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).