Rogue waves—individual, isolated waves far larger than the surrounding waves—were reported for centuries by sailors. But their stories of massive walls of water appearing in the open ocean were not corroborated until 1995 when a rogue wave struck an offshore platform. How these giant waves form is still under active research, but one leading theory is that nonlinear interactions between waves allow one wave to sap energy from surrounding waves and focus it into one much larger, short-lived wave. I first learned of rogue waves during a seminar in graduate school. At the time, this idea of nonlinear focusing had only been explored in simulation, but a few years later a research group was able to demonstrate the effect in a wave tank, as shown in the video above. Wait for the end, and you’ll notice how the rogue wave that takes down the ship is much larger than its predecessors. For more on rogue waves and their mind-boggling behavior, be sure to check my previous post on the subject. (Video credit: A. Chabchoub, N. Hoffmann, and N. Akhmediev)
Tag: nonlinear dynamics
The Fluid Dynamical Sewing Machine
Originally posted: 3 Jan 2012 Nonlinearity and chaos are important topics for many aspects of fluid dynamics but can be difficult to wrap one’s head around. But this video provides an awesome, direct example of one of the key concepts of nonlinear systems–namely, bifurcation. What you see is a thread of very viscous fluid, like honey, falling on a moving belt. Initially, the belt is moving quickly and the thread falls in a straight line. When the belt slows down, the thread begins to meander sinusoidally. With additional changes in the belt’s speed, the thread begins to coil. A multitude of other patterns are possible, too, just by varying the height of the thread and the speed of the belt. Each of these shifts in behavior is a bifurcation. Understanding how and why systems display these behaviors helps unravel the mysteries of chaos. (Video credit: S. Morris et al.)
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Liquid Mushrooms
The Rayleigh-Taylor instability can form at the interface between two liquids of different density under the influence of gravity, but a similar instability can occur in the absence of gravity. The image sequence above shows the Richtmyer-Meshkov instability, which occurs between two liquids of differing densities (regardless of their orientation) when impulsively accelerated. In this case, the experiment was conducted in a drop tower to simulate microgravity with the apparatus dropped on a spring to provide the impulse. As the instability grows, asymmetries appear. Nonlinear dynamics will amplify these distortions, eventually leading to turbulent breakdown. (Photo credit: C. Niederhaus/NASA Glenn, J. Jacobs/University of Arizona)
The Chaos of a Bouncing Droplet
This video explores chaos in a bouncing droplet. A drop of silicon oil bounces on a vibrating bath of oil; the thin layer of air injected with each bounce between the droplet and bath keeps them from coalescing. Initially, the droplet behaves like a bouncing ball, jumping once per oscillation. As the vibration amplitude increases, the droplet begins making a small jump, then a large jump, then a small jump, and so on. This is called period doubling since the droplet now jumps in a pattern with twice the period of the original and is a hallmark of nonlinear dynamical systems. Further increase in the vibration amplitude leads to chaotic bouncing and occasional ejecta. (Video credit: D. Terwagne et al.)
Honey Coiling
The liquid rope coiling effect occurs in viscous fluids like oil, honey, shampoo, or even lava when they fall from a height. The exact behavior of the coil depends on factors like the fluid viscosity, the height from which the fluid falls, the mass flow rate, and the radius of the falling jet. Here Destin of the Smarter Every Day series outlines the four regimes of liquid coiling behavior commonly observed. As with many problems in fluid dynamics the regimes are described in terms of limits, which can help simplify the mathematics. The viscous regime (2:34 in the video) exists in the limit of a small drop height, whereas the inertial regime (3:15) exists in the limit of large drop height. Many complicated physical problems, including those with nonlinear dynamics, are treated in this fashion. For more on the mathematics of the coiling effect, check out Ribe 2004 and Ribe et al. 2006. (Video credit: Destin/Smarter Every Day; submitted by inigox5)
Rogue Wave Recreated
For years, mariners have reported occurrences of rogue waves–sudden, isolated waves many times larger than the surrounding surface waves. Until 1995, when a rogue wave was first measured, debate raged as to whether such waves even existed. Scientists have since agreed that nonlinear models of wave interaction are the most likely source of the amplification necessary to create rogue waves. Since the Navier-Stokes equations that govern hydrodynamics are so difficult to solve, scientists have looked to simpler nonlinear wave equations, like the nonlinear Schroedinger equation that governs optics, to generate rogue-wave-like behavior. While the equation gives insight into how a given wave system will evolve, it is still necessary to determine what initial conditions can lead to the formation of a rogue wave. All manner of random conditions exist in the ocean, but to recreate the behavior in a simplified system, we must know which initial conditions are the right ones. Akhmediev et al presented a theoretical perspective on the initial conditions that might lead to rogue wave amplification, and now, for the first time, researchers have been able to create a rogue wave in a wave tank. That little blip that sinks the Lego pirate ship is a great accomplishment toward understanding a phenomenon whose very existence was in question less than twenty years ago. (Video credit: A Chabchoub, N Hoffmann, and N Akhmediev; via Gizmodo; for more, see APS Viewpoints and Akhmediev et al)
Viscous Fluid Falling on a Moving Belt
In this video a very viscous (but still Newtonian) fluid is falling in a stream onto a moving belt. Initially, the belt is moving quickly enough that the viscous stream creates a straight thread. As the belt is slowed, the stream begins to meander sinusoidally and ultimately begins to coil. Aside from some transient behavior when the speed of the belt is changed very quickly, the behavior of the thread is very consistent within a particular speed regime. This is indicative of a nonlinear dynamical system; each shift in behavior due to the changing speed of the belt is called a bifurcation and can be identified mathematically from the governing equation(s) of the system. (Video credit: S. Morris et al)
