numerical modeling of diffusive heat transport

8/8/2019 Numerical Modeling of Diffusive Heat Transport

1/12

Numerical modeling of diffusive heat transport across magnetic islandsand local stochastic field

Q. YuMax-Planck-Institut fr Plasmaphysik, EURATOM Association, D-85748 Garching, Germany

Received 6 February 2006; accepted 28 April 2006; published online 23 June 2006

The heat diffusion across magnetic islands is studied numerically and compared with analytical

results. For a single island, the enhanced radial heat diffusivity, r, due to the parallel transport alongthe field lines is increased over a region of about the island width w. The maximum enhanced heat

conductivity at the rational surface is proportional to w21/2 for sufficiently high values of

/, where / is the ratio between the parallel and the perpendicular heat diffusivity. For low

ratios of/, however, the maximum value ofr is proportional to w4. In a locally stochastic

magnetic field, r is again proportional to w4 for low /, which is in agreement with the

analytical results. With increasing / , r is dominated first by the additive effect of individual

islands and then by the field ergodicity. 2006 American Institute of Physics.

DOI: 10.1063/1.2206788

I. INTRODUCTION

Magnetic islands generally exist in laboratory and spaceplasmas, caused either by resonant helical magnetic field per-

turbations finite error fields in fusion devices or by thetearing mode type instabilities, driven by an unfavorable

plasma current density gradient,13

the perturbed bootstrap

current,49

or the electron temperature gradient.10,11

The ex-

istence of magnetic islands inside the plasma usually has a

very significant effect on the plasma behavior. In tokamak

plasmas magnetic islands caused by the classical or the neo-

classical tearing mode NTM have been found to lead to adegradation of plasma confinement or even to disruptions.

28

For high plasmas the onset of a magnetic field perturbation

with its corresponding rational surface being very close to

another saturated island is often observed, and a strong inter-

action between them and the subsequent change in the

plasma energy confinement are found.6,12

Extensive studies

have also been devoted to the transport across the stochastic

field boundary in stellarators and in tokamaks as produced by

an externally applied helical field.13,14

The heat transport

across magnetic islands and the stochastic magnetic field is

thus of general interest in plasma physics.

The effect of a single magnetic island on the plasma

energy confinement has been analyzed by assuming the ratio

between the parallel and the perpendicular heat conductivity,

/, to be infinite. The resulting degradation of plasma

energy confinement due to the fast parallel transport alongthe field lines is found to be determined by the island width,

the minor radius of the rational surface and the local equilib-

rium plasma pressure gradient.15

The electron temperature

perturbations due to a single island have also been studied in

the limits of a sufficiently low or high value of / for

determining the transport threshold for the onset of the

NTM.16

When the plasma region is occupied by well-

overlapping islands, the magnetic field becomes stochastic

there. For such a situation the enhanced radial transport

across the stochastic field has been extensively investigated

and is considered to be one of the possible mechanisms for

the anomalous heat transport of tokamak plasmas.1726

It is well understood that both the parallel and the per-pendicular transport are important in determining the heat

diffusion across a single island or a stochastic field, no mat-

ter if the perpendicular transport is caused by plasma colli-

sions or turbulence.1620

Despite the existing theories on the

heat transport across stochastic fields, for a finite value of

/ there is no straightforward analytical solution for the

temperature perturbations even due to a single island.16

Since

the plasma temperature changes by several orders of magni-

tude from the center to the edge in a tokamak, the parameter

/ changes even more. It is of great interest to study how

the parameter / affects the heat transport across islands

and the related plasma energy confinement degradation. In

some experimental cases one finds a local stochastic field

region due to magnetic perturbations of different helicities

with their corresponding rational surfaces being close

together.6,12

To the authors knowledge no transport theory

has been established for such a local stochastic field yet. The

understanding of the heat transport in such a case is, how-

ever, important for understanding both the corresponding

change in the plasma energy confinement as well as its effect

on plasma instabilities such as the NTMs.6,12

The drift-

tearing mode instability driven by the electron temperature

gradient is found to be particularly sensitive to the perpen-

dicular heat transport.11

The numerical calculation of the heat transport acrossmagnetic islands is usually quite challenging for a high ratio

of/, which could be in the order of 1010 or even larger

in the central region of large tokamak plasmas.16

This large

/ usually introduces an artificial perpendicular heat flux

which could be even larger than the real one. Recently a new

numerical method was developed, which has been shown to

suppress this artificial perpendicular heat flux.27

In addition

to the numerical problem, the parallel heat flux is also much

more complicated for a sufficiently high /. It is of the

classical form only in a narrow region around the rational

PHYSICS OF PLASMAS 13, 062310 2006

1070-664X/2006/136 /062310/12/$23.00 2006 American Institute of Physic13, 062310-1

Downloaded 24 Sep 2010 to 130.237.45.46. Redistribution subject to AIP license or copyright; see http://pop.aip.org/about/rights_and_permissio
http://dx.doi.org/10.1063/1.2206788http://dx.doi.org/10.1063/1.2206788http://dx.doi.org/10.1063/1.2206788http://dx.doi.org/10.1063/1.2206788http://dx.doi.org/10.1063/1.2206788http://dx.doi.org/10.1063/1.2206788


2/12

surface or the islands x-point.28

Away from this region the

heat flux is carried by free-streaming electrons and is

nonlocal.29

For such a case the flux limit form for was

used before e.g., Refs. 16 and 30, but this is only a crudeapproximation.

29

In the present paper the complicated physics of the par-

allel heat flux mentioned above will not be addressed. Our

numerical results of the diffusive heat transport across mag-

netic islands are obtained by using a constant along theminor radius. This assumption is the same as that of the

existing theories, allowing an easy comparison with

them.1620

For a single island, the enhanced radial heat dif-

fusivity r due to the parallel transport along the field lines is

found to be increased over a radial width of about the island

width w. The maximum value at the rational surface is pro-

portional to w2/1/2 for a sufficiently high value of

/. For a low ratio of / ,r is proportional to w4.

When two neighboring islands overlap, the local magnetic

field becomes stochastic. In this case, for low ratios of

/, the radial heat transport is found to be dominated by

the additive effect of individual islands. This is in agreement

with the analytical result as shown in the Appendix. Only forsufficiently high ratios of/, the radial heat transport is

dominated by the ergodicity of the magnetic field.

In Sec. II our model is described. The numerical results

and the comparison with analytical results are presented in

Sec. III, and the discussion and summary are given in Sec. V.

II. MODEL

In our model the large aspect-ratio tokamak approxima-

tion is utilized. The magnetic field B is expressed in the form

B=B0tet+B0prep +B1, where B0t and B0p are the equilib-

rium toroidal and poloidal magnetic field, respectively, andB1 =et is the perturbed helical field. B0t is assumed to

be a constant. is expressed in terms of a Fourier decom-

position,

= i

ircosmi+ ni , 1

where mi and ni are the poloidal and toroidal mode numbers

of the ith component ofB1, and and are the poloidal and

toroidal angle, respectively. The toroidal magnetic field per-

turbation is usually much smaller than the poloidal one and

is therefore neglected.

The equilibrium safety factor qr = rB0t/RB0p is cho-

sen to be the form

qr = q0er/Lq 2

to have a constant magnetic shear along the minor radius,

where q0 =0.2 and Lq =0.3a are chosen to ensure q =3 /2 and

4/3 surfaces to be inside the plasma 0ra , a: plasmaminor radius, except when mentioned elsewhere.

ir in Eq. 1 is taken as

ir = i,0r/a21 r/a2 3

to have a smooth change along the minor radius, being typi-

cal for the classical or neoclassical tearing modes.1,9

The following electron energy transport equation:

3

2ne

Te

t= ne Te + ne Te + Pr, 4

is solved, where Te is the electron temperature, ne is the

electron density and P is the heat source. Here ne , and are assumed to be constant along the minor radius for sim-

plicity, and the convective transport is neglected which is

valid for slowly changing islands such as those due to an

error field or NTMs.9,16

The heat source Pr in Eq. 4 is taken to be of the form

Pr = P01 rac1c2, 5

where c1 =2 and c2 = 16 are taken except when mentioned

elsewhere. P0 is chosen to ensure Te0r= 0 =1, where Te0 isthe equilibrium electron temperature obtained from Eq. 4by taking B1 = 0 without magnetic field perturbations.

The boundary conditions used here are Ter= a = Te0r= a and Ter= 0 =0, where the prime denotes d/dr. Thenormalization scheme is as follows: the length is normalized

to a, the magnetic field to B0t, and Te to Te0r= 0.

III. MODELING RESULTS

Equation 4 is solved numerically using the transportsubroutines of the initial value code TM1.

9TM1 is a new

version of our previous code TM for modelling the nonlinear

evolution of NTMs and their stabilization by rf wave current

drive.30

In TM1 the new numerical scheme described in Ref.

27 is applied, and both the parallel and the perpendicular

transport terms in Eq. 4 are fully implicit.In Sec. III A the heat transport across a single island is

studied, and in Sec. III B a two island case is studied. Since

the magnetic field becomes stochastic when two islands of

different helicity overlap, this allows us to investigate the

heat transport across a local stochastic field. A comparison

between the numerical and analytical results is given in Sec.

III C.

A. Heat transport across a single island

An example for the change of the radial temperature

profile due to a single m/n =3/2 island is shown in Fig. 1,

FIG. 1. Radial profiles ofTe0 dotted curve, T0/0, and the Te along the linepassing through the islands o-point and x-point with w3/2 =0.0452a , r3/2=0.604a, and /= 10

10. T0/0 is the same as Te0 for rr, and T0/0 is the

same as Te0 for rr+.

062310-2 Q. Yu Phys. Plasmas 13, 062310 2006



3/12

where the m/n =0/0 component of Te , T0/0, and the tempera-

ture profile along the line passing through the islands

o-point and x-point in the steady state are shown. The origi-

nal equilibrium temperature Te0 is shown by the dotted

curve. The q =3/2 surface is at r3/2 =0.604a, the island widthis w3/2 =0.0452a3/2,0 = 910

4aB0t, and /= 1010 is

taken. It is seen that the local Te profile becomes flattened

inside the island when viewing along the line passing

through the islands o-point due to the fast parallel transport

at a large /. Along the line passing through the islands

x-point, Te has a finite radial gradient as expected. The radial

gradient of T0/0 is the same as that of Te0 for rr, and T0/0is the same as Te0 for rr+ due to the conservation of the

perpendicular heat flux, where r =0.582a and r+ =0.627a are

the inner and outer edges of the island.

Corresponding to Fig. 1, in Fig. 2 the radial profiles of

the fundamental m/n =3/2 component and higher harmon-

ics m/n =6/4, 9/6, and 12/8 components of Te are shown.Higher harmonic temperature perturbations localize in the

island region, and their amplitudes decrease for a larger m or

n, as expected from the infinite / limit result.16

For such

a high value of/ the temperature is a constant along the

magnetic field line except in a thin layer around the islands

separatrix where the parallel heat flux does not vanish.

In Fig. 3 the radial profiles of the original equilibrium

temperature Te0 solid curve and T0/0 in steady state areshown for /= 10

12 , 1010 , 108 and 106, with the other

parameters being the same as in Fig. 1. In all our results is kept unchanged. It has been shown by analytical theory

that, T0/0 is approximately the same as Te0 as long as the

island width w is much smaller than the critical island width

wc = a/1/4an/ 8Lq1/2, where Lq = q/q and = a/R.16For infinite ratio of /, the Te profile becomes flattened

inside the island, and the decrease of T0/0 for rr is Te= Te0 T0/0 Te0 r= rsw/ 2

1/2.15

With our input parameters

one finds wc =0.63102a , 2.0102a, and 6.3102a for

/= 1010 , 108, and 106, respectively. Only for /

= 106, the island width is smaller than the critical island

width w3/2/wc =0.72, so that the difference between Te0 andT0/0 is small. A larger / leads to a more flattening T0/0profile across the island and therefore a smaller T0/0 for r

r. The T0/0 for /= 1012 is very close to that for

/= 1010, indicating that the further change in Te is small

for w3/2

/wc

1. It is seen from Fig. 3 that for describing the

large island limit of NTM,16 /= 1010 has to be taken in

the numerical calculations.9

Corresponding to Fig. 3, the radial profiles of the m/n

=3 /2 component of Te , T3/2, are shown in Fig. 4 for

/= 1010 , 108, and 106. For low /, the magnitudes

of T3/2 and T3/2 are small in the island region due to w3/2wc, corresponding to a small bootstrap current perturbation

for the NTM.16

A higher / , /= 108, leads to larger

local magnitudes of T3/2 and T3/2 . For even larger

/1010, the T3/2 profile becomes flattened near the ra-

tional surface, in agreement with the infinite / limit

result.16

Away from the island these T3/2 profiles become the

same, since Te and high harmonics of Te vanish there.Considering the conservation of the perpendicular heat

flux, q= T0= eT0/0 , where e is the effective radial

heat diffusivity in the presence of the island, one can define

an normalized effective radial heat conductivity

= e/ = Te0 /T0/0 . 6

The enhanced radial heat diffusivity due to the parallel trans-

port across the island is then given by r=e = 1. Corresponding to Fig. 3, in Fig. 5 radial profiles of are shown for /= 10

10 , 108, and 106. As expected from

Fig. 3, 1 for /= 106, and a larger / leads to a

larger . Along the minor radius has a width being about

FIG. 2. Correspondi ng to Fi g. 1, radi al profil es of the m/n

=3/2, 6/4, 9/6, and 12/8 components of Te. High harmonic temperature

perturbations are localized in the island region, and their amplitudes de-

crease for a larger m n.

FIG. 3. Radial profiles of Te0 solid and the T0/0 for /=1012 , 1010 , 108, and 106. For a low / , T0/0 is close to Te0. A larger

/ leads to a more flattening local T0/0 profile.

FIG. 4. Corresponding to Fig. 3, radial profiles of T3/2 for /= 1010 , 108, and 106. For /= 10

6 , T3/2, and T3/2 are small in the island

region. A larger /=108 leads to a larger T3/2 and T3/2 . For /

= 1010 , T3/2 profile becomes flattened near the rational surface.

062310-3 Numerical modeling of diffusive heat transport Phys. Plasmas 13, 062310 2006



4/12

the same as the island width and a maximal value at the

rational surface.

The parameter r/= 1 measures the normalizedto enhanced radial heat diffusivity due to the paralleltransport. In Fig. 6 the maximal value of log 1 at the

rational surface see Fig. 5, log 1max, is shown as afunction of log/ for the 3/2 island by the solid curve,with other parameters being the same as those of Fig. 1. The

condition w3/2 = wc,3/2 leads to log/c =6.6, where wc,3/2is the critical island width of the 3/2 island. It is seen that,

1max is proportional to / for / /c,since in this case only the quasilinear correction of T0/0 is

important when wwc, as shown by the analysis in the Ap-

pendix. For //c, however, 1max is propor-tional to /

1/2. Between these two limits there is a tran-

sition region around 10/c or w3/2 1.8wc,3/2. Thedotted curve with empty circles in Fig. 6 shows log 1max for an m/n =4/3 island, which is quite similar to that

of the 3/2 island. The 4/3 island width w4/3 =0.0450a, beingapproximately the same as w3/2, but its critical island width

wc,4/3 is smaller due to its higher toroidal mode number. The

condition w4/3 = wc,4/3 leads to log/c =6.2. The largervalue of log 1max for the 4/3 island is due to its smallerwc,4/3.

In Fig. 7 numerical results of log 1max versuslogw/wc

2 for the 3/2 island is shown by the solid curve

with circles, and the dotted curve shows the analytical result

given by Eq. A12 in the Appendix . Here in w/wc2 only

the island width w changes, /= 1010 and other param-

eters are the same as those of Fig. 1. It is seen that

1max is proportional to w4 for w/wc1 but to w2 forw/wc3. The transition region between these two slopes is

around w/wc 1.8. When w/wc1, the numerical result isthe same as the analytical one given by Eq. A12. Whenw/wc1, the parallel heat transport is along a thin layer

around the islands separatrix. The radial layer width is wc2/w

at the helical angle being the same as the islands o-point,16

where Te Te0 . Due to the islands geometry, the layer widthexpands to wc along the minor radius and to 4wc/walong the helical angle around islands x-point. Therefore,

the Te at the x-point Tex wc/wTe0 . Since Te is zero wellinside the island for w/wc1 , T0/0 , the averaged Te along

the helical angle, is proportional to Tex /wc/w2Te0 .

This explains why 1max is proportional to /1/2 andw2 in the limit w/wc

21, as shown in Figs. 6 and 7.

The degradation of the plasma energy confinement due

to an island can be described by the parameter = 0, where = 2Rne 2rT0/0dr is the total plasma thermalenergy in the presence of the island, and 0= 2Rne 2rTe0dr is that without the island. In Fig. 8

FIG. 5. Corresponding to Fig. 3, radial profiles of for /= 1010 , 108,

and 106. has a maximal value at r3/2 and a width about the island width. A

larger / leads to a larger .

FIG. 6. log 1max versus log/ for the 3/2 island solid and the 4/3island dotted. w3/2 = wc,3/2 leads to log/c =6.6. 1max/ for//c but 1max/

1/2 in the opposite limit. There is a

transition region around 10/c or w3/2 =1.8wc,3/2.

FIG. 7. Numerical results of log1 max versus logw/wc2 for a 3/2 island

with /= 1010 solid. Here in w/wc

2 only the island width w varies.

1maxw4 for w/wc1 but 1maxw

2 for w/wc1. The transition

region between these two limits is around w/wc =1.8. The dotted curve

shows the analytical results of Eq. A12.

FIG. 8. /0 versus log/ for the 3/2 island with w3/2 =0.0452asolid. increases with log/ from /c =3.810

6 up to

1010w3/2/wc =7.2. For /1010 , saturates. The /0 due to a

4/3 island dotted is larger for low / due to its smaller wc. For high/, the /0 due to the 3/2 island is larger because r3/2r4/3 and

w3/2wc.

062310-4 Q. Yu Phys. Plasmas 13, 062310 2006



5/12

/0 is shown as a function of log/ for the 3/2island by the solid curve, with other parameters being the

same as those for Fig. 1. It is seen that significantly

increases with log/ from log/c =6.6 up to 10corresponding to w/wc =1 to 7.2. For /10

10 , es-

sentially saturates. When w/wc1 , Te only slightly changes

around the island separatrix, and is mainly determined by

the Te for rr as seen from Fig. 3. Therefore, there is no

significant increase ofwith / for /1010. Us-

ing the analytical formula for the /= limit,15/0

=0.041 is obtained, in agreement with the numerical result.

The /0 due to the 4/3 island is also shown in Fig. 8 by

the dotted curve. The q =4/3 surface is at r4/3 =0.569a with

r4/3r3/2, and all the other parameters are the same as men-

tioned for Fig. 6. For small /, the /0 due to the 4/3

island is larger than that due to the 3/2 island, since the

temperature flattening inside the 4/3 island occurs at a

smaller / due to its smaller wc. For large /, the

/0 due to the 3/2 island is larger because r3/2r4/3 and

wwc.15

B. Heat transport across two islands and stochasticfield

In this part the heat transport across two islands, the

m/n =3/2 and 4/3 islands, is studied. The corresponding ra-

tional surfaces are at r3/2 =0.604a and r4/3 =0.569a, respec-

tively. When these two islands are large enough to overlap,

the local magnetic field becomes stochastic, allowing us to

study the heat transport across a local stochastic field.

In Fig. 9 an example of the local magnetic surface on the

r- plan at =0 is shown for 0 i,0/aB0t=3/2,0/aB0t=4/3,0/aB0t= 10

4. In this case the m/n =3/2 and 4/3 islands

are not large enough to overlap. Nevertheless, an additional

secondary m/n =7/5 island exists due to the coupling be-

tween the 3/2 and the 4/3 magnetic field perturbations.

Corresponding to Fig. 9, the local temperature contour is

shown in Fig. 10 for /= 1012. In addition to m/n = 3 / 2

and 4/3 islands, there is also an m/n =7/5 island in the tem-

perature contour in agreement with Fig. 9. In this case Pr= P0r/a

81 r/a28 is taken to deposit more power den-sity in the island region for viewing the 7/5 island more

clearly. The small 7/5 island can be seen in the temperature

contour only for /1010, indicating that a small sec-

ondary island is important in the transport only for a suffi-

ciently large /.

With increasing perturbation amplitude 0

, more second-

ary islands are seen and the magnetic surface first becomes

stochastic around the islands separatrix. For sufficiently

large 0, the local magnetic field becomes stochastic. Figure

11 shows such an example for 0 = 9104, leading to w3/2

=0.0452a , w4/3 =0.0450a, and w3/2 + w4/3/2 r3/2 r4/3 = 1.29, where w3/2 and w4/3 are calculated by the conven-

tional single island formula.2

Corresponding to Fig. 11, in Fig. 12 the local radial pro-

files of Te0 dotted-dashed curve and the T0/0 for /= 1010 , 108, and 106 are shown. For low / , T0/0 is close

to Te0 similar to a single island. A large / leads to a

flattening T0/0 profile across the island region.

Corresponding to Fig. 12, in Fig. 13 radial profiles oflog 1 are shown for /= 10

10 , 109 and 108. For

/= 108 ,has only two peaks around the q =3/2 and 4/3

surfaces, and its magnitude is approximately the same as that

of a single island as shown in Fig. 5. For /= 1010 addi-

tional peaks of due to the ergodicity are apparent. Com-

pared to Fig. 5 it is found that, is about 5 times larger than

that due to a single island with /= 1010, indicating that

the ergodicity leads to fast radial transport for large /.

In Fig. 14, the radial profiles of Te0 and the T0/0 for 0= 2104 , 5104, a n d 9104 are shown with /= 1010. For 0 = 210

4 , T0/0 is approximately the same as

Te0 between the two island, indicating that the local confine-

FIG. 9. The magnetic surface on the r- plane at

=0 for 0 =104. A secondary m/n =7/5 island exists in

addition to the 3/2 and 4/3 islands.




6/12

ment is not destroyed, although there are secondary islands

similar to that shown in Fig. 9. With the increase of the

island width, the two islands overlap, and the local T0/0 de-

creases. When plotting the magnetic surface with 0 = 5

104, it is found to be stochastic in the region r=0.58a

0.59a similar to Fig. 11. However, T0/0 is larger there than

that at the two rational surfaces.

To study the difference in energy confinement between

the cases with and without ergodicity, in Fig. 15 /0 is

shown as a function of log/ by the solid curve for

0 = 9

10

4

. The corresponding magnetic surface is shownin Fig. 11. The dotted curved shows the 1/0 = 3/2+4/3/0, where 3/24/3 is the due to a single

m/n = 3 / 2 4/3 island, as shown in Fig. 8. The differencebetween and 1 therefore measures the additional deg-

radation in the energy confinement due to the ergodicity of

the magnetic field. It is seen that 1 for /3107, corresponding to w3/21.7w3/2,c. This means that for

a sufficiently low /, the ergodicity does not affect the

energy confinement, and the heat transport recognizes the

individual island structure rather than the ergodicity. For

/3107 , becomes larger than 1, showing the

additional degradation of the confinement due to ergodicity.

However, such an additional degradation is not significantwith / up to 1011, which is of the same order of magni-

tude as that for the central region plasma of large

FIG. 10. Color online Corresponding to Fig. 9, thetemperature contour for /=10

12. There is an m/n

=7/5 island in the temperature contour in agreement

with Fig. 9.

FIG. 11. Same as Fig. 9 but for 0 =9104. The local

field becomes stochastic.

062310-6 Q. Yu Phys. Plasmas 13, 062310 2006



7/12

tokamaks.16

The small difference between and 1 is

due to the fact that, is mainly determined by the Te for

rr of the inner island, the 4/3 island in the present case,

and Te changes little for sufficiently large / similar to

that of a single island as seen from Fig. 3.

In Fig. 16 log 1 versus log/ is shown for 0= 9104 by the solid curve. The value of is taken at r

=0.587a where it has a local minimum value see Fig. 13.One finds 1/ in the low / limit in agreementwith Eq. A14. For /310

8w3/23.2wc,3/2 , 1 again approximately scales as /. Between these twolimits there is a transition region around /= 3

107w3/2 =1.7wc,3/2. In this region 1 increases lessthan linear with /, similar to the case of a single island

with w wc. The dotted curve in Fig. 16 shows log3/2+4/3 2, where 3/24/3 is the obtained with a single 3/24/3 island at the same radial location. The solid curve isapproximately the same as the dotted one for /3

107, as predicted by Eq. A14 that the heat transport isdetermined by the additive effects of the individual islands

for wwc. When plotting the radial profiles for the case

wwc , log 1 is found to be the same as log3/2 +4/3 2 everywhere across the island region. For sufficientlylarge /, the solid curve is larger than the dotted one,

indicating the role of the stochastic field.

With an even larger magnetic field perturbation ampli-

tude, 0 =3.6103, the local magnetic surface becomes

fully ergodic, and the results are found to be similar to those

shown in Fig. 16. In addition to the case of overlapping the

m/n =3/2 and 4/3 islands discussed above, studies have also

been carried out for the m/n =30/20 and 30/21 islands. In

this case the local magnetic field is more stochastic due to ashorter distance between the two rational surfaces, but results

similar to those shown in Figs. 15 and 16 have been found.

It is seen from the results above that, similar to the single

island case, the parameter w/wc also characterizes the heat

diffusion across a local stochastic field. For wwc, the heat

transport is determined by the additive effects of the indi-

vidual islands, and r is found as predicted by the ana-

lytical result Eq. A14. Around w =1.7wc there is a transitionregion where the additive effect of individual islands is still

important, and r only slowly increases with . For suffi-

ciently large w/wc , w/wc3 ,r nearly scales with , but the

heat diffusion is dominated by the ergodicity of the magnetic

field.

C. Comparison with analytical theories

There have been extensive theoretical studies on the heat

transport across the stochastic field due to the spacial diffu-

sion of the magnetic field lines.1724

In the collisionless re-

gime where the electron mean free path e is longer than the

FIG. 13. Corresponding to Fig. 12, radial profiles of log1 . For /=108 , only peaks around the q =3/2 and 4/3 surfaces. For /= 10

10,

additional peaks appear due to the ergodicity, and is about five times larger

than that for a single island.

FIG. 14. Radial profiles of Te0 and the T0/0 for 0 =2104 , 5104, and

9104 with /= 1010. With the increase of the island width, the local

T0/0 decreases.

FIG. 15. /0 solid and 1/0 =3/2 +4/30 dotted versuslog/ with 0 = 910

4. For low / ,=1 as predicted by Eq.

A13. For high / ,1, showing the enhanced transport due toergodicity.

FIG. 12. Corresponding to Fig. 11, radial profiles of Te0 dotted-dashed andthe T0/0 for /= 10

10 , 108, and 106. A larger / leads to a more flat-

tening local T0/0 profile.




8/12

Kolmogorov length LkLs2/k

2 DM1/3,

1720the enhanced

radial heat conductivity is shown by Rechester and Rosen-

bluth to be

17

r = DM/e = DMvTe, 7

where vTe is the electron thermal velocity,

DM = L0kbk,rB0t2

mk/q nk, 8

L0 R , bk,r is the radial magnetic field perturbation of thekth component, k is the perpendicular wave vector of the

perturbations, Ls =RR2/rq, and the summation is over k to

include contributions from all components. Stix has same

results as Eq. 7.18 In the collisional regime eLk,

r = DM/Lc, 9where Lc=Lc lnr/mLc/

1/2, Lc =R/ln/ 2, = w1 + w2/2 r1 r2 , w1 and w2 are the widths of twoneighboring islands, and r1 and r2 are the minor radius of the

corresponding rational surfaces.17

Using a fluid approach, Kadomsev and Pogutse calcu-

lated the heat transport in several regimes,19

but in no regime

was the result the same as the collisional result of Rechester

and Rosenbluth.17

Later Krommes et al. showed that the col-

lisional regime consists of three subregimes.20

With a de-

crease of L, they are

a Rechester-Rosenbluth regime:

rRR = DM/Lk, 10

being valid for k, where =L02/ , k=Lk

2/ and

= 1 /k2 . Equation 10 differs from Eq. 9 by a factor

Lc/Lk.

b Kadomsev-Pogutse regime:

rKP = DM

1/2k, 11

being valid for k.

c Fluid regime:

rF= DM/L0, 12

being valid for k.

The validities of various regimes depend on electron col-

lisionality and the level of the stochastic magnetic field

fluctuations.20,25,26

The original theory of Krommes et al. is

even more complicated.20

Since the fluid equation, Eq. 4, is utilized in our model,it is of interest to have a comparison between our numerical

results and the analytical ones in the collisional regime, Eqs.

912. As shown in the Appendix , when each individualisland width, wk, is smaller than its corresponding critical

island width, wc,k, the enhanced radial heat diffusivity is

given by Eq. A14 rather than Eq. 12, the fluid regimeresult. In this case only these magnetic field perturbations

with their rational surfaces being sufficiently close to the

observation point r have a significant contribution to r, and

the heat diffusion is essentially determined by the additive

effects of these individual islands satisfying r rs,k/wc,k2. The valid regime of Eq. A14, w

k/w

c,k,1, is also

much wider than that of Eq. 12. It is seen from Fig. 16 that,r is proportional to for low values of/ and is deter-

mined by the additive effects of these individual islands up to

w3/2/wc,3/21.7, as predicted by Eq. A14.In the transition region around w3/2 wc,3/2 , 1

slowly increases with / shown by Fig. 16. It is clear that

this is due to wk wc,k so that the single island effect is stillimportant. The Kadomsev-Pogutse regime given by Eq. 11is not found from numerical results. In fact, only for a single

island with wwc one finds r,max 1/2 see Fig. 6.

For sufficiently large / , 1 given by Fig. 16 approxi-mately scales with /, suggesting a behavior predicted by

Eq. 10.For a detail comparison with Eqs. 912, in Fig. 17 ar/L0 bk,r/B0t

2 versus log/ is shown for0 = 110

4 , 6104 , 9104 , 2.1103, and 3.0

103, with other parameters being the same as those for

Fig. 16. Here 1 = rdr/rb ra is the radial averaged rfrom ra =0.58a to rb =0.59a where the magnetic field is sto-

chastic see Fig. 11. It is seen from Fig. 17 that, for low/, is the same for different 0, as predicted by Eq.

A14 in the limit zk=0. In this limit =0.048 is obtainedfrom Eq. A14 in agreement with the numerical results. Forhigh /, the value of oscillates for 0210

3 but

approaches a nearly steady value for larger 0. This differs

FIG. 16. log 1 at r=0.587a versus log/ for 0 = 9104 solid.

For low or high /, 1/. There is a transition region around/=310

7 w3/2 =1.7wc,3/2. The dotted curve shows log3/2 +4/3 2,where 3/2 4/3 is the for a single 3/2 4/3 island. The two curves are thesame for /310

7 as predicted by Eq. A14.

FIG. 17. versus log/ for 0 = 1104 , 6104 , 9104 , 2.1

103, and 3.0103. is the same with different 0 for low / as

predicted by Eq. A14. For high / , oscillates for 02103 but

approaches a nearly steady value for larger 0.

062310-8 Q. Yu Phys. Plasmas 13, 062310 2006



9/12

from the prediction of Eq. 10 that 1 /LkDM1/3 0

2/3.

Using Eq. 10, one finds =9.28103 for 0 = 3103.

By increasing the magnetic shear by 3 times to Lq

=0.1a, in Fig. 18 versus log/ is shown for 0 =3.0103 , 4.5103, and 6.0103. In this case r3/2 =0.595a,r4/3 =0.583a and the other parameters are the same as that for

Fig. 17. It is seen that approaches a steady value for low

/ and a nearly steady value for high / and 0,

similar to those of Fig. 17. The dotted curve in Fig. 18 is the

result with a smaller magnetic shear, Lq =0.3a, and 0 =3.0

103, shown here for comparison. becomes smaller for a

larger magnetic shear, differing from the prediction of Eq.

10 that 1 /LkLs2/3. The dashed curve in Fig. 18 shows

the result of Eq. 9 for 0 =6.0103 and Lq =0.1a, which is

different from the numerical results.

Numerical calculations have also been carried out by

using magnetic perturbations with four Fourier components,m/n =30/19, 30/20, 30/21, and 30/22, and five components,

m/n =3/2, 4/3, 7/5, 10/7, and 11/8. In these cases the local

field is more stochastic due to shorter distance between ra-

tional surfaces, and similar results to Figs. 17 and 18 are

found. An example of four island case is shown in Fig. 19. In

this case the rational surfaces are at r30/19 =0.620a , r30/20=0.604a , r30/21 =0.595a, and r30/22 =0.576a, and the radial

average is taken from r=0.59a to 0.61a, where the magnetic

field is stochastic. is shown for 0 =9.0104 , 1.2

103 , 1.8103, and 2.1103. It is seen that ap-

proaches a steady value for low / and a nearly steady

value for high / and 0, similar to those of Fig. 17. The

dotted curve in Fig. 19 is the result with only the 3/2 and 4/3

island and 0 =2.1103, shown here for comparison. The

for the four island case is smaller due to higher mode num-

bers and therefore smaller critical island widths.

It is seen from Figs. 1619 that, the quasilinear resultpresented in the Appendix is confirmed by numerical results,

while the Kadomsev-Pogutse regime is not seen. As for the

Rechester-Rosenbluth regime given by Eq. 10, the scalingr

RR is found for a sufficiently large / from numeri-cal results, but the scaling with the magnetic shear and the

perturbation amplitude is different.

IV. DISCUSSION AND SUMMARY

It is clear from the above results that, w/wc is a key

parameter in determining the heat transport across a single

island. The heat diffusion consists of three regimes: a thequasilinear regime w/wc1, b the transitional regime w wc, and c the large island regime wwc. The energyconfinement degradation significantly increases with w/wcfrom w/wc =1 to 7.2. For w/wc7.2, the degradation is de-

termined by w , Te0r= rs and rs as found in the infinite/ limit.

15As for the radial gradient of the fundamental

harmonic, it takes its maximal value in the island region in

the middle range of w/wc, and the large island limit of

NTM discussed in Ref. 16 is for w/wc7.2.

For the local stochastic magnetic field due to the overlap

of two neighboring islands, the parameter w/wc is also found

to be important in characterizing the transport. Differing

from Ref. 20, we find that the heat diffusion consists of threeregimes:

a The quasilinear regime w/wc1 as shown by Eq.A14. In this regime the transport is determined by the ad-ditive effect of individual islands.

b The transitional regime w wc, where r slowly in-creases with .

c The regime wwc, where r approximately scaleswith .

Our numerical results together with the analysis in the

Appendix indicate that, the fluid regime should be replaced

by the quasilinear results, Eq. A14. The Kadomsev-Pogutseregime is not found from the numerical results as expected

from Eq. A14. It is seen from our results that, the effect ofw/wc not considered in previous analytical theories is impor-

tant in determining the transport for w/wc1 and leads to

the difference between our numerical results and Eqs. 9,11, and 12. Recent studies have shown that, the magneticfield shear plays an important in the spacial diffusion of the

field lines, and the excursions of field lines significantly dif-

fer from Brownian motions.24

It should be mentioned that, there is a difference be-

tween our numerical model and that of previously analytical

theories.1720

Our model is a local stochastic field due to the

overlap of two or several islands, while in Ref. 20 a infinite

stochastic field is assumed. It is not clear whether such a

FIG. 18. versus log/ for 0 =3.0103, 4.5103, and 6.0

103, with Lq =0.1a , r3/2 =0.595a, and r4/3 =0.583a. Other parameters are

the same as those for Fig. 17. The dotted curve is the result with Lq =0.3a

and 0 =3.0103. becomes smaller for a larger magnetic shear. The

dashed curve is obtained from Eq. 9 for 0 =6.0103.

FIG. 19. versus log/ with 0 =9.0104 , 1.2103 , 1.8103,

and 2.1103 and m/n =30/19, 30/20, 30/21, and 30/22. The dotted

curve is the result for only the 3/2 and 4/3 island and 0 =2.1103. is

smaller for the 4 island case due to the smaller wc.




10/12

difference could lead to the difference in the scaling of rwith the magnetic shear and the perturbation amplitude, al-

though r is obtained from the numerical results for asufficiently large perturbation amplitude and /. Future

calculations using a nonlocal stochastic field and including

more Fourier components of magnetic perturbations will be

helpful for a further comparison with Eq. 10.Equation A14 has an important implication on the heat

diffusion across a stochastic field where / is not highenough. For the tokamak edge parameters Te =100 eV, ne= 1019 m3 , Lq = q/q= a , R/a = 3 , n =2, and an anomalous

perpendicular heat diffusivity, =0.5 m2 /s, one finds

/=1.7108 and wc =0.03a by using the classical paral-

lel electron heat conductivity =3.16vTee. This means that

for smaller islands w0.03a the field ergodicity plays norole, and the heat diffusion is determined by the additive

effect of individual islands. Only for sufficiently large islands

or high Te, the ergodicity dominates the radial transport.

The validity of Eq. 4 and the constant assumption inour calculations should be discussed. It was shown by the

analytical theory that, the classical heat conductivity c is

valid for ke1. While for ke1 , vTe/k, where k=B0 k/B0 and k is the wave vector of the island.

28In the

lowest order k = n r rs /LqR, where rs is the minor radiusof the rational surface. With the parameters mentioned

above, ke1 leads to r rs 0.24a. For an island with itswidth w0.24a, across the island region is still by c.

Since Te is essentially a constant in the island region, the

assumption of a constant is reasonable. In the outer region

away from the island one has ke1 and vTe/k. In thisregion, however, the use of vTe/k or c will lead to thesame result: the temperature along the field lines becomes a

constant due to the fast parallel transport, since both forms of

are large enough to lead to /

1 see Fig. 4 andAppendix. Therefore, for the above case, being relevant tothe tokamak edge plasmas such as those in the dynamic er-

godic divertor and the edge stochastic field,31,32

the use ofcleads to the correct results both in the island and in the outer

region.

For a higher electron temperature and density, Te=1 keV, ne = 510

19 m3, and the other parameters being

the same as mentioned above, one finds c/=1.11010,

and ke1 leads to r rs / a0.012. Since the regionke1 is very narrow for a high Te case, a modified form of

, =c1 + 3.16ek21/2 was used before,30 which re-

duces to c in the limit ke1 and to vTe/k in the opposite

limit. However, as mentioned in the Introduction, =vTe/kis only a crude approximation for the ke1 limit.

29Future

calculations with a more exact model for the parallel heat

flux is necessary for high ratio of/.

In our work only the electron heat transport is studied.

For the ion heat transport across magnetic islands, the results

are expected to be similar. Since the parallel ion heat trans-

port are mi/me1/2 times slower than the electrons, the cor-

responding wc is mi/me1/8 times larger, leading to a factor

about 3 for a deuterium plasma. Therefore, a larger island

width is required to affect the ion temperature profile.

The heat transport is studied here with given perturbed

magnetic fields. Spontaneous growing islands are often ob-

served in tokamak experiments.3,12,33,34

Theoretically, the mi-

crotearing mode could lead to small islands.10

It has been

shown recently that the drift-tearing mode with high mode

numbers can be driven unstable by the electron temperature

gradient due to the perpendicular heat transport.11

Further

investigation on the nonlinear mode saturation is necessary

for calculating their effect on the transport.

In summary, it is found in the present paper that:

1 The heat transport across a single island is deter-mined the parameter w/wc. The normalized enhanced radial

heat diffusivity due to the island, r/, has a radial width

being about the island width and a maximal value at the

rational surface being proportional to w/wc2 for w/wc3.

While for w/wc1 , r/ w/wc4. Between these two

limits there is a transition region around w/wc =1.8. The en-

ergy confinement degradation significantly increases with

w/wc from w/wc =1 to 7.2.

2 The heat transport across a local stochastic magneticfield due to the overlap of two or several neighboring islands

is also characterized by w/wc. For wwc, the heat transport

is determined by the additive effects of the individual islands

as predicted by Eq. A14. Around w =1.7wc there is a tran-sition region where r slowly increases with , and the ad-

ditive effects of the individual islands is still important. For

sufficiently large w/wc , w/wc3 , r approximately scales

with .

3 The fluid regime should be replaced by the quasilin-ear results given by Eq. A14, and the Kadomsev-Pogutseregime is not found from the numerical results. As for the

Rechester-Rosenbluth regime, the scaling r is obtainedfrom the numerical results for wwc, but the scaling of rwith the magnetic shear and the perturbation amplitude is

different for a local stochastic field.

APPENDIX: QUASILINEAR ANALYSIS

In this appendix r is analyzed in the limit that

/1 but w/wc1. Assuming ne , and to be con-

stant, in steady state Eq. 4 becomes

Te + Te + Pr/ne = 0 . A1

When there is only one Fourier component of magnetic

perturbations, B1 =1et with

1r,, = 1rcos, A2

Te can be expressed in terms of Fourier series,

Te = T0/0r,t + Tkr,texpik + c.c./2, A3where = m+ n is the helical angle, and the summation is

over k from k=1 to infinity.

If the island width w is much smaller than the critical

island width wc, one can define a small expansion parameter

= w/wc. Taking the operator dcos / with the integra-tion from =0 to 2, it is found from Eqs. A1A3 that inthe lowest order in

k2

T1 + ikb1rT0/B0t + T1 = 0 , A4

where k =k1 B0/B0, and k1 is the wave vectors of B1. InEq. A4 the terms containing Tk with k2 are neglected

062310-10 Q. Yu Phys. Plasmas 13, 062310 2006



11/12

since they are smaller by at least an order 2.

For /1 and away from the rational surface, Eq.

A4 is dominated by the first two terms, and the outer regionsolution T1 is simply determined by

T1 = iT0/0 b1r/kB0t. A5

In the inner region near the rational surface rs with r rs rs , T1 has a large radial derivative since T1 1 /r

rs as r approaches rs, as seen from Eq. A5, so thatT1mT1/r and Eq. A4 is simplified to

/T1 k2T1 = ikTe0 b1r/B0t. A6

The balancing between the two terms on the right-hand

side of Eq. A6 leads to the critical width wc. Equation A6can be written into the form

d2f

dz2 z2f = z, A7

where z = 23/2r rs1/wc , T1 = 25/2w2/wcTsfz, and Ts

= Te0 rs. The solution of Eq. A7 is

fz = z

2

0

1

d1 21/4 exp z2/2. A8

For z1 , fz =1/z, and T1 matches to the outer regionsolution given by Eq. A5. While for z1 , fz 1.20z,and

T1 = 0.3w/wc2Tsr rs, A9

is in agreement with Ref. 16. It can be shown from Eqs.

A1A3 that Tk with k2 are 2k1 times smaller than

T1.

Similar to the procedure for obtaining Eq. A6, by tak-

ing the operator d/with the integration from =0 to 2and keeping the terms in the lowest order, the equation forT0/0 is found from Eqs. A1A3 to be

0.5b1rikT1 + b1rT0/B0t + T0/0 T0 = 0 . A10

The neglected terms in Eq. A10 are at least 2 timessmaller. Substituting T1 into Eq. A10, one finds T= T0/0 Te0 to be

Tz = w4

8wc4 Tsgz, A11

where gz = 1 +z fz. The profiles of fz and gz areshown in Fig. 20. gz approaches zero for z2.

The parameter r= 1 =Te0 /T0/0 1 measuresthe enhanced radial heat diffusivity due to the parallel trans-

port across the island. From Eq. A11 is found that

rr = w4

8wc4 gz =

b1r2 rs

2B0t2 gz. A12

Equation A12 gives the enhanced radial transport dueto a single island in the limit w/wc1, which agrees with

the numerical result as shown in Fig. 7.

When there are magnetic perturbations of different he-

licities, and each individual island width wk calculated from

the conventional single island formula is smaller than its

corresponding critical island width wc,k, then in the lowest

order Tk is found to be determined by an equation like Eq.

A6, and one obtains Tk= 25/2wk

2/wc,kTsfzk, where zk= 23/2r rs,k/wc,k and rs,k is the minor radius of the rationalsurface of the kth component. A similar analysis to Eqs.

A7A12 leads to the quasilinear result

Tz = wk4

8wc,k4

T0rgkzk A13

and

rr = wk

4

8wc,k4

gkzk = br,k

2

2B0t2

gkzk, A14

where the subscript k corresponds to the kth component of

magnetic perturbations, the summation is over k for includ-

ing contribution from all components. Equation A14 agreeswith the numerical results as shown in Figs. 1619 for wkwc,k.

EquationA14

reduces to Eq.

12

in the limit z

k= 0

except for a factor 2mk/q nk. The function gkzk in Eq.A14, however, indicates the role of wc,k. Since gkzk ap-proaches zero as zk2, only these magnetic field perturba-

tions with their rational surfaces being sufficiently close to

the observation location r have a significant contribution to

r, and r is approximately determined by the additive ef-

fects of these individual islands satisfying zk2. This is dif-

ferent from Eqs. 912 in which the DM includes the con-tribution from all resonant magnetic perturbations.

The valid regime of Eq. A14 is also very different fromthat of Eq. 12. To be self-consistent Eq. A14 requireskgkzk2, where k= /br,k/B0t

2. While Eq. 12

is valid for k, leading to kA1 andbr,k/B0t2A2, where A1 = kL0

2 br,k/B0t2 and A2

= RqLq/mL022. For typical tokamak parameters R/a

= 3 , q = 2 , m = 5 , Lq = a , rs =0.8a, and br,k/B0t= 104, one

finds A1 =5.5106 and A2 =1.810

4. Equation A14 isvalid for a much higher / than Eq. 12 since A1 106 is very small. The condition br,k/B0t

2A2 is usu-

ally satisfied for the magnetic perturbations in tokamak

plasmas.28

The Kadomsev-Pogutse regime, Eq. 11, is valid for k, corresponding to A1kA3, where A3= br,k/B0tRq

2Lq2m/r32/3. For the above parameters

one finds A3 = 0.013. Therefore, the valid regime of Eq.

FIG. 20. fz and gz versus z. gz approaches zero for z2.




12/12

A14 covers the Kadomsev-Pogutse regime, indicating thatEqs. 11 and 12 are problematic. The Rechester-Rosenbluth regime, Eq. 10, is valid for k in additionto k, leading to kA3. It is seen that the valid re-gime of Eq. A14 could well extend into the Rechester-Rosenbluth regime, especially for the case with smaller per-

turbation amplitudes in a large magnetic shear region.

1

H. P. Furth, J. Killen, and M. N. Rosenbluth, Phys. Fluids 6, 459 1963.2J. Wesson, Tokamaks Clarendon, Oxford, 1987.3

J. A. Wesson, R. D. Gill, M. Hugon, F. C. Schller et al., Nucl. Fusion 29,

641 1989.4

Z. Chang, J. D. Callen, E. D. Fredrickson et al., Phys. Rev. Lett. 74, 4663

1995.5

H. Zohm, G. Gantenbein, A. Gude, S. Gnter et al., Nucl. Fusion 41, 197

2001.6

S. Gnter, A. Gude, M. Maraschek, S. Sesnic, H. Zohm et al., Phys. Rev.

Lett. 87, 275001 2001.7

S. Gnter, A. Gude, M. Maraschek, Q. Yu, and the ASDEX Upgrade

Team, Plasma Phys. Controlled Fusion 41, 767 1999.8

R. J. La Haye, L. L. Lao, E. J. Strait, and T. S. Taylor, Nucl. Fusion 37,

397 1997.9

Q. Yu, S. Gnter, and K. Lackner, Phys. Plasmas 11, 140 2004.10

N. T. Gladd, J. F. Drake, C. L. Zhang, and C. S. Liu, Phys. Fluids 23, 1182

1980.11Q. Yu, S. Gnter, and B. D. Scott, Phys. Plasmas 10, 797 2003.12

Q. Yu, S. Gnter, K. Lackner et al., Nucl. Fusion 40, 2031 2000.13

E. Strumberger, J. Nucl. Mater. 266-269, 1207 1999.14

Ph. Ghendrih, A. Grosman, and H. Capes, Plasma Phys. Controlled Fusion

38, 1653 1996.

15Z. Chang and J. D. Callen, Nucl. Fusion 30, 219 1990.

16R. Fitzpatrick, Phys. Plasmas 2, 825 1995.

17A. B. Rechester and M. N. Rosenbluth, Phys. Rev. Lett. 40, 38 1978.

18T. H. Stix, Nucl. Fusion 18, 353 1978.

19B. B. Kadomtsev and O. P. Pogutse, Plasma Physics and Controlled

Fusion Research 1978, Proceedings of the 7th International Conference,

InnsbruckInternational Atomic Energy Agency, Vienna, 1979, Vol. 1, p.649.

20J. A. Krommers, C. Oberman, and R. G. Kleva, J. Plasma Phys. 30, 11

1983.21A. B. Rechester, M. N. Rosenbluth, and R. B. White, Phys. Rev. Lett. 42,

1247 1979.22

A. H. Boozer and R. B. White, Phys. Rev. Lett. 49, 786 1979.23

E. Vanden Eijnden and R. Balescu, Phys. Plasmas 4, 270 1997.24

M. de Rover, A. M. R. Schilham, A. Montvai, and N. J. Lopes Cardozo,

Phys. Plasmas 6, 2443 1999.25

R. J. Bickerton, Plasma Phys. Controlled Fusion 39, 339 1997.26

P. C. Liewer, Nucl. Fusion 25, 543 1985.27

S. Gnter, Q. Yu, J. Krger, and K. Lackner, J. Comput. Phys. 209, 35

2005.28

Z. Chang and J. D. Callen, Phys. Fluids B 4, 1167 1992.29

G. W. Hammett, F. W. Perkins et al., Phys. Rev. Lett. 64, 3019 1990.30

Q. Yu, S. Gnter, G. Giruzzi, K. Lackner, and M. Zabiego, Phys. Plasmas

7, 312 2000.31

R. C. Wolf, W. Biel, M. F. M. de Bock et al., Nucl. Fusion 45, 1700

2005.32

T. E. Evans, R. A. Moyer, P. R. Thomas et al., Phys. Rev. Lett. 92,

235003 2004.33

A. Gude, S. Gnter, S. Sesnic, and the ASDEX Upgrade Team, Nucl.

Fusion 39, 127 1999.34

E. D. Fredrickson, Phys. Plasmas 9, 548 2002.

062310-12 Q. Yu Phys. Plasmas 13, 062310 2006

numerical modeling of diffusive heat transport

Documents