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Discoveries in plasmas while teaching simulation
Computer Physics Communications 164 (2004) 189–194 www.elsevier.com/locate/cpc
Discoveries in plasmas while teaching simulation Plasma Theory and Simulation Group (PTSG) Charles K. (Ned) Birdsall ∗ , Edison T. Estacio EECS Department of Cory Hall, University of California, Berkeley, CA 94720-1770, USA Available online 21 July 2004
Abstract Once PC’s became ubiquitous, we have been using them for teaching plasma simulation, hands-on by instructors and by students. The transfer of skills from instructor to class has been very rapid (most desirable). However, occasionally some unanticipated results are observed with plausible explanations expected from the instructor (scary). Our examples are all onedimensional. First, we show the famous two-stream instability in a periodic model, starting either cold or warm, which does not (quite) Maxwellianize; why not? Second, we show Landau damping also in a periodic model, with what appears to be small (hence linear) excitation, but observe trapping in the wave frame; going to very small excitation the trapping diminishes and the damping rate approaches that from Landau linear theory. Lastly, we show a warm plasma bounded by two grounded metal planar walls, uniform in density at t = 0, bounded, one-dimensional. For t > 0 we observe spontaneous plasma frequency oscillations in the midplane, sheath formation at ion sound speed at both walls, trapping of electrons, and acceleration of the ions to the walls; however, we also observe an oscillatory axial current, and ‘staircasing’ of the number of electrons in time. Both can come only from some degree of asymmetry in the system. The frequency of the current is the series resonance between the sheath capacitance (almost no electrons, so vacuum) and the bulk plasma ‘inductance’ (as ωseries ωp ). 2004 Elsevier B.V. All rights reserved.
1. Two-opposing-cold-streams instability (Fig. 1) Two opposing electron streams, cold or warm (vt < 0.7vdrift) are unstable and grow in time, in a fixed ion background, in a periodic system. In a system where two streams pass each other one wavelength in one period of natural oscillation (of one of the streams), there can be exponential growth of an initial small perturbation. This instability is very old, observed in fluid (e.g., fiord flow over the ocean), * Corresponding author.
E-mail addresses: [email protected] (C.K. Birdsall), [email protected] (E.T. Estacio). 0010-4655/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cpc.2004.06.029
and with fast and slow electron streams (amplifier by Haeff [1]), or drifting electrons and non-drifting ions (theory by Pierce [6]). The field energy may grow as fast as exp(ωp t). With two opposing streams, as above, at stream velocity reversal time (t = 27 pictures), v(x) vs. x, ρ has many harmonics (new k’s), which broadens the velocity distribution, traps the streams into huge vortices (t = 126.4) with no drifts remaining, and stops the growth. Even after a long time, the overall velocity distribution (both cold and warm streams) is not Maxwellian, which might be expected from balance between drag and diffusion. Why not?
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Plasma Theory and Simulation Group (PTSG) / Computer Physics Communications 164 (2004) 189–194
Fig. 1. Two-opposing-cold-streams instability.
The model has a Poisson equation solver, and E = −∇(Φ), plus a Newton–Lorentz equation of motion, mdv/dt = qE. There is no µmv (momentum exchange) term, or drag, and no explicit diffusion term; usually these terms balancing leads to a Maxwellian. Not here! With the length L used, the fastest growing mode is mode 4. While the two f (v)’s appear Maxwellian, the phase space still shows one vortex, very long lived, not (yet) at the end of interaction.
Indeed, at t = 30, mode 4 dominates, in the vx –x phase space and in the charge density, ρ. By time 63.6, the 4 phase space vortices have merged, leaving two vortices. By time 244.8, all vortices in phase space have become just one vortex, which is very long lived. The f (v, both streams) is shown in the middle panels, with the dip in the center at v = 0. As pointed out by Wendt [7], the warm streams [sufficient (vt /vo )] tend to saturate with a double peaked f (v), close to a Penrose distribution, which is stable.
2. Two-opposing-warm-streams instability (Fig. 2) The interaction of two warm streams is shown here. At t = 0, the ratio of thermal velocity to drift velocities vt /vo = 0.4. The dispersion relation shows that the m = 4 mode has the fastest growth rate.
3. Warm plasmas; linear Landau damping, nonlinear heating (Figs. 3–5) Warm plasmas exhibit simple harmonic oscillations at the plasma frequency, ωp , as known from theory
Plasma Theory and Simulation Group (PTSG) / Computer Physics Communications 164 (2004) 189–194
191
Fig. 2. Two-opposing-warm-streams instability.
and experiments by Langmuir and Tonks in the 1920s. Much later (1946), using linear kinetic theory, L. Landau [3] found the rate that these oscillations damp in time. Both detailed analysis and physical understanding may be found in [2]. Experimental proof came in the 1960s (e.g., by Malmberg and Wharton, [4], and others). Initial wave energy (electrostatic) is absorbed by the plasma, increasing the plasma’s kinetic energy (temperature). Below are particle-in-cell (PIC) simulations using Bruce Langdon’s XES1 1d3v periodic code. On the left are logs of the electrostatic energy vs. time, all oscillating at about 1.2ωp , and damping exponentially very close to the Landau/Jackson rate; the initial displacements are x1 = 0.1, 0.01, 0.001. On the right are the particle trajectories (‘traces’) vs. x, in the wave frame; the top one shows particle trapping,
nonlinear; the middle one shows smaller trapping; the bottom one has no visible trapping. There is a small frequency shift. A hundred thousand particles used in order to maintain the damping to a long time.
4. Warm plasma between two grounded plates at t = 1e–7 (Fig. 6) Elementary 1d3v bounded warm plasma observations, short circuit, undriven. Starting from an initial uniform plasma between shorted planes, we observe: (a) formation of sheaths; followed by (b) plasma frequency potential oscillations at the central density (symmetric mode); (c) series resonance observed in the current, due to resonance between the capacitive
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Plasma Theory and Simulation Group (PTSG) / Computer Physics Communications 164 (2004) 189–194
Fig. 3. Left side: electrostatic energy vs. time, log scale, for 3 initial displacements of particles. Right side: vx –x phase space with particle tracing (for trajectories).
(almost vacuum) sheath and the inductive bulk, resonant at ωseries = ωp (center)[(2s/L)]1/2 , where s is the average sheath width and L is the diode width. This is an asymmetric mode (see [5]) and also is the cut-off frequency for asymmetric (m = 1) surface waves along the walls. (Omitted are the ion sound waves traversing the plasma and bouncing off the two sheaths/walls.)
5. Summary Teaching live and interactively can be both exciting and challenging, and have unanticipated surprises,
similar to the examples just given. An audience of theorists and experimentalists may be unaccustomed to observing what is occurring inside the plasma (as functions of space, velocity, frequency, wave number, more) as displayed with several dozen diagnostics. In order to improve communication in lecturing and teaching, we have found it advisable to show, say, 1 to 4 live diagnostics at a time, complemented by a large poster board showing, say, a dozen or two diagnostics stills. Of course, the best way is to have the audience running the same simulations, at their speed, and with their choice of diagnostics, on their computers. Go and do likewise!
Plasma Theory and Simulation Group (PTSG) / Computer Physics Communications 164 (2004) 189–194
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Fig. 4. Dispersion relation for warm plasma from Jackson 1960; Re(ω) and Im(ω) versus κλD . Simulation run with initial excitation at κλD = 0.4 (arrow).
Fig. 5. Im(ω) vs. initial displacement from uniformity for mode 1. Run with 1,000,000 particles.
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Plasma Theory and Simulation Group (PTSG) / Computer Physics Communications 164 (2004) 189–194
Fig. 6. Warm plasma between two grounded plates at t = 1e–7.
Acknowledgement This work is partially supported by Center for Plasma Theory and Computation, Directed Energy Directorate, Air Force Research Laboratory, Kirtland Air Force Base, New Mexico 87117.
References [1] A.V. Haeff, The electron-wave tube, Proc. I.R.E. (1949) 4–10.
[2] J.D. Jackson, Longitudinal plasma oscillations, J. Nucl. Energy, Part C, Plasma Phys. 1 (1960) 5. [3] L. Landau, On the vibrations of the electronic plasma, J. Phys. (U.S.S.R.) 10 (1946) 25. [4] J. Malmberg, C. Wharton, Phys. Rev. Lett. 17 (1966) 172. [5] J.V. Parker, J.C. Nickel, R.W. Gould, Resonance oscillations in a hot nonuniform plasma, Phys. Fluids 7 (1964) 1489. [6] J.R. Pierce, W.B. Hebenstreit, A new type of high frequency amplifier, Bell System Tech. J. (1949) 33–51. [7] A. Wendt, Saturation characteristics of counterstreaming warm electrons, Memo. UCB/ERL, UC Berkeley, CA, 28 May, 1985.
Discoveries in plasmas while teaching simulation Plasma Theory and Simulation Group (PTSG) Charles K. (Ned) Birdsall ∗ , Edison T. Estacio EECS Department of Cory Hall, University of California, Berkeley, CA 94720-1770, USA Available online 21 July 2004
Abstract Once PC’s became ubiquitous, we have been using them for teaching plasma simulation, hands-on by instructors and by students. The transfer of skills from instructor to class has been very rapid (most desirable). However, occasionally some unanticipated results are observed with plausible explanations expected from the instructor (scary). Our examples are all onedimensional. First, we show the famous two-stream instability in a periodic model, starting either cold or warm, which does not (quite) Maxwellianize; why not? Second, we show Landau damping also in a periodic model, with what appears to be small (hence linear) excitation, but observe trapping in the wave frame; going to very small excitation the trapping diminishes and the damping rate approaches that from Landau linear theory. Lastly, we show a warm plasma bounded by two grounded metal planar walls, uniform in density at t = 0, bounded, one-dimensional. For t > 0 we observe spontaneous plasma frequency oscillations in the midplane, sheath formation at ion sound speed at both walls, trapping of electrons, and acceleration of the ions to the walls; however, we also observe an oscillatory axial current, and ‘staircasing’ of the number of electrons in time. Both can come only from some degree of asymmetry in the system. The frequency of the current is the series resonance between the sheath capacitance (almost no electrons, so vacuum) and the bulk plasma ‘inductance’ (as ωseries ωp ). 2004 Elsevier B.V. All rights reserved.
1. Two-opposing-cold-streams instability (Fig. 1) Two opposing electron streams, cold or warm (vt < 0.7vdrift) are unstable and grow in time, in a fixed ion background, in a periodic system. In a system where two streams pass each other one wavelength in one period of natural oscillation (of one of the streams), there can be exponential growth of an initial small perturbation. This instability is very old, observed in fluid (e.g., fiord flow over the ocean), * Corresponding author.
E-mail addresses: [email protected] (C.K. Birdsall), [email protected] (E.T. Estacio). 0010-4655/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cpc.2004.06.029
and with fast and slow electron streams (amplifier by Haeff [1]), or drifting electrons and non-drifting ions (theory by Pierce [6]). The field energy may grow as fast as exp(ωp t). With two opposing streams, as above, at stream velocity reversal time (t = 27 pictures), v(x) vs. x, ρ has many harmonics (new k’s), which broadens the velocity distribution, traps the streams into huge vortices (t = 126.4) with no drifts remaining, and stops the growth. Even after a long time, the overall velocity distribution (both cold and warm streams) is not Maxwellian, which might be expected from balance between drag and diffusion. Why not?
190
Plasma Theory and Simulation Group (PTSG) / Computer Physics Communications 164 (2004) 189–194
Fig. 1. Two-opposing-cold-streams instability.
The model has a Poisson equation solver, and E = −∇(Φ), plus a Newton–Lorentz equation of motion, mdv/dt = qE. There is no µmv (momentum exchange) term, or drag, and no explicit diffusion term; usually these terms balancing leads to a Maxwellian. Not here! With the length L used, the fastest growing mode is mode 4. While the two f (v)’s appear Maxwellian, the phase space still shows one vortex, very long lived, not (yet) at the end of interaction.
Indeed, at t = 30, mode 4 dominates, in the vx –x phase space and in the charge density, ρ. By time 63.6, the 4 phase space vortices have merged, leaving two vortices. By time 244.8, all vortices in phase space have become just one vortex, which is very long lived. The f (v, both streams) is shown in the middle panels, with the dip in the center at v = 0. As pointed out by Wendt [7], the warm streams [sufficient (vt /vo )] tend to saturate with a double peaked f (v), close to a Penrose distribution, which is stable.
2. Two-opposing-warm-streams instability (Fig. 2) The interaction of two warm streams is shown here. At t = 0, the ratio of thermal velocity to drift velocities vt /vo = 0.4. The dispersion relation shows that the m = 4 mode has the fastest growth rate.
3. Warm plasmas; linear Landau damping, nonlinear heating (Figs. 3–5) Warm plasmas exhibit simple harmonic oscillations at the plasma frequency, ωp , as known from theory
Plasma Theory and Simulation Group (PTSG) / Computer Physics Communications 164 (2004) 189–194
191
Fig. 2. Two-opposing-warm-streams instability.
and experiments by Langmuir and Tonks in the 1920s. Much later (1946), using linear kinetic theory, L. Landau [3] found the rate that these oscillations damp in time. Both detailed analysis and physical understanding may be found in [2]. Experimental proof came in the 1960s (e.g., by Malmberg and Wharton, [4], and others). Initial wave energy (electrostatic) is absorbed by the plasma, increasing the plasma’s kinetic energy (temperature). Below are particle-in-cell (PIC) simulations using Bruce Langdon’s XES1 1d3v periodic code. On the left are logs of the electrostatic energy vs. time, all oscillating at about 1.2ωp , and damping exponentially very close to the Landau/Jackson rate; the initial displacements are x1 = 0.1, 0.01, 0.001. On the right are the particle trajectories (‘traces’) vs. x, in the wave frame; the top one shows particle trapping,
nonlinear; the middle one shows smaller trapping; the bottom one has no visible trapping. There is a small frequency shift. A hundred thousand particles used in order to maintain the damping to a long time.
4. Warm plasma between two grounded plates at t = 1e–7 (Fig. 6) Elementary 1d3v bounded warm plasma observations, short circuit, undriven. Starting from an initial uniform plasma between shorted planes, we observe: (a) formation of sheaths; followed by (b) plasma frequency potential oscillations at the central density (symmetric mode); (c) series resonance observed in the current, due to resonance between the capacitive
192
Plasma Theory and Simulation Group (PTSG) / Computer Physics Communications 164 (2004) 189–194
Fig. 3. Left side: electrostatic energy vs. time, log scale, for 3 initial displacements of particles. Right side: vx –x phase space with particle tracing (for trajectories).
(almost vacuum) sheath and the inductive bulk, resonant at ωseries = ωp (center)[(2s/L)]1/2 , where s is the average sheath width and L is the diode width. This is an asymmetric mode (see [5]) and also is the cut-off frequency for asymmetric (m = 1) surface waves along the walls. (Omitted are the ion sound waves traversing the plasma and bouncing off the two sheaths/walls.)
5. Summary Teaching live and interactively can be both exciting and challenging, and have unanticipated surprises,
similar to the examples just given. An audience of theorists and experimentalists may be unaccustomed to observing what is occurring inside the plasma (as functions of space, velocity, frequency, wave number, more) as displayed with several dozen diagnostics. In order to improve communication in lecturing and teaching, we have found it advisable to show, say, 1 to 4 live diagnostics at a time, complemented by a large poster board showing, say, a dozen or two diagnostics stills. Of course, the best way is to have the audience running the same simulations, at their speed, and with their choice of diagnostics, on their computers. Go and do likewise!
Plasma Theory and Simulation Group (PTSG) / Computer Physics Communications 164 (2004) 189–194
193
Fig. 4. Dispersion relation for warm plasma from Jackson 1960; Re(ω) and Im(ω) versus κλD . Simulation run with initial excitation at κλD = 0.4 (arrow).
Fig. 5. Im(ω) vs. initial displacement from uniformity for mode 1. Run with 1,000,000 particles.
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Plasma Theory and Simulation Group (PTSG) / Computer Physics Communications 164 (2004) 189–194
Fig. 6. Warm plasma between two grounded plates at t = 1e–7.
Acknowledgement This work is partially supported by Center for Plasma Theory and Computation, Directed Energy Directorate, Air Force Research Laboratory, Kirtland Air Force Base, New Mexico 87117.
References [1] A.V. Haeff, The electron-wave tube, Proc. I.R.E. (1949) 4–10.
[2] J.D. Jackson, Longitudinal plasma oscillations, J. Nucl. Energy, Part C, Plasma Phys. 1 (1960) 5. [3] L. Landau, On the vibrations of the electronic plasma, J. Phys. (U.S.S.R.) 10 (1946) 25. [4] J. Malmberg, C. Wharton, Phys. Rev. Lett. 17 (1966) 172. [5] J.V. Parker, J.C. Nickel, R.W. Gould, Resonance oscillations in a hot nonuniform plasma, Phys. Fluids 7 (1964) 1489. [6] J.R. Pierce, W.B. Hebenstreit, A new type of high frequency amplifier, Bell System Tech. J. (1949) 33–51. [7] A. Wendt, Saturation characteristics of counterstreaming warm electrons, Memo. UCB/ERL, UC Berkeley, CA, 28 May, 1985.