Kinetic models#

The bulk plasma is kinetic, out of thermal equilibirum; there can be fluid components too (not bulk).

class struphy.models.kinetic.VlasovAmpereOneSpecies(with_B0: bool = True)[source]#

Bases: StruphyModel

Vlasov-Ampère equations for one species.

Normalization:

\[\begin{align} \hat v = c \,, \qquad \hat E = \hat B \hat v\,,\qquad \hat \phi = \hat E \hat x \,. \end{align}\]

Equations:

\[\begin{split}&\frac{\partial f}{\partial t} + \mathbf{v} \cdot \, \nabla f + \frac{1}{\varepsilon} \left( \mathbf{E} + \mathbf{v} \times \mathbf{B}_0 \right) \cdot \frac{\partial f}{\partial \mathbf{v}} = 0 \,, \\[2mm] -&\frac{\partial \mathbf{E}}{\partial t} = \frac{\alpha^2}{\varepsilon} \int_{\mathbb{R}^3} \mathbf{v} f \, \text{d}^3 \mathbf{v}\,,\end{split}\]

with the normalization parameter

\[\alpha = \frac{\hat \Omega_\textnormal{p}}{\hat \Omega_\textnormal{c}}\,,\qquad \varepsilon = \frac{1}{\hat \Omega_\textnormal{c} \hat t} \,,\qquad \textnormal{with} \qquad \hat\Omega_\textnormal{p} = \sqrt{\frac{\hat n (Ze)^2}{\epsilon_0 (A m_\textnormal{H})}} \,,\qquad \hat \Omega_{\textnormal{c}} = \frac{(Ze) \hat B}{(A m_\textnormal{H})}\,,\]

where \(Z=-1\) and \(A=1/1836\) for electrons. At initial time the weak Poisson equation is solved once to weakly satisfy Gauss’ law,

\[\begin{split}\begin{align} \int_\Omega \nabla \psi^\top \cdot \nabla \phi \,\textrm d \mathbf x &= \frac{\alpha^2}{\varepsilon} \int_\Omega \int_{\mathbb{R}^3} \psi\, (f - f_0) \, \text{d}^3 \mathbf{v}\,\textrm d \mathbf x \qquad \forall \ \psi \in H^1\,, \\[2mm] \mathbf{E}(t=0) &= -\nabla \phi(t=0)\,. \end{align}\end{split}\]

Moreover, it is assumed that

\[\nabla \times \mathbf B_0 = \frac{\alpha^2}{\varepsilon} \int_{\mathbb{R}^3} \mathbf{v} f_0 \, \text{d}^3 \mathbf{v}\,,\]

where \(\mathbf B_0\) is the static equilibirum magnetic field.

Notes

  • The Control variate method for Ampère’s law is optional; in case it is enabled via the parameter file, the following system is solved:

Find \((\mathbf E, f) \in H(\textnormal{curl}) \times C^\infty\) such that

\[\begin{split}\begin{align} -\int_\Omega \mathbf F\, \cdot \, &\frac{\partial \mathbf{E}}{\partial t}\,\textrm d \mathbf x = \frac{\alpha^2}{\varepsilon} \int_\Omega \int_{\mathbb{R}^3} \mathbf F \cdot \mathbf{v} (f - f_0) \, \text{d}^3 \mathbf{v}\,\textrm d \mathbf x \qquad \forall \ \mathbf F \in H(\textnormal{curl}) \,, \\[2mm] &\frac{\partial f}{\partial t} + \mathbf{v} \cdot \, \nabla f + \frac{1}{\varepsilon} \left( \mathbf{E} + \mathbf{v} \times \mathbf{B}_0 \right) \cdot \frac{\partial f}{\partial \mathbf{v}} = 0 \,. \end{align}\end{split}\]

Propagators (called in sequence):

  1. PushEta

  2. VlasovAmpere

  3. PushVxB

class EMFields[source]#

Bases: FieldSpecies

class KineticIons[source]#

Bases: ParticleSpecies

class Propagators(with_B0: bool = True)[source]#

Bases: object

allocate_helpers()[source]#

Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)[source]#

Generate a parameter file with default options for each species, and save it to the current input path.

The default name is params_<model_name>.yml.

Parameters:

  • path (str) – Alternative path to getcwd()/params_MODEL.py.

  • prompt (bool) – Whether to prompt for overwriting the specified .yml file.

Returns:

params_path – The path of the parameter file.

Return type:

str

class struphy.models.kinetic.VlasovMaxwellOneSpecies[source]#

Bases: StruphyModel

Vlasov-Maxwell equations for one species.

Normalization:

\[\begin{align} \hat v = c \,, \qquad \hat E = \hat B \hat v\,,\qquad \hat \phi = \hat E \hat x \,. \end{align}\]

Equations:

\[\begin{split}&\frac{\partial f}{\partial t} + \mathbf{v} \cdot \, \nabla f + \frac{1}{\varepsilon} \left( \mathbf{E} + \mathbf{v} \times \left( \mathbf{B} + \mathbf{B}_0 \right) \right) \cdot \frac{\partial f}{\partial \mathbf{v}} = 0 \,, \\[2mm] -&\frac{\partial \mathbf{E}}{\partial t} + \nabla \times \mathbf B = \frac{\alpha^2}{\varepsilon} \int_{\mathbb{R}^3} \mathbf{v} f \, \text{d}^3 \mathbf{v}\,, \\[2mm] &\frac{\partial \mathbf{B}}{\partial t} + \nabla \times \mathbf{E} = 0 \,,\end{split}\]

with the normalization parameters

\[\alpha = \frac{\hat \Omega_\textnormal{p}}{\hat \Omega_\textnormal{c}}\,,\qquad \varepsilon = \frac{1}{\hat \Omega_\textnormal{c} \hat t} \,,\qquad \textnormal{with} \qquad \hat\Omega_\textnormal{p} = \sqrt{\frac{\hat n (Ze)^2}{\epsilon_0 (A m_\textnormal{H})}} \,,\qquad \hat \Omega_{\textnormal{c}} = \frac{(Ze) \hat B}{(A m_\textnormal{H})}\,,\]

where \(Z=-1\) and \(A=1/1836\) for electrons. At initial time the weak Poisson equation is solved once to weakly satisfy Gauss’ law,

\[\begin{split}\begin{align} \int_\Omega \nabla \psi^\top \cdot \nabla \phi \,\textrm d \mathbf x &= \frac{\alpha^2}{\varepsilon} \int_\Omega \int_{\mathbb{R}^3} \psi\, (f - f_0) \, \text{d}^3 \mathbf{v}\,\textrm d \mathbf x \qquad \forall \ \psi \in H^1\,, \\[2mm] \mathbf{E}(t=0) &= -\nabla \phi(t=0)\,. \end{align}\end{split}\]

Moreover, it is assumed that

\[\nabla \times \mathbf B_0 = \frac{\alpha^2}{\varepsilon} \int_{\mathbb{R}^3} \mathbf{v} f_0 \, \text{d}^3 \mathbf{v}\,,\]

where \(\mathbf B_0\) is the static equilibirum magnetic field.

Notes

  • The Control variate method for Ampère’s law is optional; in case it is enabled via the parameter file, the following system is solved:

Find \((\mathbf E, \tilde{\mathbf B}, f) \in H(\textnormal{curl}) \times H(\textnormal{div}) \times C^\infty\) such that

\[\begin{split}\begin{align} -\int_\Omega \mathbf F\, \cdot \, &\frac{\partial \mathbf{E}}{\partial t}\,\textrm d \mathbf x + \int_\Omega \nabla \times \mathbf{F} \cdot \tilde{\mathbf B}\,\textrm d \mathbf x = \frac{\alpha^2}{\varepsilon} \int_\Omega \int_{\mathbb{R}^3} \mathbf F \cdot \mathbf{v} (f - f_0) \, \text{d}^3 \mathbf{v}\,\textrm d \mathbf x \qquad \forall \ \mathbf F \in H(\textnormal{curl}) \,, \\[2mm] &\frac{\partial \tilde{\mathbf B}}{\partial t} + \nabla \times \mathbf{E} = 0 \,, \\[2mm] &\frac{\partial f}{\partial t} + \mathbf{v} \cdot \, \nabla f + \frac{1}{\varepsilon}\Big[ \mathbf{E} + \mathbf{v} \times (\mathbf{B}_0 + \tilde{\mathbf B}) \Big] \cdot \frac{\partial f}{\partial \mathbf{v}} = 0 \,, \end{align}\end{split}\]

where \(\tilde{\mathbf B} = \mathbf B - \mathbf B_0\) denotes the magnetic perturbation.

Propagators (called in sequence):

  1. Maxwell

  2. PushEta

  3. PushVxB

  4. VlasovAmpere

Model info:

class EMFields[source]#

Bases: FieldSpecies

class KineticIons[source]#

Bases: ParticleSpecies

class Propagators[source]#

Bases: object

allocate_helpers()[source]#

Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)[source]#

Generate a parameter file with default options for each species, and save it to the current input path.

The default name is params_<model_name>.yml.

Parameters:

  • path (str) – Alternative path to getcwd()/params_MODEL.py.

  • prompt (bool) – Whether to prompt for overwriting the specified .yml file.

Returns:

params_path – The path of the parameter file.

Return type:

str

class struphy.models.kinetic.LinearVlasovAmpereOneSpecies(with_B0: bool = True, with_E0: bool = True)[source]#

Bases: StruphyModel

Linearized Vlasov-Ampère equations for one species.

Normalization:

\[\begin{align} \hat v = c \,, \qquad \hat E = \hat B \hat v\,,\qquad \hat \phi = \hat E \hat x \,. \end{align}\]

Equations:

\[\begin{split}\begin{align} & \frac{\partial \tilde{\mathbf E}}{\partial t} = - \frac{\alpha^2}{\varepsilon} \int_{\mathbb R^3} \mathbf{v} \tilde f\, \textrm d^3 \mathbf v \,, \\[2mm] & \frac{\partial \tilde f}{\partial t} + \mathbf{v} \cdot \, \nabla \tilde f + \frac{1}{\varepsilon} \left( \mathbf{E}_0 + \mathbf{v} \times \mathbf{B}_0 \right) \cdot \frac{\partial \tilde f}{\partial \mathbf{v}} = \frac{1}{v_{\text{th}}^2 \varepsilon} \, \tilde{\mathbf E} \cdot \mathbf{v} f_0 \,, \end{align}\end{split}\]

with the normalization parameter

\[\alpha = \frac{\hat \Omega_\textnormal{p}}{\hat \Omega_\textnormal{c}}\,,\qquad \varepsilon = \frac{1}{\hat \Omega_\textnormal{c} \hat t} \,,\qquad \textnormal{with} \qquad \hat\Omega_\textnormal{p} = \sqrt{\frac{\hat n (Ze)^2}{\epsilon_0 (A m_\textnormal{H})}} \,,\qquad \hat \Omega_{\textnormal{c}} = \frac{(Ze) \hat B}{(A m_\textnormal{H})}\,,\]

where \(Z=-1\) and \(A=1/1836\) for electrons. The background distribution function \(f_0\) is a uniform Maxwellian

\[f_0 = \frac{n_0(\mathbf{x})}{\left( \sqrt{2 \pi} v_{\text{th}} \right)^3} \exp \left( - \frac{|\mathbf{v}|^2}{2 v_{\text{th}}^2} \right) \,,\]

and the background electric field has to verify the following compatibility condition between with background density

\[\nabla_{\mathbf{x}} \ln (n_0(\mathbf{x})) = \frac{1}{v_{\text{th}}^2 \varepsilon} \mathbf{E}_0 \,.\]

At initial time the weak Poisson equation is solved once to weakly satisfy Gauss’ law,

\[\begin{split}\begin{align} \int_\Omega \nabla \psi^\top \cdot \nabla \phi \,\textrm d \mathbf x &= \frac{\alpha^2}{\varepsilon} \int_\Omega \int_{\mathbb{R}^3} \psi\, \tilde f \, \text{d}^3 \mathbf{v}\,\textrm d \mathbf x \qquad \forall \ \psi \in H^1\,, \\[2mm] \tilde{\mathbf{E}}(t=0) &= -\nabla \phi(t=0) \,. \end{align}\end{split}\]

Moreover, it is assumed that

\[\int_{\mathbb{R}^3} \mathbf{v} f_0 \, \text{d}^3 \mathbf{v} = 0 \,.\]

Propagators (called in sequence):

  1. PushEta

  2. PushVinEfield

  3. EfieldWeights

  4. PushVxB

Model info:

class EMFields[source]#

Bases: FieldSpecies

class KineticIons[source]#

Bases: ParticleSpecies

class Propagators(with_B0: bool = True, with_E0: bool = True)[source]#

Bases: object

allocate_helpers()[source]#

Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)[source]#

Generate a parameter file with default options for each species, and save it to the current input path.

The default name is params_<model_name>.yml.

Parameters:

  • path (str) – Alternative path to getcwd()/params_MODEL.py.

  • prompt (bool) – Whether to prompt for overwriting the specified .yml file.

Returns:

params_path – The path of the parameter file.

Return type:

str

class struphy.models.kinetic.LinearVlasovMaxwellOneSpecies(with_B0: bool = True, with_E0: bool = True)[source]#

Bases: LinearVlasovAmpereOneSpecies

Linearized Vlasov-Ampère equations for one species.

Normalization:

\[\begin{align} \hat v = c \,, \qquad \hat E = \hat B \hat v\,,\qquad \hat \phi = \hat E \hat x \,. \end{align}\]

Equations:

\[\begin{split}\begin{align} & \frac{\partial \tilde{\mathbf E}}{\partial t} = \nabla \times \tilde{\mathbf B} - \frac{\alpha^2}{\varepsilon} \int_{\mathbb R^3}\mathbf{v} \tilde f\, \textrm d^3 \mathbf v \,, \\[2mm] & \frac{\partial \tilde{\mathbf B}}{\partial t} = - \nabla \times \tilde{\mathbf E} \,, \\[2mm] & \frac{\partial \tilde f}{\partial t} + \mathbf{v} \cdot \, \nabla \tilde f + \frac{1}{\varepsilon} \left( \mathbf{E}_0 + \mathbf{v} \times \mathbf{B}_0 \right) \cdot \frac{\partial \tilde f}{\partial \mathbf{v}} = \frac{1}{v_{\text{th}}^2 \varepsilon} \, \tilde{\mathbf E} \cdot \mathbf{v} f_0 \,, \end{align}\end{split}\]

with the normalization parameter

\[\alpha = \frac{\hat \Omega_\textnormal{p}}{\hat \Omega_\textnormal{c}}\,,\qquad \varepsilon = \frac{1}{\hat \Omega_\textnormal{c} \hat t} \,,\qquad \textnormal{with} \qquad \hat\Omega_\textnormal{p} = \sqrt{\frac{\hat n (Ze)^2}{\epsilon_0 (A m_\textnormal{H})}} \,,\qquad \hat \Omega_{\textnormal{c}} = \frac{(Ze) \hat B}{(A m_\textnormal{H})}\,,\]

where \(Z=-1\) and \(A=1/1836\) for electrons. The background distribution function \(f_0\) is a uniform Maxwellian

\[f_0 = \frac{n_0(\mathbf{x})}{\left( \sqrt{2 \pi} v_{\text{th}} \right)^3} \exp \left( - \frac{|\mathbf{v}|^2}{2 v_{\text{th}}^2} \right) \,,\]

and the background electric field has to verify the following compatibility condition between with background density

\[\nabla_{\mathbf{x}} \ln (n_0(\mathbf{x})) = \frac{1}{v_{\text{th}}^2 \varepsilon} \mathbf{E}_0 \,.\]

At initial time the weak Poisson equation is solved once to weakly satisfy Gauss’ law,

\[\begin{split}\begin{align} \int_\Omega \nabla \psi^\top \cdot \nabla \phi \,\textrm d \mathbf x &= \frac{\alpha^2}{\varepsilon} \int_\Omega \int_{\mathbb{R}^3} \psi\, \tilde f \, \text{d}^3 \mathbf{v}\,\textrm d \mathbf x \qquad \forall \ \psi \in H^1\,, \\[2mm] \tilde{\mathbf{E}(t=0)} &= -\nabla \phi(t=0) \,. \end{align}\end{split}\]

Moreover, it is assumed that

\[\int_{\mathbb{R}^3} \mathbf{v} f_0 \, \text{d}^3 \mathbf{v} = 0 \,.\]

Propagators (called in sequence):

  1. PushEta

  2. PushVinEfield

  3. EfieldWeights

  4. PushVxB

  5. Maxwell

Model info:

class EMFields[source]#

Bases: FieldSpecies

class KineticIons[source]#

Bases: ParticleSpecies

class Propagators(with_B0: bool = True, with_E0: bool = True)[source]#

Bases: object

class struphy.models.kinetic.DriftKineticElectrostaticAdiabatic[source]#

Bases: StruphyModel

Drift-kinetic equation for one ion species in static background magnetic field, coupled to quasi-neutrality equation with adiabatic electrons.

Normalization:

\[\hat v = \hat v_\textrm{i} = \sqrt{\frac{k_B \hat T_\textrm{i}}{m_\textrm{i}}}\,,\qquad \hat E = \hat v_\textrm{i}\hat B\,,\qquad \hat \phi = \hat E \hat x \,.\]

Equations:

\[\begin{split}&\frac{\partial f}{\partial t} + \left[ v_\parallel \frac{\mathbf{B}^*}{B^*_\parallel} + \frac{\mathbf{E}^* \times \mathbf{b}_0}{B^*_\parallel}\right] \cdot \frac{\partial f}{\partial \mathbf{X}} + \left[\frac{1}{\varepsilon} \frac{\mathbf{B}^*}{B^*_\parallel} \cdot \mathbf{E}^*\right] \cdot \frac{\partial f}{\partial v_\parallel} = 0\,. \\[2mm] - &\nabla_\perp \cdot \left( \frac{n_0}{|B_0|^2} \nabla_\perp \phi \right) + \frac{1}{\varepsilon} n_0 \left(1 + \frac{1}{Z \varepsilon} \frac{1}{T_{0}} \phi \right) = \frac 1 \varepsilon \int f B^*_\parallel \,\textnormal d v_\parallel \textnormal d \mu \,.\end{split}\]

where \(f(\mathbf{X}, v_\parallel, \mu, t)\) is the guiding center distribution and

\[\mathbf{E}^* = - \nabla \phi - \varepsilon \mu \nabla |B_0| \,, \qquad \mathbf{B}^* = \mathbf{B}_0 + \varepsilon v_\parallel \nabla \times \mathbf{b}_0 \,,\qquad B^*_\parallel = \mathbf B^* \cdot \mathbf b_0 \,,\]

and with the normalization parameters

\[\varepsilon := \frac{1}{\hat \Omega_\textrm{c} \hat t}\,,\qquad \hat \Omega_\textrm{c} = \frac{q_\textrm{i} \hat B}{m_\textrm{i}} \,.\]

Notes

  • The Control variate method in the Poisson equation is optional; in case it is enabled via the parameter file, the following Poisson equation is solved:

Find \(\phi \in H^1\) such that

\[\int \frac{n_0}{|B_0|^2} \nabla_\perp \psi \cdot \nabla_\perp \phi\,\textrm d \mathbf x + \frac{1}{Z\varepsilon^2} \int \frac{n_0}{T_{0}} \psi \phi \,\textrm d \mathbf x = \frac 1 \varepsilon \int \int \psi \, (f - f_0) B^*_\parallel \,\textrm d \mathbf x\,\textnormal d v_\parallel \textnormal d \mu \qquad \forall \ \psi \in H^1\,.\]

Propagators (called in sequence):

  1. ImplicitDiffusion

  2. PushGuidingCenterBxEstar

  3. PushGuidingCenterParallel

Model info:

class EMFields[source]#

Bases: FieldSpecies

class KineticIons[source]#

Bases: ParticleSpecies

class Propagators[source]#

Bases: object

allocate_helpers()[source]#

Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)[source]#

Generate a parameter file with default options for each species, and save it to the current input path.

The default name is params_<model_name>.yml.

Parameters:

  • path (str) – Alternative path to getcwd()/params_MODEL.py.

  • prompt (bool) – Whether to prompt for overwriting the specified .yml file.

Returns:

params_path – The path of the parameter file.

Return type:

str