[docs]
class VariationalMHD_with_q(StruphyModel):
"""Full (non-linear) visco-resistive MHD equations, with the q variable (square root of the pressure), 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.sqrt_p = 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,
rho: FEECVariable = None,
with_viscosity: bool = True,
with_resistivity: bool = True,
):
self.variat_dens = VariationalDensityEvolve()
self.variat_mom = VariationalMomentumAdvection()
self.variat_qb = VariationalQBEvolve()
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(
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_qb.variables.u = self.mhd.velocity
self.propagators.variat_qb.variables.q = self.mhd.sqrt_p
self.propagators.variat_qb.variables.b = self.em_fields.b_field
if with_viscosity:
self.propagators.variat_viscous.variables.s = self.mhd.sqrt_p
self.propagators.variat_viscous.variables.u = self.mhd.velocity
if with_resistivity:
self.propagators.variat_resist.variables.s = self.mhd.sqrt_p
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 post_allocate(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):
q = self.mhd.sqrt_p.spline.vector
gamma = self.propagators.variat_qb.options.gamma
return 1.0 / (gamma - 1.0) * Propagator.mass_ops.M3.dot_inner(q, q)
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 "variat_dens.Options" in line:
new_file += [
"model.propagators.variat_dens.options = model.propagators.variat_dens.Options(model='full_q')\n",
]
elif "variat_qb.Options" in line:
pass
elif "variat_viscous.Options" in line:
new_file += [
"model.propagators.variat_viscous.options = model.propagators.variat_viscous.Options(model='full_q')\n",
]
elif "variat_resist.Options" in line:
new_file += [
"model.propagators.variat_resist.options = model.propagators.variat_resist.Options(model='full_q')\n",
]
elif "sqrt_p.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{2 q}{\gamma - 1} \nabla q + \mathbf{B} \times \nabla \times \mathbf{B} - \nabla \cdot \left( (\mu + \mu_a(\mathbf{x})) \nabla \mathbf{u} \right) = 0
Energy-like variable:
.. math::
\partial_t q + \nabla \cdot (q \mathbf{u}) + \left( \frac{\gamma}{2} - 1 \right) q \nabla \cdot \mathbf{u} = \frac{2 q}{\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 flow variables are normalized with 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 q-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_qb_evolve.VariationalQBEvolve`
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. VariationalQBEvolve:**
{VariationalQBEvolve.__doc__}
**4. VariationalViscosity:**
{VariationalViscosity.__doc__}
**5. VariationalResistivity:**
{VariationalResistivity.__doc__}
"""
return doc
[docs]
@classmethod
def doc_long_description(cls):
r"""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."""
[docs]
@classmethod
def doc_examples(cls):
r"""Create and initialize the q-based visco-resistive MHD model:
.. code-block:: python
from struphy.models import VariationalMHD_with_q
model = VariationalMHD_with_q()
model.mhd.sqrt_p
model.em_fields.b_field
"""
[docs]
@classmethod
def doc_use_cases(cls):
r"""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"""
[docs]
@classmethod
def doc_cannot_be_used_for(cls):
r"""This model is not suitable for:
- ideal dissipation-free MHD
- kinetic or hybrid particle physics
- reduced linear perturbation problems
- pressure or entropy primitive-variable studies"""
