Formation and evolution of flapping and ballooning waves in magnetospheric plasma sheet

By adopting Lembége & Pellat’s 2D plasma-sheet model, we investigate the flankward flapping motion and Sunward ballooning propagation driven by an external source (e.g., magnetic reconnection) produced initially at the sheet center. Within the ideal MHD framework, we adopt the WKB approximation to obtain the Taylor–Goldstein equation of magnetic perturbations. Fourier spectral method and Runge–Kutta method are employed in numerical simulations, respectively, under the flapping and ballooning conditions. Studies expose that the magnetic shears in the sheet are responsible for the flapping waves, while the magnetic curvature and the plasma gradient are responsible for the ballooning waves. In addition, the flapping motion has three phases in its temporal development: fast damping phase, slow recovery phase, and quasi-stabilized phase; it is also characterized by two patterns in space: propagating wave pattern and standing wave pattern. Moreover, the ballooning modes are gradually damped toward the Earth, with a wavelength in a scale size of magnetic curvature or plasma inhomogeneity, only 1–7% of the flapping one; the envelops of the ballooning waves are similar to that of the observed bursty bulk flows moving toward the Earth.     

Numerical simulation of drift-resistive ballooning turbulence in the presence of the GA mode in the plasma edge tokamak

Drift-resistive ballooning turbulence is simulated numerically based on a quasi-three-dimensional computer code for solving nonlinear two-fluid MHD equations in the scrape-off layer plasma in a tokamak. It is shown that, when the toroidal geometry of the magnetic field is taken into account, additional (geodesic) flux terms associated with the first poloidal harmonic (∼sinθ) arise in the averaged equations for the momentum, density, and energy. Calculations show that the most important of these terms is the geodesic momentum flux (the Stringer-Windsor effect), which lowers the poloidal rotation velocity. It is also shown that accounting for the toroidal field geometry introduces experimentally observed, special low-frequency MHD harmonics—GA modes—in the Fourier spectra. GA modes are generated by the Reynolds turbulent force and also by the gradient of the poloidally nonuniform turbulent heat flux. Turbulent particle and heat fluxes are obtained as functions of the poloidal coordinate and are found to show that, in a tokamak, there is a “ballooning effect” associated with their maximum in the weak magnetic field region. The dependence of the density, temperature, and pressure on the poloidal coordinate is presented, as well as the dependence of turbulent fluxes on the toroidal magnetic field.     

Stability of global magnetohydrodynamic modes in the L-5 compact torsatron

Global magnetohydrodynamic (MHD) modes in free-boundary plasma equilibrium configurations of the L-5 six-period compact torsatron (project of the Prokhorov General Physics Institute, Russian Academy of Sciences; G. M. Batanov et al., Plasma Dev. Operat. 11, 161 (2003)) have been investigated using the CAS3D code. It is shown that global internal ideal MHD modes can be found reliably only in Mercier unstable plasmas. In all cases under study, they are intrinsically ballooning in nature, i.e., the disturbance at the inner side of the torus is significantly lower in value than that at the outer side. Besides internal modes localized near low-order rational magnetic surfaces, external modes that are localized near the free plasma boundary and are not associated with any rational magnetic surface inside the plasma (the so-called peeling modes) have been found. In contrast to a tokamak with the same aspect ratio, in which pressure-gradient edge-localized modes are ballooning in nature, the peeling modes in the 3D currentless magnetic configuration investigated in the present work are quasi-cylindrical in the flux coordinate system. The problem of plasma boundary destruction under the action of magnetic fields generated by plasma currents is also briefly discussed.     

Fluid mechanic constraints on spider ballooning

Summary Early investigations point to ballooning as an important mechanism for the dispersal of spiders. Most studies, however, have focused on collecting and interpreting observations with the aim of establishing the statistical characteristics of the dispersed population and their relation to biotic factors. With few exceptions, important physical factors that mechanically constrain the ballooning activity have been neglected or simply ignored. Especially important are various fluid mechanics phenomena that affect the initiation and maintenance of the ballooning activity. By reference to a simple mechanical model that simulates the essential drag characteristics of the ballooning spider-filament system, a region is defined, in terms of the relevant physical parameters, within which the ballooning activity can be initiated. Extension of the model allows a numerical investigation of the influence of vertical wind oscillations on the velocities and trajectories of ballooning spider-filament systems. Results are presented and discussed to illustrate the relative importance to ballooning of drag on the spider's body and on the silk filament to which it is attached.     

Ballooning propensity of canopy and understorey spiders in a mature temperate hardwood forest

1. Spiders frequently disperse and colonise habitats through ballooning, a passive aerial dispersal process. Ballooning is pre‐eminent in open habitat spider communities and its propensity can be modulated by habitat conditions and availability, and by life‐history traits such as body size, degree of specialisation, and feeding behaviour. 2. Using spiders from the canopy and understorey of a north‐temperate hardwood forest as a model system, our main objectives were to detect if foliage spiders of a mature forest disperse through ballooning, and identify life‐history traits that influence ballooning propensity. 3. Our results demonstrate that foliage spiders living in the canopy and understorey of a mature forest do balloon, and in some cases have very high ballooning propensities similarly to open field spiders. Species level models showed that small body size had a strong positive effect on ballooning for juveniles of species with large‐bodied adults, while individuals of small‐bodied species initiated ballooning regardless of size, habitat or development stage. A generalised linear mixed model indicated that small size web‐building spiders from the Retro Tibial Apophysis (RTA) and Orbicularia clades had the highest propensity for foliage spiders of this north‐temperate hardwood forest. 4. In conclusion, we provide the first demonstration that forest spiders can have high ballooning propensities and that ballooning propensity is negatively affected by body size and positively affected by the prominent use of silk to catch prey. However, spiders originating from the canopy and understorey of a north‐temperate hardwood forest did not differ in their ballooning propensities.     

Numerical simulation for heat transfer in tissues during thermal therapy

In this paper, a mathematical model describing the process of heat transfer in biological tissues for different coordinate system during thermal therapy by electromagnetic radiation is studied. The boundary value problem governing this process has been solved using Galerkin’s method taking B-polynomial as basis function. The system of ordinary differential equation in unknown time variable, thus obtained, is solved by homotopy perturbation method. The effect of thermal conductivity, antenna power constant, surface temperature, and blood perfusion rate on temperature for different coordinates are discussed. It has been observed that the process is faster in spherical symmetric coordinates in comparison to axisymmetric coordinate and faster in axisymmetric in comparison to Cartesian coordinate.     

Magnetic geometry, plasma profiles, and stability

The history of the stability of short wavelength modes, such as MHD instabilities and drift waves, has been a long and tortuous one as increasingly realistic representations of the equilibrium magnetic geometry have been introduced. Early work began with simple slab or cylindrical models where plasma profiles and magnetic shear were seen to play key roles. Then the effects of toroidal geometry, in particular the constraints imposed by periodicity in the presence of magnetic shear, provided a challenge for theory, which was met by the ballooning transformation. More recently the limitations on the conventional ballooning theory arising from effects such as toroidal rotation shear, low magnetic shear, and the presence of the plasma edge have been recognized. These have led in turn to modifications and extensions of this theory. These developments have produced a continuously changing view of the stability of the “universal” drift wave, for example. After a survey of this background, we describe more recent work of relevance to currently important topics, such as transport barriers characterized by the presence of strong rotation shear and low magnetic shear and the edge localized modes that occur in H-mode. Published in Russian in Fizika Plazmy, 2006, Vol. 32, No. 7, pp. 588–598. Based on an invited talk given at the 11th European Fusion Theory Meeting, Aix-en-Provence, France, September 2005. The text was submitted by the author in English.     

Geometric properties of plasma equilibrium near a given magnetic surface

In order to describe plasma equilibrium near a given magnetic surface, it is sufficient to specify the shape of the surface, the distribution of the magnetic field strength on it, and two profile coefficients (the derivatives of the plasma pressure and current). Geometrically, this means that all the basis vectors of the flux coordinate system should be determined on the magnetic surface. Expressions for these vectors in an invariant basis are obtained. The maximum possible value of the pressure profile coefficient consistent with equilibrium is described by a universal geometric relationship that expresses the limiting value of the torsion of the magnetic field line on the magnetic surface as a function of the curvature of the surface. The relationships obtained are used to show that the stability of a system with closed magnetic field lines is governed by perturbations of the anti-Mercier type.     

Influence of an External AC Electric Field on Plasma Turbulence in the Tokamak Near-Wall Layer

Braginskii reduced equations of two-fluid hydrodynamics are modified to take into account the presence of an external ac electric field localized in the tokamak near-wall layer. Numerical simulations show that, after reaching certain amplitude, such a field oscillating with the frequency ω ≈ ω_GAM is capable of suppressing turbulent processes. The turbulence suppression mechanism consists in a sharp decrease in the growth rate of drift-resistive ballooning instability due to the appearance of additional nonlinear terms related to the external field in the equation for the vorticity.     

Particle confinement in axisymmetric poloidal magnetic field configurations with zeros of B: Methodological note

Collisionless particle confinement in axisymmetric configurations with magnetic field nulls is analyzed. The existence of an invariant of motion—the generalized azimuthal momentum—makes it possible to determine in which of the spatial regions separated by magnetic separatrices passing through the magnetic null lines the particle occurs after it leaves the vicinity of a magnetic null line. In particular, it is possible to formulate a sufficient condition for the particle not to escape through the separatrix from the confinement region to the external region. In the configuration under analysis, the particles can be lost from a separatrix layer with a thickness on the order of the Larmor radius because of the nonconservation of the magnetic moment μ. In this case, the variations in μ are easier to describe in a coordinate system associated with the magnetic surfaces. An analysis is made of the applicability of expressions for the single-pass change Δμ in the magnetic moment that were obtained in different magnetic field models for a confinement system with a divertor (such that there is a circular null line).     

Ballooning dispersal using silk: world fauna, phylogenies, genetics and models

Aerial dispersal using silk ('ballooning') has evolved in spiders (Araneae), spider mites (Acari) and in the larvae of moths (Lepidoptera). Since the 17th century, over 500 observations of ballooning behaviours have been published, yet there is an absence of any evolutionary synthesis of these data. In this paper the literature is reviewed, extensively documenting the known world fauna that balloon and the principal behaviours involved. This knowledge is then incorporated into the current evolutionary phylogenies to examine how ballooning might have arisen. Whilst it is possible that ballooning co-evolved with silk and emerged as early as the Devonian (410-355 mya), it is arguably more likely that ballooning evolved in parallel with deciduous trees, herbaceous annuals and grasses in the Cretaceous (135-65 mya). During this period, temporal (e.g. bud burst, chlorophyll thresholds) and spatial (e.g. herbivory, trampling) heterogeneities in habitat structuring predominated and intensified into the Cenozoic (65 mya to the present). It is hypothesized that from the ancestral launch mechanism known as 'suspended ballooning', widely used by individuals in plant canopies, 'tip-toe' and 'rearing' take-off behaviours were strongly selected for as habitats changed. It is contended that ballooning behaviour in all three orders can be described as a mixed Evolutionary Stable Strategy. This comprises individual bet-hedging due to habitat unpredictability, giving an underlying randomness to individual ballooning, with adjustments to the individual ballooning probability being conferred by more predictable habitat changes or colonization strategies. Finally, current methods used to study ballooning, including modelling and genetic research, are illustrated and an indication of future prospects given.     

Nonlinear ineraction of an annular electron beam with azimuthal surface waves

A nonlinear theory is developed that describes the interaction between an annular electron beam and an electromagnetic surface wave propagating strictly transverse to a constant external axial magnetic field in a cylindrical metal waveguide partially filled with a cold plasma. It is shown theoretically that surface waves with positive azimuthal mode numbers can be efficiently excited by an electron beam moving in the gap between the plasma column and the metal waveguide wall. Numerical simulations prove that, by applying a constant external electric field oriented along the waveguide radius, it is possible to increase the amplitude at which the surface waves saturate during the beam instability. The full set of equations consisting of the waveenvelope equation, the equation for the wave phase, and the equations of motion for the beam electrons is solved numerically in order to construct the phase diagrams of the beam electrons in momentum space and to determine their positions in coordinate space (in the radial variable-azimuthal angle plane).     

Analytic examples of force-free toroidal MHD equilibria

Analytically described toroidal (axisymmetric and three-dimensional) equilibrium magnetic field configurations with a “flat” current density, j=λB (λ = const), are proposed. Such configurations are superpositions of several force-free two-dimensional configurations with plane, axial, or helical coordinate symmetry. Each of them is generated by an exact partial solution to the corresponding Grad-Shafranov equation. A variety of toroidal configurations thus obtained allows one to model topological changes of magnetic surfaces, such as magnetic axis splitting (doublets) in axisymmetric equilibrium configurations and the appearance and interaction of magnetic islands and ergodic lines in three-dimensional configurations.     

Tearing instability in a tokamak with a noncircular cross section

Using a straight-column model to describe tokamak plasma with a noncircular cross section, it is shown how to (i) find the boundary of tearing instability from the condition of existence of a magnetohydrodynamic plasma equilibrium different from that of a straight cylinder by solving a two-dimensional linear boundary-value problem with a second-order equation with respect to the flux coordinate and (ii) find the spatial structure of the tearing mode and the corresponding effective Δ' when there is only one resonance magnetic surface in the plasma for a given axial wavenumber by solving some kind of a boundary-value problem for the perturbation. The proposed approach is illustrated by numerical calculations for the case of an elliptical cross section as an example.     

Rate of excitation propagation in a reduced Hodgkins-Huxley model. III. Integrodifferential equations]

The Hodgkin - Huxley system of equations is reduced to single integral-differential equation in neglection of slow variables dynamics. Two limiting cases of fast and slow sodium activation processes are considered. The first case leads to a nonlinear differential equation for the potential, the second one - to an ordinary differential equation with a known source as a function of coordinate. Such a simplification is due to approximation of steady-state sodium activation variable with the help of Heviside function. The validity of this approximation is discussed; the corresponding error is estimated by calculation of the second approximation for the source function.     

Analysis and Optimization of Pulse Dynamics for Magnetic Stimulation

Magnetic stimulation is a standard tool in brain research and has found important clinical applications in neurology, psychiatry, and rehabilitation. Whereas coil designs and the spatial field properties have been intensively studied in the literature, the temporal dynamics of the field has received less attention. Typically, the magnetic field waveform is determined by available device circuit topologies rather than by consideration of what is optimal for neural stimulation. This paper analyzes and optimizes the waveform dynamics using a nonlinear model of a mammalian axon. The optimization objective was to minimize the pulse energy loss. The energy loss drives power consumption and heating, which are the dominating limitations of magnetic stimulation. The optimization approach is based on a hybrid global-local method. Different coordinate systems for describing the continuous waveforms in a limited parameter space are defined for numerical stability. The optimization results suggest that there are waveforms with substantially higher efficiency than that of traditional pulse shapes. One class of optimal pulses is analyzed further. Although the coil voltage profile of these waveforms is almost rectangular, the corresponding current shape presents distinctive characteristics, such as a slow low-amplitude first phase which precedes the main pulse and reduces the losses. Representatives of this class of waveforms corresponding to different maximum voltages are linked by a nonlinear transformation. The main phase, however, scales with time only. As with conventional magnetic stimulation pulses, briefer pulses result in lower energy loss but require higher coil voltage than longer pulses.     

Stabilization of ballooning modes by nonparaxial cells

An analysis is made of the effect of high-curvature stabilizing nonparaxial elements (cells) on the MHD plasma stability in open confinement systems and in confinement systems with closed magnetic field lines. It is shown that the population of particles trapped in such cells has a stabilizing effect not only on convective (flute) modes but also on ballooning modes, which govern the maximum possible β value. In the kinetic approach, which distinguishes between the effects of trapped and passing particles, the maximum possible β values consistent with stability can be much higher than those predicted by the MHD model.     

Optimal coordination and control of posture and locomotion.

This paper presents a theoretical model of stability and coordination of posture and locomotion, together with algorithms for continuous-time quadratic optimization of motion control. Explicit solutions to the Hamilton-Jacobi equation for optimal control of rigid-body motion are obtained by solving an algebraic matrix equation. The stability is investigated with Lyapunov function theory, and it is shown that global asymptotic stability holds. It is also shown how optimal control and adaptive control may act in concert in the case of unknown or uncertain system parameters. The solution describes motion strategies of minimum effort and variance. The proposed optimal control is formulated to be suitable as a posture and stance model for experimental validation and verification. The combination of adaptive and optimal control makes this algorithm a candidate for coordination and control of functional neuromuscular stimulation as well as of prostheses.     

Toroidal mirror system

A study is made of a toroidally linked mirror system with a zero rotational transform and a three-dimensional magnetic field that ensures good confinement of charged particles. A toroidally linked magnetic mirror configuration at low plasma pressures is calculated by numerically solving the isometry equation for the magnetic field to second order in the small parameter of the paraxial approximation. The calculations carried out with the VMEC code for a particular linked magnetic mirror configuration demonstrate the possibility of achieving good confinement of drifting particles. The calculated results show that it is, in principle, possible to link mirror cells into a toroidal configuration capable of providing plasma confinement at a tokamak level.     

Optimal coordination and control of posture and movements

This paper presents a theoretical model of stability and coordination of posture and locomotion, together with algorithms for continuous-time quadratic optimization of motion control. Explicit solutions to the Hamilton–Jacobi equation for optimal control of rigid-body motion are obtained by solving an algebraic matrix equation. The stability is investigated with Lyapunov function theory and it is shown that global asymptotic stability holds. It is also shown how optimal control and adaptive control may act in concert in the case of unknown or uncertain system parameters. The solution describes motion strategies of minimum effort and variance. The proposed optimal control is formulated to be suitable as a posture and movement model for experimental validation and verification. The combination of adaptive and optimal control makes this algorithm a candidate for coordination and control of functional neuromuscular stimulation as well as of prostheses. Validation examples with experimental data are provided.