Marginal stability

Abstract: We present a model for fault dynamics consisting of a uniform chain of blocks and springs pulled slowly across a rough surface. The only nonlinear element of our model is a slip-stick friction force between the blocks and the surface. We find that this model gives rise to events of all sizes. Our numerical evaluation of the distribution of earthquake magnitudes results in power-law spectrum similar to what is observed in nature. Like certain other dissipative dynamical systems, the observed large fluctuations in earthquake magnitude persist because the system is in a state of marginal stability.

Abstract: This chapter reviews the progress accomplished since the redaction of the first ITER Physics Basis (1999 Nucl Fusion 39 2137-664) in the field of energetic ion physics and its possible impact on burning plasma regimes New schemes to create energetic ions simulating the fusion-produced alphas are introduced, accessing experimental conditions of direct relevance for burning plasmas, in terms of the Alfvenic Mach number and of the normalised pressure gradient of the energetic ions, though orbit characteristics and size cannot always match those of ITER Based on the experimental and theoretical knowledge of the effects of the toroidal magnetic field ripple on direct fast ion losses, ferritic inserts in ITER are expected to provide a significant reduction of ripple alpha losses in reversed shear configurations The nonlinear fast ion interaction with kink and tearing modes is qualitatively understood, but quantitative predictions are missing, particularly for the stabilisation of sawteeth by fast particles that can trigger neoclassical tearing modes A large database on the linear stability properties of the modes interacting with energetic ions, such as the Alfven eigenmode has been constructed Comparisons between theoretical predictions and experimental measurements of mode structures and drive/damping rates approach a satisfactory degree of consistency, though systematic measurements and theory comparisons of damping and drive of intermediate and high mode numbers, the most relevant for ITER, still need to be performed The nonlinear behaviour of Alfven eigenmodes close to marginal stability is well characterized theoretically and experimentally, which gives the opportunity to extract some information on the particle phase space distribution from the measured instability spectral features Much less data exists for strongly unstable scenarios, characterised by nonlinear dynamical processes leading to energetic ion redistribution and losses, and identified in nonlinear numerical simulations of Alfven eigenmodes and energetic particle modes Comparisons with theoretical and numerical analyses are needed to assess the potential implications of these regimes on burning plasma scenarios, including in the presence of a large number of modes simultaneously driven unstable by the fast ions

Abstract: Examines finite-time stability of homogeneous systems. The main result is that a homogeneous system is finite-time stable if and only if it is asymptotically stable and has a negative degree of homogeneity.

Abstract: We investigate the correspondence between existence/stability of BPS states in type II string theory compactified on a Calabi-Yau manifold and BPS solutions of four dimensional N=2 supergravity. Some paradoxes emerge, and we propose a resolution by considering composite configurations. This in turn gives a smooth effective field theory description of decay at marginal stability. We also discuss the connection with 3-pronged strings, the Joyce transition of special Lagrangian submanifolds, and Pi-stability.