The Ocean, the Stars, and the Kitchen Sink

Sometimes a simple analogy offers the best approach to a complex problem.

By Hans Christian Von Baeyer|Tuesday, March 01, 1994
The steady wind that attracted Orville and Wilbur Wright to the Outer Banks of North Carolina produces an uncommonly reliable surf. Hour after hour, with mesmerizing monotony, the big waves roll in from the deep ocean. Each liquid mountain swells ominously as it approaches the shore, crests majestically, and breaks with a thundering crash, giving birth to a new, flat wave called a bore, a rushing wall of water with a foaming vertical front. Racing toward the shore, the bore eventually spreads and disperses, hiding the white sand under a blanket of froth.

Three years ago the gulls on the beach witnessed the curious spectacle of an awkward little amphibious craft that was daring the waves by positioning itself right under the largest breakers, where it was drenched and thrown about in a most alarming way. The boat's crew of four men seemed to be heaving mysterious boxes overboard and somehow anchoring them in the pounding surf--but unlike others who come to that lonely beach, they were neither fishing nor trying out some dangerous new sport. They were, in fact, conducting an oceanographic experiment designed to map the water's speed and depth in the vicinity of a bore. Their data would later be used to construct a mathematical description of a bore, an essential ingredient in our knowledge of the complex flow patterns at the edge of the ocean. In this particular case their effort led to an additional, and unusual, scientific payoff: it contributed to a better understanding of the roiling of hot gases on the surface of pulsating stars. Furthermore, it would turn out that the shapes of both ocean waves and star waves are mirrored in the shape of the stream from a faucet splashing into a kitchen sink. Understanding that homely, familiar sight goes a long way toward explaining what happens on the beach and far off in the unreachable depths of the galaxy.

Of course, science doesn't really explain why things are the way they are. The most we can hope for, and we ought to be grateful for achieving so much, is to connect bits of nature's grand design to each other by discovering resemblances among them. "All science," observed Jacob Bronowski, the creator of the memorable television series The Ascent of Man, "is the search for unity in hidden likenesses." When Galileo Galilei discovered the formula for the distance an object falls in a given time, he established a likeness among an infinity of objects, from the bread crumb softly dropping on the carpet to the cascades of Niagara; when Isaac Newton found an unsuspected likeness between the way Earth pulls on an apple and the way it attracts the moon, he discovered the law of universal gravitation; and when James Clerk Maxwell noticed the similarity between electric and magnetic forces, he found the key to his theory of electromagnetism. William Blake expressed the same thought more poetically when he extolled our ability "to see a world in a grain of sand." Early in this century Niels Bohr managed to equal the vast compass of Blake's vision when he likened a hydrogen atom to the solar system, but scientific analogies don't need to be that spectacular to be instructive. To see the heaving of a star in the ocean surf, and both in the kitchen sink, is sufficient proof of the power of analogy. If Bronowski's characterization of science is right, analogy, the ancient device of explaining the unknown by pointing out its resemblance to the familiar, turns out to be one of the most useful instruments in the toolbox of science.

Physicists often call analogies models, in the sense that one phenomenon mimics, or models, another. In its primary sense the word refers to manufactured models, which play a role in all civilizations. Figurines of people and animals, together with the paraphernalia of their daily lives, fill the cabinets of anthropological museums. In our own time the popularity of miniature ships, soldiers, cars, railroads, airplanes, and dinosaurs attests to the universal fascination of models. The scientific counterparts of these toys are the mechanical models of everything from the solar system to the propagation of light waves that became indispensable accessories of physics in the late nineteenth century. "I am never content until I have constructed a mechanical model of the subject I am studying," claimed the great Scottish physicist Lord Kelvin, who created the scientific scale of temperature that bears his name. "If I succeed in making one, I understand; otherwise I do not."

Later the concept was generalized to include mathematical models, which don't necessarily have to have a tangible representation. A mathematical model is an idealized theoretical description that reproduces the essential features of a phenomenon while leaving out inessential complications. Galileo's law of free-fall is a case in point: it describes the way a feather and a hammer fall in a vacuum, but it only approximates very crudely what we usually see because it ignores air resistance. The set of equations that is fed into a supercomputer to describe the interior of a star is a more sophisticated mathematical model.

The shorthand nature of mathematics often allows the same equation to describe different phenomena that display similar behavior. The oscillation of a pendulum, for example, obeys exactly the same equation as the oscillating current in certain electric circuits. When that happens, the two phenomena model each other in the third sense of the word. Physical models differ from both artificial, mechanical models and abstract, mathematical ones. Two physical systems that serve as models of each other may seem quite different in appearance, or even totally unrelated, but if they share some hidden principle of operation, the comparison of their behavior will be revealing. A good illustration is Bohr's planetary atomic model, in which electrons are forced to move in elliptical orbits around the nucleus by electrical attraction in exactly the same way that planets move around the sun under the influence of gravity--even though the solar system differs from an atom as much as the world differs from a grain of sand. The connection between ocean bores, stellar gases, and the swirl of water in the kitchen sink is a splendid example of a three-way physical model.

The common feature that unites these three disparate physical phenomena is the appearance of a velocity discontinuity--a sharp boundary where the velocity of the medium suffers an abrupt change. The velocity of water molecules just behind the bore is obviously shore bound, while the molecules in front of it are returning seaward. At the boundary, the water velocity thus exhibits an abrupt reversal from inbound to outbound. Velocity discontinuities are complex phenomena, much more difficult to capture mathematically than the graceful sinusoidal curves of the ordinary waves that occur so profusely in nature and technology; it is this very complexity that gives physical models of systems with velocity discontinuities their great value.

Fortunately, not all such systems are as inaccessible as those found in stars or even on windswept beaches. A lovely and intriguing example can be produced by turning on the faucet over a kitchen sink. (Once you know what to look for, the effect will catch your attention in many other places as well.) Turn the tap so it produces a thin, steady stream that hits the flat bottom of the sink. Around the point of impact you will notice a circle whose radius varies as you adjust the strength of the stream. Closer observation reveals that the ring marks a boundary where the water level suddenly jumps upward: the stream generates a circular, vertical wall of water. A little experimentation proves this ring to be remarkably robust. You can substitute a plate or the lid of a pan for the bottom of the sink and move it up and down in the stream or even tilt it without entirely losing the ring. What's going on?

More specifically, why does the water level jump discontinuously instead of changing smoothly, the way surfaces of liquids usually do? A physicist might add a quantitative question: What determines the radius of the circle? If the sink is smooth, what's so special about the point where the wall of water, known as the "hydraulic jump," develops?

In the past this problem has been attacked with sophisticated computations that hide the essential physical processes behind an impenetrable curtain of formulas, but last September Robert Godwin, a physicist at the Los Alamos National Laboratory in New Mexico, supplied a simple, though surprising, explanation. To begin with, he showed that the jump, like the ocean bore, does indeed entail a velocity discontinuity: at the boundary, Godwin found, the water speed drops abruptly to less than a tenth of its former value. That the speed of flow should be reduced, of course, makes sense. Since the height of the water increases suddenly at the jump, the speed must diminish correspondingly in order to preserve the net flow. (A much more pronounced example of the same reduction in speed is seen in the bathtub, where water enters from the tap in a thin, fast stream while the broad surface of the water in the tub rises at an almost imperceptible crawl.) Most of the energy of the water, incidentally, is converted from kinetic (energy of motion) to potential (stored gravitational energy) as the level rises at the stationary boundary.

But the real question remains: Why does the hydraulic jump appear in the first place? The answer involves two little-known properties of water. First, although it is far from obvious, the bottom layer of water molecules locks onto the molecules of the sink. Second, water has a small but measurable viscosity, or "stickiness," which results from the weak mutual attraction of water molecules. These two effects combine to create an invisible, paper-thin layer of water at the bottom, called the boundary layer, that refuses to participate fully in the general outward rush because it is held back by molecular forces. The speed of the molecules in this layer increases from zero at the bottom, where water meets sink, to a maximum at the top of the layer, where it exactly matches the speed of the bulk of the water flowing above it. (The boundary layer is actually a universal property of the passage of liquids and gases over solids. A familiar consequence of this peculiar behavior is the invariable coating of dust on the blades of a ceiling fan, even though one would expect them to be swept clean by the continuous rush of air. The reason is that the layer of air next to the blade is stationary--relative to the fan--so the dust calmly settles down.)

Before it reaches the jump, the water flow is smooth and easy to understand. As the water spreads out it thins, until--at some distance from the center--its depth becomes so small that it equals the microscopic thickness of the boundary layer. At that point, Godwin argued, the viscous forces suddenly become dominant; the floor of the sink grabs the water and, aided by this internal mutual attraction of water molecules, stops it. No longer able to surge forward, the water begins to pile up instead, creating a vertical wall. (Think of pushing a rug along a floor, only to have it stop, buckle, and pile up.) Guided by this reasoning, Godwin produced a theory that correctly predicts the radius and height of the hydraulic jump, given the size and speed of the incoming column.

Ocean bores and hydraulic jumps are closely related, but they differ in three obvious ways. A bore is a drop in height, instead of an upward jump; it moves, rather than remaining still; and it has nothing to do with viscosity. But that's the way it is with physical models--they usually display significant differences besides interesting similarities. It often happens, for example, that the equations describing them are identical only at the lowest level of approximation and diverge as more details are brought to bear. Thus, although Bohr's equations for the hydrogen atom resembled Newton's description of the solar system, as soon as the demands of the quantum theory of 1925 were incorporated in the theory, the analogy broke down because electrons were discovered to behave like waves, not planets. With this modification Bohr's model lost its usefulness.

Not so the analogy between ocean bores and waves in stellar gas. Both subjects are so complicated, and stars are so inaccessible, that collaboration between oceanographers and astronomers is still capable of yielding rich dividends of insight. The water bore results from the propagation of a wave onto a sloping beach from which the previous wave is receding. In 1952 the American astronomer Martin Schwarzschild suggested that something similar happens in pulsating stars, such as the one labeled W in the constellation Virgo, whose size and brightness rise and fall periodically in a 17-day cycle.

During those days when the brightness is on the increase, the dense, hot gases that make up the body of the star--overwhelmingly hydrogen--push outward at a speed of 20 miles per second or more. But they don't encounter empty space. The remnants of the previous expansion, reduced to a thin, cool gas by now, are still in the process of falling back onto the surface of the star. This tenuous outer mantle forms what is termed the atmosphere of the star. At the boundary between the upwelling gas and the returning atmosphere a shock wave develops, similar to the shock wave we hear when a sphere of high pressure spreads out from an explosion and crashes with a thunderclap into the still air of the surrounding atmosphere. If one takes the density of the gas to be the analogue of water depth, the ocean bore turns out to be a faithful model of the stellar shock. The vast, slow cosmic surf pounding the surface of a star in the bleakness of outer space is mirrored on the beaches of the Outer Banks.

And how do we know that? Except for the sun and a few of its closest neighbors, all stars look like points. It is the triumph of astronomy to be able to glean a veritable cornucopia of information from those minuscule pinpricks of light. Schwarzschild detected the signature of shock waves in a unique combination of spectral lines from W in Virgo. (The bright and dark lines in a star's spectrum are caused by the emission and absorption of particular wavelengths of light; only specific atoms, in particular energy states, can account for a given pattern of lines.) In addition to the normal starlight, there were three other sets of lines that appeared only during the growing phases of the star--one set of emission lines, and two different sets of absorption lines. At the shock front, hydrogen atoms collide with each other with abnormal violence. These collisions confer extra energy on each atom, with the result that each atom's electrons jump up to energy levels they never reach in the star itself. But the electrons don't stay there; they quickly fall back to their accustomed places while emitting light in specific, characteristic colors that allow astronomers to identify not only the elements that emitted them but also the excessive vehemence of the collisions characteristic of a shock wave.

The dark absorption lines represent light from the main body of the star that has been intercepted on its way out through the atmosphere. The gases above and below the shock front absorb light also, but they do so differently and in a manner that reveals their contrary velocities. Thanks to the Doppler effect, the color of the light absorbed by atoms approaching the star is shifted toward the blue end of the spectrum, while light absorbed by atoms fleeing the star is shifted toward the red. (The underlying principle is the same as that of a police radar gun.) Guided by this displacement, Schwarzschild could interpret the faint lines of the star's changing spectrum as though he were watching a movie of the surface.

In the intervening years many other pulsating stars have been observed, and their spectra have been measured with great precision. Schwarzschild's basic scenario has held up well, but with the accumulation of more data, new questions have arisen. Not surprisingly, stellar shock waves turn out to be as varied as ocean waves and, indeed, as the stars themselves; they can be placed into common categories, but each one is, in the end, unique. The task of deriving the observed spectra, with all their idiosyncrasies, from the other properties of the star, such as mass, composition, surface temperature, and size, represents a formidable challenge.

To provide physical insight and guidance, astronomers have turned to oceanographers for help. Thus it came about that George Wallerstein of the astronomy department of the University of Washington in Seattle began to collaborate with Steve Elgar, a physical oceanographer at Washington State University in Pullman, who directed the field trials on the other side of the continent in North Carolina. The most significant result they achieved from a comparison of their data was a graphic explanation of a peculiar pattern sometimes found in the pulsation of certain stars. Normally the brightness of the star rises and falls in a regular wavy pattern--from maximum to minimum and back again. Occasionally, however, each regular cycle is followed by a much smaller one, as though the star's intensity control was prevented from reaching its usual maximum and minimum settings. Since brightness is an indication of size, this pattern signals a stellar pulse with alternating powerful and feeble heartbeats.

A similar alternation was found on the beach and is easily understood. Sometimes an incoming wave with an abnormally large height would propagate up the beach as a large bore. The corresponding heavy backwash returning to sea would retard the next wave and cause the bore it created to be unusually feeble, allowing the next bore to be larger again-- and thus the cycle would repeat. By analogy the model explains a detail of the stellar heartbeat whose detailed dynamics would be very difficult, if not impossible, to derive directly from the basic equations of gas flow.

The general shapes of the curves of stellar brightness plotted as a function of time, and of water depth measured at a fixed spot as time elapses, resemble each other closely. Furthermore, interesting features such as the sequences of alternating strength, and double peaks that result from the simultaneous presence of two similar shocks in quick succession, are found in both traces. Of course there are many differences between the two systems as well: the compressibility, heating, and light emission of gases in the star, which have no analogues in water, are some of the most obvious. Nevertheless, Wallerstein and Elgar are confident that the similarities outweigh the differences and that their collaboration will continue to lead to the solution of some of the outstanding problems associated with stellar pulsation.

Perhaps the third partner of the analogy, the hydraulic jump, will lead to new questions. When the jump is observed on a plate held a few inches under the faucet, and the stream is fairly small, and the light falls on it just right, the surface of the water inside the ring suddenly shows a pretty pattern of concentric ripples. And then, if the plate is raised even higher, the vertical water column itself begins to develop bulges until it looks like a necklace of pearls. To date, neither the ripples nor the pearls have been studied in detail, but when they are, they may suggest the search for new phenomena on the beach, and possibly even in the pulsation of stars.

Analogy is a two-edged sword: it can guide, but it can also lead us terribly astray. The Bohr model--with its planetlike electrons--missed the true wave nature of particles. And Lord Kelvin's stubborn reliance on mechanical models prevented him from accepting the abstractions of Maxwell's theory of electromagnetism. The trick is to take advantage of the opportunities offered by analogy without falling into its traps. The most penetrating assessment of this powerful tool of modern physics was expressed by the Victorian English novelist and satirist Samuel Butler, who remarked, "Though analogy is often misleading, it is the least misleading thing we have."
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