##### Author

Zaleski, Joseph

##### Other Contributors

Lvov, Yuri, V; Kovacic, Gregor; Kramer, Peter; Onorato, Miguel;

##### Date Issued

2021-12

##### Subject

Applied mathematics

##### Degree

PhD;

##### Terms of Use

This electronic version is a licensed copy owned by Rensselaer Polytechnic Institute (RPI), Troy, NY. Copyright of original work retained by author.;

##### Abstract

We present new results in the field of Wave Turbulence Theory, applying our results in theanalysis of two specific systems: interacting surface and internal waves, and β-Fermi-Patsa-Ulam-Tsingou chains. In chapters 4–6 we consider interactions between surface and internal waves in a modelproblem: a two layer system. We analyze this simplified model with the goal of building a
tractable framework to understand how energy flows between the surface and internal wave
modes from first principles. We consider an approach using standard Wave Turbulence techniques, based on theHamiltonian structure of the equations of motion. We include the general procedure for
diagonalization of the quadratic part of the Hamiltonian with two wave types, a non-trivial
question, with our transformation being applicable to a other Hamiltonians which may share
a similar structure of nondiagonal terms. We derive the interaction coefficients between the surface and internal waves, obtainingthe coupled kinetic equations which describe the evolution of the total spectral energy of the
system. Notably, our derivation allows for both resonant and near-resonant interactions, the
latter which are important in some parameter regimes when the system is no longer weakly
nonlinear. We consider the case when the surface waves follow an ocean JONSWAP spectrum.We find that energy transfer from surface to internal waves occurs along a timescale of
hours; for our choice of parameters, we find that the energy transfers are dominated by the
specific class III resonances. We also note that internal waves oblique to the direction of the
wind generated, with a specific lobed spectrum of internal waves developing for some initial
conditions. In chapter 7–9 we consider general Hamiltonian nonlinear dispersive wave systems withcubic nonlinearity, investigating the common Wave Turbulence assumption of the Random
Phase Approximation. We show that such systems can develop anomalous correlators between phases during their evolution, despite the prescribed randomness in the initial data. To explain this phenomena theoretically, we follow a prior generalization of the Wick’sdecomposition and Wave Turbulence theory, showing that stationary solution with nonzero
anomalous correlations exists. We also describe the relationship between the anomalous correlator and the appearance
of “ghost” excitations in the spatiotemporal spectrum of the system, i.e. excitations given
by negative frequencies, in addition to the usual positive frequencies given by the linear
dispersion relation and the standard Wave Turbulence theory. We test our theory on extensive simulations of the celebrated β-Fermi-Pasta-UlamTsingou chain; numerically measured values of the anomalous correlators agree with ourtheory in the weakly nonlinear domain. We hypothesize that such excitations exist in other
systems dominated by nonlinear interactions, such as surface-gravity waves. In chapter 10 we outline a derivation for the collision integral of a kinetic equation of asystem of nonhomogenous wave turbulence. We provide a specific physical example of how
this kinetic equation may prove useful in investigating the nonlinear Schrodinger equation
with a varying potential.;

##### Description

December2021; School of Science

##### Department

Dept. of Mathematical Sciences;

##### Publisher

Rensselaer Polytechnic Institute, Troy, NY

##### Relationships

Rensselaer Theses and Dissertations Online Collection;

##### Access

Restricted to current Rensselaer faculty, staff and students in accordance with the
Rensselaer Standard license. Access inquiries may be directed to the Rensselaer Libraries.;