Building on the vertical slice model, both the extension to three-dimensional flow and the addition of Coriolis forces are important and non-trivial steps in classic wave—mean interaction theory. Specifically, in three dimensions we recover the generic 2 + 1 structure of the linear problem, in which we have two gravity-wave modes and one balanced or vortical mode controlled by the PV distribution. This leads to the generic importance of the zero-frequency PV mode for strong interactions.

The Coriolis forces lead to source terms in the horizontal momentum budgets and therefore to differences between Lagrangian and Eulerian fluxes of horizontal momentum. As noted before, this leads to the important definition of the vertical Eliassen—Palm flux or form stress in the context of wave drag computations. Once again, the zonal pseudomomentum plays a crucial role in formulating the interaction theory.

We briefly recall the governing equations and then look at the modifications of the linear dynamics, including that of pseudomomentum and its flux. This is followed by rotating three-dimensional lee waves and by a discussion of how to simplify the considerably more complicated mean-flow equations in this case. The key concept here is to focus on the vortical mode of the mean-flow response. Finally, because its intrinsic importance in idealized modelling, we discuss the vertical slice model with rotation.

Linear wave theory has a special place in applied mathematics. For example, the powerful concepts of linear wave theory, such as dispersion, group velocity or wave action conservation, are fundamental for describing the behaviour of solutions to many commonly occurring partial differential equations (PDEs). Also, whilst it is certainly not true that every linear wave problem has an explicit general solution, it is true that every linear problem can be approached by using linear thinking, i.e., by building up more complex solutions out of superpositions of simpler solutions. In some cases, this procedure can be carried to its logical conclusion and the complete general solution to a problem can be formulated as a sum over special solutions. For example, this works for PDEs with constant coefficients in a periodic domain, for which the general solution can be written as a sum of plane waves described mathematically by a Fourier series.

But even in cases where there is no explicit general solution, the possibility to develop special solutions using asymptotic methods and the ability to combine several simple solutions to form a more complex solution always deepens our understanding of the underlying problem, and such an improved understanding could then be used to aid a numerical simulation for situations of particular interest, for example. Thus time spent studying linear wave theory is time well spent.

For the revised edition

The happy occasion of the revised paperback printing made it possible to add a section on Langmuir circulations and the Craik-Leibovich instability to chapter 11. These are important and fundamental topics that ought to have been included already in the first edition. This new material also prompted significant changes in section 13.4 on the vorticity generated by breaking surface-gravity waves, which hopefully make this crucial topic more transparent. In addition, there are smaller changes such as high-lighting the amazing curl-curvature formula for wave ray tracing in a weak vortical mean flow in §4.4.3, as well as numerous small fixes and some additional references. A small number of exercises has also been added to various chapters, which hopefully will aid the educational aspects of this book.

I am exceptionally grateful to Michael McIntyre for his very detailed reading of the first edition and for his support in preparing this revised edition. Thanks are also due to Rick Salmon and William Young for their insightful suggestions and to David Tranah for his continued support at Cambridge University Press.

Finally, I would like to dedicate this edition to the memory of my father by the last words of Mahler's Lied der Erde: “Ewig, ewig”.

New York, March 2013.

The aim of this book

This book is on waves and on their interactions with mean flows such as shear flows or vortices.

It is convenient to start with a brief summary of fluid dynamics fundamentals in order to establish the mathematical notation and the physical concepts that will be used throughout this book. We will first look at the kinematics of fluid flow, especially at how to capture the evolution of material elements such as material points or lines.

This is followed by a description of perfect fluid dynamics, which is the natural point of departure for the study of flows at very high Reynolds numbers in the atmosphere and the ocean. In these flows the direct influence of viscous forces is confined to boundary layers and to sparse pockets of three-dimensional turbulence within the fluid.

The culmination of perfect fluid dynamics is Kelvin's circulation theorem and the various links of this theorem to vorticity dynamics. Indeed, as we go on it will become increasingly clear that the circulation theorem is also the key result in wave–mean interaction theory.

Flow kinematics

In continuum fluid mechanics the molecular structure of the fluid is ignored and the description of the physical state of the fluid is accomplished by specifying a finite number of flow fields as functions of position x and time t, say. How many fields are needed depends on the complexity of the fluid under consideration, but all fluid flows require a working mass and momentum budget, which leads to the definitions of the density and velocity fields.

The similarities and differences between zonal-mean theory and local averaging are illustrated nicely by the problem of wave-driven circulations in the nearshore regions on beaches. This problem is both significant in coastal oceanography as well as directly observable in everyday life, which is an attractive feature.

We first describe the classic theory of wave-driven longshore currents, which is based on zonal averaging and simple geometry, and then we consider the changes in the problem once localized wavetrains are allowed. This will lead to a discussion of vorticity generated by breaking waves and also to a consideration of vortex dynamics in a sloping domain, which are interesting fluid-dynamical topics in their own right.

We conclude with a consideration of how the long-term mean-flow behaviour may differ significantly from the predictions of classic theory in the presence of non-trivial topography features such as barred beaches.

Wave-driven longshore currents

The basic situation is as envisaged in the left panel of figure 13.1: looking down on the xy-plane ocean waves are obliquely incident from the left on a beach with a straight shoreline located at x = 0, say. The waves are refracted and turned towards the shoreline by the decreasing water depth as the shoreline is approached. To fix terminology, the x-direction is called the cross-shore direction and the y-direction is called the longshore direction.

This is the classic body of wave-mean interaction theory that has been developed extensively in atmospheric fluid dynamics since the late 1960s. Here ‘zonal symmetry’ refers to basic flows that are independent of longitude, which is a natural starting point for analysing large-scale atmospheric flows. As we know, such basic flows induce a conservation law for the zonal component of the pseudomomentum vector, and much of the classic interaction theory is focused on the interplay between zonal pseudomomentum and the zonal component of the mean velocity field.

Many interesting and powerful results are available in this theory, such as so-called ‘non-acceleration conditions’, which provide criteria for whether the presence of waves may lead to an acceleration of the zonal mean flow. Another example is the ‘pseudomomentum rule’, which makes precise the impact of wave dissipation on mean-flow acceleration.

Against these obvious successes of the classic theory must be weighed its obvious restrictions to zonally symmetric basic flows. For instance, it has been much harder to apply this theory in the ocean, where mean circulations (with few exceptions such as the Antarctic circumpolar current) are hemmed-in by the continents and therefore are manifestly not independent of longitude. This problem is compounded by the fact that many results of the zonally symmetric theory do not apply even approximately in a situation with a slowly varying mean flow. We will go beyond zonal symmetry in part THREE of this book.

We now consider wave refraction due to velocity strain and shear associated with vortical mean flows. Such refraction changes the waves' pseudomomentum field and, arguably, the central topic of wave-mean interactions outside simple geometry is how such pseudomomentum changes are related to the leading-order mean-flow response. The same question was satisfactorily answered in simple geometry by the pseudomomentum rule. However, refractive changes in the pseudomomentum do not rely in any essential way on wave dissipation or external forces, and yet they can irreversibly change the total amount of pseudomomentum in the wave field. This makes clear that the usual pseudomomentum rule of simple geometry, which equates such changes to an effective force exerted on the mean flow, must be modified.

As we shall see, the conservation law for the sum of pseudomomentum and GLM impulse is the key for understanding the wave-mean interactions in the presence of refraction. We will illustrate this by a number of examples consisting of wavepackets and confined wavetrains. The most important result is the following: if the concept of an effective mean force makes sense at all, then this force is not exerted at the location of the wavepacket, but at the location of the vortices that induce the straining field. This gives the wave—mean interactions a non-local character that was clearly absent in simple geometry, where the effective mean force was always exerted at the location of the wavepacket.

We study the Boussinesq system, which is the simplest fluid model that admits internal gravity waves. These dispersive waves are ubiquitous in the atmosphere and ocean and they owe their restoring mechanism to the stable stratification in these environments, i.e., to the fact that density decreases with altitude. Internal gravity waves are typically far too small in scale (especially vertical scale) to be resolvable within global numerical models and therefore their dynamics and their interactions with the mean flow must be parametrized (i.e., put in by hand) based on a combination of observations and theory. Consequently, the classic wave-mean interaction theory for internal gravity waves has been extensively developed and this provides a convenient starting point for us.

Boussinesq system and stable stratification

The simplest fluid model that captures the effect of stable internal stratification is the Boussinesq model, which can be derived from the Euler equations and its dissipative counterparts under the assumption of a small density contrast across the fluid, together with ∇ . u = 0. Importantly, the latter constraint filters sound waves and thereby reduces the number of degrees of freedom compared to the full Euler system.

Before writing down the governing equations we note that as a realistic model for atmospheric and oceanic flows the Boussinesq system is mostly limited by its global restriction to small density contrasts across the fluid, and that these limitations are much more severe in the atmosphere than in the ocean.

We now turn to wave—mean interactions involving Rossby waves, the peculiar vorticity waves whose linear dynamics was described briefly in §4.2.2. Unlike acoustic waves or gravity waves, the dynamics of Rossby waves is essentially linked to the layerwise advection of PV, and this gives the mathematical description of Rossby waves and of their interactions with a mean flow a very special character, including the one-way phase propagation of Rossby waves.

The easiest model in which to study this topic is the quasi-geostrophic approximation to the shallow-water equations on a β-plane. However, the results easily generalize to three-dimensional stratified flow.

Quasi-geostrophic dynamics

We have no interest in gravity waves in this chapter and therefore we use the simplest theoretical approximation that filters these waves whilst retaining the balanced flow structure of Rossby waves and shallow-water vortices. This is accomplished by the quasi-geostrophic approximation to the equations, which is essentially a nonlinear extension of the linear balanced mode. These equations use a single dynamical variable, namely the PV.

Overall, the use of PV and of balanced flow systems based on PV advection and PV inversion (such as the quasi-geostrophic system or its many variants) are key concepts in atmosphere ocean fluid dynamics. For instance, balanced models were an essential component of the first successful numerical weather forecasts. Such a direct quantitative use of balanced models is less important today, but the insights that can be gained from studying such reduced dynamical systems remain as valuable as ever.

We now embark on a journey into new theoretical territory: Lagrangian-mean theory based on particle-following averaging. This theory allows a sharper and more succinct description of material advection in fluid dynamics, which greatly simplifies the mean description of material invariants such as scalar tracers or, crucially, of vorticity and potential vorticity. Based on this, Lagrangian-mean theory is superior to Eulerian-mean theory in the description of flow dynamics to do with vorticity and circulation. This will be particularly important for wave–mean interactions outside simple geometry, where long-range mean pressure fields greatly complicate the description of the mean flow based on momentum budgets.

On the downside of Lagrangian-mean theory we need to count the increased structural complexity of the theory, the requirement to evolve particle displacements alongside with the usual flow variables, and the potential breakdown of the particle-following flow map under averaging in the case of large-amplitude waves. Thus whether Eulerian or Lagrangian averaging is more efficient can depend on the problem at hand. As before, we take the view that Eulerian and Lagrangian concepts are complementary to each other in the optimal description of all aspects of fluid motion, and this is certainly true in wave–mean interaction theory as well.

We begin with a general discussion of Lagrangian averaging and of small-amplitude Stokes corrections and then give a comprehensive introduction to the so-called generalized Lagrangian-mean theory (GLM), which formally is not restricted to small-amplitude waves.