Nonlinear Vortex Dynamics Research Articles

The nonlinear behavior of generic tail fins for small wind turbines

This paper describes analysis and measurements of the yaw response of tail fins for small wind turbines. It is based on an extension of unsteady slender body theory (USBT) to cover non-slender fins and high angles of incidence, both of which make the theory nonlinear. We provide three main additions to the substantial literature on linearized USBT for tail fins. First, USBT is extended to high angles by modeling the nonlinear vortex dynamics. Second, the restriction to slender bodies is removed by modeling the chordwise load variation. Third, we consider the effect of time-varying wind speed. The extended theory is compared to wind tunnel measurements of the yaw behavior of delta, elliptical, and rectangular tail fins without a rotor and nacelle. The fins were released from initial yaw angles of −40° and −80°; the latter is of sufficient magnitude to show the importance of the nonlinear yaw dynamics. Generally good agreement was found between the theory and measurements, and the theory was shown to be more accurate than a “polar” or quasi-steady model which uses only the lift and drag of a delta planform. Of the three planforms, the rectangular one showed the lowest accuracy in terms of frequency but the damping was accurately predicted. Overall, the results demonstrate the importance of nonlinearity in the response of a yawing tail fin, particularly for the higher aspect ratio fins at large yaw angles.

Magnetic vortex polarity reversal induced gyrotropic motion spectrum splitting in a ferromagnetic disk

We investigate the gyrotropic motion of the magnetic vortex core in a chain of a few micron-sized Permalloy disks by electrical resistance measurement with amplitude-modulated magnetic field. We observe a distinctive splitting of the resistance peak due to the resonant vortex-core motion under heightened radio frequency (RF) magnetic field excitation. Our micromagnetic simulation identifies the splitting of the resonant peak as an outcome of vortex polarity reversal under substantial RF amplitudes. This study enhances our understanding of nonlinear magnetic vortex dynamics amidst large RF amplitudes and proposes a potential pathway for spintronic neural computing thanks to their unique and controllable magnetization dynamics.

Generation Model of a Spatially Limited Vortex in a Stratified Unstable Atmosphere

This paper presents a new model for the generation of axisymmetric concentrated vortices. Thesolution of a nonlinear equation for internal gravity waves in an unstable stratified atmosphere is obtained andanalyzed within the framework of ideal hydrodynamics. The corresponding expressions describing thedependences on the radius for the radial and vertical velocity components in the inner and outer regions ofthe vortex include combinations of Bessel functions and modified Bessel functions. The proposed new nonlinearanalytical model makes it possible to study the structure and nonlinear dynamics of vortices in theradial and vertical regions. The vortex is limited in height. The maximum vertical velocity component isreached at a certain height. Below this height, radial flows converge towards the axis, and above it, an outflowoccurs. The resulting instability in the stratified atmosphere leads to an increase in the radial and verticalvelocity components according to the hyperbolic sine law, which turns into exponential growth. The characteristicgrowth time is determined by the inverse growth rate of the instability. The formation of vortices withfinite velocity components, which increase with time, is analyzed. The radial structure of the azimuthalvelocity is determined by the structure of the initial perturbation and can change with height. The maximumrotation is reached at a certain height. The growth of the azimuth velocity occurs according to a super-exponentiallaw.

Topological Charge Switch in Active Multi‐Core Fibers

AbstractTopological properties can make light field remarkably robust to various external perturbations. The ability to control and change on demand topological characteristics of light paves the way to new interesting physical phenomena and applications. Here, numerical modelling design of the device based on active multi‐core fiber that can change topological charge of the state of light has been proposed. The concept is based on the nonlinear dynamics of optical vortices in active multi‐core optical fiber with linearly coupled cores, saturated gain, and constant linear losses. Results demonstrate that the proposed system can provide change of the topological charge of vortices.

Nonlinear dynamics and dissipation of a curvilinear vortex driven by a strong time-dependent Meissner current

We report numerical simulations of large-amplitude oscillations of a trapped vortex line under a strong ac magnetic field $H(t)=H\sin\omega t$ parallel to the surface. The power dissipated by an oscillating vortex segment driven by the surface ac Meissner currents was calculated by taking into account the nonlinear vortex line tension, vortex mass and a nonlinear Larkin-Ovchinnikov (LO) viscous drag coefficient $\eta(v)$. We show that the LO decrease of $\eta(v)$ with the vortex velocity $v$ can radically change the field dependence of the surface resistance $R_i(H)$ caused by trapped vortices. At low frequencies $R_i(H) $ exhibits a conventional increases with $H$, but as $\omega$ increases, the surface resistance becomes a nonmonotonic function of $H$ which decreases with $H$ at higher fields. The effects of frequency, pin spacing and the mean free path $l_i $ on the field dependence of $R_{i}(H) $ were calculated. It is shown that, as the surface gets dirtier and $l_i$ decreases, the anomalous drop of $ R_{i}(H) $ with $H$ shifts to lower fields which can be much smaller than the lower critical magnetic field. Our numerical simulations also show that the LO decrease of $\eta(v)$ with $v$ can cause a vortex bending instability at high field amplitudes and frequencies, giving rise to the formation of dynamic kinks along the vortex. Measurements of $R_i(H)$ caused by sparse vortices trapped perpendicular to the surface can offer opportunities to investigate an extreme nonlinear dynamics of vortices driven by strong current densities up to the depairing limit at low temperatures. The behavior of $R_i(H)$ which can be tuned by varying the rf frequency or concentration of nonmagnetic impurities is not masked by strong heating effects characteristic of dc or pulse transport measurements.

Coevolving advances in animal flight and aerial robotics

Our understanding of animal flight has inspired the design of new aerial robots with more effective flight capacities through the process of biomimetics and bioinspiration. The aerodynamic origin of the elevated performance of flying animals remains, however, poorly understood. In this themed issue, animal flight research and aerial robot development coalesce to offer a broader perspective on the current advances and future directions in these coevolving fields of research. Together, four reviews summarize and 14 reports contribute to our understanding of low Reynolds number flight. This area of applied aerodynamics research is challenging to dissect due to the complicated flow phenomena that include laminar–turbulent flow transition, laminar separation bubbles, delayed stall and nonlinear vortex dynamics. Our mechanistic understanding of low Reynolds number flight has perhaps been advanced most by the development of dynamically scaled robot models and new specialized wind tunnel facilities: in particular, the tiltable Lund flight tunnel for animal migration research and the recently developed AFAR hypobaric wind tunnel for high-altitude animal flight studies. These world-class facilities are now complemented with a specialized low Reynolds number wind tunnel for studying the effect of turbulence on animal and robot flight in much greater detail than previously possible. This is particular timely, because the study of flight in extremely laminar versus turbulent flow opens a new frontier in our understanding of animal flight. Advancing this new area will offer inspiration for developing more efficient high-altitude aerial robots and removes roadblocks for aerial robots operating in turbulent urban environments.

Zero-Bias Shapiro Steps in Asymmetric Pinning Nanolandscapes

The coherent nonlinear dynamics of Abrikosov vortices in asymmetric pinning nanolandscapes is studied by theoretical modeling and combined microwave and dc electrical resistance measurements. The problem is considered on the basis of a single-vortex Langevin equation within the framework of a stochastic model of anisotropic pinning. When the distance over which Abrikosov vortices are driven during one half ac cycle coincides with one or a multiple of the nanostructure period, Shapiro steps appear in the current-voltage curves (CVCs) as a general feature of systems whose evolution in time can be described in terms of a particle moving in a periodic potential under combined dc and ac stimuli. While a dc voltage appears in response to the ac drive, the addition of a dc bias allows one to diminish the rectified voltage and eventually to change its sign when the extrinsic dc bias-induced asymmetry of the pinning potential starts to dominate the intrinsic one. This rectified negative voltage in the CVCs becomes apparent as \emph{zero-bias} Shapiro steps, which are theoretically predicted and experimentally observed for the first time.

Quantum Turbulence: Vortex Bundle Collapse and Kolmogorov Spectrum

The statement of problem is motivated by the idea of modeling the classical turbulence with a set of chaotic quantized vortex filaments in superfluids. Among various arguments supporting the idea of quasi-classic behavior of quantum turbulence, the strongest, probably, is the k dependence of the spectra of energy, \(E(k)\propto k^{-5/3}\) obtained in numerical simulations and experiments. At the same time, the mechanism of classical vs. quantum turbulence is not clarified and the source of the \(k^{-5/3}\) dependence is unclear. In this work, we concentrated on the nonuniform vortex bundles. This choice is related to the actively discussed question concerning a role of collapses in the vortex dynamics in formation of turbulent spectra. We demonstrate that the nonuniform vortex bundles, which appear in result of nonlinear vortex dynamics, generates an energy spectrum which is close to the Kolmogorov dependence \(\propto k^{-5/3}\).

IUTAM Symposium on Vortex Dynamics: Formation, Structure and Function, 10–14 March 2013, Fukuoka, Japan

IUTAM Symposium on Vortex Dynamics: Formation, Structure and Function, 10–14 March 2013, Fukuoka, Japan

Nonlinear magnetic vortex dynamics in a circular nanodot excited by spin-polarized current.

We investigate analytically and numerically nonlinear vortex spin torque oscillator dynamics in a circular magnetic nanodot induced by a spin-polarized current perpendicular to the dot plane. We use a generalized nonlinear Thiele equation including spin-torque term by Slonczewski for describing the nanosize vortex core transient and steady orbit motions and analyze nonlinear contributions to all forces in this equation. Blue shift of the nano-oscillator frequency increasing the current is explained by a combination of the exchange, magnetostatic, and Zeeman energy contributions to the frequency nonlinear coefficient. Applicability and limitations of the standard nonlinear nano-oscillator model are discussed.

DC to AC converter on Abrikosov vortices in a washboard pinning potential

The nonlinear dynamics of Abrikosov vortices in a cosine dc-biased washboard pinning potential at nonzero temperature is theoretically investigated. The problem is treated relying upon the exact solution of the Langevin equation for non-interacting vortices by using the Fokker-Planck method combined with the scalar continued fractions technique. The time variation of the local mean vortex velocity v(t) is calculated. The time voltage E(t) ∝ v(t) is predicted to oscillate with a dc current-dependent frequency and a tunable pulse shape. Formulas for v(t) are discussed as functions of dc transport current and temperature, in a wide range of the corresponding dimensionless parameters. The derived expressions can be adapted for a number of physical applications utilizing the overdamped motion of a Brownian particle in a tilted cosine potential, e.g., the resistively shunted Josephson junction model.

Double resonance response in nonlinear magnetic vortex dynamics

We present experimental evidences for the dynamical bifurcation behavior of ac-driven magnetic vortex core gyration in a ferromagnetic disk. The dynamical bifurcation, i.e., appearance and disappearance of two stable dynamical states in the vortex gyration, occurring as the amplitude of the driving Oersted field increases to ${B}_{Oe}>{B}_{Oe}^{cr}$, manifests itself in a double resonance response in the dependence of homodyne the dc-voltage signal on the frequency $\ensuremath{\omega}$ of the applied microwave current. We find that the frequency range $\ensuremath{\delta}\ensuremath{\omega}$ between the two resonance features strongly increases with the excitation power. Our analysis based on the model of a low dissipative nonlinear oscillator subject to a resonant alternating force is in good agreement with the experimental results. This allows us to determine quantitatively key parameters of magnetic vortex dynamics, i.e., the critical value of the driving Oersted field ${B}_{Oe}^{cr}$ for nonlinear dynamics to occur, the resonant frequency, and the quality factor as well as damping of the magnetic vortex gyration.

Noise-Assisted Microwave Up-conversion by Vortices in Thin-Film Superconductors with a dc-Biased Washboard Pinning Potential

So far the main theoretical basis for understanding and optimization of the microwave properties of vortices in type II superconductors has been relying upon the Coffey–Clem (CC) approach for the linear impedance at nonzero temperature. However, the CC model does not account for the non-linear response and the possibility to control it by changing the value of the dc transport current in a superconducting sample. For this reason, we have exactly solved the Langevin equation for the two-dimensional non-linear vortex dynamics in a dc bias-tilted cosine pinning potential in the presence of an ac current of arbitrary amplitude and frequency ω and have, thereby, substantially generalized the CC results. In this work we analyze the behavior of the non-linear response on kω-frequency in a wide range of dc and ac current densities, ω, and temperature. The kω-response is found to depend strongly on all these parameters, as exemplified for the third-harmonic (k=3) transformation coefficient Z 3. The parametric window for the most enhanced up-conversion is presented. The predicted effects can be experimentally verified in thin-film superconductors with some pinning potential of the washboard type.

Parametric oscillator based on nonlinear vortex dynamics in low-resistance magnetic tunnel junctions

Radiofrequency vortex spin-transfer oscillators based on magnetic tunnel junctions with very low resistance area product were investigated. A high power of excitations has been obtained characterized by a power spectral density containing a very sharp peak at the fundamental frequency and a series of harmonics. The observed behaviour is ascribed to the combined effect of spin transfer torque and Oersted-Amp\`ere field generated by the large applied dc-current. We furthermore show that the synchronization of a vortex oscillation by applying a ac bias current is mostly efficient when the external frequency is twice the oscillator fundamental frequency. This result is interpreted in terms of a parametric oscillator.

Magnetic vortex core reversal by excitation of spin waves

Micron-sized magnetic platelets in the flux-closed vortex state are characterized by an in-plane curling magnetization and a nanometer-sized perpendicularly magnetized vortex core. Having the simplest non-trivial configuration, these objects are of general interest to micromagnetics and may offer new routes for spintronics applications. Essential progress in the understanding of nonlinear vortex dynamics was achieved when low-field core toggling by excitation of the gyrotropic eigenmode at sub-GHz frequencies was established. At frequencies more than an order of magnitude higher vortex state structures possess spin wave eigenmodes arising from the magneto-static interaction. Here we demonstrate experimentally that the unidirectional vortex core reversal process also occurs when such azimuthal modes are excited. These results are confirmed by micromagnetic simulations, which clearly show the selection rules for this novel reversal mechanism. Our analysis reveals that for spin-wave excitation the concept of a critical velocity as the switching condition has to be modified.

Non-linear dynamics of semi-dilute polydisperse polymer solutions in microfluidics: effects of flow geometry

The non-linear dynamics of a semi-dilute (c/c* = 15) polydisperse polyethylene oxide (PEO) solution in microfluidics are studied experimentally using benchmark contraction–expansion flow geometries with three contraction–expansion ratios (4:1:4, 8:1:8 and 16:1:16) and two narrow channel lengths (L c/D h = 53 and 5.3, where L c is the length of the narrow channel and D h is its hydraulic diameter). Complex flows over a range of elasticity numbers (El), Weissenberg numbers (Wi) and Reynolds numbers (Re) are characterized using micro-particle image velocimetry ( $\upmu$ -PIV) and pressure drop measurements. The evolution of vortex formation and dynamics has been visualized through a step-flow-rate experiment. Various flow dynamics regimes have been quantified and are presented in a Wi–Re diagram. The experimental results reveal that the contraction ratio can result in qualitatively different vortex dynamics of semi-dilute polymer solutions in microfluidics, whereas the length of the narrow channel merely affects the dynamics at a quantitative level. A single elasticity number, if defined by the size of the narrow channel, is not sufficient to account for the effects of contraction ratio on the non-linear vortex dynamics.

Spreading of superfluid vorticity clouds in normal-fluid turbulence

In this paper, we formulate a self-consistent model of thermal superfluid dynamics. By solving it, we analyse the problem of superfluid vorticity cloud propagation in normal-fluid turbulence. We show that superfluid cloud expansion is driven by pattern-forming superfluid vortex instabilities taking place in the interface layer between the cloud's bulk and the outer undisturbed normal-fluid turbulence. The radius of the cloud increases linearly with time. Mutual friction transfers energy from the normal-fluid turbulence to the superfluid cloud, whilst damping the smallest normal-fluid turbulence motions. This damping action is much weaker than viscous dissipation effects in a corresponding pure normal-fluid turbulence. The energy spectrum of superfluid turbulence presents the k−3 scaling that characterizes the spiral superfluid vorticity patterns of normal vortex tube–superfluid vortex interactions. The corresponding k−2 pressure spectrum signifies the singular nature of superfluid vorticity. These two scalings coincide in wavenumber space with the Kolmogorov regime in the normal-fluid turbulence. We compute a fractal dimension df ≈ 1.652 for superfluid vorticity. Due to simpler underlying superfluid vortex dynamics in relation to the strongly nonlinear classical vortex dynamics, this fractal dimension is smaller than the corresponding dimension of vortex tube centrelines in classical turbulence.

Nonlinear vortex dynamics in an inhomogeneous magnetized plasma with a sheared velocity field

The long term evolution of a vortex chain generated by the nonlinear development of the Kelvin–Helmholtz instability in a magnetized two-fluid plasma with a sheared velocity field is investigated in a two-dimensional configuration. The different roles played by the in-plane magnetic field component and by the density inhomogeneity are elucidated. This investigation is of interest in understanding the plasma dynamics that arises from the nonlinear competition of different plasma instabilities involving the interplay of large and small spatial scales and, in particular, for understanding the mixing mechanism between the solar wind and the Earth's magnetospheric plasmas that occurs at the Earth's magnetospheric flanks.

Nonlinear gyrotropic vortex dynamics in ferromagnetic dots

The quasistationary and transient (nanosecond) regimes of nonlinear vortex dynamics in a soft magnetic dot driven by an oscillating external field are studied. We derive a nonlinear dynamical system of equations for the vortex core position and phase, assuming that the main source of nonlinearity comes from the magnetostatic energy. In the stationary regime, we demonstrate the occurrence of a fold-over bifurcation and calculate analytically the resonant nonlinear vortex frequencies as a function of the amplitude and frequency of the applied driving field. In the transient regime, we show that the vortex core dynamics are described by an oscillating trajectory radius. The resulting dynamics contain multiple frequencies with amplitude decaying in time. Finally, we evaluate the ranges of the system parameters leading to a vortex core instability (core polarization reversal).

Nonlinear vortex dynamics in open nonequilibrium systems with bulk mass loss and a generation mechanism for tornadoes and typhoons

Based on a general model of nonlinear vortex dynamics in open thermodynamically nonequilibrium systems with bulk or surface mass losses, an analysis is presented of the mechanism of generation of violent atmospheric vortices (tornadoes, typhoons, cyclones) associated with the formation of deep cloud systems by intense condensation of water vapor from moist air cooled below the dew point. Simple particular solutions to the Navier-Stokes equations are found that describe both axisymmetric and nonaxisymmetric incompressible vortex motions involving radial and vertical flows with viscous dissipation vanishing identically everywhere except for a thin shear layer at the boundary of the condensation region. It is shown that the nonlinear convective and local Coriolis forces generated by radial inflow in the presence of a background vorticity due to a global Coriolis force (the Earth’s rotation) accelerate the solid-body rotation in the vortex core either exponentially or in a nonlinear regime of finite-time blow-up. Due to updrafts, such a vortex is characterized by a strong helicity. This mechanism explains a number of observed properties and characteristics of the structure and evolution of tornadoes and typhoons. Upper estimates are found for the kinetic energies of violent atmospheric vortices. It is shown that increase in rotational kinetic energy of atmospheric vortices with constant vortex-core radii is consistent with energy and momentum conservation, because radial inflow continually supplies the required amount of rotational kinetic energy drawn from the ambient atmosphere to an open system.