A steadystate, semianalytical model of energetic particle acceleration in radiojet shear flows due to cosmicray viscosity obtained by Webb et al. is generalized to take into account more general cosmicray boundary spectra. This involves solving a mixed Dirichlet–Von Neumann boundary value problem at the edge of the jet. The energetic particle distribution function
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Abstract f _{0}(r ,p ) at cylindrical radiusr from the jet axis (assumed to lie along thez axis) is given by convolving the particle momentum spectrum with the Green’s function ${f}_{0}(\infty ,p\prime )$ , which describes the monoenergetic spectrum solution in which $G(r,p;p\prime )$ as ${f}_{0}\to \delta (pp\prime )$r → ∞ . Previous work by Webb et al. studied only the Green’s function solution for . In this paper, we explore for the first time, solutions for more general and realistic forms for $G(r,p;p\prime )$ . The flow velocity ${f}_{0}(\infty ,p\prime )$ =u u (r ) _{z}is along the axis of the jet (thee z axis). is independent ofu z , andu (r ) is a monotonic decreasing function ofr . The scattering time in the shear flow region 0 < $\tau {(r,p)={\tau}_{0}(p/{p}_{0})}^{\alpha}$r <r _{2}, and , where $\tau {(r,p)={\tau}_{0}(p/{p}_{0})}^{\alpha}{(r/{r}_{2})}^{s}$s > 0 in the regionr >r _{2}is outside the jet. Other original aspects of the analysis are (i) the use of cosmic ray flow lines in (r ,p ) space to clarify the particle spatial transport and momentum changes and (ii) the determination of the probability distribution that particles observed at ( ${\psi}_{p}(r,p;p\prime )$r ,p ) originated fromr → ∞ with momentum . The acceleration of ultrahighenergy cosmic rays in active galactic nuclei jet sources is discussed. Leaky box models for electron acceleration are described. $p\prime $Free, publiclyaccessible full text available November 22, 2024 
Abstract Several generalizations of the wellknown fluid model of Braginskii (1965) are considered. We use the Landau collisional operator and the moment method of Grad. We focus on the 21moment model that is analogous to the Braginskii model, and we also consider a 22moment model. Both models are formulated for general multispecies plasmas with arbitrary masses and temperatures, where all of the fluid moments are described by their evolution equations. The 21moment model contains two “heat flux vectors” (third and fifthorder moments) and two “viscosity tensors” (second and fourthorder moments). The Braginskii model is then obtained as a particular case of a one ion–electron plasma with similar temperatures, with decoupled heat fluxes and viscosity tensors expressed in a quasistatic approximation. We provide all of the numerical values of the Braginskii model in a fully analytic form (together with the fourth and fifthorder moments). For multispecies plasmas, the model makes the calculation of the transport coefficients straightforward. Formulation in fluid moments (instead of Hermite moments) is also suitable for implementation into existing numerical codes. It is emphasized that it is the quasistatic approximation that makes some Braginskii coefficients divergent in a weakly collisional regime. Importantly, we show that the heat fluxes and viscosity tensors are coupled even in the linear approximation, and that the fully contracted (scalar) perturbations of the fourthorder moment, which are accounted for in the 22moment model, modify the energy exchange rates. We also provide several appendices, which can be useful as a guide for deriving the Braginskii model with the moment method of Grad.more » « less

In Part 2 of our guide to collisionless fluid models, we concentrate on Landau fluid closures. These closures were pioneered by Hammett and Perkins and allow for the rigorous incorporation of collisionless Landau damping into a fluid framework. It is Landau damping that sharply separates traditional fluid models and collisionless kinetic theory, and is the main reason why the usual fluid models do not converge to the kinetic description, even in the longwavelength lowfrequency limit. We start with a brief introduction to kinetic theory, where we discuss in detail the plasma dispersion function $Z(\unicode[STIX]{x1D701})$ , and the associated plasma response function $R(\unicode[STIX]{x1D701})=1+\unicode[STIX]{x1D701}Z(\unicode[STIX]{x1D701})=Z^{\prime }(\unicode[STIX]{x1D701})/2$ . We then consider a onedimensional (1D) (electrostatic) geometry and make a significant effort to map all possible Landau fluid closures that can be constructed at the fourthorder moment level. These closures for parallel moments have general validity from the largest astrophysical scales down to the Debye length, and we verify their validity by considering examples of the (proton and electron) Landau damping of the ionacoustic mode, and the electron Landau damping of the Langmuir mode. We proceed by considering 1D closures at higherorder moments than the fourth order, and as was concluded in Part 1, this is not possible without Landau fluid closures. We show that it is possible to reproduce linear Landau damping in the fluid framework to any desired precision, thus showing the convergence of the fluid and collisionless kinetic descriptions. We then consider a 3D (electromagnetic) geometry in the gyrotropic (longwavelength lowfrequency) limit and map all closures that are available at the fourthorder moment level. In appendix A, we provide comprehensive tables with Padé approximants of $R(\unicode[STIX]{x1D701})$ up to the eighthpole order, with many given in an analytic form.more » « less