The stability of astrophysical jets in the linear regime is investigated by presenting a methodology to find the growth rates of the various instabilities. We perturb a cylindrical axisymmetric steady jet, linearize the relativistic ideal magnetohydrodynamic (MHD) equations, and analyze the evolution of the eigenmodes of the perturbation by deriving the differential equations that need to be integrated, subject to the appropriate boundary conditions, in order to find the dispersion relation. We also apply the WKBJ approximation and, additionally, give analytical solutions in some subcases corresponding to unperturbed jets with constant bulk velocity along the symmetry axis.

A number of observations of astrophysical jets, at different scales, have shown that jets are often non-uniform outflows in their cross-section. Their structure is believed to play an important role in their overall stability. In this work, we combine analytical methods and numerical simulations to investigate the stability of non-uniform jets originating from active galactic nuclei. We adopt a standard 'spine and sheath' model, using a fast, light inner spine and a heavier, slower outer sheath. In the first part of this work, we conduct a linear stability analysis, finding the time-scales for the growth of the instabilities and the corresponding eigenfunctions. We focus on the nature of the physical processes that dominate and drive the destabilization of configurations. In the second part, we examine the evolution of the perturbed jets through relativistic 3D numerical simulations using the PLUTO code. Starting with the eigenfunctions found in the first part as initial conditions, we derive instability growth times and evolution which are in good agreement with the linear analysis.

Aims: We study the stability properties of relativistic magnetized astrophysical jets in the linear regime. We consider cylindrical cold jet configurations with constant Lorentz factor and constant density profiles across the jet. We are interested in probing the properties of the instabilities and identifying the physical quantities that affect the stability profile of the outflows. Methods: We conducted a linear stability analysis on the unperturbed outflow configurations we are interested in. We focus on the unstable branches, which can disrupt the initial outflow. We proceeded with a parametric study regarding the Lorentz factor, the ratio of the rest mass density of the jet to that of the environment, the magnetization, and the ratio of the poloidal component of the magnetic field to its toroidal counterpart measured on the boundary of the jet. We also consider two choices for the pressure of the environment, either thermal or magnetic, and check if this choice affects the results. Additionally, we applied a WKBJ method at the radius of the jet in order to study the local stability properties. Finally, we adapted the jet configuration in Cartesian geometry and compared the planar flow results with the results of the cylindrical counterpart. Results: While investigating the stability properties of the configurations, we observed the existence of a specific solution branch, which showcases the growth timescale of the instability comparable to the light crossing time of the jet radius. Our analysis focuses on this solution. All of the quantities considered for the parametric study affect the behavior of the mode while the magnetized environments seem to hinder its development compared to the hydrodynamic equivalent. Also, our analysis of the eigenfunctions of the system alongside the WKBJ results show that the mode develops in a very narrow layer near the boundary of the jet, establishing the notion of locality for the specific solution. The results indicate that the mode is a relativistic generalization of the Kelvin-Helmholtz instability. We compare this mode with the corresponding solution in Cartesian geometry and define the prerequisites for the Cartesian Kelvin-Helmholtz to successfully approximate the cylindrical counterpart. Conclusions: We identify the Kelvin-Helmholtz instability for a cold nonrotating relativistic jet carrying a helical magnetic field. Our parametric study reveals the important physical quantities that affect the stability profile of the outflow and their respective value ranges for which the instability is active. The Kelvin-Helmholtz mode and its stability properties are characterized by the locality of the solutions, the value of the angle between the magnetic field and the wavevector, the linear dependence between the mode's growth rate and the wavevector, and finally the stabilization of the mode for flows that are ultrafast magnetosonic. The cylindrical mode can be approximated successfully by the Cartesian Kelvin-Helmholtz instability whenever certain length scales are much larger than the jet radius.

Physics-informed neural networks (PINNs) are machine learning models that integrate data-based learning with partial differential equations (PDEs). In this work, for the first time we extend PINNs to model the numerically challenging case of astrophysical shock waves in the presence of a stellar gravitational field. Notably, PINNs suffer from competing losses during gradient descent that can lead to poor performance especially in physical setups involving multiple scales, which is the case for shocks in the gravitationally stratified solar atmosphere. We applied PINNs in three different setups ranging from modeling astrophysical shocks in cases with no or little data to data-intensive cases. Namely, we used PINNs (a) to determine the effective polytropic index controlling the heating mechanism of the space plasma within 1% error, (b) to quantitatively show that data assimilation is seamless in PINNs and small amounts of data can significantly increase the model's accuracy, and (c) to solve the forward time-dependent problem for different temporal horizons. We addressed the poor performance of PINNs through an effective normalization approach by reformulating the fluid dynamics PDE system to absorb the gravity-caused variability. This led to a huge improvement in the overall model performance with the density accuracy improving between 2 and 16 times. Finally, we present a detailed critique on the strengths and drawbacks of PINNs in tackling realistic physical problems in astrophysics and conclude that PINNs can be a powerful complimentary modeling approach to classical fluid dynamics solvers.