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Workshop Programme

for period 6 - 17 September 2004

Magnetohydrodynamics of Stellar Interiors

6 - 17 September 2004

Timetable

Monday 6 September
11:30-12:35 Rosner, R (Chicago)
  Stellar magnetic activity (overview) Sem 1
15:30-16:40 Moffatt, HK (Cambridge)
  Introduction to MHD and dynamo theory I Sem 1
 

Magnetic fields in astrophysics are generated by the inductive action of turbulence in the conducting fluid medium. This turbulence is usually generated by buoyancy forces and strongly influenced by coriolis effects, and in consequence 'lacks reflexional symmetry'; in particular, the mean helicity is nonzero, i.e. there is a correlation between velocity and vorticity fields. This property in general leads to an 'alpha-effect' in the fluid, whereby magnetic field grows on length-scales large compared with the dominant energy-containing scale of the turbulence. At the same time, the turbulent diffusivity controls the growth of the field. The primary problem of mean-field dynamo theory is to obtain reliable expressions for alpha and for the turbulent diffusivity in terms of the statistical properties and the magnetic Reynolds number of the turbulence. The first lecture will be concerned with this problem.

The second lecture will focus on dynamic back-reaction effects: as the magnetic field grows by turbulent dynamo action, the Lorentz force ultimately modifies the turbulence tending to reduce both alpha and turbulent diffusivity, until some kind of equilibrium is established, this equilibrium depending on the mechanism by which energy is supplied to the turbulence. Some aspects of this problem, which is the subject of much current debate, will be considered.

 
16:40-17:05 Kleeorin, N (Beer Sheva)
  Universal mechanism of dynamo saturation:dynamics of magnetic helicity Sem 1
 

The nonlinear saturation mechanism based on the magnetic helicity evolution is discussed. It is shown that this universal mechanism is nearly independent of the form of the flux of magnetic helicity, and it requires only a nonzero flux of magnetic helicity. Different forms of the flux of magnetic helicity are discussed. We also studied a simple model for the solar dynamo in the framework of the Parker migratory dynamo, with a nonlinear dynamo saturation mechanism based on magnetic helicity conservation arguments. We found a parameter range in which the model demonstrates a cyclic behaviour with properties similar to that of Parker dynamo with the simplest form of algebraic alpha-quenching. We compared the nonlinear current helicity evolution in this model with data for the current helicity evolution obtained during 10 years of observations at the Huairou Solar Station of China. On one hand, our simulated data demonstrate behaviour comparable with the observed phenomenology, provided that a suitable set of governing dynamo parameters is chosen. On the other hand, the observational data are shown to be rich enough to reject some other sets of governing parameters. We conclude that, in spite of the very preliminary state of the observations and the crude nature of the model, the idea of using observational data to constrain our ideas concerning magnetic field generation in the framework of the solar dynamo appears promising.

 
17:05-17:30 Busse, FH (Bayreuth)
  Finite amplitude convection in rotating spherical fluid shells and its Prandtl number dependence Sem 1
Tuesday 7 September
09:00-10:10 Moffatt, HK (Cambridge)
  Introduction to MHD and dynamo theory II Sem 1
 

Magnetic fields in astrophysics are generated by the inductive action of turbulence in the conducting fluid medium. This turbulence is usually generated by buoyancy forces and strongly influenced by coriolis effects, and in consequence 'lacks reflexional symmetry'; in particular, the mean helicity is nonzero, i.e. there is a correlation between velocity and vorticity fields. This property in general leads to an 'alpha-effect' in the fluid, whereby magnetic field grows on length-scales large compared with the dominant energy-containing scale of the turbulence. At the same time, the turbulent diffusivity controls the growth of the field. The primary problem of mean-field dynamo theory is to obtain reliable expressions for alpha and for the turbulent diffusivity in terms of the statistical properties and the magnetic Reynolds number of the turbulence. The first lecture will be concerned with this problem.

The second lecture will focus on dynamic back-reaction effects: as the magnetic field grows by turbulent dynamo action, the Lorentz force ultimately modifies the turbulence tending to reduce both alpha and turbulent diffusivity, until some kind of equilibrium is established, this equilibrium depending on the mechanism by which energy is supplied to the turbulence. Some aspects of this problem, which is the subject of much current debate, will be considered.

 
10:10-10:35 Rogachevskii, I (Beer Sheva)
  Effect of differential rotation on nonlinear mean electromotive force and stellar dynamos Sem 1
 

An effect of the mean differential rotation on the nonlinear electromotive force is found. It includes a nonhelical $\alpha$ effect which is caused by a differential rotation, and it is independent of a hydrodynamic helicity. There is no quenching of this effect contrary to the quenching of the usual $\alpha$ effect caused by a hydrodynamic helicity. The nonhelical $\alpha$ effect vanishes when the rotation is constant on the cylinders which are parallel to the rotation axis. The mean differential rotation causes the "shear-current" effect. The ''shear-current" effect is associated with the $\bar{\bf W} {\bf \times} \bar{\bf J}$-term in the mean electromotive force and results in the generation of the mean magnetic field even in a nonhelical homogeneous turbulence (where $\bar{\bf W}$ is the mean vorticity caused by the differential rotation and $\bar{\bf J}$ is the mean electric current). The ''shear-current" effect changes its sign with the nonlinear growth of the mean magnetic field at some value $\bar{\bf B}_\ast$. The magnitude $\bar{\bf B}_\ast$ determines the level of the saturated mean magnetic field which is less than the equipartition field. However, there is no quenching of this effect. It is shown that the background magnetic fluctuations due to the small-scale dynamo enhance the "shear-current" effect, and reduce the magnitude $\bar{\bf B}_\ast$. When the level of the background magnetic fluctuations is larger than $1/3$ of the kinetic energy of the turbulence, the mean magnetic field can be generated due to the "shear-current" effect for an arbitrary exponent of the energy spectrum of the velocity fluctuations. These phenomena determine the nonlinear evolution of the stellar and solar large-scale magnetic fields. An effect of a uniform rotation on the nonlinear electromotive force is also studied. A nonlinear theory of the ${\bf \Omega} {\bf \times} \bar{\bf J}$ effect is developed, and the quenching of the hydrodynamic part of the $\alpha$ effect which is caused by a uniform rotation and inhomogeneity of turbulence, is found. Other contributions of a uniform rotation to the nonlinear electromotive force are also determined. All these effects are studied using the $\tau$-approximation (the Orszag third-order closure procedure). An axisymmetric mean-field dynamo is considered. Applications of these effects to the stellar and solar large-scale magnetic fields are discussed.

Related Links

 
10:35-11:00 Arlt, R (AIP, Potsdam)
  Magneto-rotational instability in the solar core and Ap star envelopes Sem 1
 

The radiative core of the Sun rotates uniformly as suggested by helioseismological observations. It also rotates as slow as the average surface rotation. The pre-main squence Sun probably rotated much faster. If surface breaking has slowed down the Sun, a fast angular-momentum transport would be necessary to reduce the rotation rate of the core, too. Pure microscopic viscosiy is not sufficient. The problem of slow-down and equalizing of the core rotation is attributed to the magneto-rotational instability. Time-scales of 10-100 mill yr are found. A similar scenario may be possible in the radiative envelopes of Ap stars. Given their shorter life-times, the redistribution of angular momentum must be an ongoing process.

 
11:25-12:30 Gough, DO (Cambridge)
  What we know about stellar interiors (overview) Sem 1
15:30-16:40 Title, AM (LMSAL, Palo Alto)
  Observations of magnetic fields on the Sun I Sem 1
 

New and old techniques allow us to observe sunspots in a range of wavelengths, to determine the associated flow systems at a range of heights, to detect oscillations and waves and to determine the strength and angular distributions of their magnetic fields. This lecture will provide a selection of examples of processes associated with the sunspot phenomena. But rather that providing a comprehensive review, the goal of the talk is to raise questions

 
16:40-17:05 Voegler, A (MPS, Katlinburg-Lindau)
  Decay of a simulated bipolar magnetic field in the solar surface layers Sem 1
 

Using MURaM - a MHD code designed for applications in the solar photosphere and convection zone, we have studied the evolution of a mixed-polarity magnetic field in the surface layers of the Sun. The simulations, which have a horizontal extent of 6 Mm x 6 Mm and contain the visible surface around optical depth unity, include a detailed treatment of non-local radiative transfer effects in the photosphere in order to allow a direct comparison with observations. We discuss the time-evolution of magnetic structures and their statistical properties, present details of the flux-cancellation process and comment on its signatures in simulated observations of our numerical model. We show that the rate at which the field decays is consitent with the expected turbulent decay time scale and discuss the dependence of the decay rate on the initial conditions.

 
17:05-17:30 Dumin, YV (IZMIRAN, Troitsk)
  Heating the stellar plasma in the transitional region from the inductive to drift freezing of magnetic field Sem 1
 

Because of a dramatic variation in the plasma density along the radius of a star, the process of magnetic field freezing occurs there in two substantially different regimes: in the lower (collisional) layers it is caused by generation of circular currents , compensating variations of the magnetic field in the co-moving (with plasma) flux tubes; whereas in the upper (collisionless) layers the magnetic field freezing is maintained by synchronism of the plasma drift, due to a divergence-free character of the magnetic field and an electric equipotentiality of the magnetic field lines [e.g. Yu.V. Dumin, Solar Sys. Res., v.32, p.323 (1998)]. Transitional region between the two regimes of freezing is a natural place for reconstruction of the current systems and, therefore, the energy release. In the particular case of the Sun (where all parameters are known with a reasonable accuracy), predicted position of the transitional region corresponds very well to the zone of sharp temperature increase in the base of the solar corona [Yu.V. Dumin, Adv. Space Res., v.30, p.565 (2002)].

 
Wednesday 8 September
09:00-10:10 Cameron, AC (St Andrews)
  Magnetic activity in rapidly rotating stars I Sem 1
10:10-10:35 Ferriz-Mas, A (Vigo)
  Influence of external flows on the stability of magnetic flux tubes in a stellar convection zone Sem 1
 

A fundamental question in connection with the dynamo mechanism is how to retain the magnetic flux in the convection zone (or below) for a time sufficiently long to be amplified by differential rotation. Magnetic buoyancy poses a serious problem, for it might lead to a rapid loss of magnetic flux from the convection zone and thus prevent the operation of the dynamo.

Apart from observations of the solar surface, there are indirect hints for the existence of a strong toroidal system of magnetic flux at the bottom of the solar convection zone. The concentration of magnetic flux into flux tubes has important consequences for its storage. In this context, the stability of toroidal flux tubes has been subject of research since the 1980's, starting with Spruit and van Ballegooijen (1982).

In this talk I will report on an ongoing investigation which extends previous research on the stability properties of a thin toroidal flux tube. A new feature is the inclusion of external velocity fields other than rotation; this is motivated by the necessity of including the effects of a meridional flow into the stability analysis.

 
10:35-11:00 Steiner, O (KISP, Freiburg)
  Connecting solar irradiance variability to the solar dynamo using the virial theorem Sem 1
 

The variability of solar radiance over a solar cycle is thought the result of a delicate balance between the blocking effect of sunspots and the positive contribution of bright plage and network faculae. Although the net effect is small, it must imply structural changes of the Sun or partial layers of it as an unavoidable consequence of the virial theorem. Using a general form of the virial theorem for continua including the magnetic field it is shown, how solar radiance variability might be connected to a deeply seated flux-tube dynamo and how this connection is established on a hydrodynamical time-scale.

Related Links

 
11:25-12:35 Title, AM (LMSAL, Palo Alto)
  Observations of magnetic fields on the Sun II Sem 1
 

New and old techniques allow us to observe sunspots in a range of wavelengths, to determine the associated flow systems at a range of heights, to detect oscillations and waves and to determine the strength and angular distributions of their magnetic fields. This lecture will provide a selection of examples of processes associated with the sunspot phenomena. But rather that providing a comprehensive review, the goal of the talk is to raise questions

 
Thursday 9 September
09:00-10:10 Cameron, AC (St Andrews)
  Magnetic activity in rapidly rotating stars II Sem 1
10:10-10:35 Tavakol, R (QMUL)
  Dynamo models and differential rotation in sun and stars Sem 1
10:35-11:00 Galloway, DJ (Sydney)
  Different limiting mechanisms for nonlinear dynamos Sem 1
 

Theoreticians often study nonlinear dynamos by postulating a specific force field designed to produce a flow which is known to give rise to an effective kinematic dynamo. The subsequent evolution is then followed numerically to determine how the dynamo equilibrates. The most studied example is the (1,1,1) ABC flow, with the supplied forcing proportional to the inverse kinetic Reynolds number. In this case, scaling arguments can be adduced which give very pessimistic estimates for the high Reynolds number performance of the dynamo. Recently Archontis (PhD thesis, 2000) found an interesting example of a dynamo where the performance is far superior, with approximately equal scaled magnetic and velocity fields which are very close to (sin z,sin x,sin y) when the kinetic and magnetic Reynolds numbers are large. Numerical results will be described which attempt to show how and why this and some similar dynamos work so well, and whether such behaviour can be expected in real astrophysical objects.

 
11:25-12:35 Proctor, MRE (Cambridge)
  Magnetoconvection I Sem 1
 

In these lectures I give a description, starting from basic concepts, of the effects of magnetic fields on convection and of convection on the fields. The principal motivation for studying such interaction is given by observations of photospheric magnetic fields, which are manifested on a wide range of scales from the largest sunspots down to small network bright points.

In the first lecture I shall discuss the fundamental kinematics and dynamics of the interaction, with emphasis on simplified models to illustrate the main ideas, including flux concentration, horizontal scale selection, the occurrence of oscillations, and local evacuation. In the second lecture I shall focus on numerical models of various types, which aim to simulate observed behaviour in the Sun.

Related Links

 
15:30-16:40 Hartmann, LW (CFA, Cambridge, Mass)
  Young stars I Sem 1
 

Young solar-type stars exhibit strongly enhanced levels of magnetic activity. In addition, a complex set of behaviors arise from the presence of circumstellar accretion disks with their own magnetic fields, along with the interaction between the stellar magnetic field and the disk. I will review the observational evidence for the star-disk-magnetosphere paradigm, discuss angular momentum regulation, and point out current theoretical challenges.

 
16:40-17:05 Matthews, PC (Nottingham)
  Localised states in magnetoconvection Sem 1
 

Convection in an imposed vertical magnetic field can lead to localised states in domains of large horizontal extent. This phenomenon can be seen as an instability of convection rolls that arises naturally from the fact that the magnetic flux through the layer is conserved. Nonlinear states in two dimensions can be found involving only a single convection cell (known as a "convecton") surrounded by almost stationary fluid, or, conversely, a single column of concentrated magnetic flux surrounded by convection cells. Similar localised solutions can be found when convection is oscillatory. In three dimensions, these states are unstable and the behaviour is usually time-dependent, but localisation can still occur.

 
17:05-17:30 Rincon, F (Toulouse)
  Large scale simulations of compressible convection Sem 1
 

We present the results of three-dimensional direct numerical simulations of fully compressible polytropic turbulent convection in a very large aspect ratio ($\lambda=42.6$) cartesian box with $1024\times 1024\times 82$ mesh points. We investigate the general properties of this flow and discuss the similarities and differences between such an idealized experiment and the known properties of photospheric convection. We particularly focus on the emergence of large-scale coherent structures in the simulation and consider the potential implications of these results for the problem of the origin of mesogranular and supergranular flows and the large scale distribution of magnetic fields at the solar surface.

 
Friday 10 September
09:00-10:10 Proctor, MRE (Cambridge)
  Magnetoconvection II Sem 1
 

In these lectures I give a description, starting from basic concepts, of the effects of magnetic fields on convection and of convection on the fields. The principal motivation for studying such interaction is given by observations of photospheric magnetic fields, which are manifested on a wide range of scales from the largest sunspots down to small network bright points.

In the first lecture I shall discuss the fundamental kinematics and dynamics of the interaction, with emphasis on simplified models to illustrate the main ideas, including flux concentration, horizontal scale selection, the occurrence of oscillations, and local evacuation. In the second lecture I shall focus on numerical models of various types, which aim to simulate observed behaviour in the Sun.

Related Links

 
10:00-10:35 Bushby, PJ (Cambridge)
  Modelling photospheric magnetoconvection in the weak field regime Sem 1
 

High resolution observations can now provide detailed information regarding the complex interactions between convective motions and magnetic fields in the solar photosphere. This problem is investigated theoretically through numerical simulation of three dimensional compressible magnetoconvection in a large aspect ratio box. Attention is focused on the regime in which the imposed vertical magnetic field is relatively weak. Turbulent convection patterns and intermittent field structures are found. For a moderately weak imposed field, large elongated magnetic structures are found, similar to those found in plage regions on the solar surface. Decreasing the total magnetic flux further causes the magnetic structures to become almost point-like, similar to those seen in the quiet Sun. Calculation of the fractal dimension allows quantitative analysis of the surface distribution of the magnetic field. Although coming from idealised simulations, these results can be closely related to solar observations.

 
10:35-11:00 Nordlund, A (Copenhagen)
  Current status of facular region and sunspot models Sem 1
11:25-12:35 Hartmann, LW (CFA, Cambridge, Mass)
  Young stars II Sem 1
 

Young solar-type stars exhibit strongly enhanced levels of magnetic activity. In addition, a complex set of behaviors arise from the presence of circumstellar accretion disks with their own magnetic fields, along with the interaction between the stellar magnetic field and the disk. I will review the observational evidence for the star-disk-magnetosphere paradigm, discuss angular momentum regulation, and point out current theoretical challenges.

 
15:30-16:40 Thomas, JH (Rochester)
  Sunspots I Sem 1
 

Sunspots provide the best test of magnetohydrodynamic theory under astrophysical conditions. Nowhere else in astrophysics is the theory confronted with such a wealth of detailed observations. Recent, remarkable advances in high-resolution observations provide us with key information that allows us to begin to assemble a coherent picture of the formation of a sunspot, its complicated magnetic and thermal structure, and associated flows and oscillations. Numerical simulations of nonlinear magnetoconvection are beginning to reproduce some of the fine structure observed in a sunspot, including umbral dots, penumbral grains, and light bridges. A new picture of penumbral structure has emerged from the observations, involving two components, with different magnetic field inclination, that remain essentially distinct over the lifetime of the spot. The darker component, in which the magnetic field is more nearly horizontal, includes "returning" magnetic flux tubes that dive back down below the solar surface near the outer edge of the penumbra. These arched flux tubes carry most of the photospheric Evershed flow, which can be attributed to siphon flows driven by pressure drops along thin flux tubes. The returning flux tubes and the curious "interlocking-comb" structure of the penumbral magnetic field can be understood to be a consequence of downward pumping of magnetic flux by the turbulent granular convection in the moat surrounding a sunspot. This robust flux-pumping mechanism, which has been demonstrated in three-dimensional numerical simulations of fully compressible convection, is an important key to understanding the formation and maintenance of the penumbra and the behavior of moving magnetic features in the moat.

Another key to understanding the structure of a sunspot is the array of characteristic oscillations observed in the umbra and penumbra, which serve as a probe of sunspot structure. The techniques of helioseismology have shown that sunspots absorb a significant fraction of the power in incident p-modes. This absorption seems to be due to a conversion of acoustic waves to slow magneto-acoustic waves that leak downward out of the p-mode cavity. Time-distance helioseismology has been used to detect flow patterns in the convection zone beneath a sunspot. A better understanding of the interaction between acoustic and magnetic-acoustic waves in realistic sunspot models is needed in order for the techniques of sunspot seismology to reach their full potential.

 
16:40-17:05 Birch, AC (Colorado Research Associates)
  Forward modeling for local helioseismology Sem 1
 

In order to interpret measurements made with local helioseismology it is crucial to understand the sensitivity of these measurements to different types of perturbations to a solar model. Local helioseismic measurements are sensitive not only to subsurface mass flows, sound-speed variations, and magnetic field, but also to local changes in the wave damping and excitation rates. I will show example calculations of the sensitivity of travel-times measured by time-distance helioseismology and phase shifts measured by acoustic holography to flows, sound speed perturbations, and local changes in the wave excitation and damping rates.

 
Monday 13 September
09:00-10:10 Scharmer, GB (Stockholm)
  Fine structure of solar magnetic fields Sem 1
10:10-10:35 Moreno-Insertis, F (IAC, Tenerife)
  Emergence of magnetic flux into the solar atmosphere: three dimensional experiments Sem 1
10:35-11:00 Brummell, NH (Boulder, CO)
  What is a magnetic flux tube? Sem 1
 

Using 3D nonlinear MHD simulations, we examine the interaction of localised velocity shear and magnetic fields. We find that buoyant magnetic structures resembling flux tubes are generated by the interaction of induced magnetic buoyancy and the shear. Depending on the parameters, the generation process can result in steady (equilibrated) structures, or cyclic and chaotic emerging structures. By integrating along magnetic field lines and constructing return maps, we examine the relationship of these spontaneously-produced structures to the preconceived concept of a flux tube. We discuss how these results impact our simple ideas of a flux tube as an object with an inside and an outside whose dynamics are constrained by this assertion.

 
11:25-12:35 Thomas, JH (Rochester NY)
  Sunspots II Sem 1
 

Sunspots provide the best test of magnetohydrodynamic theory under astrophysical conditions. Nowhere else in astrophysics is the theory confronted with such a wealth of detailed observations. Recent, remarkable advances in high-resolution observations provide us with key information that allows us to begin to assemble a coherent picture of the formation of a sunspot, its complicated magnetic and thermal structure, and associated flows and oscillations. Numerical simulations of nonlinear magnetoconvection are beginning to reproduce some of the fine structure observed in a sunspot, including umbral dots, penumbral grains, and light bridges. A new picture of penumbral structure has emerged from the observations, involving two components, with different magnetic field inclination, that remain essentially distinct over the lifetime of the spot. The darker component, in which the magnetic field is more nearly horizontal, includes "returning" magnetic flux tubes that dive back down below the solar surface near the outer edge of the penumbra. These arched flux tubes carry most of the photospheric Evershed flow, which can be attributed to siphon flows driven by pressure drops along thin flux tubes. The returning flux tubes and the curious "interlocking-comb" structure of the penumbral magnetic field can be understood to be a consequence of downward pumping of magnetic flux by the turbulent granular convection in the moat surrounding a sunspot. This robust flux-pumping mechanism, which has been demonstrated in three-dimensional numerical simulations of fully compressible convection, is an important key to understanding the formation and maintenance of the penumbra and the behavior of moving magnetic features in the moat.

Another key to understanding the structure of a sunspot is the array of characteristic oscillations observed in the umbra and penumbra, which serve as a probe of sunspot structure. The techniques of helioseismology have shown that sunspots absorb a significant fraction of the power in incident p-modes. This absorption seems to be due to a conversion of acoustic waves to slow magneto-acoustic waves that leak downward out of the p-mode cavity. Time-distance helioseismology has been used to detect flow patterns in the convection zone beneath a sunspot. A better understanding of the interaction between acoustic and magnetic-acoustic waves in realistic sunspot models is needed in order for the techniques of sunspot seismology to reach their full potential.

 
15:30-16:40 Cattaneo, F (Chicago)
  Small-scale dynamos I Sem 1
 

Small-scale dynamo action describes the generation of magnetic fields on scales comparable with, or smaller than the characteristic scale of the velocity. It is believed to occur quite naturally in turbulent fluids when the magnetic Reynolds number --- a dimensionless measure of the electrical conductivity --- is sufficiently high. In general terms dynamo action succeeds if, on average, field amplification exceeds field destruction. In a turbulent system, field generation is due to the stretching of field lines by the flow, while field destruction is due to enhanced diffusivity. In these lectures I will review some of the efforts to provide a quantitative description of these two processes. I will introduce ideas like the Lyapunov exponents and the topological entropy to measure the stretching rate, and the cancellation exponent to measure the rate of enhanced diffusion. I will also distinguish between dynamos driven by smooth velocities and dynamos driven by rough velocities, i.e. velocities that are strongly fluctuating on the scale at which magnetic reconnection occurs. Physically these two cases correspond to fluids whose magnetic Prandtl number---the ratio of the viscosity to the magnetic diffusivity---is larger (smooth case), or smaller (rough case) than unity.

 
16:40-17:05 Kosovichev, AG (Stanford)
  Helioseismic observations of magnetohydrodynamics of the solar interior Sem 1
 

Observations of the Sun's interior with the MDI instrument on board the SOHO spacecraft has provided tremendous amount of new information about basic physical MHD processes inside the Sun, formation of magnetic structures in the solar plasma and mechanisms of solar and stellar activity. In particular, the new results have revealed the deep structure of sunspots and associated complicated patterns of plasma flows, the dynamics of the emerging magnetic flux and formation of active regions, the supergranular structure and dynamics of the upper convection zone, as well as the global structures and circulation patterns in the deep interior, evolving with the activity cycle. In addition, first attempts are made to find the links between the internal dynamics and processes of magnetic energy release in the solar corona. Understanding of these results, often puzzling and counter-intuitive, represents a major challenge for MHD theories of astrophysical plasma.

 
Tuesday 14 September
09:00-10:10 Spruit, HC (MPA, Garching)
  Magnetohydrodynamics of stably stratified stars Sem 1
 

Magnetic fields can be created in stably stratified (non-convective) layers in a differentially rotating star. A magnetic instability in the toroidal field (wound up by differential rotation) replaces the role of convection in closing the field amplification loop.Tayler instability is likely to be the most relevant magnetic instability. A dynamo model is developed from these ingredients, and applied to the problem of angular momentum transport in stellar interiors. It produces a predominantly horizontal field. This dynamo process is found to be more effective in transporting angular momentum than the known hydrodynamic mechanisms. It may account for the observed pattern of rotation in the solar core.

Related Links

 
10:10-10:35 Cameron, R (MPS, Katlinburg-Lindau)
  Photospheric MHD simulations of solar pores Sem 1
 

We have performed simulations of solar pores using the MURaM -- Max-Planck Institute for Aeronomy, University of Chicago, RAdiative Magnetohydrodynamics -- code. The code is a MHD code and includes both radiative transfer and the effects of partial ionization, both of which are necessary for a realistic treatment of the photosphere. The simulations allow a direct comparison with observations to be made. We shall present such a comparison of the temperature structure, the magnetic and velocity fields, and the centre to limb variation of the intensity. Part of the value of this type of simulation is that the knowledge of the full state of the system allows the physics to be understood, and so we shall relate our comparison between the simulation and observation to the underlying physics.

 
10:35-11:00 Brandenburg, A (Nordita, Copenhagen)
  Catastrophic alpha quenching alleviated by helicity flux and shear Sem 1
 

A new simulation set-up is proposed for studying mean field dynamo action. The model combines the computational advantages of local cartesian geometry with the ability to include a shear profile that resembles the sun's differential rotation at low latitudes. It is shown that in a two-dimensional mean field model this geometry produces cyclic solutions with dynamo waves traveling away from the equator -- as expected for a positive alpha effect in the northern hemisphere. In three dimensions with turbulence driven by a helical forcing function, an alpha effect is self-consistently generated in the presence of a finite imposed toroidal magnetic field. The results suggest that, due to a finite flux of current helicity out of the domain, alpha quenching appears to be non-catastrophic -- at least for intermediate values of the magnetic Reynolds number. For larger values of the magnetic Reynolds number, however, there is evidence for a reversal of the trend and that $\alpha$ may decrease with increasing magnetic Reynolds number. Control experiments with closed boundaries confirm that in the absence of a current helicity flux, but with shear as before, alpha quenching is always catastrophic and alpha decreases inversely proportional to the magnetic Reynolds number. For solar parameters, our results suggest a current helicity flux of about 0.001 G^2/s. This corresponds to a magnetic helicity flux, integrated over the northern hemisphere and over the 11 year solar cycle, of about 10^{46}Mx^2.

Related Links

 
11:25-12:35 Cattaneo, F (Chicago)
  Small-scale dynamos II Sem 1
 

Small-scale dynamo action describes the generation of magnetic fields on scales comparable with, or smaller than the characteristic scale of the velocity. It is believed to occur quite naturally in turbulent fluids when the magnetic Reynolds number --- a dimensionless measure of the electrical conductivity --- is sufficiently high. In general terms dynamo action succeeds if, on average, field amplification exceeds field destruction. In a turbulent system, field generation is due to the stretching of field lines by the flow, while field destruction is due to enhanced diffusivity. In these lectures I will review some of the efforts to provide a quantitative description of these two processes. I will introduce ideas like the Lyapunov exponents and the topological entropy to measure the stretching rate, and the cancellation exponent to measure the rate of enhanced diffusion. I will also distinguish between dynamos driven by smooth velocities and dynamos driven by rough velocities, i.e. velocities that are strongly fluctuating on the scale at which magnetic reconnection occurs. Physically these two cases correspond to fluids whose magnetic Prandtl number---the ratio of the viscosity to the magnetic diffusivity---is larger (smooth case), or smaller (rough case) than unity.

 
15:30-16:40 Thompson, MJ (Sheffield)
  Internal rotation of the Sun I: Results from helioseismology Sem 1
 

In this first of two lectures on the Sun's internal rotation and seismic probing of solar and stellar interiors, I shall review the results obtained with helioseismology regarding the internal rotation of the Sun, and compare those inferences with the results coming out of numerical simulations.

 
16:40-17:05 Haugen, NEL (Trondheim/Cambridge)
  Numerical simulations of small scale dynamo activity; spectra and critical magnetic Reynolds numbers Sem 1
 

There is as yet no consensus on the appearance of the large Reynolds number energy spectrum of non-helical MHD turbulence without background field. We use direct numerical simulations with up to 1024^3 meshpoints, together with simulations with hyperdiffusion, in an attempt to see this large Reynolds number limit. We show that the energy spectra consist of a sub-inertial range, around the forcing wavenumber, where the magnetic energy is in sub-equipartition. At larger wavenumbers there is first a range where the magnetic energy is in super-equipartition, followed by a possible equipartition range with a k^(-5/3) slope, before we see a small bottleneck and the diffusive range at the smallest scales. We also investigate the critical magnetic Reynolds number (Rmc) for dynamo action. We find Rmc to be at a minimum for magnetic Prandtl numbers slightly above unity. For magnetic Prandtl numbers smaller than unity we find Rmc=Re^alpha, where alpha is 1/3 for magnetic Prandtl numbers larger than 1/8, and possibly 1 for smaller magnetic Prandtl numbers.

 
Wednesday 15 September
09:00-10:10 Thompson, MJ (Sheffield)
  Internal rotation of the Sun II Sem 1
 

In the first lecture I considered the results from helioseismic probing of the Sun's internal rotation. In this lecture I focus on the theoretic and analytical methods used for making inferences about the rotation of the interior of the Sun and stars from seismological observations.

 
10:10-10:35 Schekochihin, AA (Camrbridge)
  Magnetic-field generation in low-magnetic-Prandtl-number plasmas Sem 1
 

Magnetic Prandtl number Pm is a key parameter for astrophysical MHD fluids. The Pm>>1 regime is realised in high-temperature low-density plasmas of galaxies and clusters. It has been firmly established both theoretically and numerically that large-Pm turbulent plasmas can generate equipartition-level magnetic fluctuation energy via the small-scale dynamo. In numerical simulations, this regime qualitatively persists down to values of Pm~1 [astro-ph/0312046]. The plasmas in stellar (solar) convective zones and protostellar discsare denser and have low Pm. There is ample observational evidence that the solar photosphere contains large amounts of small-scale magnetic field. This field may be generated by small-scale dynamo or induced via shredding by turbulence of the large-scale ("mean") solar field --- or both. We have recently shown numerically that small-scale dynamo in the low-Pm is problematic: it either does not exist at all (i.e., there is a critical Pm_c) or requires extremely large magnetic Reynolds numbers (i.e., there is a critical Rm_c) numerically inaccessible at current resolutions [PRL 92, 054502 (2004)]. I will discuss these numerical results as well as report some new ones that improve on them. I will also discuss theoretical arguments in favour of and against the dynamo. I emphasise that there is no numerical or laboratory evidence available at present that would show that low-Pm turbulence is a dynamo, nor is there a physical scenario that would explain how such a dynamo is possible. In this context, small-scale magnetic fluctuations induced by a mean field acquire renewed relevance. While it is not possible to perform adequately resolved simulations that incorporate both the self-consistent generation of the large-scale fields and the small-scale turbulence, it is certainly possible to study the effect of an imposed mean field on the latter. I will report an extensive numerical study of the properties of induced small-scale fields (in nonhelical turbulence). Their possible role in explaining the photospeheric fields and in quenching the mean-field dynamo mechanisms will be discussed. Furthermore, these results are subject to direct comparison with experimental liquid-metal results of laboratory dynamo experiments in uncons�trained geometries (e.g., Lyon, Maryland, and Wisconsin). Finally, I will present some analytical considerations on the interaction between large- and small-scale dynamo-generated magnetic fields in the case of large scale seprations between the system size, the turbulence scale, and the magnetic dissipation scale.

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10:35-11:00 Brun, AS (Saclay)
  Dynamo action in turbulent convective shells with or without rotation Sem 1
11:25-12:35 Tobias, SM (Leeds)
  Solar and stellar dynamos I Sem 1
 

These lectures will provide an introduction to the mathematics and physics underpinning research into solar and stellar dynamos. Here I will focus on the large scale fields generated in these stars.

In the first lecture I shall review the observations of large-scale magnetic activity in the Sun and other stars, including historical and proxy data. I shall then review the current paradigms for solar dynamo action discussing the physics behind both interface and flux transport dynamos

In the second lecture I shall review in detail the attempts to model solar and stellar dynamos, discussing large-scale computations, mean-field electrodynamics and illustrative low-order models. I shall discuss the limitations of each approach, and conclude by speculating on possible future avenues of research.

 
Thursday 16 September
09:00-10:10 Tobias, SM (Leeds)
  Solar and stellar dynamos II Sem 1
 

These lectures will provide an introduction to the mathematics and physics underpinning research into solar and stellar dynamos. Here I will focus on the large scale fields generated in these stars.

In the first lecture I shall review the observations of large-scale magnetic activity in the Sun and other stars, including historical and proxy data. I shall then review the current paradigms for solar dynamo action discussing the physics behind both interface and flux transport dynamos

In the second lecture I shall review in detail the attempts to model solar and stellar dynamos, discussing large-scale computations, mean-field electrodynamics and illustrative low-order models. I shall discuss the limitations of each approach, and conclude by speculating on possible future avenues of research.

 
10:10-10:35 Ruzmaikin, AA (JPL, Pasadena)
  Non-axisymmetric solar magnetic fields Sem 1
 

Solar magnetic activity tends to cluster at ``preferred" longitudes, which indicates the involvement of non-axisymmetric large-scale (mean) magnetic fields in the process of the activity formation. We investigate the generation of non-axisymmetric modes of the mean solar magnetic field and their coupling to the axisymmetric mode.

Our dynamo model incorporates the solar rotation reconstructed by inversion of helioseismic data in the convection zone and simulated distributions of the turbulent resistivity and the mean kinetic helicity. We demonstrate first that the dynamo breaks the axial symmetry by exciting non-axisymmetric modes even when all sources of generation are axisymmetric. Then we couple axisymmetric and non-axisymmetric modes using a non-axisymmetric addition to the mean helicity. Mathematically, it is done in the kinematic approximation without the use of a non-linear quenching.

We find that this coupling of non-axisymmetric and axisymmetric modes (1) reproduces the phase relation between these modes observed in the solar cycle; (2) the non-axisymmetric modes are localized near the base of the convection zone (thus influencing the formation of active regions) and appear near 30 degrees latitude.

 
10:35-11:00 Zhang, KK (Exeter)
  Non-axisymmetric solar interface dynamos Sem 1
11:25-12:35 Fauve, S (ENS, Paris)
  Dynamo experiments I Sem 1
 

The generation of a magnetic field by a flow of liquid sodium has been observed in recent experiments (Karlsruhe, Riga). We emphasize two very interesting features displayed by these experiments. - The observed dynamo threshold is in good agreement with the one computed from the mean flow alone, i. e. neglecting turbulent fluctuations although the kinematic Reynolds number is of order 105 to 106. - On the contrary, the mean magnetic field measured above dynamo threshold is 1000 times larger than the one predicted from a laminar weakly nonlinear calculation. We first understand these two observations and give the expected scaling law for the magnetic energy above dynamo threshold.

We then consider magnetic fluctuations above dynamo threshold or in MHD turbulence and report a Kolmogorov type spectrum in the inertial range and 1/f noise at low frequency.

Finally, we discuss the effect of turbulence and rotation on dynamo onset and saturation in flows without strong geometrical constraints.

 
15:30-16:40 Glatzmaier, GA (Santa Cruz)
  Direct simulation of planetary and stellar dynamos I: methods and results Sem 1
 

The first computer simulations of convection and magnetic field generation in a 3D spherical fluid shell were made two decades ago in an attempt to understand the solar dynamo. Early on the Boussinesq equations were replaced with the anelastic equations to more realistically represent the density-stratified solar interior. Many more global models were developed during the past decade to simulate the geodynamo; however, most have employed the Boussinesq equations because of the relatively small density stratification of the Earth's fluid core. These simulations have demonstrated that thermal convection in a rotating electrically-conducting fluid shell can maintain a global magnetic field. Geodynamo simulations have produced fields that have intensity, structure and time dependence at the surface that are surprisingly similar to those of the geomagnetic field. However, one must question how realistic the dynamo mechanism is well below the surface in these simulations. The early solar convection simulations produced a surface equatorial acceleration similar to the sun's; but the internal differential rotation in those simulations was later shown not to agree with that inferred from helioseismology.

In this first talk I will briefly review some of the numerical models that have been employed in these 3D global simulations. Some example dynamo simulations for the sun, Jupiter and the Earth will be presented. However, because of insufficient computing resources, none of these simulations are strongly turbulent. The challenges for future generation models to overcome this limitation and others will be discussed in the second talk.

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16:40-17:05 Gilbert, AD (Exeter)
  Nonlinear dynamo action in rotating convection and shear Sem 1
 

(joint work with Pu Zhang, Andrew Soward, Keke Zhang).

Magnetic field amplification is studied in a rotating plane layer system. A shearing fluid flow is driven by motion of the bottom boundary and takes the form of an Ekman boundary layer. Above this layer, convection is driven. The combination of shear and convection amplifies magnetic field by means of an alpha-omega dynamo mechanism, and the saturation of magnetic fields is discussed by means of numerical simulations.

 
17:05-18:15 Fauve, S (ENS, Paris)
  Dynamo experiments II Sem 1
Friday 17 September
09:00-10:10 Glatzmaier, GA (Santa Cruz)
  Direct simulation of planetary and stellar dynamos II: future challenges Sem 1
 

The past two decades of 3D global simulations of convective dynamos have improved our understanding of how stars and planets generate their magnetic fields. However, no global dynamo simulation has yet been able to afford the spatial resolution required to simulate the turbulent convection which exists in the low-viscosity fluid interiors of these bodies. They have all employed greatly enhanced "eddy" diffusivities to crudely account for the transport and mixing by the small unresolved (subgrid scale) turbulence and to numerically stabilize the low resolution numerical solutions. Consequently convection in most numerical simulations has been laminar with spatial scales comparable to the size of the convection zone instead of being a broad spectrum of anisotropic heterogeneous turbulence. Many dynamo simulations have also ignored density stratification, which is a major source of vorticity in rotating turbulent convection. So, how robust are the results of the current models? We will not know until next-generation dynamo models are run at much higher spatial resolution and much lower viscosity, which will require more computational resources and improved numerical methods. In the mean time we can get some insight by testing 2D models, which, although lack the correct geometry, can simulate strong turbulence within a large density stratification. Current 2D tests reveal important issues regarding convective penetration, gravity waves, Alfven waves and the maintenance of differential rotation.

 
10:10-10:35 Rogers, TM (UC, Santa Cruz)
  Numerical simulations of the solar tachocline Sem 1
10:35-11:00 Schrinner, M (MPS, Katlinburg-Lindau)
  Comparison of three-dimensional geodynamo calculations with mean-field models Sem 1
 

Mean-field theory has been proven to be a useful description of stellar dynamos. It provides a conceptual understanding of the induction mechanisms that lead to the generation of large-scale magnetic field in stellar interiors. Our aim is to test the validity and reliability of mean-field theory by a comparison of three-dimensional geodynamo calculations with corresponding mean-field models. Therefore, we calculated the full \(\alpha\) and \(\beta\) tensors with the help of three-dimensional MHD-simulations. The resulting mean-field coefficients are confirmed by an independent analytical calculation that is applicable under the first-order-smoothing approximation. In a second step, we used these coefficients in a mean-field model and compared the results with three-dimensional calculations. Basic characteristics of the geodynamo benchmark-model,"case1", were reproduced by our mean-field model. Applied to this example, mean-field theory is a valid approximation. The investigation to which extent the mean-field approach is correct for more complicated dynamos and which of the mean-field coefficients are most important is work in progress.

 

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