% % FINAL Version % \documentstyle[12pt,twoside,aries]{report} \setcounter{chapter}{2} \begin{document} \newcommand{\AI}{\hbox{ARIES-I}} \newcommand{\AII}{\hbox{ARIES-II}} \newcommand{\AIII}{\hbox{ARIES-III}} \newcommand{\degC}{$^\circ$C} \newcommand{\SdegC}{$\,^\circ$C} \newcommand{\Z}{\phantom{0}} \newcommand{\D}{\phantom{.0}} \newcommand{\Y}{\phantom{$\sim$ }} \newcommand{\N}{\phantom{$-$}} \newcommand{\X}{\times} \newcommand{\LZ}{Li$_2$ZrO$_3$} % %\newcommand\gsim{\hbox{\raise 1pt\hbox{$>$\hskip -6pt\lower 5pt\hbox{$\sim$}}}} %\newcommand\lsim{\hbox{\raise 1pt\hbox{$<$\hskip -6pt\lower 5pt\hbox{$\sim$}}}} % \newcommand{\gsim}{\buildrel{\lower 5.22636pt\hbox{$\textstyle >$}}\over{\lower 3.22636pt\hbox{$\sim$}}} \newcommand{\lsim}{\buildrel{\lower 5.22636pt\hbox{$\textstyle <$}}\over{\lower 3.22636pt\hbox{$\sim$}}} \title{\bf 3. EQUILIBRIUM, STABILITY, AND TRANSPORT} \author{ \hbox to \hsize{ \hbox to 2.truein{ Y-K.~Martin Peng \hfil } \hfil \hbox to 2.truein{\hfil Charles G.~Bathke \hfil } \hfil \hbox to 2.truein{\hfil David A.~Ehst } } \\ \hbox to \hsize{ \hbox to 2.truein{ John T.~Hogan \hfil } \hfil \hbox to 2.truein{\hfil Erfan Ibrahim \hfil } \hfil \hbox to 2.truein{\hfil Stephen C.~Jardin } } \\ \hbox to \hsize{ \hbox to 2.truein{ Marc L.~Klaski \hfil } \hfil \hbox to 2.truein{\hfil James A.~Leuer \hfil } \hfil \hbox to 2.truein{\hfil Tak-Kuen Mau } } \\ \hbox to \hsize{ \hbox to 2.truein{ Dennis J.~Strickler \hfil } \hfil \hbox to 2.truein{\hfil David J.~Ward \hfil } \hfil \hbox to 2.truein{\hfil John C.~Whitson } } \\ } \date{ } \maketitle \begin{titlepage} {%\baselineskip 1.0ex \tableofcontents } \end{titlepage} \setcounter{page}{1} \chapter{EQUILIBRIUM, STABILITY, AND TRANSPORT} \section{INTRODUCTION} \label{s-3-1} The \AI\ design is a conceptual commercial reactor based on modest extrapolation from the present tokamak physics data base. For a commercial reactor, steady-state operation is preferable because of the many undesirable features of pulsed operation (such as thermal fatigue of in-vessel components and magnets and the requirement for a thermal-energy storage system), which lead to a short reactor lifetime and high costs. For an economical steady-state reactor, the recirculating power fraction should be \hbox{$<$20\%}. This can be achieved by reducing the amount of plasma current to be driven by external means ({\it i.e.,} minimizing the total plasma current and maximizing the bootstrap-current fraction) and also by using an efficient, cost-effective current-drive technique. The \AI\ design operates at a relatively high plasma aspect ratio (\hbox{$A = 4.5$}), a low plasma current (\hbox{$I_p = 10.2$~MA}), and a high on-axis magnetic field (\hbox{$B_o = 11.3$~T}). As a result, the poloidal beta is high and a high bootstrap-current fraction of 0.68 is predicted. Because of the low plasma current and high bootstrap-current fraction, only 3.3~MA of current should be driven by external means. Therefore, a steady-state reactor with relatively small current-drive power is possible. Systems-code analysis confirms that an optimum first-stability tokamak reactor operates with high aspect ratio, low current, and high bootstrap-current fraction. The key parameters of the \AI\ reactor are listed in \hbox{Table~\ref{t-3-1-1}}. \begin{table}[p] \caption{} \centering {\bf Key Parameters of the \AI\ Reactor} \vspace{1.0cm} \renewcommand{\arraystretch}{1.3} \begin{tabular}{l @{\hspace{1.truein}} c} \hline \\ Major radius, $R$ (m) & 6.75 \\ Minor plasma radius, $a$ (m) & 1.50 \\ Aspect ratio, $A$ & 4.50 \\ Elongation, $\kappa_\X$ & 1.80 \\ Triangularity, $\delta_\X$ & 0.70 \\ Magnetic field at coil, $B_c$ (T) & 21.00 \\ Magnetic field on axis, $B_o$ (T) & 11.33 \\ Plasma current, $I_p$ (MA) & 10.20 \\ Safety factor on axis, $q_o$ & 1.30 \\ Edge safety factor, $q^\ast$ & 3.90 \\ Peak electron density, $n_{e0}$ ($10^{20}$ m$^{-3}$) & 1.88 \\ Average electron density, $n_e$ ($10^{20}$ m$^{-3}$) & 1.45 \\ Separatrix electron density, $n_s$ ($10^{20}$ m$^{-3}$) & 0.87 \\ Peak electron temperature, $T_{e0}$ (keV) & 36.62 \\ Average (density-weighted) electron temperature, $T_e$ (keV) & 19.30 \\ Volume-averaged electron temperature, $\bar{T}_e$ (keV) & 17.44 \\ Toroidal beta, $\beta_t$ (\%) & 1.90 \\ Effective charge, $Z_{eff}$ & 1.65 \\ Bootstrap-current fraction, $I_{BS}/I_p$ & 0.68 \\ \\ \hline \end{tabular} \label{t-3-1-1} \end{table} The \AI\ reactor parameters are found through extensive and self-consistent iterations among magnetohydrodynamic (MHD) equilibrium and stability, transport, current-drive, and edge-physics analyses. Engineering constraints imposed by system integration for a power reactor have also been taken into account. In the areas of MHD equilibrium and stability and plasma transport, the regime of operation of the \AI\ plasma is different from current tokamak experiments in many aspects including: \begin{enumerate} \item Operating at a relatively high plasma aspect ratio (\hbox{$A \equiv 1/\epsilon = 4.5$}), a low plasma current (\hbox{$I_p = 10.2$~MA}), a high poloidal beta (\hbox{$\epsilon \beta_p = 0.7$}), and plasma current profiles characterized by on-axis safety factor, \hbox{$q_o \approx 1.3$}, and 95\% flux safety factor, \hbox{$q_{95} > 4$}. \item Operating at \hbox{$A \equiv 1/\epsilon = 4.5$} and placing the poloidal-field (PF) coils outside the toroidal-field coils leads to large stored energy in the PF coils. Minimizing this energy will place constraints on the choice of plasma elongation, $\kappa$, and triangularity, $\delta$. The MHD equilibrium analysis is reported in \hbox{Sec.~\ref{s-3-2}}. \item Operating at MHD-stability beta limit against high-$n$ ballooning and \hbox{$n = 1$} kink modes with the conductive wall at infinity and using plasma profiles consistent with current-drive (driven current density) and transport (flat density and narrow temperature profiles) analyses. The MHD $\beta$ limit is discussed in \hbox{Sec.~\ref{s-3-3}}. \item Placing the passive conductor for stabilizing the vertical motion of the plasma behind the blanket, which reduces the engineering constraints (cooling, neutron damage, {\it etc.}), leads to a reduced plasma elongation, \hbox{$\kappa_\X = 1.8$}. The vertical stability of the \AI\ plasma is considered in \hbox{Sec.~\ref{s-3-4}}. \item Operating an ignited plasma with high synchrotron radiation caused by high on-axis magnetic field, high electron temperature, and a poorly reflective first wall. Absence of deep fueling techniques leads to a flat density profile. The time-dependent transport analysis of the \AI\ plasma is presented in \hbox{Sec.~\ref{s-3-5}}. \end{enumerate} \section{MHD EQUILIBRIUM} \label{s-3-2} \subsection{Minimizing Stored Energy of Poloidal-Field Coils} \label{s-3-2-1} Operating at \hbox{$A \equiv 1/\epsilon = 4.5$} and placing the poloidal-field (PF) coils outside the toroidal-field (TF) coils leads to large stored energy in the PF coils. In this section, a prescription for choosing plasma shape (plasma elongation, $\kappa$, and triangularity, $\delta$) to minimize this stored energy is presented. Placing the \AI\ PF coils external to the TF coils permits the use of a multipole expansion technique~\cite{Reu86,Kes86} to describe accurately the PF-coil magnetic fields used in MHD equilibrium computations performed with the tokamak simulation code (TSC)~\cite{Jar86} or the HEQ code~\cite{Str89}. Limiting the multipole expansion to hexapole and lower moments minimizes the number of dependent variables required to describe the plasma shape. Using the multipole expansion technique to examine plasma equilibria parametrically for a range of plasma elongation and triangularity values resulted in a prescription for minimizing the PF-coil stored energy, $W_{PF}$, which is used in the ARIES systems code (\hbox{Sec.~2}). The TSC was modified so that the poloidal flux produced by a PF-coil set could be determined by specifying either the PF-coil locations and currents or the amplitudes of the even and odd nullapole, dipole, quadrupole, hexapole, and octopole moments, and the even decapole moment in the multipole expansion series~\cite{Reu86}. In addition, the TSC was modified to perform the inverse operation of decomposing the poloidal flux produced by a PF-coil set into the multipole components of the same truncated series. The TSC was then used to analyze parametrically the equilibrium of the interim design point~I given in \hbox{Table~\ref{t-3-2-1}}. \begin{table}[b] \caption{} \centering {\bf Fixed Parameters for TSC Parametric Equilibrium Analysis} \vspace{1.cm} \renewcommand{\arraystretch}{1.3} \begin{tabular}{l @{\hspace{2.truecm}} c c} \hline \\ Interim design point & I & II \\ Plasma current, $I_p$ (MA) & \Z7.26 & 10.5\Z \\ Toroidal field, $B_t$ (T) & 12.38 & 12.72 \\ Profile form factors \\ \quad $\alpha$ & $-3.0$\N & $-3.0$\N \\ \quad $\gamma$ & $-2.5$\N & $-3.0$\N \\ \quad $\beta_J$ & \Z3.24 & \Z1.94 \\ Major radius, $R_o$ (m) & 6.0 & \Z6.12 \\ Minor radius, $a$ (m) & \phantom{$^{(a)}$}1.0$^{(a)}$ & \Z1.36 \\ Nullapole current equivalent (MA) & \Z\Z0.351 & \Z\Z0.271 \\ Dipole current equivalent (MA) & \Z\Z0.429 & \Z\Z0.517 \\ \\ \hline \multicolumn{3}{l}{$^{(a)}$For $A = 4.5$, $a = 1.333$.}\\ \end{tabular} \label{t-3-2-1} \end{table} Since the PF coils are relatively far from the plasma, the cost of the coils to produce the higher order multipoles is high, making them undesirable. Therefore, in this analysis, the multipole expansion was limited to hexapole and lower moments, thereby reducing the number of variables. The plasma current, toroidal field, profile form factors (\hbox{Sec.~\ref{s-3-2-2}}), major and minor radii, and the nullapole and dipole moments were held fixed at the interim design values. The ratio of the hexapole to quadrupole amplitudes was selected as the dependent variable. The absolute magnitudes of these two amplitudes were determined as the minimum amplitudes that are necessary to produce a separatrix with the specified minor radius. This algorithm, subsequently, generates a set of plasma equilibria that requires only low-order multipoles which can be efficiently produced by the external PF coils. Consequently, the external field solutions have minimum PF-coil current, \hbox{$\sum \vert I_{PF} \vert$}, and, hence, minimum $W_{PF}$. Because both plasma triangularity, $\delta$, and elongation, $\kappa$, vary as the ratio of the quadrupole to hexapole moment varies, this algorithm yields the $\delta$ that minimizes the PF-coil stored energy, $W_{PF}$, for a given $\kappa$. The results of this analysis are shown in \hbox{Fig.~\ref{f-3-2-1}}. Plasmas with shapes corresponding to the region below the $\delta$-$\kappa$ correlation of \hbox{Fig.~\ref{f-3-2-1}(A)} would not have a separatrix, and those above the correlation would either have a smaller minor radius (larger $A$) or would require higher multipole moments and larger $W_{PF}$ to maintain the same minor radius. This analysis also produced the correlation between the equivalent multipolar current, which scales approximately as the square root of $W_{PF}$, and $\kappa$. This correlation, shown in \hbox{Fig.~\ref{f-3-2-1}(B)}, indicates that $W_{PF}$ increases as $\kappa$ is decreased ($\delta$ increases) for the range of \hbox{$1 < \kappa < 2.7$} considered. It was found, however, that this $W_{PF}$-$\kappa$ correlation did not have a significant impact on determining the final design values of $\kappa$ and $\delta$ (\hbox{Sec.~2}). \begin{figure}[p] \vspace*{7.1in} \caption[Fig-3-2-1] {The correlation between (A)~plasma triangularity and elongation, which minimizes the PF-coil stored energy, and (B)~the equivalent multipolar current (\hbox{$\propto W_{PF}^{1/2}$}) and elongation of the 95\% flux surface. (Analysis performed for the interim design point~I of \hbox{Table~\protect\ref{t-3-2-1}} with \hbox{$A = 4.5$} and \hbox{$A = 6$}.)} \label{f-3-2-1} \end{figure} \begin{table}[p] \caption{} \centering {\bf Equilibrium Calculations with Different Multipole Moments \\ for Interim Design Point~II of Table~\ref{t-3-2-1}} \vspace{1.cm} \renewcommand{\arraystretch}{1.2} \begin{tabular}{l c c c c c} \hline &HEQ$^{(a)}$ &BASE$^{(a)}$ &DEC$^{(b)}$ &OCT$^{(b)}$ &HEX$^{(b)}$\\ \hline \\ Major radius, $R_o$ (m) & 6.12 & 6.12 & 6.12 & 6.13 & 6.13 \\ Minor radius, $a$ (m) & 1.36 & 1.35 & 1.35 & 1.34 & 1.34 \\ Magnetic axis, $R_M$ (m) & 6.33 & 6.35 & 6.35 & 6.35 & 6.34 \\ Elongation \\ \qquad $\kappa_\X$ & 1.75 & 1.73 & 1.68 & 1.70 & 1.84 \\ \qquad $\kappa_{95}$ & 1.58 & 1.59 & 1.55 & 1.56 & 1.67 \\ Triangularity \\ \qquad $\delta_\X$ & 0.74 & 0.77 & 0.75 & 0.68 & 0.46 \\ \qquad $\delta_{95}$ & 0.48 & 0.50 & 0.49 & 0.47 & 0.36 \\ Separatrix coordinates (m) \\ \qquad $R_\X$ & 5.11 & 5.12 & 5.14 & 5.25 &5.51 \\ \qquad $z_\X$ & 2.38 & 2.39 & 2.31 & 2.32 &2.53 \\ Toroidal beta, $\beta$ (\%) & 1.84 & 1.86 & 1.94 & 1.98 & 1.86 \\ Poloidal beta, $\beta_p$ & 1.85 & 1.77 & 1.78 & 1.78 & 1.74 \\ Plasma volume (m$^3$) & 347 & 348 & 337 & 329 & 343 \\ Safety factor \\ \qquad $q_{(0)}$ & 1.59 & 1.59 & 1.54 & 1.50 & 1.58 \\ \qquad $q_{(a)}$ & 7.84 & 7.02 & 6.46 & 6.19 & 6.52 \\ Current, $\sum \vert I_{PF} \vert$ (MA) & 142 & 142 & 114 & 111 & 82 \\ Stored energy, $W_{PF}$ (GJ) & 7.57 & 7.57 & 7.78 & 8.10 & 6.80 \\ \\ \hline \multicolumn{6}{l}{$^{(a)}$PF-coil flux determined by PF-coil currents and locations using HEQ and}\\ \multicolumn{6}{l}{\phantom{$^{(a)}$}TSC (BASE) codes.}\\ \multicolumn{6}{l}{$^{(b)}$PF-coil flux determined by truncated multipole-expansion series with up to}\\ \multicolumn{6}{l}{\phantom{$^{(b)}$}decapole (DEC), octopole (OCT), and hexapole (HEX) moments.}\\ \end{tabular} \label{t-3-2-2} \end{table} To pursue further the issue of minimizing $W_{PF}$, a code (FLXCON) was developed to determine the locations and currents of a PF-coil set that would reproduce the flux pattern of a given set of multipole moments. The FLXCON code moves the coils along a specified surface while minimizing an object function defined as the sum of two terms. The first term measures the relative error with which the coils reproduce the flux of the given set of multipole moments on a test surface representative of the plasma surface and the second term measures $W_{PF}$. To illustrate the use of FLXCON, an HEQ-generated equilibrium for interim design point~II (\hbox{Table~\ref{t-3-2-1}}) was modeled with TSC using the same set of 12 PF coils with 6 current groups; the HEQ and TSC results are given for comparison in \hbox{Table~\ref{t-3-2-2}}. The small differences in $R_T$, $a$, and $\kappa_\X$ between the HEQ and TSC results are directly attributable to numerical inaccuracies due to finite grid size in both codes. The multipole decomposition of the PF-coil flux was used to generate three sets of six PF coils and six current groups labeled DEC, OCT, and HEX in \hbox{Table~\ref{t-3-2-2}} to denote the maximum multipole moment used. As the higher moments are eliminated, \hbox{$\sum \vert I_{PF} \vert$} decreases as expected, but $W_{PF}$ is lower only for the HEX case. The plasma shape, however, drifts from the base shape because the higher moments were used inefficiently to suppress $\kappa$ and to provide additional triangularity. A more efficient method to generate the same plasma shape would require less quadrupole and more hexapole moments, and will be used in the future. \subsection{Free-Boundary MHD Equilibria and Operational Requirements} \label{s-3-2-2} Given the coil distribution, reference MHD equilibria for \AI\ are computed using the HEQ code~\cite{Str89}, which calculates free-boundary solutions for a given plasma position, shape, and linked poloidal flux while minimizing the stored energy. The plasma shape is chosen to have \hbox{$\kappa_\X = 1.8$} to allow for adequate vertical stabilization (\hbox{Sec.~\ref{s-3-4}}) and \hbox{$\delta_\X = 0.7$} based on the analysis in \hbox{Sec.~\ref{s-3-2-1}} to minimize the PF-coil stored energy. The plasma pressure profile is consistent with the transport analysis of \hbox{Sec.~\ref{s-3-4}} (flat density and narrow temperature profiles), and the current profile is consistent with first-stability operation with high $\epsilon\beta_p$ and with current-drive analysis of the driven current-density profile. These constraints led to a choice of profiles that are close to the following pressure, $p$, and poloidal current-profile, $f^\prime$, functions: % \begin{eqnarray} \label{e-3-2-1} p({x}) & = & p_o \left( \frac{e^{-\alpha {x}} - e^{-\alpha}} {e^{-\alpha} - 1}\right) \:, \\ \label{e-3-2-2} f f^\prime(x) & = & \mu_o \, R^2_o \, p_o \left(\frac{1}{\beta_J - 1}\right) \left(\frac{e^{-\gamma{x}} - e^{-\gamma}}{e^{-\gamma} - 1}\right) \: , \end{eqnarray} % where $x$ is the poloidal flux normalized to 1 within the plasma. Values of \hbox{$\alpha = -1.35$}, \hbox{$\gamma = -1.35$}, and \hbox{$\beta_J = 2.29$} were chosen for the equilibrium analysis. The toroidal plasma current density is % \begin{eqnarray} \label{e-3-2-3} J_t & = & R \, p^\prime + \frac{ff^\prime}{\mu_o \, R} \: , \end{eqnarray} % where $R$ is in the direction of the major radius. \begin{figure}[p] \vspace*{7.7in} \caption[Fig-3-2-2] {Plasma-equilibrium-flux configuration and PF-coil placement.} \label{f-3-2-2} \end{figure} \begin{figure}[p] \vspace*{7.3in} \caption[Fig-3-2-3] {Profiles of (A)~pressure and toroidal current density along the major radius, and (B)~$q$ and $\bar q$ (the average-field safety factor) along the normalized poloidal flux within the plasma edge. (For plasma equilibrium of \hbox{Fig.~\protect\ref{f-3-2-2}}.)} \label{f-3-2-3} \end{figure} The poloidal-flux distribution of the intermediate \AI\ equilibrium is given in \hbox{Fig.~\ref{f-3-2-2}}. Profiles of the plasma pressure, toroidal current density, and safety factor are shown in \hbox{Fig.~\ref{f-3-2-3}}. Parameters of this intermediate equilibrium that are relevant to stability and current-drive analyses are given in \hbox{Table~\ref{t-3-2-3}}. The reference \AI\ equilibrium of \hbox{Table~\ref{t-3-1-1}} is essentially similar to this equilibrium (scaled to \hbox{$R_o = 6.75$~m}). \begin{table}[p] \caption{} \centering {\bf The Interim \AI\ Divertor MHD Equilibrium$^{\vec{(a)}}$} \vspace{1.cm} \renewcommand{\arraystretch}{1.3} \begin{tabular}{l @{\hspace{2.truecm}} r} \hline \\ Major radius, $R_o$ (m) & 7.24 \\ Minor radius, $a$ (m) & 1.61 \\ External toroidal field at $R_o$, $B_t$ (T) & 11.9 \\ Plasma current, $I_p$ (MA) & 11.3 \\ Safety factor on axis, $q_o$ & 1.24 \\ Average-field safety factor, $\bar q$ & 4.45 \\ Safety factor at 95\% flux, $q_{95}$ & 4.64 \\ Toroidal beta, $\beta$ (\%) & 1.90 \\ Poloidal beta, $\beta_p$ & 2.19 \\ Elongation at $\X$-point, $\kappa_\X$ & 1.80 \\ Elongation at 95\% flux, $\kappa_{95}$ & 1.60 \\ Triangularity at $\X$-point, $\delta_\X$ & 0.70 \\ Triangularity at 95\% flux, $\delta_{95}$ & 0.50 \\ $\X$-point location \\ \quad $R_\X$ (m) & 6.11 \\ \quad $Z_\X$ (m) & 2.90 \\ Internal inductance, $l_i$ & 0.82 \\ Stored PF-coil energy, $W_{PF}$ (GJ) $^{(b)}$ & 12.0 \\ \\ \hline \multicolumn{2}{l}{$^{(a)}$The reference equilibrium of \hbox{Table~\protect\ref{t-3-1-1}} is essentially similar}\\ \multicolumn{2}{l}{\phantom{$^{(a)}$} this equilibrium (scaled to \hbox{$R_o = 6.75$~m}).}\\ \multicolumn{2}{l}{$^{(b)}$Stored energy is higher at full $I_p$ and low $\beta$.} \end{tabular} \label{t-3-2-3} \end{table} \clearpage Analysis leading to the interim MHD equilibria provided an adjustment to the conditions that relate $I_p$ to $q$, $a$, $B_t$, and the plasma shape parameters. This adjusted relationship, used in systems studies, is % \begin{eqnarray} \label{e-3-2-4} I_p \, \bar q & = & 5 \, a \, B_t \left[ \frac{\epsilon\, (1.15 - 0.65\epsilon)}{(1 - \epsilon^2)^2} \right] \left( \frac{1 + \kappa^2_\X}{2} \right) \: , \end{eqnarray} % where $\bar q$ is the average-field safety factor using the averaged poloidal field at the plasma edge. For this class of equilibria, the edge and 95\% flux surface quantities are related through % \begin{eqnarray} \label{e-3-2-5} \frac{\delta_\X}{\delta_{95}} & = & 1.59 \, \frac{\kappa_\X}{\kappa_{95}} \: = \: (1.13 - 0.08\epsilon) \, \frac{q_{95}}{\bar q} \: = \: 1.09 \: . \end{eqnarray} Different forms of the profile functions can be used to produce equilibria nearly identical to this case in all its global parameters, as are given in \hbox{Table~\ref{t-3-2-3}}. The results of the free boundary equilibrium and the PF-coil currents do not change significantly when these different profile functions are used, as long as the global parameters remain unchanged. Since the \AI\ reactor uses noninductive methods to assist start-up of the plasma current (Sec.~12), the amount of poloidal flux linkage between the plasma and the PF coils can be chosen to reduce the PF-coil stored energy. Some flexibility exists near the condition of minimum stored energy to vary the PF-coil current with a fixed $\X$-point location: low beta and low linked flux, high beta and low linked flux, and high beta and high linked flux. The maximum current for each coil is then estimated and used in sizing its cross sections and locating the coil, as plotted in \hbox{Fig.~\ref{f-3-2-2}} (assuming an overall current density of 20~MA/m$^2$ for each coil). This data is then used as input to the engineering design of the PF-coil system (Sec.~7). \section{MHD-STABILITY BETA LIMIT} \label{s-3-3} \subsection{Dependences of the Beta Limit} \label{s-3-3-1} The first stability regime requires that all ideal MHD modes be at least marginally stable in the absence of a conducting shell beyond the plasma edge~\cite{Tro84}. While this requirement is broad in scope, it is usually adequate to examine only the high-$n$ ballooning modes and the low-$n$ (\hbox{$n = 1$}) kink modes to determine the stability beta limit. The intermediate-$n$ ballooning modes (the ``infernal'' modes) are easily avoided by retaining small gradients in the $q$ profile near the plasma axis. As an input to design trade-offs involving plasma shaping (profiles, $A$, and the $\beta$ limit), our study places an emphasis on clarifying the dependences of $\beta$ on $A$, $\kappa_{95}$, $q_o$, and $q_{95}$. We use only the traditionally successful profile functions for the analysis. This study, therefore, is limited in its scope, since several other parameters ({\it e.g.}, $\delta$ and the $q$ and pressure profiles) also affect the plasma beta limit. However, this study benefits from an extensive review of the beta limit investigation recently carried out for ITER~\cite{Tsu89} and from reviews of the large body of information in the literature. Calculations are carried out for high-$A$ (4.5 and 6.0) \AI\ plasmas using the PEST equilibrium and stability codes~\cite{Gri83} to ``fill in'' data where needed. The combined data base of the stability analysis covers a range \hbox{$A = 2.6$ to 6.0}, \hbox{$\kappa_{95} = 1.6$} to 3.2, \hbox{$q_o = 1.0$} to 2.0, and \hbox{$q_{95} \le 5$}. The pressure and safety-factor profile functions used include those optimized for the JET plasma~\cite{Tro84} and those used in the ITER studies~\cite{Tsu89} and are given by: % \begin{eqnarray} \label{e-3-3-1} p & = & p_o \, \left[ \, (1 - y^\alpha)^\gamma \, + \, p_1 \, y^\zeta \, (1 - p_2y^\eta) \right] \:, \\ \label{e-3-3-2} q & = & q_o + q_1 \, y^\lambda + q_2 \, y^{\nu} \:, \hspace*{1in}{\rm or} \\ \label{e-3-3-3} q & = & \frac{q_o}{(1 - \zeta \, y^\rho)^\sigma} \: . \end{eqnarray} % Here $y$ is the poloidal flux normalized to 95\% of the $\X$-point flux. In the pressure profile function, beta determines $p_o$, profiles are adjusted through $p_1$, and other variables are set according to \hbox{$\alpha = 1.5$}, \hbox{$\gamma = 2.5$}, \hbox{$\zeta = 3$}, \hbox{$\eta = 1.2$}, and \hbox{$p_2 = 0$} or 1. For the safety-factor profile function of \hbox{Eq.~(\ref{e-3-3-2})}, \hbox{$\lambda = 6$}, \hbox{$\nu = 2$}, and $q_o$ is chosen such that \hbox{$q_{95} = q_o + q_1 + q_2 = 3.1$} (where $q_1$ and $q_2$ are independent variables). The safety factor profile function of \hbox{Eq.~(\ref{e-3-3-2})} used for the case with \hbox{$A \approx 6$}, has \hbox{$\sigma = ln(q_o/q_{95})/ln(1 - \zeta)$}, \hbox{$\zeta=0.7$}, and \hbox{$\rho = 2$}. The shape of the 95\% flux surface is given by % \begin{eqnarray} \label{e-3-3-4} R & = & R_o + a \, \cos(\theta + \delta^\prime \sin \theta) \: , \\ \label{e-3-3-5} Z & = & \kappa_{95} \, a \, \sin \theta \: , \end{eqnarray} % where \hbox{$\delta^\prime \approx \delta_{95}$}. The 95\% flux surface is used in the stability analysis to avoid the numerical difficulties near the $\X$-point. The stability data base produced by the PEST code~\cite{Gri83} is shown in \hbox{Fig.~\ref{f-3-3-1}} as the dependence of beta limit on $\kappa_{95}$ (for $A \approx 6$) and as the dependence of Troyon factor~\cite{Tro84} limit, \hbox{$C_T{\,}={\,}\beta aB_t/I_p$} (in \%~Tm/MA), on $A$. From this result, one obtains % \begin{eqnarray} \label{e-3-3-6} C_T & \approx & 2.8 \left[ \frac{1 - 0.4 \, (\kappa_{95} - 1)^2}{(1-\epsilon)^{1.5}} \right] \: , \end{eqnarray} % which gives \hbox{$C_T \approx 3.5$} and \hbox{$\beta \approx 2.06\%$} for the reference plasma parameters in \hbox{Table~\ref{t-3-2-3}}. It is important to note that this approximate scaling has a limited basis and its use should be limited to the profiles given here and to the range of parameters indicated above. For \hbox{$A \approx 3$}, it has also been shown that this beta limit remains relatively unchanged as long as $l_i$ remains below 0.75. \begin{figure}[p] \vspace*{7.3in} \caption[Fig-3-3-1] {The stability data base produced by the PEST code~\protect\cite{Gri83}: (A)~dependence of the beta limit on $\kappa_{95}$ (for $A \approx 6$), and (B)~dependence of the Troyon factor~\protect\cite{Tro84} limit, \hbox{$C_T = \beta a B_t/I_p$} (in \% Tm/MA), on the plasma aspect ratio, $A$.} \label{f-3-3-1} \end{figure} Additional studies of the beta limit have also been carried out using polynomial profiles and with parameters encompassing the reference case: $I_p$ ranging from 16 to 8~MA, $q_{95}$ from 3 to 6, and $\beta_p$ from 1.4 to 3. The value of $C_T$ is 3.1 to 3.2 as long as $q_{95}$ is above 3.7. This result is considered conservative relative to the preceding indications. Design values of \hbox{$C_T = 3.2$} (corresponding to \hbox{$\beta = 1.9\%$}) and \hbox{$l_i = 0.74$} are therefore adopted for the \AI\ reactor. \subsection{Stability of Reference Equilibrium} \label{s-3-3-2} To ensure complete self-consistency among MHD equilibrium and stability, current drive, and transport analysis, the \AI\ reference equilibrium is found using a simple pressure function, \hbox{$p (\psi) = p_o \tilde{\psi}^\alpha$}, where \hbox{$\alpha = 1.4$} and $\tilde{\psi}$ is the normalized poloidal flux. The equilibrium flux contours and profiles of safety factor and toroidal current density of the reference equilibrium are similar to those displayed in \hbox{Figs.~\ref{f-3-2-2}} and \hbox{\ref{f-3-2-3}}. As shown in Sec.~4, this equilibrium can be generated with $\sim$100~MW of absorbed RF heating [mostly ion-cyclotron range-of-frequency (ICRF) fast-wave current drive]. The pressure profile also matches that of transport analyses. Several sequences of equilibria were tested for stability at different aspect ratios and safety factors. In the range \hbox{$4.5 \le A \le 6.0$}, it was always possible to obtain stability at Troyon ratios \hbox{$C_{T} \lsim 3.2$}. For the interim geometry of \hbox{Table~\ref{t-3-2-3}}, stable equilibria were found in the range \hbox{$1.56\% \le \beta \le 2.60\%$} as the axis safety factor varied respectively from \hbox{$q_o = 1.50$} to 1.10. All equilibria were stable to \hbox{$n = \infty$} ballooning modes (tested with the Phillips code~\cite{Phi88}) and were stable to \hbox{$n = 1$} external kinks (tested with PEST~II) with the conducting wall at infinity. Generally the edge safety factor, $q_a$, was roughly four times the axis value, and it was found necessary to avoid integer values of $q_a$ in order to ensure kink stability. \section{VERTICAL STABILITY} \label{s-3-4} Vertical stability is an important issue for \AI\ because of the moderate-to-high aspect ratio (\hbox{$A \geq 4.5$}) and the high separatrix elongation (\hbox{$\kappa_\X = 1.8$}). A toroidally continuous conducting shell is required to retard the growth of an externally excited vertical instability with a time constant ranging from an Alfv\'en time (\hbox{$\tau_A \sim 10~\mu$s}) without a shell to the electrical $L/R$ time constant of the shell, $\tau_{L/R}$. The passive stabilization provided by this shell must be augmented by an active feedback system that provides vertical stability for times \hbox{$\gsim \tau_{L/R}$}. A rigid-plasma model (PSTAB)~\cite{Leu89} and linear (NOVA-W)~\cite{War89} and non-linear tokamak-simulation-code (TSC)~\cite{Jar86} deformable-plasma models are used to estimate the conductor location and size required for passive stabilization. Time-dependent TSC simulations are used to determine the current, voltage, location and size of the feedback coils. \subsection{Passive Stabilization} \label{s-3-4-1} Without passive stabilization, a vertically displaced plasma will move vertically on an Alfv\'en time scale ($\sim$10~$\mu$s). Passive conductors must be positioned around the plasma to slow this vertical motion sufficiently to allow an active feedback system operating on a longer time scale (\hbox{$\gsim 100$~ms}) to control the vertical position of the plasma with a reasonable expenditure of reactive power (\hbox{$\lsim 10$~MVA}). Passive stability performance is measured by the stability parameter that is defined for an idealized system by % \begin{eqnarray} \label{e-3-4-1} f & \equiv & 1 + \, \frac{\tau_v}{\tau_{L/R}} \: , \end{eqnarray} % where \hbox{$\tau_v = 1/\gamma$} is the vertical-instability time constant and $\gamma$ is the growth rate. A design constraint of % \begin{eqnarray} \label{e-3-4-2} f & \geq & 1.3 \: , \end{eqnarray} % is adopted to ensure that a sufficient stability margin exists above the \hbox{$\gamma = \infty$} limit under all plasma conditions. Placement of passive (metallic) stabilizer elements close to the plasma cause severe engineering difficulties: (1)~neutron damage and activation, (2)~nuclear heating which requires active cooling, (3)~adverse effects of tritium breeding, and (4)~difficulties in assembly/disassembly of components and of maintenance. It is, therefore, desirable to place the passive elements away from the plasma, preferably behind the blanket (at a distance of $\sim\!0.6a$ from the plasma surface in \AI). A preliminary analysis of the maximum allowable distance permitted between the plasma and the passive stabilizer elements for vertical stability was performed with the PSTAB~\cite{Leu89} code. The PSTAB formulation assumes that the plasma is a massless rigid body simulated by an array of filamentary current elements. The equations describing a small vertical displacement in the presence of an array of resistive filamentary conductors simulating the passive stabilizer are linearized and solved as an eigenvalue problem. \begin{figure}[b] \vspace*{4.0in} \caption[Fig-3-4-1] {The geometry of the plasma and the passive stabilizer (shell).} \label{f-3-4-1} \end{figure} The notation used to describe the location of the passive stabilizer relative to the plasma is shown in \hbox{Fig.~\ref{f-3-4-1}}. The plasma surface is assumed to be given by % \begin{eqnarray} \label{e-3-4-3} Z & = & a \, \kappa_\X \, \sin \theta \: , \\ \label{e-3-4-4} R & = & R_o + a \, \cos(\theta + \delta_\X \, \sin\theta) \: , \end{eqnarray} % where $R_o$ is the major radius, $a$ is the minor radius, $\kappa_\X$ and $\delta_\X$ are, respectively, the separatrix elongation and triangularity, and the poloidal angle $\theta$ varies from 0 to $2\pi$. The passive stabilizer is placed on a surface parallel to the plasma surface. \begin{figure}[b] \vspace*{4.0in} \caption[Fig-3-4-2] {The stability parameter as a function of poloidal coverage of an outboard stabilizer at various radial locations ($c/a$) (\hbox{Fig.~\protect\ref{f-3-4-1}}) for \hbox{$A = 6$}, \hbox{$\kappa_{\X} = 2$}, and \hbox{$\delta_\X = 0.5$}.} \label{f-3-4-2} \end{figure} A series of calculations were performed for an \hbox{$A = 6$} plasma that was considered during the \AI\ scoping-phase activities (\hbox{$R_o = 6$~m}, \hbox{$a = 1$~m}, \hbox{$\kappa_\X = 2$}, \hbox{$\delta_\X = 0.5$}). The poloidal coverage, $p/a$, and the normalized radial location, $c/a$, of a passive outboard stabilizer were varied and the results are shown in \hbox{Fig.~\ref{f-3-4-2}}. It was found that relatively small passive stabilizers (\hbox{$p/a \sim 0.2$}) meet the stability constraint, but these conductors must be located close to the plasma (\hbox{$c/a \leq 1.1$}). Placement of the passive stabilizer behind the blanket (0.9~m away from plasma surface), even with full coverage on the outboard side, does not provide sufficient stabilization ({\it i.e.}, \hbox{$f > 1.3$}). The cost-optimized \AI\ design, however, has a lower aspect ratio of \hbox{$A = 4.5$}. Vertical stability of \hbox{$A = 4.5$} plasmas with passive stabilizer elements positioned behind the blanket, as shown in \hbox{Fig.~\ref{f-3-4-3}} (0.7~m away from the plasma surface on the inboard side and 0.9~m on the outboard side) were examined. The stability parameter as a function of the plasma elongation of such plasmas is shown in \hbox{Fig.~\ref{f-3-4-4}}. This analysis indicates that \hbox{$\kappa_\X \lsim 1.83$} is required for a stabilizer located behind the inboard and outboard blankets. \begin{figure}[p] \vspace*{3.0in} \caption[Fig-3-4-3] {The geometry of the plasma and the passive stabilizer for \hbox{$A = 4.5$} and \hbox{$\kappa_\X = 1.74$}. A similar geometry was used for the other cases in the $\kappa$ parametric study. The coil locations used in the active feedback analysis are also indicated.} \label{f-3-4-3} \end{figure} \begin{figure}[p] \vspace*{3.0in} \caption[Fig-3-4-4] {Dependence of the stability parameter on elongation for \hbox{$A = 4.5$} and stabilizer geometry shown in \hbox{Fig.~\protect\ref{f-3-4-3}}.} \label{f-3-4-4} \end{figure} The accuracy of the PSTAB rigid-plasma model results was then checked by benchmarking the \hbox{$\kappa_\X = 1.74$} case against the TSC and NOVA-W deformable-plasma models. The TSC~\cite{Jar86} performs time-dependent simulations of resistive free-boundary, axisymmetric plasmas and the associated external conductors and poloidal-field circuits, including active feedback amplifiers. Consequently, TSC is more costly to use than either PSTAB or NOVA-W. The plasma force balance in TSC is modified by scaling up the plasma mass and viscosity to maintain the plasma in force balance, while alleviating the time-scale disparity between wave and diffusion phenomena. This parameter scaling does not affect plasma bulk motion that is stable on the ideal-MHD time scale ($\mu$s), but does affect the growth rate of vertical instability. To calculate $\tau_v$, TSC is set up with the plasma and passive-stabilizer geometry shown in \hbox{Fig.~\ref{f-3-4-3}}. The PF coils are represented with a multipole series truncated with the even decapole. The simulations begin with a radial magnetic field applied for 1~$\mu$s to produce an initial $\sim$3-mm vertical displacement of the plasma. The plasma simulation is then continued until the equilibrium effects of the initial perturbation are damped out and the vertical instability asymptotically relaxes to a linear growth rate, as reported by inboard and outboard pickup coils. Corroboration of the value of $\tau_v$ predicted from the pickup coils was obtained by fitting the vertical displacement of the magnetic axis, $Z_M$, with the functional form, % \begin{eqnarray} \label{e-3-4-5} Z_M (t) & = & Z_o \, e^{t/\tau_v} \: . \end{eqnarray} This $\tau_v$ calculation was made for four values of plasma mass-enhancement factors. Results (denoted as squares) are shown in \hbox{Fig.~\ref{f-3-4-5}}. An analytically derived quadratic polynomial that yields the behavior of $\tau$ as a function of the plasma mass-enhancement factor for small values of the mass-enhancement factor was fit to the results and then extrapolated to a unity mass-enhancement factor to determine the mass independent value of $\tau_v$ reported in \hbox{Table~\ref{t-3-4-1}}. The accuracy of this extrapolation is demonstrated by fitting polynomials to the three smallest and three largest mass-enhancement-factor results displayed in \hbox{Fig.~\ref{f-3-4-5}} and reporting a range for $\tau_v$ and $f$ in \hbox{Table~\ref{t-3-4-1}}. \begin{figure}[p] \vspace*{3.4in} \caption[Fig-3-4-5] {Dependence of the vertical-stability time constant on plasma mass-enhancement factor as calculated with TSC (squares) for \hbox{$A = 4.5$}, \hbox{$\kappa_\X = 1.74$}, and the stabilizer geometry shown in \hbox{Fig.~\protect\ref{f-3-4-3}}. Also included are quadratic polynomials fitted to the three largest (solid) and three smallest (dash) mass-enhancement-factor results and extrapolated to a unity mass-enhancement factor.} \label{f-3-4-5} \end{figure} \begin{table}[p] \caption{} \centering {\bf Vertical Stability Benchmark of the Configuration Shown in \hbox{Fig.~\ref{f-3-4-3}}} \vspace{1.0cm} \renewcommand{\arraystretch}{1.3} \begin{tabular}{l @{\hspace{2.cm}} c @{\hspace{2.cm}} c } \hline & Vertical-Stability & \\ \hfil {Code} \hfil & Time Constant, $\tau_v$ (ms) & Stability Parameter, $f$ \\ \hline \\ PSTAB~\cite{Leu89} & 217 & 1.88 \\ TSC~\cite{Jar86} & $175 - 186$ & $ 1.56 - 1.60$ \\ NOVA-W~\cite{War89} & 173 & 1.55 \\ \\ \hline \end{tabular} \label{t-3-4-1} \end{table} This calculation was repeated with the NOVA-W code~\cite{War89}, a non-ideal MHD linear-stability code that was developed to calculate the linear stability of axisymmetric modes with passive resistive conductors and active feedback included in the vacuum region surrounding the plasma. The NOVA-W code is a derivative of the NOVA~\cite{Che87} code. Both codes are non-variational MHD stability codes that solve the stability eigenvalue problem for the growth rate (eigenvalue). The use of the eigenvalue formulation rather than the $\delta W$ formulation to calculate stability permits the inclusion in NOVA-W of non-ideal effects, such as wall resistivity, that would otherwise destroy the self-adjointness property underlying the $\delta W$ approach. A Green's function formulation is used to express the perturbed poloidal flux in the vacuum region in terms of the feedback currents in the vacuum region and an integral over the boundary surfaces: the plasma/vacuum interface and the surface of the resistive wall surrounding the plasma. Separate Green's equations are obtained for the vacuum region between the plasma and the resistive wall, and for the region outside the resistive wall. A thin wall approximation is used to relate the discontinuity in the normal derivative of the poloidal flux across the thin resistive wall boundary to the time derivative of the flux on the boundary. Combining the Green's equations for the different boundary surfaces yields the normal derivative of the perturbed flux at the boundary in terms of poloidal flux, which in turn is known as a function of the Fourier modes of the displacement. This approach provides the correct vacuum boundary condition needed to solve the NOVA eigenvalue equation for the growth rate and eigenfunction of the instability. An \AI\ equilibrium produced with TSC is used as input to the NOVA-W code. Special care must be used for equilibria (such as those of \AI) where the separatrix is near the plasma surface, because this condition can adversely affect the accuracy of the mapping to stability coordinates and, therefore, affect the result. Consequently, equilibria limited by the 95\% flux surface and below are used in the NOVA-W calculation and then the growth rates are extrapolated to the separatrix. The NOVA-W result for the benchmark case is given in \hbox{Table~\ref{t-3-4-1}}. Agreement between the NOVA-W result and the mean TSC result is good at 4\% for $\tau_v$ and 2\% for $f$, but agreement between the PSTAB result and the mean TSC result is 20\% for $\tau_v$ and 19\% for $f$. The differences between PSTAB and TSC result in part from rigid-{\it versus} deformable-plasma effects and from differences in the calculation of the stabilizer inductances. Also, a 20\% difference was obtained in the PSTAB and TSC calculations of $\tau_{L/R}$ with a vertically asymmetric current distribution in the stabilizer. The possibility of placing the passive stabilizer behind the shield and effectively using the vacuum vessel as the stabilizer element was also considered. To examine the feasibility of such a move, NOVA-W was used to analyze a 0.01-m-thick aluminum (\hbox{$\eta = 2.824\X 10^{-8}$~$\Omega$-m}) stabilizer shell located either behind the blanket (inner wall configuration) or behind the shield (outer wall configuration) as shown in \hbox{Fig.~\ref{f-3-4-6}}. The results indicate that the growth time for the axisymmetric mode for the inner wall configuration is \hbox{$330 \pm 5$~ms}. For the outer wall configuration, the growth time is \hbox{$44 \pm 1$~ms}. In fact, the outer wall configuration was found to be unacceptably close to the ideal stability limit, having a stability parameter \hbox{$f < 1.1$}. Therefore, only the inner vacuum-vessel wall configuration is effective in stabilizing the axisymmetric mode to a time scale that an active feedback system can handle with reasonable power levels. \begin{figure}[tb] \vspace*{3.8in} \caption[Fig-3-4-6] {The geometry of two possible locations for passive stabilization shells considered: behind the blanket (inner location) and behind the shield (outer location). The effect of tungsten divertor plates was included in both cases.} \label{f-3-4-6} \end{figure} \subsection{Active Feedback} \label{s-3-4-2} The active-feedback power requirements were determined from TSC simulations of the \hbox{$A = 4.5$} and \hbox{$\kappa_\X = 1.74$} case used in the passive stability benchmark described in the \hbox{Sec.~\ref{s-3-4-1}}. A simulation begins with the vertical coordinates of the plasma magnetic axis, $Z_M$, maintained in the equatorial plane for 0.1~s. The feedback coils are pre-programmed to initiate a 50-mm vertical displacement of $Z_M$ at 0.1~s into the simulation and to maintain that position once attained. As the plasma mass does not affect the voltage or current of the feedback coils, a large mass-enhancement factor (8,000) was used for computational expediency. \begin{figure}[p] \vspace*{7.3in} \caption[Fig-3-4-7] {The vertical position of the magnetic axis and the feedback-coil current and voltage as functions of time as calculated with TSC for \hbox{$A = 4.5$}, \hbox{$\kappa_\X = 1.74$}, the stabilizer geometry shown in \hbox{Fig.~\protect\ref{f-3-4-3}}, and feedback-coil~A in \hbox{Fig.~\protect\ref{f-3-4-3}}.} \label{f-3-4-7} \end{figure} Simulation results for feedback-coil~A (\hbox{Fig.~\ref{f-3-4-3}}) are given in \hbox{Fig.~\ref{f-3-4-7}}. A gain for driving the feedback-coil current was selected to yield a common value of \hbox{$G \sim -2.6$} for the dimensionless gain defined as the ratio of the response flux difference produced by the feedback coils to the flux difference produced by the plasma in the pickup coils. The dimensionless gain must be in the range of $-1$ to $-10$ to ensure stability and practicality of the feedback system. A value of \hbox{$G \sim -2.6$} rapidly moves the plasma to a 50-mm displacement with only an $\sim$10-mm overshoot. The gain for driving the feedback-coil voltage is set at a low value of 1~mV/A to ensure a smooth voltage response. A pickup-coil location approximately equal to the location of feedback-coil~B in \hbox{Fig.~\ref{f-3-4-3}} was used for feedback-coils~A and C. For feedback-coil~B, a pickup-coil location corresponding approximately to the location of feedback-coil~A was used to obtain a clear resolution of the flux difference produced by the plasma. \begin{figure}[b] \vspace*{3.4in} \caption[Fig-3-4-8] {The reactive power as a function of normalized radial location calculated with TSC for \hbox{$A = 4.5$}, \hbox{$\kappa_\X = 1.74$}, and the geometry shown in \hbox{Fig.~\protect\ref{f-3-4-3}}. Simulation results were scaled to a common 50-mm displacement of $Z_M$.} \label{f-3-4-8} \end{figure} The maximum reactive power occurs at the maximum vertical displacement of $Z_M$ and scales with $Z_M^2$. The reactive power requirements for the three feedback-coil locations of \hbox{Fig.~\ref{f-3-4-3}} are shown in \hbox{Fig.~\ref{f-3-4-8}} for a common 50-mm displacement of $Z_M$. The feedback coils were simulated with a \hbox{0.1-m $\X$ 0.1-m} cross section of room-temperature, copper alloy with a conductor filling fraction of 0.7. Feedback-coil~C is preferable over the other two coil locations. This coil is positioned just outside of the TF coil and is easily accessible for maintenance. The $\sim$2~MVA reactive power required to drive this coil is well within the 26~MW of recirculating power set aside for miscellaneous plant needs (\hbox{Sec.~2}). Furthermore, a 50-mm displacement could not be tolerated by the divertor and represents a maximum design constraint. \section{TRANSPORT ANALYSIS} \label{s-3-5} The 1-1/2-D, time-dependent BALDUR transport code~\cite{Sin89} was used to simulate steady-state core-plasma behavior for the \AI\ tokamak design and to study the time evolution of plasma density and temperature from a few keV ({\it i.e.}, typical of ohmic discharges) to a steady-state fusion plasma. In addition to the analyses of the approach to ignition and thermal stability, the BALDUR simulations were used to study fueling of the \AI\ plasma, the resultant density and temperature profiles, and helium ash buildup and exhaust. The parameters of the steady-state \AI\ plasma were also used to benchmark the \hbox{0-D} results of the systems code. Several physics models including an estimate for synchrotron radiation losses (\hbox{Sec.~\ref{s-3-5-1}}) and transport models (\hbox{Sec.~\ref{s-3-5-2}}) were incorporated in BALDUR for \AI\ transport analyses which are reported in this section. In addition to BALDUR simulations, the \AI\ reference equilibrium with the simple pressure profile of \hbox{$p (\psi) = p_o \tilde{\psi}^\alpha$} (\hbox{$\alpha = 1.4$} and $\tilde{\psi}$ is the normalized poloidal flux), was used to calculate and benchmark the \AI\ fusion-power performance and operating point. The 2-D flux contours for this equilibrium were used in the TRAC~II code, which computes a flux-surface-averaged power balance. This study was performed to ensure complete self-consistency among all plasma physics analyses since the density and temperature profiles used in this power balance analysis are exactly the same as those used for the current-drive and stability calculations, and these profiles are roughly consistent with the BALDUR results (although the 1-D BALDUR transport calculation cannot be exactly compared to 2-D MHD profiles). The plasma profiles used include a peaked temperature profile, \hbox{$T_e(\psi) = T_{eo} \tilde{\psi}^{\alpha_T}$}, with \hbox{$\alpha_T = 1.1$} and \hbox{$T_{eo} = 44$~keV}, and a broad density profile, \hbox{$n_e (\tilde{\psi}) = n_{eo} \tilde{\psi}^{\alpha_n}$}, with \hbox{$\alpha_n = 0.3$} and \hbox{$n_{eo} = 1.85\X 10^{20}$~m$^{-3}$}. The profile factors, $\alpha_T$ and $\alpha_n$, were chosen so that the profiles are consistent with BALDUR simulations and $T_{eo}$ and $n_{eo}$ are set so that the average temperatures and density agree with the systems code results. The alpha density was set at 10\% of the fuel ions, and an additional (nominally oxygen) impurity was added to raise $Z_{eff}$ to 1.6. The results of this simulation for the interim \AI\ design point are summarized in \hbox{Table~\ref{t-3-5-1}} and are in good agreement with systems code results and BALDUR simulations for that design point. \begin{table}[p] \caption{} \centering {\bf TRAC~II Simulation of Interim ARIES-I Parameters} \vspace{1.cm} \renewcommand{\arraystretch}{1.0} \begin{tabular}{l @{\hspace{3.truecm}} c} \hline \\ Major toroidal radius, $R_o$ (m) & 7.245 \\ Minor radius (95\%), $a$ (m) & 1.555 \\ Elongation, $\kappa_{95}$ & 1.60 \\ Triangularity, $\delta_{95}$ & 0.50 \\ Plasma volume (95\%) (m$^3$) & 523 \\ Shape factor, $S$ & 1.33 \\ Approximate first-wall area (m$^2$) & 629 \\ Plasma current, $I_p$ (MA) & 10.6 \\ On-axis toroidal field, $B_t$ (T) & 11.6 \\ Toroidal beta, $\beta$ (\%) & 1.9 \\ Troyon coefficient, $C_T$ (\% Tm/MA) & 3.2 \\ RMS beta, $\beta^*$ (\%) & 2.52 \\ Shafranov poloidal beta, $\beta_I$ & 2.4 \\ On-axis safety factor, $q_o$ & 1.3 \\ Toroidal radius of magnetic axis, $R_m$ (m) & 7.481 \\ Self-inductance, $L$ ($\mu$H) & 17.3 \\ Plasma pressure (MPA) \\ \quad Peak, $p_o$ & 2.97 \\ \quad Volume averaged, $\bar{p}$ & 1.03 \\ Electron temperature (keV) \\ \quad Peak, $T_{eo}$ & 44.2 \\ \quad Volume averaged, $\bar{T_e}$ & 17.4 \\ Electron density ($10^{20}$ m$^{-3}$) \\ \quad Peak, $n_{eo}$ & 1.85 \\ \quad Volume averaged, $\bar{n_e}$ & 1.33 \\ Fuel-ion density ($10^{20}$ m$^{-3}$) \\ \quad Peak, $n_{DTo}$ & 1.44 \\ \quad Volume averaged, $\bar{n_{DT}}$ & 1.04 \\ Effective plasma charge, $Z_{eff}$ ($\sim$1\% oxygen) & 1.6 \\ Absorbed current-drive power, $P_{CD}$ (MW) &100 \\ Fusion power, $P_f$ (MW) & 2,100 \\ Average neutron wall load (MW/m$^2$) & 2.7 \\ \\ \hline \end{tabular} \label{t-3-5-1} \end{table} \subsection{Synchrotron Radiation Effects} \label{s-3-5-1} A unique feature of the \AI\ reactor is that half of the plasma energy is radiated in the form of synchrotron radiation. The synchrotron power is high because: (1)~the first-wall material is SiC composite which is poorly reflective, (2)~high on-axis magnetic field, and (3)~high electron temperature. A large synchrotron radiation is beneficial because the heat load to the divertor plates will be reduced. On other hand, a better core-plasma confinement (conduction and convection) is required. To determine the synchrotron radiation losses, the overall reflectivity of the first wall should be calculated. The first wall of \AI\ comprises an SiC-fiber composite shell coated with a 2-mm-thick chemical-vapor-deposited SiC layer (Sec.~8). Helium coolant flows through channels inside the shell. A mixture of \LZ\ solid breeder and Be multiplier in the form of sphere-pac is located behind the first wall. The spectrum of synchrotron radiation emitted from the plasma peaks at the fifth electron-cyclotron harmonic ($\sim$1.64~THz) and extends all the way to the far infrared regime, but short of the regions of resonant and relaxation absorption by either He gas or SiC. Dissipation of the radiated power, therefore, takes place solely as a consequence of the presence of charge carriers in the SiC molecules due to gamma irradiation or the introduction of elemental impurities ({\it e.g.}, Be, B, and A$\ell$), which results in a finite electrical conductivity. Data on the electrical properties of SiC are limited, but millimeter wave measurement on water-free fused silica, for example, shows a loss tangent value of $10^{-3}$ at $\sim$400~GHz~\cite{But83}. The value of the loss tangent is a sensitive function of both frequency of the radiation and purity of the material (see also Sec.~4.5.3). Using a multi-slab model of the first wall and assuming a loss tangent in the range of $5\X 10^{-3}$ to 0.1 for SiC composite (corresponding to an equivalent conductivity of 5 to 100~mhos/m), the effective reflectivity of SiC-composite first wall is calculated to be 0.27. The divertor target plates of \AI\ are coated with tungsten which has an effective reflectivity of 0.99. Noting that the divertor-plates cover 20\% of the total first-wall area, the overall first-wall reflectivity, $\Re$, has been found to be 0.41. There are many ways to calculate the total synchrotron-power radiated from the plasma given its density and temperature profiles~\cite{Bor83}. The most rigorous approach is to solve the equation of radiative transfer in an inhomogeneous plasma, taking into account the emission, re-absorption, reflection from the wall, and transport of the radiation within the plasma. This problem was already solved by Tamor and the results are embodied in the SNECTR code~\cite{Tam75}. Subsequently, a much simpler and less time-consuming code, CYTRAN, was developed~\cite{Tam81} and satisfactorily benchmarked against SNECTR. For \AI\ applications, where a weakly reflecting wall is required, a calculational method simpler than using CYTRAN is possible. It involves the local application of the global model (LAGM) of synchrotron power loss from a homogeneous, cylindrical plasma at each point inside an inhomogeneous plasma. This so-called LAGM method was first proposed by Tamor~\cite{Tam81b} and was found to give total power loss, and associated radial cooling profiles, in reasonable agreement with those of SNECTR, particularly when the wall is weakly reflecting (\hbox{$\Re \ll 1$}). Using Trubnikov's approach~\cite{Tru79} and including the relativistic correction factor, the local radiated-power density is given by: % \begin{eqnarray} W_{sync} & = & 4.14\X 10^{-11} ( 1 - \Re )^{1/2} \, n_e^{1/2} \, T_e^{5/2} \, B_t^{5/2} \, ( 1 - \beta )^{5/4} \nonumber \\ \label{e-3-5-1} & & \X \: a^{-1/2} \, \kappa^{-1/4} \, \left( 1 + \frac{T_e}{204} \right) \, \left( 1 + \, \frac{18a \, \kappa^{1/2}}{R_o \, T_e^{1/2}} \right)^{1/2} \: , \end{eqnarray} % where SI units are used except for the electron temperature, $T_e$, which is in keV. Also, $n_e$ is the electron density, $B_t$ is the magnetic field strength, $\beta$ is the plasma beta, $\kappa$ is the plasma elongation, $R_o$ is the major radius, and $a$ is the minor radius. The expression in \hbox{Eq.~(\ref{e-3-5-1})} can be integrated over the plasma density and temperature profiles to obtain the total synchrotron-power loss, $P_{sync}$. \begin{figure}[tb] \vspace*{3.6in} \caption[Fig-3-5-1] {Radial profiles of synchrotron radiated-power density calculated by LAGM using \hbox{Eq.~(\protect\ref{e-3-5-1})} and by CYTRAN, for the reference \AI\ parameters.} \label{f-3-5-1} \end{figure} To check the accuracy of the LAGM calculation of the $W_{sync}$ profile, the reference \AI\ parameters of \hbox{Table~\ref{t-3-1-1}} were used for benchmarking against the results of \hbox{CYTRAN}. Shown in \hbox{Fig.~\ref{f-3-5-1}} are the radiated-power density profiles as a function of normalized plasma radius, calculated by LAGM using \hbox{Eq.~(\ref{e-3-5-1})} and by \hbox{CYTRAN}. The corresponding total radiated powers, $P_{sync}$, are 303 MW and 266 MW for, respectively, LAGM and CYTRAN, which compare reasonably well with the value of 221 MW calculated from the systems code. It is noted from \hbox{Fig.~\ref{f-3-5-1}} that the two profiles have very similar radial dependence, except for the difference in their local magnitudes that accounts for the 14\% difference in $P_{sync}$. At \hbox{$\Re = 0.41$}, $W_{sync}$ from CYTRAN shows a small negative value near the plasma periphery, implying net synchrotron heating due to reabsorption of the wall-reflected synchrotron power. This effect, of course, is not taken into account by the LAGM calculations. It should be pointed out here that $W_{sync}$, as calculated from \hbox{Eq.~(\ref{e-3-5-2})}, scales as \hbox{$n_{e}^{1/2} \, T_{e}^{5/2} \, B^{5/2}$} and, therefore, is strongly sensitive to the electron temperature profile and the magnetic field. In addition, since $P_{sync}$ and the alpha heating power have the same order of magnitude in \AI, an accurate evaluation of the $W_{sync}$ profile and $P_{sync}$ is essential in determining the self-consistent $n_{e}$ and $T_{e}$ profiles, and the plasma thermal equilibrium point. For this reason, the LAGM calculation of $W_{sync}$ in conjunction with \hbox{Eq.~(\ref{e-3-5-1})} is incorporated into the 1-D BALDUR transport code, which will be discussed in the following sections. \subsection{The \AI\ Transport Model} \label{s-3-5-2} The transport model that was implemented in the BALDUR transport code was adapted from Houlberg's plasma modeling work on the compact ignition torus~\cite{Hou89}. This model is purely empirical and is designed to reproduce the global empirical scalings of energy and particle confinement time. In addition to global agreement with empirical scalings, this model has been used successfully at the Princeton Plasma Physics Laboratory to predict the density and temperature profiles in certain pellet-fueled TFTR discharges~\cite{Sto89}. Houlberg has also been successful in predicting plasma density and temperature behavior on JET discharges using this transport model in the WHIST transport code~\cite{Hou89}. The Houlberg transport model is based on the following assumptions: (1)~The radial ion and electron thermal diffusivities are equal. (2)~The radial particle-diffusion coefficient, $D_{\perp}$, is equal to 1/2 the radial thermal diffusivity, $\chi_\perp$. (3)~The edge thermal diffusivity is five times that at the center. (4)~There exists an anomalous inward particle pinch for ions and electrons proportional to the particle diffusion coefficient. The radial thermal diffusivity, $\chi_{\perp} (r)$, is defined as % \begin{eqnarray} \label{e-3-5-2} \chi_{\perp} & = & \frac {a^2}{4\tau_E} \, \left( 1+ \frac{4r^2}{a^2} \right) \, \left( \frac{2\kappa^2}{1+ \kappa^2} \right) \, C_N \: , \end{eqnarray} % where $a$ is the plasma minor radius, $\kappa$ is the plasma vertical elongation, $r$ is the radial plasma coordinate, $C_N$ is a normalizing factor equal to 0.218, and $\tau_E$ is the global diffusive energy-confinement time. For \AI\ simulations, $\tau_E$ is assumed to scale as the empirical Riedel-Kaye energy-confinement time~\cite{Gol89}, % \begin{eqnarray} \label{e-3-5-3} \tau_{E,R-K} & = & 0.03 \, f_H \, a^{-0.27} \, R_o^{1.64} \, I_{p}^{1.01} \, B_t^{0.07} \, n^{-0.04} \, P^{-0.54} \, \kappa^{0.59} \, A_i^{0.5} \: , \end{eqnarray} % where $f_H$ is the confinement enhancement factor over the so-called L-mode, $R_o$ is the major toroidal radius, $I_p$ is the plasma current, $n$ is the electron density, and $P$ is the net plasma-heating power (assumed to be equal to total plasma-heating power minus the radiation losses). In \hbox{Eq.~(\ref{e-3-5-3})}, SI units are used except for $I_p$ in MA, $n$ in 10$^{19}$~m$^{-3}$, and $P$ in MW. The confinement enhancement factor, \hbox{$f_H = 2.8$}, was used in the BALDUR simulations. The anomalous inward particle pinch has been introduced by Stotler as a refinement on Houlberg's transport model to give improved agreement between BALDUR simulations and TFTR discharges in terms of density profiles. This inward pinch is given as~\cite{Sto89} % \begin{eqnarray} \label{e-3-5-4} V_{pinch} & = & - \, \frac{2 \, D_\perp\, r}{a^2} \: . \end{eqnarray} % For the \AI\ simulation work, 3/4 of this pinch is included in the transport model. This choice was based on the fact that the difference between including the full pinch and 3/4 of the pinch was less than the scatter in the empirical data base on which the transport model had been benchmarked. Therefore, there was some freedom in choosing the exact multiplying factor for the inward pinch and the lower value of the inward pinch was used. In the simulation of the \AI\ plasma, the MHD equilibrium was determined by solving the Grad-Shafranov equation in two dimensions. The \hbox{2-D} grid used to solve this differential equation was established by specifying a fixed boundary for the poloidal flux using the technique of harmonic moments~\cite{Bat89}. The Grad-Shafranov equation was solved every five seconds during the simulation using the evolving pressure and current-density profiles. As a result, the simulation involved a coupled evolution of \hbox{1-D} radial transport and \hbox{2-D} MHD equilibrium. A high-$n$ ballooning-mode transport model was included in the simulation to maintain the pressure gradient at or below the critical limit for the onset of ballooning modes. The transport model for ballooning modes is in the form of an enhancement factor, $f_\chi$, on the local $\chi_{\perp}$ equal to~\cite{Bat89} % \begin{eqnarray} \label{e-3-5-5} f_\chi & = & \left[ 1+A \left(\frac{p'}{p'_{crit}}\right) \right]^B \: , \end{eqnarray} % where $p'$ is the local pressure gradient, $p'_{crit}$ is the pressure gradient limit for high-$n$ ballooning modes, and values of \hbox{$A = 1$} and \hbox{$B = 10$} were used. The result of including this enhancement in $\chi_{\perp}$ is the local flattening of the temperature profile in regions where \hbox{$p'>p'_{crit}$}. In the \AI\ plasma simulation, the effect of including this enhancement factor on global confinement time and the average temperature is found to be small. This result also confirms that the \AI\ plasma is stable against high-$n$ ballooning modes in the bulk of the plasma. \subsection{Transport Simulation Results} \label{s-3-5-3} The transport models described in \hbox{Secs.~\ref{s-3-5-1}} and \hbox{\ref{s-3-5-2}} were incorporated into the BALDUR code. The evolution of the \AI\ plasma was simulated by starting with an ohmic-like plasma with a temperature of 5~keV on axis (typical of plasmas in ohmic equilibrium~\cite{Sin90}) and continuing to steady-state, fusion-burn conditions. For \AI\ simulations, it was assumed that the density and temperature profiles have pedestal boundary conditions such that at steady state \hbox{$T_e(a) = T_i(a) = 0.30$~keV} and \hbox{$n_e(a) = 1.0\X 10^{20}$~m$^{-3}$}, where $T_e(a)$ and $T_i(a)$ are, respectively, the separatrix electron and ion temperatures, and $n_e(a)$ is the separatrix electron density. The fast-wave current-drive system was used to heat the \AI\ plasma to ignition. It was assumed that the auxiliary heating associated with the fast-wave current drive has a parabolic profile (consistent with the current-drive calculations of fast-wave power absorption in the plasma). The 96.7~MW of auxiliary heating is divided so that 93\% of the power is delivered to electrons and 7\% to the ions. The available current drive was found to be more than adequate for heating the \AI\ plasma to ignition. Typically, the \AI\ plasma achieved ignition in about 5~s and steady state in approximately 20~s of plasma simulation time; it maintained a steady burn for the remaining 20~s of the simulation. The reference fueling scenario for the \AI\ plasma includes the injection of 2-mm-radius pellets made of a 50:50 deuterium-tritium (DT) mixture at a frequency of 3~Hz. The pellets are injected at the mid-plane. In general, it was found that for the range of pellet size and speed considered, pellets would ablate in the outer 1/3 to 1/2 of the plasma radius (corresponding to pellet speeds of 5 and 20~km/s, respectively). In the simulations, the pellet fuel ions were transported towards the center by the anomalous inward particle pinch. The density profile was found to be very broad and, without an anomalous pinch term, the density profile would have been even flatter. On the other hand, the \AI\ plasma was found to be very stable (thermal and MHD) to perturbations in the density profile. For example, \hbox{Fig.~\ref{f-3-5-2}} shows the evolution of temperature and density profiles of the \AI\ plasma fueled with large size (4-mm radius) pellets injected every eight seconds. In this simulation, the density and temperature profiles were recovered in about 2~s after each pellet injection. The transport simulation of the \AI\ plasma gives a $^4$He particle-confinement time of approximately 7.76~s. The $^4$He ash buildup in steady state is 8.6\%. This particle confinement is very sensitive to the value of the particle diffusion coefficient and to the inward pinch that is assumed in the simulation. It should be noted, however, that the assumption of \hbox{$D_\perp = 0.5 \chi_\perp$} and \hbox{$V_{pinch} = -1.5 \, D \, r/a^2$} results in plasma-core $\tau_P/\tau_E$ values that are in the empirical range for present-day tokamak experiments. \begin{figure}[p] \vspace*{7.7in} \caption[Fig-5-2] {Time evolution of the ion (A) temperature and (B) density profiles.} \label{f-3-5-2} \end{figure} \begin{table}[p] \caption{} \centering {\bf Comparison of BALDUR Simulation Results with 0-D Predictions\\ for \AI\ Plasma Parameters} \vspace{1.cm}\renewcommand{\arraystretch}{1.3} \begin{tabular}{l c c} \hline \hfil {Plasma Parameters} \hfil & BALDUR & 0-D Simulations \\ \hline \\ Current-drive power (MW) & \Z97. & \Z97. \\ Alpha power (MW) & 375. & 385. \\ Energy confinement time (s) & 2.49 & 2.53 \\ Confinement enhancement factor, $f_{H,Riedel-Kaye}$ & 2.80 & 2.52 \\ Particle confinement time (s) & 7.76 & 9.39 \\ Average electron density (10$^{20}$~m$^{-3}$) & 1.39 & 1.45 \\ Average ion density (10$^{20}$~m$^{-3}$) & 1.20 & 1.24 \\ Average electron temperature$^{(a)}$ (keV) & 18.3 & 19.3 \\ Average ion temperature$^{(a)}$ (keV) & 20.7 & 20.0 \\ Toroidal beta (\%) & 2.00 & 1.90 \\ Effective plasma charge, $Z_{eff}$ & 1.73 & 1.65 \\ Core-plasma radiation fraction, $f_{rad}$ & 0.48 & 0.49 \\ $^4$He ash concentration (\%) & 8.58 & 10.0 \\ \\ \hline \multicolumn{3}{l}{$^{(a)}$Density weighted.}\\ \end{tabular} \label{t-3-5-2} \end{table} The steady-state plasma-parameter values from the BALDUR simulations are compared to \hbox{0-D} prediction of the systems code in \hbox{Table~\ref{t-3-5-2}}. Good agreement between the 0- and \hbox{1-D} values for the \AI\ plasma parameters were found. The plasma pressure and density profiles are found to scale roughly as % \begin{eqnarray} \label{e-3-5-6} p (r) & = & p_o \left[ 1 - \left(\frac{r}{a}\right)^2 \right]^{\alpha_p} \: , \\ \label{e-3-5-7} n (r) & = & n(a) + n_o \left[ 1 - \left(\frac{r}{a}\right)^2 \right]^{\alpha_n} \: , \end{eqnarray} % with \hbox{$\alpha_p \simeq 1.4$} and \hbox{$\alpha_n \simeq 0.3$}. These profiles were used in other plasma analyses. The electron and ion temperature profiles however are not similar, the electron temperature being flatter than that of the ions. This difference in the temperature profiles is attributed to the flattening of the electron temperature profile due to a more accurate synchrotron-radiation model in the simulation (\hbox{Sec.~\ref{s-3-5-1}}). \subsubsection{Summary} \label{s-3-5-4} The \hbox{1-D} transport simulations using an empirically benchmarked tokamak plasma-transport model has confirmed that the \AI\ plasma achieves ignition using the available current-drive power and achieves steady-state burn as long as the $\tau_E$ enhancement factor of 2.8 can be attained. Judging from the improvements in the empirical $\tau_E$ values achieved in tokamaks in the last 10 years, this level of confinement appears to be a reasonable extrapolation. The \hbox{1-D} simulation results are in good agreement with \hbox{0-D} predictions. Fueling remains an issue because pellets with reasonable sizes and speeds oblate in the outer parts of the plasma, and an anomalous inward pinch appears to be necessary to transport the fuel ions towards the plasma center. The simulations also show that the Houlberg plasma-transport model not only predicts the plasma density and temperature profiles of certain TFTR and JET discharges accurately, but can also be extrapolated to power reactor conditions. It is important to note that while Houlberg's plasma-transport model predicts the density profiles used in the \AI\ plasma analyses (\hbox{$\alpha_n = 0.3$}), there exist other transport models ({\it e.g.}, the theoretical transport model of Singer~\cite{Sin91}) that predict even flatter density profiles for \AI\ parameters. Future work in this area will involve a comparison between the models of Houlberg and Singer to decide which is more relevant to reactor-grade plasmas (comparison between Singer's model predictions and empirical results has been confined to ASDEX discharges). \section{SUMMARY AND CONCLUSIONS} \label{s-3-6} The \AI\ design is a conceptual commercial reactor based on modest extrapolation from the present tokamak physics data base. The \AI\ design operates at a relatively high plasma aspect ratio (\hbox{$A = 4.5$}), a low plasma current (\hbox{$I_p = 10.2$~MA}), and a high on-axis magnetic field (\hbox{$B_o = 11.3$~T}). As a result, the poloidal beta is high and a high bootstrap-current fraction of 0.68 is predicted. Because of the low plasma current and high bootstrap-current fraction, only 3.3~MA of current should be driven by external means; therefore a steady-state reactor with relatively small current-drive power is possible. Systems-code analysis confirms that an optimum first-stability tokamak reactor operates with high aspect ratio, low current, and high bootstrap-current fraction. The key parameters of the \AI\ reactor are listed in \hbox{Table~\ref{t-3-1-1}}. The \AI\ MHD equilibrium, beta limit, vertical stability, and the plasma transport analyses have been described in this section. Generally only modest extrapolation from the near-term physics data base is required. Some particular results of these studies indicate that: \begin{itemize} \item Diverted plasma with low elongation (1.8) can be created with relatively far-removed PF coils. \item Design iterations have led to a minimum stored-energy PF-coil set. \item A value for the Troyon beta coefficient of 3.2 is sufficient for first stability operation. \item Passive stabilization of short time-scale axisymmetric instabilities ($<$100~ms) is possible with the passive stabilizer elements located behind the blanket ($\sim\!0.6a$ away from the plasma surface). \item Active stabilization of longer time-scale axisymmetric instabilities is possible with modest power demands (several MVAs). \item The first wall is weakly reflecting, which results in large synchrotron-radiation losses. 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