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):

struphy.propagators.shear_alfven_b1.ShearAlfvenB1
struphy.propagators.hall.Hall
struphy.propagators.magnetosonic_uniform.MagnetosonicUniform

Diagnostics

The following scalars are tracked during simulation:

Flow kinetic energy: en_U
Thermal pressure energy: en_p
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: en_tot
Magnetic helicity-like invariant: helicity

Example

Create and initialize a linear extended-MHD model:

from struphy.models import LinearExtendedMHDuniform

model = LinearExtendedMHDuniform()
model.em_fields.b_field
model.mhd.density
model.mhd.velocity

model.mhd.pressure