Multi-species electrostatic turbulence system with neutrals#
Solves a system of \(4+3S\) fields representing \(S\) ion species. This equation system is used by specifying
<I PROPERTY="EQTYPE" VALUE="ElectrostaticTurbulence"/>
in the <SOLVERINFO> field
Electrons are always present, and at least one ion species must be present. Explicitly the fields are:
\[\left[\omega, n_e, m_e n_e v_{\parallel e}, \frac{3}{2}p_e, n_i m_i n_i v_{\parallel i}, \frac{3}{2}p_i, ...\right]\]
The continuity equations are:
\[\frac{\partial n}{\partial t} = \nabla \cdot \left[n\left(\mathbf{v}_{\mathbf{E}\times\mathbf{B}} + \mathbf{b}v_{\parallel e} + \mathbf{v}_de\right)\right] + S_n\]
\[\frac{\partial \omega}{\partial t}\]
\[\frac{\partial}{\partial t} \left(m_e n_e v_{\parallel e}\right) = \nabla \cdot \left[m_e n_e v_{\parallel e}\left(\mathbf{v}_{\mathbf{E}\times\mathbf{B}} + \mathbf{b}v_{\parallel e} + \mathbf{v}_{de}\right)\right] - \mathbf{b}\cdot\nabla p_e - enE_{\parallel} - F_{ei}\]
\[\frac{\partial}{\partial t} \left(\frac{3}{2}p_e\right) = \nabla \cdot \left[\frac{3}{2}p_e\left(\mathbf{v}_{\mathbf{E}\times\mathbf{B}} + \mathbf{b}v_{\parallel e} + \frac{5}{2}\mathbf{v}_{de}\right)\right] - p_e\nabla\cdot\left(\mathbf{v}_{\mathbf{E}\times\mathbf{B}} + \mathbf{b}v_{\parallel e} \right) + \nabla\cdot\left(\kappa_{\parallel e}\mathbf{b}\mathbf{b}\cdot\nabla T_e\right) + S_{Ee} + W_{ei}\]
\[\frac{\partial}{\partial t} \left(m_i n_i v_{\parallel i}\right) = \nabla \cdot \left[m_i n_i v_{\parallel i}\left(\mathbf{v}_{\mathbf{E}\times\mathbf{B}} + \mathbf{b}v_{\parallel i} + \mathbf{v}_{di}\right)\right] - \mathbf{b}\cdot\nabla p_i + ZenE_{\parallel} + F_{ei}\]
\[\frac{\partial}{\partial t} \left(\frac{3}{2}p_i\right) = \nabla \cdot \left[\frac{3}{2}p_i\left(\mathbf{v}_{\mathbf{E}\times\mathbf{B}} + \mathbf{b}v_{\parallel i} + \frac{5}{2}\mathbf{v}_{di}\right)\right] - p_i\nabla\cdot\left(\mathbf{v}_{\mathbf{E}\times\mathbf{B}} + \mathbf{b}v_{\parallel i} \right) + \nabla\cdot\left(\kappa_{\parallel i}\mathbf{b}\mathbf{b}\cdot\nabla T_i\right) + S_{Ei} + S_n\frac{1}{2}m_i n_i v_{\parallel i}^2 - W_{ei} + \frac{p_i}{en_0}\nabla\cdot\left(\mathbf{J}_{\parallel}+\mathbf{J}_d\right)\]
The last two equations are duplicated for each ion species.
Finally the vorticity equation is:
Boundary Conditions#
The boundary conditions for this system of equations are as follows.
For the electron density
\[\frac{\partial n}{\partial \mathbf{n}} = \mp \frac{n}{c_s} \frac{\partial v_{\parallel i}}{\partial \mathbf{n}}\]
For parallel electron velocity
\[v_{\parallel e} = \pm c_s \exp\left(\Lambda - \frac{\phi}{T_e}\right)\]
For the electron energy
\[\frac{\partial \frac{3}{2}p_e}{\partial \mathbf{n}} = \gamma_e n_e T_e c_s\]
For parallel ion velocity
\[v_{\parallel i} = \pm c_s\]
For the ion energy
\[\frac{\partial \frac{3}{2}p_i}{\partial \mathbf{n}} = \gamma_i n_i T_i c_s\]
For vorticity
\[\omega = -\cos^2\alpha\left(\frac{\partial v_{\parallel i}}{\partial \mathbf{n}}\right)^2 \pm c_s \frac{\partial^2 v_{\parallel i}}{\partial \mathbf{n}^2}\]