J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)237–240
000–000

Interaction Between the Trapped Electrons andMagnetic Field Gradient and Curvature-Driven Drift Waves
in the Toroidal Plasmas
WANG Aike, SANUKI Heiji1, DONG Jiaqi, ZONCA Fulvio2 and ITOH Kimitaka1
Southwestern Institute of Physics, P.O.Box 432, Chengdu 610041, P.R.China
1National Institute for Fusion Science, Toki 509-5292, Japan
2
ENEA C.R.Frascati, C.P. 65, 00044 Frascati, Rome, Italy(Received: 9 December 2003 / Accepted: 12 March 2004)

Abstract
In this paper, we study the interaction between trapped electrons and magnetic field gradient and curvature(MFGC) –driven drift waves. In addition to the MFGC instabilities of long wavelength, a new unstable branch ofshort wavelength is excited after the trapped electrons are included. It is found that there exist heat pinches in thelong wavelength regime. Especially in the position of parallel wave number k// ≈ 0, the electron-wave resonancestrongly destabilizes the modes with the short wavelength, which contribute the significant positive value to ion andelectron thermal diffusivities (χi and χe) but the significant negative value to particle diffusivity (De). Finally, a set ofnonlinear equations for the MFGC mode with the trapped electron effects is derived and qualitative discussions on it
are made.
Keywords:
magnetic field gradient and curvature, trapped electron, drift wave

1. Introduction

rate from zero to infinity. Subsequently, the effects of theelectron-wave resonance on the instabilities and associatedtransport are emphasized. Finally, a set of nonlinear equations
is derived and qualitative analyses on the predicted nonlinear
processes are made, which indicate that the nonlinear
processes would lead to the minimum energy states satisfyingk// ≈ 0 with the minimum safety factor and turbulent transportlevel.

It has been realized that the turbulence in plasma is amultiple-length-scale structure [1]. In addition to the longwavelength turbulence, it has been an interesting subject toexplore the short wavelength one [2]. Recently, it becomes
especially attractive to study the interaction between thedifferent length scale structures [1,3]. Thus, it is desired toobtain a turbulence structure from zero to infinity wave
number.
In the previous work [4], we have presented a new kindof instability in the toroidal plasmas: magnetic filed gradientand curvature (MFGC)-driven one. The MGFC instability is
the long wavelength dominant one. It possesses the finite
growth rate even if the plasma pressure gradient vanishes. Inaddition, no matter what magnitude the plasma pressure
gradient is, the instability can be stabilized as long as themagnetic field gradient and curvature is zero.
In this work, we study the interaction between MFGC
modes and trapped electrons. In addition to the MFGC
instability of long wavelength, a new unstable branch of shortwavelength is excited after the trapped electrons are included.
Thus, there exist the multi-mode interactions, e.g., thosebetween the two unstable branches and between the stableand unstable branches. As a result, they lead to the present
hybrid instabilities with the wave number spectrum of growth

2. Linear instability and associatedtransport
To study the effects of trapped electrons on the MFGC
modes, we use two-fluid description in the sheared coordinatesystem ê⊥ = b̂ × r̂ [5], where r̂ is radial unit vector and b̂ =B/B is the unit vector along magnetic field B. Then, theelectron response is taken as

ne = ne 0 [1 + eφ / Te + iδ et eφ / Te ],(1a)where Te and ne are the temperature and density of electrons,respectively, –e is electron charge, and φ is electrostatic
potential. For the collisionless electron-wave resonance [6]δ et = δ 0 k⊥ ρ[bi − ηe / 2],(1b)where δ0 = (π me /2mi)1/2/k// Ln, ρ = (Te mi)1/2/eB, bi = (k⊥2 +©2004 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchCorresponding author’s e-mail: akwang@swip.ac.cn
1237

Wang A.K. et al., Interaction Between the Trapped Electrons and Magnetic Field Gradient and Curvature-Driven Drift Waveskr2)ρ2, and ηe = Ln /LTe. Similarly, we have ηi = Ln /LTi . Heremi and me are ion and electron mass, and k⊥ and kr are theperpendicular and radial wave number, respectively; Ln =–ne(∂ ne /∂ r)–1, LTj = –Tj (∂ Tj /∂ r)–1, and the subscript j = i, edenotes ion and electron species, respectively. Then, following
the previous route [4], we obtain the dispersion equation,

the trapped electron-ion temperature gradient modes [7] fromthe viewpoint of drive sources. Considering Eq. (1b), alongwith ωJ = (ene)–1k·J = bJ (ω * pe – ω * pi ), we find the growthrate of unstable modes, described by Eq. (4), are direct
proportional to the safety factor q, i.e.,γ = qε −1δ 0 (1 − ηe / 2 bi )k / / ρ (ω *pe − ω *p ) .
i[bi Ω − (1 + iδ et )(ω J − ω *pi + ω *pe )] + ω Di ∆ i = ω De ∆ e , (2a)⎡55
∆ i = ⎢ bi Ω + (1 − τ −1 + iδ et )ω Di3⎣310⎤−1+ τ ω *p ⎥ (Ω + ω *p − ω Di ) −1,ii
3⎦Obviously, the instability is the short wavelength dominant
one: the larger the wave number, the larger the growth rate.
For the short wavelength limit η e /2b i in Eq. (5) can beomitted.
In the general case with both the finite plasma pressure
gradient and the magnetic field gradient and curvature, there
exist the interaction between the two unstable branches (3)and (4) and the coupling of the stable branches with theunstable branches. As a result, they lead to the present hybrid
instabilities with the wave number spectrum of growth ratefrom zero to infinity. In particular, the electron-wave
resonance in the position of k// ≈ 0 strongly enhances thegrowth rate, as shown in Figs. 1 and 2, where k//ρ = 10–7. The
first unstable branch (solid line) in Fig. 1 is dominated byEq. (3) and becomes stable in the short-wave length regime.
The second unstable branch (dashed line) in Fig. 1 is mainly
determined by Eq. (5) and stable in the long wave length
regime. Here it should be pointed out that the fluid description
and qausineutrality used here are valid for the first branch
but only give a qualitative physical picture for the secondbranch.
Now, we turn our attention to the ion and electron heat(2b)⎡5⎤∆ e = ⎢ (2 + iδ et )ω De − ω *pe⎥ ( Ω + ω *pe⎦⎣310− ω De + ω J ) −1.
(2c)3where, τ = Ti /Te, Ω = (ω – k·Vi), Vi is the equilibrium fluid
velocity of ions; ωJ = (ene)–1k·J = bJ (ω * pe – ω * pi) [5], J is thecurrent density, and bJ = (qε –1k// /k⊥ ) – 1 with the safety factor
q and ε = r/R << 1 (r and R are the minor and major radius ofplasma, respectively); ωDj = –2k⊥Tj /eBR and ω * pj = (1 + ηj)ω *j.
Here Eq. (2a) is obtained by subtracting the continuity
equation of electrons from that of ions with the quasineutrality condition and Eqs. (2b) and (2c) come from theenergy equation of ions and electrons, respectively.
In order to understand the instability described byEq. (2), it is necessary to discuss Eq. (2) in the two specificcases. First, if the plasma pressure gradient vanishes, i.e.,
LTi–1, LTe–1, and Ln–1 all are zero, Eq. (2) reduces to [4]⎡5(1 − τ )τ + 105bi Ω 3 − bi (2 − τ )Ω 2 − ⎢33⎣
50 biτ ⎤50Ω − τ (τ + 1) = 0 ,+9 ⎥⎦
9(5)

(3)

where Ω is normalized to ωDe . Eq. (3) has generally theunstable solutions. The instabilities are driven completely bythe magnetic field gradient and curvature. The MFGC
instabilities are the long wavelength-dominated ones due tothe finite Larmor radius effect 1/bi . The previous fluid models[7] for toroidal ion temperature gradient / trapped electronmodes (ITG/TEMs) have no unstable MFGC solutions in theelectrostatic case when the plasma pressure gradient vanishes,even if the bad curvature enhances the ITG/TEM instabilities.
Compared with those model equations, the present include
both the equilibrium fluid velocity of ions and electrons andthe continuity and energy equation of electrons. Furthermore,
if the magnetic field gradient and curvature is zero, i.e., ωDi= ωDe = 0, from Eq. (2) we have the following unstable
solution for Ω :bi Ω = (1 + iδ et )(ω J + ω *pe − ω * p ).
iFig. 1(4)The instabilities originate in the coupling between the trappedelectron effects δet and the plasma total pressure gradient
LP–1 = –P –1dP/dr ∝ (ωJ + ω * pe – ω * pi), which are similar to2238

Growth rate and real frequency for two unstable
solutions of Eq. (2a) versus k⊥ ρ (from zero to 4). Solidline is unstable in the regime k⊥ ρ < 2.2; dashed line isunstable in the regime k⊥ ρ > 2.2 (see, Fig. 2). R /Ln = 15,R /LTi = 100, R /LTe = 150, τ = 1.5, q = 1.5, R /ρ = 1500, andkr ρ = 0.4.

Wang A.K. et al., Interaction Between the Trapped Electrons and Magnetic Field Gradient and Curvature-Driven Drift Waves

Fig. 2Growth rate (solid line) and real frequency (dashed line)
for unstable short-wave length branch of Eq. (2a)versus k⊥ ρ (from 400 to 500). All other parameters are
the same as Fig. 1.

Fig. 3χi (solid line), χe (dashed line), and De (dotted line)
induced by the instabilities in Fig. 1. Here χi , χe , and De
are normalized to 106 × ωDe /kr2.

Fig. 4χi (solid line), χe (dashed line), and De (dotted line)
induced by the instabilities in Fig. 2. Here χi , χe , and De
are normalized to 1016 × ωDe /kr2.

and particle transport caused by the hybrid instability. Fromthe definitions of the turbulent fluxes for ion and electron heatand particles, [8]qTi = Re δTiδv Er * = − rˆχ i ∂Ti (r ) / ∂r ,(6)

qTe = Re δTeδv Er * = − rˆχ e ∂Te (r ) / ∂r ,(7)ˆ e ∂ne (r ) / ∂r ,qne = Re δneδv Er * = − rD

(8)along with the assumption for saturated turbulenceφ˜ = 2γ / ω De kr R,(9)

we haveqTi = 2γ 35(τ − 1)/ 3 + 50 biτ / 9 − (5biτ / 3 − 1)(Ln−1 + LT−i 1)/ 2[Ω r − (Ln−1 + LTi −1 )τ / 2 + 10τ / 3]2 + γ 2rˆ
,⎧⎪5τ [Ω r − (Ln−1 + LTi −1 )τ / 2 + 10τ / 3]/ 3 ⎫⎪rˆ+ 2γ δ et ⎨1 −22⎬
−1−1⎪⎩ [Ω r − (Ln + LTi )τ / 2 + 10τ / 3] + γ ⎪⎭2(10)qTe =[( Ln−1 + LTe−1) / 2 − 10 / 3]2γ 3rˆτ [Ω r + ω J + ( Ln−1 + LTe −1) / 2 − 10 / 3]2 + γ 2−1−12γ 2 ⎧⎪ 5 [Ω r + ωJ + (Ln + LTe )/ 2 − 10 / 3]/ 3 ⎫⎪ˆδ et⎨1 ++22⎬r−1−1τ⎩⎪ Ω r + ωJ + (Ln + LTe )/ 2 − 10 / 3] + γ ⎭⎪,(12)where δ vEr is the radial fluctuation of E × B velocity, and< > refers to a time average. Here both q Ti and q Te arenormalized to T i R ω De /k r2 R 2 , q ne to n e R ω De /k r2 R 2 , thefrequencies to ωDe , and Ln , LTi , and LTe to R. Correspondingly,
the ion and electron heat and particle transport coefficients,
normalized to ωDe /kr2, are, respectively,ˆ T,χ i = qTi ⋅ rLi(14)ˆ n,De = qne ⋅ rL(15)where qTi , qTe, and qne are given by Eqs. (10), (11), and (12),respectively. Here note that, both qTi and qTe are finite when
the plasma pressure gradient vanishes. In the case, the growthrate of instability is given by Eq. (3) and correspondingly, χiand χe tend to infinite. Furthermore, the transport coefficients
χi, χe, and D e under the general case with the both finite
plasma pressure gradient and the magnetic field gradient andcurvature, are shown in Figs. 3 and 4. The present χi, χe, andD e in the short wavelength regime (bi >> ηe /2) have theopposite sign with those in the long wavelength regime (bi ≤ηe /2). Especially, the electron-wave resonance-excited shortwavelength instabilities contribute the significant positivevalues to χi and χe but the significant negative value to De(particle pinch).
In deriving the above transport coefficients, we used theassumption (9). However, the saturated value for aperturbation, e.g., φ̃ , should be decided by the corresponding
nonlinear processes. In the following, we give the nonlinear
equations with the interaction between MFGC mode andtrapped electrons.

(11)
qne = −2γ 2δ et rˆ,χ e = qTe ⋅ rˆτLTe ,

(13)

3239

Wang A.K. et al., Interaction Between the Trapped Electrons and Magnetic Field Gradient and Curvature-Driven Drift Waves

3. Nonlinear modes

transport barriers have been often observed at or near theminimum rational surface.

In what follows, we use x̂, ŷ, and ẑ coordinate system,
where x̂, ŷ, and ẑ are counterparts of r̂, ê⊥, and b̂ in thesheared coordinate system, respectively. Then, we can derive
the following set of nonlinear equations∂φ˜ k+ i(1 + iδ et )(ω J + ω * p − ω * p )φ˜ kei∂t+iω Di p˜ ik − iω De p˜ ek − [φ˜ k , ∇ ⊥ 2φ˜ k ] = 0 ,4. Summary
In this work, we studied the interaction of MFGC modes
with trapped electrons. The MFGC instabilities are the longwavelength dominant ones. After the trapped electrons areincluded, however, a new unstable branch of short wavelengthis excited, the growth rate of which is direct proportional to
the safety factor q. In the case, there exist multi-modeinteractions, e.g., that between the long and short wavelength
unstable branches. As a result, they lead to the present hybrid
instabilities with the wave number spectrum of growth ratefrom zero to infinity. It is found that both χi and χe is negative(heat pinches) while De are positive in the long wavelength
regime k⊥2ρ 2 ≤ ηe /2, and inversely, De is negative (particlepinch) but χi and χe are positive in the short wavelength
regime k⊥2ρ 2 >> ηe /2. In particular, the electron-wave
resonance in the position of k// ≈ 0 strongly destabilizes themodes in the regime k⊥2ρ 2 >> ηe /2, which contribute thesignificant positive values to χi and χe but significant negativevalue to De.
Finally, we derived a set of nonlinear equations withinteraction of MFGC modes with trapped electrons. A
qualitative discussion on the corresponding nonlinear
processes was made from the linear results and minimum
energy principle.

k⊥ 2(16)∂p˜ ik 5 2 ∂φ˜ k+ k⊥∂t3∂t55⎡⎤+i ⎢τ −1 ⎛ ω Di − ω * p ⎞ − (1 + iδ et )ω Di ⎥ φ˜ ki⎝⎠33⎣⎦
105−i (ω * p − ω Di ) p˜ ik + (φ˜ k , p˜ ik ) − (φ˜ k , ∇⊥ 2φ˜ k ) = 0, (17)i33∂p˜ ek10− i ⎛ ω J + ω * pe − ω De ⎞ p˜ ek⎝⎠3∂t
105−i ⎛ ω De − ω * pe + i δ etω De ⎞ φ˜ k + (φ˜ k , p˜ ek ) = 0 ,⎝ 3
⎠3(18)

where the spatial length scale is normalized to ρ, and velocity
to cs = (Te /mi)1/2; φ̃ , p̃i , and p̃e are the normalized fluctuations
for the electrostatic potential and ion and electron pressure,i.e., φ̃ = eδφ /Te, p̃e = δ pe /Pe, and p̃i = δ pi /Pi ; [φ̃ , f ] = δ vE ·∇fis the Poisson brackets, and the subscript k expresses theFourier component of the fluctuation with wave number k.
Eqs. (16)-(18) describe the nonlinear processes withinteraction between the MFGC modes and trapped electrons,
which are retained for the future numerical research. But,
qualitative discussion on them is also useful. As we argued
previously, the present linear instabilities have the significant
growth rates (or energy) in the short wavelength regime,
which are direct proportional to the safety factor q (or q2)
and driven by the electron-wave resonance in the position ofk// ≈ 0. Based on the minimum energy principle, we canprimarily judge that the nonlinear processes, described by
Eqs. (16)-(18), should transfer the energy of the shortwavelength modes to the long wavelength modes andmeanwhile relax the safety factor q to its minimum (even if
the present electrostatic description can not influence q).
Consequently, it is possible that φ̃ in the short wavelength
regime is almost in the same order of magnitude as that in
the long wavelength regime and thus, the turbulent transports,
induced by the different length scale turbulence,
approximately cancel each other because the present χi, χe,
and De in the short wavelength regime have the opposite signwith those in the long wavelength regime. That is, the wavewave couplings of nonlinear processes with the electron-wave
resonance in the position of k// ≈ 0 would lead to the minimum
energy states satisfying k// ≈ 0 with the minimum safety factor
and turbulent transport level. Such minimum energy states
may be candidates to explain the experimental facts that theAcknowledgements
This work is supported by the JSPS-CAS CoreUniversity Program on Plasma and Nuclear Fusion, by theNatural Science Foundation of China under Grant Nos.
10375019 and 10135020.

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