h λr 3.3175 λr 3.3663 λr 6.1863 λr 13.926 rel. err. COC rel. err. COC rel. err. COC rel. err. COC
0.2 9.364E-4 - 3.943E-3 - 1.439E-1 - 6.104E-1
-0.1 1.833E-4 2.35 2.147E-3 0.88 1.734E-4 9.70 4.484E-1 0.44 0.05 3.751E-5 2.29 1.188E-3 0.85 2.241E-5 2.95 1.599E-1 1.49 0.025 8.405E-6 2.16 6.463E-4 0.88 2.833E-6 2.98 1.120E-5 13.8 0.0125 2.081E-6 2.01 3.439E-4 0.91 3.667E-7 2.95 1.478E-6 2.92 Table B.3: Approximation of the rst four eigenvalues forεr = 0.5. We used α = 0.7 in the
simulations.
h λr'4.5339 λr '6.2503 λr'7.0371 λr '22.342 rel. err. COC rel. err. COC rel. err. COC rel. err. COC
0.2 4.559E-1 - 6.052E-1 - 6.410E-1 - 8.869E-1
-0.1 2.859E-1 0.67 4.731E-1 0.36 5.310E-1 0.27 8.512E-1 0.06 0.05 3.306E-2 3.11 2.982E-1 0.67 3.763E-1 0.50 8.033E-1 0.08 0.025 2.154E-6 13.9 7.748E-2 1.94 1.772E-1 1.09 7.406E-1 0.12 0.0125 2.608E-7 3.05 3.258E-3 4.57 5.946E-7 18.2 6.602E-1 0.17 Table B.4: Approximation of the rst four eigenvalues forεr = 0.1. We used α = 0.8 in the
simulations.
Then, the Riesz-Thorin theorem implies that kψkb H˙s(K)b ≤c hs−
3 2
K kψkH˙s(K),
where we dened H˙s(E) := [ ˙L2(E),H˙1(E)]s,2 with L˙2(E) and H˙1(E) being the subspaces of the functions of zero average in L2(E) and H1(E), respectively. We conclude using Lemma B.7.2
Lemma B.7.2. The spaces [ ˙L2(E),H˙1(E)]s and [L2(E), H1(E)]s∩L˙2(E) are identical and the induced norms are identical, i.e.,kvkH˙s(E)=kvkHs(E) for all for allv∈[L2(E), H1(E)]s∩ L˙2(E).
Proof. One can use Lemma A1 from [64] withT being the projection onto L˙2(Ω). Lemma B.7.3. The following holds for all s∈[0,1] and for allv∈Hs(Ω),
(B.7.3) X
K∈Th
kv|Kk2Hs(K)≤ kvk2Hs(Ω).
Proof. The result is evident fors= 0 ands= 1. Let us assume now that s∈(0,1). Letv be a member of Hs(Ω). Recall that
kvkHs(Ω):=
Z ∞ 0
K(t, v,Ω)2t−1−2sdt 12
, K(t, v,Ω)2 := inf
w∈H1(Ω)
kv−wk2L2(Ω)+t2kwk2H1(Ω)
.
For allt∈R+, let us denotevtthe function inH1(Ω)that minimizesK(t, v,Ω), i.e.,−t2∆vt+ t2vt+ (vt−v) = 0 over Ωwith homogeneous Neumann boundary condition. Then
X
K∈Th
kv|Kk2Hs(K)= X
K∈Th
Z ∞ 0
K(t, v|K, K)2t−1−2sdt
≤ X
K∈Th
Z ∞ 0
kv|K−vt|Kk2L2(Ω)+t2kvt|Kk2H1(K)
t−1−2sdt
= Z ∞
0
X
K∈Th
kv|K−vt|Kk2L2(K)+t2kvt|Kk2H1(K)
t−1−2sdt
= Z ∞
0
K(t, v,Ω)2t−1−2sdt:=kvk2Hs(Ω). This completes the proof.
We now state the main result of this section. It is a variant of Lemma 8.2 in [25] with the extra term kφkL2(K). Our proof slightly diers from that in [25] since the proof therein did not appear convincing to us (actually, the embedding inequality at line 9, page 2224 in [25]
has a constant that depends on the size of the cell; as result the estimate (8.11) therein is not uniform with respect toh).
Lemma B.7.4. For all k∈N and all σ ∈(0,2) there is c, uniform with respect to the mesh family, so that the following holds for all facesF ∈ Fh in the mesh, all polynomial function v of degree at most k, and all functionφ∈Hσ(K)∩H(curl, K)
(B.7.4) Z
F
(v×n)·φ≤ckvkL2(F)h−
1 2
F (hσKkφkHσ(K)+hKk∇×φkL2(K)+kφkL2(K)), where K is either one of the two elements sharing the face F.
Proof. We restrict ourselves to three space dimensions. In two space dimensions φis scalar-valued and the proof must be modied accordingly. Let K be either one of the two elements sharing the face F. Let φ the average of φ over K and let us denote ψ := φ−φ. Upon denoting v(x) =b JKTv(TK(bx)) and ψ(b x) =b JKTψ(TK(x))b , it is a standard result (see [111, 3.82]) that
Z
F
(v×n)·ψ = Z
Fb
(bv×bn)·bψ,
where bn is one of the two unit normals on Fb. Let us extend bv by zero on ∂K\b Fb; then bv ∈: H12−σ(∂K)b for all σ > 0. Note that it is not possible to have σ = 0. Now let R : H12−σ(∂K)b −→H1−σ(K)b be a standard lifting operator. There is a constant depending only onKb andσ so that
kRvkb L2(
K)b +k∇×Rb vkb H−σ(K)b ≤c(K, σ)kRb bvkH1−σ(K)b ≤c0c(K, σ)kb bvk
H12−σ(Fb), where ∇×b is the curl operator in the coordinate system ofKb. Then we have
Z
Fb
(bv×bn)·bψ
= Z
Kb
(Rbv)·∇×b ψb −ψ·b ∇×(Rb v)b
≤c
k(Rv)kb L2(
K)b k∇×b ψkb L2(
K)b +kψkb Hσ
0(K)b k∇×(Rb bv)kH−σ(K)b
≤c
k∇×b ψkb L2(
K)b +kψkb Hσ 0(K)b
kvkb
H12−σ(Fb)
≤c
k∇×b ψkb L2(
K)b +kψkb Hσ(
K)b
kvkb
H12−σ(Fb),
where we used that Hσ(K) =b Hσ0(K)b for σ ∈ [0,12). Due to norm equivalence for discrete functions over Kb and using that kJKk ≤ chK, hK/hF ≤ c and |F| ≤ ch2F in three space dimensions, where c depends of the shape-regularity constant of the mesh family and the polynomial degreek, we have
kbvk
H12−σ(F)b ≤ckvkb L2(
F)b ≤ckJKk|F|−12kvkL2(F)≤chKh−1F kvkL2(F)≤c0kvkL2(F). Using the identity (see [111, Cor. 3.58])
(∇×ψ)(TK(bx)) = 1
det(JK)JK(∇×b ψ)(b x),b we obtain
k∇×b ψkb L2(K)≤c|det(JK)|12kJK−1kk∇×ψkL2(K)≤ch
1 2
K k∇×ψkL2(K).
Since the average of ψ over K is zero, we can use Lemma B.7.1 (with an extra scaling by kJKk for ψb =JKTψ(TK)) to deduce
kψkb Hσ(
K)b ≤chσ−
1 2
K kψkHσ(K). In conclusion we have obtained the following estimate:
Z
F
(v×n)·(φ−φ)≤c hK k∇×φkL2(K)+hσKkφ−φkHσ(K)
h−
1 2
K kvkL2(F). Observing thatk1kHσ(K)≤ k1k1−σ
L2(K)k1kσH1(K)=k1kL2(K)=|K|12, we infer that kφ−φkHσ(K)≤ kφkHσ(K)+|φ||K|12
The Cauchy-Schwarz inequality yields|φ| ≤ |K|−12kφkL2(K); as a result, kφ−φkHσ(K)≤ kφkHσ(K)+kφkL2(K)≤2kφkHσ(K). Now we evaluate a bound from above onR
F(v×n)·φas follows:
Z
F
(v×n)·φ
≤ |φ||F|12kvkL2(F)≤ |K|−12kφkL2(K)|F|12kvkL2(F)
≤ckvkL2(F)h−
1 2
F kφkL2(K). The result follows by combining all the above estimates.
Lemma B.7.5. Let α∈(12,1). There is exists a constant c(α) so that (B.7.5) kukL2(Γ)≤c(α)kuk1−
1 2α
L2(Ω)kuk
1 2α
Hα(Ω), ∀u∈Hα(Ω).
Similarly, for s∈ 0,12
, there exists a constantc(s) so that, for Θ :=2(1−s)1−2s , (B.7.6) kukL2(Γ)≤c(s)kuk1−ΘHs(Ω)kukΘH1(Ω), ∀u∈H1(Ω).
Proof. We start with the standard estimate kukL2(Γ)≤ckuk
1 2
L2(Ω)kuk
1 2
H1(Ω), ∀u∈H1(Ω),
which allows us to apply Lemma B.7.6. This implies that the trace operator is a continuous linear mapping from[L2(Ω), H1(Ω)]1
2,1 to L2(Γ). Then the re-iteration lemma implies that [L2(Ω), Hα(Ω)]1
2α,1= [L2(Ω),[L2(Ω), H1(Ω)]α,2]1
2α,1 = [L2(Ω), H1(Ω)]1 2,1
[Hs(Ω), H1(Ω)]Θ,1= [[L2(Ω), H1(Ω)]s,2, H1(Ω)]Θ,1 = [L2(Ω), H1(Ω)]1 2,1
The norms being equivalent, we can write:
kukL2(Γ)≤ckuk[L2(Ω),H1(Ω)]1
2,1 ≤c(α)kuk[L2(Ω),Hα(Ω)]1
2α,1 ≤c(α)kuk1−
1 2α
L2(Ω)kuk
1 2α
Hα(Ω), kukL2(Γ)≤ckuk[L2(Ω),H1(Ω)]1
2,1 ≤c(s)kuk[Hs(Ω),H1(Ω)]Θ,1 ≤c(s)kuk1−ΘHs(Ω)kukΘH1(Ω).
Lemma B.7.6 (Lions-Petree). LetE1 ⊂E0be two Banach spaces, with continuous embedding.
Let L be a linear mapping E1 → F with F another Banach space. For s∈ (0,1), L extends to a linear mapping from [E0, E1]s,1 toF if and only if there exists C >0 such that
∀u∈E1, kLukF ≤Ckuk1−sE
0 kuksE
1. Proof. See Lemma 25.3 in Tartar [143].
Electromagnetic induction in non-uniform domains
A. Gieseckea, C. Noreb,c, F. Stefania, G. Gerbetha, J. Léoratd, F. Luddensb,e, J.-L. Guermondb,e Abstract
Kinematic simulations of the induction equation are carried out for dierent setups suit-able for the von-Kármán-Sodium (VKS) dynamo experiment. The material properties of the ow driving impellers are modeled by means of high conducting and high permeability disks in a cylindrical volume lled with a conducting uid. Two entirely dierent numer-ical codes are mutually validated by showing quantitative agreement on Ohmic decay and kinematic dynamo problems using various congurations and physical parameters.
Field geometry and growth rates are strongly modied by the material properties of the disks even if the disks are thin. In contrast the inuence of external boundary conditions remains small.
Utilizing a VKS like mean uid ow and high permeability disks yields a reduction of the critical magnetic Reynolds number Rmc for the onset of dynamo action of the simplest non-axisymmetric eld mode. However this threshold reduction is not sucient to fully explain the VKS experiment. We show that this reduction ofRmc is inuenced by small variations in the ow conguration so that the observed reduction may be changed with respect to small modications of setup and properties of turbulence.
C.1 Introduction
Magnetic elds of galaxies, stars or planets are produced by dynamo action in a homogenous medium in which a conducting uid ow provides for generation of eld energy. During the past decade the understanding of the eld generation mechanism has considerably benetted from the examination of dynamo action in the laboratory. However, realization of dynamo action in laboratories at least requires the magnetic Reynolds numberRm =U L/η (whereU
a Helmholtz-Zentrum Dresden-Rossendorf, PO Box 510119, 01314 Dresden, Germany
b Laboratoire d'Informatique pour la Mécanique et les Sciences de l'Ingénieur, CNRS, BP 133, 91403 Orsay Cedex, France
c Institut Universitaire de France, 103 Bd Saint-Michel, 75005 Paris, France
d Luth, Observatoire de Paris-Meudon, place Janssen, 92195 Meudon, France
eDepartment of Mathematics, Texas A&M University 3368 TAMU, College Station, TX 77843-3368, USA
andL represent typical velocity and length scales and η denotes the magnetic diusivity) to exceed a threshold of the order ofRmc∼10...100. From the parameter values of liquid sodium the best known liquid conductor at standard laboratory conditions (η= 1/µ0σ≈0.1 m2/s and L ≈ 1m, where µ0 is the vacuum permeability and σ the electrical conductivity) it becomes immediately obvious that self excitation of magnetic elds in the laboratory needs typical velocity magnitudes ofU ∼10m/s, which is already quite demanding. Therefore, the rst successful dynamo experiments performed by [98, 99] utilized soft-iron material so that the magnetic diusivity is reduced (this issue deserves indeed a specic study and is examined below) and the magnetic Reynolds number is (at least locally) increased. Although these experiments cannot be classied as hydromagnetic dynamos (no uid ow and therefore no backreaction of the eld on a uid motion is possible) they allowed the examination of distinct dynamical regimes manifested in steady, oscillating or reversing elds. It is interesting to note that these results did not initiate further numerical studies on induction in the presence of soft iron domains.
The eects of internal and external walls with nite permeability and conductivity have been examined in [9, 8] by analytically solving a one dimensional kinematic dynamo driven by anα-eect. A facilitation of dynamo action is obtained for increasing conductivity and/or permeability of given inner and outer walls. This threshold reduction is monotonous in the case of a stationary dynamo mode but non monotonous in the case of a time dependent dynamo due to dissipation from eddy currents induced within the container walls. The authors also assumed that a mean ow may increase the dynamo threshold due to additional dissipation.
More recently, [127] performed nonlinear simulations in a sphere with a ow driven by the counter rotation of the two hemispherical parts of the outer sphere. Their setup and geometry are only roughly representative for the VKS conguration (they also included an inner sphere made of a solid electrical insulator). They performed nonlinear simulations simultaneously varying permeability and conductivity of the external walls, applying thin wall conditions (where the wall thicknessh → 0 and the permeability µr → ∞ and conductivity σ → ∞ so that the product hµr (hσ) remains nite). Only a few runs exhibit dynamo action and their results cannot yield any general conclusion about the inuence of the wall permeability or conductivity on the dynamo threshold.
A possibility to increase the eective magnetic Reynolds number in uid ow driven dy-namo experiments arises from the addition of tiny ferrous particles to the uid medium leading to an uniform enlargement of the relative permeability [50, 43]. Since the amount of particles is limited so as to retain reasonable uid properties, the maximum uid permeability achievable by this technique isµr ≈2. The main eect found in the simulations of [43] was a reduced decay of the initial eld but not a smaller threshold (essentially because of nonmonotonous behavior of the growth rate with respect toRm).
Another type of ferromagnetic inuence on dynamo action is observed in the von-Kármán-sodium (VKS) dynamo. In the VKS experiment a turbulent ow of liquid von-Kármán-sodium is driven by two counterotating impellers located at the opposite end caps of a cylindrical domain [110].
Dynamo action is only obtained when the impellers are made of soft-iron with µr ∼ 100 [153]. Recently it has been shown in [58] that these soft-iron impellers essentially determine the geometry and the growth rates of the magnetic eld by locally enhancing the magnetic Reynolds number and by enforcing internal boundary conditions for the magnetic eld at the material interfaces. We conjecture that non-homogenous distributions of the material coecients µr and σ may support dynamo action because gradients of µr and σ modify the induction equation by coupling toroidal and poloidal components of the magnetic elds which
is essential for the occurrence of dynamo action. An example for this dynamo type has been presented in [28] where it was shown that even a straight ow without shear over an (innite) conducting plate with sinusoidal variation of the conductivity is able to produce dynamo action. However, an experimental realization of this setup would require either an unachievable large magnetic Reynolds number or rather large variations of the conductivity (>∼factor of 100 and with a mean value which should be of the order of the uid conductivity).
On the other hand, large permeability variations are more easily achievable experimentally, for instance the relative permeability of soft-iron alloys easily attains values of several thousands.
Although these dynamos are of little astrophysical relevance the experiments of Lowes and Wilkinson and in particular the rich dynamical behavior of the VKS dynamo demonstrate the usefulness of such models.
The purpose of the present work is to validate the numerical tool necessary to establish a basic understanding of the inuence of material properties on the induction process. Emphasis is given to the problem of free decay in cylindrical geometry where two disks characterized by high conductivity/permeability and their thickness are inserted in the interior of a cylindrical container lled with a conducting uid. To demonstrate the reliability of our results we use two dierent numerical approaches and show that both methods give results in agreement.
The study is completed by an application of a mean ow as it occurs in the VKS experiment in combination with two high permeability disks.