PHYSICS Computational Plasma Physics

Transcription

1 PHYSICS 78 - Computational Plasma Physics INSTRUCTOR Dr. Earl Scime (escime@wvu.edu) 93-34, ext Office hours: MW :30 3:30 and whenever door is open Rm 18 & 05 Hodges Hall Class: MWF 1:30-:0 Rm 334 Hodges Hall Anticipated instructor absences from Morgantown: Oct. 30 Nov 3 TEXTS: Plasma Physics via Computer Simulation, Birdsall and Langdon (required) Plasma Physics and Controlled Fusion, Francis F. Chen, nd Edition (required) OOPIC, Tech-X corporation ($50 for year student license, required) Elementary Numerical Analysis, Atkinson, first four chapters provided Numerical Recipes, Press et al., (recommended) COMPUTATIONAL RESOURCES: You will need regular access to a computer with a C or Fortran compiler. Matlab or a similar program could also be used. The computers in 334 Hodges are available for your use and your ID card will be added to the door lock. COURSE OBJECTIVE: To apply modern computational techniques to specific classes of plasma physics problems. The topics to be covered are: 1. Basic plasma time scales and parameters. 4. Basic structure of one dimensional particle in. Solution of dispersion relationships using cell (PIC) codes root finding methods 5. Use of the OOPIC code to investigate basic 3. Single particle tracking by solving plasma phenomena. differential equations and using numerical 6. Conceptual understanding of more advanced integration. codes, e.g. gyrokinetic, fluid. HOMEWORK/PROJECTS: The assignment sheet lists the dates of the lectures, the general topic to be covered in each lecture, the sections of the text that relate to the lecture, and the problems assigned. Most of the assignments and projects involve writing code. All code should be adequately documented and code output should be easily interpretable. Homework and projects will be collected on the dates shown on the assignment sheet. Example codes written in LabWindows (C complier) and MATLAB are available for you to gain familiarity with those software environments. GRADING: BREAKDOWN Homework Problems and Short Projects 50% First Plasma Project 5% choose by 9/1 due on 10/13 Second Plasma Project 5% choose by 10/13 due on 1/6

2 Date Chapter Planned Topics Assignment Due M 8/1 Chen 1 & Plasma time and spatial scales CPP #1 9/1 W 8/3 Cold plasma waves, computer lab example F 8/5 examples M 8/8 Atkinson Single particle motion, W 8/30 Numerical methods, errors F 9/1, Project #1 Choices Due CPP # 9/8 M 9/4 Holiday W 9/6 Newton's method F 9/8 Numerical Integration, CPP #3 9/15 M 9/11 BL 1 & Generation of distribution functions W 9/13 Approaches to plasma simulation F 9/15 CPP #4 9/ M 9/18 Chen 3, 4, 6 Finite difference equations W 9/0 Cloud in Cell codes F 9/ CPP #5 9/9 M 9/5 OOPIC Cold plasma electrostatic oscillations W 9/7 ES1-OOPIC coldplasma, Computer Lab F 9/9 OOPIC Cold beam instability CPP #6 10/6 M 10/ BL 3 & 4 Plasma and hybrid oscillations W 10/4 F 10/6 OOPIC hybrid oscillations CPP #7 10/13 M 10/9 BL 5 Cold two stream oscillations W 10/11 F 10/13 1 st project due, demonstrations CPP #8 10/7 M 10/16 OOPIC Warm two stream oscillations W 10/18 F 10/0 Landau damping OOPIC Warm two stream oscillations M 10/3 OOPIC OOPIC Landau damping W 10/5 F 10/7 CPP #9 11/10 M 10/30 APS-DPP W 11/1 APS-DPP F 11/3 APS-DPP M 11/6 Journal paper Boundaries in PIC codes W 11/8 F 11/10 CPP #10 11/17 M 11/13 Journal paper MHD basics-fluid codes W 11/15 F 11/17 Journal paper Hybrid and gyrokinetic codes CPP #11 1/1

3 M 11/0 Thanksgiving Recess W 11/ Thanksgiving Recess F 11/4 Thanksgiving Recess M 11/7 Hybrid and gyrokinetic codes W 11/9 Sheath and double layer modeling F 1/1 CPP #1 1/8 M 1/4 W 1/6 F 1/8 Project Reviews Chemistry modeling, SECOND PROJECT DUE

4 First Computational Plasma Project Topics 1. Write a code that updates the three-dimensional trajectory of an ion in an externally oscillating electric field (ω ~ Ω c ) and a static magnetic field. Use a simple timestep algorithm to solve the Lorentz force and inject a Maxwellian distribution of particles. Assume a static magnetic field along the z axis, electric field along x, and rectangular boundaries in x and y. Give the z axis a periodic boundary condition for particles and remove particles from the simulation that pass through the boundaries in x and y. Start with a Maxwellian distribution of x velocities and compare the final distribution to the initial case (you'll have to include y velocities for rotation effects, but only look at final distribution along x). Examine the effect of the timestep used on the character of the final distribution after 1000 times the ion gyrofrequency.. Poisson's equation relaxation of a D array containing surfaces at different electric potentials. Use the successive over-relaxation algorithm assuming uniform grid spacing (w = 1.5 is a good value to try). Allow for conducting surfaces such as wires or plates to be placed within your D grid structure. Relax grid until maximum update during an entire sweep is within some externally controlled value. φ est ( φ[ x + h, y] + φ[ x h, y] + φ[ x, y + h] + [ x, y ]) = 1 4 φ h φ new = ( 1 w) φold + wφest 3. Write a program to solve the lower hybrid dispersion relationship for a plasma with a density gradient per equation (61) in Chapter 3 of Stix s Waves in Plasmas text. ωpi ω pe d ωωpi( y) ωpi ωpe ω e kx 1+ + k x + k = i ω e dy i ( i ω ) ω ω γk κt e / m e ω Ω Ω Ω Ω The code should permit the user to supply a density profile with a resolution of 100 points, the wave frequency, the electron temperature, the ion mass, the parallel wavenumber, and the magnetic field strength. The code should then numerically calculate the derivative of the ion plasma frequency. The output of the code should be a plot of the perpendicular wavenumber as a function of y position. The definitions of the terms in the equation can be found in Stix.

5 4. Write a code to plot the perturbed ion distribution (due to an electrostatic wave), c a qnoϕ1 k k( y) = (v ) o y M vth f v f e e e mn, = T T i (v yx k ) Ω ( ξ ξ ) Zξ ( ξ Zξ ) 0 ' kv y o mn + ( mn + ) mn + ( mn + ) 1 k Ω π π im ( ) v ik imn y + θ m k nk Ω k= k ac a e e J ( ) e I ( ) I ( ). 8 as a function of perpendicular velocity. Note, k k vth a = β Ω Ω, ky kyvth c = β Ω Ω ' Vo = dvo / dx = shear in the parallel flow, s 1 e Z( ζ m ) ds ; π s ζ ω mω kv 0( x) and ζ m =. k vth Use parameters typical of ion cyclotron waves for both a Q-machine and the helicon source and discuss and contrast the results. Assume no shear in the parallel flow. Both C and MatLab subroutines that calculate the Zeta function are available for your use see the instructor.

6 Second Plasma Physics Project Topics (others can be negotiated) 1. Using OOPIC, model a linear, cylindrical plasma device with a uniform magnetic field. Create a density gradient in an argon along the field and see if it results in the formation of a strong potential gradient at low gas pressures [see A. Meige, R.W. Boswell, C. Charles, and M. Turner, Phys. Plasmas 1, (005) and X. Sun, C. Biloiu, A. Keesee, E. E. Scime, A. Meige, R. Boswell, and C. Charles, Observations of ion-beam formation in a current-free double-layer,' Phys. Rev. Lett. 95, (005)]. If no potential gradient forms spontaneously, inject an electron beam from one end and determine the energy and density of the beam required to create a double layer.. Using OOPIC, model a linear, cylindrical plasma device with a uniform magnetic field. Add two different oscillating electric fields to the static background magnetic field. Examine what happens when the difference in oscillating electric field frequencies is close to the particle gyrofrequency. Does the ion motion become stochastic, i.e., the ions experience large-scale diffusion in velocity space. Examine the effect of increasing neutral pressure on the ion heating. At what neutral pressure does the heating effect start to go away? 3. Write a code to calculate the normalized phase velocity as a function of propagation angle for warm plasma waves. Solve for both the fast and slow wave and use the warm plasma dielectric tensor elements as given in Swanson s book. Assume the plasma is quasi-neutral and composed of protons and electrons. Allow the user to input the magnetic field strength, the plasma density, and the wave frequency. 4. Using OOPIC, model a linear, cylindrical plasma device with a uniform magnetic field. Create a argon plasma in an applied 10 MHz RF field and a neutral gas pressure of 8 mtorr. Use realistic gas collision parameters. Investigate what happens to the equilibrium density as a function of magnetic field. Does the plasma density increase or decrease with increasing neutral pressure? What happens to the plasma density if the RF frequency is changed to the lower hybrid frequency For all of these projects you will need to turn in a copy of your input file, a written summary of your results, and examples of your results.