1. Modeling and Data Analysis Associated With Supergranulation Walter Allen

2. Summary Granulation Mesogranulation Supergranulation Modeling Supergranulation Questions to Be Answered Conclusion

3. Granulation Granulation is the roughening or structure of the Sun’s surface. Granules are congregations of gas that rise to the surface from below the photosphere. Images of granules show cascades of bright and dark areas where the dark areas are 1/4 th of an arc second across. Granules seem to oscillate in brightness with a period of 5 minutes. These oscillations help us to understand the solar interior.

4. Mesogranulation Moving up in scale in terms of diameter and depth we have polygonal structures called mesogranules. Mesogranules are hypothetical constructs and not as much are known about them. Mesogranules are perhaps groups of individual granules with a common velocity pattern.

5. Supergranulation They where first observed as 2D Doppler spectroheliograms in the 1960’s Supergranules are large scale motion of the Sun’s surface. It is thought that supergranules are formed from conventional granules. Similar to conventional granulation pictures of supergranules showed dark and light regions through line of sight velocities toward and away from observation points. The fluid motion inside and on the surface of supergranules carry the magnetic field to its boundaries. This allows the supergranule to have its characteristic polygonal shape.

6. Granulation Mesogranulation Supergranulation Width (Mm) Depth (Mm).7 5-10 25-8510 Life Time 10-16min. 2hrs. >1day Population 5,000,000 2500 Velocity (Km/sec) ~1(rms,vertical) ~2(rms,horizontal) 0.3-0.5(horizontal) 0.06(rms,vertical).161

7. Definitions Corks->Corks are a time series of magnetograms. They are composed of points that sample the speed of moving features on the surface of the Sun. Corks float above the Sun’s surface much like a cork bottle top floats on top of water. Tree’s of Fragmenting Granules (TFG)-> TFG are families of repeatedly splitting granules originating from a single granule.

8. BFI (Broadband Filter Image) From 0- 48hrs. *To the left this image shows cork Motions and TFG’s (Trees of Fragmenting Granules) through the BFI (Broadband Filter Imager) From Hinode courtesy of Th. Roudier *White patches are TFG with lifetimes Of 24hrs. Yellow patches are TFG with Lifetimes of 20hrs. Green and blue Patches last shorter than 18hrs. *Granules fragment and combine With other granules to make larger Structures.

9. BFI (Broadband Filter Image) From 0- 48hrs. *Although TFG decrease in number with time a significant fraction of long lived TFG cover the Sun’s surface. *Shorter lifetime TFG’s appear Scattered everywhere in the field of View between the longest TFG. From Hinode courtesy of Th. Roudier *TFG’s are tools used to quantify the temporal/spatial organization of solar granulation at larger scales.

10. Stokes V Sequence 0-48hrs. *To the left is a Stokes V image Showing the motion of corks on a Time scale that ranges from 0 to 48 hours. *Magnetic fields are swept from the inner workings of the super granule to its boundaries. *Horizontal flows (fluid flow) diverge from the cell center and subside at cell boundaries. From Hinode courtesy of Th. Roudier

11. Stokes V Sequence 0-48hrs. *Cork motions are advected by the Horizontal velocity field. *Magnetic fields are squeezed between TFG’s (Trees of Fragmenting Granules) and fields follow their displacement while drifting to their boundaries. From Hinode courtesy of Th. Roudier

12. *This is an image of the Stokes V Fe I at 690 nm *TFG’s composed of parts on Mesogranule scales sweep corks And are pushed by new TFG. *The sweeping of corks contribute to the formation of the larger (super- granular) scale. From Hinode courtesy of Th. Roudier Stokes V Fe I 630nm 0-48 hrs.

13. HMI (Helioseismic Magnetic Imager) At present staff at the NSO and at Stanford have been preparing for the launch of HMI aboard SDO. HMI will support objectives that include differential rotation, subsurface flows, magnetic flux in active regions and attributes of the tacholine. HMI will provide space based measurements with pixel resolution of 4096 by 4096 with spherical harmonic degree of up to

14. Advantages Pertaining to HMI It is hoped to use data from HMI to facilitate the modeling of supergranules. The key goal will be to come up with a model and use the data from HMI to populate the model. Since HMI will yield unprecedented space based observations it may prove to be the best current instrument to use.

15. A Dopplergram The Dopplergram at the left shows the supergranulation pattern

16. The Model The task is to model supergranulation. We start with a PDE which takes the form of a wave equation. Cylindrical coordinates are used and the boundary conditions facilitate the modeling of the supergranule. The shape of the supergranule is generalized to that of a hexagon.

18. Wave Equation where,

19. Separation of Variables and

20. Partial Solutions

21. Boundary Conditions: Horizontal Boundary conditions in the horizontal direction are: So that:

22. Boundary Conditions: Vertical Boundary conditions in the vertical direction are: So that: and

23. General Solution

24. Coefficients

26. The plot below is generated with a sequence of numbers for r,t and z. The main goal in the future will be to populate the model with data from HMI.

28. Surface Plot

29. Conclusions For the future we seek to fulfill the following goals: I.To extend the boundary conditions so as to model any general polygon. II.To obtain a better estimate of the depth of supergranulation. III.To determine if supergranules are directly convective or constructs of smaller structures. IV.To add to the governing wave equation additional forces such as gravitational stratification and pressure gradients. V.To take into account magneto-hydrodynamic effects. VI.To obtain clues to the origin of supergranulation. VII.To obtain a better idea of the subsurface magnetic field.