critical scales for the destabilization of the toroidal ... · figure 2. growth rate as a function...

Under consideration for publication in J. Plasma Phys. 1

Critical scales for the destabilization of thetoroidal ion-temperature-gradient instabilityin magnetically confined toroidal plasmas

Summary of work done in collaboration withJ. W. Connor1, H. Doerk2, P. Helander2 and P. Xanthopoulos2

and presented by Alessandro Zocco2 at the First JPP Conference”Frontiers in Plasma Physics”,

Abazia della Santissima Trinità SpinetaMaggio 2017

1Culham Science Centre, Abingdon, Oxon, OX14 3DB, UK2Max-Planck-Institut für Plasmaphysik, D-17491, Greifswald, Germany

(Received 11 May 2017)

We present, for the first time, a complete mathematical treatment of the resonant limitof the ion-temperature-gradient (ITG) driven instability for magnetically confined fusionplasmas with arbitrary ion Larmor radii. The analysis is performed in the local kineticlimit, thus neglecting the variation of the eigenfunction along the magnetic field line, butis valid for arbitrary geometry (i). Analytical progress is made by introducing a Padéapproximant (ii) for the ion reponse after integrating resonaces (iii). These are treated byusing a new series representation of the Owen T−function [Owen, D B (1956). "Tablesfor computing bivariate normal probabilities". Annals of Mathematical Statistics, 27,1075–1090] which exploits the properties of the incomplete Euler gamma function [Tri-comi F. G., Sulla funzione gamma incompleta, Annali di Matematica Pura ed Applicata,31, 1, 263-279 (1950), Tricomi F. G., Asymptotische Eigenschaften der unvollständigenGammafunktion, Mathematische Zeitschrift, Band 53, Heft 2, S. 136-148 (1950)] (iv).No approximation for the velocity space dependence of the particles drifts is made. Thecritical threshold for resonant destabilization of the toroidal branch of the ITG is thencalculated and compared to previous analytical results. Comparisons to direct numericalsimulations performed with the GENE code in a simple geometrical setting are also re-ported (v). The full Larmor radius resonant theory predicts higher critical gradients thanthe long wavelength resonant theory. Like results from numerical simulations, it gives acritical gradient that depends linearly with τ = Ti/Te, where Ti and Te are the ion andelectron temperature, respectively, but it underestimates the numerical results. The dis-crepancy can be attributed to the fact that analytical theories predict a real frequencyat marginality (vi) that does not depend on τ as opposed to the numerical findings (vii).This is true for all analytical theories here revisited and derived. The stability diagramin Fig. (2) of Biglari Diamond and Rosenbluth [Phys. of Fluids B 1, 109 (1989)] hasalso been reproduced, and gives virtually no dependence on τ. The results from GENE

(i) Eq. (4.1) of Sec. (4)(ii) See Eqs. (4.8)-(4.15) and (4.17) of Section (4)

(iii) See Eq. (4.8) of Section (4)(iv) This is done in Appendix (B)

(v) See Fig. (1) of Sec. (1)(vi) A discussion of the evaluation of the real frequency at marginality is in Appendix (D)

(vii) See Fig. (3) of Sec. (1)

2 A. Zocco, et al.

simulations are somewhat consistent with the fitting formula of Jenko Dorland and Ham-mett (JDH) [Phys. Plasmas, 8, 9 (2001)] (viii) but do not seem to relate to the theoryof Hahm and Tang [PPPL Report (1988)] (ix). In the limit in which the streaming termcontribution is negligible in the JDH formula (q → ∞, and/or s → 0, where q and sare the safety factor and the global shear, respectively) the analysis of the velocity spacestructure of solutions (x) allows us to characterize the role of the trapped ions in themodification of the critical threshold for destabilization of the ITG. These generate afamily of unstable trapped ion modes (TIM) below the critical threshold for destabiliza-tion of the ITG (xi). Such modes require high velocity space resolution to be adequatelyresolved. The TIM-modified critical threshold is studied as a function of τ , for severaldensity gradients (xii). For τ & 1, such threshold shows a dependence on τ. For τ . 1,the critical threshold of R/LT is proporional to the inverse density gradient scale R/Ln.A TIM-modified critical threshold is found also in the Stellarator Wendelstein 7X; herethe TIM effect is even more pronounced (xiii). The eventuality of trapped ion turbulencebelow critical ITG turbulence must therefore be assesed in devices that allow comparableTIM and ITG levels.

(viii) Fig. (1), compare JDH to GENE curves(ix) Fig. (1), compare “Romanelli” to “GENE at q = 100” curve

(x) See Figs. (7)-(8) of Sec. (1)(xi) Fig. (4) of Sec. (1)(xii) Fig. (8) of Sec. (1)

(xiii) See. Fig. (9)

Resonant destabilization of ITG mode 3

0

1

2

3

4

5

6

7

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

R/L

T| cr

it

τ

GENEJDH

Linear F itEq.(31)noFLR

Eq.(31) full FLRGENE q = 100

Romanelli

Figure 1. Critical threshold from GENE data, linear fit of GENE data, fitting formula of JDHand different local theories. The GENE data at q = 100 should be compared to the local theories.

1. Results

1.1. Tokamak preliminaries

Figure (1) shows the critical gradient for resonant destabilisation of the ion-temperature-gradient (ITG) instability from the GENE code for several ion temperatures [cross-linein Fig. (1)]. Adiabatic electrons. Cyclon Base Case type parameters in s− α geometry:s = 0.786, q = 1.4, ǫ = r/R = 0.18 R/Ln = 2, kyρi = 0.3. The critical values areextrapolated from Fig. (2), which shows the growth rates for several τ = Ti/Te. Noticethe dependence on τ in Fig. (3) (mode frequency). The linear fit of the numerical datagives R/LT = (2.45±0.07)τ+2.07±0.06,with a reduced χ2 = 6.5×10−4. The fitting JDHformula of Jenko et al. (2001) is R/LT = (1 + τ) (4/3 + 1.91 s/q) . The curve derived byRomanelli (1989) is R/LT = (1+ τ) (4/3) , which is evaluated by replacing v2⊥/2+ v2‖ →4/3(v2⊥ + v2‖) in the definition of the particles magnetic drift [see Eq. (4.4)] in order to

fit the numerical solution of Eq. (4.1) at R/Ln ≪ 1, for kyρS =√0.1 ≈ 0.32, where

ρS = ρi/√2. It should not match the numerical results for R/Ln = 2, R/LT = O(1),

even if q→ ∞ [in which case the streaming term effect measured by the factor s/q inthe JDH formula should be negligible, as it is neglected in Eq. (4.1)]. The condition fordestabilization of Biglari et al. (1989) (BDR) [ Eq. (1.1), rederived in two different waysin Appendix (D)] would give R/LT = 2.7, with virtually no dependence on τ. We muststress that we reproduced the stability diagram of Fig. (5) of Biglari et al. (1989). Theirresult, R/LT = 2.7, can be obtained only if we include an extra multiplicative factorin the definition of the magnetic drift, e.g. ωκ = 2ω∗i(Ln/R) [however, in this work wefollow Hastie & Hesketh (1981) and use ωκ = ω∗i(Ln/R)], and the frequency at marginal

4 A. Zocco, et al.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

2 4 6 8 10 12 14 16

γ/(c s/R

)

R/LT

τ = 0.1τ = 0.3τ = 0.5τ = 1

τ = 1.5τ = 2.5

Figure 2. Growth rate as a function of the temperature gradient for different τ. CBCparameters.

stability is evaluated by solving the equation[

1− ω∗iωr

+ ηiω∗iωr

(

1− 2ωr

ωκ

)]

ℜ[

Z

(√

ω

ωκ

)]

− ηiω∗i√ωrωκ

ωr = 0. (1.1)

More details are in Appendix (D).


0

0.5

1

1.5

2

2.5

2 4 6 8 10 12 14 16

ω/(c s/R

)

R/LT

τ = 0.1τ = 0.3τ = 0.5τ = 1

τ = 1.5τ = 2.5

Figure 3. Real frequency as a function of the temperature gradient for different τ. CBCparameters.

0

0.05

0.1

0.15

0.2

0.25

0.3

0 2 4 6 8 10

γ/(c s/R

)

R/LT

τ = 0.1τ = 0.3τ = 0.5τ = 1.0τ = 1.5τ = 2.5

Figure 4. Imaginary part of eigenvalue as a function of the temperature gradient for differentτ. Parameters like in Fig 2 but s = 0.1, ǫ = 0.03. The pedestal at marginality is evident.

6 A. Zocco, et al.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 2 4 6 8 10

ω/(c s/R

)

R/LT

τ = 0.1τ = 0.3τ = 0.5τ = 1.0τ = 1.5τ = 2.5

Figure 5. Real part of eigenvalue as a function of the temperature gradient for different τ.Parameters like in Fig. 3 but s = 0.1, ǫ = 0.03. A jump is evident.

-1 -0.5 0 0.5 1

v‖

0

0.1

0.2

0.3

0.4

0.5

µ

0

0.5

1

1.5

2

2.5

3

Figure 6. Ion distribution function in phase space. τ = 1, R/LT = 3, “weak” mode thatbelongs to the “pedestal” of Fig. 4 Resonance occurs inside the trapped-passing boundary.


-1 -0.5 0 0.5 1

v‖

0

0.1

0.2

0.3

0.4

0.5

µ

0

50

100

150

200

250

Figure 7. Ion distribution function in phase space. τ = 1, R/LT = 5, “strong” mode.Resonance occurs outside the trapped-passing boundary.

1.5

2

2.5

3

3.5

4

4.5

0 0.5 1 1.5 2 2.5

R/L

T| cr

it

τ

R/Ln = 1.5R/Ln = 2

Figure 8. Critical threshold from GENE data of Fig. 4 for several density gradients R/Ln.

8 A. Zocco, et al.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.5 1 1.5 2 2.5 3

γ/(c s/a

)

a/LT

W7X low mirror

Figure 9. Growth rate of microinstability in Wendelstein7X from the GENE/GIST code (fluxtube). Flat density, low mirror configuration, adiabatic electrons, radial position r/a = 0.7. Thepedestal is evident, and more pronounced than in the tokamak case.

1.2. New Frontiers in Plasma Physics: the Stellarator case

While analysing the data that generated Fig. (1), and trying to recover an asymptoticlimit where the local theory would be valid, we discovered a family of unstable modesbelow the critical threshold of the ITG [See Fig. (6)]. These modes are present at smallglobal shear s, therefore we can conjecture they could also be present in Stellarators.Numerical simulations show that this is indeed the case [see Fig. (9)]. The role of suchmodes for turbulence in complex geometries has parhaps been overlooked.

2. Essential literature and preliminary discussion

The first study of the resonant toroidal ITG was presented by Terry et al. (1982). Thefull dispersion relation is studied numerically. Our integral Ii is presented in Eq. (A9),but the authors say “While this form is exact, it is cumbersome analytically”. The authorsthen focus their analysis on the so-called “grad-B drift model” and on the R/Ln ≫ 1 limit.Romanelli (1989), instead, focuses on the more relevant flat density limit R/Ln ≪ 1. In1988, Romanelli also used the “grad-B drift model” in order to derive the mode frequencyat marginality [Eq. (25) of Ref. (Romanelli 1989)], which in turns determines the criticalgradient length for destabilization. The same year, Biglari et al. (1989) (BDR), however,anticipated that such frequency can only be evaluated numerically. We agree with BDR,as we derived analytically from our closed form the ℑ[D] = 0 equation of page 11 ofRef. Biglari et al. (1989) [See Appendices C and D]. The impossibility of determininganalytically the frequency at marginality can be traced to the use of the full expressionof the magnetic drifts. In all cases, the frequency at marginal stability does not dependon the temperature ratio τ, at odds with the numerical results here reported. Therefore,


0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.5 1 1.5 2 2.5 3

ω/(c s/a

)

a/LT

W7X low mirror

Figure 10. Frequency associated with the unstable modes of Fig. (9) evaluated for Wendel-stein7X from the GENE/GIST code (flux tube). Low mirror configuration, adiabatic electrons,radial position r/a = 0.7. A sharp jump is present.

the connection of the fitting formula for the critical gradient of Jenko et al. (2001) toprevious local resonant theories seems to be weakened by our results. The key originalresult of this study is the perhaps surprising presence of a family of unstable trapped-ion-modes below the critical threshold for ITG destabilisation. These modes modify what onewould think is the threshold of marginal microstability, especially for low global shear.The same effect manifests in stellarators, shifting the critical threshold to values up to afactor of two lower than then in the ITG case.

3. Open Discussion

[Questions][Answers]

10 A. Zocco, et al.

4. Complementary material: local kinetic limit formulation

We study the following dispersion relation

1 + τ = Ii ≡∫

d3vω − ωT

∗iω − ωdi

J20

(

k2⊥ρ2i v

2⊥) F0i

n0, (4.1)

where τ = Ti/Te, ω is the mode complex frequency, k⊥ is the wave vector, v2 = (v2‖ +

v2⊥)/v2thi, with vthi =

√

2Ti/mi the ion thermal speed, and mi and n0 the ion mass anddensity, respectively. The equilbrium distribution function is taken to be Maxwellian,

F0i =n0

(πv2thi)3/2

e−v2

. (4.2)

Two characteristic frequencies are present in Eq. (4.1)

ωT∗i = ω∗i

[

1 + ηi(

v2 − 3/2)]

, (4.3)

with ω∗i = 0.5kyρivthi/LN , ηi = LN/LT , and

ωd =(

ωκv2‖ + ωBv

2⊥/2

)

, (4.4)

where ρi = vthi/Ωci is the ion Larmor radius, ωκ =, ωB =, L−1N = n−1

0 dn0/dx, andL−1T = T−1

i dTi/dx. Equation (4.1) [explain]. We proceed with our analysis.

Some preliminary manipulations are in order. We add and subtract ωκv2‖ +ωB v

2⊥/2 at

the numerator of Ii to obtain

Ii = 2

∫ ∞

0

dv⊥v⊥J20

(

k2⊥ρ2i v

2⊥)

e−v2

⊥

− ω∗iω

(

1− 3

2ηi

)

J (0)

+ωκ − ω∗iηi

ωJ(2)⊥

+ωB/2− ω∗iηi

ωJ(2)‖ ,

(4.5)

with

J(n)⊥,‖ = 2

∫ ∞

0

dv⊥v⊥

∫ ∞

−∞dv‖v

n⊥,‖

J20

(

k2⊥ρ2i v

2⊥)

1− (ωκ

ω v2‖ +ωB

ω v2⊥/2)

e−(v2

‖+v2

⊥)

√π

. (4.6)

The first line of Eq. (4.5) gives the well known result of gyrokinetic theory

2

∫ ∞

0

dv⊥v⊥J20

(

k2⊥ρ2i v

2⊥)

e−v2

⊥ = Γ0(b), (4.7)

where b = k2⊥ρ2i /2, and Γ0(b) = I0(b) exp(−b), with I0 the modified Bessel function. We

now integrate the resonances in a way similar to that presented by Biglari Diamond andRosenbluth (Biglari et al. 1989) (BDR in the following). For J (0) we have


J (0) = −2i

∫ ∞

0

dv⊥v⊥J20

∫ ∞

−∞

dv‖√π

∫ ∞

0

dλeiλ[1−(ωκω v2

‖+ωBω v2

⊥/2)]−(v2

‖+v2

⊥)

= −2i

∫ ∞

0

dλeiλ∫ ∞

0

dv⊥v⊥J20

∫ ∞

−∞

dv‖√πe−v2

⊥(1+iλωB2ω )e−v2

‖(1+iλωκω )

= −2i

∫ ∞

0

dλeiλ∫ ∞

0

dv⊥v⊥J20

e−v2

⊥(1+iλωB2ω )

(

1 + iλωκ

ω

)1/2

= −i

∫ ∞

0

dλeiλ

(

1 + iλωκ

ω

)1/2

Γ0

(

b)

1 + iλωB

2ω

,

(4.8)

with b = b/[1 + iλωB/(2ω)]. We need

ℑ[λωκ

ω] < 1, (4.9)

ℑ[λωB

2ω] < 1, (4.10)

and

ℑ[λ] > 0 (4.11)

in order to guarantee convergence of the velocity-space and dλ integrals, respectively.Conditions (4.9)-(4.11) thus define an abscissa of convergence in the complex λ−plane

ℜ[λ] > αc ≡ ℑ[ω]−1(ℑ[λ]ℜ[ω]− |ω|2 /ωκ). (4.12)

Here we are using ωB ≡ ωκ (valid in the electrostatic case) and taking the most restrictiveof conditions (4.9)-(4.11). Similarly, we obtain

J(2)‖ = − i

2

∫ ∞

0

dλeiλ

(

1 + iλωκ

ω

)3/2

Γ0

(

b)

1 + iλωB

2ω

, (4.13)

and

J(2)⊥ = −i

∫ ∞

0

dλeiλ

(

1 + iλωκ

ω

)1/2

Γ0

(

b)

+ b[

Γ1

(

b)

− Γ0

(

b)]

(

1 + iλωB

2ω

)2 . (4.14)

We now introduce the Padé approximants

Γ0

(

b)

≈ 1

1 + b1+iλ

ωB2ω

, (4.15)

and

Γ0

(

b)

+ b[

Γ1

(

b)

− Γ0

(

b)]

≈ 1(

1 + b1+iλ

ωB2ω

)2 , (4.16)

which will allow us to perform the dλ integration. The integral J (0) becomes

J (0) = −i

∫ ∞

0

dλeiλ

(

1 + iλωκ

ω

)1/2

1

1 + iλωB

2ω + b. (4.17)

For λ → λω/ωκ, b → 0, and ωB ≡ ωκ,

12 A. Zocco, et al.

limb→0

J (0) = −iω

ωκ

∫ ∞

0

dλei

ωωκ

λ

(1 + iλ)1/21

1 + iλ2

=ω

ωκF1,1,

(4.18)

with F1,1 defined in Eq. (A2) of BDR. We notice that the change of variables λ → λω/ωκ

requires (ℜ[λ]ℑ[ω]+ℑ[λ]ℜ[ω])/ωκ > 0 for the convergence of the dλ integral. At marginalstability, for ωκ < 0 and ω < 0, this is condition (4.11).

5. Arbitrary Larmor radii solution

For b 6= 0, the analytical technique used by BDR to solve for F1,1 cannot be used toperform the integration. We prefer to proceed in a different way.

Let us consider

R = −i

∫ ∞

0

dλeiΩλ

(1 + iλ)1/2

1

1 + iλ ωB

2ωκ+ b

, (5.1)

with Ω = ω/ωκ. We change variables

iλ = t2 − 1, (5.2)

and obtain

R = −4ωκ

ωB

∫ eiπ4 ∞

1

dteΩ(t2−1)

2 ωκ

ωB(1 + b)− 1 + t2

= −4ωκ

ωBe−Ω

∫ eiπ4 ∞

1/√g

du√g

eΩgu2

1 + u2,

(5.3)

with

g = 2ωκ

ωB(1 + b)− 1. (5.4)

In the case ωκ = ωB, g = 1 + 2b > 0 ∀ b. We split the domain of integration and define

R = −4ωκ

ωBe−Ω

(

∫ eiπ4 ∞

0

du√g−∫ 1/

√g

0

du√g

)

eΩgu2

1 + u2

≡ −4ωκ

ωBe−Ω (I∞ − Ig) .

(5.5)

In appendix A and B, we show that

R = −4ωκ

ωB

e−Ω

√g

− i

2

√πZ(

√

gΩ)

− 2πe−gΩT

[

i√

2gΩ;1√g

]

, (5.6)

where Z is the plasma dispersion function (Fried & Conte 1961), and T the OwenT−function (Owen 1956). By direct comparison of Eqs. (4.17) and (5.1), we have

J (0) =ω

ωκR. (5.7)


The integral J(2)⊥ now becomes

J(2)⊥ = −i

∫ ∞

0

dλeiλ

(

1 + iλωκ

ω

)1/2

1(

1 + iλωB

2ω + b)2

= − d

dbJ (0).

(5.8)

To evaluate the integral J(2)‖ , it is convenient to consider a simple generalization of the

auxiliary integral R, namely

Rν = −i

∫ ∞

0

dλeiΩλ

(ν + iλ)1/2

1

1 + iλ ωB

2ωκ+ b

. (5.9)

Then

J(2)‖ = − lim

ν→1Ω

d

dνRν . (5.10)

The integral Rν can be related to R after some straightforward algebra. We find

Rν = −4ωκ

ωBe−νΩ

∫ eiπ4

∞

√ν/gν

du√gν

egνΩu2

1 + u2

= −4ωκ

ωB

e−νΩ

√gν

− i

2

√πZ(

√

gνΩ)

− 2πe−gνΩT

[

i√

2gνΩ;

√

ν

gν

]

, (5.11)

with

gν =2ωκ

ωB(1 + b)− ν. (5.12)

Summarizing, the local kinetic dispersion relation is

1 + τ = Γ0(b)

− ω∗iω

(

1− 3

2ηi

)

J(0)⊥

+ω∗iω

(

ωκ

ω∗i− ηi

)

J(2)⊥

+ω∗iω

(

ωB

2ω∗i− ηi

)

J(2)‖ ,

(5.13)

with

J (0) = limν→1

ω

ωκRν , (5.14)

J(2)⊥ = − d

dbJ (0), (5.15)

J(2)‖ = − lim

ν→1Ω

d

dνRν , (5.16)

with Rν defined in Eq. (5.11), Z is the plasma dispersion function, Ω = ω/ωκ, gν =2(1 + b)ωκ/ωB − ν, and

T

[

i

√

2gνω

ωκ;

√

ν

gν

]

=egν

ωωκ

2π√gν

∞∑

n=0

(−ν/gν)n

(2n+ 1)

n∑

m=0

(−gνωωκ

)m

m!. (5.17)

This expression for the Owen T−function is derived in Appendix (B).

14 A. Zocco, et al.

Appendix A. The integral I∞

For I∞, we change variables to

gΩu2 → −ξ2, (A 1)

so that, when u → eiπ4 ∞, ξ → ei(

3

4π+ arggΩ

2 )∞. A closed form of the integral can befound for an unstable mode arg(Ωg) = arg ω [2ωκ(1 + b)− ωB] /(ωκωB) ≈ π/2, for

ℜ[ω] → 0. Then, ξ → ei(3

4π+ arggΩ

2 )∞ ≈ ei(3

4π+π

4 )∞ = −∞, and I∞ becomes

I∞ = −i√Ω

∫ −∞

0

dξe−ξ2

(√gΩ)2 − ξ2

= i

√Ω

2

∫ ∞

−∞dξ

e−ξ2

2√gΩ

1

ξ +√Ω

− 1

ξ −√Ω

= − i

2

√

π

gZ(

√

gΩ)

,

(A 2)

where Z is the plasma dispersion function (Fried & Conte 1961). Analytic continuationis needed for damped modes.

Appendix B. The integral Ig and the Owen T−function

In the case of Ig, we have

Ig =

∫ 1/√g

0

du√g

eΩgu2

1 + u2. (B 1)

This integral can be written in terms of the Owen T−function (Owen 1956).

Ig =

∫ 1/√g

0

du√g

eΩgu2

1 + u2

= e−gΩ

∫ 1/√g

0

du√g

e−(i

√2gΩ)2

(

u2

2+ 1

2

)

1 + u2

= 2πe−gΩ

√g

T

[

i√

2gΩ;1√g

]

.

For analytical manipulations and numerical implementation, the most convenient wayof writing the Owen function is perhaps the following. Since

T

[

i

√

2gνω

ωκ;

√

ν

g

]

=1

2π

∫

√ν/gν

0

dtegν

ωωκ

(1+t2)

1 + t2, (B 2)


if we rewrite

T

[

i

√

2gνω

ωκ;

√

ν

g

]

=1

2π

∫

√ν/gν

0

dtegν

ωωκ

(1+t2)

1 + t2

=1

2π

∫ ∞

0

dξ

∫

√ν/gν

0

dte−ξ(1+t2)egνωωκ

(1+t2)

=1

2πegν

ωωκ

∫ ∞

0

dξe−ξ

∫

√ν/gν

0

dte−(ξ−g ωωκ

)t2

=1

2πegν

ωωκ

∫ ∞

0

dξe−ξ

√

π/4

ξ − gνωωκ

Erf

√

ν

(

ξ

gν− ω

ωκ

)

,

(B 3)

we are then able to introduce the series representation of the Error function, to obtain

T

[

i

√

2gνω

ωκ;

√

ν

g

]

=egν

ωωκ

2π√gν

∫ ∞

0

dξe−ξ∞∑

n=0

(−ν/gν)n

n!(2n+ 1)

(

ξ − gνω

ωκ

)n

=egν

ωωκ

2π√gν

∞∑

n=0

(−ν/gν)n

n!(2n+ 1)

∫ ∞

0

dξe−ξ

(

ξ − gνω

ωκ

)n

=1

2π√gν

∞∑

n=0

(−ν/gν)n

n!(2n+ 1)

∫ ∞

−gνωωκ

dye−yyn

=1

2π√gν

∞∑

n=0

(−ν/gν)n

n!(2n+ 1)

(

∫ ∞

0

dy −∫ −gν

ωωκ

0

dy

)

e−yyn

=1

2π√gν

∞∑

n=0

(−ν/gν)n

n!(2n+ 1)

[

Γ (n+ 1)− γ(n+ 1,−gνω

ωκ)

]

,

(B 4)

where

γ

(

n+ 1,−gνω

ωκ

)

(B 5)

is the incomplete gamma function (Tricomi 1950b,a). Tricomi shows that, for n integer

γ

(

n+ 1,−gνω

ωκ

)

= n!

[

1− egνωωκ

n∑

m=0

(−gνωωκ

)m

m!

]

, (B 6)

therefore, our series representation of the Owen function is

T

[

i

√

2gνω

ωκ;

√

ν

gν

]

=egν

ωωκ

2π√gν

∞∑

n=0

(−ν/gν)n

(2n+ 1)

n∑

m=0

(−gνωωκ

)m

m!. (B 7)

Appendix C. Proof of some identities

We derived the following result

R = −4ωκ

ωB

e−Ω

√g

− i

2

√πZ(

√

gΩ)

− 2πe−gΩT

[

i√

2gΩ;1√g

]

, (C 1)

which is valid ∀ b. We now prove that, for ωκ ≡ ωB,

limb→0

R = F1,1, (C 2)

16 A. Zocco, et al.

where

F1/21,1 = e−Ω

∫ Ω

−∞dz

e−z

z1/2(C 3)

is the result found by BDR (Biglari et al. 1989).Since for b → 0, g → 1, then we have

limb→0

R = 2i√πe−ΩZ

(√Ω)

+ 4πe−2ΩΦ(

i√2Ω) [

1− Φ(

i√2Ω)]

, (C 4)

where

Φ(x) =1√2π

∫ x

−∞dte−t2/2. (C 5)

It is easy to see that

Φ(x) =i

2√π

∫ Ω

−∞dz

e−z

z1/2

=i

2√πeΩF

1/21,1

(C 6)

Then

limb→0

R = 2i√πe−ΩZ

(√Ω)

+ 4πe−2ΩΦ(

i√2Ω) [

1− Φ(

i√2Ω)]

,

= 2i√πe−Ω

Z(√

Ω)

+ F1/21,1

+ F1,1.(C 7)

However, it is also true that

F1/21,1 = e−Ω

∫ Ω

−∞dz

e−z

z1/2

= −2ie−Ω

∫ iΩ1/2

−∞dte−t2

= −Z(√

Ω)

,

(C 8)

then the first two terms in equation (C 7) cancel, leaving us with

limb→0

R = F1,1, (C 9)

which is what we wanted to prove.

Appendix D. Real frequency at marginality, long wavelength limit

b → 0

In the light of the identities just proved in Appendix C, we are in the position to saythat, for b → 0,

D ≡ − (1 + τ) + Ii

= − (1 + τ) +(

1− ω∗iω

)

ΩZ2(√

Ω)

+ ηiω∗iω

(1− 2Ω)ΩZ2(√

Ω)

− 2Ω3/2Z(√

Ω)

,

(D 1)

which corresponds to D0 in Eq. (3) of BDR (Biglari et al. 1989). The real frequencyat marginality is evaluated by solving ℑ[D(ωr)] ≡ Di(ωr) = 0, where we are setting


ω = ωr + iγ, and taking γ → 0. This comes directly from Eq. (D 1)

1− ω∗iωr

+ ηiω∗iωr

(

1− 2ωr

ωκ

)

ℜ[

Z

(√

ω

ωκ

)]

= ηiω∗i√ωrωκ

. (D 2)

The solution of this equation cannot possibly be given Eq. (25) or Romanelli (1989),which was evaluated by using the “grad-B drift” model. Indeed, Eq. (D 2) is the thirdequation on the second column of page 11 of BDR (Biglari et al. 1989). Since the conditionthat determines destabilization is ℜ[D(ωr)] > 0, we see how important is to obtain thefrequency correctly. As a sanity check, one can set J2

0 → 1 in the definition of Ii andintegrate the resonances in yet another way. It shall be

Ii = 2

∫ ∞

0

dv⊥v⊥e−v2

⊥

∫ ∞

−∞

dv‖e−v2

⊥

√π

Ω− Ω∗[

1 + ηi

(

v2‖ + v2⊥ − 3/2)]

Ω−(

v2‖ + v2⊥

)

= −2

∫ ∞

0

dv⊥v⊥e−v2

⊥

Ω− Ω∗i

(

1− 3

2ηi

) Z(

√

Ω− v2⊥/2)

√

Ω− v2⊥/2

+ ηiΩ∗i

1 + 2

∫ ∞

0

dv⊥v3⊥e

−v2

⊥

Z(

√

Ω− v2⊥/2)

√

Ω− v2⊥/2

+ ηiΩ∗i2

∫ ∞

0

dv⊥v⊥e−v2

⊥

√

Ω− v2⊥/2Z

(

√

Ω− v2⊥/2

)

.

(D 3)

It is easy to see that

2

∫ ∞

0

dv⊥v⊥e−v2

⊥

Z(

√

Ω− v2⊥/2)

√

Ω− v2⊥/2= −4ie−2Ω

∫ ∞

iΩ1/2

dζe−2ζ2

Z (iζ) ≡ I1, (D 4)

2

∫ ∞

0

dv⊥v3⊥e

−v2

⊥

Z(

√

Ω− v2⊥/2)

√

Ω− v2⊥/2= 2

∫ ∞

0

dv⊥2v⊥

(

v2⊥2

− Ω+ Ω

)

e−v2

⊥

Z(

√

Ω− v2⊥/2)

√

Ω− v2⊥/2

= 2ΩI1 + 2(−4i)e−2Ω

∫ ∞

iΩ1/2

dζζ2e−2ζ2

Z (iζ)

≡ 2ΩI1 + 2I2,

(D 5)

and

2

∫ ∞

0

dv⊥v⊥e−v2

⊥

√

Ω− v2⊥/2Z

(

√

Ω− v2⊥/2

)

= −I2. (D 6)

Then

1 + τ − ηiΩ∗i = −

Ω− Ω∗i

[

1− 3

2ηi

(

1− 4

3Ω

)]

I1 − ηiΩ∗iI2. (D 7)

By writing the plasma dispersion function as an Error function of imaginary argument,one sees that the integrals I1,and I2 can be performed analytically to obtain

1 + τ = − (Ω− Ω∗i)− ηiΩ∗i (1− 2Ω)πe−2Ω[

1− Erfi2(√

Ω)

− 2iErfi(√

Ω)]

+ 2ηiΩ∗i√πΩe−Ω

[

Erfi(√

Ω)

+ i]

,

(D 8)

18 A. Zocco, et al.

whose imaginary part set to zero gives Eq. (D 2).

REFERENCES

Biglari, H., Diamond, P. H. & Rosenbluth, M. N. 1989 Toroidal ion-pressure-gradient-driven drift instabilities and transport revisited. Phys. of Fluids B 1 (1), 109–118.

Fried, B.D. & Conte, S.D. 1961 The Plasma Dispersion Function. Academic Press.Hastie, R.J. & Hesketh, K.W. 1981 Kinetic modifications to the mhd ballooning mode.

Nucl. Fusion 21 (6), 651.Jenko, F., Dorland, W. & Hammett, G. W. 2001 Critical gradient formula for toroidal

electron temperature gradient modes. Phys. Plasmas 8 (9), 4096–4104.Owen, Donald B. 1956 Tables for computing bivariate normal probabilities. The Annals of

Mathematical Statistics 27 (4), 1075–1090.Romanelli, F. 1989 Ion temperature-gradient-driven modes and anomalous ion transport in

tokamaks. Phys. Fluids B 1 (5), 1018.Terry, P., Anderson, W. & Horton, W. 1982 Kinetic effects on the toroidal ion pressure

gradient drift mode. Nucl. Fusion 22 (4), 487.Tricomi, F G 1950a Asymptotische eigenschaften der unvollständigen gammafunktion. Math-

ematische Zeitschrift 53 (2), 136–148.Tricomi, F G 1950b Sulla funzione gamma incompleta. Annali di Matematica Pura ed Applicata

31 (1), 263–279.

critical scales for the destabilization of the toroidal ... · figure 2. growth rate as a function...

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