For the understanding of planetary and stellar dynamos an overview of the major parameter dependences of convection driven dynamos in rotating spherical fluid shells is desirable. Although the computationally accessible parameter space is limited, earlier work is extended with emphasis on higher Prandtl numbers and uniform heat flux condition at the outer boundary. The transition from dynamos dominated by non-axisymmetric components of the magnetic field to those dominated by the axisymmetric components depends on the magnetic Prandtl number as well as on the ordinary Prandtl number for higher values of the rotation parameter $\tau$. The dependence of the transition on the latter parameter is also discussed. A variety of oscillating dynamos is presented and interpreted in terms of dynamo waves, standing oscillation or modified relaxation oscillations.
Deep Dive into Parameter dependences of convection driven dynamos in rotating spherical fluid shells.
For the understanding of planetary and stellar dynamos an overview of the major parameter dependences of convection driven dynamos in rotating spherical fluid shells is desirable. Although the computationally accessible parameter space is limited, earlier work is extended with emphasis on higher Prandtl numbers and uniform heat flux condition at the outer boundary. The transition from dynamos dominated by non-axisymmetric components of the magnetic field to those dominated by the axisymmetric components depends on the magnetic Prandtl number as well as on the ordinary Prandtl number for higher values of the rotation parameter $\tau$. The dependence of the transition on the latter parameter is also discussed. A variety of oscillating dynamos is presented and interpreted in terms of dynamo waves, standing oscillation or modified relaxation oscillations.
Most global magnetic fields of planets and stars are generated by thermal or compositional convection in the deep interiors of those bodies. In spite of their common origin a large variety of dynamo processes is indicated by the observations of planetary and stellar magnetic fields. This is not surprising in view of the widely varying conditions under which dynamos operate. It is thus desirable to understand the effects of the most influential parameters on convection driven dynamos. The purpose of a study of the parameter dependence of dynamos would be twofold: On the one hand properties of observed magnetic fields could be explained in terms of parameter values of the respective system. On the other hand, unknown conditions in the interior of the celestial bodies may be inferred from the spatio-temporal structures of their magnetic fields. While it is not yet possible to create convincingly detailed models of planetary dynamos and of the solar cycle, some major variations of dynamos as function of their parameters can be explored through numerical simulations.
The increasing availability in recent years of computer capacity has facilitated large scale numerical simulations of the generation of magnetic fields by convection in rotating spherical fluid shells. Because of limited numerical resolution, molecular values of material properties are usually not attainable in computer simulations and eddy diffusivities representing the effects of the unresolved scales of the turbulent velocity field must therefor be invoked for comparisons with observations. It is often assumed for this reason that the eddy diffusivities for velocities, temperature and magnetic fields are identical. The effects of turbulence on the diffusion of vector and scalar quantities differ, however, and eddy diffusivity ratios such as the effective Prandtl number and the effective magnetic Prandtl number thus do not equal unity in general. Besides the Prandtl numbers, the boundary conditions exert a strong influence on convection and its dynamo action. Both effects will be considered in this paper.
In some respects this paper represents an extension of an earlier paper (Simitev and Busse, 2005, to which we shall refer to by SB05) which has focused on the dependence of average properties of convection driven dynamos on the Prandtl number. New results will be reported in the following and the emphasis will be placed on time dependent properties and on dynamo oscillations in particular. In addition the effect of boundaries of low thermal conductivity will be studied which are often more realistic than the commonly assumed boundaries with fixed temperature. Especially in the case of the Earth the low conductivity of the mantle will lead to a uniform heat flux from the core unless the effect of mantle convection is taken into account. Since the inhomogeneity introduced by the latter is not well known it will not be considered in the present analysis. For simulations with various inhomogeneous thermal boundary conditions see the papers by Glatzmaier et al. (1999) and by Olson and Christensen (2002). If compositionally driven convection in the Earth’s core is emphasized the choice of uniform flux at the inner boundary and of fixed composition at the outer boundary is appropriate as has been assumed in the work of Glatzmaier and Roberts (1995).
After a brief introduction of the basic equations and the method of their numerical solution in section 2, some properties of convection without magnetic field will be considered in section 3. In particular the influence of fixed heat flux boundary conditions will be explored. In section 4 the onset of convection driven dynamos in fluids with different Prandtl numbers is described and in section 5 the oscillatory dynamos are interpreted in terms of the theory of dynamo waves. The influences of various boundary conditions are studied in section 6 and a concluding discussion is given in the final section 7.
We consider a rotating spherical fluid shell of thickness d and assume that a static state exists with the temperature distribution T S = T 0 -βd 2 r 2 /2. Here β = q/(3 κ c p ) and T 0 = T 1 -(T 2 -T 1 )/(1 -η), where T 1 and T 2 are the constant temperatures at the inner and outer spherical boundaries, η = r i /r o is the radius ratio of the inner r i to the outer r o radius, q is the uniform heat source density, κ its thermal diffusivity, c p is its specific heat at constant pressure and rd is the length of the position vector with respect to the center of the sphere. The gravity field is given by g = -dγr. In addition to d, the time d 2 /ν, the temperature ν 2 /γαd 4 and the magnetic flux density ν(µ̺) 1/2 /d are used as scales for the dimensionless description of the problem where ν denotes the kinematic viscosity of the fluid, ̺ its density and µ is its magnetic permeability. Since we shall assume the Boussinesq approximation material properties are regarded as constants except for the temperature
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