1. 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.

2. 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

3. 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

4. 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.

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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),

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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