Amplification of twists in magnetic flux tubes

Youra Taroyan

Department of Physics, Aberystwyth University, email: yot@aber.ac.uk

users.aber.ac.uk/djp12

Brief chronology of Alfvén waveshttp://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 atmospherePhase 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) (McLaughlin et al 2011)

Dissipation of Alfvén waves in the atmosphereResonant 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 atmosphereNonlinear mode conversion

1.5D 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 forward modellingDissipation of Alfvén waves in the atmosphere

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

Alfvén waves in flux tubes simultaneous blue and red shifts nonthermal broadening of a spectral line profile

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), McIntosh (2009, 2011) …

Is there coupling between Alfven waves and flows? How do they interact?

Questions

x=L

B0

00 ,, Acu

00 ,, Acu

x=0

0)(20

220

00

2

2

2

bu

cu

xu

xbu

xtt

b A

A ‘simple’ model

Stability analysis

Apply t -> ω Laplace transform

Connect the solutions in the + and – regions at x=L

Invert and determine the response of the system to an arbitrary perturbation

Response depends on the location of singularities in the complex ω plane

Location of singularities depends on the sign of Acuu 00

Case 1: incompressible flow 000 Acuu

0u

0u

Tran

sverse p

erturb

ation

Case 2: compressible flow 000 Acuu

0u

0u

Tran

sverse p

erturb

ation

Case 3: compressible flow 000 Acuu

0u

0u

Tran

sverse p

erturb

ation

Conclusions from the simple model

An instability exists when the flow is compressible enough

Over-reflection at the interface

No shear required

Sub-sonic and sub-Alfvenic flow

Taroyan, PRL 2008

s=0 s=L

0u

cu00u

0B- +Corona

l l

Taroyan, ApJ 2009

Taroyan, A&A 2015

Steady State

Linear twists

Stability analysis

Divide the tube into two parts: variable flow for 0<s<L and constant flow for s>L

Find numerical solutions in 0<s<L and analytical solutions in s>L

Connect the solutions at x=L

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

Flow

Magnetic field

Peter (2001)

Conclusions

Steady state derived

Stratified flux tubes with smooth flow profiles are unstable with respect to linear torsional perturbations

Observational signature: - below over-reflection height; - above

Example: Photospheric sound speed 8 km/s, Alfven speed 10km/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

𝜕𝜕𝑡∫𝑥1

𝑥2

𝑒𝑑𝑥=𝐹 (𝑥1 )−𝐹 (𝑥2 )

aa

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

Conclusions

Steady state derived

Stratified flux tubes with smooth flow profiles are unstable with respect to linear torsional perturbations

Observational signature: - below over-reflection height; - above

Example: Photospheric sound speed 8 km/s, Alfven speed 10km/s, flow speed 4 km/s, distance of reflection height 250 km: amplification factor of 100 in about 10 min

Non-linear evolution of twist studied in magnetic shock tube

Kinetic energy of the flow extracted by the twist part of which converted back into kinetic energy in the upper regions.

Enough flux generated to heat corona and possibly chromosphere.