3 4456 0334514 b -117

f

S. L. Painter J. E Lyo

-. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. '. ... . ... .. ............ ._..__ _...,,

ORNL/TM-11756 Dist. Category UC-426

Fusion Energy Division

TRANSPORT ANALYSIS OF STELLARATOR REACTORS

S. L. Painter*,t J. F. Lyon

Date published-February 1991

Prepared for submission to Nuclear f i s h

*Nuclear Engineering Department, The University of Tennessee, Knoxville.

t Permanent address: Research School of Physical Sciences, Australian National University, P.O. Box 4, Canberra, A.C.T. 2601, Australia.

Prepared for the Office of Fusion Energy

under Budget Activity No. AT 10 10 15 B

Prepared by the OAK RIDGE NATIONAL LABORATORY

Oak Ridge, Tennessee 37831-6285 managed by

MARTIN MARIETTA ENERGY SYSTEMS, INC. for the

U.S. DEPARTMENT OF ENERGY MARTIN MAPIETTA ENERGY EYblEMS LIBRAPI'S

under contract D E-AC05-840R2 1400

3 4 4 5 b 0334514 b

CONTENTS

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v 1 . INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 . PREVIOUS STELLARATOR REACTOR TRANSPORT STUDIES . . . 2 3.APPROACH . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4 . ELECTRON-ROOT AND ION-ROOT OPERATION . . . . . . . . . 9

4.1 Role of the radial electric field . . . . . . . . . . . . . . . . . 9 4.2 Electron-root operation . . . . . . . . . . . . . . . . . . . . 11 4.3 Ion-root operation . . . . . . . . . . . . . . . . . . . . . . 13 4.4 Sensitivity to the magnitude of the electric field . . . . . . . . . . 18

5 . EFFECT OF ALPHA-PARTICLE LOSSES . . . . . . . . . . . . . 20 6 . SENSITIVITY TO DEVICE PARAMETERS

AND ASSUMPTIONS . . . . . . . . . . . . . . . . . . . . . . . 22 6.1 Device parameter sensitivity . . . . . . . . . . . . . . . . . . 22 6.2 Sensitivity to profile assumptions . . . . . . . . . . . . . . . . 24 6.3 Sensitivity to anomalous transport . . . . . . . . . . . . . . . 26

7 . RESULTS WITH AN EMPIRICAL TRANSPORT SCALING . . . . . 27 8 . OPERATION WITH D-3He . . . . . . . . . . . . . . . . . . . . 30 9 . IMPLICATIONS FOR STELLARATOR REACTORS . . . . . . . . . 33

L4ppendix A . THE COLTRANE TRANSPORT CODE . . . . . . . . . . 37 Appendix B . ESTIMATES OF THE AMBIPOLAR ELECTRIC FIELD . . . 39 A4CKNOTVLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . 42 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

... 111

ABSTRACT

The performance of deuterium-tritium stellarator reactors is studied with a

new, fast one-dimensional (1-D) transport survey code that is based on the spectral

collocation method. Two operating modes with different signs of the assumed radial

electric field are identified. The operating mode with a positive electric field is

characterized by high temperatures and moderate densities, whereas the other inode

has lower ternperatures and higher densities. Both modes lead to possible reactors

that could tolerate a large alpha-particle energy loss. The sensitivity to device

parameters and to profile assumptions is cxamined. Scaling expressions useful for

parametric studies are obtained for different quantities of interest, and the 1-D

code results are compared with results derived from an empirical scaling relation.

Deuterium-helium-3 ( D3He) operation is also feasible but is more demanding. The

implications for stellarat or reactor design optimization are discussed.

V

1. INTRODUCTION

Studies of stellarator reactors have become more relevant as improvements in

toroidal physics understanding have led to optimization of stellarator concepts

[l], new experiments have started operation [2], design studies of new large next-

generation stellarators have begun [3], and the potential limitations of tokamaks

as reactors have become more appreciated [4]. Stellarators offer the possibility of

ignited, steady-state fusion reactors without the need for current drive or pulsed

loads, or the danger of current-driven disruptions. Modern stellarator reactor

designs [l, 5 , 61 have high beta capability (volume-average beta (p ) > &lo%)

and feasible coil designs and maintenance schemes. Torsatron configurations also

have a naturally occuring bundle of diverted field lines that channels particles to

the region outside the main coils and simplifies the design of the power and particle

collection systems.

Adequate confinement is probably the key issue for stellarator reactors. It is

a potential concern because of the combination of the inherent non-axisymmetry,

which leads to a population of particles trapped in the helical modulation of the

magnetic field, and the poloidal variation of the magnetic field inherent in toroidal

devices, which causes the helically trapped particles to drift radially. Direct loss of

helically trapped alpha particles can occur and the diffusive heat loss associated with

helically trapped thermal particles can be large, unless measures are taken to reduce

these losses. The thermal losses can be reduced by radial electric fields that develop

to ensure quasi-neutrality by modifying, through E x B poloidal orbit rotation, the

otherwise non-ambipolar neoclassical losses of helically trapped particles [TI . Both

alpha-particle losses and thermal losses can be reduced by magnetic configuration

optimizations that reduce the effective helical ripple or increase the effective plasma

aspect ratio [8, 91. This paper exansines stellarator reactors with regard to these

losses and assesses the acceptable limits on the neoclassical (and anomalous) losses

for low-aspect-ratio rcactors (in particular, for A, EE &/a , - 5) for a range of

assumptions. Here Ro is the major radius and u p is the average radius of these

non-circular (and non-axisyrnmetric) plasmas.

Low-aspect-ratio reactors were chosen to assess the effect of helical-ripple-

induced losses on stellarator reactor performance because the coinbination of low

aspect ratio and large helical ripple in these devices enhances these losscs. Also, a

particular type of these devices, the Compact Torsatrons [lo], has attractive physics

properties (high beta capability in the second stability regime) and attractive

1

2

reactor properties (excellent access €or blankets and maintenance), and such devices

lead to reactors that are one-third to one-half the size of other stellarator reactors

[SI. Although configuration parameters similar to those of Compact Torsatrons

are used for specific examples, the treatment and conclusions apply in general to

stellarator reactors. Specific results applying to Advanced Stellarators are also

noted. Previous stiidies of stellara,tor reactors are summarized briefly in Section 2.

Section 3 discusses the approach used in our studies. The two modes of operation

obtained, which depend on the sign of the radial electric field, are described in

Section 4. Section 5 discusses the impact of alpha-particle losses on the two modes

of operation. The sensitivity of reactor performance to device parameters, profiles,

and other assuiriptions is disciissed in Section 6, and scalings for various quantities of

interest are obtained and compared with scalings obtained from simple arguments.

Section 7 describes the results obtained when an empirical global confinement

scaling relation is used instead of the more complete transport description. Rcactors

using B 3 H e are discussed in Section 8. Finally, Section 9 discusses implications for

future stellarator reactor optimization. Appendices give more detailed information

on the one-dimensional (1-U) transport code and estimates of the neoclassical

amhipolar electric field,

2. PREVIOUS STELLARATOR REACTOR

TRANSPORT STUDIES

Studies of stellarator reactors [I, 111 are not as extensive as those of tokamak

reactors. In particular, engineering features of stellarator reactors and global models

for energy confinement have been given more attention than have assessments of

reactor performance based on the more complete l -D transport equations, and

point designs have generally been emphasized over scoping studies. Studies of the

Heliotron H ( A , 2 11.7) reactor [12] used fixed temperature and density profiles

and assumed central values to calculate the total power output and the neutron

wall loading. These stiidies were extended to lower aspect ratio (A , = 8) in the

Bcliotron I study [13], again with fixed plasma profiles and assumed pararnelcrs.

The Modular Stellarator Reactor studies [14] analysed the paranietric dependences

of modular stellarator reactors for .4, = 11-28 but assumed a tokamak (Alcator)

scaling to obtain a global energy coifineirient time. Studies at the Kharkov

Institute of Physics and 'I'echnology of the U-2M R reactor (A , = 9.4) [15]

3

used an elec tric-field-dependent expression for the neoclassical fluxes derived by

Kovrizhnykh [16] to estimate a global energy confinement time and hence the

reactor parameters. Hitchon [17] has analysed low-aspect-ratio stellarator reactors

by developing approximate analytic expressions relating magnetic configuration

properties to parameters describing the shape of simplified modular stellarator coils.

The most-developed stellarator reactor perforniance studies for large-aspect-

ratio ( A , > 10) stellarators are those of the Garching modular Advanced Stellarator

reactor concept ASRAGC ( A , = 12.5) [GI. A steady-state 1-D code with the Shaing

model for ripple-induced neoclassical diffusion [ 181 was used to calculate profiles

of density, electron temperature, and ion temperature. At the Kharkov Institute

of Physics and Technology, the large-aspect-ratio Thermonuclear Power Plant with

A, = 1’5.5-25 was analysed with a 1-D transport code using plateau and extended-

plateau transport [19]. The most-developed low-aspect-ratio ( A , < 10) stellarator reactor studies [5]

have been based on the Compact Torsatron sequence [lo]. Three helical winding

configurations were examined with A, = 7.8, 4.7, and 3.9. The WHIST 1-D

transport code [20] was modified to includc the diagonal (only VT term) version of

the Shaing model with assumed profiles for density and electric field. In addition,

Houlberg et al. [21] used the time-depcndent WHIST transport code and the Shaing

model to study a point design based on a scaleup of an A, = 7 approximation to the

Advanced Toroidal Facility (ATF) experiment [22]. The ambipolar electric field was

determined self-consistently during the reactor startup and was forced to a preset

value as the fusion powcr increased to its design value.

The present study differs from these earlier studies in several ways. The direct

loss of alpha particlcs, which was shown to be significant in Ref. [8], is taken

into account. The transport simulations summarized in Refs. [5 , 6, 211 were for

fixed designs, in contrast to this study which considers a multidimensional space of

configuration and device parameters. Greater emphasis is placed on thc sensitivities

to design variables and to modeling assumptions, similar to the approach taken in

Ref. [5 ] . The transport models employed here are, however, different from those

used in Ref. [5] in that the off-diagonal terms proportional to the density gradient

and to the radial electric field are included in addition to the usual temperature

gradient term in the heat flux equations.

3. APPROACH

A self-consistent treatment of transport in stella,rators requires solutions to the

particle balance equation and to power balance equations for the electrons and

ions. The electron and ion particle fluxes To and I?; result from a combination

of several processes, some of which are not well understood, including anomalous

(fluctuation-induced) particle losses, convectioii processes, and direct particle losses

(due to orbit loss regions). In addition, the electron and ion loss rates are not

intrinsically ambipolar, and large radial electric fields (E,) develop to insure that

the plasma remains quasi-neutral. These radial electric fields could be determined

in principle by solving the ambipolarity constraint l?,(E,j = Fi(Er) on each

flux surface. However, these solutions are sensitive to the large uncertainty in the

particle flux.

To prevent this uncerta.inty Trom propagating through the transport equations,

the particle density and radial electric field profiles are held fixed while the

electron and ion power balance equations are solved. Special attention is given

to determining the sensitivity of the results to the profile a.ssumptions. This

approach is reasonable because it should be possible to control the density profile

arid the potential profile using a variety of techniques. The density profile could be

controlled by adjusting the particle deposition profile (gas pufing, pellet injection,

neutral beam injection) and by modifying the local particle confinement (through

localized field perturbations, modification of the plasma edge conditions, etc.). It

may be possible to control the electric potentia.1 profile either directly through biased

plates or indirectly through enhancing either electron or ion losses by an appropriate

heating method (perpendicular neutral beam injection, electron cyclotron heating,

ion cyclotron heating ) I

The following parameterization was chosen to describe the density profiles used

in this study:

2 112 .(p) = no[(l P)(1- p2)>*" + P(1 -YnP ) 1

The broadness parameter P controls the ratio of the central density nJo to the

volume-averaged density (n) . Values of P < 1 result in peaked profiles, 'P E 1

yields broad density profiles, and 'P > 1 produces h o l l o ~ profiles; the parameter yn

controls the edge density. Mere p is the average radius of a flux surface divided by

ap. The three density profiles used in this paper are shown in Fig. '1. Each profile

has an edge densiky that is 10% of the central value. The broad (reference) profile

1.2

g 1.0 m Z W L3 Z 0.8 0 cc t- 8 0.6

n E 0.4 a 5 0 0.2

-I W

-..I

LI

z

0

5

ORNL-DWG 90-251 1 FED

a"= 1 -11 P = 1

p = 1/3\ a , = 4

I I I 1

0 0.2 0.4 0.6 0.8 1 .o RADIAL VARIABLE p

Fig. 1. Density profiles used in this paper.

has P = 1 and yn = 0.99, so no = 1.49(n); the peaked profile has P = 1/3, a, = 4, and yn = 0.91, so no = 2.7(n); and the hollow profile has P = 3, an =I 1, and

yn = 0.999, so no = (n). The electric potential CP is parameterized by 50,

where T i 0 is the central ion temperature. The reference electric potential is parabolic

( t = 1, s = 1). Because of the large uncertainties in the transport models, it is sufficient to use

cylindrical geometry instead of the actual flux surface geometry and to treat the

deuterons and tritons as a single species with an effective atomic mass of 2.5 amu.

The steady-state 1-D heat transport equations for the ions (i) and electrons (e) are

6

solved in integral-differential form,

The heat flux qa = q:' + qa tok + q:nom, where q,"' is the neoclassical contribution

due to the helically trapped particles developed by Shaing [18],

arid qFk is the contribution from axisyinmetric neoclassical transport (Chang-

Hinton tokamak formulation [23]), which overestimates this component of the

diffusive loss in a stellarator because of the much smaller fraction of banana-trapped

particles. Because of the large helical ripple, the q,"" ripple-loss component usually

dominates the power balance; it can be much larger than the axisymmetric neoclas-

sical and anomalous losses occurring in tokamaks. An anomalous contribution of

the form qznom = - K : ~ ~ ~ ~ V T ~ is also incliided to simulate the effect of turbulence,

10l8 m-' .s-l and rianom - 0 are the reference values. The Karioin - - 5 x e

effect of larger levels of anomalous transport is addressed in Section 6.3.

The power density pa - p a j a + fapx + pint,a, where p, is the alpha-particle

heating, px is the external heating with fa the fraction delivered to species a, and

pint represents all remaining heat sources and sinks. The form chosen for p, is

p,o(l - p2)2 . The magnitude of p,o is varied to give a desired value of the density-

averaged temperature ( T ) W/(2(n)) , where W is the plasma stored energy.

When pwo is greater than zero, it simply represents the amount of external power

required to maintain thermal equilibrium at the specified temperature. A zero value

for p,o corresponds to ignition, while negative values indicate that additional power

losses are needed to mainta.in a thermal equilibrium.

-

It is usually assumed that all alpha particles thermalize on their birth flux

surface without being lost. However, a.nalytic descriptions of the trapped-particle

orbits [24] suggest that a significant fraction of the alpha particles born on trapped

orbits can be lost when Apeha N 1, which is the situation of interest here. Here €ha

is the amplitude of the helical ripple at the plasma edge. This has been confirmed

by niiiiierical calculations of alpha-particle orbits in the full t hree-dimensiona.1 (3-D)

magnetic configurations of Compact Torsatrons [SI. Although this direct loss can

be reduced or even eliminated by modification of the magnetic configuration [8],

finite-beta effects can undo this optimization and result in the loss of nearly all

7

- .- - e-

TO ELE~TRONS

the helically trapped alpha particles [25]. Trapped alpha particles can also be lost

from Advanced Stellarator configurations (despite the fact that these configurations

were optimized to reduce the drift of trapped particles) because of stochastic drifts

induced by modular coil ripple [26]. In addition, alpha particles not initially on lost

orbits can pitch-angle scatter into the near-perpendicular (small-wll) loss region as

they slow down in energy on the background plasma. This alpha-particle energy

loss [8] is included in our calculations; an example of the magnitude of this loss and

the fraction of the energy transferred to the ions and electrons is shown in Fig. 2

as a function of electron temperature where Z e ~ = 1.5 due to oxygen has been

assumed. For T, = 20 keV, the fraction of the alpha-particle energy going to the

ions is reduced from 30% to 11% when these losses are considered. After giving up

most of their energy to the background plasma, the remaining alpha particles are

rapidly scattered into the near-perpendicular loss region; this provides a mechanism

for efficient ash removal [$I.

1 .o

W 5 0.8

w --I 2 I- C 0.6 2

-I Q 0.4

15 z 0 I= 0.2 0

3 LI

0

ORNL-DWG 90M-2522R FED

TO IONS 0

I I

0 10 20 30 40 50

T, Fig. 2. Fraction of the alpha-particle energy delivered to the electrons and ions

with (solid curves) and without (dashed curve) the velocity-space loss region.

8

The internal heat sourccs and sinks in pint include radiation losses (classical

bremsstrahlung, impurity linc radiation, and synchrotron radiation) and electron-

ion Coulomb collisions. Synchrotron radiation losses are computed using Trub-

nikov’s formulation [27] with a wall reflectivity of 90%. Impurity radiation is

calculated from Post’s coronal model [28] using an impurity fraction of

for carbon and

( 2 , ~ = 1.37 and ni/ne L=I 0.94). for iron to represcnt typical low-Z and high-Z impurities

A new transport survey code, COLTRANE (COLlocation method €or TRANs-

port Equations), was developed for this study. In COLTRANE, the spectral

collocation method is applied to the transport equations in their integral-differential

form. Each of the resulting algebraic equations represents a power balance over a

finite volume, similar to the finite-element method. This discretely conservative

property causes the global quantities to converge rapidly as the number of basis

functions is increased, so that a coarse radial grid can be used. Continuity conditions

at the elemcnt interfaces are intrinsically satisfied. Sensitivity coefficients describing

the change in the solution as external parameters appearing in the equation change

are also calculated. This allows Plasma Operating CONtours (POPCONS) [20] to

be constructed efficiently. As a result, COLTRANE requires only a few seconds

to calculate the entire (n)-(2’) Operating space on a VAX 8700 computer, and it

can be used in an iteration loop in a reactor optimization code [29]. Details of the

methods used in the l -D code are given in Appendix A.

The typical reactor case used to illustrate the various physics issues in this paper

is a single-helicity stellarator with an effective hclical ripple Eh obtained from the

field approximation B = Bo [I - (p/Ap)cos 8 + q,(p)cos(tO - M 4 ) ] , where 19 and

4 are the poloidal and toroidal angles, M is the number of field periods toroidally,

and the poloidal multipolarity e is taken to he 2. Thesc calculations can also be

applied to multiple-helicity stcllarators by using the procedures of Ref. [30] to define

an effective El,. The other assumptions are &, = 10 in, up 2 rn (so A , = 5 ) , and

on-axis field Bo = 7 T, with Eh = €hap2 with &ha = 0.2, a broad density profile

( P == l), and a parabolic potential profile with &, = f 3 (sign depending on the

case), unless otherwise nnted.

9

4. ELECTRON-ROOT AND ION-ROOT OPERATION

4.1 Role of the radial electric Aeld

Radial electric fields affect energy coi-dinement in stellarators in two ways. The

E x B poloidal orbit rotation reduces the radial excursions of helically trapped

particles and causes the transport coefficients to decrease with increasing electric

field in a manner that is roughly independent of the sign of E for large enough values

of the electric field. Radial electric fields also induce particle and hcat fluxes that

are in opposite directions for electrons and ions. The effect of the radial electric field

on the neoclassical component of the heat flux can be seen in Fig. 3, where parabolic

profiles of temperature, density, and electric potential are chosen and effective

thermal diffusivities, defined as xEff = -q,”“/(n,VT,) and evaluated at p = 0.75,

are plotted as functions of the normalized electric potential. The dcvice parameters

are those of the reference case; in Fig. 3a, (2’) = 8 keV and (n) = 3 x lo2’ mP3,

while in Fig. 3b, (7’) = 25 keV and (n ) = 7.5 x lo1’ m-’. In Fig. 3a the ions are

in the collisionless detrapping regime (x oc v), where the diffusion coefficients are

sensitive to the radial electric field, while the morc collisional electrons are at the

high-v end of the x cx 1/v regime where the diffusion coefficients are small. Here

v is the collision frequency. In the higher-temperature case shown in Fig. 3b, the

ions are deep in the low-collisionality, x cx Y regime where the transport coefficients

are small, while the electrons have just entered this regime from the l / v regime;

the electron heat flux is the dominant transport loss. The ion heat diffusivity in

Figs. 3a and 3b peaks at small negative values of the radial electric field where

the E x B and V B drifts cancel. The electron heat diffusivity in Fig. 3b peaks

at a small positive value of 50 for thc same reason. The heat diffusivities are not

symmetric about this resonance because of the V@ term in the heat flux equation.

The off-diagonal V@ term reduces the heat flux of the species having the larger

transport coefficients at the expense of the other species, with the net effect being a

reduction in the total heat flux. This leads to two very different modes of opcration,

depending on the sign of the electric field. At low collisionality, positive radial

electric fields (60 > 0) develop to retard the outward electron heat flux, leading

to the high-temperature, low-density “electron-root” mode of operation. At high

collisionality, negative radial electric fields ( E o < 0) develop to decrease the ion heat

flux, leading to the lower-temperature, high-density “ion-root” mode of operation.

10

01 E Y

ORNL-DWG 90-3215 FED I - _ -

0 1.5 3.0 4.5 -4.5 -3.0 -1.5

NORMALIZED ELECTRIC POTENTIAL, 50 Fig. 3. Effective thermal diff usivity versus normalized electric potential for an

A, = 5 reactor (a) at moderate collisionality ( ( T ) = 8 keV, (TI) = 3 x 10’’ ~ n - ~ ) and (b) at low collisionality ( ( T ) = 25 keV, ( n ) = 7.5 x 10’’ m-3).

13

For the examples discussed here, a reasonable energy confinement time ( TE 2 1 s

for up N 2 m) requires xeff 5 1 rn2-sW1 or (0 5 -3.3 in Fig. 3a and (0 2 3.4 in

Fig. 3b; (0 = f 3 are chosen as the reference values in this paper. The neoclassical

values of (0 expected and the temperatures at which the ion and electron roots are

expected to occur are estimated in Appendix B. The reference value of for the

electron root ((0 = +3) is close to the range expected neoclassically, but the (0 = -3

value for the reference ion-root case is more negative than expected neoclassically.

More negative values of l o could result from a preferential loss of ions from the

plasma, e.g. the alpha particles that are born in the near-perpendicular loss region

or that scatter into it as they slow down on the background plasma. However,

the flux of lost alpha particles is much less than the electric-field-dependent ion

and electron fluxes. It is unlikely that the dpha-particle loss modifies the potential

significantly because even a small potential change is associated with a large change

in the particle flux (analogous to the change in xeff seen in Fig. 3). Non-ambipolar

anomalous losses are likely to have more of an effect.

4.2 Electron-root operation

A POPCON plot for the reference electron-root case with = +3 is shown

in Fig. 4. The contours are those of constant auxiliary heating power, where the

O-MW contour indicates ignition. The potential operating points for a reactor are

on the ignition contour, with the exact location being determined by the desired

fusion output. The power required to heat the plasma to ignition is minimized by

choosing a startup path that passes through a saddle point in the external power

surface. This “Cordey pass” [31] occurs at low densities with electron-root operation

because of the improved neoclassical confinement associated with low collisionality

and because thc power per particle is maximized. The saddle point is not visible

in Fig. 4. The saddle-point power Psp is too small (<20 MW) to be important in

reactor design optimization for electron-root operation.

Along the nearly vertical branch,

the ignition temperature is nearly independent of the electron density. This is

characteristic of operation on the electron root and can be understood with a simple

scaling argument. With electron-root operation, ignition occurs at low collisionality

where the plasma energy losses are dominated by v-regime transport losses

(cx n2T-’/2) . The constant of proportionality depends on the machine parameters,

the electric field, and the assumed density profile. At ignition, these losses are just

The ignition contour has two branches.

2.0

.4

c? E 0 1.6 0 7 v

A

v c

i tz 1.2

n

a 21: w

0 w

M UJ > W

8 0.8

e

2 9

0.4

0

12

ORNL-DWG 90-3213 FED

0 a 16 24 32 40

DENSITY-AVERAGED TEMPERATURE, (T) (keV)

Fig. 4. POPCON plot for electron-root operation ((0 = 3) for a stellarator reactor with 1x0 = 10 m, up = 2 m, BO = 7 T, and €ha = 0.2. The auxiliary heating power contours are equally spaced in 5-MW intervals from 0 to 50 MW and in 50-MW intervals above 50 MW.

balanced by alpha-particle heating, which is approximately proportional to n2T2,

yielding an ignition temperature that is independent of density. Along the nearly

horizontal branch of the ignition contour, alpha-particle heating is balanced by

synchrotron radiation losses (a n1/2Tz /2 , including reabsorption), resulting in an

ignition density proportional to (T) ' i3 .

The temperature profiles for an ignited plasma (fusion power Pr = 3 GW,

(n ) = 6.9 x lo1' M - ~ , ( T ) = 35.6 keV) are shown in Fig. 5. The high edge ion

temperature is a result of the transport model and the assumed boundary condition;

a shorter edge-gradient length decreases this valiie. The integrated power to the

electrons within a radius p is shown in Pig. 6a and that for the ions in Fig. 6b. The

largest electron power loss (>60% at p = 1) is the net heat conduction, which is due

13

50

40

2

2 g 20

25 30 w cc 3

[I Lu

w I--

10

0

ORNL-DWG 90-321 7 FED I I I I 1

. , ,

\

, \ \

ELECTRONS '\, , , ,

\ \ ,

I \ \ I , ,

0 0.2 0.4 0.6 0.8 1 .o RADIAL VARIABLE, p

Fig. 5 . Profiles of electron and ion temperature for an ignited reactor operating in the electron-root mode (Pf = 3 GW).

to the diagonal (VT) term being slightly larger than the sum of the off-diagonal V@

term of the opposite sign and the Bn term. However, the radiation loss is larger

over most of the plasma ( p < 0.8). For the ions, the dominant term throughout

most of the plasma is conduction (due almost entirely to the off-diagonal terms),

except near the outside where the ion-to-electron heat transfer is comparable.

4.3 Ion-root operation

The only proposed operating limit [32] on the plasma density in a stellarator

increases with the square root of the input power, but it is not restrictive for a

14

200

h 100 3: 2 v

-1 00

100

50

ORNL-BWG 90-3210 FED r i i - - - - ~ / 1

--e$

\ . 1 E XT. w COND.

-1 00 0 0.2 0.4 0.6 0.8 1 .o

RADIAL VARIABLE, p

Fig. 6. Different contributions to the integrated power to (a) the electrons and (b) the ions for an ignited reactor operating in the electron-root mode ( ( n ) = 6.9 x 10'' m-', ( T ) = 35.6 keV).

reactor plasma because the alpha-particle heating increases with the square of the

density, so no limit occurs. Stellarators reactors can tlius access a high-density,

moderate-temperature operating regime. An example of this type of operating

regime can be seen in Fig. 9, which shows a POPCON plot for the reference ion-root

case with &, = -3. The main features of ion-root operation are the minirrium in the

15

ORNL-DWG 90-3214 FED 4.0

T E

3.2 0 F v

n c v

2.4 u) z W

L3 w

E w >

n

1.6

=? 2 i 3 0.8 0 >

0 0 4 a 12 16 20

DEN SIN-AVE RAG ED TEMPERATURE , (T) (keV)

Fig. 7. POPCON plot for ion-root operation ( t o = -3) for a stellarator reactor with Ro = 10 m, up = 2 m, Bo = 7 T, and &a = 0.2. The auxiliary powcr contours are shown at 20-MW intervals from 0 to 100 MW and at 100-MW intervals above 100 MW.

ignition density and the saddle point in the external power surface. The minimum

fusion power output along the ignition contour ( Pf,min) and the minimum volume-

averaged plasma beta ( (/?)min), which are important design constraints, occur near

the point of minimum ignition density ((n),). The auxiliary power required for

startup, which is also an important design constraint, can be approximated by the

steady-state saddle point power and is larger than with the electron-root mode of

operation. Simultaneous ramping of the plasma temperature and density is required

to follow the optimal startup path, similar to the startup scenario for a tokamak

react or.

Transport in different collisionality regimes dominates the power balance along

each of the two branches of the ignition contour in Fig. 7. Along the nearly vertical

16

branch, the electrons are at the high-v end of the I / v regime, where the transport

coefficients are small and the ions are just entering the v-regime from the 1/v

regime. The v-regime ion transport dominates, yielding a.n ignition contour similar

to that obtained at lower collisionality with the electron root. Bremsstrahlung and

impurity radia'tion are also important along the nearly vertical branch of the ion-

root ignition contour; they set a lower limit on the ignition temperature. Along

the high-temperature brmch of the ignition contour, the collisionality is lower, the

ions are deeper in the v-regime where their transport coefficients are reduced, a.nd

electrons are higher in the 1 / v regime, so that alpha-particle heating is balanced by

l/v-regime electron transport. Equating the l/v-transport losses (approximately

proportional to n 0 T 7 / 2 ) to the alpha-particle heating term yields an ignition density

proportional to ('I")'/'. Operating points arc thermally stable along the high-

temperature branch because the l/v-transport losses increase with temperature

faster than the alpha-particle heating.

x lo2' n1-3, (7') = 7.6 kcV) are shown in Fig. 8.

The temperature profiles for a typical ignited plasma (Pf = 3 GW, ( n ) = 2.1

profiles are The T, and

ORNL-DWG 90-3216 FED I 1

' -1 I- , - -- 1

I' I I

0 0.2 0.4 0.6 0.8 1 .o RADIAL VARIABLE, p

Fig. 8. Profiles of electron arid ion temperature for an ignited reactor operating in the ion-root mode (Pf = 3 G W).

17

closer to each other than in Fig. 5 because of the shorter equilibration time. The

integrated power to the electrons within a radius p is shown in Fig. 9a; that for the

ions is shown in Fig. 9b. For the electrons, heat conduction, radiation, and ion-

electron heat transfer are all important; at p = 1, heat conduction and radiation

are comparable. The electron heat conduction is composed of comparable ripple-

induced diagonal and off-diagonal contributions and much smaller axisymmetric

ORNL-DWG 90-321 1 FED 300 i I I I I

g 100

3” z “ 0

0 a -100

I -200

-300 I I I I 1

50

25

a -25

0 $

-50

-75

-100 L I I I I

0 0.2 0.4 0.6 0.8 1 .o RADIAL VARIABLE, p

Fig. 9. Different contributions to the integrated power to (a) the electrons and (b) the ions for an ignited reactor operating in the ion-root mode ( ( n ) = 2.1 x lo2” m-’, ( T ) = 7.6 keV).

18

neoclassical and anoriialous contributions. For the ions, the dominant total loss is

conduction, The net heat conduction changes from negative (outward heat flow) to

positive at p N 0.54. This is because the negative diagonal (07’) term in the ion

heat flux is slightly larger than the positive sum of the V@ term of the opposite

sign and the V n term in the central part of the plasma, while it is smaller tham this

positive sum in the outer part of the plasma. The near-cancellation in the ion heat

conduction for the ion root is similar to the nea.r-cancellation in the electron heat

conduction for the electron root. It is due to the opposite sisn of the electric field

in the two cases.

4.4 Sensitivity to the magnitude ofthe electric field

Figures 10 and 11 show the sensitivity of stellarator reactor performance to the

magnitude of the radial electric field. These plots show ignition contours in (n)-(7’)

ORNL-BWG 90-3212 FED 2.0

h

c? E 0 N 0 v - 1.5 A

v c

g v> z: w Q 1.0 Q w c) a K W >

2 2 0.5

3 ?

0 0 18 20 30 40 50

DENSITY-AVERAGED TEMPERATURE, (T) (keV)

Fig. 10. POPCON plot showing the effect of the electric field on the ignition contour for an electron-root stellarator reactor.

19

ORNL-DWG 90-3209 FED 4

c5

r? E 0 cu 0

" 3 A

v t

i2 E 2 n

a

? W '

3 P

cn z

W c7 r I W

z

0

7

0 5 10 15 20 DENSITY-AVERAGED TEMPERATURE, (T) (keV)

Fig. 11. POPCON plot showing the effect of the electric field on the ignition contour for an ion-root stellarator reactor.

space for various values of to. Contours of constant ( p ) , which do not depend on

the density and temperature profiles, are also shown. Contours of constant Pf may

be different for each value of Eo due to differences in the temperature profiles and in

the values of r / T , . However, these effects are small for the range of considered,

and the Pf = 3 G W curve shown approximates the actual curves. A fusion power

of 3 G W corresponds to a net electrical output of -1.2 G W for typical values of the

thermal conversion efficiency and the energy multiplication in the reactor blanket.

With electron-root operation (Fig. lo ) , ignition occurs without exceeding the

desired total fusion power for values of (0 as low as +2.5, which is close to the range

expected neoclassically, as discussed in Appendix B. Larger values of f o shift the

reactor operating point (intersection of the Pf 3 GW curve and the ignition

contour) to lower temperature and slightly higher density (and lowcr (p) ) , where

the plasma may be less susceptible tm anomalous processes. The external power

=

20

required to reach ignition is small for electron-root operation, less than 20 MW for

the range of (0 shown. The resulting beta values are also low, ( p ) 5 6%. Larger absolute magnitudes of 50 are required for ignition with ion-root

operation (Fig. 11). For the case shown, values of &, 5 -3 are required in order

to achieve ignition without exceeding the desired Pi = 3 GW. The external power

required to reach ignition is sensitive to So, ranging from 10 MW when (0 = -4 and 50 MW when (0 = -3 to 230 MW when (0 = -2.5. The magnitudes

of (0 required are larger than those expected neoclassically, indicating that some

form of external control of the radial electric field would be necessary. This would

require a steady-state powcr input, similar to the recirculating power required for

current drive in a tokamak reactor. I€ the required values of (0 can be achieved, ion-

root operation offers some advantages. Contours of constant fusion power output

cross both branches of the ignition contour, so there is a choice for the operating

point: one at moderate values of ( n ) and ( T ) and one at higher (n) and lower

( T ) . Operating points on the high-temperature branch arc thermally stable; a

decrease in the plasma temperature away from the operating point will increase the

ignition margin and drive the plasma temperature back to the equilibrium value.

The negative electric field may also reduce transport caused by plasma turbulence

P31 *

5 . EFFECT OF ALPHA-PARTICLE LOSSES

The reference assumption about alpha-particle losses used in this paper is

that all trapped alpha particles are promptly lost from the plasma region. This

assumption is consistent with calculations of alpha-particle losses in Compact

Torsatron configurations with non-zero plasma pressure [25]. These alpha-particle

losses do not cause concerns for the reactor first wall, as they do in a tokamak

reactor, because the energetic alpha particles are channelled into the region external

to the torsatron coils, where their energy can be collected in large divertor

chambers. However, the alpha-particle heating of the background plasma is reduced,

which has prompted studies of modcrate-aspect-ratio [9] and low-aspect-ratio [25]

configurations with reduced alpha-particle losses. The potential irnprovemcnts in

plasma performance associated with reducing or eliminating the alpha-particle losses

are addressed in this section.

The effect of alpha-particle losses on the ignition contours for the reference

stellarator reactor operating in the electron- a d ion-root modes is shown in Fig. 12.

21

ORNL-DWG 90-3207 FED

0 1 2 3 4

VOLUME-AVERAGED DENSITY, (n) (1 020 m-3)

Fig. 12. Effect of alpha-particle losses on the ignition contour for the reference reactor operating in the electron-root or ion-root mode: dashed curves (without losses), solid curves (all trapped particles lost), and chain-dashed curves (only scattered losses).

The space is similiar to the POPCON space of Figs 10 and 11 except that ( T ) has

been replaced by the total fusion power Pf to better show the relationship between

the plasma variable that is most easily controlled ( ( n ) ) and the quantity that is

of most interest to reactor designers (Pf). The dashed curves are ignition contours

without alpha-particle losses, the solid curves are ignition contours when all trapped

alpha particles are lost, and the chain-dashed curves are ignition contours when

only the scattering component of the alpha-particle losses is considered. The third

situation can occur when otherwise closed orbits of energetic trapped particles are

disturbed by field errors or modulas coil ripple, leading to a stochastic drift of alpha

particles occupying a small region in velocity space.

22

With electron-root operation the main effect of reducing the alpha-particle losses

is to reduce the ignition temperature. Eliminating the direct component of the

alpha-particle losses decreases the ignition temperature from ~ 3 6 keV to ~ 3 0 keV.

Eliminating both the direct and scattering losscs reduces the ignition temperature

to 222 keV. But, as can be seen from Fig. 12, the higher temperatures do not have

a practical effect on reactor operation; (n) can be decreased to compensate for the

larger Pf and (p ) associated with higher temperatures. For all three cases ]?,,in < 1 GW, (/9)min < 5%, and Psp < 20 MW.

With ion-root operation, reducing the alpha-particle losses reduces (n ) * and

makes it easier to meet constraints on Pr, Psp, and (p ) . Eliminating the prompt

component of the alpha-particle losses decreases Pf,min from 2.5 GW to 0.9 GW,

(p)mii, from 3.0% to 2.0%) and Psp from 38 MW to 25 MW. With no alpha-particle

losses, Pf,min = 0.7 GW, (p),;, = 1.6%, and Psp = 24 MW.

6. SENSITIVITY T O DEVICE P,4RAMETERS

AND ASSUMPTIONS

6.1 Device parameter sensitivity

In order to determine the sensitivity of stellarator reactor performance to the

device parameters A,, €ha, ap, and 130, POPCON plots were generated for ranges

of these parameters, for both electron- and ion-root operation, with the loss of all

trapped alpha particles. Since constraints on Pf,min, (P),in, and Psp are important

for fusion reactor design, these quantities are obvious measures of performance.

However, Constraints on these quantities are easily satisfied with electron-root

operation because ignition can occur at low density. The ignition temperature

(2') * was chosen to characterize electron-root operation. For ion-root operation,

practical constraints on Pf,min were found to be more demanding than constraints

on the other two quantities; if Pf,,nin < 3 GW, then ( p ) < 8% and Psp < 100 MW for

the range of devicc parameters considered. The minimum fusion power at ignition

Pi ,m,n was used as a measure of performance for ion-root operation.

With electron-root operation the results can be surnrnarized with a simple

power law relation,

( 5 ) (T);t ~ 3071,-0.91A -0.46 0 . 2 3 ~ 0 ~ 0

p 'ha 0 p

with an rnis error of 3.7% for (T)Et < 50 keV. The ranges considered are A, = 3 10, €ha = 0.05-0.4, up = 1-3 m, Bo = 3- 10 T, and (0 = 2.5-5. All quantities in

the equations in Sections 6, 7, and 8 are in SI units, except for the temperatures

23

(in keV), densities (in lo2' m-'), and powers (in MW). For the reference set of

assumptions, decreasing the ignition temperature from the 37-keV value given by

Eq. (5) to the 22-keV value given in Section 5 for the case without alpha-particle

losses, thereby compensating for the alpha-particle losses, would require changing

t o to 4.8, A, to 15.5, or chi, to 0.021.

Equation (5) can be understood with the same scaling argument used to

explain the shape of the ignition curve in Fig. 4. With electron-root operation,

ignition occurs at low collisionality, where the transport losses are dominated by

v-regime transport, which is proportional to , 2 T - 1 / 2 [ ~ 2 A p 1 ~ ~ l a 2 B o a ~ . These losses

are balanced by alpha-particle heating, which is approximately proportional to

n2T2. Equating the alpha-particle heating to the transport losses yields an ignition

temperature

which is in good agreement with the trends observed in the calculations.

With ion-root operation, the results are more complicated because the dominant

= -3, which transport meclianism depends on the values of (n ) , ( T ) , and t o . For

is more negative than expected neoclassically,

(7) 1.33 2 .39 2.36 -1 .53 $'?;in = 3.13 x 10'Ap B, aP

with an rms error of 5.2%. The ranges considered are A, = 4-10, e l la = 0.05-0.3,

ap = 1.5-3 m, and BO = 5-10 T.

The

minimum density at ignition is at the intersection of the two branches of the ignition

contour. Along one branch the alpha-particle heating is balanced by l / v electron

transport (ix T7/2 (0"A,24~B02a ,4 ) , yielding an ignition density proportional to

T 3 / 4 5 ~ A , ' ~ 3 ' l " B ~ 1 a p 2 . ha Along the other branch, v-regime ion transport dominates,

yielding an ignition temperature proportional to I[,-, )-4/5A-2/5 P 'ha 1 /5B0a0 0 p' The

minimum density along the ignition contour is then

These results can also be explained with a simple scaling argument.

and P f , m i n , which is approximately proportional to ( ? I ) : (T)faEAp, is

24

which is similar to the trends observed in the calculations. Reproducing the

quantitative scalings with simple arguments is difficult because errors in the (n ) ,

and ( T ) , scalings are amplified when they are used to form the Pf,min scaling.

Reproducing the &, dependence with simple scaling arguments is even more

difficult because the dominant transport mechanisms can be different for different

values of to. When &, is reduced, the ions are closer to the resonance shown in Fig.

3a, and Pf,m;n scales differently with the device parameters. For example, when

(0 = -1, which is within the range expected neoclassically (see Appendix B),

0 .61 1 2 1 2.90 -1.98 F!tLin = 5.48 x 108A; E),; B; a,

with an rms error of 3.5%" The ranges considered here axe A, = 5 10, €ha = (9.01

0.2, up = 1.5-3 m, and Bo = 5-10 T. For this value of t o , moderate A , and low

cha are required for reasonable values of Pf,min. For example, A, = 10, €ha = 0.05,

Bo = 7 T, and ap = 2 m results in Pf,min = 3.2 GW.

6.2 Sensitivity to profile assumptions

The density profile can affect reactor operation in several ways. Peaked profiles

produce more alpha-particle heating for the same average temperature and density

and result in more effective heating since the alpha-particle power is deposited

closer to the magnetic axis. Changing the density profile can also change the heat

transpcrt losses through the collisionality dependence of x or directly through the

V n ter Q in the heat transport equations. The resulting changes in the temperature

profile can either enhance or reduce these effects.

Table I summarizes the effect of the density profile shape on Pf,min, (P)min, and

Psp for the ion-root mode of reactor operation. 'rhe density profiles assumed are the

peaked, broad, and hollow profiles shown in Fig. 1 plus the corresponding profiles

with different edge densities. Other assumptions are those of the reference case.

Reactor performance is much better with the broad or hollow profile than with the

peaked density profile. The edge density has only a small effect; increasing it results

in slightly better performance with both the broad and hollow profiles.

The dramatically different results with the peaked density profile are due to

changes in the temperature profile caused by the V n component of the heat flux.

Peaking the deiisity profile causes the temperature profiles to become broader,

lowering the central temperature and raising the edge temperature. The higher

25

TABLE I. DEPENDENCE OF ION-ROOT REACTOR PARAMETERS ON

THE DENSITY PROFILE SHAPE

Broad

(P = 1)

0.1 2.8 2.5 50 0.3 2.6 1.8 37

edge temperatures greatly increase the transport coefficients for electrons, because

they are typically in the 1 /v regime, where the transport coefficients scale as T7i2. If the Vn component of the heat flux is neglected, the temperature profile is similar

to that obtained with a broad density profile, and the heat transport is reduced to

such an extent that Pf,min is reduced from 15.7 GW to 0.4 GW for the case with

10% edge density. The fact that very different results are obtained with and without

the off-diagonal terms emphasizes the need for better understanding of these fluxes,

including any anomalous contributions.

Density profiles in present stellarator experiments tend to be very broad or even

slightly hollow, Similar behavior is also seen in numerical solutions of the density

equation [34]. The improved reactor performance found when the density profile is

very broad (or hollow) suggests that high-velocity pellet injection or other methods

for peaking the density profile may not be necessary and that high recycling at the

plasma edge may not be a problem. Low-velocity pellets can be used for shallow

penetration and control of the plasma edge density, if necessary. The fact that

broad and hollow density profiles with different edge densities yield similar results

suggests that the approximations introduced by not solving the density equation

are not critical.

The sensitivity to the assumed electric field profle was also addressed. For

the reference ion-root case, changing the electric potential profile from the reference

parabolic profile to a parabolic-squared profile, while keeping the potential difference

26

between the plasma center and plasma edge constant, results in more peaked

temperature profiles, decreases Pf,,,in from 2.5 GW to 0.8 GW, and decreases the

startup power required from ~ 5 0 MW to ~ 2 5 MW. A square-root parabolic profile

increases Pf,min to 7.5 GW and increases the required startup power to 133 MW.

For the electron-root mode of operation, changing the electric potential profile from

parabolic to parabolic-squared decreases the ignition temperature from ~ 3 7 keV to

~ 2 5 keV; a square-root parabolic profile yields {T)* N 44 keV.

6.3 Sensitivity to anomdous transport

The reference value assumed for anomalous t hernial conductivity, ~ t ~ ' ~ ~ ~ - .._

5 x lo1* rn-'-s-'. Figure 13 shows the effect of anomalous transport on the ignition

contour for the reference reactor with all trapped alpha particles lost. The additional

anomalous transport has little effect on the ignition window for ion-root operation

ORNL-DWG 90-2524 FED 4

3 zi

5% 2 2 E

03 z W

m-

K O w v > - w - 4 z 3

>

e *- e 1

6

0 0 18 26 30 40 50

DENSITY-AVERAGED TEMPERATURE, (T) (keV) Fig. 13. EEcct of anomalous tra,nsport on the ignition contoiirs for stellarator

reactors.

27

because of the high densities involved. The value of Pf,min increases nearly linearly

from 2.5 GW to 5.5 GW as 6Znom increases from zero to 2 x lo2' m-'.s-'. If 3 GW

is taken as an upper limit on Pf,min, then the maximum tolerable value of 6Fom is 5 x 10'' m-'.s-'. For electron-root operation, additional anomalous transport only

affects the operating window at low density. The ignition temperature is relatively

un&ected at densities above ~ 1 . 5 x lo2' m-'. The value of Pf,m;n increases from

<1 GW to 8.4 GW, and Psp increases from <20 MW to 148 MW as 6r0" increases

from zero to 2 x IO2' rn-l-s-'. Values of of about lo2' m-lss-l can be

tolerated before Pf,min exceeds 3 GW.

7. RESULTS WITH AN EMPIRICAL TRANSPORT SCALING

Energy confinement times in present-day stellarat or experiments approximately

follow the Large Helical Device (LHD) empirical scaling [32],

0.58-0.69 0.84 2 0.75 TE = 0.17fe1,Px- ne Bo apRo

where fie is the line-averaged electron density and P, is the absorbed heating

power. Here TE is the total energy confinement time including radiation losses,

neoclassical diffusive losses, and any anomalous transport losses. An enhancement

factor fen is included to model possible confinement enhancement, such as the

factor of -2 improvement in r~ found in tokamaks in the transition from L-

mode behavior to H-mode behavior. The similarity in confinement and plasma

behavior in stellarators and tokamaks, coupled with the better external control

of the configuration properties in stellarators, suggests that such an improvement

may be possible. Other options for improved confinement in stellamtors include

operation in the second stability regime, larger radial electric fields, a n d reduced

plasma-wall interaction using divertors with remotely located target chambers. The

global LHD scaling is similar to gyro-reduced Bohm scaling [35], which is based on

a trapped-particle drift-wave turbulence model, except for the dependence on the

ion mass and plasma elongation. The LHD scaling is also similar to the global

confinement scaling obtained from ripple-induced neoclassical transport in the 1 / ~ regime except for the power dependence, which is stronger in the neoclassical

expression.

28

The LHD scaling i s rewritten here in terms of the plasma variable (2') instead

of Px,

TE LHD = 1.06 x 10- fen G .4 (12) 3 2.38 (n)0.26 ( T ) - 1 . 3 8 ~ 2 ~ 2 O P

Equation (12) is found from Eq. (11) by using the definition of the energy

confinement timc and by assuming that the density profiles in the experiments

from which the LWD scaling was inferred are roughly square-root parabolic, so that

fie '2 (4/3)(n).

The single-fluid global energy balance equation

w Pcond = 7

F

is solved for P, at fixed values of (n) and (7'). The total radiation power Prad

and alpha-particle heating power P, are found by numerically integrating over the

plasma volume using fixed temperature and density profile shapes.

A PQPCQN plot for a stellarator reactor with A , = 5 , = 0.2, Bo = 7 T, and

ap = 2 m is shown in Fig. 14. Loss of trapped alpha particles is included, the density

profile is broad, and fen = 1.5. The temperature (2' = = T,) is characterized by

a profile parameter CUT: T = To(1 - p2),T. For this example, CUT = 1 is assumed.

Because the temperature dependence of the LHD confinement scaling is similar to

that obtained from neoclassical transport in the 1 /v regime, the ignition contour in

Fig. 14 is similar to that obtained with neoclassical ion-root operation, The strong

temperature dependence makes it necessary to operate at moderate temperatures

and high densities, and magnetic fields greater than 5 T are required. The auxiliary

power required for startup is 72 MW for this case.

Even modest improvements in the LHD confinement scaling result in signifi-

cantly better reactor performance. This same effect is seen in assessments of ignited

tokamaks [36] with the typical tokama,k scalings (power laws resembling the LHD

scaling). Figure 15 shows ignition contours for the same case as in Fig. 14 for four

values of fen. The contours with fen = 1.25, 1..5, 1.75, and 2 also correspond to

fen = 1 and Bo = 9.1 T, 11.3 T, 13.6 T, and 16 T, respectively. Also shown are

the contours for h; = 3 GW and ( p } = 5% and 10%. The minimum fusion output

for the case with fen = 1.25 is 6.3 GW. As fen increases, the ignition contour shifts

to lower values of ( T I } a.nd ( T ) ? allowing operation at smaller levels of fusion power.

Enhancement factors of 21.5 axe needed for ignited operation with tota.1 fusion

4.0

- c? E E 3.2 0 F v

n S v

i k 2.4 v) Z w a a w 2 1.6 a: w > e r” 9 2 0.8

0

29

ORNL-DWG 90-321 9 FED

100 M W 1 80 MW u 60 MW

40 M W \\ 20

0 4 8 12 16 20 DENSITY-AVERAGED TEMPERATURE, (T) (keV)

Fig. 14. POPCON plot obtained with the global LHD confinement model and fen = 1.5.

powers of 3 GW or less. The startup power required is also sensitive to the value

of fen; it decreases from 140 MW at fen = 1.25 to 72 MW at fen = 1.5 and 25 MW

at fen = 2. Peaking the density or temperature profile can also improve reactor per-

formance. For example, a parabolic-squared temperature profile results in

Pf,min 1.7 GW, compared to 3.1 GW for the reference parabolic profile.

Similiar results are obtained by peaking the density profiles. Plasmas with peaked

profiles produce more fusion power for the same average density and temperature.

This shifts the fusion power and ignition contours to lower temperatures and

densities. However, the ignition contour is more sensitive because of the temperature

dependence in Eci. (12), and the minimum fusion power output at ignition is

decreased by peaking either the density or the temperature profile. This behavior

is opposite to that obtained with the neoclassical ion-root mode of operation, even

=

30

QRNL-DWG 90-3220 FED

for the

though the diffusion coefficients depend on the plasma variables in a similar manner.

The different behavior is due to the off-diagonal terms that are present in the

neoclassical models, but not in the LHD scaling model.

8. OPERATION WITH Ib3E€e

Fusion reactors using B3Me fuel would have the advantage of a larger fraction of

the fusion power released in the kinetic energy of charged particles; the technological

difficulties associated with a high neutron flux environment would be reduced.

However, higher plasma temperatures and longer energy confinement times are

required because the cross-section for the D-3He reaction is much smaller than

that of the D-T reaction and peaks at higher temperatures. D3FIe reactors have

been evaluated for tokamaks [37,38] and for the less-developed tandem mirrors [39]

31

and field-reversed configurations [40,41], but not for stellarators. Part of the reason

for this neglect was the lower (p ) potential of earlier stellarator concepts; more

recent concepts have much higher beta limits [9,10]. The favorable temperature

dependence associated with the electron-root mode of operation suggests that

stellarators operating in this mode may be able to use D-3He fuel. A major

advantage is the absence of the need to drive the large plasma current (typically

25-50 MA) required for a D-3He tokamak reactor.

The existence of a loss region for energetic particles in stellarators makes

achievement of D-3He ignition more difficult than in tokamaks. However, the loss

of energetic particle energy due to scattering into the loss region is less severe than

with D-T operation because the 15.2-MeV proton produced by the D-3He reaction

is less affected by pitch-angle scattering than the 3.52-MeV alpha particle produced

by the D-T reaction. For example, ~ 5 % of the energy of the 15.2-MeV proton is lost

due to scattering into the loss region for a D-3He plasma temperature of 80 keV. The

corresponding fraction of the energy of the 3.52-MeV alpha particle lost at a D-T

plasma temperature of 15 keV is -35%. If ignition is reached, the loss region offers

some advantages. The accumulation of fusion-product ash, which is a problem for

D-3He tokamak reactors [38], is avoided. Since tritons produced from one branch

of the D-D side reaction do not accumulate in the plasma, the 14.1-MeV neutrons

produced by the D-T reaction are avoided, making a D-3He stellarator reactor more

nearly aneutronic than a D-3He tokamak reactor.

Ignition contours for a D-3He reactor with A, = 10, €ha = 0.05, BO = 8 T,

ap = 2 m, and a wall reflectivity R, = 95% are shown for three values of Eo

in Fig. 16. Profile assumptions are the same as for the reference D-'T case, and

the loss of energetic particles is included. The high-temperature branch of the

ignition contour is determined by the balance between synchrotron radiation and

fusion-product heating, while v-regime transport is the dominant loss along the

lower-temperature branch. The ignition contours for different values of to cross

each other because larger electric fields decrease the ignition temperature as in D-T

operation, but they also increase T'/%, resulting in increased synchrotron losses.

For the three cases shown in Fig. 16, Pf,,i, = 9.9 GW and (p),in = 33% for

EO = 3, = 3.3 GW and (p),;, = 17% for Eo = 4, and Pf,m;n = 2.6 GW and (p)min = 15% for t o = 5 . Parameter scans about these D-3He cases were

performed. The ranges considered are ( 0 = 3.5-5, A, = 3-10, 6ha = 0.01-0.1,

32

ORNL-DWG 90-3208 FED

Q 20 40 68 80 100

DENSITY-AVERAGED TEMPERATURE, (T) (keV)

Fig. 16. POPCON plot showing the effect of the electric field on the ignition contour for the base D3He reactor case.

up = 1-3 m, Bo = 7-12 T, and R, = 90% to 98%. The results can be summarized

with

(15j 0 . 6 2 -1.41 0 13 0.29 2.84 1.94 p[Lin = 203(1 - Rw) t o Ap. €ha BO up

(16) (p)gin(%) = 3473(1 - ~,)0.31~;0.81~-0.47 0 . 1 7 ~ - 0 . 6 1 -0.58

p 'ha 0 ap

with rnis errors of 6.8% in .Pf,min and 4.2% in (p)rrlil1.

These numerical results can also be explained with simple heuristic arguments.

For the range of temperatures considered, the D-He3 reaction rate coefficients are

approxiinately proportional to q2 and the ignition temperature (T), scales as it

does with D-T operation (Eq. Along the high-temperature branch of the ignition curve, synchrotron losses [cx n1/2T5/2 (1 - a,) 1 / 2 g 5 / 2 ap -1/21 are balanced

(6)).

33

by fusion-product heating, resulting in an ignition density proportional to T1I3( 1 - 1/3B5/3 . p f,min and (@),in occur near the point ( (n)*,(T)*) where the Rw) 0 UP

two branches intersect,

yielding 0.53133.33 2.33

p ‘ha 0

(p)m$el oc (1 - ~w)0.33~;1.07~;0.53 0 . 2 7 ~ - 0 . 3 3 -0.33 ‘ha 0

9. IMPLICATIONS FOR STELLARATOR REACTORS

One might expect that the reduction in fusion-product heating caused by the loss

of energetic trapped particles would make it difficult to reach ignition in stellarator

reactors. However, the calculations summarized here indicate that there are two

modes of D-T operation and one mode of D-3He operation that could lead to ignition

in stellarator reactors. Once ignition is reached, a velocity-space loss region for

energetic particles is only beneficial; it prevents the accumulation of thermalized

fusion products and helps channel the fusion-product power to external chambers

without affecting the first wall.

The most important conclusions concern the role of the radial electric field.

Both the electron and ion roots can exist in reactor-relevant stellarator plasmas,

but reactors optimized for ion-root operation would be very different from those

optimized for electron-root operation.

With electron-root operation, ignition occurs at relatively high temperatures

that are relatively independent of the electron density. The principal effect of

the alpha-particle losses is to increase the ignition temperature. Constraints on

the volume-averaged plasma beta and total fusion power output are relatively

unimportant because low-density operation can be used to compensate for the high

ignition temperatures. The values of the normalized electric potential required to

reach ignition (cO N 3) are near the range expected neoclassically, even with the

loss of all trapped alpha particles. The ignition temperature is relatively insensitive

to the machine and configuration parameters.

These findings suggest that stellarators operating in the electron-root mode

The relative insensitivity of this of operation may make attractive reactors.

34

mode to the machine and device parameters allows the reactor designer access

to economically favorable regions of the design space (sma.11 Ap and Bo). Orbit

optimization techniques that reduce the effective helical ripple amplitude do not

appear to be necessary, so that torsatron configurations having relatively simple

coil systenis, improved plasma access, a.nd natural divertor action could be used.

However, the relatively high temperatures and positive electric fields may lead

to excessive anomalous transport. Confirmation of the expected confinement

improvement from large positive electric fields at low eollisionality must await larger

experiments with more heating power [3].

If the expected confinement improvement from positive electric fields is not

realized, then the ion root offers an alternative mode of operation. However,

constraints on the total fusion output at ignition are more difficult to satisfy and

more sensitive to alpha-particle losses with ion-root operation. With configuration

paranieters similias to those of torsatrons, the values of (0 needed to meet constraints

on the total fusion power are more negative than those expected neoclassically,

suggesting that some form of external control of the electric field would be necessary,

which requires a steady-state power input, as in a tokamak reactor with current

drive. Modification of the radial electric field and a consequent improvement

of the ion thermal diffusivity by perpendicular neutml beam injection have been

demonstrated in a small stellarator [42]) but have not been demonstrated yet in a

large stellarator. If the recirculating power required to maintain the negative electric

potential is too large arid electric potexitids in the neoclassical range (&, N -1) a.re

obtained, then orbit optimization techniques will be necessary to reduce the effective

helical iipple amplitude. For example, it can he seen from Eq. (10) that the effective

helical ripple must be reduced to €ha = 0.03 for a typical. stelkarator reactor with

Bo = 7 T, up = 2 m, d4p = 5 , and Pf,min = 3 GW. This value of the effective helical

ripple amplitude is a factor of 7---10 sma.ller than that for Compact Torsatrons but

is siudiar to the values of €ha for Advanced Stelhators.

If the LHD anomalous confinement sealing persists into the reactor regime, then

a confinement enhancement factor of about 1.5 is necessary. This requirement is

reduced somewhat if magnetic fields in excess of 10 T are used or if pea,ked density

or temperature profiles can be obtained.

An orbit-optimized stellarator operating in the electron-root mode might make

an attractive D-3He reactor. By combining Eqs (15) and (16), it can be seen that an effective helical ripple amplitude of 0.1 and a magnetic field strength of 7.6 T are

35

required for a stellaxator reactor with Pf,min = 4 GW, (p ) = 20%, Eo -- 4, A, = 10, and up = 2 m. Effective helical ripples smaller than 0.1 have been achieved in quasi-

helical configurations [9], and values of (p) of 20% may be possible for a stellarator

with access to the second stability regime, but achieving the two simultaneously

may be very difficult.

The transport code results presented here indicate that the electron-root mode

of operation has the potential for an attractive stellarator reactor and may even

make a D-3He reactor feasible. Ion-root operation may also be possible, but

requires either control of the electric potential or a configuration with reduced

€he Obvious extensions of this work include using more sophisticated confinement

models (solving the electric field and density equations, better models for anomalous

transport). Constraints on the totad fusion power output and plasma beta also

need to be combined with engineering and reactor component constraints in order

to quantify the impact of the various transport assumptions on the total reactor

system [29].

37

Appendix A

THE COLTRANE TRANSPORT CODE

The transport survey code COLTRANE represents a compromise between a

global (0-D) approach [43] and a time-dependent 1;-D plasma simulation code

[20,44,45]. The spectral collocation method [46] is applied to the transport equations

in their int egral-differential form. The independent variables (electron and ion

temperatures) are expanded in an N-term series of Chebyshev polynomials. The

expansion coefficients are found by requiring that the integral-differential form of the

transport equations be satisfied on a discrete set of points known as the collocation

(ordering) points,

1 - c o s ( ~ j / N ) 2

, I < j < N - l . P i =

The collocation points are the interior extrema of the highest-order basis function

and are more closely spaced near the plasma boundary, where the solution can

change rapidly. This collocation yields equations representing power balances

over finite volumes as in the finite-element method. This discretely conservative

property causes the globally averaged quantities to converge rapidly as the number

of basis functions increases. Continuity conditions at each collocation point p j are

intrinsically satisfied, in contrast to the finite-element method, which requires that

the continuity conditions be explicitly imposed.

COLTRANE treats the general linear boundary condition Ta+b(clTa/dp)+c = 0

at the plasma boundary and qa = 0 at the plasma axis. Typically, the tempcrature

gradient scale length ap[(l/T,)(dTa/dp)]-l is specified to be 10 cm for both the

electrons and ions. The two equations coming from the boundary conditions, the

N - 1 finite-volume power balances, and an auxiliary condition on (7') are solved

iteratively for the extcrnal power and the N + 1 expansion coefficients for each

species. The Jacobian matrix used in the iterative method of solution is written in

a computationally efficient form which allows it to be evaluated numerically with

O ( N + 1) operations.

The most obvious way of constructing a POPCON plot with a steady-state

code is simply to construct a fine grid in (n)-(2') space and solve the fixed

temperature problem at each grid point. A more efficient approach solves the

fixed temperature problem on a coarse grid (typically 6 x 6) and calculates linear

sensitivity coefficients describing the change in the solution as (n) and (2') change.

The sensitivity cocfficients, found directly from the discretized equations with little

38

computational effort, are used to interpolate the solution between the coa,rse grid

points.

An ignited reactor case with A, = 5 , RQ - 2 m, Bo = 5 T, and €ha r= 0.2

was used to study the convergence properties of the collocation method. The density

and electric potential profiles are both parabolic squared with a central density of

3 x lo2’ rn-’ and (0 = -2. The residual power (error in the integrated power

balance) normalized to the total alpha-particle heating is sinall over most of the

plasnia region but peaks near the plasma boundary because of interpolation errors

in the temperature gradients between collocation points. However, the nature of the

collocation method prevents this interpolation error from propagating away from

the offending point. If the spike in the residual power near the plasma boundary

is excliided, the residual power decreases rapidly with increasing number of basis

functions. The maximum relative error in the integrated power balance for 0 5 p 5 0.9 converges approximately exponentially as the number of basis functions is

increased. Only 25 basis fiunctioiis are required before the residual error reaches the

level of numerical noise in this single-precision calculation. The average quantities

converge even more rapidly. An accuracy of 1% is obtained for the average electron

and ion temperatures at ignition with 10 basis functions, and 10% accuracy is

obtained with only 5 basis functions. The rapid convergence of the global quantities

is a direct consequence of solving the equations in integral form.

39

Appendix B ESTIMATE§ OF THE AMBIPOLAR ELECTRIC FIELD

The approach taken in this work is to assume values for the radial electric field

that are roughly consistent with solutions to the ambipolarity constraint and then

to investigate the sensitivity of the results to the assumed values. Rough estimates

for the neoclassical ambipolar electric field can be obtained by numerically solving

the ambipolarity constraint with fixed temperature and density profiles, as was done

in Ref. [47]. This type of analysis is repeated here in a manner that clarifies the

dependence of the electric field on the local plasma variables and the machine and

device parameters. Explicit criteria for the existence of the electron and ion roots

are also developed.

Figure B.l shows the normalized electric field E, = - ( e / k T ) ( d @ / d p ) as a

function of a dimensionless temperature

for four stellarators ranging from present-day experiments to low-aspect-ratio

reactors. Here T, = Ti = T , C, = v , T ~ J ’ ~ / ~ , C, = t;T3I2/n, u, = v,, + uei,

and v, = q i . In this plot, E, was found numerically from the ambipolarity

constraint using the integral expressions for the particle fluxes with parabolic profiles

of temperature and density.

For large values of the normalized temperature (5 >> l), ions and electrons are

both in the collisionless detrapping rcgime (v-regime), where the particle flux can

be approximated by

where Et = r / & , V d = -T/(apZ,eBp), A, = ( l / n ) ( d ~ z / d p ) , AT = (l /T)(dT/dp),

and = AT/X,. The radial electric field is assumed to be large enough that the

E x B poloidal drift dominates the bounce-averaged B x VB poloidal drift due to

the helical modulation of B ( t o >> €ha). Positive electric fields then develop to hold

in the more collisional and more lossy electrons,

12

8

4

0

-4

-8

40

ORNI_-DWG 90-3206 FED 1 I 1 I l l 1

i?3

a lli

10-1 100 101

NORMALIZED TEMPERATURE,

Fig. B.l. Dependence of the ambipolar electric field on the normalized temperature for parabolic profiles of n and 7' at p = 0.75. The squares are for the ATF experiment (A , = 7.8 and = 0.2), and the other symbols are for the reference reactor with various values for the electron density.

For 5 << 1, the ions and electrons are both in the 1/v regime where

and negative electric fields develop to hold in the plasma ions,

E, N -A, (1 + g7) (135)

However, for a wide range of plasma conditions, the plasma ions are in the v-

regime, the more collisional electrons remain in the 1 /v regime, and the ainipolarity

41

constraint has multiple roots. Equating the electron and ion fluxes yields a cubic

equation for the normalized electric field,

The minimum temperature Tl at which the electron root exists can be estimated

by determining when Eq. (B6) has precisely two real roots. This analysis yields

where 51 is the value of 5 corresponding to 2'1 and f ( q ) = 11 + 237 + 6.551~. This

value of 51 is indicated in Fig. B.l.

To find the maximum temperature T2 at which the ion root exists, a more

accurate expression for the ion flux must be used because the transition from the

ion root to the electron root occurs when the electric field is small, and the opposite

was assumed in deriving Eq. (B6). A good estimate for the maximum temperature

at which the ion root exists can be obtained by noting that the transition from the

ion root to the electron root occurs when the E x I3 and B x VI? drifts are equal,

or E, 3i - a c h / a p . Using this value for E, in Eq. (B6) yields the approximate value

of < at which the ion root disappears,

This value is also indicated in Fig. B.l.

and T2 = 29 lev for rL = 2 x 10'' m-' at p = 0.75.

Applying Eqs (B7) and (B8) to the reference reactor case yields 7'1 = 9.7 keV

42

ACKNOWLEDGMENTS

The first author (SLP) thanks Prof. P. N. Stevens of the University of 'Tennessee,

Knoxville for his guidance and for useful discussions on numerical methods. This

research was sponsored by the US. Department of Energy, Office of Fusion Energy,

under contract DE-AC05-840R21400 with Martin Marietta Energy Systems, Inc.

43

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50

83. A. Perez Navarro, Division de Fusion, CIEMAT, Avenida Complutense

84. Laboratory for Plasma and Fusion Studies, DepaxtIrimt of Nuclear 22, E-28040 Madrid, Spain

Engineering, Seoul National University, Shinrim-dong, Gwanak-ku, Seoul 151, Korea

Box 451, Princeton, N J 08543

Sciences, Ulitsa Vavilova 38, 117924 Moscow, U.S.S.R.

Kyoto, Japan

464-01, Japan

3402, 123182 Moscow, U.S.S.R.

Madison, VE'I 53706

Federal Republic of Cierrnany

85. J. E. Johnson, Plasma Physics Laboratory, Princeton University, P.O.

86. L. M. Kovrizhnykh, Institute of General Physics, U.S.S.R. Academy of

87. T. Obiki, Plasma Physics Laboratory, Kyoto University, Gokasho, Uji,

88. A. Iiyoshi, National Institute for Fusion Studies, Chikusa-ku, Nagoya

89. V. D. Shafranov, I. V. Kurehatov Institute of Atomic Energy, P.O. Box

90. J . 1;. Shohet, Torsatron/Stellarator Laboratory, University of Wisconsin,

91. G. Grieger, Max-Planck Institut fiir Plasmaphysik, D-8046 Garching,

92. K. Matsuoka, National. Institute for Fusion Science, Chikiisa-ku, Nagoya

93. D. G. Swanson, Physics Department, Auburn University, Auburn, AE

94. R. L. Linford, Los Alaiiios iiationd T,aboratory, MS F646, P.O. Box

95. K. 1. Thornassen, L-637, T,awrence Livermore National Laboratory, P.O.

96. R. F. Gandy, Physics Dcpartment, Auburn University, Auburn, AL

97. W. Wobig, Max-Planck Institut f& Plasmaphysik, D-8046 Garching,

98. D. T. Anderson, University ~f Wisconsin, Madison, WI 53706 99. K. Kondo, Plasma Physics Laboratory, Kyoto University, Gokasho, Uji,

464-01, Japan

36849-35 11

1663, Los Alamos, NM 87545

Box 5511, Livermore, CA 91550

36849-3511

Federal Republic of Germany

Japan

Japan

(Category Distribution UC-426, Experimental Plasma Physics) (53 copies)

100. F. Sano, Plasma Physics TAaboratory, Kyoto University, Gokasho, Uji,

101-153. Given distribution as shown in OSTI-4500, Magnetic Fusion Energy