Recently, I became interested in a current debate over whether file size distributions are best modelled by a power law distribution or a lognormal distribution. In trying to learn enough about these distributions to settle the question, I found a rich and long history, spanning many fields. Indeed, several recently proposed models from the computer science community have antecedents in work from decades ago. Here, I briefly survey some of this history, focusing on underlying generative models that lead to these distributions. One finding is that lognormal and power law distributions connect quite naturally, and hence, it is not surprising that lognormal distributions have arisen as a possible alternative to power law distributions across many fields.

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A Brief History of Generative Models for Power Law and Lognormal Distributions

KamLAND has measured the flux of nu;(e)'s from distant nuclear reactors. We find fewer nu;(e) events than expected from standard assumptions about nu;(e) propagation at the 99.95% C.L. In a 162 ton.yr exposure the ratio of the observed inverse beta-decay events to the expected number without nu;(e) disappearance is 0.611+/-0.085(stat)+/-0.041(syst) for nu;(e) energies >3.4 MeV. In the context of two-flavor neutrino oscillations with CPT invariance, all solutions to the solar neutrino problem except for the "large mixing angle" region are excluded.

First Results from KamLAND: Evidence for Reactor Anti-Neutrino Disappearance

We propose a new model for the initiation of a solar coronal mass ejection (CME). The model agrees with two properties of CMEs and eruptive flares that have proved to be very difficult to explain with previous models: (1) very low-lying magnetic field lines, down to the photospheric neutral line, can open toward infinity during an eruption; and (2) the eruption is driven solely by magnetic free energy stored in a closed, sheared arcade. Consequently, the magnetic energy of the closed state is well above that of the posteruption open state. The key new feature of our model is that CMEs occur in multipolar topologies in which reconnection between a sheared arcade and neighboring flux systems triggers the eruption. In this "magnetic breakout" model, reconnection removes the unsheared field above the low-lying, sheared core flux near the neutral line, thereby allowing this core flux to burst open. We present numerical simulations that demonstrate our model can account for the energy requirements for CMEs. We discuss the implication of the model for CME/flare prediction.

https://iopscience.iop.org/article/10.1086/306563/pdf

A Model for Solar Coronal Mass Ejections

Radiation forces on small particles in the solar system

SOLAR flares are thought to be the result of magnetic reconnection — the merging of antiparallel magnetic fields and the consequent release of magnetic energy. Flares are classified into two types1: compact and two-ribbon. The two-ribbon flares, which appear as slowly-developing, long-lived large loops, are understood theoretically2–6 as arising from an eruption of a solar prominence that pulls magnetic field lines upward into the corona. As the field lines form an inverted Y-shaped structure and relax, the reconnection of the field lines takes place. This view has been supported by recent observations7–10. A different mechanism seemed to be required, however, to produce the short-lived, impulsive compact flares. Here we report observations made with the Yohkoh11 Hard X-ray Telescope12 and Soft X-ray Telescope13, which show a compact flare with a geometry similar to that of a two-ribbon flare. We identify the reconnection region as the site of particle acceleration, suggesting that the basic physics of the reconnection process (which remains uncertain) may be common to both types of flare.

A loop-top hard X-ray source in a compact solar flare as evidence for magnetic reconnection

Model of the High-Energy Phase of Solar Flares

A new method for reconstructing force-free magnetic fields from their boundary values, based on minimizing the global departure of an initial field from a force-free and solenoidal state, is presented. The method is tested by application to a known nonlinear solution. We discuss the obstacles to be overcome in the application of this method to the solar case: the reconstruction of force-free fields in the corona from measurements of the vector magnetic field in the low atmosphere.

http://iopscience.iop.org/article/10.1086/309355/pdf

An Optimization Approach to Reconstructing Force-free Fields

The problem under discussion is that of calculating magnetic field configurations in which the Lorentz force j x B is everywhere zero, subject to specified boundary conditions. We choose to represent the magnetic field in terms of Clebsch variables in the form B = grad alpha x grad beta. These variables are constant on any field line so that each field line is labeled by the corresponding values of alpha and beta. When the field is described in this way, the most appropriate choice of boundary conditions is to specify the values of alpha and beta on the bounding surface. We show that such field configurations may be calculated by a magneto-frictional method. We imagine that the field lines move through a stationary medium, and that each element of magnetic field is subject to a frictional force parallel to and opposing the velocity of the field line. This concept leads to an iteration procedure for modifying the variables alpha and beta, that tends asymptotically towards the force-free state. We apply the method first to a simple problem in two rectangular dimensions, and then to a problem of cylindrical symmetry that was previously discussed by Barnes and Sturrock (1972). In one important respect, our new results differ from the earlier results of Barnes and Sturrock, and we conclude that the earlier article was in error.

Force-free magnetic fields - The magneto-frictional method

We investigate a one-dimensional loop model for the evaporative cooling of the coronal flare plasma. The important assumptions are that conductive losses dominate radiative cooling and that the evaporative velocities are small compared with the sound speed. We calculate the profile and evolution of the temperature and verify the accuracy of our assumptions for plasma parameters typical of flare regions. The model is in agreement with soft X-ray observations on the evolution of flare temperatures and emission measures. The effect of evaporation is to greatly reduce the conductive heat flux into the chromosphere and to enhance the EUV emission from the coronal flare plasma.

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Evaporative cooling of flare plasma

Magnetic-field structures in solar active regions are expected to be substantially force-free. A method is proposed for calculating such structures by numerical methods. The method is applied to the study of the magnetic-field pattern associated with a sunspot of one polarity surrounded by a magnetic region of opposite polarity when the sunspot rotates with respect to the surrounding region. Rotation introduces a toroidal component of magnetic field, and the associated pressure leads to inflation of the magnetic field pattern. If the differential rotation exceeds about 180 deg, the force-free magnetic field has energy greater than that of an open magnetic-field configuration with the same photospheric boundary conditions. It is concluded that, beyond this point, the force-free field structure is metastable and can be converted into an open field structure by an explosive MHD instability.

Peter A. Sturrock

Papers

Model of the High-Energy Phase of Solar Flares

An Optimization Approach to Reconstructing Force-free Fields

Force-free magnetic fields - The magneto-frictional method

Evaporative cooling of flare plasma

Force-free magnetic-field structures and their role in solar activity.