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\begin{document}

\title{\bf 2. STELLARATOR PHYSICS}
\author{
\autlist{James~F.~Lyon}{James~A.~Rome}{John~L.~Johnson}
\autlist{David~T.~Anderson}{}{Paul~R.~Garabedian}
}
\date{ }
\maketitle


\begin{titlepage}
{\baselineskip=1.8ex
\tableofcontents
}
\end{titlepage}

\setcounter{chapter}{1}
\setcounter{page}{1}

\chapter {STELLARATOR PHYSICS}

\section {INTRODUCTION}
\label{s-2-1}

Both tokamaks and stellarators are based on helical (toroidal plus poloidal)
magnetic fields that produce toroidally nested, closed magnetic surfaces.  This
results in similar plasma confinement and stability properties for the two
configurations.  In tokamaks, the stabilizing toroidal field component is
produced by a toroidal set of coils (TF coils) and most of the confining
poloidal field is produced by a large toroidal current induced in the plasma.
In stellarators~\cite{Car88,Lyo90}, both components of the magnetic field
are created by currents flowing solely in external coils.  Because the poloidal
magnetic field that produces the rotational transform $\ib$ ($\ib=1/q$, where
$q$ is the safety factor) of the magnetic field is created by currents outside
the plasma, the plasma is inherently nonaxisymmetric; the plasma cross section
changes shape as it rotates around the (sometimes noncircular and nonplanar)
magnetic axis.  The plasma aspect ratio \hbox{$A_p \equiv R_0/a_p$}
is generally larger than in a tokamak because the larger toroidal effects at
smaller aspect ratio can produce field components that break the stellarator's
helical symmetry.  However, the field structure can be designed to reduce or
even eliminate (a ``quasi-helically symmetric''
configuration~\cite{Nuh88}) this finite-aspect-ratio effect.  Here
$R_0$ is the average major radius of the torus and $a_p$ is the average
radius of the noncircular last closed flux surface (LCFS).

\subsection{Stellarator Advantages and Drawbacks}
\label{s-1-3-1}

Stellarators have some important potential operational advantages as fusion
power plants.  The absence of a net plasma current leads to inherently
steady-state operation.  There is no need for continuous power input to the
plasma to maintain the magnetic configuration, so the reactor plasma can be
truly ignited rather than having a high $Q$ as in a tokamak.  Here $Q$ is the
ratio of the fusion power produced to the plasma input power.  Current drive
power is not needed, which leads to higher plant efficiency, and there would be
no current-drive-related reliability problems, which leads to higher plant
availability.  There are no dangerous current-driven disruptions or vertical
stability problems, so a thick vacuum vessel is not required for equilibrium
control or to protect against damage to it and the blanket structure.  No
pulsed magnetic or thermal loads occur, so there is no cyclic stress or
\hbox{$d\mathbf B_{\perp}/dt$} constraint on the superconducting coils.  In
addition, there is a wide range of magnetic configurations possible with
increased flexibility and control for optimization.  The plasma parameters and
profiles do not need to satisfy simultaneous (and often conflicting)
constraints on disruption avoidance, density and beta limits, bootstrap current
fraction, current drive efficiency, divertor performance, and improved
confinement.  Thus stellarators should be better able to operate in a true
ignited steady-state fashion with time-invariant plasma profiles.  The larger
aspect ratio may also allow access from both the inboard (larger central hole)
and outboard sides for easier maintenance.

Stellarators also have some potential drawbacks as fusion power plants.  The
larger aspect ratio leads to larger size.  Although the resulting lower power
density leads to longer component life, the larger size means more structure
and consequently higher cost.  The helical geometry requires more complex
modules and more costly maintenance procedures.  The smaller distance between
the plasma and the coils leads to more restricted access and a more difficult
divertor geometry, and hence more complicated maintenance and additional cost.
It is the aim of this study to assess these advantages and drawbacks.

\subsection {Design of Stellarator Coil Configurations}
\label{S-1-3-2}

The most general form of the stellarator can have currents distributed
throughout three-dimensional (3-D) space.  A subset, still very general, of all
possible stellarator configurations is a topologically toroidal system with
currents distributed on a torus, subject only to the constraint that the
current density \hbox{$\mathbf j$} be divergenceless (\hbox{$\nabla \cdot
\mathbf j = 0$}).  Divergence-free currents on a torus can close either
helically, poloidally, or toroidally.  The first case creates helical coils
(although in general they do not necessarily have large current-free gaps).
The second case creates a nonplanar modular coil set.  The toroidally-linked
case has not been studied very much.  In general, the same magnetic
configuration can be created using any of these coil configurations.  The
appropriate choice is based upon engineering design of the resulting
coil set; some magnetic configurations are more conveniently created using
modular coils and some using helical windings.

Historically, there have been two design approaches used to create specific 
stellarator configurations: starting from a parameterized coil set or 
from desired physics properties.

%\subsubsection {Starting from coils}
%\label{s-1-3-1-2}
\paragraph{Starting from coils.}

The more traditional design approach starts from the basic coil geometry, which
can be described by a set of adjustable parameters.  A set of ``goodness''
criteria is used as an optimization goal and the coil parameters are adjusted
by a multidimensional optimization code.  As each parameter is changed, the
magnetic flux surfaces are computed, the magnetohydrodynamic (MHD) equilibrium
and stability properties are evaluated, and the other components of the
goodness criteria are evaluated.  In this approach, a small change in a coil
parameter can lead to a large change in the resulting configuration so, in a
sense, this approach is mathematically unstable because a small change can
suddenly destroy the flux surfaces.  The low-aspect-ratio torsatron CT6 was the
result of applying this method using as a goodness criterion that it should
have physics properties similar to those of the Advanced Toroidal Facility
(ATF), but with a lower aspect ratio.  The flexible heliac configuration TJ-II
was optimized to have higher rotational transform, a global magnetic well, and
a higher beta limit.  It features a set of planar TF coils whose centers follow
a helical path around central circular and helical coils that link the TF set.

%\subsubsection {Starting from physics properties}
%\label{s-1-3-1-3}

\paragraph{Starting from physics properties.}
A systematic methodology~\cite{Lot90,Lot91} for choosing a stellarator
configuration with desired physics properties has been developed at the
Max-Planck-Institut f\"ur Plasmaphysik (IPP) in Garching, Germany, building on
theory and computer codes developed at New York University~\cite{Bau84}.  This
approach is based upon the fact that the plasma boundary shape and pressure
distribution are sufficient to determine the properties of the magnetic
configuration.  The boundary (LCFS) shape can be optimized to approximate an
{\it a priori} defined set of magnetic properties (such as rotational
transform, shear, magnetic well depth, and Fourier spectral content of the
magnetic field).  While neither the existence nor the uniqueness of a
stellarator configuration with the desired properties is guaranteed (for a
given set of conditions), this approach does provide a powerful methodology for
searching for stellarator configurations.

Once a desired plasma boundary shape is derived from runs of the
NSTAB~\cite{NSTAB} and TRAN~\cite{Gar92} codes, the NESCOIL code~\cite{NESCOIL}
developed by P.~Merkel, also at IPP Garching, can be used to find a set of
coils of either helical or modular type to approximately produce the desired
LCFS.  A potential problem can be solved for the current lines on a chosen
surface using the fact that the magnetic field $\mathbf B$ lies in a flux
surface.  The type of current line closure is solely dependent on the boundary
conditions applied to this potential.  This current distribution can be
discretized to simulate `coils' and the resulting error fields at the plasma
boundary produced by real coils can then be calculated to check the degree of
approximation to the desired configuration.  The shape of the current surface
can be optimized to meet a series of engineering side conditions such as
constraints on the current density and the current-line curvature, as well as
minimizing the error on the boundary, to arrive at a final model for a physical
coil set.

\clearpage

The design of the newer modular stellarators, such as the Wendelstein 7-X
(W7-X) or the Modular Helias-like Heliac (MHH), is based on analysis of the
Fourier expansion of the magnetic field strength $B$ in terms of Boozer's flux
coordinates~\cite{Boo81} in which the $\mathbf B$ lines are straight, the
Jacobian is $1/B^2$, $\psi$ is the toroidal flux, and $\vartheta$ and $\varphi$
are poloidal and toroidal angle variables.
%
\begin{eqnarray}
1/B^2 & = & \sum b_{m,n}(\psi)\cos(m\vartheta - n\varphi) \: ,
\label{e-3-1-1}
\end{eqnarray}
%
Theory shows that the Fourier coefficients $b_{m,n}$ control most physical
properties of the configuration.  Good equilibrium, stability, and transport
properties are achieved by reducing the size of the $b_{m,n}$'s for selected
values of the indices $m$ and $n$.  This allows avoiding gross changes in the
magnetic axis position or rotational transform profile as the plasma beta
increases.  The Fourier coefficients $b_{m,n}$ are in turn related to the
Fourier coefficients $r_{m,n}$ and $z_{m,n}$ in the 3-D representation of the
plasma surface,
%
\begin{eqnarray}
\label{e-3-1-2}
 r & = & \sum r_{m,n} \cos(m\theta+ nM\phi) \: , \\
\label{e-3-1-3}
 z & = & \sum z_{m,n} \sin(m\theta+ nM\phi) \: ,
\end{eqnarray}
%
where $r$, $z$, and $\phi$ are the usual cylindrical coordinates, $M$ is the
number of toroidal periods, and $\theta$ is a poloidal angle variable.  See
also \hbox{Eq.~(\ref{e-3-6-3})} below, where the coefficients $\delta_{m,n}$
are defined that have a diagonally dominant relationship to the spectrum
$b_{m,n}$, after a shift of 1 in the index $m$.

The NESCOIL method was used to determine the W7-X coil configuration using the
``helias'' optimization criteria in which finite-beta plasma currents are
minimized (to preserve the vacuum-field properties at finite beta) and the
orbit containment at high beta is improved due to the self-dug magnetic well.
The MHH configuration chosen for this Stellarator Power Plant Study (SPPS) has
features of both the helias and the heliac, and was also designed using this
method~\cite{And94}.

\subsection {Types of Stellarator Configurations}
\label{s-1-3-3}

In general, there are four types of stellarators in operation: conventional
stellarators, torsatrons (or heliotrons), heliacs, and modular stellarators.
Conventional stellarators and torsatrons are based upon helical coils.  In
conventional stellarators, these coils are in pairs with opposite currents so
that they produce no net toroidal field; additional TF coils are thus required.
In torsatrons, the helical currents flow in the same direction so no TF coils
are required.  However, depending on the winding law, torsatrons may require
large vertical field coils.  Generally, torsatrons are preferred over
conventional stellarators due to their simpler coil configuration.

Torsatron reactors~\cite{Lyo94} typically have moderate plasma aspect ratio
(\hbox{$A_p \approx 3.3-8$}), large magnetic shear (change in $\ib$
across the plasma), significant helical field ripple (up to $\sim 20\%$ at the
plasma edge), and large access between the helical windings for blankets and
maintenance.  Modular stellarator reactors~\cite{Bei93} typically have
larger plasma aspect ratio (\hbox{$A_p \geq 10$}), small shear, smaller
effective helical field ripple, blankets between the plasma and the coils, and
divertors located inside the coils.  Conventional heliacs have only received
cursory examination~\cite{Nav10} partly because of their linked coil
geometry.

\subsection {Separation of Physics and Cost Optimizations}
\label{s-1-3-4}

The performance of a particular stellarator configuration as a reactor is
determined by its physics properties (determined by a set of dimensionless
parameters -- its aspect ratio, rotational transform, shear, magnetic well
depth, and other properties of the magnetic field) and the major device
parameters: on-axis magnetic field strength, $\mathbf B_0$, and major radius,
$R_0$.  The fusion power core cost depends primarily on the mass (volume) of
its major components (blanket, shield, and coils), and hence almost entirely on
$R_0$.

The minimum size ($R_0$) for a stellarator core is determined by the minimum
distance $\Delta$ between the edge of the plasma (LCFS) and the centerline of
the nearest helical winding or modular coil because the ratio \hbox{$A_{\Delta}
\equiv R_0/\Delta$} is a constant for a given stellarator configuration.  This
is not a constraint for tokamaks because the position of the plasma edge and
the edge configuration properties, other than the toroidal field ripple, do not
depend on the proximity of the TF coils.  The helical windings must be
relatively close to the plasma edge in torsatrons to produce the edge shear
needed for MHD stabilization, and the non-planar TF coils must be relatively
close to the plasma edge in modular stellarators to permit practical coil
designs.  In both cases, an optimized magnetic configuration requires precise
shaping of the external current paths; the resulting higher-order field
components decay rapidly away from the windings, necessitating closeness to the
plasma to avoid excessively large coil currents.

The minimum size for a stellarator core is thus given by $R_0=A_{\Delta}d$,
where $d$ is the distance required for the plasma-wall separation, the first
wall, the tritium breeding blanket, the neutron shielding, the coil dewar, the
structural case facing the plasma, the half-radial depth of the coil winding
pack, and assembly clearances.  The cost of the most expensive reactor core
components (blankets, shield, and coils) is related to the mass (volume) of
these components and in turn to the plasma surface area \hbox{[$\propto
R_0a_p \propto (A_{\Delta}d)^2/A_p$]}.  The smaller values of
$A_{\Delta}$ obtained for a compact torsatron~\cite{Car88b} allow this
configuration to be a factor of 2--3 smaller in size than previous modular
stellarators; typically \hbox{$R_0 \approx 8-10$~m} versus \hbox{$R_0 \approx
20-25$~m}.  The reference MHH studied for the SPPS has $R_0=14$~m.

\subsection{Comparison of Different Stellarator Configurations}
\label{s-1-3-5}

Four stellarator coil configurations were examined in this study: (1)~CT6, a
Compact Torsatron with 2 large continuous helical windings, 6 field periods,
and low plasma aspect ratio ($A_p=3.8$); (2)~MATF~\cite{MATF}, a
torsatron with modularized helical windings, 12 field periods, and $A_{\rm
p}=10$; (3)~MHH~\cite{MATF}, a modular helias-like heliac with 4 field periods
and $A_p=8.1$; and (4) the W7-X modular helias
configuration~\cite{HELIAS} with 5 field periods and $A_p=12.2$.  The
properties of these magnetic configurations are compared with those of the
second-stability ARIES-IV tokamak reactor in Table~\ref{t-2-1-1}.

\begin{table}[t]
\caption{}
\centering
{\bf Characteristics of the Four Stellarator Configurations Studied
and the Second-Stability ARIES-IV Tokamak} \\
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{l c c c c c} 
\hline
	                               & CT6  & MATF & MHH  
                                                        & Helias & ARIES-IV \\ 
\hline 
 & \hspace{1.25cm} & \hspace{1.25cm} & \hspace{1.25cm} 
               & \hspace{1.25cm} & \hspace{1.25cm} \\
No. of toroidal field periods, $M$     & 6    & 12   & 4    & 5	   & 1 \\
No. of main field coils	               & 2    & 12   & 32   & 50   & 16 \\
Center safety factor, $q_0 = 1/\ib(0)$ & 2.9  & 2.9  & 0.91 & 1.2  & 2.0 \\
Edge safety factor, $q_a = 1/\ib(a)$   & 1.05 & 1.05 & 0.81 & 1.03 & 12.2 \\
Average coil aspect ratio, $R_0/a_c$   & 2.5  & 4.6  & 3.3  & 5.0  & 1.0 \\
Average plasma aspect ratio, $R_0/a_p$ & 3.8  & 10   & 9.7  & 13   & 2.8 \\ 
Average plasma elongation, $\kappa$    & 2.1  & 1.7  & 2.0  & 2.0  & 2.0 \\
Minimum plasma-coil \\
\quad\quad separation ratio, $R_0/\Delta$
                                       & 6.4  & 7.2  & 7.0  & 12   & 3.2 \\ 
\\ \hline
\end{tabular}
\label{t-2-1-1}
\end{table}

The best-studied stellarator reactor configurations have been the 
modular helias stellarator~\cite{HELIAS2} and the CT6 torsatron.  The CT6
configuration (shown in Fig.~\ref{f-2-1-1}) features two large continuous
helical optimized for operation in the second stability region.
The relatively large shift of the magnetic axis with increasing beta broadens
and deepens the magnetic well to achieve higher beta.  The helias configuration
(shown in Fig.~\ref{f-2-1-2}) features 50 nonplanar poloidal coils.
It has been studied extensively at IPP Garching and forms the basis for the
proposed W7-X experiment and the Helias Stellarator Reactor (HSR)
studies~\cite{HSR}.  It relies on the opposite physics optimization approach
in which the magnetic geometry does not change much with increasing beta.

The two new coil configurations developed in this study were MHH and MATF.  The
MATF configuration (shown in Fig.~\ref{f-2-1-3}) is based on a modularization
of the ATF torsatron~\cite{Lyo86}, similar to the earlier
``symmotron''~\cite{Lyo83}, but the modules are not up-down symmetric.  The
windbacks ($40^\circ$ above and below the equatorial plane) provide most of the
required vertical field.  The legs connecting the helical coil segments to the
toroidal windbacks are perpendicular to the torus, which minimizes the field
interaction with the torus and preserves most of the physics properties of the
unmodularized coil configuration.  The coil geometry allows field lines to exit
between the windings to an exterior divertor chamber.  The physics properties
of MATF are similar to those of the ATF torsatron with continuous helical
windings, except that MATF has better orbit confinement and smaller
$A_{\Delta}$.

\begin{figure}[p]
\centerline{\psfig{file=spps-2-1-1.eps,height=7.3in}}
\caption[Fig-2-1-1]
{Top and side views of the CT6 helical coils and
the last closed magnetic surface.  A set of vertical field (VF) coils on the 
inboard side (not shown) is also needed for shaping control.}
\label{f-2-1-1}
\end{figure}

\begin{figure}[t]
\centerline{\psfig{file=spps-2-1-2.eps,height=3.4in}}
\caption[Fig-2-1-2]
{An oblique view of the helias configuration showing
the nonplanar coils and the last closed flux surface.}
\label{f-2-1-2}
\end{figure}

\begin{figure}[b]
\centerline{\psfig{file=spps-2-1-3.eps,height=3.4in}}
\caption[Fig-2-1-3]
{An oblique view of the $12$ MATF coil modules.}
\label{f-2-1-3}
\end{figure}

The MHH coil set (shown in Fig.~\ref{f-2-1-4}) is a modularization of a highly
modulated heliac whose flux surfaces are helias-like, rather than bean-shaped
as in the usual heliac.  The reference MHH has four field periods with eight
coils per field period.  The MHH plasma geometry has some similarities to the
W7-X helias configuration, but MHH has 4 field periods instead of 5, less
triangularity of the flux surfaces, and a larger helical axis excursion.  As in
a helias, the plasma separatrix permits a helical divertor on the top and
bottom.  MHH's physics properties are similar to those of the helias
configuration except that it has lower plasma aspect ratio, a larger bootstrap
current, and more space between the last closed flux surface and the center of
the coil winding surface.  The value of $A_\Delta$ is smaller than in helias
because much of the rotational transform is produced by the large helical
excursion of the magnetic axis; there is also less indentation of the plasma
surface, and the required physics properties have been relaxed somewhat.
Nevertheless, numerical nonlinear stability tests with the NSTAB
code~\cite{NSTAB} indicate that the plasma should be stable at \hbox{$\langle
\beta \rangle = 5\%$} and Monte Carlo orbit calculations indicate that its
confinement properties should be similar to that for helias~\cite{Gar94p}.
Here $\langle\beta\rangle$ is the volume-average value of the plasma beta.

\begin{figure}[p]
\centerline{\psfig{file=spps-2-1-4.ps,width=6.25in,bbllx=88pt,bblly=179pt,%
            bburx=523pt,bbury=614pt,clip=}}
\vspace{3.5cm}
\caption[Fig-2-1-4]	%	950910_SP_blktcoil.ps
{A top view of the MHH configuration showing the nonplanar
coils, the blanket/shield structure, and the last closed flux surface.}
\label{f-2-1-4}
\end{figure}

\subsection {Selection of the MHH Configuration for the SPPS}
\label{s-1-3-6}

Four 1-GW(e) stellarator reactors were compared~\cite{MATF} by minimizing their
cost for the same physics and engineering constraints.  Table~\ref{t-2-1-2},
reproduced from Ref.~\cite{MATF}, compares the plasma and device parameters for
these configurationsto the corresponding values for the second-stability
tokamak reactor ARIES-IV~\cite{ARIESIV}. The stellarator configurations are:
(1) ~the 6-field-period Compact Torsatron CT6 with two large continuous helical
windings, $R_0=12$~m and $B_0=4.6$~T; (2)~the 12-field-period torsatron MATF
with 24 modular coils, $R_0=13$~m and $B_0=5.5$~T; (3)~the 5-field-period
helias with 50 modular coils, $R_0=23$~m and $B_0=4.1$~T; and (4)~the
4-field-period helias-like heliac MHH with 32 modular coils, $R_0=13$~m and
$B_0=5.3$~T.  The minimum size (hence cost) of these configurations was
determined by their $A_{\Delta}$ value and the space $d$ between the plasma and
the coils needed for blanket, shield, structure, spacing, {\it etc.}  Smaller
$R_0$ can be obtained for less-conservative blanket and shield assumptions.
Table~\ref{t-2-1-2} shows that both MATF and MHH are competitive with ARIES-IV.
The magnetic fields on axis and at the coils for both are less than those for
ARIES-IV, although the coil geometry is more complex.

\begin{table}[t]
\caption{}
\centering
{\bf Plasma and Device Parameters for Different 1-GW(e) Power
Plants~\cite{MATF}}
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{l c c c c c} 
\hline
	                               & CT6  & MATF & MHH  
                                                        & Helias & ARIES-IV \\ 
\hline 
 & \hspace{0.9cm} & \hspace{0.9cm} & \hspace{0.9cm} 
               & \hspace{0.9cm} & \hspace{1.1cm} \\
Major radius, $R_0$ (m)                   & 11.8 & 13.4 & 13.1 & 23.3 & 6.04 \\
Average plasma radius, $a_p$ (m)          & 3.1  & 1.3  & 1.4  & 1.9  & 2.2 \\
Toroidal field on axis, $B_0$ (T)         & 4.6  & 5.5  & 5.3  & 4.1  & 7.6 \\
Maximum field on coils, $B_{max}$ (T) & 14.0 & 13.7 & 15.4 & 8.7  & 15.9 \\
Average plasma density, $\langle n \rangle$
     (10$^{20}$~m$^{-3}$)                 & 1.0  & 2.5  & 2.7  & 1.4  & 2.9 \\
Average plasma temperature, $\langle T \rangle$
    (keV)                                 & 8.1  & 6.8  & 7.0  & 8.1  & 10 \\
Central ion temperature (keV)	  & 26   & 22	& 24   & 26   & 27 \\
Central electron temperature (keV) & 29 & 25  & 24   & 26   & 27 \\
Volume-average toroidal beta, $\langle \beta \rangle$
                                     & 2.9\% & 4.3\% & 5.0\% & 5.0\% & 3.4\% \\
Energy confinement time, $\tau_E$ (s) & 4.1 & 1.6 & 1.4  & 2.4  & 1.5 \\
Lackner-Gottardi confinement multiplier   & 2.0  & 2.0  & 1.5 & 1.7  & 2.5 \\
%Fraction of power radiated, $f_{rad}$ & 0.30 & 0.30 & 0.32 & 0.28 & 0.23 \\
Neutron wall loading, $\Gamma_{n}$ (MW/m$^2$)
                                          & 0.9	 & 1.8  & 1.3  & 0.5  & 2.7 \\
Fusion core mass ($10^3$ tonnes)          & 10.7 & 8.7  & 7.8  & 18.8 & 9.0 \\
Cost of electricity (mill/kWeh)$^{(a)}$   & 70   & 62   & 66   & 100  & 68 \\ 
\\ \hline
\multicolumn{6}{l}{$(a)$ In constant 1992 dollars.}\\
\end{tabular}
\label{t-2-1-2}
\end{table}

The MHH configuration was chosen for the SPPS because its helias-like features
led to good physics properties and its heliac feature, the large helical
excursion of the magnetic axis that provides most of the rotational transform,
allows the coils to be farther from the plasma, permitting a 2-m distance
between the plasma and the coils for blankets, shields, structure, clearance,
{\it etc.} for a 14-m average major radius.  Here 14~m has been chosen instead
of the 13~m in Table~\ref{t-2-1-2} to allow for additional distance between the
plasma and the centerlines of the coils.  This new configuration has the best
physics properties (beta limit, neoclassical transport, orbit losses, reduction
of bootstrap current, {\it etc.}).  However, its smaller plasma-coil separation
leads to a larger major radius (larger blanket and shield area) and
consequently higher cost.  The CT6 configuration has good access between the
coils for blankets and divertor and the large plasma-coil distance leads to
moderate-size reactors.  The large helical field ripple leads to increased
orbit losses, which it turns out are not a problem~\cite{Lyo94}.  However, the
large continuous helical coils would be difficult to fabricate and repair.  The
MATF configuration remedied most of the CT6 drawbacks and was selected as a
backup configuration.

\clearpage

\section{THE MHH CONFIGURATION}
\label{s-2-2}

\subsection {Origin of the MHH Configuration}
\label{s-2-2-1}

In a conventional stellarator, the rotational transform is provided by the
fringing field of the coils, which decays rapidly inward from the coils toward
the plasma.  If the rotational transform could be achieved by introducing a
large helical excursion of the magnetic axis, then the remaining freedom in
specifying the coil set could be devoted to improving the physics properties of
the configuration and to increasing the plasma-coil separation.  To achieve
this goal, a configuration was devised using a set of axisymmetric circular
coils (with different currents) arranged on a circular torus.  Inside these
coils, and linking them, there was a large $\ell=1$ helical coil spatially
modulated to produce flux surfaces with a cross-section that varied from
triangular to bean-shaped.  Outside the circular coils there was a second
(smaller) helix that was used to control the shear and the magnetic well.  This
configuration, a ``helias-like heliotron'', is described in more detail in
Ref.~\cite{And94}.

A modular coil set that creates the same magnetic configuration, but involves
no interlocked coils, was developed using the Garching NESCOIL
code~\cite{NESCOIL}.  This configuration~\cite{MATF}, a Modular Helias-like
Heliac (MHH), became the basis for the SPPS power-plant concept.
Figure~\ref{f-2-2-1} shows the particular MHH coil set that was chosen.  The
configuration has four field periods with eight coils per period, for a total
of 32 coils.  Because of the stellarator symmetry, there are only four
different types of coils in the coil set.  If the configuration is viewed from
the top, it resembles a square; magnetically, it resembles four linked mirrors.
However, if viewed from the side, the large vertical excursion of the coils is
evident.  The MHH modular coils are highly modulated and are close together on
the inside of the corners where the magnetic field strength is high.  This
corner region is the focus of many of the engineering difficulties faced by the
coil designers.

\begin{figure}[p]
\centerline{\psfig{file=spps-2-2-1.ps,height=7.3in,bbllx=79pt,bblly=173pt,%
            bburx=511pt,bbury=735pt,clip=}}
\caption[Fig-2-2-1]
{A top view and side view of the MHH coil set.  The width represents the size
of the conductor without the coil case.  The side view is from the inside of
the torus and shows the closeness of the corner coils at the center of the
plot.}
\label{f-2-2-1}
\end{figure}

\subsection {The MHH Magnetic Surface Geometry}
\label{s-2-2-2}

Figure~\ref{f-2-2-2} shows the outer (last closed) flux surface for the
reference MHH case.  The four-field-period nature and the pronounced helical
distortion of the flux surface is evident in the figure.  This surface is
described by the set of Fourier coefficients $r_{m,n}$ and $z_{m,n}$ in
\hbox{Table~\ref{t-2-2-1}}.  A partial symmetry in these coefficients can be
seen; for most, but not all, of the $m,n$ values, $z_{m,n} = -r_{m,n}$.

%RLM mod 3.2-2 ok
\begin{figure}[p]
\centerline{\psfig{file=spps-2-2-2.ps,height=7.5in,bbllx=93pt,bblly=244pt,%
            bburx=497pt,bbury=778pt,clip=}}
\caption[Fig-2-2-2]
{Comparison of the last closed flux surface for the reference
MHH (top) and the W7-X helias (bottom).}
\label{f-2-2-2}
\end{figure}

\begin{table}[t]
\caption{}
\centering
{\bf Fourier Coefficients for the MHH Plasma Surface} \\
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{ c @{\hspace{1.cm}} c c c c c c c }
\hline
   & \multicolumn{3}{c}{$r_{m,n}$} && \multicolumn{3}{c}{$z_{m,n}$} \\
    \cline{2-4} \cline{6-8} 
$n$  & $m=0$   & $m=1$     & $m=2$ && $m=0$    & $m=1$     & $m=2$ \\ 
\hline 
     & \hspace{1.1cm} &\hspace{1.1cm} &\hspace{1.1cm} & \hspace{0.5cm}
                            \hspace{1.1cm} &\hspace{1.1cm} &\hspace{1.1cm} \\
--3\N & 0.008   &           &       && --0.008   \\
--2\N & 0.1\Z\Z &           &       && --0.1\Z\Z \\
--1\N & 1.0\Z\Z & --0.1\Z\Z & 0.05  && --1.0\Z\Z & \N0.1\Z\Z &--0.05 \\
  0   & 6.2\Z\Z & --0.335   & 0.09  &&           & \N0.335   & --0.09 \\
  1   & 0.13\Z  & \N0.8\Z\Z & 0.07  && --0.13\Z  & \N0.8\Z\Z & \N0.15 \\ 
  2   &         & \N0.02\Z  & 0.07  &&           & \N0.02\Z  & \N0.07 \\
  3   &         & --0.02\Z  &       &&           & --0.02\Z  \\
\\ \hline
\end{tabular}
\label{t-2-2-1}
\end{table}	

Figure~\ref{f-2-2-3} shows three cuts through the magnetic surfaces; at the
beginning, at 0$^\circ$, 22.5$^\circ$, and 45$^\circ$ field period.  The inner
solid curve is the ``design'' last closed flux surface (LCFS), and the outer
solid curve is the coil winding surface (CWS).  The design LCFS agrees quite
well with the actual LCFS except for some distortion due to magnetic islands.
The helical axisof MHH is also evident.  The LCFS has some similarity to the
helias configuration, as shown in Fig.~\ref{f-2-2-2}, which compares the LCFS
for MHH with that for W7-X.  Figure~\ref{f-2-2-4} shows that the MHH
configuration has a larger helical axis excursion and less triangularity than
the W7-X configuration~\cite{MMH-W7X}.

\begin{figure}[p]
%\vspace{4.5in}
\centerline{\psfig{file=spps-2-2-3.ps,width=6.5in,bbllx=79pt,bblly=280pt,%
            bburx=533pt,bbury=749pt,clip=}}
\caption[Fig-2-2-3]
{Magnetic surfaces at three toroidal angles for the MHH configuration showing
the change from bean-shaped (at $\phi = 0^{\circ}$) to teardrop-shaped (at
$\phi = 22.5^{\circ}$) to triangular (at $\phi = 45^{\circ}$) --- the
helias-like feature --- and the rotation of the magnetic axis about the average
major radius $R_0$ --- the heliac feature.  The outer curves show the CWS cross
section at these toroidal angles.}
\label{f-2-2-3}
\end{figure}

\begin{figure}[t]
\centerline{\psfig{file=spps-2-2-4.ps,height=4.5in,bbllx=108pt,bblly=280pt,%
            bburx=483pt,bbury=771pt,clip=}}
\caption[Fig-3-2-4]
{Comparison of the edge magnetic surface for MHH (top) and the W7-X helias
(bottom) at $\phi = 0^{\circ}$ (right) and at $\phi = 45^{\circ}$ (left).  The
heavier (outer) curves are cuts at constant toroidal angles and the thinner
(inner) curves are cuts perpendicular to the magnetic field $\mathbf B$.}
\label{f-2-2-4}
\end{figure}

Figure~\ref{f-2-2-5} displays a contour plot of the distance between the LCFS
and the CWS.  Because of the contorted topology of the MHH configuration, the 
closest distance is not necessarily perpendicular to the LCFS.  The minimum 
distance occurs on the inside of the triangular cross-section, and the minimum 
gap is 2~m for $R_0=14$~m.  The gap is much larger at some other places 
(near the top and bottom in the lower part of Fig.~\ref{f-2-2-5}).

\begin{figure}[p]
\centerline{\psfig{file=spps-2-2-5.ps,height=7.1in,bbllx=50pt,bblly=36pt,%
            bburx=519pt,bbury=650pt,clip=}}
\caption[Fig-2-2-5]
{A contour plot of the distance between the LCFS and the CWS
in the $\theta-\phi$ (poloidal angle-toroidal angle) plane.  The lower part of 
the picture shows the point ($\bullet$) where the spacing is smallest (at the 
inboard side of the torus).}
\label{f-2-2-5}
\end{figure}

Because MHH looks like a square when viewed from above, the term ``major
radius'' is in some sense an ``average'' value.  We define it to be the value
of the $r_{0,0}$ coefficient in the Fourier series that describes the LCFS.  It
is more accurate to describe MHH in terms of its volumes and cross-sectional
and surface areas, since these are the quantities that can be properly
calculated, and generally, they are the quantities required for physics and
engineering calculations.  The values for MHH, calculated using curvilinear
formulas developed by S.~Hirshman, are given in Table~\ref{t-2-2-2}.  Since the
size of the MHH configuration can be scaled as appropriate, the actual size of
the reactor embodiment is determined by external factors, usually the
plasma-coil spacing.  In order to make the minimum plasma-coil distance 2~m (as
dictated by coil current density requirements and blanket and shield
thickness), the major radius was scaled to 14~meters.

\clearpage

\begin{table}[t]
\caption{}
\centering
{\bf Geometrical Characteristics of the MHH Configuration} \\
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{l c c}
\hline
                                     & $R_0=10$~m $^(a)$ & $R_0=14$~m \\
\hline \\ 
Plasma volume inside LCFS (m$^3$)                & 267.5 &  734 \\
Plasma surface (LCFS) area (m$^2$)               & 629	 & 1,233 \\
Average plasma cross section (LCFS) (m$^2$)      & 4.29	 &  8.4 \\
Volume within coil winding surface (CWS) (m$^3$) & 2,085 & 5,720 \\
Area of coil winding surface (m$^2$)             & 1,495 & 2,930 \\
Average cross-sectional area of CWS (m$^2$)      & 32.9	 & 64.4 \\
\\ \hline
\multicolumn{3}{l}{$(a)$ Value normalized to $R_0=10$~m.}\\
\end{tabular}
\label{t-2-2-2}
\end{table}

\subsection{Coil Engineering Issues}

Another set of issues centers on the engineering feasibility of the MHH coil
set.  Figure~\ref{f-2-2-6} shows the distance between the centerline of each
coil and the neighboring coil as a function of position along the
coil~\cite{MMH-W7X}. The spacing is tightest at the corners on the inside but
is an order of magnitude larger at the outside in the straight sections.  The
final coil design allows room for the coil conductors, but requires that some
of the intercoil case be eliminated in the inside corner regions.  The maximum
field in this area is also high.  Figure~\ref{f-2-2-7} shows the $|B|$ contours
for the coils and a 3-D plot of the same calculation~\cite{MMH-W7X}. The
maximum field in the coil is about 14.5~T for a central field of 5~T; dealing
with this is a major driver in the coil design.  The fact that the high-field
region in Fig.~\ref{f-2-2-7} occurs in a small region of the coil permits the
use of graded conductors.

\begin{sidewaysfigure}[p]
\centerline{\psfig{file=spps-2-2-6.ps,angle=-90,width=7.in,clip=}}
\caption[Fig-2-2-6]
{Distance from the centerline of each modular coil to the
adjacent coil as a function of the poloidal distance along the coil.  The 
curve labels indicate the coil number and the minimum distance to the nearest 
coil for $R_0 = 14$~m.  The coil perimeter is $10.05$~m.}
\label{f-2-2-6}
\end{sidewaysfigure}

\begin{figure}[p]
\centerline{\psfig{file=spps-2-2-7.ps,height=7.5in,bbllx=64pt,bblly=115pt,%
            bburx=512pt,bbury=691pt,clip=}}
\caption[Fig-2-2-7]
{$|B|$ contours and an elevation plot of $|B|$ at the outside
and inside of the coil set near the midplane.}
\label{f-2-2-7}
\end{figure}

\clearpage
\section {MHD EQUILIBRIUM AND STABILITY}
\label{s-2-3}

The MHH configuration was designed with the physics optimization techniques
that have been developed at NYU~\cite{Bau84} and at the Max-Planck-Institute
f\"ur Plasmaphysik~\cite{Lot90,Lot91}.  The studies that were used to design
the coil system employed calculations of the magnetic field using the NSTAB
equilibrium and nonlinear stability code~\cite{NSTAB}.  In particular, the
shape of the vacuum-plasma interface with $\langle\beta\rangle=0$
was adjusted to optimize the physics and the necessary coil locations and
currents were determined by means of the NESCOIL code~\cite{NESCOIL}.  This
calculation was iterated to obtain satisfactory separation between the coils
and forces on the coils.  The major difference in the approach used in the
design of W7-X and MHH is that the Garching optimization maximizes the physics
properties whereas the SPPS imposes less stringent physics constraints to
increase the space between the coils and the plasma for blankets and shields.
This has led to a configuration in which physics issues must be considered.
The most basic consideration is associated with large-scale plasma behavior,
and is usually treated with simple MHD models.  The relevent questions that
arise concern the change in the equilibrium, the MHD stability behavior, and
the bootstrap current as $\langle\beta\rangle$ is increased.

\subsection{Basis of $\mathbf \beta$ Limits in Stellarators}
\label{s-2-3-1}

As in a tokamak, the limits on $\langle\beta\rangle$ that must be
observed in a stellarator come from both the equilibrium and the stability
properties of the configuration.  Probably the actual limiting value of $\beta$
will occur when plasma currents distort the shape of the equilibrium
configuration significantly, primarily by introducing magnetic islands that
become sufficiently large or close to each other that they destroy the
confinement.  Pressure gradients in regions where the magnetic field lines have
geodesic curvature cause Pfirsch-Schl\"uter currents (which are associated with
the condition that the current be divergence-free and have a strong dependence
on the poloidal angle) and bootstrap current (associated with poloidal
viscosity causing the drifting ions to go preferentially in one direction).
These currents in turn modify the shape of the configuration.  The
modifications of the configuration can be attributed primarily to the change in
the rotational transform profile which can introduce and relocate resonant
surfaces where mode coupling can strongly distort the magnetic field lines.
Even if these modifications of the equilibrium configuration are not critical,
the plasma currents can introduce MHD instabilities.  Unlike the situation in a
tokamak, where catastrophic disruptions driven primarily by the energy
associated with the current in the toroidal direction must be controlled, the
most damaging modes in a stellarator are pressure-driven, ballooning or
interchange instabilities that create a soft $\beta$-limit by enhancing the
transport.

The helias configuration design that is being pursued in Garching 
addresses these problems by choosing an optimization that minimizes both 
Pfirsch-Schl\"uter and bootstrap currents and uses the self-dug well at finite 
$\beta$ to improve confinement.  This physics optimization results in a 
relatively small distance between the plasma and coils for the blanket and 
shield, which makes it essential to go to a larger size than is desired by 
American utility companies.  The optimization that is used for the MHH design 
emphasizes engineering considerations and reduces, but does not eliminate the 
problems associated with pressure-driven plasma currents.  The resulting 
configuration represents a reasonable compromise between the desire to 
simplify the engineering at the cost of not completely optimizing the 
physics.  Indeed, the configuration that has been adopted for this study 
appears to be only marginally satisfactory from an MHD point of view.  Some 
work on a related three-field-period configuration (Sec.~3.6.3) has shown
that these problems may be ameliorated without seriously compromising the 
engineering considerations.

\subsection{Sensitivity of the Equilibrium Configuration to $\mathbf\beta$}
\label{s-2-3-2}

The MHH vacuum-field equilibrium configuration, as calculated with the VMEC
code~\cite{VMEC}, is shown in Fig.~\ref{f-2-3-1}.  The configuration with
$\langle\beta\rangle=4.8\%$, as calculated with the VMEC code with the
plasma-vacuum interface kept fixed, is given in Fig.~\ref{f-2-3-2}.  The shape
of the configuration is not drastically changed, but the plasma currents have
shifted the magnetic axis outwards at the $\phi=0^{\circ}$ location.  The
Pfirsch-Schl\"uter current has, however, modified the shape of the rotational
transform significantly, as can be seen in Fig.~\ref{f-2-3-3}.  The fact that
an $m=1$, $n=1$ resonant surface has been introduced into the plasma region is
of particular concern.  Similar results to those presented in these figures
were obtained with the NSTAB code~\cite{NSTAB}.

\begin{figure}[p]
\centerline{\psfig{file=spps-2-3-1-a.ps,height=2.5in,bbllx=64pt,bblly=158pt,%
             bburx=540pt,bbury=634pt,clip=}}
\vspace{0.1truecm}
\centerline{\psfig{file=spps-2-3-1-b.ps,height=2.5in,bbllx=64pt,bblly=158pt,%
            bburx=540pt,bbury=634pt,clip=}}
\vspace{0.1truecm}
\centerline{\psfig{file=spps-2-3-1-c.ps,height=2.5in,bbllx=64pt,bblly=158pt,%
            bburx=540pt,bbury=634pt,clip=}}
\caption[Fig-2-3-1]
{Vacuum field magnetic surfaces for the MHH configuration as
computed with the VMEC code at $\phi=0^\circ$ (top), $\phi=22.5^\circ$
(middle), and $\phi=45^\circ$ (bottom).}
\label{f-2-3-1}
\end{figure}

\begin{figure}[p]
\centerline{\psfig{file=spps-2-3-2-a.ps,height=2.5in,bbllx=64pt,bblly=158pt,%
            bburx=540pt,bbury=634pt,clip=}}
\vspace{+0.1truecm}
\centerline{\psfig{file=spps-2-3-2-b.ps,height=2.5in,bbllx=64pt,bblly=158pt,%
            bburx=540pt,bbury=634pt,clip=}}
\vspace{+0.1truecm}
\centerline{\psfig{file=spps-2-3-2-c.ps,height=2.5in,bbllx=64pt,bblly=158pt,%
            bburx=540pt,bbury=634pt,clip=}}
\caption[Fig-2-3-2]
{Magnetic surfaces for MHH configuration with $\langle\beta\rangle=4.8\%$ as
computed with the VMEC code at $\phi=0^\circ$ (top), $\phi=22.5^\circ$
(middle), and $\phi=45^\circ$ (bottom).}
\label{f-2-3-2}
\end{figure}

\begin{figure}[tb]
\centerline{\hbox{
\psfig{file=spps-2-3-3.ps,height=4.5in,bbllx=58pt,bblly=137pt,bburx=548pt,%
       bbury=605pt,clip=}
}}
\caption[Fig-2-3-3]
{Rotational transform for the MHH configuration.  The upper curve is for
$\langle \beta \rangle = 0$ as calculated with the VMEC code; the middle curve
is for $\langle \beta \rangle = 2.4\%$ as calculated with the PIES code, and
the lower curve is for $\langle \beta \rangle = 4.8\%$ as calculated with the
VMEC code.}
\label{f-2-3-3}
\end{figure}

Both of these three-dimensional (3-D) equilibrium codes utilize a weak
formulation of the partial differential equations involved that circumvents
questions of the nonexistence of solutions by introducing current sheets at
resonant surfaces to prevent the appearance of magnetic islands.  That the
magnetic islands could produce a problem can be seen from a calculation of the
equilibrium with $\langle\beta\rangle=2.4\%$ that was done with the PIES
code~\cite{PIES} which calculates the equilibrium configuration by integrating
along the magnetic field lines and adjusting the pressure distribution on
resonant surfaces where difficulties would otherwise occur.  The results for
this calculation are shown in Fig.~\ref{f-2-3-3} and in Fig.~\ref{f-2-3-4}.  It
should be noted that islands have been introduced at several rational surfaces
and one has become sufficiently severe that it has led to a flattening of the
rotational transform.  Calculations with higher values of $\langle\beta\rangle$
with the PIES code have not been successful because of the presence of the
$m=1$, $n=1$ resonant surface in the plasma.  We believe that our difficulty
has been due to numerical problems and that the actual configuration would be
similar to that of Fig.~\ref{f-2-3-4} but with a larger island surrounding the
$\ib=1$ surface.

\begin{figure}[p]
\centerline{\psfig{file=spps-2-3-4-a.ps,height=2.5in,bbllx=72pt,bblly=180pt,%
            bburx=562pt,bbury=645pt,clip=}}
\vspace{+0.1truecm}
\centerline{\psfig{file=spps-2-3-4-b.ps,height=2.5in,bbllx=72pt,bblly=180pt,%
            bburx=562pt,bbury=645pt,clip=}}
\vspace{+0.1truecm}
\centerline{\psfig{file=spps-2-3-4-c.ps,height=2.5in,bbllx=72pt,bblly=180pt,%
            bburx=562pt,bbury=645pt,clip=}}
\caption[Fig-2-3-4]
{Magnetic surfaces for the MHH configuration with $\langle\beta\rangle=2.4\%$
as computed with the PIES code at $\phi=0^\circ$ (top), $\phi=22.5^\circ$
(middle), and $\phi=45^\circ$ (bottom).}
\label{f-2-3-4}
\end{figure}

\clearpage
\subsection {Stability Considerations}
\label{s-2-3-3}

A major advantage that the stellarator possesses over the tokamak is that it
does not have a large toroidal current that can provide energy to drive
large-scale MHD instabilities like disruptions.  Indeed, in a typical
stellarator, and in the MHH configuration, the sign of the shear in the
rotational transform is favorable even for ballooning modes, a ``reversed-shear
configuration'' that has recently been favorably regarded by tokamak designers
because of its superior stability properties.  In the analysis of the MHH
configuration, one can note that the driving forces for large-scale modes are
small and the favorable shear makes ballooning studies less important, and thus
calculate only the local stability functions $D_I$ and $D_R$ for
ideal (Mercier) and resistive modes~\cite{Gal75}.  This has been done using the
VMEC code studies of the case with $\langle\beta\rangle=4.8\%$ where the
configuration is marginally stable.  It should be expected, but has not been
shown, that localized ballooning modes will become unstable at or possibly
below this value of $\langle \beta \rangle$ because of the lessening of the
local field line shear due to the competition between the global twisting
associated with the vacuum magnetic field and the local twisting introduced by
the Pfirsh-Schl\"uter currents.  These may not be severe since they would be
centered at rational surfaces where the presence of magnetic islands would
already have degraded the confinement.  However, if such modes were serious,
they should have been observed in some of the calculations using the NSTAB
code; they did not appear and there was no indication of a flattening of the
local pressure distribution.

\subsection {Bootstrap Currents}
\label{s-2-3-4}

A major concern in both tokamak and stellarator physics concerns the bootstrap
current -- in the tokamak it is essential to maximize it with care taken to
obtain the proper spatial distribution so as to minimize the need for external
current drive; in the stellarator it is desirable to minimize this current.

When the mean-free path in the plasma is long compared to its size, then the
parallel viscosity component in Ohm's Law creates a bootstrap current.  It
arises when particles moving parallel to the magnetic field lines are displaced
from their home flux surface by a different amount than those moving
anti-parallel due to the particle drift motion.  It is possible in a
stellarator to reduce the net bootstrap current to zero (or even to reverse its
direction~\cite{Mur91}) because the bootstrap current component due to the
helical component of the magnetic field is opposite to that associated with the
toroidal component.

\clearpage

Boozer and Gardner~\cite{Boo90} have shown that this current depends on the
magnetic configuration only through the variation of the field strength on each
magnetic surface $B(\psi,\vartheta,\varphi)$ and the rotational transform
$\ib(\psi)$, and have provided a simple approximate expression that should
apply if the fraction of trapped particles is small.  Here $\psi$ is the
toroidal flux and $\vartheta$ and $\varphi$ are poloidal and toroidal angle
variables.  Estimates of the bootstrap current in the MHH configuration with
$\langle\beta\rangle=4.8\%$ indicate that this current could cause a
significant change in the rotational transform and thus seriously affect both
the equilibrium and the stability properties of the device.  Because the
magnetic configuration is fairly complicated so that the simple model of few
trapped particles may not be applicable, the SPPS has not examined the problem
in sufficient detail.  On the other hand, as discussed in the next section, it
is reasonable to believe that a relatively straightforward modification of the
configuration would solve this problem without introducing serious new
engineering difficulties.

\subsection {Configuration Improvement}
\label{s-2-3-5}

Considerations of the reference MHH stellarator configuration indicate 
that there are, or could be, problems with all three aspects of MHD: 
equilibrium, stability, and bootstrap current.  These problems can all be 
traced to the choice of the central-vacuum-field rotational transform being 
just above the value $\ib = 1$.  A configuration (described in Sec.~3.6.3)
in which the number of helical field periods is changed from $M = 4$ to 
$M = 3$, which would lower $\ib(0)$ to about $0.8$ and $\ib(a)$ to about 
$0.95$, can eliminate all of these problems.  This change actually improves 
the engineering problems concerning supporting the magnetic field coil forces 
and providing room for the blankets and shields.  Since this modified 
configuration had not been determined before a significant part of the 
engineering studies were underway, the original configuration was used for the 
study.

\section{TRANSPORT}
\label{s-2-4}

MHD considerations set a constraint on the plasma performance through 
either greatly enhanced transport (a ``soft'' beta limit) due to the onset of 
larger magnetic islands or a more severe limit due to overlapping islands.  
Below this limit, the more usual transport considerations determine the plasma 
performance.  Although there are some theoretical models for transport, 
especially at low collisionality~\cite{Has85}, performance predictions, as
in tokamaks, are better based on empirical confinement scaling ``laws''.
However, Monte-Carlo calculations using the TRAN code played a significant
role in arriving at the MHH configuration.

\subsection{Global Scaling Laws for Stellarators}
\label{s-2-4-1}

Different scalings for the global energy confinement time $\tau_E$ fit
present stellarator data: (1)~the Large Helical Device (LHD)
scaling~\cite{Sud90},
%
\begin{eqnarray}
 \tau_E^{LHD} & = & 0.17 R_0^{0.75} a_p^2 n^{0.69} B_0^{0.84} 
    P^{-0.58} \: ,
\label{e-3-4-1}
\end{eqnarray}	   
%
an empirical fit to stellarator data; (2)~the gyro-reduced Bohm 
scaling~\cite{Gol89},
%
\begin{eqnarray}
 \tau_E^{grB} & = & 0.25 R_0^{0.6} a_p^{2.4} B_0^{0.8} n^{0.6} 
    P^{-0.6} \: ,
\label{e-3-4-2}
\end{eqnarray} 					 
%
which is based on drift-wave theory; and (3) the Lackner-Gottardi 
scaling~\cite{Lac90},
%
\begin{eqnarray}
 \tau_E^{LG} & = & 0.17 R_0 a_p^2 n^{0.6} B_0^{0.8} P^{-0.6} 
    \ib^{0.4} \: , 
\label{e-3-4-3}
\end{eqnarray}
%
which fits both tokamak and stellarator data.  Here $B_0$ is the on-axis 
field, $n$ is the line-averaged electron density (in $10^{20}$~m$^{-3}$), 
$a_p$ is the average radius of the noncircular (and nonaxisymmetric in 
stellarators) last closed magnetic surface, and $P$ is the absorbed heating 
power (in MW).  All other quantities are in SI units.

Unlike the LHD scaling, the Lackner-Gottardi scaling and the gyro-reduced Bohm
scaling are dimensionally correct; that is, they are expressible in terms of
dimensionless plasma parameters.  In addition, they have the same functional
dependence on the reactor parameters $R_0$, $B_0$, $n$, and $P$, differing only
by an aspect-ratio-dependent coefficient.  Coincidentally, the Lackner-Gottardi
and LHD scalings also give nearly the same value of $\tau_E$ for typical
reactor cases, even though they have different functional dependences on the
reactor parameters.

Lackner-Gottardi scaling was monitored with a confinement improvement factor
${H'}$ similar to the H-mode confinement improvement factor for tokamaks:
$\tau_E = {H'}\tau_E^{LG}$.  Evidence from experiments and
theoretical arguments support such a confinement improvement.  However, the
improvement with the square root of the ion mass used in tokamak scaling is not
assumed in our study, and $\ib$ is evaluated at a normalized radius $\rho =
r/a_p = 2/3$, rather than at the plasma edge; incorporating the mass
dependence, as is done in the Garching studies, would reduce the confinement
multiplier needed by a factor of 1.3.

\subsection {Comparison with Tokamak Scaling}
\label{s-2-4-2}

Stellarators and tokamaks have similar energy confinement time scaling,
indicating that the underlying physics may be dominated by common toroidal
plasma physics rather than coil-geometry-specific effects.
Figure~\ref{f-2-4-1} shows the measured energy confinement time $\tau_E$
versus that calculated for gyro-reduced Bohm scaling for different
stellarators~\cite{Mur92} and for the tokamak L-mode data
base~\cite{Kay92p} where
%
\begin{eqnarray}				
 \tau_E^{grB}({stell}) & = & 0.25 B_0^{0.8} n^{0.6} P^{-0.6} 
   a_p^{2.4} R_0^{0.6} A_i^{-0.2} \: ,
\label{e-3-4-4}
\end{eqnarray}
%
for stellarators and
%
\begin{eqnarray}				
 \tau_E^{grB}({tok}) & = & 0.194 I_p^{0.8} n^{0.6} P^{-0.6} 
  a^{2.2} R_0^{0.6} A_p^{1.02} A_i^{-0.2} \: ,
\label{e-3-4-5}
\end{eqnarray}
%
for tokamaks using average tokamak quantities (plasma ellipticity $\kappa=1.4$,
average plasma aspect ratio $A_p=3$, safety factor $q_{cyl}=3$) to
connect the tokamak plasma current $I_p$ to the on-axis field $B_0$.
Here $a$ is the plasma minor radius (for tokamaks), and $A_i$ is the
average ion mass (= 2.5 for the D-T plasma).

Generally, confinement in stellarators is competitive with that in comparable
tokamaks.  However, present stellarators~\cite{Lyo90} are much smaller
($a_p=0.2-0.27$~m) than present tokamaks ($a_p=0.85-1.62$~m) and
have not yet demonstrated adequate confinement and beta at parameters that can
be extrapolated to the reactor regime. It is evident from Fig.~\ref{f-2-4-1}
that the gap between the present performance and that needed for a practical
power plant is much larger for stellarators than it is for tokamaks, typically,
only 5-10 for tokamaks. Until larger ($a_p=0.52-0.65$~m) next-generation
stellarators~\cite{Iiy90,Bei90} begin operation near the end of this decade,
assessment of the reactor potential of stellarators must rely on large
extrapolations from present experiments and commonality with tokamak results.

\begin{figure}[p]
\centerline{\psfig{file=spps-2-4-1.eps,height=7.7in}}
\caption[Fig-2-4-1]
{Energy confinement scaling for different stellarators and the
tokamak L-mode database.}
\label{f-2-4-1}
\end{figure}

\subsection {Improvement in Confinement}
\label{s-2-4-3}

Stellarators, like tokamaks, need to rely on an improved confinement regime for
an attractive reactor.  However, because of their larger plasma volume,
stellarators do not require as large an improvement factor.  For a confinement
scaling law for which $\tau_E\propto P^{-0.6}$, which applies to both
tokamaks and stellarators, a simple power balance argument shows that
\hbox{$\langle\beta\rangle \propto H^{2.5}P_E$}~\cite{Lyo92}.  Beyond a
certain point, an improvement in confinement must be accompanied by a
corresponding improvement in the beta limit, or it cannot be used.  Tokamaks
typically require ${H'} \approx 2.5$ (as for ARIES-IV), but MHH only requires
${H'} \approx 1.5$.  This is not too far from the confinement improvement seen
in W7-AS (${H'} \approx 1.3$)~\cite{Wag95} and in CHS~\cite{Toi95}, but there is
room for further improvement.  The Pfirsch-Schl\"uter and bootstrap currents
are reduced in the MHH configuration over those in present stellarators, so
there is less free energy to drive instabilities.  In addition, the effective
helical ripple of the MHH configuration is reduced over that in a conventional
stellarator because of MHH's quasi-helically symmetric features, so
ripple-induced transport at low collisionality should also be reduced.  A
confinement improvement factor of ${H'} \approx 1.5$ appears to be reasonable,
and the next-generation of stellarator experiments should aim at higher
factors, similar to the goal of advanced tokamaks.

\subsection {Density Limit}
\label{s-2-4-4}

The maximum plasma density in stellarators is not determined by a 
disruption limit as in tokamaks.  Sudo \ea~\cite{Sud90}, have 
proposed a maximum line-average density,
%
\begin{eqnarray}
  n_{max} & = & 0.25(P B_0/R_0a_p^2)^{1/2} \: ,
\label{e-3-4-6}
\end{eqnarray}
%
based on Heliotron E data.  Densities a factor of 1.3 higher than this value
have been observed in ATF.  For this study, it is assumed that the line-average
density is constrained to $<1.5n_{max}$.

\subsection {Transport Modeling}
\label{s-2-4-5}

Four different steady-state transport modeling approaches were used for the MHH
power balance studies in the SPPS.  Time-dependent calculations were not
necessary because the time scale on which the plasma moves along a controlled
operating path in the density-temperature plane is much longer than the energy
confinement time (seconds) and other faster timescale phenomena ({\it e.g.,}
tokamak ``sawteeth'') do not occur in stellarators.  The main approach, that
used in the ARIES studies and reported here, was 0-D calculations of $\langle T
\rangle$ with fixed $n_e(r)$, $n_i(r)$, $T_e(r)$, and $T_{\rm
i}(r)$ using the global Lackner-Gottardi $\tau_E$ scaling model
[Eq.~(\ref{e-3-4-3})].  Here $n_e(r)$ and $n_i(r)$ are the electron
and ion density profiles with $n(r)$ parabolic, and $\langle T \rangle$ is the
density-averaged plasma temperature with the electron and ion temperature
profiles $T_e(r)$ and $T_i(r) \propto (\rm parabolic)^{1.1}$.
Impurities are modeled by 1\% C and 0.01\% Fe.  The helium ash accumulation and
fuel ion dilution are calculated from the fusion reaction rate and an assumed
value for the ratio of the helium particle confinement time $\tau_{He}$ to
the energy confinement time $\tau_E$ at each value of $\langle n \rangle$
and $\langle T \rangle$.

Calculations using the other approaches are reported elsewhere: ($a$) 1-D
calculations of $T_e(r)$ and $T_i(r)$ with fixed $n(r)$ and $\chi
(r)$ normalized to the global Lackner-Gottardi $\tau_E$ scaling
model~\cite{Lyo94b}; ($b$) 1-D calculations of $T_e(r)$ and $T_{\rm
i}(r)$ with fixed $n(r)$, $\chi_{anom}(r)$, and $E_r(r)$ with the
Shaing-Houlberg ripple-induced neoclassical transport matrix~\cite{Gul94}; and
($c$) 1-D self-consistent calculations of $T_e(r)$, $T_i(r)$,
$n_e(r)$, $n_i(r)$, and $E_r(r)$ with fixed fueling profile for the
Shaing-Houlberg ripple-induced neoclassical transport matrix and anomalous
$\chi (r)$ and $D(r)$~\cite{Lyo94c}. Here $\chi (r)$ and $D(r)$ are the thermal
and particle diffusivity profiles, and $E_r(r)$ is the radial electric field.

Figure~\ref{f-2-4-2} shows the operating space for an MHH reactor in the POPCON
(Plasma OPerating CONtour) density-temperature ($\langle n \rangle-\langle T
\rangle$) plane~\cite{Hou82} with contours of the externally applied heating
power $P_{ext}$ required for equilibrium at a given point (the light solid
curves labeled in MW from 0 to 100~MW in 10-MW intervals and in 100-MW
intervals for auxiliary heating powers $\ge 100$~MW).  Contours of constant
$\langle \beta \rangle$ (in multiples of 5\%) and constant thermal fusion power
($P_{fus}$, in multiples of $P_{fus,op} = 1.73$~GW) are indicated by the
dotted and chain-dash curves, respectively; $\langle n \rangle$ is the
volume-averaged electron density.  The ignition contour is the heavy curve
where $P_{ext} = 0$.  The dotted curve indicates the Sudo density ``limit''
calculated from Eq.~(\ref{e-3-4-6}).  The reference assumptions in
Fig.~\ref{f-2-4-2} are $R_0 = 14$~m, $a_p = 1.63$~m, $B_0 = 5$~T, ${H'} =
1.5$, $\tau_{He}/\tau_E = 6$, and 5\% alpha-particle power lost.

\begin{figure}[tb]
\centerline{\psfig{file=spps-2-4-2.eps,height=4.5in}}
\caption[Fig-2-4-2] 
{POPCON plot for the reference MHH power plant parameters:
$R_0 = 14$~m, $B_0 = 5$~T, $\tau_{He}$/$\tau_E = 6$, and $H' = 1.5$.}
\label{f-2-4-2}
\end{figure}

The ignited MHH operating point, indicated by a ``$\bullet$'', occurs where the
$P_{fus, op} = 1.73$~GW contour crosses the ignition curve.  In
Fig.~\ref{f-2-4-2}, $P_{fus} = 1.73$~GW, corresponding to $P_E =
1$~GW for the values of thermal conversion efficiency and blanket energy
multiplication assumed in the SPPS.  The values obtained for the reference
operating point are $\langle n \rangle = 1.24 \times 10^{20}$~m$^{-3}$,
$\langle T \rangle = 11.5$~keV, $\langle \beta \rangle = 4.4\%$, $f_{He} =
5\%$, and $Z_{eff} =1.46$.  The values for $P_{fus}$ and $\langle \beta
\rangle$ vary as $\langle n \rangle$ and $\langle T \rangle$ vary along the
ignition curve.  The minimum values for the reference case in
Fig.~\ref{f-2-4-2} are $P_{fus,min} = 0.69$~GW ($P_E = 0.4$~GW)
and $\langle \beta \rangle_{min} = 2.69\%$.

The ignition contour in the POPCON plots, and consequently the operating point,
varies with the MHH reactor parameters that are assumed.  Figure~\ref{f-2-4-3}
shows POPCON plots for six different values of the Lackner-Gottardi confinement
multiplier, $H'$.  The contour curves are characterized by the same values as
in Fig.~\ref{f-2-4-2}.  As $H'$ increases from $1.3$ to $2$, the operating
point moves along the ignition curve from higher density and lower temperature
to lower density and higher temperature.  Actually, the constant-$P_{fus}$
curve remains fixed and the ignition curve changes because $P_{fus}$ is
approximately proportional to ${\langle n \rangle}^2 {\langle T \rangle}^2
R_0^3$.  The constant-$\langle \beta \rangle$ curve also remains unchanged:
$\langle \beta \rangle \propto \langle n \rangle \langle T \rangle /B_0^2$.  As
a result, the values of $P_{fus,min}$ and $\langle \beta \rangle_{min}$
along the ignition curve change as $H'$ varies.  Table~\ref{t-2-4-1} summarizes
these results as $H'$ varies from $1.2$ to $2$.  The operating point is
thermally stable to small excursions in $\langle n \rangle$ and $\langle T
\rangle$ on the right branch of the ignition curve (where $\partial \langle n
\rangle / \partial \langle T \rangle > 0$).  This occurs for ${H'} > 1.35$.
The operating point is thermally unstable for lower values of ${H'}$.  An
operating point with $P_{fus} = 1.73$~GW cannot be obtained for $H' <
1.245$ where $P_{fus,min} > 1.73$~GW.  Figure~\ref{f-2-4-4} shows the
POPCON curves for the minimum value (${H'} = 1.245$) that satisfies the
constraint that an ignited operating point exist with $P_{fus,op} =
1.73$~GW.  The $P_{fus,op} = 1.73$~GW contour does not intersect the
ignition curve for lower values of ${H'}$.

\begin{figure}[p]
\centerline{\psfig{file=spps-2-4-3.eps,height=7.4in}}
\caption[Fig-2-4-3]
{Movement of ignition contour and operating point with
confinement improvement factor $H'$: ($a$) $H' = 1.3$, ($b$)
$H' = 1.4$, ($c$) $H' = 1.5$, ($d$) $H' = 1.6$, ($e$)
$H' = 1.8$, ($f$) $H' = 2$.}
\label{f-2-4-3}
\end{figure}

\begin{table}[t]
\caption{}
\centering
{\bf MHH Reactor Parameters for Different $H'$ Assumptions
with $\tau_{He}/\tau_E = 6$} \\
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{ c c c c c c c c }
\hline \vspace{0.2cm}
${H'}_{L-G}$ 
   & $f_{He}$ & $\langle n \rangle$ (m$^{-3}$)
                                 & $\langle T \rangle$ (keV)
                                         & $\langle \beta \rangle$
                                                  & $P_{fus,min}$ (GW)
                                           & $\langle \beta_{min} \rangle$
                                                         & $P_{SP}$ (MW) \\
\hline \\
1.2\Z & ---   & ---              & ---   & ---    & 2.06 & 5.11\% &  55 \\
1.25  & 3.7\% & $2.19\X 10^{20}$ & \Z6.9 & 4.65\% & 1.69 & 4.65\% &  45 \\
1.3\Z & 4.0\% & $1.73\X 10^{20}$ & \Z8.3 & 4.44\% & 1.52 & 4.24\% &  37 \\
1.4\Z & 4.4\% & $1.41\X 10^{20}$ &  10.0 & 4.35\% & 0.96 & 3.51\% &  25 \\
1.5\Z & 5.0\% & $1.24\X 10^{20}$ &  11.5 & 4.37\% & 0.69 & 2.96\% &  18 \\
1.6\Z & 5.4\% & $1.13\X 10^{20}$ &  12.9 & 4.44\% & 0.50 & 2.53\% &  13 \\
1.8\Z & 6.4\% & $0.99\X 10^{20}$ &  15.5 & 4.65\% & 0.28 & 1.90\% & \Z8 \\
2.0\Z & 7.4\% & $0.91\X 10^{20}$ &  18.0 & 4.93\% & 0.17 & 1.47\% & \Z5 \\
&&&&&&& \\ \hline
\end{tabular}
\label{t-2-4-1}
\end{table}

\begin{figure}[b]
\centerline{\psfig{file=spps-2-4-4.eps,height=3.7in}}
\caption[Fig-2-4-4]
{POPCON plot for the reference MHH power plant with ${H'} = 1.245$.}
\label{f-2-4-4}
\end{figure}

Unlike tokamaks, stellarators do not require power input to the 
plasma once the self-heating from fusion-produced alpha particles dominates 
the power balance; at this point the plasma runs away to ignition.  In the 
POPCON plots, this occurs where $\partial P_{ext}/\partial \langle T 
\rangle < 0$; {\it i.e.}, along a curve that connects the minima of the 
$P_{ext}$ contours.  The most efficient startup path passes through the 
minimum value of $P_{ext}$ ($P_{SP}$) along this curve.  This saddle 
point in the auxiliary heating power curves is the lowest value needed to 
reach ignition; it is listed in Table~\ref{t-2-4-1} for different $H'$ values.
Below the saddle point in the $\langle n\rangle - \langle T\rangle$ plane, the 
operating path can be carefully controlled by controlling the density and the 
external heating power to allow a slow startup in which the plasma-facing 
components and blanket have time to come to thermal equilibrium.  As the 
external power input is reduced above the saddle point, additional losses 
(either through additional impurity radiation or a perturbing magnetic field) 
are needed to slow the plasma evolution to ignition or to operate at reduced 
fusion power.  One contributing factor that slows this path to ignition and 
reduces the additional losses needed for control is the buildup of helium ash, 
which increases the bremsstrahlung radiation and dilutes the D-T fuel density
(see Table~\ref{t-2-4-2}).

\begin{table}[b]
\caption{}
\centering
{\bf MHH Reactor Parameters for Different
$\tau_{He}/\tau_E$ Assumptions with $H' = 1.5$} \\
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{ c c c c c c c c }
\hline \vspace{0.2cm}
$\tau_{He}/\tau_E$ 
   & $f_{He}$ & $\langle n \rangle$ (m$^{-3}$)
                                 & $\langle T \rangle$ (keV)
                                         & $\langle \beta \rangle$
                                                  & $P_{fus,min}$ (GW)
                                           & $\langle \beta_{min} \rangle$
                                                         & $P_{SP}$ (MW) \\
\hline \\
2\Z & \Z1.7\% & $1.12\X 10^{20}$ & 11.8 & 4.12\% & 0.60 & 2.64\% & 17 \\
4\Z & \Z3.3\% & $1.18\X 10^{20}$ & 11.7 & 4.24\% & 0.63 & 2.79\% & 18 \\
6\Z & \Z5.0\% & $1.24\X 10^{20}$ & 11.5 & 4.37\% & 0.69 & 2.96\% & 18 \\
8\Z & \Z6.6\% & $1.31\X 10^{20}$ & 11.3 & 4.51\% & 0.76 & 3.15\% & 19 \\
10  & \Z8.2\% & $1.40\X 10^{20}$ & 11.1 & 4.65\% & 0.86 & 3.38\% & 20 \\
12  & \Z9.8\% & $1.50\X 10^{20}$ & 10.7 & 4.82\% & 0.97 & 3.68\% & 21 \\
15  &  12.2\% & $1.71\X 10^{20}$ & 10.1 & 5.11\% & 1.31 & 4.45\% & 22 \\
\\ \hline
\end{tabular}
\label{t-2-4-2}
\end{table}

Since $P_{fus,min} = 0.69$~GW $< P_{fus,op} = 1.73$~GW for the
reference case, it is possible to have MHH reactors with $P_E$ down to
$0.4$~GW without changing the value for $R_0$ or $B_0$.  The desired power
level is controlled through the value of $\langle n \rangle$ on the ignition
curve.  Table~\ref{t-2-4-3} shows the variation of the reference MHH reactor
parameters as $P_E$ varies from $0.5$~GW to $2$~GW.  Here $\Gamma_{\rm
n}$ is the average neutron wall loading.  Although the cost of the reactor core
would not change with $P_E$, the cost of electricity would decrease
significantly with increasing power output.  Nevertheless, $1$~GW(electric) was
chosen to allow comparison with tokamak reactors.  If the $\langle \beta
\rangle$ value is limited to $<5\%$, then a slightly larger value of $R_0$ or
$B_0$ would be needed: $B_0^4R_0^3 \propto P_E/\langle \beta \rangle^2$.

\begin{table}[tb]
\caption{}
\centering
{\bf MHH Operating Parameters for Different Values of $P_E$} \\
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{c c c c c c}
\hline
&&&&& \\
$P_E$ (GW) 
      & $\langle n \rangle$ (m$^{-3}$)
                         & $ \langle T \rangle$ (keV) 
                                 & $\langle \beta \rangle$
                                          & $f_{He}$ 
                                              & $\Gamma_{n}$~MW/m$^2$ \\
\hline \\ 
0.5\Z & $1.21\X 10^{20}$ & \Z8.4 & 3.13\% & 3.98\% & 0.52 \\
0.6\Z & $1.19\X 10^{20}$ & \Z9.3 & 3.39\% & 4.22\% & 0.62 \\
0.7\Z & $1.19\X 10^{20}$ &  10.0 & 3.64\% & 4.43\% & 0.73 \\
0.8\Z & $1.20\X 10^{20}$ &  10.6 & 3.89\% & 4.62\% & 0.83 \\
1.0\Z & $1.24\X 10^{20}$ &  11.5 & 4.37\% & 4.95\% & 1.04 \\
1.25  & $1.29\X 10^{20}$ &  12.5 & 4.93\% & 5.29\% & 1.30 \\
1.5\Z & $1.35\X 10^{20}$ &  13.3 & 5.47\% & 5.58\% & 1.56 \\
1.75  & $1.40\X 10^{20}$ &  14.0 & 5.97\% & 5.82\% & 1.82 \\
2.0\Z & $1.45\X 10^{20}$ &  14.6 & 6.45\% & 6.04\% & 2.08 \\
&&&&& \\ \hline 
\end{tabular}
\label{t-2-4-3}
\end{table}

\subsection{Plasma Heating Options}
\label{s-2-4-6}

Different heating options are available for MHH power plant startup because
external power input is only required transiently for plasma heating and not
continuously for current drive.  This power is only needed until the plasma is
ignited, so it can be discontinued afterwards.  The various options are
electron cyclotron heating (ECH), ion cyclotron heating (ICH), and neutral beam
injection (NBI).  ECH is the preferred heating option because it can be located
remotely from the plasma and connected to it via relatively small quasi-optical
waveguides, minimizing the real estate needed for this system.  The 280-GHz
frequency required for second harmonic X-mode ECH at $B_0=5$~T allows operation
up to \hbox{$n_e=2.5\X 10^{20}$~m$^{-3}$}, above the required operating-point
density.  ICH is not as practical because it requires a larger launcher exposed
to the plasma.  NBI introduces a particle and momentum source in addition to
heating the plasma and requires larger access to the plasma with the source
components exposed to the intense neutron flux from the plasma during heating
to ignition.

\section {PARTICLE AND HEAT REMOVAL}
\label{s-2-5}

\subsection{Separation of Particle and Heat Removal}
\label{s-2-5-1}

The fusion power is carried by alpha particles (with energy $E_{\alpha} =
3.5$~MeV) and neutrons ($E_{n} = 14.1$~MeV).  The neutron flux is assumed
to penetrate the first wall to be absorbed in the blanket and shield, so the
only power impacting on the first wall and divertor is that which originates in
the alpha particles.  If the total fusion power is $P_{fus} = 1.725$~GW and
the power limit on the divertor targets,$Q_{d}$, is $5$~MW/m$^2$, then the
surface area required for the divertor plates (if all the power is handled by
the divertor) is
%
\begin{eqnarray}
\label{e-3-5-1}
 S & = & [E_{\alpha}/(E_{\alpha} + E_{n})] \times [P_{fus}/Q_{
      d}] = 69~m^2 \: ,
\end{eqnarray}
%
which is about $5\%$ of the plasma surface area ($1233$~m$^2$).  In the SPPS 
study, the divertor takes up about $15\%$ of the plasma surface area, or about 
$185$~m$^2$, so in principle a factor of $2.7$ margin exists.  However, the 
exiting flux is quite non-uniform around the torus and the power is not 
uniformly distributed on the divertor plates, so it is prudent to adopt some 
additional strategy to ensure engineering feasibility.

It would be useful to spread out the alpha-particle heat load over the entire
first wall by radiating away most of the power at the edge.  Experiments in the
TEXTOR~\cite{Mes95} and TORE SUPRA~\cite{Tore95} tokamaks show that
$>80\%$ of the power can be converted to radiation by introducing a layer of
impurities ({\it e.g.}, neon) at the plasma edge.  It is assumed that this
technique can be used for MHH, and that $80\%$ of the power can be radiated to
the wall (with an average power of $0.26$~MW/m$^2$, a factor of $2$ below the
$0.5$~MW/m$^2$ power-handling capability assumed for the first wall).  The
remaining $20\%$ of the escaping alpha-particle power must be handled by the
divertor, about $68$~MW.

The particle flux calculation is more speculative because of the uncertainty in
the value of the particle confinement time, $\tau_p$.  In the SPPS study,
it is assumed that $\tau_p \sim 6\tau_E$.  In most tokamaks,
$\tau_p \sim 2\tau_E$.  In elm-free $H$-mode tokamak plasmas,
$\tau_p \sim 4\tau_E$.  In some experiments~\cite{Uck94} in the ATF
torsatron at Oak Ridge National Laboratory, biasing the edge of the plasma made
$\tau_p >> \tau_E$.  In steady-state, of course, the efflux of
particles, $\Gamma_p$, times the area of the LCFS must equal the fueling
rate, $\Phi$(particles/second).  The total number of particles in the plasma
$N_{tot}$ is given by the average density times the plasma volume, so
%
\begin{eqnarray}
 N_{tot}/\tau_p & = & ({particles \, escaping/second}) \:\: 
   = \:\: \Phi /(1 - R)
\label{e-3-5-2}
\end{eqnarray}			  
%
where $R$ is the recycling coefficient for particles at the plasma edge.

In divertor tokamaks, such as DIII-D, the recycling coefficient can be close to
zero -- most of the escaping particles go into the divertor, and very few make
it back to the plasma.  In pump limiters, typically $90\%$ of the flux hits the
limiter head and $10\%$ goes into the divertor chamber.  Of this $10\%$, about
half may make it back into the plasma, so that $R$ is about $0.95$.  The
divertor geometry in the MHH stellarator is a sort of hybrid between a divertor
and a pump limiter, so a wide range of values for $R$ could occur.

\subsection {Magnetic Geometry and Islands}
\label{s-2-5-2}

The NESCOIL code can be used to create configurations with the same LCFS, but
which can have quite different properties in the region between the LCFS and
the CWS.  This is accomplished by making small changes in the shape (Fourier
harmonics) of the CWS.

The reference MHH has small $n/m=4/3$ islands outside of the LCFS.  These
islands actually change the shape of the LCFS slightly.  There are also
variations of the MHH configuration with large islands surrounded by good
closed flux surfaces, and configurations in which this outer region is very
chaotic.  This flexibility offers many options in designing the divertor
geometry.

For example, one variation (Fig.~\ref{f-2-5-1}) shows a version of MHH
(normalized to $R_0 = 10$~m) with large well-formed islands that are surrounded
by non-ergodic closed flux surfaces.  This sort of topology is well-suited to
be used by a local island divertor (LID)~\cite{Kom95}, which is an extended
form of a pumped limiter.  The head of the divertor is placed within the island
(the black cresent-shaped region in Fig~\ref{f-2-5-2}).  The back of the island
intersects the back of the LID head (the gray region in Fig.~\ref{f-2-5-2}) at
a grazing incidence to spread the heat flux, and a second plate can be placed
just outside of the outer LCFS to close off the pumping region.

\begin{figure}[p]
\centerline{\psfig{file=spps-2-5-1.ps,height=7.3in,bbllx=43pt,bblly=79pt,%
            bburx=533pt,bbury=677pt,clip=}}
\caption[Fig-2-5-1]
{Cross section of magnetic surfaces (dotted), the nominal plasma boundary or
LCFS (inner solid curve), islands (dotted) outside the LCFS, and the CWS (outer
solid curve) at four toroidal angles in a field period.}
\label{f-2-5-1}
\end{figure}

\begin{figure}[t]
\centerline{\psfig{file=spps-2-5-2.eps,height=4.5in}}
\caption[Fig-2-5-2]
{A rough drawing of a local island divertor configuration.
The front of the head (black) is shielded from the flux.  The back of the head 
(gray) is hit by the flux coming around the back of the island at a low angle 
of incidence.}
\label{f-2-5-2}
\end{figure}

The LID concept is being tested in the Japanese helical devices CHS and LHD.
However, the operational configuration must be prespecified so that the LID
will fit into the island at the operational conditions.  As the plasma beta
increases, the configuration might change if the Pfirsch-Schl\"uter and
bootstrap currents do not remain at negligible levels.  However, for the
helias-like configuration of MHH, these currents are controlled, and the
concept may be feasible.

A second option for MHH is to utilize the escaping flux bundles in multiple
bundle divertors.  This concept was explored in the Interchangeable Modular
Stellarator at the University of Wisconsin in Madison.  By biasing a plate in
one of the escaping flux bundles, the flux can be transferred to a different
bundle.  This is important since it is impractical to collect all of the flux
bundles.  The advantage of the bundle divertor approach is that the collector
can be outside of the machine (past the modular coils).  However, the channel
for the escaping flux must be shielded on all sides, and these large orifices
take away room for the blanket.  Therefore, the bundle divertor approach may
not be appropriate for MHH.

Instead, the MHH design uses the approach proposed by Garching for the W7-X
helias.  The islands outside the LCFS are used to solve the leading edge
problem without inserting a plate entirely within the island.  The island
structure of MHH (with major radius here normalized to $9$~m) is shown in
Fig.~\ref{f-2-5-3}.  Instead of inserting the entire divertor plate within the
island, the center of the plate is inside the island, and the edges of the
plate (outside the island) are bent back so that they intercept the flux at the
desired angle, as shown in Fig.~\ref{f-2-5-4}.  The big advantage of this
scheme is that the island can move and change shape without major changes in
the divertor operation.  Indeed, the island surfaces can be slightly `fuzzy'
because the underlying connectivity topology remains even if the islands are
somewhat broken up.  The details of this design will be discussed in the next
section.

\begin{figure}[p]
\centerline{\psfig{file=spps-2-5-3.ps,height=7.5in,bbllx=14pt,bblly=57pt,%
            bburx=497pt,bbury=648pt,clip=}}
\caption[Fig-2-5-3]
{The island structure for the base case of MHH.  This particular example has
been scaled for a 9-m average major radius.}
\label{f-2-5-3}
\end{figure}

\begin{figure}
\centerline{\psfig{file=spps-2-5-4.ps,height=6.7in,clip=}}
\caption[Fig-2-5-4]
{Cross section of the last magnetic surface and the coil boundary at $\phi =
0^{\circ}$ (top) and $\phi = 22.5^{\circ}$ (bottom) indicating the divertor
plate scheme proposed for MHH.  Diffusion of field lines outside the last
closed magnetic surface is shown in the vicinity of the divertor targets
(thicker black objects) and the baffle plates (thinner black objects) at the
apexes of the MHH configuration.  The dotted curve indicates a 2-m distance
from the coil boundary.}
\label{f-2-5-4}
\end{figure}

The chief potential disadvantage of this approach is that by using divertor
plates close to the plasma, it is difficult to create a closed geometry that
can be plugged by a plasma and pumped.  The pump location is behind the
divertor plates, so either the plates must be slotted, or the neutralized
plasma must find its way around the plates to the pumps.

\subsection{Divertor Geometry}
\label{s-2-5-3}

One way of assessing the geometry of the divertor region is to calculate the
connection length of the field lines outside of the LCFS, the distance along a
field line from that point to the divertor plate.  In general, the field lines
that go close to the LCFS and that have a long connection length to the
divertor plate will carry the most plasma particles.  To calculate the
connection length topology, the region between the LCFS and the CWS was divided
into concentric shells.  Field lines were launched on each shell headed towards
the plasma.  For each field line, the minimum distance to the plasma and the
distance to the birth surface (twice the connection length) were calculated.
Fortunately, the connection length is maximized for those field lines that
approach closest to the LCFS because these field lines go around the outside of
the islands or their remnants.

\begin{figure}[p]
\vspace{1.in}
\centerline{\psfig{file=spps-2-5-5.ps,width=6.5in,bbllx=93pt,bblly=50pt,%
            bburx=512pt,bbury=440pt,clip=}}
\caption[Fig-2-5-5]
{Field lines are launched from a $\theta$-$\phi$ grid located on a phantom
surface 20\% of the way from the LCFS to the CWS.  The 3-D surface represents
the distance of closest approach of each field line to the LCFS and the lines
represent the lengths of the field lines.}
\label{f-2-5-5}
\end{figure}

Figure~\ref{f-2-5-5} shows one way of displaying this information.  Magnetic
field lines are launched on a $\theta-\phi$ grid on the phantom surface that is
$20\%$ of the way out from the LCFS to the CWS.  Note that the angle $\theta$
is the one used in the parameterization of the surface.  A $\theta = {\rm
constant}$ line rotates helically and returns to its original position after
one field period.  In this plot, the connection length is defined as the total
length of the field line from the starting point until it returns to the launch
surface.  The field lines with connection lengths above $50$~m are shown as
vertical lines whose height is proportional to the connection length.  Each
line is plotted twice -- at both the starting and ending values of $\theta$ and
$\phi$.  On the same plot, the wire-frame grid represents the distance of
closest approach of the field line to the LCFS.  Actually, the minimum value of
the flux is plotted, so the height of this surface can go slightly negative
because of the small $4/3$ island chain outside the LCFS.  The important point
to note is that the long connection-length field lines (the spikes) also
approach closest to the plasma, {\it i.e.,} they occur in the troughs of the
wire-frame surface.  Figures~\ref{f-2-5-6} and~\ref{f-2-5-7} show the same plot
for launch points $10\%$ and $15\%$ of the way out from the LCFS to the CWS.
The coincidence of the long connection lengths and close approach to the plasma
is repeated on these surfaces also.  However, the $\theta-\phi$ locations of
the longest field lines change.  Puncture plots of the long-connection-length
field lines that were launched $25\%$ of the way out from the LCFS to the CWS
(shown in Fig.~\ref{f-2-5-8}) show how the islands corral the exiting flux so
that it goes around the islands rather than through them.

\begin{figure}[p]
\vspace{1.in}
\centerline{\psfig{file=spps-2-5-6.ps,width=6.5in,bbllx=93pt,bblly=50pt,%
            bburx=512pt,bbury=440pt,clip=}}
\caption[Fig-2-5-6]
{The same plot as in Fig.~\ref{f-2-5-5}, but for field lines launched
10\% of the way out from the LCFS to the CWS.}
\label{f-2-5-6}
\end{figure}

\begin{figure}[p]
\vspace{1.in}
\centerline{\psfig{file=spps-2-5-7.ps,width=6.5in,bbllx=93pt,bblly=50pt,%
            bburx=512pt,bbury=440pt,clip=}}
\caption[Fig-2-5-7]
{The same plot as in Fig.~\ref{f-2-5-5}, but for field lines launched
15\% of the way out from the LCFS to the CWS.}
\label{f-2-5-7}
\end{figure}

\begin{figure}[p]
\centerline{\psfig{file=spps-2-5-8.ps,height=7.3in,bbllx=14pt,bblly=43pt,%
             bburx=490pt,bbury=648pt,clip=}}
\caption[Fig-2-5-8]
{Puncture plots of the long connection length field lines
launched on a grid that is 25\% of the way out from the LCFS to the CWS.  This 
plot illustrates how the presence of the small islands corrals the exiting 
flux.}
\label{f-2-5-8}
\end{figure}

The converse problem has also been solved by the Garching group for W7-X.
Starting at the plasma and work outwards, the regions of increasing field line
connection length are nested, with the longest being just outside of the LCFS.
This nesting is especially striking in the vicinity of the X-points at the tips
of the bean-shaped cross-section.

Much of the MHH divertor design has been performed by the W7-X
group~\cite{Ren95}.  Their approach to the problem is as follows:

\begin{enumerate}

\item
determine the exact position of the separatrix, especially the X-points, and
find the size of the $\ib = 4/3$ islands outside of the last closed flux
surface;

\item
investigate the field stucture outside the last closed surface;

\item 
choose the poloidal and toroidal position of segmented target and baffle
plates, with planar surface elements, in regions where the islands have a large
radial extension and provide enough space towards the first wall;

\item
adjust the minor radius position of the target plates for field line
intersection angles of about $1^{\circ}$ to $2^{\circ}$;

\item
follow magnetic field lines (with an additional artificial diffusion) that are
started inside the LCFS until they intersect the plates;

\item 
improve the size and position of the plates in a number of iterations with the
preceding step, and verify that local high power loads are avoided, especially
at the `leading edges'; and thus

\item
arrive at an interim solution, as shown in Fig.~\ref{f-2-5-9}~\cite{MMH-W7X},
with a power loading given by the deposition pattern at the target plates.

\end{enumerate}

\begin{figure}[p]
\centerline{\psfig{file=spps-2-5-9-a.ps,height=3.3in,clip=}}
\vspace*{1.25cm}
\centerline{\psfig{file=spps-2-5-9-b.ps,height=3.3in,clip=}}
\caption[Fig-2-5-9]
{A 3-D view of the proposed divertor and baffle plates in relation to the LCFS
(top) and impacts of field lines on the divertor target plates (bottom).}
\label{f-2-5-9}
\end{figure}

For this rather quick study, a coarse segmentation of the target plates was
used.  Thus the intersection pattern is not uniform, but the `leading edge'
problem is definitely avoided.  The target and baffle plates for MHH are in the
toroidal region near the apexes of the bean-shaped cross section, as is the
choice for the W7-X divertor configuration.

Four different variations of the base case were examined~\cite{MMH-W7X},
labeled MHH-A, MHH-B, MHH-C, and MHH-D.  The configuration MHH-A was selected
as the SPPS base case.  These cases differ in the current density and
cross-section of the modular coils.  There are only minor differences between
the edge structures of the four data sets, regarding the position and size of
the last closed surface or of the $\ib = 4/3$ islands.  MHH-B does not have
space for a divertor if a uniform radial width of $1.3$~m is assumed between
the bore of the winding pack and the LCFS.  The distance between the LCFS and
the coil bore is larger for MHH-C than for MHH-A, due to the different position
of the winding pack bores.  MHH-D with the smallest radial thickness of the
winding pack does not improve the spatial situation regarding a divertor, but
has rather large forces because of a very high current density ($50$~MA/m$^2$)
in the winding pack.  Reacting these very large forces and torques may lead to
unworkably large support structures.

Two main problems exist with the MHH divertor configuration, namely the rather
small radial island thickness and the tightness of the distance between the
first wall and the last closed surface, especially at the inboard side in the
middle of a field period (at the triangular cross section).  The edge flux
surfaces have a bulge in this region, shown in Fig.~\ref{f-2-2-3}, which is
caused by the presence of the X-point of the $\ib = 4/3$ islands.  This bulge
is enhanced if the number of coils per period is reduced.  In order to develop
our present interim solution towards a more useful divertor for MHH, new coils
with improved shapes need to be defined using the NESCOIL code.  The lateral
width and the radial coil height then can be optimized for given values of the
axis and peak magnetic fields, the coil current density, and the required
radial and lateral coil support.  The first problem of an increased radial
island size can be solved by redefining the current winding surface via an
iterative `fine-tuning' of the numerical NESCOIL output, with magnetic field
line tracing in order to maintain the properties of the configuration.
Obviously, further information on divertor configurations in helias-type
devices is needed from W7-AS~\cite{Fen95} and W7-X~\cite{Ren95} in order to
improve the divertor configuration needed for a reactor.

\section {IMPROVEMENTS TO THE MHH PHYSICS BASIS}
\label{s-2-6}

\subsection {Validation of the MHH Physics Basis}
\label{s-2-6-1}

Better physics understanding is needed for better extrapolation to a fusion
power plant.  The size and power (hence performance) of stellarators is not as
advanced as in tokamaks, as indicated in Fig.~\ref{f-2-6-1}.  The most common
measure of performance in tokamaks is $n \tau_E T_i(0)$.  Using the
definition of $\tau_E$ and the Lackner-Gottardi (gyro-reduced Bohm)
scaling law [Eq.~(\ref{e-3-4-3})]) gives
%
\begin{eqnarray}
 n\tau_ET_i(0) & \propto &
     V_pn^{1.2}B_0^{1.6}P^{-0.2}\ib^{0.8} \: , 
\label{e-3-6-1}
\end{eqnarray}
%
where $V_p$ is the plasma volume.  If $n$ is given by
Eq.~(\ref{e-3-4-6}), then
%
\begin{eqnarray} 
  n\tau_ET_i(0) & \propto & 
       V_p^{0.4}B_0^{2.2}P^{0.4}\ib^{0.8} \: .
\label{e-3-6-2}
\end{eqnarray}
%
Higher values of $n\tau_ET_i(0)$ are obtained in present tokamaks
because of their higher values for $V_p$, $B_0$, and $P$, rather than
better confinement properties relative to the smaller stellarators.

\begin{figure}[tb]
\centerline{\psfig{file=spps-2-6-1.eps,height=3.5in}}
\caption[Fig-2-6-1]
{Comparison of the capabilities of representative tokamaks and
stellarators in terms of the energy delivered to the plasma per pulse 
($W_{heating}$) and the stored magnetic energy inside the plasma volume 
($W_{mag}$).  The solid circles represent present and near-term 
stellarators, the open circles represent the large next-generation 
stellarators, and the solid squares represent some comparable tokamaks.}
\label{f-2-6-1}
\end{figure}

However, this is expected to change as first LHD and then W7-X come into
operation.  LHD and W7-X will study a number of confinement scaling and
transport issues needed for reactor extrapolation: ($a$) ambipolar electric
fields and their role in transport at low collisionality, ($b$) bootstrap
current minimization and compensation, ($c$) optimization of improved
confinement modes, and ($d$) attainment of several keV ion temperatures and
significantly higher values of $n \tau_E T_i$.  Also, more
information is needed from these experiments on the scaling of beta limits,
their dependence on configuration properties, and MHD behavior at more
reactor-relevant values of beta, $\langle \beta \rangle \approx 5\%$ versus the
$\langle \beta \rangle \approx 2.1\%$ obtained at present.  Experience on
steady-state particle and power handling will be obtained from W7-X and LHD.
In addition, information gained from the ergodic divertor experiments on Tore
Supra will be useful because of the similarity to the edge island structure on
MHH.  Information on the pumping efficiency with an ergodic edge and magnetic
islands is needed.  It will also be necessary to lessen the power to the
divertor plates through a radiative edge, as is being pursued on different
tokamaks and on W7-AS.

\subsection {Options for Configuration Improvement}
\label{s-2-6-2}

The same plasma configuration can be obtained by any number of 
different coil configurations.  This is achieved by moving and reshaping the 
toroidal coil winding surface using the NESCOIL code, which changes the 
properties of the region between the LCFS and the CWS.  Small changes in the 
CWS can make the field-line topology in this outer region exhibit total chaos, 
large or small magnetic islands, or good flux surfaces.  For example, 
Fig.~\ref{f-2-6-2} shows an MHH configuration which maintains the same plasma
configuration as the reference MHH configuration, but displays large islands 
outside of the LCFS.  These islands can be used for channeling flux to a 
divertor, or for shielding the leading edge of a pump-limiter head.

%RLM mod 3.6-2 ok
\begin{figure}[p]
\centerline{\psfig{file=spps-2-6-2.ps,height=7.3in,bbllx=43pt,bblly=57pt,%
            bburx=533pt,bbury=677pt,clip=}}
\caption[Fig. 3-6-2]   % fs0214.ps
{A variation of the basic MHH configuration that exhibits
large islands outside of the LCFS.  The LCFS is the same as in the base case 
-- only slight changes to the CWS produced this difference.}
\label{f-2-6-2}
\end{figure}

The freedom to vary the coil configuration is important since it can be used to
implement different types of divertor designs or to change the engineering
properties while maintaining the same physics properties.  In addition, it
should be possible to reduce the maximum field on the coils and to increase the
bend radii.  Additional space in the inside corner area and in the coil-plasma
spacing could also lead to improved reactor candidates.  Similarly, any
reduction in the non-planar excursions of the coils themselves would be
advantageous.  More optimized designs were obtained in the course of the study
after the configuration had been frozen for the engineering design summarized
in this report; {\it e.g.,} an MHH configuration with negligible plasma
currents, improved quasi-helical confinement properties, and $\langle \beta
\rangle > 6\%$~\cite{Gar94p}.  A three-field-period, eight-coil-per-period
version of MHH (described in the next section) has increased distance between
coils on the congested inside corners, more plasma-coil spacing, better MHD
properties, and lower plasma aspect ratio ($R_{0}/a_p = 6$).  Thus it should
be possible to make small modifications in the present MHH design to improve
its engineering and physics properties.

\subsection {The MHH3 Stellarator}
\label{s-2-6-3}

Subsequent to selection of the particular MHH configuration used for the
engineering design phase of the SPPS, Garabedian and Gardner~\cite{Gar95} have
continued investigations toward improvement over the reference MHH
configuration.  A new configuration, labelled MHH3, has been designed with
three field periods that has a much reduced bootstrap current and allows for
more space between the plasma and the coils.

As with the reference MHH, the plasma surface is specified by a set of Fourier
coefficients $\delta_{m,n}$ where the $r$,$z$ coordinates of the LCFS are given
by
%
\begin{eqnarray}
 r + iz & = & \sum \delta_{m,n} e^{i(mu + 3nv)} \: , 
\label{e-3-6-3}
\end{eqnarray}	
%
where $u$ and $\nu$ are poloidal and toroidal angle-like variables.  For MHH3, 
the expansion coefficients $\delta_{m,n}$ are given in Table~\ref{t-2-6-1}.

\begin{table}[tb]
\caption{}
\centering
{\bf Fourier Coefficients ($\delta_{m,n}$) for the MHH3 Plasma Surface} \\
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{l c c c c c c }			
\hline
       & $m=-3$ & $m=-2$ & $m=-1$ & $m=0$ & $m=1$      & $m=2$ \\ 
\hline 
    & \hspace{1.5cm} & \hspace{1.5cm} & \hspace{1.5cm} 
          & \hspace{1.5cm} & \hspace{1.5cm} & \hspace{1.5cm} \\
$n=-1$ &        &        &        & 0.20  &            & 0.13 \\
$n=0$  & \N0.02 &   0.02 & \N0.05 & 6.00  & \N1.\Z\Z\Z & 0.10 \\
$n=1$  &        & --0.05 & --0.50 & 0.75  & --0.15\Z   & \\
$n=2$  & --0.02 & \N0.09 & --0.09 & 0.15  & --0.035    & \\
$n=3$  &        & \N0.04 &        & 0.02  &            & \\
\\ \hline
\end{tabular}
\label{t-2-6-1}
\end{table}

Table~\ref{t-2-6-2} contrasts the geometrical properties for the two MHH
configurations.  Figure~\ref{f-2-6-3} shows a top view of the filammentary MHH3
case, and Fig.~\ref{f-2-6-4} shows the resulting flux surfaces.  This
configuration offers the possibilities of smaller reactors, easier access to
the divertor region, or reduced engineering complexities with respect to coil
crowding.  Most physical properties, including perturbative bootstrap current
effects, turn out to be even better for this new case than those of the
reference MHH design that was adopted; {\it e.g.,} it has near quasi-helical
symmetry.

Calculations with the NSTAB nonlinear stability code~\cite{NSTAB}
indicate that the equilibrium and stability limit on $\langle \beta \rangle$ 
for MHH3 is above $5\%$.  However, there is no experimental data yet above 
$\langle \beta \rangle = 1.2\%$ for modular stellarators.  Since this 
nonlinear stability test gives the expected beta limits for the ITER and TPX 
tokamaks, it is a plausible tool to apply in the SPPS project.

Monte Carlo simulations of the MHH3 reactor using the TRAN computer
code~\cite{Gar92} show that the particle confinement time $\tau_p$ scales as
$\rho_{L}^{-2.5}$, where $\rho_{L}$ is the ion gyroradius measured in units of
the plasma radius.  For $B=4$~T, \hbox{$\langle T \rangle = 10$~keV}, and
\hbox{$\langle n \rangle = 2\X 10^{20}$~m$^{-3}$}, the code estimates
$\tau_p=8$~s ($\tau_E=2$~s if $\tau_p=4\tau_E$ is assumed).

\begin{table}[t]
\caption{}
\centering
{\bf Geometrical Characteristics of the MHH and MHH3 Configurations
for the Same (2-m) Plasma to Coil-center Distance} \\
\vspace{1.0cm}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{l c c }
\hline
                                                 & MHH   & MHH3 \\
\hline \\
Average major radius (m)                         & 14    & 13.5 \\
Plasma volume (LCFS) (m$^3$)                     & 734 	 & 1,032 \\
Plasma surface area (LCFS) (m$^2$)               & 1,233 & 1,442 \\
Volume within coil winding surface (CWS) (m$^3$) & 5,720 & 7,179 \\
Area of CWS (m$^2$)                              & 2,930 & 3,158 \\
\\ \hline
\end{tabular}
\label{t-2-6-2}
\end{table}

%RLM mod 3.6-3 ok
\begin{figure}[b]
\centerline{\psfig{file=spps-2-6-3.ps,height=3.8in,bbllx=79pt,bblly=237pt%
            ,bburx=525pt,bbury=562pt,clip=}}
\caption[Fig. 3-6-3]     % augv.ps
{An oblique view of the coil set for MHH3, a three-field-period variant of the
MHH configuration.}
\label{f-2-6-3}
\end{figure}

%RLM mod 3.6-4 ok
\begin{figure}[p]
\centerline{\psfig{file=spps-2-6-4.ps,height=7.1in,bbllx=14pt,bblly=43pt,%
            bburx=493pt,bbury=648pt,clip=}}
\caption[Fig. 3-6-4]    % mhh0818.ps
{Flux surfaces for the three-field-period configuration of Fig.~\ref{f-2-6-3}
at four toroidal angles in a $120^{\circ}$ field period.  This configuration
increases the plasma-coil spacing for a given average major radius.}
\label{f-2-6-4}
\end{figure}

A primary direction for stellarator reactors is to minimize the effects of the
bootstrap current.  A new expression~\cite{Gar95}, developed by Garabedian and
Gardner for the bootstrap current, relatively track those of the present
theory, although the magnitudes of the results can differ appreciably.  This
new method permits rapid insight into how parameters of the design can be
varied to accomplish this task.  This has played a significant role in the
optimization of the MHH3 configuration.  Other variants of MHH have been found
that possess a deep magnetic well ($2\%$), low bootstrap current, and still
have relatively large plasma-coil separations and smoother coils, needed
for attractive reactors.

A possible reactor embodiment of the MHH3 has an average major radius of
$13.5$~m, a plasma radius of $2.25$~m, and a gap of $2$~m between the LCFS and
the filamentary centerline of the modular coils.  The coils themselves have
reduced curvature as compared to the MHH case.  There should be enough
flexibility to allow for corrections that might become necessary after more
detailed studies of alpha-particle containment and the divertor are made.
However, given the limited resources and time constraints of this study, the
SPPS was unable to develop this configuration.

\clearpage
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\end{thebibliography}
\end{document}

%Acknowledgments collected and moved to separate area of report -RLM
%\bigskip
%Acknowledgments

%	This work was supported by the Office of Fusion Energy, US Department
%of Energy, under contracts DE-AC05-84OR21400 with Lockheed Martin Energy
%Systems, Inc., DE-AC02-76-CHO-3073 with Princeton University,
%DE-FG02-93ER54222 with the University of Wisconsin, DE-FG02-86ER53223 with New
%York University, and DE-FG02-93ER25160, and by N.S.F. Grants DMS-8922085
%and DMS-900002P.  Some of the material in this chapter was provided by
%collaborators at the Max-Planck-Institut f\"ur Plasmaphysik in Garching,
%Germany: E.~Harmeyer, J.~Kisslinger, P.~Merkel, F.~Rau, and H.~Wobig.  We are
%also indebted to A.~H.~Boozer, F.~Bauer, H.~Gardner, K.~Gulec, S.~P.~Hirshman,
%D.~A.~Monticello, and S.~L.~Painter for helpful discussions and other
%contributions.

{\it Proc. 9th International  Conference on Plasma Physics
 and Controlled Nuclear Fusion Research},
 Baltimore, MD, 1982,
International Atomic Energy Agency, Vienna, 1983, Vol. 3, 
