Every physicist should know some fluid dynamics, and every university physics department should include the subject in its core curriculum. Those propositions can readily be justified by pointing out the usefulness of the subject – its relevance to diverse areas of contemporary research and to a vast range of problems of practical importance. What counts as much for me, however, is that most of the students I have known at Cambridge have enjoyed their limited exposure to it. The notion that the only way to arouse the enthusiasm of physicists is to teach them about quarks and black holes is in my view a myth.

I hope that this book will slightly increase the chance that future generations of physicists will be taught the subject systematically, in a way that I and my contemporaries were not. However, since newer branches of physics may continue to displace it, I have tried to write something that may be read for pleasure as well as for instruction by physicists of any age and at almost any level of sophistication who want to learn fluid dynamics for themselves. They do, of course, have many books to choose from already, but most of them were written for mathematicians or engineers. Students of all three disciplines – mathematics, physics and engineering – speak the same language and have many objectives in common, but they differ in their approach to new problems because their intuition has been honed in different ways, and they also tend to differ in what they find interesting.

An acoustic pulse propagating with a current travels faster than one propagating against the current. Ocean currents are typically of order 10 cm/s rms or less, except in strong western boundary currents such as the Gulf Stream, whereas ocean sound-speed perturbations are typically of order 5 m/s rms. Travel-time perturbations due to ocean currents are correspondingly one to two orders of magnitude smaller than travel-time signals due to sound-speed perturbations. It is nonetheless possible to measure ocean currents using acoustic techniques, by differencing the travel times of signals traveling in opposite directions. As was briefly summarized in chapter 1, travel-time signals due to sound-speed perturbations cancel in the difference travel time, leaving only the effect of currents.

Section 3.1 describes ray theory as applied to moving media. The presence of a current introduces anisotropy. Perturbation expressions for the sum and difference of reciprocal travel times are then presented in section 3.2. When the flow is in geostrophic balance, the current and sound-speed fields are related. In section 3.3, quantitative estimates of their relative sizes are made, confirming the rough orders of magnitude cited earlier.

Using a horizontal-slice approximation, section 3.4 shows that the averaging properties of acoustic travel times make acoustic techniques uniquely suited for measuring the fluid circulation by integrating around a closed contour. By Stokes's theorem, the circulation is equivalent to the areal-average relative vorticity. This result is then generalized to show that differential travel times are sensitive to the solenoidal component of the flow, from which relative vorticity can be mapped, but are not functions of the irrotational component of the flow between the transceivers, which is needed to map the horizontal flow divergence.

The idea for acoustic tomography arose abruptly. Because we can mark a clear conceptual start, the following brief chronicle of the development of ocean acoustic tomography, given from our perspective as participants, may be of interest.

Some hopes for “Monitoring the oceans acoustically” were voiced at the thirtieth anniversary of the founding of the Office of Naval Research (ONR) (Munk and Worcester, 1976). A “Preliminary report on ocean acoustic monitoring” was prepared during the JASON Summer Study (JSN-77-8) by Garwin, Munk, and Wunsch. That work was expanded into “Ocean acoustic tomography: a scheme for large scale monitoring” (Munk and Wunsch, 1979), which examined the acoustic and inverse theoretical requirements for mapping the oceans with mesoscale resolution. It concluded that “an acoustic tomographic system appears to be both practical and useful.” The name “ocean acoustic tomography” was deliberately chosen to arouse the reader's curiosity as to what it is all about. Response by the oceanographic community was varied; those with a background in inverse theory regarded the inverse problem as trivial, but found the acoustic applications to be of interest, whereas the marine acousticians were interested in the inverse problem.

Overture

Advances in several fields were prerequisites for the development of ocean acoustic tomography: an understanding of underwater sound propagation, a statistical description of oceanic processes (especially internal waves and other fine structure), and the availability of inverse methods for inference from measurements. In this section we focus on the history of the acoustic developments crucial to tomography.

The sine qua non of any inverse problem is an accurate treatment of the forward problem: to construct an acoustic arrival pattern given “the ocean.” If the forward problem cannot be solved, then the measured data cannot be inverted to reconstruct the ocean.

When we entered this field in 1978, it was under the impression that the forward problem had been solved and that our efforts could be directed toward the oceanographically interesting inverse problem. As it turned out, the forward problem had not been solved; oceanographic and acoustic fields had never been simultaneously measured with accuracy adequate for a critical test. When such measurements were finally made in the late 1980s as part of tomographic work, significant discrepancies were found between the measured and computed sound fields. The most dramatic discrepancy was traced to the equations of state for sea water; some of the commonly used sound-speed equations were shown to be in error.

That era of trouble with the forward problem came to an end with the so-called SLICE89 experiment. The forward problem is now sufficiently well understood to permit oceanographically useful applications of inverse methods. Some discrepancies remain; they appear to be associated with internal waves and other range-dependent phenomena (see chapter 4).

There is a hierarchy of techniques available to predict the propagation of acoustic pulses in the ocean, given the sound-speed field. The geometric-optics approximation (i.e., ray theory) is adequate to predict travel times in the majority of situations encountered in ocean acoustic tomography.

The fundamental experimental requirement of ocean acoustic tomography is to make precise measurements of acoustic propagation in the ocean. Any of a variety of characteristics of acoustic propagation potentially can provide useful information about the ocean through which the sound has traveled, provided that the forward problem is thoroughly understood. The task of the measurement system is to provide estimates of whatever parameters are desired, with useful precision, together with estimates of the errors of the measurements.

In this chapter we focus on the use of broadband acoustic signals to measure the impulse response of the ocean with sufficient resolution (in time and/or vertical angle) to separate individual ray arrivals. The precision with which travel times and other parameters of the individual ray arrivals can be estimated is limited by the ambient acoustic noise in the ocean. In addition, and more important in most cases, small-scale ocean variability causes fluctuations in ray amplitudes, travel times, phases, and arrival angles. Travel time is the most robust observable and is the one we emphasize in this book. The inversion of travel-time data to obtain information on the ocean sound-speed and current fields was outlined in chapter 1. Because the expected magnitude of the travel-time perturbations is O(100 ms), travel times need to be measured with a precision of a few milliseconds, corresponding to a few parts per million over 1 Mm range.

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.