Magnetohydrodynamics Adaptive Solvers in the AMROC Framework for Space Plasma Applications

Abstract

Plasma disturbances affect satellites and spacecraft and can cause serious problems to telecommunications and sensitive sensor systems on Earth. Considering the huge scale of the plasma phenomena, data collection at individual locations is not sufficient to cover this entire relevant environment. Therefore, computational plasma modelling has become a significant issue for space sciences, particularly for the near-Earth magnetosphere. However, the simulations of these disturbances present many physical as well as numerical and computational challenges. In this work, we discuss our recent magnetohydrodynamic solver, realised within the MPI-parallel AMROC (Adaptive Mesh Refinement in Object-oriented C++) framework, in which particular physical models and automatic mesh generation procedures have been implemented. A performance analysis using a selection of significant space applications validates the solvers capabilities and confirms the technical importance of our approach.

Appendix

Code Organisation

In the context of this work, the AMROC framework, as described in [15] and published online⁴ is divided into two main folders: the implementation and compilation folders. The folder vtf/amroc/amr contains the base algorithm for a numerical simulation using SAMR methods for a generic system of hyperbolic equations. The files contained in this folder specify the data structures and routines outside the scope of the simulated equations, such as mesh adaptation, mesh distribution per processor, boundary conditions, restriction and prolongation operators, etc. In particular, the function IntegrateLevel() in the file AMRSolver.h calls the numerical scheme associated with the simulated equation, implemented in the base module, using the mpass counter. For each iteration of this counter, the scheme defined in the base is computed and then the ghost cells are updated. Considering the implemented MHD solver, this counter performs three iterations, corresponding to the first Runge–Kutta (RK) step, the second RK step and the divergence cleaning step, respectively. The GLM implementation files are located in the mhd directory of the implementation folder. They contain the base virtual functions to perform a generic simulation of the MHD equations for two and three dimensions.

In special, these files contain the time evolution function Step(), called from the Generic SAMR solver, and the virtual functions called from this function. The use of virtual functions allows the definition of base functions that may be used for most of the experiments, while allowing the redefinition of these functions in the specific MHD module, if required by the studied problem. In general, the functions from the base module implement numerical operations that are independent from the problem simulated, such as flux computations, limiters and divergence cleaning routines. The problem-specific file located in the respective source directory src implements functions that are particular to each experiment. In general, this file contains initial conditions and resistivity and gravity fields. However, if necessary for the experiment, this file may contain redefinitions of virtual functions implemented in the base module. We also have for each simulation an input parameter namelist called solver.in.

Finally, the MHD module in AMROC runs scripts that already contain the commands to convert the output HDF (Hierarchical Data Format) files into binary VTK (Visualization ToolKit) files used for data visualisation in tools such as VisIt⁵[10] and ParaView⁶[1].

Geomagnetic Disturbances

To attend the interests of the geophysical community, devices to measure the geomagnetic field, designated in a general sense as magnetometers, have been installed on the ground, nowadays composing a large net spread around the world. One can find more information and specific documentation in the World Data Center for Geomagnetism, Kyoto.⁷ Also, related fundamentals on space physics are available in [42]. To quantify the level of geomagnetic disturbance occurring upon the Earth, the interested reader can survey and examine some geomagnetic disturbance indices, for instance, the index Kp for an estimated planetary disturbance behaviour, the index AE for auroral electrojet disturbance effects and the index Dst for a low latitude magnetic disturbance. In our case, we choose and present in Fig. A.1 the interplanetary magnetic field Bz, the index AE and the index Sym-H. This information can be collected effortlessly from the OMNI web service, NASA.⁸Bz is the primary variable responsible for triggering of the magnetic reconnection process (merging of the interplanetary magnetic field lines with the geomagnetic field lines), when this IMF component is a predominantly southward-oriented field (i.e. in opposition to the geomagnetic field orientation), in the frontal side, i.e. towards to the Sun, of the Earth’s magnetosphere. AE is the geomagnetic index concerning the modification of the auroral electrojet currents that produce magnetic disturbances in the higher latitudes. Sym-H is the index concerning the intensification of an equatorial, symmetrical ring electrical current (at a distance about 6–7 Earth radii) that produces magnetic disturbances in the lower latitudes. Recorded by geomagnetic indices, any geomagnetic variations link intrinsically to the electrodynamical coupling between the solar plasma and the Earth’s magnetosphere.

Fig. A.1

MAG: Geomagnetic disturbances shown by the geomagnetic indices: Auroral Electroject (AE) and Symmetric Equatorial ring current effect (Sym-H) corresponding to the period between January 16th and 18th of 2018. Indicated in the interplanetary magnetic field Bz. Letter A indicates northward-oriented field interval, B a transition value and C southward-oriented interval

From the figure, indicated in the interplanetary magnetic field Bz, the letter A identifies a corresponding maximum-value time in the northward-oriented field interval, B a time under a transition value (close to zero) and C a corresponding minimum-value time in the southward-oriented interval. There are two reasons to select this dataset region: to pick up distinct interplanetary behaviours and to be far from the simulation beginning. This procedure allows for exemplifying evolution consistency related to record inputs and tangible results. As shown in the plot, a time-coincident small intensity effect was noted in the auroral index, AE, and no effect stands out in the equatorial index, Sym-H. Shown in Fig. 7, the simulation results for the Earth’s magnetic field configuration are in physical agreement with the magnetic effects on the ground, as the physics presented and discussed, for instance, by Russell et al. [42]. The current code features provide the means for evolution analysis of the Earth’s magnetosphere in complicated scenarios, such as investigations for geomagnetically quiet conditions.