Introduction

A useful mathematical framework for studying linear and nonlinear stability is the theory of ordinary differential equations (ODEs), also known as the theory of dynamical systems when the focus is on geometric and qualitative representations of the ideas and solutions. An informal presentation of this theory was given in Chapter 1. The goal of this chapter is to give a more systematic account of it from a mathematical point of view. In particular, we shall show how to reduce the number of degrees of freedom of a problem to obtain the “normal forms” of elementary bifurcations referred to often in the previous chapters.

In spite of its restrictive nature relative to the theory of partial differential equations, the theory of dynamical systems has revealed the extraordinary richness and complexity of the types of behavior that can arise when nonlinear effects play a role. This theory originated in the work of Henri Poincaré, in particular, in his book Méthodes nouvelles de la mécanique céleste. Poincaré's ideas were further developed during the first half of the twentieth century by the Russian school of mathematics (Kolmogorov, Arnold). The discovery that a system with a small number of degrees of freedom can display unpredictable, chaotic behavior then led to a great deal of research in this area beginning in the 1960s. Qualitatively new concepts such as deterministic chaos and sensitivity to the initial conditions were introduced, and these have significantly modified our understanding of deterministic models and their use in the description of natural phenomena.

Introduction

The inertial instability of parallel flows described in the preceding chapter is associated with the existence of an inflection point in the velocity profile. This is the principal instability of parallel or quasi-parallel shear flows at large Reynolds number and far from walls or interfaces, such as mixing layers, jets, and wakes. We have seen that the order of magnitude of the growth rate is U/δ, where U is the difference of the speeds on either side of the vorticity layer of thickness δ, and that viscosity plays only a diffusive role tending to attenuate the growth rate. The instability of flow profiles without inflection points is profoundly different. Let us consider two fundamental flows: plane Poiseuille flow and boundary layer flow. Observation shows that plane Poiseuille flow is unstable beyond a certain Reynolds number. Similarly, a boundary layer on a surface becomes unstable at some distance from the leading edge. However, these two flows do not possess an inflection point, and so, ignoring viscosity, they are stable according to the Rayleigh theorem. On the other hand, the growth rate of the observed instabilities is much smaller than would be expected for an inertial instability. It is therefore clearly important to investigate the role played by viscosity, which is the goal of the present chapter. We shall see that viscosity has two effects: the expected stabilizing dissipative effect, and also a destabilizing effect.

Scaling

As a model of the fluid motion and thermodynamics of the Earth's oceans, the basic equations (1.15)–(1.19) include a formidable array of distinct types of physical processes, from divergence of molecular diffusive fluxes on scales of millimeters to coherent fluid flow on the scales of the Earth's circumference, a range of space scales of order 1010. This range of scales is comparable to that between the molecular scale and the scale of a mammal's body. Thus, in a rough sense, the problem of using these equations to understand the large-scale structure of the ocean is comparable in difficulty to using numerical computations of molecular interactions to simulate the behavior of a mammal. Numerical solution of this full set of equations is thus well beyond current computational capacities and will remain so for the foreseeable futuremore. Furthermore, these equations are sufficiently challenging that fundamental mathematical properties, such as the existence and uniqueness of solutions, are not established. Some of these basic properties remain a subject of mathematical research even for the incompressible Navier-Stokes equations, a simplified set of four equations that may be obtained from (1.15) and (1.16) by replacing the density variable ρ with a constant value ρ0.

Despite these formidable challenges, it is possible to make progress by using a combination of mathematical and physical reasoning to derive simplified, or reduced, equation sets that describe motions on the largest space and time scales of interest.

The Large-Scale Ocean Circulation

Systematic observations of the fluid properties of Earth's ocean, made primarily over the course of the last hundred years, reveal coherent features with scales comparable to those of the ocean basins themselves. These include such structures as the subtropical main thermoclines and anticyclonic gyres, which appear in all five midlatitude ocean basins, and the meridional overturning circulations that support the exchange of waters across the full meridional extent of the ocean, from the polar or subpolar latitudes of one hemisphere to the opposing high latitudes of the other. These features and motions, which prove to be connected by robust dynamical balances, constitute the large-scale circulation of the Earth's ocean.

The global field of long-term mean sea-surface temperature is dominated by the meridional gradients between the warm equatorial regions and the cold poles but contains significant zonal gradients as well (Figure 1.1). The global long-term mean sea-surface salinity field has a more complex structure (Figure 1.2), with isolated maxima in the evaporative centers of the midlatitude subtropical gyres. The global sea-surface density field computed from long-term mean temperature and salinity reflects the competing influences of temperature and salinity on density (Figure 1.3). These fields are the surface expressions of complex three-dimensional interior property fields. The downward penetration of the warm equatorial temperatures is generally limited to the upper one-fifth of the water column (Figure 1.4), while salinity perturbations are less strongly confined to shallow depths (Figure 1.5).

Domain and Boundary Conditions

With the basic large-scale-approximate equations and some of their general properties established, it is appropriate now to examine the circulation and thermocline structures that arise in specific solutions of these equations. Even for the simplest case of steady solutions, in which the flow is independent of time, this requires that a sufficient set of boundary conditions be specified. This set of boundary conditions is not unique: a variety of such conditions, which differ in physical and mathematical detail, may yield solutions of physical interest for the large-scale flow. In general, these sets of conditions for idealized models of ocean gyre structure and circulation must be chosen to represent the definitive characteristics of the general physical setting in which these large-scale features develop.

A fundamental element of the boundary conditions is the geometry specified for the ocean basin. The simplest such basin is a rectangular domain, restricted to a portion of one hemisphere, with vertical sidewalls aligned along lines of constant longitude x = {xW, xE} and latitude y = {yS, YN} and a flat bottom at the constant depth z = -H0 (Figure 5.1). Such a choice avoids the singularity of the planetary geostrophic momentum equations that arises at the equator, where the Coriolis parameter f vanishes. It also removes a complicating geometric element: the circumpolar connection that exists around the Antarctic continent in the southern hemisphere, which will be seen (Chapters 7 and 8) to have significant impact on the surface and mid-depth circulation.

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.