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.

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 equations 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 remains as valuable as ever.

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.

We are particularly interested in the behaviour of small-scale waves propagating on an inhomogeneous basic state, because this is the natural setting for unresolved waves interacting with a resolved mean flow in a numerical model.

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 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 Eliassen–Palm flux 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.

Rotating Boussinesq equations on an f-plane

We consider frame rotation in the traditional approximation, in which only the locally vertical component of the Coriolis vector is retained (cf. §4.2.2) and we work on an f-plane, in which this component is treated as constant.

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.

The beam theory of Chapter 4 and the corresponding finite element implementation was formulated in a fixed global frame of reference using the total displacements and rotations. In many cases it may be advantageous to consider the beam element with reference to a local, element-based, coordinate system. Motion of the beam then implies motion of the local frame of reference as well as deformation of the beam element within this frame. The separation of the motion of the element into two parts – a rigid body motion associated with the element-based frame of reference and a deformation of the element within this frame of reference – is called a co-rotating formulation. The co-rotating formulation has a number of advantages, provided it can be demonstrated that the tangent stiffness can be decomposed into the sum of a part associated with the rotation of the element-based frame and a part associated solely with the deformation of the element within this frame of reference. The first advantage is that displacements and rotations within the local frame of reference are small or at most moderate. Therefore, the deformation of the beam can be modeled by approximate beam theory. Secondly, the co-rotating formulation is closely associated with the idea of ‘natural modes’, advocated by Argyris et al. (1979a,b). The idea of the ‘natural modes’ is to consider any increment of the motion of an element as made up of a set of rigid body modes – typically translation and rotation – and a set of deformation modes – representing extension, bending and torsion of the beam element.

The aim of this book is to take the reader on a concentrated tour of some of the central issues of non-linear modeling and analysis of structures and solids. Traditionally, the non-linear theories of solids have been treated in books on continuum mechanics, while the questions of analysis have formed the focus of books on finite element techniques. The idea of the present book is to place the emphasis on modeling with a view to its numerical implementation right from the outset. Two guiding principles have determined the main style of the book: the story should be told in the form of concentrated chapters, each giving the central ideas of a specific aspect such as ‘finite rotations’ or ‘elasto-plastic solids’, and the reader should have the possibility of getting a feel for the numerical implementation by access and use of simple high-level implementations of the basic algorithms. A text based on these principles cannot provide exhaustive coverage, but aims at giving an interesting introduction to the basic ideas, which can then be studied elsewhere in greater detail as needed. It is hoped that the combination of a concise theoretical presentation in plain language supported by specific algorithms will make the text of interest to graduate students as well as professionals.

The book contains nine chapters: a brief introductory chapter setting the scene by use of elementary arguments, four chapters on structures, two chapters on non-linear deformation and material behavior of solids, and finally two chapters on numerical techniques for non-linear quasi-static and dynamic analysis.

The solution of non-linear problems relies heavily on numerical methods. Only few non-linear problems allow direct solution, and most often an iterative strategy must be used. A simple example of such an iterative strategy is the Newton–Raphson method described in Section 1.2. In its standard form it consists of a series of prescribed load increments combined with an iterative solution of the equilibrium equations for the corresponding displacement increments. This strategy has a number of disadvantages. In the full Newton–Raphson method each step in the iterative solution requires solution of a linearized set of equations. This may involve a very high computational effort, and therefore modified versions, in which the equations are not reformulated in each step, can be an alternative. The modified Newton–Raphson method has slower convergence, and this may offset some of the gain from the simplified solution of the equations. In addition to concerns about the numerical efficiency, the Newton–Raphson method also encounters problems in passing limit and bifurcation points.

In the Newton–Raphson method the load increment is specified at the beginning of the load step and kept constant during equilibrium iterations. This leads to lack of efficiency and possibly complications when the stiffness changes rapidly, and in particular around load limit points where the sign of the load increment changes. Several techniques have been developed to deal with this problem. Here two types of techniques will be described.

A central feature of the mechanical behavior of solids and structures is the constitutive relation connecting stresses and strains. This chapter presents the basic theory of elastic and plastic solids, including the techniques needed to implement the models in the numerical framework described in previous chapters. The theoretical basis for the material models described here has developed over the last 50 years, from the early work of Ziegler (1963), over potentials with internal variables (Rice, 1971; Hill and Rice, 1973), to a fully developed theory including conjugate variables and potentials (Halphen and Son, 1975; Germain et al., 1983). A full account of this development is outside the present scope, and the chapter is limited to elastic and elasto-plastic solids. The presentation is deliberately simplified to purely mechanical effects, leaving out e.g. thermal effects, but retaining the general structure of the formulation.

First the notion of reversible elastic deformation is introduced in Section 7.1 in connection with an internal energy potential, and the concept of conjugate variables and potentials is described. The role of stress and strain invariants and some of their properties is described and used in the formulation of isotropic elasticity. The general theory of rate-independent elasto-plasticity with internal variables is then developed in Section 7.2. The key ingredients are the internal energy potential, expressing the recoverable energy in terms of elastic strains and the primary internal variables, and a yield potential, describing the limit of the region of reversible behavior.

Kinematic non-linearity is a recognized part of continuum mechanics often termed ‘large’ or ‘finite’ displacement theory. The non-linearity arises because equilibrium is considered in the current, and initially unknown, state of the body. In order to describe finite deformation of a continuous body it is necessary to have a non-linear measure of strain and a stress definition that can be used in the deformed state. It turns out that the Green strain, introduced for axial strain in Chapter 2, can be generalized to multi-component form describing the deformation of a continuous body. This is the subject of Section 6.1.

For any continuum mechanics theory it is very desirable to use stresses and strains that satisfy some form of virtual work principle. It was demonstrated in Chapter 2 that the use of the Green strain, which was a convenient quadratic strain measure with exact invariance with respect to arbitrary rigid body motion, led to a slightly modified interpretation of the normal force N appearing in the principle of virtual work. In a similar way the use of the Green strain for a continuous body leads to a special stress definition, the second Piola–Kirchhoff stress. For small strains this stress definition has a simple physical interpretation, precisely as N in the case of a bar element. This stress is introduced in Section 6.2, and it is demonstrated how it serves as a convenient reference for other stress measures of practical importance.