MHD waves in stellar atmosphere:II. Magnetic waves

Contents

1 Introduction 2

2 MHD waves in homogeneous medium 4

3 Waves in a magnetic tube 6

4 Mode mixing and conversion of MHD waves in a stratified atmosphere 8

5 Observations of MHD waves in the solar atmosphere 11

References

• Carlsson et al. (2019), ARAA, 57, 189

• Gabriel (1976), Phil. Trans. Roy. Soc. London A, 281, 339

• Reale (2014), LRSP, 11, 4

• Edwin & Roberts (1983), Solar Phys., 88, 179

• Bogdan et al. (2003), ApJ, 599, 626

• Kudoh & Shibata (1999), ApJ, 514, 493

• Iijima & Yokoyama (2017), ApJ, 848, 38

• DeForest & Gurman (1998), ApJ, 501, L217

• De Pontieu et al. (2007), Science, 318, 1574

• Jess et al. (2009), Science, 323, 1582

• Fujimura & Tsuneta (2009), ApJ, 702, 1443

• Nakariakov et al. (1999), Science, 285, 862

• Nakariakov & Ofman (2001), A&A, 372, L53

1

1 Introduction

1.1 Structure of solar magnetic field

Figure 1 Solar photosphere and chromosphere observed by the CRISP/CHROMIS

(from review by Carlsson et al. 2019)

β =p

pm. (1)

1.2 Basic equations of MHD (re-display)

∂ρ

∂t+∇ · (ρV ) = 0 (2)

2

Figure 2 Cartoon of the magnetic field structure in solar atmosphere. (Gabriel 1976)

Figure 3 Solar corona imaged by SDO/AIA (from review by Reale 2014)

ρ∂V

∂t+ ρ(V · ∇)V = −∇p+

1

4π(∇×B)×B + ρg (3)[

∂

∂t+ V · ∇

](p

ργ

)= 0 (4)

∂B

∂t= ∇× (V ×B) (5)

∇ ·B = 0. (6)

p = RρT/µ (7)

3

2 MHD waves in homogeneous medium

2.1 Overview

2.2 Dispersion relation of MHD waves

ρ = ρ0, p = p0, V = 0, B = B0, (8)

ρ1 exp [i(k · x− ωt)] (9)

−iωρ1 + iρ0k · V 1 = 0 (10)

−iωρ0V 1 = −ikp1 +1

4π(ik ×B1)×B0 (11)

−iω(p1 − γp0ρ0

ρ1) = 0 (12)

−iωB1 = ik × (V 1 ×B0) (13)

k ·B1 = 0. (14)

ω(ω2 − k2C2A cos2 θ)[ω4 − ω2k2(C2

A + C2S) + k4C2

AC2S cos

2 θ] = 0, (15)

CS =

√γp0ρ0

(16)

CA =

√B2

0

4πρ0(17)

2.3 Characters of magnetoacoustic waves

ω2

k2=

1

2

[(C2

A + C2S)±

√(C2

A + C2S)

2 − 4C2AC

2S cos

2 θ

]. (18)

CT =

√C2

AC2S

C2A + C2

S

(19)

p1pm1

{> 0 in fast mode

< 0 in slow mode(20)

4

Figure 4 (a) (c) Phase velocity ω/k as a function of the propagation direction θ

against the magnetic field (the Friedrichs diagram). The velocity is normalized by

the sound speed. Short bars indicate the oscillating direction of velocity pertur-

bations. This direction is perpendicular to the paper for the Alfven wave. (b)(d)

Group velocity. CA/CS =√0.5 in (a)(b) and CA/CS = 2 in (c)(d).

5

2.4 Characters of Alfven wave

ω2

k2= C2

A cos2 θ (21)

B1

B0= −

(k ·B0

|k ·B0|

)V 1

CA(22)

3 Waves in a magnetic tube

3.1 Overview

3.2 Dispersion relation of waves in a magnetic tube

ρ = ρ0(ϖ), p = p0(ϖ), V = 0, B = B0(ϖ)ez. (23)

∂

∂ϖ

(p0 +

B20

8π

)= 0. (24)

ρ1 = 0, ∇ · V 1 = 0, Vz1 = 0, p1 = 0, Vϖ1 = 0, Bϖ1 = 0 (25)(∂2

∂t2− C2

A

∂2

∂z2

)Vφ1 = 0. (26)

ρ0 = ρi, p0 = pi, B0 = Bi, ϖ < a, (27)

ρ0 = ρe, p0 = pe, B0 = Be, ϖ > a (28)

ρ1 = ρ1 exp [i(nφ+ kzz − ωt)], (29)

ni

ρi(CA2i k

2z − ω2)

J ′n(nia)

Jn(nia)=

me

ρe(CA2ek

2z − ω2)

K ′n(mea)

Kn(mea)(30)

ni =√

−m20(ω, kz; ρi, pi, Bi) (31)

me =√m2

0(ω, kz; ρe, pe, Be) (32)

m20(ω, kz; ρ0, p0, B0) =

(C2Ak

2z − ω2)(C2

Sk2z − ω2)

(C2A + C2

S)(C2Tk

2z − ω2)

. (33)

CK =

√ρiCA

2i + ρeCA

2e

ρi + ρe(34)

6

Figure 5 Phase velocity of waves in a magnetic tube. (CAi = 2CSi, CSe = 0.5CSi,

and CAe = 5CSi) The meshed area does not have a localized wave modes. Solid

lines are sausage modes (n = 0) and dashed lines are kink mode (n = 1). (Edwin

& Roberts 1983)

Figure 6 Left: sausage mode (n = 0). Right: kink mode (n = 1) (Web site of the

solar wave theory group in the University of Sheffield)

7

4 Mode mixing and conversion of MHD waves in a stratified

atmosphere

4.1 Mode mixing of fast and slow magnetoacoustic waves

Figure 7 Density perturbations in a result of a two-dimensional MHD simulation

of waves in the photosphere and chromosphere. In a stratified atmosphere in a

uniform temperature, with a magnetic flux tube. Solid lines are the field lines.

The longitudinal perturbations are given at the bottom (3.5 < x[Mm] < 4.0,

z = 0). Thick white line corresponds to CA = CS. (Bogdan et al. 2003)

8

Figure 8 Result of a one-dimensional MHD simulation of waves and spicules in

the photosphere and chromosphere. (Kudoh & Shibata 1999)

Figure 9 Result of a three-dimensional radiative MHD simulation of spicules in

the photosphere and chromosphere. (Iijima & Yokoyama 2017)

9

Figure 10 Observation of propagating perturbation in the coronal polar plumes

(DeForest & Gurman 1998) by SOHO/EIT. They are interpreted as the slow-mode

magnetoacoustic waves.

Figure 11 Observation of swaying motions in spicules (De Pontieu et al. 2007) by

Hinode/SOT. They are interpreted as the kink-mode magnetoacoustic waves.

10

Figure 12 Observation of loop oscillation associated a flare in an active region

(Nakariakov et al. 1999) by TRACE. It is interpreted to be a fast-mode kink

acoustic wave.

4.2 Mode conversion from Alfven to magnetoacoustic waves

5 Observations of MHD waves in the solar atmosphere

5.1 Observations of waves in magnetic tubes and loops

5.2 Coronal Magnetic field measurement by seismology

ω

k=

2L

P≈ 1020 km/s. (35)

11