Download 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
 Help
interpret experiments
 Design
and predict new experiments
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
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
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
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,
r
r r r
F  q( E  v  B)
•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 timeindependent for each particle

Often s is chosen to be at the center of mass of the particles at
some initial time t0
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
v

X
t
Consequently, u and v are related through
r r r
r r
 u( X ( s,t),t)  v ( s,t)

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:
r r
r r
r r
r
r
r
DF ( x,t) F ( x,t) F ( x,t) dx F r



 (v  )F
r 
Dt
t
x
dt t

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 superparticles 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
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)
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.
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
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
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