[docs]
class LinearVlasovMaxwellOneSpecies(LinearVlasovAmpereOneSpecies):
"""Linearized Vlasov-Maxwell equations for one kinetic species around a Maxwellian background.
Extends :class:`~struphy.models.LinearVlasovAmpereOneSpecies` by the magnetic field perturbation (Faraday's law
and the curl of the magnetic field in Ampère's law).
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.
"""
## additional species variables
class EMFields(LinearVlasovAmpereOneSpecies.EMFields):
def __init__(self):
# declared before the parent's variables, whose init_variables() then includes it
self.b_field = FEECVariable(space="Hdiv")
super().__init__()
## additional propagators
class Propagators(LinearVlasovAmpereOneSpecies.Propagators):
"""Propagators of :class:`~struphy.models.LinearVlasovAmpereOneSpecies` plus ``maxwell`` (Faraday's law and
the curl of the magnetic field in Ampère's law).
Parameters
----------
**kwargs
Passed to :class:`LinearVlasovAmpereOneSpecies.Propagators
<struphy.models.LinearVlasovAmpereOneSpecies.Propagators>` (and from there to
:class:`Vlasov.Propagators <struphy.models.Vlasov.Propagators>`).
"""
splitting_order = ("push_eta", "push_vinE", "coupling_Eweights", "push_vxb", "maxwell")
def __init__(self, **kwargs):
super().__init__(**kwargs)
self.maxwell = MaxwellWeakAmpere()
# attribute types for IDEs and type checkers: the classes of this model, not those of the parent
em_fields: EMFields
propagators: Propagators
## abstract methods
def __init__(
self,
base_units: BaseUnits = BaseUnits(),
charge_number: int = 1,
mass_number: float = 1.0,
alpha: float = None,
epsilon: float = None,
):
# 0. store input parameters
self._store_params(locals())
# 1.-5. and initial Poisson, see LinearVlasovAmpereOneSpecies
super().__init__(base_units, charge_number, mass_number, alpha, epsilon)
def _link_propagator_variables(self):
""":meta private:"""
super()._link_propagator_variables()
self.propagators.maxwell.variables.e = self.em_fields.e_field
self.propagators.maxwell.variables.b = self.em_fields.b_field
def _create_scalars(self):
""":meta private:"""
scalars = super()._create_scalars()
scalars.pop("en_tot")
scalars["en_B"] = BilinearEnergyFEEC(self.em_fields.b_field)
scalars["en_tot"] = scalars["en_E"] + scalars["en_B"] + scalars["en_f"]
return scalars
[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_f``
- Total energy: ``en_tot``"""
[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"""
