Introduction

Particle-laden, gravity-driven flows occur in a large variety of natural and industrial situations. Typical examples include turbidity currents, volcanic eruptions, and sand-storms (see Simpson 1997 for a review). On mountain slopes, debris flows and snow avalanches provide particular instances of vigorous dense flows, which have special features that make them different from usual gravity currents. Those special features include the following:

• They belong to the class of non-Boussinesq flows since the density difference between the ambient fluid and the flow is usually very large, whereas most gravity currents are generated by a density difference of a few percent.

• Whereas many gravity currents are driven by pressure gradient and buoyancy forces, the dynamics of flows on slope are controlled by the balance between the gravitational acceleration and dissipation forces. Understanding the rheological behavior of particle suspensions is often of paramount importance when studying gravity flows on steep slope.

This chapter reviews some of the essential features of snow avalanches and debris flows. Since these flows are a major threat to human activities in mountain areas, they have been studied since the late 19th century. In spite of the huge amount of work done in collecting field data and developing flow-dynamics models, there remain great challenges in understanding the dynamics of flows on steep slope and, ultimately, in predicting their occurrence and behavior. Indeed, these flows involve a number of complications such as abrupt surge fronts, varying free and basal surfaces, and flow structure that changes with position and time.

Introduction

Volcanic systems are controlled by a wide range of fluid mechanical processes, including the subsurface migration and ensuing accumulation of molten rock in crustal reservoirs, know as magma chambers, and the subsequent explosive eruption of ash and transport high into the atmosphere (Sparks et al. 1997). The philosophy of this chapter is to develop a series of simplified physical models of the flow processes in order to gain insights about the dominant controls on the processes; many of the models have been developed based on geological field evidence and have been tested with laboratory experiments. The aim is to build understanding rather than to simulate the processes, which are extremely complex and often for which there is insufficient data for a complete characterization of the physical and chemical state of the system. We provide a brief introduction to the overall range of processes which occur, and then immerse ourselves in some of the fascinating buoyancy-driven flow processes. For a geological introduction to some of the processes and context, textbooks such as McBirney (1985) include a comprehensive description of many of the geological observations and processes.

In magma reservoirs, which may lie 5–10 km below the surface, magma is exposed to the relatively cold surrounding crust, and this may lead to cooling and crystallization of the magma, as well as melting of the surrounding crustal rock, often called country rock. In addition, new magma may be supplied to the magma reservoir leading to pressurization of the system.

Introduction

Dynamics of Water Mass Formation and Spreading

Small-scale buoyancy-driven flows, such as the overflows from marginal seas, are the main process by which the distinct water masses of the deep ocean are formed. For example, the flow of Antarctic Bottom Water (AABW) from the continental shelf down to the bottom of the Weddell Sea influences water mass properties all the way to the North Atlantic Ocean. The large range of spatial scales and mechanisms involved in the formation and spreading of these water masses poses a formidable challenge to numerical models. Legg (Chapter 5, this volume) reviews the main dense overflows of the world ocean. The width of an overflow is set either by the width of the strait or channel through which it flows (in the case of the Red Sea overflow, for example) or by the Rossby radius of deformation, which is the main dynamic scale for stratified rotating fluids. For an overflow of thickness h, with density anomaly δρ relative to the density ρ of the surrounding fluid, the reduced gravity g′ is defined as gδρ/ρ (g being the acceleration of gravity) and the Rossby radius Lr is defined as Lr = (g′h)1/2 /f with f being the Coriolis parameter. Lr decreases with latitude and its magnitude is only a few kilometers in the Nordic Seas. The dynamics of the plumes of dense water and the amount of entrainment that takes place as they descend along topographic slopes set the properties of the newly formed water masses (e.g., Chapter 5 by Legg).

Introduction

Marginal seas subject to buoyancy loss, because of their semi-enclosed geometry, are source regions for the formation of dense intermediate and bottom waters. These convective water masses generally have distinct properties relative to the open ocean and can be traced far from their formation basins. They also can transport significant amounts of heat, salt, and other tracers throughout the world ocean. The vertical circulation and meridional heat and freshwater transports are fundamental components of the oceanic circulation, and play important roles in the global climate system. Understanding how this circulation depends on the environmental parameters of the system is important if one is to better model and predict the climate system and its sensitivity to changing atmospheric conditions, such as increasing anthropogenic carbon dioxide.

The focus of this review is on the circulation and exchange resulting from surface buoyancy forcing in marginal seas. General characterisitics of the exchange between the marginal sea and the open ocean are described from eddy-resolving numerical models in idealized configurations, and the physics governing this exchange are elucidated through a combination of numerical models and simplified analytic models. Although the problems are couched in terms of marginal sea–open ocean exchange, many of the processes that emerge from this analysis are relevant to more general buoyancy-forced flows. Particular attention is paid to the dynamics involved with net vertical motions forced by surface cooling.

Introduction

Relatively fresh river or estuarine water entering the coastal ocean forms a buoyant plume that often turns anticyclonically (to the right in the Northern Hemisphere) and forms a buoyant gravity current that can flow large distances along the coast before dispersing (e.g., Mork 1981; Munchow and Garvine 1993a; Rennie et al. 1999; Royer 1981). The tendency for the buoyant water to turn and flow along the coast as a relatively narrow current is a consequence of Earth's rotation. The focus here is on two aspects relevant to buoyant gravity currents in the ocean: (1) determining the characteristics of buoyant coastal currents flowing along a sloping bottom and (2) determining the influence of wind forcing on buoyant coastal currents.

Buoyant coastal currents are important components of the circulation on most continental shelves (e.g., Simpson 1982; Hill 1998). Buoyant coastal currents also transport constituents, such as sediment, marine organisms, nutrients, and chemical pollutants large distances from their river or estuarine sources. Therefore, determining the ultimate distribution and fate of these constituents depends on understanding buoyant coastal currents and their alongshore range of influence (e.g.,Wiseman et al. 1997; Epifanio et al. 1989). Two examples of societal problems where buoyant coastal currents play an important role are hypoxia and abrupt climate change.

Hypoxia is dissolved oxygen concentrations that are reduced to a level that is detrimental to marine organisms. Hypoxia associated with nutrient transport from rivers to the coastal ocean is a global problem (Diaz 2001).

Introduction to Overflows

What Are Dense Overflows?

Dense water formed in semi-enclosed seas often has to flow through narrow straits or down continental slopes before it reaches the open ocean. These regions of dense water flowing over topography are known as dense overflows. The dense water has been formed through a variety of processes including surface cooling, the addition of salt in the form of brine from freezing pack ice in high-latitude seas, and evaporation in enclosed subtropical seas. The dense overflows are regions of significant mixing, which modifies the temperature and salinity signal of the dense water. Many of the deep water-masses of the ocean originate in these overflows and have their properties set by the mixing that occurs therein. For example, the Nordic overflows occurring in gaps in the Greenland-Iceland-Scotland Ridge (e.g., the Denmark Straits and the Faroe Bank Channel) are the source of most of the North Atlantic Deep Water (NADW), whereas Antarctic Bottom Water (AABW) is replenished by dense overflows from the Weddell and Ross seas in the Antarctic. Together these two deep water-masses are responsible for most of the deep branches of the meridional overturning circulation (MOC). Other overflows, such as the Red Sea overflow and Mediterranean outflow, contribute to important saline waters at intermediate depths. The properties of the deep and intermediate water-masses covering much of the abyssal ocean are therefore determined to a large extent by the mixing that takes place in the overflow, and hence these localized mixing regions play a significant role in influencing the large-scale ocean circulation.

Buoyancy is one of the main forces driving flows on our planet and buoyancy-driven flows encompass a wide spectrum of geophysical flows. In this book, contributions by leading world scientists summarize our present theoretical, observational, experimental, and modeling understanding of buoyancy-driven flows. These flows range from buoyant coastal currents to dense overflows in the ocean, and from avalanches to volcanic pyroclastic flows on the Earth's surface. By design, there is a strong emphasis on the ocean where a wide range of buoyancy-driven flows is observed. Buoyancy-driven currents play a key role in the global ocean circulation and in climate variability through deep-water formation. Formation of dense water usually occurs in marginal seas, which are either cooler (at high latitudes) or saltier (due to greater evaporation rates). These dense waters enter the ocean as a gravity-driven current, entrain surrounding waters as they descend along the continental slope, and modify the ocean's stratification as they become part of the global ocean circulation. Buoyancy-driven currents are also primarily responsible for the redistribution of fresh water throughout the world's oceans. In particular, buoyant coastal currents transport fresh water, heat, nutrients, sediments, biogeochemicals, pollutants, and biological organisms along many continental shelves and thus have significant impacts on ecosystems, fisheries, and coastal circulation.

The methods developed in this book can be applied rather generally to model the dynamics of coherent structures in spatially extended systems. They are gaining acceptance in many areas in addition to fluid mechanics, including mechanical vibrations, laser dynamics, nonlinear optics, and chemical processes. They are even being applied to studies of neural activity in the human brain. Numerous studies of closed flow systems have been done using empirical eigenfunctions, some of which were discussed in Section 3.7. A considerable amount of work has also been done on model PDEs for weakly nonlinear waves, such as the Ginzburg–Landau and Kuramoto–Sivashinsky equations, but this work falls largely outside the scope of this book. We do not have the abilities (or space) to provide a survey of these multifarious applications, but we do wish to draw the reader's attention to some of the other recent work on open fluid flows.

We restrict ourselves to studies in which empirical eigenfunctions are used to construct low-dimensional models and some attempt is made to analyze their dynamical behavior. There is an enormous amount of work in which the POD is applied and its results assessed in a “static,” averaged fashion. Some of this we have reviewed in Section 3.7. Yet even thus restricted, our survey cannot pretend to be complete: new applications to fluid flows are appearing at an increasing rate. We have selected five problems on which a reasonable amount of work has been done, the first of which (the jet) is a “strongly” turbulent flow.

As we near the end of our story, the reader will now appreciate that there are many steps in the process of reducing the Navier–Stokes equations to a low-dimensional model for the dynamics of coherent structures. Some of these involve purely mathematical issues, but most require an interplay among physical considerations, judgement, and mathematical tractability. While our development of a general strategy for constructing low-dimensional models has been based on theoretical developments such as the POD and dynamical systems methods, the general theory is still sketchy and, in specific applications, many details remain unresolved.

The mathematical techniques we have drawn on lie primarily in probability and dynamical systems theory. In this closing chapter we review some aspects of the reduction process and attempt to put them into context. Some prospects for rigor in the reduction process are also mentioned. This is by no means a comprehensive review or discussion of future work; instead, we have chosen to highlight a few applications of dynamical and probabilistic ideas to illustrate lines along which a general theory might be further developed.

We start by discussing some desirable properties for low-dimensional models, and criteria by which they might be judged. We then outline in Section 13.2 an a-priori short-term tracking estimate which describes, in a probabilistic context, how rapidly typical solutions of the model equations are expected to diverge from those of the full Navier–Stokes equations restricted to the model domain. Here and in the following section we view low-dimensional models as perturbations of the full evolution equations. Section 13.3 also addresses reproduction of statistics by low-dimensional models.

In the preceding nine chapters we have developed our basic tools and techniques. In this chapter and the next we illustrate their use in the derivation and analysis of low-dimensional models of the wall region of a turbulent boundary layer. First, the Navier– Stokes equations are rewritten in a form that highlights the dynamics of the coherent structures (CS) and their interaction with the mean flow. To do this, both the neglected (high) wavenumber modes and the mean flow must be modeled, unlike a large eddy simulation (LES), in which only the neglected high modes are modeled. Second, using physical considerations, we select a family of empirical subspaces upon which to project the equations. Galerkin projection is then carried out. In doing this, we restrict ourselves to a small physical flow domain, and so the response of the (quasi)local mean flow to the coherent structures must also be modeled. This chapter describes each step of the process in some detail, drawing on material presented in Chapters 2, 3, and 4. After deriving the family of low-dimensional models, in the last three sections we discuss in more depth the validity of assumptions used in their derivation. In Chapter 11 we describe use of the dynamical systems ideas presented in Chapters 6 through 9 in the analysis of these models, and interpret their solutions in terms of the dynamical behavior of the fluid flow.

Our presentation is based on a series of papers, beginning with [22] and including [24, 43, 44, 158, 161]. We have selected the boundary layer as our main illustrative example largely because we are most familiar with it.

The proper orthogonal decomposition (POD) provides a basis for the modal decomposition of an ensemble of functions, such as data obtained in the course of experiments. Its properties suggest that it is the preferred basis to use in various applications. The most striking of these is optimality: it provides the most efficient way of capturing the dominant components of an infinite-dimensional process with only finitely many, and often surprisingly few, “modes.”

The POD was introduced in the context of turbulence by Lumley in [220]. In other disciplines the same procedure goes by the names: Karhunen–Loève decomposition, principal components analysis, singular systems analysis, and singular value decomposition. The basis functions it yields are variously called: empirical eigenfunctions, empirical basis functions, and empirical orthogonal functions. According to Yaglom (see [221]), the POD was introduced independently by numerous people at different times, including Kosambi [197], Loève [215], Karhunen [183], Pougachev [285], and Obukhov [272]. Lorenz [216], whose name we have already met in another context, suggested its use in weather prediction. The procedure has been used in various disciplines other than fluid mechanics, including random variables [275], image processing [313], signal analysis [5], data compression [7], process identification and control in chemical engineering [118,119], and oceanography [286]. Computational packages based on the POD are now readily available (an early example appeared in [11]).

In the bulk of these applications, the POD is used to analyze experimental data with a view to extracting dominant features and trends – in particular coherent structures. In the context of turbulence and other complex spatio-temporal fields, these will typically be patterns in space and time. However, our goal is somewhat different.

On physical grounds there is no doubt that the Navier–Stokes equations provide an excellent model for fluid flow as long as shock waves are relatively thick (in terms of mean free paths), and in such conditions of temperature and pressure that we can regard the fluid as a continuum. The incompressible version is restricted, of course, to lower speeds and more moderate temperatures and pressures. There are some mathematical difficulties – indeed, we still lack a satisfactory existence-uniqueness theory in three dimensions – but these do not appear to compromise the equations’ validity. Why then is the “problem of turbulence” so difficult? We can, of course, solve these nonlinear partial differential equations numerically for given boundary and initial conditions, to generate apparently unique turbulent solutions, but this is the only useful sense in which they are soluble, save for certain non-turbulent flows having strong symmetries and other simplifications. Unfortunately, numerical solutions do not bring much understanding.

However, three fairly recent developments offer some hope for improved understanding: (1) the discovery, by experimental fluid mechanicians, of coherent structures in certain fully developed turbulent flows; (2) the suggestion that strange attractors and other ideas from finite-dimensional dynamical systems theory might play a rôle in the analysis of the governing equations; and (3) the introduction of the statistical technique of Karhunen– Loève or proper orthogonal decomposition. This book introduces these developments and describes how the three threads can be drawn together to weave low-dimensional models that address the rôle of coherent structures in turbulence generation.

As we shall see in Parts III and IV, the techniques of proper orthogonal decomposition and Galerkin projection can be powerful tools for obtaining low-order models that capture the qualitative behavior of complex, high-dimensional systems. However, for certain systems, the resulting models can perform poorly: even if a large fraction of energy (over 99%) is captured by the modes used for projection, the resulting low-order models may still have completely different qualitative behavior. The transients may be poorly captured, and the stability types of equilibria can even be different.

In this chapter, we present a method which can dramatically outperform projection onto traditional energy-based empirical eigenfunctions described in Chapter 3.We focus primarily (though not exclusively) on linear systems, for several reasons. Many of the pitfalls of traditional proper orthogonal decomposition can be demonstrated for linear systems, without the additional complexity of nonlinearities. Furthermore, for linear systems, one can use operator norms to quantify the difference between a detailed model and its reduced order approximation. Most importantly, for linear systems, there are established tools for performing model reduction, for instance using balanced truncation, which is described in Section 5.1. In contrast, while some modest extensions to nonlinear systems have been attempted, model reduction of nonlinear systems is still an active area of research.

The techniques described in this chapter also differ from those in Chapters 3 and 4 in that they are formulated for input–output systems. The inputs represent the external influences on the system, for instance from external disturbances, or from actuators in a flow-control setting.