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Closure models for the feedback of energetic particles on plasma turbulence

J. Pratt

TL;DR

This work addresses the challenge of modeling how energetic particles feedback on plasma turbulence, a key process in cosmic-ray acceleration, by advocating a PDF-closure framework built on Multiple Mapping Conditioning (MMC) and sparse-Lagrangian methods. It articulates how to partition species into major and minor groups, employ mapping closures, and couple stochastic particle dynamics to a fluid solver to achieve efficient, self-consistent two-way coupling. The approach extends PDF methods from reactive-flow turbulence to astrophysical plasmas and introduces a cosmic-ray adaptation that uses on-the-fly diffusion and energetic-species binning to capture spectral features like the knee and ankle. The proposed framework promises higher fidelity and scalability over traditional hybrid fluid/Fokker–Planck methods, enabling large-scale simulations of shock-turbulence interactions and cosmic-ray feedback in environments such as the heliosphere, supernova remnants, and galactic dynamos.

Abstract

Energetic particles interact with the plasma surrounding them, resonating with certain plasma waves to stabilize them while destabilizing others, and changing the character of the background turbulence in ways that have not been fully quantified or understood. Interaction with the turbulent background plasma is key to the acceleration of many types of energetic particles including high-energy cosmic rays, solar energetic particles, and pick-up ions. This is a process that would ideally be described by a kinetic model, a type of model that follows a probability distribution function (PDF) for all particles in 7-dimensional space. Because of the high dimensionality of a kinetic model, such simulations use the largest computational resources available, and are yet unable to simulate a realistic number of particles, reach the large scales necessary for astrophysical problems, or use high-precision numerical methods. Two available alternatives to kinetic plasma models have been explored: a multi-fluid model, and a hybrid fluid/Fokker-Planck model. These methods are hampered by the physical modeling of the coupling. We develop a new model, which follows the PDF for all particles; this can be viewed as a step toward physical realism above a multi-fluid MHD model, while also being more computationally efficient than a kinetic model. The equations we develop model both the background plasma and the energetic particles self-consistently. Over the last decade, similar PDF methods have been developed to a high level of sophistication to model reactive flows and turbulent combustion for engineering applications. For treatment of the feedback of the energetic particles on a background plasma, a PDF closure approach should evaluate the mean characteristics, including the density, with better statistical quality than will particle-sampling procedures.

Closure models for the feedback of energetic particles on plasma turbulence

TL;DR

This work addresses the challenge of modeling how energetic particles feedback on plasma turbulence, a key process in cosmic-ray acceleration, by advocating a PDF-closure framework built on Multiple Mapping Conditioning (MMC) and sparse-Lagrangian methods. It articulates how to partition species into major and minor groups, employ mapping closures, and couple stochastic particle dynamics to a fluid solver to achieve efficient, self-consistent two-way coupling. The approach extends PDF methods from reactive-flow turbulence to astrophysical plasmas and introduces a cosmic-ray adaptation that uses on-the-fly diffusion and energetic-species binning to capture spectral features like the knee and ankle. The proposed framework promises higher fidelity and scalability over traditional hybrid fluid/Fokker–Planck methods, enabling large-scale simulations of shock-turbulence interactions and cosmic-ray feedback in environments such as the heliosphere, supernova remnants, and galactic dynamos.

Abstract

Energetic particles interact with the plasma surrounding them, resonating with certain plasma waves to stabilize them while destabilizing others, and changing the character of the background turbulence in ways that have not been fully quantified or understood. Interaction with the turbulent background plasma is key to the acceleration of many types of energetic particles including high-energy cosmic rays, solar energetic particles, and pick-up ions. This is a process that would ideally be described by a kinetic model, a type of model that follows a probability distribution function (PDF) for all particles in 7-dimensional space. Because of the high dimensionality of a kinetic model, such simulations use the largest computational resources available, and are yet unable to simulate a realistic number of particles, reach the large scales necessary for astrophysical problems, or use high-precision numerical methods. Two available alternatives to kinetic plasma models have been explored: a multi-fluid model, and a hybrid fluid/Fokker-Planck model. These methods are hampered by the physical modeling of the coupling. We develop a new model, which follows the PDF for all particles; this can be viewed as a step toward physical realism above a multi-fluid MHD model, while also being more computationally efficient than a kinetic model. The equations we develop model both the background plasma and the energetic particles self-consistently. Over the last decade, similar PDF methods have been developed to a high level of sophistication to model reactive flows and turbulent combustion for engineering applications. For treatment of the feedback of the energetic particles on a background plasma, a PDF closure approach should evaluate the mean characteristics, including the density, with better statistical quality than will particle-sampling procedures.
Paper Structure (9 sections, 11 equations, 2 figures, 1 table)

This paper contains 9 sections, 11 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Existing models for energetic particle acceleration
  3. The benefits of PDF closures
  4. PDF methods and sparse Lagrangian models for reactive flows
  5. Major and minor reacting species
  6. Mapping closures and Mixture Models
  7. Simulation strategy for the sparse-Lagrangian MMC
  8. A new model for cosmic ray interaction with a background plasma
  9. Summary

Figures (2)

  • Figure 1.1: An illustration of the diffusive shock acceleration process by which particles gain large energies. Tennis players represent the turbulence and instabilities, generated in different ways. They repeatedly reflect particles (tennis balls) across the shock (tennis net). Tennis players are drawn in the style of the xkcd webcomic (https://xkcd.com/).
  • Figure 5.1: Possible bins in kinetic energy that could define energetic species, drawn on the cosmic ray spectrum, figure from the review of ruszkowski2023cosmic, original measurements from a variety of instruments lenok2021measurementstone2019cosmic.