2022年2⽉9⽇
GFD/CPS セミナー

⾼解像シミュレーションで迫る
太陽⾚道加速の謎
Hotta & Kusano, 2021, Nature Astronomy, 5, 1100
Hotta, Kusano & Shimada, 2022, ApJ, 933, 199,

堀⽥英之(千葉⼤学)
草野完也(名古屋⼤学)、嶌⽥遼太(東京⼤学)
1Solar rotation
Galileo's sunspot
sketch ~1620s(Galileo project)

The sun is rotating. This is first understood with the sunspot tracking.
Around 1630, we also begin to know that the sun is differentially rotating.
2Solar differential rotation

The solar differential rotation isprecisely evaluated by the
helioseismology (error < 1%).
Differential rotation is a fundamental
ingredient of solar dynamo.
We know several interesting features.
A. Fast equator, and slow pole
B. Tachocline
C. Conical profile in the middle CZ
D. Near surface shear layer
Data provided by R. Howe
Result of helioseismology
3Solar interior and convection

Convection zone

Radiation Zone

Energy generated by the nuclear fusionin the solar core is transported by
ü Radiation (inner 70%)
ü Convection (outer 30%)
The turbulent motion of the ionizedplasma can interact with magnetic field.
Anisotropic turbulence and magnetic
field causes the angular momentum
transport and results in the differential
rotation.
The turbulence in the Sun is highly
chaotic and numerical simulation is an
essential approach.
4

Solar convection calculation
L = 3.84 ⇥ 1033 erg s 1
Radiation energy flux

We have precisely evaluated theradiation energy flux from the solar
surface.
The energy flux is imposed at thebottom boundary and extracted fromthe top boundary.

Convection zone
calculation domain

The stratification is also well known by
the solar model confirmed with the
helioseismology.
There is almost no ambiguity and the“correct” calculation should lead to thecorrect flow and the magnetic field.

Radiation energy flux
(well known value)5Equations for the solar convection zone
Magnetohydrodynamics (MHD)

@⇢
@t

=

r · (⇢v)

@
(⇢v)
@t

=

r · (⇢vv)

Lorentz force

@s
⇢T
@t
@B
@t
p
<latexit sha1_base64="ZSr6xrVUPfoDnE9wHLayDckGblc=">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</latexit>=

Mass conservation
Gravityrp + ⇢g

Momentum conservation

1+ (r ⇥ B) ⇥ B + 2⇢v ⇥ ⌦4⇡⇢T (v · r)s + Qrad

Coriolis force

Entropy equationor energy conservation=

Radiationr ⇥ (v ⇥ B)
Magnetic induction equation=p(⇢, s)Equation of state (including ionization)
6

Angular momentum transport
Turbulence can transport the angular momentum. ℒ = #$! is the specificangular momentum and # = % sin ) .
Velocities are divided to mean + and perturbed +′ parts. + = + + +′Then the angular momentum conservation equation is (. is ignored)

There are two contributions of the angular momentum transport transport
by the meridional flow and turbulence (Reynolds stress)
The velocity correlation +"# +$# is important to understand the solar
differential rotation.
7

How to determine the differential rotation
Angular momentum conservation

2/0×2%

Busse, 1970

Miesch+2000

Rotational influence cause equatorward angular momentum transportand suppresses poleward one.

8

Convective conundrum
Typical "high resolution" simulation for the sun fails to
reproduce the solar like differential rotation

Ω = 3Ω⊙Ω = 2Ω⊙Ω = Ω⊙

We probably fail to reproduce the convection and/or the magnetic
field in the convection zone to construct the solar-like differential
rotation.

9

Slow rotation case (1/2)
In a slow rotation case, the convection itself is prominent. The origin of
the angular momentum transport is the radial velocity +&"!" < 0

sl

"
"#
<0

s
fa

"
"#
>0

ow

t

"!" > 0""!" "#<0

Radially inward
angular momentum
transport<latexit sha1_base64="NEWEgFE0xOtXS/4xE7t1DQQL3uQ=">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</latexit>⇢0 hvm i · rhLi =0 0r · ⇢0 hvm
v i

The angular momentum transportcauses anti-clockwise meridional flow.

10

Slow rotation case (2/2)
Important slide
Anti-clockwise meridional flow always transport the AM poleward.
The colatitudinal AM by the meridional flow is:/% +' ℒ , where ℒ = % sin ) $! .
In addition, due to the extremely low Mach number the fluid satisfies theanelastic approximation ∇ ⋅ /% 0( = 0, which lead to ∫ /% +' %7% = 0 at
constant ) surface.
Thus, the net AM transport ∫ /% +' ⟨ℒ⟩%7% isdetermined by the specific AM distribution, here
Since the solar differential rotation is not strongΩ ∼ Ω% and the deeper layer (small %) has
smaller AM ℒ ∼ % ) sin) ) Ω% .
Thus, the anti-clockwise meridional flow MUSTtransports AM poleward.
We need to suppress negative ⟨+&# +!# ⟩ to have thefast equator.
11

Scale dependence of AM flux
Radial angular momentum flux /% % sin ) +&# +!#240 Mm < A

120 Mm < A < 240 mm 60 Mm < A < 120 mm

Smaller scales tend to transport the AM radially inward
Mori & Hotta, 2023

12

Methods to reproduce the solar-like DR
Large Ro → Fast pole
Small Ro → Fast equator
1.

Increase angular velocity Ω%
Brown+2008, Nelson+2013, Hotta, 20182.

Decrease luminosity A⊙ → Decrease convection velocity
Hotta+2015

3.

Increase viscosity and/or thermal conductivity
Miesch+2000, 2006, 2008, Brun+2002, 2004, Fan+2014,
Hotta+2016

Angular velocity Ω% and luminosity A⊙ are precisely observed values
and we should not change these for solar simulations.
Large viscosity/thermal conductivity are not realistic→ One of most important problems in solar physics
13

Fugaku
Top 500 list (Nov, 2021)©RIKEN

Theoretical peak exceeds 500 PFLOPS with boost mode.
48 cores/node 158976 node = 7630848 cores
CPU: A64FX Armv8.2-A SVE (512 bit SIMD)
We use Fugaku to attack the convective conundrum!

14

R2D2 code

Radiation and RSST for Deep Dynamics
Hotta+2019, 2020a,b
ü 4th order accurate derivative (ununiform grid
applicable)
ü 4-step Runge-Kutta
ü Cartesian・Spherical (Yin-Yang,
Kageyama+2002)
ü Non-linear artificial viscosity (Rempel, 2014)
ü Entropy equation for deep convection zone
ü Realistic radiation transfer
ü OPAL EoS, Linear ⇄ Tableü Reduced Speed of Sound Technique (Hotta+2012,2015)
ü Alfven speed suppression (Rempel+2009)
In this study, abilities only for deep CZ is used.

15

Performance of R2D2 on Fugaku

Almost perfect (>99%) weak scaling is checked more than 1M cores.
48 cores/node. We can carry out "large" calculation.
3.7 107 grid update/sec/node is achieved (~3 TFLOPS/node).

16

Numerical setting
Calculation domain:
0.71E⊙ < % < 0.96E⊙
Number of grid points:
Low

96×768×1536

Middle192×1536×3072

High

384×3072×6144

Calculation：4000 days
3.5 Mstep for High case.
Statistically steady flow is obtained
around t=3000 day.
Data is averaged between 36004000 day to show the results.

Vertical velocity at % = 0.9E⊙

No explicit diffusivities with 1Ω⊙ and
1A⊙ .
17

Convection and magnetic field
Normalized entropy
G − G /G+(,

Magnetic field strength
J [kG]

18

Dependence of DR on resolution
Increase resolution
96×768×1536

Sun

192×1536×3072

384×3072×6144

While the fast equator is obtained in the
Low case (consistent with previous studies),
the High case reproduces the solar-like
differential rotation.
19

Convection and magnetic field

Increasing the resolution decreases the convection velocity and amplifies
the magnetic field. The Low case show O-./ > O(01 in all the depth, but theHigh case achieved O-./ < O(01 throughout the convection zone.
Situation is totally different.

20Analyses of calculation
We analyze the huge data (>100 TB in High case), to understandthe mechanism to maintain the solar-like differential rotation.
Key questions are:
1. Why the superequipartition magnetic field (O(01 > O-.2 )?
2. Why the equator is rotating faster?

21

Magnetic field generation (1/2)

The high resolution tends to show
inefficient stretching due to strong
magnetic field (dashed line).
The compression is increased in
High case (dotted line).
The compression is the essential
mechanism to construct thesuperequipartition magnetic field.

22

Magnetic field generation (2/2)
2D histogram between gas pressure perturbation Q3#
and magnetic pressure J ) /8S

The dashed line indicates Q3# = J ) /8S.
In High case, the most of data are on the Q3# = J ) /8S line. This result
indicates that the strong magnetic field is maintained by the gaspressure, that is, we can use the internal energy which is massive inthe Low Mach number situation in the solar convection zone.
Easy to generate the superequipartition field to the kinetic energy.
23

Angular momentum transport

Angular momentum conservation

Mean flow transport

Turbulent transport

Magnetic transport

It was thought that the turbulent correlation "$" "%" has a key role
to maintain the fast equator(e.g. Miesch+2000)

24

Latitudinal angular momentum transport
Turbulent flow
Mean flow
Magnetic field
Angular momentum
flux at ) = 45°<latexit sha1_base64="1jLWAT8WJh1Jk2W+SA9CuVWl8f0=">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</latexit>

Ftur = ⇢0 hv✓0 v 0 i,

Fmer = ⇢0 hv✓ ihLi,

hB✓ B i
Fmag =,
4⇡
Ftot = Ftur + Fmer + Fmag .

Meridional flow is responsible for the fast equator

25

Meridional ﬂow

The poleward flow around the base of the convection zone becomesprominent in high-resolution models.
Since the deep CZ is in a low Mach number situation, ∇ ⋅ /% 0 = 0 is
approximately satisfied. Thus ∫ /% +' %7% ∼ 0 at an arbitrary latitude.
The poleward meridional flow cause the equatorward meridional flowin the middle of the convection zone, which leads to net equatorward
angular momentum transport.
26

Slow rotation case (2/2)
Important slide
Anti-clockwise meridional flow always transport the AM poleward.
The colatitudinal AM by the meridional flow is:/% +' ℒ , where ℒ = % sin ) $! .
In addition, due to the extremely low Mach number the fluid satisfies theanelastic approximation ∇ ⋅ /% 0( = 0, which lead to ∫ /% +' %7% = 0 at
constant ) surface.
Thus, the net AM transport ∫ /% +' ⟨ℒ⟩%7% isdetermined by the specific AM distribution, here
Since the solar differential rotation is not strongΩ ∼ Ω% and the deeper layer (small %) has
smaller AM ℒ ∼ % ) sin) ) Ω% .
Thus, the anti-clockwise meridional flow MUSTtransports AM poleward.
We need to suppress negative ⟨+&# +!# ⟩ to have thefast equator.
27

Meridional flow in quasi-steady state

Miesch+2011

Since the differential rotation is weak and the angular momentum does
not change significantly, the topology of the meridional flow is directly
determined by the Reynolds and the Maxwell stress.
28

Angular momentum transport
by meridional flow<latexit sha1_base64="F7T7PpjqBzPuW/HPCeJkEa8sI5c=">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</latexit>0=(⇢0 hv m i · r) hLir·⇢0 hv 0m v 0 i

1+
hr · ( B m B )i4⇡

Due to the poleward meridional flow around the base of the convection
zone, the angular momentum transport by meridional flow increases
angular momentum there. What compensate it?

29

Turbulent angular momentum transport<latexit sha1_base64="F7T7PpjqBzPuW/HPCeJkEa8sI5c=">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</latexit>0=(⇢0 hv m i · r) hLir·⇢0 hv 0m v 0 i +1hr · ( B m B )i
4⇡

The turbulent angular momentum transport decelerates theequator region more in High case.
The ﬂow does not have a role to construct the poleward
meridional ﬂow at the base30

Magnetic angular momentum transport<latexit sha1_base64="F7T7PpjqBzPuW/HPCeJkEa8sI5c=">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</latexit>0=(⇢0 hv m i · r) hLir·⇢0 hv 0m v 0 i

1+
hr · ( B m B )i4⇡

In High case, the Lorentz force has a dominant role to transport theangular momentum transport to construct the fast equator. We needto investigate the correlation J"# J$# to understand the mechanism.

31

Magnetic field correlation: Radial momentum transport
Low

MiddleHigh

Negative correlation J& J! is the essential reason why wehave the solar-like fast equator. The magnetic tension
transport the angular momentum radially outward.

32

Poleward meridional flow at the base of
CZ

The poleward meridional flow at the base of CZ is caused by the radiallyoutward angular momentum transport by the magnetic field.
33Origin of negative !"! !"# (1/2)
There are two major reason to create magnetic ﬁeld correlation
1. Flow Shear

2.

Magnetic ﬁeld tends to be parallel to ﬂow. 0×. = 034

Origin of negative !"! !"# (2/2)

The magnetic field is actually inclined.

35Additional role of magnetic field

A prominent difference between the Middle and High cases are theangular velocity at the near surface equator.
The radially outward angular momentum transport by the magnetic
field helps construction36

Mechanism for the radially outward
angular momentum transport

Coriolis
force

1.
2.
3.

Magnetic tension

Coriolis force cause negative correlation +& +! , inclined flow.
The magnetic field tends to be parallel to the flow and also inclined.
The negative correlation J& J! , which transports angular momentum
radially outward via the magnetic tension.
37

The whole story
Superequipartition B
generated by compression

Poleward meridional at thebase results in strong
equatorward flow andnet equator ward angularmomentum transport.
Magnetic angular momentum
transport becomes importantCoriolis force"→ "!" "#
<0&! &# < 0Fast equator

38

Numerical convergence: "#$%×'"%%×'"%%20 M time stepG′ [erg K 43 g 43 ]J& [kG]

39

Numerical convergence:Diﬀerential rotation and meridional ﬂow

We hardly see further resolution dependence. Numerical convergence?

40

Summaryü We carry out super-high resolution
simulation for the solar convection zone
(5.4 billion grid points)
ü The solar-like differential rotation isreproduced without using any
manipulation
ü The magnetic energy is much larger
than kinetic energyü The strong magnetic field is maintainedby the internal energyü The angular momentum is transportedby the magnetic fieldü The magnetic correlation is originated by
the Coriolis force.
→ We call this process "Punching balleffect".
41