Relations Between Dynamo-Region Geometry and the Magnetic Behavior of Stars and Planets

The geo and solar magnetic fields have long been thought to be very different objects both in terms of spatial structure and temporal behavior. The recently discovered field structure of a fully convective star is more reminiscent of planetary magnetic fields than the Sun’s magnetic field [1], despite the fact that the physical and chemical properties of these objects clearly differ. This observation suggests that a simple controlling parameter could be responsible for these different behaviors. We report here the results of three-dimensional simulations which show that varying the aspect ratio of the active dynamo region can yield sharp transition from Earth-like steady dynamos to Sun-like dynamo waves. Introduction. – Observations of the magnetic fields due to dynamo activity appear to fall into two categories: fields dominated by large-scale dipoles (such as the Earth and a fully convective star), and fields whith smaller-scale and non-axisymmetric structures (such as the Sun). Moreover two kinds of different temporal behaviour have been identified so far: very irregular polarity reversals (as in the Earth), and quasi-periodic reversals (as in the Sun). Since the Earth and the Sun provide the largest database of magnetic field observations, these objects have been well studied and described in terms of alternative physical mechanisms: the geodynamo involves a steady branch of the dynamo equations, perturbed by strong fluctuations that can trigger polarity reversals, whereas the solar dynamo takes the form of a propagating dynamo wave. The signature of this wave at the Sun’s surface yields the well-known butterfly-diagram (Sunspots preferentially emerge at a latitude that is decreasing with time during the solar cycle). Modelling. – Because of their very different natures (liquid metal in one case, plasma in the other), planetary and stellar magnetic fields are studied by different communities. Non-dimensional numbers controling the dynamics of the Earth and the Sun, for example, do significantly differ (see [2,3]). As a practical matter however, the techniques as well as the typical parameters used in numerical studies of these two systems are surprisingly similar. To some extent this is due to the restricted parameter space available to present day computations. The parameter regime numerically accessible is rather remote from the actual objects. For planetary dynamos the main discrepancy relies in the rapid rotation in the momentum equation (characterized by the Ekman number), whilst for stellar dynamos it relies in solving the induction equation with weak resistive effects (characterized by high values of the magnetic Reynolds number). Yet within this restricted domain, the sharply different key characters to both geo [4] and solar [5,6] magnetic fields have been reproduced. This leads us to argue that the important parameter controlling the magnetic field behaviour is the aspect ratio of the dynamo region (i.e. the radius ratio of the inner bounding sphere to the outer bounding sphere). Indeed, in the Earth, the inert solid inner core extends to less than 40% of the core radius, whereas in the Sun, the radiative zone fills 70% of the solar radius. One expects the convective zones of stars and planets to have all possible intermediate aspect ratios, even extending to fully convective spheres. In order to isolate and understand this purely geometrical effect, we have carried out three-dimensional numerical simulations of self-excited convective dynamos in which the domain aspect ratio was slowly varied, with all other parameters held constant. The governing equations as well as parameter regimes used here were originally introduced for a geodynamo reference calculation [7]. The only distinction being the use of stress-free boundary conditions on the outer sphere of the domain, while imposing no-slip boundary conditions at the bottom of the convective re-