The center for nonlinear studies: A personal history.

The Center for Nonlinear Studies (CNLS) was an integral part of my scientific career starting as a Postdoctoral Fellow in 1983 up to my tenure as CNLS Director from 2004 to 2015. As such, I experienced a number of scientific phases of CNLS through almost four decades of foundation, evolution, and transition. Throughout this entire interval, the inspiration and influence of David Campbell guided my way.

Boundary Zonal Flow in Rotating Turbulent Rayleigh-Bénard Convection.

For rapidly rotating turbulent Rayleigh-Bénard convection in a slender cylindrical cell, experiments and direct numerical simulations reveal a boundary zonal flow (BZF) that replaces the classical large-scale circulation. The BZF is located near the vertical side wall and enables enhanced heat transport there. Although the azimuthal velocity of the BZF is cyclonic (in the rotating frame), the temperature is an anticyclonic traveling wave of mode one, whose signature is a bimodal temperature distribution near the radial boundary.

Plume dynamics in Hele-Shaw porous media convection.

Mass transport in multi-species porous media is through molecular diffusion and plume dynamics. Predicting the rate of mass transport has application in determining the efficiency of the storage and sequestration of carbon dioxide. We study a water and propylene-glycol system enclosed in a Hele-Shaw cell with variable permeability that represents a laboratory analogue of the general properties of porous media convection.

Lagrangian statistics in weakly forced two-dimensional turbulence.

Measurements of Lagrangian single-point and multiple-point statistics in a quasi-two-dimensional stratified layer system are reported. The system consists of a layer of salt water over an immiscible layer of Fluorinert and is forced electromagnetically so that mean-squared vorticity is injected at a well-defined spatial scale ri. Simultaneous cascades develop in which enstrophy flows predominately to small scales whereas energy cascades, on average, to larger scales.

Stick-slip behavior in a continuum-granular experiment.

We report moment distribution results from a laboratory experiment, similar in character to an isolated strike-slip earthquake fault, consisting of sheared elastic plates separated by a narrow gap filled with a two-dimensional granular medium. Local measurement of strain displacements of the plates at 203 spatial points located adjacent to the gap allows direct determination of the event moments and their spatial and temporal distributions. We show that events consist of spatially coherent, larger motions and spatially extended (noncoherent), smaller events.

Chaos, patterns, coherent structures, and turbulence: Reflections on nonlinear science.

The paradigms of nonlinear science were succinctly articulated over 25 years ago as deterministic chaos, pattern formation, coherent structures, and adaptation/evolution/learning. For chaos, the main unifying concept was universal routes to chaos in general nonlinear dynamical systems, built upon a framework of bifurcation theory. Pattern formation focused on spatially extended nonlinear systems, taking advantage of symmetry properties to develop highly quantitative amplitude equations of the Ginzburg-Landau type to describe early nonlinear phenomena in the vicinity of critical points.

Heat transport in the geostrophic regime of rotating Rayleigh-Bénard convection.

We report experimental measurements of heat transport in rotating Rayleigh-Bénard convection in a cylindrical convection cell with an aspect ratio of Γ=1/2. The fluid is helium gas with a Prandtl number Pr=0.7.

Local temperature measurements in turbulent rotating Rayleigh-Bénard convection.

We present local temperature measurements of turbulent Rayleigh-Bénard convection with rotation about a vertical axis. The fluid, water with Prandtl number about 6, was confined in a cell with a square cross section of 7.3×7.

Patterns in flowing sand: understanding the physics of granular flow.

Dense granular flows are often unstable and form inhomogeneous structures. Although significant advances have been recently made in understanding simple flows, instabilities of such flows are often not understood. We present experimental and numerical results that show the formation of longitudinal stripes that arise from instability of the uniform flowing state of granular media on a rough inclined plane.

Heat transport measurements in turbulent rotating Rayleigh-Bénard convection.

We present experimental heat transport measurements of turbulent Rayleigh-Bénard convection with rotation about a vertical axis. The fluid, water with a Prandtl number (sigma) of about 6, was confined in a cell with a square cross section of 7.3 x 7.

Avalanche dynamics on a rough inclined plane.

The avalanche behavior of gravitationally forced granular layers on a rough inclined plane is investigated experimentally for different materials and for a variety of grain shapes ranging from spherical beads to highly anisotropic particles with dendritic shape. We measure the front velocity, area, and height of many avalanches and correlate the motion with the area and height. We also measure the avalanche profiles for several example cases.

Flow rule of dense granular flows down a rough incline.

We present experimental findings on the flow rule for granular flows on a rough inclined plane using various materials, including sand and glass beads of various sizes and four types of copper particles with different shapes. We characterize the materials by measuring hs (the thickness at which the flow subsides) as a function of the plane inclination theta on various surfaces. Measuring the surface velocity u of the flow as a function of flow thickness h, we find that for sand and glass beads the Pouliquen flow rule u/sqrt[gh] approximately betahhs provides reasonable but not perfect collapse of the u(h) curves measured for various theta and mean particle diameter d.

Scaling in laminar natural convection in laterally heated cavities: is turbulence essential in the classical scaling of heat transfer?

We analyze heat transfer and flow properties in laminar natural convection driven by a horizontal temperature gradient in a closed cavity and propose that for the classical scaling of heat transfer turbulence does not play a decisive role. Direct numerical simulations were performed with the Rayleigh number (Ra) from 1 to 10(8) and the Prandtl number Pr = 0.71.

Rapid granular flows on a rough incline: phase diagram, gas transition, and effects of air drag.

We report experiments on the overall phase diagram of granular flows on an incline with emphasis on high inclination angles where the mean layer velocity approaches the terminal velocity of a single particle free falling in air. The granular flow was characterized by measurements of the surface velocity, the average layer height, and the mean density of the layer as functions of the hopper opening, the plane inclination angle, and the downstream distance x of the flow. At high inclination angles the flow does not reach an x -invariant steady state over the length of the inclined plane.

Physical mechanism of the two-dimensional inverse energy cascade.

We study the physical mechanisms of the two-dimensional inverse energy cascade using theory, numerics, and experiment. Kraichnan's prediction of a -5/3 spectrum with constant, negative energy flux is verified in our simulations of 2D Navier-Stokes equations. We observe a similar but shorter range of inverse cascade in laboratory experiments.

Two scenarios for avalanche dynamics in inclined granular layers.

We report experimental measurements of avalanche behavior of thin granular layers on an inclined plane for low volume flow rate. The dynamical properties of avalanches were quantitatively and qualitatively different for smooth glass beads compared to irregular granular materials such as sand. Two scenarios for granular avalanches on an incline are identified, and a theoretical explanation for these different scenarios is developed based on a depth-averaged approach that takes into account the differing rheologies of the granular materials.

Light scattering on oceanic turbulence.

Turbulent inhomogeneities of fluid flow have the effect of scattering light in near-forward angles, thus providing an opportunity to use optics to quantify turbulence. Here we report measurements of the volume-scattering function in the range of 10(-7) to 10(-3) rad using a wave-front sensing technique. The total scattering coefficient b, due to scattering on turbulent inhomogeneities, is between 1 and 10 m(-1) under typical oceanographic conditions.

Traveling waves in rotating Rayleigh-Bénard convection.

A combined analytical, numerical, and experimental study of the traveling-wave wall mode in rotating Rayleigh-Bénard convection is presented. No-slip top and bottom boundary conditions are used for the numerical computation of the linear stability, and the coefficients of the linear complex Ginzburg-Landau equation are then computed for various rotation rates. Numerical results for the no-slip boundary conditions are compared with free-slip calculations and with experimental data, and detailed comparison is made at a dimensionless rotation rate Omega=274.

Physical mechanism of the two-dimensional enstrophy cascade.

In two-dimensional turbulence, irreversible forward transfer of enstrophy requires anticorrelation of the turbulent vorticity transport vector and the inertial-range vorticity gradient. We investigate the basic mechanism by numerical simulation of the forced Navier-Stokes equation. In particular, we obtain the probability distributions of the local enstrophy flux and of the alignment angle between vorticity gradient and transport vector.