Magnetic buoyancy

Contents

1 Introduction 2

2 Magnetic buoyancy 5

3 Magnetic buoyancy instability 6

4 Non-linear studies by numerical simulations 12

References

• Acheson, 1979, Solar Physics, 62, 23

• Fukui et al., 2006, Science, 314, 106

• Machida et al., 2009, PASJ, 61, 411

• Matsumoto, 1988, PASJ, 40, 171

• Nelson et al., 2013, ApJ, 762, 73

• Newcomb, 1961, Phys. Fluids, 4, 391

• Parker, 1955, ApJ, 121, 491

• Parker, 1979, ”Cosmical Magnetic Feild” Chap 13

• Shibata et al., 1989, ApJ, 345, 584

• Tajima & Shibata, 1997, ”Plasma Astrophysics”, Section 3.2

• Toriumi & Wang (2019), Living Reviews in Solar Physics, 16-3

• Toriumi & Yokoyama, 2012, A&A, 539, A22

1

1 Introduction

1.1 Solar emerging magnetic flux

Solar active regions are areas where concentrated magnetic flux like sunspots are

observed. Such magnetic flux emerges from the interior of the sun (Fig.1). Emerging

fluxes frequently carry electric currents with them to provide free energies in active

regions. They are the sources for the eruptive phenomena, i.e. flares and coronal mass

ejections. The original location of this emergence is still under study, which is related

with the dynamo mechansim and the convection in the interior.

Figure 1 (a) A “textbook” flux emergence. Observed in visible light by Hinode

SOT, and in UV by IRIS. (from review by Toriumi & Wang, 2019) (b) Schematic

model of flux emergence (Shibata et al. 1989)

2

1.2 Stratified Atmosphere with Horizontal Magnetic Field

Suppose a stratified gas layer under a gravity supported by the gas and magnetic pres-

sures (Fig. 2). Magnetic field is horizontal in y-direction (B = Bey). The equilibrium

state is given by solving the equation of force balance,

−∇p+1

cJ ×B + ρg = 0. (1)

Figure 2 Cartoon of a stratified atmosphere with horizontal magnetic field

The Lorentz force term can be manipulated to be

1

cJ ×B = −∇

(B2

8π

)+

1

4π(B · ∇)B (2)

and, since the field lines are straight, the second term can be ignored in this force

balance. By taking a magnetic pressure

pm =B2

8π(3)

and the total pressureptot = p+ pm (4)

then the equation (1) of force balance is modified to be

−∇ptot + ρg = 0. (5)

This is just a replacement of p with ptot in the non-magnetic case.

3

Since the gravity is in z-direction(g = −gez、g > 0), the horizontal force balance in

x、y leads to the dependence of ptot only on z and eq. (5) is easily solved by using given

relations between pm and ρ. By using the equation of state,

p =R

µTρ (6)

(where µ is the mean atomic weight, and R is the gas constant) and the definition of

plasma beta

β =p

pm(7)

(5) can be integrated as

p =β

β + 1ptot =

β

β + 1ptot0 exp

(−∫

β

β + 1

µg

RTdz

)(8)

where ptot0 is a constant.

If one assumes g, T , µ and β are all constant, then

p = p0 exp(− z

H

)(9)

where p0 is the pressure at z = 0, H is the pressure scale height given as

H =

(1 +

1

β

)Hg (10)

where

Hg =RT

µg. (11)

As β → ∞, H tends to the non-magnetic scale height Hg. The scale height becomes

larger due to the existence of the magnetic field as a supporting object. Since the

temperature etc. are uniform, the density ρ has a similar distribution with the same

scale height.

ρ = ρ0 exp(− z

H

)(12)

4

2 Magnetic buoyancy

Consider an isolated flux tube in the stratified atmosphere. The tube is assumed to

be thin, i.e. the radius is much smaller than the pressure scale height. In this case,

it is approximated that the pressure pi, density ρi and field strength B are uniform

across the tube. The surrounding atmosphere has pe and ρe. The force balance in the

horizontal direction in eq. (5) gives

pi + pmi = pe (13)

where pmi = B2/(8π).

Suppose that the temperature T inside the tube is the same as the surroundings due

to, e.g., the strong heat conduction. Then, the tube has a smaller density than the

surroundings, and suffers from the buoyancy force. The force per volume fb is given as

fb = (ρe − ρi)g = (pe − pi)µg

RT=

pmi

Hg. (14)

Thus, fb > 0, i.e. it is upward. This force is called a “magnetic buoyancy” (Parker 1955).

Due to this buoyancy, such an thin flux tube in thermal balance with surroundings is

always in mechanical non-equilibrium.

5

3 Magnetic buoyancy instability

Suppose a horizontal magnetic flux sheetB = B(z)ey in a stratified atmosphere under

a gravity g = −gez (g > 0). It is possible to construct a mechanical equilibrium state.

But this equilibrium is unstable due to the magnetic buoyancy when the field-strength

gradient is steep enough (Fig.3). This is called the “magnetic buoyancy instability”.

Figure 3 Concept of the magnetic buoyancy instability: (a) interchange mode, (b)

undular mode. (from textbook by Tajima & Shibata 1997)

6

(a) Interchange mode (kh ⊥ B)

Suppose that the magnetic pressure is stronger at the lower altitude. This can lead to

a situation where the heavier gas is located above a lighter matter. By a perturbation

with kh ⊥ B (where kh is a horizontal wavenumber), field lines are interchanged without

bending (Fig. 3a). Similar to the Rayleigh-Taylor instability, this situation is unstable

for the exchange of the lighter matter with the heavier one. This mode is called the

”interchange mode”.

The condition for the growth of this mode (Newcomb 1961) is

− d

dz(lnB) >

N2

g

C2S

C2A

− d

dz(ln ρ) , (15)

where

N2 = g

[1

γp

dp

dz− 1

ρ

dρ

dz

](16)

is the Brunt-Vaisala frequency.

Let us derive (15) following Acheson (1979). One can limit the motions in a vertical

two-dimensional plane perpendicular to the field lines. Suppose a gas blob with magnetic

flux is lifted by a small distance dz. Due to the decrease of surrounding pressure, the blob

expands changing its cross section A. This can be described as ρA = (ρ+δρ)(A+δA) and

BA = (B+δB)(A+δA), where the prefix δ indicates the change in the blob. From these

relations, the density and the magnetic field changes are correlated as δB/B = δρ/ρ.

On the other hand, the total pressure is in balance between the inside and outside of

the blob as dp + BdB/(4π) = δp + BδB/(4π) where the prefix d indicates the change

in the surroundings. Due to the adiabatic change inside the blob, δp/p = γδρ/ρ. As a

result, one obtains

dp+BdB

4π=

(γp

ρ+

B2

4πρ

)δρ. (17)

To have an instability, δρ < dρ is satisfied, when

dp

dz+

B

4π

dB

dz< (C2

S + C2A)

dρ

dz. (18)

This formula can be modified to be (15) after some algebra.

7

(b) Undular mode (kh ∥ B)

In the undular mode of the magnetic buoyancy instability, field lines are bent (Fig.

3b). The gas drops along the bent field lines. The apexes of the field lines become

lighter, suffer the enhanced buoyancy force, and are brought up more. This mode is

called the “undular mode” but is more likely called the “Parker instability” (1966).

The condition for instability (Newcomb 1961) is

− d

dz(lnB) >

N2

g

C2S

C2A

. (19)

This is less-restricted limitation than that of the interchange mode because of a lack on

the condition of the density stratification.

In the undular mode, there is a critical wavelength for the growth. The wavelength

needs to be long enough to avoid the suppression of the instability by the magnetic

tension force.

Figure 4 Schematic picture of the undular mode of the magnetic buoyancy instability.

The critical length is derived as follows (Fig. 4): Assume g and T is constant. The

apex of the perturbed field lines are located at ∆z above the original height. The

dropping gas tries to settle to the new stratified equilibrium with a scale height without

the magnetic pressure, i.e.

ρi(z +∆z) ≈ ρ0(1−∆z

Hg) (20)

where Hg = RT/µg. The surrounding gas is supported by the thermal plus magnetic

pressure, i.e.

ρe(z +∆z) ≈ ρ0(1−∆z

H) (21)

8

where H = Hg(1+ β)/β、ρ0 = ρi(z) = ρe(z). Thus, the density deficit in the loop apex

leads to the buoyancy force as

fb = (ρe − ρi)g = ρ0g∆z

(1

Hg− 1

H

)= ρ0g∆z

1

βH. (22)

On the other hand, the magnetic tension force ft = B2/(4πRc) works as a restoring

one. The curvature radius of the field line can be obtained geometrically as Rc =

λ2/(8∆z). The instability grows when fb > ft, which leads to the critical condition to

the wavelength as

λ > λc = 4H

√β

1 + β. (23)

9

3.1 Linear analysis of the magnetic buoyancy instability

The linear analysis of the magnetic buoyancy instability in a stratified atmosphere

is given in, e.g., Parker (1979). The unperturbed atmosphere has a uniform tem-

perature T0 and a uniform plasma beta with uni-directional horizontal magnetic field

B = B0(z)ey. The gravity is assumed to be uniform. The sound CS and Alfven speed

CA are both constant.

The perturbations can be described by the sinusoidal functions as

∝ exp [i(k · x− ωt)] (24)

The dispersion relation of the instability is

(ω2 − k2yC2A)[ω

4 − k2(C2A + C2

S)ω2 + k2yk

2C2AC

2S]

+ 1H2 (ω

2 − k2yC2A)

[− 1

4 (C2A + C2

S)(ω2 + k2yC

2A) +

(1− 1

γ

)(C2

S

γ + C2A

)k2yC

2S

]+ 1

H2

(C2

S

γ +C2

S

2

)k2x

[C2

A

2 (ω2 + k2yC2A) +

(1− 1

γ

)C2

S(ω2 − k2yC

2A)

]= 0

(25)

where k =√

k2x + k2y + k2z and H = (C2S + C2

A/2)/g.

First, the interchange mode, i.e. ky = 0 case.

ω4−(k2x+k2z+1

4H2)(C2

A+C2S)ω

2+1

H2

(C2

S

γ+

C2S

2

)[(1− 1

γ

)C2

S +C2

A

2

]k2x = 0 (26)

The condition for the existence of an unstable solution (ω2 < 0) is (the 3rd term in the

l.h.s.) < 0, i.e.1

β< 1− γ. (27)

From the requirement of the thermodynamics, γ ≥ 1, this inequality formula can not be

satisfied at any time. This means that this equilibrium is stable to the pure interchange

mode ky = 0.

Second, the undular mode, i.e. kx = 0 case.

ω4 − (k2y + k2z +1

4H2 )(C2A + C2

S)ω2

+C2Sk

2y

[(k2y + k2z)C

2A − 1

4H2 (C2A + C2

S)C2

A

C2S+ 1

H2

(1− 1

γ

)(C2

S

γ + C2A

)]= 0

(28)

The condition for the existence of an unstable solution (ω2 < 0) is (the 3rd term in the

l.h.s.) < 0, i.e.

H2(k2y + k2z) <C2

A + C2S

4C2S

−(1− 1

γ

)(C2

S

γC2A

+ 1

). (29)

10

In order for the wavenumber (ky, kz) to be real, it is required that the r.h.s. > 0. Then

we obtain1

β>

1

4

[3γ − 4 +

√γ(9γ − 8)

]. (30)

The growth rate of undular mode with kx = kz = 0 is given in Fig. 5 as a function ky.

The maximum growth rate is approximately H/CA when the wavelength is around H.

Figure 5 Growth rate of the undular mode of the magnetic buoyancy instability.

The unperturbed state has a uniform CS and a uniform CA. kx = kz = 0. The

gas’s specific ratio is γ = 1.

11

4 Non-linear studies by numerical simulations

4.1 Galactic interstellar gas

Non-linear evolution of the magnetic buoyancy instability was studied by numerical

simulations. Two-dimensional simulations for applications to the galactic disk was per-

formed by Matsumoto et al. (1988) for the first time (Fig. 6). When β < 1, strong

downflow from the top of the loops generates shocks at the footpoints.

Figure 6 Two-dimensional MHD simulation of the Parker instability (left) log-

scaled density by gray scale, magnetic fields in solid lines and velocity in arrows.

(right) log-scaled density （Matsumoto et al. 1988）

Figure 7 Tree-dimensional MD simulation of the galactic center gas disk (a) den-

sity by color rendering and selected magnetic field lines (b) Partial close-up of

the disk, which shows an emerging loop with vertical field strength by color plot

（Machida et al. 2009）

By the radio observations of the galactic interstellar gas, it has been known that the

12

magnetic field with mean strength of µ G has been found in the galactic disk. In the

galactic central regions, the survey observations in molecular 12CO line emission, loop

like structures with strong line broadening at the loops’ “footpoints” (Fukui et al. 2006,

Fig. 8). Based on this observations, Machida et al. (2009, Fig. 7) reproduced emergent

loops in the gas disk in three-dimensional MHD simulations.

Figure 8 Radio observations of loop structures in the galactic central region by

Nagoya University NANTEN (2.6mm 12CO emission line). (A) The intensity plot

integrated over the Doppler velocity range from −180 to −90 km/s, showing loop

1 and (B) that integrated from −90 to −40 km/s, showing loop 2. (c) A contour

of the detected loops. Numbers are the Doppler velocity. (Fukui et al. 2006)

13

4.2 Solar active region formation

Fig. 9 shows examples of emerging fluxes in the solar interior and surface. Nelson et

al. (2013 panels a-c) conducted a numerical simulations for the interior from 0.72 to

0.96R⊙ achieving Ω shapes loops’ emergence. Toriumi & Yokoyama (2012) conducted

an MHD simulation from 0.92R⊙ to above surface to find a two-step emergence.

Figure 9 Numerical simulations of solar emerging fluxes. (a)-(c) Emergence of

Ω shaped loops from a toroidal flux in the interior of the sun. Note that this

simulation does not cover the surface and beyond due to the restriction of the

anelastic approximation. (d)-(f) Emergence of a flux loop in the outer part of the

convection zone and the upper atmosphere. （from review by Toriumi & Wang

2019. the originals are by Nelson et al. 2013 for (a)-(c) and Toriumi & Yokoyama

2012 for (d)-(f))

14