Article: Computational Plasma Physics

March 09, 2017   In English , numerical methods , Physics , plasma

Nowadays engineers put increasingly complex goals in front of them: to perform a thermonuclear synthesis, to land a man on Mars, to explore the solar system, to create an even faster airplane, to explore the dynamics of volcanos, etc. All of these tasks require to explore and understand a new field of physics, which is called “Plasma physics”. Due to the complexity of phenomena, in order to fulfill this goal, computational methods for computer simulations should be developed.

First, this review article gives an insight into the history of the development of plasma physics and what subfields arose during the development of the field, because it is of the most importance to understand what problems can be solved using such complicated object as plasma. Next, four main mathematical models of plasma are introduced in the article. They play the key role in the plasma physics, because they set differential equations, which specify scales on which model is correct and the mathematical complexity of the object. Computational methods are discussed later on in the article in order to give the reader all necessary information about algorithms which can be used to simulate a chosen mathematical model. At the end, one of the most commonly used computational methods, the Particle-in-Cell method, is described a little more in detail. Not only does this method plays an important role in today’s explorations, it also has implications for future prospects due to improvements in computing power and architecture.

History of plasma physics

Plasma is a dynamically complicated collection of negatively charged electrons and positively charged ions which interact with each other. It appears after an atomic decomposition when an environment’s temperature exceeds energies of an atomic ionization. One may think that plasma is a substance which rarely can be found in nature whereas it is the most common aggregation state in the universe: some scientists state that 90% of the whole Universe consists of plasma (Fitzpatrick, 2014). Moreover, it’s necessary to mention that 100% of the newborn Universe consisted of plasma. In addition, recent researches showed that Earth moves inside of the solar wind plasma. Such a close interaction with this aggregation state makes it of the greatest importance to understand its effects on our planet. Not only can plasma be found in outer space, but it also can easily be found on Earth (Fitzpatrick, 2014). Professor Fitzpatrick outlines a few phenomena which happen on Earth: “lightning, fluorescent lamps, a variety of laboratory experiments, and a growing array of industrial processes” (Fitzpatrick, 2014).

Originally, the term plasma meant “cleared blood” (oxforddictionaries.com, 2017). Irving Langmuir, the Nobel prize-winning chemist and physicist, was the first person who used this term to describe an ionized gas. He provided an analogy between physical and biological plasmas, saying that “the way blood plasma carries red and white corpuscles by the way an electrified fluid carries electrons and ions” (Fitzpatrick, 2014). The theory of plasma was developed when Langmuir tried to extend the lifetime of the filament of a tungsten-filament light-bulb. After Langmuir’s experiments, the theory of plasma spread in five other directions: electromagnetic wave propagation through nonuniform magnetized plasmas, magnetohydrodynamics, thermonuclear fusion, space plasma physics and laser plasma physics (Fitzpatrick, 2014).

Mathematical models of plasma

In order to be able to analyze physical phenomena, it’s necessary to create a mathematical model — to write mathematical equations which will accurately enough approximate the particular physical phenomenon. Mathematical models of plasma are equations of motions combined with Maxwell's Equations for electromagnetic fields. There are four main types of plasma models: N-body model, kinetic model, fluid model and hybrid kinetic/fluid model.

Fundamentally, a plasma is an N-body system of electromagnetic fields and charged particles, the dynamics of which can be described in some cases, “where transport due to short-range interactions (“collisions”) is weak, it is possible to work” (Escande, Doveil, Elskens, 2012), using just a Newton’s law of motion. For big amounts of plasma, it is impractical to solve these set of equations due to its computational complexity and the abundance of information that will be acquired after the direct integration, there will be much more information than anyone can possibly require. For these reasons different statistical models were developed: kinetic models, fluid models, and hybrid kinetic/fluid models are examples of statistical models of plasma dynamics.

Kinetic models are the most fundamental way to describe plasmas. It uses Vlasov equation, or more precisely Vlasov-Maxwell system of equations:

which is an improvement of Boltzmann’s ideas for plasma. It describes the evolution of the distribution function in 6N-dimensional phase space (Kalitkin, Kostomarov, 2009). Vlasov equation is a very accurate and complete approximation of the phenomenon but it is still most of the time unsolvable which led people to the creation of fluid models of plasma.

   
There are a lot of different fluid models of plasma, but one particular set of models are of the utmost interest to researchers, because they are “simple” relative to other models: they consider only one fluid, and at the same time they contain a lot of important macroscopic properties of a plasma: the plasma velocity field, current density, mass density, and the plasma pressure (Callen, 2003). Magnetohydrodynamics (MHD) is the simplest model of such kind, which contains equations of one-fluid model dynamics coupled with the Maxwell equations for the electromagnetic fields (Callen, 2003).

Kinetic models describe physical phenomena accurately, though it is more complex as well as computationally more intensive. On the other hand, fluid models are less computationally complex, but they can provide reasonably accurate solutions only on macroscopic scales. Nowadays, the most popular approach is to combine kinetic and fluid models into one kinetic/fluid model which treats some components of the system as a fluid, and others kinetically. It is also the most difficult approach because it requires not only a full understanding of equations of both kinetic and fluid models but also when each model can be used in every particular problem in order to get an accurate result.

Computational methods of plasma physics

After describing the most popular plasma models, it’s important to understand what types of numerical algorithms can be used to simulate the dynamics of plasma on a computer. The numerical methods in computational plasma physics closely related to computational fluid dynamics and mechanical N-body problem: fluid models of plasma are highly used for research due to good results and relative simplicity, therefore numerical plasma algorithms are mostly just an improvements of numerical fluid dynamics algorithms. 

The first and the easiest to understand approach to simulate plasma on a computer is to use N-body model. This technique allows computing full trajectories of each plasma particle -- “the first-principle dynamics” (Miloch, 2014). Such a detailed description of plasma particles movement affects exterior electric and magnetic fields and, as a consequence, these fields must be updated on every step of a simulation. Because each particle in the system interacts with every other particle in the system, the computational complexity of solving the N-body model is, which is practically unsolvable if a number of particles are too big (Miloch, 2014).

Another type of computational methods is based on solving the Vlasov-Maxwell equation, which is a nonlinear hyperbolic partial differential equation. The solution of this equation is a “Hamiltonian flow in a phase-space” (Miloch, 2014). There are different algorithms which allow solving numerically the kinetic plasma model: “e.g., semi-Lagrangian method, finite volume methods” (Miloch, 2014), though all of them are very computationally complex.

Fluid models approximate plasma dynamics on large scales and describe the evolution of macroscopic properties: density and pressure. Unlike the difficult Vlasov-Maxwell equation which can be solved only using complicated numerical algorithms and linearisation, MHD equations, which can be rewritten in the form of a first order hyperbolic partial differential equation, can be integrated using well-known Finite Difference methods like “Lax, Lax-Wendroff, leap-frog” (Miloch, 2014) as well as different Finite Volume algorithms. Finite Difference methods are divided into two categories: explicit and implicit algorithms. Explicit schemes are less computationally complicated, but they require a very strict Courant condition for a stable solution. On the other hand, implicit schemes are computationally stable, but they are more difficult in terms of programming and require more computational power. Therefore, researchers usually use hybrid explicit-implicit schemes as a “compromise between accuracy and numerical efficiency” (Miloch, 2014).

The Particle-In-Cell method

The systems that are of the interest to researchers are mostly very large in terms of the number of electrons and ions. To be able to simulate the dynamics of such systems, “super-particles” or “computer particles” are used instead of real particles. A super-particle is a computational particle which contains many real particles; it may be millions of electrons or ions in the case of a plasma simulation, therefore unlike real particles which are represented as mathematical points, super-particles should have a physical radius, because they contain a lot of real particles. Because the Lorentz force depends only on the charge to mass ratio, it is allowed to replace a number of particles with one super-particle with the same charge to mass ratio, and it will follow the same trajectory as the system of real particles (Filipič, 2008).

It’s technically impossible to solve the equations of motion and the system of Maxwell equations in the continuum space. In order to solve this problem the physical volume is divided into cells, creating a mesh, and the electromagnetic fields are getting computed only in the intersection points of this mesh (Filipič, 2008). Each cell has a local charge density which is the “number of particles in the cell divided by the volume of the same cell” (Filipič, 2008). Close interactions between particles have a low importance for a macroscopic problem, so electromagnetic fields are computed using local charge densities to compute potential and then the fields itself rather than “summing contribution of each particle” (Filipič, 2008). It’s also important to keep cell’s length in a very concrete range: it should be small enough, so the charge density separation effect can be noticed in computed fields, and at the same time large enough, so spatial cells will be bigger than the size of super-particles (Filipič, 2008).

The actual algorithm for performing a simulation consists of several steps. After reading the initial conditions, charge densities should be obtained in order to proceed with the integration of field equations. Next, the interpolation of electromagnetic fields from the mesh to the particles’ locations is performed in order to calculate to what extent “each individual grid point affects each particle” (Filipič, 2008). The last step is the integration of the equations of motion and changing the positions and velocities of each particle. After that the procedure repeats (Filipič, 2008).

Conclusion

Although plasma is an aggregate state of matter which appears very often in nature and surrounds us everywhere, it is an extremely complicated object and it requires quite an effort to simulate the dynamics of it. The four main models which approximate the dynamics of plasma are n-body, kinetic, fluid and hybrid kinetic/fluid models. All of them have their advantages and disadvantages, but fluid models of plasma, which underlie the theory of magnetohydrodynamics, are the models used the most, due to their relative simplicity and macroscopic nature. N-body models are usually very simple in terms of mathematics but very computationally complicated. The Particle-in-Cell method is an example of an algorithm which simulates the dynamics of an N-body plasma model using special methods to reduce the computational complexity. As a result, the technique allows to simulate plasma with a good accuracy, requiring an acceptable amount of time. Recent advancements in computer hardware as well as the creation of advanced parallel algorithms like Particle-in-Cell method allow to think that it will be possible to simulate large enough plasma systems which will explain real world phenomena.

Bibliography

Callen, J. (2003) “Plasma Descriptions II: MHD” in Fundamentals Of Plasma Physics, College of Engineering University of Wisconsin-Madison Press, Madison. p. 1.

Escande, D., Elskens, Y., Doveil, F. (2012) Basic microscopic plasma physics unified and simplified by N-body classical mechanics, eprint arXiv:1210.1546.

Filipič, G. (2008) "Computer simulations" in Principles of ”Particle in cell” simulations, University of Ljubljana, Faculty of mathematics and physics, p. 4-5.

Fitzpatrick, R. (2014) “Introduction” in Plasma Physics, CRC Press, Boca Raton, FL, p. 5-8.

Kalitkin, N., Kostomarov, D. (2006) Mathematical models of plasma physics (review), Matem. Mod. (vol. 18) Number 11, p. 67–94

Miloch, W. (2014) “Numerical Methods for Plasma” in Plasma Physics and Numerical Simulations, University of Oslo, p. 7-9.

Plasma, https://en.oxforddictionaries.com/definition/plasma, 2017.

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