## Mathematics & Statistics ETDs

Fall 8-24-2017

#### Abstract

This work concentrates on Langmuir wave filamentation instability in the kinetic regime of plasma and computation of Stokes wave with high precision using conformal maps.

Nonlinear effects are present in almost every area of science as soon as one tries to go beyond the first order approximation. In particular, nonlinear waves emerge in such areas as hydrodynamics, nonlinear optics, plasma physics, quantum physics, etc. The results of this work are related to nonlinear waves in two areas, plasma physics and hydrodynamics, united by concepts of instability, singularity and advanced numerical methods used for their investigation.

The first part of this work concentrates on Langmuir wave filamentation instability in the kinetic regime of plasma. In Internal Confinement Fusion Experiments (ICF) at National Ignition Facility (NIF), where attempts are made to achieve fusion by compressing a small target by many powerful lasers to extremely high temperatures and pressures, plasma is created in the first moments of the laser reaching the target and undergoes complicated dynamics. Some of the most challenging difficulties arise from various plasma instabilities that occur due to interaction of the laser beam and a plasma surrounding the target. In this work we consider one of such instabilities that describes a decay of nonlinear plasma wave (aka Langmuir wave), initially excited due to interaction of the laser beam with the plasma, into many filaments in direction perpendicular to the laser beam, therefore named Langmuir filamentation (or transverse) instability. This instability occurs in the kinetic regime of plasma,$k\lambda_D>0.2$, where $k$ is the wavenumber and $\lambda_D$ is the Debye length. The filamentation of Langmuir waves in turn leads to the saturation of the stimulated Raman scattering (SRS) \cite{GoldmanBoisPhisFL1965} in laser-plasma interaction experiments which plays an essential role in ICF experiments.

The challenging part of this work was that unlike in hydrodynamics we needed to use fully kinetic description of plasma to capture the physics in question properly, meaning that we needed to consider the distribution function of charged particles and its evolution in time not only with respect to spatial coordinates but with respect to velocities as well. To study Langmuir filamentation instability in its simplest form we performed 2D+2V numerical simulations. Taking into account that the distribution function in question was 4-dimensional function, making these simulation quite challenging, we developed an efficient numerical method making these simulations possible on modern desktop computers.

Using the developed numerical method we studied how Langmuir wave filamentation instability depends on the parameters of the Langmuir wave such as wave length and amplitude that are relevant to ICF experiments. We considered several types of Langmuir waves, including nonlinear Langmuir waves exited by external electric field as well as an idealized approximation of such Langmuir waves \cite{RoseRussellPOP2001} by a particular family of Bernstein-Greene-Kruskal (BGK) modes \cite{BernsteinGreeneKruskal1957} that bifurcates from the linear Langmuir wave. The results of these simulations were compared to the theoretical predictions \cite{RosePOP2005} in our recent papers \cite{SilantyevLushnikovRosePOP2017,SilantyevLushnikovRose2POP2017}. An alternative approach to overcome computational difficulty of this problem was considered by our research group in Ref. \cite{LushnikovRoseSilantyevVladimirovaPhysPlasmas2014}. It involves reducing the number of transverse direction in the model therefore lowering computational difficulty at a cost of lesser accuracy of the model.

The second part of this work concentrates on $2$D free surface hydrodynamics and in particular on computing Stokes waves with high-precision using conformal maps and spectral methods. Stokes waves are fully nonlinear periodic gravity waves propagating with the constant velocity on a free surface of two-dimensional potential flow of the ideal incompressible fluid of infinite depth. The increase of the scaled wave height $H/\lambda$, where $H$ is the wave height and $\lambda$ is the wavelength, from $H/\lambda=0$ to the critical value $H_{max}/\lambda$ marks the transition from almost linear wave to a strongly nonlinear limiting Stokes wave. The Stokes wave of the greatest height $H=H_{max}$ has an angle of $120^{\circ}$ at the crest \cite{Stokes1880sup}. The non-limiting Stokes waves describe ocean swell and the slow time evolution of the Stokes wave toward its limiting form is one of the possible routes to wave-breaking and whitecapping in full wave dynamics. Wave-breaking and whitecapping carry away significant part of energy and momenta of gravity waves~\cite{ZKPR2007, ZKP2009}. Formation of the limiting Stokes wave is also considered to be a probable final stage of evolution of a freak (or rogue) wave in the ocean resulting in formation of approximate limiting Stokes wave for a limited period of time with a following wave breaking and disintegration of the wave or whitecapping and attenuation of the freak wave into wave of regular amplitude \cite{ZakharovDyachenkoProkofievEuropJMechB2006,RaineyLonguet-HigginsOceanEng2006,DyachenkoNewell2016}. Thus the approach of non-limiting Stokes wave to the limiting Stokes wave has both significant theoretical and practical interests.

To obtain Stokes wave fully nonlinear Euler equations describing the flow can be reformulated in terms of conformal map of the fluid domain into the complex lower half-plane, with fluid free surface mapped into the real line. This description is convenient for analysis and numerical simulations since the whole problem is then reduced to a single nonlinear equation on the real line. Having computed solutions on the real line we extend them to the rest of the complex plane to analyze the singularities above real line. The distance $v_c$ from the closest singularity in the upper half-plane to the real line goes to zero as we approach the limiting Stokes wave with maximum hight $H_{max}/\lambda$, which is the reason for the widening of the solution's Fourier spectrum.

In this dissertation we demonstrate a new approach that allows one to overcome this difficulty. We improve performance of our numerical method drastically by introducing second conformal map that pushes the singularity higher into the upper half-plane and correspondingly shrinks the spectrum of the solution while making the computations of extremely nonlinear solutions much more efficient. Using this approach we were able to demonstrate and confirm a predicted phenomenon of intricate parameter oscillation \cite{Longuet-HigginsFoxJFM1977,Longuet-HigginsFoxJFM1978,Schwartz82-Strongly-nonlinear-waves} for strongly nonlinear Stokes waves.

Mathematics

Doctoral

#### Department Name

Mathematics & Statistics

Pavel Lushnikov

#### Second Committee Member

Alexander Korotkevich

#### Third Committee Member

Evangelos Coutsias

#### Fourth Committee Member

Jean-Claude Diels

English

#### Keywords

plasma, instability, filamentation, BGK, Langmuir wave, nonlinear waves, Stokes wave, singularity

Dissertation

COinS