Fluid model

ViscoResistiveLinearMHD

Linear visco-resistive MHD equations discretized with a variational method.

Overview

This is the linear dissipative MHD model with pressure as primitive

thermodynamic variable. It is intended for small-amplitude perturbations around an equilibrium while retaining explicit viscosity and resistivity.

This model is appropriate for:

Continuity:

\partial_{t} ρ ~ + \nabla \cdot (ρ_{0} u ~) = 0

Momentum:

\partial_{t} (ρ_{0} u ~) + \frac{1}{γ - 1} \nabla p ~ + B_{0} \times \nabla \times B ~ + B ~ \times \nabla \times B_{0} - \nabla \cdot ((μ + μ_{a} (x)) \nabla u ~) = 0

Pressure:

\partial_{t} p ~ + u ~ \cdot \nabla p_{0} + γ p_{0} \nabla \cdot u ~ = \frac{1}{γ - 1} ((μ + μ_{a} (x)) ∣\nabla u ~ ∣^{2} + (η + η_{a} (x)) ∣\nabla \times B ~ ∣^{2})

Induction:

\partial_{t} B ~ + \nabla \times (B_{0} \times u ~) + \nabla \times (η + η_{a} (x)) \nabla \times B ~ = 0

Here $μ_{a} (x)$ and $η_{a} (x)$ are artificial viscosity and resistivity coefficients.

The characteristic velocity is the Alfvén speed,

u^= v^_{A} .

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

struphy.propagators.variational_density_evolve.VariationalDensityEvolve
struphy.propagators.variational_pb_evolve.VariationalPBEvolve
struphy.propagators.variational_viscosity.VariationalViscosity (if with_viscosity is True)
struphy.propagators.variational_resistivity.VariationalResistivity (if with_resistivity is True)

Create and initialize a linear visco-resistive MHD model:

from struphy.models import ViscoResistiveLinearMHD

model = ViscoResistiveLinearMHD()
model.em_fields.b_field
model.mhd.velocity

model.mhd.pressure