A minimalist approach to the linear stability problem in fluid dynamics is developed that ensures efficiency by utilizing only the essential elements required to find the eigenvalues for given boundary conditions. It is shown that the problem is equivalent to a single first-order ordinary differential equation, and that studying the argument of the unknown complex function in the eigenvalue space is sufficient to find the dispersion relation. The method is applied to a model for relativistic magnetized astrophysical jets.

Theoretical arguments as well as observations of young stellar objects (YSO) support the presence of a diversified circumstellar environment. A stellar jet is thought to account for most of the stellar spin down and disk wind outflow for the observed high mass loss rate, thus playing a major role in the launching of powerful jets. RY Tau, for instance, is an extensively studied intermediate mass pre-main sequence star. Observational data reveal a small scale jet called microjet. Nevertheless, it is not clear how the microjet shapes the jet observed at a large scale. The goal is to investigate the spatial stability and structure of the central jet at a large scale by mixing the stellar and disk components. We mix two existing analytical self-similar models for the disk and the stellar winds to build the initial set-ups. Instead of using a polytropic equation of state, we map from the analytical solutions, the heating and cooling sources. The heating exchange rate is controlled by two parameters, its spatial extent and its intensity. The central jet and the surrounding disk are strongly affected by these two parameters. We separate the results in three categories, which show different emissivity, temperature, and velocity maps. We reached this categorization by looking at the opening angle of the stellar solution. For cylindrically, well collimated jets, we have opening angles as low as 10 degrees between 8 and 10 au, and for the wider jets, we can reach 30 degrees with a morphology closer to radial solar winds. Our parametric study shows that the less heated the outflow is, the more collimated it appears. We also show that recollimation shocks appear consistently with UV observations in terms of temperature but not density.

Context. Relativistic shocks are present in all high-energy astrophysical processes involving relativistic plasma outflows interacting with their ambient medium. While they are well understood in the context of relativistic hydrodynamics and ideal magnetohydrodynamics (MHD), there is a limited understanding of the properties related to their propagation in media characterized by finite electrical conductivity. Aims: This work presents a systematic method for the derivation and solution of the jump conditions for relativistic shocks propagating in MHD media with finite electrical conductivity. This method is applied to the numerical solution of the Riemann problem and the determination of the conditions inside the blastwave that is formed when ultrarelativistic magnetized ejecta interact with the circumburst medium during a gamma-ray burst. Methods: We derived the covariant relations expressing the jump conditions in a frame-independent manner. The resulting algebraic equations expressing the Rankine-Hugoniot conditions in the propagation medium's frame were then solved numerically. A variable adiabatic index equation of state was used in order to obtain a realistic description of the post-shock fluid's thermodynamics. This method was then employed for the solution of the Riemann problem for the case of a forward and a reverse shock, both of which form during the interaction of a gamma-ray burst ejecta with the circumburst medium. This allowed us to determine the kinematics of the resulting blastwave and the dynamical conditions in its interior. Results: Our solutions clearly depict the impact of the plasma's electrical conductivity in the properties of the post-shock medium. Two characteristic regimes are identified with respect to the value of a dimensionless parameter that has a linear dependence on the conductivity. For small values of this parameter, the shock affects only the hydrodynamic properties of the propagation medium and leaves its electromagnetic field unaffected. No current layer forms in the shock front; thus, we refer to this as the current-free regime. For large values of this parameter, the ideal MHD regime has been retrieved. We also show that the assumption of a finite electrical conductivity can lead to higher efficiencies in the conversion of the ejecta energy into thermal energy of the blastwave through the reverse shock. The theory developed in this work can be applied to the construction of Riemann solvers for resistive relativistic MHD (RR4MHD).