{'fraction_non_alphanumeric': 0.0361687876758205, 'fraction_numerical': 0.02478231748158071, 'mean_word_length': 5.821917808219178, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 2, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
On the Quasi‐linear Saturation of the Parallel Proton Firehose Instability Using a Full‐f Approach

A classic example for the application of quasi‐linear theory to electromagnetic wave‐particle interactions, the saturation of the parallel proton firehose instability, is usually considered in the long‐wavelength approximation although for β∥,p≲25 this instability is dominated by anomalous cyclotron resonance, which invalidates a macroscopic treatment (Gary et al., 1998, https://doi.org/10.1029/98JA01174). To relax the long‐wavelength approximation, Seough et al. (2015, https://doi.org/10.1063/1.4905230) solved the microscopic weak turbulence kinetic equation to model the temperature anisotropy reduction of the firehose also in the resonant regime. However, the employed moment‐kinetic approach assumes the preservation of the initially bi‐Maxwellian shape of the underlying proton velocity distribution throughout the saturation process, leading to poor results for low β∥,p. In this work, we lift the limitations of the moment‐kinetic approach and we demonstrate that allowing for distribution deformation due to anomalous cyclotron‐resonant scattering greatly improves the predictions of the kinetic quasi‐linear model except for cases of very strong firehose growth. We conclude that quasi‐linear theory can be a valid model for studying the parallel firehose saturation even in the strongly cyclotron‐resonant regime as long as the initial temperature anisotropy is not too large.