1. A Simplified Treatment of Optically Thick Radiative Transfer in Large-scale Convection Zone to Corona Models W.P. Abbett and G.H. Fisher Space Sciences Laboratory, University of California, Berkeley SHINE 2010 ABSTRACT: We present the latest in a series of numerical simulations of quiet Sun magnetic fields that extend from the upper convection zone into the low corona. We apply an efficient, simplified treatment of the physics of optically-thick radiative transfer throughout the surface layers, and investigate the effects of convective turbulence on the magnetic structure of the Sun's upper atmosphere in an initially unipolar (open-field) region. We then compare these results with earlier simulations that use an ad-hoc, parameterized treatment of surface cooling.

2. The Numerical model: We semi-implicitly solve the MHD system of conservation equations on a Cartesian mesh using a modified and improved version of the code RADMHD (Abbett 2007): The code advances the explicit portion of the MHD system by means of a third order-accurate CWENO shock capture scheme, and solves the implicit portion of the system via a Jacobian-free Newton-Krylov technique.

3. The energetics of the system: radiative transfer It is well known that in order to describe the energetics of the corona in a physical way, a model should include energy sources, Q, such as electron thermal conduction along magnetic field lines and radiative cooling in the optically-thin limit. However, at and below the photosphere, the plasma is optically thick, and a proper characterization of the radiative cooling is more challenging. The most satisfying approach would be to couple the LTE (or non-LTE) population and transfer equations to the MHD system to obtain cooling rates and intensities that could be compared directly to observations. Unfortunately, for large, active region or global-scale problems, the computational expense of these methods remain prohibitive. The parameterized “Newton cooling” approach presented in Abbett (2007), while computationally efficient, has distinct disadvantages. Namely, it is ad-hoc and unphysical, thus requiring other, more realistic simulations as a basis of calibration to get meaningful results. We therefore recently developed a more physical approximation to optically thick radiative surface cooling derived directly from the macroscopic radiative transfer equation, and have included this new radiation package into our numerical model.

4. Radiative cooling drives surface convection and is a crucial contributor to the energy balance in the region of the solar atmosphere bridging the convection zone and corona. Thus, the surface layers must be described in a physical way in order to properly understand the transport of magnetic energy and helicity into the atmosphere from below the surface. Below, we briefly describe our approach. Consider the net cooling rate for a volume of plasma at a particular location in the solar atmosphere: Here represents frequency, and solid angle. The emissivity, opacity, and specific intensity are frequency dependent, and are denoted,, and respectively. Rearranging the order of integration, and defining the source function as the ratio of the emissivity to opacity, we have: Since the source function is independent of direction, we recast the integral as: with mean intensity.

5. The formal solution for the specific intensity in the plane-parallel approximation is, where is the usual cosine angle. Then the mean intensity can be expressed as. The direction integral can be evaluated and the above expression can be recast as, where E 1 is the first exponential integral. Note that up to this point, no approximations have yet to be made, other than an assumption of a locally plane-parallel atmosphere. Now we’ll make an approximation. Note that is singular when, and that the singularity is integrable. Since E 1 is peaked around, contributions from will be centered around. Thus, we can approximate the mean intensity by:

6. . This integral can then be evaluated, giving a simple expression for the mean intensity, where E 2 is the now the second exponential integral. Note that this can be recast into the following form:,. We now return to our expression for the net cooling rate, expressing it in a slightly different form:. Substituting the approximation above into the integrand, we have.

7. Here, represents the normalization constant for the integration. The arbitrary constant appears in the exponential integral since the mean opacity used in the calculation of the optical depth scale could differ in general from the mean opacity that appears by itself in the integrand. To determine the normalization constant, we integrate our cooling function from zero to infinity in optical depth over an isothermal slab to obtain the total radiative flux. The resulting expression must be equal to the known result. This allows us to determine the normalization constant: To evaluate, we compare the detailed cooling rate depth distribution using this formulation with the cooling rate in the Bercik (2002) LTE radiative magneto-convection simulations, and conclude the best-fit value is.. We now integrate over frequency, replacing the frequency-dependent opacity by its Planck-weighted mean value. Applying the mean value theorem of Calculus, and including the exponential function in the average over frequency, we find that If we further assume LTE, the source function is simply the Planck function and

8. The advantage of this treatment lies in its simplicity. The above approximation for surface cooling, while non-linear, is trivial to calculate for each mesh element. It is certainly more physical than the ad-hoc treatment employed in Abbett (2007), since it is based on the radiative transfer equation, and is derived from an optical depth scale. Thus, our approximate cooling function takes the form Preliminary Results: We have relaxed a number of quiet Sun models at different spatial scales and resolution levels in order to understand how the new treatment cools the surface layers, and whether solar-like convective turbulence is developed and maintained in the model over hours of solar time. Each simulation includes a shallow convection zone, and extends 10 Mm out into the corona. The models are initialized with a weak vertically-directed magnetic field that is amplified in time by the convective turbulence. This configuration allows us to study the physics of open field regions, and the energization of the solar wind.

9. Shown is a comparison of the average temperature stratification through the model convection zone between the realistic radiative magneto-convection simulations of Bercik (2002) (crosses), and the RADMHD model convection zone using the new treatment of optically thick transfer (diamonds).

10. The thermodynamic stratification of a 1D symmetric, stratified atmosphere using the New RT treatment during the relaxation process. The atmosphere is not in static equilibrium, thus vertical waves and temperature variations are expected. Below the model photosphere (at z~3Mm) the atmosphere is super-adiabatically stratified, and an application of an entropy perturbation will initiate convection in the 3D domain. The difference between this initial state and that used in Abbett (2007) is that here, there are no adjustable parameters. The atmosphere naturally develops a solar-like stratification based on the boundary conditions, and the radiative cooling, thermal conduction and magnetic heating specified in the RADMHD energy equation.

11. Temperature (upper frame) and the vertical component of the magnetic field at the RADMHD quiet Sun photosphere in a 3D simulations using the new treatment for optically thick transfer. Note that the net field is unipolar, thus there is more outward-directed flux (white) than inward (black). In this simulation, the domain spans 21 X 12 X 12 Mm at a resolution of 448 X 256 X 256. The model convection zone is 2.5 Mm deep. This simulation was performed on a small cluster at SSL using 112 processors.

12. Temperature in the RADMHD model chromosphere using the new treatment for optically thick transfer. Light (dark) regions correspond to cooler (hotter) temperatures.

13. For comparison, the temperature (upper frames) and the vertical component of the magnetic field (lower frames) at three different heights (400 km below the photosphere, at the photosphere, and in the chromosphere) in the quiet Sun photosphere of Abbett (2007). This simulation was performed using the ad hoc parameterized Newton cooling described in that paper.

14. Left: The temperature at the RADMHD quiet Sun photosphere in a 3D convection zone- to-corona simulation that extends over a larger area. Here, the domain spans 24 X 24 X 12 Mm at a relatively high resolution of 512 X 512 X 256. Again, the model convection zone is 2.5 Mm deep. This simulation was performed on NASA’s “Discover” cluster and utilized 512 CPUs.

15. Left: The temperature 500 km below the surface. Dark areas correspond to cooler temperature, vortical downdrafts.

16. Temperature at the RADMHD model photosphere (left frame) and magnetic fieldlines threading the photosphere over a small sub-domain (the cyan box in the left frame indicates the approximate size of the sub-domain). This snapshot is taken from the same simulation as the previous figure, but at a later time. The following two mages show the magnetic structure at an even later time over a much larger portion of the computational domain. The gray slice represents the approximate location of the model photosphere.

19. Summary: We have significantly improved our ability to describe the thermodynamics of the lower atmosphere in the models by developing a simplified treatment of optically thick radiative cooling. This technique retains the computational efficiency of parameterized methods, but without the need for continual calibration against more realistic models where the transfer equation is solved in detail. The new method successfully reproduces the average thermodynamic stratification of smaller-scale, realistic LTE radiative magneto- hydrodynamic simulations of surface convection. We find that while the convection pattern is stable and solar-like (i.e., it successfully reproduces the distribution of granule sizes, and the turnover time characteristic of solar granulation), the cooling rate at the edges of the intergranular lanes is unphysically high, and a lower limit must be enforced. We believe this is due to two separate issues. First, certain regions in the atmosphere are not well resolved in optical depth, and the current implementation does not treat the divergence of the radiative flux in a strictly conservative fashion. This can lead to numerical error. Second, the current formalism neglects the important effects of radiative transport in the horizontal directions.

20. We are currently addressing each of these problems. The solution to the first problem is straightforward --- we have recast the radiative cooling in a flux conservative fashion using a variant of the Constrained Transport scheme of Stone and Norman (1992). This will conserve radiative flux to numerical round-off error, and mitigate excessive cooling in regions under-resolved in optical depth. Second, we are developing and testing a means to incorporate the physics of radiative transport in the horizontal directions; again, without having to take the computationally expensive step of solving the frequency- dependent transfer equation along separate ray paths. We hope to report on these results in the near future. The techniques presented here will allow us to simulate the dynamics of the convection zone-to corona system over active region spatial and temporal scales in a physical, yet computationally efficient way. Acknowledgements: This ongoing work is supported by the NASA TR&T program, and the NSF ATM program References: Abbett, W.P., “The Magnetic Connection Between the Convection Zone and Corona in the Quiet Sun,” 2007, ApJ 665, 1469. Bercik, D.J., “A Numerical Investigation of the Interaction Between Convection and Magnetic Field in a Solar Surface Layer,” PhD Thesis, 2003, Michigan State University. Stone, J.M., Norman, M.L., “ZEUS-2D: A Radiation MHD Code for Astrophysical Flows in Two Space Dimensions. II. The MHD Algorithms and Tests,”1992, ApJS 80, 791.