The calculation of rate coefficients is a discipline of nonlinear science of importance to much of physics, chemistry, engineering, and biology. Fifty years after Kramers' seminal paper on thermally activated barrier crossing, the authors report, extend, and interpret much of our current understanding relating to theories of noise-activated escape, for which many of the notable contributions are originating from the communities both of physics and of physical chemistry. Theoretical as well as numerical approaches are discussed for single- and many-dimensional metastable systems (including fields) in gases and condensed phases. The role of many-dimensional transition-state theory is contrasted with Kramers' reaction-rate theory for moderate-to-strong friction; the authors emphasize the physical situation and the close connection between unimolecular rate theory and Kramers' work for weakly damped systems. The rate theory accounting for memory friction is presented, together with a unifying theoretical approach which covers the whole regime of weak-to-moderate-to-strong friction on the same basis (turnover theory). The peculiarities of noise-activated escape in a variety of physically different metastable potential configurations is elucidated in terms of the mean-first-passage-time technique. Moreover, the role and the complexity of escape in driven systems exhibiting possibly multiple, metastable stationary nonequilibrium states is identified. At lower temperatures, quantum tunneling effects start to dominate the rate mechanism. The early quantum approaches as well as the latest quantum versions of Kramers' theory are discussed, thereby providing a description of dissipative escape events at all temperatures. In addition, an attempt is made to discuss prominent experimental work as it relates to Kramers' reaction-rate theory and to indicate the most important areas for future research in theory and experiment.

Reaction-rate theory: fifty years after Kramers

Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications

Progress in the area of MHD stability and disruptions, since the publication of the 1999 ITER Physics Basis document (1999 Nucl. Fusion 39 2137-2664), is reviewed. Recent theoretical and experimental research has made important advances in both understanding and control of MHD stability in tokamak plasmas. Sawteeth are anticipated in the ITER baseline ELMy H-mode scenario, but the tools exist to avoid or control them through localized current drive or fast ion generation. Active control of other MHD instabilities will most likely be also required in ITER. Extrapolation from existing experiments indicates that stabilization of neoclassical tearing modes by highly localized feedback-controlled current drive should be possible in ITER. Resistive wall modes are a key issue for advanced scenarios, but again, existing experiments indicate that these modes can be stabilized by a combination of plasma rotation and direct feedback control with non-axisymmetric coils. Reduction of error fields is a requirement for avoiding non-rotating magnetic island formation and for maintaining plasma rotation to help stabilize resistive wall modes. Recent experiments have shown the feasibility of reducing error fields to an acceptable level by means of non-axisymmetric coils, possibly controlled by feedback. The MHD stability limits associated with advanced scenarios are becoming well understood theoretically, and can be extended by tailoring of the pressure and current density profiles as well as by other techniques mentioned here. There have been significant advances also in the control of disruptions, most notably by injection of massive quantities of gas, leading to reduced halo current fractions and a larger fraction of the total thermal and magnetic energy dissipated by radiation. These advances in disruption control are supported by the development of means to predict impending disruption, most notably using neural networks. In addition to these advances in means to control or ameliorate the consequences of MHD instabilities, there has been significant progress in improving physics understanding and modelling. This progress has been in areas including the mechanisms governing NTM growth and seeding, in understanding the damping controlling RWM stability and in modelling RWM feedback schemes. For disruptions there has been continued progress on the instability mechanisms that underlie various classes of disruption, on the detailed modelling of halo currents and forces and in refining predictions of quench rates and disruption power loads. Overall the studies reviewed in this chapter demonstrate that MHD instabilities can be controlled, avoided or ameliorated to the extent that they should not compromise ITER operation, though they will necessarily impose a range of constraints.

https://iopscience.iop.org/article/10.1088/0029-5515/47/6/S03/pdf

Chapter 3: MHD stability, operational limits and disruptions

The ideal helical kink instability of a force-free coronal magnetic flux rope, anchored in the photosphere, is studied as a model for solar eruptions. Using the flux rope model of Titov and D?moulin as the initial condition in MHD simulations, both the development of helical shape and the rise profile of a confined (or failed) filament eruption (on 2002 May 27) are reproduced in very good agreement with the observations. By modifying the model such that the magnetic field decreases more rapidly with height above the flux rope, a full (or ejective) eruption of the rope is obtained in very good agreement with the developing helical shape and the exponential-to-linear rise profile of a fast coronal mass ejection (CME) on 2001 May 15. This confirms that the helical kink instability of a twisted magnetic flux rope can be the mechanism of the initiation and the initial driver of solar eruptions. The agreement of the simulations with properties that are characteristic of many eruptions suggests that they are often triggered by the kink instability. The decrease of the overlying field with height is a main factor in deciding whether the instability leads to a confined event or to a CME.

https://iopscience.iop.org/article/10.1086/462412/pdf

Confined and Ejective Eruptions of Kink-unstable Flux Ropes

The temporal relationship between coronal mass ejections (CMEs) and associated solar flares is of great importance to understanding the origin of CMEs, but it has been difficult to study owing to the nature of CME detection. In this paper, we investigate this issue using the Large Angle and Spectrometric Coronagraph and the EUV Imaging Telescope observations combined with GOES soft X-ray observations. We present four well-observed events whose source regions are close to the limb such that we are able to directly measure the CMEs' initial evolution in the low corona (~ 1-3 R☉) without any extrapolation; this height range was not available in previous space-based coronagraph observations. The velocity-time profiles show that kinematic evolution of three of the four CMEs can be described in a three-phase scenario: the initiation phase, impulsive acceleration phase, and propagation phase. The initiation phase is characterized by a slow ascension with a speed less than 80 km s-1 for a period of tens of minutes. The initiation phase always occurs before the onset of the associated flare. Following the initiation phase, the CMEs display an impulsive acceleration phase that coincides very well with the flares' rise phase lasting for a few to tens of minutes. The acceleration of CMEs ceases near the peak time of the soft X-ray flares. The CMEs then undergo a propagation phase, which is characterized by a constant speed or slowly decreasing in speed. The acceleration rates in the impulsive acceleration phase are in the range of 100-500 m s-2. One CME (on 1997 November 6, associated with an X9.4 flare) does not show an initiation phase. It has an extremely large acceleration rate of 7300 m s-2. The possible causes of CME initiation and acceleration in connection with flares are explored.

/pdf/on-the-temporal-relationship-between-coronal-mass-ejections-2mn3kin3m7.pdf

On the temporal relationship between coronal mass ejections and flares

We present a new approach to the theory of large-scale solar eruptive phenomena such as coronal mass ejections and two-ribbon flares, in which twisted flux tubes play a crucial role. We show that it is possible to create a highly nonlinear three-dimensional force-free configuration consisting of a twisted magnetic flux rope representing the magnetic structure of a prominence (surrounded by an overlaying, almost potential, arcade) and exhibiting an S-shaped structure, as observed in soft X-ray sigmoid structures. We also show that this magnetic configuration cannot stay in equilibrium and that a considerable amount of magnetic energy is released during its disruption. Unlike most previous models, the amount of magnetic energy stored in the configuration prior to its disruption is so large that it may become comparable to the energy of the open field.

https://iopscience.iop.org/article/10.1086/312444/pdf

A Twisted Flux Rope Model for Coronal Mass Ejections and Two-Ribbon Flares

We consider a three-dimensional bipolar magnetic field B, occupying a half-space, which is driven into evolution by the slow turbulent diffusion of its normal component on the boundary The latter is imposed by fixing the tangential component of the electric field and leads to flux cancellation We first present general analytical considerations on this problem and then construct a class of explicit solutions in which B keeps evolving quasi-statically through a sequence of force-free configurations without exhibiting any catastrophic behavior Thus, we report the results of a series of numerical simulations in which B evolves from different force-free states, the electric field on the boundary being imposed to have a vanishing electrostatic part (the latter condition is not enforced in the analytical model, and thus it is possible a priori for the results of the two types of calculations to be different) In all the cases, we find that the evolution conserves the magnetic helicity and exhibits two qualitatively different phases The first one, during which a twisted flux rope is created, is slow and almost quasi-static, while the second one is associated with a disruption, which is confined for a small initial helicity and global for a large initial helicity Our calculations may be relevant for modeling the coronal mass ejections that have been observed to occur in the late dispersion phase of an active region In particular, they may allow us to understand the role played by a twisted flux rope in these events

https://iopscience.iop.org/article/10.1086/377444/pdf

Coronal Mass Ejection: Initiation, Magnetic Helicity, and Flux Ropes. II. Turbulent Diffusion-driven Evolution

We present some preliminary results on different mathematical problems encountered in attempts to reconstruct the coronal magnetic field, assumed to be in a force-free state, from its values in the photosphere. We discuss the formulations associated with these problems, and some new numerical methods that can be used to get their approximate solutions. Both the linear constant-α and the nonlinear cases are considered. We also discuss the possible use of dynamical 3D MHD codes to construct approximate solutions of the equilibrium force-free equations, which are needed for testing numerical extrapolation schemes.

Reconstructing the solar coronal magnetic field as a force-free magnetic field

This Letter is devoted to the still open problem of the evolution of a three-dimensional coronal flux tube embedded in a low-beta ideal plasma and having its footpoints twisted by slow photospheric motions. Such a process has been simulated with a recently developed magnetohydrodynamic code. In the particular calculation reported here, the system occupies a large cubic box. The field is initially potential, being generated by an underlying horizontal dipole, and it is twisted by two vortices located on the lower face {z = 0} of the box, on both sides of the neutral line. In a first phase, the field roughly evolves quasi-statically through a sequence of force-free configurations. Thus, it enters a dynamical phase during which it suffers a very fast expansion, closely approaching after some finite time a semiopen configuration. The energy increases monotonically during all the evolution, and it tends to a limit, which is equal to about 80% of the energy of the totally open field associated with Bz.

http://iopscience.iop.org/article/10.1086/310158/pdf

Very Fast Opening of a Three-dimensional Twisted Magnetic Flux Tube

The curvature effects on the dynamics of magnetic island evolution in tokamaks are investigated both theoretically and numerically. By taking into account perpendicular and parallel heat diffusion, a new dispersion relation is derived for tearing modes that match the linear and nonlinear results. This evolution equation allows a quantitative description over the whole range of island sizes. It predicts a nonlinear instability, i.e., growing magnetic islands in linearly stable magnetic configurations. All these predictions are in excellent agreement with full tridimensional linear and nonlinear magnetohydrodynamic (MHD) computations with the latest version of XTOR [K. Lerbinger and J. F. Luciani, J. Comput. Phys. 97, 444 (1991)]. These results have important consequences on the onset of neoclassical tearing modes because they predict a resistive MHD threshold.

Jean-François Luciani

Papers

A Twisted Flux Rope Model for Coronal Mass Ejections and Two-Ribbon Flares

Coronal Mass Ejection: Initiation, Magnetic Helicity, and Flux Ropes. II. Turbulent Diffusion-driven Evolution

Reconstructing the solar coronal magnetic field as a force-free magnetic field

Very Fast Opening of a Three-dimensional Twisted Magnetic Flux Tube

Curvature effects on the dynamics of tearing modes in tokamaks