The control of the displacement of a passive particle in a point vortex flow

This work reports numerical explorations in advection of one passive tracer by point vortices living in the unbounded plane. The main objective is to find the energy-optimal displacement of one passive particle (point vortex with zero circulation) surrounded by N point vortices. The direct formulation of the corresponding control problems is presented. The restrictions are due to (i) the ordinary differential equations that govern the displacement of the passive particle around the point vortices, (ii) the available time T to go from the initial position z0 to the final destination zf, and (iii) the maximum absolute value umax that is imposed on the control variables. The latter consist in staircase controls, i.e., the control is written as a finite linear combination of characteristic functions on the real interval. The resulting optimization problems are solved numerically. The numerical results shows the existence nearly/quasi optimal control for the cases of N=1, N=2, N=3, and N=4 vortices.

Magma transport beneath mid-ocean ridges

<p>Melt transport beneath the lithosphere is elusive. With a distinct viscosity and density from the surrounding mantle, magmatic melt moves on a different time scale as the surrounding mantle. To resolve the temporal scale necessary to accurately capture melt transport in the mantle, the model simulations become numerically expensive quickly. Recent computational advances make possible two-phase numerical explorations to understand magma transport in the mantle. We review results from a suite of two-phase models applied to the mid-ocean ridges, where we varied half-spreading rate and intrinsic mantle permeability using new openly available models, with the goal of understanding melt focusing beneath mid-ocean ridges and its relevance to the lithosphere-asthenosphere boundary (LAB). Here, we highlight the importance of viscosities for the melt focusing mechanisms. In addition, magmatic porosity waves that are a natural consequence of these two-phase flow formulations. We show that these waves could explain long-period temporal variations in the seafloor bathymetry at the Southeast Indian Ridge.</p>

An Energy-Based Interaction Model for Population Opinion Dynamics with Topic Coupling

We introduce a new, and quite general variational model for opinion dynamics based on pairwise interaction potentials and a range of opinion evolution protocols ranging from random interactions to global synchronous flows in the opinion state space. The model supports the concept of topic “coupling”, allowing opinions held by individuals to be changed via indirect interaction with others on different subjects. Interaction topology is governed by a graph that determines interactions. Our model, which is really a family of variational models, has, as special cases, many of the previously established models for the opinion dynamics. After introducing the model, we study the dynamics of the special case in which the potential is either a tent function or a constructed bell-like curve. We find that even in these relatively simple potential function examples there emerges interesting behavior. We also present results of preliminary numerical explorations of the behavior of the model to motivate questions that can be explored analytically

An energy-based interaction model for population opinion dynamics with topic coupling

We introduce a new, and quite general variational model for opinion dynamics based on pairwise interaction potentials and a range of opinion evolution protocols ranging from random interactions to global synchronous flows in the opinion state space. The model supports the concept of topic “coupling”, allowing opinions held by individuals to be changed via indirect interaction with others on different subjects. Interaction topology is governed by a graph that determines interactions. Our model, which is really a family of variational models, has, as special cases, many of the previously established models for the opinion dynamics. After introducing the model, we study the dynamics of the special case in which the potential is either a tent function or a constructed bell-like curve. We find that even in these relatively simple potential function examples there emerges interesting behavior. We also present results of preliminary numerical explorations of the behavior of the model to motivate questions that can be explored analytically.