[docs]
class ViscoResistiveMHD_with_p(StruphyModel):
"""Full (non-linear) visco-resistive MHD equations, with the pressure variable, discretized with a variational method.
Parameters
----------
base_units: BaseUnits
Base units for normalization (default: BaseUnits())
mass_number: float
Mass number (in units of Proton mass) of the fluid species (default: 1.0)
with_viscosity: bool
Whether to include viscous dissipation (default: True)
with_resistivity: bool
Whether to include resistive dissipation (default: True)
"""
@classmethod
def model_type(cls) -> LiteralOptions.ModelTypes:
return "Fluid"
## species
class EMFields(FieldSpecies):
def __init__(self):
self.b_field = FEECVariable(space="Hdiv")
self.init_variables()
class MHD(FluidSpecies):
def __init__(self, mass_number: float = 1.0):
self.density = FEECVariable(space="L2")
self.velocity = FEECVariable(space="H1vec")
self.pressure = FEECVariable(space="L2")
self.init_variables(mass_number=mass_number)
class Diagnostics(DiagnosticSpecies):
def __init__(self):
self.div_u = FEECVariable(space="L2")
self.u2 = FEECVariable(space="Hdiv")
self.init_variables()
## propagators
class Propagators:
def __init__(
self,
div_u: FEECVariable = None,
u2: FEECVariable = None,
rho: FEECVariable = None,
with_viscosity: bool = True,
with_resistivity: bool = True,
):
self.variat_dens = VariationalDensityEvolve()
self.variat_mom = VariationalMomentumAdvection()
self.variat_pb = VariationalPBEvolve(div_u=div_u, u2=u2)
if with_viscosity:
self.variat_viscous = VariationalViscosity(rho=rho)
if with_resistivity:
self.variat_resist = VariationalResistivity(rho=rho)
## abstract methods
def __init__(
self,
base_units: BaseUnits = BaseUnits(),
mass_number: float = 1.0,
with_viscosity: bool = True,
with_resistivity: bool = True,
):
# 0. store input parameters
self.params = copy.deepcopy(locals())
# 1. instantiate all species
self.em_fields = self.EMFields()
self.mhd = self.MHD(mass_number=mass_number)
self.diagnostics = self.Diagnostics()
# 2. derive units (must be done after instantiating species to access charge and mass numbers)
self.setup_equation_params(base_units=base_units)
# 3. instantiate all propagators
self.propagators = self.Propagators(
div_u=self.diagnostics.div_u,
u2=self.diagnostics.u2,
rho=self.mhd.density,
with_viscosity=with_viscosity,
with_resistivity=with_resistivity,
)
# 4. assign variables to propagators
self.propagators.variat_dens.variables.rho = self.mhd.density
self.propagators.variat_dens.variables.u = self.mhd.velocity
self.propagators.variat_mom.variables.u = self.mhd.velocity
self.propagators.variat_pb.variables.u = self.mhd.velocity
self.propagators.variat_pb.variables.p = self.mhd.pressure
self.propagators.variat_pb.variables.b = self.em_fields.b_field
if with_viscosity:
self.propagators.variat_viscous.variables.s = self.mhd.pressure
self.propagators.variat_viscous.variables.u = self.mhd.velocity
if with_resistivity:
self.propagators.variat_resist.variables.s = self.mhd.pressure
self.propagators.variat_resist.variables.b = self.em_fields.b_field
# 5. define scalars to be tracked during simulation
kinetic_energy = BilinearEnergyFEEC(self.mhd.velocity, bilinear_form_name="WMMnew")
thermo_energy = FunctionScalarFEEC(self._compute_en_thermo)
magnetic_energy = BilinearEnergyFEEC(self.em_fields.b_field)
density_total = VolumeFormEnergyFEEC(self.mhd.density)
div_b_total = FunctionScalarFEEC(self._compute_tot_div_B)
self.scalars = Scalars(
en_U=kinetic_energy,
en_thermo=thermo_energy,
en_mag=magnetic_energy,
en_tot=kinetic_energy + thermo_energy + magnetic_energy,
dens_tot=density_total,
tot_div_B=div_b_total,
)
@property
def bulk_species(self):
return self.mhd
@property
def velocity_scale(self):
return "alfvén"
def allocate_helpers(self):
projV3 = L2Projector("L2", Propagator.mass_ops)
def f(e1, e2, e3):
return 1
f = xp.vectorize(f)
self._integrator = projV3(f)
self._ones = Propagator.derham.V3pol.zeros()
if isinstance(self._ones, PolarVector):
self._ones.tp[:] = 1.0
else:
self._ones[:] = 1.0
self._tmp_div_B = Propagator.derham.V3pol.zeros()
def _compute_en_thermo(self):
p = self.mhd.pressure.spline.vector
gamma = self.propagators.variat_pb.options.gamma
return Propagator.mass_ops.M3.dot_inner(p, self._integrator) / (gamma - 1.0)
def _compute_tot_div_B(self):
b = self.em_fields.b_field.spline.vector
div_B = Propagator.derham.div.dot(b, out=self._tmp_div_B)
return Propagator.mass_ops.M3.dot_inner(div_B, div_B)
# default parameters
def generate_default_parameter_file(self, path=None, prompt=True):
params_path = super().generate_default_parameter_file(path=path, prompt=prompt)
new_file = []
with open(params_path, "r") as f:
for line in f:
if "pressure.add_background" in line:
new_file += ["model.mhd.density.add_background(FieldsBackground())\n"]
new_file += [line]
else:
new_file += [line]
with open(params_path, "w") as f:
for line in new_file:
f.write(line)
[docs]
@classmethod
def doc_pde(cls):
r"""**PDEs solved by model:**
Continuity:
.. math::
\partial_t \rho + \nabla \cdot (\rho \mathbf{u}) = 0
Momentum:
.. math::
\partial_t (\rho \mathbf{u}) + \nabla \cdot (\rho \mathbf{u} \otimes \mathbf{u}) + \frac{1}{\gamma - 1} \nabla p + \mathbf{B} \times \nabla \times \mathbf{B} - \nabla \cdot \left( (\mu + \mu_a(\mathbf{x})) \nabla \mathbf{u} \right) = 0
Pressure:
.. math::
\partial_t p + u \cdot \nabla p + \gamma p \nabla \cdot u = \frac{1}{\gamma - 1} \left( (\mu + \mu_a(\mathbf{x})) |\nabla \mathbf{u}|^2 + (\eta + \eta_a(\mathbf{x})) |\nabla \times \mathbf{B}|^2 \right)
Induction:
.. math::
\partial_t \mathbf{B} + \nabla \times (\mathbf{B} \times \mathbf{u}) + \nabla \times (\eta + \eta_a(\mathbf{x})) \nabla \times \mathbf{B} = 0
Here :math:`\mu_a(\mathbf{x})` and :math:`\eta_a(\mathbf{x})` are artificial viscosity and resistivity coefficients.
"""
[docs]
@classmethod
def doc_normalization(cls):
r"""The characteristic flow speed is the Alfvén speed,
.. math::
\hat u = \hat v_A.
"""
[docs]
@classmethod
def doc_scalar_quantities(cls):
r"""**The following scalars are tracked during simulation:**
- Kinetic energy: ``en_U``
- Thermodynamic energy: ``en_thermo``
- Magnetic energy: ``en_mag``
- Total energy: ``en_tot``
- Total density: ``dens_tot``
- Divergence diagnostic: ``tot_div_B``"""
[docs]
@classmethod
def doc_discretization(cls):
"""Time integration is performed by the following propagators (in sequence):
1. :class:`~struphy.propagators.variational_density_evolve.VariationalDensityEvolve`
2. :class:`~struphy.propagators.variational_momentum_advection.VariationalMomentumAdvection`
3. :class:`~struphy.propagators.variational_pb_evolve.VariationalPBEvolve`
4. :class:`~struphy.propagators.variational_viscosity.VariationalViscosity` (if :attr:`with_viscosity` is True)
5. :class:`~struphy.propagators.variational_resistivity.VariationalResistivity` (if :attr:`with_resistivity` is True)
"""
doc = rf"""**1. VariationalDensityEvolve:**
{VariationalDensityEvolve.__doc__}
**2. VariationalMomentumAdvection:**
{VariationalMomentumAdvection.__doc__}
**3. VariationalPBEvolve:**
{VariationalPBEvolve.__doc__}
**4. VariationalViscosity:**
{VariationalViscosity.__doc__}
**5. VariationalResistivity:**
{VariationalResistivity.__doc__}
"""
return doc
[docs]
@classmethod
def doc_long_description(cls):
r"""This is the pressure-based full nonlinear dissipative MHD model. It is
the natural counterpart to the entropy- and q-based full MHD variants
when pressure is preferred as primitive thermodynamic variable."""
[docs]
@classmethod
def doc_examples(cls):
r"""Create and initialize the pressure-based visco-resistive MHD model:
.. code-block:: python
from struphy.models import ViscoResistiveMHD_with_p
model = ViscoResistiveMHD_with_p()
model.mhd.pressure
model.em_fields.b_field
"""
[docs]
@classmethod
def doc_use_cases(cls):
r"""This model is appropriate for:
- full nonlinear dissipative MHD in p-formulation
- comparison against entropy- and q-based full MHD models
- variational resistive/viscous MHD benchmarks"""
[docs]
@classmethod
def doc_cannot_be_used_for(cls):
r"""This model is not suitable for:
- ideal MHD without dissipation
- kinetic plasma physics
- reduced linear perturbation studies
- thermodynamic formulations that require entropy or q as primitive variables"""
