Cosmic ray transport in partially turbulent space plasmas with compressible magnetic turbulence

Recently a new transport theory of cosmic rays in magnetized space plasmas extending the quasilinear approximation to the particle orbit has been developed for the case of an axisymmetric incompressible magnetic turbulence. Here we generalize the approach to the important physical case of a compressible plasma. As previously obtained in the case of an incompressible plasma we allow arbitrary gyrophase deviations from the unperturbed spiral orbits in the uniform magnetic field. For the case of quasi-stationary and spatially homogeneous magnetic turbulence we derive in the small Larmor radius approximation gyro-phase averaged cosmic ray Fokker-Planck coefficients. Upper limits for the perpendicular and pitch-angle Fokker-Planck coefficients and for the perpendicular and parallel spatial diffusion coefficients are presented.

💡 Research Summary

The paper presents a comprehensive extension of a recently developed cosmic‑ray transport theory to the physically important case of compressible magnetic turbulence. The original framework (Schlickeiser 2011, “Paper 1”) was limited to axisymmetric, incompressible turbulence (δBz = 0) and assumed that particle orbits deviated only infinitesimally from the unperturbed helical motion in a uniform guide field. Here the authors relax both restrictions: (i) they allow a finite parallel magnetic‑field fluctuation δBz, thereby incorporating compressible modes that are ubiquitous in space plasmas such as the solar wind, interstellar medium, and galaxy clusters; (ii) they admit arbitrary gyrophase deviations δφ(t) from the simple Ωt rotation, treating the gyrophase as a stochastic variable independent of the turbulent fields (Corrsin‑type independence hypothesis).

Starting from the relativistic equations of motion (Eqs. 1‑4) the authors define random forces hμ(t) and hi(t) that act on the pitch‑angle cosine μ = vz/v and on the guiding‑center coordinates Xi. Under the weak‑turbulence (quasi‑linear) approximation they retain only first‑order terms in the fluctuating fields and neglect electric fields. The ensemble‑averaged Vlasov equation then reduces to a Fokker‑Planck equation (Eq. 11) where the diffusion tensor Pαβ is expressed as a time‑integral of the two‑point correlation of the random forces (Eq. 12).

A key technical advance is the inclusion of a generalized particle orbit (Eq. 13‑15) that contains an arbitrary gyrophase correction δφ(t−s). This leads to a phase factor G(ξ)=Ω ξ+δφ(ξ) which appears in all subsequent integrals. By Fourier transforming the stochastic forces and assuming quasi‑stationary, spatially homogeneous turbulence, the authors obtain compact expressions for the gyro‑averaged diffusion coefficients (Eqs. 31‑43). The coefficients involve the turbulence correlation tensor Pij(k,τ), Bessel functions Jn(Z) (with Z∝k⊥v√(1−μ²) and an integral over G), and exponential factors exp(i v μ k∥ τ). The formalism is fully general; however, to make progress they specialize to axisymmetric turbulence (Pij independent of the wave‑phase ψ) and perform the ψ‑integration, yielding simplified forms (Eqs. 47‑50).

In the axisymmetric case the perpendicular diffusion coefficients Dij are proportional to an integral over k∥ and k⊥ of J0(Z) multiplied by a combination of μ²Pij and a term proportional to (1−μ²)cos G(τ) Pzz. The mixed coefficients Diμ and Dμi contain sin G(τ) and cos G(τ) multiplied by the cross‑components Pz1, Pz2, etc. The pitch‑angle diffusion coefficient Dμμ (Eq. 50) is expressed in terms of left‑ and right‑handed polarized spectra PL L and PRR, showing explicitly how compressible fluctuations (δBz) contribute through the G(τ) phase. The authors further rewrite Dμμ using the helicity decomposition (Eqs. 51‑53), which makes clear that the presence of compressible modes introduces additional resonant‑like terms e^{±iG(τ)}.

A crucial physical insight emerges from the analysis of the τ‑integral limits. Assuming a finite decorrelation time tc, the upper limit can be extended to infinity, and the diffusion coefficients become time‑independent for t≫tc. This yields well‑defined upper bounds for the perpendicular and parallel spatial diffusion coefficients (κ⊥, κ∥) derived from the Fokker‑Planck tensor. The paper also discusses the competition between mirror forces (N0) and turbulent scattering (R0). In compressible plasmas, mirror forces associated with δBz can trap particles, but if the turbulence decorrelates rapidly (small tc) the stochastic scattering dominates, leading to efficient diffusion.

Overall, the work delivers a mathematically rigorous, physically transparent framework for cosmic‑ray transport in compressible, axisymmetric turbulence. It provides explicit formulas for all relevant diffusion coefficients, clarifies the role of gyrophase randomness, and establishes upper limits that can be directly compared with observations or numerical simulations. The results are immediately applicable to modeling cosmic‑ray propagation in the heliosphere, interstellar medium, and extragalactic environments, where compressible magnetic fluctuations are non‑negligible.