Topology and field strength in spherical, anelastic dynamo simulations

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Topology and field strength in spherical, anelastic dynamo simulations

Raphaël Raynaud, Martin Schrinner, Ludovic Petitdemange, Emmanuel Dormy

To cite this version:

Raphaël Raynaud, Martin Schrinner, Ludovic Petitdemange, Emmanuel Dormy. Topology and field strength in spherical, anelastic dynamo simulations. AGU Fall Meeting, Dec 2014, San Francisco, United States. �hal-02428473�

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Topology and field strength in spherical,

anelastic dynamo simulations

Raphaël Raynaud, Martin Schrinner, Ludovic Petitdemange, Emmanuel Dormy

raphael.raynaud@ens.fr

LRA,Département de Physique, École normale supérieure, Paris

A

BSTRACT

Dynamo action, i.e. the self-amplification of a

magnetic field by the flow of an electrically con-ducting fluid, is considered to be the main mecha-nism for the generation of magnetic fields of stars and planets. Intensive and systematic parameter studies by direct numerical simulations using the Boussinesq approximation revealed fundamental properties of these models. However, this approx-imation considers an incompressible conducting fluid, and is therefore not adequate to describe convection in highly stratified systems like stars or gas giants. A common approach to overcome this difficulty is then to use the anelastic approxi-mation, that allows for a reference density profile while filtering out sound waves for a faster numer-ical integration. We present the results of a sys-tematic parameter study of spherical anelastic dy-namo models, and compare them with previous results obtained in the Boussinesq approximation. We discuss the influence of the stratification on the field geometry and the field strength, and also compare the different scaling laws for the velocity amplitude, the magnetic dissipation time, and the convective heat flux.

1. A

NELASTIC MHD EQUATIONS

•

The reference state:

decomposition of the thermodynamics vari-ables into the sum of a steady variable corresponding to the reference atmosphere

and a convective disturbance

f = f

_a

+ f

_c.

The reference state must be in mechanical and thermal quasiequilibrium, defined by :

– hydrostatic balance

−∇P

_a

+ ρ

_a

g = 0

– “well-mixed” isentropic reference state

∇S

_a

= 0

Then,

P = P

_c

ζ

n+1

, % = %

_c

ζ

n

, T = T

_c

ζ,

with

ζ = f (r, N

_%

, n, χ)

•

Navier-Stokes equation

D

_t

v = P m

−

1 E

∇

P

0

ζ

n

+

P m

P r

Ra

S

r

2

ˆ

r

−

2 E

ˆ

z × v + F

ν

₊

1 E ζ

n

(∇ × B) × B

F

_iν

= ζ

−n

∂

_j

ζ

n

(∂

_i

v

_j

+ ∂

_j

v

_i

) −

2₃

δ

_ij

∂

_k

v

_k

•

Induction equation

∂B

∂t

= ∇ × (v × B) + ∇

2

B

•

Heat transfert equation

D

_t

S = ζ

−n−1

P m

P r

∇ · ζ

n+1

_∇S

+

Di

ζ

E

−1

ζ

−n

(∇ × B)

2

+ Q

ν

with

Q

ν

= 2

e

_ij

e

_ij

−

1₃

(∇ · v)

2

together with the constraints

∇ · (ζ

n

v) = 0

∇ · B = 0

Our numerical solver PARODY reproduces the

anelastic dynamo benchmark (Jones et al. 2011).

2. C

ONTROL PARAMETERS

The system involves 7 dimensionless numbers:

Ra =

GM d∆S

νκc

_p

∈

10

4

_{, 10}

7

_{, P r =}

ν

κ

∈ [1, 2] ,

P m =

ν

η

∈ [1, 5] , E =

ν

Ωd

2

∈

10

−3

_{, 10}

−5

_,

N

_%

= ln

%

i

%

_o

∈ [0.1, 3.5] ,

n = 2,

χ =

r

i

r

_o

∈ [0.35, 0.60] .

5. S

CALING LAWS

Field strength Velocity

Lo f_ohm1/2

=

1.58 Ra

0.35_Q

1.19 Ra

0.34_Q

Ro = 1.66 Ra

0.42 Q

Ohmic dissipation time Heat flux

τ

_diss

= 0.75 Ro

−0.76

N u

?

= 0.25 Ra

0.59_Q

3. T

HEORETICAL ISSUES

•

use of a non steady reference state

•

consistency of the different variants of the

anelastic approximation

•

averaging the heat transfert equation

ρT D

_t

S = ∇ · ρT κ

t

∇S

+ ∇ · (K∇T ) + Q

ν

4. O

UTPUT PARAMETERS

•

Scaling laws –

N u

?

= (N u

_bot

− 1)

_{P r}E –

Ra

_Q

= (N u

_bot

− 1)

_rRa E₂ 3 o P r2

– dynamo efficiency

f

_ohm

=

D/P

,

where P is the power released by buoy-ancy and D is the ohmic dissipation

•

Field topology

– kinetic energy density

E

_k

−→ Ro =

√

2E

_k

E/P m

– magnetic energy density

E

_m

−→ Lo =

√

2E

_m

E/P m

– local Rossby number

Ro

_`

= Ro

_c

`

_c

/π

`

_c stands for the mean harmonic

de-gree of the velocity component

v

_c from

which the mean zonal flow has been subtracted

`

_c

=

X

`

< (v

c

)

`

· (v

c

)

`

>

< v

_c

· v

_c

>

.

– modified tilt angle of the dipole

Θ

?

=

2 π

*r

Θ(t) −

π

2

+

t

C

ONCLUSIONS

1. dichotomy between dipolar and multipolar dynamos extends to anelastic simulations

2. equatorial dipole not related to large

N

_%

3. consistency of the scaling laws between Boussinesq and anelastic simulations

4. toward observations: M dwarfs (bistabilty), equatorial dipole

References:

- Braginsky & Roberts, 1995, GAFD, 79, 1

- Christensen, 2010, Space Sci. Rev., 152, 565 - Duarte et al., 2013, PEPI, 222, 22

- Jones et al., 2011, Icarus, 216, 120 - Schrinner et al., 2012, APJ, 752, 121 - Schrinner et al., 2013, A&A, submitted - Yadav et al., 2013, Icarus, 225, 185

6. F

IELD TOPOLOGY

•

Dipolarity Boussinesq

f

_dip vs.

Ro

_` (Schrinner et al., 2012) Anelastic

(crosses indicate the presence of a strong equatorial dipole component)

P m

vs

N

_%

hB

_r

i

_φ (E = 10−4, P r = 1, variable Ra)

vs. time and colatitude

E = 10−4, P r = 2, P m = 4 Ra = 5 × 106, N_% = 3

•

Equatorial dipole E = 10−4, P r = 1, N% = 0.1 0.2 0.4 0.6 0.8 Ra 1e7 0 1 2 3 4 5 6 Pm 0.2 0.4 0.6 0.8 Ra 1e7 0 1 2 3 4 5 6 Pm

bistability modified tilt angle

Θ

?

(dip.: circle, mult.:square) (white:0, black:1)

Equatorial cut of Vr Equatorial cut of Br

B

_r