Proceedings of ITC/ISHW2007

A model of interaction between magnetic island and drift wave
turbulence
Akihiro ISHIZAWA1) , Patrick H. DIAMOND2) and Noriyoshi NAKAJIMA1)
1)

2)

National Institute for Fusion Science, Toki 509-5292, Japan
University of California San Diego, La Jolla, California 92093-0424, USA
October 15, 2007

A modified Rutherford equation of magnetic island is introduced. The island equation includes multi-scaleinteraction with drift wave turbulence and is coupled with turbulent wave energy equation. In this model, drift
wave turbulence is prey and magnetic island corresponds to predator. The magnetic island suppresses the growthof turbulence by flattening temperature gradient because of the violation of magnetic surfaces. On the other hand,
the turbulence affects perturbed neoclassical bootstrap current in the Rutherford equation through anomaloustransport. In these interactions heat flux is fixed, and thus perpendicular thermal diffusion coefficient depends on
the turbulence energy and the island width. A stabilizing effect of the turbulence on magnetic island growth isfound and new critical island width of neo-classical tearing mode excitation is obtained.
Keywords: magnetic island, drift wave turbulence, multi-scale, NTM
DOI:

1Introduction

Recently, multi-scale interaction between microturbulence and macro-scale magnetohydrodynamic(MHD) instability has been studied extensively [1, 2, 3].
The interaction would play crucial role in the analysis ofneoclassical tearing mode (NTM), which limits the beta oftokamak plasmas[4, 5]. The NTM is driven by perturbed
neoclassical bootstrap current density inside the separatrix
of magnetic island, and it is unstable in high beta tokamak
plasma, even if the current density profile is linearly stableagainst tearing mode. Thus, NTM is a nonlinear instability
and it starts to grow when the width of island caused by an
external perturbation exceeds a threshold.
The excitation mechanism of NTM is an open problem that is divided into two parts: the threshold and thetrigger. The threshold which is called critical island widthis evaluated as follows. When magnetic island appears thepressure gradient is reduced inside the island because ofstrong heat conductivity along magnetic field. This flattening of pressure gradient reduces neoclassical bootstrap current inside the magnetic island and destabilizes the magnetic island. When the island width is very small the flattening is not completed because it is not able to overcome
perpendicular transport. Hence, competition between parallel and perpendicular heat transport determines criticalisland width [6]. The critical island width is also affectedby the polarization current [7, 8]. On the other hand,
the trigger problem is the process of producing the seedmagnetic island caused by external phenomena. Once thewidth of seed island exceeds the critical island width, thenthe island grows as NTM. The external phenomena can be
author’s e-mail: ishizawa@nifs.ac.jp

MHD modes of different helicities such as sawteeth andedge localized modes and induce the seed island throughtoroidal mode coupling [9, 10]. The external phenomena
can also be micro-turbulence because the turbulence is able
to produce seed island through nonlinear mode coupling[11].
The anomalous perpendicular transport due to drift
wave turbulence should play crucial role in evaluating critical island width. In addition, magnetic island affects theturbulence simultaneously. We need a model which describes these mutual interaction between them. In this paper, we propose a simple model of interaction between drift
wave turbulence and magnetic island. The point of ourmodel is that we fix heat flux instead of thermal diffusivity
coefficient when we evaluate temperature gradient. Then
we introduce effects of drift wave turbulence on thermal
diffusivity and introduce effect of magnetic island which
reduces the growth of turbulence by flattening temperature
gradient inside the island.
We present our model of interaction between magnetic
island and the turbulence in Sec. 2. In Sec. 3 we evaluate
critical island width by using the model. Finally we summarize results in Sec. 4.

2Model of interaction between magnetic island and drift wave turbulence

We present magnetic island equation which couples withdrift wave turbulence energy equation. The modified

Proceedings of ITC/ISHW2007
whereRutherford equation of magnetic island evolution is [4]
Ls rs
τR dw
= r s ∆0 +
δJBS ,r s dt
B0 w

(1)

where w, r s , B0 , τR , and L s are island width, minor radius of resonant surface, uniform toroidal magnetic field,
resistive diffusion time, and magnetic shear length, respectively. In this equation ∆0 is the stability parameter of tearing mode and is negative so that it is stable against tearing mode. We remark that the island width is related toperturbed magnetic field as δB = kθ B0 w2 /L s . Here we assume that density profile is uniform n0 , then the perturbed
bootstrap current is!
 1/2 n0 dT e
δJBS =,(2)δBθdrwhere , T e , and Bθ are aspect ratio, electron temperature,and poloidal magnetic field, respectively. When we calculate temperature gradientdT e
Qe=− ,drχ⊥χ⊥= χ⊥turb + χ⊥island + χneo ,(4)χ⊥turb= χ⊥turb (ε) = χ0 ε(w),!2!2kθ 2
δBw ,= χk= χk
B0Ls(5)(6)

where χ⊥turb is assumed to be proportional to turbulenceenergy ε, χk is parallel thermal diffusivity, and kθ ispoloidal wave number. Then Eq. (3) is rewritten as,dT e−Qe.
=drχ0 ε(w) + χk (w2 kθ /L s )2 + χneo(7)By using Eqs. (7) and (2) the island equation (1) is writtenas,dŵ ˆ 0=∆ +dtˆβ̂e 11 (8)Q̂e −44ŵ
ε(0) + χ̂neo ε(ŵ) + ŵ /ŵd + χ̂neo= ∆ˆ 0 +ŵ4d (ε(0) + χ̂neo ) 
β̂eQ̂e1 − ,(9)ŵ ε(0) + χ̂neoŵ4 + ŵ4d (ε(ŵ) + χ̂neo )where χ̂⊥ (ε(ŵ)) = ε(ŵ). Normalizations are ŵ = w/r s ,tˆ = t/τR , χ̂k = χk /χ0 , χ̂neo = χneo /χ0 , k̂⊥ = kθ r2s /L s , L̂ s =L s /LT 0 , B̂θ = Bθ /B0 , Q̂e = Qe LT 0 /T e χ0 , β̂e = βe  1/2 L̂ s / B̂θ,ŵd = (χ̂k k̂⊥2 )−1/4 .
In order to close interaction loop between magnetic
island and the turbulence we need equation of turbulence
wave energy[12]dε= γ(ε, w)ε − βε2 ,dt

(10)

(11)

where ( f )>0 is zero if f is negative. In this equation weinclude feedback from magnetic island to the turbulence.
The strong parallel thermal diffusion flattens pressure gradient inside the separatrix of the island. This reduces
growth rate γ by increasing length scale of temperature
gradient
1LT (ε, w)==−1 dT eT e drQ̂e
1.
LT 0 ŵ4 + ŵ4d (ε(ŵ) + χ̂neo )

(12)
(13)

When we calculate the temperature gradient we fix heat
flux again. By substituting this equation to Eq. (11) wehave equation of turbulence energy including the effect ofmagnetic island as,(3)we fix heat flux Qe , and thus perpendicular heat diffusivityχ⊥ is variable. The heat diffusivity coefficient consists ofdiffusion due to perturbed magnetic field in the presence ofmagnetic island χ⊥island , anomalous transport by drift wave
turbulence χ⊥turb , and neoclassical transport χ⊥neo as,χ⊥islandγ(ε, w) = γ0 LT 0!
11,−
LT (ε, w) LT cr >0

1 dε=
τR γ0 dtˆ
1 Q̂e ε − β̂ε2 , (14)−ŵ4 + ŵ4d (ε(ŵ) + χ̂neo ) L̂T cr >0where L̂T cr = LT cr /LT 0 and β̂ = β/γ0 . Hence, we haveobtained a closed set of equations of magnetic island andthe turbulence Eqs. (9) and (14).

3

Critical island width of NTM

We have established a model which is able to evaluate critical island width of NTM. We consider a situation that theturbulence saturates because drift frequency time, which ischaracteristic time scale of drift wave turbulence, is much
faster than the resistive diffusion time τR γ0 >> 1, where
γ0 ≈ ω∗ . Thus, we neglect left hand side of Eq. (14) andhave the equation of turbulence energy as,!
q−1 1
2+ −F(ŵ) + F(ŵ) + 4Q̂e /β̂ ,(15)ε(w) =β̂L̂cr 2
where F(ŵ) = ŵ4 /ŵ4d + χ̂neo − β̂L̂1 . Substituting this ε(w)crinto Eq. (9) we have the island equation of our model.
We show curves of island growth dŵ/d tˆ calculated byour model and by the standard modeldŵ
dtˆ=β̂e ŵ2∆ˆ 0 +ŵ ŵ2 + ŵ2d(16)in Fig. 1. Here we set parameters ∆ˆ 0 = −1, β̂e = 0.2,χ̂neo = 0.1, Q̂e = 2, χ̂k k̂⊥2 = 109 , β̂ = 0.6, and L̂T cr = 0.5.
The critical island width wcritical /r s is given by dŵ/d tˆ = 0.
The critical island width of our model is larger than thestandard model. Thus, the turbulence has a stabilizing effect on the excitation of NTM. Notice that we cannot apply our model to evaluate saturated island width because

Proceedings of ITC/ISHW2007

model implies wcritical /r s ∝ χ̂k−1/2 . Figure 3 shows curvesof plasma beta βe as a function of critical island width.
Our model implies wcritical /r s ∝ β̂e−1/3 , while the standard
model implies wcritical /r s ∝ β̂−1e , and thus our model suggests weak dependence of wcritical on βe compared to thestandard model.

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    Fig. 1 Curves of island growth dŵ/d tˆ as a function of island
width w/rs . Solid curve indicates our model and dashedcurve indicates standard model.

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We have obtained a predator-prey model of interaction between magnetic island and drift wave turbulence. The turbulence affects perturbed bootstrap current in the island
equation of NTM through anomalous perpendicular transport. When we evaluate bootstrap current we fix heat flux
and make heat diffusivity depend on the turbulence and theisland width. In order to close interaction loop between
the magnetic island and the turbulence we introduce turbulence wave energy equation including effect of magneticisland. The magnetic island makes temperature gradient
flatten inside it and reduces growth rate of the turbulence.
Our model predicts larger critical island width thanthe one by standard model. This implies that the drift wave
turbulence has stabilizing effect on magnetic island excitation of NTM. In addition we found new βe scaling of thecritical island width of NTM wcritical /r s ∝ β̂e−1/3 . In order
to compare our model with experimental observation wewould include the polarization current effect in our futurework.

  w/rs

Summary and discussion&('%$#"+-,). ,+-,+-/). ,+-/+-0). ,+-0+ * .,+*+ & .,

Log10(wcritical/rs)
Fig. 2 Curves of parallel diffusion coefficient χk as a function ofcritical island width. Solid curve indicates our model anddashed curve indicates standard model.

Log10(βe 1/2LsB0/LT0Bθ)

4      !  τR/rs·dw/dt

a large magnetic island strongly reduces turbulence energy and makes it negative in our model. For the above
parameter set the turbulence energy ε is positive whenw/r s < 0.00548. Figure 2 shows curves of parallel diffusion coefficient χk as a function of critical island width.
Our model implies wcritical /r s ∝ χ̂k−1/3 , while the standard

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Fig. 3 Curves of plasma beta βe as a function of critical island
width. Solid curve indicates our model and dashed curve
indicates standard model.