地球流体力学（GFD）
オンラインセミナー
2021年11月16日

太陽対流層内部のロスビー波と傾圧不安定波
Rossby waves and baroclinic waves
in the Sunʼs convection zone

戸次宥人（マックス・プランク太陽系研究所）
共同研究者 : Robert Cameron, Laurent Gizon, 他

Interior of the Sun and Helioseismology•

The Sun consists of radia%on zone (stable / inner 70%) and convec%on zone (unstable / outer 30%)•

Historically, acous%c (p) modes and gravity (g) modes have been extensively used to probe the interior•

Recently, lots of iner%al modes have been newly observed on the Sun, including Rossby waves•

They are expected to be used as an alternaCve tool to further probe the interior[Gizon et al. 2021, A&A Letters]© ESA

2021. 11. 16

GFD Seminar

IntroducCon (1/6)

What are Rossby waves?
•

Rossby waves are global-scale vorCcity waves in a rotaCng ﬂuid that are rooted in the conservaCon
law of potenCal vorCcity Π•Rossby waves can be classiﬁed based on the so-called β-eﬀects (restoring “force” of Rossby waves)•

A propagation direction of Rossby waves is determined by thesign of β-effect (regardless of its physical origin)
vor$city equa$on

2021. 11. 16

propaga$on direct

GFD Seminar

Introduction (2/6)

Classiﬁca>on of Rossby waves in the Sun(tradi'onal) Rossby wave
(r-modes)thermal Rossby wave

topographic Rossby wave

𝜆
ℎ$

𝜕𝜁!
≈ 𝛽! 𝑣"𝜕𝑡2Ω# sin 𝜃
𝛽! =
𝑟

•
•
•

(planetary)β-eﬀect

equatorial mode
retrograde propagaCon
non-convecCng[Rossby 1939,1940, Saio 1982]

2021. 11. 16

𝜕𝜁$≈ 𝛽% 𝑣&𝜕𝑡2Ω# sin 𝜃
𝛽% = −
𝐻'

•
•
•

𝜕𝜁$
≈ 𝛽( 𝑣&𝜕𝑡𝑑lnℎ$𝛽( = 2Ω#𝑑𝜆compressional
β-eﬀect

equatorial mode
prograde propagation
convectively-driven•••

topographic
β-eﬀect

high-laCtude mode
retrograde propagaCon
non-convecCng[e.g., Busse 1970, Miesch et al., 2008]

GFD Seminar

Introduction (3/6)

I. Tradi>onal Rossby waves (r-modes)• Near the equator, Rossby waves (r-modes) have been robustlyobserved at 3 ≤ 𝑚 ≤ 15
• They contribute a signiﬁcant fracCon of the large-scale velocity power at low laCtudes• The observed ridge can be well approximated by the sectoral mode (𝒍 = 𝒎) r-modes’ dispersion rela%on• On the other hand, the observed eigenfunc%ons show a signiﬁcant non-sectoral contribu%on

Radial vorticity power spectrum
La$tudinal eigenfunc$on of radial vor$city
sectoral mode (𝑙 = 𝑚)

[LöpPen et al. 2018]2021. 11. 16

sign changes at mid latitudes

GFD Seminar[Proxauf et al. 2020]

IntroducCon (4/6)

II. Thermal Rossby waves (columnar convec>ve modes)• Thermal Rossby waves have been reported many times in numerical simulations as north-south aligned
downflow lanes across the equator (also called Banana cells)• They are regarded as the most efficient form of convection outside the tangential cylinder• The origin of solar differential rotation is often attributed to thermal Rossby waves• However, they have NEVER been successfully detected on the Sun
Banana cell[Busse 2002][Miesch et al. 2006][Miesch et al. 2008]

2021. 11. 16

GFD Seminar

IntroducCon (5/6)

III. Topographic Rossby waves (high-la>tude modes)• Topographic Rossby waves have never been discussed in the solar context in previous literature• ObservaCons show the m=1 ﬂow feature at high-la%tudes that spirals towards the poles• Some people argue that this high-laCtude ﬂow represents the deep-seated giant cell convecCon• We will argue that the high-la%tude observa%ons can be explained in terms of topographic Rossby waves(and baroclinic instability)
Observational snapshot of the non-axisymmetric 𝑣$[Data provided by Hathaway & Upton., 2013, 2020]2021. 11. 16

GFD Seminar

IntroducCon (6/6)

Outline of this talk
• Linear analysis of Rossby waves in the Sun-

Dispersion relation and eigenfunctions of equatorial modes
Discovery of the ”mixed” Rossby modes between r-modes and thermal Rossby modes• Nonlinear rota9ng convec9on simula9on-

Eigen-modes extracted from singular-value-decomposiCon
InteracCon between turbulent convecCon and Rossby waves• Baroclinic origin of the high-la9tude ﬂows in the Sun-

Topographic Rossby waves become baroclinically unstable
Physical origin of the high-laCtude ﬂow spiralEﬀects of magneCc ﬁeld and solar dynamo

2021. 11. 16

GFD Seminar

OutlineOutline of this talk
• Linear analysis of Rossby waves in the Sun-

Dispersion relaCon and eigenfuncCons of equatorial modes
Discovery of the ”mixed” Rossby modes between r-modes and thermal Rossby modes• Nonlinear rota9ng convec9on simula9on-

Eigen-modes extracted from singular-value-decomposition
Interaction between turbulent convection and Rossby waves• Baroclinic origin of the high-latitude flows in the Sun-

Topographic Rossby waves become baroclinically unstable
Physical origin of the high-laCtude ﬂow spiralEﬀects of magneCc ﬁeld and solar dynamo

2021. 11. 16

GFD Seminar

OutlineLinear Model : Eigenmode Solver•

The system equaCons are linearized and transformed to a pseudo-eigenvalue problem[Linearized system equa1ons]

5𝑁! 𝑁"

𝜕
⇒ −𝑖𝜔
𝜕𝑡
𝜕
⇒ 𝑖𝑚
𝜕𝜙

𝜔
@s1
=
@t

v✓ @s0
r @✓

⌦1

@s1
1
+
r · (⇢0 T0 rs1 ).
@⇢0 T 0

𝑁! 𝑁"

𝜌*

𝜌*

𝑣!

𝑣!

𝑣"

=

"
𝑴(𝑚,
Ω% )

𝑣"

𝑣)

𝑣)

𝑠*

𝑠*
5𝑁! 𝑁"•

SpaCal derivaCves are evaluated with 4th-order central diﬀerence method•

Boundary condiCons (closed and stress-free) are incorporated in the diﬀerenCal operator•

The eigenvalue equaCon is solved numerically and focus on the low-frequency modes (|ℜ 𝜔 | < 2Ω# )•The solar model S is used for the background straCﬁcaCon•

We begin our analysis for inviscid, uniformly-rota%ng Sun•

Then, add the turbulent diﬀusiviCes, diﬀerenCal rotaCon, and the background entropy variaCon
Bekki et al. 2021b (to be submi3ed)

2021. 11. 16

GFD Seminar

Model : LinearLinear Results : r-modes with no radial nodes (n=0)

2021. 11. 16

GFD Seminar

Results : Linear (1/8)

Linear Results : r-modes with no radial nodes (n=0)

2021. 11. 16

GFD Seminar

Results : Linear (1/8)

Linear Results : Thermal Rossby waves (𝜻𝒛 symm)

2021. 11. 16

GFD Seminar

Results : Linear (2/8)

Linear Results : Topographic Rossby waves (𝜻𝒛 symm)[Glatzmaier & Gilman 1981]

2021. 11. 16

GFD Seminar

Results : Linear (3/8)

Linear Results : Topographic Rossby waves (𝜻𝒛 an>-symm)

2021. 11. 16

GFD Seminar

Results : Linear (4/8)

Linear Results : “Mixed” Rossby modes•

We find that the r-modes with one radial node (n=1) and the thermal Rossby waves with north-south antisymmetric 𝜻𝒛 are mixed with one another, forming the “mixed” Rossby modes•

This mode coupling can be understood in analogous to “Yanai” waves (mixed Rossby gravity waves)r-modes(n=1)

thermal Rossby(an$-symm)

2021. 11. 16

GFD Seminar

Results : Linear (5/8)

Linear Results : “Mixed” Rossby modes•

We ﬁnd that the r-modes with one radial node (n=1) and the thermal Rossby waves with north-south an%symmetric 𝜻𝒛 are mixed with one another, forming the “mixed” Rossby modes•

This mode coupling can be understood in analogous to “Yanai” waves (mixed Rossby gravity waves)r-modes(n=1)

thermal Rossby(an$-symm)

2021. 11. 16

GFD Seminar

Results : Linear (5/8)

Linear Results : “Mixed” Rossby modes•

We ﬁnd that the r-modes with one radial node (n=1) and the thermal Rossby waves with north-south an%symmetric 𝜻𝒛 are mixed with one another, forming the “mixed” Rossby modes•

This mode coupling can be understood in analogous to “Yanai” waves (mixed Rossby gravity waves)r-modes(n=1)

thermal Rossby(anti-symm)

2021. 11. 16

GFD Seminar

Results : Linear (5/8)

Eﬀect of Turbulent Diﬀusivi>es in the Sun on r-modes•

When moderate turbulent diﬀusion is added, the n=0 r-modes are more and more shi`ed downwards,
leading to a signiﬁcant deviaCon from the well-known 𝑟𝑚 dependence•This is likely due to the imbalance of the radial forces and downward diﬀusive momentum ﬂux•

Consequently, the n=0 modes peak at middle laCtudes at the surface in contrast to observaCons•

On the other hand, the n=1 modes always peak at the surface and at the equator
uniform viscosity in radius (! = const) / solar stratification(b) radial eigenfunction𝑛 = 0 eigenfunc$ons for diﬀerent turbulent diﬀusivi$es(a) meridional eigenfunction! = 10&! = 10!

! = 10"#! = 10""! = 10"$! = 10"%ℜ[#! ]ℑ[#! ]

2021. 11. 16

GFD Seminar

Results : Linear (6/8)

Eﬀect of Solar Diﬀeren>al Rota>on on r-modes•

Inclusion of differential rotation introduces viscous critical layers where the r-modes’ phase speed become
equal to the local differential rotation speed•

In the vicinity of the critical layers, strong radial flows are driven and substantial horizontal Reynolds stressare generated, leading to an equatorward angular momentum transport

𝑣#

𝑣$

𝑣%

Horizontal Reynolds stresses

r-modes(n=0)

cri8cal la8tudes

r-modes(n=1)

2021. 11. 16

GFD Seminar

Results : Linear (7/8)

Are the observed Equatorial Rossby modes n=0 or n=1?
•

In terms of the dispersion relaCon, the observed frequencies lie
between those of n=0 and n=1 modes. So, hard to disCnguish•

But in terms of the surface eigenfuncCons, the n=1 modes cangive becer explanaCons for the observaCons-

prominent peak at the equator-

sign flip at middle latitudes
surface eigenfunc$on for (leF) n=0 and (right) n=1 r-modes

observa$on[Proxauf et al. 2020]2021. 11. 16

GFD Seminar

Results : Linear (8/8)

Are the observed Equatorial Rossby modes n=0 or n=1?
•

In terms of the dispersion relation, the observed frequencies lie
between those of n=0 and n=1 modes. So, hard to distinguish•

But in terms of the surface eigenfunctions, the n=1 modes can

Wefor
argue
that the equatorial Rossby modes
give better explanations
the observations-

prominent peak atobserved
the equatoron the Sun are likely n=1 modes,-

sign ﬂip at middle la$tudes

rather than n=0 modes as normally assumed

surface eigenfunc$on for (leF) n=0 and (right) n=1 r-modes

This can have substan9al implica9ons

observa$on

because the solar Rossby waves can be
par9ally convec9ve in nature[Proxauf et al. 2020]2021. 11. 16

GFD Seminar

Results : Linear (8/8)

Rossby Waves in a Nonlinear Rotating Convection Simulation
How do these modes (such as newly-discovered “mixed” Rossby modes) behave in the nonlinear regime?

•

Full-spherical convec$on simula$on with solar-like stra$ﬁca$on from 0.71𝑅⊙ < 𝑟 < 0.96𝑅⊙•

Rota$ng at the solar rota$on rate Ω⊙ /2𝜋 = 431nHz but the luminosity is decreased by a factor of 20 to achievea solar-like diﬀeren$al rota$on (equator accelera$on)[Basic equaPons]•

Total 15 year $me series of data with a 5 day cadence is analyzed•

We perform a singular-value decomposi2on on the power spectra to ﬁlter out the Rossby modes
and to extract the eigenfunc$ons (both real and imaginary parts)

2021. 11. 16

GFD Seminar

Bekki et al. 2021a (to be submi3ed)

Model : Nonlinear

Nonlinear Simula>on : Thermal Rossby waves• In our convecCon simulaCon, thermal Rossby waves are found to be the most dominant modesin the horizontal velocity power spectrum (~10* Cmes stronger than the r-modes)• They transport signiﬁcant amount of enthalpy and angular momentum

2021. 11. 16

GFD Seminar

Results : Nonlinear (1/4)

Nonlinear Simula>on : r-modes (𝒏 = 𝟎)•

A well-deﬁned power ridge can be seen in the sectoral mode Rossby wave dispersion relaCon•

At low-m, r-mode exists globally in radius (radial node n=0)•

At higher-m, the mode tends to be strongly conﬁned near the base of the convecCon zone

2021. 11. 16

GFD Seminar

Results : Nonlinear (2/4)

Nonlinear Simula>on : r-modes (𝒏 = 𝟎)•

A well-deﬁned power ridge can be seen in the sectoral mode Rossby wave dispersion relaCon•

At low-m, r-mode exists globally in radius (radial node n=0)•

At higher-m, the mode tends to be strongly conﬁned near the base of the convecCon zone𝑙 = 𝑚 power spectrum of 𝑣" at 𝑚 = 162021. 11. 16

Radial eigenfunctions of
𝑛 = 0 r-modes at the equator

GFD Seminar

Results : Nonlinear (2/4)

”Mixed” Rossby modes found in the simula>on•

We ﬁnd two disCnct oppositely-propagaCng modes either in the 𝑙 = 𝑚 power spectrum of 𝑣+ or 𝑙 = 𝑚 + 1
spectrum of ∇ 2 𝑣, near the surface•(A) retrograde and (B) prograde mode form a conCnuous power ridge across 𝑚 = 02021. 11. 16

GFD Seminar

Results : Nonlinear (3/4)

”Mixed” Rossby modes found in the simula>on(A) The retrograde modes are idenCﬁed as r-modes with the radial node 𝑛 = 1(B) The prograde modes are identified as north-south anti-symmetric thermal Rossby wave
Mixed Rossby modes between r-modes and thermal Rossby waves robustly exist in therota%ng convec%on simula%ons
Prograde mode / an3-symmetric thermal Rossby wave

power spectrum at the surface(B) prograde mode

Retrograde mode / r-mode with one radial node (n=1)(A) retrograde mode

2021. 11. 16

GFD Seminar

Results : Nonlinear (3/4)

Nonlinear Simula>on : Topographic Rossby waves• At high latitudes, topographic Rossby waves are found to exist inside the tangential cylinder• Predominantly exist at m=1 and propagate in a retrograde direction• However, the observed spiral pattern is not reproduced
likely due to the lack of strong diﬀerenCal rotaCon and the baroclinicity (laCtudinal entropy
variaCon) in the simulaCon (to be discussed later)
power spectrum of 𝑣" at 𝑚 = 1

EigenfuncPon of 𝑚 = 1

𝑟/𝑅⨀ = 0.75

2021. 11. 16

GFD Seminar

Results : Nonlinear (4/4)

Outline of this talk
• Linear analysis of Rossby waves in the Sun-

Dispersion relaCon and eigenfuncCons of equatorial modes
Discovery of the ”mixed” Rossby modes between r-modes and thermal Rossby modes• Nonlinear rota9ng convec9on simula9on-

Eigen-modes extracted from singular-value-decomposition
Interaction between turbulent convection and Rossby waves• Baroclinic origin of the high-latitude flows in the Sun-

Topographic Rossby waves become baroclinically unstable
Physical origin of the high-laCtude ﬂow spiralEﬀects of magneCc ﬁeld and solar dynamo

2021. 11. 16

GFD Seminar

OutlineWhat is baroclinic instability?
• Baroclinic instability is a hydrodynamic instability in a stratified, rotating fluid• Accompanied by horizontal temperature gradient (instability of the thermal wind)• Ubiquitous in Earth’s atmosphere, and controls the weather at middle-high latitudes [Vallis 2006]• Leads to a formation of large-scale vortices

isobaric surfaces
isosteric surfaces≠(𝑝 = const.)(𝜌 = const.)

B
𝒈

C

∇𝑝

stableA

unstable∇𝜌©NASA Scieniﬁc visualizaPon studio∇𝑇
2021. 11. 16

GFD Seminar

IntroducCon II (1/2)

Thermal wind balance in the Sun
Ω/2𝜋•

𝜕𝑠# /𝜕𝜃

DiﬀerenCal rotaCon in the Sun is known to be baroclinic(does not follow the Taylor-Proudman’s constraint)•

Balanced by la%tudinal entropy (temperature) gradient•

The required temperature diﬀerence between pole-equatoris ≲ 10 K [Rempel 2005, Miesch et al. 2006]•[data from Larson & Schou 2018]

Historically, it has long been believed that baroclinic instability is strongly suppressed when thestratification is convectively-unstable [Knobloch & Spruit 1982,1984]•

However, recent numerical studies indicate that baroclinic instability can occur even in the presence of
convec%on [Callies & Farrari 2018]2021. 11. 16

GFD Seminar

IntroducCon II (2/2)

Linear analysis : Baroclinically-unstable modes
𝑚=1•

Without laCtudinal entropy gradient 𝜕𝑠//𝜕𝜃 , 𝒎 = 𝟏 topographic

unstable

Rossby mode is stable and does not show a spiralling pacern•

With Increasing 𝝏𝒔𝟎 /𝝏𝜽, the mode becomes unstable (growing) evenwhen the background is convecCvely stable•

The baroclinically-unstable mode show a spiralling paSern around thepoles similar to the observaCons•

stableThe dispersion rela%on agrees well with the observaCons

without 𝜕𝑠# /𝜕𝜃 (non-baroclinic)

with 𝜕𝑠# /𝜕𝜃 (baroclinic)

𝑣1

observation

𝑣1
𝑣1
Linear model at 𝑚 = 1

2021. 11. 16[Gizon et al. 2021]

GFD Seminar

Results II : LinearLinear analysis : Baroclinically-unstable modes•

Without laCtudinal entropy gradient 𝜕𝑠//𝜕𝜃 , 𝒎 = 𝟏 topographic
Rossby mode is stable and does not show a spiralling pacern•

With Increasing 𝝏𝒔𝟎 /𝝏𝜽, the mode becomes unstable (growing) evenwhen the background is convecCvely stable•

The baroclinically-unstable mode show a spiralling paSern around thepoles similar to the observaCons•

The dispersion rela%on agrees well with the observaCons

𝒎=𝟏

𝑣1(−90.1 nHz, 14.1 nHz)

2021. 11. 16

𝒎=𝟐

𝒎=𝟑

𝒎=𝟒

𝒎=𝟓

𝑣1

𝑣1

𝑣1

𝑣1

(−169.9 nHz, 23.3 nHz)(−243.1 nHz, 24.8 nHz)(−313.1 nHz, 29.2 nHz)(−381.7 nHz, 33.2 nHz)

GFD Seminar

Results II : Linear

Nonlinear Model : Full-spherical HD Mean-ﬁeld Simula>ons
mass:

Case 1Case 2

Case 3

𝜹𝟎 = −𝟏×𝟏𝟎%𝟓

𝜹𝟎 = −𝟐×𝟏𝟎%𝟓

𝜹𝟎 = −𝟒×𝟏𝟎%𝟓

more subadiabatic

motion:entropy:•

Small-scale convecPon is NOT solved (mean-ﬁeld)•

Λ-eﬀect is parameterized [Kitchatinov & Rudiger 1995]•

Weakly subadiabaPc layer is included near the base CZ
to achieve the thermal wind balance [Rempel 2005]•

We vary the subadiabaPcity in the lower CZ (𝜹𝟎 ) that
controls the baroclinicity of the system

2021. 11. 16

more baroclinic

GFD Seminar

Model II : Nonlinear

Nonlinear Results : Temporal evolution𝑣' (𝑚 ≠ 0) at the surface
H /2𝜋
Ω

𝑠(̅

Case 1𝜹𝟎 = −𝟏×𝟏𝟎%𝟓Diﬀeren1al rota1on

Case 2
𝜹𝟎 = −𝟐×𝟏𝟎%𝟓

m=1 baroclinic mode

Case 3
𝜹𝟎 = −𝟒×𝟏𝟎%𝟓2021. 11. 16

GFD Seminar

Results II : Nonlinear

Nonlinear Results : Temporal evolution𝑣' (𝑚 ≠ 0) at the surface
H /2𝜋Ω

Subcri'cal…

Case 1𝑠(̅•baroclinicity too weak•baroclinic modes are stable𝜹𝟎 = −𝟏×𝟏𝟎%𝟓𝑣'

Case 2
𝜹𝟎

Subcri'cal…

•

Moderate baroclinicity•

𝒎 = 𝟏 mode becomes unstable and develops•

Not strong enough to aﬀect the mean (𝑚 = 0)

= −𝟐×𝟏𝟎%𝟓𝑠(̅𝒎R𝟏
ﬂows (𝑬𝒎R𝟎𝐤𝐢𝐧 ≫ 𝑬𝐤𝐢𝐧 )Case 3
𝜹𝟎 = −𝟒×𝟏𝟎%𝟓

H /2𝜋
Ω

Supercri'cal!

Diﬀeren1al rota1on
m=1 baroclinic mode•

Strong baroclinicity

•

VR/
VRW
𝒎 = 𝟏 mode develops very fast (𝐸STU≳ 𝐸STU)•

Baroclinic modes transport sufficient angularmomentum and heat to affect the mean state2021. 11. 16

GFD Seminar

Results II : Nonlinear

Comparison : Observation vs. Simulation
Observa9on (SG tracking)

Simula9on (Case 3)

Data provided by Hathaway & Upton (2020)2021. 11. 16

GFD Seminar

Results II : Nonlinear

Nonlinear Results II : Power spectrum and eigenfunc>ons•

The power ridge agree well with the dispersion relaCon of the observed high-laCtude inerCal modes•

EigenfuncCons are extracted using the singular-value decomposiCon (SVD)•

The eigenfuncCons show a spiraling pacern similar to the observaCons•

Baroclinic modes transport both angular momentum and heat equatorward
Power spectrum near the surface
averaged over high-latitudes

Extracted eigenfuncPons at 𝑚 = 1

horizontal
Reynolds stresslatitudinalenthalpy flux

2021. 11. 16

GFD Seminar

Results II : Nonlinear

Nonlinear Results II : Power spectrum and eigenfunc>ons•

The power ridge agree well with the dispersion relation of the observed high-latitude inertial modes•

Eigenfunctions are extracted using the singular-value decomposition (SVD)•

The eigenfunctions show a spiraling pattern similar to the observations•

Baroclinic modes transport both angular momentum and heat equatorward
Power spectrum
near the surface
Baroclinic
instability canExtracted
be considereda mechanism
eigenfuncPons
at 𝑚 = 1 that modulates
Reynolds stressaveraged over high-laPtudes

the amplitude of the baroclinicity in the convec'on zone, and thus,
controls the amplitude of the diﬀeren'al rota'on in the Sun

Enthalpy ﬂux

2021. 11. 16

GFD Seminar

Results II : Nonlinear

The role of magne>c ﬁelds : Mo>va>on & Model•

ObservaCons imply that the amplitude of high-laCtude inerCal modes change over the solar cycle

Courtesy of Z.C.Liang•

Our mean-ﬁeld model is extended to MHD regime•

We add a strong toroidal ﬁeld as an iniCal condiCon into

entropy:

the HD simulaCon Case 3•

mo-on:

Babcock-Leighton source is switched oﬀ (decaying)

induc-on:Bekki et al. 2021c (to be submitted)

2021. 11. 16

GFD Seminar

MHD : ModelMHD Results : Suppression eﬀects by magne>c ﬁelds•

We numerically demonstrate that magneCc ﬁeld has a suppressing eﬀect for baroclinic instability [Gilman 2017]-2021. 11. 16

when 𝑩 is added (strong), baroclinic modes are suppressed
when 𝑩 decays away, baroclinic modes become prominent

GFD Seminar∆Ω and ∆𝑠% are enhanced∆Ω and ∆𝑠% are suppressed

MHD : Results (1/2)

MHD Results : Cyclic Dynamo Simulation•

Babcock-Leighton source term is now switched on (dynamo becomes cyclic)•

Amplitude of the high-la%tude baroclinic modes oscillates along with the magne%c cycle

2021. 11. 16

GFD Seminar

MHD : Results (2/2)

MHD Results : Cyclic Dynamo Simula>on•

Babcock-Leighton source term is now switched on (dynamo becomes cyclic)•

Amplitude of the high-la%tude baroclinic modes oscillates along with the magne%c cycle•

High-laCtude baroclinic modes are strong (weak) during the rise (fall) of the magneCc acCvity cycle•

Accordingly, the horizontal Reynolds stress and la%tudinal heat ﬂux are modulated with the ac%vity cycle

High-latitude power spectrum at 𝑚 = 1

Horizontal Reynolds stress

LaPtudinal heat ﬂux

2021. 11. 16

GFD Seminar

MHD : Results (2/2)

Summary• Recent interesting observations of inertial modes in the Sun [Gizon et al., 2021]-

Robust observaCons of the equatorial Rossby waves (r-modes) and high-laCtude ﬂow spirals
Thermal Rossby waves (that are repeatedly predicted in simulaCons) are NOT observed on the Sun• Mode coupling between r-modes and thermal Rossby waves-

Found both in the linear analysis and nonlinear convecCon simulaCons
Owing to this mode mixing, r-modes can parCally transport the energy and angular momentum• Baroclinic instability in the Sun-

Quite ubiquitous in the Earth’s atmosphere. Likely occur in the Sun’s convection zone as well
The observed high-latitude flow properties can be well explained by baroclinic Rossby waves
Close interaction with the solar dynamo-generated magnetic fields

•
•
•
•
•

Bekki et al., “Solar equatorial Rossby modes in a rotating convection simulation” (to be submitted to A&A)
Bekki et al., “Linear model of global-scale inertial modes in the Sun’s convection zone” (to be submitted to A&A)
Bekki and Cameron., “3D MHD mean-field simulation of solar Babcock-Leighton dynamo” (to be submitted to A&A Letters)
Bekki et al., “Baroclinic origin of the Sun’s m=1 high-latitude inertial mode” (to be submitted to A&A)
Bekki et al., “Solar cycle dependence of baroclinically-driven high-latitude inertial modes” (in prep)

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GFD Seminar

SummarySupplementaryPoten>al vor>city and β-eﬀects•

Ignoring dissipaCon, Lorentz force, and radiaCve heaCng, the potenCal vorCcity Π is materially conserved
with

•

where

is a fluid vorticity

If the spherical surface is isentropic to a good approximaCon,

In the Eulerian form,
where•is the planetary β-effectIf the xy-plane can be approximated to be isentropic,

In the Eulerian form,
where
and
2021. 11. 16

is the compressional β-eﬀect

Here, we used the integrated equaCon of
conCnuityis the topographic β-eﬀect
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Supplementary

Compressional β-eﬀect vs. Topographic β-eﬀect
For the solar internal model S (Jorgen Christensen-Dalsgaard 1996)

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SupplementaryDispersion rela>ons of the Major Rossby waves in the Sun

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SupplementaryAngular momentum transport by r-modes

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Supplementary

Eﬀects of Turbulent Diﬀusivi>es on r-modes
Typical oscillaCon period of Rossby waves at azimuthal order m is given by

Typical diﬀusive Cme scale is
For a given diﬀusivity, the criCcal azimuthal order can be deﬁned

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GFD Seminar

Supplementary

Thermal Rossby waves become convectively-unstable•

Thermal Rossby waves become unstable when the background straCﬁcaCon is changed towards
superadiabaCc (𝛿 > 0)•

PropagaCon frequencies decrease (increase) when 𝛿 > 0 (𝛿 < 0)2021. 11. 16

GFD Seminar

Supplementary

Transport proper>es by thermal Rossby waves•

PropagaCon frequencies decrease (increase)when 𝛿 > 0 (𝛿 < 0)depends on the way it couples with g-modes(addiConal restoring force)•

When the background is superadiabatic (𝛿 > 0), thermal Rossby waves transport heat and angularmomentum upwards, and equatorwards

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GFD Seminar

SupplementaryPower spectra comparison : Observa>on vs. Simula>on

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GFD Seminar

Supplementary

Singular Value Decomposition
Let 𝑞(𝑟, 𝜃, 𝜙; 𝑡) be a variable that characterizes the mode of interest.
FFT in Cme and longitude, we have 𝑞(𝑟,S 𝜃; 𝑚; 𝜔)At fixed m, we compute the equatorial spectrum 𝑞SXY (𝑟; 𝜔)
Depending on the mode of interest, we may limit (𝑟, 𝜃) domain to be focusedso that the prominent power peak exists inside the domain.
Applying SVD, the spectrum is decomposed as

Only keeping the 1st singular value 𝜎/ gives a desired
decomposiCon of
into one radii 𝑈/ and one
frequency funcCon 𝑉/
EigenfuncCons of an arbitrary variable 𝜓 are calculated as,

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GFD Seminar

Supplementary

Numerical methods : Babcock-Leighton dynamo code I
mass:motion:induc-on:entropy:@⇢1
1=
r · (⇢0 v),
@t
⇠2
@v
rp1
⇢1
1
1
= v · rv
ger + 2v ⇥ ⌦0 +(r ⇥ B) ⇥ B + r · ⇧,
@t
⇢0
⇢0
4⇡⇢0
⇢0
@B
= r ⇥ (v ⇥ B + E ⌘r ⇥ B),
@t
i
@s1
vr
1
1 h
⌘
= v · rs1 + cp+
r · (⇢0 T0 rs1 ) +(⇧ · r) · v +|r ⇥ B|2 ,@t
Hp⇢0 T 0⇢0 T 04⇡•“Mean-field” MHD equations in a 3D full-spherical shell (NOT azimuthal mean)•

Small-scale convective angular momentum transport(Λ-effect) is parameterized [Kitcha1nov & Rudiger 1995]•

Convective energy transport is implicitly assumed•

Lower CZ is set weakly subadiabatic to achieve
the thermal wind balance [Rempel 2005]
HD solver2021. 11. 16

GFD Seminar

Supplementary@⇢1=r · (⇢ v),
Numerical methods : Babcock-Leighton
dynamo code II
@t
⇠
1

2

induction:

0

@v
rp1
⇢1
1
1
= v · rv
ger + 2v ⇥ ⌦0 +(r ⇥ B) ⇥ B + r · ⇧,
@t
⇢0
⇢0
4⇡⇢0
⇢0
@B
= r ⇥ (v ⇥ B + E ⌘r ⇥ B),
@t
i
@s1
vr
1
1 h
⌘
2
= v · rs1 + cp+
r · (⇢0 T0 rs1 ) +(⇧ · r) · v +|r ⇥ B| ,@t
Hp⇢0 T 0⇢0 T 04⇡

Bipolar magnetic regions (BMRs) are produced instantaneously at the surface in response to the dynamo-generated
toroidal field near the base of CZ[step I] determine the loca$on of BMR emergence (𝜃 ∗ , 𝜙 ∗ ) that sa$sﬁes the following condi$on𝑓() 𝜃 ∗ 𝐵$ 𝜃 ∗ , 𝜙 ∗ |*+*)*+ > 𝐵,-./ = 500 G
suppress highla8tude emergencetoroidal ﬁeld
near the basethreshold field strengthfor emergence[step 2] electro-motiver force 𝓔 is set proportional to 𝐵$ 𝜃 ∗ , 𝜙 ∗ℰ0
4 ,52 6∗
=
𝛼
𝑓
(𝑟,
𝜃,
𝜙)
𝐵$ 𝜃 ∗ , 𝜙 ∗ |*+*)*+
123
2.7 6∗ℰ$
localized near the surface
Joy’s law 8lt: 𝜓 ∗ = 35° cos 𝜃 ∗[step 3] frequency of BMR emergence is determined by the
log-normal $me-delay distriobu$on with ∆9 = 𝑡 − 𝑡9 ≳ 10 days

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GFD Seminar

𝑩#

SupplementaryR-modes in the mean-ﬁeld simula>ons•

ObservaPons suggest that Rossby waves (r-modes) are importanat ingredients for large-scale dynamics [Loep1en et al. 2018]•

r-modes exist at roughly 3 ≤ 𝑚 ≤ 15 (possibly excited by non-axisymmetric Lorentz force)•

Extracted eigenfuncPon of 𝑣" peaks at the equator, changes its sign, and decays at higher laPtudes [Proxauf et al. 2020]•

r-mode’s eigenfuncPons are strongly trapped in the equatorial region by the viscous criPcal layers [Gizon et al. 2020]

Power of 𝑣" near the surface

extracted eigenfunc1on at 𝑚 = 8

rota/onrate ∆Ω𝜔=−2021. 11. 16

2 Ω! + ∆Ω
+ 𝑚∆Ω
𝑚+1

GFD Seminar

SupplementaryPhysical Picture of Baroclinic instability

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Supplementary

Numerical Simula>on of Baroclinic instability

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Supplementary