The impurity fluid model

These models contain a field (variable number 8 or var_rhoimp in the code) corresponding to the impurity density. This page applies to dedicated impurity models such as models 501/502, as well as thei modular general model 600.

Such models can be used for example for SPI simulations or MGI simulations.

Models 501/502

Important note: At present, tau_ic = 0. should be set when using models 501/502, i.e. diamagnetic effects should not be included (they are not properly implemented).

Definition of the variables

Variable number 5: In models 501/502, $\rho$ (variable number 5) is the total mass density, all species included (but neglecting the electron mass). That is:

\begin{aligned} \rho = \rho_i + \rho_{imp} \end{aligned}

where $\rho_i$ is the main ion density and $\rho_{imp}$ is the impurity density, all charge states included.

Pressure: In models 501/502, the pressure that appears naturally in the energy and momentum equations is the total pressure $p$, all species included. In model 501, we assume that all species have a common temperature $T/2$. The pressure is therefore (with non-JOREK normalizations for the moment, to avoid confusions) $p=n_{tot}T/2$, where $n_{tot}$ is the total particle density: $n_{tot}=n_{i}+n_{e}+n_{imp}$. By quasi-neutrality we have $n_e=n_i+\langle Z \rangle_{imp}n_{imp}$. Therefore, $n_{tot}=2n_i+(\langle Z \rangle_{imp}+1)n_{imp}$. Now, $n_i=\rho_i/m_i$ and $n_{imp}=\rho_{imp}/m_{imp}$, so: $n_{tot}=[2\rho_i+\frac{(\langle Z \rangle_{imp}+1)m_i}{m_{imp}}\rho_{imp}]/m_i=[2\rho+(\frac{(\langle Z \rangle_{imp}+1)m_i}{m_{imp}}-2)\rho_{imp}]/m_i$. The pressure can therefore be expressed as (now in JOREK units):

\begin{aligned} p=(\rho+\alpha_{imp}\rho_{imp})T/m_{i} \end{aligned}

where we define $\alpha_{imp} \equiv \frac{(\langle Z \rangle_{imp}+1)m_i}{2m_{imp}}-1$. Note that we will assume that $\langle Z \rangle_{imp}=\langle Z \rangle_{imp}(T)$, so that $\alpha_{imp}=\alpha_{imp}(T)$.

Variable number 8: In models 501/502, variable number 8 is $\rho_{imp}$, the impurity mass density, all charge states included.

Normalizations

Connection between SI and normalized units   Description of the quantity / Comment
$\rho_\mathrm{SI}~[\mathrm{kg/m^3}]$ $=\rho\;\rho_0$ Total mass density
$\rho_\mathrm{imp,SI}~[\mathrm{kg/m^3}]$ $=\rho_\mathrm{imp}\;\rho_0$ Impurity mass density
$n_\mathrm{i,SI}~[\mathrm{m^{-3}}]$ $=(\rho - \rho_\mathrm{imp})\;n_0$ Main ion number density
$n_\mathrm{imp,SI}~[\mathrm{m^{-3}}]$ $=\rho_\mathrm{imp}\;n_0\;\mu_{imp}$ Impurity number density
$n_\mathrm{e,SI}~[\mathrm{m^{-3}}]$ $= n_\mathrm{i,SI}\;+\;\langle Z \rangle_{imp}\;n_\mathrm{imp,SI}$ Electron number density

Having defined:

Symbol Expression Comment
$n_0~[m^{-3}]$ central_density $\cdot 10^{20}$ central_density gets a default value in preset_parameters.f90 and should be specified in the input file
$\rho_0~[kg\;m^{-3}]$ central_mass $\cdot n_0 \cdot m_\text{AMU}$ central_mass gets a default value in preset_parameters.f90 and should be specified in the input file
$\mu_\mathrm{imp}~[1]$ m_i_over_m_imp $ = m_\mathrm{i}/m_\mathrm{imp}$ m_i_over_m_imp is defined by the impurity species.

Equations

Vorticity equation (eq. 2)

The full MHD momentum equation reads:

\begin{aligned} \rho (\partial_t + \mathbf{v} \cdot \nabla) \mathbf{v} = \mathbf{j} \times \mathbf{B} - \nabla p + \nu \Delta \mathbf{v} - (\partial_t \rho + \nabla \cdot (\rho \mathbf{v})) \mathbf{v} + S_{\rho_{imp}} \mathbf{v}_{imp} \end{aligned}

where $\mathbf{v}{imp}$ is the velocity of the impurities deposited by the $S{\rho_{imp}}$ source term. Note the presence of the term $-(\partial_t \rho + \nabla \cdot (\rho \mathbf{v})) \mathbf{v}$ on the RHS. This term, which ensures momentum conservation, is not present in other JOREK models at the moment (21/08/18) but may be included in the future.

The reduced MHD vorticity equation is derived by applying the operator $\nabla\phi\cdot\nabla\times(R^2…)$ to the momentum equation. The result in the absence of the $-(\partial_t \rho + \nabla \cdot (\rho \mathbf{v})) \mathbf{v}$ term is presented here. The $-(\partial_t \rho + \nabla \cdot (\rho \mathbf{v})) \mathbf{v}$ term yields an additional term $-\nabla \cdot \left[ R^2 (\partial_t \rho + \nabla \cdot (\rho \mathbf{v})) \nabla_{pol} u \right] $ on the RHS of the vorticity equation (we neglect the contribution from $- (\partial_t \rho + \nabla \cdot (\rho \mathbf{v})) v_{   }\mathbf{B} $). In weak form, this term becomes $ \int - u^* \nabla \cdot \left[ R^2 (\partial_t \rho + \nabla \cdot (\rho \mathbf{v})) \nabla_{pol} u \right] dV$ (where $u^$ is the test function), which can be transformed into $ \int R^2 (\partial_t \rho + \nabla \cdot (\rho \mathbf{v})) (\nabla u^ \cdot \nabla_{pol} u) dV $ via an integration by parts. In order to implement this term, it is necessary to develop $\nabla \cdot (\rho \mathbf{v})$, which is done here.

Definition of current (eq. 3)

This equation remains the same as in other JOREK reduced MHD models: \begin{aligned} j = \Delta^*\psi \end{aligned}

Definition of vorticity (eq. 4)

This equation remains the same as in other JOREK reduced MHD models: \begin{aligned} \omega = \Delta_\text{pol} u \end{aligned}

Global continuity equation (eq. 5)

The global continuity equation, i.e. the equation describing the evolution of the total mass density, reads:

\begin{aligned} \partial_t \rho = - \nabla \cdot (\rho \mathbf{v}) + \nabla \cdot (D_i \nabla (\rho - \rho_{imp})) + \nabla \cdot (D_{imp} \nabla \rho_{imp}) + S_{\rho_i} + S_{\rho_{imp}} \end{aligned}

One important aspect of the model for e.g. SPI or MGI simulations is the volumetric source term $S_{\rho_{imp}}$ through which impurities are deposited. Some information on the parameterization of this term is given here.

Pressure equation (eq. 6)

The pressure equation reads something like:

\begin{aligned} \partial_t p = - \mathbf{v} \cdot \nabla p - \gamma p \nabla \cdot \mathbf{v} + Conduction + Joule - Radiation - Ionization \end{aligned}

assuming that all species have the same ratio of heat capacities $\gamma$, which is the case if we consider mono-atomic impurities and main ions (which we plan to do). We recall that here $p=(\rho+\alpha_{imp}\rho_{imp})T$ (see above).

Parallel momentum equation (eq. 7)

The parallel momentum equation is obtained by applying $\mathbf{B} \cdot (…) $ to the full MHD momentum equation (written above, in the Vorticity equation section). The result in the absence of the $-(\partial_t \rho + \nabla \cdot (\rho \mathbf{v})) \mathbf{v}$ term is presented here. To implement the latter, the approximation $\mathbf{B} \cdot \mathbf{v} \approx B^2 v_{\parallel} \approx \frac{F_0^2}{R^2} v_{\parallel} $ is used, which intuitively seems to make sense in reduced MHD. In weak form, the parallel momentum equation therefore gets a RHS term $ \int -(\partial_t \rho + \nabla \cdot (\rho \mathbf{v})) B^2 v_{parallel} $. In order to implement this term, it is necessary to develop $\nabla \cdot (\rho \mathbf{v})$, which is done here.

Impurity continuity equation (eq. 8)

The impurity continuity equation, i.e. the equation describing the evolution of the impurity mass density (all charge states included), reads:

\begin{aligned} \partial_t \rho_{imp} = - \nabla \cdot (\rho_{imp} \mathbf{v}) + \nabla \cdot (D_{imp} \nabla \rho_{imp}) + S_{\rho_{imp}} \end{aligned}

As already mentioned above, one important aspect of the model for e.g. SPI or MGI simulations is the volumetric source term $S_{\rho_{imp}}$ through which impurities are deposited. Some information on the parameterization of this term is given here.

Implementation of the equations

Atomic coefficients

Atomic coefficients are taken from the ADAS database. The "baseline" version of models 501/502 assume coronal equilibrium, but a non-equilibrium has been implemented recently (see below).

Non-equilibrium impurities

A particle based non-equilibrium model has been implemented in JOREK, compatible with 501/502 and 600. See The marker representation of collisional-radiative impurity model.