The integral equation approach to kinematic dynamo theory and its application to dynamo experiments in cylindrical geometry

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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: M.Xu@fz-rossendorf.de

Key words: Dynamo theory, Integral equations, Magnetic induction, Dynamo experi-ments

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 K´arm´an 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

∂t =∇ × (u × B) + 1

µσ∆B, ∇ · B = 0, (1)

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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 value, denoted by Rc

m, 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 elec-trically 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 elec-trical 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 ap-proach (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 homoge-neous 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}

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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 cylindri-cally symmetric problems. Then we switch over to the treatment of specific problems, including the free decay case, the ”von K´arm´an 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:}

b(r) = µσ 4π Z V (u(r0)× (B0(r0) + b(r0))× (r − r0) |r − r0_|3 dV 0 −µσλ_4π Z V A(r0₎_{× (r − r}0₎ |r − r0_|3 dV 0₋ µσ 4π Z Sφ(s 0_)n(s0₎_× r− s0 |r − s0_|3dS 0 ₍₃₎ 1 2φ(s) = 1 4π Z V (u(r0₎_{× (B} 0(r0) + b(r0))· (s − r0) |s − r0_|3 dV 0 − λ 4π Z V A(r0)· (s − r0₎ |s − r0_|3 dV 0 ₋ 1 4π Z Sφ(s 0_)n(s0₎_· s− s0 |s − s0_|3dS 0 ₍₄₎ A(r) = 1 4π Z V (B0(r0) + b(r0))× (r − r0) |r − r0_|3 dV 0 ₊ 1 4π Z Sn(s 0₎_×B0(s0) + b(s0) |r − s0_| dS0, (5)

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 con-ductivity, 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 = [ρ0cos ϕ0, ρ0sin ϕ0, z0]T,

b = [bρ, bϕ, bz]T, u = [uρ, uϕ, uz]T. (6)

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in the following series form:    b φ A   =X    bm φm Am   exp(imϕ). (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

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c |ρ0_=Ra,z=H − RaE₁m|_ρ0_=Ra,z=H)dz0

+(_−ρ0RaE_cm_|ρ=Rauz+ ρ02E1m|ρ=Rauz+ ρ0(z− z0)uρE1m|ρ=Ra)(B0ϕ+ bϕ)

+(ρ0uϕRaEcm|ρ=Ra+ ρ0RauρEsm|ρ=Ra− ρ02uϕE1m|ρ=Ra)(B0z+ bz)dρ0dz0

−Z Ra 0 φ(z− H)E m 1 |ρ=Ra,z0_=Hρ0dρ0− Z H −Hφ(E m

c |ρ=ρ0_=Ra− E₁m|_ρ=ρ0_=Ra)Ra2dz0

+ Z Ra 0 φ(z + H)E m 1 |ρ=Ra,z0_=−Hρ0dρ0− λ Z H −H Z Ra 0 (ρ 0_RaEm c |ρ=Ra

−ρ02E1m|ρ=Ra)Aρ− ρ0RaEsm|ρ=RaAϕ+ ρ0(z− z0)E1m|ρ=RaAzdρ0dz0], (12)

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.

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+ρ0(B0ϕ+ bϕ)Dcm|z0_=−Hdρ0] (14) Aϕ = 1 4π[ Z H −H Z Ra 0 −ρ 0_(z_{− z}0_)Em c (B0ρ+ bρ) + ρ0(z− z0)Esm(B0ϕ+ bϕ) +ρ0(ρEm 1 − ρ0Ecm)(B0z+ bz)dρ0dz0+ Z Ra 0 ρ 0_Dm c |z0_=H(B_0ρ+ b_ρ) −ρ0_Dm s |z0=H(B0ϕ+ bϕ)dρ0− Z H −HRa(B0z + bz)D m c |ρ0=Radz0 + Z Ra 0 ρ 0_Dm s |z0_=−H(B_0ϕ+ b_ϕ)− ρ0D_cm|_z0_=−H(B_0ρ+ b_ρ)dρ0] (15) Az = 1 4π[ Z H −H Z Ra 0 −ρ 0_ρEm s (B0ρ+ bρ) + ρ0(ρ0E1m− ρEcm)(B0ϕ+ bϕ)dρ0dz0 + Z H −HRaD m 1 |ρ0_=Ra(B_0ϕ+ b_ϕ)dz0], (16) where Dm_s (ρ, ρ0, z, z0) = Z 2π 0 sin ϕ0_{sin mϕ}0 (ρ2_{− 2ρρ}0_{cos ϕ}0_{+ ρ}02_{+ (z}_{− z}0₎2₎1₂dϕ 0 Dm_c (ρ, ρ0, z, z0) = Z 2π 0 cos ϕ0cos mϕ0 (ρ2_{− 2ρρ}0_{cos ϕ}0_{+ ρ}02_{+ (z}_{− z}0₎2₎12 dϕ0 Dm 1 (ρ, ρ0, z, z0) = Z 2π 0 cos mϕ0 (ρ2_{− 2ρρ}0_{cos ϕ}0_{+ ρ}02_{+ (z}_{− z}0₎2₎12 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 bϕ bz    = µσ[P    B0ρ+ bρ B0ϕ+ bϕ B0z + bz   − Q    φs1 φs2 φs3   − λR    Aρ Aϕ Az   ], (17) 1 2    φs1 φs2 φs3    = S    B0ρ+ bρ B0ϕ+ bϕ B0z + bz   − λT    Aρ Aϕ Az   − U    φs1 φs2 φs3   , (18)    Aρ Aϕ Az    = V    B0ρ+ bρ B0ϕ+ bϕ B0z+ bz   , (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,

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bz = [bz(i∆ρ, j∆z)]n×m, (20) φρ = [φρ(i∆ρ, j∆z)]n×m, φϕ = [φϕ(i∆ρ, j∆z)]n×m, φz = [φz(i∆ρ, j∆z)]n×m, (21) Aρ = [Aρ(i∆ρ, j∆z)]n×m, Aϕ = [Aϕ(i∆ρ, j∆z)]n×m, Az = [Az(i∆ρ, j∆z)]n×m. (22)

Combining Eqs.(17-19), we obtain

(I− µσE − µσλG)b = µσ(E + λG)B0, (23) where E = P_{− Q · (}1 2I + U) −1_{· S} ₍₂₄₎ G = Q_{· (}1 2I + U) −1_{· T · V − R · V .} ₍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_{− R}mE)· 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}

non-hermitian 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 decomposi-tion. The obtained induced magnetic field will be compared with the data measured in experiment.

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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 -0.5 0 0.5 1 x 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 z

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, Mari´e, 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: ur = − π 2 r (1− r) 2_{(1 + 2r) cos(πz)} uϕ = 4r(1− r) sin(πz/2) uz = (1− r)(1 + r − 5r2) sin(πz). (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}

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(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}

the critical magnetic Reynolds number Rc

m, layers at the top and the bottom (so-called

”lid layers”) lead to an unexpected increase of Rc

m. A similar phenomenon was also found

for the VKS experiment26_{. 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.

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0 1 2 3 4 5 6 0 5 10 15 20 25 30 35 40 45 50

Induced field/Applied field

Rm (a) Axial: meas. at r=0.5 Axial: comp. at r=0.417 0.458 0.5 0.542 0.583 0 1 2 3 4 5 6 0 5 10 15 20 25 30 35 40 45 50

Rm (c) Axial: meas. at r=0.5 Axial: comp. at r=0.417 0.458 0.5 0.542 0.583 0 1 2 3 4 5 6 0 5 10 15 20 25 30 35 40 45 50

Rm (d) Axial: meas. at r=0.5 Axial: comp. at r=0.417 0.458 0.5 0.542 0.583

Figure 3: Ratio of axial induced magnetic field bz to the applied magnetic field. Experimental results at radius r=0.5 (taken from27_{.) 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.

0 1 2 3 4 5 6 0 5 10 15 20 25 30 35 40 45 50

Rm (a) Azimuth: meas. at r=0.5 Azimuth: comp. at r=0.417 0.458 0.5 0.542 0.583 0 1 2 3 4 5 6 0 5 10 15 20 25 30 35 40 45 50

Rm (c) Azimuth: meas. at r=0.5 Azimuth: comp. at r=0.417 0.458 0.5 0.542 0.583 0 1 2 3 4 5 6 0 5 10 15 20 25 30 35 40 45 50

Rm (c) Azimuth: meas. at r=0.5 Azimuth: comp. at r=0.417 0.458 0.5 0.542 0.583

Figure 4: Same as Fig. 3, but for the azimuthal field component bϕ.

-80 -60 -40 -20 0 20 40 60 0 5 10 15 20 25 30 35 40 45 50

Angle of induced field

Rm (a) Angle: meas. at r=0.5 Angle: comp. at r=0.417 0.458 0.5 0.542 0.583 _-80 -60 -40 -20 0 20 40 60 0 5 10 15 20 25 30 35 40 45 50

Rm (c) Angle: meas. at r=0.5 Angle: comp. at r=0.417 0.458 0.5 0.542 0.583 _-80 -60 -40 -20 0 20 40 60 0 5 10 15 20 25 30 35 40 45 50

Rm (d) Angle: meas. at r=0.5 Angle: comp. at r=0.417 0.458 0.5 0.542 0.583

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 uϕkeeps 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

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(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.

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3.4 Riga experiment

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 -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 1000 1200 1400 1600 1800 2000 2200 2400 2600 Growth rate p s [1/s]

Rotation rate Ω_s [1/min] kinematic: DEA kinematic: IEA kinematic: measured saturation: measured 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 1000 1200 1400 1600 1800 2000 2200 2400 2600 F-requency f s [1/s]

Rotation rate Ω_s [1/min] kinematic: DEA

kinematic: IEA kinematic: measured saturation: measured

(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 dy-namo 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 agree-ment. 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

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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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