Hybrid model

ColdPlasmaVlasov

Hybrid cold-plasma model: a cold electron fluid coupled with a hot kinetic species and Maxwell's equations.

Overview

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.

Use cases

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

Governing equations

PDEs solved by model:

Hot Vlasov species:

\frac{\partial f}{\partial t} + v \cdot \nabla f + \frac{1}{ε _{h}} [E + v \times (B + B_{0})] \cdot \frac{\partial f}{\partial v} = 0

Cold-plasma current:

\frac{1}{n _{0}} \frac{\partial j _{c}}{\partial t} = \frac{1}{ε _{c}} E + \frac{1}{ε _{c} n _{0}} j_{c} \times B_{0}

Faraday's law:

\frac{\partial B}{\partial t} + \nabla \times E = 0

Ampère's law:

- \frac{\partial E}{\partial t} + \nabla \times B = \frac{α ^{2}}{ε _{h}} (j_{c} + \int_{R^{3}} v f d^{3} v)

where $(n_{0}, B_{0})$ denotes an inhomogeneous background.

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

\nabla \cdot E = ν \frac{α ^{2}}{ε _{h}} \int_{R^{3}} f d^{3} v

Normalization

Velocities are normalized with the speed of light and the kinetic

background is scaled as

v^= c, E^= c B^, f^= n^/ c^{3} .

The model distinguishes cold and hot cyclotron scales through $ε_{c}$ and $ε_{h}$ .

Discretization

Time integration is performed by the following propagators (in sequence):

struphy.propagators.maxwell_weak_ampere.MaxwellWeakAmpere
struphy.propagators.ohm_cold.OhmCold
struphy.propagators.jxb_cold.JxBCold
struphy.propagators.push_eta.PushEta
struphy.propagators.push_vxb.PushVxB
struphy.propagators.vlasov_ampere_coupling.VlasovAmpereCoupling

Diagnostics

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

Example

Create and initialize the cold-plasma plus Vlasov model:

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