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Stabilization of beam heated plasmas by beam modulation

Lukas Einkemmer

arXiv:2408.16888v3physics.plasm-ph

TL;DR

A beam injected into a stationary plasma can create a bump-on-tail instability, while feedback control may be difficult on the instability’s fast time scales. This paper numerically optimizes modulated beam profiles and reports substantial suppression, including stable behavior after the beam is switched off.

  • Problem

    Beam heating can create bump-on-tail features that drive instability, and feedback control may be infeasible for fast microinstabilities.

    Source: §I · p. 1

  • Method

    The authors numerically optimize time-varying beam-intensity profiles to reduce the instability without feedback, under a fixed total-intensity constraint.

    Source: §II · p. 2, §IV · p. 5, §VII · p. 8

  • Results

    Time-averaged electric energy, a measure of instability severity, is reduced by at least 75% in all studied configurations compared with its original value.

    Source: §VII · p. 8

  • Takeaways & Limitations

    The optimized control requires no feedback, tolerates beam-profile perturbations, and can leave the system stable after switch-off; its required modulation frequency scales roughly inversely with system size.

    Source: §VII · p. 8

Abstract

from arXiv · show

A constant intensity beam that propagates into a stationary plasma results in a bump-on-tail feature in velocity space. This results in an instability that transfers kinetic energy from the plasma to the electric field. We show that there are intensity profiles for the beam (found by numerical optimization) that can largely suppress this instability and drive the system into a state that, after the beam has been switched off, remains stable over long times. The modulated beam intensity requires no feedback, i.e. no knowledge of the physical system during time evolution is required, and the frequency of the modulation scales approximately inversely with system size, which is particularly favorable for large plasma systems. We also show that the results obtained are robust in the sense that perturbations, e.g. deviation from the optimized beam profiles, can be tolerated without losing the ability to suppress the instability.

I. INTRODUCTION

Beam heating can create bump-on-tail features that drive plasma instabilities, while feedback control may be too slow for their fast dynamics. This paper asks whether beam modulation can suppress the instability without feedback and seeks profiles more effective than sinusoidal modulation.

I. INTRODUCTION

  • Beam-heating bump-on-tail features can destabilize Alfvén waves and potentially degrade reactor performance.The instability transfers energy from fast particles to the electric field as the field grows toward saturation.Source: §I · p. 1
  • Feedback control may be infeasible because microinstabilities evolve on fast time scales and non-Maxwellian velocity features are often poorly measured experimentally.Data acquisition, processing, and control changes would need to keep pace with plasma dynamics.Source: §I · p. 1
  • Prior work found that fixed-frequency beam-intensity modulation near the system’s natural oscillation frequency reduces observed electric-field strength [11].Related studies examined nonlinear mode coupling [12] and sinusoidal modulation of beam velocity [13].Source: §I · p. 1
  • Earlier electric-field control studies indicated that instability suppression without feedback is possible when the initial state is known, but arbitrary field injection is difficult in practice [15, 16].The required precise knowledge of the initial deviation motivates examining the less direct control of beam modulation.Source: §I · p. 1
  • The paper studies whether beam modulation can reduce or suppress the instability without feedback and uses numerical optimization to seek profiles more effective than sinusoidal modulation.It also asks how fast the modulation must be.Source: §I · pp. 1–2

II. MODEL

The model evolves a beam-driven, initially homogeneous Maxwellian plasma using a one-dimensional non-dimensionalized Vlasov–Poisson description. A spatially Gaussian, velocity-Maxwellian beam source is modulated in time, with selected beam parameters constrained while the beam profile is optimized.

II. MODEL

  • The electron dynamics are modeled with a one-dimensional non-dimensionalized Vlasov–Poisson equation, in which the beam enters through a source term.The model treats ions as a homogeneous neutralizing background and obtains the electric field from the Poisson equation.Source: §II · p. 2
  • The beam source has a Gaussian spatial profile and a Maxwellian velocity profile, with its amplitude I(t) available for modulation.Its spatial extent is σ, its average velocity is vb, and its thermal velocity is vth,b.Source: §II · p. 2
  • Above a threshold intensity, the added velocity-distribution peak is unstable; the kinetic description identifies this through the Penrose criterion.The paper notes that the dispersion relation can be solved within this fully kinetic description.Source: §II · p. 2
  • Unlike static analyses that begin with a bump-on-tail feature, this setup starts from a current-free, spatially homogeneous Maxwellian and lets the beam dynamically create the feature and trigger instability.The described analysis concerns this evolving, beam-driven situation.Source: §II · p. 2
  • The model restricts vth,b to 0.2–0.6, fixes vb at 3.5, and constrains total beam intensity to I0tb with I0 = 0.05 while optimizing I(t), σ, and vth,b.Fixing vb holds the total energy added to the system constant.Source: §II · p. 2

III. RESULTS FOR SINUSOIDAL BEAM PROFILES

Sinusoidal beam modulation reduces bump-on-tail instability severity, with appropriately chosen frequencies scaling roughly inversely with system size. It can leave a stable velocity plateau with low electric energy after the beam is switched off.

III. RESULTS FOR SINUSOIDAL BEAM PROFILES

  • The figure of merit integrates total electric energy over a specified interval and normalizes results to the unmodulated case, balancing saturation strength against post-shutoff electric energy.The integration interval is tf = 150 for the small system and tf = 2tb for the other configurations.Source: §III · pp. 2–3
  • With an appropriately selected frequency and amplitude, a sinusoidal beam reduces instability severity by 56–78%; the reduction is not merely transient and persists after the beam is switched off.Source: §III · p. 3
  • The modulation produces a velocity plateau roughly two to five times the thermal velocity, without a bump, and the resulting distribution is stable by the Penrose criterion.The unmodulated case also forms a plateau, but with larger density variations and electric energy; modulation produces smaller density variations.Source: §III · p. 3, Figure 3 · p. 4
  • Sinusoidal modulation causes parts of the electrostatic wave to merge early, breaking up the overall structure formed during classic bump-on-tail saturation.In the classic instability, a saturated nonlinear wave traps particles and can remain stable over long times.Source: §III · p. 3
  • The sinusoidal beam’s driving frequency can be chosen roughly inversely proportional to system size, rather than near the plasma frequency.The discussion links the frequency vb/L to exciting the largest spatial wavelength; non-multiples can excite multiple modes.Source: §III · pp. 4–5

IV. RESULTS FOR GENERAL BEAM PROFILES

Numerical optimization of general beam profiles yields reductions in instability severity beyond sinusoidal modulation. The optimized profiles also show strong post-shutoff behavior, with required frequency roughly inversely proportional to system size.

IV. RESULTS FOR GENERAL BEAM PROFILES

  • Optimization varies beam intensity, width, and temperature to reduce the figure of merit, with nonnegative intensity represented by a few Fourier modes under a fixed-total-intensity constraint.The rough landscape has many local minima, motivating a global search.Source: §IV · pp. 5–6
  • For the two larger system sizes, optimized profiles reduce instability severity by approximately 75–80%, about a factor of 2 beyond sinusoidal modulation; the smallest system shows a decrease exceeding 99%.The optimized profiles also show excellent long-time behavior after the beam is switched off.Source: §IV · p. 6, Figure 7 · p. 7
  • The frequency required for suppression is roughly inversely proportional to system size.Figure 7 compares optimized profiles with the optimal sinusoidal profiles and displays the figure of merit for each candidate.Source: §IV · p. 6, Figure 7 · p. 7
  • Beam width has a relatively minor effect: even the large-system optimized profile with σ ≈ 6.26 deteriorates only slightly in figure of merit.The optimization constrained σ to at least 0.2, 0.8, and 3.2 for the small, medium, and large systems, respectively.Source: §IV · p. 6
  • The optimized solution develops a velocity-distribution plateau with almost no bump-on-tail feature, and optimized profiles merge electrostatic-wave parts more effectively.The phase-space plots show the evolution for several beam profiles at medium and large system sizes.Source: §IV · pp. 6, 8, Figure 5 · p. 6, Figure 6 · p. 7

V. ROBUSTNESS OF THE BEAM PROFILES TO PERTURBATION

The optimized beam profiles were generally robust to random perturbations of their parameters, including perturbations up to 5%, with one profile deteriorating relatively quickly.

V. ROBUSTNESS OF THE BEAM PROFILES TO PERTURBATION

  • Perturbations of up to 5% in all parameters left the figure of merit remarkably robust, except for the optimized 1 profile at intermediate system size.That profile deteriorated relatively quickly to the sinusoidal beam profile’s level.Source: §V · p. 8
  • With one exception, profiles optimized only for figure of merit proved robust even though the optimization did not account for robustness.Robustness could also be included by penalizing profiles that are not robust.Source: §V · p. 8
  • The study perturbed Table I parameters using uniformly distributed random multipliers in [1 − ε, 1 + ε]; the figure of merit remained remarkably robust through 5% perturbations in all parameters.The exception was the optimized 1 beam profile for the intermediate system size, which relatively quickly deteriorated to the sinusoidal beam profile’s level.Source: §V · p. 8

VI. METHODOLOGY

The study uses global genetic optimization to search beam-profile parameters, with batched simulations to reduce solver overhead and higher-resolution runs to check numerical independence.

VI. METHODOLOGY

  • A modified differential-evolution genetic algorithm searches globally for beam-profile parameters to avoid local minima.Candidate solutions are parameter sets that determine the beam profiles, and their figure of merit serves as fitness.Source: §VI · p. 8
  • Batched C++ Vlasov–Poisson simulations evaluate candidate profiles together, reducing the overhead of individual solver runs and using Kokkos for GPU-capable computing.The algorithm evaluates all candidate solutions in a generation at the same time.Source: §VI · p. 8
  • Optimization uses a splitting-based semi-Lagrangian scheme with ninth-order Lagrange interpolation, while plotted results use increased resolution to check numerical independence.Optimization runs use 512 spatial and 256 velocity grid points with Δt = 0.02; plotted-data runs use 1024 spatial and 512 velocity points.Source: §VI · p. 8

VII. DISCUSSION

Optimized beam modulation reduces the instability across the studied configurations and offers a feedback-free control scheme with favorable scaling and robustness. The discussion also identifies limits in the modeled heating process and plasma model.

VII. DISCUSSION

  • At least 75% lower time-averaged electric energy was achieved in every studied configuration, relative to its original value.Time-averaged electric energy measures the severity of the instability.Source: §VII · p. 8
  • The control requires no feedback, and its modulation acts on the evolving bump-on-tail feature rather than an already established one.The authors regard an already established bump-on-tail feature as more a modeling artifact than a reasonable beam-heating setup.Source: §VII · pp. 8–9
  • The required modulation frequency scales favorably with system size, making actuation favorable for large plasma systems.The modulation frequency determines how fast beam intensity must be actuated.Source: §VII · p. 8
  • The system can reach a state that remains stable after beam switch-off, with a velocity-space plateau and almost uniform spatial density.The state implies little kinetic energy transfer from the beam to the field.Source: §VII · p. 8
  • Beam-profile imperfections do not significantly diminish control performance; Figure 8 plots figure of merit against relative perturbation size using 20 perturbations and normalization to the unmodulated beam.The plotted average and standard deviation are shown for each perturbation size.Source: §VII · p. 8, Figure 8 · p. 9
  • Beam width has only a small influence on control effectiveness for widths up to 3% of the domain size.The scheme therefore does not require tight beam focusing.Source: §VII · p. 8
  • The study does not model collisional thermalization of the velocity distribution, leaving actual heating for future work.The authors infer that the post-beam state should remain stable, while noting that some spatial variation remains in simulations.Source: §VII · p. 9
  • The results use a one-dimensional model with stationary ions, while more realistic applications require more complex models and many optimization simulations.Such optimization may require complexity-reduction techniques, reduced models, or large-scale supercomputers.Source: §VII · p. 9
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