Plasma equilibrium in 3D magnetic confinement systems and soliton theory

Single-valued conformal flux (magnetic) coordinates can always be introduced on arbitrary toroidal magnetic surfaces. It is shown how such coordinates can be obtained by transforming Boozer magnetic coordinates on the surfaces. The metrics is substantially simplified and the coordinate grid is orthogonalized at the expense of a more complicated representation of the magnetic field in conformal flux coordinates. This in turn makes it possible to introduce complex angular flux coordinates on any toroidal magnetic surface and to develop efficient methods for a complex analysis of the geometry of equilibrium magnetic surfaces. The complex analysis reveals how the plasma equilibrium problem is related to soliton theory. Magnetic surfaces of constant mean curvature are considered to exemplify this relationship.     

Isodynamic axisymmetric equilibrium near the magnetic axis

Plasma equilibrium near the magnetic axis of an axisymmetric toroidal magnetic confinement system is described in orthogonal flux coordinates. For the case of a constant current density in the vicinity of the axis and magnetic surfaces with nearly circular cross sections, expressions for the poloidal and toroidal magnetic field components are obtained in these coordinates by using expansion in the reciprocal of the aspect ratio. These expressions allow one to easily derive relationships between quantities in an isodynamic equilibrium, in which the absolute value of the magnetic field is constant along the magnetic surface (Palumbo’s configuration).     

Pseudosymmetry near a magnetic surface in a plasma confinement system

The possibility is demonstrated of finding vacuum equilibrium magnetic configurations with an exactly pseudosymmetric nonparaxial boundary magnetic surface in the vicinity of which the pseudosymmetry condition is satisfied approximately. Equations are derived for calculating the boundary surface from a prescribed particular dependence of the magnetic field strength in special magnetic flux coordinates. In calculations, magnetic coordinates serve as ordinary angular coordinates, while their “magnetic” character is specified by additional integral conditions. As an example, a “tubular” orthogonal magnetic surface is calculated analytically.     

Relationship between the shape of equilibrium magnetic surfaces and the magnetic field strength

A local analysis of the magnetic field near an equilibrium magnetic surface shows that there is generally no relationship between the magnetic field strength and the shape of the surface. However, the relationship exists under additional requirements such as the absence of the toroidal current, symmetry conservation, and the conservation of the magnetic field strength distribution on the nearest surface. An equilibrium magnetic surface can be calculated by specifying three functions of two angular variables—the magnetic field strength, the periodic component of the magnetic potential, and the mean curvature of the surface.     

Local synthesis of 3D magnetic surfaces

Local synthesis of nested 3D toroidal magnetic surfaces is carried out on the basis of the general theory of surfaces by using magnetic coordinates (generally unknown a priori). An equilibrium magnetic surface is calculated by specifying two functions on the surface (the absolute value of the magnetic field and the distance to the nearest magnetic surface) and three parameters (the rotational transform, pressure gradient, and poloidal current). The choice of the parameters is restricted by the requirement that the surface should be closed toroidally. A method of synthesis of a closed magnetic surface is proposed when two functions—the absolute value of the magnetic field and the major radius—are specified. A set of harmonics of a new type of poloidally preudosymmetric configuration (a toroidal mirror with a large mirror ratio and small rotational transform) is obtained.     

Generation of Alfvén waves by a plasma inhomogeneity moving in the Earth’s magnetosphere

The generation of an Alfvén wave by an azimuthally drifting cloud of high-energy particles injected in the Earth’s magnetosphere is studied analytically. In contrast to the previous studies where the generation mechanisms associated with the resonant wave-particle interaction were considered, a nonresonant mechanism is investigated in which the wave is excited by the alternating current produced by drifting particles. It is shown that, at a point with a given azimuthal coordinate, a poloidally polarized wave, in which the magnetic field lines oscillate predominantly in the radial direction, is excited immediately after the passage of the particle cloud through this point. As the cloud moves away from that point, the wave polarization becomes toroidal (the magnetic field lines oscillate predominantly in the azimuthal direction). The azimuthal wavenumber m is defined as the ratio of the wave eigenfrequency to the angular velocity of the cloud (the drift velocity of the particles). It is shown that the amplitudes of the waves so generated are close to those obtained under realistic assumptions about the density and energy of the particles.     

On the possibility of reflection of Alfvén waves in a curvilinear magnetic field

The propagation of Alfvén waves in a plasma immersed in a curvilinear magnetic field is investigated by using a 2D model. The waves are described by a 1D equation that formally coincides with the equation for the case of a quasi-uniform straight magnetic field with a modified Alfvén velocity that takes into account the longitudinal dependence of the Lame coefficients. It is shown that toroidal and poloidal Alfvén modes depend differently on the magnetic-field geometry. In the case of a 2D plane-parallel configuration of the magnetic field, poloidal modes are efficiently reflected from regions where the magnetic field lines sharply converge or diverge. This effect can result in the formation of open-field-line Alfvén quasi-resonators.     

Pressure profiles of a plasma confined by the magnetic field of a ring current

Pressure profiles p(ψ) marginal with respect to convective instability in a toroidal tubular plasma confined by the magnetic field of an internal levitated ring current and external ring currents are studied as functions of the shape of the magnetic separatrix. Configurations are found in which the maximum plasma pressure in a finite-width layer near the plasma boundary decreases by two orders of magnitude at the expense of artificially raising the effective length (characterized by the integral ∮dl/B) of the magnetic field lines near the separatrix surface. It is shown that, in the case of a straight cylindrical tubular plasma, which is the limiting case of a toroidal configuration with an arbitrarily large aspect ratio, the sufficient condition for the plasma to be MHD stable against both convective and kink perturbations is satisfied for local values β≤0.4. __________ Translated from Fizika Plazmy, Vol. 26, No. 6, 2000, pp. 519–528. Original Russian Text Copyright ￠ 2000 by Popovich, Shafranov.     

A geometric model for initial orientation errors in pigeon navigation

All mobile animals respond to gradients in signals in their environment, such as light, sound, odours and magnetic and electric fields, but it remains controversial how they might use these signals to navigate over long distances. The Earth's surface is essentially two-dimensional, so two stimuli are needed to act as coordinates for navigation. However, no environmental fields are known to be simple enough to act as perpendicular coordinates on a two-dimensional grid. Here, we propose a model for navigation in which we assume that an animal has a simplified ‘cognitive map’ in which environmental stimuli act as perpendicular coordinates. We then investigate how systematic deviation of the contour lines of the environmental signals from a simple orthogonal arrangement can cause errors in position determination and lead to systematic patterns of directional errors in initial homing directions taken by pigeons. The model reproduces patterns of initial orientation errors seen in previously collected data from homing pigeons, predicts that errors should increase with distance from the loft, and provides a basis for efforts to identify further sources of orientation errors made by homing pigeons.     

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.     

MHD plasma stabilization in a chain of axisymmetric adiabatic open mirror cells

Conditions for convective plasma instability in a chain of axisymmetric adiabatic mirror cells with different signs of magnetic field curvature are analyzed. The boundaries of the region that can be occupied by a stable hollow plasma in a system of two connected cells—a nonparaxial simple mirror cell and a semicusp—are determined, as well as the interval of allowed values of the ratio between the pressures in the cells. Because of the large magnetic field curvature in the component cells, the safety factor that is achieved at both—external and internal—plasma boundaries in accordance with the average min-B principle can be high. It is assumed that the unperturbed pressure in each cell is almost isotropic, in which case the mirror ratio should necessarily be large. A key role in the stability of the plasma is played by its compressibility. A comparison is made between the conditions for complete plasma stabilization against arbitrary perturbations and the conditions for stability of individual cells against the global mode. The stability of the cells against the global mode is sufficient, but not necessary, for stabilizing the chain. The analysis is done by using orthogonal coordinates associated with the unperturbed magnetic field (flux variables). Numerical simulations were carried out for nonparaxial cells from a certain three-parameter family.     

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.     

Model magnetic field configurations with islands

Helical perturbations of the tokamak magnetic field can give rise to magnetic islands in the vicinity of the rational magnetic surfaces at which the pitch of the magnetic field lines coincides with that of the perturbation. The widely known relationship between the magnetic island width and the perturbation amplitude is valid under the assumptions that the island width is small in comparison to the radius of the rational surface and that the perturbation amplitude is constant in the radial direction. The latter assumption indicates that the island width is small in comparison to the radial size of the region where the perturbation current is localized. The calculations carried out for four model magnetic field configurations show that the geometry of the magnetic islands depends on the extent to which the perturbation current is localize and that the width of the magnetic islands is smaller than that calculated from the familiar relationship. The larger the perturbation amplitude, the greater this difference: it may be as large as 25% for the strong perturbations arising during disruptions. The calculations are based on the solution of the geometric problem of constructing the lines of the magnetic field determined by the given distributions of the initial current and perturbation current; the equilibrium equation is not considered. The question of the direction of the perturbation current within the island relative to the direction of the initial unperturbed current is discussed. The perturbation current flowing in an island is directed opposite to the initial current with a radially decreasing density; for this reason, such an island can naturally be called a “negative” island. Together with the formation of negative islands, the formation of “positive” ones is also considered. The latter are shown to form under the following conditions: the perturbation current density should be higher than the density of the current that produces the unperturbed field and the perturbation current itself should be localized in a sufficiently narrow radial layer. The positive islands are smaller in size than negative ones.     

Geometric properties of magnetic field lines on toroidal magnetic surfaces in the context of plasma equilibrium

An analysis of plasma equilibrium in a magnetic confinement system includes studies of how the shape of the magnetic surfaces is distorted with varying magnitude and profile of the plasma pressure. Such studies allow one, in particular, to determine the maximum β value consistent with equilibrium, β_eq, i.e., the maximum plasma pressure above which the equilibrium in a confinement system under analysis is impossible. Since the magnetic field lines form magnetic surfaces, their global relationship with equilibrium is obvious. Here, special attention is paid to a local relationship between equilibrium and geometric properties of the magnetic field lines.     

Stability of a quasi-flute mode under the Bernstein-Kadomtsev condition in a toroidal confinement system

It is shown that the growth rate of the MHD instability in toroidal configurations is slower in a situation in which the Bernstein-Kadomtsev condition is satisfied while the Mercier stability criterion is not. Under the Bernstein-Kadomtsev condition, Alfvénic Mercier modes are not excited, but quasi-flute acoustic Mercier modes develop instead. In confinement systems with closed magnetic field lines, the Bernstein-Kadomtsev condition ensures MHD stability; however, a small rotational transform produced by magnetic perturbations can give rise to a quasi-flute acoustic instability whose growth rate is proportional to the perturbation amplitude, in which case the fastest growing oscillations are those with the shortest wavelengths.     

Closed formulae to determine the angular velocity of a body-segment based on 3D measurements.

This paper suggests a simple method to determine the global coordinates of the angular velocity and the angular acceleration of a body segment determined by the coordinates of minimum three markers. There are commonly used calculations for the angular quantities basing on the "hypothesis" of planar motion. The usage of approximate methods can result in quantitative and qualitative errors that may completely disort the reality. The method mentioned here is theoretically absolutely correct and can be well used for smoothing noisy data.     

Effect of density variations along ion drift trajectories on the poloidal and toroidal rotation velocities of a collisionless tokamak plasma

The effect of plasma density variations along ion drift trajectories on the ion velocity distribution function at a given point on a tokamak magnetic surface is studied. The observed distortion of the distribution function can be interpreted as a poloidal (or toroidal) plasma rotation that is additional to the neoclassical rotation. Due to this additional rotation, the velocity of the toroidal plasma rotation is different on the low-and high-field sides of the same magnetic surface. In the case of large ion density gradients, the poloidal rotation velocity on the same magnetic surface can have different signs at different poloidal angles.     

Role of the mean curvature in the geometry of magnetic confinement configurations

Examples are presented of how the geometric notion of the mean curvature is applied to the vector of a general magnetic field and to magnetic surfaces. It is shown that the mean curvature is related to the variation of the absolute value of the magnetic field along its lines. Magnetic surfaces of constant mean curvature are optimum for plasma confinement in multimirror open confinement systems and rippled tori.     

Advances in Geometric Morphometrics

Geometric morphometrics is the statistical analysis of form based on Cartesian landmark coordinates. After separating shape from overall size, position, and orientation of the landmark configurations, the resulting Procrustes shape coordinates can be used for statistical analysis. Kendall shape space, the mathematical space induced by the shape coordinates, is a metric space that can be approximated locally by a Euclidean tangent space. Thus, notions of distance (similarity) between shapes or of the length and direction of developmental and evolutionary trajectories can be meaningfully assessed in this space. Results of statistical techniques that preserve these convenient properties—such as principal component analysis, multivariate regression, or partial least squares analysis—can be visualized as actual shapes or shape deformations. The Procrustes distance between a shape and its relabeled reflection is a measure of bilateral asymmetry. Shape space can be extended to form space by augmenting the shape coordinates with the natural logarithm of Centroid Size, a measure of size in geometric morphometrics that is uncorrelated with shape for small isotropic landmark variation. The thin-plate spline interpolation function is the standard tool to compute deformation grids and 3D visualizations. It is also central to the estimation of missing landmarks and to the semilandmark algorithm, which permits to include outlines and surfaces in geometric morphometric analysis. The powerful visualization tools of geometric morphometrics and the typically large amount of shape variables give rise to a specific exploratory style of analysis, allowing the identification and quantification of previously unknown shape features.     

Interplay between intrinsic plasma rotation and magnetic island evolution in disruptive discharges

The behavior of the intrinsic toroidal rotation of the plasma column during the growth and eventual saturation of m/n = 2/1 magnetic islands, triggered by programmed density rise, has been carefully investigated in disruptive discharges in TCABR. The results show that, as the island starts to grow and rotate at a speed larger than that of the plasma column, the angular frequency of the intrinsic toroidal rotation increases and that of the island decreases, following the expectation of synchronization. As the island saturates at a large size, just before a major disruption, the angular speed of the intrinsic rotation decreases quite rapidly, even though the island keeps still rotating at a reduced speed. This decrease of the toroidal rotation is quite reproducible and can be considered as an indicative of disruption.