Solar magneto-seismology with asymmetric waveguides Matthew Allcock - - PowerPoint PPT Presentation

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Solar magneto-seismology with asymmetric waveguides Matthew Allcock - - PowerPoint PPT Presentation

May 15, 2023 639 likes •1.29k views

Solar magneto-seismology with asymmetric waveguides Matthew Allcock and Robertus Erd elyi Outline Introduction 1 Waves on the Sun Solar magneto-seismology (SMS) A brief history SMS with asymmetric wave-guides 2 Motivation Derivation

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SLIDE 1

Solar magneto-seismology with asymmetric waveguides

Matthew Allcock

and Robertus Erd´ elyi

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SLIDE 2

Outline

1

Introduction Waves on the Sun Solar magneto-seismology (SMS) A brief history

2

SMS with asymmetric wave-guides Motivation Derivation Mode identification Amplitude ratio Minimum perturbation shift

3

Looking ahead

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SLIDE 3

Outline

1

Introduction Waves on the Sun Solar magneto-seismology (SMS) A brief history

2

SMS with asymmetric wave-guides Motivation Derivation Mode identification Amplitude ratio Minimum perturbation shift

3

Looking ahead

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SLIDE 4

Waves on the Sun

Global pressure waves (p-modes): Standing modes, Spherical harmonics with global Sun as cavity, Global and local helioseismology for inference of sub-surface flows, density, temperature. MHD waves: Propagating or standing modes, Guided by local plasma inhomogeneity, Local magneto-seismology for inference of background magnetic field strength, heat transport coefficients, density.

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Solar magneto-seismology

An overview

Observations

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Solar magneto-seismology

An overview

Observations Equilibrium parameters Wave parameters

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SLIDE 7

Solar magneto-seismology

An overview

Observations Equilibrium parameters Wave parameters Temporal parameters Spatial parameters

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SLIDE 8

Solar magneto-seismology

An overview

Observations Physical understanding Equilibrium parameters Wave parameters Temporal parameters Spatial parameters

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SLIDE 9

Solar magneto-seismology

An overview

Observations Physical understanding Equilibrium parameters Wave parameters Temporal parameters Spatial parameters Equilibrium models

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SLIDE 10

Solar magneto-seismology

An overview

Observations Physical understanding Equilibrium parameters Wave parameters Temporal parameters Spatial parameters Equilibrium models Eigenmodes

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SLIDE 11

Solar magneto-seismology

An overview

Observations Physical understanding Equilibrium parameters Wave parameters Temporal parameters Spatial parameters Equilibrium models Eigenmodes Temporal magneto-seismology Spatial magneto-seismology

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SLIDE 12

Solar magneto-seismology

An overview

Observations Physical understanding Equilibrium parameters Wave parameters Temporal parameters Spatial parameters Equilibrium models Eigenmodes Temporal magneto-seismology Spatial magneto-seismology

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SLIDE 13

Solar magneto-seismology

An overview

Observations Physical understanding Equilibrium parameters Wave parameters Temporal parameters Spatial parameters Equilibrium models Eigenmodes Temporal magneto-seismology Spatial magneto-seismology

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SLIDE 14

Solar magneto-seismology

An overview

Observations Physical understanding Equilibrium parameters Wave parameters Temporal parameters Spatial parameters Equilibrium models Eigenmodes Temporal magneto-seismology Spatial magneto-seismology

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SLIDE 15

Solar magneto-seismology

A brief history - temporal and spatial seismology 1970 1980 1990 2000

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Solar magneto-seismology

A brief history - temporal and spatial seismology 1970 1980 1990 2000 Uchida, 1970 - Moreton wavefronts;

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Solar magneto-seismology

A brief history - temporal and spatial seismology 1970 1980 1990 2000 Uchida, 1970 - Moreton wavefronts; Rosenburg 1970 - MHD waves cause pulsations in synchrotron radiation with measuable period

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SLIDE 18

Solar magneto-seismology

A brief history - temporal and spatial seismology 1970 1980 1990 2000 Uchida, 1970 - Moreton wavefronts; Rosenburg 1970 - MHD waves cause pulsations in synchrotron radiation with measuable period Zeitsev and Stepanov, 1975 - pulsations of type IV solar radio emission due to plasma cylinder oscillations

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SLIDE 19

Solar magneto-seismology

A brief history - temporal and spatial seismology 1970 1980 1990 2000 Uchida, 1970 - Moreton wavefronts; Rosenburg 1970 - MHD waves cause pulsations in synchrotron radiation with measuable period Zeitsev and Stepanov, 1975 - pulsations of type IV solar radio emission due to plasma cylinder oscillations Roberts et al. 1984 - coronal context

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SLIDE 20

Solar magneto-seismology

A brief history - temporal and spatial seismology 1970 1980 1990 2000 Uchida, 1970 - Moreton wavefronts; Rosenburg 1970 - MHD waves cause pulsations in synchrotron radiation with measuable period Zeitsev and Stepanov, 1975 - pulsations of type IV solar radio emission due to plasma cylinder oscillations Roberts et al. 1984 - coronal context Tandberg Hanssen et al. 1995 - prominence context

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Solar magneto-seismology

A brief history - temporal and spatial seismology 1970 1980 1990 2000 Uchida, 1970 - Moreton wavefronts; Rosenburg 1970 - MHD waves cause pulsations in synchrotron radiation with measuable period Zeitsev and Stepanov, 1975 - pulsations of type IV solar radio emission due to plasma cylinder oscillations Roberts et al. 1984 - coronal context Tandberg Hanssen et al. 1995 - prominence context Coronal loop oscillations imaged! Aschwanden et al. 1999

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Solar magneto-seismology

A brief history - temporal and spatial seismology 2000 2005 2010 2017

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Solar magneto-seismology

A brief history - temporal and spatial seismology 2000 2005 2010 2017 Nakariakov and Ofman 2001 - magnetic field strength estimate using period of kink standing modes

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Solar magneto-seismology

A brief history - temporal and spatial seismology 2000 2005 2010 2017 Nakariakov and Ofman 2001 - magnetic field strength estimate using period of kink standing modes Goossens et al. 2002 - damping time scales used to estimate cross-field density variation length scales

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SLIDE 25

Solar magneto-seismology

A brief history - temporal and spatial seismology 2000 2005 2010 2017 Nakariakov and Ofman 2001 - magnetic field strength estimate using period of kink standing modes Goossens et al. 2002 - damping time scales used to estimate cross-field density variation length scales Andries et al. 2005 - density stratification deduced from period ratio

  • f standing kink mode harmonics
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SLIDE 26

Solar magneto-seismology

A brief history - temporal and spatial seismology 2000 2005 2010 2017 Nakariakov and Ofman 2001 - magnetic field strength estimate using period of kink standing modes Goossens et al. 2002 - damping time scales used to estimate cross-field density variation length scales Andries et al. 2005 - density stratification deduced from period ratio

  • f standing kink mode harmonics

Verth et al. 2007 - anti-node shift due to density stratification

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SLIDE 27

Solar magneto-seismology

A brief history - temporal and spatial seismology 2000 2005 2010 2017 Nakariakov and Ofman 2001 - magnetic field strength estimate using period of kink standing modes Goossens et al. 2002 - damping time scales used to estimate cross-field density variation length scales Andries et al. 2005 - density stratification deduced from period ratio

  • f standing kink mode harmonics

Verth et al. 2007 - anti-node shift due to density stratification Erd´ elyi and Taroyan 2008 - slow sausage and kink standing modes

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SLIDE 28

Solar magneto-seismology

A brief history - temporal and spatial seismology 2000 2005 2010 2017 Nakariakov and Ofman 2001 - magnetic field strength estimate using period of kink standing modes Goossens et al. 2002 - damping time scales used to estimate cross-field density variation length scales Andries et al. 2005 - density stratification deduced from period ratio

  • f standing kink mode harmonics

Verth et al. 2007 - anti-node shift due to density stratification Erd´ elyi and Taroyan 2008 - slow sausage and kink standing modes Arregui and Asensio Ramos 2011 - Bayesian inversion

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SLIDE 29

Solar magneto-seismology

A brief history - temporal and spatial seismology 2000 2005 2010 2017 Nakariakov and Ofman 2001 - magnetic field strength estimate using period of kink standing modes Goossens et al. 2002 - damping time scales used to estimate cross-field density variation length scales Andries et al. 2005 - density stratification deduced from period ratio

  • f standing kink mode harmonics

Verth et al. 2007 - anti-node shift due to density stratification Erd´ elyi and Taroyan 2008 - slow sausage and kink standing modes Arregui and Asensio Ramos 2011 - Bayesian inversion Pascoe et al. 2013 - using gaussian and exponential damping

  • f kink modes to estimate loop density
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Solar magneto-seismology

A brief history - temporal and spatial seismology 2000 2005 2010 2017 Nakariakov and Ofman 2001 - magnetic field strength estimate using period of kink standing modes Goossens et al. 2002 - damping time scales used to estimate cross-field density variation length scales Andries et al. 2005 - density stratification deduced from period ratio

  • f standing kink mode harmonics

Verth et al. 2007 - anti-node shift due to density stratification Erd´ elyi and Taroyan 2008 - slow sausage and kink standing modes Arregui and Asensio Ramos 2011 - Bayesian inversion Pascoe et al. 2013 - using gaussian and exponential damping

  • f kink modes to estimate loop density

Magyar et al. - global dynamic coronal seismology

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SLIDE 31

Outline

1

Introduction Waves on the Sun Solar magneto-seismology (SMS) A brief history

2

SMS with asymmetric wave-guides Motivation Derivation Mode identification Amplitude ratio Minimum perturbation shift

3

Looking ahead

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SLIDE 32

Asymmetric magnetic slab

Motivation

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Asymmetric magnetic slab

Equilibrium conditions

−x0 x0 x z y

ρ1, p1, T1 ρ0, p0, T0 ρ2, p2, T2

Uniform magnetic field in the slab. Field-free plasma outside. Different density and pressure on each side.

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Asymmetric magnetic slab

Governing equations

Ideal MHD equations: Conservation of: ρD✈ Dt = −∇p − 1 µ❇ × (∇ × ❇), momentum ∂ρ ∂t + ∇ · (ρ✈) = 0, mass D Dt p ργ

  • = 0,

energy ∂❇ ∂t = ∇ × (✈ × ❇), magnetic flux

✈ = plasma velocity, ❇ = magnetic field strength, ρ = density, p = pressure, µ = magnetic permeability, γ = adiabatic index.

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SLIDE 35

Asymmetric magnetic slab

Fourier decomposition

Look for plane wave solutions of the form: vx(x, t) = ˆ vx(x)ei(kz−ωt), vy(x, t) = 0, vz(x, t) = ˆ vz(x)ei(kz−ωt), to arrive at the following ODEs:

x −x0 x0 d2ˆ vx dx2 − m2 0ˆ

vx = 0

d2ˆ vx dx2 − m12ˆ

vx = 0

d2ˆ vx dx2 − m22ˆ

vx = 0

Boundary conditions: pressure and velocity continuous across boundaries. m02 = (k2vA2 − ω2)(k2c2

0 − ω2)

(c2

0 + vA2)(k2cT 2 − ω2) ,

m1,22 = k2 − ω2 c1,22 , cT 2 = c2

0vA2

c2

0 + vA2 ,

vA = B0 √µρ0 ,

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SLIDE 36

Asymmetric magnetic slab

Eigenmodes

Dispersion relation:

ω4m02 k2vA2 − ω2 + ρ0 ρ1 m1 ρ0 ρ2 m2(k2vA2 − ω2) − 1 2m0ω2 ρ0 ρ1 m1 + ρ0 ρ2 m2

  • (tanh m0x0 + coth m0x0) = 0,

See Allcock and Erd´ elyi, 2017. m02 = (k2vA2 − ω2)(k2c2

0 − ω2)

(c2

0 + vA2)(k2cT 2 − ω2) ,

m1,22 = k2 − ω2 c1,22 , cT 2 = c2

0vA2

c2

0 + vA2 ,

vA = B0 √µρ0 ,

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SLIDE 37

Eigenmodes

Symmetric kink surface mode

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SLIDE 38

Eigenmodes

Quasi-kink surface mode

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SLIDE 39

Eigenmodes

Symmetric sausage surface mode

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Eigenmodes

Quasi-sausage surface mode

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Eigenmodes

Body modes

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Asymmetric magnetic slab

Mode identification

Superposition of symmetric kink and sausage modes...

Morton et al. 2012

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Asymmetric magnetic slab

Mode identification

Superposition of symmetric kink and sausage modes...

Morton et al. 2012

  • r asymmetric kink mode?
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Amplitude ratio

ˆ ξx(−x0) ˆ ξx(x0)

−x0 x0 x

Amplitude ratio

RA := ˆ ξx(x0) ˆ ξx(−x0) (

Top = quasi-kink Bottom = quasi-sausage)

= +

−

ρ1m2 ρ2m1 (k2vA2 − ω2)m1

ρ0 ρ1 − ω2m0

tanh

coth

  • (m0x0)

(k2vA2 − ω2)m2

ρ0 ρ2 − ω2m0

tanh

coth

  • (m0x0)
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Amplitude ratio

Parameter inversion

Parameter inversion

Observe: ω, k, x0, Ti, and RA. Solve to find: vA and hence B0.

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Amplitude ratio

Parameter inversion

Parameter inversion

Observe: ω, k, x0, Ti, and RA. Solve to find: vA and hence B0.

Analytical inversion

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Amplitude ratio

Parameter inversion

Parameter inversion

Observe: ω, k, x0, Ti, and RA. Solve to find: vA and hence B0.

Mode Approximation of k2vA2/ω2 using amplitude ratio, RA Thin slab Incompressible Low-beta Sausage 1 +

1 x0

  • RA

ρ2 ρ0m2 + ρ1 ρ0m1 RA+1

  • 1 +
  • RA

ρ2 ρ0 + ρ1 ρ0 RA+1

  • coth kx0

1 + k

  • RA

ρ2 ρ0m2 + ρ1 ρ0m1 RA+1

  • coth kx0

Kink 1 + k2x0

  • RA

ρ2 ρ0m2 − ρ1 ρ0m1 RA−1

  • 1 +
  • RA

ρ2 ρ0 − ρ1 ρ0 RA−1

  • tanh kx0

1 + k

  • RA

ρ2 ρ0m2 − ρ1 ρ0m1 RA−1

  • tanh kx0
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SLIDE 48

Amplitude ratio

Parameter inversion

Parameter inversion

Observe: ω, k, x0, Ti, and RA. Solve to find: vA and hence B0.

Numerical inversion

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Amplitude ratio

Parameter inversion

Parameter inversion

Observe: ω, k, x0, Ti, and RA. Solve to find: vA and hence B0.

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Minimum perturbation shift

−x0 x0 x −x0 x0 x −x0 x0

  • ξx

x ∆min −x0 x0

  • ξx

x ∆min

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SLIDE 51

Minimum perturbation shift

−x0 x0 x −x0 x0 x

Quasi-kink: Quasi-sausage:

∆min = 1 m0 tanh−1(D) ∆min = 1 m0 tanh−1 1 D

  • where

D = (k2vA2 − ω2)m2

ρ0 ρ2 tanh(m0x0) − ω2m0

(k2vA2 − ω2)m2

ρ0 ρ2 − ω2m0 tanh(m0x0)

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SLIDE 52

Minimum perturbation shift

Parameter inversion

Parameter inversion

Observe: ω, k, x0, Ti, and ∆min. Solve to find: vA and hence B0.

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SLIDE 53

Minimum perturbation shift

Parameter inversion

Parameter inversion

Observe: ω, k, x0, Ti, and ∆min. Solve to find: vA and hence B0.

Analytical inversion

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SLIDE 54

Minimum perturbation shift

Parameter inversion

Parameter inversion

Observe: ω, k, x0, Ti, and ∆min. Solve to find: vA and hence B0.

Mode Approximation of k2vA2/ω2 using minimum perturbation shift, ∆min Thin slab Incompressible Low-beta Quasi-sausage

ρ1 ρ0m1 (x0 + ∆min) + 1 1+(ω/kc0)2 + k2x0∆min

1 + ρ1

ρ0 tanh k(x0 + ∆min)

1 +

kρ1 m1ρ0 tanh k(x0 + ∆min)

Quasi-kink

−b±

  • b2−4ac

2a

, defined in text 1 + ρ1

ρ0 coth k(x0 + ∆min)

1 +

kρ1 m1ρ0 coth k(x0 + ∆min)

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SLIDE 55

Minimum perturbation shift

Parameter inversion

Parameter inversion

Observe: ω, k, x0, Ti, and ∆min. Solve to find: vA and hence B0.

Numerical inversion

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SLIDE 56

Minimum perturbation shift

Parameter inversion

Parameter inversion

Observe: ω, k, x0, Ti, and ∆min. Solve to find: vA and hence B0.

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SLIDE 57

Outline

1

Introduction Waves on the Sun Solar magneto-seismology (SMS) A brief history

2

SMS with asymmetric wave-guides Motivation Derivation Mode identification Amplitude ratio Minimum perturbation shift

3

Looking ahead

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SLIDE 58

Further work

Add magnetic field outside the slab to model coronal structures. See Zs´ amberger, Allcock, and Erd´ elyi, recommended for publication in ApJ.

−x0 x0 x z y

ρ1, p1, T1 ρ0, p0, T0 ρ2, p2, T2

Key result for mode identification:

There can exist quasi-symmetric modes - where the oscillations

  • f each interface have equal amplitude - even with asymmetric

background. Symmetric mode

  • =

⇒ symmetric equilibrium

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SLIDE 59

Future work

Initial value problem

−x0 x0 x z y

ρ1, p1, T1 ρ0, p0, T0 ρ2, p2, T2 Initial impulse Initial impulse

?

Determine approximate time for set-up of asymmetric modes. Compare this to typical lifetime of solar asymmetric structures. Include transition layer to investigate resonant absorption.

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Future work

Application

Apply to observations of MHD waves in, for example: Elongated magnetic bright points,

Adaptation of Liu et al., 2017, by Zs´ amberger et al.

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Future work

Application

Apply to observations of MHD waves in, for example: Elongated magnetic bright points, Prominences,

NASA

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Future work

Application

Apply to observations of MHD waves in, for example: Elongated magnetic bright points, Prominences, Sunspot light walls.

Max Planck Institute for Solar System Research

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Thank you

matthew allcock