1 Overview

particle_tracerET tracks the motion of test particles that comove with fluid flows. As magnetic field lines remain attached to the fluid elements they thread in the MHD approximation, this thorn provides essential functionality for self-consistent visualizations of the dynamics of magnetic field lines (i.e., magnetic field vector streamlines) over time in a GRMHD simulation.

Magnetic field lines can be visualized by first picking a fluid element (i.e., a point in space at which the fluid density and magnetic fields are nonzero), and then generating the vector field streamline according to the magnetic field vector.

The instantaneous velocity of these test particles is given by the velocity in the induction equation \(v^{i} = u^{i}/u^{0}\) at the location of each test particle, so this thorn tracks particle dynamics by solving the ordinary differential equations (ODEs): \begin {equation} \frac {dx}{dt} = v^{x}\;,\quad \frac {dy}{dt} = v^{y}\;,\quad \frac {dz}{dt} = v^{z}\;, \end {equation} subject to the initial conditions \begin {equation} x(0) = x_{0}\;,\quad y(0) = y_{0}\;,\quad z(0) = z_{0}\;. \end {equation} The above initial value problem is solved for each particle using an RK4 ODE integration, interpolating the velocity vector at each particle position, at each RK4 substep. The frequency of RK4 substeps generally can be far lower than that of the global simulation RK substepping, and is set to be every “update_RK4_freq” cctk_iteration’s.

The velocities \(v^{i} = u^{i}/u^{0}\) are computed using as inputs the HydroBase velocity and ADMBase spacetime metric variables. particle_tracerET stores \(v^{i}\) to MHDvx, MHDvy, and MHDvz.

Note that the velocities above differ from the HydroBase velocities, the so-called Valencia three-velocity, \(v^{i}_{(n)}\). The relationship between \(v^{i}\) and \(v^{i}_{(n)}\) is given by (see e.g., [1]) \begin {equation} v^{i} = \alpha v^{i}_{(n)} - \beta ^{i}\;. \end {equation}

Initially, particles are seeded inside a sphere of radius \(R\) located at position \((X,Y,Z)\). The parameters that correspond to \(R\), \(X\), \(Y\), and \(Z\) are given in Table 1.

Once the position of the particle is known, it is accepted by the algorithm with probability \begin {equation} P = \left [\frac {\rho (\vec {x}_{\rm p})}{\rho _{\rm max}}\right ]^{C}\;, \end {equation} where \(\rho (\vec {x}_{\rm p})\) is the density interpolated at the particle position \(\vec {x}_{\rm p}\), \(\rho _{\rm max}\) is the maximum density on the computational domain, and \(C\) is the central condensation parameter.

2 Basic usage

The following snippet shows what to add to your parfile in order to setup the thorn to seed 5000 particles inside a sphere of radius \(R=50\) centered at \((X,Y,Z)=(0,0,0)\) (default). The maximum density is specified as well, which will affect the probability \(P\) with which the particles are accepted by the algorithm. The particle positions are then updated once every 16 time steps by solving the ODEs above.

ActiveThorns = "particle_tracerET"
particle_tracerET::num_particles                        = 5000
particle_tracerET::density_max                          = 0.000959482674936154
particle_tracerET::seed_particles_inside_sphere__radius = 50
particle_tracerET::update_RK4_freq                      = 16

References

3 Parameters

outdir	Scope: private	STRING
Description: Directory for particle tracerET output files, overrides IO::out_dir
Range		Default: (none)
	An empty string to choose the default from IO::out_dir
.+	A valid directory name

4 Interfaces

General

Implements:

particle_traceret

Inherits:

grid

hydrobase

admbase

Grid Variables

4.0.1 PRIVATE GROUPS

5 Schedule

This section lists all the variables which are assigned storage by thorn WVUThorns_Diagnostics/particle_tracerET. Storage can either last for the duration of the run (Always means that if this thorn is activated storage will be assigned, Conditional means that if this thorn is activated storage will be assigned for the duration of the run if some condition is met), or can be turned on for the duration of a schedule function.

Storage

Scheduled Functions

CCTK_INITIAL (conditional)

  particle_traceret_initial

  initialization routines of the particle_traceret thorn

particle_tracerET_Initial (conditional)

  initializeallparticletracerethelpervars

  check all parameters have been set correctly; initialize particle tracer rk4 iteration counter

particle_tracerET_Initial (conditional)

  particle_traceret_file_output_routine_startup

  create directory for file output.

CCTK_ANALYSIS (conditional)

  convert_to_mhd_3velocity

  convert to mhd 3 velocity

CCTK_ANALYSIS (conditional)

  particle_traceret

  particle traceret subroutines

particle_tracerET (conditional)

  initializeparticlepositions

  initialize particle positions

particle_tracerET (conditional)

  doonerk4stepforparticletraceret

  do one rk4 step for particle traceret

particle_tracerET (conditional)

  compute_particles_four_velocities

  compute particles four velocities

particle_tracerET

  particle_traceret_file_output

  output particle traceret data to disk