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 (Donati J.-F. et al., Science, 311 (2006) 633), 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.
Deep Dive into 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 (Donati J.-F. et al., Science, 311 (2006) 633), 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 avail-able 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- gion. This choice was made in order to create a strong shear at the base of the model, and thus try to mimic the solar tachocline [8]. The inner sphere is here assumed to be insulating, and we use differential heating. The governing equations are in non-dimensional form:
where
All simulations reported here were performed keeping the following parameters constant E = 10 -3 , Ra = 100, Pr = 1, Pm = 5 . The above system is integrated in threedimensions of space (3D) using the Parody code [9]. When the inner (non dynamo generating) body occupies less than about 60% of the convective body in radius, the flow generates a dipolar field, very similar to that of the Earth. It features patches of intense flux at high latitudes and some reversed patches at low latitude, similar to the ones revealed by a downward continuation of the Earth’s field to the Core-Mantle boundary [10]. This strongly dipolar solution becomes unstable with a further increase of the aspect ratio. For an aspect ratio of 0.65 -close to that of the Sun-the strong dipole is first maintained and then strongly weakens, but dynamo action continues in a different form: that of a wavy solution with quasi-periodic reversals (Fig. 1), reminiscent of some aspects of the solar magnetic field behavior. Drifting features can be observed both on the radial field at the surface of the model (Fig. 1 &2b) and on the azimuthal (east-west) field below the surface of the model (Fig. 2c). Due to the complex nature of these fully tri-dimensional simulations, many waves can co-exist. Some of the dominant structures appear to propagate toward the equator; others propagate poleward. Reversed waves are also observed at the surface of the Sun at higher latitudes [11]. Let us stress however that the model cannot be
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