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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