18.2 Ionization Unit

The analysis of UV and X-ray observations, and in particular of spectral lines, is a powerful diagnostic tool of the physical conditions in astrophysical plasmas (e.g., the outer layers of the solar atmosphere, supernova remnants, etc.). Since deviation from equilibrium ionization may have a non-negligible effect on the UV and X-ray lines, it is crucial to take into account these effects in the modeling of plasmas and in the interpretation of the relevant observations.

In light of the above observations, FLASH contains the unit Ionize, in particular the implementation physics/sourceTerms/Ionize/IonizeMain/Nei, which is capable of computing the density of each ion species of a given element taking into account non-equilibrium ionization (NEI). This is accomplished by solving a system of equations consisting of the fluid equations of the whole plasma and the continuity equations of the ionization species of the elements considered. The densities of the twelve most abundant elements in astrophysical material (He, C, N, O, Ne, Mg, Si, S, Ar, Ca, Fe, and Ni) plus fully ionized hydrogen and electrons can be computed by this unit.

The Euler equations plus the set of advection equations for all the ion species take the following form

$${{\partial \rho} \over {\partial t}} + {\bf\nabla} \cdot \left ( \rho {\bf v} \right ) =$$ 0 (18.21)

$${\partial \rho {\bf v} \over \partial t} + {\bf\nabla} \cdot \left ( \rho {\bf v} {\bf v} \right ) + {\bf\nabla} P = \rho {\bf g}$$ (18.22)

$${\partial \rho E \over \partial t} + {\bf\nabla} \cdot \left [ \left ( \rho E + P \right ) {\bf v} \right ] = \rho {\bf v} \cdot {\bf g} \left [ + S \right ]$$ (18.23)

$${\partial n_i^Z \over \partial t} + {\bf\nabla} \cdot n_i^Z {\bf v} = R_i^Z (i=0,\ldots,Z) ,$$ (18.24)

where $\rho$ is the fluid density, $t$ is the time, ${\bf v}$ is the fluid velocity, $P$ is the pressure, $E$ is the sum of the internal energy and kinetic energy per unit mass, ${\bf g}$ is the acceleration due to gravity, $n_i^Z$ is the number density of ions of ionization level $i$ of the element $Z$, and

$$R_i^Z = N_e[n_{i+1}^Z \alpha_{i+1}^Z + n_{i-1}^Z S_{i-1}^Z - n_{i}^Z(\alpha_{i}^Z + S_{i}^Z)] ,$$ (18.25)

where $N_e$ is the electron number density, $\alpha_{i}^Z \equiv \alpha(N_e, T)$ are the coefficients of collisional and dielectronic recombination, and $S_i^Z \equiv S(N_e, T)$ are the collisional ionization coefficients of Summers(1974). Note that NSPECIES, the total number of FLASH species, will be given by

$$N_{\mathrm{spec}} = 2 + \sum_Z (Z+1) ;$$

the sum ranges over all the elements from the list above that are included in the problem, and the additional $2$ comes from the hydrogen and electron mass fractions which are automatically included by the IonizeMain subunit.

18.2.1 Algorithms

A fractional step method is required to integrate the equations and in particular to decouple the NEI solver from the hydro solver. For each timestep, the homogeneous hydrodynamic transport equations given by (18.21) are solved using the FLASH hydrodynamics solver with $R_i^Z=0$. After each transport step, the “stiff” systems of ordinary differential equations (one system per element included in the simulation) for the NEI problem

$${\partial n_i^Z \over \partial t} = R_i^Z (i=0,\ldots,Z)$$ (18.26)

are integrated. This step incorporates the reactive source terms. Within each grid cell, the above equations can be solved separately with a standard ODE method. Since this system is “stiff”, it is solved using the Bader-Deuflhard time integration solver with the MA28 sparse matrix package. Timmes (1999) has shown that these two algorithms together provide the best balance of accuracy and overall efficiency for the similar problem of nuclear burning, see Sec:burn.

Note that in the present version, the contribution of the ionization and recombination to the energy equation (the bracketed term in (18.23)) is not accounted for. Also, it should be noted that the source term in the NEI unit implementation is adequate to solve the problem for optically thin plasma in the “coronal” approximation; just collisional ionization, auto-ionization, radiative recombination, and dielectronic recombination are considered.

18.2.2 Usage

In order to run a FLASH executable that uses the ionization unit, the ionization coefficients of Summers (1974) must be contained in a file named summers_den_1e8.rates in the same directory as the executable when the simulation is run. This file is copied into the object/ directory with the Config keyword DATAFILES in the physics/sourceTerms/Ionize/IonizeMain implementation.

The Ionize unit supplies the runtime parameters described in Table 18.1.

Table 18.1: Runtime parameters used with the Ionize unit.

Variable	Type	Default	Description
tneimin	real	$1.0\times10^4$	Min nei temperature
tneimax	real	$1.0\times10^7$	Max nei temperature
dneimin	real	$1.0$	Min nei electron number density
dneimax	real	$1.0\times10^{12}$	Max nei electron number density

There are two implementations of physics/sourceTerms/Ionize/IonizeMain: the default implementation, Nei (tested using Neitest (see Sec:neitest)), and Eqi (untested in FLASH4). The former computes ion species for non-equilibrium ionization, and the latter computes ion species in the approximation of ionization equilibrium.

The Ionize unit requires that the subunit implementation Simulation/SimulationComposition/Ionize be used to set up the ion species of the fluid. The ions are defined in a file Simulation/SimulationComposition/Ionize/SpeciesList.txt, however, the Config file in the simulation directory (e.g. Simulation/SimulationMain/Neitest/Config) defines which subset of these elements are to be used.