European Conference on Computational Fluid Dynamics
ECCOMAS CFD 2006
P. Wesseling, E. Oñate and J. Périaux (Eds)
c TU Delft, The Netherlands, 2006
THE INTEGRAL EQUATION APPROACH TO KINEMATIC
DYNAMO THEORY AND ITS APPLICATION TO
DYNAMO EXPERIMENTS IN CYLINDRICAL GEOMETRY
M. Xu, F. Stefani and G. Gerbeth
Forschungszentrum Rossendorf
P.O. Box 510119, D-01314 Dresden, Germany
e-mail: [email protected]
Key words: Dynamo theory, Integral equations, Magnetic induction, Dynamo experiments
Abstract. The conventional magnetic induction equation that governs hydromagnetic
dynamo action is transformed to an equivalent integral equation system. An advantage
of this approach is that the computational domain is restricted to the region occupied
by the electrically conducting fluid and to its boundary. This integral equation approach
is applied to simulate kinematic dynamos within cylindrical geometry including the ”von
Kármán sodium” (VKS) experiment and the Riga dynamo experiment. A modified version
of this approach is utilized to investigate magnetic induction effects under the influence
of externally applied magnetic fields. The computed induced magnetic fields for the VKS
experiment show a satisfactory agreement with the experimental results.
1
INTRODUCTION
Dynamo theory is usually invoked to explain the self-excitation of cosmic magnetic
fields, including the fields of planets, stars, and galaxies1 . Dynamo action occurs if a
seed magnetic field is amplified and sustained by the flow of an electrically conducting
fluid. As long as the magnetic field is weak and its influence on the velocity field is
negligible we speak about the kinematic dynamo regime. When the magnetic field has
gained higher amplitudes the velocity field will be modified, and the dynamo enters its
saturation regime. This complicated saturation regime, which is presently under intense
debate, will not be considered in the present paper.
The usual way to describe the kinematic dynamo action is based on the induction
equation for the magnetic field B,
∂B
1
= ∇ × (u × B) +
∆B, ∇ · B = 0,
∂t
µσ
(1)
where u is the given velocity field, µ the permeability of the fluid, and σ its electrical
conductivity. The behaviour of the magnetic field B in Eq. (1) is controlled by the
1
M. Xu, F. Stefani and G. Gerbeth
magnetic Reynolds number Rm = µσLU , where L and U are typical length and velocity
scales of the flow, respectively. When the magnetic Reynolds number reaches a critical
c
value, denoted by Rm
, the dynamo action starts.
Equation (1) follows directly from pre-Maxwell’s equations and Ohm’s law in moving
conductors. In order to make this equation solvable, boundary conditions of the magnetic
field must be prescribed. In the case of vanishing excitations of the magnetic field from
outside the considered finite region, the boundary condition of the magnetic field is given
as follows:
B = O(r −3 ) as r → ∞.
(2)
Kinematic dynamos are usually simulated in the framework of the differential equation
approach by solving the induction equation (1). For spherical dynamos, as they occur in
planets and stars, the problem of implementing the non-local boundary conditions for the
magnetic field is easily solved by using decoupled boundary conditions for each degree
of the spherical harmonics. For other than spherically shaped dynamos, in particular
for galactic and some of the recent laboratory dynamos2 , the handling of the non-local
boundary conditions is a notorious problem.
The simplest way to circumvent this problem is to replace the non-local boundary
conditions by simplified local ones (vertical field condition). This is often used in the
simulation of galactic dynamos3 .
For the simulation of the cylindrical Karlsruhe dynamo experiment, the actual electrically conducting region was embedded into a sphere, and the region between the
sphere and the surface of the dynamo was virtually filled by a medium of lower electrical conductivity4,5 .
Of course, both methods are connected with losses of accuracy. In order to fully
implement the nonlocal boundary condition, Maxwell’s equations must be fulfilled in
the exterior, too. This can be implemented in different ways. For the finite difference
simulation of the Riga dynamo, the Laplace equation was solved (for each time-step) in
the exteriour of the dynamo domain and the magnetic field solutions in the interiour and
in the exterior were matched using interface conditions6 . A similar method, although
based on the finite element method, was presented by Guermond et al.7 . Another, and
quite elegant, technique to circumvent the solution in the exteriour was presented by
Iskakov et al.8,9 where a combination of a finite volume and a boundary element method
was used to circumvent the discretization of the outer domain.
An alternative to the solution of the induction equation is the integral equation approach (IEA) for kinematic dynamos which basically relies on the self-consistent treatment
of Biot-Savart’s law. For the case of steady dynamo acting in infinite domains of homogeneous conductivity, the integral equation approach had already been employed by a few
authors10,11,12,13 . For the case of finite domains, the simple Biot-Savart equation has to
be supplemented by a boundary integral equation for the electric potential14,15 . If the
2
M. Xu, F. Stefani and G. Gerbeth
magnetic field becomes time-dependent, yet another equation for the vector potential has
to be added16 .
In the present work, the integral equation approach is applied to various dynamos in
cylindrical geometry. We start with the derivation of the numerical scheme for cylindrically symmetric problems. Then we switch over to the treatment of specific problems,
including the free decay case, the ”von Kármán sodium” (VKS) experiment17,18 and the
Riga dynamo experiment19,20,21 . Two variants of the approach are presented: in the first
one, it is implemented as an eigenvalue solver to solve genuine dynamo problems. In the
second one, it is used to treat induction effects in the case of externally applied magnetic
fields.
2
MATHEMATICAL FORMULATION
Assume the electrically conducting fluid be confined in a finite region V with boundary
S, the exterior of this region filled by insulating material or vacuum. Then, dynamo and
induction processes can be described16 by the following integral equation system:
µσ Z (u(r0 ) × (B0 (r0 ) + b(r0 )) × (r − r0 ) 0
dV
4π V
|r − r0 |3
Z
Z
µσλ
A(r0 ) × (r − r0 ) 0 µσ
r − s0
0
0
dV
−
dS 0
(3)
−
φ(s
)n(s
)
×
4π V
|r − r0 |3
4π S
|r − s0 |3
Z
1
1
(u(r0 ) × (B0 (r0 ) + b(r0 )) · (s − r0 ) 0
φ(s) =
dV
2
4π V
|s − r0 |3
Z
Z
s − s0
λ
A(r0 ) · (s − r0 ) 0
1
0
0
φ(s )n(s ) ·
−
dV −
dS 0
(4)
0
3
0
3
4π V
|s − r |
4π S
|s − s |
Z
Z
1
(B0 (r0 ) + b(r0 )) × (r − r0 ) 0
1
B0 (s0 ) + b(s0 ) 0
0
A(r) =
dV +
n(s ) ×
dS , (5)
4π V
|r − r0 |3
4π S
|r − s0 |
b(r) =
where B0 is the externally applied magnetic field, b the induced magnetic field, u the
velocity field, µ the permeability of the fluid under consideration, σ the electrical conductivity, A the vector potential, and φ the electric potential. n denotes the outward
directed unit vector at the boundary S. The time dependence of all electromagnetic
fields is ∼ exp λt where the real part of λ is the growth rate, and its imaginary part the
frequency of the fields.
In the following, we focus on the cylindrical geometry. The electrically conducting fluid
is in a cylinder with the radius Ra and height 2H. Introducing the cylindrical coordinate
system (ρ, ϕ, z), we have
r = [ρ cos ϕ, ρ sin ϕ, z]T , r0 = [ρ0 cos ϕ0 , ρ0 sin ϕ0 , z 0 ]T ,
b = [bρ , bϕ , bz ]T , u = [uρ , uϕ , uz ]T .
(6)
The magnetic field b, the electric potential φ, and the vector potential A can be written
3
M. Xu, F. Stefani and G. Gerbeth
in the following series form:




b
X  bm 
 
φ
=
 φm  exp(imϕ).
 
A
Am
(7)
Under the assumption that the velocity field is axisymmetric, one can see that [bm , φm ,
Am ]T (m = 0, ±1, ±2, · · ·) are decoupled with respect to m and they only depend on the
variables (ρ, z). In the following, we always re-denote [bm , φm , Am ]T as [b, φ, A]T for
abbreviation. Therefore, Eq.(3) becomes
bρ
Z
Z
µσ H Ra
=
[
[((z − z 0 )Ecm uz − iρ0 Esm uϕ )(B0ρ + bρ ) + (−i(z − z 0 )uz Esm
4π −H 0
+iρ0 Esm uρ )(B0ϕ + bϕ ) + (i(z − z 0 )Esm uϕ − (z − z 0 )uρ Ecm )(B0z + bz )]ρ0 dρ0 dz 0
−
Z
−λ
bϕ =
Ra
0
Z
φρ02 Esm |z 0 =H dρ0 −
H
−H
Z H
Z
0
Z
H
φRa(z − z 0 )Esm |ρ0 =Ra dz 0 +
−H
Z
Ra
0
φρ02 Esm |z 0 =−H dρ0
((z − z 0 )Esm Aρ + (z − z 0 )Ecm Aϕ + ρ0 Esm Az )ρ0 dρ0 dz 0 ],
(8)
Ra
µσ
[
[(−(ρE1m − ρ0 Ecm )uϕ + i(z − z 0 )uz Esm )(B0ρ + bρ )
4π −H 0
+((ρE1m − ρ0 Ecm )uρ + (z − z 0 )uz Ecm )(B0ϕ + bϕ ) + (−(z − z 0 )uϕ Ecm
−i(z − z
+
Z
−λ
bz
Ra
Z
H
−H
H
Z
)uρ Esm )(B0z
φRa(z − z
−H
Z H
µσ
=
[
4π
0
−H
Z
Ra
0
Z
0
0
+ bz )]ρ dρ dz −
)Ecm |ρ0 =Ra dz 0
−
Z
Ra
0
Z
Ra
0
φ(ρρ0 E1m |z 0 =H − ρ02 Ecm |z 0 =H )dρ0
φ(−ρE1m |z 0 =−H + ρ0 Ecm |z 0 =−H )ρ0 dρ0
(−(z − z 0 )Ecm Aρ + (z − z 0 )Esm Aϕ + (ρE1m − ρ0 Ecm )Az )ρ0 dρ0 dz 0 ], (9)
Ra
0
0
0
[(ρ0 E1m − ρEcm )uz (B0ρ + bρ ) + iρEsm uz (B0ϕ + bϕ )
+((−ρ0 E1m + ρEcm )uρ − iρEsm uϕ )(B0z + bz )]ρ0 dρ0 dz 0 +
−λ
Z
H
−H
Z
Ra
0
Z
H
−H
φRaρEsm |ρ0 =Ra dz 0
(−ρEsm Aρ + (ρ0 E1m − ρEcm )Aϕ )ρ0 dρ0 dz 0 ],
(10)
with
E1m (ρ, ρ0 , z, z 0 ) =
Ecm (ρ, ρ0 , z, z 0 ) =
Esm (ρ, ρ0 , z, z 0 ) =
Z
Z
Z
cos mϕ0
2π
(ρ2
0
+
ρ02
+
ρ02
2π
(ρ2
0
2π
0
(ρ2
+
ρ02
4
3
dϕ0 ,
3
dϕ0 ,
3
dϕ0 .
2ρρ0
cos ϕ0
z 0 )2 ) 2
2ρρ0
cos ϕ0
−
+ (z −
0
cos mϕ cos ϕ0
−
+ (z −
sin mϕ sin ϕ0
z 0 )2 ) 2
−
2ρρ0
z 0 )2 ) 2
0
cos ϕ0
+ (z −
M. Xu, F. Stefani and G. Gerbeth
For Eq.(4), we obtain
Z
Z
1 H Ra
1
(−ρ0 ρEsm |z=H uz − ρ0 (H − z 0 )uϕ E1m |z=H )(B0ρ + bρ )
φ(s1 ) =
[
2
4π −H 0
+((−ρ0 ρEcm |z=H + ρ02 E1m |z=H )uz + ρ0 (H − z 0 )uρ E1m |z=H )(B0ϕ + bϕ )
+((ρ0 ρEcm |z=H − ρ02 E1m |z=H )uϕ + ρ0 ρEsm |z=H uρ )(B0z + bϕ )dρ0 dz 0
Z
−
H
φRa(ρEcm |ρ0 =Ra,z=H − RaE1m |ρ0 =Ra,z=H )dz 0
−H
+2.0H
Z
Ra
0
φE1m |z=H,z 0 =−H ρ0 dρ0
−λ
Z
H
−H
Z
Ra
0
ρ0 (ρEcm |z=H − ρ0 E1m |z=H )Aρ
−ρ0 ρEsm |z=H Aϕ + ρ0 (H − z 0 )E1m |z=H Az dρ0 dz 0 ],
(11)
Z H Z Ra
1
1
φ(s2 ) =
[
(−ρ0 RaEsm |ρ=Ra uz − ρ0 (z − z 0 )uϕ E1m |ρ=Ra )(B0ρ + bρ )
2
4π −H 0
+(−ρ0 RaEcm |ρ=Ra uz + ρ02 E1m |ρ=Ra uz + ρ0 (z − z 0 )uρ E1m |ρ=Ra )(B0ϕ + bϕ )
+(ρ0 uϕ RaEcm |ρ=Ra + ρ0 Rauρ Esm |ρ=Ra − ρ02 uϕ E1m |ρ=Ra )(B0z + bz )dρ0 dz 0
−
+
Z
Z
Ra
0
Ra
φ(z − H)E1m |ρ=Ra,z 0 =H ρ0 dρ0 −
φ(z +
0
02
H)E1m |ρ=Ra,z 0 =−H ρ0 dρ0
Z
H
−H
−λ
Z
φ(Ecm |ρ=ρ0 =Ra − E1m |ρ=ρ0 =Ra )Ra2 dz 0
H
−H
Z
Ra
0
(ρ0 RaEcm |ρ=Ra
−ρ E1m |ρ=Ra )Aρ − ρ0 RaEsm |ρ=Ra Aϕ + ρ0 (z − z 0 )E1m |ρ=Ra Az dρ0 dz 0 ],
(12)
Z H Z Ra
1
1
(−ρ0 ρEsm |z=−H uz + ρ0 (H + z 0 )E1m |z=−H uϕ )(B0ρ + bρ )
φ(s3 ) =
[
2
4π −H 0
+(−ρ0 ρEcm |z=−H uz + ρ02 E1m |z=−H uz − ρ0 (H + z 0 )uρ E1m |z=−H )(B0ϕ + bϕ )
+(ρ0 ρuϕ Ecm |z=−H − ρ02 uϕ E1m |z=−H + ρ0 ρuρ Esm |z=−H )(B0z + bz )dρ0 dz 0
+2.0H
Z
Ra
0
φE1m |z=−H,z 0 =H ρ0 dρ0
−RaE1m |ρ0 =Ra,z=−H )dz 0
−λ
Z
V
−
Z
H
−H
φRa(ρEcm |ρ0 =Ra,z=−H
(ρρ0 Ecm |z=−H − ρ02 E1m |z=−H )Aρ
−ρ0 ρEsm |z=−H Aϕ + ρ0 (−H − z 0 )Az E1m |z=−H dρ0 dz 0 ],
(13)
where φ(s1 ), φ(s2 ) and φ(s3 ) are the electrical potential distributions on the surfaces of
the cylinder, s1 is the surface z = H, s2 the surface ρ = Ra, s3 the surface z = −H.
Equation(5) gets the form
Aρ
1
[
=
4π
Z
H
−H
Z
Ra
0
ρ0 (z − z 0 )Esm (B0ρ + bρ ) + ρ0 (z − z 0 )Ecm (B0ϕ + bϕ )
+ρ02 Esm (B0z + bz )dρ0 dz 0 +
+
Z
H
−H
Ra(B0z +
Z
Ra
0
−ρ0 Dsm |z 0 =H (B0ρ + bρ ) − ρ0 Dcm |z 0 =H (B0ϕ + bϕ )dρ0
bz )Dsm |ρ0 =Ra dz 0
+
Z
Ra
0
5
ρ0 Dsm |z 0 =−H (B0ρ + bρ )
M. Xu, F. Stefani and G. Gerbeth
Aϕ
+ρ0 (B0ϕ + bϕ )Dcm |z 0 =−H dρ0 ]
Z
Z
1 H Ra 0
=
[
−ρ (z − z 0 )Ecm (B0ρ + bρ ) + ρ0 (z − z 0 )Esm (B0ϕ + bϕ )
4π −H 0
+ρ0 (ρE1m − ρ0 Ecm )(B0z + bz )dρ0 dz 0 +
−ρ
+
Az =
0
Z
Dsm |z 0 =H (B0ϕ
Ra
0
1
[
4π
+
Z
Z
+ bϕ )dρ −
Z
H
−H
Ra
0
ρ0 Dcm |z 0 =H (B0ρ + bρ )
Ra(B0z + bz )Dcm |ρ0 =Ra dz 0
ρ0 Dsm |z 0 =−H (B0ϕ + bϕ ) − ρ0 Dcm |z 0 =−H (B0ρ + bρ )dρ0 ]
H
−H
H
−H
0
Z
Z
Ra
0
(14)
(15)
−ρ0 ρEsm (B0ρ + bρ ) + ρ0 (ρ0 E1m − ρEcm )(B0ϕ + bϕ )dρ0 dz 0
RaD1m |ρ0 =Ra (B0ϕ + bϕ )dz 0 ],
(16)
where
Dsm (ρ, ρ0 , z, z 0 ) =
Dcm (ρ, ρ0 , z, z 0 ) =
D1m (ρ, ρ0 , z, z 0 ) =
Z
Z
Z
sin ϕ0 sin mϕ0
2π
(ρ2
0
2π
0
(ρ2
2π
0
−
2ρρ0
−
2ρρ0
cos ϕ0
1
dϕ0
1
dϕ0
ρ02
+
+ (z −
cos ϕ cos mϕ0
z 0 )2 ) 2
cos ϕ0
ρ02
z 0 )2 ) 2
0
+
+ (z −
0
cos mϕ
1
(ρ2 − 2ρρ0 cos ϕ0 + ρ02 + (z − z 0 )2 ) 2
dϕ0 .
We use equidistant grid points ρi = i × ∆r and zj = j × ∆z to discretize the intervals
[0, Ra] and [−H, H], respectively. The extended trapezoidal rule is applied to approximate
all the integrals in Eqs. (8-16). Then we obtain the following matrix equations








bρ
B0ρ + bρ
φs1
Aρ








 bϕ  = µσ[P  B0ϕ + bϕ  − Q  φs2  − λR  Aϕ ],
bz
B0z + bz
φs3
Az








φs1
Aρ
B0ρ + bρ
φ
1  s1 






φ
A
B
+
b
−
U
φ
−
λT
=
S
 s2  ,
 ϕ
 0ϕ
 s2 
ϕ
2
φs3
Az
B0z + bz
φs3




Aρ
B0ρ + bρ




 Aϕ  = V  B0ϕ + bϕ  ,
Az
B0z + bz
(17)
(18)
(19)
where the matrix elements of P, Q, R, S, T, U, and V can be read off from Eqs. (8-16).
The fields are discretized according to
bρ = [bρ (i∆ρ, j∆z)]n×m ,
bϕ = [bϕ (i∆ρ, j∆z)]n×m ,
6
M. Xu, F. Stefani and G. Gerbeth
bz
φρ
φϕ
φz
Aρ
Aϕ
Az
=
=
=
=
=
=
=
[bz (i∆ρ, j∆z)]n×m ,
[φρ (i∆ρ, j∆z)]n×m ,
[φϕ (i∆ρ, j∆z)]n×m ,
[φz (i∆ρ, j∆z)]n×m ,
[Aρ (i∆ρ, j∆z)]n×m ,
[Aϕ (i∆ρ, j∆z)]n×m ,
[Az (i∆ρ, j∆z)]n×m .
(20)
(21)
(22)
Combining Eqs.(17-19), we obtain
(I − µσE − µσλG)b = µσ(E + λG)B0 ,
(23)
where
1
E = P − Q · ( I + U)−1 · S
2
1
G = Q · ( I + U)−1 · T · V − R · V .
2
(24)
(25)
After solving the algebraic equation system (23), the induced magnetic field b can be
obtained for the magnetic induction process.
For the kinematic dynamo problem, the following generalized eigenvalue problem has
to be solved
(I − Rm E) · b = λ∗ G · b
(26)
for the given velocity field u and the magnetic Reynolds number Rm ,where λ∗ = µσλ.
3
NUMERICAL IMPLEMENTATION AND RESULTS
In the present section, the integral equation approach will be applied to various physical
problems. We start with the problem of a freely decaying magnetic field in a cylinder.
Then, we treat an analytical test flow in a cylinder. In both problems we use the QZ
algorithm22 which is a modification of the QR algorithm for the case of generalized nonhermitian eigenvalue problems.
The integral equation approach is further employed to investigate the induction effect
of the VKS experiment. The algebraic equation system is solved by the LU decomposition. The obtained induced magnetic field will be compared with the data measured in
experiment.
At the end we deal with the Riga dynamo experiment with its large ratio of height
to radius. Due to the large resulting matrices we shift here from direct matrix inversion
methods to the generalized inverse iteration method23 .
7
M. Xu, F. Stefani and G. Gerbeth
3.1
Free field decay in a finite cylinder
The simplest problem to start with is the free decay of a magnetic field in a finite
length cylinder. This example was already treated by Iskakov et al.8 . In Fig. 1 we show
the magnetic field lines of the slowest decaying eigenfield, which has the same dipolar
structure as in Fig. 8 of the paper by Iskakov et al.8 .
1
0.9
0.8
0.7
z
0.6
0.5
0.4
0.3
0.2
0.1
0
-1
-0.5
0
x
0.5
1
Figure 1: Freely decaying magnetic field in a finite cylinder with radius=height=1.
3.2
An analytical test flow
In connection with the optimization of the VKS dynamo experiment, Marié, Normand
and Daviaud (MND) had studied an analytical test flow of the same topological type as
the flow in the real experiment24,25 (flow topology s2+ t2, what means two poloidal eddies
with radial inflow in the equatorial plane, together with two counter-rotating toroidal
eddies).
The velocity field of this MND flow reads:
π
r (1 − r)2 (1 + 2r) cos(πz)
2
= 4r(1 − r) sin(πz/2)
= (1 − r)(1 + r − 5r 2 ) sin(πz).
ur = −
uϕ
uz
(27)
For the parameter which determines the ratio of toroidal to poloidal flow, we have used
the same value = 0.7259 as in the paper by Ravelet et al.25 . For the case without
side layer, the structure of the magnetic eigenfield is illustrated in Fig. 2. In Fig. 2a
two isosurfaces (25 percent and 75 percent of the maximum value) of the magnetic field
energy are displayed. In Fig. 2b the magnetic eigenfield lines are depicted. This kind of
structure of eigenfield was already discussed in25 .
8
M. Xu, F. Stefani and G. Gerbeth
(a)
(b)
Figure 2: Simulated magnetic field structure of the dominant eigenmode of the MND flow. (a) Isosurface
plot of the magnetic field energy. The outer isosurface corresponds to 25 per cent of the maximum
magnetic field energy, the inner isosurface corresponds to 75 percent. (b) Magnetic field lines.
For this MND flow we had recently shown that the existence of enveloping layers around
the dynamo may have quite ambivalent effects26 . While a side layer was shown to lower
c
the critical magnetic Reynolds number Rm
, layers at the top and the bottom (so-called
c
”lid layers”) lead to an unexpected increase of Rm
. A similar phenomenon was also found
26
for the VKS experiment . This might explain that, up to present, the VKS experiment
has not yet produced dynamo action in contrast to the expectation based on numerical
predictions that did not take into account the lid layer effect.
3.3
Induction effects in the VKS Experiment
Another sign of the sub-optimal dynamo performance of the VKS experiment is the
fact that the measured induced magnetic fields, for large Rm , are significantly weaker
than the numerically predicted ones. Using our method we will try to understand if this
effect can also be attributed to the existence of lid layers and the flow therein.
The supposed flow field generated by the propeller ”TM73”, which was identified as the
optimal flow field25 , is employed for our calculation. Some interpolations were necessary
to project this flow field onto the grids used in our code.
In the VKS experiment a static external magnetic field is applied in the tranverse
direction. The induced magnetic field was measured in the direction perpendicular to
the externally added magnetic field at the point r = 0.5 in the equatorial plane. In the
9
M. Xu, F. Stefani and G. Gerbeth
6
4
3
2
1
(a)
0
0
5
10
15
20
25
30
35
40
45
6
Axial: meas. at r=0.5
Axial: comp. at r=0.417
0.458
0.5
0.542
0.583
5
4
Induced field/Applied field
Axial: meas. at r=0.5
Axial: comp. at r=0.417
0.458
0.5
0.542
0.583
5
Induced field/Applied field
Induced field/Applied field
6
3
2
1
(c)
0
50
0
5
10
15
20
Rm
25
30
35
40
45
Axial: meas. at r=0.5
Axial: comp. at r=0.417
0.458
0.5
0.542
0.583
5
4
3
2
1
(d)
0
50
0
5
10
15
20
Rm
25
30
35
40
45
50
Rm
6
6
6
5
5
5
3
2
1
(a)
0
0
5
10
15
20
25
30
35
40
45
Azimuth: meas. at r=0.5
Azimuth: comp. at r=0.417
0.458
0.5
0.542
4
0.583
Induced field/Applied field
Azimuth: meas. at r=0.5
Azimuth: comp. at r=0.417
0.458
0.5
0.542
4
0.583
Induced field/Applied field
Induced field/Applied field
Figure 3: Ratio of axial induced magnetic field bz to the applied magnetic field. Experimental results at
radius r=0.5 (taken from 27 .) and numerical results at different radii close to r=0.5. (a) Static lid layer.
(b) Constant velocity in lid layer. (c) Constant velocity in lid layers multiplied by factor 1.5.
3
2
1
(c)
0
50
0
5
10
15
20
Rm
25
30
35
40
45
Azimuth: meas. at r=0.5
Azimuth: comp. at r=0.417
0.458
0.5
0.542
4
0.583
3
2
1
(c)
0
50
0
5
10
15
20
Rm
25
30
35
40
45
50
45
50
Rm
Figure 4: Same as Fig. 3, but for the azimuthal field component bϕ .
20
0
-20
Angle: meas. at r=0.5
Angle: comp. at r=0.417
0.458
0.5
0.542
0.583
-40
-60
-80
0
5
10
15
20
25
30
35
40
60
(c)
40
Angle of induced field
Angle of induced field
60
(a)
40
20
0
-20
Angle: meas. at r=0.5
Angle: comp. at r=0.417
0.458
0.5
0.542
0.583
-40
-60
45
50
-80
0
5
(d)
40
Angle of induced field
60
10
15
20
Rm
25
30
35
40
20
0
-20
Angle: meas. at r=0.5
Angle: comp. at r=0.417
0.458
0.5
0.542
0.583
-40
-60
45
50
-80
0
5
Rm
10
15
20
25
30
35
40
Rm
Figure 5: Same as Fig. 3, but for the angle arctan bϕ /bz .
following the induced magnetic fields near the r = 0.5 points obtained by our integral
equation approach are presented. The influence of the rotating flow in the lid layer on the
induced field is investigated. Three kinds of velocity field in the lid layer are considered.
The first one is a static lid layer. The second one is that only a rotation of the lid layer is
assumed, but uϕ keeps constant in the axial direction, its dependence on the radial variable
is the same as on the interface between the lid layer and inner part of the cylinder. The
third one has only one difference from the second case in that the amplitude of uϕ is
amplified by a factor 1.5. For these three cases, the numerical axial induced fields around
the point r = 0.5 and the experimental result at r = 0.5 on the equatorial plane are
shown in Fig. 3 for different magnetic Reynolds numbers. Fig. 4 display the azimuthal
induced magnetic fields. From these figures one can see that the third case shows a best
10
M. Xu, F. Stefani and G. Gerbeth
(a)
(b)
Figure 6: Simulated magnetic field structure of the eigenmode of the Riga dynamo experiment. (a)
Isosurface plot of the magnetic field energy. The isosurface corresponds to 20 per cent of the maximum
magnetic energy. (b) Magnetic field lines.
agreement with the experimental one. A good agreement of the axial magnetic field with
the experimental result has been achieved for the second and third cases. But for the
azimuthal magnetic field, when the magnetic Reynolds number is larger than 30, there
is still a gap between the numerical results and the experimental ones. In addition the
angles of the induced field defined as arctan bϕ /bz for these three cases are depicted in
Fig. 5.
Hence it is quite likely that it is indeed the existence of lid layers and some azimuthal
flow therein which is responsible for the unexpected under-performance of the VKS dynamo experiment.
11
M. Xu, F. Stefani and G. Gerbeth
3.4
Riga experiment
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1.2
-1.4
-1.6
2
1.8
kinematic: DEA
kinematic: IEA
kinematic: measured
saturation: measured
F-requency fs [1/s]
Growth rate ps [1/s]
In this subsection, the integral equation approach is used to re-simulate the kinematic
regime of the Riga dynamo experiment. This experiment has been optimized and analyzed
extensively within the differential equation approach (DEA) by means of a finite difference
solver6 . The values of the velocity field on the grids used in our code are obtained by
interpolating the experimental velocity field measured in a water-dummy experiment. The
influence of less conducting stainless steel walls has not been taken into account.
The computations have been carried out on a 100×20 grid in z- and r-direction. The
structure of the magnetic eigenfield is illustrated in Fig. 6. Again, Fig. 6a shows the
isosurface of the magnetic field energy (this time at 20 percent of the maximum value).
In Fig. 6b the magnetic eigenfield lines are depicted. Basically the structure is the same
as that resulting from the differential equation approach21 with a 401×64 grid in z- and
r-direction.
1.6
kinematic: DEA
kinematic: IEA
kinematic: measured
saturation: measured
1.4
1.2
1
0.8
0.6
0.4
1000 1200 1400 1600 1800 2000 2200 2400 2600
Rotation rate Ωs [1/min]
1000 1200 1400 1600 1800 2000 2200 2400 2600
Rotation rate Ωs [1/min]
(a)
(b)
Figure 7: Comparison of the IEA and DEA results for the Riga dynamo experiment, together with
experimental results. (a) Growth rate. (b) Frequency.
The dependence of the growth rate and frequency of the eigenmode of the Riga dynamo experiment on the rotation rate is shown in Fig. 7a and Fig. 7b, respectively. The
comparison with the DEA results shows that the slopes of the curves are in good agreement. However, we see that the limited grid resolution in the IEA leads to significant
shifts in the order of 5 percent towards lower rotation rates for the growth rate and of 10
per cent towards higher rotation rate for the frequency. Hence, it could be said that the
Riga dynamo experiment marks a margin of reasonable applicability of the IEA with its
need to invert large matrices which are fully occupied.
4
CONCLUDING REMARKS
In the present paper, the integral equation approach to kinematic dynamos has been
specified to cylindrical geometry. The method was examined by its application to the
12
M. Xu, F. Stefani and G. Gerbeth
free field decay and to the ”MND” flow. The comparison of the obtained results with
other methods shows a good agreement. The integral equation approach was extended
to investigate induction effects of the VKS experiment. The obtained induced magnetic
field shows a satisfactory agreement with the experimental result when the effect of the
lid layers and a certain azimuthal flow therein are taken into account. Finally, it was
applied to simulate the Riga dynamo experiment.
It can be concluded that the integral equation approach is robust and reliable and can
be used for practical purposes, although limits of its applicability are seen for the Riga
dynamo experiment with its large ratio of length to radius.
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