# Mathematical Sciences

#### Numerical Models for Inertial-Electrostatic Confinement Fusion

Nico Braukman

##### Project Summary
Inertial-Electrostatic Confinement (IEC) fusion devices trap plasma ions with a spherically symmetric electrostatic field that accelerates ions inward, where they collide and fuse. IEC devices are simple to construct, and using them to study plasma motion will help advance fusion energy research. Here, plasma motion in an IEC device is numerically simulated by solving the Vlasov-Poisson equations using the finite volume method (FVM) and finite element method (FEM). The device consists of an outer positively-charged mesh (the anode) and an inner negatively-charged mesh (the cathode). Ions are injected into the device at the anode with small initial momentum. They are accelerated along the electric field to the cathode, then continue with constant momentum toward the center of the device. The ions that do not fuse continue to the opposite side, slowing down as they move against the electric field, and eventually reach a turning point and fall back to the center for another chance to fuse. The Vlasov-Poisson equations describe the time evolution of a collisionless plasma in an electrostatic field. For a spherically symmetric system, the equations simplify to one dimension of space and one dimension of momentum. The Vlasov equation describes how the phase-space density of ions changes due to the electric field, and the Poisson equation describes how the electric field changes due to the presence of ions. In the FVM, the Vlasov equation is solved over a discretized domain by computing the flux of ions across the boundaries of each sub-cell. The electric field is computed using Gauss’ law, which gives a solution to the Poisson equation. The FVM solution computed in MATLAB shows the expected ion motion. In the FEM, the Vlasov and Poisson equations are solved in their weak forms over piecewise linear (P1) and piecewise constant (P0) finite element spaces, respectively, using the FreeFEM PDE solving software. The FEM solution shows qualitatively similar ion motion to the FVM solution, however, the term describing the injection of ions introduces errors in the FEM solution which accumulate over time.

##### Future Works
In future simulations, spectral methods for solving the Vlasov-Poisson equations will b investigated as an alternative to the FVM or FEM.

#### Testing the Relativistic-Microwave Theory of Ball Lightning with Plasma Simulations

##### Team Member(s)
Amanda Elliott

Dr. Vladislav Bukshtynov, Dept. of Mathematical Sciences, Florida Tech

Testing the Relativistic-Microwave Theory of Ball Lightning with Plasma Simulations  File Download
##### Project Summary
Ball lightning (BL) is an unexplained phenomenon reported by thousands of eyewitnesses as a fireball, a few cm to 1 m in diameter, moving unpredictably and independently of the wind, sometimes observed during lightning storms. Here a potential theory for the creation of BL is explored. BL objects are luminous, and so, must contain an energy source. It has been theorized that this energy source is a standing microwave that ionizes the air in a spherical shape. This theory is supported by the observation that the most common reported diameter for BL objects is in the same range as the wavelengths of microwaves. This plasma shell, or bubble, could contain the standing wave, but otherwise, would be a vacuum inside. This structure would also explain the two most commonly reported forms of termination of BL objects: (1) peaceful and (2) violent termination types. BL objects have been observed to terminate by either (1) dissipating silently like a gas, or (2) to explode with a violent burst of energy that has been lethal. These two termination types suggest that BL object stability is analogous to that of a bubble, and is highly dependent on the balance of internal and external pressure and surface tension. The goal of this project is to replicate previously simulated results (by researcher Hui-Chun Wu) that suggest a way for this theorized BL structure to be created at the end of a lightning strike. The open-source Particle-in-Cell (PIC) code called Smilei is used to simulate plasma in a 2D environment. These BL creation results are divided into two separate simulations. For the first simulation, the conditions are simulated at the end of a lightning leader approaching the ground. The resulting strong electric field could lead to an electron bunch with an of energy of 50 MeV, in the relativistic rang. This electron bunch is accelerated by the field toward ground, modeled as conductor, or overdense plasma. This resulted in a radial pulse of coherent transition radiation (CTR), which is the hypothetical energy source for a BL object. This CTR was successfully simulated to have a magnitude of 317 MV/m, which is close to the Wu's original simulated value of ~310 MV/m. The goal for the second simulation is to model the standing wave trapping in a spherical plasma “bubble.” For this simulation, an input standing wave pulse is modeled using a sine wave traveling within a Gaussian envelope. For this simulation, the plasma shell forms, but expands outward without stabilizing. The original results that were partially replicated here were generated by Wu’s personal PIC code and may not be reproducible using standard PIC codes that are designed for vacuum. Atmospheric molecule collisions with electrons cannot be simulated at this time scale, but the resulting energy loss can be approximated and may stabilize the plasma shell. Once a successful theory and corresponding simulation of BL creation is achieved, this will inform how to generate BL in a laboratory or during thunderstorms. The results of this research will advance lightning protection, aviation safety, and broaden our knowledge of plasma physics and confinement methods.

##### Project Objective
The goal of this project is to replicate previously simulated results that suggest a way for this theorized BL structure to be created at the end of a lightning strike. The open-source Particle-in-Cell (PIC) code called Smilei is used to simulate plasma in a 2D environment. These BL creation results are divided into two separate simulations.

##### Analysis
Particle-in-Cell (PIC) codes are commonly used for plasma simulations. They operate by initializing charged particles to a spatial grid, and calculating the resulting electromagnetic fields using a Maxwell solver. They then calculate new particle positions and velocities based on Newtonian equations of motion. The PIC code selected for this project is an open-source code called Smilei. It contains the above features that are characteristic of PIC codes, but also includes relativistic effects and solves the Vlasov-Maxwell equation for collisionless plasmas. Inherent in Smilei's equations of motion is the radiation pressure, or ponderomotive, force which is vital for simulating the plasma bubble formation.

##### Future Works
Improvements for the second simulation is ongoing. The simulation assumed vacuum conditions, and so the next step is to approximate the energy loss that electrons experience when colliding with air molecules, and apply this to the simulation. Once this is accomplished, the results can be compared with Wu's, and this theory for BL creation can be evaluated. There are ideas to expand this model to explain additional properties of BL objects. The goal is to develop a comprehensive model that explains BL creation, its stability during its lifespan, and the two most common forms of termination.