean. Cambridge, UK: Cambridge University Press; 2001:307.

Komen GI, Cavaleri L, Donelan M, Hasselmann K, Hasselmann S, Janssen P.A.E.M. In: Dynamics and Modelling of Ocean Waves. Cambridge, UK: Cambridge University Press; 1994:532.

Kudryavtsev VN, Makin VK. Aerodynamic roughness of the sea surface at high winds. Boundary-Layer Meteorology. 2007;125:289–303.

Makin VK, Kudryavtsev VN. Wind-over-waves coupling. In: Sajjadi SG, Hunt LJ, eds. Wind Over Waves II: Forecasting and Fundamentals of Applications. Chichester: Horwood Publishing; 2003:46–56.

Phillips OM. In: The Dynamics of the Upper Ocean. Cambridge, UK: Cambridge University Press; 1966:507–538.

Powell MD, Vickery PJ, Reinhold TA. Reduced drag coefficient for high wind speeds in tropical cyclones. Nature. 2003;422:279–283.

Snodgrass FE, Groves GW, Hasselmann KF, Miller GR, Munk WH, Powers WH. Propagation of ocean swell across the Pacific. Philosophical Transactions of the Royal Society of London. 1966;259:431–497.

Toba Y. Local balance in the air–sea boundary processes. Part I: On the growth process of wind waves Journal of the Oceanographical Society of Japan. 1972;28:15–26.

Young IR. In: Wind Generated Ocean Waves. Oxford, UK: Elsevier; 1999:507–538.

Relevant Website

http://www.knmi.nl/waveatlas –The KNMI/ERA-40 Wave Atlas.

Rogue Waves

K. Dysthe    University of Bergen, Bergen, Norway

H.E. Krogstad    NTNU, Trondheim, Norway

P. Müller    University of Hawaii, Honolulu, HI, USA

Introduction

The terms ‘rogue’ or ‘freak’ waves have long been used in the maritime community for waves that are much higher than expected, given the surrounding sea conditions. For the seafarer these unexpected waves represent a frightening and often life-threatening experience. There are many accounts of such waves hitting passenger and container ships, oil tankers, fishing boats, and offshore and coastal structures, sometimes with catastrophic consequences. It is believed that more than 22 supercarriers were lost to rogue waves between 1969 and 1994 (Figure 1). Rogue waves are not mariners’ tales. They have been observed and documented, most succinctly from oil platforms. Two well-studied examples are the Draupner ‘New Year’s Wave’ and the Gorm platform waves discussed below (Figure 2).

Figure 1 Locations of 22 supercarriers assumed to be lost after collisions with rogue waves between 1969 and 1994. © C. Kharif and E. Pelinovsky. Used with permission.

Figure 2 Two examples of rogue waves. ‘Gorm’ is one of the abnormal waves recorded at the Gorm field in the North Sea on 17 Nov.1984. The wave that stands out has a crest height of 11 m, which exceeds the significant wave height of 5 m by a factor of 2.2. ‘Draupner’ is the ‘New Year Wave’ recorded at the Draupner platform in the North Sea 1 Jan. 1995. The crest height is about 18.5 m and exceeds the significant wave height of 11.8 m by a factor of 1.54. Reprinted, with permission, from the Annual Review of Fluid Mechanics , Volume 40 © 2008 by Annual Reviews.

With their sometimes catastrophic impact the motivation for investigating rogue waves is clear, and the scientific community has studied the topic for some time, more intensely since 2000.

Despite these efforts, there is no generally accepted explanation or theory for the occurrence of rogue waves. There is even no consensus of how to define a rogue wave. Some of the inherent difficulties are related to the random nature on ocean waves: a wave recording will show waves of different sizes and shapes. In discussing rogue waves one introduces the notion of ‘one wave’, which is the recorded elevation over one wave period, containing one crest and one trough. One also distinguishes between the wave height H (the distance from trough to crest) and the crest height ƞcr (the distance from mean sea level to crest). Early wave statistics from the 1950s suggests that the most probable maximum wave height, HM, in a wave record containing N waves is given by

   [1]

where Hs is the significant wave height defined as four times the standard deviation of the surface elevation. (The old definition of significant wave height as the mean of the one-third largest waves, H1/3, is approximately 5% lower than Hs.) Thus, as the duration of the wave record increases, the