An investigation is made into the effect of the reduction in anomalous perpendicular electron heat transport inside the separatrix of a magnetic island chain associated with a neoclassical tearing mode in a tokamak plasma, due to the flattening of the electron temperature profile in this region, on the overall stability of the mode. The onset of the neoclassical tearing mode is governed by the ratio of the divergences of the parallel and perpendicular electron heat fluxes in the vicinity of the island chain. By increasing the degree of transport reduction, the onset of the mode, as the divergence ratio is gradually increased, can be made more and more abrupt. Eventually, when the degree of transport reduction passes a certain critical value, the onset of the neoclassical tearing mode becomes discontinuous. In other words, when some critical value of the divergence ratio is reached, there is a sudden bifurcation to a branch of neoclassical tearing mode solutions. Moreover, once this bifurcation has been triggered, the divergence ratio must be reduced by a substantial factor to trigger the inverse bifurcation.

Neoclassical tearing modes are large-scale magnetohydrodynamical instabilities that cause the axisymmetric, toroidally nested, magnetic flux-surfaces of a tokamak plasma to reconnect to form helical magnetic island structures on low mode-number rational magnetic flux surfaces.1 Island formation leads to a degradation of plasma energy confinement.2 Indeed, the confinement degradation associated with neoclassical tearing modes constitutes a major impediment to the development of effective operating scenarios in present-day and future tokamak experiments.3 Neoclassical tearing modes are driven by the flattening of the temperature and density profiles within the magnetic separatrix of the associated island chain, leading to the suppression of the neoclassical bootstrap current in this region, which has a destabilizing effect on the mode.4 The degree of flattening of a given profile (i.e., either the density, electron temperature, or ion temperature profile) within the island separatrix depends on the ratio of the associated perpendicular (to the magnetic field) and parallel transport coefficients.5 

The dominant contribution to the perpendicular transport in tokamak plasmas comes from small-scale drift-wave turbulence, driven by plasma density and temperature gradients.1 The fact that the density and temperature profiles are flattened within the magnetic separatrix of a magnetic island chain implies a substantial reduction in the associated perpendicular transport coefficients in this region. Such a reduction has been observed in gyrokinetic simulations,6–11 as well as in experiments.12–18 A strong reduction in perpendicular transport within the magnetic separatrix calls into question the conventional analytic theory of neoclassical tearing modes in which the perpendicular transport coefficients are assumed to be spatially uniform in the island region.5 

The aim of this paper is to investigate the effect of the reduction in perpendicular transport inside the separatrix of a neoclassical magnetic island chain, due to profile flattening in this region, on the overall stability of the mode. For the sake of simplicity, we shall only consider the influence of the flattening of the electron temperature profile on mode stability. However, the analysis used in this paper could be generalized, in a fairly straightforward manner, to take into account the influence of the flattening of the ion temperature and density profiles.

Consider a large aspect-ratio, low-β, circular cross-section, tokamak plasma equilibrium. Let us adopt a right-handed cylindrical coordinate system (r, θ, z) whose symmetry axis (r = 0) coincides with the magnetic axis of the plasma. The system is assumed to be periodic in the z-direction with period 2πR0, where R0 is the simulated major plasma radius. It is helpful to define the simulated toroidal angle φ=z/R0. The coordinate r serves as a label for the unperturbed (by the tearing mode) magnetic flux-surfaces. Let both the equilibrium toroidal magnetic field, Bz, and the equilibrium toroidal plasma current run in the +z direction.

Suppose that a neoclassical tearing mode generates a helical magnetic island chain, with mθ poloidal periods and nφ toroidal periods, that is embedded in the aforementioned plasma. The island chain is assumed to be radially localized in the vicinity of its associated rational surface, with minor radius rs, which is defined as the unperturbed magnetic flux-surface at which q(rs)=mθ/nφ. Here, q(r) is the safety factor profile (which is assumed to be a monotonically increasing function of r). Let the full radial width of the island chain's magnetic separatrix be W. In the following, it is assumed that ϵsrs/R01 and W/rs1.

It is convenient to employ a frame of reference that co-rotates with the magnetic island chain. All fields are assumed to depend (spatially) only on the radial coordinate r and the helical angle ζ=mθθnφφ. Let kθ=mθ/rs,qs=mθ/nφ, and ss=dlnq/dlnr|rs. The magnetic shear length at the rational surface is defined as Ls=R0qs/ss. Moreover, the unperturbed (by the magnetic island) electron temperature gradient scale-length at the rational surface takes the form LT=1/(dlnT0/dr)rs, where T0(r) is the unperturbed electron temperature profile. In the following, it is assumed that LT>0, as is generally the case in conventional tokamak plasmas.1 

The helical magnetic flux is defined as

(1)

where x=rrs, and the magnetic field perturbation associated with the tearing mode is written as δB=×(δχez). It is easily demonstrated that B·χ=0, where B is the total magnetic field.19 Hence, χ is a magnetic flux-surface label. It is helpful to introduce the normalized helical magnetic flux, ψ=(Ls/Bzw2)χ, where w=W/4. The normalized flux in the vicinity of the rational surface is assumed to take the form19 

(2)

where X=x/w. As is well-known, the contours of ψ map out a symmetric (with respect to X = 0), constant-ψ,20 magnetic island chain whose O-points lie at ζ=π, X = 0, and ψ=1 and whose X-points lie at ζ = 0, 2π, X = 0, and ψ=+1. The chain's magnetic separatrix corresponds to ψ=+1, the region inside the separatrix to 1ψ<1, and the region outside the separatrix to ψ>1. The full radial width of the separatrix (in X) is 4.

Finally, the electron temperature profile in the vicinity of the rational surface is written as

(3)

where Ts=T0(rs), and

(4)

Note that δT(X,ζ) is an odd function of X. In the following, it is assumed that w/LT1.

The steady state electron temperature profile in the vicinity of the island chain is governed by the following well-known electron energy conservation equation5,21

(5)

where

(6)

and

(7)

Here, κ and κ are the perpendicular (to the magnetic field) and parallel electron thermal conductivities, respectively. The first term on the right-hand side of Eq. (5) represents the divergence of the parallel (to the magnetic field) electron heat flux, whereas the second term represents the divergence of the perpendicular electron heat flux. [In fact, because [[δT,ψ],ψ] and 2δT/X2 are both O(1) in our normalization scheme, the ratio of the divergences of the parallel and perpendicular heat fluxes is effectively measured by (W/Wc)4.] The quantity Wc is the critical island width above which the former term dominates the latter, causing the temperature profile to flatten within the island separatrix.5 In writing Eq. (5), we have neglected any localized sources or sinks of heat in the island region. We have also assumed that κ and κ are spatially uniform in the vicinity of the rational surface. The latter assumption is relaxed in Sec. III

Consider the so-called narrow-island limit in which WWc.5 Let

(8)

Equation (5) transforms to give

(9)

We can write

(10)

where

(11)

subject to the boundary conditions T1(0,ζ)=0, and T10 as |Y|. Note that the solution (10) automatically satisfies the boundary condition (4). It follows that

(12)

where f(p) is the well-behaved solution of

(13)

that satisfies f(0)=0, and f0 as |p|. Note that f(p)=f(p). Hence, in the narrow-island limit,5 

(14)

Consider the so-called wide-island limit in which WWc.5 We can write

(15)

where T¯ and T̃ are both O(1), and

(16)

Here, is the so-called flux-surface average operator.19 This operator is defined as follows:

(17)

for 1ψ1, and

(18)

for ψ>1, where s=sgn(X),ζ0=cos1(ψ), and

(19)

Note that [A,ψ]0 for all A.

Equations (5) and (15) can be combined to give

(20)

The flux-surface average of the previous equation yields

(21)

which implies that

(22)

The previous equation can be integrated to give

(23)

which satisfies the boundary condition (4). Hence, in the wide-island limit,5 

(24)

The gradient discontinuity in δT(X,ζ) is resolved by a thermal boundary layer on the magnetic separatrix whose radial thickness is of order Wc.

The temporal evolution of the island width is governed by the so-called modified Rutherford equation, which takes the form4,5,19

(25)

where

(26)
(27)

Here, τR=μ0rs2/η(rs) is the resistive evolution timescale at the rational surface, and η(r) is the unperturbed plasma resistivity profile. Moreover, Δ<0 is the standard linear tearing stability index.20 Finally, αb=fs(qs/ϵs)β, where fs=1.46ϵs1/2 is the fraction of trapped electrons, β=μ0nsTs/Bz2, and ns is the unperturbed electron number density at the rational surface. The second term on the right-hand side of Eq. (25) parameterizes the destabilizing influence of the perturbed bootstrap current.4,5 Note that, in this paper, for the sake of simplicity, we have employed the so-called lowest-order asymptotic matching scheme described in Ref. 22. This accounts for the absence of higher-order island saturation terms in Eq. (25).

In conventional tokamak plasmas, the dominant contribution to the perpendicular electron thermal conductivity, κ, comes from small-scale drift-wave turbulence driven by electron temperature gradients.1 The fact that the electron temperature gradient is flattened within the magnetic separatrix of a sufficiently wide magnetic island chain implies a substantial reduction in the perpendicular electron thermal conductivity in this region. There is clear experimental evidence that this is indeed the case.13,14,16–18 In particular, Ref. 16 reports a reduction in κ at the O-point of the magnetic island chain associated with a typical neoclassical tearing mode by 1 to 2 orders of magnitude. Obviously, such a strong reduction in κ within the magnetic separatrix calls into question the conventional analytic model of neoclassical tearing modes, described in Sec. II, in which κ is assumed to be spatially uniform in the vicinity of the rational surface.

As a first attempt to model the reduction in κ due to temperature flattening within the magnetic separatrix of a neoclassical island chain, let us write

(28)

where κ1 and κ0 are the spatial constants, with κ1κ0. Since the mean temperature gradient outside the separatrix of a neoclassical magnetic island chain is similar in magnitude to the equilibrium temperature gradient [see Eq. (42)], it is reasonable to assume that κ0 is equal to the local (to the rational surface) perpendicular electron thermal conductivity in the absence of an island chain.

Let

(29)
(30)

be the critical island widths outside and inside the separatrix, respectively. Likewise, let

(31)
(32)

measure the ratios of the divergences of the parallel and perpendicular electron heat fluxes outside and inside the separatrix, respectively. Finally, let the parameter

(33)

measure the relative reduction of perpendicular electron heat transport within the island separatrix.

Let us adopt the following simple model:

(34)

where 0<δ1. According to this model, the degree of perpendicular transport reduction within the separatrix is controlled by the parameter ξ1, which measures the ratio of the divergences of the parallel and perpendicular electron heat fluxes inside the separatrix (see Secs. II C and II D). If ξ1 is much less than unity, then there is no temperature flattening within the separatrix, which implies that λ = 1 (i.e., there is no reduction in transport). On the other hand, if ξ1 is much greater than unity, then the temperature profile is completely flattened inside the separatrix, and the transport is reduced by some factor δ (say). The previous formula is designed to interpolate smoothly between these two extremes as ξ1 varies.

Equations (33) and (34) can be combined to give

(35)

It follows that δλ1, with ξ0=0 when λ = 1, and ξ0 as λδ. It is easily demonstrated that the function ξ0(λ) has a point of inflection when δ=δcrit=1/(1+e2)=0.1192. This point corresponds to ξ0=4δcrit=0.4768 and λ=2δcrit=0.2384.

Figure 1 shows the perpendicular electron transport reduction parameter, λ, plotted as a function of the ratio of the divergences of the parallel to perpendicular electron heat fluxes outside the island separatrix, ξ0, for various values of the maximum transport reduction parameter, δ. It can be seen that if δ>δcrit, then the ξ0λ curves are such that dξ0/dλ<0 for δλ1. This implies that λ decreases smoothly and continuously as ξ0 increases and vice versa. We shall refer to these solutions as continuous solutions of Eq. (35). On the other hand, if δ<δcrit, then the ξ0λ curves are such that dξ0/dλ>0 for some intermediate range of λ values lying between δ and 1.

FIG. 1.

The perpendicular electron transport reduction parameter, λ, plotted as a function of the ratio of the divergences of the parallel to perpendicular electron heat fluxes outside the island separatrix, ξ0. The solid, short-dashed, long-dashed, dot-short-dashed, and dotted curves correspond to δ= 0.9, 0.5, 0.2, 0.1192, and 0.01, respectively. Here, δ is the maximum transport reduction parameter. The thin dot-long-dashed curve shows the locus of points where dξ0/dλ=0 (dξ0/dλ>0 to the left of the curve and vice versa).

FIG. 1.

The perpendicular electron transport reduction parameter, λ, plotted as a function of the ratio of the divergences of the parallel to perpendicular electron heat fluxes outside the island separatrix, ξ0. The solid, short-dashed, long-dashed, dot-short-dashed, and dotted curves correspond to δ= 0.9, 0.5, 0.2, 0.1192, and 0.01, respectively. Here, δ is the maximum transport reduction parameter. The thin dot-long-dashed curve shows the locus of points where dξ0/dλ=0 (dξ0/dλ>0 to the left of the curve and vice versa).

Close modal

As illustrated in Fig. 2, the fact that if δ<δcrit, then dξ0/dλ>0 for intermediate values of λ implies that there are two separate branches of solutions to Eq. (35)—the first characterized by dξ0/dλ<0 and relatively large λ and the second characterized by dξ0/dλ<0 and relatively small λ. We shall refer to the former solution branch as the large-temperature-gradient branch [because it is characterized by a relatively large value of λ, which implies a relatively small value of ξ1 from Eq. (34) and implies a relatively large electron temperature gradient inside the separatrix from Eq. (41)] and the latter as the small-temperature-gradient branch [because it is characterized by a relatively small value of λ, which implies a relatively large value of ξ1 from Eq. (34) and implies a relatively small electron temperature gradient inside the separatrix from Eq. (41)]. The two solution branches are separated by a dynamically inaccessible branch characterized by dξ0/dλ>0. We shall refer to this branch of solutions as the inaccessible branch. Referring to Fig. 2, as ξ0 increases from zero, we start off on the large-temperature-gradient solution branch, and λ decreases smoothly. However, when a critical value of ξ0 is reached (at which dξ0/dλ=0), there is a bifurcation to the small-temperature-gradient solution branch. We shall refer to this bifurcation as the temperature-gradient-flattening bifurcation because it is characterized by a sudden decrease in the transport ratio parameter, λ, which implies a sudden decrease in the electron temperature gradient within the island separatrix. Once on the small-temperature-gradient solution branch, the control parameter ξ0 must be reduced significantly in order to trigger a bifurcation back to the large-temperature-gradient solution branch. We shall refer to this bifurcation as the temperature-gradient-restoring bifurcation because it is characterized by a sudden increase in the transport ratio parameter, λ, which implies a sudden increase in the electron temperature gradient within the island separatrix.

FIG. 2.

The solid curve shows the perpendicular electron transport reduction parameter, λ, plotted as a function of the ratio of the divergences of the parallel to perpendicular electron heat fluxes outside the island separatrix, ξ0, for δ=0.01. Here, δ is the maximum transport reduction parameter. The dot-long-dashed curve shows the locus of points where dξ0/dλ=0 (dξ0/dλ>0 to the left of the curve and vice versa). The various solution branches and bifurcations are labeled.

FIG. 2.

The solid curve shows the perpendicular electron transport reduction parameter, λ, plotted as a function of the ratio of the divergences of the parallel to perpendicular electron heat fluxes outside the island separatrix, ξ0, for δ=0.01. Here, δ is the maximum transport reduction parameter. The dot-long-dashed curve shows the locus of points where dξ0/dλ=0 (dξ0/dλ>0 to the left of the curve and vice versa). The various solution branches and bifurcations are labeled.

Close modal

Figure 3 shows the critical values of the control parameter ξ0 below and above which a temperature-gradient-flattening and a temperature-gradient-restoring bifurcation, respectively, are triggered, plotted as a function of δ/δcrit.

FIG. 3.

The upper curve shows the critical value of the ratio of the divergences of the parallel to perpendicular electron heat fluxes outside the island separatrix, ξ0, below which a temperature-gradient-flattening bifurcation is triggered, plotted as a function of δ/δcrit. The lower curve shows the critical value of ξ0 above which a temperature-gradient-restoring bifurcation is triggered, plotted as a function of δ/δcrit. Here, δ is the maximum transport reduction parameter, and δcrit=0.1192 is the critical value of δ below which bifurcations occur.

FIG. 3.

The upper curve shows the critical value of the ratio of the divergences of the parallel to perpendicular electron heat fluxes outside the island separatrix, ξ0, below which a temperature-gradient-flattening bifurcation is triggered, plotted as a function of δ/δcrit. The lower curve shows the critical value of ξ0 above which a temperature-gradient-restoring bifurcation is triggered, plotted as a function of δ/δcrit. Here, δ is the maximum transport reduction parameter, and δcrit=0.1192 is the critical value of δ below which bifurcations occur.

Close modal

Figure 4 shows the extents of the various solution branches (i.e., the continuous, large-temperature-gradient, small-temperature-gradient, and inaccessible branches) plotted in ξ0ξ1 space. It is clear that the large-temperature-gradient solution branch is characterized by ξ01 and ξ11. In other words, the region outside the island separatrix lies in the narrow-island limit, WWc0, whereas that inside the separatrix lies in the narrow/intermediate island limit, WWc1 [see Eqs. (31) and (32)]. This implies weak to moderate flattening of the temperature gradient within the separatrix. On the other hand, the small-temperature-gradient solution branch is characterized by ξ01 and ξ11. In other words, the region outside the island separatrix lies in the narrow-island limit, WWc0, whereas that inside the separatrix lies in the wide-island limit, WWc1. This implies strong flattening of the temperature gradient within the separatrix. Figure 4 suggests that bifurcated solutions of Eq. (35) occur because it is possible for the regions inside and outside the island separatrix to lie in opposite asymptotic limits (the two possible limits being the wide-island and the narrow-island limits). Obviously, this is not possible in the conventional model in which κ is taken to be spatially uniform in the island region.

FIG. 4.

The extents of the various solution branches of Eq. (35) plotted in ξ0ξ1 space. Here, ξ0 is the ratio of the divergences of the parallel to perpendicular electron heat fluxes outside the island separatrix, whereas ξ1 is the same ratio inside the separatrix.

FIG. 4.

The extents of the various solution branches of Eq. (35) plotted in ξ0ξ1 space. Here, ξ0 is the ratio of the divergences of the parallel to perpendicular electron heat fluxes outside the island separatrix, whereas ξ1 is the same ratio inside the separatrix.

Close modal

Finally, according to our simple model, the critical value of the maximum transport reduction parameter, δ, below which bifurcated solutions of the electron energy transport equation occur is 0.1192. As we have seen, there is experimental evidence for a transport reduction within the separatrix of a neoclassical island chain by between 1 and 2 orders of magnitude.16 According to our model, such a reduction would be large enough to generate bifurcated solutions.

Let

(36)

be the island temperature profile in the narrow-island limit [see Eq. (14)]. Here, ξ=(W/Wc)4. Likewise, let

(37)

be the island temperature profile in the wide-island limit [see Eqs. (23) and (24)]. Let us write

(38)

where [cf. Eq. (34)]

(39)

and

(40)

It follows that

(41)
(42)

Here, f(0)=1.1981, as determined from the numerical solution of Eq. (13).

According to Eqs. (26), (27), (41), and (42),

(43)
(44)

where

(45)
(46)
(47)
(48)

Here, the use has been made of the easily proved result

(49)

Let ψ=2k21. It follows that dψ=4kdk. In the region 0k1, we can write

(50)
(51)
(52)
(53)

On the other hand, in the region k > 1, we can write

(54)
(55)
(56)
(57)

Here, it is assumed that A is an even function of X.

Let

(58)
(59)
(60)

It follows from Eqs. (50)–(57) that in the region 0k1,

(61)
(62)

On the other hand, in the region k > 1,

(63)
(64)
(65)

Here,

(66)
(67)

are complete elliptic integrals.23 Hence, according to Eqs. (45)–(48) and (58)–(60),

(68)
(69)
(70)
(71)

Thus, Eqs. (43) and (44) yield

(72)
(73)

respectively.

The dimensionless parameter G2, appearing in the modified Rutherford equation (25), measures the destabilizing influence of the perturbed bootstrap current. Figure 5 shows G2 plotted as a function of the so-called neoclassical tearing mode control parameter

(74)

which measures the ratio of the divergences of the parallel to the perpendicular electron heat fluxes outside the island separatrix [see Eqs. (29) and (31)]. The curves shown in this figure are obtained from Eqs. (33), (35), and (73). Note that κ and κ0 are the local (to the rational surface) parallel and perpendicular electron thermal conductivities, respectively, in the absence of an island chain.

FIG. 5.

The bootstrap destabilization parameter, G2, plotted as a function of the ratio of the divergences of the parallel to perpendicular electron heat fluxes outside the island separatrix, ξ0. The solid, short-dashed, long-dashed, dot-short-dashed, and dotted curves correspond to δ= 0.9, 0.5, 0.2, 0.1192, and 0.01, respectively. Here, δ is the maximum transport reduction parameter.

FIG. 5.

The bootstrap destabilization parameter, G2, plotted as a function of the ratio of the divergences of the parallel to perpendicular electron heat fluxes outside the island separatrix, ξ0. The solid, short-dashed, long-dashed, dot-short-dashed, and dotted curves correspond to δ= 0.9, 0.5, 0.2, 0.1192, and 0.01, respectively. Here, δ is the maximum transport reduction parameter.

Close modal

It can be seen, from Fig. 5, that if the maximum transport reduction parameter, δ, is relatively close to unity (implying a relatively weak reduction in the perpendicular electron thermal conductivity inside the island separatrix when the electron temperature profile is completely flattened in this region), then the bootstrap destabilization parameter, G2, increases monotonically with increasing ξ0, taking the value of 3.492ξ01/2 when ξ01 and approaching the value of 6.140 asymptotically as ξ0.5 [These two limits follow from Eq. (73), given that ξ1ξ0 when λ1.]

According to Fig. 5, as δ decreases significantly below unity (implying an increasingly strong reduction in the perpendicular electron thermal conductivity inside the island separatrix when the electron temperature profile is completely flattened in this region), it remains the case that G2=3.492ξ01/2 when ξ01, and G26.140 as ξ0. However, at intermediate values of ξ0 [i.e., ξ0O(1)], the rate of increase of G2 with ξ0 becomes increasingly steep. This result suggests that a substantial reduction in the perpendicular electron thermal conductivity inside the island separatrix, when the electron temperature profile is completely flattened in this region, causes the bootstrap destabilization term in the modified Rutherford equation, (25), to “switch on” much more rapidly as the neoclassical tearing mode control parameter, ξ0, is increased, compared to the standard case in which there is no reduction in the conductivity.

Finally, it is apparent from Fig. 5 that if δ falls below the critical value δcrit=0.1192, then the bootstrap destabilization parameter, G2, becomes a multi-valued function of ξ0 at intermediate values of ξ0. As illustrated in Fig. 6, this behavior is due to the existence of separate branches of solutions of the electron energy conservation equation (see Sec. III B). The large-temperature-gradient branch is characterized by relatively weak flattening of the electron temperature profile within the island separatrix and a consequent relatively small value (i.e., significantly smaller than the asymptotic limit of 6.140) of the bootstrap destabilization parameter, G2. On the other hand, the small-temperature-gradient branch is characterized by almost complete flattening of the electron temperature profile within the island separatrix. Consequently, the bootstrap destabilization parameter, G2, takes a value close to the asymptotic limit of 6.140 on this solution branch. The large-temperature-gradient and small-temperature-gradient solution branches are separated by a dynamically inaccessible branch of solutions. Referring to Fig. 6, as the neoclassical tearing mode control parameter, ξ0, increases from a value much less than unity, we start off on the large-temperature-gradient solution branch, and the bootstrap destabilization parameter, G2, increases smoothly and monotonically from a small value. However, when a critical value of ξ0 is reached, there is a gradient-flattening-bifurcation to the small-temperature-gradient solution branch. This bifurcation is accompanied by a sudden increase in G2 to a value close to its asymptotic limit 6.140. Once on the small-temperature-gradient solution branch, ξ0 must be decreased by a significant amount before a gradient-restoring-bifurcation to the large-temperature-gradient solution branch is triggered. Moreover, the gradient-restoring-bifurcation is accompanied by a very large reduction in G2.

FIG. 6.

The bootstrap destabilization parameter, G2, plotted as a function of the ratio of the divergences of the parallel to perpendicular electron heat fluxes outside the island separatrix, ξ0, for δ=0.01. Here, δ is the maximum transport reduction parameter. The various solution branches and bifurcations are labeled.

FIG. 6.

The bootstrap destabilization parameter, G2, plotted as a function of the ratio of the divergences of the parallel to perpendicular electron heat fluxes outside the island separatrix, ξ0, for δ=0.01. Here, δ is the maximum transport reduction parameter. The various solution branches and bifurcations are labeled.

Close modal

The parallel electron thermal conductivity takes the form21 

(75)

in a collisional plasma, where ne is the electron number density, ve is the electron thermal velocity, and λe is the electron mean-free-path. However, in a conventional tokamak plasma, the mean-free-path λe typically exceeds the parallel (to the magnetic field) wavelength λ of low-mode-number helical perturbations. Under these circumstances, the simple-minded application of Eq. (75) yields unphysically large parallel heat fluxes. The parallel conductivity in the physically relevant long-mean-free-path limit (λeλ) can be crudely estimated as5,24

(76)

which is equivalent to replacing parallel conduction by parallel convection in the electron energy conservation equation, (5). For a magnetic island of full radial width W, the typical value of λ is nφssw/R0. Hence, in the long-mean-free-path limit, the expression for the neoclassical tearing mode control parameter (74) is replaced by

(77)

where κ=nφnevess/R0, and ne and ve are evaluated at the rational surface.

In this paper, we have investigated the effect of the reduction in anomalous perpendicular electron heat transport inside the separatrix of a magnetic island chain associated with a neoclassical tearing mode in a tokamak plasma, due to the flattening of the electron temperature profile in this region, on the overall stability of the mode. Our model (which is described in Sec. III) is fairly crude, in which the perpendicular electron thermal conductivity, κ, is simply assumed to take different spatially uniform values in the regions inside and outside the separatrix. Moreover, when the temperature profile is completely flattened within the island separatrix, κ in this region is assumed to be reduced by some factor δ, where 0<δ1. The degree of temperature flattening inside the separatrix is ultimately controlled by a dimensionless parameter ξ0 that measures the ratio of the divergences of the parallel and perpendicular electron heat fluxes in the vicinity of the island chain. Expressions for ξ0 in the short-mean-free-path and the more physically relevant long-mean-free-path limits are given in Eqs. (74) and (77), respectively. Finally, the destabilizing influence of the perturbed bootstrap current is parameterized in terms of a dimensionless quantity G2>0 that appears in the modified Rutherford equation [see Eqs. (25) and (27)]. A large value of G2 implies substantial destabilization and vice versa.

In the standard case δ = 1 (in which there is no reduction in the perpendicular electron thermal conductivity inside the island separatrix when the electron temperature profile is completely flattened in this region), the bootstrap destabilization parameter G2 increases smoothly and monotonically as the control parameter ξ0 increases, from a value much less than unity when ξ01 to the asymptotic limit 6.140 when ξ01.5 (See Section III E.)

As δ decreases significantly below unity (implying an increasingly strong reduction in the perpendicular electron thermal conductivity inside the island separatrix when the electron temperature profile is completely flattened in this region), the small-ξ0 and large-ξ0 behaviors of the bootstrap destabilization parameter remain unchanged. However, at intermediate values of the control parameter ξ0 [i.e., ξ0O(1)], the rate of increase of G2 with ξ0 becomes increasingly steep (see Fig. 5). In other words, a substantial reduction in the perpendicular electron thermal conductivity inside the island separatrix, when the electron temperature profile is completely flattened in this region, causes the bootstrap destabilization parameter G2 to “switch on” much more rapidly as the control parameter ξ0 is increased, compared to the standard case in which δ = 1 (see Section III E).

Finally, if δ falls below the critical value of 0.1192, then the bootstrap destabilization parameter, G2, becomes a multi-valued function of the control parameter ξ0, at intermediate values of ξ0. This behavior is due to the existence of separate branches of solutions of the electron energy conservation equation (see Sec. III B). The large-temperature-gradient branch is characterized by relatively weak flattening of the electron temperature profile within the island separatrix and a consequent relatively small value (i.e., significantly smaller than the asymptotic limit 6.140) of the bootstrap destabilization parameter, G2. On the other hand, the small-temperature-gradient branch is characterized by almost complete flattening of the electron temperature profile within the island separatrix. Consequently, the bootstrap destabilization parameter, G2, takes a value close to the asymptotic limit of 6.140 on this solution branch. The large-temperature-gradient and small-temperature-gradient solution branches are separated by a dynamically inaccessible branch of solutions. As the control parameter, ξ0, increases from a value much less than unity, the system starts off on the large-temperature-gradient solution branch, and the bootstrap destabilization parameter, G2, increases smoothly and monotonically from a small value. However, when a critical value of ξ0 is reached, there is a gradient-flattening-bifurcation to the small-temperature-gradient solution branch (see Fig. 6). This bifurcation is accompanied by a sudden increase in G2 to a value close to its asymptotic limit of 6.140. Once on the small-temperature-gradient solution branch, ξ0 must be decreased by a significant amount before a gradient-restoring-bifurcation to the large-temperature-gradient solution branch is triggered. Moreover, the gradient-restoring-bifurcation is accompanied by a very large reduction in G2 (see Section III E).

The behavior described in the preceding paragraph points to the disturbing possibility that a neoclassical tearing mode in a tokamak plasma could become essentially self-sustaining. In other words, once the mode is triggered and the electron temperature profile is flattened within the island separatrix, the consequent substantial reduction in the perpendicular thermal conductivity in this region reinforces the temperature flattening, making it very difficult to remove the mode from the plasma.

This research was funded by the U.S. Department of Energy under Contract No. DE-FG02-04ER-54742.

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