Hans Haucke

Mode-locking and chaos in Rayleigh—Benard convection

Abstract We have experimentally studied the transition to chaos in a thermally convecting dilute solution of 3 He in superfluid 4 He. Two natural oscillatory instabilities are found in our geometry. Mode-locking is observed over a range of Rayleigh and Prandtl number. The mode-locking steps have a measured fractal dimension in good agreement with theoretical predictions and the larger steps show Farey tree ordering. Far below chaos many features of the measurements are suggestive of the circle map. We produce a one-dimensional map showing a tangent bifurcation to a mode-locked state. An effective phase angle change across a mode-locked step is calculated from the data. Close to chaos, hysteresis and other effects are suggestive of a two-dimensional map. The fractal dimension of the attractor is measured over a range of Rayleigh number close to the chaotic onset. Our Poincare sections show the fractalization of the torus and are characterized by extremely high resolution.

Oscillatory convection in a dilute3He-superfluid4He solution

Convective instabilities in a rectangular, unity-aspect-ratio Rayleigh-Bénard cell with a solution of 1.46%3He in superfluid4He have been studied in the temperature range 0.70–1.05 K, with a corresponding Prandtl number range of 0.045<σ<0.15. The onset of stationary convection is much like that in a classical, one-component fluid. The oscillatory instability is studied by using an extremely sensitive local temperature probe. It is found that the total heat transport efficiency is suppressed by the oscillations in the entire range of Prandtl number we have studied. The local temperature probe indicates a striking difference in the oscillatory amplitude when the sense of rotation of the convective rolls is reversed. The magnitude of the convective velocity is deduced from both the initial slope of the Nusselt number near the onset of the stationary convection and the frequency of the oscillations. Determinations of the temperature dependence of the convective velocity using these two methods agree very well. The observed behavior of the oscillatory frequency and onset condition supports the theory of oscillatory convection for a classical, low-Prandtl-number, one-component fluid.

Simple differential thermometer for low temperatures using a thermocouple with a SQUID detector

Details of design and construction of a simple and rugged thermocouple differential thermometer are presented. Gold +0.03 at. % iron wire is used, with niobium leads connected to a SQUID. It is demonstrated that at 1 K the response time is about 15 ms and that the temperature sensitivity of 10−7 K with a 10‐Hz filter is Johnson‐noise limited. Additional qualities such as small size, little self‐heating, and high reproducibility make this device ideal for differential thermometry at low temperatures.

Time-dependent thermal convection in dilute solutions of 3He in superfluid 4He

Time-dependent thermal convection can occur in a unity-aspect-ratio Rayleigh-Beńard convection cell containing a dilute solution of 3He in superfluid 4He when the fluid is heated from above. Results are presented primarily for a 0.24 mole % He solution at 0.925 K. Means is provided for introducing heat at the top and separately for a central plug and an outer ring such that both are at a constant temperature gDT above the bottom. A critical temperature difference δTcfor convection can be defined above which both steady and time-dependent convection occur. The time-dependent effects include a region of δT. near δTcand characterized only by excessive noise, a region of somewhat higher δT where there are intermittent major changes in the plug heating rate with a time distribution like that for random events, and a region at still higher δT where periodic but nonsinusoidal variation of the heat flow is observed. When a long enough time, several months, has elapsed after cooling down the apparatus, time-dependent states no longer occur, and the heat flow above δTcis limited to steady convection. Briefly raising the temperature of the apparatus to 77 K is sufficient to restore the possibility of time-dependent states.

Noise-induced intermittency in the quasiperiodic regime of Rayleigh-Benard convection

Thermal measurements on a converting dilute3He-superfluid4He solution in the quasiperiodic regime show a transition from a mode-locked periodic state to chaotic time dependence via intermittency. The type of intermittency is discussed in the context of standard models of the phenomenon. In a region just below the onset of intermittency, injection of external multiplicative noise with noise amplitude above a certain threshold level induces the chaotic state. This noise-induced transition can be understood to be due to perturbations of a system with a barely stable attractor; the noise causes the system to escape the weakly attracting periodic points. We present a numerical simulation of a 1D map with external noise which explains some aspects of the noise-induced behavior, and a 2D map which has certain features of the intermittency.

Dimension and Entropy for Quasiperiodic and Chaotic Convection

High quality experimental data have been taken on a convection cell containing a dilute 3He−4He solution. We discuss some problems with the determination of dimension and entropy for experimental data, and compare the results to detailed Poincare sections. At the chaotic transition, we show the behavior of dimension and entropy as a function of Rayleigh number.

Simple differential thermometer using a thermocouple with a SQUID detector

Details of design and construction of a simple and rugged thermocouple differential thermometer are presented. Gold + 0.03 at. % iron wire is used, with niobium leads connected to a SQUID. It is demonstrated that at 1 K the response time is about 15 ms and that the temperature sensitivity of 10/sup -7/ K with a 10 Hz filter is Johnson-noise limited. 3 references.

Understanding experimentally the transition to chaos in a convecting dilute solution of 3He in superfluid 4He

After a brief review of Benard convection experiments at low temperatures, experiments with dilute solutions of 3He in superfluid 4He in small‐aspect‐ratio cells are described. In both rectangular and cylindrical geometries, two states with distinctively different thermal conductances are observed. The square of the frequency of the oscillation found in the state with higher conductance is observed to vary linearly with e=(ΔT/ΔTc)−1. The transition to chaos has been studied using a solution of 1.6 mole % 3He in a rectangular cell with an aspect ratio Γ=1.00 at 0.7 K. Techniques used to characterize the transition include the time series, power spectral density, phase space reconstruction of the trajectory, and Poincare sections. The analyses reveal clearly that the chaotic state is described by a strange attractor.

Phase space analysis of convection in a 3He-superfluid 4He solution☆

Abstract Observations have been made on thermal convection below 1 K in a dilute solution of 3He in superfluid 4He contained in a cylindrical cell of aspect ratio Γ=1.20. Complicated oscillatory phenomena were observed with a high degree of reproducibility using two temperature sensors. Phase space analysis suggests a description in terms of strange attractor dynamics.