Hybrid model

LinearMHDDriftkineticCC

Hybrid linear ideal MHD coupled with energetic ions (5D drift-kinetic) via the current-coupling scheme.

Overview

LinearMHDDriftkineticCC is the reduced-kinetic hybrid current-coupling

model for energetic ions. It is useful when gyrophase averaging is acceptable but energetic-particle feedback on linear MHD still has to be retained.

This model is appropriate for:

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 + (\nabla \times B_{0}) \times B ~ + \frac{A _{h}}{A _{b}} [\frac{1}{ϵ} n_{gc} U ~ - \frac{1}{ϵ} J_{gc} - \nabla \times M_{gc}] \times B

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

Energetic-particle drift-kinetic equation:

\frac{\partial f _{h}}{\partial t} + \frac{1}{B _{∥}^{*}} (v_{∥} B^{*} - b_{0} \times E^{*}) \cdot \nabla f_{h} + \frac{1}{ϵ} \frac{1}{B _{∥}^{*}} (B^{*} \cdot E^{*}) \frac{\partial f _{h}}{\partial v _{∥}} = 0

Energetic-particle moments:

n_{gc} = \int f_{h} B_{∥}^{*} d v_{∥} d μ

J_{gc} = \int \frac{f _{h}}{B _{∥}^{*}} (v_{∥} B^{*} - b_{0} \times E^{*}) d v_{∥} d μ

M_{gc} = - \int f_{h} B_{∥}^{*} μ b_{0} d v_{∥} d μ

where

B_{∥}^{*} = b_{0} \cdot B^{*} B^{*} E^{*} = B + ϵ v_{∥} \nabla \times b_{0} = - U ~ \times B - ϵ μ \nabla (b_{0} \cdot B)

The bulk and energetic-particle flow scales are normalized with the

bulk Alfvén speed, while the magnetic moment carries its own unit:

U^= v^= v^_{A, bulk}, μ^= A_{h} m_{H} v^_{h}^{2} / B^.

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

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