Numerical Simulation of Flaring Loops

Motivation

The post flare loops are comprised of a thermal and nonthermal distribution of particles. Yet, few if any flare simulation codes model both populations in tandem and allow them to ineract. In this work, new simulation codes were developed in order to model the thermal and nonthermal distributions and their interactions in order to better simulate the emission signatures of solar flares, and to provide a laboratory where theoretical particle acceleration mechanisms can be tested against real observations.

Analytic Loops

Below are simulated flaring loops. The geometry for the first experiments were derived analytically using Green's solution for a model post flare arcade with a current sheet. (N.B. This is often called a Syrovatskii current sheet model in the literature. However, the Syrovatskii solution is slightly different. See Bungey & Preist, 1995)

Geometry

Using the formalism developed by Bungey and Priest (1995), complex variable notation is used to map the magnetic field lines associated with Green's solution (Green, 1965) to a potential field configuration with a 2.5D current sheet. A field line below the separatrix is chosen in order to avoid the situation where B=0 along the field line which would cause a discontinuity in the area of the loop. This is physically reasonable since a field line only spends ~ 2 seconds in the current sheet (Fletcher & Martens, 1998) and it is the evolution of the post flare loop that is our subject of interest.

A contour plot of the magnetic field lines and current sheet generated by Green's solution (Green, 1965). The field line we will use to describe our loop is in red. The current sheet is drawn in blue and the units are arbitrary.

B_map.gif

The magnetic field strentgh of the space can also be determined using Bungey and Priest (1995) if a scaling factor is given. The base of the flare loop is given a magnetic field strength of 100 G in order to provide a proper scaling factor. A contour map of the magnetic field strength mapped below. The characteristic field line that we will use to describe our loop superimposed.

B_map.gif

The method described above defines a geometry in 2.5D, with the dimension coming into and out of the page being redundant. information This is used to construct a 3D representation of a post flare loop by defining a circular area at the loop base (or apex) and determing the area everywhere else by using the constant flux condition. A rendering of the 3D model is shown below.

3d_model.gif

Nonthermal Particle Beams

The beams of nonthermal particles are generated according the experiment to be carried out. The model beams are comprised of at least 2x10^4 test particles. The test particle's initial pitch-angle (the angle made by the loop's axis and the particle's velocity vector) is drawn randomly from a probability distribution. For this experiment the distribution is characterized by a single parameter, gamma_PA. The results from each test particle are multiplied by a scaling factor in order to represent a true beam on the Sun. The initial pitch-angle distribution of the nonthermal electrons injected into the loop apex for a gamma_PA=4 is shown below.

fig1.gif

The electron beams are then injected into the 3D flare loop and allowed to interact with the thermal plasma contained within. The movie below tracks the position and energies of the particles within the loop. The particles disappear as they become part of the thermal distribution.

Particle Display Movie

As the nonthermal particles collide with the thermal plasma they undergo pitch-angle scattering. The movie below shows the evolution of the pitch-angle profile.

Particle Display Movie

Thermal Plasma Response

The thermal plasma is defined by three state variables: velocity, electron number density, and energy density. The benefit of numerical models is that we can track these variables, and functions of them, as a function of position and time. The movie below tracks the pressure, electron density, velocity and temperature.

Hydrodynamics Movie

Simulation Tests

Any numerical model requires an analysis of the simulation's error. There are many sources of error in numerical modeling. There are errors related to the precision of the computer and variable types being used (eg. roundoff errors). There are errors do to limitations of the numerical techniques used to solve equations (eg. truncation error). Often, physical quantities that are an integral part of the model's equations are not well known, or approxiamted by theoretical constraints, which makes for a source of error that is nearly impossible to quatify.

Grid Test

Limitations in the number of spatial and temporal grids is also a source of error. In order to quantify the "grid error", a simulation was conducted on a variety of grids. Each simulation used an identical initial nonthermal particle beam. However, the equations governing the evolution of the particle beam have stochastic elements, which will still make for differences in solutions that are not grid dependent. A series of plots were made, one at each output time step, that illustrate the effect of spatial grid space size. A temperature and electron density measurement is made at the apex of each loop. A percent difference between the measurement made at a given grid size and its predecessor is plotted and shown below.

Grid Test 1

The upturn in the percent difference at 1100 grids was unexpected. Conventional wisdom would have the percent difference decrease as the number of grids increases. However, the percent difference remains below 8% for all time yielding an upper bound on the error in the simulation due to a finite number of grid spaces. Due to time and computer constraints, a grid size of 700 was used to conduct the current series of experiments.

Run Tests

Since the nonthermal particles are drawn from random distributions and the equations governing their behaviour contains stochastic terms, each run of a simulation will be different. The measurement of these differences can provide an upper bound for the error incurred by using a finite number of test particles to represent the entire beam, and a lower bound for the overall error.

A series of N runs were made, each regenerating a beam from the random distributions in order to better sample the parameter space. As before temperature and electron number density are measured at the loop apex. For the each run from 2 to N a running average value is made and plotted. Overplotted is the standard deviation of the average. Additional runs are added until the error is considered acceptable for the experiment to be carried out.

Run Test 1

Another useful way to determine how many runs are needed is to plot the percent error as a function of time for a given number of runs. No additional runs are necessary when the percent error goes below a predetermined threshold or when the addition of more runs no longer reduces the percent error. Below is a plot showing the percent error of a simulation as a fucntion of time.

Run Test 1

Emission Signatures

After a simulation run we have information on temperature and density for each grid cell as a function of time. The simulation also records the energy and momentum loss of nonthermal test particles as they travel through each grid cell. Using this information the emission of photons from thermal and nonthermal bremsstrahlung can be calculated. The movie below shows the total emission in the 3-6 keV band in black, the nonthermal component in green, and the thermal component in red. The lower plot shows the emission of the viewing area as a function of time. The imager simulated has a pixel size of 7'' and a Gaussian PSF with FHWM of 3 pixels.

Run Test 1