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This work reviews the present position of and surveys future perspectives in the physics of chaotic advection: the field that emerged three decades ago at the intersection of fluid mechanics and nonlinear dynamics, which encompasses a range of applications with length scales ranging from micrometers to hundreds of kilometers, including systems as diverse as mixing and thermal processing of viscous fluids, microfluidics, biological flows, and oceanographic and atmospheric flows.

Weakly-rotating turbulent Rayleigh-Bénard convection was studied experimentally and numerically. With increasing rotation and large enough Rayleigh number an abrupt transition from a turbulent state with nearly rotation-independent heat transport to another turbulent state with enhanced heat transfer is observed at a critical inverse Rossby number 1/Roc ≃ 0.4. Whereas for 1/Ro < 1/Roc the strength of the large-scale convection-roll is either enhanced or essentially unmodified depending on parameters, its strength is increasingly diminished beyond 1/Roc where it competes with Ekman vortices that cause vertical fluid transport and thus heat-transfer enhancement.PACS numbers: 47.27.te,47.32.Ef,47.20.Bp,47.27.ek Turbulence evolves either through a sequence of bifurcations, possibly passing through periodic and chaotic states [1] as in Rayleigh-Bénard (RB) convection [2] when the Rayleigh number Ra (to be defined below) is increased, or through subcritical bifurcations [3] as in pipe or Couette flow. Once the flow is turbulent, it usually is characterized by large random fluctuations in space and time and by a loss of temporal and spatial coherence. For the turbulent state common wisdom is that the large fluctuations assure that the phase-space is always fully explored by the dynamics, and that transitions between potentially different states that might be explored as a control parameter is changed are washed out.Contrary to the above, we show that sharp transitions between distinct turbulent states can occur in RB convection [4,5] when the system is rotated about a vertical axis at an angular velocity Ω. In dimensionless form the angular velocity is given by the inverse Rossby number 1/Ro = 2Ω/ βg∆/L. Here L is the height of a cylindrical RB sample, β the thermal expansion coefficient, ∆ the temperature difference between the bottom and top plate, and g the gravitational acceleration. At relatively small Ra where the turbulence is not yet fully developed, we find that the system evolves smoothly as 1/Ro is increased. However, when Ra is larger and the turbulent state of the non-rotating system is well established, we find that sharp transitions between different turbulent states occur, with different heat-transfer properties and different flow organizations. Similar sharp transitions between different states were reported recently for turbulent flows in liquid sodium [6], where the increase of the magnetic Reynolds number beyond a certain threshold led to the spontaneous creation of a mean magnetic field and where sharp bifurcations between different turbulent states were observed when a control parameter was tuned. One sees that sharp transitions, usually regarded as characteristic of low-dimensional systems, are displayed also by the fully developed turbulent flows.We present both experimental measurements and direct numerical simulations (DNS) for a sample with diameter D equal to L. They cover different but overlapping parameter ranges and thus complement each other. Where they overlap they agree very well. W...