Effects of the Second Harmonic on the GAM in Electron Scale Turbulence

0.18 MB·

by  Johan Anderson

Effects of the Second Harmonic on the Geodesic
4
Acoustic Mode in Electron Scale Turbulence
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0
2

n Johan Anderson1, Hans Nordman1 and Raghvendra Singh2,3
a
J 1 Dept. Earth and Space Sciences, Chalmers University of Technology, SE-412 96

4 G¨oteborg, Sweden
2 2 WCI, Daejon, South Korea
3
Institute for Plasma Research, Bhat, Gandhinagar, Gujarat, India 382428
]
h
E-mail: [email protected]
p
-
m
Abstract. The effects higher order harmonics have been self-consistently included
s
a in the derivation of the electron branch of the electron Geodesic Acoustic Mode (el-
l GAM)inanElectron-Temperature-Gradient(ETG)turbulencebackground. Thework
p
. is based on a two-fluid model including finite β-effects while retaining non-adiabatic
s
c ions. In solving the linear dispersion relation, it is found that the due to the coupling
i
s to the m = 2 mode the real frequency may be significantly altered and yield higher
y
values.
h
p
[

1 PACS numbers: 52.55.-s, 52.35.Ra,52.35.Kt
v
0
0
3
6
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1
0
4
1
:
v
i
X
r
a
Effects of the Second Harmonic on the GAM in Electron Scale Turbulence 2
1. Introduction
Research during recent years has provided the community with significantly increased
knowledge on the importance of coherent structures such as vortices, streamers and
zonal flows (m = n = 0, where m and n are the poloidal and toroidal mode numbers
respectively) in determining the overall transport in magnetically confined plasmas.
Zonal flows impede transport by shear decorrelation, whereas the Geodesic Acoustic
Mode (GAM) [1, 2, 3, 4, 5, 6, 7, 8, 9] is the oscillatory counterpart of the zonal flow
(m = n = 0 in the potential perturbation, m = 1, n = 0 in the perturbations in
density, temperature and parallel velocity) and thus a weaker effect on turbulence is
expected. Nevertheless experimental studies suggest that GAMs are related to the
L-H transition and transport barriers. The GAMs are weakly damped by Landau
resonances and moreover this damping effect is weaker at the edge suggesting that
GAMs are more prominent in the region where transport barriers are expected. [3]
Evidence of interactions between the turbulence driven E~ B~ zonal flow oscillation
×
or Geodesic Acoustic Mode (GAM), turbulence and the mean equilibrium flows during
this transition was found. Furthermore, periodic modulation of flow and turbulence
level with the characteristic limit cycle oscillation at the GAM frequency was present.
[2] Moreover, in Ref. [10], it was observed that GAMs are only somewhat less effective
than the residual zonal flow in providing the non-linear saturation.
For heat transport in the electron channel a likely candidate is the Electron
Temperature Gradient (ETG) mode driven by a combination of electron temperature
gradients and field line curvature effects. [11, 12, 13, 14, 15, 16] The short scale
fluctuations that determines the ETG driven heat transport do not influence ion heat
transport and is largely unaffected by the large scale flows stabilizing ion-temperature-
gradient (ITG) modes. The generation of large scale modes such as zonal flows
and GAMs is here realized through the Wave Kinetic Equation (WKE) analysis that
is based on the coupling of the micro-scale turbulence with the GAM through the
WKE under the assumptions that there is a large separation of scales in space and
time. [8, 17, 18, 19, 20, 21, 22] In non-linear gyrokinetic simulations large thermal
transport levels, beyond mixing length estimates have been observed for a long time.
[23, 13, 24, 25, 10, 26]
In recent work the el-GAM, the finite β-effects were elaborated on, and numerical
quantifications of the frequency and growth rate were given in Refs. [27, 28]. The
finite β-effects were added in an analogous way compared to the recent work on zonal
flows in Ref. [29, 30]. In particular, the Maxwell stress was included in the generation
of the el-GAM. The frequency of the el-GAM is higher compared to the ion GAM
by the square root of the ion-to-electron mass ratio (Ω (electron)/Ω (ion) m /m
q q i e
≈
where Ω (electron) and Ω (ion) are the real frequencies of the electron and ionqGAMs,
q q
respectively.). It was foundthat similar to the linear growthratethe finite β effects were
stabilizingtheGAMusingamodecouplingsaturationlevel. Furthermore, increasingthe
non-adiabaticityparameter(Λ )decreasedthegrowthratethroughalinearcontribution.
e
Effects of the Second Harmonic on the GAM in Electron Scale Turbulence 3
It isinteresting to note that in simulations, damping of the GAMdue to coupling to
higher m modes has been found. [31, 32, 33] In a careful evaluation of the contributions
from higher m modes it can be shown that they are, in general, of the order ǫ smaller.
n
However, the effect of higher harmonics is increased by the square of the safety factor
(q¯) and thus in order to evaluate the effects a more detailed study is called for.
To this end, in this work a detailed investigation of the effects of the higher
harmonics on the el-GAM driven by electron temperature gradient (ETG) modes is
presented. Wehaveutilizedatwo-fluidmodel fortheETGmodebasedontheBraginskii
equations with non-adiabatic ions including impurities and finite β - effects. [14, 16]
It is shown that the effects of the second harmonics of the density and temperature
perturbations on the linear GAM frequency and non-linear generation of the GAM,
found in Ref [27] can be significant and elevate the frequency of the el-GAM similar to
what was discovered in Ref [34].
The remainder of the paper is organized as follows: In Section II the linear ETG
mode including the ion impurity dynamics is presented. The linear el-GAM is presented
and the non-linear effects are discussed in Section III. A quantification of the effects of
the second harmonics is presented in Sec. IV and the paper is summarized in Sec. V.
2. The linear Electron Temperature Gradient Mode
We will start be giving the preliminaries of the Electron-Temperature-Gradient mode
described by a two-fluid model. The ETG mode is considered under the following
restrictions on real frequency and wavelength: Ω ω ω << Ω , k c > ω > k c .
i ⋆ e ⊥ i k e
≤ ∼
Here Ω are the respective cyclotron frequencies, ρ the Larmor radii and c = T /m
j j j j j
the thermal velocities. The diamagnetic frequency is ω k ρ c /L , k and kqare the
⋆ θ e e n ⊥ k
∼
perpendicular andtheparallelwave numbers. TheETG modelconsists ofacombination
of ion and electron fluid dynamics coupled through quasineutrality, including finite β-
effects [14, 16]. First, we will describe the electron dynamics for the toroidal ETG
mode governed by the continuity, parallel momentum and energy equations adapted
from the Braginskii fluid equations. The electron equations are analogous to the ion
fluid equations used for the toroidal ITG mode,
∂n
e
+ (n ~v +n ~v )+ (n ~v +n ~v )+ n ~v = 0, (1)
e E e ⋆e e pe e πe e ke
∂t ∇· ∇· ∇·
(cid:16) (cid:17)
3 dT
e
n +n T ~v + ~q = 0. (2)
e e e e e
2 dt ∇· ∇·
Here we used the definitions ~q = (5p /2m Ω )eˆ T as the diamagnetic heat flux,
e e e e k e
− ×∇
~v is the E~ B~ drift, ~v is the electron diamagnetic drift velocity, ~v is the electron
E ⋆e Pe
×
polarization drift velocity, ~v is the stress tensor drift velocity, and the derivative is
π
defined as d/dt = ∂/∂t + ρ c ˆe φ . A relation between the parallel current
e e k
× ∇ · ∇
density and the parallel component of the vector potential (A ) can be found using
k
e
Amp`ere’s law,
4π
2A = J . (3)
∇⊥ k − c k
e e
Effects of the Second Harmonic on the GAM in Electron Scale Turbulence 4
Taking into account the diamagnetic cancellations in the continuity and energy
equations, the Eqs. (1, 2 and 3) can be simplified and written in normalized form as
∂n ∂ 1 ∂
e 2 φ 1+(1+η ) 2 φ 2A +
− ∂t −∇⊥∂t − e ∇⊥ r∂θ −∇k∇⊥ k
e 1 ∂ e (cid:16)∂ (cid:17) e e
ǫ cosθ +sinθ φ n T =
n e e
r∂θ ∂r! − −
(cid:16) (cid:17)
(β /2) A , 2A + φ, e2φe, e (4)
− e k ∇k k ∇
h ∂ i h i
β /2 e2 e+(1+eη )(eβ /2) A + φ n T =
e −∇⊥ ∂t e e ∇y! k ∇k − e − e
(cid:16) (cid:17) (cid:16) (cid:17)
(β /2) φ n ,A +(β /2) T ,A +eφ, 2A e, e e (5)
− e − e k e e k ∇⊥ k
∂ 5h 1 i∂ h∂ 1 ∂i h i2 1 ∂ 2 ∂
T + ǫe ceosθe +sinθ e e T +e η e φ n = φ,T .
e n e e e e
∂t 3 r∂θ ∂r! r∂θ (cid:18) − 3(cid:19) r∂θ − 3∂t −h i
e e e e e e(6)
Note that similar equations have been used previously in estimating the zonal flow
generation in ETG turbulence and have been shown to give good agreement with linear
gyrokinetic calculations [14, 16]. The variables are normalized according to
φ,n,T = (L /ρ )(eδφ/T ,δn /n ,δT /T ), (7)
e n e eo e 0 e e0
(cid:16) (cid:17)
A = (2c L /β cρ )eA /T , (8)
ke e e e n e e k e0
β = 8πnT /B2, (9)
ee e 0
2L
n
ǫ = , (10)
n
R
L
n
η = . (11)
e
L
Te
Here, R is the major radius and [A,B] = ∂A1∂B 1∂A∂B is the Poisson bracket. The
∂r r ∂θ − r ∂θ ∂r
gradient scale length is defined as L = (dlnf/dr)−1.
f
−
Next, we will describe the ion fluid dynamics in the ETG mode description. In the
limitω > k c theionsarestationaryalongthemeanmagneticfieldB~ (whereB~ = B eˆ )
k e 0 k
whereas in the limit k c >> ω, k ρ >> 1 the ions are unmagnetized. In this paper
⊥ i ⊥ i
we will use the non-adabatic responses in the limits ω < k c < k c , where c = TI
⊥ I ⊥ i I mI
is the impurity thermal velocity, and we assume that Ω < ω < Ω are fulfilled forqthe
i e
ions and impurities. In the ETG mode description we can utilize the ion and impurity
continuity and momentum equations of the form
∂n
j
+n ~v = 0, and (12)
j j
∂t ∇·
∂~v
j
m n +en φ+T n = 0, (13)
j j j j j
∂t ∇ ∇
where j = i for ions and j = I for impurities. Now, we derive the non-adiabatic ion
response with τ = T /T and impurity response with with τ = T /T , respectively. We
i e i I e I
Effects of the Second Harmonic on the GAM in Electron Scale Turbulence 5
thus have
zτ
j
n = φ. (14)
j
−1 ω2/ k2c2 
− ⊥ j
e  (cid:16) (cid:17) e
Here T and n are the mean temperature and density of species (j = e,i,I), where
j j
n = δn/n , n = δn /n and φ = eφ/T are the normalized ion density, impurity
i i I I I e
density and potential fluctuations and z is the charge number of species j. Next we
e
e e
present the linear dispersion relation. Using the Poisson equation in combination with
(14) we then find
τ n /n (Z2n /n )τ
n = i i e + I e I +k2λ2 φ. (15)
e − 1−ω2/k⊥2c2i 1−ω2/(k⊥2c2I) ⊥ De!
e
Considering thee linear dynamical equations (4, 5 and 6) and utilizing Eq. (15) as in
Ref. [16] we find a semi-local dispersion relation as follows,
β
ω2 Λ + e(1+Λ ) +(1 ǫ¯ (1+Λ ))ω +
e e n e ⋆
" 2 ! −
5
k2ρ2(ω (1+η )ω ) ω ¯ǫ ω +
⊥ e − e ⋆ − 3 n ⋆
i(cid:18) (cid:19)
β 2 2
e
ǫ¯ ω ω (η )ω + ωΛ =
n ⋆ e ⋆ e
− 2 !(cid:18) − 3 3 (cid:19)
(1+Λ ) ω 5ǫ¯ ω η 2 ω 2ωΛ
c2k2k2ρ2 e − 3 n ⋆ − e − 3 ⋆ − 3 e . (16)
e k ⊥ e ω(cid:16)βe +k2ρ2(cid:17) β(cid:16)e (1+η(cid:17))ω 
2 ⊥ e − 2 e ⋆
 (cid:16) (cid:17) 
InthefollowingwewillusethenotationΛ = τ (n /n )/(1 ω2/k2c2)+τ (z n /n )/(1
e i i e − ⊥ i I eff I e −
ω2/k2c2) + k2λ2 . Here we define z z2n /n . Note that in the limit T = T ,
⊥ I ⊥ De eff ≈ I e i e
ω < k c , k λ < k ρ 1 and in the absence of impurity ions, Λ 1 and
⊥ i ⊥ De ⊥ e e
≤ ≈
the ions follow the Boltzmann relation in the standard ETG mode dynamics. Here
λ = T /(4πn e2) is the Debye length, the Debye shielding effect is important
De c e
for λ /qρ > 1. The dispersion relation Eq. (16) is analogous to the toroidal
De e
ion-temperature-gradient mode dispersion relation except that the ion quantities are
exchanged to their electron counterparts. Eq. (16) is derived by using the ballooning
mode transform equations for the wave number and the curvature operator,
2f = k2f = k2 1+(sθ αsinθ)2 f, (17)
∇⊥ − ⊥ − θ −
(cid:16) (cid:17)
1 ∂f
e e e
f = ik f , (18)
k k
∇ ≈ qR∂θ
e
ǫ fe = ǫ (ecosθ+(sθ αsinθ)sinθ)f = ǫ g(θ)f. (19)
n n n
−
The geometricalequantities will be determined usingea semi-loceal analysis by assuming
e
an approximate eigenfunction while averaging the geometry dependent quantities along
the field line. The form of the eigenfunction is assumed to be
1
Ψ(θ) = (1+cosθ) with θ < π. (20)
√3π | |
Effects of the Second Harmonic on the GAM in Electron Scale Turbulence 6
In the dispersion relation we will replace k = k , k = k and ω = ω by the
k k ⊥ ⊥ D D
h i h i
averages defined through the integrals D E
1 π s2 10 5
k2 = dθΨk2Ψ = k2 1+ π2 7.5 sα+ α2 , (21)
D ⊥E N (Ψ) Z−π ⊥ θ 3 (cid:16) − (cid:17)− 9 12 !
1 π 1
k2 = dθΨk2Ψ = , (22)
k N (Ψ) k 3q2R2
D E Z−π
1 π 2 5 5
ω = dθΨω Ψ = ǫ ω + s α = ǫ gω , (23)
D D n ⋆ n ⋆
h i N (Ψ) Z−π (cid:18)3 9 − 12 (cid:19)
1 π k2 π2 8 3
k k2k = dθΨk k2k Ψ = θ 1+s2 0.5 sα+ α2 ,
D k ⊥ kE N (Ψ) Z−π k ⊥ k 3(qR)2 3 − !− 3 4 !
(24)
π
N(Ψ) = dθΨ2. (25)
Z−π
Here we have from the equilibrium α = βq2R(1+η +(1+η ))/(2L ) and β =
e i n
8πn (T +T )/B2 is the plasma β, q is the safety factor and s = rq′/q is the magnetic
o e i
shear. The α-dependent term above (in Eq.16) represents the effects of Shafranov shift.
3. Modeling Electron Geodesic Acoustic modes
In this section we will describe the derivation of the dispersion relation for the electron
Geodesic Acoustic Modes including the m = 2 higher harmonic coupling to the m = 1
and m = 0 components. The GA mode is defined as having m = n = 0, k = 0
r
6
perturbation of the potential field and the n = 0, m = 1, k = 0 perturbation in the
r
6
density, temperatures and the magnetic field perturbations. [1, 8] In addition we will
now consider the m = 2 components of the density, temperature and magnetic field
perturbations. The GAM (q,Ω ) induced by ETG modes (k,ω) is considered under
q
the conditions when the ETG mode real frequency satisfies Ω > ω > Ω at the scale
e i
k ρ < 1 and the real frequency of the GAM fulfils Ω c /R at the scale q < k .
⊥ e q e r r
∼
We start by deriving the linear electron GAM dispersion relation following the outline
in the previous paper Ref. [27, 28], by writing the m = 1 and m = 2 equations for the
density, parallel component of the vector potential and temperature, and the m = 0 of
the electrostatic potential, respectively. Starting with the m = 0 component,
∂ ∂
2 φ(0) ǫ sinθ n(1) +T(1) = 0, (26)
−∇⊥∂t G − n ∂r eG eG
(cid:16) (cid:17)
and then the m = 1 Eqeuations, e
e
(1)
∂n ∂
eG +ǫ sinθ φ(0) n(2) +T(2) 2A(1) = 0, (27)
− ∂t n ∂r G − eG eG −∇k∇⊥ kG
e ∂ (cid:16) (cid:16) (cid:17)(cid:17)
β /2 2 A(1) e n(1e) +T(1)e e = 0, (28)
e −∇⊥ ∂t kG −∇k eG eG
(cid:16) (cid:17) (cid:16) (cid:17)
∂ 2 ∂
T(1) n(1)e e e = 0. (29)
∂t eG − 3∂t eG
e e
Effects of the Second Harmonic on the GAM in Electron Scale Turbulence 7
Finally the m = 2 Equations,
(2)
∂n ∂
eG 2A(2) ǫ sinθ n(1) +T(1) = 0, (30)
− ∂t −∇k∇⊥ kG − n ∂r eG eG
βe e 2 ∂ A(2e) n(2) +T(cid:16)(2e) e (cid:17) = 0, (31)
2 −∇⊥! ∂t kG −∇k eG eG
(cid:16) (cid:17)
∂ 5 e ∂ 2 ∂ e
(2) (1) e (2)
T + ǫ sinθ T n = 0. (32)
∂t eG 3 n ∂r eG − 3∂t eG
Using Eqs. (26) -e(32), we will derivee the lineaer GAM frequency, by obtaining a relation
(1) (1)
of the form T = C n eliminating the m = 2 components. We continue by noting
eG 0 eG
that the Eqs. (29) and (32) are symmetric in m using the Fourier representation we
e
e
find,
2 5ǫ q
(1) (1) n r (2)
T = n + sinθ T , (33)
eG 3 eG 3 Ω eG
q
e(2) 2e(2) 5ǫnqr e(1)
T = n + sinθ T . (34)
eG 3 eG 3 Ω eG
q
We will use Eqe. (33) toe derive a relationebetween the second harmonic (m = 2) of the
density perturbation expressed in terms of the first harmonic (m = 1) variables. Eqs
(30) and (31) yield,
Ω n(2) +q q2A(2) ǫ sinθ q (n(1) +T(1)) = 0, (35)
− q eG k ⊥ kG − n r eG eG
β
e +eq2 Ω A(2e) +q n(2) +T(2e) e = 0. (36)
2 ⊥! q kG k eG eG
(cid:16) (cid:17)
In order to obtain the desiered resultewe usee Eq. (36) and substitute the m = 2
temperature perturbation by Eq. (34) and we find,
5 q ǫ q
(2) k (2) n r (2)
A = n + sinθ T . (37)
kG −3 β2e +q⊥2 Ωq eG Ωq eG!
e (cid:16) (cid:17) e e
Now employ Eq. (35) and eliminate the parallel vector potential finding,
5 q2q2 ǫ q
(2) k ⊥ (2) n r (1)
n = n + sinθ T (38)
eG − 3Ωq β2e +q⊥2 eG Ωq eG!
e ǫnqr (cid:16) (2) (cid:17) (e1) e
sinθ n +T . (39)
− Ω eG eG
q
(cid:16) (cid:17)
Collecting terms and re-arrangineg, we fiend a remarkably simple relation for the second
harmonic density perturbation in terms of the m = 1 components,
ǫ q 1
(2) n r (1) (1)
n = sinθ n +T , (40)
eG − Ω C eG eG
q (cid:18) 1 (cid:19)
e 5 q2q2 e e
k ⊥
C = 1+ . (41)
1
3Ω2 βe +q2
q 2 ⊥
Now the relation between the(cid:16)m = 1(cid:17)components of temperature and density will be
determined by using Eqs. (33) and (34),
2 10ǫ q
(1) (1) n r (2)
T = n + sinθ n
eG 3 eG 9 Ω eG
q
e
e e
Effects of the Second Harmonic on the GAM in Electron Scale Turbulence 8
25ǫ2q2
+ n r sin2θ T(1). (42)
9 Ω2 eG
q
e
Collecting terms and eliminating the m = 2 density perturbation gives,
1 5 ǫ2nqr2 sin2θ
T(1) = 2 − 3Ω2qC1 n(1), (43)
eG 3 1 5ǫ2nqr2 sin2θ eG
− 3 Ω2q
e
1 5 ǫ2nqr2 sin2θe
C = 2 − 3Ω2qC1 . (44)
0 3 1 5ǫ2nqr2 sin2θ
− 3 Ω2q
We have now obtained the desired coefficient C . Note that, neglecting contributions
0
from the m = 2 couplings C = 1 and the previous relation between the density and
1
temperature is recovered. A key element in determining the dispersion relation, is the
(0) (1)
relation between the φ and the m = 1 density perturbation n , this is found in
G eG
similar way as in Refs. [27, 28] by using Eq. (27) as,
e
e
q2q2 1+C 5 ǫ2q2
n(1) 1 k ⊥ 0 + n r sin2θ
eG" − Ω2q β2e +q⊥2 3Ω2qC1 #
ǫ q
e n r sinθ φ(0) = 0, (45)
− Ω G
q
e
(2) (2)
while noting that there is a simple relation for n +T as,
eG eG
5 ǫ q
n(2) +T(2) = n r sinθ n(1). e e (46)
eG eG −3Ω C eG
q 1
We can now deeterminee the dispersion relaetion for the GAM by considering the m = 0
component in Eq. (26) and in addition employ Eqs. (28) and (45),
q2q2 1+C 5 ǫ2q2 q2 ǫ2
1 k ⊥ 0 + n r sin2θ = r n (1+C )sin2θ. (47)
" − Ω2q β2e +q⊥2 3Ω2qC1 # q⊥2 Ω2q 0
Here, we employ averaging of the sine components as sin2θ = 1/2 over the poloidal
angle θ. We note that neglecting the m = 2 contributDions thEe coefficient C = 2 and
0 3
C = 1. Note that, in the limit of vanishing temperature perturbations C would be
1 0
zero. Furthermore, the third term on the right hand side comes from the coupling to
the m = 2 component. This is to be compared to the regular GAM frequency found in
Refs. [27, 28],
5 c2 1 1
Ω2 = e 2+ . (48)
q 3R2 q¯21+βe/(2qr2)!
Here, q¯ is the safety factor. Note that the linear electron GAM is purely oscillating
analogously to its ion counterpart c.f. Ref. [4] and its frequency is decreasing with
increasing q¯. Here it is of interest to note that it is very similar to the result found
in Ref. [35]. In order for the GAM to be unstable a non-linear driving by the ETG
background is needed. The non-linear state was presented in detail in Refs. [27, 28] and
thus only the main result is given. The non-linear extension to the evolution equations
Effects of the Second Harmonic on the GAM in Electron Scale Turbulence 9
presented previously in Eqs. (4)–(6) are
∂n ∂
e 2 φ 1+(1+η ) 2 φ 2A +
− ∂t −∇⊥∂t − e ∇⊥ ∇θ −∇k∇⊥ k
e 1 ∂ e (cid:16)∂ (cid:17) e e
ǫ cosθ +sinθ φ n T =
n e e
r∂θ ∂r! − −
(cid:16) (cid:17)
+ φ, 2φ (β /2) A , e2A e, e (49)
e k k
∇ − ∇
h i ∂ h i
βe/2 e 2 +(1e+η )e(β /2) A + φ n T = φ, 2A ,
e −∇⊥ ∂t e e ∇θ! k ∇k − e − e ∇⊥ k
(cid:16) (cid:17) (cid:16) (cid:17) h i
e e e e e e(50)
∂ 5 1 ∂ ∂ 1 ∂ 2 1 ∂ 2 ∂
T + ǫ cosθ +sinθ T + η φ n = φ,T .
e n e e e e
∂t 3 r∂θ ∂r! r∂θ (cid:18) − 3(cid:19) r∂θ − 3∂t −
h i
e e e e e e(51)
Herewewill keep thenon-linear terminthem = 0component whereasalltheotherscan
be considered small due to the fact that in evaluating the non-linear terms a summation
over the spectrum is performed and that the m = 1 non-linear terms are odd and thus
yield a negligible contribution to the non-linear generation of the GAM. The non-linear
contribution to the m = 0 potential perturbations are,
∂ ∂
2 φ(0) ǫ sinθ n(1) +T(1) =
−∇⊥∂t G − n ∂r eG eG
(cid:16) (cid:17)
φ , 2φe (0) (β /2) eA , 2eA (0) = N(0). (52)
k ∇ k − e kk ∇ kk 2
Dh iE Dh iE
In order to evaluate the Maxwell stress part in Eq. (52), we will approximate the parallel
e e e e
part of the electromagnetic vector potential with the electrostatic potential through a
linear relation. The relation A = A φ is found by using the Eqs. (5), (6) and the
kk 0 k
non-adiabatic response Eq. (15) giving an approximation of the total stress of the form
e e
N(0) = (1 Ω 2) φ , 2φ (0). (53)
2 −| α| k ∇ k
Dh iE
The Ω factor is found by using Eq. (5)
α e e
2
β k (1+Λ +ϕ )
Ω 2 = e k e 0 , (54)
| α| 2 (cid:12)(cid:12)(βe/2+k⊥2)ω −(1+ηe)βekθ/2(cid:12)(cid:12)
(cid:12) (cid:12)
where ϕ0 is determined by(cid:12) the temperature equation (cid:12)
(cid:12) (cid:12)
(η 2/3) 2/3ωΛ
e e
T = − − φ = ϕ φ , (55)
ek k 0 k
ω +5/3ǫ gk
n θ
and Λ is deteremined by the non-adiabaticeresponsee condition. The expression Eq. ( 54)
e
for the magnetic flutter non-linearity is comparable to that found in Ref. [15] except
that in Eq. ( 54) the adiabatic response is taken into account. Note that Ω vanishes
α
at β = 0. The relevant non-linear terms can be approximated in the following form
e
ω
φ , 2φ 1 Ω 2 q2 k k | r|δN (~q,Ω ). (56)
k ∇⊥ k ≈ −| α| r r θ ǫ k q
Dh iE (cid:16) (cid:17) Xk 0
In order to determeine thee non-linear generation of el-GAMs by the ETG modes will use
the wave kinetic equation [8, 17, 18, 19, 4, 20, 21, 22] to describe the background short
Effects of the Second Harmonic on the GAM in Electron Scale Turbulence 10
scale ETG turbulence for (Ω ,q) < (ω,k), where the action density N = E / ω
q k k r
| | ≈
ǫ φ 2/ω . Here ǫ φ 2, is the total energy in the ETG mode with mode number k
0 k r 0 k
wh|ere| ǫ = τ + k2 |+ η|e2kθ2. We assume that for all GAMs we have q > q , with the
0 ⊥ |ω|2 r θ
following relation between δN and ∂N /∂k ,
k k0 r
(1)
∂N k q T N
δN = iq2k φ0G(Ω ) 0k + θ r eG 0k , (57)
k − r θ G q ∂k τ √η η
r i e eth
e−
~ (0)
where we have used δω = k ~v i(k q k q )φ in the wave kinetic equation and
q · Eq ≈ θ r − r θ G
the definition G(Ω ) = 1 . Here the linear instability threshold of the ETG
q Ωq−qrvgr+iγk
mode is denoted by η and is determined by numerically solving Eq. (16). Using the
eth
results from the wave-kinetic treatment we can compute the non-linear contributions to
be of the form
φ, 2φ (0) = i 1 Ω 2 q4 k k2|ωr|G(Ω ) ∂Nkφ(0). (58)
∇⊥ − −| α| r r θ ǫ q ∂k G
0 r
Dh iE (cid:16) (cid:17) X
Wenotethattheenon-lienearcontributionispurelycomplexandthuswillsoelelydetermine
thegrowthrateoftheGAMs. Thegrowthrateswill behaveinthesamemanner asfound
in Ref. [28], however the real frequency of the GAM will be modified by the m = 2
contributions.
4. Results and Discussion
Here will quantify the effect of the contributions of the higher harmonics to the real
frequency of the el-GAM by numerically solving the dispersion relation found in Eq.
(47) while comparing the results with the corresponding values found by using the Eq.
(48).

1.6
1.5
1.4
R)
/
/(ce1.3
Ωq Linear Ω
q
1.2 Linear Ωq with m=2
1.1
1
1 1.5 2 2.5
q¯
Figure 1. (color online) The linear el-GAM real frequency (with m = 2 harmonics
included in black line and without represented by the red line) normalized to (ce/R)
as a function of the safety factor q¯is shown for the parameter ηe = 4.0 whereas the
remaining parameters are ǫn = 0.909, β = 0.01, qxρe = 0.3 in the strong ballooning
limit g(θ)=1.