arXiv subjects
Publications and source records attributed to F. H. Busse.
The range of existence and the properties of two essentially different chaotic attractors found in a model of nonlinear convection-driven dynamos in rotating spherical shells are investigated. A hysteretic transition between these attractors is established as a function of the rotation parameter \ttau. The width of the basins of attraction is also estimated.
A physically consistent model of magnetic field generation by convection in a rotating spherical shell with a minimum of parameters is applied to the Sun. Despite its unrealistic features the model exhibits a number of properties resembling those observed on the Sun. The model suggests that the large scale solar dynamo is dominated by a non-axisymmetric $m=1$ component of the magnetic field.
Thermal convection in fluid layers heated from below are usually realized experimentally as well as treated theoretically with fixed boundaries on which conditions for the temperature and the velocity field are prescribed. The thermal and velocity boundary layers attached to the upper and lower boundaries determine to a large extent the properties of turbulent convection at high Rayleigh numbers. Fixed boundaries are often absent in natural realizations of thermal convection. This paper studies the properties of convection driven by a planar heat source below a cooling source of equal size immersed in an otherwise stably stratified fluid layer are studied in this paper. Unavoidable boundaries do not influence the convection flow since they are separated from the active convection layer by nearly motionless stably stratified regions. The onset of convection occurs in an inner unstably stratified region where the mean temperature gradient is reversed. But the region of a reversed horizontally averaged temperature gradient disappears at higher amplitudes of convection such that the vertical derivative of the mean temperature no longer changes its sign.
We report the results of three-dimensional numerical simulations of convection-driven dynamos in relatively thin rotating spherical shells that show a transition from an strong non-oscillatory dipolar magnetic field to a weaker regularly oscillating dipolar field. The transition is induced primarily by the effects a stress-free boundary condition. The variation of the inner to outer radius ratio is found to have a less important effect.
Some concepts used in the theory of convection-driven dynamos in rotating spherical fluid shells are discussed. The analogy between imposed magnetic fields and those generated by dynamo action is evaluated and the role of the Elsasser number is considered. Eddy diffusivities are essential ingredients in numerical dynamo simulations, but their effects could be misleading. New aspects of the simultaneous existence of different dynamo states are described.
We present results on modeling solar pores and sunspots using 2D axisymmetric magneto-hydrodynamic (MHD) simulations. These models are helpful for understanding the mechanisms of magnetic field concentration in sunspots, and the large-scale flow patterns associated with them. The simulations provide consistent, self-maintained, although not fully realistic, models of concentrated magnetic field near the solar surface. In this paper, we explore under which conditions the associated flows are converging or diverging near the surface. We find that in most cases in which a stable, pore-like concentration of magnetic field forms, a configuration with converging over diverging flow is established.
The value of the Prandtl number $P$ exerts a strong influence on convection-driven dynamos in rotating spherical shells filled with electrically conducting fluids. Low Prandtl numbers promote dynamo action through the shear provided by differential rotation, while the generation of magnetic fields is more difficult to sustain in high-Prandtl-number fluids where higher values of the magnetic Prandtl number $P_m$ are required. The magnetostrophic approximation often used in dynamo theory appears to be valid only for relatively high values of $P$ and $P_m$. Dynamos with a minimum value of $P_m$ seem to be most readily realizable in the presence of convection columns at moderately low values of $P$. The structure of the magnetic field varies strongly with $P$ in that dynamos with a strong axial dipole field are found for high values of $P$ while the energy of this component is exceeded by that of the axisymmetric toroidal field and by that of the non-axisymmetric components at low values of $P$. Some conclusions are discussed in relation to the problem of the generation of planetary magnetic fields by motions in their electrically conducting liquid cores.
Patterns of convection in internally heated, self-gravitating rotating spherical fluid shells are investigated through numerical simulations. While turbulent states are of primary interest in planetary and stellar applications the present paper emphasizes more regular dynamical features at Rayleigh numbers not far above threshold which are similar to those which might be observed in laboratory or space experiments. Amplitude vacillations and spatial modulations of convection columns are common features at moderate and large Prandtl numbers. In the low Prandtl number regime equatorially attached convection evolves differently with increasing Rayleigh number and exhibits an early transition into a chaotic state. Relationships of the dynamical features to coherent structures in fully turbulent convection states are emphasized.
The chapter begins with a description of the annulus model of rotationg convection. The basic equations for the spherical problem are then introduced and the onset of columnar convection in spherical shells is discussed. The onset of inertial mode convection is described which prevails at very low Prandtl numbers. The properties of finite amplitude convection are outlined for a wide range of Prandtl numbers. Equatorially attached convection is considered which evolves from inertial mode convection.
Possibilities and difficulties of applying the theory of magnetic field generation by convection flows in rotating spherical fluid shells to the Giant Planets are outlined. Recent progress in the understanding of the distribution of electrical conductivity in the Giant Planets suggests that the dynamo process occurs predominantly in regions of semiconductivity. In contrast to the geodynamo the magnetic field generation in the Giant Planets is thus characterized by strong radial conductivity variations. The importance of the constraint on the Ohmic dissipation provided by the planetary luminosity is emphasized. Planetary dynamos are likely to be of an oscillatory type, although these oscillations may not be evident from the exterior of the planets.
The dynamics of convecting fluids in rotating spherical shells is governed at Prandtl numbers of the order unity by the interaction between differential rotation and roll-like convection eddies. While the differential rotation is driven by the Reynolds stresses of the eddies, its shearing action inhibits convection and causes phenomena such as localized convection and turbulent relaxation oscillations. The response of the system is enriched in the case of dynamo action. Lorentz forces may brake either entirely or partially the geostrophic differential rotation and give rise to two rather different dynamo states. Bistability of turbulent dynamos exists for magnetic Prandtl numbers of the order unity. While the ratios between mean magnetic and kinetic energies differ by a factor of 5 or more for the two dynamo states, the mean convective heat transports are nearly the same. They are much larger than in the absence of a magnetic field.
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 $τ$. 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.
The theory of planetary dynamos and its applications to observed phenomena of planetary magnetism are outlined. It is generally accepted that convection flows driven by thermal or compositional buoyancy are the most likely source for the sustenance of global planetary magnetic fields. While the existence of dynamos in electrically conducting fluid planetary cores provides constraints on properties of the latter, the lack of knowledge about time dependences of the magnetic fields and about their toroidal components together with the restricted parameter regions accessible to theory have prevented so far a full understanding of the phenomena of planetary magnetism.
The onset of convection in the form of inertial waves in a rotating fluid sphere is studied through a perturbation analysis in an extension of earlier work by Zhang (1994). Explicit expressions for the dependence of the Rayleigh number on the azimuthal wavenumber are derived and new results for the case of a nearly thermally insulating boundary are obtained.
It is proposed that convection driven dynamos operating in planetary cores could be oscillatory even when the oscillations are not directly noticeable from the outside. Examples of dynamo simulations are pointed out that exhibit oscillations in the structure of the azimuthally averaged toroidal magnetic flux while the mean poloidal field shows only variations in its amplitude. In the case of the geomagnetic field, global excursions may be associated with these oscillations. Long period dynamo simulations indicate that the oscillations may cause reversals once in a while. No special attempt has been made to use most realistic parameter values. Nevertheless some similarities between the simulations and the paleomagnetic record can be pointed out.
Bistability and hysteresis of magnetohydrodynamic dipolar dynamos generated by turbulent convection in rotating spherical fluid shells is demonstrated. Hysteresis appears as a transition between two distinct regimes of dipolar dynamos with rather different properties including a pronounced difference in the amplitude of the axisymmetric poloidal field component and in the form of the differential rotation. The bistability occurs from the onset of dynamo action up to about 9 times the critical value of the Rayleigh number for onset of convection and over a wide range of values of the ordinary and the magnetic Prandtl numbers including the value unity.
A numerical model of axisymmetric convection in the presence of a vertical magnetic flux bundle and rotation about the axis is presented. The model contains a compressible plasma described by the nonlinear MHD equations, with density and temperature gradients simulating the upper layer of the sun's convection zone. The solutions exhibit a central magnetic flux tube in a cylindrical numerical domain, with convection cells forming collar flows around the tube. When the numerical domain is rotated with a constant angular velocity, the plasma forms a Rankine vortex, with the plasma rotating as a rigid body where the magnetic field is strong, as in the flux tube, while experiencing sheared azimuthal flow in the surrounding convection cells, forming a free vortex. As a result, the azimuthal velocity component has its maximum value close to the outer edge of the flux tube. The azimuthal flow inside the magnetic flux tube and the vortex flow are prograde relative to the rotating cylindrical reference frame. A retrograde flow appears at the outer wall. The most significant convection cell outside the flux tube is the location for the maximum value of the azimuthal magnetic field component. The azimuthal flow and magnetic structure are not generated spontaneously, but decay exponentially in the absence of any imposed rotation of the cylindrical domain.
A convection-driven MHD dynamo in a rotating spherical shell, with clearly defined structural elements in the flow and magnetic field, is simulated numerically. Such dynamos can be called deterministic, in contrast to those explicitly dependent on the assumed properties of turbulence. The cases most interesting from the standpoint of studying the nature of stellar magnetism demonstrate the following features. On a global scale, the convective flows can maintain a ``general'' magnetic field with a sign-alternating dipolar component. Local (in many cases, bipolar) magnetic structures are associated with convection cells. Disintegrating local structures change into background fields, which drift toward the poles. From time to time, reversals of the magnetic fields in the polar regions occur, as ``new'' background fields expel the ``old'' fields.