In this chapter we introduce the concept of an ensemble average, which allows one to form averages for time-dependent processes. One such ensemble average statistical measure is the autocovariance (or autocorrelation) function. It gives information about the average time dependence of a process. The Fourier transform of the autocovariance, in turn, describes the frequency contents of the process. For two random functions of time one can define cross covariance between values of the two functions at different times. The Fourier transform of the cross covariance with respect to delay time gives the cross-spectral density. When these measures are independent of the choice of time origin, the processes are stationary. We shall look at examples of how one can derive a propagation speed from cross covariances or cross spectra and also see how one can find decay times and other properties of a random process. A useful application of spectra and covariance functions is to the relationship between input and output statistical measures for a linear system. From observations of the excitation and the response one is able to draw conclusions about the dynamics of a system. If one knows the system dynamics and some of the statistical properties of the input, one can find the statistical properties of the output, and vice versa.

Many fluid flows can be approximated by linear systems of equations. This means that, in turn, some flows may react to excitation in ways that we can analyze, especially if the excitation is weak.

An example of a linear response of a flow field to turbulence is the emission of acoustic waves from a turbulent jet, as first analyzed by Lighthill (1952) and discussed in Chapter 10. Flows that respond to excitation by divergent oscillation are unstable; such flows are discussed in Chapter 7.

Correlations and spectra depend upon the second moments of a joint probability density. In order to relate correlations and probability distributions, this chapter also outlines some of the elements of probability theory, including the central limit theorem and the normal distribution.

As an illustration of a non-normal distribution the log-normal distribution is presented.

Ensemble averages

A random function is a function that cannot be predicted from its past. An example of a random function of space and time is the velocity field in a turbulent jet.

Fluid flow turbulence is a phenomenon of great importance in many fields of engineering and science. It presents some of the most difficult problems both in the fundamental understanding of its physics and in applications, many of which are still unresolved. Turbulence and related areas have therefore continued to be subjects of intensive research over a period that has lasted for more than a century, and the interest in this field shows no signs of abatement.

In recognition of the need for helping graduate students prepare for their own research in this and related areas of fluid dynamics, a course with the cover title was started by one of us (E. M.-C.) some 20 years ago. Our joint efforts in producing a set of notes for this course has resulted in the present book. The course and its subject matter has evolved over this time period of teaching a mixed group of students from all fields of engineering and from many areas of science, including astrophysics, physics, chemistry, applied mathematics, meteorology, oceanography, and occasionally biology and physiology. With students of such widely different backgrounds we could not assume much commonality in preparation beyond the basics. Hence we found it necessary to start each topic at a fundamental level, and very few concepts could be borrowed from common professional experiences. Many of the students in the course were looking for a thesis topic or needed more insight into turbulence in support of their ongoing research. Discussions with students have resulted in the start of successful research subjects in many instances.

The main aim of the book is to give the students the background enabling them to follow the literature and understand current research results. The book stresses fundamental concepts and basic methods and approaches, although attempting to introduce some recent ideas that we think will prove important in future work on turbulence and related fields. The flavor of a course based on this book will be strongly dependent on the instructor and on the emphasis and the examples of research results chosen for presentation, since the book in itself is not a complete course. Reading of the literature and monographs are also needed.

In addition to correcting misprints and errors in the text, the equations, and the figures of the first edition, we also have further clarified points that have proved difficult for students. We also have benefited from reviews of the book and made other additions and changes as needed.

We added in Chapter 8 a short description of a simplified model for the temporal and spatial evolution of three-dimensional disturbances in a strong mean shear, which we thought might give some theoretical framework for the study of bursting in the near-wall region of a turbulent boundary layer. We also have added a short chapter (Chapter 12) on numerical modeling of turbulence, the lack of which many reviewers pointed to as a shortcoming of the first edition.

A few reviewers have questioned the need to include stability and wave motions in an introductory book on turbulence. In our view, research on hydrodynamic instability has contributed significantly to our understanding of how turbulence is created and maintained. The work in the new field of nonlinear dynamical systems and their chaotic behavior has added further insights showing, for example, that nonlinear waves may show chaotic behavior.

Additions notwithstanding, we have tried strenuously to retain the compactness of the book. It is intended to be a graduate-level introduction and overview of the subject suitable for a one-term course.

Flows of fluids of low viscosity may become unstable when large gradients of kinetic and/or potential energy are present. The flow field set up by the instability generally tends to smooth out the velocity and temperature differences causing it. The available kinetic or potential energy released by the instability may be so large that transition to a fully developed turbulent flow occurs.

Transition is influenced by many parameters. An important one is the level of preexisting disturbances in the fluid; a high level would generally cause early transition. Another cause for early transition in the case of wall-bounded shear flows is surface roughness. The manner in which transition occurs may also be very sensitive to the detailed flow properties.

For shear flows the basic nondimensional flow parameter measuring the tendency toward instability and transition is the Reynolds number; for high Re values, kinetic energy differences can be released faster into fluctuating motion than viscous diffusion will have time to smooth them out. For a heated fluid subject to gravity the Rayleigh number is the main stability parameter.

Of crucial importance for the tendency of a flow to become unstable and go through transition is the detailed distribution of mean velocity and/or temperature in the field. The analysis that follows is intended to illustrate this.

Although the flow processes involved in instability and transition might at a first glance appear to have only a slight resemblance to those observed in fully developed turbulence, they are nevertheless related to it in important ways. In a gross sense turbulence may be regarded as a manifestation of flow instability occurring randomly in space and time. The linear instability problem is the simplest flow model incorporating the interaction between unsteady fluctuations and a background shear or density distribution. With the aid of nonlinear instability theory one may also possibly be able to clarify some of the mechanisms whereby turbulence is maintained.

Instability to small disturbances

Because of the mathematical difficulties in the analysis of flow instability, only idealized cases for which the basic fluid flow properties vary with one spatial coordinate can be analyzed in a reasonably simple manner.

Introduction

This is an outline of some applications of finite-dimensional inverse theory to ocean modeling. The objective is ot to offer a comprehensive discussion of every application and its consequences; rather it is to introduce several concepts in a relatively simple setting:

• an incomplete ocean model, based on physical laws but possessing multiple solutions;

• measurements of quantities not included in the original model but related through additional physical laws;

• inequality constraints on the model fields or the data;

• prior estimates of errors in the physical laws and the data; and

• analysis of the level of information in the system of physical laws, measurements and inequalities.

Much of this material is well covered in mathematical texts, geophysical monographs and scientific review articles. Thus the presentation is brief and directed towards subsequent application of these concepts in more complex settings. However actual oceanographic studies are discussed, and tutorial problems are posed.

The β-spira

A major objective of physical oceanography in the 1970's was the exploration of the dynamics of mesoscale eddies and their influence on large-scale ocean circulation (Robinson, 1983). It was therefore something of a surprise when, in 1977, Stommel & Schott showed that the vertical structure of large-scale horizontal velocity fields could be explained using simple equations expressing geostrophy and mass balance. A compelling aspect of their study was the use of data in order to complete and then test their calculations. The original paper (Stommel & Schott, 1977) is somewhat cryptic.

Inverse methods combine oceanic observations with theoretical models of ocean circulation. The methods lead to

• estimates of oceanic fields from sparse data, guided by physical laws;

• estimates of meteorological forcing fields;

• estimates of parameters in the physical laws;

• designs for oceanic observing systems;

• resolution of mathematically ill-posed modeling problems; and

• tests of scientific hypotheses.

The rapid development of inverse methods in physical oceanography over the last decade has been greatly influenced by the work of meteorologists and solid-earth geophysicists; the growing interest of oceanographers in inverse methods is largely in response to the arrival of great volumes of data collected by artificial earth satellites. Collected articles may be found in conference proceedings edited by Anderson & Willebrand (1989), and by Haidvogel & Robinson (1989). Even greater volumes of data are anticipated from planned, future missions. The emergence of inverse theory as a scientific tool has provided a major stimulus to numerical modeling of ocean circulation, which has moved beyond the infancy of thought-experiments but has only tentatively entered the maturity of operational forecasting long occupied by meteorologists. The extraordinary recent gains in computer performance will enable theoretical oceanographers, who have come to inverse theory relatively lately, to consider applying the most elegant and powerful inverse methods to their most complex models. (While this preface was being written, a computer manufacturer announced a 128-gigaflops, 32-gigabyte machine.)

Introduction

A generalized inverse estimate of ocean circulation is a field which nearly solves an ocean circulation model and which nearly fits a finite set of data. It is intuitively obvious that if several of the data were collected at almost the same place or at almost the same time, yet had widely differing values, then fitting the data closely would be very difficult. Any field which did fit would tend to fluctuate wildly in an extended neighborhood of the observing sites. However, if the data were assumed to possess significantly large errors, then a relatively well behaved field might be able to come consistently close to the data: see Fig. 6.1.1. Nevertheless it would be concluded that there is redundancy in the data.

The concepts “almost the same place or time,” “significantly large error” and “well behaved” are defined relative to the scales and amplitudes of the circulation, as characterized by the stream-function, for example. In a generalized inverse estimate the scales and amplitudes are those of the initial errors, the dynamical errors, and the internal scales of the dynamics. The initial and dynamical information is contained in covariances such as A and Q in §5.6, and in the dynamical operator L appearing in §5.2. The measurements are completely described by the functionals C, and the measurement error covariance matrix such as, also defined in §5.6.

It is shown in this chapter that for linear dynamics, the redundancies in the observing system or “antenna” may be determined from these parameters and operators.

Introduction

An essential feature of the inverse theory described in the preceding chapters has been linearity, which allows explicit expressions for the solutions of the various inverse problems. The expressions permit complete analyses of the analytical and statistical properties of the estimates. The one striking exception has been the evolution equation for the state error covariance in the Kalman filter. The equation is nonlinear, being of matrix-Riccati form, as a consequence of the requirement for a sequential approximation to the fixed-interval smoother. However, the filter estimate of the state does depend linearly on the prior estimates of forcing and initial conditions, and on the data.

The linearity of an inverse theory may be violated in many ways.

Nonlinear dynamics

The complexity of fluid motion is fundamentally due to advection, which takes a nonlinear form in the Eulerian description of flow. While there are a few classes of interesting and important oceanic motions which are adequately described by linearized dynamics, or at most linearized advection, such as tides, coastal trapped waves, equatorial interannual variability and Sverdrup flow, the importance of advection is unavoidable for most oceanic circulation. The classical examples are the western boundary currents of the subtropical gyres, the highly variable extension regions of the boundary currents, the meandering Antarctic Circumpolar Current, and equatorial mesoscale variability. The dynamics of a model may also be nonlinear as a consequence of turbulence closures expressed in terms of shear-dependent diffusivi-ties. Ocean models which include chemical tracers or biological fields may be nonlinear owing to chemical reactions, or to predation or to grazing. Only linear measurement functional have been considered thus far.

Spatial structure

The Kalman filter (KF) was originally developed for simple dynamical systems, such as the few ordinary differential equations representing the motion of a projectile. Ocean models are represented by partial differential equations, which are equivalent to infinite systems of ordinary differential equations. These are known as “distributed parameter systems” in the engineering literature (e.g., Aziz, Wingate & Balas, 1977). There is a question of convergence, equivalent to determining whether the KF estimate of the state (here, the ocean circulation) is physically realizable. Unsatisfactory estimates are obtained even at modest spatial resolution if realizability conditions are not met. Typically, the estimates are strongly influenced by the data only in the immediate neighborhood of the measurement site. In that case, the approximate estimation procedure has served little purpose. The range of influence of the data is correctly determined by the scales of the dynamics and also those of the system noise covariance. A small range may be the consequence of realistic choices for the scales, in which case even exact KF estimation would serve little purpose. The objective of this section is to analyze the relationship between the range of influence of the data (that is, the spatial scales of the Kalman gain) and the scales of the dynamics, the system noise, and, to a lesser extent, the initial noise. The discussion follows Bennett & Budgell (1987, 1989).

To expedite the analysis of scales, the single-layer quasi-geostrophic model (3.3.1) will be used in conjunction with the choice (i) for boundary conditions, namely, periodicity of all fields in the x-and y-directions.