Modulational instability

Vaupotic N, Copic M (2005) Polarization modulation instability in liquid crystals with spontaneous chiral symmetry breaking. Phys Rev E 72 031701-1-031701-4... 

Abstract Pattern formation is a widespread phenomenon observed in different physical, chemical and biological systems on varions spatial scales, including the nanometer scale. In this chapter discussed are the universal features of pattern formation pattern selection, modulational instabilities, structure and dynamics of domain walls, fronts and defects, as well as non-potential effects and wavy patterns. Principal mathematical models used for the description of patterns (Swift-Hohenberg equation, Newell-Whitehead-Segel equation, Cross-Newell equation, complex Ginzburg-Landau equation) are introduced and some asymptotic methods of their analysis are presented. 

If the wavenumber of the roll solution satisfies the condition K < 0 (i.e. A < 1), a transverse modulational instability takes place with respect to disturbances with k] = 0, ky 0. It is also called the zigzag instability. 

Kini [29] considered the electric field-induced static modulated structures of nematic liquid crystals. The electric field E was applied parallel to the sample, and the initial uniform director orientation was tilted with respect to the sample boundaries in the plane normal to E. The formation of modulated structures was shown to be favored when a stabilizing magnetic field H of sufficient strength was applied along the initial director orientation. This type of modulated instability was observed in experiment [30]. 

However, the modulational instability is a quasi-resonant interaction process, i.e., wave numbers and frequencies satisfy the following conditions ... 

M. Onorato, A. R. Osborne, M. Serio, L. Cavaleri, C. Brandini and C. T. Stansberg, Extreme waves, modulational instability and second order theory Wave flume experiments on irregular waves, Eur. J. Mech. B/Fluids 25(5), 586-601 (2006). 

For M = 1, we have structures periodic in one dxrtciion, stripes, in our 2D space. The nature of the solutions and of the bifurcations are as in ID. However the stability problem now also implies testing the stability of the stripes with respect to the formation of patterns of other symmetries. As shown in Section 4, other wavenumber modulational instabilities may also occur as compared to the ID situation. 

External relief valves should be of the full-flow, non-chattering (modulating) style. In oil systems with relatively low pressure levels, the hydraulic-type relief valve becomes attractive because the plug lift is smooth, and instability during lifting is eliminated. The valves should be located as close to the pump as possible to provide fast reaction time. 

One such index is line frequency, which provides indications of instability. Modulations, or harmonics, of line frequency may indicate the motor s inability to find and hold magnetic center. Variations in line frequency also increase the amplitude of the fundamental and other harmonics of running speed. 

The sections to follow describe the most common machine-train failure modes critical speeds, imbalance, mechanical looseness, misalignment, modulations, process instability, and resonance. 

ICF flame motion during cyclic modulation of the flow <3> = 0.8, v = 1.87 = m/s,/= 150Hz, and v = 0.15m/s. (Adapted from Candel, S., Durox, D., and Schuller, T., Flame interactions as a source of noise and combustion instabilities, AIAA paper 2004—2928,10th AlAAl CEAS Aeroacoustics Conference, Manchester, U.K., May 2004. With permission.)... 

A second application of current interest in which widely separated length scales come into play is fabrication of modulated foils or wires with layer thickness of a few nanometers or less [156]. In this application, the aspect ratio of layer thickness, which may be of nearly atomic dimensions, to workpiece size, is enormous, and the current distribution must be uniform on the entire range of scales between the two. Optimal conditions for these structures require control by local mechanisms to suppress instability and produce layer by layer growth. Epitaxially deposited single crystals with modulated composition on these scales can be described as superlattices. Moffat, in a report on Cu-Ni superlattices, briefly reviews the constraints operating on their fabrication by electrodeposition [157]. 

Fig. 3. Solubility of silk proteins in solution as a function of time. Low solubility corresponds to protein aggregation. The fast and slow aggregations are observed in vitro (Dicko et al., 2004a), whereas the stable helical conformation (storage structure) is observed in vivo (Dicko et al., 2004b,d). This illustrates the inherent instability of silk protein in solution and shows the /(-sheet polymorph structure as the most stable form. In other words, the spiders actively control and modulate the unavoidable silk protein aggregation prior to fiber formation.

To reduce spurious signals due to drifts of the EPR line setting arising from mechanical and thermal instabilities, double coding of the ENDOR information is often employed23). Normally a low-frequency Zeeman modulation (30-300 Hz) is applied while the rf field is frequency or amplitude modulated at frequencies of about 1-30 kHz. This modulation scheme, however, has two major disadvantages ... 

The first simulations of the collapsar scenario have been performed using 2D Newtonian, hydrodynamics (MacFadyen Woosley 1999) exploring the collapse of helium cores of more than 10 M . In their 2D simulation MacFadyen Woosley found the jet to be collimated by the stellar material into opening angles of a few degrees and to transverse the star within 10 s. The accretion process was estimated to occur for a few tens of seconds. In such a model variability in the lightcurve could result for example from (magneto-) hydrodynamic instabilities in the accretion disk that would translate into a modulation of the neutrino emission/annihilation processes or via Kelvin-Helmholtz instabilities at the interface between the jet and the stellar mantle. 

At the stability boundary, ion motion is in resonance with this modulation voltage, and thus ion ejection is facilitated. Axial modulation basically improves the mass-selective instability mode of operation. 

---