Transfer of energy from large to small scales in turbulent flows is described as a flux of energy from small wave numbers to large wave numbers in the spectral representation of the Navier–Stokes equation (1.17). The problem of resolving the relevant scales in the flow corresponds in the spectral representation to determining the spectral truncation at large wave numbers. The effective number of degrees of freedom in the flow depends on the Reynolds number. The Kolmogorov scale η depends on Reynolds number as η ∼ Re−¾ (1.11), so the number of waves N necessary to resolve scales larger than η grows with Re as N ∼ η−3 ∼ Re9/4. This means that even for moderate Reynolds numbers ∼ 1000, the effective number of degrees of freedom is of the order of 107. A numerical simulation of the Navier–Stokes equation for high Reynolds numbers is therefore impractical without some sort of reduction of the number of degrees of freedom. Such a calculation with a reduced set of waves was first carried out by Lorenz (1972) in the case of the vorticity equation for 2D turbulence.

The idea is to divide the spectral space into concentric spheres, see Figure 3.1. The spheres may be given exponentially growing radii kn = λn, where λ > 1 is a constant. The set of wave numbers contained in the nth sphere not contained in the (n − 1)th sphere is called the nth shell.

Fluids have always fascinated scientists and their study goes back at least to the ancient Greeks. Archimedes gave in “On Floating Bodies” (c. 250 BC) a surprisingly accurate account of basic hydrostatics. In the fifteenth century, Leonardo da Vinci was an excellent observer and recorder of natural fluid flows, while Isaac Newton experimented with viscosity of different fluids reported in Principia Mathematica (1687); it was his mechanics that formed the basis for describing fluid flow. Daniel Bernoulli established his principle (of energy conservation) in a laminar inviscid flow in Hydrodynamica (1738). The mathematics of the governing equations was treated in the late eighteenth century by Euler, Lagrange, Laplace, and other mathematicians. By including viscosity the governing equations were put in their final form by Claude-Louis Navier (1822) and George Gabriel Stokes (1842) in the Navier–Stokes equation. This has been the basis for a vast body of research since then.

The engineering aspects range from understanding drag and lift in connection with design of airplanes, turbines, ships and so on to all kinds of fluid transports and pipeflows. In weather and climate predictions accurate numerical solutions of the governing equations are important. In all specific cases when the Reynolds number is high, turbulence develops and the kinetic energy is transferred to whirls and waves on smaller and smaller scales until eventually it is dissipated by viscosity. This is the energy cascade in turbulence.

Fully developed turbulence is the notion of the general or universal behavior in any physical situation of a violent fluid flow, be it a dust devil or a cyclone in the atmosphere, the water flow in a white-water river, the rapid mixing of the cream and the coffee when stirring in a coffee cup, or perhaps even the flow in gigantic interstellar gas clouds. It is generally believed that the developments of these different phenomena are describable through the Navier–Stokes equation with suitable initial or boundary conditions. The governing equation has been known for almost two centuries, and a lot of progress has been achieved within practical engineering in fields like aerodynamics, hydrology, and weather forecasting with the ability to perform extensive numerical calculations on computers. However, there are still fundamental questions concerning the nature of fully developed turbulence which have not been answered. This is perhaps the biggest challenge in classical physics. The literature on the subject is vast and very few people, if any, have a full overview of the subject. In the updated version of Monin and Yaglom's classic book the bibliography alone covers more than 60 pages (Monin & Yaglom, 1981).

The phenomenology of turbulence was described by Richardson (1922) and quantified in a scaling theory by Kolmogorov (1941b). This description stands today, and has been shown to be basically correct by numerous experiments and observations.

Chaos can be observed in simple nonlinear Hamiltonian systems. This is a dynamical system governed by Hamilton's equations where the energy is conserved, such as a physical pendulum or double pendulum. The phase space portrait of the trajectories of this kind of system can, even with few degrees of freedom, be very complicated. The phase space flow fulfils Liouville's theorem, which states that phase space volume is conserved. Another type of chaotic dynamics can arise in non-autonomous systems, like the Duffing oscillator, where a simple nonlinear system is influenced by an external periodic force. A third kind of chaotic system is nonlinear dissipative systems, such as the Lorenz (1963) model, which has only three degrees of freedom. The Lorenz model was derived from the set of ordinary differential equations describing development of wave amplitudes in the spectral representation of Rayleigh–Bernard convective flow. The Lorenz model is equivalent to a spectral truncation where only the first three wave numbers are represented. The phase space portrait of dissipative systems is different from that of Hamiltonian systems because the energy dissipation implies a shrinking of phase space volume. The dynamics of such a system is described in phase space by strange attractors. Strange attractors are sets in phase space of states un which are invariant with respect to the dynamical equation. This means that an initial state un(0) belonging to the attractor will develop along a trajectory which will stay within the attractor.

Intermittency in turbulence is a topic which has been actively investigated for several decades, and a major part of Frisch's book (1995) is devoted to the subject.

Dynamical systems are often characterized by long quiescent periods interrupted by bursts of activity. This kind of dynamics is called intermittent. A way of quantifying this could be by high pass filtering the dynamical signal. If the signal has purely Gaussian statistics, which would be natural for a system of many degrees of freedom, high pass filtering is a linear operation and the high pass filtered signal would be Gaussian as well. If the high pass signal differs from the Gaussian by having heavier tails, it is intermittent. Thus intermittency could be formally defined by the deviation from Gaussian statistics. In this case, intermittency could be a sign of dynamics not merely governed by simple statistics given by the central limit theorem or equilibrium statistical mechanics. In the case of a turbulent velocity field, high pass filtering roughly corresponds to extracting information on velocity differences below or at some cutoff length scale. As described previously, the statistics of velocity increments in turbulence is found not to be Gaussian. The self-similarity of the flow assumed in K41 theory is not valid and there will be corrections to the scaling exponents for the moments of the velocity increments as expressed in (1.64).

Kolmogorov's lognormal correction

The dynamical origin of the deviation from the K41 value for the scaling exponents is a very non-uniform distribution of the energy dissipation.

Introduction

So far we have focused mainly upon the dynamics of waves in uniform and stationary ambient fluids. In the case of interfacial waves, we assumed that horizontal boundaries, if present, were flat so that the depth of the ambient was constant. Since interfacial wave speeds are a function of depth the question arises as to how the waves propagate in a medium that, for example, gets shallower approaching a beach. Likewise, for internal waves we have usually assumed that the background stratification is uniform. But we have seen that the stratification varies vertically in both the atmosphere and ocean. Except in the consideration of unstable shear flows, for the most part we have also neglected the presence of background winds and currents. In this chapter we examine how waves propagate in media that are non-uniform in the sense that the depth or stratification and the background flow varies.

We begin with the general treatment of small-amplitude wave propagation in non-uniform, but slowly varying media. This is known as ray theory. A simplified set of equations arising from ray theory assumes the motion is two-dimensional and steady, and that the background varies in only one spatial dimension. The theory is applied to study surface and interfacial waves approaching a beach and to examine internal waves approaching critical and reflection levels. In a somewhat different mathematical approach, the study of tunnelling examines the partial transmission and reflection of small-amplitude internal waves through weakly stratified regions.

Why write a book on internal gravity waves when so many other books cover the subject already? The textbooks listed in the appendix include at least some discussion of internal gravity waves. Some focus upon interfacial waves, which are internal gravity waves at interfaces; some focus upon internal waves, which exist in continuously stratified fluid. Different books emphasize different dynamics such as mechanisms for generation, propagation in non-uniform media, nonlinear evolution and stability. Textbooks on geophysical fluid dynamics (e.g. Gill (1982), Vallis (2006)) understandably devote only a chapter to the subject because, although internal waves are non-negligible in their influence upon global weather and ocean circulation patterns, they are by no means dominant. Internal waves are noise, if sometimes irritatingly loud. Textbooks on the theory of waves and instability (e.g. Whitham (1974), Lighthill (1978), Drazin and Reid (1981), Craik (1985)) examine how non-uniform media and nonlinearity affect the evolution of interfacial and internal waves. But these books can be daunting to graduate students lacking strong mathematical backgrounds. Textbooks on stratified fluid dynamics (e.g. Turner (1973), Baines (1995)) help to provide physical insight into the dynamics of internal gravity waves through a combination of theory and laboratory experiments, though sometimes without providing the mathematical details. Some textbooks are devoted to the subject of internal gravity waves (e.g. Miropol'sky (2001), Nappo (2002), Vlasenko et al. (2005)), but these focus either on atmospheric or oceanic waves.

Introduction

In this chapter we examine the various mechanisms through which interfacial waves and internal waves can be created. Broadly speaking, the waves can be generated either by solid bodies or by disturbances within the fluid such as convection, imbalance of large-scale flows, gravity currents and turbulence. Here we focus upon generation by solid bodies, the theory for which is better established.

In a uniformly stratified fluid, we have already seen that an oscillating body creates a cross-shaped pattern of waves. Here, we study this problem in more detail by examining the structure of the wave beams that emanate from a vertically oscillating cylinder and sphere. Although this mechanism has little bearing on geophysical flows, it is amenable to study in laboratory experiments which can test the limitations of theory applied to this seemingly simple circumstance.

More realistic is the study of uniform and tidal flow over topography. This subject has been well studied in theory and by way of experiments, and the results have been compared with ample observational data. Because much of this work is discussed elsewhere, only the salient points will be presented here.

Oscillating bodies

In Section 3.3 we saw that any body oscillating with frequency ω in uniformly stratified fluid with buoyancy frequency N0 creates a cross-shaped pattern of internal waves if ω < N0. Here we examine how the structure of the wave beams is affected by the shape of the body in two and three dimensions.

Introduction

Outside of theoretical interest in their peculiar properties, one of the main motivations for understanding the dynamics of internal gravity waves is that they occur naturally in the atmosphere and oceans. In the atmosphere, through transporting horizontal momentum from the ground upwards, internal waves influence wind speeds and consequently the thermal structure of the atmosphere. By contrast, internal waves in the ocean are primarily important as they affect mixing through the transport of energy. Although internal waves do not play a dominant role in the evolution of weather and climate, their influence is non-negligible: numerical simulations that do not include the effects of internal waves predict wind speeds and temperature in the atmosphere incorrectly and they do not account for the observed levels of turbulent diffusivity in the oceans. At the mesoscale in the atmosphere internal wave breaking is a source of clear-air turbulence and in the ocean, internal solitary waves influence biological activity over continental shelves through mass transport and mixing.

This chapter begins with a brief introduction to stratified fluids and internal gravity waves and then gives an overview of the structure of the atmosphere and oceans with mention of internal gravity wave phenomena in these fluids. In the following sections, we derive the equations describing the motion and thermodynamics of fluids and then make approximations relevant to internal gravity wave dynamics. The derivations are sometimes heuristic, aiming to provide physical intuition rather than emphasizing rigour.