Tokamak magnetic field lines described by simple maps

The magnetic field line structure in a tokamak can be obtained by direct numerical integration of the field line equations. However, this is a lengthy procedure and the analysis of the solution may be very time-consuming. Otherwise we can use simple two-dimensional, area-preserving maps, obtained either by approximations of the magnetic field line equations, or from dynamical considerations. These maps can be quickly iterated, furnishing solutions that mirror the ones obtained from direct numerical integration, and which are useful when long-term studies of field line behavior are necessary (e.g. in diffusion calculations). In this work we focus on a set of simple tokamak maps for which these advantages are specially pronounced.

Integrable maps with non-trivial topology: application to divertor configurations

We explore a method for constructing two-dimensional area-preserving, integrable maps associated with Hamiltonian systems, with a given set of fixed points and given invariant curves. The method is used to find an integrable Poincare map for the field lines in a large aspect ratio tokamak with a poloidal single-null divertor. The divertor field is a superposition of a magnetohydrodynamic equilibrium with an arbitrarily chosen safety factor profile, with a wire carrying an electric current to create an X-point. This integrable map is perturbed by an impulsive perturbation that describes non-axisymmetric magnetic resonances at the plasma edge. The non-integrable perturbed map is applied to study the structure of the open field lines in the scrape-off layer, reproducing the main transport features obtained by integrating numerically the magnetic field line equations, such as the connection lengths and magnetic footprints on the divertor plate.

Divertor map with freedom of geometry and safety factor profile

An explicit, area-preserving and integrable magnetic field line map for a single-null divertor tokamak is obtained using a trajectory integration method to represent equilibrium magnetic surfaces. The magnetic surfaces obtained from the map are capable of fitting different geometries with freely specified position of the X-point, by varying free model parameters. The safety factor profile of the map is independent of the geometric parameters and can also be chosen arbitrarily. The divertor integrable map is composed of a nonintegrable map that simulates the effect of external symmetry-breaking resonances, so as to generate a chaotic region near the separatrix passing through the X-point. The composed field line map is used to analyze escape patterns (the connection length distribution and magnetic footprints on the divertor plate) for two equilibrium configurations with different magnetic shear profiles at the plasma edge.

Chaotic transport in reversed shear tokamaks

For tokamak models using simplified geometries and reversed shear plasma profiles, we have numerically investigated how the onset of Lagrangian chaos at the plasma edge may affect the plasma confinement in two distinct but closely related problems. Firstly, we have considered the motion of particles in drift waves in the presence of an equilibrium radial electric field with shear. We have shown that the radial particle transport caused by this motion is selective in phase space, being determined by the resonant drift waves and depending on the parameters of both the resonant waves and the electric field profile. Moreover, we have shown that an additional transport barrier may be created at the plasma edge by increasing the electric field. In the second place, we have studied escape patterns and magnetic footprints of chaotic magnetic field lines in the region near a tokamak wall, when there are resonant modes due to the action of an ergodic magnetic limiter. A non-monotonic safety factor profile has been used in the analysis of field line topology in a region of negative magnetic shear. We have observed that, if internal modes are perturbed, the distributions of field line connection lengths and magnetic footprints exhibit spatially localized escape channels. For typical physical parameters of a fusion plasma, the two Lagrangian chaotic processes considered in this work can be effective in usual conditions so as to influence plasma confinement. The reversed shear effects discussed in this work may also contribute to evaluate the transport barrier relevance in advanced confinement scenarios in future tokamak experiments.

Nontwist symplectic maps in tokamaks

We review symplectic nontwist maps that we have introduced to describe Lagrangian transport properties in magnetically confined plasmas in tokamaks. These nontwist maps are suitable to describe the formation and destruction of transport barriers in the shearless region (i.e., near the curve where the twist condition does not hold). The maps can be used to investigate two kinds of problems in plasmas with non-monotonic field profiles: the first is the chaotic magnetic field line transport in plasmas with external resonant perturbations. The second problem is the chaotic particle drift motion caused by electrostatic drift waves. The presented analytical maps, derived from plasma models with equilibrium field profiles and control parameters that are commonly measured in plasma discharges, can be used to investigate long-term transport properties. (C) 2011 Elsevier B.V. All rights reserved.

Analytical solutions for Tokamak equilibria with reversed toroidal current

In tokamaks, an advanced plasma confinement regime has been investigated with a central hollow electric current with negative density which gives rise to non-nested magnetic surfaces. We present analytical solutions for the magnetohydrodynamic equilibria of this regime in terms of non-orthogonal toroidal polar coordinates. These solutions are obtained for large aspect ratio tokamaks and they are valid for any kind of reversed hollow current density profiles. The zero order solution of the poloidal magnetic flux function describes nested toroidal magnetic surfaces with a magnetic axis displaced due to the toroidal geometry. The first order correction introduces a poloidal field asymmetry and, consequently, magnetic islands arise around the zero order surface with null poloidal magnetic flux gradient. An analytic expression for the magnetic island width is deduced in terms of the equilibrium parameters. We give examples of the equilibrium plasma profiles and islands obtained for a class of current density profile. (C) 2011 American Institute of Physics. [doi: 10.1063/1.3624551]

Kinetic ion effect on geodesic acoustic Alfven modes in tokamaks

Using a quasitoroidal set of coordinates with coaxial circular magnetic surfaces, the Vlasov equation is solved for collisionless plasmas, and the dielectric tensor is found for large aspect ratio tokamaks in a low frequency band. Taking into account q-profile and charge separation parallel electric field, it is found that the Alfven wave continuum is deformed by ion geodesic effects producing continuum minimum at the rational magnetic surfaces. Low frequency geodesic ion induced Alfven waves are found below the continuum minimum where collisionless damping has a gap for Maxwell distribution. In kinetic approach, the ion thermal motion defines the geodesic effect but the mode frequency is strongly corrected due to parallel motion of electrons.

Escape patterns of chaotic magnetic field lines in a tokamak with reversed magnetic shear and an ergodic limiter

The existence of a reversed magnetic shear in tokamaks improves the plasma confinement through the formation of internal transport barriers that reduce radial particle and heat transport. However, the transport poloidal profile is much influenced by the presence of chaotic magnetic field lines at the plasma edge caused by external perturbations. Contrary to many expectations, it has been observed that such a chaotic region does not uniformize heat and particle deposition on the inner tokamak wall. The deposition is characterized instead by structured patterns called magnetic footprints, here investigated for a nonmonotonic analytical plasma equilibrium perturbed by an ergodic limiter. The magnetic footprints appear due to the underlying mathematical skeleton of chaotic magnetic field lines determined by the manifold tangles. For the investigated edge safety factor ranges, these effects on the wall are associated with the field line stickiness and escape channels due to internal island chains near the flux surfaces. Comparisons between magnetic footprints and escape basins from different equilibrium and ergodic limiter characteristic parameters show that highly concentrated magnetic footprints can be avoided by properly choosing these parameters. (c) 2008 American Institute of Physics.

Kinetic effect on low frequency Alfven continuum in tokamaks

Using a quasi-toroidal set of coordinates in plasmas with coaxial circular magnetic surfaces, the Vlasov equation is solved, and dielectric tensor is found for large aspect ratio tokamaks in a low frequency band. Taking into account the q-profile and drift effects, Alfven wave continuum deformation by geodesic effects is analyzed. It is shown that the Alfven continuum has a minimum defined by the ion thermal velocity at the rational magnetic surfaces q(s)=-M/N, where M and N are the poloidal and toroidal mode numbers, respectively, and the parallel wave number is zero. Low frequency global Alfven waves are found below the continuum minimum. In hot ion plasmas, the geodesic term changes sign, provoking some deformation of Alfven velocity by a factor (1+q(2))(-1/2), and the continuum minimum disappears. (C) 2008 American Institute of Physics.

Patient matching in treatment for alcohol dependence: is the null hypothesis still alive and well?

In the light of Project MATCH, is it reasonable to accept the null hypothesis that there are no clinically signi® cant matching effects between patient characteristics and cognitive± behaviour therapy (CBT), motivational enhancement therapy (MET) and Twelve-Step facilitation therapy (TSF)? The Project MATCH investigators considered the null hypothesis but preferred the alternative hypothesis that further analysis may reveal combinations of patient and therapist characteristics that show more substantial matching effects than any of the variables that they have examined to date.1

A null mutation in the inflammation-associated S100 protein S100A8 causes early resorption of the mouse embryo

S100A8 (also known as CP10 or MRP8) was the first member of the S100 family of calcium-binding proteins shown to be chemotactic for myeloid cells. The gene is expressed together with its dimerization partner S100A9 during myelopoiesis in the fetal liver and in adult bone marrow as well as in mature granulocytes. In this paper we show that S100A8 mRNA is expressed without S100A9 mRNA between 6.5 and 8.5 days postcoitum within fetal cells infiltrating the deciduum in the vicinity of the ectoplacental cone. Targeted disruption of the S100A8 gene caused rapid and synchronous embryo resorption by day 9.5 of development in 100% of homozygous null embryos. Until this point there was no evidence of developmental delay in S100A8(-/-) embryos and decidualization was normal. The results of PCR genotyping around 7.5-8.5 days postcoitum suggest that the null embryos are infiltrated with maternal cells before overt signs of resorption. This work is the first evidence for nonredundant function of a member of the S100 gene family and implies a role in prevention of maternal rejection of the implanting embryo. The S100A8 null provides a new model for studying fetal-maternal interactions during implantation.

Accentuated osteoclastic response to parathyroid hormone undermines bone mass acquisition in osteonectin-null mice

Matricellular proteins play a unique role in the skeleton as regulators of bone remodeling, and the matricellular protein osteonectin (SPARC, BM-40) is the most abundant non-collagenous protein in bone In. the absence of osteonectin, mice develop progressive low turnover osteopenia, particularly affecting trabecular bone. Polymorphisms in a regulatory region of the osteonectin gene are associated with bone mass in a subset of idiopathic osteoporosis patients, and these polymorphisms likely regulate osteonectin expression. Thus it is important to determine how osteonectin gene dosage affects skeletal function. Moreover, intermittent administration of parathyroid hormone (PTH) (1-34) is the only anabolic therapy approved for the treatment of osteoporosis, and it is critical to understand how modulators of bone remodeling, such as osteonectin, affect skeletal response to anabolic agents. In this study, 10 week old female wild type, osteonectin-haploinsufficient, and osteonectin-null mice (C57Bl/6 genetic background) were given 80 mu g/kg body weight/day PTH(1-34) for 4 weeks. Osteonectin gene dosage had a profound effect on bone microarchitecture. The connectivity density of trabecular bone in osteonectin-haploinsufficient mice was substantially decreased compared with that of wild type mice, suggesting compromised mechanical properties. Whereas mice of each genotype had a similar osteoblastic response to PTH treatment, the osteoclastic response was accentuated in osteonectin-haploinsufficient and osteonectin-null mice. Eroded surface and osteoclast number were significantly higher in PTH-treated osteonectin-null mice, as was endosteal area. In vitro studies confirmed that PTH induced the formation of more osteoclast-like cells in marrow from osteonectin-null mice compared with wild type. PTH treated osteonectin-null bone marrow cells expressed more RANKL mRNA compared with wild type. However, the ratio of RANKL:OPG mRNA was somewhat lower in PTH treated osteonectin-null cultures. Increased expression of RANKL in response to PTH could contribute to the accentuated osteoclastic response in osteonectin(-/-) mice, but other mechanisms are also likely to be involved. The molecular mechanisms by which PTH elicits bone anabolic vs. bone catabolic effects remain poorly understood. Our results imply that osteonectin levels may play a role in modulating the balance of bone formation and resorption in response to PTH. (c) 2008 Elsevier Inc. All rights reserved.

Familial corticosteroid-binding globulin deficiency due to a noval null mutation: Association with fatigue and relative hypotension

Corticosteroid-binding globulin is a 383-amino acid glycoprotein that serves a hormone transport role and may have functions related to the stress response and inflammation. We describe a 39-member Italian-Australian family with a novel complete loss of function (null) mutation of the corticosteroid-binding globulin gene. A second, previously described, mutation (Lyon) segregated independently in the same kindred. The novel exon 2 mutation led to a premature termination codon corresponding to residue -12 of the procorticosteroid-binding globulin molecule (c.121G->A). Among 32 family members there were 3 null homozygotes, 19 null heterozygotes, 2 compound heterozygotes, 3 Lyon heterozygotes, and 5 individuals without corticosteroid-binding globulin mutations. Plasma immunoreactive corticosteroid-binding globulin was undetectable in null homozygotes, and mean corticosteroid-binding globulin levels were reduced by approximately 50% at 18.7 ± 1.3 µg/ml (reference range, 30–52 µg/ml) in null heterozygotes. Morning total plasma cortisol levels were less than 1.8 µg/dl in homozygotes and were positively correlated to the plasma corticosteroid-binding globulin level in heterozygotes. Homozygotes and heterozygote null mutation subjects had a high prevalence of hypotension and fatigue. Among 19 adults with the null mutation, the systolic blood pressure z-score was 12.1 ± 3.5; 11 of 19 subjects (54%) had a systolic blood pressure below the third percentile. The mean diastolic blood pressure z-score was 18.1 ± 3.4; 8 of 19 subjects (42%) had a diastolic blood pressure z-score below 10. Idiopathic chronic fatigue was present in 12 of 14 adult null heterozygote subjects (86%) and in 2 of 3 null homozygotes. Five cases met the Centers for Disease Control criteria for chronic fatigue syndrome. Fatigue questionnaires revealed scores of 25.1 ± 2.5 in 18 adults with the mutation vs. 4.2 ± 1.5 in 23 healthy controls (P < 0.0001). Compound heterozygosity for both mutations resulted in plasma cortisol levels comparable to those in null homozygotes. Abnormal corticosteroid-binding globulin concentrations or binding affinity may lead to the misdiagnosis of isolated ACTH deficiency. The mechanism of the association between fatigue and relative hypotension is not established by these studies. As idiopathic fatigue disorders are associated with relatively low plasma cortisol, abnormalities of corticosteroid-binding globulin may be pathogenic.

Characterizations of power indices based on null player free winning coalitions

In this paper, we characterize two power indices introduced in [1] using two different modifications of the monotonicity property first stated by [2]. The sets of properties are easily comparable among them and with previous characterizations of other power indices.

Repairs of databases with null values

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