Fluid model

ViscoResistiveDeltafMHD

Delta-f visco-resistive MHD equations discretized with a variational method.

Overview

ViscoResistiveDeltafMHD is the dissipative perturbative MHD model using

pressure as thermodynamic variable. It is intended for departures from a background equilibrium that are not strictly linear but still naturally formulated as perturbations.

Use cases

This model is appropriate for:

perturbative nonlinear MHD around a prescribed equilibrium
dissipative MHD verification with background fields
comparing linear and delta f variational formulations

Governing equations

PDEs solved by model:

Continuity:

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

Momentum:

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

Pressure:

\partial_{t} p ~ + u ~ \cdot \nabla (p ~ + p_{0}) + γ (p ~ + 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 ~ + 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.

Normalization

The velocity scale is Alfvénic:

u^= v^_{A} .

Discretization

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

struphy.propagators.variational_density_evolve.VariationalDensityEvolve
struphy.propagators.variational_momentum_advection.VariationalMomentumAdvection
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)

Diagnostics

The following scalars are tracked during simulation:

Kinetic energy: en_U
Thermal energies: en_thermo, en_thermo_l1
Magnetic energies: en_mag_1, en_mag_2, en_mag_l1
Total energies: en_tot, en_tot_l1

Example

Create and initialize the visco-resistive delta f MHD model:

from struphy.models import ViscoResistiveDeltafMHD

model = ViscoResistiveDeltafMHD()
model.mhd.pressure

model.em_fields.b_field