Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Solving Ordinary Differential Equations

Astrophysical magnetic fields and nonlinear dynamo theory

Coronal mass ejections (CMEs) are the largest-scale eruptive phenomenon in the solar system, expanding from active region-sized nonpotential magnetic structure to a much larger size. The bulk of plasma with a mass of ∼ 1011,1013 kg is hauled up all the way out to the interplanetary space with a typical velocity of several hundred or even more than 1000 km s−1, with a chance to impact our Earth, resulting in hazardous space weather conditions. They involve many other much smaller-sized solar eruptive phenomena, such as X-ray sigmoids, filament/prominence eruptions, solar flares, plasma heating and radiation, particle acceleration, EIT waves, EUV dimmings, Moreton waves, solar radio bursts, and so on. It is believed that, by shedding the accumulating magnetic energy and helicity, they complete the last link in the chain of the cycling of the solar magnetic field. In this review, I try to explicate our understanding on each stage of the fantastic phenomenon, including their pre-eruption structure, their triggering mechanisms and the precursors indicating the initiation process, their acceleration and propagation. Particular attention is paid to clarify some hot debates, e.g., whether magnetic reconnection is necessary for the eruption, whether there are two types of CMEs, how the CME frontal loop is formed, and whether halo CMEs are special.

/pdf/coronal-mass-ejections-models-and-their-observational-basis-18wrfif7lq.pdf

Coronal Mass Ejections: Models and Their Observational Basis

A numerical model of isotropic homogeneous turbulence with helical forcing is investigated. The resulting flow, which is essentially the prototype of the α2 dynamo of mean field dynamo theory, produces strong dynamo action with an additional large-scale field on the scale of the box (at wavenumber k = 1; forcing is at k = 5). This large-scale field is nearly force free and exceeds the equipartition value. As the magnetic Reynolds number Rm increases, the saturation field strength and the growth rate of the dynamo increase. However, the time it takes to build up the large-scale field from equipartition to its final superequipartition value increases with magnetic Reynolds number. The large-scale field generation can be identified as being due to nonlocal interactions originating from the forcing scale, which is characteristic of the α-effect. Both α and turbulent magnetic diffusivity ηt are determined simultaneously using numerical experiments where the mean field is modified artificially. Both quantities are quenched in an Rm-dependent fashion. The evolution of the energy of the mean field matches that predicted by an α2 dynamo model with similar α and ηt quenchings. For this model an analytic solution is given that matches the results of the simulations. The simulations are numerically robust in that the shape of the spectrum at large scales is unchanged when changing the resolution from 303 to 1203 mesh points, or when increasing the magnetic Prandtl number (viscosity/magnetic diffusivity) from 1 to 100. Increasing the forcing wavenumber to 30 (i.e., increasing the scale separation) makes the inverse cascade effect more pronounced, although it remains otherwise qualitatively unchanged.

/pdf/the-inverse-cascade-and-nonlinear-alpha-effect-in-3x1j634gex.pdf

The inverse cascade and nonlinear alpha-effect in simulations of isotropic helical hydromagnetic turbulence

Images of the X-ray corona near the solar disk's center were examined for large, transient brightenings of the type known to be associated with Hα filament eruptions and coronal mass ejections. Many of the brightenings were sigmoid (S-shaped). The measured ratios of length to width of the sigmoid features are shown to be consistent with ratios predicted by a simple model based on a kinked and twisted magnetic flux rope. Many of the studied sigmoid brightenings evolved into arcades of bright loops. Such arcades are often associated with coronal mass ejections, and it is suggested that the cause of the ejections is an MHD helical kink instability in the Hα filament/coronal arcade complexes. Reverse-S brightenings outnumbered forward-S brightenings by six to one in the northern hemisphere. Forward-S brightenings were similarly predominant in the south. This hemispherical segregation suggests that the magnetic fields in the transient features are systematically twisted. Globally, the implied distribution of magnetic helicity is similar to the distribution that Martin et al. discovered in quiescent Hα filaments.

https://iopscience.iop.org/article/10.1086/310118/pdf

Evidence for Helically Kinked Magnetic Flux Ropes in Solar Eruptions

In the theories of solar magnetism, kinetic and magnetic helicities, which arise as a consequence of the rotation of the Sun, play a key role The dynamo for the main field is assumed to operate in the convection zone The solar rotation also may be the ultimate cause for the generation of dc electric currents in the atmosphere, needed as the energy source for flares Then in the atmosphere the electric current helicity, HC = B · ▽ × B, which is a pseudo-scalar quantity, should be antisymmetric about the equatorial plane An inspection of 16 active regions, for which HC has been estimated by using extrapolation of measured photospheric magnetic fields, leads to the result that the electric current helicity is predominantly negative in the northern and positive in the southern hemisphere The helicity of the large-scale currents generated according to standard dynamo theory by the alpha effect in the convection zone is just opposite in sign Current generation due to rotational motions of sunspots and other magnetic elements in accordance with the global differential rotation, ie, counter-clockwise in the northern and clockwise in the southern hemisphere, however, can explain the rule found Also in some alternative dynamo models for the global field, in which the dynamo operates at the base of the convection zone, the large-scale current helicity generated by the alpha effect has the sign needed

Electric current helicity in the solar atmosphere

At first it is shown that a magnetic field being force-free, i.e. satisfying ▽ × B = αB, with α = constant (α ≠ 0) in the whole exterior of the Sun cannot have a finite energy content and cannot be determined uniquely from only one magnetic field component given at the photosphere. Then the boundary value problem for a semi-infinite column of arbitrary cross section is solved by a Green's function method.

https://link.springer.com/content/pdf/10.1007/BF00157267.pdf

Determination of constant α force-free solar magnetic fields from magnetograph data

It is shown that the \ensuremath{\alpha} effect of mean-field magnetohydrodynamics, which consists of the generation of a mean electromotive force along the mean magnetic field by turbulently fluctuating parts of velocity and magnetic field, is equivalent to the simulaneous generation of both turbulent and mean-field magnetic helicities, the generation rates being equal in magnitude and opposite in sign. In the particular case of statistically stationary and homogeneous fluctuations this implies that the \ensuremath{\alpha} effect can increase the energy in the mean magnetic field only under the condition that magnetic helicity is also accumulated there. \textcopyright{} 1996 The American Physical Society.

Nature of the α effect in magnetohydrodynamics

Context. Reliable measurements of the solar magnetic field are restricted to the phoptosphere. As an alternative to measurements, the field in the higher layers of the atmosphere is calculated from the measured photospheric field, mostly under the assumption that it is force-free. However, the magnetic field in the photosphere is not force-free. Moreover, most methods for the extrapolation of the photospheric magnetic field into the higher layers prescribe the magnetic vector on the whole boundary of the considered volume, which overdetermines the force-free field. Finally, the extrapolation methods are very sensitive to small-scale noise in the magnetograph data, which, however, if sufficienly resolved numerically, should affect the solution only in a thin boundary layer close to the photosphere. Aims. A new method for the preprocessing of solar photospheric vector magnetograms has been developed that, by improving their compatibility with the condition of force-freeness and removing small-scale noise, makes them more suitable for extrapolations into three-dimensional nonlinear force-free magnetic fields in the chromosphere and corona. Methods. A functional of the photospheric field values is minimized whereby the total magnetic force and the total magnetic torque on the considered volume above the photosphere, as well as a quantity measuring the degree of small-scale noise in the photospheric boundary data, are simultaneously made small. For the minimization, the method of simulated annealing is used and the smoothing of noisy magnetograph data is attained by windowed median averaging. Results. The method was applied to a magnetogram derived from a known nonlinear force-free test field to which an artificial noise had been added. The algorithm recovered all main structures of the magnetogram and removed small-scale noise. The main test was to extrapolate from the noisy photospheric vector magnetogram before and after the preprocessing. The preprocessing was found to significantly improve the agreement of the extrapolated with the exact field.

https://www.aanda.org/articles/aa/pdf/2007/46/aa8454-07.pdf

Preprocessing of solar vector magnetograms for force-free magnetic field extrapolation

The field lines of closed magnetic structures above the photosphere define a mapping from the photosphere to itself. This mapping is discontinuous, and the field line connectivity to the boundary can change discontinuously in response to continuous changes of field strength and direction, if field lines either end in a singular point of the field or are tangential to the photosphere at one end. Whereas the general existence of singular points is questionable, the field has typically a cell structure due to the presence of segments of the zero line of the photospheric longitudinal field on which the transversal field is directed from negative (pointing into the Sun) to positive fields. The cell boundaries are made up of field lines which all touch the photosphere on one of these line segments. Within each of the cells the field line mapping is continuous. When during a slow evolution a substantial part of a coronal loop or of an arcade has passed from one cell into another a fast dynamic instability may set in which was previously prevented by the anchoring of field lines in the dense photosphere.

Norbert Seehafer

Papers

Electric current helicity in the solar atmosphere

Determination of constant α force-free solar magnetic fields from magnetograph data

Nature of the α effect in magnetohydrodynamics

Preprocessing of solar vector magnetograms for force-free magnetic field extrapolation

On the magnetic field line topology in solar active regions