The Azimuth Project
Nonlinear science (changes)

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Contents

Idea

To have an Azimuth springboard to this increasingly growing field, which currently encompasses:

just to mention the major research subfields. There is also research on combinations of these subfields and combinations with traditional physical modeling according to Lam. For example the Active walker model in pattern formation (AWM). Nonlinear Schrodinger equations in monster waves.

Details and environmental scientific usage

Capsule History

Many of these areas have arised from questions in dynamics of earth and its atmosphere like three-body collisions which is what got Poincare’ to spin-off research in topology, and later physics researchers n-body simulations. Lorenz who found strange attractors in the climate model he used.

Towards instability with laminar flow:

Here is an animation done with Palabos - of the Kelvin-Helmholtz instability:

<iframe frameborder='0' height='315' src='http://www.palabos.org/demos/movies/palabos_kelvin_helmholtz_particles_z.flv' width='560'></iframe>

iframe width=“560” height=“315” src=“http://www.palabos.org/demos/movies/palabos_kelvin_helmholtz_particles_z.flv” frameborder=“0” allowfullscreen></iframe>

Lord Kelvin and Dr Helmholtz with their instability theory as an precursor and predictor of turbulence. Dr May who tried to make a discretized version of the logistic equation and found chaos in the logistic map.

Mandelbrot who finds fractals in many places in both natural and anthropogenic phenomena.You know you are dealing with a nonlinear system if the output is not proportional to the input data.

Chaos

Applied or originated in some research fields like climate modeling,

Fractals and multifractals

Complex adaptive systems

Pattern Formation

Solitons

They are almost ubiquitous and exists in size from 10 710 1010^{-7} - 10^10. Used in modeling Rossby solitons and they might also be important in rogue waves like the Draupner wave. But they are not that suitable to tsunami/tidal wave modeling

References

Abstract: We have numerically calculated chaotic waves of the focusing nonlinear Schrodinger equation (NLSE), starting with a plane wave modulated by relatively weak random waves. We show that the peaks with highest amplitude of the resulting wave composition (rogue waves) can be described in terms of exact solutions of the NLSE in the form of the collision of Akhmediev breathers?.

Abstract: Large-amplitude internal solitary waves (or ‘‘solitons’’) occurring in packets near the shelf break in the Bay of Biscay are well-documented and understood. The presence of similar features has now also been reported in the central Bay, E150 km from the nearest shelf break topography. The present paper analyses available remote-sensing synthetic aperture radar (SAR) data from the ERS satellites in this region. By doing so, we are able to provide convincing support for the hypothesis that these waves, instead of having travelled along the thermocline from the shelf break, are instead generated locally in the central Bay by the surfacing of a beam of internal tidal energy originating from the shelf break. This reinforces the results of a previous independent study, while at the same time providing a much more extensive investigation than was then possible. We have also exploited the large swath width (100 km) and high spatial resolution (100m100m) of the SAR to examine for the first time the full surface structure of the internal waves in the

central Bay, which are found to have a mean wavelength of 1.35 km, and a mean along-crest ‘‘coherence’’ length of 21.55 km.

We propose initial conditions that could facilitate the excitation of rogue waves. Understanding the initialconditions that foster rogue waves could be useful both in attempts to avoid them by seafarers and in generating highly energetic pulses in optical fibers