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.     

Islands and ergodicity of the plasma current in toroidal magnetic confinement systems

Magnetic and current structures arising due to resonant perturbations of an equilibrium current-carrying magnetic configuration are analyzed using the Hamiltonian formalism. Special attention is paid to axisymmetric tokamak and pinch configurations. It is shown that, due to the very different dependences of the magnetic and current rotational transforms on the plasma pressure, the resonances (islands) of the magnetic field may not coincide with those of the current. The perturbed force-free equilibrium of a cylindrical pinch in which the field and current islands overlap is analyzed. The long-lived ribbon structures observed in the JET tokamak are explained as a manifestation of a force-free magneto-current island.     

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.     

Plasma equilibrium in axisymmetric open divertor configurations

The equilibrium of a plasma with isotropic pressure in a periodic divertor configuration with a poloidal magnetic field is calculated. The issue of how the plasma equilibrium changes as the parameter β≡8πp/B ² increases is considered for a fairly representative class of pressure profiles p(ψ) (where ψ is the flux coordinate). It is shown that the plasma can be in equilibrium up to β values (in terms of the vacuum magnetic field at the divertor axis) on the order of unity.     

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.     

Role of the plasma interaction with magnetic field in the “missing power” problem

It is shown that, at rapid changes of the heating power, the magnetically confined equilibrium plasma almost completely absorbs the injected energy, so that its only small part goes to the magnetic field. The result is obtained within the standard MHD theory with use of exact consequences of the force-balance equations in toroidal geometry. It is assumed that, when heated, the plasma evolves from one equilibrium state to another with the magnetic field frozen-in. Another constraint is the conservation of the toroidal magnetic flux in the plasma-wall vacuum gap. It is shown that the plasma interaction with magnetic field (which is traditionally neglected in the analysis of heat transport in tokamaks and stellarators) is the natural mechanism of fast redistribution of energy in the plasma, observed in some experiments in these devices at switchon of powerful heat sources.     

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.     

Concluding remarks: historical perspective and the future of island biogeography theory

MacArthur and Wilson’s equilibrium theory revolutionized the field of island biogeography and, to a large degree, ecology as well. The theory, which quickly became the ruling paradigm of island biogeography, has changed little over the past three decades. It has not kept pace with relevant theory and our growing appreciation for the complexity of nature, especially with empirical findings that species diversity on many islands: 1) is not in equilibrium; 2) is influenced by differences in speciation, colonization, and extinction among taxa; and 3) is influenced by differences among islands in characteristics other than area and isolation. The discipline of biogeography, itself, is in a state of disequilibrium. We may again be about to witness another paradigm shift, which will see the replacement of MacArthur and Wilson’s theory. Wherever this shift may take us, we are confident that the next generation of biogeographers will still look to islands for insights into the forces that shape biological diversity.     

Magnetic surfaces in an axisymmetric torus

A method is developed for specifying the boundary equilibrium magnetic surface in an axially symmetric torus by using the absolute values of the magnetic field B = B _s (θ) and the gradient of the poloidal flux ||?Ψ| = |?Ψ| _s (θ) in a special flux coordinate system. By setting two surface constants (e.g., the safety factor q and dp/dΨ) and matching the absolute values of the magnetic field and the flux gradient on a closed magnetic surface, it is possible to find all equilibrium magnetic functions (including n · ? ln B and the local shear s) and all constants (including the toroidal current J and the shear dμ/dΨ) on this surface. Such a non-traditional formulation of the boundary conditions in solving the stability problem in an axisymmetric torus allows one to impose intentional conditions on plasma confinement and MHD stability at the periphery of the system.     

Island biogeography theory explains the genetic diversity of a fragmented rock ptarmigan (Lagopus muta) population

The island biogeography theory is one of the major theories in ecology, and its applicability to natural systems is well documented. The core model of the theory, the equilibrium model of island biogeography, predicts that species diversity on an island is positively related to the size of the island, but negatively related by the island's distance to the mainland. In recent years, ecologists have begun to apply this model when investigating genetic diversity, arguing that genetic and species diversity might be influenced by similar ecological processes. However, most studies have focused on oceanic islands, but knowledge on how the theory applies to islands located on the mainland (e.g., mountain islands, forest islands) is scarce. In this study, we examined how the size and degree of isolation of mountain islands would affect the genetic diversity of an alpine bird, the rock ptarmigan (Lagopus muta). Within our study area, we defined the largest contiguous mountain area as the mainland, while smaller mountains surrounding the mainland were defined as islands. We found that the observed heterozygosity (H_o) was significantly higher, and the inbreeding coefficient (F_is) significantly lower, on the mainland compared to islands. There was a positive significant relationship between the unbiased expected heterozygosity (H_n.b.) and island size (log km²), but a negative significant relationship between H_o and the cost distance to the mainland. Our results are consistent with the equilibrium model of island biogeography and show that the model is well suited for investigating genetic diversity among islands, but also on islands located on the mainland.     

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).     

Predicting community structure in snakes on Eastern Nearctic islands using ecological neutral theory and phylogenetic methods

Predicting species presence and richness on islands is important for understanding the origins of communities and how likely it is that species will disperse and resist extinction. The equilibrium theory of island biogeography (ETIB) and, as a simple model of sampling abundances, the unified neutral theory of biodiversity (UNTB), predict that in situations where mainland to island migration is high, species-abundance relationships explain the presence of taxa on islands. Thus, more abundant mainland species should have a higher probability of occurring on adjacent islands. In contrast to UNTB, if certain groups have traits that permit them to disperse to islands better than other taxa, then phylogeny may be more predictive of which taxa will occur on islands. Taking surveys of 54 island snake communities in the Eastern Nearctic along with mainland communities that have abundance data for each species, we use phylogenetic assembly methods and UNTB estimates to predict island communities. Species richness is predicted by island area, whereas turnover from the mainland to island communities is random with respect to phylogeny. Community structure appears to be ecologically neutral and abundance on the mainland is the best predictor of presence on islands. With regard to young and proximate islands, where allopatric or cladogenetic speciation is not a factor, we find that simple neutral models following UNTB and ETIB predict the structure of island communities.     

How island size and isolation affect bee and wasp ensembles on small tropical islands: a case study from Kepulauan Seribu,Indonesia

Aim To study the effects of isolation and size of small tropical islands on species assemblages of bees (superfamily Apoidea) and wasps (superfamily Vespoidea). Location Twenty islands in the Kepulauan Seribu Archipelago off the coast of west Java, Indonesia. The size of surveyed islands ranges between 0.75 and 41.32 ha; their distance from the coast of Java varies between 3 and 62 km. Methods Field work was conducted from February to May 2005. Bees and wasps were caught with a sweep net during sampling units of 15 min, continuing until four consecutive samples revealed no new species. Total species richness was quantified by the estimators Chao 2, first‐order jackknife and Michaelis–Menten. The software binmatnest was used to test for nestedness of species assemblages. Similarities of species composition between islands were quantified by Sørensen’s similarity index. Results Eighty‐two species were recorded on the 20 surveyed islands. Species richness declined with increasing isolation of islands from the source area, Java. Although the size of the largest island exceeded that of the smallest island by a factor of almost 60, island size only very weakly affected species richness of bees; no effect of island size was found for wasps. Mean body size of species decreased with increasing island isolation. Nestedness of island faunas was only weakly developed. Species composition of both superfamilies was affected by island isolation, but not by island size. Main conclusions While the species–isolation relationship on the very small islands of Kepulauan Seribu followed the prediction of MacArthur and Wilson’s equilibrium theory, the absence of a species–area relationship indicated a weak ‘small‐island effect’, at least in wasps. The combination of an only weakly developed pattern of nested species subsets, the shift in species compositions and the decline of mean body size with increasing island isolation from the source area indicates that biotic interactions and different species traits contribute to the shaping of communities of bees and wasps within the archipelago. The potential of biotic interactions for generating distribution patterns of species within the archipelago is also emphasized by the observed restriction of some species with apparently high dispersal abilities to outer islands.     

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.     

Kinetic theory of the positive column of a low-pressure discharge in a transverse magnetic field

The influence of a transverse magnetic field on the characteristics of the positive column of a planar low-pressure discharge is studied theoretically. The motion of magnetized electrons is described in the framework of a continuous-medium model, while the ion motion in the ambipolar electric field is described by means of a kinetic equation. Using mathematical transformations, the problem is reduced to a secondorder ordinary differential equation, from which the spatial distribution of the potential is found in an analytic form. The spatial distributions of the plasma density, mean plasma velocity, and electric potential are calculated, the ion velocity distribution function at the plasma boundary is found, and the electron energy as a function of the magnetic field is determined. It is shown that, as the magnetic field rises, the electron energy increases, the distributions of the plasma density and mean plasma velocity become asymmetric, the maximum of the plasma density is displaced in the direction of the Ampère force, and the ion flux in this direction becomes substantially larger than the counter-directed ion flux.     

Local surface equilibrium equations for currentless magnetic configurations

Invariant local surface equilibrium equations are derived that interrelate the absolute value of the magnetic field B, the absolute value of the gradient of the magnetic flux |?Φ|, the local shear s, and the plasma pressure on nested equilibrium magnetic surfaces in currentless configurations. Examples of applying these equations to analysis of symmetric and isodynamic equilibria are considered.     

Oceanic island biogeography through the lens of the general dynamic model: assessment and prospect

The general dynamic model of oceanic island biogeography (GDM) has added a new dimension to theoretical island biogeography in recognizing that geological processes are key drivers of the evolutionary processes of diversification and extinction within remote islands. It provides a dynamic and essentially non‐equilibrium framework generating novel predictions for emergent diversity properties of oceanic islands and archipelagos. Its publication in 2008 coincided with, and spurred on, renewed attention to the dynamics of remote islands. We review progress, both in testing the GDM's predictions and in developing and enhancing ecological–evolutionary understanding of oceanic island systems through the lens of the GDM. In particular, we focus on four main themes: (i) macroecological tests using a space‐for‐time rationale; (ii) extensions of theory to islands following different patterns of ontogeny; (iii) the implications of GDM dynamics for lineage diversification and trait evolution; and (iv) the potential for downscaling GDM dynamics to local‐scale ecological patterns and processes within islands. We also consider the implications of the GDM for understanding patterns of non‐native species diversity. We demonstrate the vitality of the field of island biogeography by identifying a range of potentially productive lines for future research.     

Island,archipelago and taxon effects: mixed models as a means of dealing with the imperfect design of nature's experiments

A major aim of island biogeography has been to describe general patterns of species richness across islands and to identify the processes responsible. Data are often collected across many islands; with larger datasets providing increased statistical power and more accurate parameter estimates. However, there is often structure in observational data, violating an assumption of linear models that each datum is independent. In island biogeography this structure may take the form of an island, archipelago or taxon being represented by multiple data points. We survey recent papers in this field and find that these forms of non‐independence are a common feature. Most authors addressed this problem by conducting separate analyses for each archipelago, taxon or combination of the two, but a better tool for dealing with non‐independence and structure in data, the mixed model, already exists. We demonstrate the advantages of a mixed model approach by applying it to a well‐known dataset that spans 134 observations of single island endemic (SIE) richness across 39 islands, four archipelagos and four taxa. Taking island area and age into account, SIE richness varies substantially more among archipelagos than it does among islands or taxa. We find that SIE richness rises with island age on the Azores and Galapagos, while on the Canaries and Hawaii SIE richness initially rises with age but later declines on older islands. Our analyses demonstrate three advantages to island biogeography of applying a mixed modelling approach: 1) structure in the data is controlled for; 2) the variance among islands, archipelagos and taxa is estimated; 3) all the data can be included in a single model, making it possible to test whether trends are general across all archipelagos and taxa or are idiosyncratic.     

Formation of electrode sheaths during the acceleration of a magnetized plasma

A study is made of the motion of a plasma with a frozen-in magnetic field along the electrode surfaces in the direction transverse to the magnetic field. A one-dimensional problem of an electrode sheath is formulated in which all of the quantities depend only on the coordinate orthogonal to the electrode surface. Viscous plasma heating, plasma cooling via heat conduction, and other kinetic effects are taken into consideration. Account is also taken of the effect of plasma acceleration and of the related current that is transverse to the electrode surfaces and, due to the Hall effect, carries the magnetic flux away from the cathode and toward the anode. Solving the one-dimensional problem with a constant electric current and constant magnetic field shows that, in a sheath that forms near the cathode, the solution becomes self-similar, the plasma mass grows linearly, and the electron magnetization parameter remains unchanged. It is found that the anode sheath cannot be described in the magnetohydrodynamic approximation, according to which the plasma density in the sheath rapidly vanishes, while the current through the sheath remains constant. This difficulty can be overcome by incorporating some of the nonhydrodynamic effects (primarily, electron dispersion), thereby making the problem physically correct. Solving the problem numerically shows that a decrease in the plasma density in the anode sheath due to the Hall effect gives rise to additional significant plasma acceleration.