Nonlinear dynamos: A complex generalization of the Lorenz equations

Plane nonlinear dynamo waves can be described by a sixth order system of nonlinear ordinary differential equations which is a complex generalization of the Lorenz system. In the regime of interest for modelling magnetic activity in stars there is a sequence of bifurcations, ending in chaos, as a stability parameter D (the dynamo number) is increased. We show that solutions undergo three successive Hopf bifurcations, followed by a transition to chaos. The system possesses a symmetry and can therefore be reduced to a fifth order system, with trajectories that lie on a 2-torus after the third bifurcation. As D is then increased, frequency locking occurs, followed by a sequence of period-doubling bifurcations that leads to chaos. This behaviour is probably caused by the Shil'nikov mechanism, with a (conjectured) homoclinic orbit when D is infinite.     

Ion-acoustic shock waves with negative ions in presence of dust particulates

Dust acoustics shock waves have been investigated experimentally in a homogeneous unmagnetized dusty plasma device containing negative ions. When the negative ion density larger than a critical concentration ‘rc’ negative shock waves were observed instead of positive shock waves. Again when it is nearly equal to ‘rc’ both positive and negative shock waves propagate. The experimental findings are compared with modified KdV-Burgers equation. The velocity of the shock waves are also measured and compared with the numerical integration of modified KdV-Burgers equation.     

Rogue waves, dissipation, and downshifting

We investigate the effects of dissipation on the development of rogue waves and downshifting by adding nonlinear and linear damping terms to the one-dimensional Dysthe equation. Significantly, rogue waves do not develop after the downshifting becomes permanent. Thus in our experiments permanent downshifting serves as an indicator that damping is sufficient to prevent the further development of rogue waves. Using the inverse spectral theory of the NLS equation, simulations of the damped Dysthe equation for sea states characterized by JONSWAP spectrum consistently show that rogue wave events are well-predicted by proximity to homoclinic data, as measured by the spectral splitting distance δ. The cut off distance δ_cutoff decreases as the strength of the damping increases, indicating that for stronger damping the JONSWAP initial data must be closer to homoclinic data for rogue waves to occur.     

Flow helicity in a simplified statistical model of a fast dynamo

The role of kinetic helicity in small-scale fast dynamo action is investigated by employing a simple statistical model for the underlying flow with statistics that are Gaussian distributed, temporally delta-correlated and spatially homogeneous and isotropic. In order to focus on small-scale dynamo action we restrict our attention to flows possessing no net kinetic helicity. With the help of a diagrammatic technique and a numerical calculation we show that the dynamo growth rate is independent of the kinetic helicity as the magnetic Reynolds number R_m → ∞. It is indicated that the latter enhances the growth of the magnetic energy only for finite R_m.     

The Klein-Gordon equation as a limiting case of a suitably hyperbolized schrödinger equation

We obtain a hyperbolic equation whose discontinuity waves are all exceptional and propagate with velocity λ. When λ → ∞ or λ=c, this equation becomes identical to the Schrödinger equation and to the Klein-Gordon equation respectively. We also show that λ is related to the dispersion relationE(p).     

History and results of the Riga dynamo experiments

On 11 November 1999, a self-exciting magnetic eigenfield was detected for the first time in the Riga liquid sodium dynamo experiment. We report on the long history leading to this event, and on the subsequent experimental campaigns which provided a wealth of data on the kinematic and the saturated regime of this dynamo. The present state of the theoretical understanding of both regimes is delineated, and some comparisons with other laboratory dynamo experiments are made. To cite this article: A. Gailitis et al., C. R. Physique 9 (2008).     

Metric-torsion decay of non-adiabatic chiral helical magnetic fields against chiral dynamo action in bouncing cosmological models

Metric-torsion effects on chiral massless fermions are investigated in the realm of the adiabatic amplification of cosmological magnetic fields (CMFs) in a general relativistic framework and in the framework of Einstein–Cartan (EC) bouncing cosmologies. In GR the chiral effect is proportional to the Hubble factor and the solution of the dynamo equation leads to an adiabatic magnetic field, while in Einstein–Cartan bouncing cosmology we have non-adiabatic magnetic fields where the breaking of adiabaticity is given by a torsion term. Using a EWPT magnetic field of the order of \(B_{\text {seed}}\sim {10^{24}}\) G at 5 pc scale, we obtain a CMF in EC of the order of \(10^{-10}\) G, which is still able to seed a galactic dynamo which amplifies this field up to galactic magnetic fields of four orders of magnitude, which is a mild dynamo. In the case of massive chiral fermions it is shown that torsion actually attenuated the convective dynamo term in the dynamo equation obtained from the QED of an electron–positron pair \(e^{-}e^{+}\). Chiral effects on general relativity may lead to strong magnetic fields of the order of \(\sim {10^{18}}\) G at the early universe resulting from pure metric effects. Strong magnetic fields of the order of \(B_{\text {metric}-\text {torsion}}\sim {10^{8}}\) G may be obtained from very strong seed fields. At 1 Mpc scale of the present universe a galactic dynamo seed of the order of \(10^{-19}\) G is found. It is shown in this paper that chiral dynamo effects in the expanded universe can be obtained if one takes into account the speed of the cosmic plasma.     

Bifurcations and the stability of the surface envelope solitons for a finite-depth fluid

The dynamics of the quasi-monochromatic surface gravitational waves in a finite-depth fluid is studied for the case where the product of the wavenumber by the depth of the fluid is close to the critical value k _cr h ≈ 1.363. Within the framework of the Hamiltonian formalism, the general nonlinear Schrödinger equation is derived. In contrast to the classical nonlinear Schrödinger equation, this equation involves the gradient terms to the four-wave interaction, as well as the six-wave interaction. This equation is used to analyze the modulation instability of the monochromatic waves, as well as the bifurcations of the soliton solutions and their stability. It is shown that the solitons are stable and unstable to finite perturbations for focusing and defocusing nonlinearities, respectively.     

Linear and nonlinear magnetohydrodynamic waves in twisted magnetic flux tubes

Dynamics of axisymmetric Alfvén and slow magnetoacoustic waves in weakly twisted magnetic flux tubes is considered. Linear dispersion relations for the waves are derived and analyzed in the presence of the twisting. The weakly nonlinear dynamics is shown to be governed by the Korteweg-de Vries equation. Nonlinear slow body waves appear as a narrowing of tube in a low β plasma and widening of tube, when β ⪢ 1. Nonlinear Alfvén torsional waves appear as a widening (β > 1) and narrowing (β < 1) of tube, accompanied by the local increase of the tube twisting.     

Extreme waves statistics for the Ablowitz-Ladik system

We examine statistics of waves for the problem of modulation instability development in the framework of discrete integrable Ablowitz-Ladik (AL) system. Modulation instability depends on one free parameter h that has the meaning of the coupling between the nodes on the lattice. For strong coupling h ? 1, the probability density functions (PDFs) for waves amplitudes coincide with that for the continuous classical nonlinear Schrödinger equation; the PDFs for both systems are very close to Rayleigh ones. When the coupling is weak h ～ 1, there appear highly localized waves with very large amplitudes, that drastically change the PDFs to significantly non-Rayleigh ones, with so-called “fat tails” when the probability of a large wave occurrence is by several orders of magnitude higher than that predicted by the linear theory. Evolution of amplitudes for such rogue waves with time is similar to that of the Peregrine solution for the classical nonlinear Schrödinger equation.     

Stability criterion for projective synchronization in three-dimensional chaotic systems

Projective synchronization, in which the state vectors synchronize up to a scaling factor, has recently been observed in coupled partially linear chaotic systems (Lorenz system) under certain conditions. In this Letter, we present a stability criterion that guarantees the occurrence of the projective synchronization in three-dimensional systems. By applying the criterion to two typical partially linear systems (Lorenz and disk dynamo), it shows that only some parameters play the key role in influencing the stability. Projective synchronization only happens when σ>−1 for the Lorenz and μ>0 for the disk dynamo.     

Asymptotic stability of solitary waves

We show that the family of solitary waves (1-solitons) of the Korteweg-de Vries equation $$\partial _t u + u\partial _x u + \partial _x^3 u = 0 ,$$ is asymptotically stable. Our methods also apply for the solitary waves of a class of generalized Korteweg-de Vries equations, $$\partial _t u + \partial _x f(u) + \partial _x^3 u = 0 .$$ In particular, we study the case wheref(u)=u ^p+1/(p+1),p=1, 2, 3 (and 3<p<4, foru>0, withf∈C ⁴). The same asymptotic stability result for KdV is also proved for the casep=2 (the modified Korteweg-de Vries equation). We also prove asymptotic stability for the family of solitary waves for all but a finite number of values ofp between 3 and 4. (The solitary waves are known to undergo a transition from stability to instability as the parameterp increases beyond the critical valuep=4.) The solution is decomposed into a modulating solitary wave, with time-varying speedc(t) and phase γ(t) (bound state part), and an infinite dimensional perturbation (radiating part). The perturbation is shown to decay exponentially in time, in a local sense relative to a frame moving with the solitary wave. Asp→4^?, the local decay or radiation rate decreases due to the presence of aresonance pole associated with the linearized evolution equation for solitary wave perturbations.     

Nonplanar Electron - Acoustic Shock Waves with Superthermal Hot Electrons

Nonplanar electron-acoustic shock waves having superthermal hot electrons are investigated with two temperature electrons model in unmagnetized plasma. Using reductive perturbation method, Korteweg-de Vries-Burgers (KdVB) equation is obtained in the cylindrical/spherical coordinates. Dissipation effect is introduced in the model by means of kinematic viscosity term. On the basis of the solutions of KdVB equation, variation of shock waves features (amplitude, velocity and width) with different plasma parameters are analysed. KdV-Burgers equation always leads to monotonic solitons and no oscillatory peak may appear. The combined effect of particle density (α), superthermal parameter (κ), electron temperature ratio (??) and kinetic viscosity (η₀) is numerically studied, and it is observed that these parameters significantly change the properties of the shock waves in nonplanar geometry especially in spherical coordinates. Results could be helpful to analyse the soliton features in laboratory as well as in the space environments.     

General Schrödinger equation for whistler waves propagating along a magnetic field

The general Schrödinger equation (GSE) for whistler waves with their group velocity directed along an external magnetic field is derived. The “mean” wave vector of the wave beam may be parallel to or have an angle Θ = arccos(2ω/ω_c) with the magnetic field. Applications of GSE to the whistler propagation in density ducts are considered. The results are important for the problem of the self-focusing of whistler waves.     

The VKS experiment: turbulent dynamical dynamos

The VKS experiment studies dynamo action in the flow generated inside a cylinder filled with liquid sodium by the rotation of coaxial impellers (the von Kármán geometry). We report observations related to the self-generation of a stationary dynamo when the flow forcing is symmetric, i.e. when the impellers rotate in opposite directions at equal angular velocities. The bifurcation is found to be supercritical, with a neutral mode whose geometry is predominantly axisymmetric. We then report the different dynamical dynamo regimes observed when the flow forcing is asymmetric, including magnetic field reversals. We finally show that these dynamics display characteristic features of low dimensional dynamical systems despite the high degree of turbulence in the flow. To cite this article: VKS Collaboration, C. R. Physique 9 (2008).     

Nonlinear waves in elastic media

We ask about the possible existence of solitary waves in infinite, homogeneous, isotropic, elastic media. Namely, can a nonlinear localized wave packet propagate without altering its shape in such materials? We consider one- dimensional propagation both of body and surface waves. In the first case we show, under rather general assumptions, that if a wave packet propagates without altering its shape it must, of necessity, be a solution of a linear wave equation and in this sense, (body) solitary waves do not exist. Surface solitary waves may however exist: a model equation is derived in which nonlinear and dispersive effects balance each other to allow for waves-both periodic and solitary-of constant shape. It is conceivable they are of some relevance in seismology.     

The fourth-order nonlinear Schrödinger equation for the envelope of Stokes waves on the surface of a finite-depth fluid

The method of multiple scales is used to derive the fourth-order nonlinear Schrödinger equation (NSEIV) that describes the amplitude modulations of the fundamental harmonic of Stokes waves on the surface of a medium-and large-depth (compared to the wavelength) fluid layer. The new terms of this equation describe the third-order linear dispersion effect and the nonlinearity dispersion effects. As the nonlinearity and the dispersion decrease, the equation uniformly transforms into the nonlinear Schrödinger equation for Stokes waves on the surface of a finite-depth fluid that was first derived by Hasimoto and Ono. The coefficients of the derived equation are given in an explicit form as functions of kh (h is the fluid depth, and k is the wave number). As kh tends to infinity, these coefficients transform into the coefficients of the NSEIV that was first derived by Dysthe for an infinite depth.     

Products of distributions and singular travelling waves as solutions of advection-reaction equations

In this paper, we restrict our attention to the advection-reaction equation u _t + [?(u)] _x = ??(u), where ? and ?? are entire functions. Conditions for the propagation of a distributional wave profile are presented and the wave speed is evaluated. As an example, we prove that, under certain conditions, the propagation of delta-waves in models ruled by the diffusionless Burgers-Fisher equation is possible and compute the speeds of propagation of these waves. In the same setting, the propagation of travelling waves with the shape of a C ¹-function with one jump discontinuity is also studied. These results will be easily explained by our theory of distributional products and are based on a rigorous and consistent concept of a solution that we have already introduced in previous works.     

SIMPLIFIED DISPERSION RELATIONSHIPS FOR IN-VACUO PIPES

The dispersion characteristics of a pipe offer a way to gain physical insight into its dynamic behaviour. Whilst these can be found in the literature they are generally calculated by numerically solving the characteristic equation. In this paper, a simplified characteristic equation of an in vacuo pipe is presented and from this analytical expressions for the wavenumbers for the circumferential modes below the ring frequency are derived. It is shown that before waves cut on and propagate, they change from being decaying standing waves at low frequencies to being nearfield waves. A simplified expression is also determined for the cut on frequencies of the n?2 circumferential modes. Simulations are presented to validate the results against some established theories of pipe vibration.     

Obliquely Propagating Electron Acoustic Shock Waves in Magnetized Plasma

Obliquely propagating electron acoustic shock waves in plasma with stationary ions, cold and superthermal hot electrons are investigated in magnetized plasma. Employing reductive perturbation method, Korteweg-de Vries-Burgers equation (KdVB) is derived in the small amplitude approximation limit. The analytical and numerical calculations of the KdVB equation show the variation of shock waves structure (amplitude, velocity, and width) with different plasma parameters. Particle density (α), superthermal parameter (κ), electron temperature ratio (??), kinetic viscosity (η₀), obliqueness (k_z), and strength of magnetic field (ω_c) significantly modify the properties of the shock waves structures. The present investigation is useful to understand dissipative structures observed in space or laboratory plasma where multielectrons population with superthermal electrons are prevalent.