Magnetic compressibility and ion-temperature-gradient-driven microinstabilities in magnetically confined plasmas

The electromagnetic theory of the strongly driven ion-temperature-gradient (ITG) instability in magnetically confined toroidal plasmas is developed. Stabilizing and destabilizing effects are identified, and a critical $\beta_{e}$ (the ratio of the electron to magnetic pressure) for stabilization of the toroidal branch of the mode is calculated for magnetic equilibria independent of the coordinate along the magnetic field. Its scaling is $\beta_{e}\sim L_{Te}/R,$ where $L_{Te}$ is the characteristic electron temperature gradient length, and $R$ the major radius of the torus. We conjecture that a fast particle population can cause a similar stabilization due to its contribution to the equilibrium pressure gradient. For sheared equilibria, the boundary of marginal stability of the electromagnetic correction to the electrostatic mode is also given. For a general magnetic equilibrium, we find a critical length (for electromagnetic stabilization) of the extent of the unfavourable curvature along the magnetic field. This is a decreasing function of the local magnetic shear

Differential equations for the cuspoid canonical integrals

Differential equations satisfied by the cuspoid canonical integrals I_n(a) are obtained for arbitrary values of n≥2, where n−1 is the codimension of the singularity and a=(ɑ_1,ɑ_2,...,ɑ_(n−1)). A set of linear coupled ordinary differential equations is derived for each step in the sequence I_n(0,0,...,0,0) →I_n(0,0,...,0,ɑ_(n−1)) →I_n(0,0,...,ɑ_(n−2),ɑ_(n−1)) →...→I_n(0,ɑ_2,...,ɑ_(n−2),ɑ_(n−1)) →I_n(ɑ_1,ɑ_2,...,ɑ_n−2,ɑ_(n−1)). The initial conditions for a given step are obtained from the solutions of the previous step. As examples of the formalism, the differential equations for n=2 (fold), n=3 (cusp), n=4 (swallowtail), and n=5 (butterfly) are given explicitly. In addition, iterative and algebraic methods are described for determining the parameters a that are required in the uniform asymptotic cuspoid approximation for oscillating integrals with many coalescing saddle points. The results in this paper unify and generalize previous researches on the properties of the cuspoid canonical integrals and their partial derivatives

Measuring Well-Being: A Review of Instruments

Interest in the study of psychological health and well-being has increased significantly in recent decades. A variety of conceptualizations of psychological health have been proposed including hedonic and eudaimonic well-being, quality-of-life, and wellness approaches. Although instruments for measuring constructs associated with each of these approaches have been developed, there has been no comprehensive review of well-being measures. The present literature review was undertaken to identify self-report instruments measuring well-being or closely related constructs (i.e., quality of life and wellness) and critically evaluate them with regard to their conceptual basis and psychometric properties. Through a literature search, we identified 42 instruments that varied significantly in length, psychometric properties, and their conceptualization and operationalization of well-being. Results suggest that there is considerable disagreement regarding how to properly understand and measure well-being. Research and clinical implications are discussed

High-m Kink/Tearing Modes in Cylindrical Geometry

The global ideal kink equation, for cylindrical geometry and zero beta, is simplified in the high poloidal mode number limit and used to determine the tearing stability parameter, $\Delta^\prime$. In the presence of a steep monotonic current gradient, $\Delta^\prime$ becomes a function of a parameter, $\sigma_0$, characterising the ratio of the maximum current gradient to magnetic shear, and $x_s$, characterising the separation of the resonant surface from the maximum of the current gradient. In equilibria containing a current "spike", so that there is a non-monotonic current profile, $\Delta^\prime$ also depends on two parameters: $\kappa$, related to the ratio of the curvature of the current density at its maximum to the magnetic shear, and $x_s$, which now represents the separation of the resonance from the point of maximum current density. The relation of our results to earlier studies of tearing modes and to recent gyro-kinetic calculations of current driven instabilities, is discussed, together with potential implications for the stability of the tokamak pedestal.Comment: To appear in Plasma Physics and Controlled Fusio

Object-Oriented Paradigms for Modelling Vascular\ud Tumour Growth: a Case Study

Motivated by a family of related hybrid multiscale models, we have built an object-oriented framework for developing and implementing multiscale models of vascular tumour growth. The models are implemented in our framework as a case study to highlight how object-oriented programming techniques and good object-oriented design may be used effectively to develop hybrid multiscale models of vascular tumour growth. The intention is that this paper will serve as a useful reference for researchers modelling complex biological systems and that these researchers will employ some of the techniques presented herein in their own projects

A characterization of (I,J)-regular matrices

Let I, J be two ideals on N which contain the family Fin of finite sets. We provide necessary and sufficient conditions on the entries of an infinite real matrix A = (a(n, k)) which maps I-convergent bounded sequences into J-convergent bounded sequences and preserves the corresponding ideal limits. The well-known characterization of regular matrices due to Silverman-Toeplitz corresponds to the case I = J = Fin. Lastly, we provide some applications to permutation and diagonal matrices, which extend several known results in the literature

Study of the spectral properties of ELM precursors by means of wavelets

The high confinement regime (H-mode) in tokamaks is accompanied by the occurrence of bursts of MHD activity at the plasma edge, so-called edge localized modes (ELMs), lasting less than 1 ms. These modes are often preceded by coherent oscillations in the magnetic field, the ELM precursors, whose mode numbers along the toroidal and the poloidal directions can be measured from the phase shift between Mirnov pickup coils. When the ELM precursors have a lifetime shorter than a few milliseconds, their toroidal mode number and their nonlinear evolution before the ELM crash cannot be studied reliably with standard techniques based on Fourier analysis, since averaging in time is implicit in the computation of the Fourier coefficients. This work demonstrates significant advantages in studying spectral features of the short-lived ELM precursors by using Morlet wavelets. It is shown that the wavelet analysis is suitable for the identification of the toroidal mode numbers of ELM precursors with the shortest lifetime, as well as for studying their nonlinear evolution with a time resolution comparable to the acquisition rate of the Mirnov coils