Ghost Interaction of Breathers

Mutual interaction of localized nonlinear waves, e.g., solitons and modulation instability patterns, is a fascinating and intensively-studied topic of nonlinear science. Here we report the observation of a novel type of breather interaction in telecommunication optical fibers, in which two identical breathers propagate with opposite group velocities. Under controlled conditions, neither amplification nor annihilation occurs at the collision point and most interestingly, the respective envelope amplitude, resulting from the interaction, almost equals another envelope maximum of either oscillating and counterpropagating breather. This ghost-like breather interaction dynamics is fully described by an N-breather solution of the nonlinear Schrödinger equation.

A Nonlinear Multiscale Theory of Atmospheric Blocking: Eastward and Upward Propagation and Energy Dispersion of Tropospheric Blocking Wave Packets

In this paper, a nonlinear multiscale interaction model is used to examine how the planetary waves associated with eddy-driven blocking wave packets propagate through the troposphere in vertically varying weak baroclinic basic westerly winds (BWWs). Using this model, a new one-dimensional finite-amplitude local wave activity flux (WAF) is formulated, which consists of linear WAF related to linear group velocity and local eddy-induced WAF related to the modulus amplitude of blocking envelope amplitude and its zonal nonuniform phase. It is found that the local eddy-induced WAF reduces the divergence (convergence) of linear WAF in the blocking upstream (downstream) side to favor blocking during the blocking growth phase. But during the blocking decay phase, enhanced WAF convergence occurs in the blocking downstream region and in the upper troposphere when BWW is stronger in the upper troposphere than in the lower troposphere, which leads to enhanced upward-propagating tropospheric wave activity, though the linear WAF plays a major role. In contrast, the downward propagation of planetary waves may be seen in the troposphere for vertically decreased BWWs. These are not seen for a zonally uniform eddy forcing. A perturbed inverse scattering transform method is used to solve the blocking envelope amplitude equation. It is found that the finite-amplitude WAF represents a modified group velocity related to the variations of blocking soliton amplitude and zonal wavenumber caused by local eddy forcing. Using this amplitude equation solution, it is revealed that, under local eddy forcing, the blocking wave packet tends to be nearly nondispersive during its growth phase but strongly dispersive during the decay phase for vertically increased BWWs, leading to strong eastward and upward propagation of planetary waves in the downstream troposphere.

The stability of a slowly diverging swirling jet

The spatial evolution of small-amplitude unsteady disturbances of an axisymmetric swirling jet is examined theoretically. The slow axial divergence of the jet mean flow is accounted for by using the method of multiple scales and a consistent solution for both the mean flow and unsteady disturbance is derived. Previous work by Lu & Lele (1999) has considered the leading-order analysis, in which the modal eigenvalues are determined from locally parallel theory, but the key feature of our analysis is the solution of the next-order secularity condition for the axial variation of the wave-envelope amplitude.The swirling jet profile sustains two types of instability waves: the Kelvin–Helmholtz instability associated with axial shear, and a centrifugal instability which arises due to a decrease in circulation with radial distance. The evolution of the disturbance axial wavenumber and envelope amplitude with downstream distance is calculated. Numerical results show that the growth of the centrifugal mode is significantly curtailed as a result of a rapidly decaying envelope amplitude. The shear instability is significantly more amplified by the addition of swirl.The general solution for the disturbance envelope amplitude breaks down at so-called turning points. This is found to occur for a series of neutral propagating modes. A rescaling in the vicinity of the turning point shows that the amplitude in this region is governed by a parabolic cylinder equation. The modal amplitude is seen to decay very significantly through this turning point, even though the mode is neutral to leading order.

Sunspot Number Series Envelope and Phase

The sunspot number series R( t) from 1700 to date is found to be representable by R( t) = I Jf' {Re( E( t) exp[i {wo t + ( t) I]) + U( t) 1 I, where Wo is the angular frequency corresponding to a period of 22 years, E(t) is the instantaneous envelope amplitude, (t) is the instantaneous phase of a complex time-varying analytic function, U(t) is an undulation of low amplitude and period about 30 (22-year) cycles and jy is a nonlinear operator whose main effect is to introduce a small amount of third harmonic (period about 7 years). The justification for the 22-year period is the known fact that the observable sunspot magnetic fields reverse polarity every 11 years or so at the time of sunspot minimum; the undulation has been demonstrated, and its period determined, in fossil records discovered by Williams; and the third harmonic is an expected consequence of minor nonlinearity in the dependence of the arbitrarily defined R( t) on the physical cause of sunspots. The algebraic representation is established by the Hilbert transform method of forming a complex analytic function as proposed by Gabor. The method reveals three obscuring features that may be alleviated as follows: use the alternating series R� (t) in which alternate II-year cycles take opposite signs, remove the third harmonic, and subtract the undulation. These justifiable steps remove artificial components, such as sum and difference frequencies, that are gratuitously and nonlinearly introduced by conventional Fourier analysis as applied to the rectified, or absolute, value of the 22-year oscillation. Then a complex envelope E( t) exp {i ( t)j can be discerned whose intrinsic behaviour can be studied to reveal statistics that bear on the physical origin of the solar cycle. The results favour a deep monochromatic oscillator whose influence is propagated to the observable surface via a time-varying medium. The r.m.s. value of the component of E(t) is 0�4 of the mean and the characteristic time is a century. Frequency analysis of the envelope does not support a 78-year period in the modulation noticed by Wolf. Both the statistical frequency distribution of the amplitude E( t) and its spectrum are subject to refinement by analysis of fossil solar records. The results do not favour the theory that the 22-year period is set by the natural frequency of a resonator with characteristic damping subject to random turbulent excitation. Also disfavoured is the theory of energy release at intervals determined by a relaxation process. Correlation has been found between the phase departure ~(t) from linear and envelope amplitude and attributed to propagation of the magnetic .cycles through a time-varying, such as a convecting, medium. A correlation not depending on Hilbert transform analysis is predicted between the reciprocal cycle length and envelope amplitude and found to� exist. Variability of the sunspot cycle length can be viewed as a Doppler shift due to propagation in a time-varying medium and the Wolf modulation then represents the concomitant intensity change. Agreement has been found between E(t) and '(t) but not explained. If the explanation is dispersion in the propagation of the assumed magnetic flux waves then there is a mode of oscillation. that has the characteristics required for the undulation U( t). Extra buoyancy possessed by the magnetic field of strong cycles accounts for the fast rise time of strong cycles.

Complex seismic trace analysis

The conventional seismic trace can be viewed as the real component of a complex trace which can be uniquely calculated under usual conditions. The complex trace permits the unique separation of envelope amplitude and phase information and the calculation of instantaneous frequency. These and other quantities can be displayed in a color‐encoded manner which helps an interpreter see their interrelationship and spatial changes. The significance of color patterns and their geological interpretation is illustrated by examples of seismic data from three areas.