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3D numerical simulations of MHD waves in solar flux tube

3D numerical simulations of MHD waves in solar flux tube. Viktor Fedun , Robertus Erd é lyi. The University of Sheffield, Department of Applied Mathematics v.fedun@sheffield.ac.uk. Waves in the Solar Atmosphere Numerics (SAC) Geometry, equilibrium and equations

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3D numerical simulations of MHD waves in solar flux tube

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  1. 3D numerical simulations of MHD waves in solar flux tube Viktor Fedun, Robertus Erdélyi The University of Sheffield, Department of Applied Mathematics v.fedun@sheffield.ac.uk

  2. Waves in the Solar Atmosphere Numerics (SAC) Geometry, equilibrium and equations Magnetic field in MHD model 3D full MHD simulations Conclusions Outline of the talk

  3. Waves in the Solar Atmosphere Recent high-resolution ground-based observations (ROSA, SST) provide clear evidence for the existence of oscillations driven by magnetic twist in flux tubes. These torsional oscillations are associated with Alfven waves. It is of particular interest to study the excitation and propagation of torsional Alfven waves into the upper, magnetised atmosphere because they can channel photospheric energy into the corona.

  4. Numerics 1D-3D MHD Our developed code SAC (Sheffield Advanced Code) is used to carry out our simulations. The code can solve the full system if ideal hydrodynamic or magnetohydrodynamic equations in one, two or three dimensional Cartesian geometry. where denotes the total pressure.

  5. Numerics 1D-3D MHD CD4 ! The derivatives can be represented as their central difference approximations: Good, precise, easy to manage analytically. However, numerically unstable.

  6. Numerics 1D-3D MHD What we are going to use is called hyperdiffusion(numerical diffusion, sub-grid diffusion): not like flux limiter (intrinsic property of numerical scheme), but an additional term in the MHD equations Successfully filters the solution from numerical instabilities

  7. Neo-classical MHD system: Shelyag, S., Fedun, V., Erdélyi, R. A&A, 2008

  8. Where background pressures

  9. Numerical examples I (1D-HD/MHD) Our method work pretty well for solving many kinds of MHD/HD problems Brio-Wu MHD Shock Tube The initial left and right states are given by and This is a good problem to test wave properties of a particular MHD solver, because it involves two fast rarefaction waves, a slow compound wave, a contact discontinuity and a slow shock wave.

  10. Numerical examples II (2D-MHD) The snapshots of Orszag-Tang (Orszag and Tang, 1979) vortex problem show the temperature for the normalized time 0 to 10, simulated with 512x512 grids covering 2π-by-2π space. This problem is a simple two-dimensional classic test for MHD codes.

  11. Numerical examples III (2D-MHD)

  12. Numerics MHD • We have started by considering a case with realistic temperature stratification. • The temperature profile of our model is based on the VALIIIC(Vernazza et al 1981) model atmosphere below the transition regionphere above it. A – a dark point within a cell B – the average cell center C – the average quiet Sun D - the average network E – a bright network element F – very bright network element

  13. For a planar photosphere unbounded above, the scalar potential is by analogy to Coulomb’s law. The magnetic field current improvement Extrapolation Relaxations methods

  14. The magnetic field current improvement We used a self-similar non-potential magnetic field configuration For Cartesian coordinate system in 3D geometry Schlüter & Temesváry 1958; Schüssler & Rempel 2005; Cameron et al. 2008 The magnetic field constructed in this way is divergence-free by definition. where B0z describes the decrease in the vertical component of magnetic field towards the top of the model, andfis the function defining how the magnetic field opens up with height.

  15. The magnetic field current improvement

  16. The background

  17. Alfvén Waves in the Lower Solar Atmosphere Magnetic bright point group analysed by David B. Jess, MihalisMathioudakis,Robert Erdélyi, Philip J. Crockett, Francis P. Keenan, Damian J. Christian (2009)withSwedish Solar Telescope (SST). Photosphere Chromosphere Simultaneous images in the (left) Hα continuum (photosphere) and (right) Hα core (chromosphere) obtained with the SST. The conglomeration of bright points within the region we investigated is denoted by a square of dashed lines. The scale is in heliocentric coordinates where 1 arc sec 725 km. A wavelength-versus-time plot of the H profile showing the variation of line width at full-width half-maximum as a function of time. The arrows indicate the positions of maximum amplitude of a 420-s periodicity associated with the bright-point group. TorsionalAlfvén waves incompressible so can be detected by periodic spectral line broadening.

  18. Detection of Alfvén waves Chromosphere Expanding magnetic flux tube sandwiched between photospheric and chromospheric intensity images obtained with the SST, undergoing a torsionalAlfvénic perturbation and generating a wave that propagates longitudinally in the vertical direction. At a given position along the flux tube, the Alfvénic displacements are torsional oscillation Photosphere TorsionalAlfvén waves incompressible so can be detected by periodic spectral line broadening.

  19. Torsional wave driver Convectively Driven Vortex Flows in the SunBonet et al. (2008). The movie shows BPs swirling around intergranular points where several dark lanes converge. These motions are reminiscent of the bathtub vortex flows. Logarithmic spiral (solid lines) that fits the trajectories of six observed BPs (symbols) SST observed small scale rotational motion of magnetic bright points, counterclockwise (+) and clockwise (o) TorsionalAlfvén waves may be generated all over the photosphere!

  20. Torsional wave driver Small-scale swirl events in the quiet Sun chromosphere,S.Wedemeyer-Böhmand L. Rouppe van derVoort, A&A, 2009 The time series show a chaotic and dynamic scene that includes spatially confined “swirl” events. These events feature dark and bright rotating patches, which can consist of arcs, spiral arms, rings or ring fragments. They exhibit Doppler shifts of −2 to −4 km/s but sometimes up to −7 km/s, indicating fast upflows. The diameter of a swirl is usually of the order of 2′′. At the location of these swirls, the line wing and wide-band maps show close groups of photospheric bright points that move with respect to each other. Conclusions. A likely explanation is that the relative motion of the bright points twists the associated magnetic field in the chromosphereabove.

  21. Numerics 3D MHD The driver The half width of the Gaussian Before looking at the propagation of the real signal, we considered a 30 second driver. This driver is well below the acoustic cut-off period at any point in our atmosphere, and therefore allows us to look at the simple case of strong propagation. Next :

  22. Numerics3D (source)

  23. Numerics 3Dtube Driver period 30 sec

  24. Numerics 3D tube Driver period 30 sec zoom More example movies are at Solar Theory Grop (SWAT) website at http://swat.group.shef.ac.uk/simulations.html

  25. Numerics 3D tube Driver period 30 sec zoom

  26. Numerics 3D tube Driver period 120 sec, zoom

  27. Numerics 3D tube Driver period 120 sec

  28. Numerics 3D tube Driver period 120 sec

  29. 2.72 mHz 2.49 mHz 2.22 mHz 2.09 mHz Magneto-seismology / Frequency distribution Since the torsionalAlfvén waves can be generated independently on each magnetic surface, for the first time we can resolve the frequency as a function of radius in the chromospheric flux tube! Now we know: TORSIONAL wave driver can cause such a distribution!!!

  30. Magneto-seismology / Multiple driver Superposition of drivers

  31. Magneto-seismology / Multiple driver

  32. Magneto-seismology Slice at 0.5 Mm Slice at 1.Mm

  33. Magneto-seismology Reconstruction of the magnetic field

  34. Thank you !

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