[docs]
class ColdPlasmaVlasov(StruphyModel):
"""Hybrid cold-plasma model: a cold electron fluid coupled with a hot kinetic species and Maxwell's equations.
Parameters
----------
base_units: BaseUnits
Base units for normalization (default: BaseUnits())
thermal_charge_number: int
Charge number of the cold (fluid) electron species (default: -1)
thermal_mass_number: float
Mass number of the cold (fluid) electron species (default: 1/1836)
hot_charge_number: int
Charge number of the hot (kinetic) electron species (default: -1)
hot_mass_number: float
Mass number of the hot (kinetic) electron species (default: 1/1836)
thermal_alpha: float, optional
Dimensionless parameter: cold-species plasma frequency / cyclotron frequency. If None, computed from units and charge/mass numbers.
thermal_epsilon: float, optional
Normalized cyclotron period of the cold species. If None, computed from units and charge/mass numbers.
hot_epsilon: float, optional
Normalized cyclotron period of the hot species. If None, computed from units and charge/mass numbers.
"""
@classmethod
def model_type(cls) -> LiteralOptions.ModelTypes:
return "Hybrid"
## species
class EMFields(FieldSpecies):
def __init__(self):
self.e_field = FEECVariable(space="Hcurl")
self.b_field = FEECVariable(space="Hdiv")
self.phi = FEECVariable(space="H1")
self.init_variables()
class ThermalElectrons(FluidSpecies):
def __init__(
self,
charge_number: int,
mass_number: float,
alpha: float,
epsilon: float,
):
self.current = FEECVariable(space="Hcurl")
self.init_variables(
charge_number=charge_number,
mass_number=mass_number,
alpha=alpha,
epsilon=epsilon,
)
class HotElectrons(ParticleSpecies):
def __init__(
self,
charge_number: int,
mass_number: float,
epsilon: float,
):
self.var = PICVariable(space="Particles6D")
self.init_variables(
charge_number=charge_number,
mass_number=mass_number,
epsilon=epsilon,
)
## propagators
class Propagators:
def __init__(self):
self.maxwell = MaxwellWeakAmpere()
self.ohm = OhmCold()
self.jxb = JxBCold()
self.push_eta = PushEta()
self.push_vxb = PushVxB()
self.coupling_va = VlasovAmpereCoupling()
## abstract methods
def __init__(
self,
base_units: BaseUnits = BaseUnits(),
thermal_charge_number: int = -1,
thermal_mass_number: float = 1 / 1836,
hot_charge_number: int = -1,
hot_mass_number: float = 1 / 1836,
thermal_alpha: float = None,
thermal_epsilon: float = None,
hot_epsilon: float = None,
):
# 0. store input parameters
self.params = copy.deepcopy(locals())
# 1. instantiate all species
self.em_fields = self.EMFields()
self.thermal_elec = self.ThermalElectrons(
thermal_charge_number,
thermal_mass_number,
thermal_alpha,
thermal_epsilon,
)
self.hot_elec = self.HotElectrons(
hot_charge_number,
hot_mass_number,
hot_epsilon,
)
# 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()
# 4. assign variables to propagators
self.propagators.maxwell.variables.e = self.em_fields.e_field
self.propagators.maxwell.variables.b = self.em_fields.b_field
self.propagators.ohm.variables.j = self.thermal_elec.current
self.propagators.ohm.variables.e = self.em_fields.e_field
self.propagators.jxb.variables.j = self.thermal_elec.current
self.propagators.push_eta.variables.var = self.hot_elec.var
self.propagators.push_vxb.variables.ions = self.hot_elec.var
self.propagators.coupling_va.variables.e = self.em_fields.e_field
self.propagators.coupling_va.variables.ions = self.hot_elec.var
# 5. define scalars to be tracked during simulation
electric_energy = BilinearEnergyFEEC(self.em_fields.e_field)
magnetic_energy = BilinearEnergyFEEC(self.em_fields.b_field)
current_energy = BilinearEnergyFEEC(
self.thermal_elec.current,
bilinear_form_name="M1ninv",
normalization=self.thermal_elec.equation_params.alpha**2,
)
particle_energy = KineticEnergyPIC(
self.hot_elec.var,
normalization=self.hot_elec.equation_params.alpha**2,
)
self.scalars = Scalars(
en_E=electric_energy,
en_B=magnetic_energy,
en_J=current_energy,
en_f=particle_energy,
en_tot=electric_energy + magnetic_energy + current_energy + particle_energy,
)
# initial Poisson (not a propagator used in time stepping)
hot_alpha = self.hot_elec.equation_params.alpha
hot_epsilon = self.hot_elec.equation_params.epsilon
particles_to_grid = ParticlesToGrid(
self.hot_elec.var,
"H1",
Pyccelkernel(accum_kernels.charge_density_0form),
)
self.initial_poisson = PoissonSolve(
rho=particles_to_grid,
rho_coeffs=hot_alpha**2 / hot_epsilon,
)
self.initial_poisson.variables.phi = self.em_fields.phi
@property
def bulk_species(self):
return self.thermal_elec
@property
def velocity_scale(self):
return "light"
def allocate_helpers(self):
"""Solve initial Poisson equation.
:meta private:
"""
logger.info("\nINITIAL POISSON SOLVE:")
# use control variate method
particles = self.hot_elec.var.particles
particles.update_weights()
self.initial_poisson.allocate()
# Solve with dt=1. and compute electric field
logger.info("\nSolving initial Poisson problem...")
self.initial_poisson(1.0)
phi = self.initial_poisson.variables.phi.spline.vector
Propagator.derham.grad.dot(-phi, out=self.em_fields.e_field.spline.vector)
logger.info("... Done.")
## 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 "coupling_va.Options" in line:
new_file += [line]
new_file += ["model.initial_poisson.options = model.initial_poisson.Options()\n"]
elif "saving_params = " in line:
new_file += ["\nbinplot = BinningPlot(slice='e1', n_bins=128, ranges=(0.0, 1.0))\n"]
new_file += ["saving_params = SavingParameters(binning_plots=(binplot,))\n\n"]
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:**
Hot Vlasov species:
.. math::
\frac{\partial f}{\partial t} + \mathbf{v} \cdot \nabla f + \frac{1}{\varepsilon_\textnormal{h}} \Big[ \mathbf{E} + \mathbf{v} \times \left( \mathbf{B} + \mathbf{B}_0 \right) \Big] \cdot \frac{\partial f}{\partial \mathbf{v}} = 0
Cold-plasma current:
.. math::
\frac{1}{n_0} \frac{\partial \mathbf{j}_\textnormal{c}}{\partial t} = \frac{1}{\varepsilon_\textnormal{c}} \mathbf{E} + \frac{1}{\varepsilon_\textnormal{c} n_0} \mathbf{j}_\textnormal{c} \times \mathbf{B}_0
Faraday's law:
.. math::
\frac{\partial \mathbf{B}}{\partial t} + \nabla \times \mathbf{E} = 0
Ampère's law:
.. math::
-\frac{\partial \mathbf{E}}{\partial t} + \nabla \times \mathbf{B} = \frac{\alpha^2}{\varepsilon_\textnormal{h}} \left( \mathbf{j}_\textnormal{c} + \int_{\mathbb{R}^3} \mathbf{v} f \, \text{d}^3 \mathbf{v} \right)
where :math:`(n_0, \mathbf{B}_0)` denotes an inhomogeneous background.
At initial time the Poisson equation is solved once to weakly satisfy the Gauss law:
.. math::
\nabla \cdot \mathbf{E} = \nu \frac{\alpha^2}{\varepsilon_\textnormal{h}} \int_{\mathbb{R}^3} f \, \text{d}^3 \mathbf{v}
"""
[docs]
@classmethod
def doc_normalization(cls):
r"""Velocities are normalized with the speed of light and the kinetic
background is scaled as
.. math::
\hat v = c,\qquad \hat E = c \hat B,\qquad \hat f = \hat n / c^3.
The model distinguishes cold and hot cyclotron scales through
:math:`\varepsilon_\mathrm{c}` and :math:`\varepsilon_\mathrm{h}`."""
[docs]
@classmethod
def doc_scalar_quantities(cls):
r"""**The following scalars are tracked during simulation:**
- Electric field energy: ``en_E``
- Magnetic field energy: ``en_B``
- Cold-current energy: ``en_J``
- Hot-particle kinetic energy: ``en_f``
- Total energy: ``en_tot``"""
[docs]
@classmethod
def doc_discretization(cls):
"""Time integration is performed by the following propagators (in sequence):
1. :class:`~struphy.propagators.maxwell_weak_ampere.MaxwellWeakAmpere`
2. :class:`~struphy.propagators.ohm_cold.OhmCold`
3. :class:`~struphy.propagators.jxb_cold.JxBCold`
4. :class:`~struphy.propagators.push_eta.PushEta`
5. :class:`~struphy.propagators.push_vxb.PushVxB`
6. :class:`~struphy.propagators.vlasov_ampere_coupling.VlasovAmpereCoupling`
"""
doc = rf"""**1. MaxwellWeakAmpere:**
{MaxwellWeakAmpere.__doc__}
**2. OhmCold:**
{OhmCold.__doc__}
**3. JxBCold:**
{JxBCold.__doc__}
**4. PushEta:**
{PushEta.__doc__}
**5. PushVxB:**
{PushVxB.__doc__}
**6. VlasovAmpereCoupling:**
{VlasovAmpereCoupling.__doc__}
"""
return doc
[docs]
@classmethod
def doc_long_description(cls):
r"""ColdPlasmaVlasov is an electromagnetic hybrid model that keeps a cold
fluid response for the thermal electrons and a kinetic PIC description
for a hot species. It is designed for problems where the energetic
population matters kinetically but the bulk response can still be
treated as cold."""
[docs]
@classmethod
def doc_examples(cls):
r"""Create and initialize the cold-plasma plus Vlasov model:
.. code-block:: python
from struphy.models import ColdPlasmaVlasov
model = ColdPlasmaVlasov()
model.em_fields.e_field
model.em_fields.b_field
model.thermal_elec.current
model.hot_elec.var
"""
[docs]
@classmethod
def doc_use_cases(cls):
r"""This model is appropriate for:
- hybrid electromagnetic problems with one hot kinetic species
- energetic-particle interaction with a cold background plasma
- reduced-cost alternatives to fully kinetic multi-population models
- benchmarks of fluid-kinetic current coupling"""
[docs]
@classmethod
def doc_cannot_be_used_for(cls):
r"""This model is not suitable for:
- fully warm-fluid or pressure-anisotropic background dynamics
- all-species kinetic simulations
- collision operators or detailed dissipative closures
- electrostatic-only reductions where magnetic evolution is irrelevant"""
