Month: May 1998

The instability of Stokes waves (steady propagating waves on the surface of an ideal fluid with infinite depth) represents a fundamental challenge in the realm of nonlinear science. A team of researchers recently identified the origin of breaking oceanic waves in a recent publication in the Proceedings of the National Academy of Sciences.

The research includes two graduates from the UNM Department of Mathematics and Statistics: former students Sergey Dyachenko (now an assistant professor in the Department of Mathematics at the University of Buffalo) and Anastasiya Semenova (now a postdoctoral researcher in the Department of Applied Mathematics at the University of Washington). The team also included Professor Bernard Deconinck from the Department of Applied Mathematics at the University of Washington and UNM Distinguished Professor Pavel Lushnikov.

Steady propagating surface gravity waves, discovered in the 19th century by Stokes, are the key structure of ocean swells easily seen from beaches, airplanes, and ocean liners. Lushnikov explains, “A Stokes wave is a surface gravity wave that propagates in the ocean with a constant velocity and is spatially periodic in the direction of propagation.”

The dominant instability of these waves depends on their steepness. Lushnikov further explains, “We studied the instability of large amplitude gravity waves on the surface of the ocean. The tallest such waves have eluded analysis, and their dynamics remains largely unexplored which motivated our study.”

Since the 1960s, the Benjamin-Feir or modulational instability has dominated the dynamics of small-amplitude waves, resulting in a slow variation of the swell. The team demonstrated that, for steeper waves, another instability caused by disturbances localized at the wave crest significantly surpasses the growth rate of the modulational instability.

“Benjamin-Feir or modulational instability characterizes the growth of disturbances of small amplitude Stokes waves. That instability has the spatial scale greatly exceeding the spatial period of Stokes wave. Disturbances in that case grow on the temporal scales greatly exceeding the temporal period of Stokes wave,” said Lushnikov. “We used mathematical techniques of conformal mappings and developed a new matrix-free approach to address the large-scale eigenvalue problem. These tools allow us to reveal the nature of the instability of Stokes waves.”

These dominant localized disturbances are either co-periodic with the Stokes wave or have twice its period. In either case, the nonlinear evolution of the instability leads to rapid formation of plunging breakers destroying steep waves. This phenomenon explains why long-propagating ocean swell consists of small-amplitude waves. The breaking of oceanic waves provides a key mechanism for the exchange of energy between atmosphere and oceans, significantly affecting the global climate dynamics.

“The sun heats the atmosphere, which then transfers energy to the oceans, heating them. The direct transfer of energy through thermal conductance and friction between air and water are relatively small. Instead, if the surface gravity wave breaks, it transfers most of its kinetic energy into small spatial scales eventually transforming that energy into heating of the ocean water. Global climate dynamics are determined by the interaction of atmosphere and oceans.”

The research team successfully uncovered the pivotal mechanism driving the captivating spectacle of large ocean wave breaking. Through their rigorous investigation, they’ve identified the crucial role played by fast wave breaking in shaping the composition of long-propagating ocean swells. This newfound insight sheds light on why these expansive swells are primarily comprised of small-amplitude waves. By delving into the heart of this phenomenon, the team has not only revealed a fundamental aspect of wave dynamics but also deepened the understanding of the intricate forces at play with Earth’s ocean.

Lushnikov concludes, “We found the dominant mechanism of the fascinating phenomenon of breaking of large ocean waves. Fast wave breaking found by us explains why long propagating ocean swell consists of small-amplitude waves.”

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Credit of the article given to Dani Rae Wascher, University of New Mexico

Let’s lay down some definitions:

• Let FF be a field and let EE be an extension of FF. An element α∈Eα∈E is algebraicover FF if there is a polynomial f(x)f(x) in F[x]F[x] such that f(α)=0f(α)=0 (i.e. α∈Eα∈E is an algebraic element if it is the root of some polynomial with coefficients in FF).
• Let FF be a field and let EE be an extension of FF. If K⊂EK⊂E is a subset of EE which contains all elements which are algebraic over FF, then KK is actually a subfieldof EE and an algebraic extension of FF. We call KK the algebraic closure of FF and denote it by ¯¯¯¯FF¯. [1]

It’s a fact that an algebraic closure ¯¯¯¯FF¯ exists for every field FF (and is actually unique up to isomorphism). So we can draw a containment picture like the one above.

Those familar with some topology and/or analysis will notice that such a “field tower” is suggestive of a vaguely analogous result: given a topological space XX we can always (assuming some conditions about XX, namely it being locally-compact Hausdorff) stick an open set VV and its closure ¯¯¯¯VV¯ between a certain compact set KK and open set UU:  K⊂V⊂¯¯¯¯V⊂U.K⊂V⊂V¯⊂U.

Now, don’t buy too much into the analogy. I only mention this topological result to motivate the fact that the closure of a set and the algebraic closure of a field do indeed convey the same concept: wholeness. It seems then that we can view algebraic elements as the mathematical cousins of limit points of sequences of real numbers. Why? Because, topologically speaking, what is the closure of a set? The collection of limit points of that set, right? So in particular, when we let our topological space be RR, the set of real numbers (with the usual topology) and consider the subset {xn}∞n=1{xn}n=1∞ – some sequence of real numbers,

• we say x∈Rx∈R is a limit pointof {xn}{xn} if for every ϵ>0ϵ>0 there is an n∈Nn∈N such that |xn−x|<ϵ|xn−x|<ϵ.

Then in light of our comments above, we can make the analogous statement for a subset of polynomials {fn(x)}⊂F[x]⊂E[x]{fn(x)}⊂F[x]⊂E[x]:

• We say α∈Eα∈E is an algebraic element(over FF) if there is an n∈Nn∈N such that fn(α)=0fn(α)=0**.

Notice there’s no need for an approximation by ϵϵ in the second bullet. Why? Well, imagine placing a “metric” dd on E[x]E[x] by d:E[x]×E[x]→Ed:E[x]×E[x]→E via***

(So intuitively, f(x)f(x) is far away from αα if αα is not a root, but if αα is a root of f(x)f(x), then f(x)f(x) and αα are just as close as they can be.) In this way, the distance between an algebraic element and its corresponding polynomial is precisely 0. So in this case there’s no need to approximate a distance of zero by an arbitrarily small ϵϵ-ball – we have zero exactly!

And thus we have stumbled upon another insight into one of the main differences between analysis and algebra: you know the adage –

Analysts like inequalies; algebraists like equalities!

Digging Deeper

It would be interesting to see if there’s something in the language of category theory which allows one to see that closure of an algebraic field and closure of a topological set really are the same. Now I don’t know much about categories, but as one of my classmates recently suggested, we might want to look for a functor from the category of fields to the category of topological spaces such that the operation of closure is equivalent in each. In this case, perhaps it’s more appropriate to relate an algebraic closure to the completion of a topological space, as opposed to its closure. Admittedly, I’m not sure about all the details, but I think it’s worth looking into!

Footnotes:

* This is actually a bit deceiving. How we measure “closeness” really depends on the topology of the space we’re working on. For example, we can place the ray topology on RR so that the open sets are intervals of the form (a,∞)(a,∞) for a∈Ra∈R. Then in the strict definition of a limit point we see that -763 is a limit point of the interval (0,1)(0,1) even though it’s “far away”!

** Okay okay… there’s no reason to assume an arbitrary collection of polynomials is countable. I really should write FF for some family of polynomials in which case this statement would read “…if there is some f∈Ff∈F such that f(α)=0f(α)=0.” But bear with me for analogy’s sake.

*** I put “metric” in quotes here because as defined dd is not a metric in the strict sense of the word. Indeed, we don’t have the condition d(f(x),α)=0d(f(x),α)=0 if and only if “f(x)=αf(x)=α” since the latter is like comparing apples and oranges! But it would be interesting to see if we could place looser version of a metric on a polynomial ring. For instance, the way I’ve defined dd here, an open ball centered at αα would correspond to all polynomials in E[x]E[x] which have αα as a root! This idea seems to be related to Hilbert’s Nullstellensatz and the Zariski topology.

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*Credit for article given to Tai-Danae Bradley*