A Route to Magnetic Field Reversals: An Example of an ABC-Forced Nonlinear Dynamo

We are investigating numerically the nonlinear behaviour of a space-periodic MHD system with ABC forcing. Most computations are performed for magnetic Reynolds numbers increasing from 0 to 60 and a fixed kinematic Reynolds number, small enough for the trivial solution with a zero magnetic field to be stable to velocity perturbations. At the critical magnetic Reynolds number for the onset of instability of the trivial solution the dominant eigenvalue of the kinematic dynamo problem is real. In agreement with the bifurcation theory new steady states with non-vanishing magnetic field appear in this bifurcation. Subsequent bifurcations are investigated. A regime is detected, where chaotic variations of the magnetic field orientation (analogous to magnetic field reversals) are observed in the temporal evolution of the system.     

KS－熵与混沌层次

有两种不同类型的混沌层次：一种得之于映射维数的增加；另一种则由不变环面分叉产生。通过计算相应的ＫＳ－熵，我们发现前者只是一种几何现象，后者则是本质的混沌层次，即伴随有混沌性的增加。     

Mixed-Parity Solutions in a Mean-Field Dynamo Model

We consider the effect of including both dipole and quadrupole parities in the previous mean-field model of Hollerbach and Jones (1995), which considered dipole parity only. Allowing for both parities, we find that the onset of dynamo action occurs at ₀ 6, in the form of a purely quadrupolar dynamo wave. A symmetry-breaking bifurcation then occurs at ₀ 11, beyond which the solutions are of mixed parity. The quadrupolar component still oscillates about a zero time-average, but the dipolar component about a non-zero average. For even greater ₀ we obtain an unconnected upper-branch solution. In sharp contrast to the HJ95 pure-parity upper branch, however, this mixed-parity upper branch is steady-state rather than periodic. Although it does not appear to be possible to connect these two upper branches by any simple sequence of bifurcations, we nevertheless suggest how aspects of the mixed-parity branch may help in understanding features of the previous pure-parity branch.     

From the circular to the spatial elliptic restricted three-body problem

We deal with the study of the spatial restricted three-body problem in the case where the small particle is far from the primaries, that is, the so-called comet case. We consider the circular problem, apply double averaging and compute the relative equilibria of the reduced system. It appears that, in the circular problem, we find not only part of the equilibria existing in the elliptic case, but also new ones. These critical points are in correspondence with periodic and quasiperiodic orbits and invariant tori of the non-averaged Hamiltonian. We explain carefully the transition between the circular and the elliptic problems. Moreover, from the relative equilibria of elliptic type, we obtain invariant 3-tori of the original system.     

Nonlinear Convection in a Rotating Annulus with a Finite Gap

Thermal convection in a fluid-filled gap between the two corotating, concentric cylindrical sidewalls with sloping curved ends driven by radial buoyancy was first studied by Busse (Busse, F.H., "Thermal instabilities in rapidly rotating systems", J. Fluid Mech . 44 , 441-460 (1970)). The annulus model captures the key features of rotating convection in full spherical geometry and has been widely employed to study convection, magnetoconvection and dynamos in planetary systems, usually in connection with the small-gap approximation neglecting the effect of azimuthal curvature of the annulus. This article investigates nonlinear thermal convection in a rotating annulus with a finite gap through numerical simulations of the full set of nonlinear convection equations. Three representative cases are investigated in detail: a large-gap annulus with the ratio of the radii ( s i and s o ) of the sidewalls ξ = s i / o s = 0.1, a medium-gap annulus with ξ = 0.35 and a small-gap annulus with ξ = 0.8. Near the onset of convection, the effect of rapid rotation through the sloping ends forces the first (Hopf) bifurcation in the form of small-scale, steadily drifting rolls (thermal Rossby waves). At moderately large Rayleigh numbers, a variety of different convection patterns are found, including mixed-mode steadily drifting, quasi-periodic (vacillating) and temporally chaotic convection in association with various temporal and spatial symmetry-breaking bifurcations. Our extensive simulations suggest that competition between nonlinear and rotational effects with increasing Rayleigh number leads to an unusual sequence of bifurcation characterized by enlarging the spatial scale of convection.     

The Lissajous transformation III. Parametric bifurcations

We examine a parametric family of cubic perturbed 1-1 resonant harmonic oscillators with an aim to understanding the phase flows of the reduced system. Variation of the parameters leads the system through five bifurcations of different types. Some bifurcations are due to passage through cases of discrete symmetry or integrability. A conjecture correlating degenerate equilibria in reduced systems with integrability is modified and reinforced.     

Magnetic field generation by convective flows in a plane layer: the dependence on the Prandtl numbers

Investigation of magnetic field generation by convective flows is carried out for three values of kinematic Prandtl number: P = 0.3, 1 and 6.8. We consider Rayleigh–Bénard convection in Boussinesq approximation assuming stress-free boundary conditions on horizontal boundaries and periodicity with the same period in the x and y directions. Convective attractors are modelled for increasing Rayleigh numbers for each value of the kinematic Prandtl number. Linear and non-linear dynamo action of these attractors is studied for magnetic Prandtl numbers P _m ≤ 100. Flows, which can act as magnetic dynamos, have been found for all the three considered values of P, if the Rayleigh number R is large enough. The minimal R, for which of magnetic field generation occurs, increases with P. The minimum (over R) of critical P_m for magnetic field generation in the kinematic regime is admitted for P = 0.3. Thus, our study indicates that smaller values of P are beneficial for magnetic field generation.