Fluid model

ViscoResistiveMHD_with_q

Full (non-linear) visco-resistive MHD equations, with the q variable (square root of the pressure), discretized with a variational method.

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Overview

This model is the full nonlinear q-based dissipative MHD formulation.

It is useful when the square-root pressure variable is preferred in the variational discretization while still retaining full nonlinear magnetic dynamics.

Use cases

This model is appropriate for:

full nonlinear dissipative MHD in q-form
comparison against the p- and entropy-based full models
variational MHD studies where q is numerically advantageous

Governing equations

PDEs solved by model:

Continuity:

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

Momentum:

\partial_{t} (ρ u) + \nabla \cdot (ρ u \otimes u) + \frac{2 q}{γ - 1} \nabla q + B \times \nabla \times B - \nabla \cdot ((μ + μ_{a} (x)) \nabla u) = 0

Energy-like variable:

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

Induction:

\partial_{t} B + \nabla \times (B \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 flow variables are normalized with the Alfvén speed:

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_qb_evolve.VariationalQBEvolve
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
Thermodynamic q-energy: en_thermo
Magnetic energy: en_mag
Total energy: en_tot
Total density: dens_tot
Divergence diagnostic: tot_div_B

Example

Create and initialize the q-based visco-resistive MHD model:

from struphy.models import ViscoResistiveMHD_with_q

model = ViscoResistiveMHD_with_q()
model.mhd.sqrt_p

model.em_fields.b_field