Spiral Solutions Research Articles

Isolated Spiral Solutions of the Ginzburg–Landau Equation

Using finite element simulation, we study the main features of rotating wave solutions of the cubic complex Ginzburg–Landau equation. To focus on the characteristics of the waves themselves, we have used a circular domain to avoid the effects of irregular boundaries; we have inhibited the formation of defects using an Archimedean wave centered at the origin as the initial condition, and we have chosen a domain size long enough to contain several spiral arms but not so long as to promote long-wave instabilities. This allows us to focus on the geometric features of the solutions often overlooked in traditional works with random initial conditions in large domains. We show with our simulations that the convective and absolute stabilities differ from those predicted for plane waves. Likewise, we show that the appearance of spirals and anti-spirals can occur in any of the quadrants of the parameter space and depends fundamentally on the initial conditions. Finally, regarding the stability of spiral waves against sideband disturbances, we show that the persistence of two-dimensional spiral waves, if fulfilled, does not correspond to the Eckhaus criterion. Our simulations allow us to corroborate the idea that two-dimensional spirals depend on the size of the domain and that they require new analytical results whose trends are identified in this work.

Chiral Modulations in Non-Heisenberg Models of Non-Centrosymmetric Magnets Near the Ordering Temperatures

The structure of skyrmion and spiral solutions, investigated within the phenomenological Dzyaloshinskii model of chiral magnets near the ordering temperatures, is characterized by the strong interplay between longitudinal and angular order parameters, which may be responsible for experimentally observed precursor effects. Within the precursor regions, additional effects, such as pressure, electric fields, chemical doping, uniaxial strains and/or magnetocrystalline anisotropies, modify the energetic landscape and may even lead to the stability of such exotic phases as a square staggered lattice of half-skyrmions, the internal structure of which employs the concept of the “soft” modulus and contains points with zero modulus value. Here, we additionally alter the stiffness of the magnetization modulus to favor one- and two-dimensional modulated states with large modulations of the order parameter magnitude. The computed phase diagram, which omits any additional effects, exhibits stability pockets with a square half-skyrmion lattice, a hexagonal skyrmion lattice with the magnetization in the center of the cells parallel to the applied magnetic field, and helicoids with propagation transverse to the field, i.e., those phases in which the notion of localized defects is replaced by the picture of a smooth but more complex tiling of space. We note that the results can be adapted to metallic glasses, in which the energy contributions are the same and originate from the inherent frustration in the models, and chiral liquid crystals with a different ratio of elastic constants.

Spiral shocks induced in a galactic gaseous disk: Hydrodynamic understanding of observational properties of spiral galaxies

Context. We investigate the properties of spiral shocks in a steady, adiabatic, non-axisymmetric, self-gravitating, mass-outflowing accretion disk around a compact object. Aims. We obtained the accretion-ejection solutions in a galactic disk and applied them to spiral galaxies in order to investigate the possible physical connections between some observational quantities of galaxies. Methods. We considered the self-gravitating disk potential to examine the properties of the galactic gaseous disk. We obtained spiral shock-induced accretion-ejection solutions following the point-wise self-similar approach. Results. We observed that the self-gravitating disk profoundly affects the dynamics of the spiral structure of the disk and the properties of the spiral shocks. We find that the observational dispersion between the pitch angle and shear rate and between the pitch angle and star formation rate in spiral galaxies contains some important physical information. Conclusions. There are large differences among the star formation rates of galaxies with similar pitch angles. These differences may be explained by the different star formation efficiencies caused by distinct galactic ambient conditions.

Formation of spiral structures in rich-hydrogen air flames at elevated pressures

Travelling combustion fronts demonstrate the appearance of a variety of instabilities and structures and thus the revealing of the mechanisms of these phenomena is important both in theoretical and practical aspects. In this work, the mechanism of formation of spiral nonlinear wave structures in the process of propagation of a combustion wave in a rich hydrogen-air mixture at elevated pressure is studied analytically and numerically. A detailed model of the hydrogen oxidation reaction is reduced to a system of partial differential equations for the evolution of H, HO2, O2 and temperature profiles describing both low-temperature and high-temperature reactivity. It is shown to be able to successfully reproduce the characteristics of the diffusive-thermal pulsations emerging with the increase of pressure. A model for the dynamics of the low temperature flame region is separated from the reduced model by using a number of assumptions and appeared to be similar to the Sal'nikov model. It is shown that for the parameter values at which spiral waves are observed, the latter model turns to excitable regime and generates the spiral solutions. Thus it is concluded that the low temperature oxidation processes are responsible for emergence of the spiral structures at the combustion front. We think that the current work is an important step towards understanding the mechanisms of formation of spiral wave structures in expanding combustion fronts.

Transition of spiral solutions according to the time and space steps discretization of reaction-diffusion system of FitzHugh-Nagumo type

Spiral solutions or spiral waves can be found in many natural systems. Spiral waves were observed in studies about the potential in brain and heart cells. Their appearance in the human heart is a presentation of arrhythmia. The paper showed how to create spiral solutions of diffusion-reaction system of FitzHugh-Nagumo type and the transition of spiral solutions according to the time step and space step discretization of finite difference method. Decreasing the value of space step discretization makes the spiral wave grow bigger, but if the value of time step discretization is increased at the same given space step, the finite difference method will be explosive, meaning that spiral wave no longer exists.

A note on special solutions for magnetohydrodynamics system in the van der Waals fluid

The present article deals with the study of special flows such as circulatory, radial, and spiral flows, respectively, for the magnetohydrodynamics system with the van der Waals fluid. Employing the method of hodograph transformation, the solutions have been obtained in the (u,v)-plane. An inversion of the spiral solution back to the (x,y)-plane is possible under a specific requirement, i.e.,bˆ2>2aρ, where bˆ represents Alfven speed and a is the van der Waals parameter.

The Effect of a Spiral Density Wave on the Galaxy’s Rotation Curve, as Applied to the Andromeda Galaxy (M31)

The rotational velocity curve, which is the circular velocity profile of the stars and gas in a spiral galaxy as a function of their distance from the galactic center, plays an important role in the kinematic and dynamic investigation of spiral galaxies. There are observations of approximately flat rotation curves (RC) at large distances that have introduced mass discrepancy between the theoretically derived RC and the observed one. In this paper, we derive a rotational velocity expression using a nonlinear spiral density wave solution for the surface mass density (SMD) within the disk. We show that the proposed nonlinear spiral solution is able to support the observed flat rotational velocity curve for large distances with no mass deficiency. The aim of the paper is to confirm the crucial importance of the mass distribution on the rotation curve profile. Although the model is limited by the fluid description of the galactic disk, it provides an improved rotational velocity expression and a rotation curve with no mass discrepancy in the outer part of the disk due to the inclusion of the spiral mass distribution. The disk mass has not been averaged within the exponential disk approximation, but it rather follows the observed spiral pattern given by the analytical solution of the nonlinear equation. The M31 galaxy has been chosen as the closest and well mapped spiral galaxy, similar in many aspects to our host galaxy, in order to apply a rotational velocity expression that accounts for nonlinear effects and derive RC. The obtained result can have a strong influence on large-scale gravity dynamics, as well.

Current-driven spiral domain wall in a ferrimagnet near the magnetization compensation point

A spiral domain wall emerges when the Dzyaloshinskii-Moriya vector is along the easy axis. While the ferromagnetic spiral wall has been well studied, its ferrimagnetic counterpart has nevertheless not been explored. Here, using the collective variable approach, we derive the spiral domain-wall solutions in a ferrimagnetic nanostrip with a longitudinal easy axis and a bulk Dzyaloshinskii-Moriya interaction. At the magnetization compensation point, the wall rotates and propagates steadily in a wide current range. The (angular) velocity of this wall increases almost linearly with the experimentally feasible currents. Also, the wall exhibits a relativistic-like contraction with increasing velocity. Near this point, the wall moves smoothly with its velocity linearly depending on the current below the Walker breakdown. Above this critical current, all the internal parameters of the wall, such as the azimuthal angle, width, and spiral pitch, as well as their rates of change, oscillate periodically. The wall moves forward with its velocity varying periodically. These different behaviors at or near the compensation point are ascribed to the tunable demagnetization effect. The adjustability of FiM parameters, combining with the twist induced by the Dzyaloshinskii-Moriya interaction, can effectively change the DW's propagation and rotation.

Logarithmic spiral solutions of the Kopell-Howard lambda-omega reaction-diffusion equations.

Our investigation of logarithmic spirals is motivated by disparate experimental results: (i) the discovery of logarithmic spiral shaped precipitate formation in chemical garden experiments. Understanding precipitate formation in chemical gardens is important since analogous precipitates form in deep ocean hydrothermal vents, where conditions may be compatible with the emergence of life. (ii) The discovery that logarithmic spiral shaped waves of spreading depression can spontaneously form and cause macular degeneration in hypoglycemic chick retina. The role of reaction-diffusion mechanisms in spiral formation in these diverse experimental settings is poorly understood. To gain insight, we use the topological shooting to prove the existence of 0-bump stationary logarithmic spiral solutions, and rotating logarithmic spiral wave solutions, of the Kopell-Howard lambda-omega reaction-diffusion model.

Two Dimensional Subsonic and Subsonic-Sonic Spiral Flows Outside a Porous Body

In this paper, we investigate two dimensional subsonic and subsonic-sonic spiral flows outside a porous body. The existence and uniqueness of the subsonic spiral flow are obtained via variational formulation, which tends to a given radially symmetric subsonic spiral flow at far field. The optimal decay rate at far field is also derived by Kelvin’s transformation and some elliptic estimates. By extracting spiral subsonic solutions as the approximate sequences, we obtain the spiral subsonic-sonic limit solution by utilizing the compensated compactness. The main ingredients of our analysis are methods of calculus of variations, the theory of second-order quasilinear equations and the compensated compactness framework.

Exact ground states and domain walls in one dimensional chiral magnets

We determine exactly the phase structure of a chiral magnet in one spatial dimension with the Dzyaloshinskii-Moriya (DM) interaction and a potential that is a function of the third component of the magnetization vector, n3, with a Zeeman (linear with the coefficient B) term and an anisotropy (quadratic with the coefficient A) term, constrained so that 2A ≤ |B|. For large values of potential parameters A and B, the system is in one of the ferromagnetic phases, whereas it is in the spiral phase for small values. In the spiral phase we find a continuum of spiral solutions, which are one-dimensionally modulated solutions with various periods. The ground state is determined as the spiral solution with the lowest average energy density. As the phase boundary approaches, the period of the lowest energy spiral solution diverges, and the spiral solutions become domain wall solutions with zero energy at the boundary. The energy of the domain wall solutions is positive in the homogeneous phase region, but is negative in the spiral phase region, signaling the instability of the homogeneous (ferromagnetic) state. The order of the phase transition between spiral and homogeneous phases and between polarized (n3 = ±1) and canted (n3 ≠ ±1) ferromagnetic phases is found to be second order.

On some smooth symmetric transonic flows with nonzero angular velocity and vorticity

This paper concerns the structural stability of smooth cylindrically symmetric transonic flows in a concentric cylinder. Both cylindrical and axi-symmetric perturbations are considered. The governing system here is of mixed elliptic–hyperbolic and changes type and the suitable formulation of boundary conditions at the boundaries is of great importance. First, we establish the existence and uniqueness of smooth cylindrical transonic spiral solutions with nonzero angular velocity and vorticity which are close to the background transonic flow with small perturbations of the Bernoulli’s function and the entropy at the outer cylinder and the flow angles at both the inner and outer cylinders independent of the symmetric axis, and it is shown that in this case, the sonic points of the flow are nonexceptional and noncharacteristically degenerate, and form a cylindrical surface. Second, we also prove the existence and uniqueness of axi-symmetric smooth transonic rotational flows which are adjacent to the background transonic flow, whose sonic points form an axi-symmetric surface. The key elements in our analysis are to utilize the deformation-curl decomposition for the steady Euler system to deal with the hyperbolicity in subsonic regions and to find an appropriate multiplier for the linearized second-order mixed type equations which are crucial to identify the suitable boundary conditions and to yield the important basic energy estimates.

Metasurface for Near-Field Wireless Power Transfer With Reduced Electric Field Leakage

Wireless power transfer is a breakthrough technology which can be used in all aspects of humans daily life. Here, a bi-layer metasurface as a transmitter for near-field wireless power transfer is proposed and studied. The novelty and advantage of the proposed metasurface is the spatial separation of the electric and magnetic near fields. Magnetic fields responsible for power transfer are sufficiently high on top of the metasurface whereas the electric fields are almost completely confined between two layers of the metasurface. These unique properties have been obtained due to the special metasurface design based on two orthogonal layers of resonant wires immersed in high-permittivity background. The theoretical and experimental study reveal the quasi-uniform magnetic field distribution over the metasurface dimensions of 40×40 cm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> that makes it suitable for wireless power transfer via resonant magnetic coupling to one or several receivers placed above it. Compared with a conventional planar spiral coil solution, the specific absorption rate of the proposed metasurface is reduced by 47 times, which enables to greatly increase the allowable transferred power without violating the safety regulation and reducing the efficiency.

Evolving galactic dynamos and fits to the reversing rotation measures in the halo of NGC 4631

Rotation measure (RM) synthesis maps of NGC 4631 show remarkable sign reversals with distance from the minor axis in the northern halo of the galaxy on kpc scales. To explain this new phenomenon, we solve the dynamo equations under the assumption of scale invariance and search for rotating logarithmic spiral solutions. Solutions for velocity fields representing accretion onto the disk, outflow from the disk, and rotation-only in the disk are found that produce RM with reversing signs viewed edge-on. Model RM maps are created for a variety of input parameters using a Faraday screen and are scaled to the same amplitude as the observational maps. Residual images are then made and compared to find a best fit. Solutions for rotation-only, i.e. relative to a pattern uniform rotation, did in general, not fit the observations of NGC 4631 well. However, outflow models did provide a reasonable fit to the magnetic field. The best results for the specific region that was modelled in the northern halo are found with accretion. Since there is abundant evidence for both winds and accretion in NGC 4631, this modelling technique has the potential to distinguish between the dominant flows in galaxies.

A trio of heteroclinic bifurcations arising from a model of spatially-extended Rock–Paper–Scissors

One of the simplest examples of a robust heteroclinic cycle involves three saddle equilibria: each one is unstable to the next in turn, and connections from one to the next occur within invariant subspaces. Such a situation can be described by a third-order ordinary differential equation (ODE), and typical trajectories approach each equilibrium point in turn, spending progressively longer to cycle around the three points but never stopping. This cycle has been invoked as a model of cyclic competition between populations adopting three strategies, characterised as Rock, Paper and Scissors. When spatial distribution and mobility of the populations is taken into account, waves of Rock can invade regions of Scissors, only to be invaded by Paper in turn. The dynamics is described by a set of partial differential equations (PDEs) that has travelling wave (in one dimension) and spiral (in two dimensions) solutions. In this paper, we explore how the robust heteroclinic cycle in the ODE manifests itself in the PDEs. Taking the wavespeed as a parameter, and moving into a travelling frame, the PDEs reduce to a sixth-order set of ODEs, in which travelling waves are created in a Hopf bifurcation and are destroyed in three different heteroclinic bifurcations, depending on parameters, as the travelling wave approaches the heteroclinic cycle. We explore the three different heteroclinic bifurcations, none of which have been observed in the context of robust heteroclinic cycles previously. These results are an important step towards a full understanding of the spiral patterns found in two dimensions, with possible application to travelling waves and spirals in other population dynamics models.

Analytic solutions for calcium ion fertilisation waves on the surface of eggs.

The evolution of calcium fertilisation waves on the cortex of amphibian eggs can be described by a nonlinear reaction-diffusion process on the surface of a sphere. Here, we use the nonclassical symmetry technique to find an exact analytic solution that describes the evolution of the calcium concentration. The solutions presented compare well with published experimental results. The analytic solution can be used to give insight into the processes governing the fertilisation wave, such as the flow of calcium ions from the sperm entry point. By finding a spiral solution to an approximate equation linearised near saturation, we also demonstrate how solutions with other properties may be constructed using this technique.

Analytical Cartesian solutions of the multi-component Camassa-Holm equations

Here, we give the existence of analytical Cartesian solutions of the multi-component Camassa-Holm (MCCH) equations. Such solutions can be explicitly expressed, in which the velocity function is given by u = b (t) + A(t)x and no extra constraint on the dimension N is required. The advantage of our method is that we turn the process of analytically solving MCCH equations into algebraically constructing the suitable matrix A(t). As the applications, we obtain some interesting results: 1) If u is a linear transformation on x ∈ ℝN , then p takes a quadratic form of x. 2) If A = f(t)I + D with DT = −D, we obtain the spiral solutions. When N = 2, the solution can be used to describe “breather-type” oscillating motions of upper free surfaces. 3) If \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A = ({\ extstyle{{{{\\mathop \\alpha \\limits^. }_i}} \\over {{\\alpha _i}}}})N \ imes N,$$\\end{document}, we obtain the generalized elliptically symmetric solutions. When N = 2, the solution can be used to describe the drifting phenomena of the shallow water flow.

Relativistic jets and event horizons for a kinetic spiral solution

Relativistic jets and event horizons for a kinetic spiral solution

Aircraft planar trajectories in crosswind navigation: some hypergeometric solutions

We study the planar motion of a self-propelled object against a withstanding flow, such as wind or sea current, whose speed is capable of perturbing the trajectory of the object, heading it off its path. Seven cases are dealt with, corresponding to fairly general wind types and all leading to nonlinear ordinary differential equations for the object motion. The first two cases concern motions described in Cartesian coordinates. They are followed by motions under a cyclonic wind, analysed in a polar reference frame; the wind is modelled as a logarithmic or hyperbolic spiral and closed-form solutions are obtained by means of the Gauss hypergeometric function and the Lerch transcendent. Further wind instances, treated in a Cartesian or in a polar reference system, require some special functions, such as the Lambert function, while finding the motion time law leads to a trinomial equation, in the most interesting situation of a mobile object that is faster than the wind. In the last case of a horizontal hyperbolic wind, time law and trajectory are computed in a polar reference frame by means of the Lambert function.

Turbulent Helicity in the Atmospheric Boundary Layer

We consider the assumption postulated by Deusebio and Lindborg (J Fluid Mech 755:654–671, 2014) that the helicity injected into the Ekman boundary layer undergoes a cascade, with preservation of its sign (right- or alternatively left-handedness), which is a signature of the system rotation, from large to small scales, down to the Kolmogorov microscale of turbulence. At the same time, recent direct field measurements of turbulent helicity in the steppe region of southern Russia near Tsimlyansk Reservoir show the opposite sign of helicity from that expected. A possible explanation for this phenomenon may be the joint action of different scales of atmospheric flows within the boundary layer, including the sea-breeze circulation over the test site. In this regard, we consider a superposition of the classic Ekman spiral solution and Prandtl’s jet-like slope-wind profile to describe the planetary boundary-layer wind structure. The latter solution mimics a hydrostatic shallow breeze circulation over a non-uniformly heated surface. A 180°-wide sector on the hodograph plane exists, within which the relative orientation of the Ekman and Prandtl velocity profiles favours the left rotation with height of the resulting wind velocity vector in the lowermost part of the boundary layer. This explains the negative (left-handed) helicity cascade toward small-scale turbulent motions, which agrees with the direct field measurements of turbulent helicity in Tsimlyansk. A simple turbulent relaxation model is proposed that explains the measured positive values of the relatively minor contribution to turbulent helicity from the vertical components of velocity and vorticity.