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Non-MHD Models

Non-MHD Models. Wendy Mata University of California, Los Angeles GEM Summer Workshop Student tutorial June 16, 2013. Simulations are an important tool in scientific research. Computer simulations are carried out to: Understand the consequences of fundamental physical laws

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Non-MHD Models

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  1. Non-MHD Models Wendy Mata University of California, Los Angeles GEM Summer Workshop Student tutorial June 16, 2013

  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

  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

  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 aLagrangianframe are tracked in continuous phase space, whereas moments of the distribution are computed simultaneously onEulerian(stationary) mesh points

  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

  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 theEuleriansystem 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 fields, 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

  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

  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)

  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.

  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

  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

  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. 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 Summary

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