Fluid model

LinearExtendedMHDuniform

Linear extended MHD with zero-flow equilibrium, for uniform background conditions only.

Overview

LinearExtendedMHDuniform extends the linear MHD model with Hall

dynamics for a uniform equilibrium. It is targeted at wave propagation and mode studies where the equilibrium is simple but finite Hall corrections are important.

Use cases

This model is appropriate for:

linear Hall-MHD wave studies
uniform-equilibrium benchmarks for extended MHD
comparison against ideal-MHD dispersion in the Hall-corrected regime

Governing equations

PDEs solved by model:

Continuity:

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

Momentum:

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

Pressure:

\frac{\partial p ~}{\partial t} + \frac{5}{3} p_{0} \nabla \cdot U ~ = 0

Induction:

\frac{\partial B ~}{\partial t} - \nabla \times (U ~ \times B_{0} - \frac{1}{ε} \frac{\nabla \times B ~}{ρ _{0}} \times B_{0}) = 0

Normalization

All velocities are normalized by the Alfvén speed,

U^= v^_{A},

and Hall effects enter through $ε = 1/ (Ω^_{c} t^)$ .

Discretization

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

Diagnostics

The following scalars are tracked during simulation:

Flow kinetic energy: en_U
Mean pressure perturbation (linear diagnostic): en_p
Thermal perturbation energy: en_thermal
Magnetic perturbation energy: en_B
Background pressure energy: en_p_eq
Background magnetic energy: en_B_eq
Total magnetic energy: en_B_tot
Total perturbation energy (kinetic + magnetic + thermal): en_tot
Magnetic helicity-like invariant: helicity

Examples

Whistler and ion-cyclotron waves in Hall MHD

Broadband noise in a uniform magnetized plasma excites the two circularly polarized branches into which the Hall term splits the shear Alfvén wave: the whistler, whose frequency grows like k², and the ion-cyclotron wave, which saturates at the ion cyclotron frequency. Their power spectra and phase velocities are compared with the analytic Hall-MHD dispersion relation. The initial background has $\mathbf{B}_0=\mathbf{e}_z,\qquad n_0=1,\qquad p_0=0.25,\qquad \epsilon=1,$ with coefficient-noise amplitude $0.1$ in all three velocity components and periodic length $L_z=60$ .