soledge3x

Version 0.9.0

Edge plasma code

Conservative equations

Soledge3X follows drift ordering fluid equations. It solves for every charged species mass, parallel momentum and energy balance. Current balance is also solved to compute the electric potential. It its present form, the code is electrostatic and assumes a fixed magnetic field. The conservation equations take the following generic form:

Mass balance:

$$\frac{\partial n}{\partial t} + \vec{\nabla}\cdot\left(n \vec{v} \right) = S_n$$ with:

  • $n$ the density of the plasma species
  • $\vec{v}$ the plasma species velocity that can be decomposed as $\vec{v} = v_\parallel \vec{b} + \vec{v}_E + \vec{v}^\star + \vec{v}_p + \vec{v}_D$.
    • $v_\parallel$ is the projection along the magnetic field (unit vector $\vec{b}=\vec{B}/B$).
    • $\vec{v}_E$ is the “E cross B” drift: $$\vec{v}_E = \frac{\vec{E}\times\vec{B}}{B^2}$$
    • $\vec{v}^\star$ is the diamagnetic drift $$\vec{v}^\star = \frac{\vec{B}\times\vec{\nabla}p}{Z e n B^2}$$ where $Z$ is the number of charge of the plasma species
    • $\vec{v}_p$ is the polarization drift
    • $\vec{v}_D$ is an effective velocity associated with diffusive processes (that can be used to describe turbulent transport for reduced models, typically when the code is used as a transport code) $$\vec{v}_D = -D \frac{\vec{\nabla} n}{n}$$
  • $S_n$ is a volumic source of particle due to ionization/recombination processes

For electrons, the mass balance is not solved and one uses quasineutrality assumption: $$n_e = \sum_i Z_i n_i$$

Parallel momentum balance:

$$\frac{\partial n v_\parallel}{\partial t} + \vec{\nabla}\cdot\left(n v_\parallel \vec{v} \right) = -\nabla_\parallel p + \left( \vec{\nabla}\cdot \overline{\overline{\tau}} \right) \cdot \vec{b} + n Z e E_\parallel + R_\parallel + \vec{\nabla} \cdot \left( n \nu \vec{\nabla}_\perp v_\parallel \right) + S_v$$ with:

  • $p$, the static pressure $p=n T$ where $T$ denotes the species temperature
  • $\overline{\overline{\tau}}$ is Braginskii stress tensor (containing parallel viscosity, classical cross-field viscosity and gyro-viscous tensor)
  • $E_\parallel$ is the parallel electric field
  • $R_\parallel$ is the parallel projection of the friction force resulting from collisions with other species
  • $\nu$ is the “anomalous” cross-field viscosity (that can be used in reduced models for turbulence when the code is used as a transport code)
  • $S_v$ is the volumic parallel momentum source due to ionization/recombination processes

Electron parallel momentum is not solved. Electron parallel velocity is obtained from parallel current definition: $$j_\parallel = -n_e v_{\parallel,e} + \sum_i Z_i n_i v_{\parallel,i}$$ The current being computed from current balance.

Energy balance:

$$\frac{\partial \mathcal{E}}{\partial t} + \vec{\nabla}\cdot\left(\mathcal{E} \vec{v} + p v_\parallel \vec{b} + v_\parallel \overline{\overline{\tau}} \cdot \vec{b} + q_\parallel \vec{b} \right) = n Z e v_\parallel E_\parallel + v_\parallel R_\parallel + \vec{\nabla} \cdot \left( n v_\parallel \nu \vec{\nabla}_\perp v_\parallel + n \chi \vec{\nabla}_\perp T \right) + Q + S_\mathcal{E}$$ with

  • $\mathcal{E}$ the internal + parallel kinetic energy: $$\mathcal{E} = \frac{3}{2} n T + \frac{1}{2} m n v_\parallel^2$$
  • $q_\parallel$ is the parallel heat flux assuming a conductive closure: $$q_\parallel = - \kappa_\parallel \nabla_\parallel T$$
  • $\chi$ is the “anomalous” cross-field heat conduction (that can be used in reduced models for turbulence when the code is used as a transport code)
  • $Q$ is an energy exchange term due to collisions in between species
  • $S_\mathcal{E}$ is the volumic energy source due to ionization/recombination processes.

Current balance:

$$\vec{\nabla} \cdot \vec{j} = 0$$ with $$\vec{j} = j_\parallel \vec{b} + \vec{j}^\star + \vec{j}_p + \vec{j}_D$$ where

  • $j_\parallel$ is the parallel current given by generalized Ohm’s law: $$j_\parallel = \sigma_\parallel \left(-\nabla_\parallel \phi + \frac{1}{e n_e} ( \nabla_\parallel p_e + R_{\parallel, eT} ) \right)$$ $\sigma_\parallel$ being plasma parallel conductivity, $\phi$ the electrostatic potential and $R_{\parallel, eT}$ the collision thermal force on electrons. This general Ohm’s law comes from the electon parallel momentum balance where the electron inertia is neglected. The value of plasma conductivity comes from the collisional friction force.
  • $\vec{j}^\star$ is the diamagnetic current: $$\vec{j}^\star = \sum_{e,i} Z e n \vec{v}^\star$$
  • $\vec{j}_p$ is the polarization current: $$\vec{j}_p = \sum_i Z e n \vec{v}_p$$ where the polarization velocity is given for each ion species by $$n \vec{v}_p = -\partial_t \vec{\omega} - \vec{\nabla} \cdot ( \vec{v}^0 \otimes \vec{\omega} )$$ with:
    • $\vec{v}^0 = v_\parallel \vec{b} + \vec{v}_E + \vec{v}^\star$
    • $\vec{\omega} = -(\vec{b}/\omega_c)\times(n \vec{v}^0)$ that can also be expressed: $$\vec{\omega} = \frac{m}{Ze B^2} \left( n\vec{\nabla}_\perp \phi + \frac{1}{Ze} \vec{\nabla}_\perp p \right)$$

The divergence of the polarization current can be rewritten introducing the vorticity $\Omega = \vec{\nabla} \cdot (\sum_i Z \vec{\omega})$: $$\vec{\nabla}\cdot \vec{j}_p = -\partial_t \Omega - \vec{\nabla} \cdot \left( \sum_i Z \vec{\nabla}\cdot (\vec{v}^0 \otimes \vec{\omega} ) \right)$$

  • The diffusion current $\vec{j}_D$ is a regularization current acting as vorticity diffusion $\vec{j}_D = \zeta \vec{\nabla}_\perp \Omega$.

Finally, the current balance can be rewritten in the form of a vorticity equation: $$\frac{\partial \Omega}{\partial t} + \vec{\nabla} \cdot \left( \sum_i Z \vec{\nabla}\cdot (\vec{v}^0 \otimes \vec{\omega} ) \right) = \vec{\nabla}\cdot(\zeta \vec{\nabla}_\perp \Omega) + \vec{\nabla}\cdot (j_\parallel \vec{b} + \vec{j}^\star)$$

Boundary conditions on the wall

Bohm-Chodura boundary conditions apply on the wall setting parallel velocity, namely $$ | (v_\parallel \vec{b} + \vec{v}^E + \vec{v}^\star) \cdot \vec{n}{wall} | \ge | c_s \vec{b} \cdot \vec{n}{wall} | $$ where $c_s$ is the species sound speed: $$c_s = \sqrt{\frac{T_i + Z T_e}{m_i}}$$

For heat flux, sheath transmission factors link energy flux ($\phi_E$) and particle flux ($\phi_n$): $$\phi_{E,BC} = \left(\gamma T + \frac{1}{2} m v_\parallel^2 \right) \phi_{n,BC}$$

For the current, one has $$j_{BC} = j_{sat} \left(1 - \exp \left( \Lambda - \frac{\phi}{T_e} \right) \right)$$ where

  • $j_{sat}$ is the saturation current: $$j_{sat} = \sum_i Z e \phi_{n,BC}$$
  • $\Lambda$ is the sheath potential drop factor
Last updated on 11 Mar 2025
Published on 11 Mar 2025