Fluid-kinetic hybrid models

The bulk plasma is fluid, but there is also at least one kinetic species (e.g. energetic particles).

class struphy.models.hybrid.LinearMHDVlasovCC

Bases: StruphyModel

Hybrid linear MHD + energetic ions (6D Vlasov) with current coupling scheme.

Normalization:

\[\hat U = \hat v = \hat v_\textnormal{A} \,, \qquad \hat f_\textnormal{h} = \frac{\hat n}{\hat v_\textnormal{A}^3} \,.\]

Equations:

\[\begin{split}\begin{align} \textnormal{MHD}\,\, &\left\{\,\, \begin{aligned} &\frac{\partial \tilde{\rho}}{\partial t}+\nabla\cdot(\rho_0 \tilde{\mathbf{U}})=0\,, \\[2mm] \rho_0 &\frac{\partial \tilde{\mathbf{U}}}{\partial t} + \nabla \tilde p =(\nabla\times \tilde{\mathbf{B}})\times\mathbf{B}_0 + \mathbf{J}_0\times \tilde{\mathbf{B}} \color{blue} + \frac{A_\textnormal{h}}{A_\textnormal{b}} \frac{1}{\varepsilon} \left(n_\textnormal{h}\tilde{\mathbf{U}}-n_\textnormal{h}\mathbf{u}_\textnormal{h}\right)\times(\mathbf{B}_0+\tilde{\mathbf{B}}) \color{black}\,, \\[2mm] &\frac{\partial \tilde p}{\partial t} + (\gamma-1)\nabla\cdot(p_0 \tilde{\mathbf{U}}) + p_0\nabla\cdot \tilde{\mathbf{U}}=0\,, \\[2mm] &\frac{\partial \tilde{\mathbf{B}}}{\partial t} = \nabla\times(\tilde{\mathbf{U}} \times \mathbf{B}_0)\,,\qquad \nabla\cdot\tilde{\mathbf{B}}=0\,, \end{aligned} \right. \\[2mm] \textnormal{EPs}\,\, &\left\{\,\, \begin{aligned} &\quad\,\,\frac{\partial f_\textnormal{h}}{\partial t}+\mathbf{v}\cdot\nabla f_\textnormal{h} + \frac{1}{\varepsilon} \left[\color{blue} (\mathbf{B}_0+\tilde{\mathbf{B}})\times\tilde{\mathbf{U}} \color{black} + \mathbf{v}\times(\mathbf{B}_0+\tilde{\mathbf{B}})\right]\cdot \frac{\partial f_\textnormal{h}}{\partial \mathbf{v}} =0\,, \\[2mm] &\quad\,\,n_\textnormal{h}=\int_{\mathbb{R}^3}f_\textnormal{h}\,\textnormal{d}^3 \mathbf v\,,\qquad n_\textnormal{h}\mathbf{u}_\textnormal{h}=\int_{\mathbb{R}^3}f_\textnormal{h}\mathbf{v}\,\textnormal{d}^3 \mathbf v\,, \end{aligned} \right. \end{align}\end{split}\]

where \(\mathbf{J}_0 = \nabla\times\mathbf{B}_0\) and

\[\varepsilon = \frac{1}{\hat \Omega_{\textnormal{c,hot}} \hat t}\,,\qquad \textnormal{with} \qquad\hat \Omega_{\textnormal{c,hot}} = \frac{Z_\textnormal{h}e \hat B}{A_\textnormal{h} m_\textnormal{H}}\,.\]

Propagators (called in sequence):

1. CurrentCoupling6DDensity

2. ShearAlfven

3. CurrentCoupling6DCurrent

4. PushEta

5. PushVxB

6. Magnetosonic

Model info:

class EMFields

Bases: FieldSpecies

class MHD

Bases: FluidSpecies

class EnergeticIons

Bases: ParticleSpecies

class Propagators

Bases: object

allocate_helpers()

Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)

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.hybrid.LinearMHDVlasovPC(turn_off: tuple[str, ...] = (None,))

Bases: StruphyModel

Hybrid linear MHD + energetic ions (6D Vlasov) with pressure coupling scheme.

Normalization:

\[\hat U = \hat v =: \hat v_\textnormal{A, bulk} \,, \qquad \hat f_\textnormal{h} = \frac{\hat n}{\hat v_\textnormal{A}^3} \,,\qquad \hat{\mathbb{P}}_\textnormal{h} = A_\textnormal{h}m_\textnormal{H}\hat n \hat v_\textnormal{A}^2\,,\]

Implemented equations:

\[\begin{split}\begin{align} \textnormal{MHD} &\left\{ \begin{aligned} &\frac{\partial \tilde{\rho}}{\partial t}+\nabla\cdot(\rho_0 \tilde{\mathbf{U}})=0\,, \\ \rho_0 &\frac{\partial \tilde{\mathbf{U}}}{\partial t} + \nabla \tilde p + \frac{A_\textnormal{h}}{A_\textnormal{b}} \nabla\cdot \tilde{\mathbb{P}}_{\textnormal{h},\perp} =(\nabla\times \tilde{\mathbf{B}})\times\mathbf{B}_0 + \mathbf{J}_0\times \tilde{\mathbf{B}} \,, \qquad \mathbf{J}_0 = \nabla\times\mathbf{B}_0\,, \\ &\frac{\partial \tilde p}{\partial t} + \nabla\cdot(p_0 \tilde{\mathbf{U}}) + \frac{2}{3}\,p_0\nabla\cdot \tilde{\mathbf{U}}=0\,, \\ &\frac{\partial \tilde{\mathbf{B}}}{\partial t} - \nabla\times(\tilde{\mathbf{U}} \times \mathbf{B}_0) = 0\,, \end{aligned} \right. \\[2mm] \textnormal{EPs}\,\, &\left\{\,\, \begin{aligned} &\quad\,\,\frac{\partial f_\textnormal{h}}{\partial t} + (\mathbf{v} + \tilde{\mathbf{U}}_\perp)\cdot \nabla f_\textnormal{h} + \left[\frac{1}{\epsilon}\, \mathbf{v}\times(\mathbf{B}_0 + \tilde{\mathbf{B}}) - \nabla \tilde{\mathbf{U}}_\perp\cdot \mathbf{v} \right]\cdot \frac{\partial f_\textnormal{h}}{\partial \mathbf{v}} = 0\,, \\ &\quad\,\,\tilde{\mathbb{P}}_{\textnormal{h},\perp} = \int \mathbf{v}_\perp\mathbf{v}^\top_\perp f_\textnormal{h} d\mathbf{v} \,, \end{aligned} \right. \end{align}\end{split}\]

where

\[\epsilon = \frac{\hat \omega}{2 \pi \, \hat \Omega_{\textnormal{c,hot}}} \,,\qquad \textnormal{with} \qquad\hat \Omega_{\textnormal{c,hot}} = \frac{Z_\textnormal{h}e \hat B}{A_\textnormal{h} m_\textnormal{H}}\,.\]

Propagators (called in sequence):

1. PushEtaPC

2. PushVxB

3. PressureCoupling6D

4. ShearAlfven

5. Magnetosonic

Model info:

class EnergeticIons

Bases: ParticleSpecies

class EMFields

Bases: FieldSpecies

class MHD

Bases: FluidSpecies

class Propagators(turn_off: tuple[str, ...] = (None,))

Bases: object

allocate_helpers()

Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)

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.hybrid.LinearMHDDriftkineticCC(turn_off: tuple[str, ...] = (None,))

Bases: StruphyModel

Hybrid linear ideal MHD + energetic ions (5D Driftkinetic) with current coupling scheme.

Normalization:

\[\hat U = \hat v =: \hat v_\textnormal{A, bulk} \,, \qquad \hat f_\textnormal{h} = \frac{\hat n}{\hat v_\textnormal{h} \hat \mu \hat B} \,,\qquad \hat \mu = \frac{A_\textnormal{h} m_\textnormal{H} \hat v_\textnormal{h}^2}{\hat B} \,.\]

Equations:

\[\begin{split}\begin{align} \textnormal{MHD} &\left\{ \begin{aligned} &\frac{\partial \tilde{\rho}}{\partial t}+\nabla\cdot(\rho_{0} \tilde{\mathbf{U}})=0\,, \\ \rho_{0} &\frac{\partial \tilde{\mathbf{U}}}{\partial t} - \tilde p\, \nabla = (\nabla \times \tilde{\mathbf{B}}) \times \mathbf{B} + (\nabla \times \mathbf B_0) \times \tilde{\mathbf{B}} + \frac{A_\textnormal{h}}{A_\textnormal{b}} \left[ \frac{1}{\epsilon} n_\textnormal{gc} \tilde{\mathbf{U}} - \frac{1}{\epsilon} \mathbf{J}_\textnormal{gc} - \nabla \times \mathbf{M}_\textnormal{gc} \right] \times \mathbf{B} \,, \\ &\frac{\partial \tilde p}{\partial t} + \nabla\cdot(p_0 \tilde{\mathbf{U}}) + \frac{2}{3}\,p_0\nabla\cdot \tilde{\mathbf{U}}=0\,, \\ &\frac{\partial \tilde{\mathbf{B}}}{\partial t} - \nabla\times(\tilde{\mathbf{U}} \times \mathbf{B}) = 0\,, \end{aligned} \right. \\[2mm] \textnormal{EPs}\,\, &\left\{\,\, \begin{aligned} \quad &\frac{\partial f_\textnormal{h}}{\partial t} + \frac{1}{B_\parallel^*}(v_\parallel \mathbf{B}^* - \mathbf{b}_0 \times \mathbf{E}^*)\cdot\nabla f_\textnormal{h} + \frac{1}{\epsilon} \frac{1}{B_\parallel^*} (\mathbf{B}^* \cdot \mathbf{E}^*) \frac{\partial f_\textnormal{h}}{\partial v_\parallel} = 0\,, \\ & n_\textnormal{gc} = \int f_\textnormal{h} B_\parallel^* \,\textnormal dv_\parallel \textnormal d\mu \,, \\ & \mathbf{J}_\textnormal{gc} = \int \frac{f_\textnormal{h}}{B_\parallel^*}(v_\parallel \mathbf{B}^* - \mathbf{b}_0 \times \mathbf{E}^*) \,\textnormal dv_\parallel \textnormal d\mu \,, \\ & \mathbf{M}_\textnormal{gc} = - \int f_\textnormal{h} B_\parallel^* \mu \mathbf{b}_0 \,\textnormal dv_\parallel \textnormal d\mu \,, \end{aligned} \right. \end{align}\end{split}\]

where

\[\begin{split}\begin{align} B^*_\parallel = \mathbf{b}_0 \cdot \mathbf{B}^*\,, \\[2mm] \mathbf{B}^* &= \mathbf{B} + \epsilon v_\parallel \nabla \times \mathbf{b}_0 \,, \\[2mm] \mathbf{E}^* &= - \tilde{\mathbf{U}} \times \mathbf{B} - \epsilon \mu \nabla (\mathbf{b}_0 \cdot \mathbf{B}) \,, \end{align}\end{split}\]

with the normalization parameter

\[\epsilon = \frac{1}{\hat \Omega_\textnormal{c,hot} \hat t} \,, \qquad \hat \Omega_\textnormal{c,hot} = \frac{Z_\textnormal{h} e \hat B}{A_\textnormal{h} m_\textnormal{H}} \,.\]

Propagators (called in sequence):

1. PushGuidingCenterBxEstar

2. PushGuidingCenterParallel

3. CurrentCoupling5DGradB

4. CurrentCoupling5DCurlb

5. CurrentCoupling5DDensity

6. ShearAlfvenCurrentCoupling5D

7. Magnetosonic

Model info:

class EnergeticIons

Bases: ParticleSpecies

class EMFields

Bases: FieldSpecies

class MHD

Bases: FluidSpecies

class Propagators(turn_off: tuple[str, ...] = (None,))

Bases: object

allocate_helpers()

Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)

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.hybrid.ColdPlasmaVlasov

Bases: StruphyModel

Cold plasma hybrid model.

Normalization:

\[\hat v = c\,,\qquad \hat E = c \hat B \,,\qquad \hat f = \frac{\hat n}{c^3} \,.\]

Equations:

\[\begin{split}&\frac{\partial f}{\partial t} + \mathbf{v} \cdot \, \nabla f + \frac{1}{\varepsilon_\textnormal{h}}\Big[ \mathbf{E} + \mathbf{v} \times \left( \mathbf{B} + \mathbf{B}_0 \right) \Big] \cdot \frac{\partial f}{\partial \mathbf{v}} = 0 \,, \\[2mm] \frac{1}{n_0} &\frac{\partial \mathbf j_\textnormal{c}}{\partial t} = \frac{1}{\varepsilon_\textnormal{c}} \mathbf E + \frac{1}{\varepsilon_\textnormal{c} n_0} \mathbf j_\textnormal{c} \times \mathbf B_0\,, \\[2mm] &\frac{\partial \mathbf B}{\partial t} + \nabla\times\mathbf E = 0\,, \\[2mm] -&\frac{\partial \mathbf E}{\partial t} + \nabla\times\mathbf B = \frac{\alpha^2}{\varepsilon_\textnormal{h}} \left( \mathbf j_\textnormal{c} + \int_{\mathbb{R}^3} \mathbf{v} f \, \text{d}^3 \mathbf{v} \right) \,,\end{split}\]

where \((n_0,\mathbf B_0)\) denotes a (inhomogeneous) background and

\[\alpha = \frac{\hat \Omega_\textnormal{p,cold}}{\hat \Omega_\textnormal{c,cold}}\,, \qquad \varepsilon_\textnormal{c} = \frac{1}{\hat \Omega_\textnormal{c,cold} \hat t}\,, \qquad \varepsilon_\textnormal{h} = \frac{1}{\hat \Omega_\textnormal{c,hot} \hat t} \,.\]

At initial time the Poisson equation is solved once to weakly satisfy the Gauss law:

\[\begin{align} \nabla \cdot \mathbf{E} & = \nu \frac{\alpha^2}{\varepsilon_\textnormal{h}} \int_{\mathbb{R}^3} f \, \text{d}^3 \mathbf{v}\,. \end{align}\]

Propagators (called in sequence):

1. Maxwell

2. OhmCold

3. JxBCold

4. PushVxB

5. PushEta

6. VlasovAmpere

class EMFields

Bases: FieldSpecies

class ThermalElectrons

Bases: FluidSpecies

class HotElectrons

Bases: ParticleSpecies

class Propagators

Bases: object

allocate_helpers()

Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)

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