Astron. Nachr. / AN 331, No. 9/10, 883 – 889 (2010) / DOI 10.1002/asna.201011419 The interior of the Sun in 3-D: Beyond the spherical Sun picture S. Mathis ⋆ Laboratoire AIM, CEA/DSM-CNRS-Universit´ e Paris Diderot, IRFU/SAp Centre de Saclay, F-91191 Gif-sur-Yvette, France Received 2010 Sep 9, accepted 2010 Sep 13 Published online 2010 Nov 11 Key words magnetohydrodynamics (MHD) – Sun: evolution – Sun: helioseismology – Sun: interior – Sun: magnetic fields – Sun: rotation With all the results now obtained by global helioseismology, it is clear that we have to go beyond the classical spherical modelling of the Sun. In this way, it is now necessary to draw a picture that takes into account internal dynamical processes such as rotation, magnetic field, and waves from the core to the surface. This is a challenging task since such mechanisms involve length and time-scales that differ from several orders of magnitude and impact on the solar behaviour both on dynamical and secular times. In this review, I will thus describe the present state of the art in the modelling of the internal dynamics of the Sun. c  2010 WILEY-VCH Verlag GmbH& Co. KGaA, Weinheim 1 Introduction To simulate the dynamical processes in a star in full details would require including length scales and time scales span- ning many orders of magnitude. This is clearly not feasible, even with the most powerful computers. Either one chooses to describe what occurs on a dynam- ical time scale, such as a convective turnover time, or one focuses on the long time evolution where the typical time is either the Kelvin-Helmholtz time or that characterising the dominant nuclear reactions. The same is true for the length scales. First, in the vertical direction, we have to take the resolution that adequately represents the steepest gradients that develop during the evolution. Moreover, in the horizon- tal direction, the resolution that allows to describe convec- tive structures and turbulence has to be chosen. This is the reason why it is nowadays necessary to use and couple 1-D, 2-D, and 3-D models to get a global picture of macroscopic MHD transport processes in stellar (solar) interiors. In this short review, we report the state of the art of the modelling of both convective envelope and radiative core of our Sun (and solar-type stars) aimed to explain its magnetic activity and its internal differential rotation revealed by he- lioseismology (Garcia et al. 2007, Fig. 1). First, we present the common method used both on dynamical and secular time scales, namely the spectral expansion of MHD equa- tions. Next, we focus on the convective envelope dynami- cal processes with describing couplings between differential rotation, meridional circulation, turbulence, and magnetic field that leads to the observed dynamo action with empha- sizing the peculiar role of the tachocline. Then, we give a global picture of transport and mixing processes operating in the radiation zone. Finally, we conclude on the necessity ⋆ Corresponding author: stephane.mathis@cea.fr Fig. 1 (online colour at: www.an-journal.org) Internal angular velocity in the Sun revealed by helioseismology. It is conical in the convective envelope (0.02 M⊙), uniform in the radiative core (0.98 M⊙), the transition layer, the tachocline, being very thin (≤ 0.04 R⊙). Adapted from Garcia et al. (2007), courtesy “Sci- ence”. to obtain integrated models of the Sun from its core to its surface. 2 Modelling 2.1 Preliminary definitions First, the macroscopic velocity field is expanded V (r,t)= r sin θ Ω(r, θ, t)  e ϕ +˙ r  e r + U M (r, θ, t)+ u (r, θ, ϕ, t) , (1) c  2010 WILEY-VCH Verlag GmbH& Co. KGaA, Weinheim