**Research Interests**
My research interests center on nonlinear dynamics of dissipative systems. These focus on bifurcation theory, particularly in systems with symmetries, transition to chaos in such systems, low-dimensional behavior of continuous systems and the theory of nonlinear waves. Applications include pattern formation in fluid systems, reaction-diffusion systems, and related systems of importance in geophysics and astrophysics. I am also interested in the theory of turbulent transport and the theory of turbulence.
**Current Projects**
Pattern formation in large aspect-ratio systems: In many problems in physics it is convenient to ignore the presence of distant boundaries. In the theory of pattern formation in continuous systems (e.g. onset of convection in a fluid heated from below) one finds that even distant boundaries can have surprisingly important effects. This is because the boundaries break continuous symmetries (translations/rotations) present in the unbounded system. I am trying to understand under what conditions can sidewall effects be treated perturbatively (the easy case) and when not (the interesting case). This is particularly relevant when the pattern-forming instability produces propagating waves.
Complex dynamics due to broken symmetries: Even the loss of discrete symmetries can lead to unexpected behavior. For example, the fact that the Earth’s rotation breaks reflection symmetry in planes through the rotation axis is responsible for the pervasive drift of weather patterns. The origin of complex behavior due to forced symmetry breaking can be understood using group-theoretic techniques coupled to bifurcation theory.
When the resulting waves interact with spatial inhomogeneities novel dynamical phenomena result. These are associated with absolute instability and produce, for example, chaotic waves in one part of the domain that coexist with laminar waves in another, separated by a shock-like structure. Phenomena of this type are expected to have wide applicability to flows in rotating systems (including dynamo theory) or ones with through-flow and they appear to explain the break-up of spiral waves in certain chemical reactions.
Strongly nonlinear patterns: In fluid systems with strong restraints (for example rapidly rotating systems, or a strong imposed magnetic field) the flow in one or more directions is inhibited. As a result the motion is simplified and asymptotic techniques can be used to cast the pattern formation problem into the form of a nonlinear eigenvalue problem. In the case of convection the solution of this problem determines the heat flux across the fluid layer for a given applied temperature difference. This approach promises to be extremely useful in studies of geophysical and astrophysical flows which are almost always highly nonlinear, and has already led to a simplified description of rapidly rotating turbulence which agrees well with full 3d simulations and experiments.
Parametric instabilities of vibrating flows: A fluid layer vibrated vertically breaks up into a pattern of surface waves. These waves interact with large scale flows which they themselves generate. A description of this interaction is a challenging task but promises to provide an explanation for the dynamics observed in a number of experiments both on this system and on vibrated soap films.
Thin liquid films: Patterns generated by the breakup of thin liquid films on a horizontal substrate, and those generated by flowing films over inclined substrates represent a major challenge in fluid mechanics. I am also working to understand the dynamics of contact lines in vibrating systems, and the pinning and depinning of driven contact lines on structured substrates.
Spatially localized states: Spatially localized states, such as the recently identified {\it convectons} in binary fluid convection, or the spatially localized oscillations known as {\it oscillons} in vibrated granular media represent classes of states whose properties can be understood using a technique known as spatial dynamics. These results are now being extended to two and three spatial dimensions.
Dynamics of traveling pulses: The partial differential equations describing calcium dynamics in cells support traveling pulses. The appearance and disappearance of these states can be understood using singular perturbation methods for excitable equations of FitzHugh-Nagumo type, coupled with the theory of global bifurcations.
Selected Publications
M. Higuera, E. Knobloch, and J. M. Vega. Dynamics of nearly inviscid Faraday waves in almost circular containers. Physica D 201, 83-120 (2005).
O. Batiste and E. Knobloch. Simulations of oscillatory convection in 3He-4He mixtures in moderate aspect ratio containers. Phys. Fluids 17, 064102 (2005).
E. Knobloch and K. Julien. Saturation of the magnetorotational instability. Phys. Fluids 17, 094106 (2005).
U. Thiele, J. M. Vega, and E. Knobloch. Long-wave Marangoni instability with vibration. J. Fluid Mech. 546, 61-87 (2006).
M. Sprague, K. Julien, E. Knobloch and J. Werne. Numerical simulation of an asymptotically reduced system for rotationally constrained convection. J. Fluid Mech. 551, 141-174 (2006).
J. Burke and E. Knobloch. Localized states in the generalized Swift-Hohenberg equation. Phys. Rev. E 73, 056211 (2006).
K. Julien, E. Knobloch, R. Milliff, and J. Werne. Generalized quasi-geostrophy for spatially anisotropic rotationally constrained flows.
J. Fluid Mech. 555, 233-274 (2006).
O. Batiste, E. Knobloch, A. Alonso, and I. Mercader. Spatially localized binary fluid convection. J. Fluid Mech. 560, 149-158 (2006).
U. Thiele and E. Knobloch. Driven drops on heterogeneous substrates: Onset of sliding motion. Phys. Rev. Lett. 97, 204501 (2006).
A. Yochelis, J. Burke and E. Knobloch. Reciprocal oscillons and nonmonotonic fronts in forced nonequilibrium systems. Phys. Rev. Lett. 97, 254501 (2006). |