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. A proper history of CNLS encompassing all of the many contributors to the CNLS story is beyond my means or purpose here. Instead, I present the history as I experienced it. I emphasize the main scientific accomplishments achieved at CNLS over more than 40 years, but I will also attempt to describe and quantify the attributes that made and continue to make the Center for Nonlinear Studies a special institution of remarkable impact and longevity. Throughout its existence, CNLS owes much to the enduring legacy of David Campbell who laid down the foundations and operating principles that have made it so successful.

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. The BZF width is found to scale like Ra^{1/4}Ek^{2/3} where the Ekman number Ek decreases with increasing rotation rate.

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. The interface between the fluids, tracked using an optical shadowgraph technique, is used to determine the mass transport rate, the spatial separation of solutal plumes, and the velocity and width characteristics of those plumes. One finds that the plume dynamics are closely related to the mass transport rate.This article is part of the themed issue 'Energy and the subsurface'.

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. Lagrangian correlations and one- and two-point displacements are measured for random initial conditions and for initial positions within topological centers and saddles. Some of the behavior of these quantities can be understood in terms of the trapping characteristics of long-lived centers, the slow motion near strong saddles, and the rapid fluctuations outside of either centers or saddles. We also present statistics of Lagrangian velocity fluctuations using energy spectra in frequency space and structure functions in real space. We compare with complementary Eulerian velocity statistics. We find that simultaneous inverse energy and enstrophy ranges present in spectra are not directly echoed in real-space moments of velocity difference. Nevertheless, the spectral ranges line up well with features of moment ratios, indicating that although the moments are not exhibiting unambiguous scaling, the behavior of the probability distribution functions is changing over short ranges of length scales. Implications for understanding weakly forced 2D turbulence with simultaneous inverse and direct cascades are discussed.

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. Solitons, mathematically precise localized nonlinear wave states, were generalized to a larger and less precise class of coherent structures such as, for example, concentrated regions of vorticity from laboratory wake flows to the Jovian Great Red Spot. The combination of these three ideas was hoped to provide the tools and concepts for the understanding and characterization of the strongly nonlinear problem of fluid turbulence. Although this early promise has been largely unfulfilled, steady progress has been made using the approaches of nonlinear science. I provide a series of examples of bifurcations and chaos, of one-dimensional and two-dimensional pattern formation, and of turbulence to illustrate both the progress and limitations of the nonlinear science approach. As experimental and computational methods continue to improve, the promise of nonlinear science to elucidate fluid turbulence continues to advance in a steady manner, indicative of the grand challenge nature of strongly nonlinear multi-scale dynamical systems.

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. The noncoherent events have a probability distribution of event moment consistent with an M(-3/2) power law scaling with Poisson-distributed recurrence times. Coherent events have a log-normal moment distribution and mean temporal recurrence. As the applied normal pressure increases, there are more coherent events and their log-normal distribution broadens and shifts to larger average moment.

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. The range of control parameters for Rayleigh numbers 4×10^{9}

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 7.3×7.3 cm(2) and a height of 9.4 cm. Temperature fluctuations and boundary-layer profiles were measured for Rayleigh numbers 1×10(7)

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 7.3 cm2 and a height of 9.4 cm. Heat transport was measured for Rayleigh numbers 2 x 10(5)

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. The form of the stripes depends critically on the mean density of the flow with a robust form of stripes at high density that consists of fast sliding pluglike regions (stripes) on top of highly agitated boiling material--a configuration reminiscent of the Leidenfrost effect when a droplet of liquid lifted by its vapor is hovering above a hot surface.

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. As the shape irregularity of the grains is increased, there is a dramatic qualitative change in avalanche properties. For rough nonspherical grains, avalanches are faster, bigger, and overturning in the sense that individual particles have down-slope speeds u p that exceed the front speed uf as compared with avalanches of spherical glass beads that are quantitatively slower and smaller and where particles always travel slower than the front speed. There is a linear increase of three quantities: (i) dimensionless avalanche height, (ii) ratio of particle to front speed, and (iii) the growth rate of avalanche speed with increasing avalanche size with increasing tan theta r where theta r is the bulk angle of repose, or with increasing beta P, the slope of the depth averaged flow rule, where both theta r and beta P reflect the grain shape irregularity. These relations provide a tool for predicting important dynamical properties of avalanches as a function of grain shape irregularity. A relatively simple depth-averaged theoretical description captures some important elements of the avalanche motion, notably the existence of two regimes of this motion.

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. In the laminar steady flow regime with the Ra approximately from 10(3) to 10(7), power-law scalings of heat transfer and maximum velocity with Ra have exponents of 0.31 and 0.54, respectively. The scalings agree well with results obtained in turbulent Rayleigh-Bernard convection, turbulent convection in laterally heated cavities and laminar convection in inclined enclosures, etc., which, with some simple physical arguments and reviews of the literature, leads us to propose that turbulence is not essential for the classical near 1/3 power-law scaling of Nu.

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. Improved collapse is obtained for sand and glass beads by using a recently proposed scaling of the form u/sqrt[gh]=betahtan2theta/hstan2theta1 where theta1 is the angle at which the hs(theta) curves diverge. Measuring the slope beta for ten different sizes of sand and glass beads, we find a systematic, strong increase of beta with the divergence angle theta1 of hs. Copper materials with different shapes are not well described by either flow rule with u approximately h3/2.

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. Our theory predicts, and the data confirm, that inverse cascade results mainly from turbulent stress proportional to small-scale strain rotated by 45 degrees. This "skew-Newtonian" stress is explained by the elongation and thinning of small-scale vortices by large-scale strain which weakens their velocity and transfers their energy upscale.

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. For low volume flow rates, a transition was detected between dense and very dilute (gas) flow regimes. We show using a vacuum flow channel that air did not qualitatively change the phase diagram and did not quantitatively modify mean flow velocities of the granular layer except for small changes in the very dilute gaslike phase.

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. The numerical calculations of turbulent volume-scattering functions compare well with the laboratory measurement. These results suggest that optical measurements at small angles are affected by turbulence-related scattering, and their effects can be well modeled with numerical calculations.

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. It is found that the inclusion of the more realistic no-slip boundary conditions for the top and bottom surfaces brings the numerical linear stability analysis into better agreement with the experimental data compared with results using free-slip top/bottom boundary conditions. Some remaining discrepancies may be accounted for by the finite conductivity of the sidewall boundaries.