\documentclass[12pt]{article}
\author{David Alves}

\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{dirtytalk}
\usepackage[a4paper, total={6.5in, 8.75in}]{geometry}
\usepackage{forest}
\usepackage{listings}
\usepackage{skak}
\usepackage{tikz}
\usepackage{titling}
\usepackage{wrapfig}
\usepackage{xcolor}

\usetikzlibrary{decorations.pathreplacing}
\usetikzlibrary{patterns}

\pgfdeclarepatternformonly{bigger north west lines}{\pgfqpoint{-2pt}{-2pt}}{\pgfqpoint{8pt}{8pt}}{\pgfqpoint{6pt}{6pt}}%
{
  \pgfsetlinewidth{1pt}
  \pgfpathmoveto{\pgfqpoint{0pt}{6pt}}
  \pgfpathlineto{\pgfqpoint{6.2pt}{-0.2pt}}
  \pgfusepath{stroke}
}

\def\multichoose#1#2{\ensuremath{\left(\kern-.3em\left(\genfrac{}{}{0pt}{}{#1}{#2}\right)\kern-.3em\right)}}

\newcommand{\ProblemStatement}[1]{
\subsection*{Problem Statement}
#1
\subsection*{Solution}
}

% If uncommented, next line hides problem statements
%\renewcommand{\ProblemStatement}[1]{}

\title{Math 142 Problem Set 6}
\author{David Alves}
\date{2016-10-13}

\begin{document}
\pagenumbering{gobble}

\begin{center}
\large \thetitle \\
\theauthor \\
\thedate
\end{center}

\subsection*{Sources}

    \begin{itemize}
    \item http://tex.stackexchange.com and https://www.sharelatex.com for help with \LaTeX
    \item Wikipedia and math.stackexchange.com for a refresher on implication and logical equivalence.
    \end{itemize}

\section{Disjoint Tuples}
\ProblemStatement{
Fix positive integers $n$ and $k$, and let $S = [n]$. Find the number of $k$-tuples $(T_1, T_2, \ldots, T_k)$ of subsets $T_i$ such that $T_i$ are pairwise disjoint.
}

There are $(k+1)^{n}$ different $k$-tuples of pairwise-disjoint subsets of $S$.

\begin{proof}
An element $e \in S$ can be present in at most one of the sets $T_1, T_2, \ldots, T_k$ since if $e$ was present in two sets then those two sets would not be disjoint. It's also possible that $e$ is not present in any of the sets in the $k$-tuple. Therefore there is a bijection between $k$-tuples of pairwise-disjoint subsets of $S$ and placing $n$ distinguishable balls into $k+1$ distinguishable bins (one bin per element in the $k$-tuple and one for \say{not present in the tuple}). This gives $(k+1)^{n}$ different $k$-tuples of pairwise-disjoint subsets of $S$.
\end{proof}

\section{Bipartite Graphs}
\ProblemStatement{
Call a graph \textbf{bipartite} if its vertices can be written as a disjoint
union $A \cup B$ such that edges only go between $A$ and $B$, and never between
two vertices in $A$ or two vertices in $B$. Show that a graph $G$ is bipartite if and only if there is \textbf{not} an odd-length cycle, namely a sequence of vertices $v_1, v_2, \ldots, v_k$ where $(v_1, v_2), (v_2, v_3), \ldots, (v_{k-1}, v_k), (v_k, v_1)$ are all edges in $G$ and $k$ is odd.
}

Let B(G) denote the statement that G is a bipartite graph and O(G) denote the statement that there exists a subgraph of G which is an odd-length cycle. In order to show that for all $G$,$B(G) \iff \neg O(G)$, we must show that for all $G$, $B(G) \implies \neg O(G)$ and for all $G$, $\neg O(G) \implies B(G)$.

\subsubsection*{Proof that for all G, $B(G) \implies \neg O(G)$}

\begin{proof}
We prove this by contradiction. Assume that (for all $G$, $B(G) \implies \neg O(G)$) is false. We can rewrite $\neg(B(G)\implies \neg O(G))$ as $\neg(\neg B(G)\vee \neg O(G))$  which simplifies to $B(G) \wedge O(G)$. Therefore to prove our assumption, we must show that there exists a graph $G$ such that $B(G)$ and $O(G)$ are true.

Let $G$ be a graph such that $B(G)$ and $O(G)$ are both true. Then there exists a vertex sequence $v_1, v_2, \ldots, v_k$ such that $k$ is odd and $(v_1, v_2), (v_2, v_3), \ldots, (v_{k-1}, v_k), (v_k, v_1)$ are edges in $G$, and disjoint vertex sets $A$ and $B$ exist such that all vertices are in $A$ or $B$. Without loss of generality, let $v_1 \in A$. Then $v_2 \in B$ since we know that edge $(v_1, v_2)$ exists and $v1 \in A$. Similarly $v_3$ must be in $A$ because $v_2 \in B$ and edge $(v_2, v_3)$ exists. This gives an alternating sequence of $A$ and $B$ memberships $v_1 \in A, v_2 \in B, v_3 \in A, v_4 \in B, \ldots$ such that $v_i \in A$ if and only if $i$ is odd. This means that $v_k \in A$ since $k$ is odd, which means that $(v_k, v_1)$ does not exist since $v_1$ and $v_k$ are both in $A$, but $O(G)$ states that $(v_k, v_1)$ exists. Thus no graph $G$ exists such that $B(G) \wedge O(G)$ , so our assumption must be false and  $B(G) \implies \neg O(G)$ for all $G$.
\end{proof}

\subsubsection*{Proof that for all G, $\neg O(G) \implies B(G)$}
\begin{proof}
We prove this by contradiction. Assume that (for all G, $\neg O(G) \implies B(G)$) is false. We can rewrite $\neg(\neg O(G)\implies B(G))$ as $\neg(O(G) \vee B(G))$  which simplifies to $\neg B(G) \wedge \neg O(G)$. Therefore to prove our assumption, we must show that there exists a $G$ such that $B(G)$ and $O(G)$ are false.

Arbitrarily number the vertices in $G$ as $v_1, v_2, \ldots, v_n$. Divide all vertices into disjoint sets $A$ and $B$ as follows:

    \[
    v_k \in
    \begin{cases}
      A:& d(v_1, v_k) \text{ even (including when } k = 1 \text{ since }d(v_1, v_1) = 0)\\
      B:& d(v_1, v_k) \text{ odd}\\
    \end{cases}
    \]
    where $d(v_i, v_j)$ is defined as the shortest-path distance from $v_i$ to $v_j$. We know that after doing this there must be some $v_i, v_j \in G$ such that $(v_i, v_j) \in G$ and $v_i, v_j$ are both in the same set because otherwise we have shown that $G$ is bipartite. But if $v_i$ and $v_j$ are in the same set then $d(v_1, v_i)$ and $d(v_1, v_j)$ are either both even or both odd, so their sum is even. Since $(v_i, v_j)$ exists in the graph, there is an odd-length cycle:
    \[
%        (v_i, v_j) + \text{ path from } v_j \text{ to } v_1 + \text{ path from v_1 to v_i })
        (v_i, v_j) + \text{ (path from } v_j \text{ to } v_1) + \text{ (path from } v_1 \text{ to } v_i)
    \]
    This contradicts our assumption that there are no odd-length cycles in $G$, so there is no graph $G$ such that $\neg B(G) \wedge \neg O(G)$, so our assumption must be false and therefore $\neg O(G) \implies B(G)$ for all $G$.
\end{proof}

\section{Fibonacci Squares}
\ProblemStatement{
Prove the identity that $\sum_{i=0}^{n} (a_i)^2 = a_na_{n+1}$ where $a_n$ is the $n$-th Fibonacci number. For example, \[
    1^2 + 1^2 + 2^2 + 3^2 + 5^2 = 40 = 5 * 8
\]
}

\begin{proof}
We prove this by induction. Let $p(n)$ be the statement $\sum_{i=0}^{n}
(a_i)^2 = a_na_{n+1}$. For $n = 0$, we have $(a_0)^2 = a_0a_1$ or $1^2 = 1 \times 1$, so $p(0)$ is true. Next we show that for $n \geq 0$ if $p(n)$ is true, then $p(n+1)$ is true. $p(n)$ is

\begin{equation}
    a_0^2 + a_1^2 + \ldots + a_{n-1}^2 + a_n^2 = a_na_{n+1}
\end{equation}
while $p(n+1)$ is
\begin{equation}
    a_0^2 + a_1^2 + \ldots + a_n^2 + a_{n+1}^2 = a_{n+1}a_{n+2}
\end{equation}
We can then substitute equation 1 into equation 2, giving
\[
     a_na_{n+1} + a_{n+1}^2 = a_{n+1}a_{n+2}
\]
dividing by $a_{n+1}$ gives
\[
     a_n + a_{n+1} = a_{n+2}
\]
which is true because it is the definition of the Fibonacci series. Therefore if $p(n)$ is true then $p(n+1)$ is true, which completes the proof.

\end{proof}

\begin{center}
\begin{tikzpicture}

\draw[line width=1pt] (0,0) rectangle (-1,1);
\draw[line width=1pt] (0,1) rectangle (-1,2);
\draw[line width=1pt] (-1,0) rectangle (-3,2);
\draw[line width=1pt] (0,2) rectangle (-3,5);
\draw[line width=1pt] (0,0) rectangle (5,5);

\node at (-.5,.5) {$1^2$};
\node at (-.5,1.5) {$1^2$};
\node at (-2,1) {$2^2$};
\node at (-1.5,3.5) {$3^2$};
\node at (2.5,2.5) {$5^2$};
\draw [decorate,decoration={brace,amplitude=10pt,raise=4pt},yshift=0pt]
(-3,0) -- (-3,5) node [black,midway,xshift=-0.8cm] {5};
\draw [decorate,decoration={brace,amplitude=10pt,raise=4pt},yshift=0pt]
(-3,5) -- (5,5) node [black,midway,yshift=0.8cm] {8};

\end{tikzpicture}
\end{center}

\section{Trillary vs. Hump}
\ProblemStatement{
Suppose there are $2n$ anonymous votes to be tallied for Trillary vs. Hump in an important election, and you're looking at the votes one by one. Call a \emph{voting sequence} the sequence of votes you are seeing. (For example, there are four voting sequences for $n=1: TH, TT, HT, HH$, using abbreviations for candidate names). For each $n$, count the number of voting sequences there are such that both
\begin{itemize}
\item The votes are tied at the end with $n$ votes each.
\item At no point during your one-by-one examination of the votes is Hump ever ahead.
\end{itemize}
}

The number of voting sequences is the $n$\textsuperscript{th} Catalan number, $C_n$.
\begin{proof}
As you examine the votes, plot the partial results as follows:
\begin{itemize}
\item Start at (0,0)
\item As you examine each vote, increase the $x$ value by 1. If it is a vote for Trillary, increase the $y$ value by one, otherwise decrease the $y$ value by one.
\item Connect the position before the current vote with the position after the current vote by a straight line.
\end{itemize}
The $y$ value for any $x$ is the number of votes that Trillary is leading by after the first $x$ votes have been counted. Since we are given that Hump is never ahead, the plot will never go below the $x$-axis. Since we are given that the votes are tied at the end, the final position will be $(2n,0)$. Thus there is a bijection between sequences of votes in this problem and the number of paths from $(0,0)$ to $(2n,0)$ such that each step of the path moves by $(+1, +1)$ or $(+1, -1)$ and the path never goes below $x=0$. Therefore the number of voting sequences is the $n$\textsuperscript{th} Catalan number, $C_n$.
\end{proof}
\begin{center}
\begin{tikzpicture}
%\shade[bottom color=white, top color=blue,opacity=.2](-5,0) rectangle (5,5);
%\shade[top color=white, bottom color=red,opacity=.2](-5,0) rectangle (5,-5);
\draw[step=1.0,black,thin,color=black,opacity=.35] (-5,-3) grid (5,5);
\node (S) at (-5, 0) {};
\node (E) at (5, 0) {};
\node (T) at (-5, 5) {};

\draw[pattern=bigger north west lines, pattern color=red!35] (-5,0) rectangle (5,-3) node [midway,fill=white] {Hump can never be ahead};

\draw [-,line width=.9] (-5,-3) -- (-5,5) node [pos=.85,align=center,xshift=2em] {$\uparrow$\\Trillary\\Lead};
\draw [-,line width=.9] (-5,0) -- (5,0) node [midway,yshift=1em] {\# of votes examined $\rightarrow$};

\draw [->,line width=1.75,blue] (-8,1.5) -- (-7,2.5) node[midway, black, yshift=-2em] {1 Trillary Vote};
\draw [->,line width=1.75,red]  (-8,0.5) -- (-7,-.5) node[midway, black, yshift=-2em] {1 Hump Vote};

\draw [->,line width=1.75,blue] (-5,0) -- (-4,1);
\draw [->,line width=1.75,red] (-4,1) -- (-3,0);
\draw [->,line width=1.75,blue] (-3,0) -- (-2,1);
\draw [->,line width=1.75,blue] (-2,1) -- (-1,2);
\draw [->,line width=1.75,red] (-1,2) -- (0,1);
\draw [->,line width=1.75,blue] (0,1) -- (1,2);
\draw [->,line width=1.75,blue] (1,2) -- (2,3);
\draw [->,line width=1.75,red] (2,3) -- (3,2);
\draw [->,line width=1.75,red] (3,2) -- (4,1);
\draw [->,line width=1.75,red] (4,1) -- (5,0);
\node [black] at (-5.6,0) {$(0,0)$};
\node [black] at (5.7,0) {$(2n,0)$};

\end{tikzpicture}
\end{center}

\section{Computers \& Printers}
\ProblemStatement{
We have 10 distinguishable computers and 3 distinguishable printers. Each computer must be linked to exactly one printer. But if a printer has 8 or more computers linked to it, it will overheat. How many assignments of computers to printers are there such that we have no overheating?
}

There are 58446 ways to connect 10 printers to 3 printers without overheating.

\begin{proof}
There are three choices of printer for each computer, giving $3^{10}=59049$ possible assignments by the product principle. The total number of assignments is equal to the number of non-overheating assignments plus the number of assignments where a single printer has 8, 9, or 10 computers attached to it by the sum principle. Therefore we can subtract the overheating assignments from the total number of assignments to get the non-overheating assignments.

Three assignments cause overheating by connecting all 10 computers to the same printer. For connecting nine computers to a single printer, there are $\binom{10}{9} = 10$ ways to choose the 9 computers multiplied by 3 choices for the printer. For each of these there are two printer choices for the remaining computer, resulting in a total of $10 \times 3 \times 2 = 60$ assignments which overheat due to 9 computers on a single printer. Lastly, for 8 computers connected to a single printer there are $\binom{10}{8}=45$ ways to choose the 8 computers, 3 ways to choose the printer, and $2^2=4$ choices for the other two computers.

Therefore we have a total of
\[
    3^{10} - 3 - \binom{10}{9}\times3\times2 - \binom{10}{8}\times3\times2^2 = 58446
\]
ways to connect 10 computers to 3 printers without overheating.

\end{proof}

\section{Time Spent \& Thoughts}

How about some harder bijections / proofs that are related to the stuff we've been doing in class? This problem set seemed like four easy problems from class and one very hard problem unrelated to what we've done (the bipartite graphs / odd cycles proof). They all took about 15-30 minutes except that one, which took about 3-4 hours.

\end{document}


\chapter{Protein Models}
\label{chap:protein}

\section{Clementi G\={o}}
\label{sec:protein_cc_go}

Clementi \emph{et al.} proposed an off-lattice G\={o} model and successfully
applied it to small fast-folding proteins, getting consistent folding pathways
with experiments~\cite{Clementi2000}.  This model has been used by several
following works showing its capability of studying the protein folding reaction
mechanism~\cite{Hoang2000, Koga2001}.

\subsection{Topology}
\label{subsec:protein_cc_go_top}

In this minimal G\={o} model, each amino acid residue in a given protein is
represented by a single CG bead put at the C\textsubscript{$\alpha$} atom, with
unified mass $m_{i} = 1$.

\note{} In the protein G\={o} model simulations, people usually use the reduced
unites, \emph{i.e.}, $m=1, \epsilon=1, k_B=1$.  However, when the G\={o} model
is used for protein together with other CG biomolecular models such as the
3SPN.2C DNA, the user has to be careful about the units.

\subsection{Potentials}
\label{subsec:protein_cc_go_potential}

The Clementi G\={o} model includes several terms:
\begin{equation}
  \label{eq:protein_cc_go_all}
  U = U_{bond} + U_{ang} + U_{dih} + U_{nonlocal}
\end{equation}

\subsubsection{Bond}
\label{sec:protein_cc_go_potential_bond}

\begin{smallpage}{Clementi G\={o} bond potential}<white>
  \begin{equation}
    \label{eq:protein_cc_go_local_bond}
    U_{bond} = \sum_{i}^{bonds} k_b (r_i - r_{i,0})^2
  \end{equation}
  \tcblower
  \begin{itemize}
  \item Bonds involved:
    \begin{itemize}
    \item C\textsubscript{$\alpha$}-C\textsubscript{$\alpha$}
    \end{itemize}
  \item $r_{i, 0}$: $r_{i}$ in the native structure of protein.
  \item $k_b = 100.0$.
  \end{itemize}
\end{smallpage}

\subsubsection{Angle}
\label{sec:protein_cc_go_potential_angle}

\begin{smallpage}{Clementi G\={o} bond angle potential}<white>
  \begin{equation}
    \label{eq:protein_cc_go_local_angle}
    U_{ang} = \sum_{i}^{angles} k_\theta (\theta_i - \theta_{i,0})^2.
  \end{equation}
  \tcblower
  \begin{itemize}
  \item Angles involved:
    \begin{itemize}
    \item C\textsubscript{$\alpha$}-C\textsubscript{$\alpha$}-C\textsubscript{$\alpha$}
    \end{itemize}
  \item $\theta_{i, 0}$: $\theta_{i}$ in the native structure of protei.
  \item $k_\theta = 20.0$.
  \end{itemize}
\end{smallpage}

\subsubsection{Dihedral Angle}
\label{sec:protein_cc_go_potential_dihedral_angle}

\begin{smallpage}{Clementi G\={o} dihedral angle potential}<white>
  \begin{equation}
    \label{eq:protein_cc_go_local_dihedral_angle}
    U_{dih} = \sum_{i}^{dihedrals} k_{\phi}^{(n)} \big[ 1+\cos\big(n( \phi_i - \phi_{ i,0 } )\big) \big].
  \end{equation}
  \tcblower
  \begin{itemize}
  \item Angles involved:
    \begin{itemize}
    \item C\textsubscript{$\alpha$}-C\textsubscript{$\alpha$}-C\textsubscript{$\alpha$}-C\textsubscript{$\alpha$}
    \end{itemize}
  \item $\phi_{i, 0}$: native value shifted by $\pi/n$.
  \item $k_\phi^{(1)} = 1.0$.
  \item $k_\phi^{(3)} = 0.5$.
  \end{itemize}
\end{smallpage}

\note{} The dihedral angle potential in GENESIS has the form of $k_{\phi} \big[
1+\cos(n\phi - \delta)\big]$, where we should set $\delta = n\phi_0 + \pi$.

\subsubsection{Nonlocal Interactions}
\label{sec:protein_cc_go_potential_nonlocal}

\begin{smallpage}{Clementi G\={o} native contact potential}<white>
  \begin{equation}
    \label{eq:protein_cc_go_native_contact}
    U_{contact} = \sum_{i}^{native contact} \varepsilon^{(n)} \big[ 5 \big( \frac{\sigma_{ij}}{r_{ij}} \big)^{12} - 6 \big( \frac{\sigma_{ij}}{r_{ij}} \big)^{10}\big].
  \end{equation}
  \tcblower
  \begin{itemize}
  \item Angles involved:
    \begin{itemize}
    \item C\textsubscript{$\alpha$}-C\textsubscript{$\alpha$}-C\textsubscript{$\alpha$}-C\textsubscript{$\alpha$}
    \end{itemize}
  \item $\phi_{i, 0}$: native value shifted by $\pi/n$.
  \item $k_\phi^{(1)} = 1.0$.
  \item $k_\phi^{(3)} = 0.5$.
  \end{itemize}
\end{smallpage}


\subsection{Group means and the grand mean}

\subsubsection{Introduction}


\clearpage
\subsection{Expressions (with Arrays)} % (fold)
\label{sub:expressions_with_arrays_}

Expressions allow you to read values from the elements of an array. To get an elements value you must supply the name of the array, and the index of the element you want to read.

\begin{figure}[h]
   \centering
   \includegraphics[width=\textwidth]{./topics/arrays/diagrams/ExpressionWithArray}
   \caption{You can read the values back from an array}
   \label{fig:expression-with-arrays}
\end{figure}

\mynote {
\begin{itemize}
  \item Expression is the \textbf{term} given to the code that calculates values within your Statements.
  \item You can read the values of elements of an array in an expression.
  \item The index values used to access the individual elements of an array are expressions themselves.
  \item Arrays are similar to variables in expressions, the expression reads the value from the element of the array.
  \item The index you supply determine which value is read.
\end{itemize}
}

% subsection expressions_with_arrays_ (end)


In this chapter, a brief derivation of the reduced fluid models used in the included publications is presented. The derivations start from the Braginskii fluid equations whose assumptions and validity for SOL plasmas are discussed. Applying drift reduction, Bohm-normalization and a number of approximations, results in the reduced two-fluid model, equivalent to the two-dimensional fluid models used in Paper III and IV. By applying interchange normalization this model will be further modified to the idealized interchange model, used in Paper I.

\section{Braginskii fluid equations}
The Braginskii fluid model is derived by taking successive velocity moments of the kinetic Boltzmann equation and applying a collisional closure. Each moment depends on the next higher order and therefore require additional assumptions to obtain a closure for the model. The Braginskii equations describe the evolution of the three lowest order fluid moments. The assumptions and the formulation of this closure are presented in \cite{braginskii}. The standard Braginskii fluid equations describing the evolution of the particle density $n_\alpha$, fluid velocity $\textbf{u}_\alpha$ and temperature $T_\alpha$ for particle species $\alpha$ are given by
\begin{equation}\label{brag_1}
	\frac{\partial n_\alpha}{\partial t} + \nabla \cdot (n_\alpha \textbf{u}_\alpha) = 0,
\end{equation}
\begin{equation}\label{brag_2}
	m_\alpha n_\alpha \left(\frac{\partial}{\partial t} + \textbf{u}_\alpha \cdot \nabla\right)\textbf{u}_\alpha = - \nabla p_\alpha - \nabla \cdot \Pi_\alpha +Z_\alpha e n_\alpha\left(\textbf{E} + \textbf{u}_\alpha \times \textbf{B}\right) + \textbf{R}_\alpha ,
\end{equation}
\begin{equation}\label{brag_3}
	\frac{3}{2}n_\alpha \left(\frac{\partial}{\partial t} + \textbf{u}_\alpha \cdot \nabla\right)T_\alpha + p_\alpha \nabla \textbf{u}_\alpha = - \nabla \cdot \textbf{q}_\alpha - \Pi_\alpha : \nabla \textbf{u}_\alpha + Q_\alpha.
\end{equation}

Here, $\alpha$ determines the particle species, i.e., electrons and ions, $m$ the particle mass, $p$ the pressure, $Z_\alpha e$ the particle charge, $\textbf{R}$ the friction force, $\Pi$ the viscous stress tensor, $\textbf{q}$ the heat flux, : the tensor inner product and $Q$ the frictional interspecies heating and energy exchange.%, and $S$, $\textbf{S}$ and $W$ represent local density, momentum and energy sources, respectively.
%In the following, we will focus on the first two moments as all models used in the included papers are isothermal.

The Braginskii equations are only applicable if certain assumptions for the modeled system are valid. Applying fluid equations requires that the distribution of particle velocities is close to Maxwellian, i.e., the time scale of relaxation back to a Maxwellian must be shorter than the characteristic time scales of the modeled system. If this condition is fulfilled, the system is referred to as collisional. %This can be expressed as $\omega_c \ll \nu_{ei}$ and $\omega_c \ll \nu_{ii}$ where $\omega_c$ stands for the characteristic frequency of the modeled system and $\nu_{\alpha\beta}$ for the average collision frequency between particle species $\alpha$ and $\beta$. In addition, $l_\parallel \gg \lambda_e$ and $l_\perp \gg \lambda_i$ where $l_\parallel$ and $l_\perp$ represent the smallest length scales of the system parallel and perpendicular to the magnetic field and $\lambda_\alpha$ is the mean free path for a particle species $\alpha$, given by $\lambda_\alpha = v_{\mathrm{th},\alpha}/\nu_{\alpha\beta}$ with the thermal velocity defined as $v_{\mathrm{th},\alpha}= \sqrt{T_\alpha/m_\alpha}$. \textcolor{red}{Ask Fulvio about  i and e}
In addition to being collisional, a plasma must be strongly magnetized to be adequately described by the Braginskii equations. This implies that the particles complete many gyrations between collisions, setting an upper limit for the collisionality of the plasma. %This condition can be formulated as $\nu_{ie} \ll \Omega_e$ and $\nu_{ii} \ll \Omega_i$ where the gyration frequency is defined as $\Omega_\alpha = eB/m_\alpha$, and $l_\parallel \gg \rho_e$ and $l_\parallel \gg \rho_i$ with the gyration radius given by $\rho_\alpha = v_{\mathrm{th},\alpha}/\Omega_\alpha$. These conditions define the range of time and length scales in which the Braginskii model is valid.

In summary, the phenomenon we want to model needs to satisfy the following conditions in order to be well described by the Braginskii equations:
\begin{equation}
	L_\perp \gg \rho_\alpha,
\end{equation}
\begin{equation}
	L_\parallel \gg \lambda,
\end{equation}
\begin{equation}
	\tau \gg \tau_c \gg \Omega_\alpha^{-1}.
\end{equation}
In these expressions $L_\perp$ stands for the characteristic size of the modeled phenomena perpendicular to the magnetic field, $\rho_\alpha$ is the gyration radius for species $\alpha$, $L_\parallel$ the parallel size of the system, $\lambda$ the collisional mean free path, $\tau$ the characteristic time of the problem, $\tau_c$ the collision time and $\Omega_\alpha$ the gyration frequency of the referred particle species \cite{militellobook}.

For the further derivation using drift reduction it will be useful to quantify the magnetization. We thereby define the magnetization parameter $\delta$ as
\begin{equation}
	\delta = \frac{\rho_\alpha}{L_\perp}.
\end{equation}
The magnetization can be equivalently expressed in the temporal domain by
\begin{equation}
	\delta = \frac{\nu_{ie}}{\Omega_i}
\end{equation}
where $\nu_{ie}$ stands for the collisional frequency between ions and electrons.

For both electrons and ions the magnetization parameter is $\delta \ll 1$ for a fully magnetized plasma.

\section{Drift reduction}
The Braginskii model given by \Eqsref{brag_1} - \eqref{brag_3} is very general, making modeling of SOL plasmas with the presented equations relatively inefficient. A more suitable description of plasma phenomena in the SOL can be derived by simplifying the presented model with an approach called drift ordering. Since turbulence and filaments in the SOL, evolve with velocities much lower than the plasma sound speed $c_s = \sqrt{(T_e+T_i)/m_i}$ we apply the ordering
\begin{equation}
	\textbf{u}_\perp \sim \frac{\rho_\alpha}{L_\perp} c_s \sim \delta c_s.
\end{equation}
This ordering assumes that the transverse electric fields are small, resulting in the perpendicular electric field being substantially electrostatic. This is a direct consequence of the $\textbf{E}\times\textbf{B}$ velocity being a factor $\delta$ smaller than sound speed and Faraday's law \cite{militellobook}. We can now determine the perpendicular part of the momentum equation, given by \Eqref{brag_2}, by taking the cross product with $\textbf{B}$ resulting in
\begin{equation}
	\textbf{u}_{\alpha,\perp} = \frac{\textbf{E}\times\textbf{B}}{B^2} - \frac{\nabla p_\alpha \times \textbf{B}}{e_\alpha n_\alpha B^2} - \frac{m_\alpha d \textbf{u}_\alpha/d t \times \textbf{B}}{e_\alpha B^2} - \frac{\nabla \cdot \Pi_\alpha\times\textbf{B}}{e_\alpha n_\alpha B^2} + \frac{\textbf{R}_\alpha \times \textbf{B}}{e_\alpha n_\alpha B^2},
\end{equation}
where we used $d/dt = \partial/\partial t + (\textbf{u}_\alpha\cdot \nabla)$ and assumed single charge particle species, i.e., $Z_\alpha e = e_\alpha$. The terms in this expression display the fluid drifts occurring in the system, namely from left to right: the $\textbf{E}\times\textbf{B}$ drift; the diamagnetic drift; the polarization drift; the viscous drift and the collisional drift. From this expression we can determine the dominant drifts and thereby simplify the model.

As mentioned previously, the electric drift velocity is of $\mathcal{O}(\delta)$ compared to the plasma sound speed:
\begin{equation}
	\textbf{u}_E = \frac{\textbf{E}\times\textbf{B}}{B^2} \sim \delta c_s.
\end{equation}
Similarly, the diamagnetic drift is also of $\mathcal{O}(\delta)$ since
\begin{equation}
	\textbf{u}_{\mathrm{dia}} = -\frac{\nabla p_\alpha \times \textbf{B}}{e_\alpha n_\alpha B^2} \sim \frac{n_\alpha T_\alpha B}{L_\perp e_\alpha n_\alpha B^2} \sim \frac{T_\alpha}{L_\perp \Omega_\alpha m_\alpha} \sim \delta c_s.
\end{equation}
The polarization drifts for both ions and electrons are smaller in comparison, as can be shown by
\begin{equation}
	\textbf{u}_{\mathrm{pol},i} = \frac{m_i d \textbf{u}_i/d t \times \textbf{B}}{e_i B^2}  \sim \delta^3 c_s,
\end{equation}
and
\begin{equation}
	\textbf{u}_{\mathrm{pol},e} \sim \frac{m_e}{m_i} \delta^3 c_s.
\end{equation}
For the viscous drift we use Bragniskii's approximation for the perpendicular component of the viscous stress tensor $\Pi_\alpha \sim (p_\alpha/\Omega_\alpha) \nabla v_{\alpha}$ which shows that this term is of $\mathcal{O}(\delta^3)$ since
\begin{equation}
	\textbf{u}_{\mathrm{vis},i} = \frac{\nabla \cdot \Pi_\alpha\times\textbf{B}}{e_\alpha n_\alpha B^2} \sim \frac{nT \delta c_s}{e_i n_i B L_\perp^2 \Omega} \sim \delta^3 c_s,
\end{equation}
and
\begin{equation}
	\textbf{u}_{\mathrm{vis},e} \sim \frac{m_e}{m_i}\delta^3  c_s,
\end{equation}
respectively. Lastly, we need to find an approximation for the collisional drift. For this we use $\textbf{R}_\perp = e n\,\textbf{J}_\perp/\sigma_\perp$ for the perpendicular momentum transfer from electron-ion friction and $\sigma_\perp = ne^2\nu_{ei}/m_e$. From the ordering follows $\textbf{J}_\perp \sim en\delta c_s$ which leads to the approximation of the frictional drift
\begin{equation}\label{diffusion}
	\textbf{u}_{\mathrm{fri}} =\frac{\textbf{R}_\alpha \times \textbf{B}}{e_\alpha n_\alpha B^2} \sim \frac{ne\delta c_s}{B \sigma_\perp} \sim \frac{m_e}{m_i}\frac{\nu_{ei}}{\Omega}\delta c_s.
\end{equation}
For SOL conditions we can typically assume that $\nu_{ei}/\Omega\sim \delta$ so that the collisional drift is of  $\mathcal{O}(\delta^2)$.

This ordering reveals that the dominant perpendicular drifts are the electric and the diamagnetic drifts as all other drifts are at least one order of magnitude smaller. By substituting the remaining drifts into the Braginskii equation we can rewrite the electron density equation in a simpler form,
\begin{equation}\label{n_e}
	\frac{\partial n_e}{\partial t} + \nabla \cdot\left[n_e\left(\textbf{u}_E + \textbf{u}_{\mathrm{dia},e}+ \textbf{u}_{e \parallel}\right)\right] = 0.
\end{equation}
Since the plasma is quasi-neutral, i.e.\ $n_e\simeq n_i\simeq n$, this equation is used to describe the evolution of the total plasma density $n$. \Eqref{n_e} is usually manipulated to
\begin{equation}\label{dens}
	\frac{\partial n}{\partial t} + \textbf{u}_E \cdot \nabla n = -\nabla\cdot\left(n\textbf{u}_{\parallel,e}\right) + \left(\frac{1}{e}\nabla p_e - n \nabla \phi\right)\cdot\nabla\times \left(\frac{\textbf{b}}{B}\right).
\end{equation}
%As we can see, the diamagnetic drift does not give rise to plasma advection as this term is not a drift of guiding centers but a consequence of counting particle motion through an area in the fluid picture, leading to so called diamagnetic cancellation inside the divergence terms. In non-uniform magnetic fields, this cancellation results in a non-divergence free term $\nabla \times (\textbf{b}/B)$, referred to as the curvature operator, which will be discussed later.

Instead of explicitly deriving separate continuity equations for electrons and ions, we can utilize quasi-neutrality and charge conservation to derive an equation for the fluid velocity, which will prove to be very handy. For this we use $\nabla \cdot \textbf{J} = 0$ with $\textbf{J}= e n (\textbf{u}_i - \textbf{u}_e)$. Inserting all drifts that give rise to a net current results in
\begin{equation}\label{diff_J}
	\nabla \cdot \left(\textbf{J}_{\mathrm{dia}} + \textbf{J}_{\mathrm{pol}} +\textbf{J}_{\mathrm{vis}}+ \textbf{J}_{\parallel}\right) = 0.
\end{equation}
In the following we only include the leading order drifts in the ion polarization velocity and neglect the electron polarization drift entirely due to the small electron mass. The sum of the ion polarization and viscous drifts using the lowest order solution
of the perpendicular momentum equation and the parallel velocity $\textbf{u}_0 = \textbf{u}_{0\perp} + \textbf{u}_{i\parallel}$ is then given by
\begin{equation}
	\textbf{u}_{\mathrm{pol},i} + \textbf{u}_{\mathrm{vis},i} = \textbf{b}\times \frac{1}{enB}\left[m_i n \left(\frac{\partial}{\partial t} + \textbf{u}_{i0} \cdot \nabla\right)\textbf{u}_{i0} + \nabla \cdot \Pi_{i0}\right],
\end{equation}
where $\Pi_{i0}$ is the viscous stress tensor calculated with $\textbf{u}_{i0}$ and
\begin{equation}
	\textbf{u}_{0\perp} = \textbf{b}\times \frac{1}{B}\left(\nabla \phi + \frac{1}{en} \nabla p_i \right).
\end{equation}
Inserting this into \Eqref{diff_J} leads to the drift-reduced charge conservation equation  \cite{militellobook}
\begin{multline}\label{charge_conservation}
	-\nabla \cdot \left\{\textbf{b}\times\frac{1}{B} \left[m_i n\left(\frac{\partial}{\partial t} + \textbf{u}_{i0}\cdot \nabla\right)\textbf{u}_{i0} + \nabla\cdot \Pi_{i0}\right]\right\} =\\ \nabla\cdot\textbf{J}_\parallel + \nabla \left(p_i + p_e\right) \cdot \nabla\times \left(\frac{\textbf{b}}{B}\right),
\end{multline}
which will be further simplified in the following.
\section{Further approximations and simplifications}
A number of additional approximations are applied to simplify the model and make it efficiently numerically solvable.

First of all, we apply the {electrostatic} approximation, and thereby neglect all time derivatives of $\textbf{B}$ and calculate the electric field $\textbf{E}$ directly from the electric potential. Under this approximation, \Eqref{charge_conservation} can be expressed in the more readable form
\begin{multline}\label{electrostatic}
	m_{i}\nabla\cdot\left[\frac{n}{B} \frac{d_0}{dt}\left(\frac{\nabla_\perp\phi}{B} + \frac{\nabla_\perp p_i}{enB}\right) \right] - \nabla\cdot \left(\textbf{b}\times \nabla\cdot \Pi_0\right) =
	\\ \nabla\cdot\textbf{J}_\parallel + \nabla\left(p_i + p_e\right)\cdot \nabla\times \left(\frac{\textbf{b}}{B}\right),
\end{multline}
where we used $d_0/dt = \partial/\partial t + (\textbf{u}_{i0}\cdot \nabla)$. In addition, we neglected spatial non-uniformity of $\textbf{B}$, which will be discussed in further detail later. Studies of electromagnetic effects on plasma blob-filament transport showed that these effects in high temperature or high beta plasmas suppress the resistive drift wave turbulence in filaments \cite{lee2015electromagnetic,hoare2019dynamics} but will not be considered in the following.

We can further simplify \Eqref{electrostatic} by applying scale separation for the plasma density, so that $\nabla n \sim \nabla n_0 + \nabla \widetilde{n} \sim 1/L_n + k \widetilde{n}$, where the particle density has been separated into a background $n_0$ and a fluctuation $\widetilde{n}$. $L_n$ stands for the characteristic scale length for the background density and $k$ for the wave number for the particle density fluctuations. Dividing by $n_0$ leads to $\nabla \ln{n} \sim 1/L_n + k \widetilde{n}/n_0$. We now assume that $1/kL_n \ll 1$ and $\widetilde{n}/n_0 \ll 1$. The latter assumption is the so called {thin layer} or {Boussinesq} approximation where we assume that the density perturbations are small compared to the equilibrium. This assumption is hardly justified since relative fluctuations in the SOL can be of order unity as discussed in the previous chapter. This approximation is, however, commonly used since it makes the numerical integration of \Eqref{electrostatic} significantly more efficient. By introducing a generalized vorticity,
\begin{equation}
	\varpi = \nabla\cdot\left(\nabla_\perp \phi + \frac{\nabla_\perp p_i}{en}\right),
\end{equation}
we can now simplify the first term in \Eqref{electrostatic} to
\begin{equation}
	m_{i}\nabla\cdot\left[\frac{n}{B} \frac{d_0}{dt}\left(\frac{\nabla_\perp\phi}{B} + \frac{\nabla_\perp p_i}{enB}\right) \right] \approx \frac{m_i n}{B^2}\frac{d_0\varpi}{dt}.
\end{equation}
From this expression, $\varpi$ can be relatively easily inverted, especially when assuming that ions are cold, leading us to the next approximation.

For the remaining derivation we will assume {small ion temperature}, $T_i \ll T_e$, simplifying the equations significantly. This is a restrictive assumption, as experimental measurements indicate that the ion temperature is higher than the electron temperature in the SOL \cite{kovcan2007ion,kocan2012ion}. Numerical simulations incorporating finite ion temperature have shown that the coherency of filaments is increased \cite{Ahmed_ions}. However, since the simplified model still captures the fundamental dynamics in the SOL, this approximation is commonly used to reduce the model complexity.

As for the electrons, all models in the included publications and manuscripts assume {isothermal electrons}. This assumption simplifies the model drastically, as it makes \Eqref{brag_3} obsolete. Numerical simulations of isolated filaments with dynamic electron temperature have shown that thermal effects lead to a strong increase in the filament propagation in the poloidal direction and reduce the net radial propagation. These effects arise from the electron temperature dependence of the sheath currents, which will be discussed later in this chapter \cite{walkden2016dynamics}.

Next, we will define the geometry of the magnetic field. For the whole simulation domain, we assume {straight magnetic field} lines with {constant} field strength. We need to make one exception to this assumption, as no curvature term would remain in a completely homogeneous field. As there would be no drive for filament motion without this term, it is required to capture some effects of curvature in the model. With the use of vector algebra presented in \cite{militellobook} we can write the curvature term from \Eqsref{dens} and \eqref{electrostatic} as
\begin{equation}
	\nabla\times\left(\frac{\textbf{b}}{B}\right) = 2\frac{\textbf{b}\times{\boldsymbol\kappa}}{B} + \frac{\mu_0\left(\textbf{J}_\parallel - \textbf{J}_\perp\right)}{B^2},
\end{equation}
where we introduced the curvature vector $\boldsymbol\kappa = (\textbf{b}\cdot\nabla)\textbf{b}$. Note that one unit of the term $\textbf{b}\times\boldsymbol\kappa/B$ originates from the magnetic gradient and one from the curvature. The second term on the right hand side can be neglected due to charge conservation \cite{militellobook}. The magnetic field in a tokamak can be approximated to lowest order to be purely toroidal and falling radially with $1/R$. In a cylindrical coordinate system $(R,\Phi,Z)$ the toroidal magnetic field is therefore
\begin{equation}
	\textbf{B} = \frac{B_0R_0}{R}\boldsymbol{\widehat{\Phi}}.
\end{equation}
In a slab geometry with Z being replaced with the binormal direction $y$ which is perpendicular to $\widehat{\textbf{R}}$ and $\boldsymbol{\widehat{\Phi}}$ this motivates the definition of the curvature operator
\begin{equation}\label{curvature}
	\mathcal{K}(u) = \nabla\times \left(\frac{\textbf{b}}{B}\right)\cdot\nabla u\approx -\frac{2}{B_0R_0}\frac{\partial u}{\partial y}.
\end{equation}
%In this expression $R_0$ is defined as the sum of the major and minor radius and $B_0$ the magnetic field strength at that radial position.

Despite arguing that the frictional drift is negligible in \Eqref{diffusion} one typically retains an approximation of this term due to numerical reasons. We therefore add this term to \Eqref{dens} as
\begin{equation}\label{diffusion}
	\nabla\cdot \left(n_e \textbf{u}_{\textrm{fri}}\right) \approx -\nabla\cdot \left(D_n\nabla_\perp n_e\right)\approx-D_n\nabla^2_\perp n_e,
\end{equation}
where we introduced the density diffusion coefficient $D_n$ which we assumed to be spatially constant. Similarly, one can derive the diffusion term for \Eqref{electrostatic} from its ion viscosity term, since we can use the approximation
\begin{equation}
	\nabla\cdot \Pi_i = -m_in\mu_\omega\nabla_\perp^2\textbf{u}_E,
\end{equation}
where $\mu_\omega$ stands for the effective cross-field kinematic viscosity of the ions. Inserting the electric drift and taking the divergence results in the diffusion term for $\nabla_\perp^2\phi$ as
\begin{equation}
	\nabla\cdot\left(\frac{m_i \mu_\omega}{eB^2} \nabla_\perp\nabla_\perp^2\phi\right).
\end{equation}
The diffusion coefficients can be approximated from classical or neo-classical diffusion such as presented in \cite{fundamenski2007dissipative}, or are chosen for numerical accuracy and stability.

Arguably the starkest simplification of the presented models in this thesis is the restriction to only two dimensions, the plane perpendicular to \textbf{B}. The parallel closure of the model equations is different for closed and open field lines, i.e., whether the simulation domain is located in the SOL or in the edge region. Since the parallel direction plays an important role in the SOL for particle and current dissipation as plasma flows along the magnetic field lines towards the divertor plates, a suitable approximation for the parallel losses is required. This closure is achieved by integrating over the parallel direction where the so-called sheath boundary conditions come into play. In the initial transient period where the plasma vessel is filled and the cold wall surface electrically neutral, electrons will strike the surface at a higher rate than the ions due to their higher speed. This charges the vessel walls negatively which impedes further electron flow towards the surface and results in a thin sheath at material surfaces. Here, the ions shield the electric potential of the surface and the sheath extends a few Debye lengths, $\lambda_D = \sqrt{\epsilon_0T_e/n_ee^2}$, outwards from the surface into the plasma. In this region quasi neutrality is violated since the ion density is higher than the electron density, $n_i > n_e$. The electric current density drawn by the vessel walls is governed by the influx of electrons and ions at the sheath surface. It depends on the potential $\phi$ at the sheath entrance and can be written as
\begin{equation}\label{parallel_flux}
	\textbf{J}_\parallel|_{\mathrm{sheath}} = en_{se}c_s \left[1 - \text{e}^{\Lambda-e{\phi}/{T_e}}\right],
\end{equation}
with the plasma density at the sheath edge $n_{se}$, the acoustic speed $c_s$ and the floating potential $\Lambda=\mathrm{ln}\sqrt{m_i/2\pi m_e}$. The first term in the parenthesis is due to the ion flux and the second due to the electron flux \cite{stangeby2000plasma}. %Note that while the ion flux at the sheath edge is given by $n_{se}c_s$, the electron flux is dependent on the potential drop in the sheath \cite{stangeby2000plasma}.
We can now take the average of the parallel dimension in a slab geometry with $\textbf{B} = B\, \widehat{\textbf{z}}$,
\begin{equation}\label{v_parallel}
\frac{1}{L_\parallel} \int_{-L_\parallel/2}^{L_\parallel/2} \nabla_\parallel \cdot \textbf{J}_\parallel dz,
\end{equation}
and use \Eqref{parallel_flux} as the boundary conditions. The first term on the right hand side of \Eqref{dens} can be handled analogously for the parallel electron velocity.

Paper III includes a core region in the simulation domain, requiring a different closure for the parallel dynamics. In this model we include resistivity in the parallel component of the electron momentum equation neglecting inertia, i.e.,
\begin{equation}\label{par}
	en\frac{\partial\phi}{\partial z} - T_e\frac{\partial n}{\partial z} + \chi enJ_\parallel=0,
\end{equation}
where the resistivity is given by $\chi = m\nu_{ei}/n_e e^2$. %We neglect the parallel ion momentum since the ions remain to lowest order stationary due to their high inertia.
Rearranging \Eqref{par} for $J_\parallel$ and taking the parallel derivative results in \cite{bellan2008fundamentals,meyer_thesis}
\begin{equation}\label{resistivity}
	\nabla_\parallel \cdot \textbf{J}_\parallel = \frac{T_e}{e\chi}\frac{\partial^2}{\partial z^2}\left(\mathrm{ln}\,n_e - \frac{e\phi}{T_e}\right).
\end{equation}
From this we can take the average of the parallel dimension by integrating over $z$, resulting in the desired 2D model equations. A systematic analysis of the dimensionality of scrape-off layer turbulence is presented in \cite{nicholas_dim,Nicholas2021comparing}.

\section{Reduced two-fluid model}

Since the Braginskii fluid model is only valid in a specific range of time and length scales it seems natural to normalize all physical variables to values that are characteristic for the modeled system. We will first discuss the so-called Bohm normalization where we normalize the spatial and temporal units by $\rho_s$ and $\Omega_i$, respectively, i.e.,
\begin{equation}
	\nabla \rightarrow \nabla' = \rho_s\nabla, \,\,\, \frac{\partial}{\partial t} \rightarrow \frac{\partial}{\partial t'} = \frac{1}{\Omega_i}\frac{\partial}{\partial t}.
\end{equation}
Here, $\rho_s$ stands for the ion sound Larmor radius defined as $\rho_s = \sqrt{T_e m_i}/eB$.
We normalize the remaining variables with their characteristic values for SOL conditions $N$ and $T_0$ as
\begin{equation}
	n \rightarrow n' = \frac{n}{N},\,\,\, T_e \rightarrow T' = \frac{T_e}{T_0},\,\,\,\phi \rightarrow \phi' = \frac{e\phi}{T_0}.
\end{equation}
From these expressions we can define the characteristic magnitude for the density source, diffusion coefficients and effective gravity drive as
\begin{equation}
	S\rightarrow S' = SN\Omega_i,\,\,\,D_n\rightarrow D_n' = D_{\textrm{Bohm}}D_n,\,\,\,\mu_\omega\rightarrow D_\Omega' = D_{\textrm{Bohm}}\mu_\omega,\,\,\,g = \frac{2\rho_s}{R},
\end{equation}
where the collisional diffusion is defines as $D_{\textrm{Bohm}}= \rho_s^2\Omega_i$. Applying this normalization to \Eqref{dens} and dropping the dash sign, inserting the curvature operator from \Eqref{curvature} and adding the diffusion term of \Eqref{diffusion} results in the electron density equation
\begin{equation}
	\frac{\mathrm{d}n}{\mathrm{d}t} + g \left(\frac{\partial n}{\partial y} - n\frac{\partial\phi}{\partial y}\right) = D_n \nabla_\perp^2 n + S_n + \Bigg \langle \nabla_\parallel \left(n{u}_{e\parallel} \right)\Bigg \rangle_\parallel,
\end{equation}
where the advective derivative is given by $\mathrm{d}/\mathrm{d}t = \partial/\partial t + \textbf{u}_E\cdot \nabla_\perp$ and $\textbf{u}_E = - \nabla_\perp\phi\times\textbf{B}/B^2$ is the $\textbf{E}\times\textbf{B}$ drift. Here, $\langle\cdot\rangle_\parallel$ refers to the average over the parallel dimension. We also added the density source term $S_n$. Performing the same kind of operations on \Eqref{electrostatic} after applying the Boussinesq approximation results in the vorticity equation
\begin{equation}
	\frac{\mathrm{d}\nabla_\perp^2 \phi}{\mathrm{d}t} + \frac{g}{n}\frac{\partial n}{\partial y} = D_{\Omega}\nabla_\perp^4 \phi + \Bigg \langle \frac{1}{n} \nabla_\parallel J_{\parallel} \Bigg \rangle_\parallel,
\end{equation}
where we introduced $D_{\Omega}$  as the collisional dissipation term representing viscosity. Averaging over the parallel direction after inserting \Eqref{v_parallel} and \Eqref{resistivity} can be expressed as
\begin{subequations}
	\begin{gather}
		\Bigg \langle \nabla_\parallel \left(n{u}_{e\parallel} \right) \Bigg \rangle_\parallel = - \eta(x) n\,\exp(\Lambda-\phi) + \chi(x)( \widehat{\phi} - \widehat{n} ) ,
		\\
		\Bigg \langle \frac{1}{n} \nabla_\parallel J_{\parallel} \Bigg \rangle_\parallel =  \eta(x) \left[ 1-\exp(\Lambda-\phi) \right] + \chi(x)( \widehat{\phi} - \widehat{n} ),
	\end{gather}
\end{subequations}
where the spatially fluctuating electron density $\widehat{n}$ and plasma potential $\widehat{\phi}$ are defined as $\widehat{n}=n-\langle{n}\rangle_y$ and $\widehat{\phi}=\phi-\langle{\phi}\rangle_y$ and $\langle{\cdot}\rangle_y$ refers to the flux surface average. Note, that we neglect $1/n$ in the plasma conductivity term for the vorticity equation, since we assume the plasma density to have small relative fluctuation levels in the edge region and $n \sim 1$. We redefined $\chi = \left(\rho_s/L_\parallel\right)^2(m_i/m_e)(\Omega_s/\nu_{ei})$ as the normalized parallel plasma conductivity where we used $\nabla_\parallel^2 \rightarrow -k^2_\parallel\simeq-L_\parallel^{-2}$ and introduced $\eta=\rho_s/L_\parallel$ as the normalized sheath dissipation coefficient. $\nu_{ei}$ stands for the collision frequency between electrons and ions given by
\begin{equation}
	\nu_{ei} = \frac{\textrm{log}\Lambda_c e^4Z^2n_i}{6\sqrt{2}\pi^{3/2} \epsilon_0^2\sqrt{m_e}T_e^{3/2}},
\end{equation}
and the Coulomb logarithm is approximately \cite{militellobook}
\begin{equation}
	\textrm{log}\Lambda_c\approx 18 - \textrm{log}\left[\left(\frac{n_e}{10^{19}}\right)^{1/2}\left(\frac{T_e}{10^3e}\right)^{-3/2}\right].
\end{equation}
These parameters depend on the radial position, the sheath dissipation term only occurs in the SOL and the plasma conductivity term is finite in the plasma edge region. A schematic illustration of these two regions in the simulation domain is shown in \Figref{Fig:model}.
\begin{figure}[t]
	\centering
	\includegraphics[width=10cm]{figures/model.eps}
	\caption{Schematic illustration of the edge and scrape-off layer region in the simulation domain. The position of the plasma
source term (gray shaded) and the border between edge and SOL (dashed vertical line) are indicated \cite{decristoforo2021numerical}.}
	\label{Fig:model}
\end{figure}

This model is equivalent to the one used in Paper III. Paper IV utilizes a slightly simpler model placing the whole domain in the SOL by choosing $\eta = \text{constant}$ and $\chi = 0$, and discarding $\Lambda$ in the sheath dissipation term. A list of representative machine parameters relevant for the reduced two-fluid models is presented in Table \ref{tab:machine_parameters}. Since each device performs a range of experiments with slightly different configurations, these parameters might vary. The presented parameters are consistent with those used in the numerical simulations in the included references. The radial position of the SOL is indicated by the sum of the major and minor radius and the parallel connection length is estimated as $L_\parallel = \pi q_{95} R$, with $q_{95}$ as the safety factor  at the 95\% poloidal magnetic flux surface, if not explicitly stated in the references. It should also be noted, that the definition of the connection length varies from source to source as it may refer to the whole poloidal length or only to the lenth between the outboard midplane to the outer divertor plates.

\begin{table}[h!]
	\begin{center}
		\caption{Machine parameters.}
		\label{tab:machine_parameters}
		\begin{tabular}{c|c c c c c|c}
			 & $n_e[\mathrm{m}^{-3}]$ & $T_e[\mathrm{eV}]$ & $B[\mathrm{T}]$ & $L_\parallel[\mathrm{m}]$ & $R+r[\mathrm{m}]$ & reference\\
			\hline
			MAST & $8\times 10^{18}$ & 40 & 0.5 & 30 & 1.5 & \cite{easy2016investigation}\\
			C-Mod & $1.4\times 10^{19}$ & 23 & 4.5 & 20 & 0.9 & \cite{terry2003observations}\\
			TCV & $5\times 10^{18}$ & 25 & 1.45 & 15 & 1.13  & \cite{nespoli2017blob,wensing2019solps}\\
			KSTAR & $7\times 10^{17}$ & 35 & 2.0 & 26 & 2.3 & \cite{bak2015investigation} \\
			AUG & $8\times 10^{18}$ & 40 & 2.5 & 25 & 2.15 & \cite{militello2011multi} \\
			JET & $2\times 10^{19}$ & 45 & 3.45 & 25 & 4.21 & \cite{fundamenski2007dissipative}\\
			NSTX & $6\times 10^{18}$ & 13 & 0.25 & 20 & 1.53 & \cite{terry2003observations}
		\end{tabular}
	\end{center}
\end{table}
The plasma parameters calculated for these device parameters are presented in Table \ref{tab:plasma}.
\begin{table}[h!]
	\begin{center}
		\caption{Physical parameters.}
		\label{tab:plasma}
		\begin{tabular}{c|c c c c}
		 & $\rho_s[\mathrm{m}]$ &$\Omega_i[\mathrm{s}^{-1}]$&$c_s[\mathrm{ms}^{-1}]$\\
			\hline
			MAST & $1.8\times 10^{-3}$ & $2.4\times 10^{7}$ & $4.4\times 10^{4}$\\
			C-Mod& $1.5\times 10^{-4}$ & $2.2\times 10^{8}$ & $3.3\times 10^{4}$\\
			TCV & $5.0\times 10^{-4}$ & $6.9\times 10^{7}$ & $3.5\times 10^{4}$\\
			KSTAR & $4.3\times 10^{-4}$ & $9.6\times 10^{7}$ & $4.1\times 10^{4}$\\
			AUG & $3.7\times 10^{-4}$ & $1.2\times 10^{8}$ & $4.4\times 10^{4}$\\
			JET & $2.8\times 10^{-4}$ & $1.7\times 10^{8}$ & $4.6\times 10^{4}$\\
			NSTX & $2.1\times 10^{-3}$ & $1.2\times 10^{7}$ & $2.5\times 10^{4}$\\
		\end{tabular}
	\end{center}
\end{table}
Based on these values, we can estimate the input parameters for the reduced two-fluid model shown in Table \ref{tab:input}. Note that the diffusion and viscosity coefficients for the model are not presented as they are chosen arbitrarily high for numerical stability in the included publications.
\begin{table}[h!]
	\begin{center}
		\caption{Input parameters.}
		\label{tab:input}
		\begin{tabular}{c|c c c }
			&  $g$ & $\eta$ & $\chi$ \\
			\hline
			MAST & $2.4\times 10^{-3}$ &  $6.1\times 10^{-5}$ & $2.7 \times 10^{-4}$\\
			C-Mod& $3.4\times 10^{-4}$ &  $7.7\times 10^{-6}$ & $1.0 \times 10^{-5}$\\
			TCV & $8.8\times 10^{-4}$ &  $3.3\times 10^{-5}$ & $1.9 \times 10^{-4}$ \\
			KSTAR & $3.7\times 10^{-4}$ & $1.6\times 10^{-5}$ & $6.8 \times 10^{-4}$ \\
			AUG & $3.4\times 10^{-4}$ & $1.5\times 10^{-5}$ & $7.7 \times 10^{-5}$ \\
			JET & $1.3\times 10^{-4}$ & $1.1\times 10^{-5}$ & $3.1 \times 10^{-5}$ \\
			NSTX & $2.7\times 10^{-3}$ & $1.0\times 10^{-4}$ & $1.1 \times 10^{-4}$ \\
		\end{tabular}
	\end{center}
\end{table}

\section{Idealized interchange model}
A minimal model for SOL plasma dynamics in the cross-field plane can be obtained by ignoring parallel dynamics entirely and applying the so called interchange normalization. We start again with \Eqref{dens} and \Eqref{electrostatic}, use the curvature operator given by \Eqref{curvature}, and include the diffusion terms. The perpendicular components then take the form
\begin{subequations}
	\begin{gather}
		\left(\frac{\partial}{\partial t} + \frac{1}{B}\widehat{\textbf{z}}\times \nabla\phi\cdot\nabla\right) n - \frac{2n}{B_0 R_0}\frac{\partial \phi}{\partial y} + \frac{2T_e}{eB_0R_0}\frac{\partial n}{\partial y} = D_n\nabla_\perp^2 n,
		\\
		 \left(\frac{\partial}{\partial t} + \frac{1}{B}\widehat{\textbf{z}}\times \nabla\phi\cdot\nabla\right)\nabla_\perp^2\phi + \frac{2c_s^2}{R_0n}\frac{\partial n}{\partial y} = D_\Omega\nabla_\perp^4 \phi.
	\end{gather}
\end{subequations}
Under the interchange normalization, length scales are normalized by the characteristic length $l$ of the system, time scales by the ideal interchange rate $\gamma = \sqrt{g/l}$ with $g=2c_s^2/R_0$ and the plasma density and electrostatic potential accordingly, i.e.,
\begin{equation}
	\nabla \rightarrow \nabla' = l\nabla, \,\,\, \frac{\partial}{\partial t} \rightarrow \frac{\partial}{\partial t'} = \frac{1}{\gamma}\frac{\partial}{\partial t}, \,\,\,n \rightarrow n' = \frac{n}{N},\,\,\,\phi \rightarrow \phi' = \frac{\phi}{\gamma B_0l^2}.
\end{equation}
Inserting these expressions into the equations for plasma density and vorticity results in
\begin{subequations}
	\begin{gather}
		\left(\frac{\partial}{\partial t'} + \widehat{\textbf{z}}\times \nabla'\phi'\cdot\nabla'\right) n' - 2n'\frac{l}{R_0}\frac{\partial \phi'}{\partial y'} + \frac{\gamma}{\Omega_i}\frac{\partial n'}{\partial y'} = \frac{D_n}{\gamma l^2}{\nabla'}_\perp^2 n',
		\\
		\left(\frac{\partial}{\partial t'} + \widehat{\textbf{z}}\times \nabla'\phi'\cdot\nabla'\right){\nabla'}_\perp^2\phi' + \frac{1}{n'}\frac{\partial n'}{\partial y'} = \frac{D_\Omega}{\gamma l^2}{\nabla'}_\perp^4 \phi'.
	\end{gather}
\end{subequations}
We neglect the term resulting from the compression of the electric drift since its prefactor is $l/R_0 \ll 1$. The term resulting from the compression of the diamagnetic drift will also be neglected in the continuity equation as previous work has shown that it has a negligible contribution to the cross-field dynamics and since $\gamma/\Omega_i \ll 1$ \cite{Garcia_thesis}. In addition we introduce the normalized particle diffusion and viscosity coefficients
\begin{equation}
	\kappa = D_n/\gamma l^2 \,\,\,\text{and}\,\,\, \mu = D_\phi/\gamma l^2,
\end{equation}
neglect $1/n$ in front of the interchange term as we apply the Boussinesq approximation and drop the dash notation to receive the minimal model for plasma convection
\begin{subequations}
	\begin{gather}
		\left(\frac{\partial}{\partial t} + \widehat{\textbf{z}}\times \nabla\phi\cdot\nabla\right) n = \kappa\nabla_\perp^2 n,
		\\
		\left(\frac{\partial}{\partial t} + \widehat{\textbf{z}}\times \nabla\phi\cdot\nabla\right)\nabla_\perp^2\phi + \frac{\partial n}{\partial y} = \mu\nabla_\perp^4 \phi.
	\end{gather}
\end{subequations}
The emphasis of this model lies on reducing the complexity and the number of free parameters as drastically as possible without losing the capability of modeling plasma advection self-consistently. This model has also been used in the past to describe buoyancy-driven convection in a fluid confined between two horizontal plates and heated from below. The model, named the Rayleigh-Bénard convection model after the original experimental work of Henri Bénard \cite{benard1901tourbillons} and the first analytical work on this model of Lord Rayleigh \cite{rayleigh1916lix}, has become a paradigm to investigate nonlinear phenomena due to its rich dynamics \cite{decristoforo2020intermittent, busse1978non,siggia1994high,bodenschatz2000recent,kadanoff2001turbulent,ahlers2009heat}. The normalized particle diffusion and viscosity coefficients in the presented formulation are related to the Rayleigh and Prandtl numbers as $R = 1/\kappa\mu$ and $R=\mu/\kappa$, which are typically used as model parameters for the Rayleigh-Bénard model. We use this model in Paper I.


\documentclass[12pt]{amsart}
\usepackage{geometry} % see geometry.pdf on how to lay out the page. There's lots.
\usepackage{bsymb}
\usepackage{unitb}
\usepackage{calculational}
\usepackage{ulem}
\usepackage{hyperref}
\normalem
\geometry{a4paper} % or letter or a5paper or ... etc
% \geometry{landscape} % rotated page geometry

% See the ``Article customise'' template for some common customisations

\title{}
\author{}
\date{} % delete this line to display the current date

%%% BEGIN DOCUMENT
\begin{document}

\maketitle
%\tableofcontents

%\section{}
%\subsection{}

\begin{machine}{m0}

\newset{\Pcs}

%\hide{
	\begin{align*}
\variable{		pc : \Pcs \pfun \Int}
	\end{align*}
%}

\begin{align*}
\dummy{	p,p_0,p_1: \Pcs}
\end{align*}

\begin{align*}
\dummy{	N,k: \Int}
\end{align*}

\with{functions}
\with{sets}

\newevent{step}
\begin{align*}
&	\progress{prog0}{\true}{N \le pc.p}
\refine{prog0}{disjunction}{prog1,prog2}{ with }
&	\progress{prog2}{pc.p \le N}{N \le pc.p}
\\&	\progress{prog1}{N \le pc.p}{N \le pc.p}
\refine{prog1}{implication}{}{}
&	pc.p \le N \3\mapsto N \le pc.p \tag{\ref{prog2}}
\refine{prog2}{induction}{prog3}{ over \var{pc.p}{up}{N} }
&	\progress{prog3}{pc.p = k \land pc.p \le N}{(k < pc.p \land pc.p \le N) \lor N \le pc.p}
\refine{prog3}{PSP}{prog4,saf0}{}
&	\safety{saf0}{k \le pc.p \land pc.p \le N}{N \le pc.p}
\\ &	\progress{prog4}{pc.p = k}{\neg k = pc.p}
\refine{prog4}{discharge}{tr0}{}
&	\transientB{step}{tr0}{ \index{p}{p' = p} }{ pc.p = k }\textbf{ is transient}
\end{align*}

\begin{align*}
\cschedule{step}{c0}{\true} \\
\evassignment{step}{a0}{ pc' \2= pc \2| p \fun (pc.p + 1) }
\end{align*}

\begin{align*}
\indices{step}{	p: \Pcs}
\end{align*}

%\begin{proof}{step/CO/saf0}
%	\begin{free:var}{k}{k}
%	\begin{free:var}{p}{p_0}
%	\begin{by:cases}
%		\begin{case}{hyp0}{p_0 = p}
%		\easy
%		\end{case}
%		\begin{case}{hyp1}{\neg p_0 = p}
%		\begin{calculation}
%			k \le pc'.p_0
%		\hint{=}{  \ref{a0} }
%			k \le (pc \1 | p \1\fun pc.p \0+ 1).p_0
%		\hint{=}{  \eqref{hyp1} }
%			k \le pc.p_0
%		\end{calculation}
%		\end{case}
%	\end{by:cases}
%	\end{free:var}
%	\end{free:var}
%\end{proof}

\begin{proof}{step/TR/tr0}
%	\begin{free:var}{k}{k}
%	\begin{free:var}{p}{p_0}
\begin{align}
	\assume{hyp1}{ pc' &\1= pc \1| p \fun pc.p\0+1 } \\
	\goal{\neg ~ pc'.p  &= pc.p}
%	\goal{
\end{align}
	\easy
%	\begin{subproof}{hyp0}
%	\begin{calculation}
%		pc'.p = pc.p
%	\hint{=}{ \ref{hyp1} }
%		(pc \1| p \fun pc.p + 1).p = pc.p
%	\hint{=}{ function application }
%		pc.p + 1 = pc.p
%	\hint{=}{ function application }
%		\false
%	\end{calculation}
%	\end{subproof}
%	\begin{by:cases}
%		\begin{case}{hyp0}{p_0 = p}
%		\easy
%		\end{case}
%		\begin{case}{hyp1}{\neg p_0 = p}
%		\begin{calculation}
%			k \le pc'.p_0
%		\hint{=}{  \ref{a0} }
%			k \le (pc \1 | p \1\fun pc.p \0+ 1).p_0
%		\hint{=}{  \eqref{hyp1} }
%			k \le pc.p_0
%		\end{calculation}
%		\end{case}
%	\end{by:cases}
%	\end{free:var}
%	\end{free:var}
\end{proof}

\begin{align*}
\invariant{inv0}
{	&\qforall{p_0,p_1}{}{pc.p_0 \le pc.p_1 + 1} } \\
\initialization{init0}
{	&\quad pc = \qfun{p}{}{0} }
\end{align*}

\begin{proof}{step/INV/inv0}
	\begin{free:var}{p_0}{p_0}
	\begin{free:var}{p_1}{p_1}
	\begin{by:cases}
		\begin{case}{hyp0}{p = p_1}
		easy ...
		\easy

		\end{case}
		\begin{case}{hyp1}{\neg ~ p = p_1 \land p = p_0}
\begin{calculation}
		pc'.p_0 \le pc'.p_1 + 1
	\hint{=}{ \ref{a0} and \eqref{hyp1} }
		pc.p + 1 \le pc.p_1 + 1
	\hint{=}{ arithmetic }
		pc.p \le pc.p_1
	\hint{=}{ \ref{grd0} }
		\true
\end{calculation}
		\end{case}
		\begin{case}{hyp2}{\neg ~ p = p_1 \1\land \neg~p = p_0}
		\easy
		\end{case}
	\end{by:cases}
	\end{free:var}
	\end{free:var}
\end{proof}

\begin{align*}
\evguard{step}{grd0}
{	\qforall{p_0}{}{ pc.p \le pc.p_0 } }
\end{align*}

\refines{m1}

\end{machine}

\begin{machine}{m1}

\refines{m0}

\end{machine}

\end{document}


Surrogate models were developed to resolve computational limitations in the analysis of massive datasets by replacing a resource-expensive procedure with a much cheaper approximation
\cite{Sondergaard2003}. They are especially useful in applications where
numerous evaluations of an expensive procedure are required over the same or
similar domains, e.g.~in the parameter optimisation of a theoretical model. The
term ``metamodel'' proves especially meaningful in this case, when the surrogate
model approximates a computational process which is itself a model for a
(perhaps unknown) physical process~\cite{Myers2002}. There exists a spectrum
between ``physical'' surrogates which are constructed with some contextual
knowledge in hand, and ``empirical'' surrogates which are derived purely from
samples of the underlying expensive model.

In this work, we develop \textcolor{red}{a family of empirical surrogate models} for the tritium breeding
ratio (TBR) in an inertial confinement fusion (ICF) reactor. The expensive model
that our surrogate model approximates is a Monte Carlo (MC) neutronics
simulation, Paramak~\cite{paramak}, which returns a prediction of the TBR for a given
configuration of a spherical ICF reactor. Although more expensive 3D parametric models exist, we chose the Paramak simulation for its preferable speed in dataset generation in order to most fully demonstrate our methods. \textcolor{red}{We quantify the success of several of our best-performing surrogate models by studying their accuracy and prediction time. We further propose an adaptive sampling algorithm (QASS) suitable for reducing the quantity of expensive samples needed to train our surrogate models.}

\begin{figure}[!ht]
  \centering
    \begin{tikzpicture}
    \fill[green!40!white]  (0,0) circle (3cm);
    \fill[blue!40!white]  (0,0) circle (2cm);
    \fill[white!40!white]  (0,0) circle (1.9cm);
    \fill[red!40!white]  (0,0) circle (0.1cm);
    \end{tikzpicture}

    \caption{Diagram of the simple sphere geometry (not to scale) where the
	blanket is \fcolorbox{white}{green}{\rule{0pt}{6pt}\rule{6pt}{0pt}}, the
first wall is \fcolorbox{white}{blue}{\rule{0pt}{6pt}\rule{6pt}{0pt}} and the
neutron point source is \fcolorbox{white}{red}{\rule{0pt}{6pt}\rule{6pt}{0pt}}. \textcolor{red}{
Blanket and first wall thickness, as well as their material and structural
properties, are adjustable parameters of the simulation that are later optimized
(see~\Tref{tbl:params} for complete parameter listing).}}
    \label{fig:model_diagram}
\end{figure}

Paramak facilitates simulation via \textcolor{red}{an} OpenMC neutronics workflow that is enclosed in
a portable Docker container, which conveniently exposes an HTTP API using the
Python~3 \texttt{flask} package. \textcolor{red}{Within this setup, we employ} a Muir energy distribution \cite{openmcmuir}\footnotemark[1]
around~\SI{14.06}{\mega\electronvolt} to approximate a Deuterium-Tritium (D-T)
plasma neutron source. Illustrated in~\Fref{fig:model_diagram}, the simulated \textcolor{red}{reactor}
geometry was left adjustable \textcolor{red}{in order to study its influence on the TBR}. Nuclear data for simulation was extracted from the following sources, in order of preference: \textcolor{red}{FENDL 3.1d \cite{fendl31d}; JEFF 3.3 \cite{jeff33}; and ENDF/B-VII.1 \cite{2011ii}}.
To maintain model-agnostic approach, variance reduction (VR) techniques
were not used to accelerate the MC neutronics simulation~\cite{Kleijnen2013}.
It should be noted that depending on application, VR may constitute a viable
alternative to the presented work.

For the remainder of~\Sref{sec:introduction}, we will define the TBR and further motivate our
research. In~\Sref{sec:methodology} we will present our methodologies for the comparison
testing of a wide variety of surrogate modelling techniques, as well as defining \textcolor{red}{an add-on} adaptive sampling procedure QASS. After delivering the
results of these approaches in~\Sref{sec:results}, we will give our final conclusions and
recommendations in~\Sref{sec:conclusion}.

\footnotetext[1]{A bug in the Muir distribution involving erronous normal
sampling was recently uncovered (see
\url{https://github.com/openmc-dev/openmc/pull/1670}), but is disregarded in the present \textcolor{red}{methods-focused} work.}

\subsection{Problem Description}
\label{sec:problemdescription}

Nuclear fusion technology relies on the production and containment of an
extremely hot and dense plasma containing enriched Hydrogen isotopes. The current frontier generation of fusion reactors, such as the Joint European Torus (JET) and the
under-construction \textcolor{red}{ITER}, make
use of both tritium and deuterium fuel. While at least one deuterium atom occurs for every \num{5000} molecules of naturally-sourced water, and may be easily distilled, tritium is extremely rare in nature. \textcolor{red}{Tritium} may be produced indirectly through irradiation of heavy water
(\DDO) during nuclear fission, but only at very low rates which could
never sustain industrial-scale fusion power.

\textcolor{red}{Modern} D-T reactors rely on tritium breeding blankets, specialized
layers of material which partially line the reactor and produce tritium upon
neutron bombardment, e.g.~by
\begin{eqnarray}
	\isotope[1][0]{n} + \hspace{3pt} \isotope[6][3]{Li}
	&\longrightarrow \hspace{3pt}
	\isotope[3][1]{T} + \hspace{3pt}\isotope[4][2]{He} \\
	\isotope[1][0]{n} + \hspace{3pt} \isotope[7][3]{Li}
	&\longrightarrow \hspace{3pt}
	\isotope[3][1]{T} + \hspace{3pt} \isotope[4][2]{He} + \hspace{3pt} \isotope[1][0]{n}.
\end{eqnarray}%
%
\textcolor{red}{The TBR is defined as the ratio of tritium produced per source neutron}, whose description in Paramak is facilitated by \textcolor{red}{two} classes of parameters
(exhaustively listed in~\Tref{tbl:params}). While the geometry of a given
reactor is described by continuous parameters, material selections are specified
by discrete categorical parameters. For all parameters, we have attempted to cover the full theoretical range of values even where those values are practically infeasible with current technology (e.g. packing fractions close to 1). Simulating broadly around typical values of parameters \textcolor{red}{improves the accuracy of the model nearer to typical values}, and also aids in demonstrating the robustness of constructed models.

In our work, we set out to produce a \textcolor{red}{fast TBR surrogate model}, which takes the same input parameters \textcolor{red}{as the MC model used in Paramak and approximates its output} with the greatest achievable regression performance, while also minimising the required quantity of expensive-model samples \textcolor{red}{needed for training}. \textcolor{red}{This represents a significant step forwards in computational fusion-reactor design, as any speed-up achieved in TBR evaluation directly informs a speed-up in numerical optimization of reactor parameters, although such optimization is beyond the scope of the present work.}

\begin{table}[t]
	\setlength\tabcolsep{2pt}
	\renewcommand{\arraystretch}{0.95}
	\caption{\label{tbl:params}Input parameters supplied to Paramak and surrogates in alphabetical order. Groups of fractions marked\textsuperscript{\textdagger
		\textdaggerdbl} are independently required to sum to 1.}
	\begin{indented}
	\item[]
		\begin{tabular}{l|ll}
		\toprule
		{} & Parameter name \textcolor{red}{(abbreviation)} & Domain\\
		\midrule
		\parbox[t]{2mm}{\hspace{-2pt}\multirow{12}{*}{\rotatebox[origin=c]{90}{Blanket}}}
		   & Breeder fraction\textsuperscript{\textdagger} & $[0,1]$\\
		   & Breeder \isotope[6]{Li} enrichment fraction & $[0,1]$\\
		   & Breeder material \textcolor{red}{(BBM)} & $\{\text{Li}_2\text{TiO}_3, \text{Li}_4\text{SiO}_4\}$\\
		   & Breeder packing fraction & $[0,1]$\\
		   & Coolant fraction\textsuperscript{\textdagger} & $[0,1]$\\
		   & Coolant material \textcolor{red}{(BCM)} & $\{\text{D}_2\text{O}, \text{H}_2\text{O}, \text{He}\}$\\
		   & Multiplier fraction\textsuperscript{\textdagger} & $[0,1]$\\
		   & Multiplier material \textcolor{red}{(BMM)} & $\{\text{Be}, \text{Be}_{12}\text{Ti}\}$\\
		   & Multiplier packing fraction & $[0,1]$\\
		   & Structural fraction\textsuperscript{\textdagger} \textcolor{red}{(BSM)} & $[0,1]$\\
		   & Structural material & $\{\text{SiC}, \text{eurofer}\}$\\
		   & Thickness & $[0,500]\text{ cm}$\\
		\midrule
		\parbox[t]{2mm}{\hspace{-2pt}\multirow{6}{*}{\rotatebox[origin=c]{90}{First wall}}}
		   & Armour fraction\textsuperscript{\textdaggerdbl} & $[0,1]$\\
		   & Coolant fraction\textsuperscript{\textdaggerdbl} & $[0,1]$\\
		   & Coolant material \textcolor{red}{(FCM)} & $\{\text{D}_2\text{O}, \text{H}_2\text{O}, \text{He}\}$\\
		   & Structural fraction\textsuperscript{\textdaggerdbl} & $[0,1]$\\
		   & Structural material \textcolor{red}{(FSM)} & $\{\text{SiC}, \text{eurofer}\}$\\
		   & Thickness & $[0,20]\text{ cm}$\\
		\bottomrule
		\end{tabular}
	\end{indented}
\end{table}