SWMF/M-FLAMPA

M-FLAMPA, part of the Space Weather Modeling Framework (SWMF) developed by the University of Michigan, enables users to solve the acceleration and transport processes of solar energetic particles (SEPs) along multiple interplanetary magnetic field lines originating from the Sun. The model is a high-performance extension of the original FLAMPA code, which simulates SEP distribution along a single field line.

M-FLAMPA solves for the gyrotropic SEP distribution function. The code takes advantage of the fact that particles stay on the same magnetic field line and, therefore, the distribution function may be treated as a function of the distance along the field lines, rather than a 3D vector. Additionally, coefficients in the governing equations depend solely on background plasma parameters and their Lagrangian derivatives. This crucial property reduces the problem of particle acceleration in a 3D magnetic field to a set of independent 1D problems on continuously evolving Lagrangian grids. In other words, each field line in the model is treated separately, resulting in a perfectly parallel algorithm.

M-FLAMPA is directly coupled with the AWSoM-R model via an advanced coupling algorithm within the SWMF. This technique seamlessly connects field lines between the two distinct computational domains, where lines are extracted based on a concurrently updated solution of solar wind parameters. The integrated model traces magnetic field lines from the AWSoM-R models to find the area that is covered by field lines originating from a given area of interest. Each field line is represented by a Lagrangian grid that moves with the background plasma in a time-dependent manner. The relevant data at the location of the grid points are transferred to M-FLAMPA, which in turn calculates the evolution of the energetic particle population by solving the governing kinetic equations.

As the time-accurate simulation begins, AWSoM-R and M-FLAMPA run simultaneously. At each time step, the evolving magnetic field lines and plasma properties are extracted from the AWSoM-R solutions, along which the particle distribution function is solved. Additionally, novel mathematical approaches are applied to the extracted magnetic field lines to sharpen the shock wave front, enhancing the efficiency of the DSA process.

Recently, a new particle-number-conserving scheme, Poisson Bracket Scheme, was developed and implemented in M-FLAMPA to solve the Parker diffusion equation. Using the Poisson Bracket Scheme, the prediction for SEP fluxes will not be contaminated by fake particle productions or disappearances due to approximation errors at high spatial gradients near the shock wave front.

The coupled CME and SEP simulations are performed on a steady-state solar wind background generated by SA-AWSoM-R (Stream-Aligned AWSoM-R). SA-AWSoM-R enforces perfect alignment between the plasma velocity (u) and magnetic field (B) in the co-rotating frame, effectively nullifying the electric field (E = −u × B = 0) and Poynting flux in the induction and energy equations, while retaining the Lorentz (Ampère) force in the momentum equation. This approach preserves physically valid steady-state solutions, such as the Parker spiral, without the misalignment artifacts (e.g. “u-shaped” or “v-shaped” field lines) that can arise from numerical diffusion in full MHD simulations. Using SA-AWSoM-R is critical for accurately tracing magnetic connectivity from observers back to solar structures and for modeling energetic particle transport in space weather forecasting. It is therefore used to provide the steady-state solar wind background for coupled CME and SEP runs.

The figure below illustrates the 2D distribution of energetic protons with energies greater than 10 MeV on the 1 AU sphere. The Earth's location is marked with a white solid circle, and the flare location on the Sun is marked with a purple solid circle. It also shows the time evolution of the proton flux at the Earth's location.

An example run from the Run-on-Request system at CCMC can be found here.

This model needs to couple with the AWSoM-R SC and IH domains to pull magnetic fields and other plasma properties to compute the acceleration. The user can specify the parallel diffusion coefficients far upstream of the shock and the injection coefficients (or scaling factor in Zhao et al., 2024) that determines the number of seed particles that are injected into the shock acceleration process. The seed population energy spectra index can also be determined by the user, with a default value of 5 (p-5). The user can also specify the number of field lines and the area that the field lines cover in the carrington longitudes and latitudes.

Outputs of the M-FLAMPA model include: 1) the time evolving proton flux at energies >10 MeV, >30 MeV, >50 MeV, along individual M-FLAMPA field lines at 1 AU, 2) the time evolving 2D proton flux distribution on the 1AU sphere at the same energies as above. Upon request, more detailed output, including the full distribution function of protons (10 keV - 1 GeV), and the differential intensities of proton at energies corresponding different spacecraft (e.g., ACE, WIN, SOHO, PSP, SoLO, BepiColombo) can be provided.