# Difference between revisions of "Applied/ACMS/absF14"

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The determination of scaling laws for heat transport in Rayleigh-Benard convection is a long-standing problem. Rigorous upper bounds for heat transport have been derived in various mathematical and physical settings. In recent years, researchers have harnessed the power of computers to determine scaling laws in turbulent Rayleigh-Benard convection. In the present work we take a different approach and determine scaling laws for steady, unstable solutions that optimize heat transport. In this talk, the essential ideas and history behind the problem at hand are reviewed. Following this, a quick description of a numerical algorithm that has been developed to find the unstable, coherent solutions is presented. New results on optimal scaling laws at various values of the Prandtl number are discussed. Based on these observations, and with the rigorous upper bounds in mind, conclusions are drawn and implications for scaling laws are discussed. | The determination of scaling laws for heat transport in Rayleigh-Benard convection is a long-standing problem. Rigorous upper bounds for heat transport have been derived in various mathematical and physical settings. In recent years, researchers have harnessed the power of computers to determine scaling laws in turbulent Rayleigh-Benard convection. In the present work we take a different approach and determine scaling laws for steady, unstable solutions that optimize heat transport. In this talk, the essential ideas and history behind the problem at hand are reviewed. Following this, a quick description of a numerical algorithm that has been developed to find the unstable, coherent solutions is presented. New results on optimal scaling laws at various values of the Prandtl number are discussed. Based on these observations, and with the rigorous upper bounds in mind, conclusions are drawn and implications for scaling laws are discussed. | ||

+ | |||

+ | === Mary Silber (Northwestern) === | ||

+ | |||

+ | ''Spontaneous Pattern Formation in Models of Vegetation in Dryland Ecosystems'' | ||

+ | |||

+ | Semi-arid ecosystems frequently show striking patterns of vegetation cover when they are stressed by reduced precipitation. It has been suggested, on the basis of mathematical ecological models, that there may be a standard sequence to the patterns that form as the water input to the ecosystem decreases. Specifically, the vegetation cover goes from gap patterns, to labyrinthine patterns, to spot patterns as precipitation decreases. We examine this proposal from the perspective of equivariant bifurcation theory, and show that this is not the only generic possibility. |

## Revision as of 10:57, 13 November 2014

## Contents

# ACMS Abstracts: Fall 2014

### Agisilaos Athanasoulis (Leicester)

*Semiclassical regularization for ill-posed classical flows: microlocal coarse-graining beyond Wigner measures*

Wigner measures (WMs) have been successfully used as a parameter-free tool to provide homogenised descriptions of wave problems. Notable applications are the efficient simulation of large linear wave fields, and the painless resolution of linear caustics. However, their applicability to non-linear problems has been very limited.

In this talk we discuss the role of smoothness of the underlying flow as a limiting factor in the applicability of WMs. Non-smooth flows are ill-posed for measures, and new phenomena are possible in that regime. For example, single wavepackets may be "split" cleanly into several new wavepackets. We introduce a modification of the WM approach, and show that it can capture successfully some of these new phenomena. These results include joint work with T. Paul, I. Kyza and Th. Katsaounis.

The main idea behind this regularised scheme can be used to setup a unifying framework for several different approaches developed in the last few years. Some ideas about the extension of this framework to non-linear problems are also discussed.

### Dongbin Xiu (Utah)

*Uncertainty quantification algorithms for large-scale systems*

Abstract: The field of uncertainty quantification (UQ) has received an increasing amount of attention recently. Extensive research efforts have been devoted to it and many novel numerical techniques have been developed. These techniques aim to conduct stochastic simulations for very large-scale complex systems. Although remarkable progresses have been made, UQ simulations remains challenging due to their exceedingly high simulation cost for problems at extreme scales. In this talk I will discuss some of the recent developed UQ algorithms that are particularly suitable for extreme-scale simulations. These methods are (1) collocation based, such that they can be directly applied to systems with legacy simulation codes; and (2) capacity based, such that they deliver the (near) optimal simulation accuracy based on the available simulation capacity. In another word, these methods deliver the best UQ simulation results based on any given computational resource one can afford, which is often very limited at the extreme scales.

### Stanislav Boldyrev (UW)

*Recent results on magnetohydrodynamic turbulence*

Abstract: Magnetic plasma turbulence is observed over a broad range of scales in the natural systems such as the the solar corona, the solar wind, and the interstellar medium, where the Reynolds numbers far exceed the Reynolds numbers achievable in numerical experiments. Much attention is therefore drawn to the universal scaling properties of small-scale fluctuations, which can be reliably measured in the simulations and extrapolated to astrophysical scales. The recent numerical and phenomenological results on the scaling and structure of magnetohydrodynamic turbulence will be discussed.

### Erik Bollt (Clarkson)

*Shape coherence and finite-time curvature evolution*

Abstract: Mixing, and coherence are fundamental issues at the heart of understanding fluid dynamics and other non-autonomous dynamical systems. Here we define shape coherent sets from which we show that tangency of finite time stable foliations (related to a forward time perspective) and finite time unstable foliations (related to a “backwards time" perspective) serve a central role. We develop zero-angle curves, meaning non-hyperbolic splitting curves, by continuation from the implicit function theorem. Furthermore, slowly evolving curvature corresponds to shape coherence and therefore we introduce a finite-time curvature (FTC) evolution field to indicate the existence of shape coherent sets in a direct and graphical manner. Simple examples as well as from meteorology and oceanography.

### Roseanna Zia (Cornell)

*A micro-mechanical study of coarsening and rheology of colloidal gels: Cage building, cage hopping, and Smoluchowski’s ratchet*

Abstract: Reconfigurable soft solids such as viscoelastic gels have emerged in the past decade as a promising material in numerous applications ranging from engineered tissue to drug delivery to injectable sensors. These include colloidal gels, which microscopically comprise a scaffoldlike network of interconnected particles embedded in a solvent. Network bonds can be permanent or reversible, depending on the nature and strength of interparticle attractions. When attractions are on the order of just a few kT, bonds easily rupture and reform. On a macroscopic scale, bond reversibility allows a gel to transition from solidlike behavior during storage, to liquidlike behavior during flow (e.g., injection or shear), and back to solidlike behavior in situ. On a microscopic scale, thermal fluctuations of the solvent are occasionally strong enough to break colloidal bonds, temporarily allowing particles to migrate and exchange neighbors before rebonding to the network, leading to structural evolution over time. Prior studies of colloidal gels have examined evolution of length scales and dynamics such as decorrelation times. Left open were additional questions such as how the particle-rich regions are structured (liquidlike, glassy, crystalline), how restructuring takes place (via bulk diffusion, surface migration, coalescence of large structures), and the impact of the evolution on rheology. In this talk I discuss these themes as explored in our recent dynamic simulations. We find that the network strands comprise a glassy, immobile interior near random-close packing, enclosed by a liquidlike surface along which the diffusive migration of particles drives structural coarsening. We show that coarsening is a three-step process of cage forming, cage hopping, and cage arrest, where particles migrate to ever-deeper energy wells via “Smoluchowski’s ratchet.” Both elastic and viscous high-frequency moduli are found to scale with the square-root of the frequency, similar to the perfectly viscoelastic behavior of non-hydrodynamically interacting, purely repulsive dispersions. But here, the behavior is elastic over all frequencies, with a quantitative offset between elastic and viscous moduli, which owes its origin to the hindrance of diffusion by particle attractions. Propagation of this elasticity via the network gives rise to age-stiffening as the gel coarsens. This simple phenomenological model suggests a rescaling of the moduli on dominant network length scale that collapses moduli for all ages onto a single curve. We propose a Rouse-like theoretical model and, from it, derive an analytical expression that predicts the effects of structural aging on rheology whereby linear response can be determined at any age by measurement of dominant network length scale—or vice versa.

### Dan Hu (SJTU)

*Optimization, adaptation, and initiation of biological transport networks*

Blood vessel systems and leaf venations are typical biological transport networks. The energy consumption for such a system to perform its biological functions is determined by the network structure. In the first part of this talk, I will discuss the optimized structure of vessel networks, and show how the blood vessel system adapts itself to an optimized structure. Mathematical models are used to predict pruning vessels in the experiments of zebra fish. In the second part, I will discuss our recent modeling work on the initiation process of transport networks. Simulation results are used to illustrate how a tree-like structure is obtained from a continuum adaptation equation system, and how loops can exist in our model. Possible further application of this model will also be discussed.

### Tim Chumley (Iowa State)

*Random billiards and diffusion*

We introduce a class of random dynamical systems derived from billiard maps and study a certain Markov chain derived from them. We then discuss the interplay between the billiard geometry and stochastic properties of the random system. The main results presented investigate the affect of billiard geometry on a diffusion process obtained from an appropriate scaling limit of the Markov chain.

### Alina Chertock (NCSU)

*Pedestrian flow models with slowdown interactions*

In this talk, I will discuss one-dimensional models for the behavior of pedestrians in a narrow street or corridor. I will first formulate a stochastic cellular automata model with explicit rules for pedestrians moving in two opposite directions. Coarse-grained mesoscopic and macroscopic analogs will then be carefully derived leading to the coupled system of PDEs for the density of the pedestrian traffic. The obtained first-order system of conservation laws is only conditionally hyperbolic and thus higher-order nonlinear diffusive corrections resulting in a parabolic macroscopic PDE model will be introduced. Finally, I will present a number of numerical experiments comparing and contrasting the behavior of the microscopic stochastic model and the resulting coarse-grained PDEs for various parameter settings and initial conditions. These numerical experiments demonstrate that the nonlinear diffusion is essential for reproducing the behavior of the stochastic system in the nonhyperbolic regime.

### David Sondak (UW)

*Effect of Prandtl number on optimal scaling laws in Rayleigh-Benard convection*

The determination of scaling laws for heat transport in Rayleigh-Benard convection is a long-standing problem. Rigorous upper bounds for heat transport have been derived in various mathematical and physical settings. In recent years, researchers have harnessed the power of computers to determine scaling laws in turbulent Rayleigh-Benard convection. In the present work we take a different approach and determine scaling laws for steady, unstable solutions that optimize heat transport. In this talk, the essential ideas and history behind the problem at hand are reviewed. Following this, a quick description of a numerical algorithm that has been developed to find the unstable, coherent solutions is presented. New results on optimal scaling laws at various values of the Prandtl number are discussed. Based on these observations, and with the rigorous upper bounds in mind, conclusions are drawn and implications for scaling laws are discussed.

### Mary Silber (Northwestern)

*Spontaneous Pattern Formation in Models of Vegetation in Dryland Ecosystems*

Semi-arid ecosystems frequently show striking patterns of vegetation cover when they are stressed by reduced precipitation. It has been suggested, on the basis of mathematical ecological models, that there may be a standard sequence to the patterns that form as the water input to the ecosystem decreases. Specifically, the vegetation cover goes from gap patterns, to labyrinthine patterns, to spot patterns as precipitation decreases. We examine this proposal from the perspective of equivariant bifurcation theory, and show that this is not the only generic possibility.