第6回 DTAシンポジウム「星形成を軸に俯瞰する磁場の役割とその観測的検証」

太陽の大スケールダイナモにおける小スケールダイナモの役割
Hotta et al., 2016, Science, 351, 1427

千葉大学
堀田英之
Hideyuki Hotta1

Magnetic field is frequently coherent
and large-scale
Plasma in the universe
frequently constructs highlevel turbulence with large
Reynolds numbers.
Magnetic field in smallest
scale has the largest growthrate for the small-scale
dynamo.
In addition, large-scalefeather is expected to bedestructed with the turbulent
motionPalmeirim+2013, molecular cloud

How can they construct suchlarge-scale magnetic fields.
Hideyuki Hotta

Fletcher+2011
Galaxy2Sunspot and solar cycle(Courtesy of Okamoto)(SDO/HMI)

Average daily sunspot area (% of visible hemisphere)

Hideyuki Hotta

3

Butterfly diagram

Latitude

11 year solar cycle is filled with ordered rules

ü Sunspot latitude migrates equatorward during a cycle.
ü Large-scale dipole field reverses every 11 years.
Hideyuki Hotta

4

Hale-Nicholsonʼs law
1996-2008

E-++-

2008-

WE+--+W

More than 90% of sunspot pairs obey a coherent polarity rule.
This indicates large-scale magnetic field in the solar interior.
Hideyuki Hotta

5

Convection generates magnetic field
Convection
zone

Radiation
Zone

The solar convection zone, theouter 30% of the solar interior,is filled with the turbulent
convection due to the energy
input from the radiation zone.
The turbulent motion of theionized plasma amplifies themagnetic field → dynamo
We need to understand thedetails of the turbulence andmagnetic field to understandthe 11 year cycle problem.

Hideyuki Hotta

6

Order from Chaos(Hinode/SOT)How can we makelarge-scale featurefrom such smallscales.

(Rempel, 2014, Numerical Calculation)

Hideyuki Hotta7

Solar physics is lucky
L = 3.84 ⇥ 1033 erg s 1

Convection zone
Calculation region

Thanks to detailed solar
observation (mainlyhelioseismology) and the solar
standard model, we know thedetails of the energy flux, andthe stratification (density,pressure, and temperatures) in
the solar convection zone.

Energy flux
(The value is well known)

That means that input
parameters for calculations arewell determined and ideally we
can calculate the turbulence andmagnetic field properly with nicesupercomputers.

Hideyuki Hotta8

Convection can produce global field
5 times larger than solar rotation

The Astrophysical Journal, 731:69 (19pp), 2011 April 10

disrupt the convection at the mid-latitudes. In co
dynamo case D5 the differential rotation is substan
in both radial and latitudinal angular velocity coevident in Figure 3(a), the convective cells fill iquite completely.
In our prior hydrodynamic simulations of cyounger suns we reported on localized nests o(Brown et al. 2008), with those most prominent arotation rates. Though there is some modulation w
Figure 1.in
Magneticwreaths achieved
case D3 (PaperI) and this(b) casethe equatorial
rollin (a)amplitudes
here,modu
D5 (this paper). Shown are time–latitude plots of mean(Brown+2010)
toroidal (longitudinal)
in either caseD5scalingor H5thanin ourmagnetic extreme
field ⟨Bφ ⟩ at mid-convection
zone, withvaluesindicated.
Case preD3 buildsrotating
persistent, time-independent
the wreaths
achieved in casesimulationswreathsof butstellarconvection(e.g.,
D5 undergo quasi-regular reversals of polarity (three shown here, with roughly
Brown
et al.
2008).
difference
appears to bea 1500 dayperiod). The
dynamic
range Thisof the color
bars is indicated.
backgroundstratification
feedbacks
from ther(A color versionof thisfigureis available
in the andonline
journal.)Recent 3D calculations are able to produce
globalscale
magnetic
near the top of the domain. Here, we have attemp
field and cycle (Ghizaru+2010, Brown+2010,
Racine+2011,
regionof ofinfluence
the unresolved
SGS
the twothe
strongwreaths
oppositelyof
directedpolarities havewhichcarries fluxconnectivity
out the topof theKäpylä+2012, Masada+2013, Fan+2014,
Warnecke+2015,
substantial
cross-equatorial(Figure2(a)).domain
This (cross-equatorial
flux appears
play an importantthe2008). Thisthinnertothermalboundaryrole
hasin larger
Karak+2015).
polarityareversals
thatareobserved.
steeper profile of the background entropy gradie
Unlike the wreaths of case D3, here magnetism fills the entire
thus slightly higher Rayleigh numbers and radi
convection zone including the polar regions (Figure 2(b)). On
In our broader
studytheofwreathsrapidlynearrotating
dynamtheir high-latitude(polar) edges,
the equator
found tothat
strongly
localized
activenests ofandconve
are connectedmagneticstructuresof weaker
amplitude
oppositepolarity
atthepolarcaps.
These
polar
structures
areHideyuki Hottapossible in dynamo simulations at the most
9 rapid

High resolution destroys the order
Most solar global dynamo models are carried out in low
resolutions, i.e., small-scale dynamo is inefficient.
Higher resolution kills the large-scale features.
Diffusivity：η=2.6x1012 cm2 s-1

Diffusivity：η=1.2x1012 cm2 s-1(Nelson+2014)

In small-scale dynamos, stretching is the essential process and thesmallest scale has largest linear growth rate.
How can the sun maintain the large-scale field with very smallviscosity(O(1 cm2 s-1)) and magnetic diffusivity(〜O(104 cm2 s-1))?
Hideyuki Hotta

10

1.0

Low resolution
High resolution

0.8
Ekin/E /EkinEmagmag(Hotta+2015)

0.6

0.4

0.2
0.0
0.75

0.80

0.85
x/Rsun0.90

0.95High resolution Low resolution

Magnetic energy/Kinetic energy
vertical v

vertical B

The magnetic energy reaches more than 90% (0.95Beq) of kinetic
energy at the convection zone in High resolution, while low resolution
achieved in other studies can maintain 5% of kinetic energy.
Hideyuki Hotta

11Comparison with MURaM

Beq, BRMS [kG]2015

xmax=0.99Rsun Δx=700 km
xmax=0.96Rsun Δx=700 km
MURaM Δx=32 km (Rempel, 2014)

Beq

105Brms

00.75

0.80

0.85r/RO•

0.90

0.95

1.00

Vx and Bx from xmax=0.99Rsun

MURaM calculation is nicely consistent with the solar
observation (e.g., Hinode). At the bottom of MuRAM calculation(x=0.97Rsun), Brms=3300 G(MuRAM) and 2500 G(Ours). Although
our calculation can be influenced by lower top boundary andresolution. The agreement is good, i.e., we can assume solar
convection zone filled with SSD!
Hideyuki Hotta12

Summary of introduction
In global calculations, higher resolution tends to show lower
energy of mean (global scale) magnetic field. Small-scale
dynamo should be suppressed.
Photospheric magnetic field requires efficient (almostequipartition) small-scale dynamo deep in the convection zone
in order to be consistent with the solar obsevation.
Large-scale dynamo

Theoretically (in numerical calculation), they cannot coexist.
Both are necessary to explain observations.
Small-scale dynamo
Hideyuki Hotta13

New method for increasing resolutionr · (⇢0 v) = 0@⇢1=@t! ⇢ = ⇢0 + ⇢1r · (⇢v)
2⇠

In order to avoid severe constraint by fast sound wave in thesolar convection zone, the anelastic approximation (infinite
speed of sound), is broadly used. But, frequent global
communications in parallel computer is needed and scaling islimited.
We develop reduced speed of sound technique (RSST), in which
the speed of sound is reduced (moderate speed of sound andeasily scales in massive super computer.)
The validity is confirmed in Hotta+2012 (see also Hotta+2015)
Hideyuki Hotta

14

Achievement
192x128x128 grid/node (weak)

Performance to the peak(%)

1012

cell updates/sec

96x64x64 grid/node (weak)
96x64x64 grid/node

10

11@82944 node (strong)

1010

109
103

4510
10
No. of Cores

1030

20

10

60
103

104
105
No. of Cores106

Our code is super efficient. Scales up to 663,552 core (full Kcomputer) and shows about 24% efficiency to the theoretical peak!
Hideyuki Hotta

15

Achievement

Hotta+2014

Miesch+2008
Hideyuki Hotta

16

Further high resolution

Low diffusivity

We check if the tendency that the high-resolution kills large-scale
magnetic field continues even in extremely high resolutions?
Do we miss some important physics?
Cases

Low̲DNr Nθ Nφη, ν [cm2 s-1] Note

64x192x384

Medium 64x192x384
High

256x768x1536

1x1012

Fan+2014

N/A
N/A

Initially we put 100 G antisymmetric toroidal magnetic field andvery small perturbation on entropy. Then calculate 50 years.
The case name “D” has explicit viscosity and magnetic diffusivity,
without “D”, only numerical diffusivity exists.
Hideyuki Hotta18

vrBr

Resolution comparison
Low̲D
ν=κ=1 1012 cm2 s-1

Low diffusivity

Medium64x192x384

High
256x768x1536

vr at
0.95R
Bφ at
0.72R
Coherent magnetic field is generated in Low̲D (Fan+2014). Wheneffective resolution is increased (Medium), large-scale magnetic
field seems disappeared. In further high resolution (High), the largescale magnetic field seems generated again. We need to seebutterfly diagram to check the tendency properly.
Hideyuki Hotta20

Large-scale magnetic field and cycle<Emag>: mean magnetic energy<Bφ> at r=0.72Rsun

Less diffusiveLow̲D
ν=η=1 1012 cm2 s-1<Emag>=2.9 104 erg cm-3

Medium64x192x384<Emag>=1.3 104 erg cm-3

High
256x768x1536<Emag>=2.4 104 erg cm-3

In the highest resolution, coherent large-scale magnetic field isgenerated even in high Reynolds numbers.
Hideyuki Hotta21

Why high resolution creates large-scale?
In the highest resolution, the smallscale dynamo is very efficient andsmall-scale magnetic energy
exceeds small-scale kinetic energy.

Ekin and Emag [erg cm-3]

107
106
105
104
103
1

As a result, the Lorentz force
suppresses the small-scale
turbulent motion which originallydestroyed large-scale feature.

MediumHigh
10100
spherical harmonic degree l

Solid：Kinetic energy
Dotted：Magnetic energy

Therefore, the large-scale
magnetic field can be maintainedeven in high resolutions.

Hideyuki Hotta

22Summary

Previous calculations indicate that high resolution, i.e., lowdiffusivities, reduces the large-scale magnetic energy.
(Nelson+2013)
In these calculations small-scale Lorentz feedback is not effective,
due to low resolution.
3D high-resolution MHD calculation shows a recovery of largescale magnetic field, with suppressing the small-scale
turbulence by small-scale Lorentz feedback.

Hideyuki Hotta23