UCB-SSL Plans for Next Year Joint CCHM/CWMM Workshop, July 2007 W.P. Abbett, G.H. Fisher, and B.T. Welsch

RADMHD: Modeling the combined convection zone-to-corona system: The code solves 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),

Modeling the combined convection zone-to-corona system in a physically self-consistent way: 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)

Modeling the combined convection zone-to-corona system: 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): 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.

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: 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,. 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:

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. 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 ~ 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 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

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.

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

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)

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 B z 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

log B log J log βBzBz log B

Characteristics of the Quiet Sun model atmosphere: Note: Above movie is not a timeseries!

PLANS: A Focus on the Physics of Active Region Flux Emergence…. Extend the RADMHD quiet Sun simulations to active region spatial scales using our new allocation on NASA’s Discover cluster at Goddard Space Flight Center Once the large-scale quiet Sun model has energetically relaxed, introduce active region strength magnetic fields from below by (1) introducing a magnetic flux tube directly into the portion of the domain representing the convection zone (similar to the more idealized simulations of, e.g., Magara 2004, Manchester 2004, Archontis 2007); (2) introduce magnetic flux into the domain from below using previous ANMHD calculations of buoyant Ω-loops in the deep interior (Abbett 2000, 2004); (3) introduce an interacting pair of twisted flux ropes into the RADMHD domain below the visible surface, and study the magnetic topology of the corona as one system emerges into another;

The Physics of Flux Emergence…. (4) study the effects of magnetic flux emergence at small scales (e.g., quiet- Sun fields generated by surface convection or ephemeral active regions), and the effect of small scale flux emergence on the large scale magnetic topology of the model corona; (5) follow the evolution of the model active region after emergence, study the decay process and magnetic connectivity as convective turbulence interacts with loop footpoints; (6) test and validate our inversion techniques and boundary driving schemes by driving RADMHD model corona with synthetic magnetograms and comparing the resultant model coronae against the self-consistent calculations.

RADMHD Flux Emergence: Computational Requirements The most computationally intensive portion of the project is to relax the convectively unstable portion of the large-scale domain. We reduce the computational cost by (1) relaxing periodic sub-domains on our local cluster, (2) filling the blocks of the larger grid with these solutions, and (3) introducing an entropy perturbation throughout the large-scale domain to break the symmetry. We find that this process takes fewer processor hours than if we simply perturb large-scale horizontally-invariant, super-adiabatically stratified background atmospheres and allow the magnetoconvection to develop from scratch. This is still an expensive process. By comparison, the AR emergence timescale is quite rapid, and the emergence runs are relatively inexpensive (though we choose to remain constrained by the magnetosonic wavespeed in the corona)!