Chapter 5
Short Wavelength Modes
5.1
Introduction
In short wavelength regime (k⊥ ρi )2 > 1 (cross field wavelength smaller than the ion Larmor radius), the
toroidal ITG mode tends to be stabilized because ion dynamics tends to be adiabatic,
ni → −
eφ
n0 ,
Ti
(k⊥ ρi )2 À 1.
However, the slab ITG mode persists in toroidal geometry provided the toroidicity is not too strong, Ln /R .
0.15. The ion Larmor radius ρi and the electron skin depth c/ω pe are related through
ρi ω pe
=
c
r
mi 4πnTi
=
me B 2
r
mi
β,
2me i
where β i = 8πnTi /B 2 is the ion β factor. In deuterium plasma with β i = 1%, the ratio is approximately
4.3. As the wavelength decreases or k⊥ ρi increases, the lower unstable edge of electron mode should appear
at k⊥ ' ωpe /c. In this regime, neither ions nor electrons are adiabatic. In addition, stability analysis
must be done in terms of electromagnetic mode equation as evident from the appearance of the skin depth.
Further decrease in the wavelength brings in the regime k⊥ λDe ' 1 where λDe is the electron Debye length.
It is noted that in tokamaks, Ωe & ω pe holds in general, and mode with k⊥ λDe ' 1 satisfies k⊥ ρe < 1.
The well known mode in this regime is the electron temperature gradient (ETG) mode. Charge neutrality
breaks down in the ETG mode in practical tokamak discharges characterized by λDe > ρe . In this case, the
maximum growth rate of the ETG mode becomes strongly dependent on the β e factor, not because of finite
β destabilization but because of charge non-neutrality.
The maximum growth rate of the ETG mode is of the order of the electron transit frequency kk vT e . This
is another major difference from the ITG mode in which ion transit frequency is subdominant, |ω| > kk vT i .
1
The electron thermal diffusivity due to the ETG mode is much larger than
χe '
vT e ρ2e
=
LT
r
me cs ρ2s
,
mi LT
which would be expected if the ITG and ETG modes were completely dual, namely, if charge neutrality holds
in both modes, ions are adiabatic in the ETG mode, as electrons are in the ITG mode, and |ω| > kk vT i ,
ω > kk vT e hold respectively in the ITG and ETG mode. However, as noted, charge neutrality breaks down
in the ETG mode which is characterized by ω < kk vT e . The duality does not hold in tokamaks, and the
following electron thermal diffusivity emerges from the ETG mode,
qvT e
χe '
LT
5.2
µ
c
ω pe
¶2
p
βe.
Local Kinetic Dispersion Relation
We continue to employ the gyro-kinetic equations subject to the conditions that ω ¿ Ωi (¿ Ωe ) and plasma
nonuniformity scale lengths be much larger than the ion gyro radius. The maximum frequency and growth
p
rate of interest does not exceed the electron transit frequency ω T e = Te /me /qR. The condition ω ' ωT e
¿ Ωi becomes
ρe ¿
me
qR,
mi
where ρe is the electron Larmor radius which is of the order of 10−4 m or less. In the RHS,
me
mi qR
' 10−3
m. Therefore, the condition ω ¿ Ωi is satisfied with a large margin. The second condition ρi ¿ LT e is also
well satisfied even in the internal transport barrier (ITB) characterized by steep density and temperature
gradient. In the ETG mode, ions are essentially adiabatic,
ni ' −
eφ
n0 ,
Ti
particularly in the regime where the growth rate peaks. As will be shown, this occurs at k⊥ ' 0.7kDe
where kDe is the electron Debye wavenumber. However, in the lower end of the k spectrum, the wavelength
approaches the electron skin depth, k⊥ ' ω pe /c, where ions are not adiabatic. Note that c/ω pe is comparable
with the ion Larmor radius ρi . In order to cover the entire spectrum of the ITG and ETG modes satisfactorily,
fully kinetic, electromagnetic ion and electron responses without the assumption of adiabatic ions or electrons
must be employed.
As in Chapter 4, we assume that the unperturbed distribution functions are Maxwellian. The perturbed
ion and electron distribution functions are:
eφ
ω+ω
b ∗i
fi = − fMi +
J2
Ti
ω − kk vk + ω
b Di 0
2
µ
k⊥ v⊥
Ωi
¶³
vk ´ e
φ − Ak
fMi ,
c
Ti
(5.1)
fe =
eφ
ω−ω
b ∗e
fMi −
J2
Te
ω − kk vk − ω
b De 0
µ
k⊥ v⊥
Ωe
¶³
vk ´ e
φ − Ak
fMe ,
c
Te
(5.2)
where φ is the scalar potential and Ak is the vector potential parallel to the magnetic field,
ω
b ∗i,e (v2 ) =
·
µ
¶¸
cTi,e
mi,e v2 3
−
1
+
η
[∇(ln n0 ) × B] · k⊥ ,
i,e
eB 2
2Ti,e
2
cmi,e
eB 3
ω
b Di,e (v) =
µ
¶
1 2
v⊥ + vk2 (∇B × B) · k⊥ ,
2
(5.3)
(5.4)
J0 is the Bessel function, and kk is gradient operator along the magnetic field. The magnetosonic perturbation
is ignored in light of low β tokamak discharges.
The Poisson’s equation,
∇ φ = −4πe
Z
(fi − fe ) dv,
(5.5)
4πe
=−
c
Z
vk (fi − fe ) dv,
(5.6)
2
and the parallel Ampere’s law,
∇2⊥ Ak
yields the following electromagnetic dispersion relation,
#
µ
½
¾"
¶2
³ ω ´2
³ ω ´2
k
pe
pi
2
Fe2 + 2
Fi2
− τ (1 − Fi0 )
k⊥ + 2
Fe0 − 1 −
c
c
kDe
=2
where
Fej =
Fij =
³ ω ´2 µ
pe
c
*µ
*µ
vk
vT e
¶j
vk
vT i
¶j
Fe1 +
r
τ me
Fi1
mi
¶2
,
+
ω−ω
b ∗e
2
J (Λe ) ,
ω−ω
b De − kk vk 0
+
ω+ω
b ∗i
2
J (Λi ) ,
ω+ω
b Di − kk vk 0
(5.7)
(5.8)
(5.9)
h· · ·i indicating averaging over the velocity with Maxwellian weighting. The norms of the differential operators based on a simple trial eigenfunction φ (θ) = 1 + cos θ, |θ| ≤ π are:
µ
¶
­ 2®
5 2
π 2 − 7.5 2 10
2
s − sα + α ,
k⊥ = kθ 1 +
3
9
12
hω D i = 2εn ω∗
µ
¶ D E
1
5
2 5
+ s − α , kk2 =
f (s, α),
3 9
12
3(qR)2
(5.10)
(5.11)
where
f(s, α) =
1 + (π2 /3 − 0.5)s2 − 8sα/3 + 3α2 /4
.
1 + (π2 /3 − 2.5)s2 − 10sα/9 + 5α2 /12
(5.12)
The validity of the local dispersion relation has been checked by comparing the mode frequency ω = ω r + iγ
3
of the ITG with that those found from the method based on integral equations.
5.3
Integral Equation Approach
Although the local kinetic dispersion relation is expected to describe low frequency modes in tokamaks
qualitatively, its validity and accuracy should be checked against a more rigorous approach based on integral
equations in the ballooning space. In particular, the norm of the parallel gradient operator kk has to be
justified through comparison between eigenvalues found from the local kinetic dispersion relation and from
nonlocal analysis. For long wavelength modes including ITG and kinetic ballooning modes, the validity of the
local kinetic dispersion relation has been well established. More recently, nonlocal analysis has been carried
out of shorter wavelength (skin size) drift mode. As the wavelength becomes even shorter, charge neutrality
breaks down and the set of integral equations are to be modified to implement the Poisson’s equation for
the scalar potential.
We again consider a high temperature, low β tokamak discharge with eccentric circular magnetic surfaces.
The frequency regime of interest is ω bi < ω . ω be , where ω bi(e) is the trapped ion (electron) bounce frequency.
The magnetosonic perturbation (A⊥ ) is ignored in light of the low β assumption and we employ the twopotential (φ and Ak ) approximation to describe electromagnetic modes. The basic field equations are the
Poisson’s equation,
and the parallel Ampere’s law,
£
¤
∇2 φ = −4πe ni (φ, Ak ) − ne (φ, Ak ) ,
∇2⊥ Ak = −
4π
J (φ, Ak ),
c k
(5.13)
(5.14)
where the density perturbations are given in terms of the perturbed velocity distribution functions fi and
fe as
ni =
Z
fi dv,
and the parallel current by
Jk = e
Z
ne =
Z
fe dv,
vk (fi − fe )dv.
(5.15)
(5.16)
The distribution functions fi and fe are given by
eφ
fMi + gi (v, θ)J0 (Λi ),
Ti
(5.17)
eφ
fMe + ge (v, θ)J0 (Λe ),
Te
(5.18)
fi = −
fe =
4
where gi,e are the nonadiabatic parts that satisfy
µ
¶
³
vk (θ) ∂
vk ´ e
+ω+ω
b Di gi = (ω + ω
i
b ∗i )J0 (Λi ) φ − Ak
fMi ,
qR ∂θ
c
Ti
(5.19)
µ
¶
³
vk ´ e
vk (θ) ∂
+ω−ω
b De ge = −(ω − ω
b ∗e )J0 (Λe ) φ − Ak
fMe .
i
qR ∂θ
c
Te
(5.20)
Here, θ is the extended poloidal angle, φ is the scalar potential, Ak is the parallel vector potential, J0 is the
Bessel function with argument Λi,e = k⊥ v⊥ /ω ci,e , and qR is the connection length.
For circulating particles, gj (j = i, e) can be integrated as
vk > 0,
gj+
ej fMj
= −i
Tj
Z
vk < 0,
gj−
ej fMj
= −i
Tj
Z
θ
−∞
∞
θ
where
0 ¯qR¯ iβ j
dθ ¯ ¯ e
vk
(ω −
0 ¯qR¯ −iβ j
dθ ¯ ¯ e
vk
β j (θ, θ0 ) =
Z
θ
θ0
ω
b ∗j )J0 (Λ0j )
(ω −
Ã
ω
b ∗j )J0 (Λ0j )
!
Ak (θ ) ,
(5.21)
!
(5.22)
eiσβ(θ,θ ) γ σ dθ0 ,
(5.23)
0
φ(θ ) −
Ã
0
φ(θ ) +
¯ ¯
¯v ¯
k
c
¯ ¯
¯vk ¯
c
0
0
Ak (θ ) ,
qR
¯ ¯ [ω − ω
b Dj (θ00 )]dθ00 .
¯vk ¯
For trapped particles with turning points θ 1 and θ2 (θ 2 > θ1 ), the solution is
gσ =
0
0
eiσβj (θ,θ )sin(θ−θ )
2 sin[β(θ, θ 0 )]
Z
θ2
θ1
Z
³
´
0
0
e−iβ(θ2 ,θ )sgn(θ) + eiβ(θ2 ,θ )sgn(θ) dθ 0 − iσ
θ
0
θ1
where σ = sgn(vk ),
γ σ = γ φ + σγ A ,
(5.24)
ej qR
¯ ¯ (ω − ω∗j )J0 (Λj )φ(θ0 )fMj ,
Tj ¯vk ¯
¯ ¯
¯vk ¯
ej qR
¯
¯
Ak (θ 0 )fMj .
γ A = − ¯ ¯ (ω − ω∗j )J0 (Λj )
Tj vk
c
γφ =
(5.25)
(5.26)
Since for electrons, β(θ2 , β 1 ) is of order of ω/ω be ¿ 1 where ω be is the electron bounce frequency, trapped
electron response may be approximated by
geσ
1
'
2β(θ2 , θ 1 )
Z
θ2
θ1
0
0
(γ φ + iβ(θ2 , θ )γ A )dθ − i
Z
θ
γ A dθ0 .
(5.27)
θ1
In this analysis, we ignore trapped ions since the frequency regime of interest is at least of order of the
ion transit frequency. Substitution of perturbed distribution functions into charge neutrality and parallel
5
Ampere’s law yields
∇2 φ = −4π
X
j
¶
µ
Z
£ +
¤
ej
ej − φ +
gj (θ) + gj− (θ) J0 (Λj )dv ,
Tj
∇2⊥ Ak (θ) = −
where
R
dv = 2π
R∞
0
v⊥ dv⊥
R∞
0
Z
£
¤
4π X
ej vk gj+ (θ) − gj− (θ) J0 (Λj )dv,
c j
(5.28)
(5.29)
dvk . This system of inhomogeneous integral equations can be solved by
employing the method of Fredholm in which the integral equations are viewed as a system of linear algebraic
equations.
5.4
Short Wavelength ITG Modes
It is generally conjectured that the ITG mode should be deactivated in the regime (k⊥ ρi )2 À 1 since ion
dynamics tends to be adiabatic. This is not the case if ωDi < |ω| < ω ∗i where ω Di is the ion magnetic drift
frequency and ω ∗i is the ion diamagnetic drift frequency. Consider the ion perturbation in slab geometry
(ωDi = 0) with negligible ion transit frequency |ω| À kk vT i ,
or its integral
·
µ
¶¸
eφ
ω+ω
b ∗i 2
eφ
Mv2 3
J0 (Λi ) fMi , ω
b ∗i = ω ∗i 1 + ηi
−
fi = − fMi +
,
Ti
ω
Ti
2Ti
2
ni ' −
·
¸
µ
¶
eφ
ω + ω∗i −bi
eφ
∂
e I0 (bi )
n0 +
n0 , ω ∗i = ω ∗i 1 + ηi Ti
, bi = (k⊥ ρi )2 .
Ti
ω
Ti
∂Ti
(5.30)
(5.31)
If |ω| ¿ ω ∗i , the ion density perturbation in the limit bi À 1 approaches
·
µ
¶¸
p
vT i /Ln
1
eφ
n0 , vT i = Ti /M,
ni ' −1 + √
1 − ηi
2
Ti
2πω
(5.32)
which is far from adiabatic. If |ω| ¿ kk vT e (electron transit frequency), the following stable mode emerges,
ω'
vT i /Ln
√
2 2π
µ
¶
1
τ
, τ = Te /Ti .
1 − ηi
2
1+τ
(5.33)
In short wavelength regime, the mode frequency becomes non-negligible compared with the electron transit
frequency and electron Landau damping can destabilize the mode.
In toroidal geometry, stability analysis of short wavelength ITG mode requires use of the integral equations
in order to implement the ion and electron transit effects (ion and electron Landau damping) in a satisfactory
manner. This has been done recently in A. Hirose, Phys. Rev. Lett. 92, 025001 (2004). Main findings
are as follows. (a) In the slab limit (small toroidicity εn = Ln /R), a strong temperature gradient driven
6
ion mode persists in the regime bi À 1. The instability requires both ηi and ηe above critical values. (b)
Toroidicity has significant stabilizing influence on the mode. Stabilization occurs for Ln /R & 0.15. (c) The
p
instability is driven by magnetic shear and the growth rate is approximately proportional to |s| where s
is the shear parameter, either positive or negative. (d) As in the case of long wavelength η i mode in the
regime bi < 1, the mode is stabilized by a modest α, the ballooning parameter. And, (e) trapped electrons
have no significant influence on the mode in short wavelength regime.
Figure 5-1: Mode frequency (a) and growth rate (b) normalized by cs /Ln as functions of bs = (kθ ρs )2
when τ = Te /Ti = 1, εn = Ln /R = 0.1, η i = η e = 2.5, me /mi = 1/1837 (hydrogen), s = 1.5, q = 1.5,
β i = β e = 0.1 %.
Figure 1 shows the mode frequency normalized by cs /Ln as a function of bs = (kθ ρs )2 = kθ2 Te /mi Ω2i
when εn = Ln /R = 0.1, η e = η i = 2.5, s = q = 1.5, Ti = Te , β i = β e = 0.1 % and mi /me = 1837 (hydrogen
discharge). Negative frequency ω r < 0 indicates propagation in the ion diamagnetic direction. The first peak
7
in the growth rate at small bs (' 0.5) is the conventional long wavelength toroidal ηi mode. It is deactivated
as bs increases due to finite ion Larmor radius effect. However, the growth rate exhibits a second peak at
shorter wavelength bs ' 6. For the parameters assumed, bs = 6 corresponds to kθ c/ωpe ' 2.6. (Note that
when β i = β e ,
µ
kθ c
ω pe
¶2
= 2bs
me
,
mi β e
where me /mi is the electron/ion mass ratio and β e is the electron beta factor β e = 8πn0 Te /B 2 .) The mode
frequency ωr above bs = 2 becomes constant. In terms of the ion acoustic transit frequency ω s = cs /qR, the
normalized frequency is ωr /ωs ' −9. Therefore, parallel ion dynamics should not play a role. For electrons,
p
|ω| ' ωDe ' 0.2ω T e where ω T e = Te /me /qR is the electron transit frequency. Electron parallel resonance
(Landau damping) is thus expected to be important. Since ωr < 0, there is no resonance with the electron
magnetic drift ω De . Therefore, the instability is largely due to electron parallel resonance (Landau damping).
Too large toroidicity (ω De = 2εn ω ∗e ) should deactivates the instability because ion dynamics tends to be
adiabatic.
=
Figure 5-2: (a) Dependence of γ/ωs (ω s = cs /qR) on εn = Ln /R. (b) Dependence of γ/ω ∗e on ηi = η e .
Other parameters are: bs = 6, ηe = η i = 2.5, τ = 1, s = 1.5, q = 1.5, β i = β e = 0.1 %.
Dependence of the growth rate on the toroidicity εn = Ln /R and the temperature gradient (ηi = ηe assumed)
is shown in Fig. 2 (a) and (b), respectively. As expected, the instability is deactivated at large toroidicity
8
(εn & 0.15) . The instability persists in the limit εn → 0 which clearly indicates that the mode is of slab nature. Stabilization of the predominantly slab mode by toroidicity may be seen from the toroidal counterpart
of Eq. (1) ,
fi = −
eφ
ω+ω
b ∗i 2
eφ
fMi +
J (Λi ) fMi .
Ti
ω+ω
b Di 0
Ti
(5.34)
Since ω
b Di /ω ∗i ' 2εn = 2Ln /R, for large εn (toroidicity), the ion density perturbation
ni
eφ
n0 +
Ti
eφ
' − n0 +
Ti
' −
·
¸
ω + ω ∗i −bi
eφ
e I0 (bi )
n0
ω + ω Di
Ti
ω + ω∗i
1 eφ
√
n0 ,
ω + ωDi 2πbi Ti
tends to be adiabatic in short wavelength regime if |ω| < ω∗i , ω Di , and the ηi slab mode is stabilized by
toroidicity.
The critical temperature gradient is approximately ηcr ' 1.5 when εn = 0.1 and ηi = ηe . Behaviour of
the growth rate when ηi and ηe are varied independently has also been investigated. When ηi = 2.5, the
critical electron temperature gradient is ηe,cr ' 0.7 which is smaller than that of ions, ηi,cr ' 1.3 when
ηe = 2.5. It is evident that both η i and ηe above respective thresholds are required simultaneously for the
instability and the instability is of hybrid nature in the sense that neither electrons nor ions are adiabatic.
Dependence of the growth rate on the magnetic shear parameter s is shown in Fig. 3. Shear is destabilizing
in both positive and negative regimes and the dependence of the growth rate on s may be approximated by
p
γ ∝ const. |s|. It is evident that magnetic shear, either positive or negative, is required for this particular
instability. The magnetic shear s enters the stability analysis through the magnetic drift frequency ω D and
kk . Positive (negative) shear enhances (reduces) toroidicity. Since in the instability of concern, toroidicity
³
p ´
is relatively unimportant (it reduces the growth rate), the dependence of the growth rate on s γ ∝ |s|
mainly originates from the shear dependence of the parallel gradient, kk , which depends on the magnitude of
s. The instability therefore has features of the “universal mode” and is primarily driven by electron parallel
Landau damping.
Dependence of the mode frequency and growth rate on the safety factor q has also been investigated.
The instability is deactivated in the region q . 0.7 and q & 3.2. Stabilization at small q may be interpreted
as due to too large an electron transit frequency and stabilization at large q is due to finite β (or α, the
ballooning parameter) stabilization, since the ballooning parameter,
α=
q2 R
[(1 + ηe ) β e + (1 + ηi ) β i ] ,
Ln
(5.35)
rapidly increases with the safety factor q. α stabilization is similar to that in the conventional ITG mode
which is caused by coupling of electron dynamics to the magnetic perturbation Ak .
9
Figure 5-3: Dependence of the growth rate on magnetic shear. (a) s > 0 and (b) s < 0.bs = 6 and other
parameters are unchanged from Fig. 1.
5.5
Stability of the ETG Mode
In this Section, we investigate stability of modes with even shorter wavelength in the regime k⊥ ρe . 1 where
ρe is the electron Larmor radius. Ions tend to be adiabatic in such regime and we are primarily concerned
with the ETG mode. In tokamaks, ρe < λDe (or Ωe > ωpe ) generally holds. Therefore it is not appropriate
to assume charge neutrality. If charge neutrality does not hold, that is, if the term (k/kDe )2 is not negligible,
there arises apparent dependence of eigenvalue ω on plasma β (or plasma density) even in electrostatic limit.
The electrostatic dispersion relation of the ETG mode with adiabatic ions is
1+τ +
µ
k
kDe
¶2
=
¿
À
ω−ω
b ∗e
2
J (Λe ) ,
ω−ω
b De − kk vk 0
τ=
Te
.
Ti
(5.36)
The charge nonneutrality factor (k/kDe )2 ' (k⊥ /kDe )2 and the electron finite Larmor radius parameter
(k⊥ ρe )2 are related through
µ
k⊥
kDe
¶2
= (k⊥ ρe )2
2 Te
,
β e mc2
(5.37)
where Te /mc2 is the normalized electron temperature. Even in the electrostatic mode equation (and resultant
dispersion relation), β e has to be specified because the ballooning parameter α = αi + αe is one of the
10
parameters to characterize plasma equilibrium. The electron FLR parameter k⊥ ρe is of course the key
parameter in gyro-kinetic formulation. Therefore, when charge neutrality does not hold, the normalized
temperature Te /mc2 has to be specified together with various other dimensionless parameters. For a given
electron temperature, charge nonneutrality is evidently more enhanced at lower plasma density. Since the
term (k/kDe )2 is stabilizing, it is expected that the growth rate of the ETG mode becomes dependent on
β e (the plasma density). The growth rate of the ETG mode with charge neutrality (k⊥ /kDe )2 ¿ 1 and
negligible electron transit frequency ω À kk vT e is approximately given by
γ'
p
ηe ω ∗e ωDe /τ.
(5.38)
Charge non-neutrality reduces the growth rate as
γ'
r
ηe ω ∗e ω De
τ + (k⊥ /kDe )2
.
(5.39)
In short wavelength regime (k⊥ /kDe )2 > τ, the growth rate approaches
r
βe
,
γ'c
LT R
being proportional to
(5.40)
p
βe
To demonstrate the importance of charge nonneutrality in the ETG mode in terms of the roots of the local
kinetic electromagnetic dispersion relation, we show in Fig. 4 the dependence of mode frequency and growth
rate, both normalized by the electron transit frequency ω T e = vT e /qR, on the normalized perpendicular
wavenumber, de = (kθ /kDe )2 , for three values of β, β e = β i = 0.1%, 0.2% and 0.5% when Te = Ti = 10 keV
¡
¢
Te /me c2 ' 0.02 . Other parameters assumed are: Ln /R = 0.2, s = 1, q = 2, ηe = ηi = 2, mi /me = 1836
(hydrogen). The growth rate peaks at be ' 0.02 when β e = 0.1% and at be = 0.07 when β e = 0.5%. The
maximum growth rate increases with β e (actually with the plasma density) as expected. The plots in Fig. 5
clearly indicate that the maximum growth rate occurs approximately at a constant value of (kθ /kDe )2 ' 0.5
when the plasma density and electron temperature are varied. In this case, the conventional normalization
by the electron Larmor radius, be = (kθ ρe )2 , is not convenient because be at the maximum growth rate shifts
as β e is varied. It is then evident that in the ETG mode, the Debye length appears as an important scale
length.
It is noted that mode frequency ω is of the order of the electron transit frequency, ωT e = vT e /qR. This is
the major difference from the long wavelength ITG mode in which the ion transit frequency is subdominant,
|ω| > vT i /qR. Electron parallel Landau damping thus plays a major role in the ETG mode. Since kk ' 1/qR,
the mode frequency is expected to be sensitively dependent on the safety factor q. This is shown in Fig. 5
for Te = 10 keV. A relatively small value of β e = 0.2% is chosen to keep the ballooning parameter α, which
11
Figure 5-4: Mode frequency ω r /ω T e (solid line) and growth rate γ/ωT e (dotted line) as functions of de =
(kθ /kDe )2 when Te = Ti = 10 keV, s = 1, τ = 1, εn = 0.2, η i = η e = 2, q = 2, mi /me = 1836. β e (= β i ) is
scanned from 0.1 to 0.5%.
is proportional to q 2 , below the limit of drift reversal. The maximum growth rate γ/ω T e increases with q in
a manner approximately proportional to q 2 , γ max /ω T e ∝ q 2 . Since ω T e = vT e /qR, the unfolded growth rate
is proportional to q.
Dependence on the magnetic shear parameter s has also been investigated. Shear is destabilizing as in
the case of the short wavelength ITG mode discussed in the preceding Section. When q = 2 (α = 0.6 for the
assumed parameters), the critical shear parameter is scr ' −0.5 and when q = 3 (α = 1.35), scr ' 0. The
critical shear parameter is in qualitative agreement with the condition ω De > 0. Negative shear is stabilizing
in contrast to the case of the ITG mode. From these observations, it is evident that the ETG mode is driven
by both toroidicity and electron parallel resonance.
12
Figure 5-5: q dependence of (ω r + iγ) /ω T e when Te = Ti = 5 keV, β e = β i = 0.2%, s = 1, τ = 1, εn = 0.2,
ηi = ηe = 2, mi /me = 1836.
5.6
Mixing Length Estimate of χe
In this Section, mixing length estimate of the electron thermal diffusivity,
χe =
ω 2r
γ3
1
2 ,
+ γ 2 k⊥
(5.41)
is presented for the ETG mode. (The short wavelength ITG mode is stabilized by toroidicity and may not
2
be dangerous in practical tokamak discharges.) Fig. 6 shows χe in units of (vT e /qR) /kDe
when β e = 0.2%,
Te = Ti = 5 keV and 10 keV. The safety factor q is scanned between q = 2 and 4. The maximum value of
χe is proportional to q 2 but inversely proportional to the temperature. Since ωT e ∝ 1/q, this suggests that
the electron thermal diffusivity has the following scaling,
χe ∝
qvT e
.
n0
13
Results of scanning β e , electron temperature Te , and the safety factor q can be summarized by the following
electron thermal diffusivity,
χe '
qvT e
LT
µ
c
ω pe
¶2
p
βe.
(5.42)
Further studies will be needed to go beyond the simple mixing length estimate for the diffusivity. In particular, whether the ETG mode can drive a zonal flow and its consequence on thermal transport remains an
open question.
2
Figure 5-6: χe in units of ωT e /kDe
vs. de = (kθ /kDe )2 when β e = β i = 0.2% and q = 2, 3, 4. In (a),
Te = Ti = 5 keV and in (b), Te = Ti = 10 keV.
5.7
Conclusions
In this Chapter, recent investigations of short wavelength collisionless temperature gradient modes in tokamak geometry are reviewed. To summarize the features of the short wavelength ITG mode with (k⊥ ρi )2 À 1,
it has been shown that the instability is driven by parallel electron Landau damping as is the universal mode.
The mode is of slab nature and toroidicity has stabilizing effects. The growth rate rapidly decreases with
the toroidicity parameter εn = Ln /R and the instability is operative only when the density gradient is steep,
14
εn . 0.15. As in the case of the conventional long wavelength toroidal ITG mode, the instability is subject
to finite β (or α, the ballooning parameter) stabilization. The mode is insensitive to trapped electrons.
To summarize the findings made of the ETG mode, it has been shown that for practical tokamak discharges with ωpe < Ωe , charge neutrality breaks down and a natural normalization of the wavenumber for
the ETG mode is (k/kDe )2 , rather than (kρe )2 . The lower cutoff of the ETG mode occurs at k⊥ & ω pe /c.
The maximum electron thermal diffusivity occurs at (k/kDe )2 ' 0.1, namely, at wavelength longer than that
corresponding to the maximum growth rate, (k/kDe )2 ' 0.5.
15