In the preceding chapters, a number of asymptotic methods were introduced for the approximate solution of nonlinear flow problems. In many of the cases considered so far, including all of the problems of the two preceding chapters, the asymptotic limiting process produced a simplification of the full nonlinear problem by restricting the geometry of the flow domain to one in which certain terms in the equations could be neglected because the length scales in some direction (or directions) become very large compared with the length scales in other directions. In retrospect, even the exact unidirectional flow problems of Chap. 3 can often be regarded as a first approximation of some more general problem in which the geometry reduces to a unidirectional form in the limit as a ratio of two length scales vanishes, e.g., the “Dean” problem of Chap. 4, which reduces to the unidirectional Poiseuille flow problem in the limit as the ratio of the tube radius to the radius of curvature of the tube in the axial direction goes to zero. In some cases, this disparate ratio of length scales was true everywhere in the flow domain, and then the asymptotic solution was found to be “regular”; e.g., the Dean problem or the eccentric cylinder problem of Chap. 5. In others, the region with a small length-scale ratio was restricted to a local part of the overall flow domain, and in these cases, the asymptotic approximation was of the “singular” type, in which the simplified form of the governing equations is valid only locally, and the resulting approximate solution must be “matched” to a solution of the unapproximated equations that are valid elsewhere.

Although the full Navier–Stokes equations are nonlinear, we have studied a number of problems in Chap. 3 in which the flow was either unidirectional so that the nonlinear terms u · ∇u were identically equal to zero or else appeared only in an equation for the cross-stream pressure gradient, which was decoupled from the primary linear flow equation, as in the 1D analog of circular Couette flow. This class of flow problems is unusual in the sense that exact solutions could be obtained by use of standard methods of analysis for linear PDEs. In virtually all circumstances besides the special class of flows described in Chap. 3, we must utilize the original, nonlinear Navier–Stokes equations. In such cases, the analytic methods of the preceding chapter do not apply because they rely explicitly on the so-called superposition principle, according to which a sum of solutions of linear equations is still a solution. In fact, no generally applicable analytic method exists for the exact solution of nonlinear PDEs.

The question then is whether methods exist to achieve approximate solutions for such problems. In fluid mechanics and in convective transport problems there are three possible approaches to obtaining approximate results from the nonlinear Navier–Stokes equations and boundary conditions.

First, we may discretize the DEs and boundary conditions, using such formalisms as finite-difference, finite-element, or related approximations, and thus convert them to a large but finite set of nonlinear algebraic equations that is suitable for attack by means of numerical (or computational) methods.

A BRIEF HISTORICAL PERSPECTIVE OF TRANSPORT PHENOMENA IN CHEMICAL ENGINEERING

“Transport phenomena” is the name used by chemical engineers to describe the subjects of fluid mechanics and heat and mass transfer. The earliest step toward the inclusion of specialized courses in fluid mechanics and heat or mass transfer processes within the chemical engineering curriculum probably occurred with the publication in 1923 of the pioneering text Principles of Chemical Engineering by Walker, Lewis, and McAdams. This was the first major departure from curricula that regarded the techniques involved in the production of specific products as largely unique, to a formal recognition of the fact that certain physical or chemical processes, and corresponding fundamental principles, are common to many widely differing industrial technologies.

A natural outgrowth of this radical new view was the gradual appearance of fluid mechanics and transport in both teaching and research as the underlying basis for many of the unit operations. Of course, many of the most important unit operations take place in equipment of complicated geometry, with strongly coupled combinations of heat and mass transfer, fluid mechanics, and chemical reaction, so that the exact equations could not be solved in a context of any direct relevance to the process of interest. Hence, insofar as the large-scale industrial processes of chemical technology were concerned, even at the unit operations level, the impact of fundamental studies of fluid mechanics or transport phenomena was certainly less important than a well-developed empirical approach (and this remains true today in many cases).

In Chap. 4 we explored the consequences of a weak departure from strict adherence to the conditions for unidirectional flow; namely, the effect of slight curvature in flow through a circular tube. For that case, the centripetal acceleration associated with the curved path of the primary flow was shown to produce a weak secondary motion in the plane orthogonal to the tube axis. In this chapter we consider another class of deviations from unidirectional flow that occur when the boundaries are slightly nonparallel.

If the boundaries of the flow domain are not parallel, the magnitude of the primary velocity component must vary as a function of distance in the flow direction. This not only introduces a number of new physical phenomena, as we shall see, but it also means that the Navier–Stokes equations cannot be simplified following the unidirectional flow assumptions of Chap. 3, and exact analytical solutions are no longer possible. In this chapter, we thus consider only a special limiting case, known as the “thin-gap” limit, in which the distance between the boundaries is small compared with the lateral gap width. In this case, we shall see that we can obtain approximate analytical solutions by using the asymptotic and scaling techniques that were introduced in the preceding chapter.

The resulting theory, at the leading order of approximation, is applicable to a number of important phenomena. There are two generic classes of problems.

Although the solution methods of the preceding chapter are very useful and allowed us to derive a number of interesting results, they were all based on transforming the creeping-flow and continuity equations into a single higher-order differential for the streamfunction so that the classical methods of eigenfunction expansions can be used to obtain a general representation of the solution. This approach will work whenever the geometry of the boundaries and the form of any imposed flow are consistent with either a 2D or axisymmetric velocity and pressure field. We applied it to some representative 2D problems, as well as to problems involving the motions of spherical or near-spherical particles, bubbles, and drops in axisymmetric applied flows. There are, of course, other problems involving axisymmetric bodies in an axisymmetric flow – for example, any body of rotation in which the rotation axis is parallel to the axis of symmetry of the undisturbed flow. These problems can all be formulated in the same way, and, provided the geometry of the body is coincident with a coordinate surface in some orthogonal coordinate system, the same procedures can be followed.

In spite of the fact that there are actually quite a large number of axisymmetric problems, however, there are many important and apparently simple-sounding problems that are not axisymmetric.

We are concerned in this book primarily with a description of the motion of fluids under the action of some applied force and with convective heat transfer in moving fluids that are not isothermal. We also consider a few analogous mass transfer problems involving the convective transport of a single solute in a solvent.

It is assumed that the reader is familiar with the basic principles and equations that describe these processes from a continuum mechanics point of view. Nevertheless, we begin our discussion with a review of these principles and the derivation of the governing differential equations (DEs). The aim is to provide a reasonably concise and unified point of view. It has been my experience that the lack of an adequate understanding of the basic foundations of the subject frequently leads to a feeling on the part of students that the whole subject is impossibly complex. However, the physical principles are actually quite simple and generally familiar to any student with a physics background in classical mechanics. Indeed, the main problems of fluid mechanics and of convective heat transfer are not in the complexity of the underlying physical principles, but rather in the attempt to understand and describe the fascinating and complicated phenomena that they allow. From a mathematical point of view, the main problem is not the derivation of the governing equations that is presented in this second chapter, but in their solution. The latter topic will occupy the remaining chapters of this book.

Abstract

As more high-resolution observations become available, our view of ocean mesoscale turbulence more closely becomes that of a “sea of eddies.” The presence of the coherent vortices significantly affects the dynamics and the statistical properties of mesoscale flows, with important consequences on tracer dispersion and ocean stirring and mixing processes. Here we review some of the properties of particle transport in vortex-dominated flows, concentrating on the statistical properties induced by the presence of an ensemble of vortices. We discuss a possible parameterization of particle dispersion in vortex-dominated flows, adopting the view that ocean mesoscale turbulence is a two-component fluid which includes intense, localized vortical structures with non-local effects immersed in a Kolmogorovian, low-energy turbulent background which has mostly local effects. Finally, we report on some recent results regarding the role of coherent mesoscale eddies in marine ecosystem functioning, which is related to the effects that vortices have on nutrient supply.

Introduction

The ocean transports heat, salt, momentum and vorticity, nutrients and pollutants, and many other material and dynamical quantities across its vast spaces. Some of these transport processes are at the heart of the mechanisms of climate variability and of marine ecosystem functioning. In addition, a large portion of the available data on ocean dynamics are in the form of float and drifter trajectories. These provide a Lagrangian view of the ocean circulation which is not always easy to disentangle.

This book has been motivated by the recent surge in the density and availability of Lagrangian measurements in the ocean, recent mathematical and methodological developments in the analysis of such data to improve forecasts and transport characteristics of ocean general circulation models, and numerous applications to dispersion of biological species. Another source of motivation has been the Lagrangian Analysis and Prediction of Coastal and Ocean Dynamics (LAPCOD) workshops (www.rsmas.miami.edu/LAPCOD/meetings.html).

The main purpose of this book is to conduct a review of Lagrangian observations, analysis and assimilation methods in physical and biological oceanography, and to present new methodologies on Lagrangian analysis and data assimilation, and new applications of Lagrangian stochastic models from biological dispersion studies. Some of the chapters included in this volume were presented at LAPCOD workshops, while others have been specifically written for this collection. Given the size of the Lagrangian field, the present work cannot be considered as an exhaustive effort, but one which is aimed to cover many of the central research topics. It was our intent to maintain a good balance between historical and state-of-the-art developments in Lagrangian-based observations, theory, numerical modeling and analysis techniques.

This book seems to be a first of its kind because the central theme is the Lagrangian viewpoint for studying the transport phenomena in oceanic flows. Another unique and timely aspect of this book is its multidisciplinary nature with contributions from experimentalists, theoreticians, and modelers from diverse fields such as physical oceanography, marine biology, mathematics, and meteorology.

Introduction

In the last 20 years, the deployment of surface and subsurface buoys has increased significantly, and the scientific community is now focusing on the development of new techniques to maximize the use of these data. As shown by Davis (1983, 1991), oceanic observations of quasi-Lagrangian floats provide a useful and direct description of lateral advection and eddy dispersal. Data from surface drifters and subsurface floats have been intensively used to describe the main statistics of the general circulation in most of the world ocean, in terms of mean flow structure, second-order statistics and transport properties (e.g. Owens, 1991; Richardson, 1993; Fratantoni, 2001; Zhang et al., 2001; Bauer et al., 2002; Niiler et al., 2003; Reverdin et al., 2003). Translation, swirl speed and evolution of surface temperature in warm-core rings, which are ubiquitous in the oceans, have also been studied with floats by releasing them inside of the eddies (Hansen and Maul, 1991). Trajectories of freely drifting buoys allow estimation of horizontal divergence and vertical velocity in the mixed layer (Poulain, 1993). Also, data from drifters allows investigation of properties and statistics of near-inertial waves, which provide much of the shear responsible for mixing in the upper thermocline and entrainment at the base of the mixed layer (Poulain et al., 1992). Drifters have proved to be robust autonomous platforms with which to observe ocean circulation and return data from a variety of sensors.

Introduction

The prediction particle trajectories in the ocean is of practical importance for problems such as searching for objects lost at sea, tracking floating mines, designing oceanic observing systems, and studying ecological issues such as the spreading of pollutants and fish larvae (Mariano et al., 2002). In a given year, for example, the US Coast Guard (USCG) performs over 5000 search and rescue missions (Schneider, 1998). Even though the USCG and its predecessor, the Lifesaving Service, have been performing search and rescue operations for over 200 years, it has only been in the last 30 years that Computer Assisted Search Planning has been used by the USCG. The two primary components are determining the drift caused by ocean currents and the movement caused by wind. The results presented in this review are motivated by the drift estimation problem.

A number of authors (e.g., Aref, 1984; Samelson, 1996) have shown that prediction of particle motion is an intrinsically difficult problem because Lagrangian motion often exhibits chaotic behavior, even in regular and simple Eulerian flows. In the ocean, the combined effects of complex time-dependence (Samelson, 1992; Meyers, 1994; Duan and Wiggins, 1996) and three-dimensional structure (Yang and Liu, 1996) are likely to induce chaotic transport. Chaos implies strong dependence on the initial conditions, which are usually not known with great accuracy, so that the task of predicting particle motion is often extremely difficult.

Introduction

A complete description of a dynamical system must include information about two things: its state and its kinetics. The first part defines its condition or state at some instant in time, but nothing about its motion. The latter does the opposite, it tells us how the system is evolving, but nothing about its state. Thus, for a full understanding of a dynamical system, we need information on both. If we consider the ocean as such a system, its state would be determined by the distribution of mass while the kinetics of the system would be given by the distribution of currents. Since the birth of modern oceanography, we have developed an increasingly accurate picture of the state of the ocean, more specifically the distribution of heat and salt: the two properties that determine the mass field and hence the internal forces acting on it. Progress has been much slower – and more recent – with respect to a corresponding description of the kinetics of the ocean. Indeed, our view of the ocean circulation is still incomplete and depends to a significant extent upon assumptions about its internal dynamics in order to estimate ocean currents from the observed mass field. We have employed this methodology out of convenience and necessity because for a very long time we did not have the tools to observe the ocean in motion directly.

Introduction

Plankton have inhabited the Earth's oceans for hundreds of millions of years as evidenced by the fossil record. The exterior covering of identifiable dinoflagellates, for example, are well preserved in Mesozoic rock strata. Pelagic diatoms possess siliceous frustules with identifiable species dating from early Cretaceous sediments (see Falkowski et al., 2004). Given the extent of fossil plankton, it is apparent that a drifting mode of life has been a successful means for survival in the sea for much of life's history.

With the importance of plankton in marine ecosystems, it is surprising that biological oceanographers have only recently begun to use drifting, or more formally Lagrangian, techniques. However, as with other aspects of biological oceanography, the Lagrangian ‘tools’ for studying plankton are relatively recent, and have often followed technique development by physical oceanographers and engineers. The main goal of this chapter is to summarize how biological oceanographers have applied Lagrangian and related methods to further our understanding of oceanic plankton distributions and dynamics, as well as biogeochemical processes. Our target audience is physical oceanographers and mathematicians who will hopefully gain some benefit from this exercise, while biological oceanographers may also be encouraged to further consider Lagrangian approaches in their field studies. We include studies on bacterio-, phyto-, zoo-, and ichthyoplankton and discuss the advances made in specific sub-disciplines of biological oceanography through the use of Lagrangian techniques. This review is timely in that new, low power sensors are now being adapted for deployments on a variety of Lagrangian platforms.

Introduction

We produce the population dynamics of a stage structured population, where the stages are defined by sharp biological events (egg hatching, molt, adult emergence, beginning and end of oviposition, death), by means of a stochastic individual-based model that simulates the life histories of its individuals (Judson, 1994; Berec, 2002; Buffoni et al., 2002; Buffoni et al., 2004). Aspects of the life history of an individual, such as survival probabilities, development rates and egg production, depend on its “status,” on the population size, and on external factors such as the environmental conditions (e.g. physical factors, food availability). In general, the status of an individual can be identified by means of a number of physiological variables or biometric descriptors, which describe the behavior of an individual in a given situation, and define its physiological age. The physiological age of an individual is generally described only by a variable. Here the status of an individual is individuated by its stage and its physiological age in the stage. The physiological age is defined as the percentage of development for non-reproductive individuals, and as the percentage of the potential reproductive effort for an adult female. The life history is obtained by the time evolution of the status of an individual, from birth to death, following its development and, when the individual is an adult female, the production of eggs.

In this chapter, a collection of “favorite trajectories” from various authors are presented.

While Lagrangian data analysis uses an extensive array of sophisticated tools, including classical statistics, dynamical system theory, stochastic modelling, assimilation techniques, and many others, visual inspection of individual trajectories still plays an important role, providing the first and often fundamental glimpse of the underlying dynamics. Often, for Lagrangian investigators, looking at trajectories gives the first intuition, then leading to the use of sophisticated and appropriate analysis. Trajectories tell the story of the journey of drifters and floats, and these stories are often complex and fascinating.

In the following sections, a number of investigators take us in the various world oceans, including Atlantic, Pacific and regional Seas, from the Poles to the Tropics, telling us the stories of their favorite trajectories and giving us their intuition and physical insights.

Mesoscale eddies in the Red Sea outflow region

In 2001–2002, 50 RAFOS floats were released at the core depth (∼ 650 m) of Red Sea Outflow Water (RSOW) in the Gulf of Aden (northwestern Indian Ocean) as part of the Red Sea Outflow Experiment (REDSOX). The objective was to determine how warm, saline RSOW spreads from its source at the southern end of Bab al Mandeb Strait to the open Indian Ocean. Our hypothesis was that either boundary undercurrents or submesoscale coherent vortices (SCVs like Meddies, but here called “Reddies”) were the main transport mechanisms for RSOW.

The use of a particle-following or Lagrangian perspective to follow the dynamics of life in the sea is explored from a variety of perspectives. The discussion begins with the consideration of the energetics of marine organisms and a demonstration that mean field models fail to adequately describe the life of large marine fishes in the sense that they require sizable, > 100–1000 X aggregation of prey over the average biomass density in the ocean. In place of a mean field model in time a structured population model where populations are dependent on space, time, age, and their metabolism is derived. Having introduced the structured model it is then argued that it is impractical to use such a model except in a Lagrangian frame. Methods for coupling these models in a Lagrangian description of the marine environment are then discussed. This section of the manuscript ends with an appraisal of the amount of spatial aggregation required to support large pelagic fishes such as swordfish and tunas. The second portion of the paper goes on to provide examples of trajectories in different marine environments including boundary currents, mesoscale eddy fields and fronts, and the coastal environment. An emphasis on the dynamics of trajectories at various trophic levels provides insights on aggregation mechanisms and rates.

The last two sections introduce methods of modeling population structure with Lagrangian trajectories.

Introduction

As illustrated throughout this book, Lagrangian data can provide us with a unique perspective on the study of geophysical fluid dynamics, particle dispersion, and general circulation. Drifting buoys, floats, and even a crate-full of rubber ducks or athletic shoes lost in mid-ocean (Christopherson, 2000) may be used to gain insights into ocean circulation. All Lagrangian instruments will be referred to as “drifters” hereafter for simplicity. Because movement of a drifter tends to follow that of a water parcel, the primary attributes of Lagrangian measurements are (i) horizontal coverage due to dispersion in time, (ii) that many of the observed variables obey conservation laws approximately over some lengths of time, and (iii) their ability to trace circulation features such as meanders and vortices at a wide range of spatial scales. Due mainly to inherently irregular spatial distributions, the Lagrangian measurements must first be interpolated for most applications. As we will see, the design of interpolation and mapping schemes that can preserve the Lagrangian attributes is often non-trivial.

To observe finer dynamical details of oceanic and coastal phenomena and to forecast drifter trajectories more accurately (for search-and-rescue operation, spill containment, and so on), Lagrangian data afford a particularly informative and novel perspective if they are combined with a dynamical model, rather than mapped by a standard synoptic-scale interpolation procedure which can smear some details at smaller and faster scales. Data assimilation can be viewed as a methodology for imposing dynamical consistency upon observed data for the purpose of space-time interpolation.

Introduction

The study of Newton's second law of motion is a natural basis for all fluid dynamical problems and the Eulerian form of this law is the basis (in addition to the conservation of mass) of Euler (or Navier–Stokes) equations. Despite its primordial importance in Geophysical Fluid Dynamics (GFD, hereafter) the application of Newton's second law of motion to the rotating spherical earth is commonly done only briefly as an addendum to the fluid dynamical problems. A detailed analysis of these equations as applied to the motion of particles on the rotating spherical earth is the subject of the present paper, which summarizes the advances made in the subject in recent years. In particular a comparison between the dynamics on the β-plane and on the sphere will be carried out in order to highlight the ramifications of the inconsistent approximations made in transforming the spherical geometry to a planar one on the β-plane.

The complexity of the spherical geometry is the culprit behind the development of GFD in Cartesian coordinates. Several semi-analytical studies in spherical coordinates were published in the 1960s and 1970s (a review of these works can be found in Moura, 1976) but more recent studies on a sphere are mostly numerical. Recent discussions of the balance between acceleration, the Coriolis force and pressure gradient forces on the elliptical Earth, as well as the subtleties of the Coriolis force itself there, are given in Durran (1993) and Persson (1998).