第27回理論懇シンポジウム「理論天文学・宇宙物理学と境界領域」

天体乱流研究と太陽内部物理学
c.f.,Masada & Sano (2014a), PASJ
Masada & Sano (2014b), ApJL
Masada & Sano (2015a) in prep.
[Masada, Takiwaki & Kotake (2015), ApJL]

Masada & Sano (2015b) in prep.

政田洋平（神戸大学）
共同研究者：佐野孝好（大阪大学）

Hinode SOT

太陽黒点・太陽蝶形図の起源の理解を目指して

Eventually the①heirarchy
大局性・赤道反対称性
must culminate in motions
周期性（約22年周期）
large enough ②
to sensethespherical geometry and③ 緯度方向のマイグレーション
rotation

Giant Cells!

⌘複雑なMHD乱流系
・強い密度成層・コリオリ力・差動回転, 子午面流・対流安定層・高Re & 高ReM

なぜ太陽ダイナモ機構を研究するのか？ ~天体乱流研究のプロトタイプ~【基礎物理学的意義】

太陽ダイナモ：乱流中で生じる磁場の（局所から巨視的スケールへの）逆カスケード現象
乱流の対称性の破れがもたらす磁場の自発的組織化の帰結 (後述)

→ 自然界の自己組織化現象の背後に潜む共通原理に対する理解.

【天文学的意義】

乱流：大規模構造から惑星形成に至る宇宙のあらゆる時空間・エネルギースケールで普遍的.
乱流によって生成される磁場は天体のダイナミクスに大きな影響.
観測可能な太陽は天体乱流の物理の『天然の実験場』.

→

天体の活動性や形成・進化の理解に直結.

【太陽地球物理学的意義】

太陽黒点の起源：
・太陽活動の中長期変動（数百年〜数千年スパン）

・太陽活動の中長期変動と地球の気候変動との密接な関連 (e.g., Gray et al. 2010)
・近年の太陽活動の異常 (e.g., McComas et al. 2013)→ 太陽–地球環境と生命の46億年の共進化とその未来に対する理解の深化.

太陽内部熱対流の物理~ 乱流場と平均場の相互作用の理解を目指して ~上昇流 = 発散流

（時計回り）

Ω
δT > 0#
δP > 0下降流 = 収束流Ωambient（反時計回り）
Vz

δT < 0#δP < 0Top View

ambient
コリオリ力によって !
時計回りの運動が励起→ 〈δvr δvh〉 ≠ 0お!
→ ヘリシティ!
レイノルズ応力broader & slower!
cellular upflow★ 密度成層

強い下降流
弱い上昇流

★ 太陽内部熱対流：

コリオリ力によって!
反時計回りの運動が励起Side View

buoyancy braking- 激しい乱れ（乱流）- 対称性の破れ (成層・回転)

収束流
発散流

narrower & faster!
downflow lane
コリオリ力

ヘリシティ
レイノルズ応力• 角運動量輸送→ Ω効果 (平均場)• 運動学的ヘリシティ→ α効果 (乱流場)

回転成層対流によるダイナモをシンプルなモデルで理解するΩ効果（速度シアーによる磁場増幅）が無くても大局的磁場はできる

1. 回転成層対流によるダイナモ~ 「大局的」磁場生成の物理機構 ~Masada & Sano (2014a) PASJ
Masada & Sano (2014b) ApJL

乱流渦
乱流渦

場
磁
局的

大

乱流渦
乱流渦
乱流渦

大局的磁場

= 乱流渦スケール以上のコヒーレントな磁場構造

乱流磁場= 乱流渦スケール以下の間欠的でもつれた磁場構造

Simple Model of Turbulent Convection and Dynamo[3層ポリトロープモデル: 上部放射層（冷却層）, 中部対流層 & 下部放射層]★ 圧縮性MHD (回転系)
上部：開放境界
下部：完全導体境界

Ro = vrms/2Ω0d~ 0.03 @対流層

Pr = 1.2, Pm = 4,
Ra = 4×106

上下の密度差は実際の太陽と比
べると小さい

冷却層

冷却層

Ω

(対流安定)

g

対流層

対流

層

放射

層

放射層

(対流安定)★差動回転（Ω効果）は存在しない（ 水平方向に対称性の破れが無い）.

★上下非対称性を持つ対流構造（→ 実効的ヘリシティ）.

激しく乱れた対流中での磁場形成

0.3R

sun

sun
R
7
0.

0.5

z (d

周期的極性反転をともなう大局的磁場の生成
z (depth)

1.00.0(a) 〈Bx〉h

0.5

1.0

(b) 〈By〉h0z (depth)

100

200

1.2 13)(seealso-1.2
Kapyla+13; Favier & Bushby
t/τ
cv

300

400

η/η0=0.5

0.0

500

600

700

800

0.5

上下安定層
有り

1.0
1.5
2.0
0.0

z (depth)

Masada & Sano (2014a)(c) 〈Bx〉h0.5

η/η0=1.0

1.0
1.5

2.0

(d) 〈By〉h

0

200

400

600

800

t/τcv

1000

-1.5

1200

1400

1.5

z (depth)0.0

Fig. 2. 0.5
Time-depth diagram of the horizontally-averaged horizontal magnetic field. Panels (a) and (b) [(c) and (d)]0 x〉h(a) 〈Bdemonstrate⟨Bx ⟩h and ⟨By ⟩h normalized by Beq for the model A [model B]. The orange and blue tones denote the
1.0
0.0
positive and negative strengths of the magnetic field.

z (depth)0.5

z (depth)η/η =2.0

1.0

安定層無し

are adopted(b)for〈By〉hboth models. The total grid size is 256 (in x) ×256 (in y) × 64 (in z) for the model
1.0
0
100
400
500
600
700
800
A, and 256 (in x) ×256
(in y) 200
× 128 (in 300z) for the t/τ
model
B.-1.2A
small random
perturbation
1.2 is added to
cv
0.0
the velocity
and magnetic fields when the calculation starts.
0.5

3.1.

(c) 〈Bx〉h Results
Simulation
2.0
1.5

0.0

z (depth)

3.

0.5

1.0
Properties
of Convective Dynamo
1.5(d) 〈By〉h● 激しい乱流が支配する対流層で大局的磁場が形成.
● 大局的磁場と乱流磁場のエネルギーは同等.
● 極性反転と動径方向へのマイグレーション.

2.0
After
the convective
motiona saturated
state att ≃ 250τcv1400
for both0200
400 sets in, the600 system reaches
800
1000
1200
エネルギー
-1.5
1.5
cv
models. The mean convective velocity is evaluatedt/τ
there
as ucv = 0.017 (0.019), providing Beq = 0.045
注入のスケール(0.045), Co = 47 (42) and τcv = 58.8 (52.6) for the model A (model B). Since a sufficient scalex & y magnetic field. Panels (a) and (b) [(c) and (d)]Fig. 2. Time-depth diagram of the horizontally-averaged horizontal
separation between
the convective eddies and the box scale is known as a necessary ingredient for the
運動エネルギースペクトル
demonstrate ⟨Bx ⟩h and ⟨By ⟩h normalized by Beq for the model A [model B]. The orange and blue tones denote thelarge-scale
dynamoBrandenburg&field.
Subramanian 2005; Käpylä et al. 2009), we have chosen
positive andnegative (e.g.,strengthsof the magnetic> 40), yielding small convective cells relative to the box scale.
the relatively rapid rotation (Co ∼are adopted
forin
both
models.
Thedistribution
total grid size
is 256(invelocityx) ×256in(iny)horizontal
× 64 (in z)
for theShown
Figure1 is theof theradialtheplaneat zmodel= zmA,
and(a)
256(in40τ
x) ×256(in y) × 128 (in z) for the (対流層のみのモデルの極性反転周期は3層モデルの1/3)
model
B. A small
perturbation
added
to
when
t =
A [panels(d) random
and (e) arethose for is
the
model
cv and (b) t = 400τcv for the model
theandandmagnetic
thecalculation
starts.
B]. velocity
The darker
lighterfields
toneswhen
depictdownflowandupflow velocities. In Figure 1(c) and (f), thetemporal evolutions of two-dimensional kinetic energy spectra for the models A and B are shown.
3.
Simulation Results
Note that a two-dimensional Fourier spectrum of the kinetic energy at the each depth is projected22
onto
one-dimensionalwavenumber
k 2 = kx
+ kyand then is averaged over the convection zone and3.1. a Properties
of Convective
Dynamo
さらに単純化
磁気エネルギースペクトルthe time span
shown in the figure legend. The different lines correspond to different time spans. The
After the convective motion sets in, the system reaches a saturated state at t ≃ 250τcv for bothhorizontal axis is normalized by kc = 2π/W .
models. The mean convective velocity is evaluated there as ucv = 0.017 (0.019), providing Beq = 0.045
In the kinematic phase of the model A (left top), we can find the appearance of large-scale(0.045), Co = 47 (42) and τcv = 58.8 (52.6) for the model A (model B). Since a sufficient scale● B B の間の位相のズレは π/2.
● 安定層の有無は大局的磁場の生成に影響しない.
一方、極性反転周期には影響.

● ダイナモ機構は何か？
直接数値計算（DNS）とカップルさせた1D平均場モデル

y et al.
d (e.g.,∂t∂⟨Brateh ⟩ 2rate
ain
sequencestarswithwithdiﬀerent
diﬀerentagerotation
Chaplinetal.
definedby andsequencestars
age
andthusthusdiﬀerent
diﬀerent
rotation(e.g.,
Chaplin= ∇ × [α⟨Bh ⟩ − (η + ηt )∇ × ⟨Bh
"
#
γρν
ν
g d where∇−∇∂t
CZ(Bx , Bythe) isunderstanding
the horizontal
magnetich(2)=mulationtandemwith
withthethe advanced
advanced
observation
help
Pr =
, Pm =, Ra = will
,Bdeepening
DNSとカップルさせた平均場ダイナモモデル
lationin intandemobservation
will
help
deepeningtheunderstanding
κ
η
χ ν
H
z α and η represent the α-eﬀ
nent.
TheBcoeﬃcients
where
=
(B
,
B
magnetiRZh era.xy ) is thet horizontalordynamics
andstellardynamoactivities
intheastroseismology
where
ν
,
η
,
and
κ
are
the
viscosity,
magnetic
diffusivdynamics and stellar dynamo activities in the astroseismology
era. diﬀusivity. Here all the te
the turbulent magnetic
0

0

0

0

4

0

0

0

0 0

ad

p

0

ity, and thermal conductivity, respectively. Here∇ − ∇Thenent.
coeﬃcients α and ηt represent the α-e
ad is
the superadiabatic temperature gradient with ∇ad = 1 − 1/γ,
theturbulentmagnetic diﬀusivity. Here all the∇ = (∂ ln T /∂ ln P), and Hp is the pressure scaleheight.
The（場の量を平均場と乱流場に分ける）
variables ρ, ∇, and Hp in equation (2) are evaluatedlated attothe the mean flow andz the mean(b)verticalh h
mid-convection zone of the depth zm = (z2 − z1 )/2.
ηt0 from the si
droppedbecause of ⟨u⟩ = ⟨Bz∝⟩=h hh h The volume
h haverage intthe convection
0 zone andh horizonh
theare denotedby tsingle angularwithhsub-hh h tal average hh
0 brackets
results.
All the variables, except the magnetic dscript “v" and subscript “h", respectively. The time-average ofη, havetime and z dependences.
teach spatial mean is denotedby additional angularbrackets.
乱流起電力がダイナモを担う
平均流れ場によるダイナモ項はゼロ
relative strength of rotation to the convection is !
measuredwith of the vertical velocity The
The
evolutions of the α and ηt are aﬀected by. 1.— Snapshotwhent = 400τcv . The
by the Coriolis number Co = 2Ω0 d/ucv , where ucv ≡ ⟨⟨u2z ⟩⟩v∝α
blue)
toneSnapshot
denotes
downflowThe
field The convective
back-reaction of the mean-fieldon the dy
ismeanmagnetic
convective
turn-overFig. 1.—the vertical
when
t=
t of
h (upflow).
h velocityz thehh 400τvelocity.
h hlinear
cv . Theare (blue)
visualized
the halfdownflow
volume. (upflow).
time and The
the equipartition
fieldfieldstrength
are defined, respecred
toneindenotes
magnetic!
eﬃcients.
Todenotes
take it the
self-consistently
intocompone
accoun2 ⟩⟩ hTime
seriesoftheverticalprofiles
of
(a)
⟨B
⟩
and
(b)
⟨B
⟩.
The
orange(blue)
tone
positive(negative)
乱流α効果(磁場増幅)
乱流による移流
乱流磁気拡散
t
h
h
z
h
h
h
x
y
tively,
by
τ
≡
d/u
and
B
≡
⟨⟨µ
ρu
.
cv
cv
eq
0
v
2
ines2 are visualized
in the half volume.
dynamical α-quenching
and algebraic η-que
All the variables are assumed to be periodic thein the horizonmagnetic
field.
v
2v
tal directions. Stress-free boundary conditions are used in the k 2
h
t 0
h
!
as ⟨B⟩2v /⟨B 2 ⟩v = 0.6 in the convectionzone.
In
contrast2
vertical direction for the velocity. Perfect conductor∂αand verti- t c
αk −2ηt µ0 ⟨J · Bh ⟩k 2 α⟨Bh ⟩ −eq
cal
field
boundary
conditions
are
used
for
the
magnetic
field
atto the horizontal component, the vertical field does not= −2ηt kc+ t k
the bottom and top boundaries, respectively. While ∂ta constant2Bηtshow any coherent signatures,is fullybye,visdominatedz and energy
flux
imposed on the bottom
boundary,
the internal
2 −1eq(by
FOSA)
is fixed on the top boundary.
k! FIG. 1.— (a) Modelh setup 2and"volume
fluctuating component, i.e., ⟨Bz ⟩ =energy
0.
−1 visualization of vertical velocitytMThe
fundamental
equationsaresolvedbythesecond-order
t
⟨B
⟩
2
h
the fiducial
model with Pm = 4. The red (blue) tone
k when t = 400τcv for 2Theoscillatory
Godunov-type finite-difference
scheme which employsan
apv large-scale field shows a well-regulated
eq
η
=
η
1
+
Re
, red dashed, blue dotted lines
denotes downflow(upflow). (b) Black solid,
t
M
t
2
•
乱流場の統計平均プロファイルをDNSの結果から直接抽出→
乱流起電力を決める.
′
proximate
MHD
Riemannsolver(Sanoetal.
1998).
The
magBeq average of the kinetic helicity,
denote the time and horizontal
Heft , and u2rms
behavior.
The mean field componentisthestrongestat
3
The normalizations are their extremum values.
netic field is evolved with CMoC-CT method (Clarke 1996).
′
6
h
0
•
大局的磁場が乱流場に及ぼすbackreaction(非線形効果)も考慮(quenchingmodel).
Non-dimensionalparameters
of
Pr
=
1.4
and
Ra
=
4
×
10
,
around z3 − z = 1.2 and seems to propagate from there
Jc all= ∇downflow
× Bh /µ&asBrandenbu
acquiresa negative vorticity
a consequence ofconstant angular velocity of Ω0 = 0.4 are withadopted for
0 (Blackman
to top and base of thez convection and
zone.
The
polarityisthe Coriolis force acting on the converging motion, yielding
the models. The modelscv
with different magnetic Prandtl numwherekischaracteristic
wavenumber
of
c
a the
negativehelicity inhthe upper convection
zone. In contrast
bers
Pm
=
2,
4
and
8
are
simulated.
Thefiducialmodel
withthen gradually reversed over the period of ∼ 200τcv . It
near thebase fof thezone, the downflow
Pm = 4 is same as the model B in MS14a. The
spatial resvective
eddies,andis convectiona coeﬃcientwhichis decelcons noteworthyx that there
is
a
phase
diﬀerenceofabout
eratedanddivergedbynegativebuoyancy,
resultinginaposiolution of (Nx , Ny , Nz ) = (256, 256, 128) is adopted for all theysaturationfieldstrength.
Inthefollowing,
weh to thetive helicity. The non-zero gradient of the kinetic helicity is uaruns. A small
random perturbation is added
π/2 between ⟨Bx ⟩ and ⟨By ⟩. Thesimulation
observedoscillatory
h large-scalet in our models.
h (2002).
for thedynamovelocity and magnetic fields when the calculation
starts.
suggested
bysource
Blackman& Brandenburg
behavior is reminiscent of the solar butterfly
diagram
Regardless the magnitude of the magnetic diffusivity, os3. SIMULATION RESULTSk
In the second
ordercorrelationcillatorylarge-scale
magnetic
fields approximation
are spontaneously orgakalthough there is a diﬀerence in the propagation
directiontk
The systemcv
reaches an equilibrated stateBy
typically
aftercoeﬃcientsthein thebulkandof the ηconvection
zone.
The bytime-depth diathe ynized
informationof theDNS,
husing
x
namo
α
are
given
k
t
magnetic
diffusion
time
(see
MS14a
in
detail).
Thequantities
gramof
horizontally-averaged
horizontal
magnetic
field ⟨Bx ⟩hbetween the simulated field and the sunspot field.
wedeterminet
ucv , Co, Beq and τcv depend less on the magnetic diffusivity
is shown in Figure 2. Panels (a), (b) and (c) correspond to the
Since there is no physical mechanism
forthesymmetry
information
kc withz Pm
x =y2, 4 andx8, respectively.
y z
c 1
k
and are evaluated there as 0.02, 40, 0.045 and 50, respectively
modelsThe orangeandxyαk (z) = −τ∂x uythe⟩⟩positive
+ ⟨⟨u
− τ
for all themodels.
blue
tones zdenoteandnegative
field=strengthsc [⟨⟨ux∂y uz ⟩⟩]
braking in the horizontal directions,
the
flow
is
ofmeanfluctuating
3
The volume visualization in Figure 1a is the vertical velocThe time is normalized by τcv . Note that ⟨By ⟩h shows the simk
22absent in our simulation, i.e., ⟨u⟩ ity=when
0. t In
contrast,
thevelocityfield= 400τ
model. The red (blue) tone
ilar behavior with1⟨Bx ⟩h but has a phase difference of abou
cv for the fiducial
c from(c.f.,tk
2z it (see2rms
z cOssendlijver+02)
denotes
downflow(upflow)
region. The convective ηmotionis = τπ/2
MS14a).
(z)
⟨⟨u
⟩⟩
=
τ
u
mean kinetic helicity naturally arises
fromtheup-downc The zspatiotemporalcstructuretrms ,of the large-scale magnetic
characterized by cellular upflows surrounded by networksof3
narrowerand faster downflows
in the saturated state. Since
field is similar among the models with different magnetic difasymmetry in the convective motion
as shown
in Figurec
is no symmetry breaking in the horizontal directions,fusivity: it is the strongest at the middle of the convection
3a. The solid line is the verticalthere
profile
of
the
mean
where
τ
is
the
correlationtime.
cthe mean horizontal shear flow is absent in our models.
zone andpropagates from there
to topTheand basereplacemof the convec-

lated to the mean flow and the mean vertical fi★ 平均場ダイナモ方程式:
droppedbecauseof ⟨u⟩ = ⟨B ⟩ = 0 from the sim∂⟨B ⟩magnetic diﬀ∂⟨B ⟩= ∇ × [⟨v ⟩ × ⟨B ⟩ + E − η ∇ ×results.
⟨B ⟩ All] the variables, except the(11)∂t = ∇ × [⟨u ⟩ × ⟨B ⟩ + E − η ∇η,×have⟨Btime⟩ ] and z dependences.
(11)

∂t

The evolutions of the α and η are aﬀected by tlinear back-reaction of the mean-field on the dyna
To take it self-consistently(12)into account,
E = α⟨B ⟩ + γe × ⟨B ⟩ − η∇ × ⟨B eﬃcients.
⟩ ,the ⟩dynamical
α-quenching and algebraic(12)η-quenE = α⟨B ⟩ + γe × ⟨B ⟩ − η∇ × ⟨B,!
B⟩ /⟨B ⟩ = 0.6 in the convection zone. In contrast∂αα⟨B ⟩ − η µ ⟨J · B ⟩ α − αhe horizontal component, the vertical field does not= −2η k
+
α効果
〜
相関時間
×
運動学的ヘリシティ
inany
thecoherent
bulk oftheconvection
max- bywhat
regulates
the oscillation
cycleof the mean∂t computations
Bcarriedη /ηTrefereefor constructive
comments.
Numerical
were
on field.
wanonymous
signatures,and zone.
is fullyThedominated
an-fieldstrengthreachesan. The
explorethe underlying
dynamomechanism,! of Japan.
" work waswe constru
uatingcomponent,i.e.,⟨B⟩order=National
0. of×B2乗速度
乱流磁気拡散〜相関時間
beUniv.,
and
Cray
XC30
at
Astronomical
ObservatoryThis
onymousrefereefor constructive
comments.
Numerical computations
carried on⟨B dynamo⟩ were model.
he
and
total
field
strengths
is evaluatedmean-field
he meanlarge-scale
field showsa well-regulated
oscillatorya one-dimensionalη = η 1 + Re,
KAKENHI
grant
Nos.
24740125and
20260052.
B
B ⟩ =The
0.6meaninCraythefieldconvection
the variables
into This
horizontal
avior.
component
is In
thecontrast
strongestat By dividing
Univ.,
andXC30
at zone.
National
Astronomical
Observatoryof Japan.
work mean
was anizontalverticalfield doesfrom
not therefluctuating components, as u = ⟨u⟩ + u and B
nd z −component,z = 1.2 andthe
seems
to propagate
AKENHI
grant
Nos.
24740125and
20260052.
=∇
× B the/µ horizontal(Blackmanaverage& Brandenburg
coherent
signatures,and is fully
dominatedby
B , Jandtakingof the iopand baseof the convectionzone.
The polarity
is⟨B⟩ +with
where
k
is
the
characteristic
wavenumber
reversedg graduallycomponent,
i.e., ⟨Bover⟩ =the0. period of ∼ 200τ . Itduction equation, the mean-field dynamo equationofis tho1DMFequation
for
B
(t,z)
:
vective
eddies,andfisacoeﬃcient
which
contr
★
DNS
REFERENCES
oteworthy
that
there
is a phase
diﬀerenceof abouttained
ge-scale
field
shows
a well-regulated
oscillatory
saturation field strength. In the following, we us
between⟩ and⟨B ⟩. TheobservedoscillatoryThemean⟨Bfieldcomponentis thestrongestat∂⟨B ⟩ by Blackman & Brandenburg (2002).
suggestedavior
solar butterfly
diagram= ∇ × [α⟨B ⟩ − (η + η )∇ × ⟨B ⟩] , (− M.,
z =is 1.2
and seems
tothe
propagatefrom 110
thereel,&reminiscent
Wicht,
J. of
2007,
Icarus,190,
REFERENCESIn
the∂t second order correlation approximation,is aconvection
diﬀerence inthe propagation
doughbasethereof thezone.
The polaritydirectionisnamo
coeﬃcients
αtheandη are given
by compween
thesimulated
field
and
the
sunspotfield.
where
B
=
(B
,
B
)
is
horizontal
magneticually
reversedovertheperiod
of
∼
200τ
.
It
H.M.&2001,
MNRAS,
324,Icarus,
498 190, 110
M.,
Wicht,
J.
2007,
nce there
no physicalmechanism
for oftheabout
symmetrynent. The coeﬃcients α and η represent the α-eﬀect
an
1
rthy
that isthere
is a phase
diﬀerenceα (z)= −τ ⟨⟨u∂ u + uHere∂ u all⟩⟩ the= −terms
τ Hr
king
in
the
horizontal
directions,
the
mean
flow
is
the
turbulentmagnetic
diﬀusivity.
een
⟨B
⟩
and
⟨B
⟩.
The
observedoscillatory3bonneau, P., Racine, E., & Smolarkiewicz, P.K. 2013, Sol. Phys., 282, 335
inour simulation,= 0. Indiagram
contrast, thelated to the mean flow and theM.
2001,
MNRAS,
324,⟨u⟩498
isnt reminiscentof the i.e.,
solarbutterfly
1 mean vertical field aη (z) =ofτ ⟨u⟩
⟨⟨u =
⟩⟩ ⟨B
= ⟩τ=u0 from, the simulatinherekinetic
helicity naturally
arises fromdirection
the up-downdropped because
is M.S.,a diﬀerence
inA.S.,
the propagationiesch,
Brun,&
Toomre,
J.
2006,
ApJ,
648,
L157
3
mmetryintheconvective
motionas
shown
inFigureresults.
Allthevariables,
exceptthe magnetic diﬀusivi
he simulated
field and E.,
the &sunspotfield.
nneau,P.,
Racine,Smolarkiewicz,
P.K.
2013,
Sol.
Phys.,
282,
335
The
solid
line ismechanism
the verticalprofileof the meanη, have
whereis thecorrelation time. The replacemen
timeτandz dependences.
ere
is
no
physicalforthesymmetry
urlburt,N.E.,& Toomre,J. 1996, ApJ,where
473, the494
tic helicity
definedby H = ⟨⟨u·(∇×u)⟩⟩,by u is allowed here because ⟨u⟩ = 0 in our simu

– 12 –

平均場モデルの非線形解 vs. DNS(Masada & Sano 2014b)– 5 –depth (z)

0.0
0.4
0.8
1.2
1.6
2.0
0

stratification than the actual solar stratification. Since the stronger stratificati
the magnetic dynamo properties, we will study the eﬀect of the density stratific(a) DNSofthe
DR profile in the MHD regime (c.f., Käpylä et al. 2011, 2014; Karak et a
MF
model
200

400

600

800

t/τcv

1000
-0.05

1200

0.05

0.0

depth (z)The recent development of the astroseismology opens up the way to study
in the solar-type main sequencestars with diﬀerent age and thus diﬀerent rota
– 6 –(b) MF
2010). Computer
simulation in tandem with the advanced observation will help
–6–t/τof the stellar interior dynamics and stellar dynamo activities in the astroseismo
0.03
0.4

0.8

1.2

1.6

2.0
0

1

3

2

4

diff

5

6

7

-0.04

0.04

Fig.
model
0.02 2.— Time-depth diagram of ⟨Bx ⟩h for the reference model in (a) the DNS and (b) the MF√
0.01
coupledwith the DNS. The orange (blue) tone denotes the positive (negative) ⟨Bx ⟩h normalized by dg0 ρ0 .
0
The horizontal dashed lines show the interface between the convection zone and the stable zones.
-0.01
h h
-0.02
h h
h h
t
0
h
-0.03
electromotive force (e.g., Ossendrijver et al. 2002). The coeﬃcients α, γ, and η represent the α-eﬀect,
turbulent pumping, and turbulent magnetic diﬀusivity, respectively. All the terms related to ⟨u⟩h and ⟨Bz ⟩hFig. 3.— The time series of ⟨Bx ⟩v and ⟨By ⟩v for the reference model. The cyan [orange] solid line denotes
can be ignored in considering the symmetry of the system. All the variables, except
for η0 , depend on the
乱流起電力がダイナモを担う！
⟨Bx ⟩v [⟨By ⟩v ] in the DNS. The red dashed and blue dash-dotted lines are those in the MF model. The time
Orange:B
indepthDNS,
Cyan: By in DNS
xturbulent
time (t)(z).
is normalizedbytheandmagnetic
diﬀusion
time.

∂⟨B ⟩
= ∇ × [⟨v ⟩ × ⟨B ⟩ + E − η ∇ × ⟨B ⟩h∂t
E = α⟨B ⟩ + γe × ⟨B ⟩ − η∇ × ⟨B ⟩ ,Red: Bxseries
in dynamo
MF,
Blue
B
MFthe
Fig. 3.—
The timeof ⟨Bxdescribed⟩v and :⟨B⟩inmodel.
The
denotes h hv forh category.
hcyan [orange]
zThe MFsolid
h hlineTheMF
byyy equation(2)reference
falls intotthe
α2 -type
theorypredicts
2 = −dz/d ln⟨⟨ρ⟩⟩⟨Bx ⟩v [⟨Biny our⟩vthat] inthetheαHDNS.
reddashedanddash-dotted
lines
those inthe (e.g.,
MF model.
The timemodel,
where
the density
scaleheight.
Here,blue
the subscript“k” refers
to thewith
d modeThecanh isgeneratea large-scale
magnetic
field
an areoscillatory
natureBaryshnikova&
乱流α効果(磁場増幅)
乱流による移流
乱流磁気拡散unquenched coeﬃcient, which is calculated from DNS results of the saturated convective turbulence.
is normalized
by the 1987;
turbulent
diﬀusiontime.
Shukurov
Rädlermagnetic& Bräuer 1987;
Brandenburg
et al. 2009). A key ingredient for the oscillatory mode
In the first-order smoothing approximation (FOSA), the unquenched coeﬃcients αk , γk and ηk inis the nonuniformity of the α-eﬀect, which can arise naturally as an outcome of rotating stratified convection
anisotropic forms are given by (e.g., Käpylä et al. 2006, 2009b),2

in the stellar interior. Using the rigidly rotating system studied here, the α dynamo wave was excited,
αk (z) = −τc [⟨⟨uz ∂x uy ⟩⟩h + ⟨⟨ux ∂y uz ⟩⟩h ] ≡ −τc Heﬀ ,
(7)
which
propagates
only inthe depth
direction. However, as
shown by Käpylä et al. (2013b), in the global
in our model,
whereHγdk (z)==−dz/d2 ln⟨⟨ρ⟩⟩2 h is the density scale height. Here, the subscript “k” refers to the−τc ∂z ⟨⟨uz ⟩⟩h ≡ −τc ∂z urms ,(8)
it canwhich
travel also
in 2 the latitudinal
direction
due ofto thethe strong
antisymmetryof theα-eﬀect across2
unquenchedsystem,
coeﬃcient,calculated
from DNS
results
saturatedconvective
turbulence.
ηk (z) = τc ⟨⟨uis(9)z ⟩⟩h ≡ τc urms ,（対流層中部から上下方向に伝搬）
the equator.

We thank the anonymous referee for constructive comments. Numerical c● DNSの磁場の時空間進化の特徴を平均場モデルで再現：
π-Computer at Kobe Univ., and Cray XC30
at National Astronomical Observat● 時空間進化パターン→ α効果駆動型のダイナモ波の伝搬
where τ is the correlation time, H is the eﬀective helicity, and uis the root-mean-square velocity. The
In thefirst-ordersupported
smoothing approximation(FOSA), theunquenched
coeﬃcientsand
αk , γk and ηk in
by
JSPS
grant
Nos.
24740125
vertical profilesof Hdynamo-generated
and u
in the reference DNS
model
are shown KAKENHI
in Figure1b by solidforce
and dashedTheMF
producesa Lorentz
that tends
to “quench”
the turbulent20260052.
motions andanisotropic● パターンだけではなく、磁場の振幅と反転周期(~
乱流磁気拡散時間)も定量的に一致.
lines,forms
respectively.are given by (e.g., Käpylä et al. 2006, 2009b),
control the nonlinear evolution and saturation of the system. Since there is no definitive model to describe
c

eﬀ

eﬀ

rms

2
rms

● Bx & Byの間の位相のズレ π/2も定量的に再現. 反転周期は乱流磁気拡散時間程度.

The correlation time should be zero in the top cooling and bottom stable layers since the convective
the magnetic quenching
eﬀect(e.g.,
Rogachevskii&⟨⟨u
Kleeorin
2001; Blackman & Brandenburg 2002) as yet,(7)uStrouhal+turbulence is not fully developed;αthus
αk = γ k == ηk = −τ
0 there.
Assuming
theis unityk (z)c [⟨⟨uz ∂xy ⟩⟩h number
x ∂yinuz ⟩⟩h ] ≡ −τc Heﬀ ,we adopt
themodels,α-quenching, algebraic γ- and η-quenching of
the convection
zone (St= τc uprototypicalprofile of τc which
is given 2
by are the dynamicalrms kc = 1), the vertical
2
γ
(z)
=
−τ
∂
⟨⟨u
⟩⟩
≡
−τ
∂
u
,
(8)
k
c z rms!
"#$ !c z " z #$h
the catastrophic-type;1
z − zbzt − zτc (z) =1 + erf
1 +2
erf
,
(10)
2
4uk
kc
η
(z)
= h τc ⟨⟨u
,
(9)
rms
"
h h≡ τc u
z ⟩⟩!
rms
2
∂α
α⟨B
⟩
−
η
(∇
×
⟨B
⟩
)
·
⟨B
⟩
α
−
α
h
h
h
h
h
k
2
h
where zi (i = t, b) represents the location of the boundaries
betweenregionswith and without
fully developed
= −2η
+
,
(4)
kk
c
2
B
Re
M
turbulence.
We define zt and zb time,as the ∂tdepth
Hachievesthemaximumandminimum
values,where τc isthe correlation
Hwhereistheeﬀectivehelicity,
and
u
is
the
root-mean-squarevelocity.
The
eq
eﬀ
eﬀ
rms

REFERENCES→ 激しい対流そのもの (乱流α効果) が大局的ダイナモを担う.

2. 回転成層対流によるダイナモ~ 極性反転周期を決める物理と表面活動現象との繋がり ~Masada & Sano (2015a) in prep.

★ parameter study(a). 計算ボックス幅(b). 磁気拡散率(c). 放射層の厚み(d). 冷却層の厚み
パラメータ依存性から物理を理解する

極性反転周期のスケーリング則 (c.f, Masada & Sano 2015 in prep.)
35000

35000

τcyc = τdiff

25000
20000
15000
dRZ,t: 表面冷却層の厚み
W: ボックス幅
η0: 微視的磁気拡散係数

10000
5000

τcyc ∝ τAlf

30000

0

τcyc [cycle period]

τcyc [cycle period]

30000

25000
20000
15000

VA ≡ BM/(4πρcz)1/2
where2
BM ≡ 〈〈Bx〉〉cz+ 〈〈Bx〉〉2cz

10000
5000

0

5000 10000 15000 20000 25000 30000 35000

052dRZ,t W/η0

★ スケーリング則 (1)：

15

20

25

30

dRZ,t /VA
※ dRZ,t：表面冷却層の厚み

τcyc = τdiff 【τdiff ≡ 2dRZ,tW/η0 (~ 微視的磁気拡散時間) 】→ 表面冷却層（対流安定）の磁気拡散時間と関連！！

★ スケーリング則 (2)：10τcyc ∝ τAlf 【τAlf ≡ dRZ,t/VA (Alfven横断時間) 】

ココ35

ダイナモ極性反転の定性描像（1）● 反転周期のスケーリング則：

τcyc ~ τdiff
ダイナモ反転周期を決める物理の定性的描像

τcyc ∝ τAlfvenτcyc ~ τdiff,tMasada and Sano
●4 平均場モデルを使った解析から得た知見：

see MS14b

τcyc = τdiff,t ~ dcz2/〈ηt〉cz〈ηt〉cz ・・・対流層の乱流磁気拡散.

F IG . 3.— The time series of ⟨Bx ⟩v and ⟨By ⟩v for the reference model. The cyan [orange] solid line denotes ⟨Bx ⟩v [⟨By ⟩v ] in the DNS. The red dashed and blue
dash-dotted lines are those in the MF model. The time is normalized by the turbulent magnetic diffusion time.

0.4
0.8
1.2
1.6

dcz・・・対流層の厚み0.0depth (z)depth (z)

0.0

0.4
0.8
1.2
1.6

(平均磁場の非線形バックリアクションも考慮)ダイナモ極性反転の定性描像（2）electromotive force (e.g., Ossendrijver et al. 2002). Th

turbulent pumping, and turbulent magnetic diﬀusivity, recan be ignored in considering the symmetry of the systemtime (t) and depth (z).
磁場の極性反転The MF dynamo磁気ヘリシティロス
described by equation (2) falls intthat the α2 mode can generate a large-scale magnetic fiel
Shukurov 1987; Rädler &Bräuer 1987; Brandenburg et a
開放境界
is the nonuniformity of the α-eﬀect, which can arise naturin thestellar interior. Using the rigidly rotating system
上部安定層がある場合、ダイナモ生成
which propagates only in the depth direction. However,
磁場は拡散以外に外に抜ける術が無い
system, it can travel also in the latitudinal direction due
the equator.cyc
diff ※安定層には乱流が無い

τ

BMの強度が決まる＊うまく説明できていない点

・安定層が無い時はどう決まるのか？

~ τ

ので微視的磁気拡散

強い制限がかかるMF produces a Lorentz force
The dynamo-generated
control the nonlinear evolution and saturation of the syst
the magnetic quenching eﬀect (e.g., Rogachevskii & Klee
we adopt the prototypical models, which are the dynam
対流層の乱流磁気拡散が調節
the catastrophic-type;
!
2
cyc
diff
diff,t∂α2 α⟨Bh ⟩h − η (∇ ×
= −2ηk kc
∂t
Be2
γk
γ =
,
2
quenchingの式の例
1 + ReM ⟨Bh ⟩2h /Beq
ηk
η =
,
t
, M
2
1 + ReM ⟨Bh ⟩2h /Beqτ

η ∝ f(v B )

~ τ

~τ

彩層・コロナ加熱問題との関わりelectromotive
force (e.g., Ossendrijver et al. 2002). Th★彩層・コロナ加熱とダイナモの繋がり
Hinode/SOT

turbulent pumping, and turbulent magnetic diﬀusivity, re★CME(フレア)とダイナモの繋がり
can be ignored in considering the symmetry of the system
・どうやって磁場を散逸させるか？
time (t)and depth (z).

・どうやって磁気ヘリシティを抜くか？

The MF dynamo described by equation (2) falls intthat the α2 mode can generate a large-scale magnetic fiel
Shukurov 1987; Rädler & Bräuer 1987;
示唆Brandenburg et ais the nonuniformity of the α-eﬀect, which can arise naturin thestellar interior. Using the rigidly rotating system
上部安定層がある場合、ダイナモ生成
which propagates only in the depth direction. However,
磁場は拡散以外に外に抜ける術が無い
system, it can travel also in the latitudinal direction due
the equator.cyc
diff ※安定層には乱流が無い

τ

BMの強度が決まる
磁場生成の鍵 = 自転についても依存性の調査が必要~ τ

ので微視的磁気拡散

強い制限がかかるMF produces a Lorentz force
The dynamo-generated
control the nonlinear evolution and saturation of the syst
the magnetic quenching eﬀect (e.g., Rogachevskii & Klee
we adopt the prototypical models, which are the dynam
対流層の乱流磁気拡散が調節
the catastrophic-type;
!
2
cyc
diff
diff,t∂α2 α⟨Bh ⟩h − η (∇ ×
= −2ηk kc
∂t
Be2
γk
γ =
,
2
quenchingの式の例
1 + ReM ⟨Bh ⟩2h /Beq
ηk
η =
,
t
, M
2
1 + ReM ⟨Bh ⟩2h /Beqτ

η ∝ f(v B )

~ τ

~τ

3. 議論~ 緯度方向の磁場のマイグレーションの起源と現実の太陽への拡張 ~磁気ヘリシティロスの別ルート★ 恒星では表面と赤道面を介した南北半球間の磁気ヘリシティ損失の可能性.
90N

北半球
表面

M

のH
消失

赤道面を介した
南北半球間の磁気
ヘリシティ損失0.2

赤道面

Masada et al. 2013

EQ

M

0.0

0

(v̄r2 + v̄θ2 )1/2 /vrms<Ω̄(r, θ)/Ω0から

H

0.3 0.4 0.5 0.6 0.7 0.8 0.9

North: Hm < 0,South: Hm > 0
1 (e.g., Hagino &Sakurai 2004)

30N

30S
90S
還流
リコネクション

赤道面

H

M

<

0

北半球

西暦

南半球

還流

南半球

単純に彩層・コロナでの磁気ヘリシ
ティ損失だけでは物理は決まらない.

緯度方向への磁場のマイグレーション★テストモデル：磁場の境界条件を変更＊upper ：Open B.C., bottom：Closed B.C.
＊horizontal：Periodic B.C.
境界条件を変更＊upper ：Open B.C., bottom：Closed B.C.
＊ y：Periodic B.C. x : Open & Closed B.Cs.

動径方向に加えて緯度方向のマイグレーション（ただし極向き）

開放境界緯度方向への磁場のマイグレーション• 太陽ではBr & Bφが赤道反対称. 一方で、Bθ は赤道対称 = 赤道が開放境界の役割.

赤道を介した磁気ヘリシティロス・球殻系でのダイナモでは赤道での磁気ヘリシティロス（Mitra+10）.

・グローバルな効果でα効果駆動型のダイナモ波は緯度方向に伝搬する.
0≦x≦4はDNSの結果. -4≦x≦0はその結果のBφの極性を反転させたもの.

Bφ

1.2

2.0

By/BeqLatitude (x)

4.0

0.0
-2.0
-1.2

-4.0
0

100

200

300

400

500

TIME (τ/τcv)

600

700

800

900

1000

より現実的な太陽内部構造モデルへの拡張 (Masada & Sano 2015b)@z = z1@z = z2@z = z3★dbottom/dtop = 700z1(太陽の0.7Rsun-0.991Rsunに相当)
1000

100

z3

ρ

depth

z2

10

longitude

2dcz

完全圧縮性方程式を解いている

対流ダイナモは起こるのか？
-2dcz

近似無し.

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1z (depth)まとめ＊太陽黒点と太陽蝶形図の起源の解明へ向けて● 太陽ダイナモは天体乱流研究のプロトタイプ.
● 2つの基本的ダイナモ効果：α効果とΩ効果.
● Ω 効果の存在しない対流ダイナモ計算.
● 大局的磁場の生成と周期的極性反転を発見.
● 平均場モデルでα効果の存在を定量的に実証.
● 2つのスケーリング則の発見：τcyc ∝ τdiff, τAlfven
35000● ダイナモと彩層・コロナ加熱との繋がりを示唆.

30000● 緯度方向の磁気ヘリシティロスが緯度方向のマイグレーションを生む–5–
彩層・コロナ加熱問題⇄ 太陽ダイナモ問題

depth (z)

0.0
0.4
0.8
1.2
1.6
2.0
0

τcyc ~ τdiff

25000
20000
15000
10000

(a)
200

400

600

800

t/τcv

1000
-0.05

1200
0.05

0.0

depth (z)τcyc [cycle period]● 反転周期は上部安定層の物理と深い関わり.

5000

0.4
0.8

0

1.2
1.6
2.0
0

(b)
1

2

3

4

t/τdiff

5

6
-0.04

0

7
0.04

5000 10000 15000 20000 25000 30000 35000

2dRZ,t W/η0

天体乱流研究への応用① ~降着円盤ダイナモとの関連~
Shi et al. (2010)・α効果によるダイナモは元々は降着円盤で調べられてきた (Baryshnikova & Shukurov 1987).
・赤道面上下で運動学的ヘリシティが逆転.
・赤道面を介した磁気ヘリシティロスが存在.
・Ω機構が無くても、振動型のダイナモ解が得られる.

・降着円盤シミュレーションで見られる蝶形図(上図)も実は磁気浮上起源ではなくダイナモ波パターン？
(see Gressel 2010)

天体乱流研究への応用② ~太陽コロナ加熱と超新星爆発機構との関連~(ˀʸƟ˲ʃ‫ٶ‬වɻʃɾɘ)

密度差が重要ρ (g/cm33))
ρ(g/cm
2 2
10
10
10-1

chromosphere
遷移層（彩層）
Solar
Wind
太陽風
B2 →
ρT
coronaコロナ層

ρv2 → B2

熱伝導鈴木健さん（名大）の資料改変(スケールは適当)

10-10
-23

10

0
0

7
7
10
10

1066
10

10-7

10-16

T (K)
T(K)

6000

6000

optically-thick

optically-thin1
1 1.003
1.003

光球

215 R
Rsun
sun
215
(1AU)
(1AU)

ଊфʂ‫֊ڶ‬ʂࣘɿ‫ۿ‬фʺ˴ˑƅਫ๐೴
・熱源から離れるのに温度は上昇：不思議な状態 → コロナ加熱問題
௯‫(ٺ‬યअ)ɟʝ๸ʠʟʂɿƅф୞ʃࣘࢭ: ɞɟɫɾࣨੁ
・密度の高いところで運動エネ → 磁気エネ
(ダイナモ問題)‘ʺ˴ˑю௯෻੕’【磁場の生成】
・密度の低いところで磁気エネ → 内部エネ (コロナ加熱問題)【磁場の散逸】

天体乱流研究への応用② ~太陽コロナ加熱と超新星爆発機構との関連~

定在衝撃波の背後（黄色部分）にいかにエネルギーを注入できるか⌘ PNS has smaller Ra, Re than those of the Sun:

高密度領域の膨大なエネルギーをいかに低密度領域に落とすか？

(太陽ダイナモよりもPNS dynamoの方が現実的な計算が可能)- Ra = 1010 (Rasun = 1015) (c.f., Thompson & Duncan 1993)- ReM = 1017 (ReMsun = 1013)- Re = 104 (Resun = 1012)⌘ The magnetic field sustainable by the PNS dynamo:
B = (4πρvcv2) ~ 1015 G (ρ/1013 [g/cm3])(vcv/108 [cm/s])2

太陽のコロナ加熱問題, outflow
駆動問題と大きな違いは無いτcv = LMLT /vcv = 104 ~ 105/108 = 0.1 ~ 1 msec
(sufficiently enough time for the convective dynamo)