University of California, Los Angeles Non-MHD Models Wendy Mata University of California, Los Angeles GEM Summer Workshop Student tutorial June 16, 2013 I decided to focus on one particular nonMHD model, the particle-in-cell method (PIC) b/c many of the other non-MHD models share the same theory and technical implementation and the PIC method is relatively simple.

Simulations are an important tool in scientific research Computer simulations are carried out to: Understand the consequences of fundamental physical laws Help interpret experiments Design and predict new experiments *2(can provide detailed information which is difficult to measure

MHD Models Strengths: Work surprisingly well for global modeling (large scale transport and structure) allows you to model the global magnetospheric dynamics in a way that's computationally tractable Describes dominant large scale communication (by Alfven waves) Cost: over-simplification of the physics Weaknesses: Not very good at “microphysics” (reconnection) Not very good at physics that is on the scale of a particle gyroradius • Magnetic flux transport is based on ions only • No drift physics Simple energy equation No drift physcis-no hall term Simple energy equation: isotropic pressure and no heat flux

Particle-in-cell (PIC) method PIC method refers to a technique used to solve a certain class of partial differential equations In this method, individual particles in a Lagrangian frame are tracked in continuous phase space, whereas moments of the distribution are computed simultaneously on Eulerian (stationary) mesh points Mesh points are a set of finite elements used to represent a geometric object for modeling or analysis.

Basic Technical Implementation The PIC method is relatively intuitive and straightforward to implement Typically includes the following procedures: Integration of the equations of motion Interpolation of charge and current source terms to the field mesh Computation of the fields on mesh points Interpolation of the fields from the mesh to the particle locations PIC method is susceptible to error from discrete particle noise For, quantities such as density which must be integrated over a certain area or volume. As the number of particles approaches infinity, things get smoother, but when there are only a few particles, they are not smooth. In other words using discrete particles rather than a continuous fluid leads to a certain kind of statistical "noise" in your simulation. And as you increase the number of particles, that goes away because you get closer to the fluid/continuum approximation

Basic Technical Implementation Electrons, ions, neutrals, and molecules can be treated with the PIC method The set of equations associated with PIC methods: The Lorentz force as the equation of motion, solved in the pusher or particle mover of the code, Maxwell’s equations for determining the electric and magnetic fields, calculated in the (field) solver

Lagrangian Frame In continuum mechanics, the Lagrangian system of the flow or of the displacement field is to observe the motion following an individual particle as it moves through space and time. e.g. sitting in a boat and drifting down a river

Eulerian Frame In the Eulerian system we look at the motion that focuses on specific locations in the space through which the fluid flows as time passes e.g. sitting on the bank of a river and watching the water pass a fixed point

Mathematical Formulation for Lagrangian and Eulerian Frame In the Eulerian system of the field, the quantities are depicted as a function of fixed position x and time t. The velocity is described as u(x,t) On the other hand, in the Lagrangian system, all particles are represented by some vector field s, where s is time- independent for each particle Often s is chosen to be at the center of mass of the particles at some initial time t0 *last( accounts for the possible changes of the geometry over time and for easily parametrizing the velocity v(s,t) of the particles

Mathematical Formulation for Lagrangian and Eulerian Frame In the Lagrangian system the velocity v(s,t) is related to the position X(s,t) of particles by Consequently, u and v are related through Within a chosen coordinate system, s and x are referred to as the Lagrangian coordinates and Eulerian coordinates of the motion

Mathematical Formulation for Lagrangian and Eulerian Frame In the Lagrangian and Eulerian systems, the kinematics and dynamics of the field are related by the convective derivative: This tells us that the total rate of some vector function F as the particles move through a field described by its Eulerian specification u is equal to the sum of the local rate of change and convective rate of change of F

Super-particles The real systems studied are often extremely large in the number of particles In order to make simulations efficient, so-called super-particles are used A super-particle is a computational particle that represents many real particles; it may be millions of electrons or ions Because the Lorentz force depends only on the charge to mass ratio, a super-particle will follow the same trajectory as a real particle would

Solving the particle mover (equations of motion) Even with super-particles, the number of simulated particles is usually very large (>105) Thus, the mover is required to be of high accuracy and speed and much effort is spent on optimizing the different schemes For solving the particle mover

Solving the particle mover (equations of motion) The schemes used for the particle mover can be split into two categories, implicit and explicit solvers Implicit solver calculate the particle velocity from the already updated fields Explicit solvers use only the old force from the previous time step Two frequently uses schemes are the leapfrog method (explicit) and the Boris scheme (implicit) Explicit solver…and are therefore simpler and faster, but require a smaller time step

The Field Solver The most commonly used methods for solving Maxwell’s equations (or more generally, partial differential equations (PDE) belong to one of the following three categories: Finite difference methods (FDM) Finite element methods (FEM) Spectral methods

The Field Solver -FDM With the FDM, the continuous domain is replaced with a discrete grid of points, on which the electric and magnetic fields are calculated. Derivatives are then approximated with differences between neighboring grid-point values and thus PDEs are turned into algebraic equations

The Field Solver - FEM Using FEM, the continuous domain is divided into a discrete mesh of elements. The PDEs are treated as an eigenvalue problem and initially a trial solution is calculated using basis functions that are localized in each element. The final solution is then obtained by optimization until the required accuracy is reached

The Field Solver - Spectral Methods Also spectral methods transform the PDEs into an eigenvalue problem, but this time the basis functions are high order and defined globally over the whole domain. The domain remains continuous. Again, a trial solution is found by inserting the basis functions into the eigenvalue equation and then optimized to determine the best values of the initial trial parameters. *1)such as the fast fourier transform (FFT),

Particle and Field Weighting The origin of the name ‘particle-in-cell’ stems from how plasma macro-quantities are assigned to simulation particles, that is, the particle weighting Particles can be situated anywhere on the continuous domain, but macro-quantities are calculated only on the mesh points, just as the fields are To obtain the macro-quantities, one assumes that the particles have a given “shape” determined by the shape function The shape function has to satisfy the following conditions: Space isotropy Charge conservation Increasing accuracy (convergence) for higher-order terms In addition, to ensure that the forces acting on particles are self-consistently obtained

Particle and Field Weighting The fields obtained from the field solver are determined only on the grid points and can’t be used directly in the particle mover to calculate the force acting on particles, but have to be interpolated via the field weighting

Particle and Field Weighting Calculating macro-quantities from particle positions on the grid points and interpolating fields from grid points to particle positions has to be consistent since they both appear in Maxwell’s equations Above all, the field interpolation scheme should conserve momentum This can be achieved by choosing the same weighting scheme (shape function) for particles and field and by ensuring the appropriate space symmetry

Accuracy and Stability Conditions the time step and the grid size must be well chosen, so that the shortest time and length scale phenomena are properly resolved in the problem In addition, time step and grid size have also an impact on the speed and accuracy of the code

Accuracy and Stability Conditions two important conditions regarding the grid size Δx and the time step Δt should be fulfilled in order to ensure the stability of the solution: Δx<3.4λD, Δt≤2ωpe-1 Debye length: the measure of a charge carrier's net electrostatic effect in solution, and how far those electrostatic effects persist These two criteria can be derived considering the harmonic oscillations of a one-dimensional unmagnetized plasma Small ∆ t is required for stability in explicit schemes Large ∆ t can be achieved with implicit schemes

Accuracy and Stability Conditions The latter condition is strictly required but practical considerations related to energy conservation suggest to use a much stricter constraint where the factor 2 is replaced by a number one order of magnitude smaller The use of Δt ≤ 0.1 ωpe-1 is typical Not surprisingly, the natural time scale in the plasma is given by the inverse plasma frequency and the length scale by the Debye length

Summary In plasma physics applications, PIC methods consist of following the trajectories of charged particles in self-consistent electromagnetic/electrostatic fields computed on a fixed mesh Within plasma physics, PIC simulation has been used to study laser-plasma interactions, electron acceleration and ion heating in the auroral ionosphere,, magnetic reconnection, a