Beyond Braginskii

Transcollisional extensions

The Braginskii closure is strictly valid only in the collisional limit $\lambda_{mfp}=c_s \tau \ll L$, i.e. when the collisional mean-free path is much shorter than any relevant macroscopic length scale (such as gradient lengths). This is typically the case in the divertor, but never in the confined region of a tokamak or stellarator. At vanishing collisionality, one should use the (gyro-)kinetic model. But those typically have performance problems, on the other hand, at high collisionality. Thus, in the SOL it seems more efficient to use a fluid model, but we'd like to extend its applicability to somewhat lower collisionalities to be also useful in the edge of the confined plasma.

The collisional closure terms R and Q go to zero for vanishing collisionality and are thus less problematic. The viscosity G as well as conductive heat fluxes $q_\parallel^{e,i}$, on the other hand, diverge as $T^{5/2}$. The largest and most problematic term is $q_\parallel^e$, $\chi_{\parallel e}/\chi_{\parallel i} \approx 20-60$ and $\eta_{\parallel i} \approx \chi_{\parallel i}$. Therefore we discuss heat conductivity extensions first, and turn later also to ion viscosity.

The trivial solution

A simple solution is to introduce a heat flux limiter. The most simple form implemented in GRILLIX is $$\begin{equation} \tilde{\chi}_{\parallel e,i} = \min( \chi_{\parallel e,i}, \chi_{\parallel \mathrm{lim}}^{e,i}), \end{equation}$$ with an arbitrary maximum allowed heat conductivity $\chi_{\parallel \mathrm{lim}}^{e,i}$.

The free streaming heat flux limiter

A handwaving physical explanation for why the Braginskii heat conductivity might need to be limited can be found in [Stangeby, chapter 26]. Chapter 26.3 also points out that a similar limiter might be applicable for the parallel viscosity.

Essentially, in a fluid only small deviations from a Maxwellian are allowed for the velocity distribution of the particles. Hence, although the heat is carried mostly by the high velocity tail of the particle distribution, it seems reasonable that if the fluid description is applicable at all, the maximum allowed heat flux should be a fraction of the free streaming heat flux $$\begin{equation} q_\mathrm{FS} = \alpha n v_\mathrm{T} T, \end{equation}$$ with thermal velocity $v_\mathrm{T} = \sqrt{T/m}$ and $\alpha < 1$. For the one-way Maxwellian heat flux density, $\alpha=0.5$.

The actual heat flux can then be limited in a continuous manner by $$\begin{align} \frac{1}{\tilde{q}} = \frac{1}{q_\mathrm{Braginskii}} + \frac{1}{q_\mathrm{FS}} = \frac{q_\mathrm{Braginskii} + q_\mathrm{FS}}{q_\mathrm{Braginskii} \cdot q_\mathrm{FS}} \end{align}$$ or more conveniently $$\begin{align} \tilde{\chi}_{\parallel e,i} = \chi_{\parallel e,i} \left( 1 + \frac{|q_\mathrm{Braginskii}|}{q_\mathrm{FS}} \right)^{-1} = \chi_{\parallel e,i} \left( 1 + \frac{\chi_{\parallel e,i}|\nabla_\parallel T|}{\alpha nv_\mathrm{T}T} \right)^{-1} \label{eq:free-streaming-limiter} \end{align}$$

This expression is particularly useful in the SOL, when steep parallel gradients comparable with the experiment can be only achieved by (locally) limiting the heat conductivity. For the electrons, $\alpha_e \approx 0.2$ is suggested in the SOLPS-ITER manual from kinetic calculations, and $\alpha_e \in (0.1, 0.3)$ is often used [David Coster, Alex Chankin, D. Tskhakaya]. For the ion heat conductivity, the situation is less clear, but from computational concerns the situation is also more relaxed.

A heat flux limiter for periodic systems

It is not clear whether this solution is already well suited for the confined region. In the SOLPS-ITER manual, chapter "B.2 Problems for a fluid description of the edge plasma", it is suggested to take the periodicity of the system into account. The recommended formula (B.63) for the electrons translates to $$\begin{equation} \tilde{\chi}_{\parallel e} = \chi_{\parallel e} \left( 1 + \frac{\lambda_\mathrm{HC}}{q R_0} \right)^{-1}, \end{equation}$$ with the mean free path of tail electrons $$\begin{equation} \lambda_\mathrm{HC} [\mathrm{m}] = 7.5 \times 10^{16} \frac{T_e^2 [\mathrm{eV}]}{n [\mathrm{m}^{-3}]}, \end{equation}$$ the major machine radius $R_0$ and the safety factor $q$. \vspace{\baselineskip}

This formula can be derived by simply approximating $$\begin{equation} \frac{|\nabla_\parallel T_e|}{T_e} \approx \frac{1}{q R_0} \label{eq:periodicity-approximation} \end{equation}$$ in equation \eqref{eq:free-streaming-limiter} - this can be done for both electrons and ions. We then obtain for the electrons $$\begin{align} \tilde{\chi}_{\parallel e} = \chi_{\parallel e} \left( 1 + \frac{\chi_{\parallel e}}{\alpha_e nv_\mathrm{T} R_0 q} \right)^{-1} = \chi_{\parallel e} \left( 1 + \frac{3.16 \sqrt{T_e/m_e} \tau_e}{\alpha_e R_0 q} \right)^{-1} = \chi_{\parallel e} \left( 1 + \frac{\lambda_\mathrm{HC}}{R_0 q} \right)^{-1} \end{align}$$ with $\alpha_e = 0.48$ and coulomb logarithm $\Lambda = 13$. Hence, for both electrons and ions, with the periodicity approximation \eqref{eq:periodicity-approximation}, we can define the limiter $$\begin{equation} \tilde{\chi}_{\parallel e,i} = \chi_{\parallel e,i} \left( 1 + \frac{\chi_{\parallel e,i}}{\alpha_{e,i} nv_\mathrm{T_{e,i}} R_0 q} \right)^{-1}, \label{eq:free-streaming-limiter-periodic} \end{equation}$$ with different thermal speeds for electrons and ions.

Normalization

In the GRILLIX normalization, we have $$\begin{align} \chi_{\parallel e} &= 3.16 \frac{n_0 T_{e0} \tau_{e0}}{m_e} \hat{T}_e^{5/2} = n_0 c_{s0} R_0 \chi_{\parallel 0}^e \hat{T}_e^{5/2}, \\ \chi_{\parallel i} &= 3.9 \frac{n_0 T_{i0} \tau_{i0}}{m_i}\hat{T}_i^{5/2} = n_0 c_{s0} R_0 \chi_{\parallel 0}^i \hat{T}_i^{5/2}. \end{align}$$ Then, with $\hat{T} = e^\xi$, the free-streaming flux limiter \eqref{eq:free-streaming-limiter} can be translated to $$\begin{align} \tilde{\chi}_{\parallel 0}^e = \chi_{\parallel 0}^e \left( 1 + \frac{\chi_{\parallel 0}^e \sqrt{\mu}}{\alpha_e} \frac{\hat{T}_e^2 |\nabla_\parallel \xi_e|}{\hat{n}} \right)^{-1}, \label{eq:free-streaming-limiter-normalized-e} \\ \tilde{\chi}_{\parallel 0}^i = \chi_{\parallel 0}^i \left( 1 + \frac{\chi_{\parallel 0}^i}{\alpha_i \sqrt{\zeta}} \frac{\hat{T}_i^2 |\nabla_\parallel \xi_i|}{\hat{n}} \right)^{-1}, \label{eq:free-streaming-limiter-normalized-i} \end{align}$$ and the periodicity-free-streaming flux limiter \eqref{eq:free-streaming-limiter-periodic}, with $|\hat{\nabla}_\parallel \hat{\xi}| \approx 1 / q$, turns out to be $$\begin{align} \tilde{\chi}_{\parallel 0}^e = \chi_{\parallel 0}^e \left( 1 + \frac{\chi_{\parallel 0}^e \sqrt{\mu}}{\alpha_e} \frac{\hat{T}_e^2}{\hat{n}q} \right)^{-1}, \label{eq:free-streaming-limiter-periodic-normalized-e} \\ \tilde{\chi}_{\parallel 0}^i = \chi_{\parallel 0}^i \left( 1 + \frac{\chi_{\parallel 0}^i}{\alpha_i \sqrt{\zeta}} \frac{\hat{T}_i^2}{\hat{n}q} \right)^{-1}. \label{eq:free-streaming-limiter-periodic-normalized-i} \end{align}$$

Neoclasical ion viscosity corrections

\label{sec:neocl-visc-corrections}

The Braginskii ion viscosity coefficient in the viscous function $G$, which is equal to the trace of the ion viscous stress tensor $P_i$ and enters the equations for vorticity, parallel momentum and ion temperature, is only valid in the collisional Pfirsch-Schlüter (PS) regime. However, in the confined region, where collisionality is lower, the plasma regime shifts: The plasma edge of tokamaks typically operates in the plateau regime and the deep confined region operates in the banana regime. The PS regime is only reached in the highly collisional SOL. In order to improve the validity of the fluid closure at low collisionality two neoclassical corrections are implemented in GRILLIX, one based on the ion momentum and one based on the ion heat flux terms occuring occuring in the viscous function $G$. This section deals with the ion momentum part of $G$,

$$\begin{equation} G_\mathrm{flow} = - \eta_\mathrm{i0} \left( \frac{2}{B^{3/2}}\nabla\cdot\left(u_\parallel B^{3/2}\mathbf{b}\right) - \frac{1}{2}C(\phi) - \frac{1}{2en_e}C(p_i) \right) \, , \label{eq:G_flow} \end{equation}$$

where the ion viscosity coefficient $\eta_\mathrm{i0}$ is now corrected according to Rozhansky2009 and Hirshman1981 (see fig. 1 on p. 40), such that it fits neoclassical calculations across all three regimes. The Braginskii ion viscosity coefficient $\eta_\mathrm{i0}$ is replaced by

$$\begin{equation} \tilde{\eta}_\mathrm{flow} = \eta_\mathrm{i0} \frac{1}{1 + \epsilon^{-3/2}\nu_*^{-1}} \frac{1}{1 + \nu_*^{-1}}, \label{eq:neoclassical_ion_momentum_viscosity} \end{equation}$$ with the inverse aspect ratio $\epsilon\approx0.5/1.65$ (for ASDEX) and the collisionality parameter

$$\begin{equation} \nu_*^{-1} = \frac{\epsilon^{3/2}}{R_0 q} \sqrt{\frac{2T_\mathrm{i}}{m_\mathrm{i}}} \tau_i, \label{eq:normalized_neocl_coll_time} \end{equation}$$

where $\tau_i$ is the Braginskii ion collision time. The whole expression in general, and the geometric parameters $\epsilon$ and $R_0 q$ in particular, are very approximate and could be refined in future, in particular for diverted equilibria.

In normalized GRILLIX units the neoclassical ion viscosity coefficient is given by

$$\begin{equation} \eta_{i0}\hat{T}_i^{5/2} \rightarrow \tilde{\eta} = \eta_{i0}\hat{T}_i^{5/2} \frac{1}{1 + \epsilon^{-3/2}\nu_*^{-1}} \frac{1}{1 + \nu_*^{-1}}. \label{eq:normalized_neoclassical_ion_momentum_viscosity} \end{equation}$$

Neoclasical ion heat viscosity extension

This section deals with the ion heat flux based neoclassical correction to the trace $G$ of the ion viscous stress tensor $P_i$, following the approach taken by [Rozhansky et al, B2SOLPS5.2 in 2009]. With this $G$ is extended as follows

$$\begin{equation} G = - \underbrace{ \eta_{\mathrm{flow}} \left[ \frac{2}{B^{3/2}}\nabla\cdot\left(u_\parallel B^{3/2}\mathbf{b}\right) - \frac{1}{2}C(\phi) - \frac{1}{2en_e}C(p_i) \right]}_{=G_{\mathrm{flow}}} - \underbrace{ \eta_{\mathrm{heat}} \left[ \frac{2}{n_e T_i B^{3/2}}\nabla\cdot\left(q_{\parallel,i}B^{3/2}\mathbf{b}\right) - \frac{5}{4}C(T_i) \right]}_{=G_{\mathrm{heat}}} \, , \label{eq:G_flow_plus_G_heat} \end{equation}$$

with

$$\begin{equation} \eta_{\mathrm{heat}} = 0.75 \, \alpha_{\mathrm{neo}} n_e T_i \tau_i \, , \label{eq:eta_heat} \end{equation}$$

where

$$\begin{equation} \alpha_{\mathrm{neo}} = \frac{8}{15} (k^T - 1) \frac{1}{1 + \nu_*^{-1}} \frac{1}{1 + \epsilon^{-3/2} \nu_*^{-1}} \end{equation}$$

and

$$\begin{equation} k^T = \frac{ -0.17 + 1.05\sqrt{\nu_*\sqrt{2}} + 2.7\left(\nu_*\sqrt{2}\right)^2\epsilon^3 }{ 1 + 0.7\sqrt{\nu_*\sqrt{2}} + \left(\nu_*\sqrt{2}\right)^2\epsilon^3 }. \end{equation}$$

For high temperatures, i.e. low collisionality, the neoclassical viscosity coefficients $\eta_\mathrm{flow}$ and $\eta_\mathrm{heat}$ vary significantly in amplitude and temperature dependence from the original Braginskii viscosity. At low temperatures the neoclassicale viscosities are consistent with Braginskii, as can be seen in the image below. Note that the heat viscosity is zero around $T_e\approx150\mathrm{eV}$ and it is chosen to plot its absolute value to avoid a double logarithmic x-axis.

TODO: Landau fluid