A Linear Stability Analysis on the Onset of Thermal
Convection of a Fluid with Strongly
Temperature-dependent Viscosity in a Spherical Shell
Masanori Kameyama
August 21, 2010

Supplementary material for GFD Seminar

August 21, 2010. – slide 1

Acknowledgement
MCK thanks Drs. Hiroki Ichikawa, Takashi Nakagawa (ETHZ), Shijie
Zhong (UC Boulder) and David A. Yuen (Univ. Minnesota) for fruitful
discussion and encouragement. MCK also greatly acknowledges thoroughsupport from the Global COE program from the Ministry of Education,
Culture, Sports, and Technology (MEXT) of Japan, and financial
support from Grant-in-Aid for Scientific Research (C) from Japan Society
for the Promotion of Science (#19540452).

Supplementary material for GFD Seminar

August 21, 2010. – slide 2➢Acknowledgement
Introduction➢transitions (1)➢transitions (2)➢transitions (3)➢questions➢what we do
Numerical Model
Result 1: isoviscous
Results 2 : η(T ) ina planar layer

Introduction

Results 3 : η(T ) ina spherical shell
Discussion andConcluding Remarks
Linear Stability
Analysisanalytically for shell

Supplementary material for GFD Seminar

August 21, 2010. – slide 3

Transitions in convective regimes (1)➢Acknowledgement

Changes in vertical flow structures due to η(T )Introduction➢transitions (1)➢transitions (2)➢transitions (3)➢questions➢what we do

weak(Solomatov, 1995)
temperature-dependence of viscosity
strongNumerical Model
Result 1: isoviscous
Results 2 : η(T ) ina planar layer
Results 3 : η(T ) ina spherical shell

T

Discussion andConcluding Remarks
Linear Stability
Analysisanalytically for shell

ψ
rη

102

103

104

105

Note: The viscosity of the hottest fluid ηb is kept constant, yielding the Rayleighnumber of Rab = 107 (where Rab is defined with ηb ).
Supplementary material for GFD Seminar

August 21, 2010. – slide 4

Transitions in convective regimes (2)➢Acknowledgement
Introduction➢transitions (1)➢transitions (2)➢transitions (3)➢questions➢what we do
Numerical Model
Result 1: isoviscous
Results 2 : η(T ) ina planar layer

Changes in vertical flow structures(Solomatov, 1995)

Three convective regimes are identified depending on thetemperature-dependence of viscosity:❏ “Whole-Layer” (WH) mode
(for weak dependence)

thin cold thermal boundary layer is involved
into convection at depth.

Results 3 : η(T ) ina spherical shell
Discussion andConcluding Remarks
Linear Stability
Analysisanalytically for shell❏ Transitional mode (for moderate dependence)❏ “Stagnant-Lid” (ST) mode
(for strong dependence)

thick and stiff cold thermal boundary layer(or “lid”) is NOT involved into convection
at depth.
Supplementary material for GFD Seminar

August 21, 2010. – slide 5

Transitions in convective regimes (3)➢Acknowledgement

Changes in horizontal flow structures due to η(T )(figures from Yoshida and Kageyama, 2005)

Introduction➢transitions (1)➢transitions (2)➢transitions (3)➢questions➢what we do

8
7

Results 3 : η(T ) ina spherical shell
Discussion andConcluding Remarks
Linear Stability
Analysisanalytically for shell106

No Convection

narrow cells

5log10rη

Results 2 : η(T ) ina planar layer

Rab
107

6

Numerical Model
Result 1: isoviscousrη

4wide cells"Sluggish-Lid"3106

thermalanomalies
hot (yellow)
cold (blue)2
narrow
cells

1
0

2

3

4 5 6
log10Rab

7

8

104

Horizontal length scale of convection changes from narrow
to wide, and then from wide to narrow ones with increasing
temperature-dependence of viscosity.
Supplementary material for GFD Seminar

August 21, 2010. – slide 6

Questions to be addressed in this study➢Acknowledgement
Introduction➢transitions (1)➢transitions (2)➢transitions (3)➢questions➢what we do
Numerical Model
Result 1: isoviscous❏ What is the relation between the changes in vertical andhorizontal flow structures ?
❏ How convection looks when viscosity is strongly

temperature-dependent ?
Is it always characterized by convection cells of small
horizontal length scales beneath cold stiff lids ?

Results 2 : η(T ) ina planar layer≡?

Results 3 : η(T ) ina spherical shell
Discussion andConcluding Remarks
Linear Stability
Analysisanalytically for shell❏ If not, under which conditions does convection

simultaneously have cells of large horizontal length scales
and cold stiff lids ?
∈Supplementary material for GFD Seminar? and/or∈?

August 21, 2010. – slide 7

What we do in this study➢Acknowledgement
Introduction➢transitions (1)➢transitions (2)➢transitions (3)➢questions➢what we do
Numerical Model

Linear stability analysis on the onset of thermal convectionof a fluid with strongly temperature-dependent viscosity ina spherical shell geometry.
❏ to classify the flow patterns of spherical shell convection,from the changes in the vertical flow structures,

Result 1: isoviscous
Results 2 : η(T ) ina planar layer
Results 3 : η(T ) ina spherical shell
Discussion andConcluding Remarks
Linear Stability
Analysisanalytically for shell

1. onset of convection in a planar layer❏ to develop a criterion for transition into ST regime

2. onset of convection in a spherical shell❏ to identify the transition into ST regime by using same

criterion
Supplementary material for GFD Seminar

August 21, 2010. – slide 8➢Acknowledgement
Introduction
Numerical Model➢model➢equations
Result 1: isoviscous
Results 2 : η(T ) ina planar layer
Results 3 : η(T ) ina spherical shell

Numerical Model

Discussion andConcluding Remarks
Linear Stability
Analysisanalytically for shell

Supplementary material for GFD Seminar

August 21, 2010. – slide 9

Model
➢Acknowledgement

Geometry

Introduction
Numerical Model➢model➢equations(a) planar layerz(b) spherical shell

T=Ts

Result 1: isoviscous
Results 2 : η(T ) ina planar layerroLResults 3 : η(T ) ina spherical shell
Discussion andConcluding Remarks
Linear Stability
Analysisanalytically for shell

r

T=Ts

T=Tb

0

T=Tb

ri

γ ≡ ri /ro (< 1)❏ Onset of thermal convection driven by basal heating(i.e., no internal heat sources)❏ Exponentialof viscosity temperature-dependence

T − Tsη ∝ exp −E
Tb − Ts❏ top/bottom boundaries are free-slip (F) or rigid (R)Supplementary material for GFD Seminar

August 21, 2010. – slide 10

Basic Equations (dimensionless)➢Acknowledgement❏ Equation of heat transport

∂T
2
=
∇
T
+v
·
∇T
{z
}
|{z}
|
∂t

Introduction
Numerical Model➢model➢equations
Result 1: isoviscous
Results 2 : η(T ) ina planar layer
Results 3 : η(T ) ina spherical shell
Discussion andConcluding Remarks
Linear Stability
Analysisanalytically for shell

advection

conduction❏ Equation of continuity (incompressible fluid)∇·v =0❏ Equations of motion (force balance)

0 = −∇p + ∇ · [η (∇ ⊗ v + v ⊗ ∇)] + RaT eg❏ Constitutive Equation (temperature-dependence of viscosity)η = η1/2 exp −E T − 12

where Ra : Rayleigh number (defined with η1/2 , not with ηb )E : temperature-dependence of viscosity
In this study, linearized equations are solved for infinitesimal
perturbations in T and v.

Supplementary material for GFD Seminar

August 21, 2010. – slide 11➢Acknowledgement
Introduction
Numerical Model
Result 1: isoviscous➢growth/decay➢growth rate
➢Critical state➢in spherical shell
Results 2 : η(T ) ina planar layer
Results 3 : η(T ) ina spherical shell

Results 1: Onset of convection ofisoviscous fluid

Discussion andConcluding Remarks
Linear Stability
Analysisanalytically for shell

Supplementary material for GFD Seminar

August 21, 2010. – slide 12

Convective Instability: Grow or Not ?
➢Acknowledgement
Introduction
Numerical Model
Result 1: isoviscous➢growth/decay➢growth rate
➢Critical state➢in spherical shell

Convective instability grows up for sufficiently Rayleighnumber Ra
thermal buoyancy
Ra ∝viscous resistance × thermal diffusion

Results 2 : η(T ) ina planar layer
Results 3 : η(T ) ina spherical shell
Discussion andConcluding Remarks

Ra = 1000

Linear Stability
Analysisanalytically for shell

Ra = 600
Supplementary material for GFD Seminar

August 21, 2010. – slide 13

Temporal evolution of perturbation➢Acknowledgement
Introduction

Growth rate a of convective instability (perturbation)
3000

Rayleigh mumber

30

Results 2 : η(T ) ina planar layer

2000

0

Result 1: isoviscous➢growth/decay➢growth rate
➢Critical state➢in spherical shell

10

Numerical Model

20

10

1000
0

Results 3 : η(T ) ina spherical shell
Discussion andConcluding Remarks
Linear Stability
Analysisanalytically for shell

-10

0
0

1

2

3

4

5

6

−1

wavenumber ∝ (plume spacings)

a > 0 for growth,a < 0 for decay,
a = 0 for neutral.

Growth rate a becomes:❏ larger for larger Rayleigh number Ra❏ largest for intermediate wavenumber K of perturbation

Supplementary material for GFD Seminar

August 21, 2010. – slide 14

Critical Rayleigh number

Introduction
Numerical Model
Result 1: isoviscous➢growth/decay➢growth rate
➢Critical state➢in spherical shell
Results 2 : η(T ) ina planar layer

Critical Rayleigh number (Rac ): Ra which gives a = 0.
3000

critical Rayleigh mumber➢Acknowledgement

2000

1000

Results 3 : η(T ) ina spherical shell
Discussion andConcluding Remarks
Linear Stability
Analysisπ4657.51134 ≃ 27
42.2212608 ≃ √π2

0
0

1

2

3

wavenumber

4

5

6

a > 0 for growth,a < 0 for decay,
a = 0 for neutral.

analytically for shell

The perturbation with absolute minimum of Rac (≡ Rac0 ) ismost important. (because it is destabilized most easily !)Supplementary material for GFD Seminar

August 21, 2010. – slide 15

Critical Rayleigh number in spherical shells
thin shell (γ = 0.95)➢Acknowledgement

Result 1: isoviscous➢growth/decay➢growth rate
➢Critical state➢in spherical shell

γ=0.95

Linear Stability
Analysis

γ=0.55r1 = 20/9
r0 = 11/9

3000

2000

1000

1000

657.85875
0Discussion andConcluding Remarks

r1 = 20
r0 = 19

2000

Results 2 : η(T ) ina planar layer
Results 3 : η(T ) ina spherical shell4000
critical Rayleigh mumber

Numerical Model

3000
critical Rayleigh mumber

Introduction

thick shell (γ = 0.55)

10

20

30

40 50 60 70 80
spherical harmonic degree

711.94985

90

100

0

1

2

3

4
5
6
7
8
9
spherical harmonic degree1011Note: Length scale is normalized so as to take r1 − r0 ≡ 1.

Growth of perturbation depends on

analytically for shell❏ Rayleigh number Ra❏ spherical harmonic degree ℓ of perturbation❏ thickness (or aspect ratio γ) of spherical shell
Supplementary material for GFD Seminar

August 21, 2010. – slide 16➢Acknowledgement
Introduction
Numerical Model
Result 1: isoviscous
Results 2 : η(T ) ina planar layer➢Rac for η(T )➢Rac0 Kc0 (1)➢Rac0 Kc0 (2)➢analytically (1)➢analytically (2)➢analytically (3)➢empirically (1)➢empirically (2)Results 2: Onset of convection oftemperature-dependent viscosity fluidin a planar layer

Results 3 : η(T ) ina spherical shell
Discussion andConcluding Remarks
Linear Stability
Analysisanalytically for shell

Supplementary material for GFD Seminar

August 21, 2010. – slide 17

Variations in critical Rayleigh numbers

Introduction
Numerical Model
Result 1: isoviscous
Results 2 : η(T ) ina planar layer➢Rac for η(T )➢Rac0 Kc0 (1)➢Rac0 Kc0 (2)➢analytically (1)➢analytically (2)➢analytically (3)➢empirically (1)➢empirically (2)Results 3 : η(T ) ina spherical shell3000
critical Rayleigh number➢Acknowledgement

2000

1000

0

0

variations in critical
Rayleigh number Rac
and wavenumber K
with increasing temperature dependence ofviscosity E.

E=0
E=2
E=4
E=6
E=8
E=10
1

2

3
4
wavenumber

5

6

E = ln(ηmax /ηmin )Discussion andConcluding Remarks❏ Variations of Rac on K depend on E,

Linear Stability
Analysis❏ Absolute minimum of Rac (≡ Rac0 ) and corresponding K

analytically for shell(≡ Kc0 ) are also dependent on E.

Supplementary material for GFD Seminar

August 21, 2010. – slide 18

Critical Rayleigh number and wavenumber (1)Rac0 against E➢Acknowledgement
2500

6

Results 2 : η(T ) ina planar layer➢Rac for η(T )➢Rac0 Kc0 (1)➢Rac0 Kc0 (2)➢analytically (1)➢analytically (2)➢analytically (3)➢empirically (1)➢empirically (2)Results 3 : η(T ) ina spherical shell
Discussion andConcluding Remarks
Linear Stability
Analysisanalytically for shell

critical Rayleigh number

Numerical Model

R/R

5

2000

critical wavenumber

Introduction

Result 1: isoviscous

Kc0 against E

R/F
1500

F/R

1000

4

R/R
3

R/F
F/R
2

F/F

F/F

500

1
0

2

4

6

8

E=ln(ηmax/ηmin)

10

12

14

0

2

4

6

8

10

12

14

E=ln(ηmax/ηmin)

Three regimes can be identified depending on E:
E . 1.5: Rac0 → and Kc0 →1.5 . E . 8.3 : Rac0 ր and Kc0 → or ց (depending on b.c.)E & 8.3: Rac0 ց and Kc0 ր Stagnant-Lid mode

Supplementary material for GFD Seminar

August 21, 2010. – slide 19

Critical Rayleigh number and wavenumber (2)Rac0 against E➢Acknowledgement
2500

6

Results 2 : η(T ) ina planar layer➢Rac for η(T )➢Rac0 Kc0 (1)➢Rac0 Kc0 (2)➢analytically (1)➢analytically (2)➢analytically (3)➢empirically (1)➢empirically (2)Results 3 : η(T ) ina spherical shell
Discussion andConcluding Remarks
Linear Stability
Analysisanalytically for shell

critical Rayleigh number

Numerical Model

R/R

5

2000

critical wavenumber

Introduction

Result 1: isoviscous

Kc0 against E

R/F
1500

F/R

1000

4

R/R
3

R/F
F/R
2

F/F

F/F

500

1
0

2

4

6

8

E=ln(ηmax/ηmin)

10

12

14

0

2

4

6

8

10

12

14

E=ln(ηmax/ηmin)

For very strong temperature-dependence (large E),❏ differences in top surface boundary conditions become

negligibly small.
✏ almost zero motion even with free-slip top boundaries.

Supplementary material for GFD Seminar

August 21, 2010. – slide 20

Analytical Criterion for ST-mode (1)➢Acknowledgement

Following the idea by Stengel et al. (1982) ....

Introduction
Numerical Model
Result 1: isoviscous
Results 2 : η(T ) ina planar layer➢Rac for η(T )➢Rac0 Kc0 (1)➢Rac0 Kc0 (2)➢analytically (1)➢analytically (2)➢analytically (3)➢empirically (1)➢empirically (2)

z

T=Ts

z

1

stagnant lid

1

ẑ

0

ẑ

T=Tb

0

1 T

Results 3 : η(T ) ina spherical shell
Discussion andConcluding Remarks
Linear Stability
Analysisanalytically for shell

In ST regime, convection begins in a basal sublayer, not in
an entire layer.
ˆ local to the
For sufficiently large E, the Rayleigh number Ra
basal sublayer exceeds Ra for the entire layer.

Supplementary material for GFD Seminar

August 21, 2010. – slide 21

Analytical Criterion for ST-mode (2)Introduction
Numerical Model
Result 1: isoviscous
Results 2 : η(T ) ina planar layer➢Rac for η(T )➢Rac0 Kc0 (1)➢Rac0 Kc0 (2)➢analytically (1)➢analytically (2)➢analytically (3)➢empirically (1)➢empirically (2)Results 3 : η(T ) ina spherical shell
Discussion andConcluding Remarks
Linear Stability
Analysis

Consider the ratio of sublayer’sˆ to Ra, in order to confirm
Rathe transition into ST regime
at E ≃ 8.

z

T=Ts

z

1

stagnant lid

1

ẑ
0

ẑ
T=Tb

0

1 T

3

 

ˆ
thickness
×
T
-gap
1 ẑ
Ra
4
= ẑ exp E
=
−
Ra
2 2
viscosity
1.2
1.0
0.8

ˆ
Ra/Ra

➢Acknowledgement

E=6
E=7
E=8
E=9
E=10

When E > 8 there exists a range
ˆ > Ra.
of ẑ (< 1) which allows Ra

0.6
0.4
0.2

analytically for shell
0.0
0.0

0.2

0.4

0.6

ẑ
Supplementary material for GFD Seminar

0.8

1.0

Convective instability tends togrow only in a basal sublayer with
ˆ > RaRa⇒ “Stagnant-Lid” mode
August 21, 2010. – slide 22

Analytical Criterion for ST-mode (3)2500

Introduction
Numerical Model
Result 1: isoviscous
Results 2 : η(T ) ina planar layer➢Rac for η(T )➢Rac0 Kc0 (1)➢Rac0 Kc0 (2)➢analytically (1)➢analytically (2)➢analytically (3)➢empirically (1)➢empirically (2)Results 3 : η(T ) ina spherical shell
Discussion andConcluding Remarks

critical Rayleigh number➢Acknowledgement

R/R
2000

R/F
1500

F/R

1000

F/F
500
0

2

4

6

8

E=ln(ηmax/ηmin)

10

12

14

Above analytical consideration
also suggests that Ra for theentire layer should be maximum at the transition into ST
regime.

The plots for free-slip top surfaces (F/F and F/R) havemaxima at around E = 8.3.

Linear Stability
Analysisanalytically for shell⇒ Transition into ST regime occurs at E ≃ 8.3 for a planar
layer.

Supplementary material for GFD Seminar

August 21, 2010. – slide 23

Empirical Criterion for ST-mode (1)vz (z + δz)➢Acknowledgementz + δzIntroduction
Numerical Model
Result 1: isoviscous
Results 2 : η(T ) ina planar layer➢Rac for η(T )➢Rac0 Kc0 (1)➢Rac0 Kc0 (2)➢analytically (1)➢analytically (2)➢analytically (3)➢empirically (1)➢empirically (2)Results 3 : η(T ) ina spherical shell
Discussion andConcluding Remarks
Linear Stability
Analysisanalytically for shell

Considering where descendingflows are amplified most intensively ....

zvz (z)

Here we define a quantity ∆h as a function of height z
defined with,∂vz∂vx ∂vyvz (z) − vz (z + δz) ≃ −
δz
δz =
+
∂z
∂x
∂y
|
{z
}
∝∆h

The value of |∆h | becomes locally maximum at a height z
where a vertical flow is amplified most intensively.

Supplementary material for GFD Seminar

August 21, 2010. – slide 24

Empirical Criterion for ST-mode (2)1.0➢Acknowledgement
Introduction

Result 1: isoviscous
Results 2 : η(T ) ina planar layer➢Rac for η(T )➢Rac0 Kc0 (1)➢Rac0 Kc0 (2)➢analytically (1)➢analytically (2)➢analytically (3)➢empirically (1)➢empirically (2)Results 3 : η(T ) ina spherical shell
Discussion andConcluding Remarks
Linear Stability
Analysisanalytically for shell

0.8

normalized height

Numerical Model

E=0
E=3
E=5
E=7
E=8
E=8.3
E=9

0.6

Vertical profiles of |∆h | for various E
As for a local maximum otherthan at z = 0,

0.4

0.2

❏ Where does it occur?

0.0
10

-2

10

-1

10

0

|∆h(z)/∆h(z=1)|

10

1

10

2

❏ How large is it ?

The local maximum of |∆h | occurs at deeper position withincreasing E.
At transition into ST regime (E = 8.3), the local maximum
becomes more than 10 times larger than |∆h | at z = 1.
⇒ to be used as a criterion for a spherical shell convection.

Supplementary material for GFD Seminar

August 21, 2010. – slide 25➢Acknowledgement
Introduction
Numerical Model
Result 1: isoviscous
Results 2 : η(T ) ina planar layer
Results 3 : η(T ) ina spherical shell➢η(T ) γ = 0.95
➢η(T ) γ = 0.55
➢empirically (1)➢empirically (2)➢empirically (3)Results 3: Onset of convection oftemperature-dependent viscosity fluidin a spherical shell

Discussion andConcluding Remarks
Linear Stability
Analysisanalytically for shell

Supplementary material for GFD Seminar

August 21, 2010. – slide 26

Critical Rayleigh number and wavenumber (1)Numerical Model
Result 1: isoviscous
Results 2 : η(T ) ina planar layer

γ=0.95γ=0.95
100

2500

R/R

2000

Results 3 : η(T ) ina spherical shell➢η(T ) γ = 0.95
➢η(T ) γ = 0.55
➢empirically (1)➢empirically (2)➢empirically (3)Discussion andConcluding Remarks
Linear Stability
Analysisanalytically for shellℓc0 against E

110

3000

critical harmonic degree

Introduction

Thin spherical shell (γ = 0.95)
Rac0 against E
critical Rayleigh number➢Acknowledgement

R/F

1500

F/R

1000

90
80
70

R/R
60

R/F
50

F/R

F/F
40

500

F/F

30
0

2

4

6

8

10

12

14

0

2

E=ln(ηmax/ηmin)

4

6

8

10

12

14

r1 = 20
r0 = 19

E=ln(ηmax/ηmin)

Quite similar to the case in a planar layer:❏ Three regimes (small E, moderate E, and large E),❏ Maxima in the plots of Rac0 for F/F and F/R at E ≃ 8.3,
❏ smooth changes in Rac0 and ℓc0 .

Supplementary material for GFD Seminar

August 21, 2010. – slide 27

Critical Rayleigh number and wavenumber (2)Numerical Model
Result 1: isoviscous
Results 2 : η(T ) ina planar layer

10

5000

γ=0.55
F/R

4000

3000

Results 3 : η(T ) ina spherical shell➢η(T ) γ = 0.95
➢η(T ) γ = 0.55
➢empirically (1)➢empirically (2)➢empirically (3)Discussion andConcluding Remarks
Linear Stability
Analysisanalytically for shell

γ=0.55

R/R
critical harmonic degree

Introduction

Thick spherical shell (γ = 0.55)
Rac0 against Eℓc0 against E
critical Rayleigh number➢Acknowledgement

F/F
R/F

2000

R/R

5

R/F

F/F

1000

0

F/R

0
0

2

4

6

8

E=ln(ηmax/ηmin)

10

12

14

0

2

4

6

8

10

12

14

r1 = 20/9
r0 = 11/9

E=ln(ηmax/ηmin)

Three regimes (small E, moderate E, and large E) exist, but ...
❏ Maxima in the plots of Rac0 for F/F and F/R at E ≃ 10,(but NOT the transition into ST-mode)❏ discontinuous changes in Rac0 and ℓc0 .

(mostly due to small horizontal extent)
Supplementary material for GFD Seminar

August 21, 2010. – slide 28

Empirical Criterion for ST-mode (1)vr (r + δr)➢Acknowledgementr + δrIntroduction
Numerical Model
Result 1: isoviscous
Results 2 : η(T ) ina planar layer
Results 3 : η(T ) ina spherical shell➢η(T ) γ = 0.95
➢η(T ) γ = 0.55
➢empirically (1)➢empirically (2)➢empirically (3)Discussion andConcluding Remarks
Linear Stability
Analysisanalytically for shell

Suppose again where descending flows are amplified most intensively in a spherical shell.

rvr (r)

Here we consider a quantity ∆h as a function of radius rdefined with,1 ∂ 2
r vr (r) − (r + δr) vr (r + δr) ≃ − 2 (r vr )r2 δr
r ∂r

1 ∂
1 ∂vφ
=
r2 δr
(sin θvθ ) +r sin θ ∂θr sin θ ∂φ
|
{z
}
2

2

∝∆h

Note the definition of ∆h different from previous one.
Supplementary material for GFD Seminar

August 21, 2010. – slide 29

Empirical Criterion for ST-mode (2)(a) thin shell(γ = 0.95)➢Acknowledgement
Introduction
Numerical Model

1.0

1.0

Discussion andConcluding Remarks
Linear Stability
Analysisanalytically for shellnormalized height r-r0

Results 3 : η(T ) ina spherical shell➢η(T ) γ = 0.95
➢η(T ) γ = 0.55
➢empirically (1)➢empirically (2)➢empirically (3)

0.8

0.6

normalized height r-r0

γ=0.95E=0 (l=43)
E=5 (l=36)
E=7 (l=41)
E=8 (l=53)
E=8.3 (l=57)
E=9 (l=64)

Result 1: isoviscous
Results 2 : η(T ) ina planar layer(b) thick shell(γ = 0.55)

0.4

0.2

γ=0.55E=0 (l=3)
E=3 (l=3)
E=8 (l=2)
E=9.1 (l=3)
E=9.2 (l=4)
E=10 (l=5)

0.8

0.6

0.4

0.2

0.0

0.0
10

-2

10

-1

10

0

10

1

10

2

|∆h(r)/∆h(r=r1)|

10

-2

10

-1

10

0

10

1

10

2

|∆h(r)/∆h(r=r1)|

Based on similar criterion with a planar layer, the transition
into ST-mode occurs❏ at E ≃ 8.3 for γ = 0.95 (as in a planar layer)❏ at E ≃ 9.2 for γ = 0.55

Supplementary material for GFD Seminar

August 21, 2010. – slide 30

Empirical Criterion for ST-mode (3)➢Acknowledgement

10.5
Analytical
Empirical

Introduction

Result 1: isoviscous
Results 2 : η(T ) ina planar layer
Results 3 : η(T ) ina spherical shell➢η(T ) γ = 0.95
➢η(T ) γ = 0.55
➢empirically (1)➢empirically (2)➢empirically (3)Discussion andConcluding Remarks
Linear Stability
Analysisanalytically for shell

10.0

threshold values of E

Numerical Model

9.0

The variations against the aspect ratio of the spherical shell γof the values of Ec for the transition into the ST regimes.

8.5

blue : empirical estimate

9.5(from ∆h )
8.0
0.2

0.4
0.6
0.8
aspect ratio of spherical shell γ

1.0

red : analytical estimate(although not shown in detail)In general, smaller γ (thicker shell) needs larger E for thetransition into ST regimes.
❏ because of larger temperature contrast between ascendingflows and surroundings ?
Supplementary material for GFD Seminar

August 21, 2010. – slide 31➢Acknowledgement
Introduction
Numerical Model
Result 1: isoviscous
Results 2 : η(T ) ina planar layer
Results 3 : η(T ) ina spherical shell
Discussion andConcluding Remarks➢other bodies?
➢γ = 0.55 (1)➢γ = 0.55 (2)➢elongated ST➢Future directions➢

Discussion and Concluding Remarks

Linear Stability
Analysisanalytically for shell

Supplementary material for GFD Seminar

August 21, 2010. – slide 32

Application to mantle convection in other bodies?
➢Acknowledgement
Introduction
Numerical Model
Result 1: isoviscous
Results 2 : η(T ) ina planar layer
Results 3 : η(T ) ina spherical shell
Discussion andConcluding Remarks➢other bodies?
➢γ = 0.55 (1)➢γ = 0.55 (2)➢elongated ST➢Future directions➢
Linear Stability
AnalysisPlanetary Data
Mercury
Radius
0.38
Mass
0.055
Density5430.
[kg/m3 ]
MoI
0.34
Rc/Rp
0.8

Ec

8.4

Venus
0.95
0.815
5250.

Earth
1
1
5515.

Mars
0.54
0.107
3940.

Moon
0.27
0.012
3340.

?
0.55?

0.33550.546

0.36620.5

0.3905
0.25

9.2

9.2

9.3

> 10.5analytically for shell

Supplementary material for GFD Seminar

August 21, 2010. – slide 33

Regime diagram for Earth’s mantle: Revisited (1)(b)  (rη = 106 )➢Acknowledgement(a)Introduction
8

Result 1: isoviscous

7

Results 2 : η(T ) ina planar layer

6

Results 3 : η(T ) ina spherical shell5

Linear Stability
Analysisanalytically for shelllog10rη

Discussion andConcluding Remarks➢other bodies?
➢γ = 0.55 (1)➢γ = 0.55 (2)➢elongated ST➢Future directions➢

γ=0.55stif

No Convection

flid

Numerical Modelnarrowcells

4(c) N (rη = 104 )wide cells
32

narrow cells
1
0
2

3

4

5

6

7

8

log10Rab

The curve of critical Rayleigh number Rab and viscosity
contrast rη ≡ exp(E) has a bend near × (rη = 104 ≃ e9.2 ).
⇒ demonstrates significance of present estimate

Supplementary material for GFD Seminar

August 21, 2010. – slide 34

Regime diagram for Earth’s mantle: Revisited (2)(b)  (rη = 106 )➢Acknowledgement(a)Introduction
8

Result 1: isoviscous

7

Results 2 : η(T ) ina planar layer

6

Results 3 : η(T ) ina spherical shell5

Linear Stability
Analysisanalytically for shelllog10rη

Discussion andConcluding Remarks➢other bodies?
➢γ = 0.55 (1)➢γ = 0.55 (2)➢elongated ST➢Future directions➢

γ=0.55stif

No Convection

flid

Numerical Modelnarrowcells

4(c) N (rη = 104 )wide cells
32

narrow cells
1
0
2

3

4

5

6

7

8

log10Rab

For the value of rη = 104 at the transition into ST regime,
convection has horizontally-elongated cells.
wide convection cells beneath cold stiff lids at rη ≃ 104 ?

Supplementary material for GFD Seminar

August 21, 2010. – slide 35

ST-mode of convection with elongated cells➢Acknowledgement
Introduction“Elongated-ST” mode
(figure and movie modified from Kameyama and Ogawa, 2000)

Numerical Model
Result 1: isoviscous
Results 2 : η(T ) ina planar layer

Rab = 6 × 106 , rη = 104 , width/height=3

Results 3 : η(T ) ina spherical shell
Discussion andConcluding Remarks➢other bodies?
➢γ = 0.55 (1)➢γ = 0.55 (2)➢elongated ST➢Future directions➢≃

Linear Stability
Analysis❏ horizontally-elongated convection cell

analytically for shell❏ minor descending plumes from base of cold lid✏ cold lid is stiff enough to prevent minor instabilities

from penetrating upward.
Supplementary material for GFD Seminar

August 21, 2010. – slide 36

Future directions➢Acknowledgement
Introduction

How come the convective flows with wide cells beneath cold
stiff lids in mantles of terrestrial planets ?

Numerical Model
Result 1: isoviscous
Results 2 : η(T ) ina planar layer
Results 3 : η(T ) ina spherical shell
Discussion andConcluding Remarks➢other bodies?
➢γ = 0.55 (1)➢γ = 0.55 (2)➢elongated ST➢Future directions➢
Linear Stability
Analysisanalytically for shellIn particular,
what mechanisms cause convection cells of large horizontal
length scales ?
Hopefully, graduate students will address:❏ the effects of material properties (other than viscosity) ?

✏ thermal expansivity, thermal conductivity, .... ?
❏ the effects of chemical heterogeneity (and surface tectonics) ?

✏ If ascending plumes are “anchored” by chemical “piles”

in the lowermost mantle ?
Supplementary material for GFD Seminar

August 21, 2010. – slide 37➢Acknowledgement
Introduction
Numerical Model
Result 1: isoviscous
Results 2 : η(T ) ina planar layer
Results 3 : η(T ) ina spherical shell
Discussion andConcluding Remarks

Linear Stability Analysis

Linear Stability
Analysis➢Linearization (1)➢Linearization (2)analytically for shell

Supplementary material for GFD Seminar

August 21, 2010. – slide 39

Linearization (1)Introduction
Numerical Model
Result 1: isoviscous
Results 2 : η(T ) ina planar layer
Results 3 : η(T ) ina spherical shell
Discussion andConcluding Remarks
Linear Stability
Analysis➢Linearization (1)➢Linearization (2)analytically for shell

Split all quantities into the sum of❏ reference state (denoted by overbars)❏ infinitesimal perturbation (denoted by primes)

T = T + T ′,v = v + v′,

p = p + p′

Choice of reference state❏ v = 0 (motionless)❏ 0 = ∇2 T

top

(1-D steady heat

conduction)
d2 T
0 = dz 2 forlayer,
 a planard
0 = r12 dr
r2 dTdrical shell,in a planar layer

for a spher-in a spherical shell

T ∝ r−1bottomSupplementary material for GFD Seminar

T ∝zheight➢Acknowledgement0

temperature

1

August 21, 2010. – slide 40

Linearization (2)➢Acknowledgement
Introduction
Numerical Model
Result 1: isoviscous
Results 2 : η(T ) ina planar layer
Results 3 : η(T ) ina spherical shell
Discussion andConcluding Remarks
Linear Stability
Analysis➢Linearization (1)➢Linearization (2)Derive linearized equation for infinitesimal perturbations
Dropping the second-order terms yields❏ Equation of heat transport∂T ′+ v ′ · ∇T = ∇2 T ′∂t❏ Equation of continuity (incompressible fluid)∇ · v′ = 0❏ Equations of motion (force balance)

0 = −∇p′ + ∇ · [η (∇ ⊗ v ′ + v ′ ⊗ ∇)] + RaT ′ eg

analytically for shell

Supplementary material for GFD Seminar

August 21, 2010. – slide 41➢Acknowledgement
Introduction
Numerical Model
Result 1: isoviscous
Results 2 : η(T ) ina planar layer
Results 3 : η(T ) ina spherical shell
Discussion andConcluding Remarks
Linear Stability
AnalysisAnalytical Estimate for Transition
into ST regime in a spherical shell

analytically for shell➢analytically (1)➢analytically (2)➢analytically (3)➢analytically (4)➢analytically (5)Supplementary material for GFD Seminar

August 21, 2010. – slide 42

Analytical Criterion for ST-mode (1)thin shell (γ = 0.95)➢Acknowledgement

Result 1: isoviscous
Results 2 : η(T ) ina planar layer
Results 3 : η(T ) ina spherical shell
Discussion andConcluding Remarks
Linear Stability
Analysisanalytically for shell➢analytically (1)➢analytically (2)➢analytically (3)➢analytically (4)➢analytically (5)5000

γ=0.95γ=0.55

R/R
F/R

critical Rayleigh number

Numerical Model

3000

critical Rayleigh number

Introduction

thick shell (γ = 0.55)

2500

4000

R/R

2000

3000

R/F

1500

F/F
R/F

2000

F/R

1000

1000

F/F

500

0
0

2

4

6

8

E=ln(ηmax/ηmin)

10

12

14

0

2

4

6

8

10

12

14

E=ln(ηmax/ηmin)

In contrast to a planar case, the maxima of Rac0 against E
do not necessarily indicate the transition into ST regimes.

Supplementary material for GFD Seminar

August 21, 2010. – slide 43

Analytical Criterion for ST-mode (2)➢Acknowledgement
Introduction

By using appropriately defined Rayleigh number Ra∗ , we will
explore the analytical estimate for the transition.

Numerical Model
Result 1: isoviscous
Results 2 : η(T ) ina planar layer
Results 3 : η(T ) ina spherical shell

Key assumption:(a) r∗ and T∗ (in entire layer) (b) r2∗ and T2∗ (in lower sublayer)
r1

r1

r2

analytically for shell➢analytically (1)➢analytically (2)➢analytically (3)➢analytically (4)➢analytically (5)heightLinear Stability
AnalysisheightDiscussion andConcluding Remarks

r*

r0

r2*

T*
0

temperature

1

r0

T2
0

T2*
temperature

1

The temporal evolution of perturbation is assumed to besensitive to the viscosity of T = T∗ at r = r∗ where theradial temperature gradient is equal to an averagetemperature gradient across the entire layer.

Supplementary material for GFD Seminar

August 21, 2010. – slide 44

Analytical Criterion for ST-mode (3)(a) r∗ − r0 against γ➢Acknowledgement

0.5

Result 1: isoviscous
Results 2 : η(T ) ina planar layer
Results 3 : η(T ) ina spherical shell
Discussion andConcluding Remarks

0.5"representative" temperature T*

Numerical Model"representative" height r*-r0

Introduction

0.4

Linear Stability
Analysisanalytically for shell➢analytically (1)➢analytically (2)➢analytically (3)➢analytically (4)➢analytically (5)(b) T∗ against γ

0.3

Tav

0.4

0.3

0.2
0.2

0.4

0.6

0.8

aspect ratio of spherical shell γ

Supplementary material for GFD Seminar

1.0

0.2

0.4

0.6

0.8

1.0

aspect ratio of spherical shell γ

August 21, 2010. – slide 45

Analytical Criterion for ST-mode (4)(a) γ = 0.55

➢Acknowledgement

γ=0.55
1.0

Result 1: isoviscous

E=9
E=9.5
E=10
E=10.5
E=11

Ra2*/Ra*

0.8

0.6

0.4

1.0

rayleigh-ratio

Numerical Model

Results 3 : η(T ) ina spherical shell

1.2

1.2

Introduction

Results 2 : η(T ) ina planar layer(b) planar layer

0.8
0.6
0.4

Discussion andConcluding Remarks

0.2

0.2

Linear Stability
Analysis0.0
0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

0.0
0.0

analytically for shell➢analytically (1)➢analytically (2)➢analytically (3)➢analytically (4)➢analytically (5)aspect ratio of sublayer γ2Supplementary material for GFD Seminar

E=6
E=7
E=8
E=9
E=10

0.2

0.4
0.6
sublayer-height

0.8

1.0

August 21, 2010. – slide 46

Analytical Criterion for ST-mode (5)Numerical Model
Result 1: isoviscous
Results 2 : η(T ) ina planar layer
Results 3 : η(T ) ina spherical shell
Discussion andConcluding Remarks
Linear Stability
Analysisanalytically for shell➢analytically (1)➢analytically (2)➢analytically (3)➢analytically (4)➢analytically (5)3000(b) Rac0∗ for γ = 0.55
3000

γ=0.95
F/F
F/R
R/F
R/R

2500

2000

γ=0.55
F/F
F/R
R/F
R/R

critical Rayleigh number Rac0*Introduction(a) Rac0∗ for γ = 0.95
critical Rayleigh number Rac0*➢Acknowledgement

2500

2000

1500

1500

1000

1000

500

500
0

2

4

6

8

E=ln(ηmax/ηmin)

Supplementary material for GFD Seminar

10

12

14

0

2

4

6

8

10

12

14

E=ln(ηmax/ηmin)

August 21, 2010. – slide 47

参考文献
•
•

•
•

•

Solomatov, V. S. 1995: Scaling of temperature- and stress-dependent
viscosity convection, Physics of Fluids, Volume 7, Issue 2, pp.266-274
A. Kageyama, M. Yoshida, 2005: Geodynamo and mantle convection
simulations on the Earth Simulator using the Yin-Yang grid, J. Phys:
Conference Series, Vol.16, pp.325—338
Stengel, K. C. Oliver, D. S. Booker, J. R. 1982: Onset of convection in avariable-viscosity fluid, J. Fluid Mech. 120, pp.411-431
Kameyama, M. Ogawa, M. 2000: Transitions in thermal convection
withstrongly temperature-dependent viscosity in a wide box, Earth Planet.
Sci. Lett., 180, pp.355–367
Kameyama, M. Ichikawa, H. 2009: Onset of Thermal Convection of a Fluid
with Strongly Temperature-dependent Viscosity in a Spherical Shell,
American Geophysical Union, Fall Meeting 2009,