Free Access
Volume 508, Number 1, December II 2009
Page(s) 9 - 16
Section Astrophysical processes
Published online 08 October 2009

Online Material

Appendix A: Benchmarks for the code used

Here we present some benchmark tests for the computer code that we used and show how the crititical dynamo numbers and dynamo periods for the models of Sects. 3.1 and 3.2 converge for an increasing number of modes taken into account.

A.1 Free decay modes

First we test the accuracy of the implementation of the exterior boundary conditions and the speed of convergence. If we neglect all the dynamo effects in Eqs. (2) and (3), only simple isotropic diffusion remains and the equations take the form

$\displaystyle \frac{\partial A}{\partial t} = \frac{\partial^{2}A}{\partial x^{...
...artial}{\partial\theta}\frac{1}{\sin\theta}\frac{\partial A}{\partial\theta} ~,$   (A.1)
$\displaystyle \frac{\partial B}{\partial t} = \frac{1}{x}\frac{\partial^{2}\lef...
...\frac{1}{\sin\theta}\frac{\partial\left(\sin\theta B\right)}{\partial\theta} ~,$   (A.2)

where for simplicity the magnetic diffusivity has been assumed to be homogeneous and has been set equal to unity. The equations for the poloidal and toroidal parts of the field are decoupled here. The eigenmodes to Eqs. (A.1) and (A.2) are the free decay modes, exponentially decaying $\propto \exp{\lambda_i t}$, where the $\lambda_i$ are the eigenvalues of the Laplacian operator for the considered domain under the imposed boundary conditions; these eigenvalues are all real and negative. For the test, we consider the case of a full sphere (rather than a spherical shell) surrounded by vacuum, for which the free decay modes can be determined analytically and are well documented in the literature (see, e.g., Moffatt 1978; Backus et al. 1996). For that purpose, the potential functions A and Bare written as
$\displaystyle A\left(x,\theta,t\right) = {\rm e}^{{\displaystyle
\lambda t}}\su...
\sin\theta ~ S_{nm}^{(A)}\left(x\right)P_{m}^{1}\left(\cos\theta\right),$   (A.3)
$\displaystyle B\left(x,\theta,t\right) = {\rm e}^{\displaystyle{\lambda t}}\sum_{n}\sum_{m}B_{nm}S_{n}^{(B)}\left(x\right)P_{m}^{1}\left(\cos\theta\right),$   (A.4)

                               $\displaystyle S_{nm}^{(A)}\left(x\right)$ = $\displaystyle x\left(P_{2n+1}\left(x\right)\!-\!P_{1}\left(x\right)\frac{\left(2n\!+\!1\right)\left(2n\!+\!2\right)\!+\!2\left(m\!+\!1\right)}{2m+4}\right),$ (A.5)
$\displaystyle S_{n}^{(B)}\left(x\right)$ = $\displaystyle x\left(P_{2n+1}\left(x\right)-P_{1}\left(x\right)\right),$ (A.6)

where the radial variable, x, varies in the interval $\left[0,1\right]$; the transformation to the variable $\xi$ (cf. Eqs. (21) and (22) in Sect. 2) is not used here. By the choice of the basis functions given by Eqs. (A.5) and (A.6) the exterior vacuum conditions are satisfied and the regularity of the fields at the origin is ensured. This set of basis functions differs from that used in our calculations for the spherical shell, but the general structure of the code and the solution algorithms are not changed.

The dependence of the solutions to Eqs. (A.1) and (A.2) on radius is given analytically in terms of the spherical Bessel functions $j_n(x)\propto \left(1/\sqrt{x}\right)J_{n+1/2}(x)$ (where Jn+1/2 is the ordinary Bessel function of half-integer order n+1/2), and the decay rates, $-\lambda_n$, are given by the squares of the zeros of the functions jn-1 for the poloidal and jn for the toroidal modes. The slowest decaying (largest scale) poloidal mode decays with the rate $-\lambda_1^{(A)}=\pi^2$, the slowest decaying toroidal mode with the rate $-\lambda_1^{(B)}\approx 4.4934^2$; the corresponding eigenfunctions are $S_1^{(A)}(x,\theta)\propto j_1\left(\lambda_1^{(A)}x\right)\sin^2\theta$and $S_1^{(B)}(x,\theta)\propto j_1\left(\lambda_1^{(B)}x\right)\sin\theta$. These two decay modes are used for the test. The $\theta$ dependences of their potential functions are given by the first terms (with m=1) in the latitudinal expansions on the right-hand sides of Eqs. (A.3) and (A.4). (The potential functions A and B differ from the potentials S and T in the poloidal-toroidal decomposition $\vec{B} =\nabla\times(\vec{r}\times\nabla S) + \vec{r}\times\nabla T$as normally used in non-axisymmetric cases, see, e.g., Moffatt (1978) and Backus et al. (1996). For our axisymmetric case, one has $\partial S/\partial\theta = A/\sin\theta$ and $\partial T/\partial\theta = B$. The angular dependence of both S for the slowest decaying poloidal mode and T for the slowest deacaying toroidal mode is given by the spherical surface harmonic $Y_1^0(\cos\theta)\propto \cos\theta$, in agreement with the $\theta$ dependences of A and B as given above.)

Table A.1 shows the convergence of the eigenvalues and of the corresponding eigenvectors for our numerical scheme. Similar to Livermore & Jackson (2005), the eigenvectors are scaled so that $S_1^{(A)}\left(x=1,\theta=\pi/2\right)=1$ and $S_1^{(B)}\left(x=0.5,\theta=\pi/2\right)=1$, and the errors are measured as $E\left(\lambda\right)=\left\vert\lambda_{{\rm true}}-\lambda_{{\rm num}}\right\vert$and $E\left(\vec{B}\right)=\int_V\left\vert\vec{B}_{{\rm true}}-\vec{B}_{{\rm num}}\right\vert^{2}{\rm d}V$. The number of modes in the radial basis, N, is varied, while in the latitudinal basis just the first mode is taken into account. The convergence is seen to be exponential in both the poloidal and toroidal cases.

Table A.1:   Convergence of the eigenvalues and eigenvectors of the slowest decaying poloidal and toroidal modes$^{\rm a}$.

A.2 Test case B of Jouve et al. (2008)

The next test case is taken from Jouve et al. (2008), who presented a comparitative benchmark study of different numerical codes for axisymmetric mean-field solar dynamo models in spherical geometry. Here we consider their test case B, which is a pure $\alpha\Omega$ dynamo in a spherical shell with sharp gradients of the turbulent magnetic diffusivity and the strength of the $\alpha $ effect at the bottom of the convection zone; for details we refer to Jouve et al. (2008). The potential functions A and B are expanded according to Eqs. (21)-(26) in Sect. 2, and the integration domain is now radially bounded by $x_{\rm b} = 0.65$ at the bottom and $x_{\rm t} =1$ at the top.

In Jouve et al. (2008), the strength of the $\alpha $ effect is regulated by a dynamo number, $C_\alpha$. The different codes are compared by indicating in tables the critical $\alpha $-effect dynamo number, $C_{\alpha}^{{\rm crit}}$, at which exponentially growing solutions appear, and the corresponding oscillation frequency, $\omega = 2\pi/T$. In addition, butterfly diagrams and the evolution of the fields in the meridional plane are shown. Our values of $C_{\alpha}^{{\rm crit}}$ and $\omega$ for different spectral resolutions are given in Table A.2, and Fig. A.1 shows the temporal evolution of the toroidal and poloidal parts of the field (i.e., of the unstable eigenmode) at the critical dynamo number. The values in Table A.2 are in best agreement with those given in the corresponding table, Table 3, of Jouve et al. (2008). Similarly, the evolution shown in Fig. A.1 is apparently identical to that shown in the corresponding figure, Fig. 7, of Jouve et al. (2008); the same applies to the simulated butterfly diagrams (not shown here).

\end{figure} Figure A.1:

As Fig. 6 ( top and middle), but for test case B of Jouve et al. (2008) at the critical $\alpha $-effect dynamo number with a spectral resolution of $16 \times 16$ modes.

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\end{figure} Figure A.2:

Convergence of the critical dynamo numbers (left) and associated dynamo periods (right) for the models of Sects. 3.1 and 3.2. N is the total number of modes taken into account. Calculations were done for resolutions of $6\times 6$, $6\times 8$, $8\times 10$, $10\times 14$, $16\times 18$, $15\times 22$, and $14\times 30$ modes in the radial and latitudinal bases, respectively (in addition, the dipolar symmetry was taken into account, so that the highest latitudinal resolution is actually 50). Solid lines refer to the $\delta ^{(\Omega )}\Omega $ dynamo model and dashed lines to the $\delta ^{(W)}\Omega $ dynamo model, and blue color (i.e., the lower/upper curve pair in the left/right panel) corresponds to $u_0=15~{\rm m/s}$ and red color to $u_0=25~{\rm m/s}$.

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Table A.2:   Test case B of Jouve et al. (2008)$^{\rm a}$.

A.3 Convergence of crititical dynamo numbers and dynamo periods for the models of Sects. 3.1 and 3.2

Figure A.2 shows the convergence of the critical dynamo numbers (where the first dipolar mode becomes unstable) and associated dynamo periods for the $\delta ^{(\Omega )}\Omega $ dynamo model considered in Sect. 3.1 and for the $\delta ^{(W)}\Omega $ dynamo model considered in Sect. 3.2. The amplitude of the meridional flow is $u_0=15~{\rm m/s}$ and $u_0=25~{\rm m/s}$; $u_0=15~{\rm m/s}$ is the value we used most, and $u_0=25~{\rm m/s}$ is the highest meridional-flow amplitude that we considered, corresponding to the largest magnetic Reynolds number in the study. High Reynolds numbers are known to cause numerical problems.

Appendix B: Definitions of the functions f $_{\rm i}^{({\rm a})}$ and f $_{\rm i}^{({\rm d})}$

Here we give the definitions of the functions fi(a) and fi(d) that are used in the representation of the turbulent electromotive force $\vec{\mathcal{E}}$. For details of the calculations we refer to Pipin (2008).

\begin{eqnarray*}f_{1}^{(a)} & = & \frac{1}

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