The problem of ocean acoustic tomography is to infer from precise measurements of travel time, or of other properties of acoustic propagation, the state of the ocean traversed by the sound field. The tomographic method was introduced by Munk and Wunsch (1979) in direct response to the demonstration in the 1970s that about 99% of the kinetic energy of the ocean circulation is associated with features that are only about 100 km in diameter, called the mesoscale. Measuring and understanding the behaviors of both the mesoscale and the larger-scale features associated with the general circulation present a formidable sampling task. Not only are the flow elements very compact spatially, but also they have long time scales (order 100 days). To produce statistically significant measurements of the fluid behavior, even in an area as compact as 1 Mm × 1 Mm (1 megameter = 1000 km), about 1% of an ocean basin, requires several full-time vessels or several hundred fixed moorings. One is accordingly led to the technology of sound propagation to measure the properties of the fluid between moorings.

Ocean acoustic tomography takes advantage of the facts that (i) travel time and other measurable acoustic parameters are functions of temperature, water velocity, and other parameters of oceanographic interest and can be interpreted to provide information about the intervening ocean using inverse methods, and (ii) the ocean is nearly transparent to low-frequency sound, so that signals can be transmitted over distances of many thousands of kilometers.

Over drinks in the Cosmos Club in 1979, Athelstan Spilhaus, who had perfected the bathythermograph for measuring temperature profiles to predict the ranges at which submarines could be detected acoustically, held forth that it should be done the other way around: the measured sonar transmission should serve to determine the ocean temperature field. Unknown to Spilhaus, we were in Washington to persuade the Office of Naval Research and the National Science Foundation to fund an experiment to do just that.

In seismology, the inversion of travel times to map the interior of the Earth has been the time-honored procedure, since the Earth is not readily accessible to direct intrusive measurements. In medicine, intrusive methods are viewed with some reluctance (at least on the part of the patient), and this has led to the development of computed tomographic inverse methods using X-rays. In contrast, the oceans are accessible to direct intrusive measurements; the limits are set by the availability of costly platforms for adequate sampling. Unlike the seismological and medical applications, ocean time variability is an essential component, and the requirements for sampling in space and time are severe. With only a few research vessels plying the world's oceans, it is not surprising that the first century of oceanography had a strong climatological flavor.

It came as a great shock in the 1960s that the oceans, like the atmosphere, had an active weather at all depths. The storms within the sea are called eddies.

Climatological sound-speed profiles and predicted acoustic arrival patterns for selected locations worldwide (fig. B.1) are displayed in this appendix. The locations are from a regular grid spaced at 15° increments in latitude and 20° increments in longitude. [Worcester and Ma (in press) provide results for all grid locations exceeding 2000 m depth.] The atlas is organized by location, beginning at 75°N and proceeding southward. At each latitude, results are presented in order of longitude, proceeding westward from the prime meridian. The interpretation of the plots presented here is discussed at length in section 2.16. Each panel is described briefly, proceeding counterclockwise from the bottom left.

The sound-speed profiles (bottom left) were computed from annual-average climatological temperature and salinity data due to Levitus (1982), using the Del Grosso (1974) sound-speed equation. The Levitus climatology is a horizontally smoothed picture of the ocean, so the results do not properly represent the behavior to be expected in frontal regions.

Acoustic normal mode functions 1 and 7 computed for 70 Hz are displayed at bottom center. The amplitude normalization is arbitrary. The group velocity for each mode is given immediately below the mode function.

Time fronts in τ, z-space for a fixed range of 500 km (bottom right) show the arrival structure for a source on the sound-channel axis, when one exceeding 100 m depth exists, or for a source at 100 m depth.

The discussion so far has dealt with a range-independent (RI) sound channel. But the ocean certainly varies horizontally and is always range-dependent (RD); one of the chief goals of tomography is to derive its range-averaged (RA) properties.

The RD treatment will vary depending on whether the scale of the horizontal variations is larger, comparable to, or very much smaller than the ray-loop range (typically 50 km). The term “adiabatic range dependence” is defined to apply to the case of small fractional variation over a ray loop. Variations on a gyre scale can accordingly be treated by the adiabatic approximation, assuming there are no sharp frontal surfaces.

The term “loop resonance” applies to ray travel-time perturbations due to ocean perturbations with horizontal scales equal to the ray-loop scale, or to a fraction of the loop scale. This includes mesoscale activity (which accidentally has a scale comparable to the loop scale) and ranges down to the longer components in the internal wave spectrum. Cornuelle and Howe (1987) have shown that measured travel-time perturbations associated with loop resonance can provide some RD information for even a single source-receiver transmission path.

Internal waves are generally included among the small-scale processes for which the forward problem yields estimates of the variance and other statistical properties of the travel time (Flatté et al., 1979). These estimates are required for inversion of the measured data set. In turn, the measured variances can provide useful information about the small-scale structure (Flatté and Stoughton, 1986).

Introduction

The preceding chapters have demonstrated that a variety of measurable acoustic features, including ray travel time, amplitude, and inclination, mode group velocity, and carrier phase, are integral functions of the oceanic sound-speed field. As discussed in previous chapters, sound-speed is intimately related to the oceanic density field, which is, in turn, a dynamic variable related to the oceanic flow field. Under many circumstances, knowledge of the density field alone is adequate to compute the oceanic flow field to a high degree of approximation. Reciprocal tomographic measurements are direct weighted averages of the flow field in the plane of the source and receiver. Thus, determinations of C and u carry immediate implications for the ocean circulation and must be consistent with known physics.

The forward problem has been presented in detail: Given C (or S) and u, and the characteristics of a sound source, compute the detailed structure of the signal as recorded at a receiver of known characteristics. This problem is labeled “forward” mainly as a reflection of its connection to the classic problem of finding solutions to the wave equation.

The “inverse” problem demands calculation of the ocean properties, C and/or u, given the measured properties of the arriving signal. At this stage, the problem becomes a matter of intense oceanographic interest.

Oceanographers are mostly familiar with point value data (e.g., a current meter reading or a thermometer measurement). In contrast, tomographic data are weighted integrals through the oceanic field.

Introduction

In the previous chapter it was shown how solutions of the Navier-Stokes equations could be constructed. Uniqueness of those solutions requires more regularity, however, than that which follows directly from their construction via the Galerkin approximations. In this chapter we shall begin to see how much regularity is needed to ensure smoothness of solutions of the Navier-Stokes equations. The minimum requirements can be reached for the 2d problem, but the problem remains open for the 3d case.

This chapter is devoted to the statement and proof of what will be referred to as the ladder theorem for the Navier-Stokes equations on ω = [0, L]d with periodic boundary conditions and zero mean, and a discussion of its consequences in both 2d and 3d. This will enable us to relate the evolution of a seminorm of solutions of the Navier-Stokes equations, containing a given number of derivatives, to one containing a lower number of derivatives. In sections 6.3 and 6.4, it is shown how the ladder leads to the identification of length scales in the solutions. Subsequently, section 6.5 contains those estimates that can be gleaned via the ladder from the 2d and 3d Navier-Stokes equations where no assumptions have been made. Finally, to show how forcing fields can be handled differently from the static spatial forcing f(x) of previous chapters, section 6.6 briefly shows how a ladder may be derived for thermal convection.

To derive the ladder theorem, it is necessary to introduce the idea of seminorms which contain derivatives of the velocity field higher than unity.

Introduction

In this chapter we show how the dimension of the global attractor ℕ can be estimated for the Navier-Stokes equations. The approach is an extension of that developed in Chapter 4 for ordinary differential equations where it was shown that if N-dimensional volume elements in the system phase space contract to zero, then the attractor dimension dL(ℕ) must be bounded by N. For partial differential equations the technical chore remains the same; namely, to derive estimates on the spectrum of the linearized evolution operator, linearized around solutions on the attractor, and to perform this operation in some function space instead of an a priori finite dimensional phase space. As we saw in Chapter 4 in the context of the Lorenz equations, this requires some knowledge of the location of the attractor, i.e., a priori estimates on the solutions. This approach is pursued in section 9.2 which deals with the 2d Navier-Stokes equations. It was shown in Chapter 7 that a global attractor si exists in this case, and we have good control of the solutions on the attractor. It turns out that the result for periodic boundary conditions is quite sharp, within logarithms of both the conventional heuristic estimate for the number of degrees of freedom in a 2d turbulent flow and rigorous lower bounds.

The 3d Navier-Stokes equations on a periodic domain are the concern of section 9.3. The lack of a regularity proof for this case results in some uncertainty concerning the very existence of a compact attractor. To achieve any formal estimate of the attractor dimension it is necessary to assume that H1 remains bounded for all t.

Introduction

The Navier-Stokes equations of fluid dynamics are a formulation of Newton's laws of motion for a continuous distribution of matter in the fluid state, characterized by an inability to support shear stresses. We will restrict our attention to the incompressible Navier-Stokes equations for a single component Newtonian fluid. Although they may be derived systematically from the microscopic description in terms of a Boltzmann equation, albeit with some additional fundamental assumptions, in this chapter we present a heuristic derivation designed to illustrate the elements of the physics contained in the equations.

Euler's equations for an incompressible fluid

First we consider an ideal inviscid fluid. The dependent variables in the so-called Eulerian description of fluid mechanics are the fluid density ρ(x, t), the velocity vector field u(x, t), and the pressure field ρ(x, t). Here x ∈ Rd is the spatial coordinate in a d-dimensional region of space (d typically takes values 2 or 3, with a default value of 3 in this chapter). An infinitesimal element of the fluid of volume δ V located at position x at time t has mass δm = ρ(x,t)δV and is moving with velocity u(x,t) and momentum δmu(x,t). The normal force directed into the infinitesimal volume across a face of area nda centered at x, where n is the outward directed unit vector normal to the face, is —np(x, t)δa. The pressure is the magnitude of the force per unit area, or normal stress, imposed on elements of the fluid from neighboring elements. These definitions are illustrated in Figure 1.1.

This book is not meant to be a review or a reference work, nor did we write it as a research monograph. It is not a text on fluid mechanics, and it is not an analysis course book. Rather, our goal is to outline one specific challenge that faces the next generation of applied mathematicians and mathematical physicists. The problem, which we believe is not widely appreciated in these communities, is that it is not at all certain whether one of the fundamental models of classical mechanics, of wide utility in engineering applications, is actually self-consistent.

The suspect model is embodied in the Navier-Stokes equations of incompressible fluid dynamics. These equations are nothing more than a continuum formulation of Newton's laws of motion for material “trying to get out of its own way.” They are a set of nonlinear partial differential equations which are thought to describe fluid motions for gases and liquids, from laminar to turbulent flows, on scales ranging from below a millimeter to astronomical lengths. Only for the simplest examples are they exactly soluble, though, usually corresponding to laminar flows. In many important applications, including turbulence, they must be modified and matched, truncated and closed, or otherwise approximated analytically or numerically in order to extract any predictions. On its own this is not a fundamental barrier, for a good approximation can sometimes be of equal or greater utility than a complicated exact result.

The issue is that it has never been shown that the Navier-Stokes equations, in three spatial dimensions, possess smooth solutions starting from arbitrary initial conditions, even very smooth, physically reasonable initial conditions. It is possible that the equations produce solutions which exhibit finite-time singularities.