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\begin{center}
{\LARGE \bf
Plasma Profile and Shape Optimization  \\
for the Advanced Tokamak Power Plant, ARIES-AT}
\end{center}
\bigskip
\bigskip
\begin{center}
{\Large \bf
C. E. Kessel\footnote[1]{Princeton Plasma Physics Laboratory, P.O. Box 451,
Princeton, NJ 08543 USA},
T. K. Mau\footnote[2]{Center for Energy Research, University of California San Diego,
9500 Gilman Drive, La Jolla, CA 93037 USA},
S. C. Jardin\footnotemark[1], F. Najmabadi\footnotemark[2]}
\end{center}
\bigskip
\bigskip

\begin{abstract}
An advanced tokamak plasma configuration is developed based on equilibrium,
ideal MHD stability, bootstrap current analysis, vertical stability and
control, and poloidal field coil analysis.
The plasma boundaries used in the analysis are forced to coincide with the
99\% flux surface from the free-boundary equilibrium.
Using an accurate bootstrap
current model and external current drive
profiles from ray tracing calculations
in combination with optimized pressure profiles, $\beta_{N}$ values above
7.0 have been obtained.  The minimum current drive
requirement is found to lie at a lower $\beta_{N}$ of 5.4.
The external kink mode is stabilized by a
tungsten shell located at 0.33 times
the minor radius and a feedback system.
Plasma shape optimization has led to an elongation of 2.2 and triangularity
of 0.9 at the separatrix.  Vertical stability could be achieved by
a combination of tungsten shells located at 0.33 times the minor radius
and feedback control coils located behind the shield.  The poloidal
field coils were optimized in location and current, providing
a maximum coil current of 8.6 MA.
These developments have led to a simultaneous reduction in the power
plant major radius and toroidal field.
\end{abstract}

\section{Introduction}
\hspace{3ex}
The simultaneous achievement of high $\beta$ (at high plasma current), high
bootstrap fraction, and the transport suppression consistent with these features
was first shown in ref\cite{Kes1}.  An overview of experimental and theoretical
results, excluding the most recent, was given in ref\cite{Jar1}.
The reversed shear configuration for the
tokamak has the potential to be an economical power plant \cite{Jar1}, and its
features, referred to as the advanced tokamak, are being pursued in several
tokamak experiments\cite{Bec}-\cite{Gru}.
Previous work reported in ref\cite{Jar1} obtained attractive $\beta$ values and
reasonable current drive requirements.  However, it is of interest to further
improve such configurations to understand the potential benefits and identify
the highest leverage research areas.
The present studies will show that accurate bootstrap models are
necessary to determine MHD stability and current drive requirements.
In addition, requiring the plasma boundary to coincide with the 99\%
flux surface of the free-boundary plasma enforces consistency and
has led to higher $\beta$ limits.  The plasma pressure profile is
optimized to provide high ballooning $\beta$ limits and bootstrap
current alignment simultaneously.  Plasma shaping has been utilized,
within engineering constraints,
to increase the $\beta$, both through increases in $\beta_{N}$
($\beta_{N}$=$\beta$aB$_{T}$/I$_{p}$, a = minor radius, B$_{T}$ =
toroidal field, and I$_{p}$ = plasma current) and
plasma current.
Vertical stability and control analysis have found a reasonable
solution for the passive stabilizer and feedback control power.
Poloidal field (PF) coil optimization has been performed and meets
all engineering constraints.

\section{Equilibrium and Stability Studies}
\hspace{3ex}
The fixed boundary flux-coordinate equilibrium code JSOLVER \cite{Del1} is used
with 257 flux surfaces and 257 theta points from 0 to $\pi$.  The n=$\infty$ ballooning
stability is analized with BALMSC \cite{Gre1}, and PEST2
\cite{Gri1} is used for low-n external kink stability.  For all cases reported
here, a conducting wall is assumed to stabilize the external kink modes,
and this wall location is determined.  In addition, $n=0$
vertical stability is assessed with Corsica \cite{Han1}.

The final optimized reversed shear plasma is shown in Fig. 1, with various equilibrium
profiles, and global plasma parameters are given in Table 1, under
ARIES-AT, where they are compared to ARIES-RS\cite{Jar1}.  The plasma
boundaries used in the fixed-boundary equilibrium calculations are taken from
free-boundary equilibria at the 99\% poloidal magnetic flux surface, in which
$q_{95}$ is kept $\geq$ 3.0.  We
attempt to use a flux surface as close to the diverted plasma separatrix as
possible, limited by the ability of the fixed-boundary equilibrium and
ideal MHD stability codes to handle the angular boundary near the X-point.
This approach can have a strong impact on the calculated stable $\beta$ values
due to progressively stronger plasma shaping as one approaches the
separatrix.  Plotted in Fig. 2 are the elongation and triangularity
for a series of flux surfaces from the 95\% to 99.5\%.
The cases shown have a plasma internal self-inductance (li) of 0.46.
Both elongations and triangularities would increase with lower li,
and decrease with higher li.
Later the impact of stronger shaping will be discussed.
In addition,
this provides strict consistency between the PF coil generated free-boundary
equilibrium and the fixed-boundary equilibria, which is important due to
the non-analytic boundaries that are generated with actual coils.

In order to calculate free-boundary equilibria for the advanced plasmas being
considered, it was necessary to make modifications to the methods used.
The free-boundary equilibrium equation to be solved is given by
the Grad-Shafranov equation,
\begin{equation}
\Delta^{*} \psi = -\mu_{o} R j_{\phi}
\end{equation}
\begin{equation}
j_{\phi} = -R \frac{dp}{d\psi} - \frac{1}{2 \mu_{o} R} \frac{dg^{2}}{d\psi}
\end{equation}
where $\psi$ is the poloidal flux function (equal to the actual poloidal
flux divided by $2 \pi$), $j_{\phi}$ is the toroidal current
density, $p$ is the pressure, and $g$ is the toroidal field function (equal to
$RB_{\phi}$).  For reversed shear plasmas at high pressure the two terms
in the definition of $j_{\phi}$ have opposite signs cancelling to some degree.
Physically, the toroidal current density is low in the plasma core due to the
hollow current profile, but is also shifting to the outboard side as the pressure
increases.  This cancellation between the two terms is critical to obtaining
proper force balance, and is not easily achieved for arbitrary choices of the
functions $p(\psi)$ and $g(\psi)$.  In order to produce these equilibria and
have them represent the desirable fixed boundary configurations with high
$\beta$ and high bootstrap current fractions, we recast the toroidal current
density in terms of the pressure and parallel current density,
\begin{equation}
j_{\phi} = -R \frac{dp}{d\psi} \left( 1 - \frac{B_{\phi}^{2}}{\langle B^{2}
\rangle} \right) + B_{\phi} \frac{\langle j \cdot B \rangle}{\langle
B^{2} \rangle}
\end{equation}
The first term is the combination of the Phirsch-Schluter and diamagnetic currents,
and the second term is any driven parallel current that exists (i.e. ohmic, bootstrap,
RFCD).
Now both terms are positive and don't require cancellation at high pressure,
and specifiying the parallel current density is how
fixed boundary equilibria are calculated in JSOLVER.
This specification of the $j_{\phi}$ requires
flux surface averages to be calculated at each iteration, which slows the
calculation down, however, with this approach the pressure, current, and
safety factor profiles can be accurately represented between fixed and
free-boundary equilibria.  In addition, one can obtain difficult equilibria
not accessible with the original formulation.

There are two types of fixed boundary equilibrium calculations used, target equilibria and
self-consistent equilibria.  Target equilibria have the total parallel
current density and pressure profile prescribed.  These are used to scan MHD
stability more efficiently for plasma shape and profile optimization.
The bootstrap current profile is calculated\cite{Kes2} at
the end of the equilibrium calculation, and is monitored for alignment with
the prescribed current profile.
Self-consistent equilibria have only the pressure profile and the
parallel current density profiles from current drive sources (i.e. FWCD and LHCD)
prescribed.  The bootstrap current density is calculated at each iteration
of the equilibrium, which is added to the current drive sources to provide
the total parallel current density.  These equilibria are used to develop
final configurations, by prescribing the required current drive profiles
and interating with the actual current drive deposition profiles from ray tracing
calculations reported in ref\cite{Jar3}.

\section{Current Profiles and Bootstrap Current}
\hspace{3ex}
The parallel current density profile used in target equilibrium calculations
is given by the following form,
\begin{equation}
\frac{\langle j \cdot B \rangle}{\langle B \cdot \nabla \phi \rangle} =
j_{o} \left( 1 - \hat{\psi} \right) + j_{1} \frac{d^{2} \hat{\psi}^{a_{j}}
( 1 - \hat{\psi} )^{b_{j}}}{(\hat{\psi} - \hat{\psi}_{o}) + d^{2}}
\end{equation}
where $\hat{\psi}$ is the normalized poloidal flux that is zero at the magnetic
axis and 1.0 at the plasma edge.
The parameters are chosen to generate a hollow current profile.  The
particular form is chosen by considering the ability to reproduce the profile
shape with bootstrap current and the need for external current drive.
The location of the $q_{min}$ (minimum safety factor) is chosen as close
to the plasma edge
as possible to obtain high ballooning $\beta$-limits, while avoiding the
degradation of the magnetic shear near the plasma edge, and to stay within
external current drive limitations (i.e. LH current penetration).

For the self-consistent equilibrium calculations only the external current
drive profiles are input, and they are described by terms that are the same
as the second term in Eqn. 4.  An example of these appears in Fig. 1,
where the parallel current density is shown, and the on-axis FWCD
and off-axis LHCD profiles are shown.
The bootstrap current is calculated self-consistently in Fig. 1.

It was found by comparing full collisional bootstrap calculations\cite{Kes2}
to the single ion collisionless calculation\cite{Hir1}, that the external
current drive required in the two cases is quite different.
A comparison was made where two
equilibria with $\approx$ 100\% bootstrap current,
all other plasma parameters the same, and $\beta_{N}$ of 5.35,
shown in Fig. 3.
The collisionless case ignores the effect of decreasing temperature near
the plasma edge, which would strongly reduce the bootstrap current there,
and has an overall shift of the current profile outward.
This causes $q_{min}$ to be further out in minor radius, which for a fixed
pressure profile, will give a higher ballooning $\beta$-limit.  The
collisionless model is misleading because it indicates that no
off-axis current drive is required to achieve ballooning stability at
this $\beta_{N}$.  The collisional model, on the other hand, would
require off-axis current drive of about 10\% of the total plasma current
to obtain ballooning stability at this $\beta_{N}$.
The difference in the location of $q_{min}$ for the two bootstrap models
is critical to assessing the ideal $\beta$-limit and the magnitude of
off-axis current drive.  The use of a collisionless model should be avoided
in configuration development because of its overly optimistic predictions
for bootstrap current near the plasma edge.

The bootstrap current calculation requires knowledge of the pressure profile
and either the density or temperature profile.  In the present work
particle and energy transport
calculations were not done.  Rather, the density and temperature profiles were
constrained to have their dominant gradients in the vicinity of the
$q_{min}$ consistent with the presence of an
internal transport barrier (ITB)\cite{Bec}-\cite{Gru}.
This approach was chosen due to the widely varying characteristics observed
on various reversed shear tokamak experiments, with the exception of
this dominant gradient.
Beyond this, the profiles are chosen to optimize the bootstrap current
alignment, and typical profiles for these plasmas are shown in Fig. 1.
Ref\cite{Jar3} presents a comparison of the assumed profiles and those from
GLF23 theoretical predictions indicating reasonable agreement.
The density and temperature profiles assumed in this study show gradients that
are spread out more than is observed experimentally, and may require some form
of ITB control to achieve, but this is beyond the scope of the present work.

\section{Pressure Profile Optimization}
\hspace{3ex}
The pressure profile is described by the following,
\begin{equation}
p(\psi) = p_{o} \left[ c_{1} (1-\hat{\psi}^{b_{1}})^{a_{1}} +
c_{2} (1 - \hat{\psi}^{b_{2}})^{a_{2}} \right]
\end{equation}
where the coefficients are chosen to reduce the pressure gradient in the
region outside the $q_{min}$ to improve ballooning stability, and to provide
bootstrap current alignment to reduce the magnitude of external current drive.
The pressure gradient is zero at the plasma edge for these profiles,
making these typical of an L-mode edge.
Previously\cite{Jar1}, the pressure was parameterized by a simpler form,
\begin{equation}
p(\psi) = p_{o} (1-\hat{\psi}^{b_{1}} )^{2}
\end{equation}
which was restrictive in optimizing both the ballooning stability and
bootstrap current simultaneously.

A sequence of
pressure profiles was determined, using Eqn. 5,
that had progressively higher $\beta$-limits,
due to a progressively smaller pressure gradient in the ballooning unstable
region.  This is shown in Fig. 4, where the pressure gradient is plotted as
a function of poloidal flux.
The region of ballooning instability is noted, and the remaining part of the
pressure profile is made to provide bootstrap alignment.
The smaller pressure gradients near the plasma edge,
to increase $\beta$, led to smaller bootstrap
current there causing higher external current to be required to provide stability.
In addition, the highest $\beta$'s led to excessive bootstrap current in the
plasma core, which requires a broader density profile to eliminate it, exacerbating
the current drive problem near the plasma edge.
These two factors
cause the current drive requirement to begin increasing as $\beta$ rises
sufficiently high.
At lower $\beta$ values there is insufficient bootstrap current to provide
a large bootstrap fraction, within our transport constraints, leading to
large current drive requirements.

The full stabilization of the external kink mode requires either plasma
rotation\cite{Bon1} or feedback control\cite{Liu1}.  Plasma rotation is difficult
to provide for reactor size plasmas so the feedback approach is taken.
For this approach the shell is necessary to slow the mode growth rate to
time scales which the feedback coils can respond with reasonable power.
This is discussed in ref\cite{Jar3}.
If no transport constraints are imposed, one can find very broad temperature
and very peaked density profile solutions that can provide the bootstrap
current at low $\beta$, but these are considered unphysical.
Due to these effects the required externally driven current
as a function of $\beta_{N}$ shows a minimum, and this is shown in Fig. 5.
From ray tracing calculations\cite{Jar3} the
$\beta_{N}$ of 6.0 was chosen as the best tradeoff between maximizing $\beta$
and minimizing current drive power.

\section{Plasma Shape Optimization}
\hspace{3ex}
The plasma triangularity and elongation have a strong impact on the ideal
MHD stability.  In addition to providing higher $\beta_{N}$ values they also
allow higher plasma current to be driven in the plasma for $q_{95}$
constrained to be $\geq$ 3.0.

The plasma triangularity has been limited by the need to provide
a sufficiently long slot divertor on the inboard side to obtain a detached
plasma and radiate the power there.
In a power plant, the divertor slot must cut through blanket and shielding,
which allows neutrons to penetrate close to the superconducting
toroidal field magnet.  This
sets a limit on the triangularity (or the angle that the separatrix flux
line can make with the inboard wall) to provide neutron sheilding.
However, recently both experimental and theoretical
results indicate that the inboard plasma detachment
can be obtained without a slot, although a short slot is still used to
disperse the heat load.
This allows us to take advantage of higher triangularities.  To assess the
benefits, a scan of the triangularity was done, for two elongations.  For
this scan the plasma boundary is given analytically, the edge safety factor
is held fixed at 3.5, the aspect ratio is 4.0, and one of the pressure profiles
described above was used.  Shown in Fig. 6 is the $\beta_{N}$ and $\beta$ as
a function of triangularity, for $n=\infty$ ballooning modes.
Included is the result given earlier\cite{Jar1},
with a more restrictive pressure profile, Eqn. 6.  It is clear that
$\beta_{N}$ improves with higher triangularity, but a rollover occurs
beyond a triangularity of 0.65.  However, the $\beta$ continues to rise due
to the continued increase in the plasma current.  The combination of higher
elongation creates an even stronger increase in $\beta$.

The plasma elongation is limited by the $n=0$ vertical instability, through
the ability to provide conducting structure close enough to the plasma and
a vertical position feedback system with reasonable power.  For power plant
designs the location of a conducting material is limited to lie outside
the blanket, and typically resides in the region between the blanket and shield.
The plasma elongation can substantially increase $\beta$ due to a
$(1+\kappa^{2})$ scaling.  Recently, optimization of the blanket\cite{Elg1}
has allowed the conducting structure to be moved closer to the plasma,
and actually be in the blanket, although,
it must exist in a very high temperature environment ($\geq$ 1000$^{o}$C).
The benefits of increasing the plasma elongation above the values previously
found\cite{Jar1} for a conducting structure behind the blanket
were examined.
As for the triangularity scan, analytic plasma boundaries were used, with
a range of triangularities between 0.4 and 0.85, aspect ratio of 4.0,
the edge safety factor fixed at 3.5, and
one of the pressure
profiles described by Eqn. 5.  Shown in Fig. 7 are $\beta_{N}$ and $\beta$
as a function of elongation, for $n=\infty$ ballooning modes.
The $\beta_{N}$ shows a decrease with increasing elongation at lower
triangularity, and the opposite at higher triangularity.  Although, at
higher triangularity the increase in $\beta_{N}$ turns over and begins
to decrease.  The point where the $\beta_{N}$ increase turns over moves
to higher elongations as the triangularity is increased.
As was seen for the triangularity scan, the $\beta$ values
continue to increase regardless of the structure in the $\beta_{N}$ versus
elongation curves, because the increase in plasma current is so strong.
The slope of the increase in $\beta$ versus elongation continues to improve
with higher triangularity over the whole range.

The low-n external kink stability was also analized, and is shown in
Fig. 8.  Here the marginal wall location is given as a function of
elongation for toroidal mode numbers from 1 to 6.  The wall locations
are only resolved to 0.025 times the minor radius.  Higher toroidal mode
numbers require closer walls for stabilization, and all mode number
wall locations move much closer to the plasma as the elongation exceeds
a value of 2.3.  The triangularity for this case was fixed at 0.7.
In order to observe the impact of triangularity on the kink mode wall
stabilization, three other values were analized; 0.4, 0.55, and 0.85.
Shown in Fig. 9 are the marginal wall locations for toroidal mode
number n=1, as a function of elongation.  It is clear that
higher triangularity results in marginal wall locations that are farther
from the plasma at the lower elongations, however, the stability at the
highest elongations is rapidly degrading.
Shown in Fig. 10 are the marginal wall locations for toroidal mode
numbers n=1-5, at a fixed elongation, and varying triangularity, showing
that the improvement with triangularity persists at higher n.
As was pointed out in ref\cite{Kes1}, when a stablizing wall is present
there is typically a toroidal mode number greater than 1
that is the most limiting
to $\beta_{N}$.  Although the curves in Fig. 8 do not indicate a minimum,
the analysis for a triangularity of 0.7, and elongation of 2.2 was extended
to toroidal mode numbers up to 9 to resolve this limiting mode.
Shown in Fig. 11 is the marginal wall location as a function of toroidal
mode number, at two different $\beta_{N}$ values, with $\beta_{N}$
corresponding to the final plasma configuration.  As the pressure is
increased the $\beta$-limiting toroidal mode number moves to higher
values and the wall must move closer to the plasma to stablize all
n.

Based on the plasma shape analysis, values for the elongation and
triangularity at the separatrix were chosen to be 2.2 and 0.9,
respectively.  This integrated the ballooning and kink stability
behavior, requiring a shell for the kink mode stabilization at
0.33 times the minor radius, and avoiding the rapid degradation
in kink stability at higher elongations.  As will be shown in the
next section, the shell for vertical stability is placed at the
same location, and therefore provides full coverage on the plasma
outboard side as required for kink stabilization.
The plasma elongation choice was also dependent
on vertical stability analysis that follows.

\section{Axisymmetric Stability}
\hspace{3ex}
The plasma elongation is ultimately limited by the vertical instability,
through the ability to locate conducting structure sufficiently close to
the plasma and provide a feedback control system with reasonable power.
Due to the high leverage of plasma elongation in increasing $\beta$,
scans were done to determine the plasma growth rate, and stability factor
as a conducting shell is moved progressively further away from the plasma,
for various plasma elongations.
The stability factor is defined as,
\begin{equation}
f_{s} = 1 + \tau_{g} / \tau_{L/R}
\end{equation}
where $\tau_{g}$ is the vertical instability growth time, and $\tau_{L/R}$
is the longest up-down asymmetric time constant of the surrounding structure.
The conducting shell used in the analysis
is vertical on the inboard side, and approximately
contouring the plasma boundary on the outboard side, with gaps at the
top and bottom for the divertor.  Fig. 12 shows this generic structure model,
which is toroidally continuous.  For the curves shown in Fig. 13, the conductor
was tungsten, 0.035 m thick, with a resistivity of 8$\times$10$^{-8}$ ohm-m.
In addition, the plasma has a $\beta_{p}$ of 0.25 and $l_{i}$ of 0.8, typical
of plasmas during rampup which are the most unstable.
Fig. 13 shows that as the plasma elongation is increased the growth rate
increases, and that the shell must be located closer to the plasma to
provide any influence on the plasma.  As the shell is moved further away,
initially the growth rate changes slowly, but later begins increasing rapidly.
Where this curve asymptotes is called the critical ideal wall location, and
the stability factor is approaching 1.0.  If the shell is located outside
this location, it will not influence (slow down) the vertical instability
even if it were superconducting.  If the shell is located at this location
or closer to the plasma, and it were superconducting it would stabilize the
plasma.  However, actual structural materials are resistive, so a more
useful shell location is the critical resistive wall location.  This is defined
as the wall location that provides a stability factor of 1.2, and
coincides with a shell location at the knee in the growth rate versus
shell distance curve.
Fig. 13 shows the stability factor for the same plasma elongations and
wall distances with $f_{s}$=1.2 denoted on the plot.

The typical location for a conducting shell in power plant designs
is between the blanket and the
shield, which is roughly at a normalized distance of 0.45 times the minor
radius, the precise value depending on the design.
However, recent optimizations of the blanket for neutronics has allowed
the conductor to be located inside the blanket, closer to the plasma.  This
location is found to be about 0.3-0.35 times the minor radius.  For a shell
at this location, elongations at the separatrix
up to 2.2-2.3 can be considered, as opposed to those in ref\cite{Jar1}
which could not exceed 1.9.
The shells used in the scoping studies above are not practical
since they surround the plasma
and will adversely affect the neutronics.  The shells are reduced in size
by removing the section closest to the midplane, which has the weakest affect
on vertical stability.  This is done until the vertical stabilization provided
by the shells starts to significantly degrade.
The dark lines in Fig. 12 through the inboard and outboard
shells show how much of the shells is removed for a final shell design,
which in this case is $\pm$60$^{o}$ on the outboard measured from the
midplane.  The inboard shell is reduced so that it has the same vertical
extent as the outboard plate.

Once the plasma elongation is chosen, $\kappa$(X-point)=2.2 in this case,
and the final shell design is provided,
more detailed vertical stability analysis is done to examine the impact
of pressure and current profile on the growth rate.  In addition, the vertical
control calculations are done with this configuration to determine the maximum
current and voltage expected during control, using the Tokamak Simulation
Code\cite{Jar2}.
It should be noted that since the stabilizing shells are located in the
blanket they must operate at high temperatures and can not be actively
cooled.  The operating temperature turned out to be about 1100$^{o}$C.
The resulting shell was 4.0 cm thick, and had a resistivity of about
35.0$\times$10$^{-8}$ ohm-m.
Shown in Fig. 14 is the vertical instability growth rate as a function of
$l_{i}$ and $\beta_{p}$, with the final reference plasma denoted by the $\times$.
Also shown is the feedback control power (the product of peak current
and peak voltage) for a random disturbance with 0.01 m RMS displacement of the
plasma vertically, giving 30 MVA for the peak feedback power.
Again, the reference plasma is denoted.  The feedback power
that can be tolerated determines the maximum vertical instability growth
rate that is allowed, which is 45 /s.
Consequently, the range of plasma pressure and current
profiles that can be produced at full elongation is limited, and
not to exceed the maximum growth rate.
Plasmas with pressure and current profiles outside the range can be produced
but only with lower elongation (which reduces the growth rate) or lower plasma
current (which reduces the feedback power for a given growth rate).
It should be noted that approximately 85\% of the power for vertical
position control is reactive, and therefore can be recovered with
a suitable energy storage system, so that this power is not included
in the recirculating power for the power plant.

\section{Poloidal Field Coil Optimization}
\hspace{3ex}
The poloidal field (PF) coil currents are determined to force the plasma
boundary to pass through specified points in space, the desired outboard
and inboard major radii, and produce a zero poloidal field at the desired X-point.
This is accomplished with a least-squares solution.  Otherwise the
plasma boundary is allowed to take on the shape that minimizes the
coil stored energy.  In addition, the PF coil locations are optimized
to minimize an energy measure equal to the sum of the coil major radius
times the coil current squared, which is found to scale with the coil
cost.
This is done by surrounding the plasma by a large number of PF coils,
avoiding regions where coils can not exist (i.e. outboard midplane
for maintenance).  The coils are
eliminated one by one and
the resulting increase in the coil energy measure is determined.
The coil that increases
this measure the least is eliminated, and the process repeated with the
remaining coils until a certain number of coils is obtained or the
coil energy measure begins increasing rapidly.  Fig. 15 shows the coil
energy measure as function of the number of coils, and it begins to
increase strongly below about 14 coils.  It should be noted that there
are 7 coils used to provide a thin solenoid on the inboard side, for
inductive startup and plasma shaping, that are not included in the
elimination.  The final PF coil arrangement is shown in Fig. 15.  The
maintenance requires that entire sectors be removed through the
outboard midplane, which forced the PF coils to be above 5.0 m.
The maximum current of 8.6 MA occurs in the largest radius coil,
and Table 2 shows the coil locations and currents.


\section{Conclusions}
\hspace{3ex}
The advanced tokamak plasma configuration has the potential to
provide a high $\beta$ and high bootstrap current fraction, resulting
in a more compact economical power plant.  The present work has
utilized high resolution equilibrium and ideal MHD stability calculations
in combination with a full velocity space bootstrap treatment to
analize the impact of pressure profile and plasma shape optimization.
In addition, plasma boundaries used in fixed boundary equilibria are
provided by the 99\% flux surface of the corresponding free-boundary equilibria.

Pressure profile optimization allowed the $\beta_{N}$ for ballooning
instabilities to be maximized, while simultaneously providing bootstrap
current alignment.  Optimization of the plasma shape allowed the
$\beta_{N}$ to be increased, but more importantly allowed the plasma
current to be increased, leading to a $\beta$ nearly twice that in
previous studies\cite{Jar1}.
It was found that the minimum current drive requirement did not occur
at the maximum $\beta$ due to a combination of bootstrap underdrive
near the plasma edge and overdrive near the plasma center.
The external kink mode is assumed to be stabilized by a combination of
a conducting shell, located at 0.33 times the minor radius, and
a feedback control system with coils located behind the shield.
The vertical instability is slowed down
by a conducting shell also located at 0.33 times the minor radius, and
a feedback control coils located behind the shield.
The resulting plasma configuration has a separatrix elongation and triangularity
of 2.2 and 0.9, respectively.
The maximum $\beta$ is 10\%, with $q_{95}$ $\geq$ 3.0, resulting in
a self-driven (bootstrap + diamagnetic + Phirsch-Schluter) current
of 91\%.  In order to provide margin between the operating pressure
and the ideal MHD limit, the $\beta$ is reduced to 90\% of its maximum
value.
The physics improvements noted here have made a significant contribution
to the reduction in the power plant cost of electricity (COE) reported
elsewhere\cite{Mil1}, by increasing $\beta$ and reducing the external
current drive power.

\clearpage

\begin{thebibliography}{99}

\bibitem{Kes1}C. E. Kessel, J. Manickam, G. Rewoldt, and W. Tang, Improved
Plasma Performance in Tokamaks with Negative Magnetic Shear, Phys. Rev. Lett.,
72 (1994) 1212.

\bibitem{Jar1}S. C. Jardin, C. E. Kessel, C. G. Bathke, D. A. Ehst, T. K. Mau,
F. Najmabadi, T. W. Petrie, and the ARIES Team, Physics Basis for a Reversed
Shear Tokamak Power Plant, Fus. Engr. and Design, 38 (1997) 27.

\bibitem{Lit1}X. Litaudon, et al., Stationary Magnetic Shear Reversal
Experiments in Tore Supra, Plas. Phys. Control. Fusion, 38 (1996) 1603.

\bibitem{Gor1}C. Gormezano, High Performance Tokamak Operation Regimes,
Plas. Phys. Control. Fusion, 41 (1999) B367.

\bibitem{Bec}A. Becoulet, et al., Performance and Control of Optimized
Shear Discharges in JET, Nuc. Fus., 40 (2000) 1113.

\bibitem{Ish1}S. Ishida and the JT-60 Team, JT-60U High Performance Regimes,
Nuc. Fus., 39 (1999) 1211.

\bibitem{Oik}T. Oikawa and the JT-60 Team, High Performance and Prospects
for Steady State Operation of JT-60U, Nuc. Fus., 40 (2000) 1125.

\bibitem{Cha}V. S. Chan, et al., DIII-D Advanced Tokamak Research
Overview, Nuc. Fus., 40 (2000) 1137.

\bibitem{Pet1}C. C. Petty, et al., Advanced Tokamak Physics in DIII-D,
Plas. Phys. Control. Fusion, 42 (2000) B75.

\bibitem{Wol1}R. C. Wolf, et al., Stationary Advanced Scenarios with
Internal Transport Barrier on ASDEX Upgrade, Plas. Phys. Control. Fusion,
41 (1999) B93.

\bibitem{Gru}O. Gruber, et al., Steady State H-mode and Te$\approx$Ti
Operation with Internal Transport Barriers in ASDEX Upgrade, Nuc. Fus.,
40 (2000) 1145.

\bibitem{Del1}J. Delucia, S. C. Jardin, and A. M. M. Todd, An Iterative
Metric Method for Solving the Inverse Tokamak Equilibrium Problem,
J. Comp. Phys., 37 (1980) 183.

\bibitem{Gre1}J. M. Greene and M. S. Chance, The Second Region of Stability
Against Ballooning Modes, Nucl. Fusion, 21 (1981) 453.

\bibitem{Gri1}R. C. Grimm, R. L. Dewar, and J. Manickam, Ideal MHD
Stability Calculations in Axisymmetric Toroidal Coordinate Systems,
J. Comp. Phys., 49 (1983) 94.

\bibitem{Han1}S. W. Haney, L. D. Pearlstein, R. H. Bulmer, and
J. P. Friedberg, Vertical Stability Analysis of Tokamaks Using a
Variational Procedure, Plas. Phys. Reports, 23 (1997) 789.

\bibitem{Kes2}C. E. Kessel, Bootstrap Current in a Tokamak, Nuc. Fus.,
34 (1994) 1221.

\bibitem{Jar3}S. C. Jardin, C. E. Kessel, T. K. Mau, et al., Physics Basis for
the Advanced Tokamak Power Plant, Fus. Engr. and Design, this issue.
Fus. Engr. and Design, this issue.

\bibitem{Hir1}S. P. Hirshman, Finite Aspect Ratio Effects on the
Bootstrap Current in Tokamaks, Phys. Fluids, 31 (1988) 3150.

\bibitem{Elg1}L. Elguebaly, et al., Neutronics Analysis for ARIES-AT,
Fus. Engr. and Design, this issue.

\bibitem{Bon1}A. Bondeson and D. J. Ward, Stabilization of External Modes in
Tokamaks by Resistive Walls and Plasma Rotation, Phys. Rev. Lett., 72 (1994) 2709.

\bibitem{Liu1}Y. Q. Liu, A. Bondeson, C. M. Fransson, et al.,
Feedback Stabilization of Non-Axisymmetric
Resistive Wall Modes in Tokamaks.  Part 1: Electromagnetic Model,
Phys. Plas., 7 (2000) 3681.

\bibitem{Jar2}S. C. Jardin, N. Pomphrey, and J. Delucia, Dynamic Modeling
of Transport and Position Control of Tokamaks, J. Comp. Phys.,
66 (1986) 481.

\bibitem{Mil1}R. Miller, et al., Systems Analysis for ARIES-AT, Fus. Engr.
and Design, this issue.

\end{thebibliography}

\clearpage

\begin{table}
\caption{}
\centering
{\large \bf ARIES-RS\cite{Jar1} and ARIES-AT Global Parameters}
\vspace{1.cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{l c c}
\hline
 & ARIES-RS & ARIES-AT  \\
\hline
$I_{p}$ (MA)  & 11.3 &12.8 \\
$B_{T}$ (T) & 7.98 & 5.86 \\
R (m)  & 5.52 & 5.20 \\
a (m)  & 1.38 & 1.30  \\
$\kappa^{*}$ & 1.70 & 2.15 \\
$\delta^{*}$ & 0.50 & 0.78  \\
$\kappa$ (Xpt) & 1.90 & 2.20 \\
$\delta$ (Xpt) & 0.70 & 0.90  \\
$\beta_{p}$  &  2.29 & 2.28  \\
$\beta$ (\%)  &  4.98 & 9.07  \\
$\beta_{\ast}$ (\%) & 6.18 & 11.0 \\
$\beta_{N}$ (\%) & 4.84 & 5.40 \\
$\beta_{N}^{max}$ (\%) & 5.35 & 6.00 \\
$q_{o}$ (axis)  & 2.80 & 3.50  \\
$q_{min}$ (minimum) & 2.50 & 2.40 \\
$q_{e}$ (edge)$^{*}$  & 3.52 & 3.70    \\
$I_{bs}$ (MA)  & 10.0 & 11.4  \\
$I_{\nabla p}/ I_{p}$  & 0.91 & 0.91 \\
$I_{CD}$ (MA) & 1.15 & 1.25 \\
$q_{\ast}$   &   2.37 & 1.85 \\
$l_{i}(3)$  & 0.42 & 0.29  \\
$n_{o}/\langle n \rangle$ & 1.36 & 1.34 \\
$T_{o}/\langle T \rangle$ & 1.98 & 1.72 \\
$p_{o}/\langle p \rangle$ & 2.20 & 1.93 \\
$(b/a)_{kink}$ & 0.25 & 0.33 \\
\hline
*value corresponds to  & & \\
fixed boundary equilibrium & &
\end{tabular}
\end{table}

\clearpage

\begin{table}
\caption{}
\centering
{\large \bf ARIES-AT Poloidal Field Coil Parameters}
\vspace{1.cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{l @{\hspace{2cm}} c @{\hspace{1cm}} c @{\hspace{1cm}} c}
\hline
coil \# & R(m) & Z(m) & I(MA)   \\
\hline
1  & 2.25 & 0.25 & -0.620  \\
2 & 2.25& 0.75 & -1.053 \\
3 & 2.25 & 1.25 & -1.513 \\
4 & 2.25 & 1.75 & -0.665  \\
5 & 2.25 & 2.25 & -0.665 \\
6 & 2.25 & 2.75 & 1.184  \\
7 &  2.25 & 3.25 & 3.360  \\
8 &  3.25 & 5.75  & 6.348  \\
9 & 3.75 & 6.00 & 6.518 \\
10 & 5.25 & 6.30 & 4.810 \\
11 & 5.75 & 6.25 & 3.643  \\
12 & 7.50 & 5.65 & -3.276 \\
13 & 8.00 & 5.40 & -5.877 \\
14 & 8.50 & 5.10 & -8.624  \\
\hline
\end{tabular}
\end{table}

\clearpage


Figure 1 -- {Equilibrium profiles for ARIES-AT, showing the plasma
pressure, safety factor, parallel current, temperature, density, and poloidal
flux.}
\vspace{0.5cm}

Figure 2 -- {Variation of the plasma elongation and triangularity
as one approaches the separatrix, where the elongation is 2.2 and triangularity
is 0.9.}
\vspace{0.5cm}

Figure 3 -- {Comparison of collisionless (a) and collisional bootstrap (b)
formulations for nearly 100\% bootstrap fraction, showing that the collisionless
form provides for a minimum safety factor that is closer to the plasma edge,
resulting in no current drive requirement.}
\vspace{0.5cm}

Figure 4 -- {A series of pressure gradient profiles as a function of
poloidal flux showing how
the gradient is reduced in the ballooning unstable region to obtain successively
higher $\beta_{N}$ values, while keeping bootstrap alignment.}
\vspace{0.5cm}

Figure 5 -- {Total externally driven current required as a function
of $\beta_{N}$, showing that a minimum exists, and that the highest $\beta$ is
not associated with the lowest current drive.}
\vspace{0.5cm}

Figure 6a -- {$\beta_{N}$ as a function of the plasma triangularity.}
\vspace{0.5cm}

Figure 6b -- {$\beta$ as a function of the plasma triangularity.}
\vspace{0.5cm}

Figure 7a -- {$\beta_{N}$ as a function of the plasma elongation.}
\vspace{0.5cm}

Figure 7b -- {$\beta$ as a function of the plasma elongation.}
\vspace{0.5cm}

Figure 8 -- {Stabilizing wall location as a function of plasma elongation
with the triangularity fixed at 0.7, and for toroidal mode number from 1-6.}
\vspace{0.5cm}

Figure 9 -- {Stabilizing wall location as a function of plasma elongation
for the n=1 kink mode, with various plasma triangularities.}
\vspace{0.5cm}

Figure 10 -- {Stabilizing wall location as a function of toroidal mode
number and various plasma triangularities, with fixed plasma elongation of 2.2.}
\vspace{0.5cm}

Figure 11 -- {Stabilizing wall location as a function of toroidal mode
number for two values of $\beta_{N}$, showing the shift of the limiting mode
number as the pressure increases.}
\vspace{0.5cm}

Figure 12 -- {Example of the generic structure used in the vertical stability
calculations, and the final structure used in the design.}
\vspace{0.5cm}

Figure 13a -- {Vertical instability growth rate versus the distance of a
tungsten shell, normalized to the minor radius, for various plasma elongations.}
\vspace{0.5cm}

Figure 13b -- {Vertical stability factor as function of the distance of a
tungsten shell, normalized to the minor radius, for various plasma elongations.}
\vspace{0.5cm}

Figure 14a -- {Vertical instability growth rate as a function of the plasma
internal self-inductance, li(3), and $\beta_{p}$, for the final design vertical
stabilizing structure.}
\vspace{0.5cm}

Figure 14b -- {Feedback power for vertical position control as a function
of the vertical instability growth rate, for the final vertical
stabilizing structure.}
\vspace{0.5cm}

Figure 15a -- {Poloidal field coil energy as function of the number of coils,
showing that the energy increases as the number of coils decreases.}
\vspace{0.5cm}

Figure 15b -- {Layout of the optimized poloidal field coils.}
\vspace{0.5cm}

\clearpage

\begin{figure}
\centering
\epsfig{figure=figat1x.epsi}
\caption{}
\label{1}
\end{figure}
\clearpage

\begin{figure}
\centering
\epsfig{figure=figat2x.epsi}
\caption{}
\label{2}
\end{figure}
\clearpage

\begin{figure}
\centering
\epsfig{figure=figat3x.epsi}
\caption{}
\label{3}
\end{figure}
\clearpage

\begin{figure}
\centering
\epsfig{figure=figat4x.epsi}
\caption{}
\label{4}
\end{figure}
\clearpage

\begin{figure}
\centering
\epsfig{figure=figat5x.epsi}
\caption{}
\label{5}
\end{figure}
\clearpage

\begin{figure}
\centering
\epsfig{figure=figat6ax.epsi}
\caption{}
\label{6a}
\end{figure}
\clearpage

\begin{figure}
\centering
\epsfig{figure=figat6bx.epsi}
\caption{}
\label{6b}
\end{figure}
\clearpage

\begin{figure}
\centering
\epsfig{figure=figat7ax.epsi}
\caption{}
\label{7a}
\end{figure}
\clearpage

\begin{figure}
\centering
\epsfig{figure=figat7bx.epsi}
\caption{}
\label{7b}
\end{figure}
\clearpage

\begin{figure}
\centering
\epsfig{figure=figat8x.epsi}
\caption{}
\label{8}
\end{figure}
\clearpage

\begin{figure}
\centering
\epsfig{figure=figat9x.epsi}
\caption{}
\label{9}
\end{figure}
\clearpage

\begin{figure}
\centering
\epsfig{figure=figat10x.epsi}
\caption{}
\label{10}
\end{figure}
\clearpage

\begin{figure}
\centering
\epsfig{figure=figat11x.epsi}
\caption{}
\label{11}
\end{figure}
\clearpage

\begin{figure}
\centering
\epsfig{figure=figat12x.epsi}
\caption{}
\label{12}
\end{figure}
\clearpage

\begin{figure}
\centering
\epsfig{figure=figat13ax.epsi}
\caption{}
\label{13a}
\end{figure}
\clearpage

\begin{figure}
\centering
\epsfig{figure=figat13bx.epsi}
\caption{}
\label{13b}
\end{figure}
\clearpage

\begin{figure}
\centering
\epsfig{figure=figat14ax.epsi}
\caption{}
\label{14a}
\end{figure}
\clearpage

\begin{figure}
\centering
\epsfig{figure=figat14bx.epsi}
\caption{}
\label{14b}
\end{figure}
\clearpage

\begin{figure}
\centering
\epsfig{figure=figat15ax.epsi}
\caption{}
\label{15a}
\end{figure}
\clearpage

\begin{figure}
\centering
\epsfig{figure=figat15bx.epsi}
\caption{}
\label{15b}
\end{figure}
\end{document}
