The Magnetic Connection Between the Sun’s Corona and Convective Interior

W.P. AbbettSpace Sciences Laboratory, UC Berkeley

Nov. 2007, Rice Univ. P & A Colloquium

Motivation All solar activity – variations in energy released by the Sun in the form of electromagnetic radiation or energetic particles – is mediated by the Sun’s magnetic field

Solar activity arises as a result of the coupling of the solar magnetic field to the rotating, turbulent plasma of the Sun’s convective envelope

Thus, to understand and predict solar activity, we must understand the magnetic and energetic connection between the solar interior and atmosphere

Image credit: Hinode JAXA / NASA

Motivation

A quantitative description of the magnetic connection between the convection zone and corona will greatly improve our understanding of the

Image credit: Hinode JAXA / NASA

and much more…….

origin, evolution, and decay of active region magnetic fields

physics of the solar cycle and convective surface dynamo

CME initiation mechanism, and the physics of eruptive events

coronal heating mechanism and irradiance variations

transport of magnetic flux, energy, and helicity into the corona

origin and source of the solar wind

Approach

1. Analyze observational data

Above the visible surface, we can obtain measurements of atmospheric emission and energetic particles over an impressive range of wavelengths and energies. These data can then be used to infer, e.g., the

structure and strength of the magnetic field in the solar atmosphere

thermodynamic characteristics of the magnetized plasma

plasma flows at and above the visible surface

Where direct observations are not possible – e.g., below the visible surface and on the far side of the Sun – we must rely upon helioseismology to provide information about the

sub-surface structure, flow pattern, and differential rotation rate

presence of active regions on the Sun’s far side

Hinode is a Japanese mission developed and launched by ISAS/JAXA, with NAOJ as domestic partner and NASA and STFC (UK) as international partners. It is operated by these agencies in co-operation with ESA and NSC (Norway). Movie courtesy of Marc DeRosa, LMSAL

Hinode G-band images of the solar photosphere

Hinode LOS magnetograms of the photospheric magnetic field

Hinode is a Japanese mission developed and launched by ISAS/JAXA, with NAOJ as domestic partner and NASA and STFC (UK) as international partners. It is operated by these agencies in co-operation with ESA and NSC (Norway). Movie courtesy of Marc DeRosa, LMSAL

Hinode Ca II H images of the chromosphere

Hinode is a Japanese mission developed and launched by ISAS/JAXA, with NAOJ as domestic partner and NASA and STFC (UK) as international partners. It is operated by these agencies in co-operation with ESA and NSC (Norway). Movie courtesy of Marc DeRosa, LMSAL

Hinode XRT soft X-ray images of the low corona

Hinode is a Japanese mission developed and launched by ISAS/JAXA, with NAOJ as domestic partner and NASA and STFC (UK) as international partners. It is operated by these agencies in co-operation with ESA and NSC (Norway). Movie courtesy of Marc DeRosa, LMSAL

LASCO images of the global corona during a CME

Movie courtesy of LMSAL, TRACE & LASCO consortia

Approach

2. First principles forward modeling

Utilize idealized theoretical and numerical models to approximate the physics of the solar magnetic field in regions of the interior and atmosphere where such idealizations apply.

the tachocline and convective overshoot layer

the optically thick convective interior well below the visible surface

the convectively unstable surface layers and the low atmosphere

the optically thin, magnetically-dominated corona

Historically, numerical investigations of the Sun’s magnetic field have focused separately on several physically-distinct regions:

Image credit: NASA

Approach

3. Combine observational data with models

Investigate the physics of the solar atmosphere and interior by directly incorporating observations of the magnetic field and flows into models. This coupling can be performed at varying levels of sophistication depending on the desired objective. Examples include

The use of magnetograms and force-free extrapolations to study the buildup of, e.g., free magnetic energy in the corona

Studies of the flux of helicity and magnetic energy into the atmosphere from below the surface using sequences of magnetograms and velocity inversion techniques

The use of magnetogram sequences and velocity inversion techniques to provide the necessary boundary conditions for numerical models of the solar corona

The use of 3D flow fields from helioseismic inversions and data assimilation techniques to produce models with improved forecasting capability

Image credit: SOHO / MDI

Outline

2. First principles forward modeling

3. Combine observational data with models

Most recent work modeling the magnetic connection between the solar interior and atmosphere has involved studies of the emergence of active regions. I’ll therefore briefly discuss

the standard picture of the origin and evolution of active regions

quantitative models of the sub-surface evolution of active regions

quantitative models of active region fields in the solar corona

the challenge of modeling the convection zone-to-corona system

recent quantitative models of the combined system: active region emergence, ephemeral active regions, and the magnetic field of the quiet Sun

1. Analyze observational data

Characteristics of Active Region Magnetic Fields

SOHO EIT Image of coronal plasma at ~1.3 MK MDI magnetogram from May 11, 2000 Mean tilt of active regions vs. latitude (Fisher et al. 1995)

Most active regions emerge as simple bipoles

Leading polarities of active regions in a given hemisphere are the same, and oppose those of the opposite hemisphere

Active region bipoles are oriented nearly parallel to the E-W direction (Hale’s Law 1919)

On average, the leading polarity of an active region is positioned closer to the equator than the trailing polarity

The mean tilt angle of active regions increases with latitude (Joy’s Law)

The standard cartoon picture of active region magnetic fields below the surface

Since most active regions emerge as simple bipoles, these structures can be interpreted as the tops of large Omega-shaped magnetic flux tubes anchored deep in the convection zone.

Similarly, active regions exhibiting non-Hale configurations (e.g., delta spots) can be interpreted as twisted, or writhed flux tubes

Beyond the cartoon – the “thin flux tube” description of active region magnetic fields below the surface

Solve an equation of motion for a flux tube moving in a field-free background model convection zone assuming that

as the tube moves, it retains its identity and does not disperse or fragment

the tube’s cross-section is small relative to all other relevant length scales of the problem

quasi-static pressure balance is maintained across the diameter of the tube at all times

ABOVE: FB refers to the magnetic buoyancy force, FT the force due to magnetic tension, FC the Coriolis force, and FD the force resulting from aerodynamic drag (ρe and ρi refer to the gas density external to the tube, and in the tube’s interior respectively)

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Successes of the thin flux tube model

provided an estimate of the magnetic field strength of the toroidal layer at the base of the convection zone

provided a physical description of “Joy’s Law”

provided a physical basis for asymmetric spot motions

explained the dispersion of tilt versus active region size

described the physical basis of morphological asymmetries in active regions

is currently the basis of the only viable theory of the origin of twist in active regions (the “Sigma effect” of Longcope et al. 1998)

Image from Caligari et al. 1995

Beyond the thin flux tube model: MHD simulations of the interior

Anelastic simulations of Fan et al. (1998) Boussinesq simulations of Longcope et al (1996)

2D results show that without substantial fieldline twist (far more than is, on average, observed), flux tubes fragment and are unable to reach the surface

3D MHD simulations of sub-surface active region magnetic fields

Relaxing the axisymmetric assumption resolves the apparent paradox. In 3D only a modest amount of twist is required for the flux rope to remain cohesive

Fieldline twist is relatively unimportant: what matters is the axial field strength relative to the kinetic energy density of strong downdrafts (Fan et al. 2003):

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Fan et al. (2003) Abbett et al. (2000)

3D MHD global simulations of active region magnetic fields below the visible surface (Fan 2007)

Parameter space exploration of e.g., different initial field strengths and twists in a stratified model convection zone

Jouve et al. (2007) are performing similar global simulations with the ASH code (Brun et al. 2004) in a turbulent background state

3D MHD simulations of the solar transition region and corona

Top row: SAIC model (from their website), Bottom row: image from Gudiksen & Nordlund 2005 (left) and Abbett & Fisher 2003 (right)

The surface layers and low atmosphere

Left: 3D model of a flux rope interacting with surface flows from Cheung 2007 Right: 3D simulation of surface magnetoconvection from Bob Stein’s website

These types of models are highly realistic, since they include the LTE radiative transfer equation in the MHD system. Like the coronal models of the previous slide that include optically thin cooling and thermal conduction in their MHD energy equation, the results of these models can be directly compared with observational data.

fig_gband_moviep.mov

The different types of numerical models discussed so far …

Sub-surface MHD Surface radiative-MHD MHD model coronae

Spatial coverage: individual active regions to global models of the convective envelope

Spatial coverage: surface layers (~5 Mm below the surface out into the low chromosphere)

Spatial coverage: transition region (>3 x 105 K) to the corona. Individual active regions to the global corona

Numerical techniques: TFT, anelastic MHD

Fully-compressible MHD with the LTE radiative transfer equation included in the system

Fully-compressible MHD with the optically thin radiative cooling and thermal conduction

Applicability: from ~10 Mm below the surface to the tachocline.

Applicability: “anywhere” Applicability: the optically thin low-density plasma of the corona

Computationally efficient Computationally expensive “Moderately” computationally expensive

Putting it all together: an early attempt

From Abbett & Fisher 2003

Putting it all together: more self-consistent, idealized simulations of flux emergence

Left: Magara (2004) ideal MHD AR flux emergence simulation as shown in Abbett et al. 2005

Right: Manchester et al. (2004) BATS-R-US MHD simulation of AR flux emergence

10

Modeling the combined convection zone-to-coronasystem in a physically self-consistent way:

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Closure relation: a non-ideal equation of state obtained through an inversion of the OPAL tables (Rogers 2000),

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Modeling the combined convection zone-to-coronasystem in a physically self-consistent way:

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The source term in the energy equation,

must include the important physics believed to govern the evolution of the combined system. In the corona, this includes

• radiative cooling (in the optically thin limit),• the divergence of the electron heat flux, • a coronal heating mechanism (if necessary).

In the lower atmosphere at and above the visible surface,• radiative cooling (optically thick)

Below the surface in the deeper layers of the convective interior• radiative cooling (in the optically thick diffusion limit)

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Numerical techniques and challenges:

A dynamic numerical model extending from below the photosphere out into the corona must:

• span a ~ 10 - 15 order of magnitude change in gas density and a thermodynamic transition from the 1 MK corona to the optically thick, cooler layers of the low atmosphere, visible surface, and below;

• resolve a ~ 100 km photospheric pressure scale height while simultaneously following large-scale evolution (we use the Mikic et al. 2005 technique to mitigate the need to resolve the 1 km transition region scale height characteristic of a Spitzer-type conductivity);

• remain highly accurate in the turbulent sub-surface layers, while still employing an effective shock capture scheme to follow and resolve shock fronts in the upper atmosphere

• address the extreme temporal disparity of the combined system

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RADMHD: Numerical techniques and challenges

• For the quiet Sun: we use a semi-implicit, operator-split method.

• Explicit sub-step:Explicit sub-step: We use a 3D extension of the semi-discrete method of Kurganov & Levy (2000) with the third order-accurate central weighted essentially non-oscillatory (CWENO) polynomial reconstruction of Levy et al. (2000).

• CWENO interpolation provides an efficient, accurate, simple shock capture scheme that allows us to resolve shocks in the transition region and corona without refining the mesh. The solenoidal constraint on B is enforced implicitly.

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RADMHD: Numerical techniques and challenges

• For the quiet Sun: we use a semi-implicit, operator-split method

• Implicit sub-step:Implicit sub-step: We use a “Jacobian-free” Newton-Krylov (JFNK) solver (see Knoll & Keyes 2003). The Krylov sub-step employs the generalized minimum residual (GMRES) technique.

• JFNK provides a memory-efficient means of implicitly solving a non-linear system, and frees us from the restrictive CFL stability conditions imposed by e.g., the electron thermal conductivity and radiative cooling.

The thermodynamic structure of the model is controlled by the energy source terms, the gravitational acceleration and the applied thermodynamic boundary conditions. No stratification is imposed a priori.

RADMHD: Modeling the combined convection zone-to-corona system:

The quiet Sun magnetic field in the model chromosphere

Magnetic field generated through the action of a convective surface dynamo.

Fieldlines drawn (in both directions) from points located 700 km above the visible surface.

Grayscale image represents the vertical component of the velocity field at the model photosphere.

The low-chromosphere acts as a dynamic, high-β plasma except along thin rope-like structures threading the atmosphere, connecting strong photospheric structures to the transition region-corona interface.

Plasma-β ~ 1 at the photosphere only in localized regions of concentrated field (near strong high-vorticity downdrafts

From Abbett (2007)

Flux submergence in the quiet Sun and the connectivity between an initially vertical coronal field and the turbulent convection zone

From Abbett (2007)

Average horizontal magnetic field as a function of height above the surface in the quiet Sun model atmosphere

Horizontally averaged magnetic field strength at the visible surface is ~108 G, and drops to ~ 37 G in the low chromosphere. Corresponding values for the vertical component: ~71 G at the surface, falling to ~15 G in the low chromosphere.

The maximum values are roughly similar, ranging from ~1 kG at the surface to ~275 G in the low atmosphere

In the corona, the field becomes more vertically oriented, and drops to ~2.5 G on average

Reverse granulation

• A brightness reversal with height in the atmosphere is a common feature of Ca II H and K observations of the quiet Sun chromosphere.

• In the simulations, a temperature (or convective) reversal in the model chromosphere occurs as a result of the p div u work of converging and diverging flows in the lower-density layers above the photosphere where radiative cooling is less dominant.

Gas temperature and Bz at ~ 700 km above and below the model photosphere

From Abbett (2007)

Flux cancellation and the effects of resolution:

The Quiet Sun magnetic flux threading the model photosphere over a 15 minute interval. Grid resolution ~ 117 X 117 km Average unsigned flux per pixel: 34.5 G

Simulated noise-free magnetograms reduced to MDI resolution (high-resolution mode) by convolving the dataset with a 2D Gaussian with a FWHM of 0.62” or 459 km. Average unsigned flux per pixel is now: 19.9 G

Simulated noise-free magnetograms reduced to Kitt Peak resolution. FWHM of the Gaussian Kernel is 1.0” or 740 km. Average unsigned flux per pixel: 15.0 G

Observed unsigned flux per pixel at Kitt Peak: 5.5 G

How force-free is the quiet Sun atmosphere?

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Characteristics of the quiet Sun model atmosphere:

Note: Above movie is not a timeseries!

Recent progress and (near) future plans:

120 Mm

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Increase domain size. Below: Starting state for a 128 node run on NASA’s Discover cluster. Symmetry will be broken by a random entropy perturbation below the surface.

Release beta version of Radmhd-1.0 under GPLv2 for further validation and testing by independent groups

Perform AR emergence simulations once the background state is dynamically and energetically relaxed

Significantly extend the coronal portion of the domain using a non-uniform mesh

Summary

Great progress has been made throughout the past decade in our ability to model the dynamic evolution of the solar magnetic field in three separate physically-distinct regions in the Sun: the convective interior, the low atmosphere, and the upper transition region and corona

Recently, substantial progress has been made in the effort to model the upper convection zone and the low-corona in a physically self-consistent way and in an efficient manner within a single computational volume

New numerical techniques have been applied to the study of dynamo-generated quiet Sun magnetic fields. This study will soon be extended to active region spatial scales. It is then possible to investigate

the large-scale emergence, evolution and decay of active regions

the numerics of data driving and data assimilation

the coupling of sub-surface, and coronal codes to RADMHD through publicly available computational frameworks