לוח שנה משולב

The complex pattern of river networks has inspired decades of studies. However, the evolution and the dynamics of a growing channel remain elusive. Here we show that the principle of local symmetry, a concept originating in fracture mechanics, explains the path followed by growing streams fed by groundwater. Although path selection does not by itself imply a rate of growth, we additionally show how local symmetry may be used to infer how rates of growth scale with water flux. Our methods are applicable to other problems of unstable pattern formation, such as the growth of hierarchical crack patterns and geologic fault networks, where dynamics is not well understood.

Turbulent flows are ubiquitous in nature, present in the atmosphere, the oceans, in industrial flows and also in one's own bathtub. From an abstract point of view, turbulence is an elemental problem in out-of-equilibrium statistical mechanics. The flow is driven out of equilibrium by forcing and dissipation acting on disparate scales, forming a chaotic motion that spans many interacting scales. Particles placed in a turbulent flow are therefore driven by an out-of-equilibrium fluctuating medium. I will discuss how the breaking of time reversibility of the flow manifests itself in the dynamics of such particles, focusing on tracers following the turbulent velocity field. I will present exact results for time irreversibility of pair dynamics in incompressible as well as compressible flows. For the latter there is an unexpected jump in the dynamics when time is reversed. For the former, I will describe the existence of an all time statistical conservation law for pair dispersion at small scales. In two dimensional or Hamiltonian flows, this conservation law is extended to an exact relation for the probability distribution function of the finite-time Lyapunov exponent. I will show that it can be interpreted as a fluctuation relation in phase space. Lastly, I will review how time irreversibility can be measured for a single particle and will discuss the application of this idea to a simple model of turbulence flow