The difficulty in writing a ‘how-to’ book on numerical methods is to find a form which is accessible to people from various scientific backgrounds. When we started this project, hierarchical N-body techniques were deemed to be ‘too new’ for a book. On the other hand, a few minutes browsing in the References will reveal that the scientific output arising from the original papers of Barnes and Hut (1986) and Greengard and Rohklin (1987) is impressive but largely confined to two or three specialist fields. To us, this suggests that it is about time these techniques became better known in other fields where N-body problems thrive, not least in our own field of computational plasma physics. This book is therefore an attempt to gather everything hierarchical under one roof, and then to indicate how and where tree methods might be used in the reader's own research field. Inevitably, this has resulted in something of a pot-pourri of techniques and applications, but we hope there is enough here to satisfy the beginners and connoisseurs alike.

Tree Construction

We have seen in the preceding chapter that in grid-based codes the particles interact via some averaged density distribution. This enables one to calculate the influence of a number of particles represented by a cell on its neighbouring cells. Problems occur if the density contrast in the simulation becomes very large or the geometry of the problem is very complex.

So why does one bother with a grid at all and not just calculate the interparticle forces? The answer is simply that the computational effort involved quite dramatically limits the number of particles that can be simulated. Particularly with 1/r-type potentials, calculating each particle–particle interaction requires an unnecessary amount of work because the individual contributions of distant particles is small. On the other hand, gridless codes cannot distinguish between near-neighbours and more distant particles; each particle is given the same weighting.

Ideally, the calculation would be performed without a grid in the usual sense, but with some division of the physical space that maintains a relationship between each particle and its neighbours. The force could then be calculated by direct integration while combining increasingly large groups of particles at larger distances. Barnes and Hut (1986) observed that this works in the same way that humans interact with neighbouring individuals, more distant villages, and larger states and countries. A resident of Lower-Wobbleton, Kent, England, is unlikely to undertake a trip to Oberfriedrichsheim, Bavaria, Germany, for a beer and to catch up on the local gossip.

In Chapter 2 we saw the basic workings of the tree algorithm. Now we will discuss some methods that can be used to optimise the performance of this type of code. Although most of these techniques are not specific to tree codes, they are not always straightforward to implement within a hierarchical data structure. It therefore seems worthwhile to reconsider some of the common tricks of the N-body trade, in order to make sure that the tree code is optimised in every sense – not just in its N log N scaling.

There are basically two points of possible improvement:

• Improvement of the accuracy of the particle trajectory calculation by means of higher order integration schemes and individual timesteps. This is especially important for problems involving many close encounters of the particles, that is, ‘collisional’ problems.

• Speedup of the computation time needed to evaluate a single timestep by use of modern software and hardware combinations, such as vectorisation, and special-purpose hierarchical or parallel computer architecture.

Individual Timesteps

For most many-body simulations one would like the total simulated time T = ntΔt (where nt is the number of timesteps) to be as large as possible to approach the hydrodynamic limit. However, the choice of the timestep Δt has to be a compromise between this aim and the fact that as Δt increases, the accuracy of the numerical integration gets rapidly worse.

The hierarchical tree method can not be adapted only for Monte Carlo applications: It can also be modified to perform near-neighbour searches efficiently. This means that the tree algorithm could also have applications for systems with short-range or contact forces. Hernquist and Katz (1989) first showed how the tree structure can be used to find near neighbours through range searching. Following their method, the near-neighbour search is performed the following way.

Consider a system in which only neighbours lying within a distance h will interact with the particle i in question. For the near-neighbour search this sphere is enclosed in a cube whose sides are of length 2h. The tree is built the usual way and the tree search starts at the root. The tree search is performed in a very similar way to the normal force calculation of Section 2.2 by constructing an interaction list. The main difference is that the s/d criterion is substituted by the question: ‘Does the volume of the search cube overlap with the volume of the pseudoparticle presently under consideration?’

If there is no overlap, this branch of the tree contains no near neighbours and is not searched any further. However, if there is an overlap, the cell is subdivided into its daughter cells and the search continues on the next highest level. If the cell is a leaf – meaning there is only one particle in the cell – one has to check whether it actually lies within the radius h of particle i.

Today, the study of The Structure of the Sun is one of the most exciting and rapidly evolving fields in physics. Helioseismology has provided us with a new tool to measure the physical state of the interior of a star, our Sun. This technique is successful to a depth of 0.7 R⊙ (i.e. 0.3 R⊙ from the centre). Deeper than this, observational data has been scarce. However, data are now becoming available from Earth-bound helioseismic networks (GONG, TON, IRIS, BISON,…) and from experiments on board SOHO (GOLF, MDI, VIRGO). These should allow the spectrum of gravity modes for the Sun to be determined, and thus the physical state of the solar core.

This book provides an up-to-date and comprehensive review of our current understanding of the Sun. Each chapter is written by a world expert. They are based on lectures given at the VIth Canary Islands Winter School on Astrophysics. This timely conference brought together leading scientists in the field, postgraduates and recent postdocs students. The aim was to take stock of the new understanding of the Sun and to focus on avenues for fruitful future research. Eight lecturers, around 60 students, and staff from the IAC met in the Hotel Gran Tinerfe in Playa de las Américas (Adeje, Tenerife) from the 5th to the 16th of December, 1994. It was a fortnight of intense and enjoyable scientific work.

INTRODUCTION

This is almost an impossible task, to summarize the subject of Global Changes in the Sun so I must apologize in advance for limiting the scope of these lectures to issues that have been choosen, in part, because of personal interests. I hope that the references provide the reader with footpoints from which to explore a larger set of questions which bear on this subject.

Here we will not discuss the very long, evolutionary, timescales over which the sun changes, nor will we explore the fast changes associated with flares and other transient phenomena. While these discussions depend on some results from MHD models of the solar magnetic cycle, we will not be concerned with the MHD mechanism. These lectures will not address the questions needed to understand local physical models that describe, for example, granulation. On the other hand we will describe some of the physical problems that connect the small-scale behavior of the sun to its global properties. By “global property” I mean an observable that is connected by physically important timescales to the entire sun: limb shape and brightness, largescale magnetic field, oscillation frequencies, solar luminosity, and solar irradiance are all examples of global properties.

Here we are interested in understanding the deviations of the sun from some standard one-dimensional static stellar model. This is a subject we can hardly approach for other stars, and for the sun it is difficult because the physics of magnetic fields and convection are linked over a wide range of spatial and temporal scales.

ABSTRACT

The X-ray Solar Physics Satellite Yohkoh has provided us with a number of new findings about the high temperature and high energy processes occurring in solar flares, in active regions, and in the background corona. According to these new findings, hot and dense corona above active regions seem to be maintained, at least in part, with the injections of already heated mass along the magnetic loops from the footpoint below. The outermost loops of the magnetic structures of these active regions are expanding away almost continuously in the case of “active” active regions. These give us quite a different and lively picture about the active region corona compared with a previous static picture with steady heating that we had based on the previous low cadence observations. New clues to the mechanism of flares, which were hidden thus-far in the yet fainter and relatively short stages before the start of flares, have been revealed by the wide-dynamic range, high cadence observations with the scientific instruments aboard Yohkoh. Those preflare signatures and their changes containing essential information about the mechanism of flares, now allow us to pursue truer understanding about the flare mechanism. The same merits of Yohkoh (wide-dynamic range and high-cadence observations) have shown us for the first time in its full form the highly dynamical behavior of the faint background corona, together with the influence of the changes in active regions sometimes exerting overwhelming effects on the surrounding corona.

FOREWORD

Broadly speaking, the inverse problem is the inverse of the forward problem. In the case of contemporary helioseismology, the forward problem is usually posed as that of determining the eigenfrequencies of free oscillation of a theoretical model of the sun. That problem is discussed by Christensen-Dalsgaard in this volume. I call inverting that problem the ‘main’ inverse problem. It is the one that I shall be discussing almost exclusively in this chapter. But also included in the forward problem must be the theoretical modelling of the oscillations as they really occur in the sun, forced, we believe, predominantly by the turbulence in the convection zone, and modulated by their nonlinear interactions with other modes of oscillation and by the perturbations they induce to the very convection that drives them, through variations in the turbulent fluxes of heat and momentum. The inverse of that problem is to derive from the fluid motion of the visible layers in the atmosphere of the sun, which I presume to be ‘observed’, estimates of the frequencies that the modes would have had had they not been disturbed by the other forms of motion. The outcome of that prior inversion provides the data for the main inverse problem.

This chapter is entitled: Testing solar models …. By ‘solar models’ is meant any theoretical description of the sun that we might have in mind.

INTRODUCTION

The present chapter addresses the forward problem, i.e., the relation between the structure of a solar model and the corresponding frequencies. As important, however, is the extent to which the frequencies reflect the physics and other assumptions underlying the model calculation. Thus in Section 2 I consider some aspects of solar model computation. In addition, the understanding of the diagnostic potential of the frequencies requires information about the properties of the oscillations, which is provided in Section 3. Section 4 investigates the relation between the properties of solar structure and the oscillations by considering several examples of modifications to the solar models and their effects on the frequencies, while Section 5 considers further analyses of the observed frequencies. Finally, the prospects of extending this type of work to other stars are addressed in Section 6.

A more detailed background on the theory of solar oscillations was given, for example, by Christensen-Dalsgaard & Berthomieu (1991), Gough (1993), and Christensen-Dalsgaard (1994). For other general presentations of the properties of solar and stellar oscillations see, e.g., Unno et al. (1989) and Gough & Toomre (1991).

A little history

The realization that observed frequencies of solar oscillation might provide information about the solar interior goes back at least two decades. Observations of fluctuations in the solar limb intensity (Hill & Stebbins 1975; Hill, Stebbins & Brown 1976), and the claimed detection of a Doppler velocity oscillation with a period close to 160 minutes (Brookes, Isaak & van der Raay 1976; Severny, Kotov & Tsap 1976) provided early indications that global solar oscillations might be detectable and led to the first comparisons of the reported frequencies with those of solar models (e.g. Scuflaire et al. 1975; Christensen-Dalsgaard & Gough 1976; Iben & Mahaffy 1976; Rouse 1977).

ANALYSIS TOOLS AND THE SOLAR NOISE BACKGROUND

When we observe solar oscillations, we are concerned with measuring perturbations on the Sun that are almost periodic in space and time. The periodic waves that interest us are, however, embedded in a background of broadband noise from convection and other solar processes, which tend to obscure and confuse the information we want. Also (and worse), the “almost-periodic” nature of the waves leads to problems in the interpretation of the time series that we measure. Much of the subject of observational helioseismology is thus concerned with ways to minimize these difficulties.

Fourier Transforms and Statistics

A common thread runs through all of the analysis tricks that one plays when looking at solar oscillations data, and indeed through many of the purely instrumental concerns as well: this thread is the Fourier transform. The reason for this commonality is, of course, that we are dealing with (almost) periodic phenomena – either the acoustic-gravity waves themselves, or the light waves that bring us news of them. Since many of the same notions will recur repeatedly, it is worth taking a little time (and boring the cognoscente) to review some of the most useful properties of Fourier transforms and power spectra. In what follows, I shall simply state results and indicate some of the more useful consequences. We shall see below that even when the Big Theorems of Fourier transforms do not apply, (as with Legendre transforms, for instance), analogous things happen, so that the Fourier example is a helpful guide to the kind of problems we may have.