Simulations of the Quiet Sun Magnetic Field: From the Upper Convection Zone into the Corona William P. Abbett Space Sciences Laboratory, Univ. of California, Berkeley 7.10 ABSTRACT: We present the latest in a series of simulations designed to directly investigate whether the magnetic field generated by a convective dynamo in the upper convection zone can account for some of the observed properties of the Quiet Sun magnetic field and atmosphere. The simulations are performed using a new numerical code capable of evolving a model solar atmosphere that extends from the upper convection zone into the low corona. This code is a parallel, semi-implicit solver capable of accommodating many of the spatial and temporal disparities intrinsic to this combined system.

Where does the Quiet Sun magnetic field come from? For decades, the prevailing view was that the Quiet Sun network was a consequence of the decay and dispersal of active region magnetic fields --- a view reinforced by the success of flux transport models in reproducing the global magnetic field over multiple solar cycles. In this picture, if there were no active regions, there would be no Quiet Sun magnetic field of any consequence! Recently, observational evidence has begun to suggest that the Quiet Sun magnetic field might not be dominated by active region sources. For instance, Title & Schrijver (1998) found from high resolution observations with SOHO/MDI that small-scale flux was replaced every forty hours, and that there may be even more magnetic flux emerging on small scales in the Quiet Sun than in active regions.

In addition, theoretical evidence has begun to mount that turbulent convective motions can efficiently generate small-scale magnetic fields. For example, Cattaneo (1999) first demonstrated the “convective dynamo” using 3D MHD simulations in the Boussinesq approximation. Bercik et al. (2005) performed similar calculations in a stratified medium using the anelastic approximation, and showed that the convective dynamo could generate enough magnetic flux to account for both the Quiet Sun magnetic flux and X-ray levels at solar minimum. While providing important insights, the Boussinesq and anelastic approximations are limited in that they only apply to layers well below the solar photosphere. Here, we present preliminary results from a new, more realistic numerical model of the Quiet Sun convection zone and corona.

A less idealized model of the combined convection zone to corona system must numerically solve the following system: The energy source term Q must include: Optically thin radiative cooling and anisotropic thermal conduction in the upper atmosphere and corona An empirical coronal heating mechanism if resistive dissipation alone is insufficient to heat the model corona (e.g., a volumetric heating proportional to |B| in the corona consistent with the empirical constraint of Pevtsov et al. 2003). Optically thick radiative cooling at and below the surface layers

The Challenges: 1. There are significant spatial and temporal disparities in the system. For instance, the small-scale magnetic features observed at the surface evolve at the granular turnover time, while plasma in the corona evolves much more rapidly. Significant topological change can occur suddenly in the corona without a noticeable, corresponding change at the photosphere. 2. The computational domain must span vastly different physical regimes. Below the surface lies a relatively dense, turbulent, high-β (the ratio of gas to magnetic pressure) plasma with strong magnetic fields organized into isolated structures. In contrast, the corona is a field-filled, low-density, magnetically dominated plasma (though in the Quiet Sun, β often exceeds unity in weak field regions away from the many isolated concentrations of magnetic flux). In addition, flow speeds in the convection zone and below the surface are typically below the characteristic sound and Alfven speeds, while the chromosphere, transition region, and corona tend to be shock- dominated.

A Numerical Solution: Here, we solve the MHD system of equations semi-implicitly on multiple processors over a domain-decomposed mesh. The non-linear, conservative portion of the system is treated explicitly using the semi-discrete central method of Kurganov & Levy (2000) with a third order three-dimensional CWENO polynomial reconstruction. This provides an efficient shock capture scheme, and allows for the treatment of e.g., steep transition region gradients without requiring adaptive mesh refinement. The remaining source terms, and the resistive and viscous contributions to the induction and momentum equations are solved implicitly using a memory- saving “Jacobian-free” Newton-Krylov method (see Knoll & Keyes 2003). It is important to note (since we employ a central scheme) that we solve the constrained MHD system --- an implicit parabolic correction is applied to the MHD induction equation to ensure that the magnetic field remains divergence-free.

The Treatment of Radiative Transfer: 1. In the optically thin upper atmosphere, radiative cooling is given by ( n e and n h refer to the electron and hydrogen number densities respectively). The radiative cooling curve Λ(T) is generated from version 4.2 of the CHIANTI atomic database. 2. Deep in the model convection zone, radiative heating and cooling is calculated in the optically thick diffusion limit, with a temperature and density-dependent Kramer's opacity given by. 3. For this study, we treat the optically thick radiative cooling of the surface layers in a parameterized fashion. We mimic surface losses by adding a low temperature chromospheric extension to the optically thin radiative cooling curve. The extension is calibrated against the realistic simulations of Bercik (2002), where the LTE transfer equation is solved in detail. This ensures that the run of superadiabaticity with depth in the our model convection zone is sufficient to initiate and maintain solar-like turbulent convection.

Generating the Quiet Sun atmosphere: With all the radiative source terms included, we first relax a reduced-size, 1D- symmetric background stratification (in a periodic domain with closed vertical boundaries) by artificially damping net vertical oscillations. The vertical extent of the domain (in this case) is 7.5 Mm, with the visible surface located 2.5 Mm above the lower boundary. We then extend the atmosphere to a 30x30x7.5 Mm 3, relatively low resolution 256x256x64 cube, eliminate the artificial damping, and break the 1D symmetry by adding random internal energy perturbations throughout the domain. Since the model convection zone is superadiabatically stratified, convective turbulence begins immediately. Once initiated, the domain is allowed to dynamically and energetically relax to a steady state. Note that at this point, the model corona remains a cold, nearly evacuated region. Finally, we introduce an energetically unimportant magnetic “seed” field into the model convection zone, and activate the remaining energetic source terms. The convective dynamo begins to operate, magnetic energy within the domain increases with time, and magnetic flux is pushed into the corona. The model corona heats up as the magnetic field becomes stronger, and the convective dynamo is allowed to saturate.

Left: log(Temperature) in degrees Kelvin along a vertical slice positioned at the center of the domain. Right: log(Density) (cgs) along the same vertical slice. The background stratification (and fluctuations about that background level) are apparent. This stratification is not imposed, rather it is the natural result of the choice of boundary conditions, and the characterization of the radiative transfer.

Top row: Vertical momentum along horizontal slices at different depths in the 30x30x7.5 Mm 3 domain. From left to right: ~1Mm below the visible surface, the photosphere, the upper chromosphere, and the low corona. Bottom row: the vertical component of the magnetic field along the same horizontal slices. The second and third frames can be thought of as a simulated LOS magnetogram at disk center --- note the difference between the photospheric and chromospheric magnetogram in the simulated Quiet Sun.

Visualizing the convective dynamo: Contours of the magnetic field strength along a horizontal slice ~1Mm below the surface (left), at the photosphere (right), and along a vertical slice through the portion of the domain representing the convective interior (below).

Along the limb: Vertical slices through the atmosphere from the photosphere out to the low corona (5 Mm above the visible surface). From top to bottom: 1. log |B| along the slice extending through the model chromosphere, transition region and low corona; bright regions represent relatively strong fields, dark regions represent weaker fields. 2. The magnetic flux piercing the vertical slice; light regions indicate flux directed toward the observer, dark regions indicate flux directed away from the observer. 3. The vertical component of the magnetic field along the same slice. 4. A visualization of the logarithm of the plasma β (the ratio of gas to magnetic pressure) along the same vertical slice. Blue regions correspond to magnetically dominated plasma, and green and orange regions correspond to areas where β>1.

Above: Another visualization of log(β), this time across the entire vertical extent of the domain. Again, blue regions represent low β regions in the atmosphere, and red orange and yellow regions represent high-β regions in both the interior and atmosphere. Below: A contour of the logarithm of the current density, log |J| (from a different vertical slice). Blue and black shades represent weak or negligible current densities.

Above: |J| along horizontal slices from (left to right) ~1Mm below the visible surface, the photosphere, the upper chromosphere, and the corona. Middle: Contours of the gas density at those same heights. Below: The log of the gas density along a vertical slice just above the photosphere.

The vertical component of the current at the model photosphere (left), and in the model chromosphere (right).

Magnetic fieldlines drawn from a horizontal slice positioned in the chromosphere

Discussion: We have presented a preliminary set of 3D MHD simulations of the Quiet Sun, convection zone and solar atmosphere in order to address the question: Can magnetic fields generated by convective dynamo account for some of the observed properties of the Quiet Sun chromosphere, transition region and corona? The complete answer to this question requires further study. The simulations presented here are preliminary --- although the convection zone is relaxed, the convective dynamo has yet to fully saturate. Current field strengths in the photosphere and corona remain somewhat weaker than one might expect (at the current time, in the Quiet Sun model corona ranges between 0.1 and 1 G, and photospheric field strengths in concentrated downdrafts average ~1000 G).

Nonetheless, we can draw some general conclusions. 1. 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 1MK. However certain unrealistic features in synthetically generated X-ray images can occur with this type of treatment. 2. If a non current-dependent, constant coefficient of magnetic resistivity is used, Joule heating in our Quiet Sun model atmosphere is insufficient to maintain a sufficiently hot corona in the absence of any additional empirically-based source term. Synthetic SXT Image (AlMg response function)

3. A relatively persistent current layer is formed as the atmosphere transitions from a high-β to a magnetically dominated layer in the model transition region. 4. 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 magnetograms taken at the model photosphere and chromosphere are substantively different, and will yield substantively different results if used as a basis for a force-free or potential field extrapolation. 5. The convective dynamo remains an efficient means of generating magnetic flux in a fully-compressible, stratified model convection zone. A magnetic “carpet” is generated and maintained without first requiring the dispersal of a model active region REFERENCES: Bercik, D. J. 2002, Ph.D. Thesis Bercik, D. J., Fisher, G. H., Johns-Krull, C. M., & Abbett, W. P. 2005, ApJ, 631, 529 Cattaneo, F., 1999, ApJL, 515, L39 Knoll, D. A., & Keyes, D. E. 2003, J. Comp. Physics, 193, 357 Kurganov, A., & Levy, D. 2000, SIAM J. Sci. Comput., 22, 1461 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 Title, A. M. & Schrijver, C. J., 1998, in ASP Conf. Ser. 154: Tenth Cambridge Workshop on Cool Stars, Stellar Systems and the Sun, ed. R. A. Donahue & J. A. Bookbinder (San Francisco: ASP), p 345