Evolution of an axisymmetric magnetic field in a
differentially rotating radiative zone
M. Gaurat, L. Jouve, F. Lignières
IRAP-OMP
Toulouse
MHD Days
3 december 2014 Potsdam
M. Gaurat, L. Jouve, F. Lignières
Evolution of an axisymmetric magnetic field in a differentially rotating radiative zone
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Introduction
Interaction axisymmetric magnetic field/differential rotation with a
density gradient but no meridional flows → 2D MHD numerical
simulations
Previous numerical works on this subject
Explain the solid body rotation of the Sun’s radiative zone :
Charbonneau & MacGregor (1992,1993), Rüdiger & Kitchatinov
(1996), Spada et al. (2010), ...
Aim of our work
Study the stability of various magnetic configurations → systematic
study of the characteristics of the magnetic field (topology, Bϕ /Bp )
on short time scales (Alfvén time)
M. Gaurat, L. Jouve, F. Lignières
Evolution of an axisymmetric magnetic field in a differentially rotating radiative zone
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Governing equations
Assumptions
Axisymmetry, no meridional flows, µ and η uniform, Bp time-independent
Governing equations
−
→→
−
∂Bϕ
1
1
=
r
sin
θ(
B
.
∇)Ω
+
∆
−
B
p
∂t
Lu
r 2 sin 2 θ ϕ
−
→
→
−
Pm
1
ρr sin θ ∂Ω
∆−
∂t = r sin θ (Bp . ∇)(r sin θBϕ ) + Lu
Lu =
tη
tAp
Bϕ
Bp
α ΩΩAp
=
Lundquist number, Pm =
where ΩAp =
tη
tν
1
r 2 sin 2 θ
(r sin θ Ω)
magnetic Prandtl number
B
√p
R 4πρ
+IC (Bϕ = 0 and Ω) / BC
−
→
Dipolar Bp
n=3 polytropic profile for ρ
M. Gaurat, L. Jouve, F. Lignières
Evolution of an axisymmetric magnetic field in a differentially rotating radiative zone
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The different cases studied
Dimensionless number
Lu = [10 ; 102 ; 103 ; 104 ; 105 ]
Pm = 1
Profile of ρ
ρc /ρ0 = 1; 560; 7.103 ; 2.104
3, 5.104 ; 7.104 ; 105 ; 3.105 ; 106
Initial profile of Ω
4 radial profiles
2 cylindrical profiles
BC on Ω at the inner boundary
no-slip : Ω = cste
stress-free : ∂Ω/∂r = 0
Spatial grid mesh
128x128 ; 256x256 ; 400x400 ; 512x512
Time step
from 10−3 to 10−8 tAp
M. Gaurat, L. Jouve, F. Lignières
Evolution of an axisymmetric magnetic field in a differentially rotating radiative zone
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A typical magnetic configuration obtained
M. Gaurat, L. Jouve, F. Lignières
Evolution of an axisymmetric magnetic field in a differentially rotating radiative zone
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Asymptotic behaviours : Lu and ρ
6
3.0
max(Bϕ /Bp )
5
2.5
4
tmax
2.0
3
1.5
2
1.0
1
0
100
102
103
Lu
104
105
0.5 0
10
106
10
101
102
1
103
Lu
104
105
106
max(Bϕ /Bp ) ×
q
ρ0
ρc
10
101
1
tmax ×
10-1
10-2 -1
10
100
q
ρ0
ρc
100
10-1
100
101
102
103
ρc /ρ0
104
105
106
107
10-2 -1
10
100
101
102
103
ρc /ρ0
104
105
106
M. Gaurat, L. Jouve, F. Lignières
Evolution of an axisymmetric magnetic field in a differentially rotating radiative zone
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Modeling and comparison to the simulations
Perturbations propagate as Alfvén
−
→
waves along Bp
⇒ 2D problem → 1D problem
Estimate of max(Bϕ /Bp ) and
tmax
Choose
R L ds the field line in which
is maximal
0 vAp
→ gives directly tmax
→ then solve 1D problem to get
max(Bϕ /Bp )
Example of a simple solution
Bϕ
Bp (t
θ∆Ω
= tmax ) = r sinvAp
cos( π2 Ls )
→ gives predictions in good agreement
with the simulations (Gaurat et al. in
prep.)
M. Gaurat, L. Jouve, F. Lignières
Evolution of an axisymmetric magnetic field in a differentially rotating radiative zone
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Instability of the magnetic configurations
Likely to be subject to MRI or Tayler instability → depends on
Bϕ
Ω/ΩAϕ (Jouve et al. 2015), where ΩAϕ = r √4πρ
Unstable cases (with Ogilvie 2007 dispersion relation)
Ω/ΩAϕ = 10
σMRI = qΩ
2
where q = dlnΩ
dlnr
Ω/ΩAϕ = 1
σTI = ΩAϕ
M. Gaurat, L. Jouve, F. Lignières
Evolution of an axisymmetric magnetic field in a differentially rotating radiative zone
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Instability of the magnetic configurations
Background axisymmetric field oscillates with a frequency ≈ ΩAp
To modify the spatial structure of the field : ΩσAp 1
MRI :
σ
ΩAp
=
q Ω
2 ΩAp
Ω
TI : ΩσAp = ΩAϕ
= α ΩΩAp where α depends on the physical
Ap
parameters → given by the simulations and also by the model
⇒
Ω
ΩAp
1
M. Gaurat, L. Jouve, F. Lignières
Evolution of an axisymmetric magnetic field in a differentially rotating radiative zone
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Conclusion
Numerous magnetic configurations of a wound-up field
obtained with 2D simulations
Extrapolation to realistic density profiles
Simple model gives good predictions of max(Bϕ /Bp ) and tmax
Possibility to test the stability of the magnetic configurations
obtained and to calculate the growth rate of the instability
Possible application
Explain the dichotomy observed in the distribution of the magnetic
field of intermediate-mass stars
M. Gaurat, L. Jouve, F. Lignières
Evolution of an axisymmetric magnetic field in a differentially rotating radiative zone
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