Andrew Tworkowski

In-out intermittency in partial differential equation and ordinary differential equation models

We find concrete evidence for a recently discovered form of intermittency, referred to as in-out intermittency, in both partial differential equation (PDE) and ordinary differential equation (ODE) models of mean field dynamos. This type of intermittency [introduced in P. Ashwin, E. Covas, and R. Tavakol, Nonlinearity 9, 563 (1999)] occurs in systems with invariant submanifolds and, as opposed to on-off intermittency which can also occur in skew product systems, it requires an absence of skew product structure. By this we mean that the dynamics on the attractor intermittent to the invariant manifold cannot be expressed simply as the dynamics on the invariant subspace forcing the transverse dynamics; the transverse dynamics will alter that tangential to the invariant subspace when one is far enough away from the invariant manifold. Since general systems with invariant submanifolds are not likely to have skew product structure, this type of behavior may be of physical relevance in a variety of dynamical settings. The models employed here to demonstrate in-out intermittency are axisymmetric mean-field dynamo models which are often used to study the observed large-scale magnetic variability in the Sun and solar-type stars. The occurrence of this type of intermittency in such models may be of interest in understanding some aspects of such variabilities. (c) 2001 American Institute of Physics.Employing some recent results in dynamics of systems with invariant subspaces we find evidence in both truncated and full axisymmetric mean-field dynamo models of a recently discovered type of intermittency, referred to as in-out intermittency. This is a generalised form of on-off intermittency that can occur in systems that are not skew products. As far as we are aware this is the first time detailed evidence has been produced for the occurrence of a particular form of intermittency for such deterministic PDE models and their truncations. The specific signatures of this form of intermittency make it possible in principle to look for such behaviour in solar and stellar observations. Also in view of its generality, this type of intermittency is likely to occur in other physical models with invariant subspaces.

Effects of boundary conditions on the dynamics of the solar convection zone

Recent analyses of the helioseismic data have produced evidence for a variety of interesting dynamical behaviour associated with torsional oscillations. What is not so far clear is whether these oscillations extend all the way to the bottom of the convection zone and, if so, whether the oscillatory behaviour at the top and the bottom of the convection zone is different. Attempts have been made to understand such modes of behaviour within the framework of nonlinear dynamo models which include the nonlinear action of the Lorentz force of the dynamo generated magnetic field on the solar angular velocity. One aspect of these models that remains uncertain is the nature of the boundary conditions on the magnetic field. Here by employing a range of physically plausible boundary conditions, we show that for near-critical and moderately supercritical dynamo regimes, the oscillations extend all the way down to the bottom of the convection zone. Thus, such penetration is an extremely robust feature of the models considered. We also find parameter ranges for which the supercritical models show spatiotemporal fragmentation for a range of choices of boundary conditions. Given their observational importance, we also make a comparative study of the amplitude of torsional oscillations as a function of the boundary conditions.

Robustness of truncated α Ω dynamos with a dynamic α

In a recent work (Covas et al., 1996), the behaviour and the robustness of truncated α Ω dynamos with a dynamic α were studied with respect to a number of changes in the driving term of the dynamic α equation, which was considered previously by Schmalz and Stix (1991) to be of the form ∼ AΦBΦ. Here we review and extend our previous work and consider the effect of adding a quadratic quenching term of the form α|B|2. We find that, as before, such a change can have significant effects on the dynamics of the related truncated systems. We also find intervals of (negative) dynamo numbers, in the system considered by Schmalz and Stix (1991), for which there is sensitivity with respect to small changes in the dynamo number and the initial conditions, similar to what was found in our previous work. This latter behaviour may be of importance in producing the intermittent type of behaviour observed in the Sun.

Mean Field Dynamos with Algebraic and Dynamic α-Quenchings

Calculations for mean field dynamo models (in both full spheres and spherical shells), with both algebraic and dynamic α-quenchings, show qualitative as well as quantitative differences and similarities in the dynamical behaviour of these models. We summarise and enhance recent results with extra examples.Overall, the effect of using a dynamic α appears to be complicated and is affected by the region of parameter space examined.

Redistribution of energy by vertical oscillations in the solar atmosphere

The effect of vertical oscillations with periods between 90 s and 300 s on a solar atmosphere governed by heat conduction and radiation loss is examined. The effect is found to be primarily a redistribution, rather than a net addition or subtraction, of energy within the low corona, mainly by long period (180 to 300 s) oscillations. The redistribution of energy is found to affect the time-averaged temperature and density profiles of such an atmosphere, particularly in the low corona. The amount of energy redistributed is found to increase with increasing period.