Edgar Knobloch

Homoclinic snaking: structure and stability.

The bistable Swift-Hohenberg equation exhibits multiple stable and unstable spatially localized states of arbitrary length in the vicinity of the Maxwell point between spatially homogeneous and periodic states. These states are organized in a characteristic snakes-and-ladders structure. The origin of this structure in one spatial dimension is reviewed, and the stability properties of the resulting states with respect to perturbations in both one and two dimensions are described. The relevance of the results to several different physical systems is discussed.

Nonlinear periodic convection in double-diffusive systems

We study two examples of two-dimensional nonlinear double-diffusive convection (thermohaline convection, and convection in an imposed vertical magnetic field) in the limit where the onset of marginal overstability just precedes the exchange of stabilities. In this limit nonlinear solutions can be found analytically. The branch of oscillatory solutions always terminates on the steady solution branch. If the steady solution branch is subcritical this occurs when the period of the oscillation becomes infinite, while if it is supercritical, it occurs via a Hopf bifurcation. A detailed discussion of the stability of the oscillations is given. The results are in broad agreement with the largeramplitude results obtained previously by numerical techniques.

Symmetries and pattern selection in Rayleigh-Bénard convection

Abstract This paper describes the process of pattern selection between rolls and hexagons in Rayleigh-Benard convection with reflectional symmetry in the horizontal midplane. This symmetry is a consequence of the Boussinesq approximation, provided the boundary conditions are the same on the top and bottom plates. All possible local bifurcation diagrams (assuming certain non-degeneracy conditions) are found using only group theory. The results are therefore applicable to other systems with the same symmetries. Rolls, hexagons, or a new solution, regular triangles, can be stable depending on the physical system. Rolls are stable in ordinary Rayleigh-Benard convection. The results are compared to those of Buzano and Golubitsky [1] without the midplane reflection symmetry. The bifurcation behavior of the two cases is quite different, and a connection between them is established by considering the effects of breaking the reflectional symmetry. Finally, the relevant experimental results are described.

Spatially localized structures in dissipative systems: open problems

An apparatus for measuring the static friction properties of materials. The apparatus has a means for applying shear force to a sled and table interface in a smooth and repeatable manner. The apparatus operates without operator interference in terms of placing a load on the material to be tested. The apparatus contains a shear type load cell which operates at the same location in the apparatus at which the shear force is applied to a contact interface and at which static friction is generated.

Oscillations in double-diffusive convection

We have studied the transition between oscillatory and steady convection in a simplified model of two-dimensional thermosolutal convection. This model is exact to second order in the amplitude of the motion and is qualitatively accurate for larger amplitudes. If the ratio of the solutal diffusivity to the thermal diffusivity is sufficiently smell and the solutal Rayleigh number, R,, sufficiently large, convection sets in as overstable oscillations, and these oscillations grow in amplitude as the thermal Rayleigh number, R,, is increased. In addition to this oscillatory branch, there is a branch of steady solutions that bifurcates from the static equilibrium towards lower values of RT; this subcritical branch is initially unstable but acquires stability as it turns round towards increasing values of RT. For moderate values of R, the oscillatory branch ends on the unstable (subcritical) portion of the steady branch, where the period of the oscillations becomes infinite. For larger values of Rs a birfurcation from symmetrical to asymmetrical oscillations is followed by a succession of bifurcations, at each of which the period doubles, until the motion becomes aperiodic at some finite value of R,. The chaotic solutions persist as R, is further increased but eventually they lose stability and there is a transition to the stable steady branch. These results are consistent with the behaviour of solutions of the full two-dimensional problem and suggest that perioddoubling, followed by the appearance of a strange attractor, is a characteristic feature o€ double-diffusive convection.

Spatially localized binary-fluid convection

Multiple states of spatially localized steady convection are found in numerical simulations of water–ethanol mixtures in two dimensions. Realistic boundary conditions at the top and bottom are used, with periodic boundary conditions in the horizontal. The states form by a mechanism similar to the pinning region around a Maxwell point in variational systems, but are located in a parameter regime in which the conduction state is overstable. Despite this the localized states can be stable. The properties of the localized states are described in detail, and the mechanism of their destruction with increasing or decreasing Rayleigh number is elucidated. When the Rayleigh number becomes too large the fronts bounding the state at either end unpin and move apart, allowing steady convection to invade the domain. In contrast, when the Rayleigh number is too small the fronts move inwards, and eliminate the localized state which decays into dispersive chaos. Out of this state spatially localized states re-emerge at irregular times before decaying again. Thus an interval of Rayleigh numbers exists that is characterized by relaxation oscillations between localized convection and dispersive chaos.

Convective and absolute instabilities of fluid flows in finite geometry

Abstract Dynamics of linear and nonlinear waves in driven dissipative systems in finite domains are considered. In many cases (for example, due to rotation) the waves travel preferentially in one direction. Such waves cannot be reflected from boundaries. As a consequence in the convectively unstable regime the waves ultimately decay; only when the threshold for absolute instability is exceeded can the waves be maintained against dissipation at the boundary. Secondary absolute instabilities are associated with the break-up of a wave train into adjacent wave trains with different frequencies, wave numbers and amplitudes, separated by a front. The process of frequency selection is discussed in detail, and the selected frequency is shown to determine the wave number and amplitude of the wave trains. The results are described using the complex Ginzburg-Landau equation and illustrated using a mean-field dynamo model of magnetic field generation in the Sun.

Convection in a rotating cylinder. Part 1 Linear theory for moderate Prandtl numbers

The onset of convection in a uniformly rotating vertical cylinder of height h and radius d heated from below is studied. For non-zero azimuthal wavenumber the instability is a Hopf bifurcation regardless of the Prandtl number of the fluid, and leads to precessing spiral patterns. The patterns typically precess counter to the rotation direction. Two types of modes are distinguished: the fast modes with relatively high precession velocity whose amplitude peaks near the sidewall, and the slow modes whose amplitude peaks near the centre. For aspect ratios Γ≡d/h of order one or less the fast modes always set in first as the Rayleigh number increases; for larger aspect ratios the slow modes are preferred provided that the rotation rate is sufficiently slow

Transitions to chaos in two-dimensional double-diffusive convection

The partial differential equations governing two-dimensional thermosolutal convection in a Boussinesq fluid with free boundary conditions have been solved numerically in a regime where oscillatory solutions can be found. A systematic study of the transition from nonlinear periodic oscillations to temporal chaos has revealed sequences of period-doubling bifurcations. Overstability occurs if the ratio of the solutal to the thermal diffusivity tau is less than 1 and the solutal Rayleigh number Rs is sufficiently large. Solutions have been obtained for two representative values of tau. For tau = 0.316, R(s) = 10,000, symmetrical oscillations undergo a bifurcation to asymmetry, followed by a cascade of period-doubling bifurcations leading to aperiodicity, as the thermal Rayleigh number R(T) is increased. At higher values of R(T), the bifurcation sequence is repeated in reverse, restoring simple periodic solutions. As R(T) is further increased more period-doubling cascades, followed by chaos, can be identified. Within the chaotic regions there are narrow periodic windows, and multiple branches of oscillatory solutions coexist. Eventually the oscillatory branch ends and only steady solutions can be found. The development of chaos has been investigated for tau = 0.1 by varying R(T) for several different values of R(s). When R(s) is sufficiently small there are periodic solutions whose period becomes infinite at the end of the oscillatory branch. As R(s) is increased, chaos appears in the neighborhood of these heteroclinic orbits. At higher values of R(s), chaos is found for a broader range in R(T). A truncated fifth-order model suggest that the appearance of chaos is associated with heteroclinic bifurcations.

Amplitude equations for travelling wave convection

New asymptotically exact amplitude equations are derived for a dissipative system near a Hopf bifurcation. Unlike the usual coupled complex Ginzburg-Landau equations these are valid for O(1) group velocities.