In this appendix we shall derive a system of depth-averaged conservation equations, the classic Saint-Venant equations (de Saint-Venant, 1871), which give a good description of the dynamics of a fluid flow when the spatial scale λ of the variations in the longitudinal direction x is large compared to the flow depth or thickness, h. These equations do not explicitly involve the nature of the flowing medium, and so they are valid, under certain conditions stated explicitly below, for a fluid flow either laminar or turbulent, as well as for a granular flow. The equations can be obtained by two equivalent methods: by depth-averaging the local conservation equations in the transverse direction, or by writing down the conservation laws for a control volume corresponding to a slice of fluid of thickness dx. Here we shall use the latter method.

Outflow from a slice of fluid

Let us consider a flow in the x-direction, inclined at an angle θ relative to the horizontal, between a bed located at y = yb(x, t), where y is the transverse direction, and a free surface at y = (yb + h)(x, t), where h is the local flow depth (Figure A.1).

We define the control volume as the slice of fluid between the two planes at x and x + dx. At any point on the boundary of the control volume, let n be the exterior normal, u the fluid velocity and w the speed of the boundary.

Introduction

An instability typical of a parallel flow was demonstrated in an experiment performed by Osborne Reynolds (1883) and repeated by Thorpe (1969). A horizontal long tube is filled carefully with a layer of water lying on top of a layer of heavier colored brine (salt water), as sketched in Figure 4.1a. The tube is suddenly tipped several degrees: the brine falls and the water rises, creating a shear flow which displays an inflection point near the interface (Figure 4.1b). In a few seconds a wave of sinusoidal shape develops at the interface of the two fluids, and leads to regular co-rotating vortices, as shown in Figure 4.2. These vortices are a manifestation of the Kelvin–Helmholtz instability. This instability owes its origin to the inertia of the fluids; viscosity plays only a minor role, tending to attenuate the growth of the wave only slightly by momentum diffusion.

Another manifestation of the Kelvin–Helmholtz instability on a much larger atmospheric scale is illustrated in Figure 4.3. An upper air layer flows over a lower layer moving more slowly, with different humidity and temperature. The thin layer of clouds formed at the interface displays billows which are the exact analogs of those in Figure 4.1. Figure 4.4 shows another example, in a mixing layer formed between a flow of water on the left and a flow of water and air bubbles on the right.

For over a century now, the field of hydrodynamic instabilities has been constantly and abundantly renewed, and enriched by a fruitful dialogue with other fields of physics: phase transitions, nonlinear optics and chemistry, plasma physics, astrophysics and geophysics. Observation and analysis have been stimulated by new experimental techniques and numerical simulations, as well as by the development and adaptation of new concepts, in particular, those related to asymptotic analysis and the theory of nonlinear dynamical systems. Ever since the observations of Osborne Reynolds in 1883, there has been unflagging interest in the fundamental problem of the transition to turbulence. This topic has been given new life by concepts such as convective instabilities, transient growth, and by the recognized importance of unstable nonlinear solutions. New problems have emerged, such as flows involving fluid–structure interactions, granular flows, and flows of complex fluids – non-Newtonian and biological fluids, suspensions of particles, bubbly flows – where constitutive laws play an essential role.

This book has been written over the course of 10 years of teaching postgraduate students in fluid dynamics at the University of Toulouse. It is intended for any student, researcher, or engineer already conversant with basic hydrodynamics, and interested in the questions listed above.

Introduction

As we have discussed in earlier chapters, a dissipative physical system in a uniform, stationary state can become linearly unstable when a control parameter R exceeds a critical value Rc. This occurs, for example, in the instability of a fluid layer heated from below governed by the Rayleigh number, or in the instability of plane Poiseuille flow governed by the Reynolds number. The instability can be manifested as the appearance of a stationary, spatially periodic structure (Rayleigh–Bénard convection rolls, for example), or as a growing traveling wave (Tollmien–Schlichting waves). This type of situation was studied in Chapter 8 for spatially confined systems, or systems with imposed periodicity, where the dynamics can be reduced to a system of differential equations for the amplitudes of a few spatial harmonics. When the physical system is spatially extended, that is, when its size is large compared to the wavelength of the periodic structure, the wave number spectrum tends to become continuous, and spatial modulations of the amplitudes can arise. The appropriate formalism for describing these modulations is that of envelope equations.

In the present chapter we shall present this formalism and study the conditions for saturation of the primary instability arising at R = Rc, as well as for secondary instabilities which arise when the bifurcation parameter exceeds a second threshold. First we study the case where the periodic structure is stationary, that is, where the eigenvalue of the linearized system crossing the imaginary axis at threshold R = Rc is real, i.e., the bifurcation is of the saddle–node or pitchfork type for a system possessing the reflection symmetry x → -x.

Introduction

A useful mathematical framework for studying linear and nonlinear stability is the theory of ordinary differential equations (ODEs), also known as the theory of dynamical systems when the focus is on geometric and qualitative representations of the ideas and solutions. An informal presentation of this theory was given in Chapter 1. The goal of this chapter is to give a more systematic account of it from a mathematical point of view. In particular, we shall show how to reduce the number of degrees of freedom of a problem to obtain the “normal forms” of elementary bifurcations referred to often in the previous chapters.

In spite of its restrictive nature relative to the theory of partial differential equations, the theory of dynamical systems has revealed the extraordinary richness and complexity of the types of behavior that can arise when nonlinear effects play a role. This theory originated in the work of Henri Poincaré, in particular, in his book Méthodes nouvelles de la mécanique céleste. Poincaré's ideas were further developed during the first half of the twentieth century by the Russian school of mathematics (Kolmogorov, Arnold). The discovery that a system with a small number of degrees of freedom can display unpredictable, chaotic behavior then led to a great deal of research in this area beginning in the 1960s. Qualitatively new concepts such as deterministic chaos and sensitivity to the initial conditions were introduced, and these have significantly modified our understanding of deterministic models and their use in the description of natural phenomena.

Introduction

The inertial instability of parallel flows described in the preceding chapter is associated with the existence of an inflection point in the velocity profile. This is the principal instability of parallel or quasi-parallel shear flows at large Reynolds number and far from walls or interfaces, such as mixing layers, jets, and wakes. We have seen that the order of magnitude of the growth rate is U/δ, where U is the difference of the speeds on either side of the vorticity layer of thickness δ, and that viscosity plays only a diffusive role tending to attenuate the growth rate. The instability of flow profiles without inflection points is profoundly different. Let us consider two fundamental flows: plane Poiseuille flow and boundary layer flow. Observation shows that plane Poiseuille flow is unstable beyond a certain Reynolds number. Similarly, a boundary layer on a surface becomes unstable at some distance from the leading edge. However, these two flows do not possess an inflection point, and so, ignoring viscosity, they are stable according to the Rayleigh theorem. On the other hand, the growth rate of the observed instabilities is much smaller than would be expected for an inertial instability. It is therefore clearly important to investigate the role played by viscosity, which is the goal of the present chapter. We shall see that viscosity has two effects: the expected stabilizing dissipative effect, and also a destabilizing effect.

Scaling

As a model of the fluid motion and thermodynamics of the Earth's oceans, the basic equations (1.15)–(1.19) include a formidable array of distinct types of physical processes, from divergence of molecular diffusive fluxes on scales of millimeters to coherent fluid flow on the scales of the Earth's circumference, a range of space scales of order 1010. This range of scales is comparable to that between the molecular scale and the scale of a mammal's body. Thus, in a rough sense, the problem of using these equations to understand the large-scale structure of the ocean is comparable in difficulty to using numerical computations of molecular interactions to simulate the behavior of a mammal. Numerical solution of this full set of equations is thus well beyond current computational capacities and will remain so for the foreseeable futuremore. Furthermore, these equations are sufficiently challenging that fundamental mathematical properties, such as the existence and uniqueness of solutions, are not established. Some of these basic properties remain a subject of mathematical research even for the incompressible Navier-Stokes equations, a simplified set of four equations that may be obtained from (1.15) and (1.16) by replacing the density variable ρ with a constant value ρ0.

Despite these formidable challenges, it is possible to make progress by using a combination of mathematical and physical reasoning to derive simplified, or reduced, equation sets that describe motions on the largest space and time scales of interest.

The Large-Scale Ocean Circulation

Systematic observations of the fluid properties of Earth's ocean, made primarily over the course of the last hundred years, reveal coherent features with scales comparable to those of the ocean basins themselves. These include such structures as the subtropical main thermoclines and anticyclonic gyres, which appear in all five midlatitude ocean basins, and the meridional overturning circulations that support the exchange of waters across the full meridional extent of the ocean, from the polar or subpolar latitudes of one hemisphere to the opposing high latitudes of the other. These features and motions, which prove to be connected by robust dynamical balances, constitute the large-scale circulation of the Earth's ocean.

The global field of long-term mean sea-surface temperature is dominated by the meridional gradients between the warm equatorial regions and the cold poles but contains significant zonal gradients as well (Figure 1.1). The global long-term mean sea-surface salinity field has a more complex structure (Figure 1.2), with isolated maxima in the evaporative centers of the midlatitude subtropical gyres. The global sea-surface density field computed from long-term mean temperature and salinity reflects the competing influences of temperature and salinity on density (Figure 1.3). These fields are the surface expressions of complex three-dimensional interior property fields. The downward penetration of the warm equatorial temperatures is generally limited to the upper one-fifth of the water column (Figure 1.4), while salinity perturbations are less strongly confined to shallow depths (Figure 1.5).

Domain and Boundary Conditions

With the basic large-scale-approximate equations and some of their general properties established, it is appropriate now to examine the circulation and thermocline structures that arise in specific solutions of these equations. Even for the simplest case of steady solutions, in which the flow is independent of time, this requires that a sufficient set of boundary conditions be specified. This set of boundary conditions is not unique: a variety of such conditions, which differ in physical and mathematical detail, may yield solutions of physical interest for the large-scale flow. In general, these sets of conditions for idealized models of ocean gyre structure and circulation must be chosen to represent the definitive characteristics of the general physical setting in which these large-scale features develop.

A fundamental element of the boundary conditions is the geometry specified for the ocean basin. The simplest such basin is a rectangular domain, restricted to a portion of one hemisphere, with vertical sidewalls aligned along lines of constant longitude x = {xW, xE} and latitude y = {yS, YN} and a flat bottom at the constant depth z = -H0 (Figure 5.1). Such a choice avoids the singularity of the planetary geostrophic momentum equations that arises at the equator, where the Coriolis parameter f vanishes. It also removes a complicating geometric element: the circumpolar connection that exists around the Antarctic continent in the southern hemisphere, which will be seen (Chapters 7 and 8) to have significant impact on the surface and mid-depth circulation.