# Applied/ACMS/absS17

### Chung-Nan Tzou (UW)

*Optimal mixing of buoyant jets and plumes in stratified fluids: theory and experiments*

We present results from an experimental and theoretical study of the influence of ambient fluid stratification on buoyant miscible jets and plumes. Given a fixed set of jet/plume parameters, and an ambient fluid stratification sandwiched between top and bottom homogenous densities, a theoretical criterion is identified showing how step-like density profiles constitute the most effective mixers within a broad class of stable density transitions. This is assessed both analytically and experimentally, respectively by establishing rigorous a priori estimates on generalized Morton-Taylor-Turner (MTT) models, and by studying a critical phenomenon determined by the distance between the jet/plume release heights with respect to the depth of the ambient density transition. For fluid released sufficiently close to the background density transition, the buoyant jet fluid escapes and rises indefinitely. For fluid released at locations lower than a critical depth, the buoyant fluid stops rising and is trapped indefinitely. We develop a mathematical formulation providing rigorous estimates on MTT models, by establishing nonlinear jump conditions and an exact critical-depth formula in good quantitative agreement with the experiments. Our mathematical analysis provides rigorous justification for the critical trapping/escaping criteria, first presented in Caulfied and Woods (1998), within a class of algebraic density decay rates. Further, the analysis uncovers surprising differences between the Gaussian and Top-hat profile turbulent entrainment closures concerning initial mixing of the jet and ambient fluid. Laboratory experimental results and comparisons with the theory will be discussed.

### Molei Tao (GaTech)

*Numerical methods for identifying hyperbolic periodic orbits and characterizing rare events in nongradient systems*

We consider differential equations perturbed by small noises. The goal is to quantify what noises can do and possibly also utilize them. More specifically, noise-induced dynamics are understood by maximizing transition probability characterized by Freidlin-Wentzell large deviation theory. In gradient systems (i.e., reversible thermodynamics), metastable transitions were known to cross separatrices at saddle points. We investigate nongradient systems (which may no longer be reversible), and show a very different type of transitions that cross hyperbolic periodic orbits. Numerical tools for both identifying such periodic orbits and computing transition paths are described. If time permits, I will also discuss how these results may help design control strategies.

### Benoit Perthame (University of Paris VI)

*Models for neural networks; analysis, simulations and behaviour*

Neurons exchange informations via discharges, propagated by membrane potential, which trigger firing of the many connected neurons. How to describe large networks of such neurons? What are the properties of these mean-field equations? How can such a network generate a spontaneous activity? Such questions can be tackled using nonlinear integro-differential equations. These are now classically used in the neuroscience community to describe neuronal networks or neural assemblies. Among them, the best known is certainly Wilson-Cowan's equation which describe spiking rates arising in different brain locations.

Another classical model is the integrate-and-fire equation that describes neurons through their voltage using a particular type of Fokker-Planck equations. Several mathematical results will be presented concerning existence, blow-up, convergence to steady state, for the excitatory and inhibitory neurons, with or without refractory states. Conditions for the transition to spontaneous activity (periodic solutions) will be discussed.

One can also describe directly the spike time distribution which seems to encode more directly the neuronal information. This leads to a structured population equation that describes at time $t$ the probability to find a neuron with time $s$ elapsed since its last discharge. Here, we can show that small or large connectivity leads to desynchronization. For intermediate regimes, sustained periodic activity occurs. A common mathematical tool is the use of the relative entropy method.

This talk is based on works with K. Pakdaman and D. Salort, M. Caceres, J. A. Carrillo and D. Smets.