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.

Introduction

Most major ocean currents have dynamics which are significantly nonlinear. This precludes the ready development of inverse methods along the lines described in Chapter. Accordingly, most attempts to combine oceans models and measurements have followed the practice in operational meteorology: measurements are used to prepare initial conditions for the model, which is then integrated forward in time until further measurements are available. The model is thereupon re-initialized. Such a strategy may be described as sequential. It is clearly the only choice for prediction (that is, genuine forecasting in real time), but the relative simplicity of the approach has led to its adoption for smoothing as well. In the latter situation, measurements are available in some fixed interval, and a best estimate is required at each time t in the interval.

Sequential estimation techniques have become known to meteorologists and oceanographers as data assimilation. There has been extensive development of data-assimilation methods in meteorology, and it is fortunate for oceanographers in particular that the methods are now comprehensively described in the text by Daley (1991). The meteorological problem is especially difficult. First, the synoptic time scale in middle latitudes is only a few days; thus predictions of the synoptic scale must be prepared within a few hours of receipt of the measurements in order to be of any value. This creates great stress on the computing resource.

Introduction

The conventional introduction to partial differential equations uses examples from classical physics: potentials, diffusion and waves. The partial differential equations (pde's) only describe the local shape of solution surfaces in space or space-time; a unique, global determination of the solution requires that some of its values be provided at least on certain faces of the domain in space or space-time. That is, initial and boundary conditions are needed. The choice of initial conditions is usually simple, with causality being the guide. The thermodynamic or mechanical state of the system must be completely specified at an instant. Boundary conditions, on the other hand, make manifest the nature of the rest of the universe. In particular the familiar conditions of Dirichlet, Neumann and Robin (Jackson, 1962) indicate both the type of surrounding medium and the time interval of interest. For example, the Dirichlet condition for the diffusion equation reveals the presence of a reservoir of so great a capacity that its level does not change as the reservoir is emptied or filled by diffusive exchange with the system of interest. On the other hand, the homogeneous Neumann condition indicates that the surrounding medium is so effectively insulating that no significant diffusion takes place during the time interval of interest. The Robin or mixed condition is no such idealization of a simple property. By specifying a diffusive flux proportional to the state of the system, the condition models a more complex transfer process, involving both the system and the rest of the universe.

Introduction

The quasi-geostrophic circulation models considered in Chapters 3-8 contain mechanisms of fundamental importance, such as baro-tropic instability, baroclinic instability (Pedlosky, 1987) and fronto-genesis (Stone, 1966). Nevertheless, only the more complex primitive-equation models (Lorenz, 1967) can reasonably be expected to have a close resemblance to major ocean current systems. The many assumptions underlying the simpler models do not hold in the real ocean. Consider, for example, the Antarctic Circumpolar Current system (ACC). A meridional section of the density field is shown in Fig. 9.1.1 (Patterson & Whitworth, 1990). Zonal geostrophic velocities of several tens of centimeters per second may be inferred, implying a Rossby number as small as. However, the bathymetry has changes of order unity in the meridional direction alone, as do the depths of the isopycnal surfaces. Moreover, in some of the eddies and rings associated with the ACC, the Rossby number may approach a value of 10-1 (Bryden, 1983). Thus for varying reasons it may be concluded that the horizontal field of velocity in the ACC is significantly divergent, and therefore poorly represented in the quasi-geostrophic approximation. Similar remarks may be made about the Gulf Stream (Stommel, 1960) and about the Kuroshio (Stommel & Yoshida, 1972). Equatorial ocean dynamics are fundamentally non-geostrophic, even on seasonal or even interannual time scales, owing to the significance of Kelvin waves (Philander, 1990).

Introduction

Conventional ocean modeling consists of solving the model equations as accurately as possible, and then comparing the results with observations. While encouraging levels of qualitative agreement have been obtained, as a rule there is significant quantitative disagreement owing to many sources of error: model formulation, model inputs, computation and the data themselves. Computational errors aside, the errors made both in formulating the model and in specifying its inputs usually exceed the errors in the data. Thus it is unsatisfactory to have a model solution which is uninfluenced by the data. In the spirit of the inverse methods in Chapter 1, the approach which is developed here finds the ocean circulation providing the best fit simultaneously to the model equations and to the data. The best fit is defined in a weighted least-squares sense, with weights reflecting prior estimates of the various standard errors. Once unknown errors are explicitly included in the model equations and the data, the problem of finding the circulation is underdetermined, and so the least-squares fit may be regarded as a generalized inverse of the combined dynamics and observing system.

Finding the generalized inverse may also be regarded as a smoothing problem. The smoothing norm involves the differential operators for the model equations as well as derivatives of, or covariances for, the errors or residuals in the equations. Indeed, it is shown that the generalized inverse is equivalent to Gauss-Markov smoothing in space and time, based on the space-time covariance of model solutions forced by random fields having prescribed covariances.

The general circulation

Introduction

The general circulation of the oceans is an essential component of the thermodynamic system which determines global climate. The contributions of the oceans to the poleward fluxes of heat and water, for example, are clearly significant if not yet reliably known (Lorenz, 1967). It is widely accepted that modeling has improved our understanding of the general ocean circulation, but the objectives of ocean modeling are evolving along with the models themselves. Goal 1 of the World Ocean Circulation Experiment (WOCE) restates the grand objective of physical oceanography:

(WOCE, 1988). So far models have been developed by exploring the consequences of adding ever more physics and ever more detail. The earliest developments showed that westward intensification in subtropical gyres could be attributed to a combination of the β-effect with flow at high Reynolds number (Stommel, 1948; Munk, 1950). The nonlinear effect of vorticity advection was shown by Bryan (1963) to lead to time-dependent flow on the §-plane at high Reynolds number, even if the flow is steadily forced. An imposed mean density stratification allows the Lorenz cycle of energy exchange between mean and eddy forms of available potential energy and kinetic energy (Holland & Lin, 1975a,b). Stratification, determined internally by thermodynamics, is found to develop plausible thermohaline structure in response to reasonably representative mean surface fluxes of heat, salt and momentum (Bryan & Cox, 1968a,b; Bryan & Lewis, 1979).

The object of the appendix It is clear that elastoplasticity problems are not easily amenable to analytical methods, but for a few exceptions as in the case of the spherical envelope in Chapter 6. In particular, the elastoplastic borderline separating the region where the material still behaves elastically and the already plasticized region is an unknown in such problems. For complex geometries then a numerical implementation seems necessary (Chapter 11). However, the few cases that admit analytical solutions are typical of a methodology of which any student and practitioner of elastoplasticity must be aware. We have thus selected four examples, the first in plane strain (the wedge problem), the second in torsion, the third exhibiting a complex loading and the fourth accounting for anisotropy in a composite material.

Elastoplastic loading of a wedge

General equations

A wedge of angle β < π/2 is made of an isotropic elastoplastic material, satisfying Hooke's law in the elastic regime and Tresca's criterion without hardening at the yield limit. On its upper face it is subjected to a pressure p which increases with time (Fig. A3.1). We look first for the fully elastic solution and then for the elastoplastic solution in which the plasticized zone progresses until the whole wedge has become plastic. The solution of this problem in the elastoplastic framework is due to Naghdi (1957) – see also Murch and Naghdi (1958) and Calcotte (1968, pp. 158–64).

The object of the chapter In the absence of plastic strain, the problem of brittle fracture by extension of cracks can be presented in a thermodynamic framework, analogous to that of elastoplasticity. This means that the fracture criterion (or the criterion of crack propagation) replaces the plasticity criterion. One important notion is the notion of mechanical field singularity (displacement, stresses).

Introduction and elementary notions

We are interested in the problem of fracture, a phenomenon that occurs, more or less violently, under monotonic loading (whereas fatigue concerns cyclical loading). More specifically, we are interested in the problem of cracking, that is, the progagation of macroscopic cracks (of size of the order of one millimetre), whereas the beginning of cracking belongs to the microscopic and to the metal analyses which will not be examined here. (Microscopic cracks are one cause of damage – see Chapter 10.) The aim of this study is to arrive at a formulation of the crack-propagation laws, based upon fracture criteria and the definition of the conditions that may insure resistance to this fracture. We are certainly aware of the interest that such a subject implies for industry; it suffices to think about aeronautical engines and nuclear installations. Actually, our main interest is brittle fracture, that is, the kind that occurs without considerable plastic strain (i.e. the separation mechanism of crystallographic facets through cleavage), whereas ductile rupture is produced by different mechanisms accompanied by great plastic strains).