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Introduction Modelling Axial dipoles Conclusion

Dipolar dynamos in stratified systems

Raphaël Raynaud, Ludovic Petitdemange, Emmanuel Dormy

LRA-LERMAÉCOLE NORMALE SUPÉRIEURE

Stellar and Planetary DynamosGöttingen – May 27, 2015

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Introduction Modelling Axial dipoles Conclusion

Table of contents

1 Introduction

2 ModellingGoverning equationsSet upMethods

3 Axial dipoles

4 Conclusion

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Introduction Modelling Axial dipoles Conclusion

Different magnetospheric configurations

Axisymmetric (m = 0) and non-axisymmetric (m = 1) configurations

Ridley, V. A., Jovimagnetic secular variation, Doctoral thesis, 2013

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Introduction Modelling Axial dipoles Conclusion

Table of contents

1 Introduction

2 ModellingGoverning equationsSet upMethods

3 Axial dipoles

4 Conclusion

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Introduction Modelling Axial dipoles Conclusion

Governing equations

How to model the fluid flow ?

The Convective Approximationsretain the essential physics with a minimum complexity

filter out sound waves

The Boussinesq Approximation

∇· (u)= 0

The Boussinesq benchmark(Christensen et al. 2001)

homogeneous mass distribution

g ∝ r

The Anelastic Approximation

∇· (ρa u)= 0

The anelastic benchmark(Jones et al. 2011)

central mass distribution

g ∝ 1/r2

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Introduction Modelling Axial dipoles Conclusion

Governing equations

How to model the fluid flow ?

The Convective Approximationsretain the essential physics with a minimum complexity

filter out sound waves

The Boussinesq Approximation

∇· (u)= 0

The Boussinesq benchmark(Christensen et al. 2001)

homogeneous mass distribution

g ∝ r

The Anelastic Approximation

∇· (ρa u)= 0

The anelastic benchmark(Jones et al. 2011)

central mass distribution

g ∝ 1/r2

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Introduction Modelling Axial dipoles Conclusion

Governing equations

The anelastic approximation

Polytropic reference state adiabatically stratified

Ta =w(r), ρa =wn, Pa =wn+1 (1)

Resulting system

∇· (ρaV)= 0 (2)

DtV=−∇(Pc

ρa

)−αSScg+Fν/ρa (3)

ρaDtSc +∇·(

Iq

Ta

)= Q

Ta+ Iq ·∇T−1

a (4)

with

Fν viscous forceQ viscous heatingIq heat flux

Braginsky, S. I. & Roberts, P. H., 1995, GAFD, 79, 1

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Introduction Modelling Axial dipoles Conclusion

Governing equations

Mean field approach for the heat transfer equation

Solve the equation for average quantitiesf = ⟨f ⟩+ f t (5)

Turbulent fluctuations create an entropy flux term

ISt = ρa⟨StVt ⟩∝∇⟨S⟩ (6)

Strong simplification

the usual molecular term Iq =−k∇T is then neglected in theheat transfer equation

=⇒ temperature is removed from the problem !

Drawbacksargument only valid far from the onset of convection

P2 of thermodynamics not under warranty anymore

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Introduction Modelling Axial dipoles Conclusion

Governing equations

MHD equations

Dimensionless system

∂v∂t

+v ·∇v = Pm[− 1

E∇ P ′

wn + PmPr

RaSr2 er − 2

Eez ×v

+Fν+ 1E wn (∇×B)×B

], (7)

∂B∂t

= ∇× (v×B)+∇2B , (8)

∂S∂t

+v ·∇S = w−n−1 PmPr

∇·(wn+1∇S

)+Di

w

[E−1w−n(∇×B)2 +Qν

], (9)

∇· (wnv) = 0 , (10)

∇·B = 0 . (11)

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Introduction Modelling Axial dipoles Conclusion

Set up

A simplified model

Set upA perfect gas in a rotating sphericalshell with constant? kinematic viscosity ν? turbulent entropy diffusivity κ? magnetic diffusivity η

Convection is driven by an imposedentropy difference ∆S between theinner and outer spheres

Boundary conditionsfixed entropy

stress-free b. c. for the velocity field

insulating b. c. for the magnetic field9 / 21

Introduction Modelling Axial dipoles Conclusion

Methods

Systematic parameter study (∼ 250 models)

Seven control parameters

Rayleigh number Ra GMd∆Sνκcp

O(106)

magnetic Prandtl number Pm ν/η O(1)Prandtl number Pr ν/κ 1Ekman number E ν/(Ωd2) 10−4

aspect ratio χ ri/ro 0.35polytropic index n 1/(γ−1) 2number of density scale heights N% ln(%i/%o) ≤ 3

The anelastic version of PaRoDy reproduces the anelastic dynamobenchmark (Jones et al. 2011).

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Introduction Modelling Axial dipoles Conclusion

Table of contents

1 Introduction

2 ModellingGoverning equationsSet upMethods

3 Axial dipoles

4 Conclusion

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Introduction Modelling Axial dipoles Conclusion

Stability domain of the dipolar branch N% = 0.1

0 5 10 15 20 25 30 35

Ra/Rac

0

1

2

3

4

5

6

7

Pm

Dipolar dynamos as a function of Ra/Rac and Pm12 / 21

Introduction Modelling Axial dipoles Conclusion

Stability domain of the dipolar branch N% = 0.5

0 5 10 15 20 25 30 35

Ra/Rac

0

1

2

3

4

5

6

7

Pm

Dipolar dynamos as a function of Ra/Rac and Pm13 / 21

Introduction Modelling Axial dipoles Conclusion

Stability domain of the dipolar branch N% = 1.5

0 5 10 15 20 25 30 35

Ra/Rac

0

1

2

3

4

5

6

7

Pm

Dipolar dynamos as a function of Ra/Rac and Pm14 / 21

Introduction Modelling Axial dipoles Conclusion

Stability domain of the dipolar branch N% = 2.0

0 5 10 15 20 25 30 35

Ra/Rac

0

1

2

3

4

5

6

7

Pm

Dipolar dynamos as a function of Ra/Rac and Pm15 / 21

Introduction Modelling Axial dipoles Conclusion

Stability domain of the dipolar branch N% = 2.5

0 5 10 15 20 25 30 35

Ra/Rac

0

1

2

3

4

5

6

7

Pm

Dipolar dynamos as a function of Ra/Rac and Pm16 / 21

Introduction Modelling Axial dipoles Conclusion

Critical magnetic Reynolds number

∀N% , Rmc ∼ 102

0.0 0.5 1.0 1.5 2.0 2.5N%

101

102

103

Rm

0 5 10 15 20 25 30

Ra/Rac

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

N%

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Introduction Modelling Axial dipoles Conclusion

Role of inertia: local Rossby number

N% = 0.5 N% = 1.5 N% = 2.0

0.02 0.04 0.06 0.08 0.10 0.12 0.14Ro`

0.0

0.2

0.4

0.6

0.8

1.0

f dipax

0.02 0.04 0.06 0.08 0.10 0.12 0.14Ro`

0.0

0.2

0.4

0.6

0.8

1.0

f dipax

0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16Ro`

0.0

0.2

0.4

0.6

0.8

1.0

f dipax

Ro` =Roc `c/π

as a function of radius for

dipolar (solid lines) and

multipolar (dashed lines)

dynamos at (N% = 2.5,Pm = 2)

(thin lines) and (N% = 3,Pm = 4)

(thick lines).

0.4 0.6 0.8 1.0 1.2 1.4 1.6

r

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0.11

0.12

Ro? l

Ra = 5.4× 106;N% = 2.5

Ra = 6.4× 106;N% = 2.5

Ra = 7.4× 106;N% = 2.5

Ra = 9.0× 106;N% = 2.5

Ra = 8.0× 106;N% = 3.0

Ra = 9.0× 106;N% = 3.0

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Introduction Modelling Axial dipoles Conclusion

Role of inertia: local Rossby number

N% = 0.5 N% = 1.5 N% = 2.0

0.02 0.04 0.06 0.08 0.10 0.12 0.14Ro`

0.0

0.2

0.4

0.6

0.8

1.0

f dipax

0.02 0.04 0.06 0.08 0.10 0.12 0.14Ro`

0.0

0.2

0.4

0.6

0.8

1.0

f dipax

0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16Ro`

0.0

0.2

0.4

0.6

0.8

1.0

f dipax

Ro` =Roc `c/π

as a function of radius for

dipolar (solid lines) and

multipolar (dashed lines)

dynamos at (N% = 2.5,Pm = 2)

(thin lines) and (N% = 3,Pm = 4)

(thick lines).

0.4 0.6 0.8 1.0 1.2 1.4 1.6

r

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0.11

0.12

Ro? l

Ra = 5.4× 106;N% = 2.5

Ra = 6.4× 106;N% = 2.5

Ra = 7.4× 106;N% = 2.5

Ra = 9.0× 106;N% = 2.5

Ra = 8.0× 106;N% = 3.0

Ra = 9.0× 106;N% = 3.0

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Introduction Modelling Axial dipoles Conclusion

Snapshots

N% = 1.5,Pm = 0.75,Ra/Rac = 5 N% = 2.5,Pm = 2,Ra/Rac = 3.4B

r(r=

r o)

Vr

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Introduction Modelling Axial dipoles Conclusion

Table of contents

1 Introduction

2 ModellingGoverning equationsSet upMethods

3 Axial dipoles

4 Conclusion

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Introduction Modelling Axial dipoles Conclusion

ResultsDipolar solutions can be observed at high N%, provided high enoughPm are considered.The critical Rm of the dipolar branch seems scarcely affected by theincrease of N%.The higher N%, the faster is reached the critical Ro` above whichinertia causes the collapse of the dipole. This explains why dipolardynamos become confined in the parameter space.The scarcity of dipolar solutions for substantial N% would thus rathercome from the restriction of the parameter space being currentlyexplored (because of computational limitations), rather than anintrinsic modification of the dynamo mechanisms.

References

Schrinner, Petitdemange, Raynaud & Dormy, 2014, A&A, 564, A78Raynaud, Petitdemange & Dormy, 2014, A&A, 567, A107Raynaud, Petitdemange & Dormy, 2015, MNRAS, 448:2055–2065

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