1 Numerical Modeling of Diffusive Heat Transport Across Magnetic Islands and Local Stochastic Field

The heat diffusion across magnetic islands is studied numerically and compared with analytical results. For a single island, the enhanced radial heat diffusivity, χr, due to the parallel transport along the 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 w2(χ‖χ⊥)1∕2 for sufficiently high values of χ‖∕χ⊥, where χ‖∕χ⊥ is the ratio between the parallel and the perpendicular heat diffusivity. For low ratios of χ‖∕χ⊥, however, the maximum value of χr is proportional to w4χ‖. In a locally stochastic magnetic field, χr is again proportional to w4χ‖ for low χ‖∕χ⊥, which is in agreement with the analytical results. With increasing χ‖∕χ⊥,χr is dominated first by the additive effect of individual islands and then by the field ergodicity.


I. Introduction
Magnetic islands generally exist in laboratory and space plasmas, caused either by resonant helical magnetic field perturbations (finite error fields) in fusion devices or by the tearing mode type instabilities, driven by an unfavorable plasma current density gradient [1][2][3], the perturbed bootstrap current [4][5][6][7][8][9], or the electron temperature gradient [10,11].The existence 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 neoclassical tearing mode (NTM) have been found to lead to a degradation of plasma confinement or even to disruptions [2][3][4][5][6][7][8].For high β 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 interaction 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, χ || /χ ⊥ , to be infinite.The resulting degradation of plasma energy confinement due to the fast parallel transport along the field lines is found to be determined by the island width, the minor radius of the rational surface and the local equilibrium 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 χ || /χ ⊥ 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 [17--26].
It is well understood that both the parallel and the perpendicular transport are important in determining the heat diffusion across a single island or a stochastic field, no matter if the perpendicular transport is caused by plasma collisions or turbulence [16][17][18][19][20].Despite the existing theories on the heat transport across stochastic fields, for a finite value of χ || /χ ⊥ 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 magnitude from the center to the edge in a tokamak, the parameter χ || /χ ⊥ changes even more.It is of great interest to study how the parameter χ || /χ ⊥ 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 author's 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 however 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 perpendicular heat transport [11].
The numerical calculation of the heat transport across magnetic islands is usually quite challenging for a high ratio of χ || /χ ⊥ , which could be in the order of 10 10 or even larger in the central region of large tokamak plasmas [16].This large χ || /χ ⊥ 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 χ || /χ ⊥ .It is of the classical form only in a narrow region around the rational surface or the island's x-point [28].Away from this region the heat flux is carried by free-streaming electrons and is non-local [29].For such a case the "flux limit" form for χ || was used before [e.g., 16,30], but this is only a crude approximation [29].
In the present paper the complicated physics of the parallel heat flux mentioned above will not be addressed.Our numerical results of the diffusive heat transport across magnetic islands are obtained by using a constant χ || along the minor radius.This assumption is the same as that of the existing theories, allowing an easy comparison with them [16][17][18][19][20].For a single island, the enhanced radial heat diffusivity χ 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 proportional to w 2 (χ || χ ⊥ ) 1/2 for a sufficiently high value of χ || /χ ⊥ .For a low ratio of χ || /χ ⊥ , χ r is proportional to w 4 χ || .When two neighboring islands overlap, the local magnetic field becomes stochastic.In this case, for low ratios of χ || /χ ⊥ , 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 for sufficiently high ratios of χ || /χ ⊥ , the radial heat transport is dominated by the ergodicity of the magnetic field.
In the following Section II our model is described.The numerical results and the comparison with analytical results are presented in Section III, and the discussion and summary are given in Section V.

II Model
In our model the large aspect-ratio tokamak approximation is utilized.The magnetic field B is expressed in the form B=B 0t e t +B 0p (r)e p +B 1 , where B 0t and B 0p are the equilibrium toroidal and poloidal magnetic field, respectively, and B 1 =∇ψ×e t is the perturbed helical field.B 0t is assumed to be a constant.ψ is expressed in terms of a Fourier decomposition, where m i and n i are the poloidal and toroidal mode numbers of the ith component of B 1 , and θ and φ are the poloidal and toroidal angle, respectively.The toroidal magnetic field perturbation is usually much smaller than the poloidal one and is therefore neglected.
The equilibrium safety factor q(r)=rB 0t /(RB 0p ) is chosen to be the form q(r) = q 0 e r/L q (2) to have a constant magnetic shear along the minor radius, where q 0 =0.2 and L q =0.3a are chosen to ensure q=3/2 and 4/3 surfaces to be inside the plasma (0≤r≤a, a: plasma minor radius), except when mentioned elsewhere.
. ψ i (r) in Eq. ( 1) is taken as to have a smooth change along the minor radius, being typical for the classical or neoclassical tearing modes [1,9].
The following electron energy transport equation is solved, where T e is the electron temperature, n e is the electron density and P is the heat source.
Here n e , χ || and χ ⊥ are assumed to be constant along the minor radius for simplicity, 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 P(r) in Eq. ( 4) is taken to be of the form where c 1 =2 and c 2 =16 are taken except when mentioned elsewhere.P 0 is chosen to ensure T e0 (r=0)=1, where T e0 is the equilibrium electron temperature obtained from Eq. ( 4) by taking B 1 =0 (without magnetic field perturbations).
The boundary conditions used here are T e (r=a)=T e0 (r=a) and T e ′ (r=0)=0, where the prime denotes d/dr.The normalization scheme is as followings: the length is normalized to a, the magnetic field to B 0t , and T e to T e0 (r=0).

III. Modelling results
Equation ( 4) is solved numerically using the transport subroutines of the initial value code TM1 [9].TM1 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 the following Part A the heat transport across a single island is studied, and in Part 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 Part 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, where the m/n=0/0 component of T e , T 0/0 , and the temperature profile along the line passing through the island's o-point and x-point in the steady state are shown.
The original equilibrium temperature T e0 is shown by the dotted curve.The q=3/2 surface is at r 3/2 =0.604a, the island width w 3/2 =0.0452a (ψ 3/2,0 =9×10 -4 aB 0t ), and χ || /χ ⊥ =10 10 is taken.It is seen that the local T e profile becomes flattened inside the island when viewing along the line passing through the island's o-point due to the fast parallel transport at a large χ || /χ ⊥ .
Along the line passing through the island's x-point, T e has a finite radial gradient as expected.
The radial gradient of T 0/0 is same as that of T e0 for r<r -, and T 0/0 is the same as T e0 for r>r + 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 harmonics (m/n=6/4, 9/6 and 12/8 components) of T e 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 χ || /χ ⊥ limit result [16].For such a high value of χ || /χ ⊥ the temperature is a constant along the magnetic field line except in a thin layer around the island's separatrix where the parallel heat flux does not vanish.
It is seen from Fig. 3 that for describing the "large island " limit of NTM [16], χ || /χ ⊥ =10 10 has to be taken in the numerical calculations [9].
Considering the conservation of the perpendicular heat flux, q ⊥ =-χ ⊥ T 0 ′ =-χ e T 0/0 ′ , where χ e is the effective radial heat diffusivity in the presence of the island, one can define an normalized effective radial heat conductivity The enhanced radial heat diffusivity due to the parallel transport across the island is then given by χ r =χ e -χ ⊥ =χ ⊥ (χ-1).Corresponding to Fig. 3, in Fig. 5 radial profiles of χ are shown for χ || /χ ⊥ =10 10 , 10 8 and 10 6 .As expected from Fig. 3, χ≈1 for χ || /χ ⊥ =10 6 , and a larger χ || /χ ⊥ leads to a larger χ.Along the minor radius χ has a width being about the same as the island width and a maximal value at the rational surface.
In Fig. 7 numerical results of log(χ-1) max versus log(w/w c ) 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/w c ) 2 only the island width w changes, χ || /χ ⊥ =10 10 and other parameters are the same as those of Fig. 1.It is seen that (χ-1) max is proportional to w 4   for w/w c <1 but to w 2 for w/w c >3.The transition region between these two slopes is around w/w c ≈1.8.When w/w c <1, the numerical result is the same as the analytical one given by Eq.
(A12).When w/w c >>1, the parallel heat transport is along a thin layer around the island's separatrix.The radial layer width is w c 2 /w at the helical angle being the same as island's o-point [16], where T e ′ ≈T e0 ′ .Due to the island's geometry, the layer width expands to w c along the minor radius and to ∆ζ~(4w c /w) along the helical angle around island's x-point.Therefore, the T e ′ at the x-point T ex ′ ~(w c /w)T e0 ′ .Since T e ′ is zero well inside the island for w/w c >>1, T 0/0 ′ , the averaged T e ′ along the helical angle, is proportional to T ex ′ ∆ζ/π~(w c /w) 2 T e0 ′ .This explains why (χ-1) max is proportional to (χ || /χ ⊥ ) 1/2 and w 2 in the limit (w/w c ) 2 >>1, as shown in Figs. 6 and 7.
The degradation of the plasma energy confinement due to an island can be described by the parameter ∆β=(β 0 -β), where β=2πRn e ∫2πrT 0/0 dr is the total plasma thermal energy in the presence of the island, and β 0 =2πRn e ∫2πrT e0 dr is that without the island.In Fig. 8   10 .Using the analytical formula for the χ || /χ ⊥ =∞ limit [15], ∆β/β 0 =0.041 is obtained, in agreement with the numerical result.The ∆β/β 0 due to the 4/3 island is also shown in Fig. 8 by the dotted curve.The q=4/3 surface is at r 4/3 =0.569a with r 4/3 <r 3/2 , and all the other parameters are the same as mentioned for Fig. 6.For small χ || /χ ⊥ , the ∆β/β 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 χ || /χ ⊥ due to its smaller w c .For large χ || /χ ⊥ , the ∆β/β 0 due to the 3/2 island is larger because r 3/2 >r 4/3 and w>>w c [15].

B. Heat transport across two islands and stochastic field
In this part the heat transport across two islands, the m/n=3/2 and 4/3 islands, is studied.
The corresponding rational surfaces are at r 3/2 =0.604a and r 4/3 =0.569a, respectively.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 addition to m/n=3/2 and 4/3 islands, there is also a m/n=7/5 island in the temperature contour in agreement with Fig. 9.In this case P(r)=P 0 (r/a) 8 [1-(r/a) 2 ] 8 is taken to deposit more power density 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 χ || /χ ⊥ >10 10 , indicating that a small secondary island is important in the transport only for a sufficiently large χ || /χ ⊥ .
In Fig. 14 the radial profiles of T e0 and the T 0/0 for ψ 0 =2×10 -4 , 5×10 -4 and 9×10 -4 are shown with χ || /χ ⊥ =10 10 .For ψ 0 =2×10 -4 , T 0/0 ′ is approximately the same as T e0 ′ between the two island, indicating that the local confinement 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 T 0/0 ′ decreases.When plotting the magnetic surface with ψ 0 =5×10 -4 , it is found to be stochastic in the region r=0.58a-0.59a similar to Fig. 11.However, T 0/0 ′ 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 ∆β/β 0 is shown as a function of log(χ || /χ ⊥ ) by the solid curve for ψ 0 =9×10 -4 .
The corresponding magnetic surface is shown in Fig. 11.The dotted curved shows the ∆β 1 /β 0 =(∆β 3/2 +∆β 4/3 )/β 0 , where ∆β 3/2 (∆β 4/3 ) is the ∆β due to a single m/n=3/2 (4/3) island, as shown in Fig. 8.The difference between ∆β and ∆β 1 therefore measures the additional degradation in the energy confinement due to the ergodicity of the magnetic field.It is seen that 7 , corresponding to w 3/2 <1.7w 3/2,c .This means that for a sufficiently low χ || /χ ⊥ , the ergodicity does not affect the energy confinement, and the heat transport recognizes the individual island structure rather than the ergodicity.For χ || /χ ⊥ >3×10 7 , ∆β becomes larger than ∆β 1 , showing the additional degradation of the confinement due to ergodicity.However, such an additional degradation is not significant with χ || /χ ⊥ up to 10 11 , which is of the same order of magnitude as that for the central region plasma of large tokamaks [16].
The small difference between ∆β and ∆β 1 is due to the fact that, ∆β is mainly determined by the ∆T e for r<r -of the inner island, the 4/3 island in the present case, and ∆T e changes little for sufficiently large χ || /χ ⊥ similar to that of a single island as seen from Fig. 3.
With an even larger magnetic field perturbation amplitude, ψ 0 =3.6×10 -3 , 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 a shorter 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/w c also characterizes the heat diffusion across a local stochastic field.For w<<w c , the heat transport is determined by the additive effects of the individual islands, and χ r ∝χ || is found as predicted by the analytical result Eq. (A14).Around w=1.7w c there is a transition region where the additive effect of individual islands is still important, and χ r only slowly increases with χ || .For sufficiently large w/w c , w/w c >3, χ r nearly scales with χ || , 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 diffusion of the magnetic field lines [17][18][19][20][21][22][23][24].In the collisionless regime where the electron mean free path λ e is longer than the Kolmogorov length [17,20], the enhanced radial heat conductivity is shown by Rechester and Rosenbluth to be [17] where v Te is the electron thermal velocity, L 0 ≈πR, b k,r is the radial magnetic field perturbation of the kth component, k ⊥ is the perpendicular wave vector of the perturbations, L s =Rq 2 /rq′, and the summation is over k to include contributions from all components.Stix has got the same results as Eq. ( 7) [18].In the collisional regime λ e <L k , where L cδ =L c ln[(r/mL c )(χ || /χ ⊥ ) 1/2 ], L c =πR/ln(π∆/2), ∆=(w 1 +w 2 )/(2|r 1 -r 2 |), w 1 and w 2 are the widths of two neighboring islands, and r 1 and r 2 are the minor radius of the corresponding rational surfaces [17].
The validities of various regimes depend on electron collisionality 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.
(9)- (12).As shown in the Appendix, when each individual island width, w k , is smaller than its corresponding critical island width, w c,k , the enhanced radial heat diffusivity is given by Eq. (A14) rather than Eq. ( 12), the Fluid regime result.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 χ r , and the heat diffusion is essentially determined by the additive effects of these individual islands satisfying (r-r s,k )/w c,k <2.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, χ r is proportional to χ || for low values of χ || /χ ⊥ and is determined by the additive effects of these individual islands up to w 3/2 /w c,3/2 <1.7, as predicted by Eq. (A14).
In the transition region around w 3/2 ~wc,3/2 , (χ-1) slowly increases with χ || /χ ⊥ as shown by Fig. 16.It is clear that this is due to w k ~wc,k so that the single island effect is still important.
An example of four island case is shown in Fig. 19.In this case the rational surfaces are at r 30/19 =0.620a, r 30/20 =0.604a, r 30/21 =0.595a, and r 30/22 =0.576a, and the radial average is taken from r=0.59a to 0.61a where the magnetic field is stochastic.κ is shown for ψ 0 =9.0×10 -4 , 1.2×10 -3 , 1.8×10 -3 and 2.1×10 -3 .It is seen that κ approaches a steady value for low χ || /χ ⊥ and a nearly steady value for high χ || /χ ⊥ and ψ 0 , similar to those of Fig. 17.The dotted curve on Fig. 19 is the result with only the 3/2 and 4/3 island and ψ 0 =2.1×10 -3 , shown here for comparison.The κ for the four island case is smaller due to higher mode numbers and therefore smaller critical island widths.
It is seen from Figs. 16-19 that, the quasi-linear result presented 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 scaling χ r RR ~χ|| is found for a sufficiently large χ || /χ ⊥ from numerical results, but the scaling with the magnetic shear and the perturbation amplitude is different.

IV. Discussion and summary
It is clear from above results that, w/w c is a key parameter in determining the heat transport across a single island.The heat diffusion consists of three regimes: (a) the quasi-linear regime w/w c <1, (b) the transitional regime w~w c and (c) the "large island" regime w>>w c .
The energy confinement degradation significantly increases with w/w c from w/w c =1 to 7.2.
For w/w c >7.2, the degradation is determined by w, T e0 ′ (r=r s ) and r s as found in the infinite χ || /χ ⊥ limit [15].As for the radial gradient of the fundamental harmonic, it takes its maximal value in the island region in the middle range of w/w c , and the "large island" limit of NTM discussed in Ref. [16] is for w/w c >7.2.
For the local stochastic magnetic field due to the overlap of two neighboring islands, the parameter w/w c is also found to be important in characterizing the transport.In different to the Ref. [20], we find that the heat diffusion consists of three regimes: (a) The quasi-linear regime w/w c <1 as shown by Eq. (A14).In this regime the transport is determined by the additive effect of individual islands.
(b) The transitional regime w~w c , where χ r slowly increases with χ || .
Our numerical results together with the analysis in the Appendix indicate that, the Fluid regime should be replaced by the quasi-linear results Eq. (A14).The Kadomsev-Pogutse regime is not found from the numerical results as expected from Eq. (A14).It is seen from our results that, the effect of w/w c not considered in previous analytical theories is important in determining the transport for w/w c ≤1 and leads to the difference between our numerical results and Eqs. ( 9), ( 11) and (12).Recent studies have shown that, the magnetic field shear plays an important in the spacial diffusion of the field lines, and the excursions of field lines significantly differ from Brownian motions [24].
It should be mentioned that, there is a difference between our numerical model and that of previously analytical theories [17][18][19][20].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 difference could lead to the difference in the scaling of χ r with the magnetic shear and the perturbation amplitude, although χ r ~χ|| is obtained from the numerical results for a sufficiently large perturbation amplitude and χ || /χ ⊥ .Future calculations using a non-local stochastic field and including more Fourier components of magnetic perturbations will be helpful for a further comparison with Eq. ( 10).
Eq. ( A14) have an important implication on the heat diffusion across a stochastic field where χ || /χ ⊥ is not high enough.For the tokamak edge parameters T e =100ev, n e =10 19 m -3 , L q =q/q′=a, R/a=3, n=2, and an anomalous perpendicular heat diffusivity, χ ⊥ =0.5m 2 /s, one finds χ || /χ ⊥ =1.7×10 8 and w c =0.03a by using the classical parallel electron heat conductivity χ ||c =3.16v Te λ e .This means that for smaller islands (w<0.03a) the field ergodicity plays no role, and the heat diffusion is determined by the additive effect of individual islands.Only for sufficiently large islands or high T e , the ergodicity dominates the radial transport.
The validity of Eq. ( 4) and the constant χ || assumption in our calculations should be discussed.It was shown by the analytical theory that, the classical heat conductivity χ ||c is valid for k || λ e <1.While for k || λ e >1, χ || ≈v Te /k || , where k || =B 0 •k/|B 0 | and k is the wave vector of the island [28].In the lowest order k || =n|r-r s |/(L q R), where r s is the minor radius of the rational surface.With the parameters mentioned above, k || λ e >1 leads to |r-r s |>0.24a.For an island with its width w<0.24a, across the island region χ || is still by χ ||c .Since T e is essentially a constant in the island region, the assumption of a constant χ || is reasonable.In the outer region away from the island one has k || λ e >1 and χ || ≈v Te /k || .In this region, however, the use of χ || ≈v Te /k || or χ ||c will lead to the same result: the temperature along the field lines becomes a constant due to the fast parallel transport, since both forms of χ || are large enough to lead to χ || /χ ⊥ >>1 (see Fig. 4 and Appendix).Therefore, for the above case, being relevant to the tokamak edge plasmas such as those in the dynamic ergodic divertor and the edge stochastic field [31,32], the use of χ ||c leads to the correct results both in the island and in the outer region.
For a higher electron temperature and density, T e =1kev, n e =5×10 19 m -3 , and the other parameters being the same as mentioned above, one finds χ ||c /χ ⊥ =1.1×10 10 , and k || λ e >1 leads to |r-r s |/a>0.012.Since the region k || λ e <1 is very narrow for a high T e case, a modified form of 1/2 was used before [30], which reduces to χ ||c in the limit k || λ e <<1 and to v Te /k || in the opposite limit.However, as mentioned in the Introduction, χ || =v Te /k || is only a crude approximation for the k || λ e <<1 limit [29].Future calculations with a more exact model for the parallel heat flux is necessary for high ratio of χ || /χ ⊥ .
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 transport are (m i /m e ) 1/2 times slower than the electron's, the corresponding w c is (m i /m e ) 1/8 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 observed in tokamak experiments [3,12,33,34].Theoretically, the micro-tearing 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 determined the parameter w/w c .The normalized enhanced radial heat diffusivity due to the island, χ r /χ ⊥ , has a radial width being about the island width and a maximal value at the rational surface being proportional to (w/w c ) 2 for w/w c >3.While for w/w c <1, χ r /χ ⊥ ∝(w/w c ) 4 .Between these two limits there is a transition region around w/w c =1.8.The energy confinement degradation significantly increases with w/w c from w/w c =1 to 7.2.
(2) The heat transport across a local stochastic magnetic field due to the overlap of two or several neighboring islands is also characterized by w/w c .For w<w c , the heat transport is determined by the additive effects of the individual islands as predicted by Eq. (A14).Around w=1.7w c there is a transition region where χ r slowly increases with χ || , and the additive effects of the individual islands is still important.For sufficiently large w/w c , w/w c >3, χ r approximately scales with χ || .
(3) The Fluid regime should be replaced by the quasi-linear results given by Eq. (A14), and the Kadomsev-Pogutse regime is not found from the numerical results.As for the Rechester-Rosenbluth regime, the scaling χ r ~χ|| is obtained from the numerical results for w>>w c , but the scaling of χ r with the magnetic shear and the perturbation amplitude is different for a local stochastic field.
Eq. (A12) gives the enhanced radial transport due to a single island in the limit w/w c <<1, which agrees with the numerical result as shown in Fig. 7.
When there are magnetic perturbations of different helicities, and each individual island width w k calculated from the conventional single island formula is smaller than its corresponding critical island width w c,k , then in the lowest order T k is found to be determined by an equation like Eq. (A6), and one obtains T k =-2 -5/2 (W k 2 /W c,k )T s ′f(z k ), where z k =2 3/2 (r-r s,k )/W c,k and r s,k is the minor radius of the rational surface of the kth component.A similar analysis to Eqs. (A7)-(A12) leads to the quasi-linear result where the subscript k corresponds to the kth component of magnetic perturbations, the summation is over k for including contribution from all components.Eq. (A14) agrees with the numerical results as shown in Figs.[16][17][18][19] for w k <w c,k .
Eq. (A14) reduces to Eq. ( 12) in the limit z k =0 except for a factor 2δ(m k /q-n k ).The function g k (z k ) in Eq. (A14), however, indicates the role of W c,k .Since g k (z k ) approaches zero as z k >2, only these magnetic field perturbations with their rational surfaces being sufficiently close to the observation location r have a significant contribution to χ r , and χ r is approximately determined by the additive effects of these individual islands satisfying z k <2.This is different from Eqs. ( 9)- (12) in which the D M includes the contribution from all resonant magnetic perturbations.