Magneto-acoustic waves in asymmetric solar waveguides Progress in - PowerPoint PPT Presentation

Magneto-acoustic waves in asymmetric solar waveguides Progress in spatial magneto-seismology Matthew Allcock and Robertus Erd´ elyi

The layers of an onion

Magnetohydrodynamic waves Ubiquitous in the solar atmosphere Credit: NASA, SDO

Magnetohydrodynamic waves Diagnosing information about solar plasma Observations

Magnetohydrodynamic waves Diagnosing information about solar plasma Wave parameters Observations Equilibrium parameters

Magnetohydrodynamic waves Diagnosing information about solar plasma Wave Temporal parameters parameters Spatial Observations parameters Equilibrium parameters

Magnetohydrodynamic waves Diagnosing information about solar plasma Wave Temporal parameters parameters Spatial Observations parameters Equilibrium parameters Physical understanding

Magnetohydrodynamic waves Diagnosing information about solar plasma Wave Temporal parameters parameters Spatial Observations parameters Equilibrium parameters Physical Equilibrium understanding models

Magnetohydrodynamic waves Diagnosing information about solar plasma Wave Temporal parameters parameters Spatial Observations parameters Equilibrium parameters Physical Equilibrium Eigenmodes understanding models

Magnetohydrodynamic waves Diagnosing information about solar plasma Wave Temporal parameters parameters Spatial Temporal Observations parameters magneto-seismology Equilibrium Spatial parameters magneto-seismology Physical Equilibrium Eigenmodes understanding models

Magnetohydrodynamic waves Diagnosing information about solar plasma Wave Temporal parameters parameters Spatial Temporal Observations parameters magneto-seismology Equilibrium Spatial parameters magneto-seismology Physical Equilibrium Eigenmodes understanding models

Magnetohydrodynamic waves Diagnosing information about solar plasma Wave Temporal parameters parameters Spatial Temporal Observations parameters magneto-seismology Equilibrium Spatial parameters magneto-seismology Physical Equilibrium Eigenmodes understanding models

Magnetohydrodynamic waves Diagnosing information about solar plasma Wave Temporal parameters parameters Spatial Temporal Observations parameters magneto-seismology Equilibrium Spatial parameters magneto-seismology Physical Equilibrium Eigenmodes understanding models

Slab structures on the Sun Max Planck Institute for Solar System Research BBSO/NJIT

Equilibrium conditions ρ 1 , p 1 , T 1 ρ 2 , p 2 , T 2 ρ 0 , p 0 , T 0 z y x 0 x − x 0 Uniform magnetic field in the slab. Field-free plasma outside. Different density and pressure on each side.

Governing equations Ideal MHD equations: Conservation of: ρ D ✈ D t = −∇ p − 1 µ ❇ × ( ∇ × ❇ ) , momentum ∂ρ ∂ t + ∇ · ( ρ ✈ ) = 0 , mass � p � D = 0 , energy D t ρ γ ∂ ❇ ∂ t = ∇ × ( ✈ × ❇ ) , magnetic flux ✈ = plasma velocity, ❇ = magnetic field strength, ρ = density, p = pressure, µ = magnetic permeability, γ = adiabatic index.

Governing equations Ideal MHD equations: Conservation of: ρ D ✈ D t = −∇ p − 1 µ ❇ × ( ∇ × ❇ ) , momentum ∂ρ ∂ t + ∇ · ( ρ ✈ ) = 0 , mass � p � D = 0 , energy D t ρ γ ∂ ❇ ∂ t = ∇ × ( ✈ × ❇ ) , magnetic flux ✈ = plasma velocity, ❇ = magnetic field strength, ρ = density, p = pressure, µ = magnetic permeability, γ = adiabatic index.

Asymmetric slab modes Dispersion relation: ω 4 m 02 k 2 v A 2 − ω 2 + ρ 0 ρ 0 m 2 ( k 2 v A 2 − ω 2 ) m 1 ρ 1 ρ 2 � ρ 0 � − 1 m 1 + ρ 0 2 m 0 ω 2 m 2 (tanh m 0 x 0 + coth m 0 x 0 ) = 0 , ρ 1 ρ 2 m 02 = ( k 2 v A 2 − ω 2 )( k 2 c 2 0 − ω 2 ) ω 2 m 1 , 22 = k 2 − 0 + v A 2 )( k 2 c T 2 − ω 2 ) , c 1 , 22 , ( c 2 c 2 0 v A 2 B 0 c T 2 = v A = 0 + v A 2 , , c 2 √ µρ 0 See Allcock and Erd´ elyi, 2017.

Asymmetric slab modes Dispersion relation: ω 4 m 02 k 2 v A 2 − ω 2 + ρ 0 ρ 0 m 2 ( k 2 v A 2 − ω 2 ) m 1 ρ 1 ρ 2 � ρ 0 � − 1 m 1 + ρ 0 2 m 0 ω 2 m 2 (tanh m 0 x 0 + coth m 0 x 0 ) = 0 , ρ 1 ρ 2 m 02 = ( k 2 v A 2 − ω 2 )( k 2 c 2 0 − ω 2 ) ω 2 m 1 , 22 = k 2 − 0 + v A 2 )( k 2 c T 2 − ω 2 ) , c 1 , 22 , ( c 2 c 2 0 v A 2 B 0 c T 2 = v A = 0 + v A 2 , , c 2 √ µρ 0 See Allcock and Erd´ elyi, 2017.

Slab eigenmodes Symmetric kink surface mode

Slab eigenmodes Quasi-kink surface mode

Slab eigenmodes Symmetric sausage surface mode

Slab eigenmodes Quasi-sausage surface mode

Slab eigenmodes Body modes

Amplitude ratio ˆ ˆ ξ x ( − x 0 ) ξ x ( x 0 ) x x 0 − x 0 Amplitude ratio ˆ ξ x ( x 0 ) Top = quasi-kink R A := ( Bottom = quasi-sausage ) ˆ ξ x ( − x 0 ) � tanh � ( k 2 v A 2 − ω 2 ) m 1 ρ 1 − ω 2 m 0 ρ 0 ( m 0 x 0 ) � + � ρ 1 m 2 coth � tanh � = ( k 2 v A 2 − ω 2 ) m 2 ρ 2 − ω 2 m 0 − ρ 2 m 1 ρ 0 ( m 0 x 0 ) coth

Minimum perturbation shift x x x 0 x 0 − x 0 − x 0 � � ξ x ξ x ∆ min ∆ min x 0 x x 0 x − x 0 − x 0

Minimum perturbation shift x x x 0 x 0 − x 0 − x 0 Quasi-kink: Quasi-sausage: � 1 � ∆ min = 1 ∆ min = 1 tanh − 1 ( D ) tanh − 1 m 0 m 0 D ( k 2 v A 2 − ω 2 ) m 2 ρ 2 tanh( m 0 x 0 ) − ω 2 m 0 ρ 0 where D = ( k 2 v A 2 − ω 2 ) m 2 ρ 2 − ω 2 m 0 tanh( m 0 x 0 ) ρ 0

Solar magneto-seismology Parameter inversion Observe : ω , k , x 0 , T i , and R A or ∆ min . Solve to find: v A and hence B 0 .

Solar magneto-seismology Parameter inversion Observe : ω , k , x 0 , T i , and R A or ∆ min . Solve to find: v A and hence B 0 .

Future work Diagnose magnetic field parameters using observations of MHD waves in magnetic structures in the solar atmosphere, for example: Elongated magnetic bright points , Adaptation of Liu et al., 2017, by N. Zs´ amberger

Future work Diagnose magnetic field parameters using observations of MHD waves in magnetic structures in the solar atmosphere, for example: Elongated magnetic bright points , Prominences , NASA

Future work Diagnose magnetic field parameters using observations of MHD waves in magnetic structures in the solar atmosphere, for example: Elongated magnetic bright points , Prominences , Sunspot light walls . Max Planck Institute for Solar System Research

matthew allcock