# Applied/ACMS/absF20

## Contents

# ACMS Abstracts: Fall 2020

### Nick Ouellette (Stanford)

Title: Tensor Geometry in the Turbulent Cascade

Abstract: Perhaps the defining characteristic of turbulent flows is the directed flux of energy from the scales at which it is injected into the flow to the scales at which it is dissipated. Often, we think about this transfer of energy in a Fourier sense; but in doing so, we obscure its mechanistic origins and lose any connection to the spatial structure of the flow field. Alternatively, quite a bit of work has been done to try to tie the cascade process to flow structures; but such approaches lead to results that seem to be at odds with observations. Here, I will discuss what we can learn from a different way of thinking about the cascade, this time as a purely mechanical process where some scales do work on others and thereby transfer energy. This interpretation highlights the fundamental importance of the geometric alignment between the turbulent stress tensor and the scale-local rate of strain tensor, since if they are misaligned with each other, no work can be done and no energy will be transferred. We find that (perhaps surprisingly) these two tensors are in general quite poorly aligned, making the cascade a highly inefficient process. Our analysis indicates that although some aspects of this tensor alignment are dynamical, the quadratic nature of Navier-Stokes nonlinearity and the embedding dimension provide significant constraints, with potential implications for turbulence modeling.

### Harry Lee (UW Madison)

Title: Recent extension of V.I. Arnold's and J.L. Synge's mathematical theory of shear flows

Abstract: A viscous extension of Arnold’s non-viscous theory ([1]) for 2D wall-bounded shear flows is established ([3]). One special form of our linearized viscous theory recaps the linear perturbation’s enstrophy (vorticity) identity derived by Synge in 1938 ([2]). For the first time in literature, we rigorously deduced the validity of Synge’s identity under nonlinear dynamics and relaxed wall conditions. Furthermore, we discovered a new ‘weighted’ enstrophy identity.

To illustrate the physical relevance of our identities, we quantitatively investigated mechanisms of linear instability/stability within the normal modal framework. We observed a subtle interaction between a critical layer and its adjacent boundary layer, which governs stability/instability of a flow. We also proposed a boundary control scheme that transitions wall settings from no-slip to free-slip, through which the 2D base flow was stabilized quickly at an early stage of the transition. Effectiveness of such boundary control scheme for 3D shear flows is yet to be tested by DNS/experiments.

Apart from physics, I shall also talk about the potential of using our nonlinear enstrophy identity to generate rigorous bounds on flow stability.

References:

[1] V. I. Arnold. Conditions for the nonlinear stability of the stationary plane curvilinear flows of an ideal fluid. Doklady Akademii Nauk, 162:975–978, 1965. URL: https://doi.org/10.1007/978-3-642-31031-7_4.

[2] F. Fraternale, L. Domenicale, G. Staffilani, and D. Tordella. Internal waves in sheared flows: Lower bound of the vorticity growth and propagation discontinuities in the parameter space. Physical Review E, 97:063102, 2018. URL: https://doi.org/10.1103/PhysRevE.97.063102.

[3] H. Lee and S. Wang. Extension of classical stability theory to viscous planar wall-bounded shear flows. Journal of Fluid Mechanics, 877:1134– 1162, 2019. URL: https://doi.org/10.1017/jfm.2019.629.

### Spencer Smith (Mount Holyoke)

Title: Braids on a lattice and maximally efficient mixing in active matter systems

In active matter systems, energy consumed at the small scale by individual agents (like microtubules, bacteria, or birds) gives rise to emergent flows at large scales. Often these flows are chaotic and effectively mix the surrounding medium. In two dimensions, this mixing can be quantified by the topological entropy of the braids formed from the intertwining motion of particle trajectories. It is natural to ask how large this topological entropy, suitably normalized, can get, and what braiding patterns achieve this. For small numbers of particles on a line, or particles on an annulus, braids with topological entropies related to the golden and silver ratios respectively are maximal. Surprisingly, these braids arise in an active matter system: active nematic microtubules confined to an annulus have topological defects that move in trajectories compatible with the silver braid. However, it is unknown what braiding pattern of particles on the plane maximizes topological entropy in an analogous manner. We will investigate this issue in spatially periodic braids defined on planar lattices. Using a newly developed algorithm, we will give numerical evidence for a candidate planar lattice braiding pattern with maximal topological entropy. Using the version of this algorithm for arbitrary flows, we will also highlight a curious mixing phenomenon in the Vicsek active matter model.

### Zhizhen Jane Zhao (UIUC)

Title: Exploiting Group and Geometric Structures for Massive Data Analysis

Abstract: In this talk, I will introduce a new unsupervised learning framework for data points that lie on or close to a smooth manifold naturally equipped with a group action. In many applications, such as cryo-electron microscopy image analysis and shape analysis, the dataset of interest consists of images or shapes of potentially high spatial resolution, and admits a natural group action that plays the role of a nuisance or latent variable that needs to be quotient out before useful information is revealed. We estimate the pairwise group-invariant distance and the corresponding optimal alignment. We then construct a graph from the dataset, where each vertex represents a data point and the edges connect points with small group-invariant distance. In addition, each edge is associated with the estimated optimal alignment group. Inspired by the vector diffusion maps proposed by Singer and Wu, we explore the cycle consistency of the group transformations under multiple irreducible representations to define new similarity measures for the data. Utilizing the representation theoretic mechanism, multiple associated vector bundles can be constructed over the orbit space, providing multiple views for learning the geometry of the underlying base manifold from noisy observations. I will introduce three approaches to systematically combine the information from different representations, and show that by exploring the redundancy created across irreducible representations of the transformation group, we can significantly improve nearest neighbor identification, when a large portion of the true edge information are corrupted. I will also show the application in cryo-electron microscopy image analysis.

### Matthias Morzfeld (Scripps & UCSD)

Title: What is Bayesian inference, why is it useful in Earth science and why is it challenging to do numerically?

Abstract: I will first review Bayesian inference, which means to incorporate information from observations (data) into a numerical model, and will give some examples of applications in Earth science. The numerical solution of Bayesian inference problems is often based on sampling a posterior probability distribution. Sampling posterior distributions is difficult because these are usually high-dimensional (many parameters or states to estimate) and non-standard (e.g., not Gaussian). In particular a high-dimension causes numerical difficulties and slow convergence in many sampling algorithms. I will explain how ideas from numerical weather prediction can be leveraged to design Markov chain Monte Carlo (MCMC) samplers whose convergence rates are independent of the problem dimension for a well-defined class of problems.

### Jingwei Hu (Purdue)

Title: A new stability and convergence proof of the Fourier-Galerkin spectral method for the spatially homogeneous Boltzmann equation

ABstract: Numerical approximation of the Boltzmann equation is a challenging problem due to its high-dimensional, nonlocal, and nonlinear collision integral. Over the past decade, the Fourier-Galerkin spectral method has become a popular deterministic method for solving the Boltzmann equation, manifested by its high accuracy and potential of being further accelerated by the fast Fourier transform. Albeit its practical success, the stability of the method is only recently proved by Filbet, F. & Mouhot, C. in [Trans.Amer.Math.Soc. 363, no. 4 (2011): 1947-1980.] by utilizing the "spreading" property of the collision operator. In this work, we provide a new proof based on a careful L2 estimate of the negative part of the solution. We also discuss the applicability of the result to various initial data, including both continuous and discontinuous functions. This is joint work with Kunlun Qi and Tong Yang.

### Dan Vimont (UW-Madison, AOS)

Title: Advances in Linear Inverse Modeling for Understanding Tropical Pacific Climate Variability

Abstract: The El Nino / Southern Oscillation (ENSO) phenomenon in the tropical Pacific Ocean is the most energetic climatic phenomenon on Earth for interannual to decadal time scales, with substantial societal and environmental impacts around the world. Despite a well-developed theory for why ENSO events occur several aspects of ENSO variability are still poorly understood, including (1) why individual ENSO events tend to evolve with different spatial structures, (2) why ENSO events tend to be positively skewed (toward El Niño events rather than La Niña events), and (3) the role of deterministic dynamics vs. stochastic forcing in influencing ENSO growth and variance. In this talk, I will present recent work using a suite of Linear Inverse Models (LIMs) in which a linear dynamical operator (including state dependent noise, or cyclo-stationary dynamics) is derived from an existing set of observations. These LIMs can be used to (1) diagnose physical processes that cause growth toward a pre-defined spatial structure, (2) investigate how state-dependent (local) correlated additive and multiplicative noise (CAM-Noise) generates higher order moments (in a linear system forced by gaussian noise), and (3) the role of seasonality in generating ENSO variability and predictability. The talk will focus on development of the linear inverse model and on the application of the models in dynamical system analyses.

### Sam Punshon-Smith (Brown)

Title: Scalar mixing and the Batchelor spectrum in stochastic fluid mechanics

Abstract: In 1959, George Batchelor predicted that a passively advected scalar in a fluid, when the scalar diffusivity is much lower than the fluid viscosity, should display a power spectral density like 1/|k| over an appropriate inertial range. Extending this result beyond Batchelor's simple example of a "pure straining flow" has proven to be a challenge despite the robust nature of the spectrum in a variety of more general physical settings and numerical experiments. In this talk, I will discuss a recent proof of a version of Batchelor's prediction for a variety of random ergodic fluid motions, including the stochastic Navier-Stokes equations on T^2 at fixed Reynolds number. We will see how the spectrum emerges as a consequence of the uniform-in-diffusivity chaotic mixing property of fluid motion, a non-trivial property that makes crucial use of the random motion and the associated uniformity of the chaotic behavior of Lagrangian trajectories.

### Yimin Zhong (Duke)

Title: Quantitative PhotoAcoustic Tomography (PAT) with simplified PN approximation

Abstract: In this talk, I will first introduce the physical and biomedical background of the quantitative photoacoustic tomography (qPAT). The quantitative step has been traditionally using the diffusion approximation to solve but fails at many scenarios in practice. In recent years, more and more researches start to use the transport model to study this problem. However there are still some open problems relating to the uniqueness and stability estimates. We will try to study the qPAT with the simplified PN approximation which is regarded as a more accurate approximation than the simplest diffusion approximation. I will show that the uniqueness and stability estimates under this formulation. Numerical experiments are performed to validate the theory.