Structure-Preserving MHD-Driftkinetic Discretization for Wave-Particle Interactions

Abstract

We present a structure-preserving discretization of the hybrid magnetohydrodynamics (MHD)-driftkinetic system for simulations of low-frequency wave-particle interactions. The model equations are derived from a variational principle, assuring energetically consistent couplings between MHD fluids and driftkinetic particles. The spatial discretization is based on a finite-element-exterior-calculus (FEEC) framework for the MHD and a particle-in-cell (PIC) method for the driftkinetic. A key feature of the scheme is the inclusion of the non-quadratic particle magnetic moment energy term in the Hamiltonian, which is introduced by the guiding-center approximation. The resulting discrete Hamiltonian structure naturally organizes the dynamics into skew-symmetric subsystems, enabling balanced energy exchange. To handle the non-quadratic energy term, we develop energy-preserving time integrators based on discrete gradient methods. The algorithm is implemented in the open-source Python package $\texttt{STRUPHY}$. Numerical experiments confirm the energy-conserving property of the scheme and demonstrate the capability to simulate energetic particles (EP) induced excitation of toroidal Alfvén eigenmodes (TAE) without artificial dissipation or mode filtering. This capability highlights the potential of structure-preserving schemes for high-fidelity simulations of hybrid systems.

Figures (8)

• Figure 1: Commuting diagram of the de Rham complex. The top row depicts the continuous sequence of differential-form spaces forming an exact sequence, while the bottom row shows the corresponding finite-dimensional spaces. Projection operators commute with the differential operators, ensuring that the discrete sequence preserves the exactness of the continuous de Rham complex.
• Figure 2: Time evolution of the relative error in total energy for different simulation parameters: (a) kinetic thermal velocity v_th, (b) number of particles per cell (ppc), (c) time step size Delta t and (d) mesh distortion factor alpha. Results from the discrete gradient scheme (solid lines) are compared with the fourth-order Runge-Kutta scheme (dashed lines).
• Figure 3: Growth rates of TAEs (left) and corresponding mode frequencies (right) as functions of EP temperature, obtained from simulations with (blue) and without (red) toroidal Fourier filter.
• Figure 4: The time evolution of magnetic (blue) and MHD kinetic (green) energies with (left) and without (right) toroidal Fourier filter. The growth rate is estimated from linear fits (orange dashed lines) during the linear growth phase (shaded gray areas).
• Figure 5: The radial mode structures of the perturbed radial MHD velocity (left) and magnetic field (right) for toroidal mode $n=6$ during the linear growth phase $(t=300\,T_A)$. The four dominant poloidal harmonics are plotted $(m=9,10,11,12)$. Solid and dashed lines correspond to the results with and without the toroidal Fourier filter, respectively.