Presenter:       Dmitriy Zhigunov

Title:                Exact coherent structures and non-universality in the direct cascade in two-dimensional turbulence

Date:               Friday, December 8, 2023

Time:               3:00 p.m.

Location:        Howey N110 / MS Teams

Committee members:

Dr. Roman Grigoriev, School of Physics, Georgia Institute of Technology (adviser)

Dr. Michael Schatz, School of Physics, Georgia Institute of Technology

Dr. Predrag Cvitanović, School of Physics, Georgia Institute of Technology

Dr. Alexander Blumenthal, School of Mathematics, Georgia Institute of Technology

Dr. Luca Dieci, School of Mathematics, Georgia Institute of Technology

Abstract:

Turbulence is the most important problem in physics and applied mathematics, with applications ranging from astrophysics, to engineering, to plumbing, and more. Turbulent flows are often characterized by the presence of various cascades, which transport various quantities across length scales. This dissertation focuses on turbulence confined in two-dimensions, which has two cascades, an inverse (energy) cascade that moves energy towards increasingly larger scales, and a direct (enstrophy) cascade that moves the enstrophy towards smaller scales.

The energy cascade leads to the formation of large-scale vortices, which often take up the largest length allowed by the domain. The first part of this dissertation flows focuses on the dynamics of these large scale vortices, where we find that these vortices behave for substantial time intervals like specific solutions of Euler equation. These solutions are in many ways analogous to recurrent solutions of the Navier-Stokes equation which are often referred to as exact coherent structures. On the other hand, these solutions have a number of properties which distinguish them from their Navier-Stokes counterparts, such as the fact that they exist in continuous, multiparameter families.

At the same time, the classical theory of the direct cascade by Kraichnan, Leith, and Batchelor fails to predict the proper scaling of the enstrophy spectrum found in numerical simulations and experiments. This discrepancy is often attributed to the presence of large-coherent vortices. We will provide a physically interpretable mechanism for the direct cascade that recovers KLB predictions in the absence of large-scale vortices, but leads to deviations in their presence. Finally, we return directly to the large-scale dynamics we explain in the first section, and investigate exactly how properties of the large-scale flow affect the scaling of the enstrophy spectrum.