Hybrid model

LinearMHDVlasovCC

Hybrid linear MHD coupled with energetic ions (6D Vlasov) via the current-coupling scheme.

Overview

LinearMHDVlasovCC is a current-coupling hybrid model for small

perturbations around an MHD equilibrium. It is intended for energetic particle effects on linear MHD waves and instabilities without the cost of a fully kinetic background plasma.

Use cases

This model is appropriate for:

linear energetic-particle coupling to MHD modes
current-coupling hybrid verification
reduced-cost studies of fast-ion effects on wave propagation

Governing equations

PDEs solved by model:

MHD continuity:

\frac{\partial ρ ~}{\partial t} + \nabla \cdot (ρ_{0} U ~) = 0

MHD momentum:

ρ_{0} \frac{\partial U ~}{\partial t} + \nabla p ~ = (\nabla \times B ~) \times B_{0} + J_{0} \times B ~ + \frac{A _{h}}{A _{b}} \frac{1}{ε} (n_{h} U ~ - n_{h} u_{h}) \times (B_{0} + B ~)

MHD pressure:

\frac{\partial p ~}{\partial t} + (γ - 1) \nabla \cdot (p_{0} U ~) + p_{0} \nabla \cdot U ~ = 0

MHD induction:

\frac{\partial B ~}{\partial t} = \nabla \times (U ~ \times B_{0}), \nabla \cdot B ~ = 0

Energetic-particle Vlasov equation:

\frac{\partial f _{h}}{\partial t} + v \cdot \nabla f_{h} + \frac{1}{ε} [(B_{0} + B ~) \times U ~ + v \times (B_{0} + B ~)] \cdot \frac{\partial f _{h}}{\partial v} = 0

Energetic-particle moments:

n_{h} = \int_{R^{3}} f_{h} d^{3} v, n_{h} u_{h} = \int_{R^{3}} f_{h} v d^{3} v

where $J_{0} = \nabla \times B_{0}$ .

Normalization

Both fluid and particle velocities are normalized with the Alfvén

speed,

U^= v^= v^_{A}, f^_{h} = n^/ v^_{A}^{3} .

The hot-species cyclotron parameter is $ε = 1/ (Ω^_{c, hot} t^)$ .

Discretization

Time integration is performed by the following propagators (in sequence):

Diagnostics

The following scalars are tracked during simulation:

MHD kinetic energy: en_U
Thermal pressure energy: en_p
Magnetic energy: en_B
Energetic-particle energy: en_f
Total energy: en_tot
Lost particles: n_lost_particles

Examples

Alfvén wave with kinetic-ion current coupling

A shear-Alfvén wave exchanges energy with a dilute population of kinetic ions. The LinearMHDVlasovCC hybrid model evolves the bulk plasma as a fluid and the energetic ions as particles, coupled through their current. Follow the wave and compare energy transfer with the total-energy conservation error. The fluid starts with $u_x(z,0)=0.05\sin(\pi z/5),$ in $\mathbf{B}_0=\mathbf{e}_z$ , $n_0=1$ . The energetic-ion Maxwellian has $n_h=0.1$ , $\mathbf{u}_h=0$ and isotropic thermal speed $v_{\mathrm{th},h}=0.5$ .