Amplification of twists in magnetic flux tubes Youra

Amplification of twists in magnetic flux tubes Youra Taroyan Department of Physics, Aberystwyth University, email: yot@aber. ac. uk users. aber. ac. uk/djp 12

Brief chronology of Alfvén waves http: //www. plasma-universe. com 1942: Alfvén theorises the existence of electromagnetic-hydromagnetic waves: "If a conducting liquid is placed in a constant magnetic field. . . a kind of combined electromagnetic-hydrodynamic wave is produced. " 1947: Alfvén suggests heating of the solar corona by these waves 1949: Laboratory experiments by Lundquist produce such waves in magnetised mercury 1949: Enrico Fermi uses Alfvén waves in his theory of cosmic rays. 1958: Alfvén waves detected in the ionosphere after a nuclear test explosion …

Basic properties of Alfvén waves • Result from the competing effects between magnetic tension and plasma inertia • Incompressible (no variations in pressure, density or magnetic field strength) not easy to detect! • Able to carry large amounts of energy along field lines • Propagation speed • In non-uniform media may couple linearly or non-linearly to other types of waves

Dissipation of Alfvén waves in the atmosphere Phase mixing • Heyvartes & Priest (1983): generation of small transverse scales as Alfven waves propagate in an inhomogeneous magnetic field • Coronal heating, wind acceleration: Browning (1991), Hood et al (1997), Ruderman et al (1998) • A diverging magnetic field enhances the efficiency of phase mixing, whereas gravitational stratification diminishes the mechanism (De Moortel et al 2000). • In open structures dissipation occurs only within several solar radii (Parker 1991, Ofman & Davila 1995) • Non-linear Alfven waves phase mixing visco-resistive heating bulk flows (outflows) (Mc. Laughlin et al 2011)

Dissipation of Alfvén waves in the atmosphere Resonant absorption • Ionson (1978), Lee & Roberts (1986), Goossens et al (1992), Ofman & Davila (1994), Erdelyi et al (1995), Ruderman et al (1997), Ruderman & Roberts (2002), Soler et al (2011)

Dissipation of Alfvén waves in the atmosphere Nonlinear mode conversion 1. 5 D simulations of loop heating by Antolin & Shibata (2010): • The regimes under which Alfven wave heating produces hot and stable coronae are found to be rather narrow; • Independently of the photospheric wave amplitude and magnetic field, a corona can be produced and maintained only for long (>80 Mm) loops. • Explanation: necessary distance for shock formation ~ wavelength but the wave has barely the distance to propagate 1 wavelength before reaching the other footpoint

Footprints and Dissipation of Alfvén waves in the atmosphere forward modelling Alfvén wave turbulence • Hollweg (1986), Buchlin et al (2007), van Ballegooijen et al (2011): Alfven waves that travel along the flux tube, reflect due to gradients in Alfven speed, and generate turbulence via nonlinear wave–wave interactions.

Observational context Ø Alfvén waves in flux tubes simultaneous blue and red shifts nonthermal broadening of a spectral line profile Ø Doschek et al (1976), Dere & Mason (1993), Doyle et al (1998), Chae (1998)

Observational context Jess et al. (2009) studied H-alpha absorption profiles with SST and found FWHM oscillations with an amplitude of 3 km/s accompanied by a blueshift of 23 km/s.

Observational context Marsch et al. (2009)

Examples of other similar observations: Xia et al. (2003, 2004), Mc. Intosh (2009, 2011) … Questions Ø Is there coupling between Alfven waves and flows? How do they interact?

A ‘simple’ model B 0 x=L

Stability analysis Ø Apply t -> ω Laplace transform Ø Connect the solutions in the + and – regions at x=L Ø Invert and determine the response of the system to an arbitrary perturbation Ø Response depends on the location of singularities in the complex ω plane Ø Location of singularities depends on the sign of

Case 1: incompressible flow Transverse perturbation

Case 2: compressible flow Transverse perturbation

Case 3: compressible flow Transverse perturbation

Conclusions from the simple model Ø An instability exists when the flow is compressible enough Ø Over-reflection at the interface Ø No shear required Ø Sub-sonic and sub-Alfvenic flow Taroyan, PRL 2008

- + Corona s=0 s=L Taroyan, Ap. J 2009

Taroyan, A&A 2015

Steady State

Linear twists

Stability analysis Ø Divide the tube into two parts: variable flow for 0<s<L and constant flow for s>L Ø Find numerical solutions in 0<s<L and analytical solutions in s>L Ø Connect the solutions at x=L Ø Solve the resulting numerical dispersion relation and find the complex frequencies, i. e. , determine stability of the system to an arbitrary twist

Alfven speed Subsonic flow Supersonic flow Magnetic field

Alfven speed Magnetic field Flow

Peter (2001)

Conclusions Ø Steady state derived Ø Stratified flux tubes with smooth flow profiles are unstable with respect to linear torsional perturbations Ø Observational signature: - below over-reflection height; - above Ø Example: Photospheric sound speed 8 km/s, Alfven speed 10 km/s, flow speed 4 km/s, distance of reflection height 250 km: amplification factor of 100 in about 10 min

• Nonlinear Evolution of the twists?

Rankine-Hugoniot Conditions http: //wonka. physics. ncsu. edu/pub/VH-1/bproblems. php Twisting a magnetic shock tube

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Chae et al. (1998), De Pontieu et al. (2014)

Conclusions Ø Steady state derived Ø Stratified flux tubes with smooth flow profiles are unstable with respect to linear torsional perturbations Ø Observational signature: - below over-reflection height; - above Ø Example: Photospheric sound speed 8 km/s, Alfven speed 10 km/s, flow speed 4 km/s, distance of reflection height 250 km: amplification factor of 100 in about 10 min Ø Non-linear evolution of twist studied in magnetic shock tube Ø Kinetic energy of the flow extracted by the twist part of which converted back into kinetic energy in the upper regions. Ø Enough flux generated to heat corona and possibly chromosphere.