[docs]
class VariationalPerturbativeMHD(StruphyModel):
"""Perturbative (delta-f) or 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)
linear: bool
Whether to solve the linearized equations, i.e. to drop the terms that are nonlinear in the perturbations
(default: False)
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.pt3 = FEECVariable(space="L2")
self.bt2 = FEECVariable(space="Hdiv")
self.init_variables()
## propagators
class Propagators:
def __init__(
self,
div_u: FEECVariable = None,
u2: FEECVariable = None,
pt3: FEECVariable = None,
bt2: FEECVariable = None,
rho: FEECVariable = None,
linear: bool = False,
with_viscosity: bool = True,
with_resistivity: bool = True,
):
self.variat_dens = VariationalDensityEvolve()
if linear:
self.variat_pb = VariationalPBEvolve(div_u=div_u, u2=u2, pt3=pt3, bt2=bt2)
else:
self.variat_mom = VariationalMomentumAdvection()
self.variat_pb = VariationalPBEvolve(pt3=pt3, bt2=bt2)
if with_viscosity:
self.variat_viscous = VariationalViscosity(rho=rho)
if with_resistivity:
# the linear model has no heating; the nonlinear one heats the pressure (full_p)
self.variat_resist = VariationalResistivity(rho=rho, pt3=pt3 if linear else None)
## abstract methods
def __init__(
self,
base_units: BaseUnits = BaseUnits(),
mass_number: float = 1.0,
linear: bool = False,
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,
pt3=self.diagnostics.pt3,
bt2=self.diagnostics.bt2,
rho=self.mhd.density,
linear=linear,
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
if not linear:
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")
magnetic_energy_1 = BilinearEnergyFEEC(self.em_fields.b_field)
magnetic_energy_2 = BilinearEnergyFEEC(self.diagnostics.bt2, right_variable="b2", normalization=2.0)
thermo_energy = FunctionScalarFEEC(self._compute_en_thermo)
thermo_energy_l1 = FunctionScalarFEEC(self._compute_en_thermo_l1)
magnetic_energy_l1 = BilinearEnergyFEEC(self.em_fields.b_field, right_variable="b2", normalization=2.0)
self.scalars = Scalars(
en_U=kinetic_energy,
en_thermo=thermo_energy,
en_mag_1=magnetic_energy_1,
en_mag_2=magnetic_energy_2,
en_tot=kinetic_energy + thermo_energy + magnetic_energy_1 + magnetic_energy_2,
en_tot_l1=thermo_energy_l1 + magnetic_energy_l1,
en_thermo_l1=thermo_energy_l1,
en_mag_l1=magnetic_energy_l1,
)
@property
def bulk_species(self):
return self.mhd
@property
def velocity_scale(self):
return "alfvén"
@property
def propagator_variants(self) -> dict:
"""The values of the option ``model`` of the variational propagators that match ``linear``."""
if self.params["linear"]:
return {
"variat_dens": "linear",
"variat_pb": "linear",
"variat_viscous": "linear_p",
"variat_resist": "linear_p",
}
return {
"variat_dens": "perturbative",
"variat_pb": "perturbative",
"variat_viscous": "full_p",
"variat_resist": "full_p",
}
def post_allocate(self):
# the propagator options must match the flag `linear`
for name, variant in self.propagator_variants.items():
prop = getattr(self.propagators, name, None)
if prop is not None and prop.options.model != variant:
raise ValueError(
f"{self.__class__.__name__}(linear={self.params['linear']}) needs "
f"{name}.Options(model='{variant}'), got model='{prop.options.model}'."
)
# L2 dofs of sqrt(g), the linear form of the integral over the domain of a 3-form:
# p . vol_dofs = int p^3 d eta = int p dx (the pressure linear form of VariationalPBEvolve, up to -1/(gamma-1))
projV3 = L2Projector("L2", Propagator.mass_ops)
self._vol_dofs = projV3.get_dofs(Propagator.domain.jacobian_det)
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):
pt3 = self.propagators.variat_pb.pt3.spline.vector
gamma = self.propagators.variat_pb.options.gamma
return pt3.inner(self._vol_dofs) / (gamma - 1.0)
def _compute_en_thermo_l1(self):
p = self.mhd.pressure.spline.vector
gamma = self.propagators.variat_pb.options.gamma
return p.inner(self._vol_dofs) / (gamma - 1.0)
# default parameters
def generate_default_parameter_file(self, path=None, prompt=True):
params_path = super().generate_default_parameter_file(path=path, prompt=prompt)
variants = self.propagator_variants
linear = self.params["linear"]
new_file = []
with open(params_path, "r") as f:
for line in f:
name = next((n for n in variants if f"{n}.Options" in line), None)
if line.startswith("model = ") and linear:
new_file += [line.replace("(base_units=base_units)", "(base_units=base_units, linear=True)")]
elif name is not None:
new_file += [
f"model.propagators.{name}.options = model.propagators.{name}.Options(model='{variants[name]}')\n",
]
elif "pressure.add_background" in line and linear:
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:**
The unknowns are the perturbations :math:`(\tilde\rho, \tilde p, \tilde{\mathbf B})` with respect to a
zero-flow MHD equilibrium :math:`(\rho_0, p_0, \mathbf B_0)`, and the velocity :math:`\mathbf u`.
Continuity:
.. math::
\partial_t \tilde{\rho} + \nabla \cdot \left( (\tilde{\rho} + \rho_0) \tilde{\mathbf{u}} \right) = 0
Momentum:
.. math::
\partial_t \left( (\tilde{\rho} + \rho_0) \tilde{\mathbf{u}} \right) + \nabla \cdot \left( (\tilde{\rho} + \rho_0) \tilde{\mathbf{u}} \otimes \tilde{\mathbf{u}} \right) + \nabla \tilde{p} + \mathbf{B}_0 \times \nabla \times \tilde{\mathbf{B}} + \tilde{\mathbf{B}} \times \nabla \times \mathbf{B}_0 + \tilde{\mathbf{B}} \times \nabla \times \tilde{\mathbf{B}} - \nabla \cdot \left( (\mu + \mu_a(\mathbf{x})) \nabla \tilde{\mathbf{u}} \right) = 0
Pressure:
.. math::
\partial_t \tilde{p} + \tilde{\mathbf{u}} \cdot \nabla (\tilde{p} + p_0) + \gamma (\tilde{p} + p_0) \nabla \cdot \tilde{\mathbf{u}} = (\gamma - 1) \left( (\mu + \mu_a(\mathbf{x})) |\nabla \tilde{\mathbf{u}}|^2 + (\eta + \eta_a(\mathbf{x})) |\nabla \times \tilde{\mathbf{B}}|^2 \right)
Induction:
.. math::
\partial_t \tilde{\mathbf{B}} + \nabla \times \left( (\tilde{\mathbf{B}} + \mathbf{B}_0) \times \tilde{\mathbf{u}} \right) + \nabla \times (\eta + \eta_a(\mathbf{x})) \nabla \times \tilde{\mathbf{B}} = 0
Here :math:`\mu_a(\mathbf{x})` and :math:`\eta_a(\mathbf{x})` are artificial viscosity and resistivity coefficients.
**Linear model** (``linear=True``): the terms that are nonlinear in the perturbations are dropped,
.. math::
&\partial_t \tilde{\rho} + \nabla \cdot (\rho_0 \tilde{\mathbf{u}}) = 0\,,
\\[2mm]
&\partial_t (\rho_0 \tilde{\mathbf{u}}) + \nabla \tilde{p} + \mathbf{B}_0 \times \nabla \times \tilde{\mathbf{B}} + \tilde{\mathbf{B}} \times \nabla \times \mathbf{B}_0 - \nabla \cdot \left( (\mu + \mu_a(\mathbf{x})) \nabla \tilde{\mathbf{u}} \right) = 0\,,
\\[2mm]
&\partial_t \tilde{p} + \tilde{\mathbf{u}} \cdot \nabla p_0 + \gamma p_0 \nabla \cdot \tilde{\mathbf{u}} = 0\,,
\\[2mm]
&\partial_t \tilde{\mathbf{B}} + \nabla \times (\mathbf{B}_0 \times \tilde{\mathbf{u}}) + \nabla \times (\eta + \eta_a(\mathbf{x})) \nabla \times \tilde{\mathbf{B}} = 0\,.
In the linear model the viscous and resistive heating is not fed back into the pressure (the dissipative
propagators with ``model="linear_p"`` update only :math:`\tilde{\mathbf{u}}` and :math:`\tilde{\mathbf{B}}`).
"""
[docs]
@classmethod
def doc_normalization(cls):
r"""The velocity scale is Alfvénic:
.. 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``
- 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``"""
[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` (if :attr:`linear` is False)
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)
The option ``model`` of the variational propagators is ``"perturbative"`` (``"full_p"`` for the dissipative
ones), or ``"linear"`` (``"linear_p"``) if :attr:`linear` 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"""VariationalPerturbativeMHD 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; with ``linear=True`` it solves linear MHD around the equilibrium."""
[docs]
@classmethod
def doc_examples(cls):
r"""Create and initialize the visco-resistive perturbative MHD model and its linear version:
.. code-block:: python
from struphy.models import VariationalPerturbativeMHD
model = VariationalPerturbativeMHD()
linear_model = VariationalPerturbativeMHD(linear=True)
model.mhd.pressure
model.em_fields.b_field
"""
[docs]
@classmethod
def doc_use_cases(cls):
r"""This model is appropriate for:
- perturbative nonlinear MHD around a prescribed equilibrium
- linear MHD waves and instabilities (``linear=True``)
- comparing linear and perturbative variational formulations"""
[docs]
@classmethod
def doc_cannot_be_used_for(cls):
r"""This model is not suitable for:
- full-f MHD evolution without a reference equilibrium
- kinetic or particle-resolved plasma effects
- entropy- or q-based thermodynamic formulations"""
