Hybrid model

LinearMHDVlasovPC

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

Overview

LinearMHDVlasovPC is the pressure-coupling counterpart to the

current-coupling hybrid model. It is targeted at linear problems where energetic-particle pressure anisotropy is the relevant feedback channel on the bulk MHD dynamics.

Use cases

This model is appropriate for:

linear pressure-coupling hybrid studies
energetic-particle pressure feedback on MHD modes
verification of PushEtaPC and pressure-coupling operators

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 ~ + \frac{A _{h}}{A _{b}} \nabla \cdot P ~_{h, ⊥} = (\nabla \times B ~) \times B_{0} + J_{0} \times B ~

J_{0} = \nabla \times B_{0}

MHD pressure:

\frac{\partial p ~}{\partial t} + \nabla \cdot (p_{0} U ~) + \frac{2}{3} p_{0} \nabla \cdot U ~ = 0

MHD induction:

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

Energetic-particle Vlasov equation:

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

Perpendicular pressure tensor:

P ~_{h, ⊥} = \int v_{⊥} v_{⊥}^{⊤} f_{h} d v

Normalization

Fluid and hot-particle velocities are normalized with the bulk Alfvén

speed. The kinetic pressure tensor is scaled consistently with $A_{h} m_{H} n^v^_{A}^{2}$ , and the hot cyclotron parameter is $ε$ .

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 kinetic energy: en_f
Total energy: en_tot
Lost particles: n_lost_particles

Examples

Hybrid MHD-kinetic energy exchange

Energetic ions drifting near the Alfvén speed exchange energy with a shear-Alfvén wave through Struphy’s pressure-coupling scheme, while the total energy stays conserved. The energetic-ion Maxwellian is specified by $n_h=0.1,\qquad u_{z,h}=1,\qquad v_{\mathrm{th},z,h}=0.3,$ with transverse thermal speeds $v_{\mathrm{th},x,h}=v_{\mathrm{th},y,h}=1$ . The bulk has $\mathbf{B}_0=\mathbf{e}_z$ , $n_0=1$ and transverse velocity coefficient-noise amplitude $10^{-3}$ .