3D magnetic field reconstruction in VKS experiment - Iramis

Transcription

3D magnetic field reconstruction in VKS experiment - Iramis
3D magnetic eld reconstruction in VKS experiment
through Galerkin Transforms
J. Boisson1 and B. Dubrulle 1
SPEC/IRAMIS/CEA Saclay, and CNRS (URA 2464), 91191 Gif-sur-Yvette Cedex,
France
Abstract.
We present a method for 3D magnetic eld reconstruction based on
Galerkin Transforms.
We test it over synthetic elds and real solenoidal (velocity)
elds, measured in a water experiment. Our study shows that reliable reconstructions
are possible provided the probes are suciently sampled and located in shifted
congurations.
A preliminary application of our method is performed over results
obtained in the VKS dynamo experiment [8]. We show that the stationary dynamos
obtained with counter-rotating impellers are mainly axisymmetric,
with a non-
axisymmetric part that decreases with increasing Reynolds numbers.
Most of the
azimuthal energy is localized near the (iron) impellers, conrming their importance in
the dynamo process.
PACS numbers: Magnetic elds measurement of, 07.55.Ge; Magnetic elds generation
of, 07.55.Db; Galerkin method, 02.70.Dh; Inverse problems, 02.30.Zz
Keywords:
Dynamo Inversion, Galerkin
3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms
2
1. Introduction
The magnetic eld is an essential ingredient of most natural objects and numerous
campaigns of observations or experiments are devoted to its measurement or modeling.
A problem often encountered in this process is the sparsity of measurements : only a
limited number of measurements, at a limited number of positions are usually available.
Specic techniques must then be implemented on the data to reconstruct the magnetic
eld outside the measurements points, so as to get information about its geometry,
intensity, direction, ... Classical examples are reconstruction of the terrestrial or Solar
magnetic eld. In the rst case, data are rather numerous at the surface of the earth and
reconstruction is achieved by tting a set of basis functions, usually spherical harmonics
or spherical caps, to magnetic data.
In the case of the Sun, data are sparser, and
the reconstruction can usually only be achieved through physical assumptions.
For
example, Solanki et al [1] use magnetic data issued from vector spectropolarimetry of
the He I 1083 nm multiplet, formed near the coronal base and assume that the He
line is formed along tubes of magnetic ux as the active region emerges through the
chromosphere into the corona. The He line emission then forms in coronal-like loops.
With this assumption a 3D magnetic eld can be constructed that can be compared
with extrapolations from deeper layers. Similar problematics can be found in laboratory
experiments. For example, electromagnetic velocity probes usually require measuring
the distribution of the 3D magnetic eld in the whole measuring volume out of the
probe. Clever assumptions about exciting currents and numerical solving of a Laplace
equation allow for reconstruction of the 3D magnetic eld from a mere measurement
of the distribution of the normal component of the magnetic eld on the surface of
the probe [2]. In the present paper, we focus on the case of the von Karman Sodium
(VKS) experiment, a laboratory facility designed to studying the turbulent uid dynamo
eect, i.e. the spontaneous generation of a magnetic eld from turbulent motion of liquid
sodium. In that experiment, the magnetic eld measurements are localized only along
a few magnetic sensors, thereby avoiding large disturbances of the measurements onto
the ow of liquid sodium. The reconstruction of the full 3D magnetic eld from these
sparse measurements therefore implies technical processing of the magnetic data. Here,
we describe a solution inspired from geophysical methods, involving tting a set of
suitable Galerkin basis functions to magnetic data.
2. Method
2.1. Geometry
The setup corresponding to the VKS experiment is sketched in Figure 1 [13].
cylindrical vessel with a xed aspect ratio of
h/R = 1.8,
where
R (h)
A
denotes the
radius (height) of the vessel which is lled with liquid sodium. We consider a cylindrical
coordinate system
R
(r, ϕ, z)
such that the axial vessel extension lies along
as the length scale, and introducing
L = h/2R = 0.9,
z.
Choosing
the variation ranges of the
j
3D magnetic eld reconstruction in VKS experiment
through Galerkin Transforms
k
3
octobre
2007
Figure 1. (a) Sketch of the VKS experiment geometry including the probe locations.
m, n
The
z
o
axis of the cylindrical coordinates is taken as the
x
axis of the experiment, i.e.
the axial extension of the cylinder. The gap in between the inner and the outer cylinder
is lled with sodium at rest. The inner container radius
R
is taken as length scale.
(r, ϕ, z) inside the experimental vessel are
therefore (0 ≤ r ≤ 1, −π ≤ ϕ ≤ +π, −L ≤ z ≤ +L). Our magnetic eld reconstruction
however also extends beyond the vessel boundaries, i.e. for r > 1 thanks to suitable
dimensionless cylindrical polar coordinates
boundary conditions (see below).
The uid is stirred by two independent rotating so-called TM73 impellers [16, 15].
The impellers are tted with curved blades, and can be rotated independently at
frequencies
F1
and
F2 .
Magnetic measurements are performed using 3D Hall probes, and recorded with
q
a National Instruments PXI digitizer.
q’
Both a single-point (three components) probe
(hereafter label G-probe) and custom-made arrays of 10 (three components) probes
(hereafter called SM-array) are used.
The array is made from Sentron 2SA-1M Hall
sensors located every 28 mm along a line. The dierent possible locations of the probes
are sketched in Fig. 1.
2.2. Brief overview
A given VKS experiment provides us with a set of measurements of the (three
components of the) magnetic eld, at
N
dierent sparse locations.
To reconstruct
the magnetic eld in the whole space, we expand it on a set of well chosen (see below)
Galerkin functions
Gi :
B(r, ϕ, z) =
M
X
bi Gi (r, ϕ, z),
(1)
i=1
where
M
is the number of modes of the reconstruction.
Then, by imposing that the
expansion matches the three components of the measured elds at the
points, we get a set of
bi ).
3N
equations with
M
N
measurements
unknown (the coecients of the expansion
Inverting the systems thus provides the values of the
bi , and thus, the reconstruction
3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms
4
at that resolution. Of course, one cannot hope to reconstruct the eld with a resolution
M < 3N to get good results). Besides,
possible non uniqueness of solution suggest to use M 3N in order to allow for
robustness of the inversion through redundancy. Since N is typically of the order to
1 to 10 (41 at most), this means that only a small number of Galerkin modes will
larger than the number of measurements (i.e.
be allowed in the reconstruction procedure. To get meaningful results, it is therefore
mandatory to choose an optimal Galerkin basis, such that a small number of modes
captures the large scale structure of the experimental elds.
This imposes that the
Galerkin basis satises both the solenoidal condition and the boundary conditions of
the magnetic eld. We therefore now describe how to implement these constraints on
the choice of the basis, following the recent work of [3, 4].
2.3. Boundary conditions
Through the cylindrical coordinate system, the system is naturally periodic in the
direction.
ϕ
Boundary conditions must however be specied in the vertical and radial
directions.
2.3.1. Vertical boundaries z = ±L
As in pseudo-spectral simulations [15], the inversion
procedure can be greatly simplied if one considers a periodized version of the problem,
in which the magnetic eld is assumed periodic in the axial direction. It is however not
possible to use as the unit cell the interval
not naturally periodic over this cell.
z ∈ [−L; L]
since the magnetic eld is
To avoid discontinuities and Gibbs phenomena
(artifact oscillations of the reconstructed magnetic eld near the discontinuity), we
have thus chosen to use as a unit cell the whole range
the original physical domain.
z ∈ [−L; 3L],
which is twice
Doing so yields magnetic eld components which are
continuous everywhere.
2.3.2.
Radial boundary r = 1
The boundary conditions at
r = 1
are set by
the condition that both the magnetic eld components and the derivative
continuous at
∂r Br
r = 1 -the last condition stems from the solenoidal nature of B -,
are
i.e. they
match continuously the exterior domain solution. Many numerical methods to deal with
the r=1 boundary conditions were developped. These schemes are based on boundary
elements formalism [24], on integral equation [26] or on spectral nite elements [25].
Here we chose a simpler way to manage with these boundary conditions. In the sequel,
we consider three main possibilities:
• External insulating:
If the medium surrounding the uid is insulating, we get the
following condition for the exterior solution:
∇ × B = 0,
∇.B = 0,
⇒ ∆B = 0
(2)
In the case of spatially homogeneous magnetic permeability we are considering, the
magnetic eld must also be continuous and nite over all space, and vanish for
r → ∞.
3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms
•
= ∞) wall at r = 1. In that
Bz = Bφ = 0 at r = 1. This
Ferromagnetic (µr
zero at
r > 1,
and that
5
case, we assume that the eld is
case mimics the situation where
the uid is surrounded by a ferromagnetic wall.
for
• General case.
Br and Bφ .
In that case, we do not specify the boundary conditions at
r=1
2.4. Galerkin basis
2.4.1. General Case
Following [3, 4], we rst switch from usual cylindrical components
B = (Br , Bϕ , Bz ) to the new components B = (B+ , B− , Bz ) such that B± = (Br ±
iBϕ )/2. Due to the periodic conditions in ϕ and z , it is natural to introduce the general
decomposition over eigenmodes of the Helmoltz operator as:



nmk
P
π/2L
B+
C
J
(µ
r)
N
M
m+1
nmk
X  +


 X X
nmk
B =  B−  =
C
J
(µ
 −
m−1
nmk r)  exp(imϕ + ikz),
n=1 m=−M k=−P π/2L
Bz
Cznmk Jm (µnmk r)

(3)
n−m−k ∗
nmk
= (C±,z
C±,z
)
to enforce reality of the eld and the numbers µnmk have to
nmk
be chosen so as to respect boundary conditions. The coecient C±,z are the spectral
with
coecients of our decomposition and are uniquely associated to our initial eld
then write symbolically the value of
B
in the Galerkin space as
GT (B)
B.
We
(for Galerkin
Transform) with

C+nmk


GT (B) =  C−nmk  .
Cznmk

For completeness, we introduce a similar decomposition for a scalar eld
Ψ=
N
M
X
X
(4)
Ψ
as
P π/2L
X
Qnmk Jm (µnmk r) exp(imϕ + ikz),
(5)
n=1 m=−M k=−P π/2L
so that we have
GT (Ψ) = Qnmk .
2.4.2. Useful formulae
(6)
Our decomposition allows for interesting representation of usual
derivatives, that are generalization of similar operations in the Fourier space. Indeed,
we have for a scalar eld:
GT (∆Ψ) = −(µ2nmk + k 2 )Qnmk ,


− 21 µnmk Qnmk


GT (∇Ψ) =  21 µnmk Qnmk  .
ikQnmk
(7)
3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms
6
In a similar way, we have for a vector eld:
GT (∇ · B) = µnmk C+nmk − C−nmk + ikCznmk ,


i
nmk
nmk
−
kC
µ
C
nmk z
+

 2
GT (∇ × B) =  2i µnmk Cznmk + kC−nmk  .
−iµnmk C+nmk + C−nmk
(8)
These simple formulae show that once the eld has been reconstruct, it is easy to perform
accurate derivative operations by working in the Galerkin space. Moreover, the formulae
can be used to select useful special Galerkin basis for magnetic eld reconstructions.
2.4.3. Useful basis
General Galerkin transform of an arbitrary vector eld requires
3 independent coecient for each (nmk). In the case of a solenoidal eld, we must
nmk
− C−nmk + ikCznmk = 0 for any (nmk), so that only two independent
have µnmk C+
coecients remain.
In the sequel, we describe even simpler cases, with only one
independent coecient, that are appropriate for elds and boundary conditions usually
met in our von Karman experiment.
2.4.4. Beltrami case
In this case, the Galerkin transform writes


GT (B) = Dnmk 
Dnmk are
p
2
± µnmk + k 2 . This
where the
1
2
1
2

(λ± − k)

(λ± + k)  ,
−iµnmk
complex numbers such that
∗
Dnmk = Dn−m−k
(9)
and
λ± =
corresponds to a decomposition into Beltrami waves,
polarization given by the sign of
with
λ.
To guarantee some interesting orthogonality
th
conditions, we further take µnmk as the n
root of Jm . The corresponding eld is
solenoidal, and such that
Bz = 0
at
r = 1.
2.5. Marié Normand Daviaud (MND) eld case
There are two cases depending on the axial wave number.
case k 6= 0
where the


c+ Km+1 /Jm+1 (µnmk )


GT (B) = Dnmk  c− Km−1 /Jm−1 (µnmk )  ,
Km /Jm (µnmk )
Dnmk
µnmk
∗
Dnmk = Dj−m−k
, Km expressed through
√
Km = Km (|k|), c± = i|k|/k 2, and the
are complex numbers such that
the modied Bessel function of order
parameter
(10)
is determined as the n
m
th
as
root of the equation:
2|k| Km (|k|)
Km−1 (|k|) Km+1 (|k|)
=
−
µ Jm (µ)
Jm−1 (µ)
Jm+1 (µ)
(11)
3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms
For each
k,
there is an innite set of real positive values
µnmk ,
7
which we truncate to
N,
the number of radial modes in the Galerkin expansions (3). Marié et al. [3] determined
the roots of (11) for
m=1
and discrete values of the axial wave number (kπ/2L). The
roots of (11) for higher values of
m,
up to
m ≤ 5,
were computed later by Leprovost
k the successive roots of (11) take increasingly large values as
n increases (µ(n−1)mk < µnmk ). As an example, for m = 1 and k = 1 their values range
between µ111 = 2.217 and µn11 = 32.22 for n = 20.
[7]. For a given value of
case k = 0
In that case, there are two dierent families of modes.
The rst family has shape


GT (B) = Dnm0 
where
µnm0
th
is the n
root of
1
µ /Jm+1 (µnm0 )
2m nm0
1
µ /Jm−1 (µnm0 )
2m nm0
0


,
(12)
Jm−1 .
The second family has shape:

0


GT (B) = Dnm0 
0
,
1/Jm (µ̄nm0

associated with a dierent set of eigenvalues
µ̄nm0
which are the zeros of
This Galerkin basis is solenoidal and matches at
∆B = 0.
(13)
r = 1
Jm .
with a eld such that
This case is therefore suitable to describe a medium surrounded by an
insulating uid.
2.6. Ferro case
For this case, we choose:


−µnmk (k − γ)


GT (B) = Dnmk  µnmk (k − γ)  ,
µ2nmk
(14)
∗
where the Dnmk are complex numbers such that Dnmk = Dn−m−k and with
th
the n
root of Jm and γ is solution of the dispersion equation:
γ=k
Jm+1 (µnmk ) + Jm−1 (µnmk )
.
Jm+1 (µnmk ) − Jm−1 (µnmk )
In this case, the eld is solenoidal, and
the boundary at
r = 1.
Bz = Bφ = 0
at
r = 1,
µnmk
being
(15)
i.e. is perpendicular at
This case therefore describes the case when the uid is enclosed
inside a ferromagnetic boundary.
3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms
8
2.7. Inverse problem
In the three cases we considered above, the nal Galerkin expansion for the magnetic
eld at a given resolution
coecients‡.
To
(N, M, P )
compute
Galerkin expansion of
B
these
requires
2N (M + 1)(2P + 2)
coecients,
one
matches the measured eld
writes
Bmes ,
the
independent real
condition
that
the
at dierent probes location
(rmes , ϕmes , zmes ):
B(rmes , ϕmes , zmes ) = Bmes .
(16)
This procedure can be recast formally into the equation:
Aij Xj = Yi ,
where
Xj , j = 1....2N (M + 1)(2P + 2)
(17)
is the vector formed with the real part and
the imaginary part of the unknown complex coecients
Dimk , Yi , i = 1....3Nmes
is the
vector formed by each of the component of the measured magnetic eld at each probe
location, and
locations.
Aij
is the matrix formed by the Galerkin basis evaluated at the probe
Inverting formally (17), one then obtains the unknown coecient of the
Galerkin expansion through:
Xi = A−1
ij Yj ,
where
A−1
ij is the pseudo-inverse of Aij .
(18)
Indeed, to ensure robustness of the procedure, we
choose in the sequel the resolution such that
2N (M +1)(2P +2) ≤ 3Nmes (see discussion
above). The problem is therefore overdetermined so that only pseudo-inverse is dened
(in the sense that there is an innity of solutions for (18)). We have tested two Matlab
procedures for the pseudo-inverse resulting in two sets of solutions that are both solutions
in the least square sense (i.e. minimizing the norm of
norm of
of
Xi (/
Xi
Aij Xi − Yj ):
one minimizing the
(pinverse function) and one minimizing the number of non zero component
function).
In addition, we have also implemented a standard regularization
procedure based on Tikhonov method using Matlab codes developed by Prof.
Per
Christian Hansen [5] and provided in http://www2.imm.dtu.dk/ pch/Regutools/. All
three procedures give similar results for smooth elds (for example time-averaged elds).
The regularization method allows elimination of spurious noise in the case of rough elds
(such as instantaneous elds). We have therefore privileged this method in the sequel.
To test the method and its sensitivity to the dierent parameters, we use two types of
elds: i) synthetic elds, constructed from the Galerkin basis; ii) real elds, issued from
Stereoscopic Particle Image Velocimetry performed on a turbulent von Karman water
ow, in a half scale cylindrical vessel with respect to the VKS experiment [17].
‡
(M + 1) takes into account
Dnmk , the factor 2P + 2 takes
The factor 2 comes from the complex nature of the coecients, the factor
the reality of the eld, thereby imposing a symmetry conditions on the
into account the existence of two families of modes at
k = 0.
3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms
3
2
1
5
Figure
9
4
2.
Projection of synthetic elds at dierent resolution
meridional plane passing through the symmetry axis.
component
Bφ
(N, M, P )
in a
The out of plan (toroidal)
is coded with color and the in-plane (poloidal) component
(Br , Bz
is coded with arrows (this representation is used in the whole article). Top left: N=1,
M=0, P=0; Top right : N=1, M=1,P =1; bottom left : N=2, M=1, P=1; bottom right
: N=3, M=2, P=1. We represent the 5 possible probes locations in the top left gure.
3. Tests on synthetic elds
3.1. Description of synthetic elds
The synthetic elds are constructed from Eq. (3) by xing a resolution
then choosing the 2N(M+1)(2P+2) independent
(N, M, P )
and
Dnmk so that the energy is concentrated
in the lowest modes and decreases with increasing mode number- the idea being to
i
mimic a "realistic" experimental eld. In practice, we set Dnmk = (−1) /i where
i = |j| + |m| + |k|. Example
resolution (N, M, P ).
of such synthetic elds in provided in Fig. 2, for dierent
The complexity of the synthetic eld topology increases with the resolution. The
synthetic measurements are extracted from these elds by computing the value of the
eld at some selected location. In the sequel, we test various reconstruction by using
"measurements" performed at the location given by VKS magnetic probes. There are
three probes in the vertical plane (probes 1 2 3), and three again in the central plane
(probes 1 4 5), at the location of the velocity mixing layer in the exact contra-rotation
3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms
10
regime. The location of these probes is reported in Fig. 2 for illustration and clarity.
3.2. Diagnostic
To quantify the quality of the reconstruction, we dene scalar quantities inspired from
classical residue denition as follow. We rst divide our available measures in two
j
distinct parts: i) a large set (X = {rmeas }), including about 95 percent of the measures,
j
that is kept to reconstruct the eld; ii) a second, smaller set (Y = {rtest }), that is used
to compute "residue-tests".
From the rst set
X,
we dene a classical residue characterizing the quality of the
reconstruction from the measures in set
X
as:
v
u N 3
j
j
i
uX X (B i (rmeas
) − Breconst
(rmeas
))2
real
r=t
j
i
(Breal
(rmeas
))2
j
i
with
i = r, φ, z
coordinates and
j = 1, N
(19)
the number of the measure in
X.
This residue
quanties the ability of the reconstruction to t the observed magnetic eld with the
set of measures
X.
Y , we dene another residue based on the measures not involved
in the magnetic eld reconstruction. This residue is named Residue test and is dened
j
j
like in Eq. (19) by substituting rmeas with rtest .
Finally, we can dene a third residue based on energy as: ∆E = (Esynth −
Ereconst )/Esynth . We refer to it as the Energy residue.
From the second set
3.3. Inuence of mode numbers
We rst explore the inuence of the mode number on the reconstruction.
Clearly,
we cannot get more reliable information than present in the eld to be reconstructed.
Therefore, the number of modes used in the reconstruction
Nr
cannot be larger than
3Nmes . Also, in order to get stable results, it is interesting
that Nr = (1 − fr )3Nmes where 0 < fr < 1 quanties the
the total number of measures
to use some redundancy, so
percentage of redundancy of the data.
In order to get a hint about typical admissible of values for
fr ,
we use a given
(N, M, P ) synthetic eld with varying Nr = 2N (M + 1)(2P + 2) modes and extract
3Nmes = 120 synthetic measures at the VKS sensor positions (10 sensors measuring
three components per probe on the 2 - 3 - 4 - 5 probes represented in gure 2).
then rst reconstruct the eld from the synthetic measure with the same set
We
(N, M, P )
of modes, the necessary test which have driven the elaboration of the reconstruction
method. We are not representing the results here, however in all the cases, the residue
−15
are very low (r ∼ 10
) i. e. the reconstruction with the same set of modes than the
initial eld always reproduces well the eld at the measurement points.
residue is fairly low until
Nr ∼ 50 (fr = 0.6)
The energy
where the reconstructed energy starts to
3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms
11
1
6
10
0.9
4
0.8
10
Energy Residue
0.7
Residue
0.6
0.5
0.4
0.3
2
10
0
10
−2
10
−4
10
0.2
−6
10
0.1
0
0
10
20
30
40
50
Modes Number
60
70
0
80
10
20
30
40
Modes Number
50
60
70
Figure 3. Evolution of residue (bottom left) and of the energy residue (bottom right)
with the magnetic eld modes number. The
(4, 2, 2)
initial eld with
144
modes is the
in the top left gure, the ltered version is represented in the top middle gure, the
reconstruction
(3, 1, 1)
with
48
modes is plotted on the top right gure.
diverge form the synthetic eld energy. Therefore, the degree of redundancy is necessary
to reconstruct the eld in this case must be at least
The
previous
situation
is
however
quite
60
per cent.
ideal.
In
real
experiments,
the
magnetic eld has a (presumably fairly large) unknown number of modes and robust
reconstruction attempts use a lower number of modes than in the experimental eld.
To test this eect, we choose an synthetic eld with a fairly large number of modes
((4, 2, 2) resulting in
with
Nr = 144
modes) and reconstruct it with dierent sets of modes,
Nr < 96.
Results are presented in gure 3.
We see that the residue decreases fairly
monotonously along with the energy residue until
Nr ≈ 20(fr = 0.8).
Above this, the
energy residue starts diverging, while the residue continues to decrease until
Nr = 40
(fr
= 0.67), where it saturates towards a value of the order of 0.5 (with big oscillations).
Note that the oscillations are mainly due to reconstructions that have either M = 2 or
P = 2. These modes are dicult to capture in our measurement set-up, since we have a
maximum of three values in both φ and z directions (see gure 2). Thus, introducing a
number of modes superior to one in these directions can lead to systematic error coming
φ coordinates. Keeping in
mind this restriction, and focusing on reconstruction with M ≤ 1, P ≤ 1, we get a
"best" reconstruction with (3, 1, 1) modes conguration, corresponding to Nr = 48 and
δE ∼ 0.03 shown in gure 3. One sees that the reconstruction captures the "large-
from the low number of measures points along the
z
and
scale" structure of the initial eld. To quantify further this, we may also compare the
12
Residue
Energy Residue
3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms
Sensors Number
Sensors Number
Figure 4.
Evolution of residue (right gure) and energy residue (left gure) of the
reconstruction with varying sensors number
Ns .
Ns /4 measurements
have 32 modes.
There are
probes. Here, both the initial and the reconstructed eld
per
reconstruction with a ltered version of the initial eld, computed using a sgolay lter
with a length scale of the order of a tenth of the cylinder radius.
One sees that the
reconstruction captures even ner details than the coarse version.
3.4. Inuence of sensors numbers
In this section, we explore the dependence of the reconstruction quality with the number
of sensors (varying
Nmes
number of measurements points).
For this, we consider a
(2, 1, 1) synthetic eld and reconstruct it with a varying number of measurements 3Nmes .
In order to minimize the inuence of probes location, we always consider the same
number of sensor
Ns /4
on each of the four probes (probes 2, 3, 4 and 5 reported in
Ns . The total
number of measurements is then 3Nmes = 3Ns . In the sequel, we vary Ns /4 from 1 to
4. The synthetic elds is reconstructed with a (2, 1, 1) set of modes, with Nr = 32.
The inuence of Ns over the reconstruction is given in gure 4. The residue is
−15
almost constant, maintaining very low values (r ∼ 10
). The energy residue ∆E is
more sensitive to Ns and takes very large values for Ns ≤ 12, i.e. a total measurement
number less than 36 .
This corresponds precisely to a range of values for which
Nr = 32 > 3(1 − 0.67)Nmes , conrming the importance of keeping at least 60 per
cent of redundancy. Above the value Ns = 12, the energy residue becomes quite low
−5
(∼ 10 ) and the reconstruction quality is excellent.
gure 2). The number of measurements is then increased by increasing
3.5. Inuence of probe location
We have worked so far assuming that the measurements are performed at the present
VKS probe location. (see gure 2). It is however interesting to check the inuence of
the probe location onto the reconstruction, so as to potentially optimize experimental
measurements. The question is then how to optimize the probes location to get maximal
quality of magnetic eld reconstruction with minimal impact on the experimental
3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms
2
1
5
Shifted (3,5)
Central (1,5)
Axial (3,2)
Residus
Energy
res.
3
Residus
Energy
res.
3
5
13
20
10
10
10
−5
10
Energy Residue
Shifted (3,5)
Residue
Axial (3,2)
Central (1,5)
−10
10
0
10
−10
10
Shifted (3,5)
Axial (3,2)
Central (1,5)
−20
10
−15
10
−30
10
0
10
20
30
40
Modes Number
50
60
70
0
10
20
30
40
Modes Number
50
60
70
Figure 5. Top gure: Example of a synthetic eld with indication of the three probes
conguration (axial, central, shifted ). Bottom: evolution of residue (left) and energy
residue (right) with number of initial eld modes (equal to the number of reconstructed
modes) depending on the three sets of probes location axial, central and shifted.
sodium ow. Such impact is decreased with decreasing the number of probes and with
a probe conguration respecting as much as possible the symmetries of the ow. We
thus consider now measurements performed with only two probes. Given the present
experimental conguration, three dierent sets of symmetric probe locations can then
be constructed. They are reported in gure 5.
The three conguration are:
i) the
axial conguration
with two probes in the
same vertical plane symmetric with respect to the meridional plane; ii) the
conguration
with two probes in the meridional plane of the cylinder (z
angles diering by
dierent
φ angles.
π;
iii) the
shifted conguration
= 0)
with two probes at dierent
central
at two
z
and
We consider 10 sensors on each probes, and reconstruct the elds with
the same increasing number of modes for both the synthetic eld and the reconstructed
eld.
The residue behavior for the three congurations is given in gure 5. Residue in
the central conguration are large and decreases with the mode number, while it is
small or close to zero for axial and shifted congurations. The energy residue for the
axial and central conguration increases signicantly with the mode number. Thus the
reconstruction in these geometry is not reliable with only two probes. In contrast, the
shifted conguration provides fairly reasonable results. We focus on this conguration
14
Residue
Residue
3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms
Energy res.
Noise
Energy res.
Noise
Noise
Noise
Figure 6. Evolution of residue and energy residue with the noise intensity. On the
left panel, the reconstructed and the initial eld have the same set of modes; On the
right panel the reconstructed has a lower number of modes than the initial eld.
from now on.
3.6. Inuence of noise
So far, we have performed reconstruction using "perfect" measures, matching exactly the
synthetic eld. In real experiments, various noise sources pollute the measurements, and
it is important to test the sensitivity of the reconstruction to measurement noise. We
therefore extract articial measurements from a synthetic
a
shifted conguration
Anoise = Rnoise /Rmeas
(4, 2, 2)
eld (144 modes) in
and superimpose to them a white noise of increasing amplitude
before reconstructing the eld.
We consider two cases :
i) a
reconstruction with the same number of modes as the synthetic eld, and varying noise
amplitude. Results are provided on the left panel of gure 6. ii) a reconstruction using
an under sampled number of modes (Nr
= 48
for
(3, 1, 1)
experimental conditions- and varying noise amplitude.
-to remain close to actual
Results are provided on the
right panel of gure 6.
In all cases, we observe a decrease of the reconstruction quality with increasing
noise.
The impact is larger for the under-sampled case but grows less rapidly.
The
energy residue however stays close from the initial energy. Our conclusion is that the
energy measurements from reconstruction techniques are reliable up to 20 percent even
in the presence of noise.
From our series of tests, we obtained two main conditions for reliable reconstruction:
shifted probe conguration and number of reconstruction mode smaller than number of
measurements. Furthermore, we observed that residue by itself is not a reliable test of
the reconstruction quality, since we can get very good residue and poor energy residue.
Since in experimental elds, we dot not have access to energy of the eld (i.e. we cannot
compute energy residue), we shall use residue test to discriminate the reconstruction
quality.
3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms
15
1
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.8
0.6
0.4
r
0.2
0
−0.2
−0.4
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.6
−0.8
−0.8
−1
−0.8
−0.6
−0.4
−0.2
0
z
0.2
0.4
0.6
−1
−0.8
0.8
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
−1
−0.8
0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
Figure 7. On the left gure, a real Von Karman water velocity eld (F=16Hz); on
the middle gure, the ltered version; and on the right gure, the reconstructed eld.
The out of plan (toroidal) component
Vφ
is coded with color (red are positive values
while blue are negative values) and the in-plane (poloidal) component (Vr ,
Vz ) is coded
with arrows. The 3 probes of 10 sensors used to reconstruct the eld are represented
in grey on the left gure.
4. Test on a real eld
In this section we just focus on testing the reconstruction method over of an experimental
velocity eld measured through a Stereoscopic Particle Image Velocimetry (SPIV) on a
Von Karman water ow in counter-rotation at a frequency of 16 Hz [23]. This velocity
eld corresponds to the mean sodium ow involved in the VKS experiment if one neglect
the magnetic eld retroaction e.g. this experimental eld is used in cinematic dynamo
study. The SPIV provides the three velocity components in a meridional plane passing
through the axis of rotation. The time average velocity eld is shown in Fig. 7. It is
obviously a fairly large scale axisymmetric eld, with four recirculating cells. To test the
reconstruction, we extract
30
articial measurements from three ctive probes located
at the locations of the VKS experiment, shown in Fig.
7.
eld using the inversion procedure described in section 2.
We then reconstruct this
However in order to take
into account the only hydrodynamic ow structure of water, we thus use the Galerkin
basis based on Beltrami waves (such that
rot(V ) = λV ,
and
Vz = 0
at
basis presented in section 2.4.4 respects the inertial nature of the ow.
r = 1).
This
Indeed from
the minimization of the energy at xed angular momentum and helicity ref. [22] shows
that the ow vorticity and the velocity are aligned everywhere and the mean ow is
a Beltrami state.
Therefore the reconstructed eld is incompressible (div V =0) and
presents boundary layers.
Since the ow is axisymmetric, we consider only cases with
of the reconstruction is provided in Fig. 7, for
is 0.26 while the energy residue is 0.25.
(5, 0, 1)
M = 0.
The result
modes. The associated residue
Comparing with the full experimental eld,
we see that the main topology of the velocity eld is well captured: the cell limits are
close to reality and their amplitudes near the centre are well reproduced. However, the
positions of the maxima of each cells is not well reproduced and the velocity amplitude
of in-plane components is overestimated near the disks.
To check whether this is an
eect of "under-resolution", we also made a comparison with a ltered-version of the
3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms
16
experimental eld, using a sgolay lter, so as to keep only largest scale. One sees that
indeed the position of the maxima of each cell is closer to that of the ltered eld.
Note that we reconstructed this topology from an axial conguration of probes (the
only one available in this situation) which is not as reliable as a shifted conguration,
as proved in Section 3.5.
From this test over an experimental eld, we conclude that the reconstruction is able
to extract general behavior (energy and global topology) of a given eld from a restrain
number of measures. Smaller scale details like the exact maxima positions, or the eld
amplitudes near the boundaries, require a much larger number of measurements, that
are for the moment unavailable in the VKS experiment.
5. Application to the VKS experiment
In this section we nally apply the reconstruction method to the magnetic eld in a
particular VKS experiment.
We consider a set up with two iron impellers.
is surrounded by sodium at rest in an outer cylinder (radius
604 mm) delineated by a copper cylinder.
in [20].
Rc = 289
The ow
mm, length
This conguration is discussed at length
In exact counter-rotation, it leads to a statistically stationary dynamo for
F1 = F2 > 16Hz .
Measurement are made through three probes:
one probe with
10 sensors, located at position 5 in gure 2 (SM1), one probe with 8 sensors at the
position 2 (SM2) and a Gaussmeter (1 sensor) in position 3 (G). Due to failure of
some experimental probe, we can consider only a restricted number of reliable measures
for reconstruction.
Here, we focus on 7 exact contra-rotation regimes at frequencies
(F1
= F2 = 12, 17, 18, 19, 20, 21, 22Hz ) corresponding to a magnetic Reynolds number
Rm = 0.6µ0 σ(F1 + F2 )/2πRC2 at temperature 120o C Rm = 24, 34, 36, 38, 40, 42, 44.
The last six regimes in this study corresponds to dynamo regime.
For each regime,
measurements are made during 60 seconds at frequency of 2000 Hz.
In order to
extract a general large-scale behavior, we average in time the measurements at each
frequency. According to results of previous section, our measurement number prevents
reconstruction of the magnetic eld with more than 54 modes.
We then reconstruct the eld with a non-axisymmetric set of modes :
(2, 1, 1)
corresponding to 32 modes (the maximum number of modes we can reconstruct the
magnetic eld with). We compute the residue and the residue test taken on one of the
sensors of SM1 (as dened in section 3.2). These residue for dierent Rm are shown in
Fig. 8. The
(2, 1, 1)
set of modes gives a residue
∼ 0.2
and a residue test
∼ 0.15
for
the dynamo regimes. This set of mode provides a reasonable t of the measurement, as
seen in Fig. 8. Its corresponding total energy as a function of
The total energy growth is roughly quadratic in
Rm
is given in Fig. 8.
Rm.
A rst striking result of our reconstruction is that the time averaged dynamo eld
is not purely axisymmetric:
m = 1
mode is needed to t the measurements.
is probably due to the presence of the
3
This
probes that break the axisymmetry of the
experiment,while the presence of the dominant m=0 already pointed in [13] is the
3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms
0.25
SM1 reconstruction
SM1
SM2 reconstruction
SM2
10
Residue
Residue test
17
Br
5
0
0.2
0.4
0.6
r
0.8
1
1.2
0.2
0.4
0.6
0.8
1
1.2
0.8
1
1.2
0.2
20
10
Bphi
Residue and Residue test
−5
0
0
−10
0
0.15
r
Bz
20
10
0
0.1
32
34
36
38
40
42
0
44
0.2
0.4
0.6
r
Rm
5000
8
4500
7
4000
6
5
3000
Em0/Em1
Total Energy
3500
2500
2000
4
3
1500
2
1000
1
500
0
32
34
36
38
40
42
0
32
44
34
36
Rm
Figure 8.
38
40
42
44
Rm
Top left:
reconstruction residue (in blue) and residue test (in red) for
dierent magnetic Reynolds number in a VKS2 experiment; Top right: comparison
between reconstructed eld (lines) and real probes measures (crosses); one sees that
the reconstruction captures well the behavior.
Bottom left:
total energy of the
reconstruction as a function of magnetic Reynolds number; Bottom right:
ratio of
the energy of m=0 modes on the energy of m=1 modes as a function of the magnetic
Reynolds number. We also add the error bars extracted from the residue.
subject of many interpretations, [27] [28] suggest a simple
α -ω
dynamo model where
the helical nature of the velocity is ejected by the impellers or [29] where the soft iron
impellers favour the growth of m=0 mode against the m=1 mode.
To quantify this
m = 0 mode to the energy of
Rm: the dynamo becomes more
result, we show in Fig. 8, the ratio of the energy of the
the
m=1
mode. We see that this ratio increases with
m = 1 does not disappear and still contains
largest Rm: for Rm = 34, Em=0 /Em=1 = 1.8;
and more axisymmetric. However the mode
a signicant amount of energy at the
Rm = 44, Em=0 /Em=1 = 5.4.
The corresponding dynamo eld structure is given in Fig. 9 at
Rm = 34
close to
the dynamo onset. It is mainly an axial dipole, with concentration of azimuthal energy
near the iron propellers. Taking a closer look at this phenomenon through iso-density
of energy surface at 25 % of the maximum energy (Fig. 9), we observe that the energy
distribution is asymmetric, with more energy close to the left disk than close to the
right disk.
At the present stage, we do not know whether this is an artifact of the
reconstruction due to the lack of measures or if this reects an actual breaking of the
3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms
18
40
1
30
0.8
0.6
20
0.4
10
r
0.2
0
0
−0.2
−10
−0.4
−0.6
−20
−0.8
−30
−1
−0.6
−0.4
−0.2
0
z
0.2
0.4
0.6
−40
Figure 9. Left: Total reconstructed magnetic eld in a VKS experiment at
just above dynamo onset.
The out of plan (toroidal) component
color and the in-plane (poloidal) component
(Br , Bz
Bφ
Rm = 34,
is coded with
is coded with arrows; Right:
Corresponding stream lines and energy density at 25% of the maximum
40
1
30
0.8
0.6
20
0.4
10
r
0.2
0
0
−0.2
−10
−0.4
−0.6
−20
−0.8
−30
−1
−0.6
−0.4
−0.2
0
z
0.2
0.4
0.6
−40
m = 0 of the reconstructed magnetic eld in a VKS experiment
Bφ is
the in-plane (poloidal) component (Br , Bz is coded with arrows;
Figure 10. Left: Mode
at
Rm = 34,
just above dynamo onset. The out of plan (toroidal) component
coded with color and
Right: Corresponding stream lines and energy density at 25% of the maximum.
axial symmetry across the z=0 plane, generated by the dynamo. More measurements
would be needed.
To try and determine the nature of the two modes, we plot them separately in Fig.
10 for the
m=0
m = 1 mode. The m = 0 mode is clearly
The m = 1 mode has a meridional dipole
mode and in Fig. 11 for the
an axial dipole, as already discussed in [13].
component, localized near the propeller which concentrates the largest energy. It does
not however not resemble the neutral mode predicted by kinematic dynamos based on
the mean velocity ow [3], conrming that the present dynamo is mainly generated by
velocity uctuations [21].
To quantify further how the repartition between the
m = 0
and
m = 1
mode
proceeds during dynamo instability, we apply a reconstruction procedure during the
3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms
19
40
1
30
0.8
0.6
20
0.4
10
r
0.2
0
0
−0.2
−10
−0.4
−0.6
−20
−0.8
−30
−1
−0.6
−0.4
−0.2
0
z
0.2
0.4
0.6
−40
m = 1 of the reconstructed magnetic eld in a VKS experiment
Bφ is
the in-plane (poloidal) component (Br , Bz is coded with arrows;
Figure 11. Left: Mode
at
Rm = 34,
just above dynamo onset. The out of plan (toroidal) component
coded with color and
Right: Corresponding stream lines and energy density at 25% of the maximum.
14
1
Em0/(Em0+Em1)
Em1/(Em0+Em1)
Normalised Energy
0.9
12
Em0/(Em0+Em1)
Em1/(Em0+Em1)
Normalised Energy
0.8
0.7
10
0.6
0.5
0.4
8
0.3
0.2
6
0.1
0
0
10
20
30
40
50
time (s)
60
70
80
90
100
4
2
0
0
10
20
30
Figure 12. Total Energy,
40
50
time (s)
m=0
and
60
m=1
70
80
90
100
modes relative energies as a function of
time during the initial growth of the magnetic eld (Rm
evolution at early time, to capture the crossover behavior.
= 36).
Insert: zoom of the
3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms
growth of the magnetic eld.
Rm = 32,
20
For this, we consider dynamical measurements at
during a growth stage, and apply a moving average over 0.1 second upon
all measurements. We then reconstruct in time the magnetic eld in order to get the
mode topology during the growth. The result is given in Fig. 12 where the normalised
magnetic energy measured in the experiment is compared with the relative energy ratios
Em=0 /(Em=0 +Em=1 ) and Em=1 /(Em=0 +Em=1 ) obtained through the reconstruction. In
this measurement, the experiment is tuned at a value of Rm larger than critical at t = 0.
The magnetic eld then increases in time due to the dynamo instability. One sees that
at the very early stage of the growth, the mode
crossover with the mode
Then, after about
30
m = 0
m=1
dominates, with an occasional
corresponding to a localized magnetic energy burst.
seconds, the magnetic eld growth accelerates, corresponding
to a crossover between the
m = 1
and
m = 0
mode, that predominates after this
stage. Note further that the magnetic eld growth proceeds in roughly three steps: one
between
t=0
and
t = 30
where the magnetic eld is small, with only one energy burst
m = 0 and m = 1 mode). A second
stage, corresponding to the crossover between the m = 0 and m = 1 mode, with larger
uctuations, but still moderate values of the magnetic eld. A third stage, from t = 60,
where the energy explodes and the dynamo settles, with a predominance of the m = 0
(corresponding to the rst crossover in between the
mode.
This interesting step-by-step dynamo onset suggests the the dynamo mechanism is
non-trivial, and may involve the action of the
m = 1
as a catalyst.
Further analysis
with more measurements will be necessary to clarify this point.
6. Discussion
We have presented a method of reconstruction for 3D magnetic eld reconstruction
based on Galerkin Transforms and tested it over synthetic elds and real solenoidal
(velocity) elds, measured in a water experiment.
Our study shows that reliable
reconstructions are possible provided the probes are suciently sampled and located in
shifted congurations. The present sets of available measurements in VKS experiments
prevent reconstruction of modes higher than
m = 1.
In that respect, it would be
interesting to develop azimuthal belt probes around the cylinder in order to extract
m > 1
modes without perturbing the ow with intrusive sensors.
To illustrate the
possibility of our method, we have applied it to an old VKS experimental conguration.
Due to the fairly low available number of measurements, we could only obtain rough
results regarding the dynamo geometry, conrming previous results.
In future VKS
dynamo campaigns, as many as four probes with 10 sensors each will be available. We
have therefore good hope that reconstruction in such a case will be more reliable, and
that we can draw interesting results concerning the magnetic eld geometry during
dynamical events (reversals, periodic cycles, extinction, bursts,...
improving our understanding of turbulent dynamos.
[14]) that can help
3D magnetic eld reconstruction in VKS experiment through Galerkin Transforms
21
Acknowledgements
JB has received nancial support from the ANR VKS-Dynamo. We thank S. Aumaitre
and F. Daviaud for comments and discussions. We acknowledge our colleagues of the
VKS team with whom the experimental data used here have been obtained.
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