[docs]
class LinearVlasovMaxwellOneSpecies(LinearVlasovAmpereOneSpecies):
"""Linearized Vlasov-Maxwell equations for one kinetic species around a Maxwellian background.
Parameters
----------
base_units: BaseUnits
Base units for normalization (default: BaseUnits())
charge_number: int
Charge number (in units of the positive elementary charge) of the species (default: 1)
mass_number: float
Mass number (in units of Proton mass) of the species (default: 1.0)
alpha: float, optional
Dimensionless parameter: plasma frequency / cyclotron frequency. If None, computed from units and charge/mass numbers.
epsilon: float, optional
Normalized cyclotron period: 1 / (cyclotron frequency × time unit). If None, computed from units and charge/mass numbers.
with_B0: bool
Whether to include the effect of a background magnetic field B0 (default: True)
with_E0: bool
Whether to include the effect of a background electric field E0 (default: True)
"""
@classmethod
def model_type(cls) -> LiteralOptions.ModelTypes:
return "Kinetic"
## 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 KineticIons(ParticleSpecies):
def __init__(
self,
charge_number: int = 1,
mass_number: float = 1.0,
alpha: float = None,
epsilon: float = None,
):
self.var = PICVariable(space="DeltaFParticles6D")
self.init_variables(
charge_number=charge_number,
mass_number=mass_number,
alpha=alpha,
epsilon=epsilon,
)
## propagators
class Propagators:
def __init__(
self,
with_B0: bool = True,
with_E0: bool = True,
):
self.push_eta = PushEta()
if with_E0:
self.push_vinE = PushVinEfield()
self.coupling_Eweights = EfieldWeightsCoupling()
if with_B0:
self.push_vxb = PushVxB()
self.maxwell = MaxwellWeakAmpere()
## abstract methods
def __init__(
self,
base_units: BaseUnits = BaseUnits(),
charge_number: int = 1,
mass_number: float = 1.0,
alpha: float = None,
epsilon: float = None,
with_B0: bool = True,
with_E0: bool = True,
):
# 0. store input parameters
self.params = copy.deepcopy(locals())
# 1. instantiate all species
self.em_fields = self.EMFields()
self.kinetic_ions = self.KineticIons(
charge_number,
mass_number,
alpha,
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(with_B0=with_B0, with_E0=with_E0)
# 4. assign variables to propagators
self.propagators.push_eta.variables.var = self.kinetic_ions.var
if with_E0:
self.propagators.push_vinE.variables.var = self.kinetic_ions.var
self.propagators.coupling_Eweights.variables.e = self.em_fields.e_field
self.propagators.coupling_Eweights.variables.ions = self.kinetic_ions.var
if with_B0:
self.propagators.push_vxb.variables.ions = self.kinetic_ions.var
self.propagators.maxwell.variables.e = self.em_fields.e_field
self.propagators.maxwell.variables.b = self.em_fields.b_field
# 5. define scalars to be tracked during simulation
electric_energy = BilinearEnergyFEEC(self.em_fields.e_field)
magnetic_energy = BilinearEnergyFEEC(self.em_fields.b_field)
particle_energy = FunctionScalarPIC(
self._compute_en_w,
self.kinetic_ions.var,
)
self.scalars = Scalars(
en_E=electric_energy,
en_B=magnetic_energy,
en_w=particle_energy,
en_tot=electric_energy + magnetic_energy + particle_energy,
)
# initial Poisson (not a propagator used in time stepping)
self.initial_poisson = PoissonSolve()
self.initial_poisson.variables.phi = self.em_fields.phi
[docs]
@classmethod
def doc_pde(cls):
r"""**PDEs solved by model:**
Linearized Ampère's law:
.. math::
\frac{\partial \tilde{\mathbf{E}}}{\partial t} = \nabla \times \tilde{\mathbf{B}} - \frac{\alpha^2}{\varepsilon} \int_{\mathbb{R}^3} \mathbf{v} \tilde{f} \, \textrm{d}^3 \mathbf{v}
Linearized Faraday's law:
.. math::
\frac{\partial \tilde{\mathbf{B}}}{\partial t} = -\nabla \times \tilde{\mathbf{E}}
Linearized Vlasov equation:
.. math::
\frac{\partial \tilde{f}}{\partial t} + \mathbf{v} \cdot \nabla \tilde{f} + \frac{1}{\varepsilon} \left( \mathbf{E}_0 + \mathbf{v} \times \mathbf{B}_0 \right) \cdot \frac{\partial \tilde{f}}{\partial \mathbf{v}} = \frac{1}{v_{\text{th}}^2 \varepsilon} \tilde{\mathbf{E}} \cdot \mathbf{v} f_0
where :math:`Z=-1` and :math:`A=1/1836` for electrons. The background distribution function :math:`f_0` is a uniform Maxwellian
.. math::
f_0 = \frac{n_0(\mathbf{x})}{\left( \sqrt{2 \pi} v_{\text{th}} \right)^3} \exp \left( - \frac{|\mathbf{v}|^2}{2 v_{\text{th}}^2} \right)
and the background electric field has to verify the following compatibility condition between with background density
.. math::
\nabla_{\mathbf{x}} \ln (n_0(\mathbf{x})) = \frac{1}{v_{\text{th}}^2 \varepsilon} \mathbf{E}_0
At initial time the weak Poisson equation is solved once to weakly satisfy Gauss' law,
.. math::
\int_\Omega \nabla \psi^\top \cdot \nabla \phi \, \textrm{d} \mathbf{x}
&= \frac{\alpha^2}{\varepsilon} \int_\Omega \int_{\mathbb{R}^3} \psi \, \tilde{f} \, \text{d}^3 \mathbf{v} \, \textrm{d} \mathbf{x}
\qquad \forall \ \psi \in H^1
\\[2mm]
\tilde{\mathbf{E}}(t=0) &= -\nabla \phi(t=0)
Moreover, it is assumed that
.. math::
\int_{\mathbb{R}^3} \mathbf{v} f_0 \, \text{d}^3 \mathbf{v} = 0
"""
[docs]
@classmethod
def doc_normalization(cls):
r"""The normalization matches the linear Vlasov-Ampère model:
.. math::
\hat v = c,\qquad \hat E = \hat B \hat v,\qquad \hat\phi = \hat E \hat x.
The model retains the same :math:`\alpha` and :math:`\varepsilon`
parameters while adding magnetic perturbation dynamics."""
[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``
- Perturbation particle energy: ``en_w``
- Total energy: ``en_tot``"""
[docs]
@classmethod
def doc_discretization(cls):
"""Time integration is performed by the following propagators (in sequence):
1. :class:`~struphy.propagators.push_eta.PushEta`
2. :class:`~struphy.propagators.push_vin_efield.PushVinEfield` (if :attr:`with_E0` is True)
3. :class:`~struphy.propagators.efield_weights_coupling.EfieldWeightsCoupling`
4. :class:`~struphy.propagators.push_vxb.PushVxB` (if :attr:`with_B0` is True)
5. :class:`~struphy.propagators.maxwell_weak_ampere.MaxwellWeakAmpere`
"""
doc = rf"""**1. push_eta.PushEta:**
{PushEta.__doc__}
**2. push_vin_efield.PushVinEfield:**
{PushVinEfield.__doc__}
**3. efield_weights_coupling.EfieldWeightsCoupling:**
{EfieldWeightsCoupling.__doc__}
**4. push_vxb.PushVxB:**
{PushVxB.__doc__}
**5. propagators.maxwell.Maxwell:**
{MaxwellWeakAmpere.__doc__}
"""
return doc
[docs]
@classmethod
def doc_long_description(cls):
r"""LinearVlasovMaxwellOneSpecies is the electromagnetic extension of the
linear delta-f Vlasov model. It is intended for weakly perturbed kinetic
electromagnetic waves and instabilities around a prescribed equilibrium."""
[docs]
@classmethod
def doc_examples(cls):
r"""Create and initialize the linear Vlasov-Maxwell model:
.. code-block:: python
from struphy.models import LinearVlasovMaxwellOneSpecies
model = LinearVlasovMaxwellOneSpecies()
model.em_fields.e_field
model.em_fields.b_field
model.kinetic_ions.var
"""
[docs]
@classmethod
def doc_use_cases(cls):
r"""This model is appropriate for:
- linear electromagnetic kinetic wave studies
- delta-f verification of linear Vlasov-Maxwell coupling
- weakly nonlinear regimes where equilibrium perturbations stay small"""
[docs]
@classmethod
def doc_cannot_be_used_for(cls):
r"""This model is not suitable for:
- strongly nonlinear fully kinetic plasma dynamics
- multi-species electromagnetic coupling
- problems without a well-defined linearization background
- collisional kinetic closures"""
