Spatio-temporal waves

The studies of Ertl and co-workers showed that the reason for self-oscillations [142, 145, 185-187] and hysteresis effects [143] in CO oxidation over Pt(100) in high vacuum ( 10 4 Torr) is the existence of spatio-temporal waves of the reversible surface phase transition hex - (1 x 1). The mathematical model [188] suggests that in each of the phases an adsorption mechanism with various parameters of CO and 02 adsorption/desorption and their interaction is realized, and the phase transition is modelled by a semi-empirical method via the introduction of discontinuous non-linearity. Later, an imitation model based on the stochastic automat was used [189] to study the qualitative characteristics for the dynamic behaviour of the surface. 

The cores of the spiral waves need not be stationary and can move in periodic, quasi-periodic or even chaotic flower trajectories [42, 43]. In addition, spatio-temporal chaos can arise if such spiral waves break up and the spiral wave fragments spawn pairs of new spirals [42, 44]. 

The local dynamics of tire systems considered tluis far has been eitlier steady or oscillatory. However, we may consider reaction-diffusion media where tire local reaction rates give rise to chaotic temporal behaviour of tire sort discussed earlier. Diffusional coupling of such local chaotic elements can lead to new types of spatio-temporal periodic and chaotic states. It is possible to find phase-synchronized states in such systems where tire amplitude varies chaotically from site to site in tire medium whilst a suitably defined phase is synclironized tliroughout tire medium 51. Such phase synclironization may play a role in layered neural networks and perceptive processes in mammals. Somewhat suriDrisingly, even when tire local dynamics is chaotic, tire system may support spiral waves... 

S. Jabubith, H. H. Rotermund, W. Engel, A. von Oertzen, G. Ertl. Spatio-temporal concentration patterns in a surface reaction Propagation of standing waves, rotating spirals and turbulence. Phys Rev Lett 65 3013-3016, 1990. 

What Is Interferometry (1.3) Interferometry deals with the physical phenomena which result from the superposition of electromagnetic (e.m.) waves. Practically, interferometry is used throughout the electromagnetic spectrum astronomers use predominantly the spectral regime from radio to the near UV. Essential to interferometry is that the radiation emerges from a single source and travels along different paths to the point where it is detected. The spatio-temporal coherence characteristics of the radiation is studied with the interferometer to obtain information about the physical nature of the source. 

Fig. 13.6. Pulse splitting cycle Single central pulse undergoes splitting. The split-off-pulses act a scatterers , which concentrate most of the energy in the spatio-temporal spectrum around loci that support diffractionless wave-forms. This provides the energy for a new central pulse, and the cycle repeats. . . ...

The Orr- Sommerfeld equation can be solved either as a temporal or as a spatial instability problem. For disturbance field created as a consequence of a localized excitation inside boundary layers, the temporal growth of the disturbance field is not realistic. It has been observed phenomenologically that for attached flows, instability is usually of a convective t3rpe and obtaining solution by spatial analysis is the appropriate one. In chapter 4, we will note that even for such a problem there can be spatio-temporally growing wave-fronts that dominate in attached boundary layers that are noted to be spatially stable. Such a problem is not evident for flows those are spatially unstable. The monograph by Betchov Criminale (1967) specifically talks about temporal growth of disturbances in shear layers and the readers are referred there for detailed expositions. 

Presented time accurate solution can be termed appropriately as the correct receptivity solution, as compared to its idealization in the signal problem. Later on, the results of this solution process is considered to look at the cases of spatially stable systems , those actually admit spatio-temporally growing wave-fronts-as given in Sengupta et al. (2006, 2006a). What is apparent for all spatially unstable cases is that there are no differences between the signal problem and the actual time-dependent problem-as two solutions shown in Fig. 2.23 match up to a certain distance-with the streamwise distance over which the match is seen stretches with time. 

To understand the spatio-temporal growth of waves, few cases were considered in Sengupta et al. (2006), marked as A, B, C and D in Fig. 4.2, with respect to the neutral curve shown in the Re — wo)-plane for the leading eigenmode. 

The essential difference between these and the cases shown in Fig. 4.3 is that the latter have three modes, while C possesses a single mode and D possesses two modes. The frontrunner in Fig. 4.3 is due to interactions of multiple stable modes. In the absence of multiple modes- as for the point C- no such forerunner is seen in Fig. 4.4. Again for the point D, there are only two stable modes that create spatio-temporally growing wave front. Thus, for fluid dynamic systems presence of a minimum of two stable modes is necessary to produce a spatio-temporally growing wave front, when the least stable mode is spatially damped. 

Further properties of spatio-temporally growing wave-front were studied in Sengupta et al. (2006a). It was investigated by exciting the Blasius boundary layer at a frequency that corresponds to the point on branch 11 of the neutral curve at wq = 0.1307 for Re = 1000. Computed streamwise perturbation velocity at different time instants are shown in Fig. 4.7. 

S. Alonso, F. Sagues, and J. M. Sancho. Excitability transitions and wave dynamics under spatio-temporal structured noise. Ph ys. Rev. E, 65 066107, 2002. 

In nature, spatio-temporal patterns in excitable media occur in seemingly unlimited variety. As early as 1946 Wiener and Rosenblueth [81] introduced the concept of excitable media to explain the propagation of electrical excitation fronts in the heart. Waves of electrical activity in the heart muscle assist its rhythmic contractions. The presence of spiral waves can indicate dangerous fibrillation. This is one of the motivations why the dynamics and control of spiral waves are studied. Furthermore spiral waves are typical, almost ubiquitous, patterns in excitable media see [88] in this volume. 

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