Simulations and Theory in FZD, February 25, 2010
Theoretical and Numerical Problems
of Magnetohydrodynamics
Frank Stefani
Institute of Safety Research, Department Magnetohydrodynamics
with thanks to:
Gunter Gerbeth, André Giesecke, Uwe Günther, Thomas Gundrum,
Thomas Wondrak, Mingtian Xu, Agris Gailitis (Riga), Karl-Heinz
Rädler (Potsdam), Günther Rüdiger (Potsdam), Rainer Hollerbach
(Leeds), Oleg Kirillov (Darmstadt), Sasa Kenjeres (Delft)...
Outline of the talk
• Motivation: Magnetic fields in the cosmos...and in the lab
• Basic theory: Induction equation and Navier-Stokes equation
• Simulations: From 3D to 0D...and slowly back
• Prospects: New experiments and numerical challenges
Motivation: Cosmic magnetic fields...
...are self-excited by the hydromagnetic dynamo effect
...help in cosmic structure formation
(stellar systems, black holes) by virtue of the magnetorotational instability (MRI) in accretion disks
Motivation: Cosmic magnetism in the liquid metal lab...
Dynamo experiments: Riga,
Karlsruhe, Cadarache, Madison, Maryland, Perm...
1
H1
H2
F
2
H3
3m
3
H4
4
H5
H6
5
0.8 m
MRI related experiments: Maryland, Princeton, Socorro, Grenoble, Rossendorf
Basic theory: What is left out...?
...A LOT !!!...sorry
• ...plasma foundation of MHD (kinetic models, Boltzmann (Vlassov)
equation, Landau’s collision integral, Particle-in-cell techniques...)
• ...derivation of the electrical conductivity of liquid metals (Ziman’s
formula, Born approximation, electron-ion pseudo-potentials...)
We restrict ourselves to the one-fluid description of MHD, and take the
conductivities as given!
Basic theory: Induction equation
j:
E:
σ:
electric current density;
electric field;
electrical conductivity;
B:
v:
µ0 :
magnetic field
velocity of the fluid
magnetic permeability
Pre-Maxwell equations:
Ampère’s law
Faraday’s law:
Magnetic field is source free:
Ohm’s law in moving conductors:
=⇒ Induction equation:
∇ × B = µ0(j + Ḋ)
∇ × E = −Ḃ
∇·B=0
j = σ(E + v × B)
Ḃ = ∇ × (v × B) + µ10σ ∆B
Basic theory: Induction equation
• Induction equation describes two competing effects
– Diffusion (decay) of magnetic field: Ḃ ∼
1
µ0 σ ∆B
– ”Stretch, Twist and Fold” (production): Ḃ ∼ ∇ × (v × B)
• Ratio of production to decay: ∼ magnetic Reynolds number
Rm := µσvL
• Rm must be large enough (Rm > 10) for self-excitation to occur
Example: Rm = 10 means for liquid sodium: vL = 1 m2/s
Basic theory: The interaction between B and v
• Induction equation for evolution of B under the influence of v:
1
Ḃ = ∇ × (v × B) +
∆B
µ0 σ
• Navier-Stokes-equation for evolution of v, including the (back-)reaction
of B on v
v̇ + (v · ∇)v = −
1
∇p
+
(∇ × B) × B + ν∆v + fextern
ρ
µ0 ρ
Basic theory: Dimensionless parameters
Reynolds:
Magnetic Reynolds:
Re
Rm
Hartmann:
Ha
Magnetic Prandtl:
Rossby:
Ekman:
Pm
Ro
Ek
= RV /ν
= RV µ0σ = Re Pm
q
σ
= BR ρν
= νµ0σ =: ν/λ
= 21 Ωr dΩ
dr
= ν/(2ΩR2)
Basic theory: Self-excitation
B0
ω
vxB
jxB v
j
0
B1
Inverse amplification B /B
Basic theory: The disk dynamo as a simple model
1
0.8
0.6
0.4
0.2
0
Self−excitation
↓
0
0.2 0.4 0.6 0.8 1
Normalized rotation rate ν L/R
Basic theory: Self-excitation in homogenous media
Right hand rule (a). α-effect (b). α2-dynamo (c)
(b)
(a)
(c)
B
B10
v
I
I2
I1
B 1+
B
I
B2+
B
B20
Basic theory: From kinematic to saturated regime
Basic theory: Some distinctions for numerical treatment
• Degree of coupling: Kinematic, inductionless, or fully coupled system
• Spatial dimensions: 0D (dispersion relation, WKB), 1D (e.g. in r), 2D
(e.g. in r, z), 3D (e.q. in r, ϕ, z)
• Treatment of small scales: Direct numerical simulations, Mean-field models, hyperdiffusivities, turbulence models (Large eddy simulations (LES),
Reynolds averaged Navier-Stokes (RANS))
• Numerical methods: Finite differences, Finite elements, Finite volumes,
Spectral elements, Integral equations, Boundary element methods
Simulations: The state of the art
Roberts and Glatzmaier (Nature 1995): First fully 3D simulation of an
Earth’s magnetic field reversals
Simulations: The state of the art
Kageyama et al. (Nature 2008): Simulation on a 2 × 511 × 514 × 1538
grid on 4096 processors (Earth Simulator): Vorticity ωz for Ek=2.3 10−7
(left) and Ek=2.6 10−7 (right)
Simulations: Stepping back from 3D to 0D
Simulation (MRI): Accretion disks in the cosmos
Jets
Black hole
Credit:
NASA/ESA
(Hubble)
Jet
Accretion
disk
Black Hole
Companion
Accretion disc
Credit:
NASA
Supermassive BH
Jet
X-ray binaries
Young star
(HH 30)
Accretion
disk
Young stars
Simulations (MRI): The problem with Rayleigh’s criterion
• Central problem: How do accretion disks work?
Central object (Star, Black Hole)
• Necessary outward angular momentum transport is not explainable by molecular viscosity.
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Accretion disk
• Turbulence could explain it.
However: Kepler-rotation is hydrodynamically stable (angular
momentum increases outward)!
Ω (r)~r −3/2
2 1/2
L(r)= Ω(r)r ~r
Simulations (MRI): A (too) simple spring model
• In Keplerian disks, inner masses orbit more rapidly than outer mass
Magnetic field acts similar as
a massless spring
• Inner masses slow down =⇒ reduce
angular momentum =⇒ move to
lower orbit
• Outer masses speed up =⇒ increase angular momentum =⇒ move
to higher orbit
• Spring tension increases =⇒ runaway process
Ω
=
i
r: Ω
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• Balbus/Hawley 1991: Revealed the astrophysical
relevance of MRI.
ha
ek
dr
an
Ch
v−
ho
ik
el
V
• Flow between inner and
outer cylinders with radii
ri and ra and angular frequencies Ωi and Ωo
3/2
ro
o
Ω
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:
always
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stable
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r
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i i
MRI
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as
• E.P. Velikhov, July 1,
1958, at Riga MHD conference: prove of MRI for
Taylor-Couette flow.
o
Simulations (MRI): Some history
Ωi
Simulations (MRI): Standard MRI and helical MRI
• Experiments on MRI with large Rm (pure axial fields) in Maryland, Princeton, New Mexico,
Moscow
• Hollerbach and Rüdiger (2005): Helical MRI: Bz
replaced by Bz + Bϕ: Recrit ∼ 103 instead of 106
• Potsdam ROssendorf Magnetic
Experiment (PROMISE)
InStability
• The ”only” Problem: Axial current ∼ 5000 Amp.
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0000
1111
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0000
1111
0000
1111
0000
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0000
1111
0000
1111
0000
1111
0000
1111
0000 B
1111
0000
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ϕ
0000
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0000
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Ωo Bz
Simulations (MRI): PROMISE 2
Outer ring
Inner ring
Inner rings rotate with the
inner cylinder, outer rings
with the outer cylinder
Simulations (MRI): Numerics with increasing complexity
• Local analysis (WKB): Liu et al. 2006, HMRI characterized as a weakly
destabilized inertial oscillation
• 1D global analysis: Hollerbach and Rüdiger 2005, Rüdiger et al. 2005,
Priede et al. 2007, own computations
• 1D global analysis with complex wavenumbers: Priede and Gerbeth 2009,
convective versus absolute instability (vanishing group velocity)
• 2D simulations: Szklarski and Rüdiger 2006, Szklarski 2007, 2008, Liu et
al. 2007, 2008
Simulations (MRI): PROMISE 2 - Increasing Icoil (Ha)
fin = 0.06 Hz. µ = 0.27, Irod = 8000 A
Fingering of angular momentum. Simulation by Jacek Szklarski.
Simulations (MRI): Comparison measurements/numerics
fin = 0.06 Hz. µ = 0.26, Icoil = 60 A. Irod = 6000 A
Measurement
Simulation (J. Szklarski)
For many more results see PRL 2006, NJP 2007, PRE 2009
Simulations (MRI): An strange paradox
R. Hollerbach 2008
Rayleigh line
– Continuous and monotonic transition between
standard MRI (SMRI)
and helical MRI (HMRI)
– HMRI is a destabilized
inertial oscillation while
SMRI is a destabilized
magneto-Coriolis wave
log(Re c )
• β = Bφ(ri)/Bz
Simulations (MRI): Dispersion relation from WKB
γ 4 + a1γ 3 + a2γ 2 + (a3 + ib3)γ + a4 + ib4 = 0
a1
=
a2
=
a3
=
a4
=
b3
=
2
„
√
Pm + √
1
Pm
«
,
a21
2
2
2
2
+ 2(1 + Ha∗ ) + 4β ∗ Ha∗ + 4Re∗ Pm(1 + Ro),
4
√
∗2
∗2
∗2
∗2
a1 (1 + Ha ) + 2a1β Ha + 8Re (1 + Ro) Pm,
“
”
∗2 2
∗2
∗2
∗2
∗2
∗2
1 + Ha
+ 4β Ha + 4Re + 4Re Ro(PmHa + 1),
∗
∗2
−8β Ha
∗√
Re
Pm,
∗
∗2
b4 = −4β Ha
∗
Re (2 + (1 − Pm)Ro).
Simulations (MRI): Resolving the paradox
HMRI appears at an EXCEPTIONAL POINT of the SMRI branch with the
inertial wave branch, at small but finite Pm (Kirillov and F.S., APJ 2010)
Simulations (MRI): Resolving the paradox
Another view on the exceptional point
Simulations (MRI): Resolving the paradox
Still, SMRI and HMRI are continuosly connected!
Simulations: Going from 0D to 1D
Simulations (reversals): Polarity reversals during the last
150 Myr
• Reversal rate between nearly zero (during superchrons) and ∼4/Myr
presently. Last reversal occurred 780 kyr ago.
Mean number of field reversals per Million years
Sequence of normal (black) and reversed (white) polarity
Simulations (reversals): Asymmetry, Clustering,
Stochastic resonance
Last five reversals
• Slow decay, but very
fast recreation (∼ 5
kyr) of dipole with opposite polarity
Brunhes-Matuyama
Upper Jaramillo
Lower Jaramillo
Upper Olduvai
Lower Olduvai
Average
10
VADM [1022 A m2]
• Saw-tooth shape of reversals
8
6
4
2
from: Valet et al. 2005
0
-80
-60
-40
-20
0
Time from transition [kyr]
20
Simulations (reversals): Questions to address
• Is there a generic and universal mechanism of reversals?
• What is a minimum model to explain the three reversal features?
• How to tailor dynamo experiments so that they show reversals?
• Can reversal features be used to constrain essential parameters
of the geodynamo?
PRL 2005, EPSL 2006, ..., Inverse Probl. 2009
Simulations (reversals): A simple reversal model
• Mean-field dynamo source is the helical turbulence parameter part α
• Induction equation: Ḃ = ∇ × (αB) + (µ0σ)−1 ∆B. Decomposition in
toroidal and poloidal field components: B = −∇ × (r × ∇S) − r × ∇T
• α is assumed spherically symmetric (this is not true for the Earth!) =⇒
Two partial differential equations for sl(r, t) and tl(r, t)
∂sl
∂t
∂tl
∂t
=
=
1 ∂2
l(l + 1)
(rs
)
−
sl + α(r, t)tl ,
l
2
2
r ∂r
r
1∂
∂
d
l(l + 1)
(rtl) − α(r, t) (rsl) −
[tl − α(r, t)sl] .
r ∂r ∂r
dr
r2
Simulations (reversals): Implementing back-reaction
• Algebraic α-quenching with averaged magnetic energy
α(r, t) = C
1+Q
2s21 (r,t)
r2
+ r12
αkin
∂(rs1 (r,t))
∂r
2
+ t21(r, t)
+ξ1(t) + ξ2(t) r 2 + ξ3(t) r 3 + ξ4(t) r 4 ,
• Noise: hξi(t)ξj (t + t1)i = D 2(1 − |t1|/τ )Θ(1 − |t1|/τ )δij .
• C - normalized dynamo number, D - noise amplitude, Q - quenching
parameter, τ - noise correlation time
Simulations (reversals): Relaxation oscillations
van der Pol oscillator
∂ 2y
∂t2
=
µ(1 − y 2 )
∂y
−y
∂t
equivalent to:
∂y
∂t
∂z
∂t
=
z
=
µ(1 − y )z − y
Spherically symmetric α2 dynamo with αquenching and noise ξ(r, τ ) (intensity D)
∂sl
∂τ
∂tl
∂τ
=
=
2
α(r, τ )
=
1 ∂2
l(l + 1)
(rs
)
−
sl + α(r, τ )tl ,
l
r ∂r 2
r2
„
«
1 ∂
∂
∂
(rtl ) − α(r, τ ) (rsl )
r ∂r ∂r
∂r
−
l(l + 1)
[tl − α(r, τ )sl ]
r2
C
αkin (r)
+ ξ(r, τ )
0
1 + Emag (r, τ )/Emag
2
y’
1
0
-1
-2
µ=0
µ=1
µ=2
-3
-4
-2
-1.5
-1
-0.5
0
y
0.5
1
1.5
2
0.8
0.6
0.4
α 2 dynamo
3
1
s’(r=1)
4
van der Pol oscillator
Simulations (reversals): Relaxation oscillations
0.2
0
-0.2
-0.4
-0.6
C=6.8
C=7.0
C=7.237
-0.8
-1
-0.4 -0.3 -0.2 -0.1
0
s(r=1)
0.1
0.2
0.3
0.4
10
15
25
20
25
0
5
10
15
2
0
-2
0
5
10
15
20
t
20
t
35
40
µ=2
30
35
40
µ=5
t
2
0
-2
30
25
30
35
40
µ=20
25
30
35
40
van der Pol oscillator
s(r=1)
0
0.4
0.2
0
-0.2
-0.4
0.4
0.2
0
-0.2
-0.4
1
0.5
0
-0.5
-1
0
0
2
2
2
4
6
t
C=7.2
t
C=7.237
4
6
4
6
8
10
8
10
8
10
8
10
α 2 dynamo
5
20
t
s(r=1)
15
-0.1
s(r=1)
10
2
0
-2
0
y
5
C=6.8
0
s(r=1)
y
0
y
0.1
µ=0
2
0
-2
van der Pol oscillator
y
Simulations (reversals): Relaxation oscillations
t
C=7.24
0
2
4
6
t
αkin(r) = C · (1 − 6r 2 − 5r 4)
Simulations (reversals): Spectral features
1.4
40
1.2
20
Growth rate
1
α(r)
0.8
0.6
0.4
0 n=1
-20 n=2
-40
n=3
-60
n=4
0.2
-80
0
-100
0
0.2
0.4
0.6
0.8
1
r
Constant α along r yields ....
n=5
0
2
n=6
4
6
8
C
... a purely real spectrum
10
Simulations (reversals): Spectral features
15
40
10
20
Growth rate
5
α(r)
0
-5
0 n=1
-20 n=2
-40
n=3
-15
-60
n=4
-20
-80
-25
-100
-10
0
0.2
0.4
0.6
0.8
r
Varying α along r yields ....
1
n=5
0
0.5
n=6
1
C
n=7
1.5
... also complex eigenvalues
(remember Uwe Günthers talk on PT-symmetric QM !)
2
Growth rate
Simulations (reversals): The basic mechanism
Noise triggered
Unstable
jump
fix point
Stable
fix point
0
y
r
o
t
la
il
c
Os
y
or
t
a
l
Sign change
and field
recovery
il
c
s
o
−
n
No
Self−accelerating
field decay
−o
Non
sc
ry
illato
Rapid
α −quenching
Exceptional point
Simulations (reversals): Parameter fit for the geodynamo
Three functionals to be minimized simultaneously:
1. Mean deviation from averaged asymmetric reversal curve (Valet 2005)
2. Mean deviation from clustering curve (Cande/Kent 95, Carbone 2005)
3. Mean deviation from residence time distribution curve (Consolini/De
Michelis 2003)
Simulations (reversals): Parameter fit for the geodynamo
• Downhill simplex method is used to determine the following parameters:
1. C - Dynamo strength; C/Ccrit - Supercriticality
2. δ - Shift parameter
3. ǫ - Relative strength of periodic forcing (Milankovic cycle 95 kyr)
4. D - Noise intensity
5. τdif f - Effective diffusive time scale: τdif f = µ0σturbR2 (β-effect)
Simulations (reversals): Parameter fit for the geodynamo
• Various results depending on relative weight of individual functionals
• Parameters of three versions resulting from the downhill simplex inversion.
Version 1
Version 2
Version 3
C
57.0
96.8
147.8
δ
-0.03
-0.16
-0.18
Ccrit
13.0
8.99
8.73
C/Ccrit
4.4
10.8
16.9
D
6.3
8.0
7.8
Td
86.8 kyr
63.7 kyr
55.1 kyr
ǫ
0.133
0.123
0.094
#/Myr
2.05
3.07
2.91
VADM [1022 A m2]
Simulations (reversals): 1. Asymmetry
8
7
6
5
4
3
Real
Version 1
2
Version 2
1
Version 3
0
-60 -50 -40 -30 -20 -10
0
Time from transition [kyr]
10
20
Simulations (reversals): 2. Clustering
1
P(h>H)
0.8
0.6
Real
0.4 Version 1
Version 2
0.2 Version 3
Poisson
0
0
0.2
0.4
0.6
H
0.8
1
Probability density [1/Myr]
Simulations (reversals): 3 - Residence time distribution
4
Real
Version 1
Version 2
Version 3
3
2
1
0
0
0.2
0.4
0.6
0.8
1
τ [Myr]
Stochastic resonance with Milankovich cycle of Earth’s orbit eccentricity
Simulations (reversals): Best results for ...
• Supercriticality of the geodynamo: Factor 10
• Relative strength of periodic forcing: around 10 per cent
• Diffusion time: 64 kyr, i.e reduction by a factor 3.5 compared to 225 kyr
resulting from molecular conductivity. This is in rough agreement with
recent numerical simulations (Schrinner et al. 2007).
Simulations (reversals): Conclusions for reversals
• Reversals are (very likely...) noise triggered relaxation oscillations in the
vicinity of exceptional points of the spectrum of the dynamo operator.
• Highly supercritical dynamos self-tune into a reversal-prone state (”edge
of chaos”?, ”self-organized criticality”?, ”punctuated equilibrium”?).
• Asymmetry, clustering, and stochastic resonance can be used as proxies
for constraining essential parameters of the geodynamo.
• The estimated conductivity reduction, if anisotropic, would be important
for dynamo simulations (selection of axial or equatorial dipole, etc.)
Simulations: Going from 1D to 2D and 3D
Simulations (dynamo experiments): Riga
• 2 m3 sodium
• Propeller (1). Helical
flow (2). Back flow
(3). Sodium at rest
(4).
1
H1
H2
F
H3
3m
• Two motors (200
kW) produce flow up
to 20 m/s.
2
3
H4
4
H5
H6
5
0.8 m
Simulations (dynamo experiments): Riga
• Optimized flow (maximum helicity!)
• Self-excitation at propeller rotation rate of ∼ 1900 rpm.
• Field structure rotates slowly (1-2 Hz)
around vertical axis.
• Axial field component: up to 120 mT
Simulations (Riga): A simple saturation model
• Weak-field dynamo: Linearization of Navier-Stokes-equation, Perturbation δv (and δp) with respect to measured unperturbed velocity v̄
∂δv
∇δp
1
+ (v̄ · ∇)δv + (δv · ∇)v̄ = −
+
(∇ × B) × B + ν∆δv .
∂t
ρ
µ0 ρ
• Inviscid approximation
• Restrictions to m = 0 mode in the Lorentz force and to m = 1 mode in
the magnetic field
• Radial and axial Lorentz force components are assumed to be absorbed
into pressure (to a large extend)
Simulations (Riga): A simple saturation model
• Focus on azimuthal component (cannot be absorbed into pressure)
(v̄z
1
∂
)δvφ =
[(∇ × B) × B]φ
∂z
µ0 ρ
• Fix the field strength to the measured excess power (via energy balance)
d
dt
Z
B2
dV = −
2µ0
Z
j2
dV −
σ
Z
v · (j × B) dV .
• Insert the perturbed velocity field into induction equation solver and
compare the output with observational facts
Simulations (Riga): Computed Lorentz force
3
20 40 60 80 100
3
fz[kN/m ]: -14-12-10 -8 -6 -4 -2 0 2 4 6 8 10 12 14
3
fp[kN/m ]: -6
-4
1
1
0
0
0
-1
z
1
z
z
fr[kN/m ]: -100 -80 -60 -40 -20 0
-1
-2
0
2
4
6
8
10
12
-1
0 0.05 0.1 0.15 0.2
0 0.05 0.1 0.15 0.2
0 0.05 0.1 0.15 0.2
r
r
r
frLorentz
fzLorentz
fϕLorentz
Simulations (Riga): Braking of vφ
• Hence, the differential rotation is decreased (Lenz’s
rule !)
Vp[m/s]: -12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
dVp[m/s]: -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2
Vp[m/s]: -12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
1
1
0
0
0
-1
z
1
z
z
• The azimuthal component
is braked in the inner cylinder and accelerated in the
back-flow cylinder.
-1
-1
0 0.05 0.1 0.15 0.2
0 0.05 0.1 0.15 0.2
0 0.05 0.1 0.15 0.2
r
r
r
v̄φ
dvφ
vφ
Simulations (Riga): saturation model
• Simultaneous solution of 2D-induction equation
∂B
1 2
= ∇ × (v(t) × B) +
∇ B
∂t
µ0 σ
and the simplified equation for δvφ:
1
∂
[(∇ × B(t)) × B(t)]φ
(v̄z )δvφ(t) =
∂z
µ0 ρ
• Gailitis et al.: Comptes Rend. Phys. 9, 721-728 (2008), Stefani et al.:
ZAMM 88, 930-954 (2008), Stefani et al.: Theor. Comp. Fluid Dyn.
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1.2
1.8
1.6
Kinematic: numerical
Kinematic: measured
Saturated: numerical (old)
Saturated: numerical (new)
Saturated: measured
-1.4
-1.6
1000 1200 1400 1600 1800 2000 2200 2400 2600 2800
Rotation rate [1/min]
Frequency [1/s]
Growth rate [1/s]
Simulations (Riga): Comparison Experiment/Simulation
1.4
1.2
1
0.8
0.6
0.4
0.2
Kinematic: numerical
Kinematic: measured
Saturated: numerical (old)
Saturated: numerical (new)
Saturated: measured
1000 1200 1400 1600 1800 2000 2200 2400 2600 2800
Rotation rate [1/min]
Good correspondence with measured growth rate and frequency in the kinematic and saturation regime
Simulations (dynamo experiments): Cadarache (VKS)
VKS experiment (Von Karman sodium): Helical flow with two poloidal and two toroidal eddies driven by counter-rotating impellers
Lid layers
90
200
80
360
Side layer
Simulations (dynamo experiments): Cadarache (VKS)
Simulation for an analytical test flow
Isosurface of magnetic energy
Magnetic field lines
Simulations (dynamo experiments): Cadarache (VKS)
• Detrimental role of rotating
layers behind impellers on the
m = 1 mode (EJM/B 2006)
• Using soft iron impellers
(µrel ∼ 70) self-excitation was
achieved (Monchaux et al.
2007)
Simulations (dynamo experiments): Cadarache (VKS)
Very nice field reversals (Berhanu et al. 2007), and extremely rich dynamics
Simulations (dynamo experiments): Cadarache (VKS)
• Recent attempts to explain low Rmcrit ∼ 31 and axial dipole (How to
bypass Cowling’s theorem???)
– Non-linear back-reaction producing non-axisymmetric flow components (Gissinger et al., PRL 2008)
– α − Ω dynamo with α concentrated at impellers (Laguerre et al., PRL
2008). However: Problem with normalization of α...
– Role of azimuthally drifting vortices
– Role of soft-iron impellers ???
Simulations (dynamo experiments): Cadarache
Take into account the correct geometry and the soft-iron impellers
Simulations (dynamo experiments): Cadarache (VKS)
We have to solve the generalized induction equation, including µrel and
the helical turbulence parameter α:
∂B
1
B
= ∇ × v × B + αB −
∇×
∂t
µ0 σ
µrel
New Finite-Volume/Boundary element code: A. Giesecke et. al.
Magnetohydrodynamics 2009, PRL 2010
Simulations (dynamo experiments): VKS
Rm=30, α=−1.5 Rm=70, α=0 Rm=30, α=0
growing m=0 growing m=1 decaying m=0
Simulations (dynamo experiments): VKS
• The soft-iron impellers
work strongly in favour of
the m=0 mode
• Still, some small amount of
helical turbulence is needed
to excite it (A. Giesecke et
al., Phys. Rev. Lett. 2010)
Prospects: New experiments and numerical challenges
Prospects: Azimuthal MRI (AMRI)
00
11
µ=0.25
00
11
00
11
00
11
00
11
µ=0.26
00
11
00
11
00
11
µ=0.27
00
11
00
11
00
11
µ=0.28
00
11
00
11
00
11
Hollerbach et al. PRL 2010
Prospects: Tayler instability
Filling tube
0−8000 A
Copper
0
1
0
electrode 1
0
1
Copper
electrode
GaInSn
75 cm
10 cm
Measuring electrodes
0
1
Water
cooling
6
µB=500
5
2
5.5
Ifluid [kA]
UDV sensor
5
4.5
4
3.5
3
0
0.05 0.1 0.15 0.2 0.25 0.3
00
00
11
rin 11
/rη out
00
11
00
11
Prospects: Spherical Couette
V
111111
000000
000000
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000000
111111
000000
111111
000000
111111
1111111111
0000000000
0000000000
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0000000000
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0000000000
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0000000000
1111111111
UDV
6 cm
Bz
111111111
000000000
000000000
111111111
000000000
111111111
000000000
111111111
UDV
000000000
111111111
GaInSn
00
11
00
11
00
11
00
11
00
11
00000
11111
00000
11111
00000
11111
00000
11111
stable
00
11
00
11
Ha
00
11
00
11
V
UD
1111111111111111
0000000000000000
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
0000000000000000
1111111111111111
111111
000000
000000
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000000
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000000
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unstable
00000
11111
00000
11111
00000
11111
00000
11111
unstable
Recrit
18 cm
000000
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000000
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0−0.2 Hz
000000
111111
UD
00
11
11
00
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
Hollerbach 2009
Prospects: Alfvén wave experiment at HLD
Ampoule with liquid
Rubidium
(hmmmm!!!)
Interaction of shear and compressional Alfvén
waves: Much more numerics needed...
Prospects: DRESDYN, MRI+Tayler in sodium
Prospects: DRESDYN, Precession experiment
• Goal: Large experiment
• J. Léorat (Meudon): Water experiment in cylinder
(ATER)
• A. Tilgner: Phys. Fluids
crit
2005, Rm
around 200
(based on unstable modes)
• Much more numerical studies needed before experiment can be designed
Some review papers
• Gailitis A., Lielausis, O., Platacis, E., Gerbeth, G., Stefani, F. (2002):
Colloquium: Laboratory experiments on hydromagnetic dynamos, Rev.
Mod. Phys. 74, 973-990
• Stefani, F., Gailitis, A., Gerbeth, G. (2008): Magnetohydrodynamic
experiments on cosmic magnetic fields, ZAMM 88, 930-954
• Stefani, F., Giesecke, A., Gerbeth, G. (2009): Numerical simulations
of liquid metal experiments on cosmic magnetic fields, Theor. Comp.
Fluid Dyn. 23, 405-429