mother nature

Nature is a collective idea, and though its essence exist in each individual of the species, can never its perfection inhabit a single object.

Henri Fuseli

Sometimes – on the really slow days – I wonder what Mother Nature might say about the applied mathematics community.

For her, it’s all so easy. So effortless.

I imagine her day starts from deep within the planet’s core. It’s awfully hot here, but she doesn’t mind. Casually, she waves her hand and just like that, convection currents drive the heat from the blistering hot core, through the mantle, and towards the surface. That’s the simple version, anyways. It’s much more complicated, of course, but she doesn’t like to show off.

Now, she turns her attention to the oceans and lakes. By now, it’s all clockwork. First, she nods her head and small and seemingly insignificant effects appear: the capillary ripples that form when the wind blows gently across a cottage lake; currents that sweep the eggs of a certain tropical fish downstream; the V-shape patterns that flutter from behind a ship in water.

Then, yawning slightly, she tends to the more global problems at hand: tsunamis that can devastate thousands, or ocean currents that displace great rivers of hot or cold water, ultimately affecting the weather.

Oh right. The weather. Now she rises out of the waters and towards the heavens. Will there be a flood in China? Maybe a hurricane for the Pacific Rim? A snowstorm in Canada? Sun for the British?

She hasn’t quite decided yet.

Everything is potentially connected: What’s happening below the surface affects what happens on the surface, affects what’s happening in the atmosphere, affects us. There’s are gargantuan number of candidate tools she uses, ranging from the very large and very blunt, like the rotation of the Earth or gravitational effects due to the Moon and Sun, to her surgical, quantum-level implements.

All this she does with uncanny grace and ease.

Now there’s us. The applied mathematicians. How many of us are out there? Thousands? Tens of thousands? Maybe a hundred thousand if you bring in the multi-disciplinary gang of physicists, biologists, engineers, and what not.

And so we’re all here on Planet Earth, right? Scrambling around like teams of freakishly organized mice. Mice with lab coats and blackboards and textbooks. We’re in awe of Mother Nature, of course, but we still try to explain her. We try to apply our mathematics to make sense of the Universe.

So how close do we get?

A Worthy Problem

Now before we dive headlong into the deep and murky waters of maths, we need to start with a problem. For this, we turn to the study of fluids.

When the governing equations of classical fluids are written out in their most general and unadulterated form, they are more or less unsolvable. As a first step, we must make certain simplifying assumptions about the nature of the physical system under consideration.

So we ask ourselves: Is the fluid thick and viscous, like gooey honey dripping from a spoon. Or is it playful and lively, like water sloshing in a backyard pool? Does the fluid behave like a compressible gas, like the air billowing over the wings of a fighter jet?

Once we’ve answered these most questions, the previously impregnable can then be simplified. Water, for instance, is mostly inviscid and incompressible, and so governed by the Euler equations:

What these equations and symbols mean isn’t particularly important. What is important is the realization that even here – even for these simplified representations of Nature – there is usually no hope of producing exact solutions; we must instead turn to developing mathematical approximations.

Asymptotic Approximations

The idea behind asymptotic approximations is to break the solution into more manageable pieces, each piece helping to produce a better and better approximation.

The first piece will describe the system — the ocean, for example – in some idealized state. To this, we may add a second piece representing a small perturbation to our idealized state. And if desired, a third piece is then added, representing yet another, even smaller perturbation. And so on to infinity.

Here, the Greek letter epsilon, ε represents a small perturbation. It can represent, for example, the height of the waves, the speed of the free stream, the amount of surface tension, or some other effect we can assume to be small.

These sums or series are called asymptotic approximations because they are expected to be exact in the asymptotic limit that ε tends to zero. That’s the idealized state.

The amazing thing with these approximations is that they tend to be very good. So good, in fact, that calculating only one or two terms provides an excellent approximation to reality.

The Devil’s Divergence

the devil

Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever.

neils abel

Now sometimes, what we get is a convergent series. That is, as we include more and more terms, the sum approaches some finite value. Intuitively, convergence seems to be a desirable trait; after all, if we’re to represent some physical process using an infinite series, wouldn’t we like for the whole thing to add up properly?

Unfortunately, what happens far more often is that in the limit the perturbations get smaller (ε tends to zero), the series is divergent: This manifests itself by the approximation getting better and better as more terms are included, up to some minimum, after which all hell breaks loose and the error shoots off to infinity.

Now think of the paradoxical nature of these divergent solutions: How correct is it to use a series that essentially adds up to nonsense, in order to describe a perfectly sensible phenomenon?

In 1828, the mathematician Niels Abel labeled the divergent series as an “invention of the devil”. That’s a sign of how strong some mathematicians felt about the whole mess.

Despite this, the theory of asymptotics flourished in the 20th century and today, asymptotical approximation is one of the most important and practical tools available to researchers; upon hearing about Abel’s condemnation, most scientists simply shrug. Dangerous and deceitful as they may be, they provide such stupendous approximations that it’s hard to avoid their use.

The Low-Speed Paradox

In 1968, naval architect T.F. Ogilvie noticed several peculiarities with asymptotic predictions of steady two-dimensional slow flows over an obstruction. This describes, for example, flows over a bumpy ocean floor or over a step in a channel.

For one thing, Ogilvie noticed that the approximations predicted a waveless surface, but despite the speed of the stream being small, we would still expect waves to form downstream. Why wasn’t the asymptotic approximation picking up these waves?

More disconcertingly, he noted that previous researchers had derived asymptotic expansions in which the asymptotic expansions seemed to ‘re-order’ as the speed of the stream tends to zero.

So for example, at moderate speeds, a single term in the asymptotic expansion might appear to be an adequate approximation. But then for lower speeds, one would need to include a second term to achieve the desired accuracy. For still lower speeds, the third-order terms would become important. And so on.

This was completely backwards! We expected these approximations to hold in the limit the speed of the stream tends to zero – in other words, an approximation valid for moderate speeds should be equally valid for low speeds. What’s going on?

This oddity Ogilvie observed — which would later be referred to as the “Low-Speed Paradox” — turns out to be nothing more than the devil’s divergence at work.

Today, using computers, we can calculate as many terms in the asymptotic expansion as desired – five, ten, a thousand…it doesn’t matter. Every time, the asymptotic approximations yield a waveless surface (and eventually diverge if enough terms are taken).

So, not yet disheartened, we return to our computer and design a program that solves for the exact numerical solutions of the equations. Now we see a different story: while the asymptotic solution seems correct initially, after the water passes the object, small ripples appear on the free-surface.

What, then, is the nature of these small ripples? Why don’t they appear in the asymptotic approximations?

Waves through a Mist

In the late 19th century, the mathematical physicist Gabriel Stokes noticed this same phenomenon when he applied asymptotics to study the formation of rainbows.

“As [the solution] passes through the critical value,” Stokes exclaimed, “the inferior term enters as it were into a mist.”

This ‘mist’ Stokes referred to – this critical value where the asymptotics all seemed to change – is now known as the Stokes Line. And across this line, Stokes discovered that a small exponential would attach itself to the solution:

This idea that an asymptotic expansion can change suddenly (but smoothly) across a critical line is now known as the Stokes’ Phenomenon.

Clearly, something similar must be going on in our problem.

Why does it Diverge?

Everything up until now has been related: the divergence, the inexactitude, Stokes’ phenomenon, the small exponential add-on. Everything.

Here is the key idea: While these devilish series may diverge and are thus a source of inexactitude, if we can somehow explain the reason for their divergence, we can perhaps justify their use, hell, we can even be improved them.

The root cause of their divergence can be succinctly explained by re-tracing our steps back to the very first asymptotic approximation. In the case of flow over a step, the first approximation revealed a singularity (a division by zero) at the corner of the step.

In applying our asymptotics, we have substituted a singular (ill-defined) function in order to represent a perfectly well behaved phenomenon. There are no singularities in the water surface, of course, but our mathematical description has forced this artifact upon the system.

And unfortunately, this innocent property spreads: It can be shown that singularities found in the early terms multiply; as more and more terms from the series are taken, the singularity gets worse and worse until – like stars being sucked into a growing black hole – the behaviour of the high order terms are entirely dominated by the early singularities.

Going Beyond all Orders

However, following this newfound understanding of the divergence of asymptotic series’, mathematicians have sought to also decode hidden information found in the divergent tails of the asymptotic sequence.

Divergence, they’ve realized, is perhaps not as fearsome and unholy as they once thought and by carefully extracting the behaviour of the later order terms of the series, a great deal of information can be gleaned.

In the 1950s, the physicist R.B. Dingle showed that the amazing fact that the form of the late terms of nearly all divergent expansions obeyed a universal form – “a factorial over power”:

The form of this expression reveals the two crucial aspects of asymptotics we’ve discussed: Not only does one have to contend with the factorial growing larger and larger, but each additional term increases the order of the previous singularity.

Now, this revelation allows us to predict the magnitude of the Stokes Switching – the small waves that are flicked on as one passes through the mist.

The process is fairly lengthy, but here is the basic idea: We begin calculating the terms in the regular asymptotic expansion up to the ‘optimal truncation’. At this point, the error in the approximation is exponentially small. Now we assume Dingle’s universal form for the expansion, and go beyond all orders. We zoom in close to examine the mist and see what changes as one crosses the Stokes line.

This process, known as exponential asymptotics (or asymptotics beyond-all-orders) is the necessary component of dealing with the Devil’s divergence.

But Wait!

So we explained that in order to describe nature’s mysterious, mathematicians and scientists have had to use divergent (badly-behaved) series approximations. Although these approximations often give superb results, we are nonetheless representing perfect phenomenons using a far-from perfect representations. “Inventions of the devil”, said Niels Abel.

Then we showed that while divergence is bad, there’s no need to disregard it. By using certain mathematical tools, we can unravel the divergence, and reveal what details were lost using the approximations.

“But wait,” cry the philosophers.

This Stokes’ Line — this switching-on of a hidden exponential across the Mist — you know that it’s not actually real, right? You do realise that, right?

The Stokes Phenemonon and the Stokes Line is wholly unphysical. There is of course no such line in reality. It is a mathematical invention. A band-aid to stop the bleeding of a theory so that we can produce better approximations.

How, then is Mother Nature so effortless in her mathematizing and yet we’re confined to making nasty deals with the devil, then squirming our way out of trouble?

In De Natura Deorum, Cicero wrote that, “Things perfected by nature are better than those finished by art”.

Perhaps this is simply an example of our inability to describe Nature in all its glory and perfection.