1. Using plasma dynamics to determine the strength of a prominence's magnetic fields GCOE Symposium 2013 @ Kyoto University Andrew Hillier

2. What is a Quiescent Prominence? ~10 Mm Image: Quiescent prominence observed on 2007/10/03 01:56 UT in the Ca II H line (3968.5 Å) Temperature: 6000K~10,000K (Tandberg-Hanssen 1995) Number Density: 10 10 ~10 11 cm -3 (eg Labrosse 2010 & Hirayama 1986) Magnetic field strength: 3~30 G (Leroy 1989) Ionisation fraction: ~0.2 at centre (Gunar et al 2008)

3. Prominences and Space Weather Prominence eruption on August 31, 2012 observed by the Solar Dynamics Observatory satellite (courtesy of NASA)

4. How Well Do We Understand Quiescent Prominences? ~10 Mm Image: Quiescent prominence observed on 2007/10/03 01:56 UT in the Ca II H line (3968.5 Å) Magnetic field strength: 3~30 G (Leroy 1989) But only ~15 prominences have had their magnetic field measured (to my knowledge) But we need to know the field strength to be able to model prominences, discuss there dynamics etc

5. The Plumes in Prominences Fig: Prominence observed in Hα on 8 th Aug 2007 using Hinode SOT Courtesy of T. Berger First observed by Stellmacher & Wiehr 1973 Rediscovered by Berger et al 2008 & De Toma et al 2008

6. The Plumes Created by the Magnetic Rayleigh-Taylor Instability The plumes (fingers of low density material rising through the dense prominence material) were hypothesized to be created by the Rayleigh- Taylor instability by Berger et al 2008 & 2010 Key Point 1: plumes have an elliptical head Key Point 2: Constant rise velocity (10 – 30 km/s) Image: Quiescent prominence observed on 2007/10/03 03:30 UT in the Ca II H line (3968.5 Å)

7. The Plumes Created by the Magnetic Rayleigh-Taylor Instability Simulations by Hillier et al (2012) investigated the 3D mode of the magnetic Rayleigh- Taylor instability in a prominence model Key Point 3: Creates filamentary structure aligned with Magnetic field

8. Using The Key Points to Make a Model Key Point 1: plumes have an elliptical head (change coordinates to make a circle) Key Point 2: Constant rise velocity (10 – 30 km/s) (Change reference frame) Key Point 3: Creates filamentary structure aligned with Magnetic field (Makes it like a tube)

9. Flow around a circular cylinder This has now reduced to a classic fluid dynamics problem Using the assumptions of invisicid, irrotational and incompressible it is possible to calculate the potential flow around a circular cylinder Potential HD flow around a circular cylinder – Source Wikipedia

10. Compression at Top of Plumes For some plumes we see a thick, bright hat. As the emission of prominences is mainly scattering, this is showing higher density regions Image: Left - Quiescent prominence observed on 2007/10/03 02:56 UT in the Ca II H line (3968.5 Å). Right – Zoomed image of plume Plume rises Material is compressed High total pressure drives material out the way

11. Mathematical model for the Compression Image: Compressible MHD flow round a circular cylinder. Magnetic field into screen We can use a classic solution of flow around a circular cylinder + MHD (Horizontal field only) +Compressibility correction to get the density distribution (van Dyke 1975).

12. How can this be used? By modelling the intensity in terms of density, the compression at the top of the plume can be calculated. This will allow for the plasma beta to be solved for.

13. Estimate of Prominence Plasma Beta – Calculating Plume Size and Velocity The dimension of the plume head (needed for normalisation) are a~900km and b~1700km The rise velocity is

14. Estimate of Prominence Plasma Beta – Fitting Intensity to calculate β Assuming that the emission is only proportional to the density we can fit to solve for M *, giving an estimate of the plasma beta of for

15. Conclusions We now have a new way to estimate the plasma beta of quiescent prominences using the Rayleigh-Taylor plumes Application to one prominence gives the plasma β as β=0.47 – 1.13 for γ=1.4 – 1.7. There are many potential improvements that can be made, that will improve the accuracy AND the amount of information we can extract from the prominence For greater detail, please see: Hillier, Hillier & Tripathi (2012) ApJ, 761, 106

17. Setting for Simulations Kippenhahn-Schlűter prominence model (Priest 1982) Buoyant tube put in centre of prominence to make it unstable and a velocity perturbation in the y direction to excite interchange of magnetic field Ideal MHD used (grid 90*150*400) Length normalised to pressure scale height Fig: Mass density (colour) and field lines (contour) of prominence model. A is x-z cut and B is y-z cut (y boundary is symmetric)

18. Movie: Temporal evolution of instability in x=0 plane. Colour shows density, arrows show velocity 2D Density Slice of simulation Swirling, vortex like structures formed once instability is initiated Reach height of approx 6Mm Upflows: ~ 6 km/s (approximately constant) Width of upflows inversely cascades from ~100 km to ~1Mm Makes threads in the prominence material

19. Evolution shown in 3D (1)Rise of cavity releases the magnetic tension, flattening the field lines. Instability starts on small scale (2)Multiple plumes formed, plume magnetic field begins to move through the prominence (3)Magnetic field lines glide passed each other in an interchange process Fig: Temporal evolution of instability in 3D, lines represent magnetic field with density isosurface

20. Application to Simulation Results To check the data, first we revise the axis to give a circular head. Note there is no density increase at the top of the plume

21. Application to Simulation Results – Velocity Around Plume Head Velocities along curve shown in previous slide (both simulated – solid, and predicted -dashed)

22. Application to Simulation Results – Matching Density Distribution Integrating the density along the x- axis shows the increase in column density at the head of the plume The above figure shows the simulated density along the slit and predicted density

23. Application to Simulation Results – Calculating χ 2 By calculating the χ 2 for fits to the density profile for different values of plasma β, we can show that the smallest χ 2 corresponds to the simulation plasma β of ~0.55

24. Can we model the Bright Emission? For some plumes we see a thick, bright hat. As the emission of prominences is mainly scattering, this is showing higher density regions Image: Left - Quiescent prominence observed on 2007/10/03 02:56 UT in the Ca II H line (3968.5 Å). Right – Zoomed movie of plume

25. What if the Magnetic Field is Vertical If the field is vertical, then the compression doesn’t occur at the head of the plume Rarefaction occurs instead It is hard to understand the observations of the plume if the prominence field is vertical Courtesy of Roger Scott, Montana State University

26. Estimate of Prominence Plasma Beta – β as a Function of γ

27. Necessary improvements for the Model Deal with projection effects and magnetic field that is not along the line of sight (use velocity equations combined with observed Doppler shifts) Include shear between the plume magnetic field and the prominence magnetic field to give direction of the magnetic field (use most unstable mode of magnetic Rayleigh-Taylor instability under shear and the with the observed plume width) Improved model for emission (there must be a way to improve my simple model for the emission – Suggestions Please!)