## A solar mean field dynamo benchmark

- L. Jouve, A. S. Brun, +9 authors A. G. Kosovichev
- 2008

- 2008

Recently, Jouve et al [2] published the paper that presents the numerical benchmark for the solar dynamo models. Here, I would like to show a way how to get it with help of computer algebra system Maxima. This way was used in [4] to test some new ideas in the large-scale stellar dynamos. What you need are the latest version of Maxima-5.16.3 (preferable compiled against the fastest lisps like sbcl of cmucl-sse2) and some knowledge of the global (spectral) methods to solve the PDE eigenvalue problem. For the quite comprehensive introduction to these methods please look at the book by John Boyd [1]. The basic steps to solve the problem are: 1. the mathematical formulation (equation+boundary conditions) 2. choice the basis function and project equations to the basis 3. find matrices (apply some integration procedure in case of Galerkin method) 4. apply linear algebra The whole consideration is divided for two cases. As the first case we explore the largest free decay modes in the sphere which is submerged in vacuum. In this problem the all dynamo effects are neglected. As the second case I test the αΩ dynamo in the solar convection zone with the tachocline included. Lets consider the spherical geometry. The evolution of the axisymmetric large-scale magnetic field (LSMF), B = e φ B + curl Ae φ r sin θ , (where r is radius, θ-co-latitude, e φ-the unit azimuthal vector) in the turbulent media subjected to the differential rotation in the spherical shell can be described with equations: ∂B ∂t = 1 r ∂ (Ω, A) ∂ (r, θ) + 1 r ∂ rE θ ∂r − ∂E r ∂θ , (1) ∂A ∂t = r sin θE φ , (2) In equations above, the turbulent contribution is expressed through the components of the mean electromotive force (MEMF) E = u × b , where u , b are the small-scale fluctuated velocity and magnetic field respectively, Ω = Ω (r, θ)-the given angular velocity distribution. For the sake of simplicity we restrict consideration to the case of αΩ dynamo with isotropic turbulent diffusion. Hence, we have