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3D numerical simulations of MHD waves in solar flux tubeThe University of Sheffield, Department of Applied Mathematicsv.fedun@sheffield.ac.ukViktor Fedun, Robertus Erdlyi

Israel, 10-17 April 2010, Dynamical Processes in Space PlasmasSolar Physics & Space Plasmas Research CentreOutline of the talkWaves in the Solar AtmosphereNumerics (SAC)Geometry, equilibrium and equationsMagnetic field in MHD model3D full MHD simulationsConclusions

Israel, 10-17 April 2010, Dynamical Processes in Space PlasmasSolar Physics & Space Plasmas Research CentreRecent 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.Waves in the Solar AtmosphereIsrael, 10-17 April 2010, Dynamical Processes in Space PlasmasSolar Physics & Space Plasmas Research CentreIsrael, 10-17 April 2010, Dynamical Processes in Space PlasmasOur 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.Numerics 1D-3D MHD Solar Physics & Space Plasmas Research CentreIsrael, 10-17 April 2010, Dynamical Processes in Space PlasmasNumerics 1D-3D MHD CD4 ! The derivatives can be represented as their central difference approximations: Good, precise, easy to manage analytically. However, numerically unstable.

Solar Physics & Space Plasmas Research CentreIsrael, 10-17 April 2010, Dynamical Processes in Space PlasmasWhat 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 equationsSuccessfully filters the solution from numerical instabilitiesNumerics 1D-3D MHD Solar Physics & Space Plasmas Research CentreIsrael, 10-17 April 2010, Dynamical Processes in Space Plasmas

Neo-classical MHD system:Shelyag, S., Fedun, V., Erdlyi, R. A&A, 2008 Solar Physics & Space Plasmas Research CentreIsrael, 10-17 April 2010, Dynamical Processes in Space Plasmas

Where background pressures

Solar Physics & Space Plasmas Research CentreIsrael, 10-17 April 2010, Dynamical Processes in Space PlasmasOur method work pretty well for solving many kinds of MHD/HD problemsNumerical examples I (1D-HD/MHD)

Brio-Wu MHD Shock TubeThis 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.

The initial left and right states are given by andSolar Physics & Space Plasmas Research CentreIsrael, 10-17 April 2010, Dynamical Processes in Space PlasmasNumerical 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 2-by-2 space. This problem is a simple two-dimensional classic test for MHD codes.

Solar Physics & Space Plasmas Research CentreIsrael, 10-17 April 2010, Dynamical Processes in Space PlasmasNumerical examples III (2D-MHD)

Solar Physics & Space Plasmas Research CentreIsrael, 10-17 April 2010, Dynamical Processes in Space PlasmasNumerics 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 cellB the average cell centerC the average quiet SunD - the average networkE a bright network elementF very bright network elementSolar Physics & Space Plasmas Research CentreIsrael, 10-17 April 2010, Dynamical Processes in Space Plasmas

The magnetic field current improvementFor a planar photosphere unbounded above, the scalar potential isby analogy to Coulombs law.

Relaxations methods Extrapolation

Solar Physics & Space Plasmas Research Centre

Israel, 10-17 April 2010, Dynamical Processes in Space PlasmasThe magnetic field current improvementFor Cartesian coordinate system in 3D geometry

We used a self-similar non-potential magnetic field configurationwhere B0z describes the decrease in the vertical component of magnetic field towards the top of the model, and f is the function defining how the magnetic field opens up with height. The magnetic field constructed in this way is divergence-free by definition.Schluter & Temesvry 1958; Schussler & Rempel 2005; Cameron et al. 2008Solar Physics & Space Plasmas Research CentreIsrael, 10-17 April 2010, Dynamical Processes in Space PlasmasThe magnetic field current improvement

Solar Physics & Space Plasmas Research CentreThe backgroundIsrael, 10-17 April 2010, Dynamical Processes in Space Plasmas

Solar Physics & Space Plasmas Research CentreIsrael, 10-17 April 2010, Dynamical Processes in Space PlasmasAlfvn Waves in the Lower Solar Atmosphere

Photosphere ChromosphereA 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.

Magnetic bright point group analysed by David B. Jess, Mihalis Mathioudakis, Robert Erdlyi, Philip J. Crockett, Francis P. Keenan, Damian J. Christian (2009) with Swedish Solar Telescope (SST).Torsional Alfvn waves incompressible so can be detected by periodic spectral line broadening.

Simultaneous images in the (left) H continuum (photosphere) and (right) H 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.Solar Physics & Space Plasmas Research CentreIsrael, 10-17 April 2010, Dynamical Processes in Space PlasmasDetection of Alfvn waves Chromosphere

PhotosphereTorsional Alfvn waves incompressible so can be detected by periodic spectral line broadening.

Expanding magnetic flux tube sandwiched between photospheric and chromospheric intensity images obtained with the SST, undergoing a torsional Alfvnic perturbation and generating a wave that propagates longitudinally in the vertical direction. At a given position along the flux tube, the Alfvnic displacements are torsional oscillationSolar Physics & Space Plasmas Research CentreIsrael, 10-17 April 2010, Dynamical Processes in Space PlasmasTorsional wave driverLogarithmic spiral (solid lines) that fits the trajectories of six observed BPs (symbols)

The movie shows BPs swirling around intergranular points where several dark lanes converge. These motions are reminiscent of the bathtub vortex flows.Convectively Driven Vortex Flows in the Sun Bonet et al. (2008).

Torsional Alfvn waves may be generated all over the photosphere!SST observed small scale rotational motion of magnetic bright points, counterclockwise (+) and clockwise (o) Solar Physics & Space Plasmas Research Centre

Israel, 10-17 April 2010, Dynamical Processes in Space PlasmasTorsional wave driverSmall-scale swirl events in the quiet Sun chromosphere, S.Wedemeyer-Bhm and L. Rouppe van der Voort, 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 chromosphere above. Solar Physics & Space Plasmas Research CentreIsrael, 10-17 April 2010, Dynamical Processes in Space PlasmasNumerics 3D MHDThe 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 :

Solar Physics & Space Plasmas Research CentreIsrael, 10-17 April 2010, Dynamical Processes in Space PlasmasNumerics 3D (source)

Solar Physics & Space Plasmas Research CentreIsrael, 10-17 April 2010, Dynamical Processes in Space Plasmas

Numerics 3D tube Driver period 30 secSolar Physics & Space Plasmas Research CentreIsrael, 10-17 April 2010, Dynamical Processes in Space PlasmasMore example movies are at Solar Theory Grop (SWAT) website at http://swat.group.shef.ac.uk/simulations.html

Numerics 3D tube Driver period 30 sec zoomSolar Physics & Space Plasmas Research CentreIsrael, 10-17 April 2010, Dynamical Processes in Space PlasmasNumerics 3D tube Driver period 30 sec zoom

Solar Physics & Space Plasmas Research CentreIsrael, 10-17 April 2010, Dynamical Processes in Space Plasmas

Numerics 3D tube Driver period 120 sec, zoom

Solar Physics & Space Plasmas Research CentreIsrael, 10-17 April 2010, Dynamical Processes in Space PlasmasNumerics 3D tube Driver period 120 sec

Solar Physics & Space Plasmas Research CentreIsrael, 10-17 April 2010, Dynamical Processes in Space PlasmasNumerics 3D tube Driver period 120 sec

Solar Physics & Space Plasmas Research CentreMagneto-seismology / Frequency distribution

Israel, 10-17 April 2010, Dynamical Processes in Space Plasmas

2.72 mHz2.49 mHz2.22 mHz 2.09 mHzSince the torsional Alfvn 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!!!Solar Physics & Space Plasmas Research CentreIsrael, 10-17 April 2010, Dynamical Processes in Space Plasmas

Magneto-seismology / Multiple driverSuperposition of driversSolar Physics & Space Plasmas Research CentreIsrael, 10-17 April 2010, Dynamical Processes in Space PlasmasMagneto-seismology / Multiple driver

Solar Physics & Space Plasmas Research Centre

Israel, 10-17 April 2010, Dynamical Processes in Space Plasmas

Slice at 0.5 MmSlice at 1.Mm

Magneto-seismologySolar Physics & Space Plasmas Research CentreIsrael, 10-17 April 2010, Dynamical Processes in Space Plasmas

Reconstruction of the magnetic fieldMagneto-seismologySolar Physics & Space Plasmas Research CentreIsrael, 10-17 April 2010, Dynamical Processes in Space PlasmasThank you !

Solar Physics & Space Plasmas Research Centre

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Vx = Ae-

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r 2 e-

z-z 0( )2

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Vy = Ae-

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r 2 e-

z-z 0( )2

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r2 = x 2 + y 2