Understanding the Photospheric and Near-Photospheric Magnetic Field A View of the Past, Present, and Future of First-Principles Magnetic Field Modeling at the Photosphere and Below George Fisher, SSL UC Berkeley

Goal of talk - Stimulate thinking and discussion of how the HMI/AIA instruments can best improve our knowledge of magnetic field structure and evolution near the photosphere. Three main topics: A review of what Thin Flux Tube models have taught us about the origins of active region magnetic fields and the observed properties of active regions A biased and incomplete survey of what 3-D MHD simulations of the solar interior have taught us about photospheric magnetic fields A glimpse into the future of magnetic field modeling in the solar atmosphere and interior

During the 1990s, most of the theoretical understanding about the origin of active regions as observed as sunspot groups and as bipolar regions in full disk magnetograms came from “thin flux tube” models. What motivates this approach, and what is it exactly?

Why we think of active regions as flux tubes. (this shape of flux tube is known as an Ω-loop)

Thin flux tube models assume that the dynamics of a flux tube can be described by the forces acting on a thin 1-D tube imbedded in a 3-D model of the solar interior

The Coriolis force deflects tube toward the poles as it tries to rise radially The action of the Coriolis Force in thin flux tube models provides one possible explanation for the observed equatorial paucity of active regions

The limits of the active region latitude belts provide strong constraints on the initial magnetic configuration and magnetic field strength, according to thin flux tube calculations If B > 10 5 G, no low-latitude equilibria possible If B < 3x10 4 G, poleward motion too great for observed latitude distributions If active region flux tubes are initially toroidal, field strength is constrained.

What are active region tilts and what is Joy’s Law?

Thin flux tube models do an excellent job of explaining Joy’s law. The tilt comes from the Coriolis force acting on the rising, expanding plasma in an emerging, initially toroidal flux loop.

Torque balance between magnetic tension and Coriolis forces determines the amount of tilt in an emerging active region (for northern hemisphere)

Fisher, Fan & Howard (1995) found from an analysis of spot group data that the predicted flux dependence of the tilt angle was consistent with the data. (Since this work was done, however, Tian & Liu (2003) have updated these results with magnetic fluxes instead of polarity separations, allowing for a more direct comparison with the theory. Those results show a less clear agreement between theory and observation.)

Not only are there tilts, but quite significant fluctuations of tilt Analysis of ~24,000 spot groups shows tilt dispersion is not a function of latitude, but is a function of d, with  ~ d -3/4.

The observed variation of Δ  with d suggests convective turbulence as a possible mechanism for tilt fluctuation

Tilt fluctuations computed using tube dynamics perturbed by convective turbulence can explain the observed variation of Δα with d.

Asymmetric spot motions can be explained by asymmetric shapes in emerging Ω-loops Asymmetric shapes in the emerging loops originate from Coriolis forces that act to preserve angular momentum Panels (a), (b), (c) correspond to field strengths at the base of the convection zone of 30, 60, and 100 kG respectively (Fan & Fisher Sol. Phys. 166, 17) Caligari Moreno-Insertis & Schüssler (1995) suggested that the emergence of these asymmetric loops will result in faster apparent motion of the “leading” spot group polarity c.f. the “following” polarity, a well known observational phenomenon.

The Coriolis force is a possible explanation for asymmetries in the morphology of active regions

Field strength asymmetries could lead to morphological differences between leading and following polarity

How twisted are typical active regions? Where does active region twist come from?

where (Longcope & Klapper 1997; Longcope, Fisher & Pevtsov 1998)

What is the physical meaning of the source term  ?  depends only on the motion of tube axis. For a thin flux tube: H =  2 (Tw+Wr). ( Conservation of magnetic helicity H,where Tw is “twist”, and Wr is “writhe”.) Therefore,  exchanges writhe (Wr) with twist (Tw).

Could flux tube writhing account for observed levels of active region twist? Possible sources of writhing: “Joy’s Law” tilts of active regions during emergence is one possibility, but is too small…

Writhing by convective turbulence is another possibility… Develop a tractable model of convective turbulence that includes kinetic helicity Solve equations of motion and twist evolution for a flux tube rising through such a turbulent medium Such a model was explored by Longcope et al. (1998)

The writhing of initially untwisted flux tubes by convective motions containing expected levels of kinetic helicity leads to a twist distribution with latitude that is consistent with observations

The life cycle of an active region must somehow transition between an emerging Ω-loop and active region decay, as described by passive flux transport models. A new idea for this transition has been proposed by Schüssler & Rempel. 1.Active region below the surface is an emerging Ω-loop 2.As the magnetic flux breaks through the photosphere, sunspots form and the initial coronal magnetic field is established 3.As the plasma in the spots cools and sinks, and the buoyant plasma from below emerges, the upper parts of these flux tubes are blown apart and are then controlled by convective motions. Passive flux transport models then describe the surface evolution of the active region field Images courtesy of Loraine Lundquist

What sub-surface thin flux-tube models have told us about the origin of active regions Active regions originate from a toroidal field at the base of the convection zone, whose sign changes across the equator, with a field strength in the range of 2x10 4 – 10 5 G. The paucity of active regions near the equator could result from deflection toward the poles by the Coriolis force (low B), or from a lack of stable solutions (high B) Hale’s law and Joy’s law (active region orientation) can be reproduced with thin flux tube models, in which Coriolis forces balance magnetic tension The dependence of active region tilt on AR size might be explained by a balance between Coriolis forces and magnetic tension The dispersion of tilt versus active region size can be understood by the forces acting on a flux tube perturbed by convective motions Asymmetric spot motions (leading vs following) can be explained by the asymmetric shapes of Ω-loops. The asymmetric shapes result from the Coriolis force. The morphological asymmetry (the leading side being more compact than following side) of active regions may be explained by a field strength asymmetry in emerging Ω-loops, driven indirectly by Coriolis forces The observed helicity distribution with latitude of active regions can be explained by expected levels of kinetic helicity in convective motions that act to writhe magnetic flux tubes during their emergence toward the surface. Future global MHD models of magnetic fields in the solar interior need to include spherical geometry and rotational effects, including Coriolis forces.

3-D MHD simulations of magnetic fields in the solar interior can describe physics not that cannot be addressed with thin flux tube models. In the late 1990s, computers and numerical techniques became powerful enough to address problems of real significance for the Sun, rather than highly idealized “toy” problems. MHD simulations can show us how active region scale flux tubes evolve in a model convection zone that is actually convecting MHD simulations have shown how kink unstable magnetic flux tubes may be able to explain many observed properties of island δ-spot active regions MHD simulations have shown directly how a small-scale disordered magnetic dynamo can be driven by convective motions

3D-MHD models of flux emergence confirm the asymmetric shape of the  loop predicted by thin flux tube models (Fig. 2 from Abbett Fisher & Fan 2001)

Active Region Fields in a Convectively Unstable Background State Q: What happens to an active region flux tube in a convection zone? From Abbett et al. 2004

Active Region Fields in a Convectively Unstable Background State From Fan et al Q: What are the conditions for the tube to retain its cohesion? Fieldline twist is relatively unimportant: what matters is the axial field strength relative to the kinetic energy density of strong downdrafts:

Island δ-spot active regions can be understood as twisted, kinking flux tubes (Linton, Fan) Properties of δ-spot regions: Sunspot umbrae of opposite polarity in a common penumbra Strong shear along neutral line Active region rotates as it emerges Large and frequent flares and CMEs Kinked geometry explains rotation, shear along neutral line Flares/CMEs might be explained by reconnection between the 2 legs of the intertwined loop structure

On small scales, the solar magnetic field appears, evolves, and disappears over very short time scales Is the small scale magnetic field on the Sun and other stars the lint from the clothes in the solar washing machine, or is it generated by its own dynamo mechanism?

We have performed our own simulations of small-scale magnetic fields driven by convective turbulence in a stratified model convection zone without rotation, starting from a small seed field. The magnetic energy grows by 12 orders of magnitude, and saturates at a level of roughly 7% of the kinetic energy in convective motions. This simulation took about 6 CPU months of computing time.

What does the generated magnetic field look like? Here is a movie showing “magnetograms” of the vertical component of the field in 2 slices of the atmosphere, near the bottom and near the top:

Here is a snapshot showing volume renderings of the entropy and the magnetic field strength in the convective dynamo simulation at a time after saturation:

This movie shows the time evolution of a volume rendering of the magnetic field strength in the convective dynamo after saturation has occurred

How do we connect our simulation results to real data for the Sun and stars? We must first convert the dimensionless units of the anelastic MHD code to real (cgs) units corresponding to the convective envelopes of real stars: (1) demand that stellar surface temperature and density match those of model stellar envelopes, (2) Demand that the convective energy flux in the simulation match the stellar luminosity divided by the stellar surface area. We use mixing length theory to connect energy flux to the unit of velocity in the simulation. After applying these assumptions, we can scale a single simulation to the convective envelopes of main-sequence stars from spectral types F to M. To convert magnetic quantities from the simulations to observable signatures, we use the empirical relationship between magnetic flux and X-ray radiance (from Pevtsov et al) to predict surface X-ray fluxes for main-sequence stars

Quantitative studies of magnetic dynamos on other stars requires a quantitative knowledge of the relationship between magnetic fields and “activity” indicators such as X-ray flux: (Pevtsov et al. 2003, ApJ 598, 1387)

So how does the convective dynamo model compare to observed X-ray fluxes in main-sequence stars? The convective dynamo model does an excellent job of predicting the lower limit of X-ray emission for slowly rotating stars, and for predicting the amount of magnetic flux observed on the Quiet Sun during solar minimum.

Many Simulations of the Plasma just below the solar photosphere now Include a great deal of physical realism, including 3-D radiation transfer and a realistic equation of state. This allows for the self-consistent Formation of cool micropores at magnetic flux concentrations, as seen in this simulation from Dave Bercik’s (2002) thesis.

Simulation of a magnetic plage region using the MURaM code by Vögler et al (2005). This code solves the 3D MHD equations and non-grey LTE RT equations in 3D for the convection zone and photosphere.

What is the future of first-principles magnetic field modeling near the solar photosphere? A correct description of the physics of magnetic field evolution of the solar atmosphere must self- consistently couple very different regions of the solar atmosphere. Presently, there are 2 approaches to this problem: (1) Develop coupled models of the different regions, which communicate across a code- code interface (2) Implement numerical techniques that can accommodate greatly different physical conditions

This figure shows Simulations of the Quiet Sun using Abbett’s new code, AMPS. These simulations extend from the upper convection zone, through the photosphere, a simplified chromosphere, transition region, and a corona.

AMPS (the Adaptive MHD Parallel Solver) was designed from the outset to use the Paramesh domain-decomposition libraries, allowing for an efficient MPI/AMR environment. The code uses a semi-implicit technique -- Newton-Krylov formalism is used to evolve the troublesome energy equation source terms implicitly, and the semi-discrete formalism of Kurganov & Levy 2000 (with 3 rd order CWENO interpolation) is employed as the shock-capture scheme and is used to explicitly advance the continuity, induction, and momentum equations. Lower left illustrates how Paramesh divides the computational domain into sub-regions, each handled by a separate processor. Lower right shows Orszag-Tang vortex and MHD blast tests with AMR.

Questions What triggers the eruption of active regions from the base of the convection zone? Instability, secular heating, convective overshoot, or something else? Most active regions exhibit only small amounts of twist, and are consistent with a tube lying initially at the base of the convection zone with no twist. How then do the island δ-spot regions acquire so much twist? How is the free energy from sub-surface fields transported into the corona? What is the magnetic connection between different active regions? Are active regions magnetically connected to each other in the dynamo region, or are they all separate? How do we relate “active longitudes” and “active nests” to a magnetic picture of the large-scale dynamo region at the base of the convection zone? Is the magnetic flux that gives birth to active regions in a smooth, slab-like geometry, or is the flux already pre-existing in the form of tubes? What happens when active region flux tubes collide in the solar interior? What happens when a new active region emerges into an old one? How do active region flux tubes interact with the small scale field in the Quiet Sun? What is the 3D analogue of the surface flux transport models? How does the following polarity from decaying, emerged active regions return to the dynamo regions? What happens to the magnetic roots of an emerged active region once the active region begins decaying? Can we infer the sub-surface structure of an active region by studying its surface evolution? (Try this for AR 8210!) Can we better predict the emergence of new active regions before it happens, either from helioseismic observation, or from a better knowledge of the physics of magnetic evolution below the photosphere?