Cutoff period for slow magnetoacoustic waves in coronal

Cut-off period for slow magnetoacoustic waves in coronal plasma structures A. Afanasyev in collaboration with V. Nakariakov Institute of Solar-Terrestrial Physics Siberian Branch of Russian Academy of Sciences Irkutsk, Russia Centre for Fusion, Space and Astrophysics University of Warwick, UK

from Curdt et al. (2008, A&A) from S. Anfinogentov www. sdo. gsfc. nasa. gov www. cfa. harvard. edu

Longitudinal waves observed in EUV • are observed in coronal loops and plumes (De. Forest & Gurman, Ap. J 1998; Berghmans & Clette, Sol. Phys. 1999), De Moortel et al. , A&A 2000) • are interpreted as slow magnetoacoustic waves (Ofman, Nakariakov et al. , Ap. J 1999, A&A 2000); Nakariakov et al. , A&A 2000) • Periods of waves are from several minutes to several tens of minutes: – 3 and 5 min -- in coronal loops (De Moortel et al. , 2002) – 10 -12 min -- in coronal plumes (Ofman et al. , Ap. J 1997) – 29, 53 and 75 min -- in coronal diffuse structures (Yuan et al. , 2011 • Long-period oscillations in radio band

Study of the long-period wave propagation in the solar atmosphere • Acoustic waves (e. g. , Sutmann et al. (1998) and references therein) • Magnetic nature of waves and plasma structuring should be taken into account • Tube waves • Defouw, 1976; Rae&Roberts, 1982; Hasan& Kalkofen, 1999; Musielak&Ulmscneider, 2003 • They obtained the Klein-Gordon equation for the case of the constant Alfven speed • Thomas (1982) – horizontal field • Roberts (2006) – uniform vertical magnetic field

Aim of study We consider the propagation of long-period slow magnetoacoustic waves in coronal fieldaligned plasma structures, taking into account: • waveguide properties of such structures • magnetic fluctuations in slow-mode waves • variation of the background magnetic field with height

Thin Flux Tube Approximation • Thin tube: R << L • p(r, z) = p 0(z) + p 1(z) r + p 2 r 2 + … • Ferriz-Mas & Schüssler showed that p(r, z) = p 0(z) + p 2 r 2 + … for scalars and zcomponents of vectors Br(r, z) = Br 1(z) r +Br 3(z) r 3 + … for r-components of vectors • Pressure balance condition → external medium • Substituting these expansions into the set of MHD equations, we obtain the 1 D problem

Set of ideal MHD equations

Linear waves • Waves in an isotermal (T = const) and stratified by gravity plasma cylinder

Wave equation – Brunt-Väisälä frequency

Limiting cases 1. If VA >> c 0, c. T c 0 : plasma motions in slow-mode waves become longitudinal, like in sound waves, so the effect of the magnetic structuring vanishes e. g. , Sutmann et al. , 1998 2. B 0 = const: Roberts, RSPT-A 2006

Klein-Gordon equation

Cut-off frequency

Limiting cases 1. VA >> c 0: 2. B 0 = const (Roberts, 2006): 3. VA=const: Defouw, 1976; Rae&Roberts, 1982; Musielak&Ulmscneider, 2003

Cut-off period calculations • Exponentially divergent magnetic field: B 0 = B 00 exp(-z/L), n 0 = n 00 exp(-z/H), L – magnetic filed scale height H – density scale height • Plasma parameters and wave characteristics: Temp = 1. 4 MK, n 00 = 5*108 cm-3 , γ = 5/3 • B 0 = (0. 5, 1. 5, 5, 10) G • VA = 43, 130, 433, 866 km/s vs. c 0 = 175 km/s • L = (0. 2, 0. 5, 1, 2, 5) H

L = 0. 2 H (red), 0. 5 H (purple), 1 H (black), 2 H(green), 5 H(blue) Beta > 1, B 0 = 0. 5 Va = 43, c 0 = 175 Beta < 1, B 0 = 5 Va = 433, c 0 = 175 Beta ~ 1, B 0 = 1. 5 Va = 130, c 0 = 175 Beta << 1, B 0 = 10 Va = 866, c 0 = 175

Discussion • Longer-period compressive waves in the corona: – Sych&Nakariakov, Sol. Phys 2008 -- 15 -min oscillations in sunspots magnetospheres – Ofman et al. , Ap. J 1997 -- possible fluctuations on longer timescales (20 -50 minutes) high above the limb (1. 9 -2. 45 RS) – Miyamoto et al. , Ap. J 2014 – at 1. 5 -20. 5, quasi-periodic density disturbances were detected, with the period ranging from 100 to 2000 s – Gupta et al. , A&A 2012 – spectroscopic observations of propagating disturbances of ≈14. 5 min period in a coronal hole – Krishna Prasad et al. , Ap. J 2014 – waves of 12, 17 and 22 min period in active region loop fans • can be associated with the trailing wake oscillating at the coronal cut-off frequency Flux tube divergence: – Marsh et al. , Apj 2011 -- area divergence has the dominant effect over thermal conduction on oscillations with longer periods (12 minutes) traveling along cool loops – Deforest et al. , Sol. Phys 1997 –superradial expansion of plumes

Conclusions • We have constructed the model describing the dynamics of longitudinal waves in vertical fieldaligned coronal plasma structures • We have derived the equation describing the propagation of linear long-wavelength slowmode MHD waves in magnetic flux tubes filled with an isothermal stratified plasma • The cut-off period for longitudinal waves is found to vary with height, decreasing significantly in the low-beta plasma as well as in the plasma with the beta of the order of unity.

Conclusions • The long-period oscillations observed in the solar corona can be due to the trailing wakes of broadband perturbations of the coronal plasma • The depressions in the cut-off period profiles can affect the propagation of longitudinal waves along coronal plasma structures towards the higher corona

Conclusions • The long-period oscillations observed in the solar corona can be due to the trailing wakes of broadband perturbations of the coronal plasma • The depressions in the cut-off period profiles can affect the propagation of longitudinal waves along coronal plasma structures towards the higher corona Спасибо!

Tube waves Equations for the uniform tube give the classical wave equation for tube wave B Vgroup = c. T k B c 0 VA R << L k┴ >> kll Long-wavelength approximation

Long-period waves in stratified media • Dispersion effects come into play: cut-off for harmonic waves, and spreading and trailing wake for pulses • Nature of the dispersive behaviour of acoustic waves was studied by Lamb (1932) • Wave equation can be reduced to the Klein. Gordon equation • Cut-off frequency

Thin flux tube equations Pressure balance → = const

Linear waves • Eliminating the variables s 1, p 1, v, we have