Two-Stream Instability

Problem Description

The two-stream instability is a rapidly growing collision-less plasma instability arising from small charge imbalances. A local imbalance leads to the acceleration or deceleration of particles in its vicinity, which in turn leads to an even stronger imbalance. One setup that allows to easily observe the instability is two counter-streaming beams of identical charge in a periodic system. The advantage of this configuration is that the generated plasma wave becomes a standing wave, thus allowing to easily observe the formation of the phase space vortices.

In this example, we use two electron streams. At t=0 the streams have drift velocities of +-1.e7 m/s. In order to accelerate the onset of the instability, the two particle beams are given a small sinusoidal perturbation in velocity space.

Input File Features

The simulation setup consists of an electromagnetic field and two particle species, one for each of the two counter propagating electron particle streams. Each electron stream is given a drift velocity and an additional velocity perturbation. For diagnostic purposes there is also output for species momentum, species energy and field energy histories.

Running the Simulation

Start VorpalComposer and select File -> Clone Example. Highlight Solving Classical Physics Problems and then select Next. Highlight the Two-Stream Instability and then select Choose. Create a new folder and then select Choose.

Alternatively, save the VORPAL input file, twostream.pre, and open in VorpalComposer.

The file should be displayed in the right pane of the Setup window. Click on the Save and Process Setup button in the lower right corner. Proceed to the run window as instructed. To run the file, click on the Run button in the lower left corner of the window. You can see the real time output of the run in the right pane.

Viewing the Output

Once instructed after the run has completed, proceed to the Visualize window to view the results. Load in the data files as instructed.

To view the phase space of the particles, switch to the Phase Space tab. Set the X-axis to the _0 component (x position) and the Y-axis to the _1 component (x velocity) of the electrons and click Draw. You can now step through the data by using the slider at the bottom to observe the evolution of the instability. To plot the second beam, click on Enable Second Plot and choose the same components for the other species. Click Draw. You can change the colors of the beams to more easily identify them.

Results

The figure shows plots of the evolution of the instability over time, the formation of the phase space vortex and finally the thermalization of the plasma. After sufficient simulation time, the plasma exhibits a thermal distribution.

Evolution of the instability through time. Plots are shown at dumps 1, 33, 66 and 100.

For a more quantitative analysis, VORPAL output can be imported into a broad range of popular data analysis packages. In this example we used IDL in order to extract the different energy contributions as seen in the figure. Both figures can be directly correlated to those shown by Birdsall and Langdon [1].

The script, written for IDL, may be found here.

The field energy is obtained from the field energy history as inserted via the input file block:

<History EFieldEnergy>
  kind = fieldEnergy
  fields = [varEmField.YeeElecField varEmField.YeeMagField]
</History>

Obtaining the thermal and drift energy requires post-processing of the particle history data. The drift energy is given by the mean particle energy, and can be found by taking the total species momentum history for each species, dividing by the electron mass and the total number of real particles in the simulation to get an average velocity. The drift energy is calculated as 1/2mv².

The thermal energy is obtained via the standard deviation from the drift energy and can be calculated by subtracting the drift energy from the total species energy.

Post-Processed results showing the Drift, Field, and Thermal energies versus time.

Further Experiments

Streaming instabilities are an incredibly rich phenomena in plasma physics and VORPAL offers simple mechanisms to investigate this richness. For example, the instability is an electrostatic instability, thus instead of running an electromagnetic simulation, VORPAL can also be run in electrostatic mode. All that needs to be done is to change the field type from EM to ES. Another important aspect of instabilities is their growth rate during the linear phase, which often can be predicted by theory. This analysis requires examination of the time evolution of individual modes, another task that can be accomplished in the user’s favorite data analysis tool.

Additional experiments worth investigating are:

• Adding a finite temperature to the two streams

• The same simulation, but with opposite charge signs in the different

  streams, leading to the so-called Buneman-Instability

• Setting the drift velocity of one stream to zero, resulting in the

  bump-on-tail instability

References

[1] C K Birdsall & A B Langdon. Plasma Physics Via Computer Simulation. Bristol and Philadelphia: Institute of Physics Publishing Ltd, 1991. Print.