A Perturbation-Theoretical Perspective

The basic point of view around which the development in this text has been organized is that of perturbation theory. In general, perturbation theory allows the replacement of a difficult or intractable problem with a simpler approximation that includes the leading-order terms in an expansion of the original variables in powers of a suitable small parameter. In the present case, a formal expansion was not necessary because the effective leading-order terms for large-scale ocean circulation dynamics could be identified by direct scaling of terms in the fundamental equations. These leading-order terms—the perturbation theory for the large-scale ocean circulation–are the planetary geostrophic equations (2.102)–(2.107).

The planetary geostrophic theories of large-scale circulation described in this text provide deep, quantitative insight into the large-scale physical structure of the ocean. For example, the reduced-gravity and ventilated thermocline models (Chapter 5) give persuasive explanations for the basic structure of the upper main subtropical thermocline, including the characteristic downward and westward slope of thermocline isosurfaces (Figure 1.7). Similarly, the Sverdrup interior solution (3.36) for the depth-integrated wind-driven transport may be favorably compared with interior geostrophic transports determined from hydrographic measurements.

Given a solution of an approximate problem constructed using perturbation theory, one would like to know how accurately it represents the solution of the original problem. If the original problem cannot be solved theoretically by other means, as is the case with large-scale ocean circulation, then the comparison must be made to observations of the system under consideration.

The Geostrophic Constraint

Suppose now that the basin geometry is altered from the simple, closed domain considered in the preceding chapters to contain a representation of the circumpolar connection that exists around Antarctica in the Southern Ocean. The simplest such representation is a periodic, reentrant zonal channel in the southern portion of an otherwise closed southern hemisphere domain, with width comparable to that of the opening between South America and Antarctica at Drake Passage (Figure 7.1). Even if the channel is partly blocked by a sill along the domain boundary at some depth above the basin bottom, this is a highly idealized geometry: in the Southern Ocean, much of the latitude-depth cross section of the Drake Passage opening is obscured at other longitudes by islands or shallow bottom topography, which can be anticipated to have a strong influence on both the circumpolar and the meridional flow. Nonetheless, this geometry allows the exploration of several fundamental elements of the effect of the circumpolar connection on the large-scale flow.

Let the periodic zonal channel representing the circumpolar connection cover the latitude range [y-, y+] and the depth range [-Hs, 0], where Hs ≤ H0 is the depth of the sill at x = xE or x = xW. In the channel opening, periodic boundary conditions apply, and the points at x = xE and x = xW are identified, so that ϕ(x = xE) = ϕ(x = xW) for any dynamical variable ϕ.

The purpose of this text is to give a concise but comprehensive introduction to the basic elements of the theory of large-scale ocean circulation as it is currently understood and established. The intended audience is graduate students and researchers in the fields of oceanic, atmospheric, and climate sciences and other geophysical scientists, physicists, and mathematicians with a quantitative interest in the planetary fluid environment.

When I first began to study the physics of ocean circulation, it was the intrinsic scientific interest of the subject that was most apparent and appealing to me. Since that time, evidence has grown strong that human activities are substantially modifying the Earth's climate, with long-term effects that threaten to significantly disrupt the environmental structures on which human life and civilization depend. This troubling development brings a new imperative to the study of the ocean's large-scale circulation as this circulation and its interactions with the atmosphere and cryosphere play a clearly important, but still poorly understood, role in the global climate system. Although the ocean components of most numerical climate models are based on the primitive equations, the dynamics that they represent are essentially those of the planetary geostrophic equations described here, because of the necessarily coarse horizontal resolution of climate-model computational grids. Thus, the present material should be of particular interest to climate dynamicists.

The text is based on lecture notes that accumulated over roughly the last decade, during which I regularly taught a core graduate physical oceanography course on the theory of large-scale ocean circulation.

Meridional Overturning Circulations

An essential element of the circulation of the Earth's ocean is the large-scale overturning flow that spans its full meridional reach, from the high southern to the high northern latitudes. To represent this flow, the single-hemisphere basin must be extended across the equator so that it reaches both the high southern and the high northern latitudes. Such a double-hemisphere rectangular basin, again with a circumpolar connection at the high southern latitudes, may be considered a simple model of the Atlantic sector of the world ocean (Figure 8.1). The extreme southern and northern latitude of the model basin, y = yS and y = YN, will now be in the southern and northern hemisphere, respectively. The western and eastern boundaries remain at x = xW and x = xE, with periodic boundary conditions for the circumpolar connection at y- < y < y+, z > -Hs.

This idealized basin retains the basic structure of the Atlantic basin, including the important circumpolar connection in the Southern Ocean. Despite neglecting many potentially significant features, including the second existing connection through the Bering Strait and the Arctic and Pacific Oceans as well as all the other complexities of seafloor topography and coastal and island geometry, it provides a useful starting point for the exploration of fundamental aspects of the large-scale overturning circulation. In such a simplified two-hemisphere basin, the circulation patterns that are naturally restricted to a single hemisphere, such as the subtropical and subpolar gyres and the circumpolar current, can be anticipated to retain the basic features and character that they possess in a single-hemisphere basin.

Goals and examples of sequence analysis

Sequences of data, either in space or in time, appear all the time in ocean research. You may have a time series of measurements at a location (e.g. sediment trap data, or ocean surface temperature), a series of stations along a hydrographic section, or isotope measurements on a long sediment core. For the sake of simplicity (initially) we shall discuss only regularly sampled data; that is, samples taken at identical intervals in space or time. The analysis becomes more difficult and more complicated when we discuss irregularly spaced samples, but the principles are similar and best understood in terms of the simplest case first. What do we hope to achieve in the analysis of data sequences? There are as many reasons (or perhaps more) as there are data sequences. The next subsections outline briefly some of the major conceptual motivations.

Searching or testing for structure or periodicities

Within a single data set you might be looking or testing for changes in a system due to periodic forcing, for example the effect of seasonal changes on biological production, or the effect of lunar tides on shell-fish contamination. This may be extended to spatial regularity as well, in that you may be looking for evidence of large-scale Kelvin waves (rapidly propagating variations of the thermocline depth) on dissolved nutrients near convergence zones in the ocean.

Until now we have concentrated on what may be loosely termed “data analysis methods”. In some respects, this is a form of modeling in that we are attempting to interpret our data within the context of some intrinsic model of how our data should behave, whether it be assuming the data follow an underlying probability distribution, vary as a function of some other variables, or exhibit some periodic behavior as a function of time. We hope you are beginning to see that all of these methods share common mathematical and algorithmic roots, and we want you to realize that many of these tools will come in handy as we now embark on a more model-intensive course.

Before doing so, we want to outline the basic aspects of model design, implementation, and analysis. Selecting the most accurate and efficient algorithms and developing robust and usable MATLAB code is important, but most of your intellectual energies should be directed at the design and analysis steps. Moreover, although correct design is vital to any successful modeling effort, developing the tools to efficiently analyze model output is just as important. It is critical to assessing the mechanics of how a model is performing as well as ultimately understanding the underlying system dynamics and how well a model compares to observations.

Basic principles

Constructing numerical models of marine systems usually involves setting up a series of partial differential equations, specifying boundary conditions and then “running the model”. Your purpose may be to establish the value of parameters (e.g. rates of reaction or the magnitude of some property), estimate fluxes, or make some prediction about the future state of the system. Although you can sometimes choose a physical problem that is simple enough to be modeled with analytic solutions (an example would be Munk's 1966 “Abyssal recipes” model; Chapter 13), more often than not you will encounter situations where the processes or the geometry of the system are too complex to allow analytic solutions.

Don't get us wrong; analytic solutions are nice. They can often provide you with a nice conceptual, intuitive feel for how the system responds, especially in an asymptotic sense. However, for realistic geometries, you will find that the few analytical solutions provided in many books are infinite series solutions. Be very, very careful when dealing with those series solutions. Pay particular attention to the assumptions made in deriving the solutions, to the conditions under which they ought to be applied, and especially to convergence issues.

Onward to the next dimension

Although one-dimensional models provide useful insight into basic biogeochemical processes, we are forced to admit that the world is made of more than one spatial dimension. The addition of an extra dimension to a model often does more than “fill space”, but rather imbues the model with behavior that is qualitatively different from its lower-dimensional analogue. The opportunity presented by the extra dimension is that more interesting, and perhaps more “realistic” phenomena may be modeled. This opportunity brings with it challenges, however, that are not just computational in nature. The choices of model geometry, circulation scheme, and boundary conditions become more complicated. Seemingly innocuous choices can have subtle or profound effects on how your model behaves. Moreover, matching model results to observations often requires decisions about whether features result from intrinsic processes of interest, or are mere artifacts of the choices made in model configuration.

For instructional purposes, we'll stick to a genre called gyre models which, as you might guess, are characterized by a quasi-circular flow on a plane. Such models have utility in the subtropics – at least that's where we'll be dwelling here – but can be used in many other parts of the ocean.

We treat one-dimensional models of sedimentary systems separately in this book because of the added complication that they contain two phases – solid material and pore waters – that not only can interact biogeochemically, exchanging chemicals, but also can move in relation to one another. In fact, with a reference system fixed at the sediment–water interface, the solid phase is actually moving owing to a combination of sedimentation (addition of material at the interface) and compaction. If you're thinking that this makes the construction of models a little more complicated, those are exactly our sediments!

We will be talking about the process of diagenesis, i.e. the sum total of all processes that bring about changes to sediments after they have been deposited on the seafloor. This includes everything from bioturbation through chemical transformation to compaction and pore water extrusion. The general topic of diagenesis extends to even longer timescale processes that include metamorphism and weathering of sedimentary rocks after uplift, but we will focus on Early Diagenesis (Berner, 1980), which encompasses changes that occur at or near the sedimentary surface or in the upper portion of the sedimentary column.

Why scientific visualization?

Throughout this book we have used a number of MATLAB's graphical capabilities as tools to monitor the progression of our mathematical and numerical travails, to demonstrate some characteristic of our results, or to reveal underlying relationships in data. Our emphasis now will be on the basic process of scientific visualization and providing you with some advice on how to effectively use (and not abuse) the many tools available to you. You may think that scientific visualization is an easy and natural thing to do, especially given the relatively powerful and reasonably intuitive tools built into MATLAB and other “point and click” packages so readily available. However, in the many years that we have been attending conferences, reading journals, and perusing text-books, we have encountered some ghastly instances of computer graphics abuse (or more to the point, abuse of the poor viewer/reader). This is a shame, because invariably the presenter has worked hard, often under difficult circumstances, to acquire scientific data, execute a model, or discover an erstwhile hidden relationship … only to fail to communicate the final result effectively. After all, isn't communication the final end-product of all our scientific endeavors?

We could also regale you with the awe-inspiring size of today's huge data sets, but rest assured that tomorrow's will be even more impressive.

Rationale

Our main objective in studying one-dimensional, open-ocean advection–diffusion models is pedagogical. The fact that they have relatively simple analytical solutions makes them a useful starting point for studying ocean models. In fact, you may find yourself turning to these more idealized representations as a tool for building intuition about the behavior of more complex models. That is, you might build a “model of your model” to explore what is happening within it. Perhaps more important to the student of modeling, they represent an elegant example of how we can use spatial distributions to illuminate underlying physical and biogeochemical dynamics.

They're not really considered “state-of-the-art”, having been extensively exploited by geochemists starting in the 1950s and used by others many decades before that. In truth, there are few parts of the ocean that can be regarded as truly satisfying the assumptions and requirements of this class of model. Even then it is highly debatable how generalizable the parameters derived from such models really are to the rest of the world. However, it is instructive to think of the abyssal ocean in terms of simple one-dimensional balances because it helps build intuition about open-ocean processes. Certainly it is an interesting historical stage in the evolution of geochemical ocean modeling, and has much to offer as a learning tool for understanding the process of ocean modeling.