A Domain-Hybridized Plasma Model Using Discontinuous Galerkin Finite Elements #
Models #
Keep in mind that the behavior of a plasma is primarily determined by the ratio between a spatial length of interest and the following reference lengths. The appropriate model for describing a plasma depends on the regime defined by these lengths.
| Reference length | |
|---|---|
| Degree of magnetization | Larmor radius \( r_L \) |
| Charge separation | Debye length \( \lambda_D \) |
| Collisionality | Mean free path \( \lambda_{mfp} \) |
Continuum Kinetic Model #
The continuum kinetic model is the most complete description, taking into account the forces acting upon each particle:
\[\dv{\vec v_i}{t} = \frac{q_i}{m_i} \left( \vec E^M + \vec v_i \cross \vec B^M \right)^\prime\]where \( M \) signifies the microscopic fields generated by particles and the \( \prime \) signifies the force on particle \( i \) excludes the self-force. Given an ensemble of \( \overline{N_\alpha} \) particles, the density in phase space is
\[N_\alpha (\vec x, \vec v, t) = \sum_{1 \leq i \leq \overline{N_\alpha}} \delta [\vec x - \vec x_i (t) ] \delta [\vec v - \vec v_i(t)]\]Taking the total derivative of \( N_\alpha \) we get the Klimontovich equation
\[\dv{N_\alpha (\vec x, \vec v, t)}{t} = \pdv{N_{\alpha} (\vec x, \vec v, t)}{t} + \vec v \cdot \pdv{N_\alpha (\vec x, \vec v, t)}{\vec x} + \frac{q_\alpha}{m_\alpha} \left( \vec E^M + \vec v_i \cross \vec B^M \right)^\prime \cdot \pdv{N_\alpha(\vec x, \vec v, t)}{\vec v} = 0\]A tractable kinetic model evolves the smooth probability distribution function \( f_\alpha \) of species \( \alpha \)
\[\pdv{f_\alpha}{t} + v_i \pdv{f_\alpha}{x_i} + \frac{q_\alpha}{m_\alpha} (E_i + \epsilon_{ijk} v_j B_k) \pdv{f_\alpha}{v_i} = \left. \pdv{f_\alpha}{t} \right| _C\]Or, in conservative form:
\[\pdv{f_\alpha}{t} + \pdv{}{x_i} (v_i f_\alpha) + \pdv{}{v_i} \left[ \frac{q_\alpha}{m_\alpha} (E_i + \epsilon_{ijk} v_j B_k) f_\alpha \right] = \left. \pdv{f_\alpha}{t} \right| _C\]5N-moment Model #
We take velocity-space moments of the Boltzmann equation to yield conservation laws for density, momentum, and energy
Continuity equation #
\[\pdv{\rho _{\alpha}}{t} + \pdv{(\rho v_{\alpha i})}{x_i} = 0\]Momentum equation #
\[\pdv{\rho_\alpha v_{\alpha i}}{t} + \pdv{(\rho_\alpha v_{\alpha i} v_{\alpha j} + P_{\alpha i j})}{x_j} = \frac{q_\alpha \rho_\alpha}{m_\alpha} (E_i + \epsilon_{ijk} v_{\alpha j} B_k) + \sum_{\beta \neq \alpha} R_{\alpha \beta_i}\]where \( R_{\alpha \beta_i} \) represents an interspecies momentum exchange (collisions) between species \( \beta \neq \alpha \).