The American Institute of Physics journal “Physics of Fluids” sponsors a ‘gallery of fluid motion’ every year (http://pof.aip.org/gallery_of_fluid_motion) and without fail, the selected images are breathtakingly amazing. Our images don’t make the cut.

Fortunately, we can display our own gallery of fluid motion right here. The following images were all taken at Cape Hattaras, North Carolina, part of the Outer Banks, infamous for the number of shipwrecks caused by sandbars located all along the coast.

Let’s discuss waves- specifically, ocean waves that break onto the beach. This is a ubiquitious phenomenon (especially if you include lakes, not to mention waves formed by sloshing around a tank), and yet the mathematics behind these structures is hideously complex.

It’s not clear why, but fluid mechanics (and its parent discipline, continuum mechanics) is essentially absent from the Physics curriculum. The mathematics are well established- classical field theory, differential geometry , etc- and the physical concepts are well posed (homogeneous/multiphase systems, thermodynamics, conservation laws, etc.) but nearly all the interesting work has been left to mathematicians and engineers (who have produced a stunningly beautiful body of knowledge). Stoker’s “Water Waves” is an especially good reference.

The problem of waves falls under the general problem of ‘interfacial phenomena’ and ‘free surface flows’, which are the red-headed stepchildren of continuum mechanics- the fluid-fluid boundary is no longer specified but is free to deform, so the interface shape is now part of the overall problem to be solved. Also, the no-slip boundary condition no longer holds. And as we will see, even the hypercomplicated problem of a free surface dividing two homogeneous Newtonian fluids is simple compared to the typical complexities of real ocean waves.

For now, consider the interface between water and air as a two-dimensional surface, referred to as a Gibbs dividing surface, possessing physical properties independent of either bulk material. The curvature of the surface is proportional to the pressure jump across the surface, for example. Also, when we say ‘fluid’, we generally mean the seawater, as opposed to the atmosphere.

Let’s start with an image of the ocean surface far from the shore:

This surface is dominated by long-wavelength waves (long waves) parallel to the beach covered by fairly uniform short-wavelength ripples. The dynamics of this image are fairly well understood- the wavespeed, dispersion relations, the overall velocity field within the bulk fluids, etc. – because the height of the waves is much smaller than the depth of the fluid, and the depth of the fluid is constant over many wave periods. Perturbative approaches are well suited to this problem and have produced useful results. For example, the speed of the wave is sqrt(gh), where h is the height of a wave and ‘g’ is 9.8 m/s^2.

However, as the waves approach the shore, the fluid depth changes rapidly, resulting in ocean fluid ‘piling up’ behind the wavefront and eventually breaking over the top, which is more visually appealing:

The upper image is useful to get a sense of scale- the front wave is about 3 feet high, while the rear wave is closer to 6 feet. For now, we will only discuss these ‘medium’ waves- of height 3-6 feet. The lower image is of a 3-foot wave, and displays a very uniform ‘phenotype’- very regular height and cylindrical profile, justifying 2-D treatments of the problem. Let’s look at the waves from a more oblique angle:

In these images, it’s possible to see what looks like ‘viscous fingering’- the wavefront divides into multiple ‘fingers’, which break independently.

Now let’s focus on the crest of a wave- it turns out that the height of the crest is proportional to the depth of the water- the larger wave, further out, is in deeper water. The wave crests (‘breaks’) when the slope becomes infinite (vertical). Short waves break sooner than long waves, and breaking occurs faster as the uniform ‘background’ fluid velocity decreases- when a wave breaks and retreats (negative velocity), the following wave breaks much sooner. Where the wave crests, and how high the wave is at cresting, tells us information about the spatial variations in ocean depth.

A note about how these images (and the remainder) were taken. Primarily, a 400/2.8 (or 800/5.6 using the 2x tele) lens was used, and the optimal lighting conditions are usual for capturing fluid motion- lots of light (direct sunlight), ‘harsh lighting’ (lots of contrast), and fast shutter speeds- 1/1000s or faster. Consequently, the lens was used at full aperture, resulting is a narrow depth of field- it was tricky to get the wave in focus, but when it is in focus , each droplet of water can produce a little flash of light like a star. It’s worth looking at these images full-scale, there is a lot of detail.

We should now pause and ask about the origin of the fluid velocity- the motive force, so to speak. The origin of tidal flow is due to the moon, but what is the origin of ocean waves?

The source of waves is wind: air flowing over the ocean surface. No air, no waves. The waves propagating to shore were produced by wind far offshore- thus the character of waves tells us about the weather conditions far offshore- stormy, calm, etc. The direction of airflow is not really correlated with ocean currents, thus another complication is that the problem is intrinsically 3-dimensional. Yep, no interesting physics here…

So, let’s add wind: it should be clear that some images were taken when the offshore weather was relatively calm, while others were taken when a storm was present- the waves are bigger and more closely spaced. Here is what happens when an on-shore 30 mph wind is added, blowing from the shore into the water, as a storm came onshore:

The waves appear much different- less cresting, flatter waves.

Upon closer examination of the images, additional features appear- the waveheight varies along the crest, for example, and in addition to the long-wavelength variation of wave height, there is a short-wavelength feature as well, and the top of the cresting wave takes on a ‘glassy’ appearance. As the wave crests, the crests merge at semi-regular intervals to form a ‘seam’.

What about larger waves, 10-15 feet in height. Do they appear similar or different?

What about the splash as the wave falls back onto the water?

What if we allow the mechanical properties of one of the fluids to change- foam formed by the waves can build up and mix with the fluid, changing the ocean fluid from a simple Newtonian fluid into a visco-elastic fluid- note the thicker ‘ropey’ structure.

Clearly, ocean waves display a complex behavior that is difficult to model. And we haven’t yet discussed the sand under the water- it’s a granular material that can flow, and in fact forms structures that look like dunes, in an orientation parallel to the waves. So we have, in general, a time-dependent 3-D mechanics problem involving a homogenous fluid (air), an inhomogenous fluid (the ocean water/foam) and a granular material (sand). If we wanted, we could add thermodynamics to the problem- to model solar heating, for example.

Let’s see how the computer graphics special effects folks do:(from http://hal.inria.fr/UNIV-GRENOBLE1/inria-00537490)

Not bad for calm water- but recall, that first image up top is very amenable to modeling. Here’s a still from “Surf’s Up”: (http://news.cnet.com/8301-13772_3-9862945-52.html)

Also not bad (Sony *did* get an Oscar nomination for this, after all)- but it’s not clear how much of the animation was based on physics and how much was smart illustration (http://library.imageworks.com/)

Here it’s clear that the model can’t compare to reality- the wave surfaces are *way* too smooth. In order to get realistic waves, Ang Lee used a giant tank for his movie, “Life of Pi”:

Illustrators can do a better job of capturing the dynamics, but omit a lot of detail:

That said, it’s important to remember that we can simplify the problem and extract out useful information- the 2D approximation for long waves is one simplification. Modeling ocean waves, while useful for various industries, can be applied to other problems as well- for example, airflow within the lung. Within the airway there is a homogeneous fluid (air), viscoelastic fluid (mucus/airway surface liquid), and a deformable soft matter layer (epithelial tissue). The air can be seeded with particulate matter (airbourne contamination), and we can model where those particles are likely to accumulate in the airway. We can also model the effects of adding a surfactant, often used as a medical intervention for premature newborns who don’t have a well-developed lung.