Waves in Geophysical Fluids

The giant tsunami that occurred in the Indian Ocean on 26th December 2004 draws attention to this natural phenomenon. The given course of lectures deals with the physics of the tsunami wave propagation from the source to the coast. Briefly, the geographical distribution of the tsunamis is described and physical mechanisms of their origin are discussed. Simplified robust formulas for the source parameters (dimension and height) are given for tsunamis of different origin. It is shown that the shallow-water theory is an adequate model to describe the tsunamis of the seismic origin; meanwhile for the tsunamis of the landslide or explosion origin (volcanoes, asteroid impact) various theories (from linear dispersive to nonlinear shallow-water equations) can be applied. The applicability of the existing theories to describe the tsunami wave propagation, refraction, transformation and climbing on the coast is demonstrated. Nonlinear-dispersive effects including the role of the solitons are discussed. The practical usage of the tsunami modeling for the tsunami forecasting and tsunami risk evaluation is described. The results of the numerical simulations of the two global tsunamis in the Indian Ocean induced by the catastrophic Krakatau eruption in 1883 and the strongest North Sumatra earthquake in 2004 are given.

There has been much interest in freak or rogue waves in recent years, especially after the Draupner “New Year Wave” that occurred in the central North Sea on January 1st 1995. From the beginning there have been two main research directions, deterministic and statistical. The deterministic approach has concentrated on focusing mechanisms and modulational instabilities, these are explained in Chap. 3 and some examples are also given in Chap. 4. A problem with many of these deterministic theories is that they require initial conditions that are just as unlikely as the freak wave itself, or they require idealized instabilities such as Benjamin-Feir instability to act over unrealistically long distances or times. For this reason the deterministic theories alone are not very useful for understanding how exceptional the freak waves are. On the other hand, a purely statistical approach based on data analysis is difficult due to the unusual character of these waves. Recently a third research direction has proved promising, stochastic analysis based on Monte-Carlo simulations with phase-resolving deterministic models. This approach accounts for all possible mechanisms for generating freak waves, within a sea state that is hopefully as realistic as possible. This chapter presents several different modified nonlinear Schrödinger (MNLS) equations as candidates for simplified phase-resolving models, followed by an introduction to some essential elements of stochastic analysis. The material is aimed at readers with some background in nonlinear wave modeling, but little background in stochastic modeling. Despite their simplicity, the MNLS equations capture remarkably non-trivial physics of the sea surface such as the establishment of a quasi-stationary spectrum with ω⁻⁴ power law for the high-frequency tail, and nonlinear probability distributions for extreme waves. In the end we will suggest how often one should expect a “New Year Wave” within the sea state in which it occurred.

The main physical mechanisms responsible for the formation of huge waves known as freak waves are described and analyzed. Data of observations in the marine environment as well as laboratory experiments are briefly discussed. They demonstrate that freak waves may appear in deep and shallow waters. The mathematical definition of these huge waves is based on statistical parameters, namely the significant height. As linear models of freak waves the following mechanisms are considered: Wave-current interaction and dispersion enhancement of transient wave packets. When the nonlinearity of the water waves is introduced, these mechanisms remain valid but should be modified and new mechanisms such as modulational instability and soliton-collisions become good candidates to explain the freak wave occurrence. Specific numerical simulations were performed within the framework of classical nonlinear evolution equations: The Nonlinear Schrödinger 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 freak wave phenomenon.

A novel fully nonlinear, rapid method for computations of ocean surface waves in three dimensions is outlined in this chapter. The essential step is to use Fourier transform to invert the integral equation over the ocean surface that solves the Laplace equation. This leads to a relation for the normal velocity of the free surface that is useful for iterations. This relation has a global contribution that is obtained by FFT and local contribution that is evaluated by rapidly converging integrals in the horizontal plane. The global part, evaluated by FFT, captures the most essential parts of the wave field. Together with an efficient time integration of the prognostic equations, where the linear part is integrated analytically and a time variable step size control is used for the nonlinear part, this results in a highly rapid computational strategy. Methods for efficient nonlinear wave generation and absorption are outlined. Conservation of various quantities of the wave field and convergence are discussed. Fully nonlinear computations of very large (rogue) wave events at sea, like the Camille and Drauper waves, are compared to laboratory measurements of the waves using Particle Image Velocimetry. The method is used to exemplify the generation, propagation and shoaling of very long wave phenomena like tsunamis, and class I and class II instabilities in water of infinite and finite depth. Both steady and oscillatory crescent wave patterns are predicted up to breaking. Competition between class I and class II instabilities is discussed.

This chapter describes the observations and modelling of very large internal waves in the ocean. We begin with a brief description of the dead-water phenomenon explained by Nansen, and the internal tides discovered by Pettersson. We continue by describing the very large oceanic solitary internal waves of depression observed in the field from 1965 and onwards. The main point of the mathematical analysis is to model the formation and propagation properties of the waves. Tidal generation of internal undular bores and solitary waves of amplitude comparable to the surface layer thickness at rest are exemplified by numerical simulations. The fully nonlinear and fully dispersive mathematical and numerical modelling is found to reproduce the wave motion taking place in the ocean, including the excursions of the mixed upper layer as large as 4–5 times the level at rest, as in the COPE experiment. The interface method — derived in two and three dimensions — is found to compare well with laboratory measurements of the waves using Particle Image Velocimetry. A fully nonlinear theory that accounts for the continuously stratified motion within the pycnocline is derived. The method is used to support experiments on internal wave breaking governed by shear. The latter model is also useful to compute internal wave motion when the upper part of the ocean is linearly stratified. The fully nonlinear modelling is put into perspective by deriving in parallel the weakly nonlinear Korteweg-de Vries, Benjamin-Ono and intermediate long wave equations. Recent observations in deep water reveal significant internal wave motion and corresponding strong bottom currents of magnitude about 0.5 m/s where the pycnocline meets the shelf slope. Future directions of internal wave research are indicated.

Different approaches to the study of internal waves in the ocean are analyzed. Generation of internal tides over submarine ridges is considered on the basis of numerical models and measurements in the ocean. Energy fluxes from submarine ridges exceed many times the fluxes from continental slopes because the dominating part of the tidal flow is directed parallel to the coastline. Submarine ridges if normal to the tidal flow form an obstacle that can cause generation of large internal waves. Internal tides are extreme when the depth of the ridge crest is comparatively small with respect to the surrounding depths. Energy fluxes from most submarine ridges were estimated. They account for approximately one fourth of the total energy loss from the barotropic tides. Model estimates were compared with the measurements on moorings at 30 study regions in the oceans. Combined calculations and measurements result in a map of global distribution of internal tide amplitudes. The study is extended to the Arctic region. Extreme internal tides were recorded near the Mascarene Ridge in the Indian Ocean, Mid-Atlantic Ridge in the South Atlantic, Great Meteor bank and the Strait of Gibraltar.