The heat-transport enhance- ment due to rotation, proportional to the Nusselt

1. Sharp transitions between different turbulent statesGuenter Ahlers, University of California-Santa Barbara, DMR 0702111 Turbulence, by virtue of its vigorous fluctuations, is expected to sample all of phase space over wide parameter ranges. This viewpoint implies that there should not be any sharp transitions (known as bifurcations) between different turbulent states. Recently we studied a turbulent cylindrical sample of fluid heated from below and cooled from above while rotated about its vertical axis at various angular velocities (proportional to the inverse Rossby number 1/Ro). To our surprise we found a sharp bifurcation between two turbulent states1 at a rotation rate 1/Roc, as reflected in the heat transport (proportional to the Nusselt number Nu, see bottom left figure). Above 1/Roc the heat transport was enhanced by the rotation.2 The enhancement of Nu was due to the formation of vortex tubes, as illustrated by the top two figures. The tubes extract warm fluid from the bottom and cold fluid from the top boundaries of the sample. The mystery was that the enhancement started only after a finite 1/Roc was reached. We expect that the vortex-tube density in an infinite system grows linearly from zero with 1/Ro. Based on our experience in the field of pattern formation, we postulated that the vortex density must vanish at the wall of the finite system. On the basis of this one can write a model equation known as a Ginzburg-Landau (GL) equation. This equation predicts that the vortex density can grow in space only gradually from zero near the wall, and that a finite vortex density in the interior can only be attained after the rotation has reached a certain level. This finite-size effect predicted by the model reproduces the experimental observation 1/Roc ~ 1/G displayed in the lower right figure, as well as a number of other features of the system. To our kowledge the use of GL equations, which has been so successful in other fields like pattern formation and superconductivity, is new in the field of turbulence. 1.) “Transitions between turbulent states in rotating Rayleigh-B\'enard convection”, R.J.A.M. Stevens, J.-Q. Zhong, H.J.H. Clercx, G. Ahlers, and D. Lohse, Phys. Rev. Lett. 103, 024503 (2009). 2.) “Prandtl-, Rayleigh-, and Rossby-number dependence of heat transport in turbulent rotating Rayleigh-B\'enard convection”, J.-Q. Zhong, R.J.A.M. Stevens, H.J.H. Clercx, R. Verzicco, D. Lohse, and G. Ahlers, Phys. Rev. Lett. 102 , 044502 (2009). 3.) “Finite-size effects lead to supercritical bifurcations in turbulent rotating Rayleigh-B\'enard convection”, S. Weiss, R.J.A.M. Stevens, J.-Q. Zhong, H.J.H. Clercx, D. Lohse, and G. Ahlers, Phys. Rev. Lett., submitted. 4.) “Heat transport and the large-scale circulation in rotating turbulent Rayleigh-Benard convection”, J.-Q. Zhong and G. Ahlers, J. Fluid Mech., in print Constant-temperature surfaces, obtained from direct numerical simulation of turbulent convection2. Left: No rotation. Right: With rotation (1/Ro = 3.3). The rotation forms vertical vortex tubes which extract additional fluid from the boundaries and thus enhance the heat transport. The heat-transport enhance- ment due to rotation, proportional to the Nusselt number Nu, divided by Nu without rotation, as a function of the rate of rotation as represented by the inverse Rossby number 1/Ro.4 The critical inverse Rossby number 1/Roc where the bifurcation from one turbulent state to another takes place, as a function of the inverse aspect ratio 1/G (G=diameter/height) of the cylindrical sample3. One sees that 1/Roc ~ 1/G.

2. Sharp transitions between different turbulent statesGuenter Ahlers, University of California-Santa Barbara, DMR 0702111 Rotation has a strong influence on turbulent convection in many natural processes. Earth’s rotation is an important factor in the formation of hurricanes. The rotation of the Sun affects the heat transport through the outer quarter of its radius and thus the climate on Earth. Global ocean currents are driven in part by convection and are influenced by Earth’s rotation. The net result is a global flow pattern that controls the climate. Earth’s rotation influences convection in its outer core (which consists mostly of molten iron and generates Earth’s magnetic field) and thus influences the magnetic field in which we live. We developed a very fruitful collaboration with scientists in Twente, The Netherlands (the group of Professor Detlef Lohse) who do theoretical modeling, and direct numerical (computer) simulations (DNS) of the equations of motion, which correspond to experiments carried out in our laboratory. Many details not accessible to experiment can be learned from DNS, but wide parameter ranges accessible to experiment can not yet be explored numerically with available computers. This fruitful collaboration has led to four recent publications in Physical Review Letters, one in Phys. Rev. E, one in the Journal of Fluid Mechanics, and a long article in Reviews of Modern Physics. The Sun Satellite view of a hurricane. Global ocean currents