Galina V. Ustyugova

The paper proposes a computational algorithm for the numerical simulation of two-dimensional magnetohydrodynamic (MHD) flows, using a symmetry analyzer as an element of the numerical method. The algorithm is based on a finite-volume Godunov-type scheme with an approximate solution of the Riemann problem for calculating flows. A polar mesh is used, but the momentum and magnetic induction equations are approximated in a Cartesian coordinate system. It is assumed that the flows are either predominantly homogeneous plane-symmetric, or predominantly axisymmetric with respect to the grid’s axis. To reconstruct vector variables in the computational cells, a symmetry analyzer is used, which makes it possible to locally classify the flow as one of the indicated types. Depending on the type of the vector field, the corresponding components are used for its reconstruction.

The paper proposes a symmetry analyzer as an element of the computational algorithm for the numerical integration of two-dimensional equations of ideal gas dynamics. A symmetry analyzer is an algorithm that allows using grid data to give a preference to a component (in the present work, Cartesian or polar) of a vector field for its reconstruction on the cell interfaces of a computational grid and subsequent calculation of fluxes of conservative variables. A computational algorithm is constructed using a polar-type computational grid and including a symmetry analyzer. The algorithm is easily transferred to the three-dimensional cylindrical type of computational grids.

We investigate the properties of magnetized stars in the propeller regime using axisymmetric numerical simulations. We modelled the propeller regime for stars with realistically large magnetospheres (5-7 stellar radii), so that our results could be applied to different types of magnetized stars, including Classical T Tauri stars, cataclysmic variables, and accreting millisecond pulsars. A wide range of propeller strengths has been studied, from very strong propellers (where the magnetosphere rotates much more rapidly than the inner disk) to very weak propellers (where the magnetosphere rotates only slightly faster than the inner disc. In both propellers, matter is accumulated at the inner disc for the majority of the time, while episodes of accretion onto the star and ejection into the wind are relatively brief. The efficiency of the propeller, which characterizes the part of inner disk matter flowing into the wind, strongly depends on the fastness parameter omega_s: propeller efficiency increases with omega_s. The properties of the winds are different in strong and weak propellers. In the strong propellers, matter is accelerated rapidly above the escape velocity and flows at a relatively small opening angle of 40-45 degrees. In the weak propellers, matter may flow faster or slower than the escape velocity and at a large opening angle of 60-70 degrees. A star-disk system loses energy and angular momentum. A part of the rotational energy of the star is ejected to the magnetically-dominated (Poynting flux) jet, which is only present in the strong propellers. The other part of the energy flows from the inner disk into a propeller-driven wind. A star spins down partly due to the flow of angular momentum from the star to the corona or to the Poynting flux jet along the open field lines, and partly due to the flow of angular momentum to the inner disk along the closed field lines.

We describe a Godunov-type magnetohydrodynamic (MHD) code based on the Miyoshi and Kusano (2005) solver which can be used to solve various astrophysical hydrodynamic and MHD problems. The energy equation is in the form of entropy conservation. The code has been implemented on several different coordinate systems: 2.5D axisymmetric cylindrical coordinates, 2D Cartesian coordinates, 2D plane polar coordinates, and fully 3D cylindrical coordinates. Viscosity and diffusivity are implemented in the code to control the accretion rate in the disk and the rate of penetration of the disk matter through the magnetic field lines. The code has been utilized for the numerical investigations of a number of different astrophysical problems, several examples of which are shown.

We present the first results of a global axisymmetric simulation of accretion onto rotating magnetized stars from a turbulent, MRI-driven disk. The angular momentum is transported outward by the magnetic stress of the turbulent flow with a rate corresponding to a Shakura-Sunyaev viscosity parameter alpha\approx 0.01-0.04. The result of the disk-magnetosphere interaction depends on the orientation of the poloidal field in the disk relative to that of the star at the disk-magnetosphere boundary. If fields have the same polarity, then the magnetic flux is accumulated at the boundary and blocks the accretion which leads to the accumulation of matter at the boundary. Subsequently, this matter accretes to the star in outburst before accumulating again. Hence, the cycling, `bursty' accretion is observed. If the disc and stellar fields have opposite polarity, then the field reconnection enhances the penetration of the disk matter towards the deeper field lines of the magnetosphere. However, the magnetic stress at the boundary is lower due to the field reconnection. This decreases the accretion rate and leads to smoother accretion at a lower rate. Test simulations show that in the case of higher accretion rate corresponding to alpha=0.05-0.1, accretion is bursty in cases of both polarities. On the other hand, at much lower accretion rates corresponding to alpha < 0.01, accretion is not bursty in any of these cases. We conclude that the episodic, bursty accretion is expected during periods of higher accretion rates in the disc, and in some cases it may alternate between bursty and smooth accretion, if the disk brings the poloidal field of alternating polarity. We find that a rotating, magnetically-dominated corona forms above and below the disk, and that it slowly expands outward, driven by the magnetic force.

We outline recent progress in understanding the accretion of plasma to rotating magnetized stars obtained from global axisymmetric (2D) and 3D magnetohydrodynamic (MHD) simulations in three main areas: (1.) Formation of jets from disk accretion onto rotating magnetized stars: From simulations where the viscosity and magnetic diffusivity within the disk are described by alpha models, we find long-lasting conical outflows/jets from the disk/magnetosphere boundary in both the case where the star is slowly rotating and where it is rapidly rotating (the “propeller regime”). Most of the mass flux in the outflows is in a hollow cone but inside this cone there is a low-density high-velocity magnetically dominated flow along the open polar field lines of the star. The outflows occur under conditions where the poloidal magnetic flux of the star is bunched up by the accretion disk near the disk/magnetosphere boundary. Recent simulations show that the conical outflows become well-collimated for axial distances of ≲ 20 times the inner disk radius. Exploratory 3D simulations show that conical winds are axisymmetric about the rotational axis (of the star and the disk), even when the dipole field of the star is significantly misaligned. (2.) Formation of intrinsically one-sided jets from disk accretion to rotating magnetized stars: There is strong observational evidence for an asymmetry between the approaching and receding jets from a number of young stars. We discuss the first MHD simulations of the formation asymmetric or one-sided jets arising from disk accretion to a rotating star with an asymmetric (dipole plus quadrupole) magnetic field. (3.) Global axisymmetric and 3D simulations of the magnetorotational instability (MRI) in disk accretion onto magnetized stars: In the axisymmetric simulations we observe cases where there is episodic or quasi-periodic burst of accretion similar to that observed in one X ray source. In 3D MHD simulations of accretion onto stars with tilted dipole fields using our Godunov-type code based on the “cubed sphere” grid we find that the density distribution is much less smooth than in the case of the laminar accretion flow described by α–viscosity. Instead, large turbulent cells dominate the flows and are strongly elongated in the azimuthal direction.

We investigate the launching of outflows from the disk-magnetosphere boundary of slowly and rapidly rotating magnetized stars using axisymmetric and exploratory 3D magnetohydrodynamic (MHD) simulations. We find long-lasting outflows in both cases. (1) In the case of slowly rotating stars, a new type of outflow, a conical wind, is found and studied in simulations. The conical winds appear in cases where the magnetic flux of the star is bunched up by the disk into an X-type configuration. The winds have the shape of a thin conical shell with a half-opening angle 30-40 degrees. The conical winds may be responsible for episodic as well as long-lasting outflows in different types of stars. (2) In the case of rapidly rotating stars (the "propeller regime"), a two-component outflow is observed. One component is similar to the conical winds. A significant fraction of the disk matter may be ejected into the winds. A second component is a high-velocity, low-density magnetically dominated axial jet where matter flows along the opened polar field lines of the star. The jet has a mass flux about 10% that of the conical wind, but its energy flux (dominantly magnetic) can be larger than the energy flux of the conical wind. The jet's angular momentum flux (also dominantly magnetic) causes the star to spin-down rapidly. Propeller-driven outflows may be responsible for the jets in protostars and for their rapid spin-down. The jet is collimated by the magnetic force while the conical winds are only weakly collimated in the simulation region.

A new type of wind - a conical wind - has been discovered in axisymmetric magnetohydrodynamic simulations of the disk-magnetosphere interaction in cases where the magnetic field of the star is bunched into an X-type configuration. Such a configuration arises if the effective viscosity of the disk is larger than the effective diffusivity, or if the accretion rate in the disk is enhanced. Conical outflows flow from the inner edge of the disk into a narrow shell with half-opening angle of 30-45 degrees. The outflow carries about 0.1-0.3 of the disk mass accretion rate and part of the disk's angular momentum. The conical winds are driven by the gradient of the magnetic pressure which exists above the disk due to the winding of the stellar magnetic field. Exploratory 3D simulations show that conical winds are symmetric about rotation axis of the disk even if the magnetic dipole is significantly misaligned with the disk's rotation axis. Conical winds appear around stars of different periods. However, in the case of a star in the "propeller" regime, an additional - much faster component appears: an axial jet, where matter is accelerated up to very high velocities at small distances from the star by magnetic pressure force above the surface of the star. The simulations are done in dimensionless units and are applicable to a variety of the disk-accreting magnetized stars: young stars, white dwarfs, neutron stars, and possibly black holes. For the case of young stars, conical winds and axial jets may appear in different cases, including Class I young stars, classical T Tauri stars, and EXors. In EXors periods of enhanced accretion may lead to the formation of conical winds which correspond to the outflows observed from these stars.

We investigate outflows from the disk-magnetosphere boundary of rotating magnetized stars in cases where the magnetic field of a star is bunched into an X-type configuration using axisymmetric and full 3D MHD simulations. Such configuration appears if viscosity in the disk is larger than diffusivity, or if the accretion rate in the disk is enhanced. Conical outflows flow from the inner edge of the disk to a narrow shell with an opening angle 30-45 degrees. Outflows carry 0.1-0.3 of the disk mass and part of the disk's angular momentum outward. Conical outflows appear around stars of different periods, however in case of stars in the "propeller" regime, an additional - much faster component appears: an axial jet, where matter is accelerated up to very high velocities at small distances from the star by magnetic pressure force above the surface of the star. Exploratory 3D simulations show that conical outflows are symmetric about rotational axis of the disk even if magnetic dipole is significantly misaligned. Conical outflows and axial jets may appear in different types of young stars including Class I young stars, classical T Tauri stars, and EXors.