Showing papers by "Friedrich H. Busse published in 2009"

Bistability and hysteresis of dipolar dynamos generated by turbulent convection in rotating spherical shells

Abstract: 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.

Prandtl number dependence of convection driven dynamos in rotating spherical fluid shells

Abstract: 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.

Parameter dependences of convection driven dynamos in rotating spherical fluid shells

Abstract: 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.

Inertial convection in rotating fluid spheres

Abstract: 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.

Oblique roll instability in inclined buoyancy layers

Abstract: When an inclined plate is heated in a stably stratified fluid, a buoyancy-driven boundary-layer flow will be generated. According to previous studies, such a layer is mainly subjected to two instability mechanisms: the transverse traveling (TS) waves and the stationary longitudinal (SL) rolls. In present paper we identify a novel oblique roll (OR) instability. The oblique rolls are more unstable than SL rolls in fluids with Prandtl numbers P of the order unity or less for all angles of inclination χ with respect to the horizontal except those close to 0°. The OR mode has smaller critical Grashof numbers than the TS waves in some regions of the P – χ -parameter space. This feature is surprising because the transverse TS wave has been reported as the most unstable infinitesimal disturbances in boundary-layer flows. In comparison with TS waves, oblique rolls have a lower critical frequency, which coincides with the Brunt–Vaisala frequency of internal waves when the surface temperature is fixed and the tilt angle is large. Another important feature of the OR mode is that its amplitude decreases less rapidly with distance from the plate than the amplitudes of SL rolls and TS waves. As a result, OR mode disturbances penetrate into the stably stratified region for nearly three times the thickness of the boundary layer. Further investigation reveals that in low Prandtl number fluids with a uniform-heat-flux boundary the preference for onset of oblique rolls instead of TS waves is even stronger than in the case of a fixed temperature at the plate.

Convection in Rotating Spherical Fluid Shells

Abstract: 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.

Homologous onset of double layer convection.

Abstract: The onset of convection in two superimposed fluid layers of the same height is considered. It is found that the neutral curve for R(a) for the onset Rayleigh number R in dependence on the wave number a is an invariant of a multidimensional parameter space of property ratios of the system even though the corresponding convection solutions may vary strongly with these property ratios. For each neutral curve R(a) two manifolds of solutions are found one of which can be understood on the basis of symmetry properties of the system, while the other does not exhibit simple symmetry features. In particular the neutral curves R(a) for various single Rayleigh-Benard convection layers are shown to correspond to two two-dimensional manifolds of solutions. Analytical expressions for the latter are derived in the case of outer stress-free boundary conditions.

Bistability and hysteresis of dipolar dynamos generated by turbulent convection in rotating spherical shells

Abstract: 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.

Dynamos of giant planets

Abstract: 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.

Onset of inertial magnetoconvection in rotating fluid spheres.

Abstract: The onset of convection in the form of magneto-inertial waves in a rotating fluid sphere permeated by a constant axial electric current is studied through a perturbation analysis. Explicit expressions for the dependence of the Rayleigh number on the azimuthal wavenumber are derived in the limit of high thermal diffusivity. Results for the case of thermally infinitely conducting boundaries and for the case of nearly thermally insulating boundaries are obtained.

Magneto-inertial convection in rotating fluid spheres

Abstract: The onset of convection in the form of magneto-inertial waves in a rotating fluid sphere permeated by a constant axial electric current is studied through a perturbation analysis. Explicit expressions for the dependence of the Rayleigh number on the azimuthal wavenumber are derived in the limit of high thermal diffusivity. Results for the cases of thermally infinitely conducting and of nearly thermally insulating boundaries are obtained.

Patterns of convection in rotating spherical shells

Abstract: 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.