The Dynamic Evolution of Quiet Sun Magnetic Fields in the Solar Atmosphere

W.P. Abbett, Space Sciences Laboratory, Univ. of California, Berkeley

abbett@ssl.berkeley.edu

AAS-SPD, May 2007

ABSTRACT: We present the latest results from a series of three-dimensional MHD simulations of the Quiet Sun magnetic field. The computational domain extends from the upper convection zone out into the corona, and includes the highly-stratified layers of the photosphere, chromosphere, and transition region. Our study focuses on the following questions: Can the magnetic field generated by a convective surface dynamo heat a model corona to soft X-ray emitting temperatures? Do physical processes such as resistive and viscous dissipation supply the necessary heating throughout the model atmosphere, or is an additional empirically-based heating mechanism required? What is the magnetic connection between the flux threading the model photosphere, and that filling the model corona? Can the magnetic field in the dynamic models be successfully reproduced by static extrapolations?

91:09

Here, we present a first step toward this goal:

Results from a recently developed code capable of quantitatively describing the physical connection between the Quiet Sun upper convection zone and low-corona.

Motivation:

Understanding the physics of the solar magnetic field ---

• from its generation in the turbulent, differentially-rotating interior • to the ultimate release of magnetic energy in the atmosphere in the form of radiation, solar wind acceleration, flares or CMEs ---

requires

• an understanding of the dynamic, energetic and magnetic connection between the convective interior and corona.

Characteristics of the convective interior:• The convection zone is high-density, high-β, optically-thick, turbulent plasma.

• Magnetic fields entrained within convective flows at and below the visible surface evolve slowly compared to the coronal field.

MHD models of the deep interior (from Abbett et al. 2004 and Bercik et al. 2005)

Characteristics of the solar corona:• The corona is a low-density, low-β, optically-thin, hot plasma

• Plasma within coronal loops evolves rapidly compared to sub-surface structures

• The corona can store energy over long periods of time, but can then undergo sudden, rapid, and dramatic changes as magnetic energy is released.

A comparison of potential field (left column), and non-linear force-free field extrapolations (middle column) to a Magara (2004) ideal MHD simulation of active region-scale flux emergence without convective turbulence (right column).The fieldlines of the top (bottom) row are generated from a synthetic magnetogram positioned in the model chromosphere (photosphere). Image from Abbett et. al. (2004).

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

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The resistive fully-compressible MHD system of equations:

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

),( epp

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

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2

4Buu

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)

Q

Modeling the combined convection zone-to-corona system:

Bcr QQQQ

We represent the source term as follows:

In order to extend the spatial domain to active region scales, we choose not to solve the optically-thick LTE transfer equation to obtain an expression for surface cooling, . Instead, we choose to approximate this cooling in a way that successfully reproduces the average stratification and solar-like convective turbulence of the more realistic simulations of Bercik (2002) and Stein et al. (2003):

Q

rQ

332211 QQQQr TTQ

TeeQ

TnnQ

R

he

),(

),(

)(

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where

and and represent dimension-less envelope functions that restrict each term to the appropriate range of densities or depths in such a way as to avoid sharp cutoffs. is obtained from the CHIANTI atomic database.

321 ,,

)(T

The structure of the transition region and corona depend strongly on the remaining non-radiative terms in : the divergence of the electron heat flux,

Modeling the combined convection zone-to-corona system:

Bcr QQQQ

2|| )(B

TTQc BB

and an additional coronal heating rate (if necessary). We employ an empirically-based description of coronal heating consistent with the observed relationship between unsigned magnetic flux and the power dissipated in the atmosphere by a coronal heating mechanism, .

BQ

cdVQB

The RHS of this equation represents the Pevtsov et al. (2003) power law relationship between X-ray luminosity and unsigned magnetic flux at the photosphere . If we choose a simple heating function of the form (consistent with Lundquist et al. 2007), we arrive at an empirically-based form of coronal heating consistent with Pevtsov’s Law:

cLxBBQB /

BdV

BcQB

The calibrated radiative source term in , coupled with a constant radiative flux lower boundary condition (on average) maintains the super-adiabatic stratification necessary to initiate and sustain convection.

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.

rQ Bcr QQQQ

Modeling the combined convection zone-to-corona system:

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.

RADMHD: Numerical techniques and challenges

• The MHD system is solved on an adaptive, domain-decomposed mesh.

Note: With our numerical techniques, AMR is not needed to simulate the Quiet Sun. However, RADMHD has the capability of interfacing with the PARAMESH libraries (MacNeice et al. 2000) to provide an adaptive framework.

• Spatial disparities of the combined convection zone-to-corona system are addressed via the CWENO explicit scheme, the domain decomposition strategy, and AMR capability if necessary.

• Temporal disparities of the combined convection zone-to-corona system are addressed via the JFNK implicit scheme. Pre-conditioning is an essential requirement if one wishes to rapidly relax atmospheres by significantly exceeding the CFL limit.

• Boundary conditions of the Quiet Sun simulations: Periodic in the transverse directions, constant radiative flux in through a closed lower boundary, open coronal boundary

On to the Results (Finally!) ……

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)

Convective reversal

• 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?

Temperature

Chr

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BJlog )/(cos BJBJ

log B

log J

log β Bz log B

Characteristics of the Quiet Sun model atmosphere:

log |J X B|

Summary and conclusions to date:

• It is possible to efficiently simulate the upper convection zone, photosphere, chromosphere, transition region and low-corona within a single computational domain

• If we use an empirically based coronal heating mechanism consistent with the Pevtsov et al. (2003) relationship between X-ray emission and magnetic flux observed at the surface, magnetic fields generated from a convective dynamo are sufficient to heat the corona to 1 MK.

• The model chromosphere exhibits a convective reversal --- plasma above cell centers tends to be cooler than the average temperature at that height, while plasma above the photospheric intergranular lanes is generally hotter. In the models, this is simply due to the work of converging and diverging flows in the relatively low density layers above the photosphere where the radiative source terms are less dominant.

up

Summary and conclusions to date:

• Analysis of magnetograph data at relatively low, one arcsecond resolution can lead to a significant underestimate of the total amount of unsigned magnetic flux threading the Quiet Sun photosphere.

• A persistent current layer is formed as the atmosphere transitions from a dynamic, high-β regime (on average) to the more magnetically-dominated transition region and corona.

• The Quiet Sun model corona is dynamic, and is not everywhere force-free or even low-β. There are regions of both relatively strong and exceptionally weak field high in the model corona.

• Synthetic vector magnetograms generated at the model photosphere and chromosphere are substantively different, and will yield substantively different results if used as a basis for force-free or potential field extrapolations. Neither extrapolation technique will reproduce the twisted horizontal structures present just above the model photosphere (and their associated magnetic connectivity) in the dynamic, often non-force-free layers of the low chromosphere.

Summary and conclusions to date:

• On average, the bulk of the unsigned magnetic flux resides below the visible surface. Of the flux that threads the surface, most remains entrained in the plasma, and turns over within a local chromospheric pressure scale height. Thus, there is a steep decline in the average amount of unsigned magnetic flux with height above the surface.

• The magnetic connectivity in the dynamic region between the photosphere and upper transition region is complex. This region is characterized by the presence of both twisted and non-twisted β ~ 1 horizontally-directed magnetic structures that thread the chromosphere and connect relatively distant concentrations of magnetic flux in the photosphere with the transition region footpoints of coronal structures.

• A non-ideal equation of state is necessary to obtain more realistic ratios of magnetic to gas pressure at, and just above, the model photosphere. However, an ideal treatment is capable of reproducing many of the global characteristics of the non-ideal atmospheres.

References:

Abbett, W. P., 2007, ApJ, in press.Bercik, D., 2002, Ph. D. thesis, Michigan State UniversityBercik, D. J., Fisher, G. H., Johns-Krull, C. M., & Abbett, W. P., 2005, ApJ, 631, 529Knoll, D. A., & Keyes, D. E., 2003, J. Comp. Physics, 193, 357Kurganov, A., & Levy, D., 2000, SIAM J. Sci. Comput., 22, 1461Levy, D., Puppo, G., & Russo, G., 2000, SIAM J. Sci. Comput., 22, 656Lundquist, L. L., Fisher, G. H., McTiernan, J. M., & Leka, K. D., 2007 ApJ, submitted.MacNeice, P., Olson, K. M., Mobarry, C., deFainchtein, R., & Packer, C., 2000, Computer Phys. Comm., 126, 330Magara, T., 2004, ApJ, 605, 480Mikic, Z., Linker, J. A.,Titov, V., Lionello, R., & Riley, P., 2005, SHINE Workshop, July 11-15, Kona HI Rogers, F. J., 2000, Physics of Plasmas, 7, 51 Stein, R. F., Bercik, D., & Nordlund, A, 2003, ASP Conf. Ser. 286: Current Theoretical Models and Future High Resolution Solar Observations: Preparing for ATST, 286, 121 Pevtsov, A. A., Fisher, G. H., Acton, L. W., Longcope, D. W., Johns-Krull, C. M., Kankelborg, C. C., & Metcalf, T. R. 2003, ApJ, 598, 1387