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\begin{document}
\begin{center}
{\huge \bf
Physics Basis for the Advanced}
\end{center}
\begin{center}
{\huge \bf
Tokamak Fusion Power Plant}
\end{center}
\begin{center}
{\huge \bf
ARIES-AT}
\end{center}
\bigskip

\bigskip
\begin{center} {\Large \bf S.~C.~Jardin\footnote[1]{Princeton Plasma
Physics Laboratory, P.O. Box 451, Princeton, NJ 08543},
C.~E.~Kessel\footnotemark[1],
T.~K.~Mau\footnote[2]{Fusion Energy Research Program, University of
California, San Diego, 9500 Gilman Dr., La Jolla, CA 92093},
R.~L.~Miller\footnotemark[2],
F.~Najmabadi\footnotemark[2], V.~S.~Chan\footnote[3]{General Atomics,
P.O. Box 85608, San Diego, CA
92186}, M.~S.~Chu\footnotemark[3], L.~L.~Lao\footnotemark[3], 
T.~Petrie\footnotemark[3], P.~Politzer\footnotemark[3],
H.~E.~St.John\footnotemark[3], P.~Snyder\footnotemark[3], 
G.~M.~Staebler\footnotemark[3], A.~D.~Turnbull\footnotemark[3], 
P.~West\footnotemark[3]}
\end{center}

\bigskip
\bigskip

\begin{abstract} The advanced tokamak is
considered as the basis for a fusion power plant.  The
ARIES-AT design has an aspect ratio of $A \equiv R/a = 4.0$,
an elongation and triangularity of $\kappa = 2.17, \delta =0.84$
(evaluated at the separatrix surface), a toroidal beta
of $ \beta = 9.2 \% $ (normalized to the vacuum toroidal
field on at the plasma center), which corresponds to a
normalized beta of $\beta_N \equiv 100 \times \beta / (I_P (MA)/a(m)B(T)) 
= 5.4$.  These beta values are chosen to be 10\% below the ideal MHD 
stability limit. The bootstrap-current fraction is $f_{BS} \equiv I_{BS}/I_{P}
 =  0.91$.  This leads to a design with total plasma current $I_P = 12.8 MA$,
and toroidal field of $ 11.1 T $ (at the coil edge) and $ 5.8 T $
(at the plasma center). The major and minor radii are 5.2 and 1.3 m.
The effects of H-mode edge gradients and the 
stability of this configuration to non-ideal modes is analyzed.  The
current drive system consists of ICRF/FW for on-axis current drive and
a Lower Hybrid system for off-axis.  Transport projections are presented using
the GLF23 model.  The approach to power and particle exhaust using both
plasma core and scrape-off-layer radiation is presented.
 
  \end{abstract}

\bigskip
\bigskip
\bigskip
\clearpage

\section{Introduction} %1.0

This, and the companion paper~\cite{Kessel01}, describe the
physics basis and physics optimization studies performed for the ARIES-AT
fusion power plant study.   ARIES-AT is an advanced tokamak, as was
ARIES-RS~\cite{Jardin97}.   As such, the tokamak q-profile has a ``reversed
shear'' property where it is a local maximum on axis, decreases outward
to a local minimum, and then increases to the plasma boundary.  This
configuration is favored for its properties that (1) it remains stable to
ideal MHD ballooning and internal kink modes at very
large values of the plasma $\beta \equiv 2 \mu_0 p / B_{Vac}^2$, (where
$B_{Vac}$ is the vacuum toroidal field evaluated at the plasma center),
(2) it implies a plasma current density profile peaked off-axis that is
consistent with a large bootstrap-current fraction $f_{BS} \sim 1$, and
(3) it is consistent with favorable transport properties near and interior
to the shear reversal region.

There are many similarities to this design and to ARIES-RS.  The primary
difference is that we have carried the optimizations further in this design
to produce a more aggressive, and hence more attractive design, by considering
a wider class of pressure and current profiles and by allowing the plasma
region to extend further to the X-point singularity in the confining magnetic
field.

We have also increased the depth of the analysis of the plasma configuration
over that which was done for ARIES-RS.   In Section~\ref{s2.0} we consider
the effects of edge gradients and current values expected for H-mode
plasmas.  We also analyze the stability requirements for the resistive
wall mode (RWM), the neoclassical tearing mode (NTM) and the edge
localized modes (ELMs).

The current drive systems are analyzed in Section~\ref{s4.0}.  This
includes an analysis and design specifications for the baseline current
drive system of ion cyclotron resonant frequency (ICRF) and lower hybrid
wave (LHW) heating and current drive, but also analysis of backup systems
of high harmonic fast wave (HHFW) and neutral beam injection (NBI).  The
primary function of these systems is to drive plasma current where it 
needed from plasma stability and transport considerations.  However, the
heating and current drive systems are also used during plasma startup,
and also produce plasma rotation which affects both the MHD stability
and the transport properties of the configuration.

The transport properties of the device are analyzed in Section~\ref{s5.0}.
Here we apply modern transport analysis tools and the GLF23 transport
model to analyze the consistency of the plasma profiles with respect
to evolution due to transport processes.  As part of this analysis,
we consider the effects of the driven rotation on plasma transport.

The power and particle exhaust properties of the plasma are analyzed
in Section~\ref{s6.0}.  Here we consider the radiative and conductive
heat losses from the plasma and edge regions, and the effect of 
impurity doping.  The engineering constraints on the maximum allowable
heat fluxes on the first wall and divertor regions put severe constraints
on the allowable solutions.

In Section~\ref{s7.0} we summarize the plasma operating regime parameters
in terms of a plasma operating contour (POPCON) diagram, and present
an analysis of the startup requirements.  The main results of this
study are summarized in Section~\ref{s8.0}. 

\section{Plasma Equilibrium and Stability} %2.0
\label{s2.0}

The optimization procedure which led to the specification of the baseline 
pressure and current profiles is described in~\cite{Kessel01}.  We summarize
the results of that optimization in Sec.~\ref{s2.1}.  In the following
sections we consider the effects of H-mode profiles on the plasma
stability (Sec.~\ref{s2.2}), and extend the stability analysis to include
the resistive wall mode (RWM)(Sec.~\ref{s2.3}), the neoclassical tearing 
mode (NTM)(Sec.~\ref{s2.4}), and the edge localized modes (ELMS)
(Sec.~\ref{s2.5}).

\subsection{Baseline Equilibrium and Profiles}
\label{s2.1}

The baseline plasma profiles
have zero pressure gradient and zero current density at the plasma edge, 
making them typical of L-mode edge plasmas.  The pressure profile
for the reference equilibrium is given by,

\begin{equation}
p(\psi)=p_0\left[ c_1( 1 - \hat{\psi}^{b_1})^{a_1} +  
                  c_2( 1 - \hat{\psi}^{b_2})^{a_2} \right]
\end{equation} 
with $ a_1 = 2.75, b_1 = 2.25, c_1 = 0.55$ and $ a_2 = 2.00, 
b_2 = 1.00, c_2 = 0.45, p_0=2.467 \times 10^6 $.  The parallel current
current density profile is a self-consistent combination of the bootstrap
current and the externally driven current profiles from fast wave and lower
hybrid current drive.

In the baseline fixed-boundary equilibrium and stability studies, the plasma-vacuum
boundary is the 99\% flux surface from the free-boundary equilibrium.
Although the plasma boundary can not be described analytically, it can be
parameterized in terms of a major and minor radius $R = 5.2m,
a = 1.30 m$, an elongation $\kappa = 2.15$ and a
triangularity $\delta = 0.78$.

As described in~\cite{Kessel01}, these parameters have been chosen to maximize
the plasma $\beta$ while maintaining stability to ideal MHD modes, and to
provide a high bootstrap fraction equilibrium that is
compatible with the current drive systems described in Section~\ref{s4.0}.
The equilibrium has a hollow current profile, with the location of the
$q_{min}$ (minimum safety factor) close to the edge.


\subsection{Effects of H-mode Like Pressure and Current Profiles} %old 2.6
\label{s2.2}

The equilibrium and stability analysis described in Ref.~\cite{Kessel01}
and in Sec.~\ref{s2.1} was based on standard pressure and current profile
functions that go smoothly to zero at the plasma edge, with zero
pressure derivative there.  These profiles are more typical of L-mode
plasmas than of H-mode.  Because ARIES-AT requires the  energy confinement
times typical of an H-mode, we have extended the stability analysis
to examine the effects of the edge gradients typical of H-mode plasmas.


\subsubsection{H-Mode Equilibrium} %2.6.1

The primary difference between the plasma profiles defined in Sec.~\ref{s2.1}
and the H-Mode profiles defined here are the values of the pressure and
current density near the plasma edge.  Because the
stability of the H-mode profiles depends sensitively on the interplay
of the strong shaping near the edge of a divertor plasma and the steep
gradients there, we have recomputed the free-boundary diverted
equilibrium with H-mode-like pressure and current profiles using the EFIT 
code.  These profiles were based on the reference L-mode limiter equilibrium.
This equilibrium~\cite{g00509.0*540} forms the basis for the stability
analysis in the remainder of this section. 

In this proceedure, the pressure and the current profiles from the L-mode 
equilibrium are first fit to a set of polynomial basis functions. H-mode 
like equilibria are then generated by perturbing the outer region of the 
L-mode equilibrium obtained using the fitted profiles while keeping the core 
as similar to the L-mode case as possible.  The edge pressure gradient and 
current density are allowed to be 
finite.  This is illustrated in Figure~\ref{f2-6-1},
where the two free pressure and poloidal current functions, 
$P^{\prime}(\psi)$  and $FF^{\prime}(\psi)$,
the safety factor $ q(\psi)$, and the flux surfaces for the L- and H-mode 
like equilibria are compared.  This particular H-mode like equilibrium has 
$P^{\prime}(1)\sim 0.2 P^{\prime}(0)$ ,
$\langle J_T \rangle (1) \sim 0.38 J_{ave}$, $\beta_T \sim 9.7\% $
, and  $\beta_N \sim 5.9$.  Because of the finite edge pressure gradient, the 
H-mode case has a slightly higher $\beta$ value.  

\begin{figure}[ht]
\begin{center}
%note:  bounding box is lower left and upper right coordinates
%       ( with 72 Postscript units per inch )
\includegraphics*[bb=-20 159 653 652,angle=0,height=5.0 in,width=!]
{./f2-6-1.ps}\caption{Comparison of the pressure and poloidal current 
functions $p^{\prime}(\psi)$ and $FF^{\prime}(\psi)$, the safety factor 
profile $q(\psi)$, and the flux surfaces for a H-mode like equilibrium 
(dashed curves) against those for the L-mode base equilibrium (solid curves).}
\label{f2-6-1}
\end{center}
\end{figure}


\subsubsection{Effect on Ballooning and Kink Stability} % 2.6.2

Ideal stability analyses indicate that stability against the low $n = 1, 2$ 
modes are relatively insensitive to the presence of the X-point and the 
broader H-mode like pressure and current profiles with.  The locations of the 
conducting wall required for stabilization against these modes remain 
similar. This is summarized in Table~\ref{t2.6-1}.

\begin{table}
\centering
\begin{tabular}{|l|l|l|} \hline

                  &  Marginal Wall     & Marginal Wall  \\
                  &  $n=1$             & $n=2$          \\ \hline
L-Mode Limiter    &  $1.525$           & $1.450$        \\
$\beta_N=5.6$&                    &                \\ \hline
L-Mode Divertor   &  $1.600$           & $1.475$        \\
$\beta_N=5.6$&                    &                \\ \hline
H-Mode Divertor   &  $1.550$           & $1.475$        \\
$\beta_N=5.8$&                    &                \\ \hline
H-Mode Divertor   &  $1.575$           & $1.450$        \\
$\beta_N=5.9$&                    &                \\ \hline
\end{tabular}
\caption{ Comparison of the critical wall location for stabilization against
the $n=1,2$ ideal modes for the L-and H-mode like equilibria.}
\label{t2.6-1}

\end{table}

Stability against the high $n$ ideal ballooning modes, which are limiting 
the L-mode like reference limiter equilibrium, are actually improved by the 
H-mode like profiles, even though the pressure gradients and $\beta$ value
are higher.  With sufficiently high edge pressure gradient and bootstrap 
current, the H-mode like equilibria have second stability access for 
ballooning modes over essentially the entire plasma volume.  This is 
illustrated in Figure~\ref{f2-6-2} for the H-mode like equilibrium shown in 
Figure~\ref{f2-6-1}.

\begin{figure}[ht]
\begin{center}
\includegraphics*[bb=-10 169 653 652,angle=0,height=4.0 in,width=!]
{./f2-6-2.ps}
\caption{Comparison of Ballooning stability for the L- and H-mode like 
equilibria.}
\label{f2-6-2}
\end{center}
\end{figure}

\subsection{Stability to the Resistive Wall Mode (RWM) and its Stabilization 
by Plasma Rotation} % old 2.7
\label{s2.3}
%
The proposed ARIES equilibria~\cite{g00509.0*540} was tested for its
stability to the Resistive Wall Mode (RWM) and stabilization of the RWM by 
plasma rotation was evaluated by using the MARS stability 
code~\cite{Bondeson??}.  MARS is an eigenvalue code, which solves for
the full MHD perturbation equations with the MHD mode frequency as the 
eigenvalue and the perturbed MHD quantities (perturbed plasma displacement, 
magnetic field and pressure) as eigenfunctions.  One of the most important 
input quantities in the present study is the plasma rotation profile. 
For simplicity, we assume that the plasma has a constant(rigid) rotation 
frequency profile across its cross-section. The model for the damping of 
the toroidal momentum used is the sound wave damping model.
In this model, there is a force which damps the 
perturbed toroidal motion of the mode according to the formula

$$F_{SD}=-\kappa_{\parallel} \sqrt{\pi} | k_{\parallel} v_{th_i} | \rho 
\vec{v} \cdot \hat{b} \hat{b}. $$

Here $\kappa_{\parallel} $ is a numerical coefficient with a value of 
$1.77$ to model the ion Landau damping process, $ k_{\parallel}$ is the 
parallel wave number $(m-nq)/R$, $v_{th_i}$ is the ion thermal velocity, 
$\rho$ is the mass density, $\vec{v}$ is the perturbed plasma velocity and 
$\hat{b}$ is the unit vector of the equilibrium magnetic field.


\subsubsection{RWM with Toroidal Mode Number $n=1$}
\label{s2.7.1}
The critical rotation frequency for stabilization of the
RWM in equilibria~\cite{g00509.0*540} is found to be between $0.07$ to 
$0.08$ of the Alfven transit frequency. No stability window is found for 
rotation frequency below $0.07$. The stability window for the location of 
the resistive wall is between $1.3$ and $1.45$ of the plasma radius when the 
plasma rotation frequency is at $0.08$ of Alfven frequency. This window 
widens to between $1.075$ to $1.45$ of the plasma radius when the rotation 
frequency is increased to $0.09$ of the Alfven frequency. Here, the Alfven  
frequency is defined as $$f_{av}=\frac {v_A} {R}. $$
Here, $R$ is the plasma center, taken to be $5.19$ $m$ and $v_A$ is 
$$v_A = \frac {B_{vac}} {\sqrt{\mu_0 \rho_0}} $$ 
and $B_{vac}$ is $5.92 T$.   A uniform density 50-50\% $D-T$ plasma 
with electron density $ {2.2 \times 10^{20}}/{m^3} $ is used and the 
value of $f_{av}$ is computed to be ${1.06 \times 10^6} /{ sec}$. The 
resistive wall time has been taken to be $5000 \tau_{a}$ or $5$ msec. When 
the rotation frequency is increased beyond $.09$ of Alfven frequency, the 
growth rate normalized to the wall time is reduced below $0.1$. It would then 
have a time scale longer than $50$ msec, or a frequency less than $20$ Hz.

These results are illustrated in the following figures. Shown in 
Figure~\ref{f2-7-1} is the mode amplitude of the various poloidal Fourier 
harmonics of the radial magnetic perturbations as a function of $\sqrt{V}$, 
with $V$ being the volume enclosed by the flux surface.  It is observed that 
this mode has a global structure.  Shown in Figure~\ref{f2-7-2} is the 
growth rate of the resistive wall mode as a function of the location of the 
resistive wall. Each curve has a different plasma rotation frequency 
relative to the resistive wall. The same equilibrium is used here as in 
Figure~\ref{f2-7-1}.


\begin{figure}[ht]
\begin{center}
\includegraphics*[bb=95 213 567 643,height=4.5in,clip=]{./f2-7-1.ps}
\caption{ Amplitudes of the poloidal harmonics of the perturbed magnetic
field $\delta B_{\psi}$ of a stabilized resistive wall mode. Here
$\delta B_{\psi}$ is plotted as a function of $\sqrt{V}$ for
equilibria~\cite{g00509.0*540}for a uniformly rotating plasma with 
$\Omega=.08$ measured in units of Alfven toroidal transit frequency and with 
$r_w=1.35r_p$ and toroidal mode number $n=1$}
%
\label{f2-7-1}
\end{center}
\end{figure}

\begin{figure}[ht]
\begin{center}
\includegraphics*[bb=103 417 396 719,height=4.5in,clip=]{./f2-7-2.ps}
%\caption{Stability Window of RWM vs Wall Location }
\caption{ Stability window in $r_w$ for the $n=1$ resistive wall
 mode for equilibrium~\cite{g00509.0*540}. Plotted are growth rates
of the ideal and resistive wall modes vs. $r_w$. The curve labelled
ideal is the growth rate of the ideal external kink with the scales
multiplied by 200. Other curves are labeled by the rotation
frequency of the plasma with respect to the resistive wall
 measured in units of the toroidal Alfven transit frequency.
The growth rates of the resistive wall
modes are multiplied by $\tau_w$.}
%
\label{f2-7-2}
\end{center}
\end{figure}

\subsubsection{RWM with Toroidal Mode Number $n=2$}
\label{s2.7.2}
A similar study has also been performed for the $n=2$ mode. It is found 
that the rotation required for the $n=2$ mode is substantially reduced from 
that for the $n=1$ mode. This reduction factor is close to 2, as
shown in Figure~\ref{f2-7-3}. In this figure growth rates of the resistive 
wall mode with the plasma rotating at different rotation frequencies are 
plotted as a function of the location of the resistive wall. Note that there 
is an opening up of the stability window first at 
$r_w = 1.5 $ at around rotation frequency of $0.03$ of the Alfven frequency. 
When the rotation frequency is increased to $0.05$ of the Alfven transit 
frequency, this window widens to between $r_w=1.32$ and $r_w=1.5$.

\begin{figure}[ht]
\begin{center}
\includegraphics*[bb=103 417 396 719,height=4.5in,clip=]{./f2-7-3.ps}
\caption{Stability window in $r_w$ for the $n=2$ resistive wall
 mode for equilibrium~\cite{g00509.0*540}. Plotted are growth rates
of the ideal and resistive wall modes vs. $r_w$. The curve labeled
ideal is the growth rate of the ideal external kink with the scales
multiplied by 200. Other curves are labeled by the rotation
frequency of the plasma with respect to the resistive wall
 measured in units of the toroidal Alfven transit frequency.
The growth rates of the resistive wall
modes are multiplied by $\tau_w$.}
\label{f2-7-3}
\end{center}
\end{figure}
The reason of this reduction is the increased sound
wave damping due to the increased number of resonant flux surfaces. The 
corresponding mode structure of the $n=2$ resistive wall mode is shown
in Figure~\ref{f2-7-4}. The mode structure is still quite global.
\begin{figure}[ht]
\begin{center}
\includegraphics*[bb=95 213 567 643,height=4.5in,clip=]{./f2-7-4.ps}
\caption{ Amplitudes of the poloidal harmonics of the perturbed magnetic
field $\delta B_{\psi}$ of a stabilized resistive wall mode.
Here $\delta B_{\psi}$ is plotted  as a function of $\sqrt{V}$ for 
equilibrium \cite{g00509.0*540} for a uniformly rotating plasma with 
$\Omega=.08$ measured in units of Alfven toroidal transit frequency and with 
$r_w=1.35r_p$ and toroidal mode number $n=2$}
\label{f2-7-4}
\end{center}
\end{figure}

\subsubsection{Stability to the RWM through use of Intelligent Shell Feedback}

It was found in Sections~\ref{s2.7.1} and \ref{s2.7.2} that 
the rotation frequency required for the stabilization of the RWM 
is over a few percent of the Alfven wave transit frequency.  This amount of
rotation would be difficult to maintain for the ARIES type of
tokamak.  Therefore, the alternative approach of active feedback 
stabilization of the RWM has also been studied.   In the
intelligent shell feedback scheme, external coil currents are utilized to 
make the resistive wall appear almost ideally conductive to the plasma. This 
approach has been formulated for a finite $\beta$ plasma in toroidal geometry
by extending the ideal MHD energy principle and implemented numerically by 
coupling GATO~\cite{GATO??} with VACUUM~\cite{Chance99}. Application of this 
formulation to ARIES geometry\cite{g00509.0*540}, employing 
``flux conserving intelligent  coils'' located on the resistive wall, 
indicated that the resistive wall can be made to be $90\%$ effective to 
$n=1$ resistive wall modes by covering the resistive wall with $7$ segments 
of poloidal coils of equal poloidal coverage. Here, the effectiveness of the 
coil arrangement is defined as 
$$Eff = \frac { \delta W - \delta W_{\eta_w=\infty}} {\delta W_{\eta _w=0}
    - \delta W_{\eta _w=\infty}} $$   
For the present configuration, $\delta W_{\eta_w=0}$ is approximately 0, 
i.e. the mode is marginally stable with an ideal wall, and 
$\delta W_{\eta _w=\infty}$ is around -0.35. These results suggest that the
reference baseline case (which is at 90\% of the ideal wall limit) is
stable to the n=1 RWM's with an intelligent shell.


%xxxxxxxxxxxxx


\subsubsection{Evaluation of Resistive Wall Mode Control Coil Requirements 
for ARIES-AT} % old 2.7.3
The $n=1$ resistive wall mode (RWM) is an ideal kink which occurs when beta 
exceeds the no-wall beta limit but is less than the ideal-wall beta limit.  
As any real wall is not ideal but resistive with an effective wall time 
$\tau_W$, the RWM can grow but is slowed to a growth rate 
$\gamma \leq \tau_W^{-1}$ and to a rotation frequency 
$\omega \leq \tau_W^{-1}$ by the resistive wall.  The 
RWM manifests itself as an $n=1$ radial field $\tilde{B}_R$ coming through 
the wall. The slowing down of the mode growth makes stabilization tractable 
by feedback control with an external (to the vacuum vessel) coil which 
opposes the change in $\tilde{B}_R$ at the wall, i.e., acts to 
make the wall behave as a perfect conductor~\cite{Boozer99,Cates00}.  
Experiments in the tokamaks HBTX at Columbia University and DIII-D at 
General Atomics are implementing such a feedback control.  The DIII-D vacuum 
vessel and control coils (as of now) are shown in Figure~\ref{f2-73-1}.

\begin{figure}[ht]
%\includegraphics*[bb=103 417 396 719,height=4.5in,clip=]{./f2-73-1.ps}
\includegraphics*[height=4.5in,clip=]{./f2-73-1.ps}
\caption{Existing geometry of DIII-D ``C-Coil'' and vacuum vessel (courtesy Jim Bialek).}
\label{f2-73-1}
\end{figure}

A conceptual feedback coil for ARIES-AT is shown in Figure~\ref{f2-73-2}.
It is based on the ``C-Coil'' in DIII-D~\cite{Garofalo99}.  Assuming the 
copper vertical stabilizing shells act to inhibit $n=1$ flux penetration, 
the feedback coil must oppose any $n=1$ flux leakage through the 
vacuum vessel at the outboard midplane.  It is assumed that an array of 
integrated ``saddle loop'' sensors will be placed on the inside vessel wall 
to detect the mode and act in the feedback loop.  The coil top and bottom 
are at   $Z_C=\pm1.45 m, R_C=7.5 m$, to subtend a simple poloidal angle of 
$60^{\circ}$ from the plasma axis; this is about 1/2 a poloidal wavelength 
for the dominant $m=3$ component.  A set of eight ``windowpane'' coils, 
subtending $45^{\circ}$ wide each toroidally, covers the torus and fits into 
the $N=16$ symmetry of the TF-coil.  These would be hooked up into four pair 
with $180^{\circ}$ opposite coils connected in anti-series for $n$ odd with 
$n=1$ dominant.  Thus four independent power supplies are required.  (A 
configuration of sixteen-$22.5^{\circ}$ wide coils making up an equivalent 
ensemble might be desirable to avoid port obstructions.)  The present design
only considers modes with n=1.  Resistive Wall Modes with $ n > 1$ have not been
observed in DIII-D.

\begin{figure}[ht]
\centering
%\includegraphics*[bb=103 417 396 719,height=4.5in,clip=]{./f2-73-2.ps}
\includegraphics*[height=5.5in,clip=]{./f2-73-2.ps}
\caption{Cross-section of ARIES-AT power core configuration showing the proposed RWM control coil.}
\label{f2-73-2}
\end{figure}


In DIII-D, modes are detectable at the level  $\left|\tilde{B}_R\right| 
\approx 1 G$ in a discharge with axial toroidal field $B_{T0}$ of $2 T$ or 
about $5 \times 10^{-5} $of $B_{T0}$ .  The C-Coil in DIII-D can produce 
field at the sensor up to 50 times this.  Thus each (of four pair) RWM coils 
in ARIES-AT must make about 
$50 \times 5 \times 10^{-5}\times 5.9 \times10^{4} G \approx 150 G $
at the outboard midplane vacuum vessel.  The necessary current is about
$I_C \approx \pi Z_C  \left|\tilde{B}_R\right|/ \mu_0 = 50 $ kA-turns.  
The number of turns and bandwidth of the four independent power supplies 
depends on the necessary frequency $\omega$ which in turn depends on the 
wall time $\tau_W$ .  Neglecting the radially thick Li-Pb blankets of 
relatively high resistivity $(1.2\mu\Omega-m)$ which have Li-Pb in SiC 
channels and also neglecting the copper vertical stabilizing shells, one 
assumes the same geometry and material vacuum vessel as DIII-D with 
$\tau_W$ scaled up by size squared from 5 msec in DIII-D to about 20 msec 
in ARIES-AT.   Requiring a linear amplifier with a voltage capable of 
driving $\pm I_C / N$ up to $\omega\tau_W \approx 3$
$(\omega / 2\pi \approx 25  Hz)$ sets $V_C \approx \omega L (I_C / N)$  
where $L$  is  the inductance of one pair in anti-series, $N$  is the number 
of turns, we neglect the coil resistance and any cabling impedance, and we 
neglect any current induced in the vessel.  Now  $L\approx N^2\mu_0R_C$ 
per pair and $ V_C \approx 3 N \mu_0R_CI_C/\tau_W \approx 300 V $
for $N=4$  with $I_C/N \approx 13 kA$ or $0.5V_C(I_C/N) \approx 2 MW$ 
each of four supplies (linear amplifiers).  Careful design to minimize 
cabling impedance and coil resistance is assumed here.  Such linear 
amplifiers are of order of what is operational on DIII-D.

Requirements for the coil and the power supplies/linear amplifiers are 
summarized in Table~\ref{t2.73-1}.  A further requirement on the linear 
amplifiers is negligible phase shift up to $ \omega \tau_W \approx 10$ so 
that the closed loop feedback system does not go unstable.

\begin{table}
\centering
\begin{tabular}{|l|l|} \hline

\# coils                                & $16^\circ - 22.5^\circ$ wide in $\phi$, $60^\circ$ wide in $\theta$, outboard \\ \hline
$N$ turns                              & 4                                                             \\ \hline
$R_C,Z_C$                              & $7.50 m,\pm 1.45 m$                                           \\ \hline
$\left|\tilde{B}_R\right|$ at vessel   & $150 G$                                                       \\ \hline
$I_C$                                  & 50 kA-turns                                                   \\ \hline
$\omega\tau_W$                         & $\leq 3$ (so $f\leq 25 Hz$ for $\tau_W \approx 20$ msec)      \\ \hline
Resistance $R$                         & $17 m\Omega$ (each of four sets, 2.6 cm OD Cu turns)          \\ \hline
$L_C = N^2\mu_0R_C$                    & $9 N^2 \mu H$                                                 \\ \hline
$I_C/N$                                & 13 kA from each supply (of four)                              \\ \hline
$V_C = 3 N \mu_0R_CI_C/\tau_W$         & 300 volts from each supply (of four)                          \\ \hline
$(I_C/N)V_C/2$                         & 2 MW from each supply (of four), reactive power               \\ \hline
$(I_C/N)^2R/2$                         & 1.5 MW (each of four), dissipated power                       \\ \hline
$\delta_T/\delta_t$                      & $1.5^\circ C$/sec (no cooling assumed)                        \\ \hline
\end{tabular}
\caption{ RWM Feedback Coil Requirements}
\label{t2.73-1}

\end{table}

An additional use of the RWM Coil would be for error field correction, 
the original purpose of the C-Coil in DIII-D.  Asymmetries in winding and 
positioning PF and TF coils produce resonant $n=1$ magnetic field 
non-axisymmetry which can slow rotation, destabilize $n=1$ RWMs, or 
lead to locked modes.  As in the DIII-D C-Coil, the proposed 
RWM Coil can be operated in a duplex fashion to both correct ``dc'' error 
field and to feedback stabilize ``ac'' resistive wall modes.

\subsection{Neoclassical Tearing Mode Control Requirements} % old 2.8
\label{s2.4}
The neoclassical tearing mode (NTM) island results when: (1) the free 
energy $\Delta ^{\prime}$ available in the current profile is negative, 
i.e., stabilizing, (2) the bootstrap current is helically perturbed 
reinforcing the perturbation, and (3) the metastable plasma without an 
island, is sufficiently perturbed above a threshold so that the island 
grows and saturates~\cite{LaHaye00} .  

In an AT plasma without sawteeth or fishbones, ELMs are a likely cause of 
seed perturbations to excite resonant $q=m/n$ mode such as $m/n = 5/2$ at 
$\rho = 0.92$ in ARIES-AT~\cite{g00509.0*540}.  The large bootstrap 
current can produce a very large island if excited and allowed to grow.  
However, current profile control with radially localized off-axis 
co-electron cyclotron current drive (ECCD) or lower hybrid current drive
(LHCD) could suppress the island by 
either replacing the ``missing'' bootstrap current~\cite{Grantenbein00} or 
by making $\Delta ^{\prime}$ more negative~\cite{Pletzer98}.

Stability is determined by analyzing the modified Rutherford equation 
including the effect of replacing the ``missing'' bootstrap current,
$J_{rf} / J_{bs}$ term, and with 
the ``polarization'' threshold $\omega_{pol}$ term~\cite{LaHaye00}:

\begin{equation}
\frac{\tau_R}{r}\frac{dw}{dt} 
 = \Delta^{\prime} r + \sqrt{\epsilon} (L_q / L_p)
   \beta_{\theta} [ \frac{r}{w} - \frac{r w_{pol}^2}{w^3} 
 - \frac{8 q r \delta}{\pi^2w^2}
                   \frac{\eta J_{rf}}{J_{bs}} ].
\label{e2.4.1} 
\end{equation} 

Here $J_{bs}= \sqrt{\epsilon} p / (L_q B_{\theta)} $ , 
$J_{rf} = I_{rf}/(2\pi r\delta) $ , $\delta$ is  the full
radial width half maximum of the rf current, and 
$\eta=\eta_0 / (1+2\delta^2/w^2) $ with perfection position on 
island assumed.  The unstable region is bounded in $\beta_{\theta} - w$ 
space  for $\dot{w} \equiv 0 $ , as given by

\begin{equation}
\beta_{\theta} = \frac{-\Delta^{\prime} r}{\sqrt{\epsilon}(L_q/L_p)}/ [\frac{r}{w} - \frac{rw_{pol}^2}{w^3}
               - \frac{8qr\delta}{\pi^2w^2}\frac{\eta J_{rf}}{J_{bs}}]
\end{equation}

 
Evaluation of the unstable region for the ARIES-AT equilibrium is shown in 
Figure~\ref{f2-8-1}.  For $m/n = 5/2$ , $\Delta^{\prime} r = -10 $  is very 
stable~\cite{Gianakon??}.  However, without rf, an unacceptably large island, 
$w/r \sim 0.5$  at $\beta_{\theta} = 2.3$ , could be excited.

The feasibility of using radially localized ECCD to replace the missing 
bootstrap current, at fixed $\Delta^{\prime}$ , is also shown in 
Figure~\ref{f2-8-1}.  This is seen to be ineffective even for 
$I_{rf}/I_p = 0.1$ (assuming $\delta = \sqrt{3}w_{pol} \approx 6 cm < w$  
for good efficiency of $\eta = 0.5$ ) as $J_{rf}/J_{bs}= 0.3$  
and about $J_{rf}/J_{bs}\approx 1 \sim 2 $ is 
needed, this corresponds to a large ECCD requirement. 
($I_{rf}/I_p \approx 0.3 \sim 1.0 $ ).

Allowing the equilibrium current density to be modified by the radially 
localized ECCD or LHCD so as to make $\Delta^{\prime}$  more negative, improves
the situation greatly.  As shown in Figure~\ref{f2-8-1}, it requires  
$\Delta^{\prime} r < -50 $ to limit the mode at $\beta_{\theta} = 2.3 $.
The feasibility of this is currently being investigated.

\begin{figure}[ht]
\includegraphics*[height=4.5in,clip=]{./f2-8-1.ps}
\caption{Unstable region $\dot{w} > 0$ , shrinks slightly at $\Delta^{\prime} r = -10$ with a very large
rf current applied (10\% of $I_P$ ) or more effectively if $\Delta^{\prime} r$  is made more negative.}
\label{f2-8-1}
\end{figure}

It seems clear that a combination of current profile modification and
a replacement of the missing bootstrap current is most effective.
Future work should include: (1) equilibrium constructions with local current 
density bumps such as in Reference~\cite{Pletzer98} for evaluation of 
$\Delta^{\prime}$ with ``TEAR'' or PEST-III, 
(2) analysis of the validity of Equ.~\ref{e2.4.1} under the ARIES-AT
conditions,  (3) evaluation of rf power and launcher 
requirements for a multi-MA off-axis ECCD system, and (4) analysis of the
feasibility of producing the required localized current profile modifications
with a LHCD system.

\subsection{Edge Stability and Edge Localized Modes (ELMs)} % old 2.9
\label{s2.5}

In addition to low toroidal mode number (low-$n$) global kink modes and 
high-$n$ ballooning instabilities, high performance AT plasmas can also be 
subject to edge localized MHD instabilities in the intermediate range of 
toroidal mode numbers ($3<n<30$).  These instabilities are driven by a 
combination of the sharp pressure gradients and the resulting high bootstrap 
current densities in H-mode type edge profiles.  The characteristic mode 
structure shows a combination of ballooning, peeling, and kink-like 
features.  Depending on radial mode width and other characteristics, these 
intermediate-$n$ edge instabilities can be associated with small benign 
ELMs, large ELMs which pose potential heat load issues for the inner wall 
and divertor, and broader edge instabilities which can inhibit core 
performance.

An analysis of intermediate-$n$ edge instabilities has been carried out 
using the ELITE MHD stability code~\cite{Wilson99}.  ELITE solves the edge 
ballooning equations, which incorporate the effects of edge 
current and a proper treatment of the plasma-vacuum boundary. The standard L-mode 
ARIES-AT equilibrium has been analyzed with ELITE and 
found to be stable to edge modes over a wide range of intermediate mode 
numbers ($10<n<25$).  This result is consistent with expectations, because 
the equilibrium has low current density and pressure gradients in the edge 
region.

The H-mode ARIES-AT equilibrium is much closer to edge instability 
boundaries.  Though the equilibrium is stable to pure ballooning modes, the 
additional ``peeling'' drive provided by the finite edge current drives this 
equilibrium close to the marginal point for intermediate $n$ stability.  
ELITE finds that modes with a radial structure similar to that 
shown in Figure~\ref{f2-9-1} are marginal at certain intermediate values of 
$n\sim15$.  These localized instabilities may lead to ELMs as the edge 
gradients are driven up by transport from the core.  

While much recent progress has been made in understanding ELMs, determination
of the precise ELM structures and dynamics to be expected in ARIES-AT will 
require further advances in physics understanding.

\begin{figure}[ht]
\includegraphics*[height=4.5in,clip=]{./f2-9-1.ps}
\caption{The radial eigenmode structure of a marginally stable $n=16$ mode 
is shown as function of normalized poloidal flux.  The mode structure is 
calculated by the ELITE 
ideal MHD stability code using an H-mode ARIES-AT equilibrium.}
\label{f2-9-1}
\end{figure}

\section{Current Drive} %4.0 to be supplied by TKM
\label{s4.0}

\subsection{\bf Overview}
\label{s-4-1}

One of the major physics considerations when designing ARIES-AT is 
how to maintain the high-performance target equilibrium with
a minimal amount of external power, which enhances
the attractiveness of the power plant.  At the present level
of physics understanding, the key lies in our ability to access
that class of equilibria where a large portion ($\sim$90\%) of the 
plasma current is self-driven due to the neoclassical bootstrap 
effect.  Target equilibrium with high values of ($\beta_{N}$) 
and with high bootstrap fraction have been identified 
in Section 2.  These equilibrium require external current drive
to maintain the required current profile.

We have used a reference ARIES-AT equilibrium to determine the current
drive requirements for a number of operating points.  This equilibrium
has the following parameters:  $R\,=\,5.2\,$m, $a\,=\,4\,$m,
$\kappa\,=2.2$, $\delta\,=\,0.84$, $B_{o}\,=\,5.9\,$T, 
$I_{p}\,=\,13.2\,$MA, $\beta\,=\,9.1\%$, and $\beta_{N}\,=\,5.4$.
In Fig.~\ref{f-4-1}, the current drive scenario is shown for a design point
with the plasma parameters: $n_{eo}\,=\,2.93\times 10^{20}\,$m$^{-3}$,
$n_{o}/<n>\,=\,1.36$, $T_{eo}\,=\,26.3\,$keV, $<T_{e}>_{n}\,=\,18.25\,$keV,
and $Z_{eff}\,=\,1.9$ using Neon as the impurity species.
Two radio frequency (RF) systems are used to drive the seed currents 
in order to maintain the target equilibrium current profile.  Plotted
are the intrinsic and externally driven components of the plasma 
current in the toroidal direction.  About 91\% of the plasma current
is self-driven, comprising the bootstrap current (87\%) and
the diamagnetic current (4\%).  The bulk (1.1$\,$MA) of the seed current is 
driven in the outer part of the plasma ($\rho > 0.8$) by 
waves in the lower hybrid (LH) frequency range, while a small 
on-axis component (0.15$\,$MA) is driven by fast magnetosonic waves
in the ion cyclotron range of frequencies (ICRF).  As indicated
in Fig.~\ref{f-4-1} the target equilibrium current profile is
well reproduced by the sum of the intrinsic and RF-driven currents.
In this optimized scenario, the total RF power requirement is determined 
to be 41.6$\,$MW, with 36.9$\,$MW in the LH system and 4.7$\,$MW delivered
by the ICRF system.
\begin{figure}[ht]
\includegraphics*[height=4.5in,clip=]{./cur1.eps}
%\vspace*{12.0cm}
\caption[Fig-4-1]
 {Toroidal components of the boostrap (B), diamagnetic (D), 
RF-driven (C) current densities vs. normalized
toroidal flux for a typical
ARIES-AT current drive scenario.  The equilibrium current profile (E)
is well reproduced by the driven current (T).}
\label{f-4-1}
\end{figure}
  
In the remainder of this section, we discuss the rationale for 
selecting the two RF current drive schemes for ARIES-AT, and how we 
determine the power requirement in each system.  During the early stages 
of this design study, we have also studied alternate ways to drive the seed
currents, namely, high-harmonic fast waves (HHFW), neutral beam injection (NBI)
and electron cyclotron (EC) waves.  The power requirements for these 
techniques will be compared to the reference systems
in the context of possible backup application to ARIES-AT. 


\subsection{\bf ICRF Fast Wave System for On-Axis CD}
\label{s-4-2}
  
The requirement for on-axis seed current drive in an ARIES-AT equilibrium
is quite modest, with about 0.15$\,$MA (or 1.1\% of $I_{p}$)
to be driven within the region $\rho < 0.6$, where $\rho$ is the
normalized toroidal flux.  Because the required power will be modest,
the current drive efficiency is not a crucial consideration.  However,
a broad deposition profile peaked on axis will be required, and the 
usual criteria of high source efficiency, adequate experimental data base and 
auxiliary heating role during startup have been taken into consideration.  

The two most viable wave candidates for this purpose would be ICRF
fast waves and EC waves.  On-axis current drive has been demonstrated on 
a number of tokamaks for both ICRF~\cite{Pet99} and EC~\cite{Luc99} 
techniques.  
Fast wave has a natural tendency to focus
towards the density peak on axis where the wave energy is absorbed and 
a current is driven.  It uses conventional technologies in
the form of tetrodes and coaxial transmission lines, with a projected
power delivery efficiency of 75\%.  On the other hand, by a suitable
choice of frequency (e.g., $f = 2f_{ce}$ on axis), EC wave can drive current
on axis in a localized fashion.  However, considerable development in the 
gyrotron source technology~\cite{Thu97} will be needed in order to 
reach steady-sate power efficiencies close to those found in the 
ICRF system.  For ARIES-AT, we have 
retained ICRF fast waves as the reference on-axis current driver as
in previous design studies of ARIES-I~\cite{Mau94} and ARIES-RS~\cite{Jardin97}.
This selection is based primarily on the projected higher power efficiency
of the ICRF system.  As its system performance continues to improve, the
EC current drive technique should deserve serious consideration in
the next generation of tokamak power plant studies.

Fast waves drive a current by pushing electrons along the magnetic
field line when the Landau resonance condition: $\omega-k_{||}v_{||}=0$
is satisfied, where $\omega$ is the wave radian frequency, and $k_{||}$
and $v_{||}$ are the wavenumber and electron velocity, respectively, along
the total magnetic field direction.  When wave power is launched 
preferentially in one toroidal direction, electrons are accelerated
asymmetrically giving rise to a net driven toroidal current.  To
optimize the current-drive (CD) efficiency, we need to choose wave
parameters that ensure strong electron damping in one radial transit
while minimizing or avoiding competing wave absorption by ions.

A prudent choice of the wave frequency is the key to avoiding absorption
by thermal fuel ions and by energetic alpha particles.  Shown in 
Fig.~\ref{f-4-2} are the cyclotron frequencies in the first few
harmonics range for deuterium, tritium and helium along the mid-plane
major radius.  Noting that the single-pass electron absorption will
be strong in the ARIES-AT regime, the range of acceptable frequencies 
is determined by the absence of cyclotron resonance on the out-board side 
of the magnetic axis while allowing for ion resonances towards the 
in-board edge of the plasma cross section.  As displayed in Fig.~\ref{f-4-2},
four frequency candidates, namely, 22, 68, 96 and 135$\,$MHz, have been
identified.  We further note that our choice of the folded waveguide
as the launching structure and the need to minimize the radial thickness
of the resonant cavity in order to fit into the blanket region dictate
higher frequency (shorter wavelength) operation.  These engineering
constraints lead us to the
choice of 96$\,$MHz as the operating frequency for the fast wave
system, which places the fourth tritium harmonic outside the out-board
edge and the second deuterium (helium) and third tritium harmonics far beyond 
the magnetic axis on the in-board side.  The 22$\,$MHz option is not
selected because the required antenna size will be incompatible with 
fusion core constraints and the data base is sparse, even though this
option totally avoids ion absorption.  On the other hand, 
for the 135$\,$MHz option, it is 
found that electron damping is so strong due to the relatively high
electron $\beta$ that the deposition and driven 
profile become very broad and even hollow, making it
unsuitable for on-axis seed current drive.     
\begin{figure}[ht]
\includegraphics*[height=4.5in,clip=]{./freq1.eps}
%\vspace*{12.0cm}
\caption[Fig-4-2]
 {Cyclotron frequencies for D, T and He over the mid-plane major
radius for the first few harmonics denoted by number in front of
species.  Dashed horizontal lines indicate frequency options for
fast wave current drive.}
\label{f-4-2}
\end{figure}

The CURRAY ray tracing code~\cite{Mau92} is then used to determine the launch
location on the out-board edge and the launched wave spectrum to
obtain the best match of the driven current profile to that of the
central seed current.  We used fifteen rays to represent the
launched wave spectrum with a cosine-squared dependence along the
poloidal extent of the waveguide mouth.   We found that launching from 
the out-board mid-plane with a spectrum peaked at $N_{||}=2$ gives the best
matching, as indicated in Fig.~\ref{f-4-1}.  Since the wave model
used in CURRAY is linear, by adjusting the incident power level,
we determine 4.7$\,$MW of power is required to drive
the 0.15$\,$MA on-axis seed current, thus yielding a CD efficiency of 
0.032$\,$A/W.  We have not attempted matching the current profiles
exactly because it will result in an unrealistic antenna spectrum.  
In Fig.~\ref{f-4-3}, we show the wave power
propagation pattern, represented by the fifteeen rays, projected onto
the minor cross section and when viewed from the top of the torus.
Strong single-pass absorption is evident from Fig.~\ref{f-4-3}(a)
and the wave is confined to the vicinity of its toroidal launch
location.  
\begin{figure}[ht]
\includegraphics*[height=4.5in,clip=]{./rays.eps}
%\vspace*{12.0cm}
\caption[Fig-4-3]
 {Fast wave ray trajectories projected on (a) minor cross section, and
(b) top view of torus.}
\label{f-4-3}
\end{figure}

In the CURRAY code, electron absorption is calculated 
with the inclusion of electron Landau damping and transit-time magnetic
pumping, while fuel ion absorption is computed for the harmonic cyclotron
damping processes.  For the ARIES-AT parameters, the ray trajectories
are calculated using the cold ion approximation, while the ion damping
model includes the fully kinetic finite-Larmor-radius effect.  
Cyclotron damping by the energetic alpha particle damping is calculated 
using an equivalent 
Maxwellian temperature for the slowing down distribution function in 
the absence of RF-induced velocity diffusion.  The corresponding density
profile is evaluated from the local fusion reaction rate.  
In the example case presented here, it is found that 99.4\%
of the launched power is aborbed by electrons, with the rest
going to alpha particles and tritium.  So there is essentially no ion
absorption of the wave in this reference scenario.
 
\subsection{\bf Lower Hybrid System for Off-Axis CD}
\label{s-4-3}

For the reference ARIES-AT equilibrium presented in Sec.~\ref{s-4-1}, 
the bulk of the seed current must be driven off axis,
$0.8<\rho<1$, with a profile peaked at $\rho\approx 0.85$.  We have
examined a variety of radio frequency CD options including LH waves,
EC waves and high harmonic fast waves (HHFW).  Of these, lower hybrid
waves have been studied most extensively, with a large experimental data 
base~\cite{Pey99} that includes demonstation of off-axis drive to 
reach a reverse-shear
configuration on Tore Supra and sustainment of hour-long discharges on
TRIAM-1M~\cite{Ito99}.  In reactor-grade plasmas, the wave typically 
propagates into the plasma and becomes totally damped before reaching 
the plasma core.  Because of the strength of the damping, the deposition 
profile tends to be highly localized in space, making it quite suitable 
for profile control.
Since the wave is resonantly absorbed by super-thermal electrons
($\omega/k_{||}v_{e}\simeq 3$), its CD efficiency is highest amongst
all RF techniques.  For off-axis drive, because of
the paucity of trapped electrons in the resonant velocity spectrum, 
degradation in CD efficiency due to neoclassical effects will be minimal.

Fast waves in the high harmonic regime ($f/f_{ci}\sim 20$) can also be 
considered for off-axis drive as suggested in ARIES-RS~\cite{Jardin97}.
Data base for 
this wave is very sparse, even as recent HHFW heating results on
the NSTX spherical tokamak~\cite{Kay00} were reported, indicating
electron heating in the plasma $\beta$ range of
5-10\%.  The wave damps on electrons with $\omega/k_{||}v_{e}\sim 2$,
thus allowing deeper penetration into the plasma core than LH waves.  
However, in the core region, energetic alphas are a source of competing 
wave damping mechanism which effectively reduces absorption by electrons.
Also, there is only modest degradation in CD efficiency due to particle 
trapping.  The deposition profile for a single launched
$k_{||}$ wave spectrum will be broad making it difficult for detailed
profile control.  Our calculations for ARIES-AT indicated that, for
the highest CD efficiency, the current would be driven in the mid-plasma
region, i.e., $\rho < 0.8$, which is not where the off-axis seed
current is located.  This point will be further explored in a 
following subsection.

We have also investigated the possibility of using electron cyclotron
waves to drive off-axis current on ARIES-AT.  This has recently been
demonstrated on DIII-D.~\cite{Luc99}  As previously mentioned,
sources (gyrotrons) in the EC regime are comparatively less efficient than
those (klystrons) in the LH and HHFW frequency ranges, which makes the
EC wave less attractive, particularly when a substantial amount of
current needs to be driven.  In addition, for off-axis drive near the 
outer plasma periphery, where the trapped particle population is high, 
the EC current drive efficiency will likely be strongly degraded.  
Based on this qualitative argument, we have excluded EC waves from 
consideration for off-axis drive on ARIES-AT.

We determine the LH wave system requirements by running the CURRAY code
at the slow branch of the wave dispersion relation.  
Because the off-axis seed current has a relatively 
broad profile, we found that it is necessary to launch a number of wave 
spectra, each centered at a different $N_{||}$, in order to achieve the 
desired profile.  Usually, the
wave frequency is determined by requiring total elimination of energetic
alpha particle interaction.  This usually leads to a relatively high
frequency and short wavelength, particularly for edge CD with high
$N_{||}$, making cooling of the corresponding waveguide 
grille launcher extremely difficult.  Here we argue that since 
current is to be driven
in the outer part of the plasma, where the energetic alpha population
should be low due to the low fusion reaction rate and to depletion from
ripple-induced drift and/or Larmor orbit losses, we may lower the
frequency requirement without suffering large wave absorption through the
alphas.  As a result, we are able to use 3.6 GHz for those lower-$N_{||}$ 
spectra that penetrate deepest into the plasma and 2.5 GHz for
those that drive currents just inside the separatrix.

First, we generate a core accessibility diagram in terms of the
window of accessible local $N_{||}$ along
the midplane major radius, using the prescribed $n_{e}$ and $T_{e}$
profiles for the equilibrium, as shown in
Fig.~\ref{f-4-4}.  For the LH wave to be accessible to
a radial location inside the plasma, the local $N_{||}$ must
satisfy the condition: 
$[1+(f_{pe}/f_{ce})^{2}]^{1/2}+f_{pe}/f_{ce}\ <\ N_{||}\ <\
7/T_{e}^{1/2}$(keV).  For a typical ARIES-AT equilibrium, we found that
the LH wave can penetrate as deep as $\rho\approx 0.8$, according to this
simple 1-D slab model.  This turns out to agree well with ray tracing
calculations even when the LH rays propagate off the midplane.  
\begin{figure}[ht]
\includegraphics*[height=4.5in,clip=]{./access.eps}
%\vspace*{12.0cm}
\caption[Fig-4-4]
 {Lower hybrid accessibility $N_{||}$ window for a typical ARIES-AT
equilibrium, and evolution of $N_{||}$ along LH ray paths for
the launched spectra given in Table~\ref{t-4-1}.}  
\label{f-4-4}
\end{figure}

To determine the required launched LH spectrum, we scan a range of
launched $N_{||}$ and poloidal locations on the outboard edge in order to
maximize the radial penetration and CD efficiency.  Wave absorption by 
energetic alphas is modeled using the unmagnetized approach, where the 
resonance condition is given by $v_{\alpha}\ =\ \omega/k$.  Launching
lower $N_{||}$ results in power deposition and current drive in
deeper radial locations, and vice versa.  However, at the
deepest deposition location, we found a small amount (~1\%) of alpha particle
wave absorption, and none as the deposition moves radially outward
as the launched $N_{||}$ is increased.  For the reference scenario,
we found that it is necessary to launch a broad $N_{||}$ spectrum in
order to drive the off-axis seed current profile as indicated in
Fig.~\ref{f-4-1}.  Details of the launched spectra are 
given in Table~\ref{t-4-1}, with the driven 
current profiles for the five $N_{||}$ components and their sum displayed in 
Fig.~\ref{f-4-5}.  
\begin{figure}[ht]
\includegraphics*[height=4.5in,clip=]{./LHCD.eps}
%\vspace*{12.0cm}
\caption[Fig-4-5]
 {Current profiles driven by various $N_{||}$ components of the 
launched spectrum.}
\label{f-4-5}
\end{figure}

\begin{table}
\caption{}
\centering
{\large \bf Lower hybrid current-drive system parameters}
\vspace{1.cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{c c c c c c}
\hline
Freq. (GHz) & $N_{||}$ & $\theta^{o}$ & I/P (A/W) & Power (MW)
& $I_{cd}/I_{seed}$  \\
\hline
3.6 & 1.65 & -90 & 0.053 & 3.06 & 0.15 \\
3.6 & 2.0 & -90 & 0.049 & 4.40 & 0.2 \\
3.6 & 2.5 & -90 & 0.039 & 8.22 & 0.3 \\
3.6 & 3.5 & -90 & 0.024 & 8.87 & 0.2 \\
2.5 & 5.0 & -90 & 0.013 & 12.39 & 0.15 \\
\hline
\end{tabular}
\label{t-4-1}
\end{table}

The evolution of $N_{||}$ along the ray trajectories are shown in 
Fig.~\ref{f-4-4}.  It is clear that radial accessibility of the five
launched spectra agrees well with the prediction from the 1-D slab
model.  In particular, the ray started with the lowest $N_{||}$(=1.65)
shows relatively little upshift in magnitude as it propagates inward
until it encounters a mode conversion point at the lower accesibility
limit, from where it turns back before total absorption by electrons.  For
higher launched $N_{||}$, the wave energy propagates inward and
gets absorbed well before it reaches mode conversion zone.  

The electron absorption is calculated assuming weak RF-induced 
diffusion in velocity space, i.e.,
the velocity distribution is taken to be Maxwellian near the resonant
region, $v_{||}/v_{e}\approx 3$.  This assumption can be justified by
noting that the local density is very high 
($n_{e}\sim 0.6\times10^{20}\,$m$^{-3}$), and
the temperature is relatively low ($T_{e}\sim 5\,$keV), so that
thermalization via collisions can be strong.  We further check
the electron damping strength assuming strong RF-induced diffusion
by setting the normalized diffusion coefficient, $D_{rf}/D_{coll}\approx 0.5$,
for which we can approximate the local damping decrement by 
$\gamma_{e}\simeq \gamma_{eo}/(1+\sqrt{2}x_{e}^{3})$,~\cite{Bon84} where
$\gamma_{eo}$ is the Maxwellian damping decrement and 
$x_{e}=\omega/k_{||}v_{e}$.  We found that, for the case of launched 
$N_{||}=1.65$ in Fig.~\ref{f-4-4}, the penetration depth remains
the same, and the deposition profile is peaked slightly inward.
Thus, the LH analysis that we performed should be adequate for the
present purpose of assessing the CD system requirement.

To conclude this section, we point out that off-axis LH current drive 
offers the prospect of stabilizing neo-classical tearing modes (NTM) which
can degrade the achievable $\beta$ limit.  The growth of the island width 
associated with these modes can be suppressed by localized current drive
in the vicinity of a rational-q surface.  This process has been demonstrated
on COMPASS-D~\cite{War99}, where LH current drive at modest levels of power 
(60-70$\,$kW) was shown to have a strong stabilizing effect on the 
m=2, n=1 neo-classical tearing mode.  Reduction of the (2,1) mode amplitude
was clearly observed during the LH pulse accompanied by a rise in
the plasma $\beta_{p}$, and the mode grew back to its original amplitude 
after the LH pulse.  The neo-classical nature of the mode was inferred from
reasonable agreement of the measured island width with that from
an acceptable analytical fit.  A detailed analysis of NTM stabilization
by LH waves for ARIES-AT is beyond of the scope of the present work,
and it should be a subject of future research.
   
\subsection{\bf Backup Current Drive Scenarios}
\label{s-4-4}

In the previous two sections, we have examined a class of
of very high $\beta$ reverse shear equilibria where 
specific plasma profiles have been prescribed to ensure that no
penetration of the LH waves inside $\rho\approx 0.8$ is required.
Under these conditions, only two RF current-drive systems will be
required, as contrasted with three systems required on ARIES-RS.
As the contraints on the profiles are relaxed, we found that seed
current will need to be driven in the mid-radius region, i.e.,
$\rho<0.8$, where the LH wave is found to be inaccessible.  In
this situation, either a third RF system in the high-harmonic frequency
regime can be invoked, or a neutral beam injection system can be used,
in place of LH waves, to drive the entire off-axis seed current.

\subsubsection{\bf Three RF Systems}
\label{s-4-4-1}

The bootstrap current alignment in an equilibrium is sensitive to
the pressure, density and temperature profiles, and to the effective
charge of the plasma.  Current mis-alignment can be caused by
mild variations of the profiles and plasma charge from the target
values which, when un-checked, will result in the equilibrium evolving 
towards a less desirable state.  Therefore, backup systems may be 
necessary to provide current drive in a broader region than is possible 
with the baseline CD systems, in order to continue maintaining the
equilibrium.  We will illustrate this concept by an example.  

Shown in Fig.~\ref{f-4-6} is a plot of the seed current radial profile
for the same equilibrium as in Sec.~\ref{s-4-1}, but with $Z_{eff}$=2.0,
$n_{eo}=2.74\times 10^{20}\,$m$^{-3}$, $T_{eo}=28.2\,$keV, 
$<T_{e}>_{n}=19.3\,$keV, and $n_{e}$, $T_{e}$ profiles adjusted to
eliminate bootstrap overdrive.  We note that the off-axis seed current
is now extended to $\rho=0.75$, and a third CD system is required to
drive the seed current in the region $0.75<\rho <0.8$.  We propose to
use the fast wave in the high-harmonic frequency range ($f/f_{ci}\sim20$)
for this purpose, since it penetrates deeper than LH waves because of its
relatively weaker electron damping.  However, the high-harmonic fast
wave is susceptible to strong energetic alpha damping in this 
mid-radius plasma region as we will show.  

The current drive scenario for this case, therfore, consists of three
RF systems: (1) ICRF fast wave for central drive, (2) HHFW for mid-radius
drive and (3) LH waves for off-axis drive.  The required RF system
parameters are given in Table~\ref{t-4-2}, and the driven currents are
displayed also in Fig.~\ref{f-4-6}, reproducing reasonably well the
target seed current profile.  For the launched HHFW power of 20.1$\,$MW, 
about 34\% is absorbed by energetic alpha particles, which are modeled
as an equivalent thermal species in the CURRAY code.  Lowering $N_{||}$
tends to increase the alpha absorption fraction, and vice versa,
with moderate shifts in the driven current peak location.  The
chosen value of $N_{||}$(=2.5) corresponds to the highest CD efficiency
while driving currents in the required location.
\begin{figure}[ht]
\includegraphics*[height=4.5in,clip=]{./hhfw.eps}
%\vspace*{12.0cm}
\caption[Fig-4-6]
 {Alignment of RF driven current profile with that of the seed
current.  Current profiles driven by the three RF systems are
also shown.}
\label{f-4-6}
\end{figure}

\begin{table}
\caption{}
\centering
{\large \bf Parameters for three CD system scenario}
\vspace{1.cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{l c c c c c }
\hline
System &Freq. (GHz) & $N_{||}$ & $\theta^{o}$ & I/P (A/W) & Power (MW)\\
\hline
ICRF/FW & 0.07 & 2.0 & -15 & 0.034 & 4.0 \\
HHFW & 2.5 & 0 & 0.019 & 20.1 \\
LH & 4 & 1.7 & -90 & 0.057 & 2.94 \\
& 4 & 2.0 & -90 & 0.052 & 3.24 \\
& 4 & 2.5 & -90 & 0.041 & 5.43 \\
& 4 & 4.0 & -90 & 0.021 & 8.17 \\
\hline
\end{tabular}
\label{t-4-2}
\end{table}

\subsubsection{\bf Neutral Beam System}
\label{s-4-4-2}

For the case shown in the previous subsection, we may also consider
replacing HHFW and LH waves with a single tangential neutral beam
injection (NBI) system to drive the off-axis seed current.
The primary motivation for investigating NBI current drive is
that the beam-imparted toroidal momentum also drives a significant
amount of rotation that may contribute to stabilizing the external
kink mode.  As shown in Fig.~\ref{f-4-7}, the deepest penetration 
required for off-axis drive is $\psi=0.35$ where $\psi$ is the 
normalized poloidal flux, which translates to $\rho=0.75$ as in
Fig.~\ref{f-4-6}.  Using the NFREYA neutral beam deposition code~\cite{Fow79},
we found that a beam energy of 120$\,$keV at a pivot angle of
70$^{o}$ is sufficient for the penetration.  With an injected
power of 44$\,$MW, a driven current profile that matches the off-axis
seed is obtained, at a CD efficiency of 0.026$\,$A/W.  Together with
the on-axis fast wave CD power of 4$\,$MW, the total current-drive
power is 48$\,$MW.
In principle, the required NBI power should be even higher because
the NFREYA analysis here has not taken the trapped particle 
effect into account in calculating the fast ion current.  Nevertheless
this investigation indicates that if NBI were used, the CD power would
be higher than using three RF systems.  
\begin{figure}[ht]
\includegraphics*[height=4.5in,clip=]{./beam.eps}
%\vspace*{12.0cm}
\caption[Fig-4-7]
 {Alignment of beam-driven current profile with that of the seed
current.  Equilibrium and plasma conditions are the same as in
Fig.~\ref{f-4-6}.}
\label{f-4-7}
\end{figure}

\subsection{\bf Current Drive Efficiency Scalings}
\label{s-4-5}

While the MHD equilibrium and stability studies in Subsections~2 and 3
provide the optimum range of plasma performance for ARIES-AT, current
drive analysis discussed in this section is aimed at providing the range of
CD power required to sustain the equilibrium at various operating
points, from which an optimum design point can be deduced.  The key
parameter, or figure of merit, for consideration is the normalized
current-drive efficiency, defined as $\gamma_{B}=\bar{n}_{e}I_{p}R/P_{CD}$,
where $\bar{n}_{e}$, $I_{p}$, $R$ and $P_{CD}$ have units of
$10^{20}\,$m$^{-3}$, MA, m, and MW, respectively.  Note that in this
definition, the contribution from the bootstrap current is taken into
account, even though there is no CD power associated with it.  The usual
definition of CD efficiency in the literature is recovered by
$\gamma_{CD}=(1-f_{BS})\gamma_{B}$.  

We note that $\gamma_{B}$ is essentially a function of
temperature $T_{e}$ and plasma $Z_{eff}$, after the density and major radius
are factored out.  The efficiency of the current drive is dependent on
$T_{e}$ and $Z_{eff}$, while the bootstrap fraction is governed by the 
aspect ratio, $\beta_{p}$, $Z_{eff}$ and details of the plasma profiles.
For a fixed equilibrium, self-consistent variation of the ($n,T$) 
profiles and $Z_{eff}$ results in varying $f_{BS}$ and current drive power
requirements.  This gives rise to a scaling of $\gamma_{B}$ with respect
to $T_{e}$ and $Z_{eff}$ for the same equilibrium.  
The role of $Z_{eff}$ dependence is important in determining the 
optimum core radiation power fraction required to lower the divertor
surface heat flux without compromising the overall power plant 
performance.

We have investigated two approaches to obtaining a CD efficiency 
scaling for use in the ARIES-AT systems analysis to determine
an optimum operating point.  In the
first approach, we simply vary the ($n,T$)
profiles and $Z_{eff}$ self-consistently from a reference plasma
parameter set for a fixed equilibrium, while avoiding localized bootstrap
current overdrive.  LH and ICRF power requirements are then determined
for a range of relevant $T_{e}$ and $Z_{eff}$ values.  As such, 
the bootstrap fraction varies for each of these points and may not 
be optimum.  In Fig.~\ref{f-4-8}, we show the resultant normalized 
CD scalings versus $T_{e}$ for $Z_{eff}$=1.6,
1.7 and 1.8.  As can be seen, the CD efficiency
is peaked at $Z_{eff}$=1.7, drops slightly at $Z_{eff}$=1.6, but is
drastically reduced at $Z_{eff}$=1.8.  It is not surprising that
$\gamma_{B}$ optimizes at $Z_{eff}=1.7$ since the bootstrap alignment
is optimized at this value of the effective charge.  It is further
noted that for $Z_{eff}$=1.8, three RF systems (ICRF+HHFW+LH)
are actually required to drive the seed currents, leading to a 
substantially lower $\gamma_{B}$ value. 
\begin{figure}[ht]
\includegraphics*[height=4.5in,clip=]{./CDscale1.eps}
%\vspace*{12.0cm}
\caption[Fig-4-8]
 {Normalized current-drive scalings with plasma temperature and effective
charge for a typical ARIES-AT equilibrium.}
\label{f-4-8}
\end{figure}

Reduction of the divertor heat load often dictates that the plasma
operates at $Z_{eff}\sim 1.8-2.0$.  Therefore, it is crucial that
at each operating point considered the bootstrap alignment is optimized
to minimize the CD power required.  Thus, a series of test equilibria
have been obtained which maintain the same global plasma parameters
such as $\beta$, size and shape, but which have optimum bootstrap
current alignment requiring only two RF current-drive systems (ICRF+LH).  
The resulting CD scalings with $T_{e}$ for three values of $Z_{eff}$
are shown in Fig.~\ref{f-4-9}.  It is clear from the figure
that $\gamma_{B}$ is higher than that shown in Fig.~\ref{f-4-8} because of 
the optimized bootstrap alignment, and only two RF systems are used.  
Again the CD efficiency is peaked at $Z_{eff}$ and drops off from there.
This set of CD efficiency scalings has been incorporated into the systems
analysis to generate the reference design point for ARIES-AT.
\begin{figure}[ht]
\includegraphics*[height=4.5in,clip=]{./CDscale2.eps}
%\vspace*{12.0cm}
\caption[Fig-4-9]
 {Normalized current-drive scalings with plasma temperature and effective
charge for similar equilibria with optimized bootstrap alignment.}
\label{f-4-9}
\end{figure}





\section{Transport Analysis and Energy Confinement}  % 5.0
\label{s5.0}
\subsection{Transport Studies Using GLF23} % 5.1

Previous reactor design studies have predicted the transport for the
device by doing a ``0D'' fit to a database of existing tokamaks.   However,
in recent years an alternative method has become available 
with the advent of drift-wave based transport models. For the ARIES-AT 
design the GLF23 drift-wave based model~\cite{Waltz97} has been used to 
project the transport.  This model has been shown to be about as accurate 
as the empirical scaling in predicting the global energy confinement time 
for a database of tokamak discharges~\cite{Kinsey**} which included both 
L-mode and H-mode discharges.  The GLF23 transport model takes no 
fitting parameters from experiment. 
It has been constructed to fit to a set 
of theoretical calculations. The model computes the linear growth rates for 
drift-wave instabilities (ion temperature gradient modes ITG, trapped 
electron modes TEM, and electron temperature gradient modes ETG). The 
growth rates are found using trial wavefunctions and the approximate 
gyro-Landau fluid (GLF) equations~\cite{Hammett90}. The trial wavefunctions 
were chosen to give a good fit to the exact gyro-
kinetic linear growth rates~\cite{Kotschenreuther95} over a range of local 
plasma parameters. The linear eigenmodes are then used in a quasilinear 
calculation of the fluxes (energy, particle, viscous stress) due to 
turbulence. A model for the saturated turbulence fluctuation level is 
used to give a fit to a limited set of nonlinear GLF turbulence 
simulations~\cite{Beer97}.  For this study the edge values are at the
separatrix and do not influence the results much.  Often, for ELMing H-mode,
we take the boundary at $\rho = 0.9$ and the ``edge'' value is then important.

Thus, the model does not use any fitting parameter to experiment but is only 
fit to more exact theoretical calculations. In principle the GLF23 model has 
greater predictive power when extrapolating to new tokamaks than empirical 
scaling laws. However, the limited range of parameters over which the model 
was fit to theory limits the range of extrapolation which is possible. These 
limitations will be brought up as required in discussing the GLF23 
modeling results for ARIES-AT.  

        Three cases will be discussed. The ion and electron temperatures and 
the toroidal and $E\times B$ velocities are evolved but the density profile 
is held fixed. Thus, the three cases involve variations of the density 
profile. The first has a peaked density profile close to 
that used for the bootstrap current calculation in the MHD equilibrium.  
The surprising thing about this case is that the energy confinement is better 
than the empirical H-mode scaling but there is no edge (or core) transport 
barrier due to $E\times B$ velocity shear. The reduced transport is predicted 
to be due to the large Shafranov shift and the negative magnetic shear.  
This case results in a very good agreement between the MHD stability 
optimized pressure profile and the pressure profile computed from GLF23. 

The second and third cases have flatter density profiles. The main difference 
between them is the boundary condition used for the $E\times B$ velocity at 
the separatrix. For all three cases the fusion power and neutral beam 
deposition was computed using the ONETWO transport code. The steady sate 
temperature profiles were then computed using the XPTOR code, which has 
special numerical methods designed for the GLF23 model. These temperature 
profiles were then used to recompute the sources in ONETWO.  This iteration 
procedure converges quickly. The fusion power was reduced if necessary by 
changing the ion isotope mixture in order to keep the total plasma stored 
energy below the design target. The toroidal rotation profile was also 
computed assuming 50MW of 120Kev co-injected neutral beams with the torque 
density computed in ONETWO. 


\begin{figure}[ht]
\includegraphics*[height=4.5in,clip=]{./fig11.eps}
\caption{Case 1 ion temperature profiles used in the MHD equilibrium (blue) 
computed with the GLF23 model (green) and computed with GLF23 with Shafranov 
shift effects turned off (red).}
\label{f5.1-2-1a} %Fig.1a.
\end{figure}


\begin{figure}[ht]
\includegraphics*[height=4.5in,clip=]{./fig12.eps}
\caption{Electron temperature profiles used in the MHD equilibrium (blue) 
computed with the GLF23 model (green) and computed with GLF23 with Shafranov 
shift effects turned off (red).} 
\label{f5.1-2-1b} %Fig.1b.
\end{figure}


The first case is displayed in Figures~\ref{f5.1-2-1a},\ref{f5.1-2-1b},
\ref{f5.1-2-1c},and
\ref{f5.1-2-1d}. The ion (a) and electron (b) temperatures are
computed to be very close to the temperature profile used in the optimized MHD 
equilibrium. The density profile (c) is peaked. Because the pressure profile 
is close to the high bootstrap fraction equilibrium the bootstrap current is 
well aligned with the total current density (d). The effective ion thermal 
diffusivity is well above the Chang-Hinton neoclassical value over most of 
the profile, except very near the magnetic axis. Thus, 
there is no transport barrier with neoclassical ion thermal transport. In 
fact, the $E\times B$ velocity shear is small compared to the computed ITG 
growth so the plasma is not even close to a transport barrier 
threshold~\cite{Waltz95}. The third curve in Figures~\ref{f5.1-2-1a} 
and \ref{f5.1-2-1b} are the computed temperature profiles without the 
Shafranov shift. 

The Shafranov shift is the main cause of the improved 
transport in this case. The Shafranov shift causes an increase in the 
temperature gradient threshold for the drift waves but does not reduce the 
transport all the way to neoclassical. This has the advantage of giving 
improved transport without a localized steep gradient for MHD instabilities 
to feed off of. The fusion alpha particle heating for a 50\% DT mixture was 
610MW . This had to be reduced to 370MW (with 53MW radiated) in order to 
keep below the target beta. This result is indeed promising 
but must not be taken at face value. The GLF23 model is based on shifted 
circle equilibria in the low beta approximation. These have a much weaker 
Shafranov shift than the ARIES-AT equilibrium. It is known that the GLF23 
model overestimates the Shafranov shift stabilization for large values 
(MHD $\alpha > 1$). The GLF23 calculations were also done in the 
electrostatic approximation. In the present calculations the Shafranov shift 
was multiplied by a factor 0.3. This was needed in order to avoid a 
complete suppression of the drift-wave transport (second stability). When 
all of the drift-wave transport is suppressed the only transport in the model 
for electrons is electron neoclassical. A runaway electron temperature thus 
occurs when all of the drift waves are suppressed. Note that $E\times B$ 
velocity shear generally cannot suppress ETG modes due to their high growth 
rates so the electron transport remains above neoclassical in an $E\times B$
shear induced barrier. The Shafranov shift suppresses all drift waves but may 
not give a second stability regime for drift-waves, as it does for MHD 
ballooning modes. 



\begin{figure}[ht]
\includegraphics*[height=4.5in,clip=]{./fig13.eps}
\caption{Equilibrium q-profile (red) and electron density profile (blue).}
\label{f5.1-2-1c} %Fig1c.
\end{figure}


\begin{figure}[ht]
\includegraphics*[height=4.5in,clip=]{./fig14.eps}
\caption{Equilibrium total current (red), bootstrap current for TOQ equilibrium (blue) and 
bootstrap current for GLF23 computed temperature profiles (green).}
\label{f5.1-2-1d} %Fig1d. 
\end{figure}

The supression of drift wave growth rates for $\hat{S} \leq 0$
is well established experimentally.  The GLF23 model may or may not be a 
faithful representation of this effect.  This is the subject of present day
research and has not yet been subjected to adaquate experimental testing.
Experiments on the physics of the Shafranov shift effect on turbulence in 
tokamaks are needed to test the theory. The peaking of the electron 
temperature near the magnetic axis in Figure~\ref{f5.1-2-1b} is the cause of 
the excess bootstrap current in this region (Figure~\ref{f5.1-2-1d}).
Experience with negative magnetic shear discharges in various tokamaks has 
shown a strong flattening of the electron temperature near the axis. The 
cause of this excess transport is unknown but it has been shown to not be 
due to linear drift-wave instabilities~\cite{Stallard99}. 


The second case is shown in Figures~\ref{f5.1-2-2a},\ref{f5.1-2-2b},
\ref{f5.1-2-2c}, and \ref{f5.1-2-2d}. This case used a modified MHD equilibrium
with a non-zero current density at the separatrix. This has little effect on 
the transport calculations. The density profile Figure~\ref{f5.1-2-2c} was 
also taken to be broader than the first case. This change has a significant 
effect on the computed ion (a) and electron (b) temperature profiles. 
The pressure was below the target beta even with a 50\%  DT ion mixture 
producing 432MW of fusion alpha heating (70MW radiated). The bootstrap 
alignment was also degraded (Figure~\ref{f5.1-2-2c}). This case has a 
boundary condition that the $E\times B$ velocity at the separatrix be equal 
to the value obtained from neoclassical theory just inside the separatrix. 
This is effectively a condition on the poloidal velocity. This boundary 
condition did not result in a significant $E\times B$ velocity shear at the 
separatrix. Thus, there is no transport barrier at the edge in this case. 


\begin{figure}[ht]
\includegraphics*[height=4.5in,clip=]{./fig15.eps}
\caption{Case 2 ion temperature profiles used in the MHD equilibrium (blue) 
computed with the GLF23 model (green).}
\label{f5.1-2-2a} %Fig.2a. 
\end{figure}



\begin{figure}[ht]
\includegraphics*[height=4.5in,clip=]{./fig16.eps}
\caption{Case 2 electron temperature profiles used in the MHD equilibrium (blue) 
computed with the GLF23 model (green).}
\label{f5.1-2-2b} %Fig.2b. 
\end{figure}


\begin{figure}[ht]
\includegraphics*[height=4.5in,clip=]{./fig17.eps}
\caption{Equilibrium q-profile (red) and electron density profile (blue).}
\label{f5.1-2-2c} %Fig.2c.
\end{figure}


\begin{figure}[ht]
\includegraphics*[height=4.5in,clip=]{./fig18.eps}
\caption{Equilibrium total current (red), bootstrap current for TOQ equilibrium (blue) and 
bootstrap current for GLF23 computed temperature profiles (green).}
\label{f5.1-2-2d} %Fig.2d.
\end{figure}


The proper separatrix boundary condition is a delicate and complex issue. 
However, one can ask what the solution would look like for a case that is 
more like an H-mode with an edge layer of suppressed transport. This case is 
shown in Figures~\ref{f5.1-2-3a},\ref{f5.1-2-3b},\ref{f5.1-2-3c}, and 
\ref{f5.1-2-3d}. The $E\times B$ velocity was set to -20Km/sec at the separatrix. 
This produced significant $E\times B$ velocity shear not just at the 
separatrix but well into the plasma edge. The $E\times B$ velocity is not
large enough to completely suppress the ITG mode but it does greatly reduce 
the edge transport by increasing the threshold temperature gradient for the 
ITG mode. The steepening of the edge temperatures is clearly seen comparing 
Figures~\ref{f5.1-2-2a} and \ref{f5.1-2-2b} and Figures~\ref{f5.1-2-3a} and 
\ref{f5.1-2-3b}.

The edge density was also lowered in this case, Figure~\ref{f5.1-2-3c}, but 
this did not by itself generate enough $E\times B$ shear.  The large edge 
bootstrap current for this case is similar to that found for H and VH-modes 
in present tokamaks. Experience with these regimes leads one to 
expect that this current profile would probably be unstable to edge kink 
modes. This is sometimes a benign condition, resulting only in edge localized 
modes which do not seriously degrade energy confinement. The $E\times B$ 
shear layer extends far from the edge more like a VH-mode in this 
case~\cite{Jackson91}. The transport is far too good for the size of this 
reactor design. The fusion power flow needed to sustain this beta was only 
160MW (with 86MW) radiated. Nevertheless, the ion thermal transport is still 
above neoclassical except near the axis. A redesign of the reactor would be 
required in order to take advantage of this regime. 


\begin{figure}[ht]
\includegraphics*[height=4.5in,clip=]{./fig19.eps}
\caption{Case 3 ion temperature profiles used in the MHD equilibrium (green) computed 
with the GLF23 model (blue).}
\label{f5.1-2-3a} %Fig.3a.
\end{figure}

\begin{figure}[ht]
\includegraphics[height=4.5in]{./fig20.eps}
\caption{Case 3 electron temperature profiles used in the MHD equilibrium (green) 
computed with the GLF23 model (blue).}
\label{f5.1-2-3b} %Fig.3b.
\end{figure}


\begin{figure}[ht]
\includegraphics*[height=4.5in,clip=]{./fig21.eps}
\caption{Case 3 electron density profile.}
\label{f5.1-2-3c} %Fig.3c.
\end{figure}


\begin{figure}[ht]
\includegraphics*[height=4.5in,clip=]{./fig22.eps}
\caption{Equilibrium total current density (blue) and bootstrap current for GLF23 
computed temperature profiles (blue).}
\label{f5.1-2-3d} %Fig.3d. 
\end{figure}

In summary, the GFL23 transport model does predict global energy confinement 
which exceeds the ARIES-AT design requirement if the density profile is 
peaked. The computed pressure profile is close to the one found for optimum 
MHD stability. The energy confinement improvement is primarily due to 
Shafranov shift stabilization of drift-waves. The turbulent transport is 
reduced but not completely suppressed by the large Shafranov shift. The 
theoretical basis of this effect is well founded. The exact level of 
transport suppression predicted by GLF23 is questionable however, since the 
Shafranov shift and beta are outside the range of the model's fit to theory. 
This work points to the need to try and produce plasmas in current tokamaks 
which have large Shafranov shifts without large $E\times B$ shear. Even in 
discharges with large enough $E\times B$ shear to quench ITG modes, the effect
of the Shafranov shift on ETG modes could be studied. 

Unexpectedly, the ARIES-AT plasma is quite unlike the better known regimes in 
current tokamaks, which owe their improved transport mostly due to $E\times B$
velocity shear and have relatively small Shafranov shifts. The high 
beta-poloidal regime~\cite{Koide96} is the most similar to ARIES-AT. The 
requirement of high bootstrap fraction has pushed the design in this 
direction. If the density profile is broad with a sharp edge, the target 
energy confinement could only be reached with some help from $E\times B$ 
velocity shear. It is not possible to achieve sufficient toroidal rotation 
for transport improvement or resistive wall mode stability with 
a practical amount of neutral beam power. A method for changing the 
separatrix $E\times B$ velocity (say by differential puffing and pumping) 
could help induce an edge transport barrier. The GLF23 model predicts far too 
high an energy confinement for this case however and the bootstrap current is 
too far to the edge. This plasma is more like the VH-mode in present tokamaks. 

\subsection{Toroidal Rotation Transport Study Using NBI}  %5.2A

The torque density due to 50Mw of 120Kev neutral beams injected in the 
direction of the plasma current has been computed by ONETWO. The toroidal 
rotation due to this torque is computed using GLF23 in the XPTOR code. It 
has been estimated in Sec.~\ref{s2.3} that an $E\times B$ rotation speed of 
at least several percent of the Alfven velocity, or about 500 km/sec at 
the $q=3$ surface is needed to prevent the resistive wall mode 
for the L-mode edge design point (case 1). The $q=3$ surface is very near to 
the edge $\rho=0.98$. This makes it impractical to achieve such a large 
rotation speed since the separatrix toroidal rotation is small and the 
rotation shear is limited by the large edge transport. The $E\times B$ 
velocity at the $q=3$ surface is predicted to be 5km/sec for case 1.  The 
toroidal rotation due to the 50MW of co-NBI was computed for this case but it 
made almost no difference at the q=3 surface. Very little shear is 
generated due to the large momentum diffusivity which for ITG modes is 
comparable to the ion thermal diffusivity. Thus we conclude that toroidal 
momentum injection by NBI is not a practical means of suppressing resistive 
wall modes in ARIES-AT.

%----------------------------------------------------------
%

\subsection{Estimate of Toroidal Rotation Driven by ICRF Minority Heating}%5.2b

Motivated by observations in Alcator C-Mod that plasmas heated by the 
minority ion-cyclotron heating can cause appreciable co-current toroidal 
rotation~\cite{Rice00}, a mechanism is proposed by~\cite{Perkins00} to 
explain the results. This mechanism can drive rotation in tokamak plasmas 
even though the ICRH introduces negligible angular momentum.  It has 
two elements: First, angular momentum transport is governed by a diffusion 
equation with a non-slip boundary condition at the plasma edge.  Second, 
Monte-Carlo calculations show that energized particles will provide a torque 
density source which has a zero volume integral but separated positive and 
negative regions. With such a source the rotation frequency profile is given 
by

\begin{equation}
\Omega(\phi) = \int d \phi \frac{T}{qV^{\prime}} <nMR^2\chi\ddot{U}\psi^2>
\end{equation}

where $T$ is the volume-integrated torque up to flux-surface $\phi$, and 
$\chi$ is the momentum diffusion coefficient. When the resonance is on the 
low-field side, the Monte-Carlo calculation predicts co-current rotation 
consistent with observation. The rotation becomes more counter and reverses 
sign for high-field side resonance according to the simulation, but this has 
not been observed experimentally. Because of this, using this mechanism to 
control rotation should be considered speculative at present. Nevertheless, 
it is of interest to estimate the power requirement for ARIES-AT to produce 
adequate rotation to stabilize resistive wall modes (RWM).

The power requirements for ARIES-AT for two cases corresponding to two 
different resonance locations (low-field side) are shown in Table~\ref{t4.3}.
Figure~\ref{f5-2B-1} shows the rotation profile with the ICRF resonance at 
       $r/a=0.34$.

\begin{table}
\label{t4.3}
\centering
\begin{tabular}{|l|l|l|}\hline
Resonance Location      & $r/a=0.1$ & $r/a=0.34$ \\ \hline

$V(0)/V_A$ & $2.1\times10^{-3}$ & $1.1\times 10^{-3}$ \\ 
           &$P(MW)\tau_E(S)$ &$ P(MW)\tau_E(S)$ \\ \hline

\end{tabular}
\end{table}



\begin{figure}  % Fig. 1
\includegraphics*[height=4.5in,clip=]{./f5-2b-1.ps}
\caption{Rotation profile with the ICRF resonance at $r/a=0.34$ }
\label{f5-2B-1}
\end{figure}

In Sec.~\ref{s2.3} we have estimated the rotation requirement for 
stabilization of RWM to be around 0.1 of the Alfven speed $V_A$.  From our 
result and assuming that the energy confinement time $\tau_E$  is $\approx 1$ 
second, the ICRH power required would be 50-100 MW. Finally, note that the
efectiveness of this mechanism to produce rotational drive is significantly
reduced when the resonant location is at $r/a > 0.5$.

\section{Power and Particle Exhaust}  %6.0
\label{s6.0}

\subsection{General Considerations} %6.1

A major concern in the  designing the ARIES-AT tokamak is excessive heat 
loading on its first wall from both electromagnetic (EM) radiated power and  
direct particle heating.  In the following, we examine the intensity and 
poloidal variation of this heat loading,  based on our present understanding 
of how these heat loading sources are distributed.  Specifically, in 
sections~\ref{s6-1-1} and~\ref{s6-1-2} we discuss the methodologies used in 
evaluating the radiative and particle  contributions to total heat 
flux on the vessel surfaces. In section~\ref{s6-2} we use these methodologies 
to evaluate the heat flux and power loading under various operating scenarios.
In section~\ref{s6-3} we review the efficacy of achieving the required 
radiated power from inside the core plasma, based on the most recent 
transport modeling.

\subsubsection{Radiated Power From Electromagnetic Sources} %s6-1-1
\label{s6-1-1}
An estimate of the poloidal radiated heat flux distribution along the first 
wall of the ARIES-AT vacuum vessel can be evaluated using the RADLOAD program.
(The source and distribution of this radiated power is discussed in 
section~\ref{s6-3} of this report.)  RADLOAD follows a ``multifilament''
approach, where the incident power loading on a wall surface element 
($Q_{EM}$) from a radiating source distribution is given by:

\begin{equation}
Q_{EM} = \int_{V} \epsilon_R \frac{\vec{r}\cdot\hat{n}}{4\pi r^3} dV
\end{equation}

where  $\epsilon_R$ is the radiating emissivity of the filament,
$\vec{r}$ is the vector from the wall segment to the radiating filament,
and $\hat{n}$ is the unit normal vector out of the wall surface element. 

The RADLOAD program assumes toroidal symmetry. The above integral is 
performed numerically over that part of the radiating volume visible to a 
given surface wall element. To do this,  the radiating region is divided 
into ``filaments'' extending around the machine in the toroidal direction 
with each filament denoted by its own set of poloidal coordinates.  The 
limits of integration for each filament are determined by testing whether or 
not the line-of-sight along the filament is visible  to a particular wall 
element.  The heat flux $Q_{EM}$ is  found by summing over the contribution 
of each filament. 


\begin{figure}  % Fig. 1
\includegraphics*[width=5.5in]{./f6-12-1.ps}
\caption{The poloidal locations of the wall segments are shown. Also shown 
are the flux surfaces used to define the ``onion rings''  needed in 
determining the radiating source function.}
\label{f6-12-1}
\end{figure}


Figure~\ref{f6-12-1} divides the vessel wall into ``segments''  for which we 
will evaluate the radiated heat flux (both peak and average). For example, in 
order to estimate the spatial distribution of the radiating power source from 
inside the core plasma, we apply an ``onion-ring'' approach on the equilibrium
shape shown in Figure~\ref{f6-12-1}. The plasma core is divided into four 
regions, each region lying between two specified flux surfaces   and 
having (roughly) similar volumes.  The total radiated power produced would 
be $\epsilon_R$ times the volume between any two (dashed) flux surfaces. To 
calculate the contribution of that ring to the incident radiated power flux 
for a specific poloidal location on the vessel wall, we assume that the power 
produced by the ring is distributed uniformly along a flux surface  
intermediate to the two flux surfaces which defines that specific 
plasma region. Typically, there are 200-300 radiating elements per region 
which must be individually evaluated. This method is repeated for each region.


\subsubsection{Particle Heating in the Divertors} %s6-1-2
\label{s6-1-2}
We assume that the radial profile of the heat flux in the scrape-off layer 
(SOL) has an exponential form. An expression for the peak heat flux at the 
divertor (along with ``base case'' values used in the calculation) can be 
written: 

\begin{equation}
\label{e6-12-2}
Q_{div,S} \approx \frac{ P_{heat} \times (1 - f_{rad}) 
              \times f_{outboard/total} \times f_{\nabla B/total}
                       \times (1 - f_{pfr}) \times \sin{\alpha}}
                       {2\pi\times R_S\times f_{exp}\times\lambda_{q_{||}}}
\end{equation}

where $Q_{div,S}$  is the peak heat flux at the divertor strike point,
$P_{heat}$ is the total heating power,
$R_S$  is the major radius of the outer (inner) divertor strike point, 
$\lambda q_{||}$is the midplane  parallel heat flux scrape-off width,
$f_{exp}$ is the flux expansion at the outer (inner) divertor target,
$\alpha$ is the angle between the divertor incline and the separatrix, 
$f_{rad}$ is the ratio of total radiated power to total input power, 
$f_{outboard/total}  (f_{inboard/total})$ is the ratio of power flowing into 
the outboard scrape-off layer (SOL)  to the power flowing into the outboard 
(inboard) SOL$f_{\nabla B/total}$ is the ratio of power striking the outboard 
divertor in the grad-B direction to the total power striking both the upper 
and the lower outboard divertors, and $f_{pfr}$ is the fraction of power 
flowing into the private flux region.

The peak heat flux predictions of Eq.~\ref{e6-12-2} have been compared with 
DIII-D data and generally found to be  within 20\% of the measured peak heat 
flux at the outboard divertor strike point in attached plasmas. 
The values we use of the quantities in Eq.~\ref{e6-12-2} are based on best 
estimates and extrapolations of available experimental data. For example:
$\lambda_{q_{||}}$: A recent study by~\cite{Loarte99} examined the ITER 
divertor power deposition database of L- and H-mode discharges and derived 
multi-machine scaling laws for heat flux widths based on machine geometry. L-
mode scaling of the heat flux width, based on ``measured divertor power'' 
formulation~\cite{Loarte99}, i.e., 

\begin{equation} %Equ. (3)
\label{e6-12-3}
\lambda_q(m) = 6.6 \times10^{-4} \times R(m)^{1.21} \times q_{95}^{0.59} 
\times 
\bar{n}_e^{0.54} \times Z_{eff}^{0.61} \times P_{div}(MW)^{-0.19}
\end{equation}

can range from $\approx 1.4-2.1 cm$ using ARIES-AT parameters. However, 
$\lambda_q$ is defined by~\cite{Loarte99} as an ``integral power width'', 
instead of the width of the steepest part of the exponential decay nearest 
the separatrix, as we use in Equ.~\ref{e6-12-2} above. We can convert 
$\lambda_q$ to $\lambda_{q_{||}}$ by dividing Equ.~\ref{e6-12-3} by 1.8. Thus,
$\lambda_{q_{||}}$ for our analysis is roughly 0.8 -1.2 cm in L-mode. For 
H-mode, the physics of the scrape-off layer is more complicated (e.g., ELMing)
and Ref.~\cite{Loarte99} arrives at the  expression:

\begin{equation}  %  Equ. 4
\label{e6-12-4}
\lambda_q (m)  = 5.2 \times 10^{-3} \times P_{div}(MW)^{0.44} \times q_{95}^{0.57} \times B_T(T)^{-0.45}
\end{equation}


Equ.~\ref{e6-12-4} suggests that $\lambda_q$ increases with input power. 
Again, using ARIES-AT parameters, $\lambda_q \approx 3.8$ cm, or 
$\lambda_{q_{||}} \approx 2.1$ cm.  In sum, based on the most recent set of 
heat flux width scaling laws, we might estimate a $\lambda_{q_{||}}$  range
of roughly 0.8 - 2.1 cm for the L- and H-mode possibilities for ARIES-AT.  
The  scaling laws for Equ.\ref{e6-12-3} and Equ.\ref{e6-12-4} were developed 
for single-null geometries. Recent studies on DIII-D, however, suggest that 
single-null and double-null $\lambda_{q_{||}}$ are comparable, at least in 
H-mode~\cite{Petrie00}.
 
$f{outboard/total}$: An estimate of this parameter can be obtained from simple 
geometric arguments by noting (1) that the ratio of the outboard plasma 
surface area to the inboard plasma surface area is $\approx 3:1$ for ARIES-AT 
and (2) that the ratio of the radial temperature gradient on the outboard 
side  to that of the inboard side of the plasma is $\approx 2.7:1$. These 
imply that the ratio of the power flowing into the outboard SOL would be 
roughly eight times that flowing into the inboard SOL. This, in turn, would 
mean that $f_{outboard/total}\approx 0.89$. 

$f_{pfr}$: The power ``spillover'' into the private flux region has been 
observed on several tokamaks. For DIII-D, this value is typically 0.10-0.15.

$ f_{\nabla B/total}: $ This value is sensitive to the precise magnetic 
balance near the double-null configuration. In fact, the peak heat flux can 
be equalized only if the DN is biased slightly  toward the direction opposite 
the $\nabla B$ ion drift~\cite{Petrie00}.  In this study, we assume that a 
slight offset from ``true'' DN  balance, such that  $f_{\nabla B/total}  = 0.5$.

\subsection{Divertor and Wall Heat Loads} % 6.2
\label{s6-2}

In this section we apply the methods discussed above to possible operating 
scenarios for  ARIES-AT.  

\subsubsection{Application to the most conservative case}  % 6.2.1
The parameters used in Equ.~\ref{e6-12-2} are summarized in Table~\ref{t6-12-1}
and will be referred to as ``most conservative'' case parameters. The 
divertor legs in this design are inclined at an angle $(\alpha)$ of $10^o$ 
relative to the divertor surfaces in the slot in order to increase the 
effective wetted area and minimize the heat flux. For this ``most conservative''
case, we assume that the 30\% of radiated power source (mostly bremsstrahlung)
is uniformly distributed inside the main plasma. We estimate the peak heat 
flux under the inboard and outboard divertor legs by assuming that the 
radiated power from the scrape-off layer and divertors are negligible. This 
gives an  upper bound estimate of the peak heat flux on the divertors, since 
including SOL and divertor radiated power would result in a reduced heat flux. 



\begin{figure}
\includegraphics*[width=5.5in]{./f6-12-2.ps}
\caption{Estimates of the peak heat flux to the inboard (a) and outboard 
(b) divertors are shown as a function of the core radiated fraction, given 
three plausible parallel heat flux scrape-off widths.}
\label{f6-12-2} % Fig. 2
\end{figure}


Figure~\ref{f6-12-2} shows the peak heat flux for both inboard and outboard
divertors as a function of frad of the core plasma, assuming three scrape-off 
widths.  If we take  $\lambda_{q_{||}} = 1.2 cm$  (which would be about right 
for an L-mode discharge and somewhat conservative for an H-mode discharge) 
and then use the ``most conservative'' case parameters described in 
Table~\ref{t6-12-1}, we estimate  for a total heating power (Pheat) of
373 MW that $Q_{div,peak}$ for the inboard divertor (``I. D.'' segment in 
Figure~\ref{f6-12-1} ) as $ \approx 3.3 MW/m^2$ and for the outboard divertor 
(``O. D.'' segment in Figure~\ref{f6-12-1}) as $\approx 13.7 MW/m^2 $.

\begin{table}
\label{t6-12-1}
\caption{``Most Conservative''  Case Parameters} 
\centering
\begin{tabular}{|l|l|l|} \hline

                               &Inboard Leg(s) & Outboard Leg(s)  \\ \hline
$f_{rad}$(uniform distribution)& 0.30          & 0.30             \\ \hline
$\lambda_{ q_{||}}$ (cm)       & 1.2           & 1.2              \\ \hline
$f_{exp}$                      & 2.5           & 4.0              \\ \hline
$f_{pfr}$                      & 0.1           & 0.1              \\ \hline
$R_s (m)$                      & 3.6           & 4.4              \\ \hline
$f_{outboard/total}$           &0.11           & 0.89             \\ \hline
$f_{\nabla B/total}$           & 0.5           & 0.5              \\ \hline
$\alpha (^0)$                  & 10            & 10               \\ \hline
$Q_{div,S} (MW/m^2)$           & 3.3           & 13.7             \\ \hline
\end{tabular}
\end{table}


The average heat fluxes on these inboard and outboard divertor segments are 
$\approx 0.8 MW/m^2$ and $\approx 7.1 MW/m^2$, respectively; the power flow 
``spillover'' into the private flux region of the divertor produces an 
average particle heat flux of $\approx 0.5 MW/m^2$. The peak particle heat 
fluxes in the upper divertor are identical in their mirror locations in 
the lower divertor. 

\begin{figure}  % Fig. 3
\centering
\includegraphics*[width=5.5in]{./f6-12-3.ps}
%\includegraphics*[height=4.5in,clip=]{./f6-12-3.ps}
\caption{The poloidal distribution of the average and peak radiative heat 
fluxes are shown for the ``most conservative'' case   No contribution from the 
divertor or scrape-off is assumed, in line with our (very) conservative 
assumptions. The radiated power density is constant inside the core plasma}
\label{f6-12-3}
\end{figure}


Figure~\ref{f6-12-3}  shows the poloidal distribution of the average 
(and peak) core-radiated heat flux (QEM) at each of the segments shown in 
Figure~\ref{f6-12-1}. While there is a pronounced poloidal distribution to 
the wall heat flux, their values are considerably less than the  
$0.45 MW/m^2$ peak heat flux limit set by cooling concerns for non-divertor 
first wall locations.  

What constitutes an ``acceptable'' peak divertor heat flux is problematical. 
ARIES-AT engineering considerations presently set this limit at 
$\approx 5 MW/m^2$, which is much less than the $13.7 MW/m^2$ from our 
(very) conservative estimate in this section.  In the following section, we 
examine two less conservative scenarios that lower divertor heat flux values 
to more acceptable levels.

\subsubsection{Radiating mantle approach} % 6.2.2
Much of the physics in the boundary and SOL regions of  single-null (SN) 
divertors, particularly the role of $E \times B$ poloidal drifts in the 
divertor~\cite{Rognlien99,Boedo00},  has only recently come under the 
detailed scrutiny of transport modeling codes, such as UEDGE~\cite{Rognlien94}.
How these poloidal drifts  affect asymmetries in divertor heat flux and 
particle flux~\cite{Petrie00} or the ``enrichment''  of radiating impurity 
ions in double-null  (DN) divertors is speculative at the present time.  
Hence, it is prudent to first examine the efficacy of 
relying on the better-understood  radiative processes occurring in the core 
plasma.  Moreover, with radiating power from inside the core plasma, we not 
only avoid the complexities and uncertainties introduced by DN divertor 
physics, but also spread the plasma power more evenly around the vessel 
structure.  From previous experience , we have found the ``radiating mantle'' 
concept effective in this regard, i.e.,  injected impurity ions radiate power 
predominantly from edge of the main plasma.
  
Impurity injection has other advantages also. ``Large'' tokamaks, such as JET 
and JT-60U, have demonstrated that good energy confinement is consistent at
higher densities under certain impurity injection scenarios.  The lower 
plasma edge temperatures which occur in a radiating mantle,  particularly 
with  the ion component,  reduces the ion temperatures in the SOL, which in 
turn, reduces sputtering at the divertor plates from ``hot'' ions.

In this section we discuss one possibility based on the radiating mantle 
concept: the minimum radiated power for which the peak heat flux in the 
divertors is ``manageable'', i.e., $\approx5 MW/m^2$, under radiating mantle 
conditions. To do this, we again take $\lambda_{q_{||}} = 0.012 m$ and  
$frad \approx 0.75$ (Figure~\ref{f6-12-2}),  resulting in $Q_{div,S}$ 
under the outboard leg(s) $\approx 5 MW/m^2$ and under the inboard leg(s) 
$\approx 1.2 MW/m^2$; the average heat fluxes on these inboard (``I. D.''), 
``dome'', and outboard (``O. D.'') segments are $\approx 0.3 MW/m^2$, 
$\approx 0.2 MW/m^2$, and $\approx 2.3 MW/m^2$, respectively. 

The 75\% core radiation requirement is about two and a half times that of the 
nominal case. In a parallel study using the MIST impurity transport code to 
predict impurity behavior inside the core plasma (described in 
section~\ref{s6-3}),   a modest amount of argon impurities added to the 
plasma was successful in achieving this 75\% radiated power requirement with 
a radiating mantle (argon case: $C_V = -0.1$). The resulting average and peak 
poloidal  heat flux on the vessel walls are shown in Figure~\ref{f6-12-4}.
At all poloidal locations, both the average and peak heat flux due to 
radiated power is = 0.9 MW/m2  with highest heat fluxes on the outboard 
vessel wall. Finally, as an upper bound estimate to wall heating by 
radiation, if it were possible to radiate all 373 MW of power  under a 
``mantle'' scenario, we estimate  that the highest radiative heat 
flux values would be  $ \leq 1.2 MW/m^2$ on the outboard wall. 


\begin{figure} % Fig. 4
\centering
\includegraphics*[width=5.5in,clip=]{./f6-12-4.ps}
\caption{The poloidal distribution of the average and peak radiative heat 
fluxes are shown for the argon ``mantle'' case, as described in the text. 
As with the ``nominal'' case, no contribution from the divertor or scrape-off 
is assumed.}
\label{f6-12-4}
\end{figure} 


Engineering considerations for cooling the (non-divertor) vacuum vessel wall 
of ARIES-AT, however, dictate that the average heat flux not exceed 
$0.3 MW/m^2$ (or, equivalently, that the peak heat flux not exceed 
$0.45 MW/m^2$). This implies that the largest $f_{rad}$ of core radiated 
power which would not violate this engineering criterion is 0.36. This 
limitation on the maximum radiated power from inside the core plasma 
means that the peak heat fluxes on the ``I.D'' and ``O.D'' segments can 
only be reduced to $3.0 MW/m^2$ and $12.5 MW/m^2$, respectively. 

\subsubsection{Radiative divertor approach} %6.2.3
As discussed above, plasma edge/SOL transport codes, like UEDGE, are only 
now beginning to realistically address double-null divertor physics issues. 
However, it is still possible to get a handle on the amount of power that 
might be radiated away  in the SOL and divertors.  In a previous study of  
the ARIES-RS divertors, for example, a simplified radiative divertor approach
provided a plausible solution to the divertor heat flux problem  by seeding 
the divertor with a neon impurity, although this required 
several favorable assumptions in the divertor physics~/cite{Jardin97}. 

In this report, we again look at this radiative divertor approach, but this 
time we choose argon as the seeding impurity. Like neon, argon can be 
expected to radiate reasonably well  under ARIES-AT divertor and scrape-off 
layer conditions.  Unlike neon, however, argon radiates  much more efficiently
in the edge region of the ARIES-AT plasma, enhancing radiative power from the 
core plasma mantle. In turn, a radiating plasma mantle relaxes the 
``enrichment'' requirement (i.e., the ratio of $n_{eAR}/n_e$ in the divertor to 
that in the core) for the radiating impurity in the  divertor.  In order to 
reduce the total (particle plus radiative) peak heat flux in the divertors 
to $\approx 5-6 MW/m^2$,  we require $ \approx 160 MW$ of radiated power in 
the scrape-off layer and divertors (i.e., $f_{rad,div}\approx 0.4$), in 
addition to the core radiative fraction of 0.36. Note:  this implies that
$Q_{div,peak} = 4.0 MW/m^2$ in an attached plasma case; choosing this value 
to be well under the $5 MW/m^2$ limit is necessary, since there will be a 
contribution to the heat flux from the radiated power in the divertors.

As an initial estimate of radiated power in the divertor, we use a 
simplified  UEDGE simulation~\cite{Rensink00} of an ARIES-AT-like edge plasma 
case. Among the simplifying assumptions are: (1)  up/down symmetric solutions
in the upper and lower divertors, (2)  no $\nabla B$ or  poloidal  
$E \times B$ drifts included in this modeling, (3) an orthogonal coordinate 
system, i.e., the flux surfaces intersect the divertor structure at normal 
incidence, and  (4) inboard and outboard strike points are isolated from each 
other, e.g., what occurs at the inboard divertor does not affect the outboard 
divertor.
  
The $n_e (n_{e,sep} = 0.5\times10^{20} m^{-3})$ and 
$T_e (T_{e,sep} =  340 eV)$ radial profiles in the scrape-off 
and divertor regions are determined self-consistently  for a hydrogenic 
ARIES-AT plasma, where the electron and ion diffusivities are  $  0.5 m^2/s $
and the particle diffusivity is $0.33 m^2/s$. We then estimate the radiated 
power from the argon present by assuming a fixed-fraction of argon in the 
SOL/divertor $ (n_{eAR}/n_{e} \approx 0.0026 )$, which, in turn, produces 
$ \approx 160 MW$ of radiated power in the SOL/divertors. Under these 
restrictions, virtually all the radiated power occurs between the divertor 
tiles and the respective X-points, particularly nearest the divertor floor 
(ceiling).  We should point out that $ n_{eAR}/n_e \approx 0.0026$ does not 
necessarily mean that this ratio will also be the concentration of argon in 
the core plasma. For example, recent ``Puff and Pump'' experiments in DIII-D 
\underline{single-null} diverted plasmas have shown that an enrichment of 
$\approx 3$ is possible, even when a significant amount of argon is 
injected~\cite{Wade98}. Interestingly, the measured fraction  
$n_{eAR}/n_e (\approx 0.0035)$ in the divertor is comparable with our 
estimated requirement for ARIES-AT.

For the next step of complexity,  we wanted to allow UEDGE to seek a more 
self-consistent solution , i.e., fix the ratio $n_{eAR}/n_e$ in the scrape-off
layer and divertor and then find the self-consistent solution in the  SOL and 
divertor. Unfortunately, it is more difficult to obtain self-consistent 
fixed-fraction solutions because numerical issues such as grid resolution and 
possible unstable plasma behavior, e.g., MARFE's, can arise for 
strongly radiating detached divertor plasmas.  These simulations require more 
manpower and computational resources for a solution.  The most realistic 
simulation of radiating impurities in UEDGE would use the multi-charge-state 
model and compute the self-consistent spatial  distribution of the impurities 
as a result of plasma transport and atomic processes.  Such simulations have 
been carried out for the ITER reference  design~\cite{Perkins99}. These 
solutions are not available at this time. We believe,  however, that
detachment of both the inboard and outboard divertor legs along the 
separatrix would occur for argon fractions of only a few tenths of a percent. 

From  engineering considerations, the peak heat flux should not exceed $0.45 
MW/m^2$ on the (non-divertor) vacuum vessel surfaces and should be 
$\approx 5 MW/m^2$ on the divertor wall segments. In the likelihood that both 
inboard and outboard separatrix strike points are detached, the peaked heat 
flux formulas used in Equ.~\ref{e6-12-2} overestimates the actual peak heat 
flux.  Hence, under  detached conditions, our calculation of the total peaked 
heat flux (below) would be overestimates.

In order to evaluate the contribution of the radiated power from the ARIES-AT  
divertors to wall heating, we take the radiating divertor source to be 
distributed uniformly under the X-points between the separatrix flux surface 
and the flux surface 1-cm outside the separatrix (as measured from the 
midplane).  Figure~\ref{f6-12-5} shows the combined poloidal distribution of 
the radiated power from the core plasma $(f_{rad,core} \approx 0.36$ and 
divertor plasma $(f{rad,div}\approx 0.43)$, as determined from RADLOAD. The 
total heat flux on any element of the vacuum vessel wall is the sum of the 
radiative $(Q_{rad})$ and the particle $(Q_{div})$ contributions. The 
``worst case'' scenario occurs when the peak in the radiative contribution 
and particle flux contribution coincide on the ``O.D.'' segment. 
Under such a conservative scenario,  the sum of the peak radiative heat flux 
contribution to the ``O.D'' segment $(\approx 1.8 MW/m^2$ in 
Figure~\ref{f6-12-1}) and particle heat flux
contribution $(\approx 4.0 MW/m^2)$ is roughly $5.8 MW/m^2$.


\begin{figure}
\centering
\includegraphics*[width=5.5in,clip=]{./f6-12-5.ps}
\caption{The poloidal distribution of the average and peak radiative heat 
fluxes are shown for the argon ``radiative divertor'' case, as described in 
the text. Thirty six percent of the power input is radiated in the core and 
43\% of the power input is radiated in the divertor. Outside the divertor, 
the radiative heat flux to the vessel walls does not exceed $0.45 MW/m^2.$}
\label{f6-12-5}
\end{figure}

\subsubsection{Summary and Conclusion} %6.2.4
We considered reducing high peak heat flux at the divertors by radiating away  
power from  inside the core plasma and/or in the divertors.   First, in order
to minimize over reliance on several speculative and largely unexplored 
aspects of DN divertor physics (e.g., poloidal drifts and impurity 
entrainment in the SOL and divertor), we took a conservative approach 
of ignoring any radiated power from the SOL and divertor region while 
concentrating on core radiated power. Although the radiating away the power 
inside the core plasma has the advantage of dissipating the power more evenly 
on the vessel wall,  wall-cooling considerations in present ARIES-AT 
design  restricted the usefulness of this approach to  a relatively low 
radiating fraction, i.e., $f_{rad,core} = 0.36$. (In this regard, future work 
in raising the cooling capacity of the non-divertor vacuum vessel walls in 
order to allow a higher frad,core should be considered.)  In order to reduce 
the peak divertor heat flux to the $5-6 MW/m^2$ range, we were required to 
radiate about 40\% of the heating power in the  divertor, i.e., $ f_{rad,div} 
\approx 0.4$. 


\subsection{Core Plasma Impurity Radiation in ARIES-AT: L-mode Edge Case} % 6.3
\label{s6-3}

\subsubsection{Overview} % 6.3.1
The power originating in the core plasma, i.e. the sum of fusion alpha 
particle power and auxiliary heating and current drive power, is exhausted as 
electromagnetic radiation from the core plasma or as particle kinetic energy 
transported across the separatrix to the scrape off layer and then along the 
open field lines to the divertor. To alleviate the thermal loads on the 
divertor surfaces in contact with the open filed lines, it is desirable 
to radiate power from the core plasma.  Bremsstrahlung and synchrotron 
radiation arises naturally from the core plasma.  Bremsstrahlung can be 
enhanced by the injection of rare gas impurities.  Impurities will also 
radiate due to plasma excitation of line radiation.  In this section we 
discuss the bremsstrahlung and impurity line radiation induced by rare gas 
impurities intentionally injected into the core plasma.  The engineering 
constraints due to the ARIES-AT main chamber first wall thermal design limit 
the total core radiation to about 140 MW.  The estimated synchrotron radiation
loss to the first wall is about 18 MW, leaving about 120 MW of desired 
impurity radiation. We will show calculations of Ne, Ar, and Kr radiation for 
the L-mode edge case of ARIES-AT using anomalous transport and a case for 
Ar using neoclassical transport.  In these cases, the bremsstrahlung 
radiation alone is sufficient to produce the required 120 MW.  An even higher 
level of radiation can be achieved from Ar and Kr due to additional line 
radiation.

\subsubsection{Anomalous Transport Modeling Using MIST} % 6.3.2

The MIST model~\cite{Hulse83} has been used to estimate the impurity radiation
induced using injected rare gases. The MIST code is a 1D trace impurity 
transport model.  Steady state impurity density profiles, and the resulting 
radiation profiles are determined for given electron temperature and density 
profiles and transport coefficients. This version of the MIST code assumes a 
circular cross section plasma.  However, when the total radiation is 
calculated from the modeled radiation profile, the actual plasma shape from 
the EFIT solution~\cite{g00509.0*540} is used.

The electron density profile being considered has an exponential fall-off 
beyond the separatrix.  The central electron density is determined from the 
total density,$ 6.306 \times 10^{20} m^{-3}$, correcting for the impurity ion 
fractions, e.g.$ f_{Ne} = 0.0075$. For these calculations the presence of 
helium has been ignored. The impact is small, only a slight reduction in the 
calculated bremsstrahlung radiation from the background plasma.  The electron 
density profile is shown in Figure~\ref{f6-3-1}. 
 
\begin{figure} % 1
\centering
\includegraphics*[width=5.5in,clip=]{./f6-3-1.ps}
\caption{The ARIES L-mode edge density profile used in the core impurity 
radiation study.}
\label{f6-3-1}
\end{figure}

The electron temperature profile, shown in Figure~\ref{f6-3-2}, is obtained 
from the EFIT pressure profile and the total density profile (same shape as 
the electron density profile with $n_0 = 6.306 \times 10^{20} m^{-3})$.  
The EFIT pressure profile goes to zero at the separatrix. To allow for a 
finite SOL temperature, a constant temperature of 300 eV is added to the 
$T_e $ profile for $\rho < 1.0$, with an exponential fall off for 
$\rho > 1.0$.  

In this study we have used a constant impurity fraction as a boundary 
condition at the central axis.  For neon, we have chosen the fraction of 
0.0075 assumed in the systems study.  For argon and krypton, the central 
fractions used produced a somewhat higher radiated power yet lower core 
$Z_{eff}$ and core dilution.

We have also chosen to use a spatially constant particle diffusion 
coefficient, $D = 10^4$ $cm^2/s$. Because we are using a fixed central 
impurity fraction, and using steady state solutions, the calculated radiation 
profiles are only weakly dependent on the value of D.  The impurity density 
profile shape does significantly effect the calculated total radiation.  
For the MIST modeling we have used the $C_V$ coefficient to set the particle 
pinch velocity,

\begin{equation}
                                v_p = C_V (D/n_e)\frac{dn_e}{dr}
\end{equation}

where $v_p$ is the local particle convective velocity.  For $C_V = 1.0$, 
the impurity density profile has the same shape as the electron density 
profile. The values of $D = 10^4 cm^2/s$ and $C_V = 1.0$ are consistent 
with measurements on DIII-D in L-mode plasmas when anomalous transport 
dominates.

 
\begin{figure} % 2
\centering
\includegraphics*[width=5.5in,clip=]{./f6-3-2.ps}
\caption{The ARIES L-mode edge ion and electron temperature profile used in 
the core impurity radiation study.}
\label{f6-3-2}
\end{figure}

In Figures~\ref{f6-3-3a},\ref{f6-3-3b},\ref{f6-3-3c}, and \ref{f6-3-3d}
radial profiles of impurity density, volume emissivity for all ions,
$Z_{eff}$ , and volume integrated power using argon with a $C_V = 1.0$ are 
shown.  The radiation includes both the line and bremsstrahlung radiation 
from each ionization state. Since Ar is fully stripped over most of the 
volume of the confined plasma, the bulk of the radiation is due 
to bremsstrahlung.
 
\begin{figure}  %3a
\includegraphics*[width=5.5in]{./f6-3-3a.ps}
\caption{Argon ion density profiles calculated by MIST for $C_V = 1.0 $. The 
dashed vertical line is at the location of the seperatrix.}
\label{f6-3-3a}
\end{figure}
 
 \begin{figure}  %3b
\includegraphics*[width=5.5in]{./f6-3-3b.ps}
\caption{Argon volume emissivity profile for the density profile shown in 
Fig.~\ref{f6-3-3a}}
\label{f6-3-3b}
\end{figure}

\begin{figure}  %3c 
\includegraphics*[width=5.5in]{./f6-3-3c.ps}
\caption{Zeff from Argon for the density profile shown in Fig.~\ref{f6-3-3a}}
\label{f6-3-3c}
\end{figure}

\begin{figure}  %3d 
\includegraphics*[width=5.5in]{./f6-3-3d.ps}
\caption{The radiated power escaping the core plasma as a function of minor radius.} 
\label{f6-3-3d} 
\end{figure}

The solid line is bremsstrahlung and line radiation from argon, the dashed 
line is bremsstrahlung from the primary deuterium ions.

Figures~\ref{f6-3-4a} and \ref{f6-3-4b} show the radial profiles of krypton 
ion density and volume integrated power using krypton with a $C_V = 1.0$. 
The central $Z_{eff}$ for Kr has been reduced to 1.5 (compared to 1.6 for Ar),
and fuel dilution is also lower, yet the total radiated power fraction is 
higher, about 0.5.  This is due to the fact that krypton is not fully ionized 
in the ARIES-AT plasma, and therefore produces more line radiation.


\begin{figure} %4a
\includegraphics*[width=5.5in]{./f6-3-4a.ps}
\caption{Krypton ion density profiles in ARIES-AT modeled using MIST with 
$C_V = 1.0 $ and a fixed core fraction of 0.0004.  Krypton is not fully 
ionized in the core plasma and can produce line radiation.}
\label{f6-3-4a}
\end{figure}

\begin{figure} %4b
\includegraphics*[width=5.5in]{./f6-3-4b.ps}
\caption{The radiated power escaping the core plasma as a function of minor 
radius for krypton impurity injection.  The solid line is bremsstrahlung and 
line radiation from krypton, the dashed line is bremsstrahlung from the 
primary deuterium ions.}
\label{f6-3-4b}
\end{figure}
%\end{document}

\subsubsection{Neoclassical Impurity Transport Modeling using STRAHL} % 6.3.3

Inside the transport barrier in ARIES-AT we expect that anomalous transport 
should be reduced to very low levels and neoclassical transport should 
dominate.  Impurity transport measurements on DIII-D have indicated that 
neoclassical affects are important during enhanced confinement 
operation~\cite{Wade00}.  We have modeled neoclassical transport for argon 
in the ARIES-AT L-mode edge plasma using the STRAHL 
code~\cite{Behringer87,Peeters99}.

The resulting effective transport coefficients are shown in 
Figures~\ref{f6-3-5a} and \ref{f6-3-5b}. The neoclassical pinch velocity is 
seen to bbe outward in the central region of the plasma.  This outward pinch 
results from the nearly flat density profile and a peaked temperature profile.
Near $\rho_{pol}$ of about 0.7, the density gradient is 
strong, and the neoclassical pinch is then negative. These transport 
coefficients result in argon density profiles shown in Figure~\ref{f6-3-6}.  
The peaking of the density near $\rho = 0.6$ is expected from the pinch 
velocity profile shown in Figure~\ref{f6-3-5b}.   The resulting Zeff profile 
isshown in Figure~\ref{f6-3-7}.
 
\begin{figure} %5a
\includegraphics*[width=5.5in]{./f6-3-5a.ps}
\caption{ The total diffusion coefficient used in the STRAHL modeling of 
neoclassical Ar transport in the ARIES-AT L-mode edge plasma is shown as 
curve 5.   A small anomalous term of $0.05 m^2/s$ is assumed inside the 
transport barrier.  Outside the transport barrier in the L-mode edge an 
anomalous contribution of $ 0.5 m^2/s$ is assumed.}
\label{f6-3-5a}
\end{figure}

\begin{figure} %5b
\includegraphics*[width=5.5in]{./f6-3-5b.ps}
\caption{The profile of the pinch velocity in the neoclassical model of 
Ar transport in the ARIES-AT l-mode edge plasma.  An anomalous pinch of 
$-0.2 m/s$ is assumed outside of the transport barrier.  Inside the transport 
barrier, no anomalous pinch is used.}
\label{f6-3-5b} 
\end{figure}

\begin{figure}
\includegraphics*[width=5.5in]{./f6-3-6.ps}
\caption{Argon density profiles predicted by the neoclassical model in the 
STRAHL code.}
\label{f6-3-6}
\end{figure}


 
\begin{figure}
\includegraphics*[width=5.5in]{./f6-3-7.ps}
\caption{ The Zeff profile from argon predicted by using the neoclassical model in 
STRAHL for the AREIS-AT L-mode edge.}
\label{f6-3-7}
\end{figure}


\subsubsection{Conclusions concerning Impurity Radiation} %6.3.4

The MIST and STRAHL modeling results for impurity radiation induced by 
injecting a rare gas into the ARIES-AT L-mode edge plasma are summarized in 
Table\ref{t6-3-1}.  In all cases, the requirements for the overall ARIES-AT 
system design are met.  These results indicate that the higher Z rare gases 
are better for the reactor grade plasmas, as they can meet the radiated power 
requirements at lower Zeff and lower fuel dilution.  


\begin{table}
\centering
\caption{Modeling results for rare gas impurity radiation in ARIES-AT: 
L-mode edge.  $_{Prad}$ includes bremsstrahlung from both the impurity and 
the fuel ions inside the separatrix.  When calculating the radiation 
fraction, an additional 18 MW of synchrotron radiation is added.  The alpha 
heating plus auxiliary power is assumed to be 360 MW.
(Note:  Core $Z_{eff}$ does not include the helium content)}
\vspace{5 mm}
\label{t6-3-1}
\begin{tabular} {|l|l|l|l|l|l|l|} \hline
Impurity & Z & $C_V$ &Core     & Core      & $P_{rad}(MW)$ & Radiation \\
         &   &       &Fraction & $Z_{eff}  $ & $\rho < 1.0$& Fraction \\ \hline

     Neon& 10&    1.0&   0.0075&        1.7&            120&     0.38 \\ \hline
    Argon& 18&    1.0&    0.002&        1.6&            140&     0.44 \\ \hline
    Argon& 18&    Neo&   0.0023&        1.7&            157&     0.49 \\ \hline
  Krypton& 36&    1.0&   0.0004&        1.5&            165&     0.51 \\ \hline
\end{tabular}
\vspace{5 mm}
\end{table}


The first wall engineering restriction to keep the core plasma radiation to 
below 140 MW (or about 36% of the total=alpha+aux thermal power), while 
maintaining low total conducted power to the divertor 
strike plates, implies a need to radiate a large amount of power from the 
divertor plasma (see section~\ref{s6-2}).  These 
modeling results indicate that with an improved main chamber first wall 
thermal design, the use of krypton as a core plasma radiator can 
significantly reduce the divertor radiation requirements.

The dynamics of feedback control of the radiated power using rare gas 
injection have not been investigated.  The large size, and near neoclassical 
transport levels mean that the impurity confinement times are long.  Control 
of core impurities using the ``puff and pump'' technique on DIII-D has been 
demonstrated~\cite{Wade98}.  Detailed studies of the stability of the burning 
plasma with feedback control of the rare gas radiated power are needed.


\section{Plasma Operating Regime and Startup} %7.0
\label{s7.0}
\subsection{POPCON analysis} %7.1
\label{s-1}
A steady-state power balance is enforced to establish the ARIES-AT
plasma operating point.  Given the input of $P_{CD} \simeq$ 37 MW of rf power
for current drive (CD), the ARIES-AT plasma operates slightly sub-ignited.
At a density-weighted, volume-average temperature 
$T_{i} \simeq T_{e} \simeq$ 18 keV, the Lawson parameter, $n_i \tau_E$, is 
$3.42 \times 10^{20}$ s/m$^{3}$, as the product of
the volume-average ion density, $n_i = 1.71 \times 10^{20}$ m$^{-3}$, and the
energy confinement time, $\tau_E = 2.0 s$.

The plasma heating power is $P_{PH} = P_{\alpha} + P_{CD}$,
where $P_{\alpha} \simeq 351$ MW is the self-heating power from alpha-particle
fusion products.  These input powers are balanced by transport losses
across the plasma separatrix, $P_{TR} \simeq 274$ MW and radiation (primarily
bremsstrahlung, $P_{BR} \simeq 55$ MW, line radiation, $P_{LINE} \simeq 43$
MW, and cyclotron radiation, $P_{CY} \simeq 19$ MW).  The core radiation
fraction is $f_{RAD} \simeq 0.30$ and $P_{TR} = P_{PH}(1-f_{RAD})$.

Once $\tau_E$ is known, a comparison can be made to any of a number
of empirical scaling relations~\cite{Ite89,Uck89,Ite99},
with the caveat that the most appropriate relation for the
ARIES-AT regime is not yet established experimentally.
It is emphasized that a scaling law is not needed to derive any of the
ARIES design points; $\tau_E$ is determined from the specification of
target net electrical power output, $P_E$, and plasma gain, $Q_p$,
that is set by the current-drive power requirement.

Recast in SI units,
the L-mode ITER89-P scaling~\cite{Yus90,Pos90,Kay97} is given by
%
\begin{eqnarray}
\label{e-4}
\tau^{89P}_E & = & 3.8033\times 10^{-6} I_{p}^{0.85} n_e^{0.1} B_{\phi 0}^{0.2}
  a^{0.3} R_T^{1.2} \kappa_\times^{0.5} A_i^{0.5} [P_{TR}]^{- 0.5} \: ,
\end{eqnarray}
%
where $A_i$ is the atomic mass (2.5 for a nominal 50:50 D-T fuel mixture).
This scaling relation is used here as a representative example.

The value of $\tau_E$ required for a design point is compared to the
various empirical scaling laws by means of a confinement multiplier,
\hbox{$H^j \equiv \tau_E/\tau^j_E$}, where the superscript $j$
denotes the particular scaling of interest.
The energy-confinement-time multiplier, $H^j$, for the ARIES-AT relative
to the ITER89-P scaling is $H^j = 2.65$.

A plasma operating contour (POPCON)~\cite{Hou82} plot has been generated
for the ARIES-AT using the ITER89-P scaling relation.  The results
assuming the ITER89-P relation are shown in \hbox{Fig.~\ref{f-7-1}}.
The plasma operating space is depicted in terms of density and temperature.
The plasma power balance equation is solved, including a term for
auxiliary heating power, $P_{AUX}$.  Using 10-MW increments, the
POPCON plot shows contours of constant values of $P_{AUX}$ with the
ignition threshold ({\it i.e.}, $Q_p \rightarrow \infty$) indicated by
the first dashed contour.  The ignition contour itself shows that the
minimum density for ignition is $n_i \simeq 1.71 \times 10^{20}$ m$^{-3}$
near $T_{i} \simeq 12$ keV.  An ignited plasma to the left of the
minimum of this contour is thermally unstable.

The ARIES-AT design point optimization moves to somewhat higher temperature
to reduce costs associated with the CD system while reducing the
fusion power density a little.  The modest $P_{CD}$ input moves
the ARIES-AT steady-state operating point to the position indicated
by the open square.
Thermally stable operation near $T_{i} \simeq 18$ keV is suggested by
this result; other scaling relations might shift this zone.

The product $nT$ is proportional to the plasma beta, $\beta = \beta_{N}
[I_{p}/(a_p B_{\phi 0})]$.  The band defined by $\beta_{N} = 5-6$ is
depicted by the dash-dot isoquants and the full-up values of $I_{p}$,
$a_p$, and $B_{\phi 0}$ for the ARIES-AT.  A dotted curve denoting
fixed fusion power, $P_F = 2,000$ MW in this $n-T$ space for the
ARIES-AT plasma volume is also shown.  The fusion power of the ARIES-AT
itself is $\sim 1,760$ MW.

Figure~\ref{f-7-1} also shows that a start-up transient with increasing
plasma density and temperature passing through the saddle region
(Cordey pass)
near $T_i \simeq 7$ keV requires auxiliary heating power, $P_{AUX}$,
of about 25 MW.
%
\begin{figure}[p]
%centerline{\hbox{
%psfig{file=./popat5.ps,height=5.0in,bbllx= 0pt,bblly=158pt,%
%      bburx=634pt,bbury=592pt,clip=t}
\includegraphics*{./popat5.ps}
\caption[Fig-1]
{ARIES-AT Plasma Operating Contour (POPCON) diagram.  Solid countours
indicate auxiliary heating power,$P_{AUX}$, in increments of 10 MW
until ``ignition'' ({\it i.e.}, $Q_p \rightarrow \infty$)
at the dashed contours.  The dash-dot isoquants
denote $\beta_{N} = 5-6$.  The dotted curve traces fixed fusion power,
$P_F = 2,000$ MW.
The steady-state ARIES operting point is denoted by the open square.}
\label{f-7-1}
\end{figure}

\subsection{Plasma Non-inductive Startup}  % 7.2
\subsubsection{Inputs \& Model Equilibrium}
The work described in this section was done using a TD spreadsheet (v. 11.0). 
This is a 0-D tokamak simulation, which includes calculations of bootstrap 
current and fusion power based on prescribed density, temperature, and 
current (q) profiles.  The basic parameters used in this study were:

\vspace{5 mm}
\begin{tabular}{llll} 
$a = 1.300 m$  &  $\kappa = 1.950$  &  $B_{T0} = 5.927 T$ & $q(0) = 3.345$ \\
$R = 5.200 m$  &  $\delta = 0.850$  &  $I = 12.972 MA$    &                 \\
\end{tabular}
\vspace{5 mm}


Note that $\kappa$ and $\delta$ have been changed slightly so that the plasma 
volume and $q_{95}$ match the reference case.  Other parameters assumed are 
(these also agree with the reference):

%\vspace{5 mm}
\begin{center}
\begin{tabular}{lll}
$<n_e> = 2.230 \times 10^{20} m^{-3}$&$ n_{e0}/<n_e> = 1.344$                     &      $N_{Greenwald} = 0.91 $ \\
$<T_e> = 15.833 keV$                 &       $T_{e0}/<Te> = 1.964$                & \\
$n_{He}/n_e = 0.190$                 &       $n_{Ar}/n_e = 1.80 x 10^{-3}$        &    $Z_{eff} = 1.93$  \\
$\tau{Ei}/\tau_{Ee} = 1$             &  $\epsilon\beta_p = 1 (\beta_p \leq 4.0)$  &  $\beta_N \leq 5.44$ \\
\end{tabular}
\end{center}
\vspace{5 mm}

Note that $N_{Greenwald} < 1$, even though the other numbers match.  Here 
$$N_{Greenwald} = \langle n_e \rangle /(I/\pi a^2).$$
These numbers give:

%\vspace{5 mm}
\begin{center}
\begin{tabular} {lll}
$\beta_N = 5.44$        &  $\beta_p = 1.94$   &                  \\ 
$\beta = 9.15\% $       &   $W = 628 MJ$      &                  \\
$P {\alpha} = 350 MW$   &   $H_{98y1} = 1.29$ & $H_{89p} = 3.24$ \\
\end{tabular}
\end{center}
\vspace{5 mm}

These agree with the reference except that the reference case has 
$H_{89p} = 2.64$ and $H_{98H} = 1.42$.  Because the bootstrap calculation 
does not include geometric effects such as the Shafranov shift, the calculated
bootstrap current is multiplied by 0.85 in order to match the reference 
equilibrium.  This leads to an underestimate of the bootstrap current at low 
beta.  The external heat sources are from the reference: 33 MW of LH power 
and 3 MW of FW.  The calculated current from these sources is 1.14 MA, in 
agreement with the reference (1.12 MA). 

\subsubsection{Rampup Scenario}
With only 36 MW of available external power, the most likely way to make the 
ramp-up work is to provide for control of the confinement, allowing H98y1 = 2.
Fixed quantities are a, $R, \kappa, \delta, B, q(0), P_{LH}, P_{FW}$, 
and the normalized profiles of density and temperature.  The density is 
varied with the current at fixed NG.  The ramp-up is started arbitrarily at 
$I_p = 0.5 MA$.  It appears that with LHCD, the current could be started from 
zero, but the required density variation was not considered.  

There are three phases to this rampup.  For $0 \leq t \leq 52 s$, 
the plasma is at the assumed $\beta_p = 4$ limit.  During this time, 
$H_{98y1}$ is increased from 0.7 to 2.0.  At 52 seconds, $I_p = 2.6 MA$.  
For $52 \leq t \leq 886 s$, $H_{98y1}$ is fixed at 2.0, $\beta_p$ falls below 
the maximum, and $\beta_N$ increases to 5.4, the maximum allowed.  At 886 s, 
$I_p = 7.9 MA$.  Finally, for $886 \leq t  \leq 5000 s$, $\beta_N = 5.4$, 
and $H_{98y1}$ and $\beta_p$ fall to their final values.  The total time 
required to bring the plasma to the operating point is roughly 1 1/2 hours.

\section{Summary}  %8.0
\label{s8.0}

ARIES-AT is based on a reversed-shear plasma configuration as was ARIES-RS,
but the ARIES-AT $\beta$ is 9.2\%, almost twice the 4.96\% of ARIES-RS.  The
increased $\beta$ has been used in the design to reduce size of the device
somewhat, and to reduce the toroidal field from $8.0 T$ in ARIES-RS to $5.8 T$.

Explicit analysis of plasma pressure and current profiles typical of H-mode
plasmas showed that these profiles, which are characterized by non-zero
pressure gradients and current density at the edge, do not reduce the 
ideal MHD $\beta$ limits.  The resistive wall mode stability has been
analyzed.  Calculations show that a stable region for this mode exists
in the presence of plasma rotation velocities of $ 7-10\% V_A$.  It was
concluded that it is not feasible to maintain these high rotation velocities
in a reactor environment, and thus an active feedback system is employed
for mode control, requiring about $10 MW$ of power.

The neoclassical tearing mode (NTM) is identified as a problem.  Stabilization
of this mode by using radially localized ECCD to replace the missing
bootstrap current appears to require prohibitively large ECCD power, and
future work will concentrate on methods to leverage this by modifying the
background current profile (ie, make $\Delta^{\prime}$ more negative).

The analysis of the stability of intermediate mode number ideal MHD modes
appears to be consistent with the occurrence of ELMs in the baseline design.

Transport analysis has been carried out to analyze the consistency of the 
standard profiles in ARIES-AT.  It is found that the large Shafranov shift
is a strong stabilizing factor.  The GLF23 model predicts global energy
confinement that exceeds the ARIES-AT design requirements if the density
profile is sufficiently peaked.  The possibility of increasing the energy
confinement through driven sheared rotation from NBI or ICRF was looked
at but considered to be not practical for a device with ARIES-AT parameters.

Methods for dispersing the heat loads from ARIES-AT were analyzed.  The
final design radiates 36\% of the exhaust power (about 140 MW)from the core. 
Another 40\% of the heating power was radiated in the scrape-off layer (SOL)
mostly in the vicinity of the divertor region.
 
It is recognized that the ARIES-AT is a very aggressive design.  Many of the
physics design parameters are outside of those now routinely obtained in
tokamaks.  However, this is presented as a vision of the ``ultimate potential''
of a tokamak if the difficulties described in this section can be overcome.

%end of summary

\begin{thebibliography}{9}

\bibitem{Kessel01} C.~E.~Kessel,T.~K.Mau, S.~C.~Jardin, F.~Najmabadi, 
``Plasma Profile and Shape Optimization Studies for Advanced Tokamak Power 
Plants,ARIES-AT'', Fusion Engineering and Design, this issue (2001)

\bibitem{Jardin97}  S.~C.~Jardin, C.~E.~Kessel, C.~G.~Bathke, D.~E.~Ehst,
T.~K.~Mau,F.~Najmabadi, T.~W.~Petrie, ``Physics basis for a reversed shear
tokamak power plant'', Fusion Engineering and Design, {\bf 38} (1997) 27


\bibitem{g00509.0*540}EFIT equilibrium reference number g00509.0*540,
General Atomics Company, San Diego, California

\bibitem{Bondeson??} A.~Bondeson,G.~Vlad,H.~Lutjens,``Resistive toroidal
stability of internal kink modes in circular and shaped tokamaks'',
Physics of Fluids B. Plasma Physics, {\bf 4} (1992) 1889

\bibitem{GATO??}  L.~C.~Bernard,F.~J.~Helton,R.~W.~Moore, ``GATO:  An MHD Stability
Code for Axisymmetric Plasmas with Internal Separatrices'',
Comput. Phys. Commun. {\bf 24} (1980) 377

\bibitem{Chance99} M.~S.~Chance,``Vacuum calculations in azimuthally
symmetric geometry'',Physics of Plasmas,{\bf 4} (1997) 2161

\bibitem{Boozer99} A.H. Boozer, ``Feedback equations for the wall
modes of a rotating plasma'', Phys. Plasmas 6, 3180 (1999).

\bibitem{Cates00} C. Cates, M. Shilev, M.E. Mauel, G.A. Navratil, D. Maurer, 
      S. Mukherjee, D. Nadle, and A. Boozer, ``Supression of resistive wall instabilities
with distributed independently controlled active feedback coils'', Phys. Plasmas 7, 3133 (2000).

\bibitem{Garofalo99} A.M. Garofalo, A.D. Turnbull, E.J. Strait, M. E. Austin, 
      J. Bialek, et al., ``Stabilization of the external kink and control
of the resistive wall mode in tokamaks'', Phys. Plasmas 6, 1893 (1999).

\bibitem{LaHaye00} R.J. La Haye, R.J. Buttery, S. Guenter, G.T. A. Huysmans, M. Marascheck, and H.R.
Wilson, ``Dimensionless scaling of the critical beta for onset of a neoclassical tearing mode'',
Phys. Plasmas 7, 3349 (2000).

\bibitem{Grantenbein00} G. Grantenbein, H. Zohm, G. Giruzzi, S. Günter, F. Leuterer, et al., 
``Complete suppression of NTM with ECRH CD in ASDEX Upgrade'',Phys. Rev. Lett. 85, 1242 (2000).

\bibitem{Pletzer98} A. Pletzer and F.W. Perkins, ''Stabilization of Neoclassical tearing 
modes Using a Continuous Localized Current Drive,'' Phys. Plasmas {\bf 6} (1999) p. 1589

\bibitem{Gianakon??}(calculated from code ``TEAR,'' courtesy of T.Gianakon, LANL)

\bibitem{Wilson99} H.R.Wilson, R.L.Miller, et al , ``Ideal MHD stability of the tokamak high-confinement 
mode edge region'', Phys. Plas. 6 1925 (1999)

\bibitem{Pet99} C.~C.~Petty, F.~W.~Baity, J.~S.~deGrassie, et al.,
``Fast Wave Current Drive in H-Mode Plasmas,'' {\it Nucl.\ Fusion} 
{\bf 39} (1999) 1421.

\bibitem{Luc99} T.~C.~Luce, , Y.~R.~Lin-Liu, J.~Lohr, et el.,
``Generation of Localized Non-Inductive Current by Electron Cyclotron 
Waves on the DIII-D Tokamak,'' {\it Phys.\ Rev.\ Lett.} {\bf 83} (1999)
4550.

\bibitem{Thu97} M.~Thumm, ``Recent Development of High Power Gyrotrons
and Windows for EC Wave Applications,'' {\it AIP Conf.\ Proc.} {\bf 403}
(1997) 183.

\bibitem{Mau94} T.~K.~Mau, D.~A.~Ehst, D.~J.~Hoffman, ``The Radio-Frequency
Curent-Drive System for the ARIES-I Tokamak Power Reactor,'' {\it Fusion
Eng.\ and Design} {\bf 24} (1994) 205.

\bibitem{Mau92} T.~K.~Mau, S.~C.~Chiu, R.~W.~Harvey, ``Modelling of Fast
Wave Current Drive in Standard and Second-Stability Bootstrapped
Reactor Plasmas,'' Abst.\ EPS Top.\ Conf. on {\it Radiofrequency Heating
and Current Drive of Fusion Devices}, Brussels, Vol.\ {\bf 16E} (1992)
181.

\bibitem{Pey99} Y.~Peysson, ``Status of Lower Hybrid Current Drive,''
{\it AIP Conf.\ Proc.} {\bf 485} (1999) 183.

\bibitem{Ito99} S.~Itoh, K.~N.~Sato, K.~Nakamura, et al., ``Recent Progress 
on High Performance Steady State Plasmas in the Superconducting Tokamak
TRIAM-1M,'' {\it Nucl.\ Fusion} {\bf 39} (1999) 1257.


\bibitem{Kay00} S.~M.~Kaye, ``Physics Results from the National Spherical 
Torus Experiment,'' {\it Bull.\ Am.\ Phys.\ Soc.} {\bf 45} (2000) 118;
paper to be published in {\it Phys.\ Plasmas} (2001).

\bibitem{Bon84} P.~Bonoli, ``Linear Theory of Lower Hybrid Heating,''
MIT Report PFC/RR-84-5 (1984).

\bibitem{War99} C.~D.~Warrick, M.~Valovic, B.~Lloyd, et al., ``The
Confinement and Stability of Quasi-Stationary High-$\beta$ Plasmas on
COMPASS-D,'' {\it 26th EPS Conf.\ on Contr.\ Fusion and Plasma Phys.}
Maastricht (1999) 153.

\bibitem{Fow79} R.~H.~Fowler, J.~A.~Holmes, J.~A.~Rome, ``NFREYA - 
A Monte Carlo Beam Deposition Code for Noncircular Tokamak Plasmas,''
{\it Oak\ Ridge\ National\ Laboratory\ Report} {\bf ORNL/TM-6845} (1979).

\bibitem{Cox94} M.~Cox, G.~Counsell, M.~R.~O'Brien, et al., ``Heating
and Current Drive in Tight Aspect Ratio Tokamaks,'' Abst.\ 21st EPS
Conf.\ {\it Controlled Fusion and Plasma Phys.}, Montpellier (1994),
vVol.\ {\bf II}, p.\ 1130.

\bibitem{Waltz97} Waltz, R. E., Staebler, G. M., Dorland, W., Hammett, 
G. W., Kotschenreuther, M., and Konings, J.A., ``A gyro-Landau-fluid 
transport model'', Phys. Plasmas 4 (1997) 2482.

\bibitem{Kinsey**} Kinsey, J.E., Waltz, R.E., Schissel, D.P., 
Proc. of the 24th Euro. Conf. on Contr. Fusion and Plasma Phys., 
Berchtesgaden, Germany (European Physical Society, P-III, 1081) (1997)

\bibitem{Hammett90} Hammett, G. and Perkins, F., ``Fluid moment models for 
Landau Damping with application to the ITG instability'', Phys. Rev. 
Lett. 64 (1990) 3019.

\bibitem{Kotschenreuther95} Kotschenreuther, M., Rewoldt, G., and Tang, 
W.M., ``Comparison of initial value and eigenvalue codes for kinetic toroidal 
plasma instabilities'', Comput. Phys. Commun. 88, 128 (1995)

\bibitem{Beer97} Beer, M.A., Hammett, G.W., Rewoldt, G., Synakowski, E.J., 
Zarnstorff, M.C., et al., ``Gyrofluid simulations of turbulence suppression 
in reversed-shear experiments on TFTR'', Phys. Plasmas 4 (1997) 1792.

\bibitem{Waltz95} Waltz, R. E., Kerbel, G. D., Milovich, and J., Hammett, 
G. W., ``Advances in the simulation of toroidal gyro-Landau fluid model 
turbulence'', Phys. Plasmas {\bf 2} 2408 (1995).

\bibitem{Stallard99} Stallard, B.W., Greenfield, C.M., Staebler, G.M., Rettig, 
C.L., Chu, M.S. et al., ``Electron heat transport in improved confinement discharges in DIII-D'',
Phys. Plasmas {\bf 6} (1999) 1978.

\bibitem{Jackson91} Jackson,G.L., Winter, J., Taylor, T.S., Burrell, K.H., DeBoo, J.C., et al., 
``Regime of very high confinement in the Boronized DIII-D tokamak'', Phys. Rev. Lett. {\bf 67} (1991) 3098.

\bibitem{Koide96} Koide Y., et al. , ``Internal transport barrier with improved confinement in the 
JT-60U tokamak'', Plasma Phys. Control. Fusion {\bf 38} (1996) 1011.

\bibitem{Rice00} Rice, J.E., et al ``Observations of Toroidal Rotation Suppression with ITB Formation 
in ICRF and Ohmic H-mode Alcator C-Mod Plasmas'' IAEA Fusion Conference, 
Sorrento, Italy, October, 2000.

\bibitem{Perkins00} Perkins, F.W., et al ``Generation of Plasma Rotation in a Tokamak by Ion-Cyclotrn 
Absorption of Fast Alfven Waves'' IAEA Fusion Conference, Sorrento, Italy, October, 
2000.

\bibitem{Loarte99} A. Loarte, et. al., ``Multi-machine scaling of the divertor peak heat flux and width 
for L-mode and H-mode discharges'', J. Nucl. Mater., {\bf 266-269} (1999) 587.

\bibitem{Petrie00} T.W. Petrie, et. al., ``The Effect of Divertor Magnetic Balance on H-mode 
performance in DIII-D'', Presented at the 14th International Conference on Plasma-Surface 
Interactions in Controlled Fusion Devices, Rosenheim, Germany, May 21-26, 2000 (to 
be published in J. Nucl. Mater.).

\bibitem{Rognlien99} T.D. Rognlien, et. al., ``Influence of ExB and 
$\nabla B$ drift terms in 2D edge/SOL transport simulations'', J. Nucl. 
Mater. {\bf 266-269} (1999) 654.

\bibitem{Boedo00} J.A. Boedo, et al., ``Electric field-induced plasma convection in tokamak divertors'',
Phys. Plasmas {\bf 7} (2000) 1075.
 
\bibitem{Rognlien94}T.~D.~Rognlien, P.~N.~Brown, R.~B.~Campbell , et. al., 
Contrib.Plasma Phys. {\bf 34} (1994)  362.


\bibitem{Rensink00}M. Rensink and T. Rognlien (private communication, 2000).

\bibitem{Wade98} M.R. Wade, et al., ``Impurity enrichment studies with induced scrape-off layer flow 
on DIII-D'',Nucl. Fusion, {\bf 38} (1998) 1839.


\bibitem{Perkins99}F.~W.~Perkins, et. al, ``ITER Physics Basis'', Nucl. 
Fusion {\bf 39} (1999) 2391.

\bibitem{Hulse83}R.A. Hulse, Nuclear Technology/Fusion ``Numerical studies of impurities in fusion 
plasmas'',{\bf 3}, 259 (1983)

\bibitem{Behringer87} K. Behringer, ``Description of the impurity transport code STRAHL'', 
JET-R(87)08, JET Joint Undertaking, Culham (1987)

\bibitem{Peeters99}A.G. Peeters, ``Reduced Charge State Equations that describe 
Pfirsch-Schluter impurity transport in tokamak plasma'', Physics of Plasmas (1999)

\bibitem{Wade00}M.R. Wade, W.A.Houlberg, L.R.Baylor,  `` Experimental 
Confirmation of Impurity Convection Driven by the Ion Temperature Gradient in 
Toroidal Plasmas'', Phys. Rev. Lett., {\bf 84} 282 (2000)

\bibitem{Ite89} ``ITER Conceptual Design Interim Report,''ITER Documentation 
Internal Series No. 7, International Atomic Energy Agency, Vienna (1989).

\bibitem{Uck89} N.~A.~Uckan, ``Tokamak Confinement Projections and 
Performance Goals,''{\it Fusion Technol.} {\bf 15} (March 1989) 391.

\bibitem{Ite99} F.~W.Perkins, et. al. ``ITER Physics Basis,'' 
{\it Nuclear Fusion}, {\bf 39}, 12 (1999) Chapter 2, Table 5. p. 2208.

\bibitem{Yus90} P.~N.~Yushmanov, T.~Takizuka, K.~S.~Riedel, 
O.~J.~W.~F.~Karduan, J.~G.~Cordey, S.~M.~Kaye, and D.~E.~Post, 
``Scalings for Tokamak Energy Confinement,''

\bibitem{Pos90} D.~E.~Post, K.~Borrass, J.~D.~Callen, S.~A.~Cohen, 
J.~G.~Cordey, F.~Engelmann, {\it et al.}, ``ITER Physics,'' International 
Thermonuclear Experimental Reactor documentation series \hbox{No. 21} 
(November 1990).

\bibitem{Kay97} S.~M.~Kaye, {\it et al.}, ``ITER L Mode Confinement Database,''
{\it Nuclear Fusion}, 37, 9 (1997) 1303.

\bibitem{Hou82} W.~A.~Houlberg, S.~E.~Attenberger, and L.~M.~Hively, 
``Contour Analysis of Fusion Reactor Plasma Performance,'' 
{\it Nuclear Fusion}, 22, 7 (1982) 935.

\end{thebibliography}

\end{document}


