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 Theory & Simulation

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# Theory and Simulations

A variety of methods, often in combination, are used:

## Analytical methods

• Basic analytical techniques are used often in combination with analytical programs such as Maple, which can be used both for exploring the possibilities of analytical treatments and for carrying out or checking the results of lengthy analyses.
• The reduction of the formulation of a physical problem to a well known equation such as that of van der Pohl or Korteweg-de Vries enables the use of an extensive literature and results from other problems leading to the same equation. Special methods can then be used e.g. inverse scattering theory for the solution of the modified Korteweg-de Vries equation for weak double layers.

## Mathematical models

The theoretical analysis of a real situation can be simplified in a mathematical model:
• The geometry can be simplified by imposing symmetries that preserve the essential properties. Even one dimensional representations, which are relatively easy to deal with, can lead to important insights.
• Numerical methods using Matlab, Femlab and related programs can be applied.

## Equivalent circuits

The physics of complex experimental and natural physical systems in space and astrophysics can be investigated by representing separate parts by circuit elements coupled by connecting currents. Examples are:
• The magnetosphere, aurora.
• Solar physics, prominences and solar flares.

## Simulations

Particle in cell (PIC) codes are used to simulate plasmas on a basic level. These codes follow the motion of a large number of super-particles (up to several million) each of which represents a huge number of physical particles. The electric and magnetic fields are found using interpolated values of current and particle densities at the points of a grid defining the cells. Super-particle motion across the cells is followed using interpolated values of the electric and magnetic fields at the grid points. Typical applications simulate the following:
• Electrostatic instabilities in a uniform current carrying plasma using periodic boundary conditions.
• A simulated plasma region coupled to an external circuit through suitable boundary conditions e.g. to represent an imposed current (external inductance) or external potential in studies of double layer formation.
• Coupling to external waves e.g. the inclusion of Alfvén wave impedance, electromagnetic radiation from plasma current fluctuations.
• Simulations of plasma flow across a curved magnetic field with boundary conditions defining the properties of inflowing plasma.

 Published by: School of Electrical EngineeringInfomaster, infomaster@spp.ee.kth.se Last updated: 2013-06-18