D. H. Mackay

Models of the Large-Scale Corona. I. Formation, Evolution, and Liftoff of Magnetic Flux Ropes

The response of the large-scale coronal magnetic field to transport of magnetic flux in the photosphere is investigated. In order to follow the evolution on long timescales, the coronal plasma velocity is assumed to be proportional to the Lorentz force (magnetofriction), causing the coronal field to evolve through a series of nonlinear force-free states. Magnetofrictional simulations are used to study the formation and evolution of coronal flux ropes, highly sheared and/or twisted fields located above polarity inversion lines on the photosphere. As in our earlier studies, the three-dimensional numerical model includes the effects of the solar differential rotation and small-scale convective flows; the latter are described in terms of surface diffusion. The model is extended to include the effects of coronal magnetic diffusion, which limits the degree of twist of coronal flux ropes, and the solar wind, which opens up the field at large height. The interaction of two bipolar magnetic regions is considered. A key element in the formation of flux ropes is the reconnection of magnetic fields associated with photospheric flux cancellation at the polarity inversion lines. Flux ropes are shown to form both above the external inversion line between bipoles (representing type B filaments) and above the internal inversion line of each bipole in a sigmoid shape. It is found that once a flux rope has formed, the coronal field may diverge from equilibrium with the ejection of the flux rope. After the flux rope is ejected, the coronal field once again relaxes down to an equilibrium. This ability to follow the evolution of the coronal fields through eruptions is essential for future full-Sun simulations in which multiple bipoles are evolved for many months or years.

Mean Field Model for the Formation of Filament Channels on the Sun

The coronal magnetic field is subject to random footpoint motions that cause small-scale twisting and braiding of field lines. We present a mean field theory describing the effects of such small-scale twists on the large-scale coronal field. This theory assumes that the coronal field is force free, with electric currents flowing parallel or antiparallel to magnetic field lines. Random footpoint motions are described in terms of diffusion of the mean magnetic field at the photosphere. The appropriate mean field equations are derived, and a numerical method for solving these equations in three dimensions is presented. Preliminary results obtained with this method are also presented. In particular the formation of filament channels is studied. Filament channels are regions where the coronal magnetic field is strongly aligned with the underlying polarity inversion line in the photosphere. It is found that magnetic flux cancellation plays an important role in the formation of such channels. Various models of the coronal field are presented, including some in which the axial field is assumed to originate from below the photosphere. The models reproduce many of the observed features of filament channels, but the observed hemisphere pattern of dextral and sinistral channels remains a mystery.

Exploring the physical basis of solar cycle predictions : flux transport dynamics and persistence of memory in advection versus diffusion-dominated solar convection zones.

The predictability, or lack thereof, of the solar cycle is governed by numerous separate physical processes that act in unison in the interior of the Sun. Magnetic flux transport and the finite time delay that it introduces, specifically in the so-called Babcock-Leighton models of the solar cycle with spatially segregated source regions for the α- and Ω-effects, play a crucial rule in this predictability. Through dynamo simulations with such a model, we study the physical basis of solar cycle predictions by examining two contrasting regimes, one dominated by diffusive magnetic flux transport in the solar convection zone, the other dominated by advective flux transport by meridional circulation. Our analysis shows that diffusion plays an important role in flux transport, even when the solar cycle period is governed by the meridional flow speed. We further examine the persistence of memory of past cycles in the advection- and diffusion-dominated regimes through stochastically forced dynamo simulations. We find that in the advection-dominated regime this memory persists for up to three cycles, whereas in the diffusion-dominated regime this memory persists for mainly one cycle. This indicates that solar cycle predictions based on these two different regimes would have to rely on fundamentally different inputs, which may be the cause of conflicting predictions. Our simulations also show that the observed solar cycle amplitude-period relationship arises more naturally in the diffusion-dominated regime, thereby supporting those dynamo models in which diffusive flux transport plays a dominant role in the solar convection zone.

A Method to Determine the Heating Mechanisms of the Solar Corona

One of the paradigms about coronal heating has been the belief that the mean or summit temperature of a coronal loop is completely insensitive to the nature of the heating mechanisms. However, we point out that the temperature profile along a coronal loop is highly sensitive to the form of the heating. For example, when a steady state heating is balanced by thermal conduction, a uniform heating function makes the heat flux a linear function of distance along the loop, while T7/2 increases quadratically from the coronal footpoints; when the heating is concentrated near the coronal base, the heat flux is small and the T7/2 profile is flat above the base; when the heat is focused near the summit of a loop, the heat flux is constant and T7/2 is a linear function of distance below the summit. It is therefore important to determine how the heat deposition from particular heating mechanisms varies spatially within coronal structures such as loops or arcades and to compare it to high-quality measurements of the temperature profiles. We propose a new two-part approach to try and solve the coronal heating problem, namely, first of all to use observed temperature profiles to deduce the form of the heating, and second to use that heating form to deduce the likely heating mechanism. In particular, we apply this philosophy to a preliminary analysis of Yohkoh observations of the large-scale solar corona. This gives strong evidence against heating concentrated near the loop base for such loops and suggests that heating uniformly distributed along the loop is slightly more likely than heating concentrated at the summit. The implication is that large-scale loops are heated in situ throughout their length, rather than being a steady response to low-lying heating near their feet or at their summits. Unless waves can be shown to produce a heating close enough to uniform, the evidence is therefore at present for these large loops more in favor of turbulent reconnection at many small randomly distributed current sheets, which is likely to be able to do so. In addition, we suggest that the decline in coronal intensity by a factor of 100 from solar maximum to solar minimum is a natural consequence of the observed ratio of magnetic field strength in active regions and the quiet Sun; the altitude of the maximum temperature in coronal holes may represent the dissipation height of Alfven waves by turbulent phase mixing; and the difference in maximum temperature in closed and open regimes may be understood in terms of the roles of the conductive flux there.

The Sun's Global Photospheric and Coronal Magnetic Fields: Observations and Models

In this review, our present day understanding of the Sun’s global photospheric and coronal magnetic fields is discussed from both observational and theoretical viewpoints. Firstly, the large-scale properties of photospheric magnetic fields are described, along with recent advances in photospheric magnetic flux transport models. Following this, the wide variety of theoretical models used to simulate global coronal magnetic fields are described. From this, the combined application of both magnetic flux transport simulations and coronal modeling techniques to describe the phenomena of coronal holes, the Sun’s open magnetic flux and the hemispheric pattern of solar filaments is discussed. Finally, recent advances in non-eruptive global MHD models are described. While the review focuses mainly on solar magnetic fields, recent advances in measuring and modeling stellar magnetic fields are described where appropriate. In the final section key areas of future research are identified.

Modelling the Global Solar Corona II: Coronal Evolution and Filament Chirality Comparison

Abstract This paper considers the hemispheric pattern of solar filaments using newly developed simulations of the real photospheric and 3D coronal magnetic fields over a six-month period, on a global scale. The magnetic field direction in the simulation is compared directly with the chirality of observed filaments, at their observed locations. In our model the coronal field evolves through a continuous sequence of nonlinear force-free equilibria, in response to the changing photospheric boundary conditions and the emergence of new magnetic flux. In total 119 magnetic bipoles with properties matching observed active regions are inserted. These bipoles emerge twisted and inject magnetic helicity into the solar atmosphere. When we choose the sign of this active-region helicity to match that observed in each hemisphere, the model produces the correct chirality for up to 96% of filaments, including exceptions to the hemispheric pattern. If the emerging bipoles have zero helicity, or helicity of the opposite sign, then this percentage is much reduced. In addition, the simulation produces a higher proportion of filaments with the correct chirality after longer times. This indicates that a key element in the evolution of the coronal field is its long-term memory, and the build-up and transport of helicity from low to high latitudes over many months. It highlights the importance of continuous evolution of the coronal field, rather than independent extrapolations at different times. This has significant consequences for future modelling such as that related to the origin and development of coronal mass ejections.

Inferring X-ray coronal structures from Zeeman–Doppler images

We have modelled the X-ray emission from the young rapid rotator AB Doradus (Prot = 0.514 d) using as a basis Zeeman‐Doppler maps of the surface magnetic field. This allows us to reconcile the apparently conflicting observations of a high X-ray emission measure and coronal density with a low rotational modulation in the X-ray band. The technique is to extrapolate the coronal field from the surface maps by assuming the field to be potential. We then determine the coronal density for an isothermal corona by solving hydrostatic equilibrium along each field line and scaling the surface plasma pressure with the surface magnetic pressure. We set the density to zero along those field lines that are open and those where at any point along their length the plasma pressure exceeds the magnetic pressure. We then calculate the optically thin X-ray emission measure and rotational modulation for models with a range of coronal densities. Although the corona can be very extended, much of the emission comes from high-latitude regions close to the stellar surface. Since these are always in view as the star rotates, there is little rotational modulation. We find that emission measures in the observed range 10 52.8 ‐ 10 53.3 cm −3 can be reproduced with densities in the range 10 9 ‐10 10.7 cm −3 for coronae at temperatures of 10 6 ‐10 7 K.

The Evolution of the Sun's Open Magnetic Flux – II. Full Solar Cycle Simulations

In this paper the origin and evolution of the Suns open magnetic flux is considered by conducting magnetic flux transport simulations over many solar cycles. The simulations include the effects of differential rotation, meridional flow and supergranular diffusion on the radial magnetic field at the surface of the Sun as new magnetic bipoles emerge and are transported poleward. In each cycle the emergence of roughly 2100 bipoles is considered. The net open flux produced by the surface distribution is calculated by constructing potential coronal fields with a source surface from the surface distribution at regular intervals. In the simulations the net open magnetic flux closely follows the total dipole component at the source surface and evolves independently from the surface flux. The behaviour of the open flux is highly dependent on meridional flow and many observed features are reproduced by the model. However, when meridional flow is present at observed values the maximum value of the open flux occurs at cycle minimum when the polar caps it helps produce are the strongest. This is inconsistent with observations by Lockwood, Stamper and Wild (1999) and Wang, Sheeley, and Lean (2000) who find the open flux peaking 1–2 years after cycle maximum. Only in unrealistic simulations where meridional flow is much smaller than diffusion does a maximum in open flux consistent with observations occur. It is therefore deduced that there is no realistic parameter range of the flux transport variables that can produce the correct magnitude variation in open flux under the present approximations. As a result the present standard model does not contain the correct physics to describe the evolution of the Suns open magnetic flux over an entire solar cycle. Future possible improvements in modeling are suggested.

New Results in Modeling the Hemispheric Pattern of Solar Filaments

New results in modeling the hemispheric pattern of solar filaments through magnetic flux transport and magnetofrictional simulations are presented. The simulations consider for the first time what type of chirality forms along the polarity inversion line lying in between two magnetic bipoles as they interact. Such interactions are important for filament formation, as observations by F. Tang show that the majority of filaments form in between bipolar regions rather than within a single magnetic bipole. The simulations also include additional physics of coronal diffusion and a radial outflow velocity at the source surface, which was not included in previous studies. The results clearly demonstrate for the first time not only the origin of the dominant hemispheric pattern but also why exceptions to it may occur. The dominant hemispheric pattern may be attributed to the dominant range of bipole tilt angles and helicities in each hemisphere. Exceptions to the hemispheric pattern are found to only occur in cases of no initial helicity or for helicity of the minority type in each hemisphere when large positive bipole tilt angles (α > 20°) are used. As the simulations show a clear dependence of the hemispheric pattern and its exceptions on observational quantities, this may be used to check the validity of the results. Future programs to consider this are put forward.

THE INFLUENCE OF THE MAGNETIC FIELD ON RUNNING PENUMBRAL WAVES IN THE SOLAR CHROMOSPHERE

We use images of high spatial and temporal resolution, obtained using both ground- and space-based instrumentation, to investigate the role magnetic field inclination angles play in the propagation characteristics of running penumbral waves in the solar chromosphere. Analysis of a near-circular sunspot, close to the center of the solar disk, reveals a smooth rise in oscillatory period as a function of distance from the umbral barycenter. However, in one directional quadrant, corresponding to the north direction, a pronounced kink in the period–distance diagram is found. Utilizing a combination of the inversion of magnetic Stokes vectors and force-free field extrapolations, we attribute this behavior to the cut-off frequency imposed by the magnetic field geometry in this location. A rapid, localized inclination of the magnetic field lines in the north direction results in a faster increase in the dominant periodicity due to an accelerated reduction in the cut-off frequency. For the first time, we reveal how the spatial distribution of dominant wave periods, obtained with one of the highest resolution solar instruments currently available, directly reflects the magnetic geometry of the underlying sunspot, thus opening up a wealth of possibilities in future magnetohydrodynamic seismology studies. In addition, the intrinsic relationships we find between the underlying magnetic field geometries connecting the photosphere to the chromosphere, and the characteristics of running penumbral waves observed in the upper chromosphere, directly supports the interpretation that running penumbral wave phenomena are the chromospheric signature of upwardly propagating magneto-acoustic waves generated in the photosphere.