Under consideration for publication in J. Plasma Phys. Critical scales for the destabilization of the toroidal ion-temperature-gradient instability in magnetically confined toroidal plasmas Summary of work done in collaboration with J. W. Connor, H. Doerk, P. Helander and P. Xanthopoulos and presented by Alessandro Zocco at the First JPP Conference Frontiers in Plasma Physics, Abazia della Santissima Trinità Spineta Maggio 7 Culham Science Centre, Abingdon, Oxon, OX4 3DB, UK Max-Planck-Institut für Plasmaphysik, D-749, Greifswald, Germany Received May 7 We present, for the first time, a complete mathematical treatment of the resonant limit of the ion-temperature-gradient ITG driven instability for magnetically confined fusion plasmas with arbitrary ion Larmor radii. The analysis is performed in the local kinetic limit, thus neglecting the variation of the eigenfunction along the magnetic field line, but is valid for arbitrary geometry i. Analytical progress is made by introducing a Padé approximant ii for the ion reponse after integrating resonaces iii. These are treated by using a new series representation of the Owen T function [Owen, D B 956. "Tables for computing bivariate normal probabilities". Annals of Mathematical Statistics, 7, 75 9] which exploits the properties of the incomplete Euler gamma function [Tricomi F. G., Sulla funzione gamma incompleta, Annali di Matematica Pura ed Applicata, 3,, 63-79 95, Tricomi F. G., Asymptotische Eigenschaften der unvollständigen Gammafunktion, Mathematische Zeitschrift, Band 53, Heft, S. 36-48 95] iv. No approximation for the velocity space dependence of the particles drifts is made. The critical threshold for resonant destabilization of the toroidal branch of the ITG is then calculated and compared to previous analytical results. Comparisons to direct numerical simulations performed with the GENE code in a simple geometrical setting are also reported v. The full Larmor radius resonant theory predicts higher critical gradients than the long wavelength resonant theory. Like results from numerical simulations, it gives a critical gradient that depends linearly with τ = T i /T e, where T i and T e are the ion and electron temperature, respectively, but it underestimates the numerical results. The discrepancy can be attributed to the fact that analytical theories predict a real frequency at 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 diagram in Fig. of Biglari Diamond and Rosenbluth [Phys. of Fluids B, 9 989] has also been reproduced, and gives virtually no dependence on τ. The results from GENE i Eq. 4. of Sec. 4 ii See Eqs. 4.8-4.5 and 4.7 of Section 4 iii See Eq. 4.8 of Section 4 iv This is done in Appendix B v See Fig. of Sec. vi A discussion of the evaluation of the real frequency at marginality is in Appendix D vii See Fig. 3 of Sec.
A. Zocco, et al. simulations are somewhat consistent with the fitting formula of Jenko Dorland and Hammett JDH [Phys. Plasmas, 8, 9 ] viii but do not seem to relate to the theory of Hahm and Tang [PPPL Report 988] ix. In the limit in which the streaming term contribution is negligible in the JDH formula q, and/or ŝ, where q and ŝ are the safety factor and the global shear, respectively the analysis of the velocity space structure of solutions x allows us to characterize the role of the trapped ions in the modification of the critical threshold for destabilization of the ITG. These generate a family of unstable trapped ion modes TIM below the critical threshold for destabilization of the ITG xi. Such modes require high velocity space resolution to be adequately resolved. The TIM-modified critical threshold is studied as a function of τ, for several density gradients xii. For τ, such threshold shows a dependence on τ. For τ, the critical threshold of R/L T is proporional to the inverse density gradient scale R/L n. A TIM-modified critical threshold is found also in the Stellarator Wendelstein 7X; here the TIM effect is even more pronounced xiii. The eventuality of trapped ion turbulence below critical ITG turbulence must therefore be assesed in devices that allow comparable TIM and ITG levels. viii Fig., compare JDH to GENE curves ix Fig., compare Romanelli to GENE at q = curve x See Figs. 7-8 of Sec. xi Fig. 4 of Sec. xii Fig. 8 of Sec. xiii See. Fig. 9
R/LT crit 7 6 5 4 3 Resonant destabilization of ITG mode 3 GENE JDH LinearFit Eq.3noFLR Eq.3fullFLR GENEq = Romanelli..4.6.8..4.6 τ Figure. Critical threshold from GENE data, linear fit of GENE data, fitting formula of JDH and different local theories. The GENE data at q = should be compared to the local theories.. Results.. Tokamak preliminaries Figure shows the critical gradient for resonant destabilisation of the ion-temperaturegradient ITG instability from the GENE code for several ion temperatures [cross-line in Fig. ]. Adiabatic electrons. Cyclon Base Case type parameters in ŝ α geometry: ŝ =.786, q =.4, ǫ = r/r =.8 R/L n =, k y ρ i =.3. The critical values are extrapolated from Fig., which shows the growth rates for several τ = T i /T e. Notice the dependence on τ in Fig. 3 mode frequency. The linear fit of the numerical data givesr/l T =.45±.7τ+.7±.6, with a reducedχ = 6.5 4. The fitting JDH formula of Jenko et al. is R/L T = +τ4/3+.9ŝ/q. The curve derived by Romanelli 989 is R/L T = +τ4/3, which is evaluated by replacing v /+v 4/3v + v in the definition of the particles magnetic drift [see Eq. 4.4] in order to fit the numerical solution of Eq. 4. at R/L n, for k y ρ S =..3, where ρ S = ρ i /. It should not match the numerical results for R/L n =, R/L T = O, even if q [in which case the streaming term effect measured by the factor ŝ/q in the JDH formula should be negligible, as it is neglected in Eq. 4.]. The condition for destabilization of Biglari et al. 989 BDR [ Eq.., rederived in two different ways in Appendix D] would give R/L T =.7, with virtually no dependence on τ. We must stress that we reproduced the stability diagram of Fig. 5 of Biglari et al. 989. Their result, R/L T =.7, can be obtained only if we include an extra multiplicative factor in the definition of the magnetic drift, e.g. = i L n /R [however, in this work we follow Hastie & Hesketh 98 and use = i L n /R], and the frequency at marginal
4 A. Zocco, et al. γ/cs/r.8.7.6.5.4.3 τ =. τ =.3 τ =.5 τ = τ =.5 τ =.5.. 4 6 8 4 6 R/L T Figure. Growth rate as a function of the temperature gradient for different τ. CBC parameters. stability is evaluated by solving the equation {[ i i +η i ] [ r R Z r r More details are in Appendix D. ] η i i r } r =..
Resonant destabilization of ITG mode 5.5 τ =. τ =.3 τ =.5 τ = τ =.5 τ =.5 /cs/r.5.5 4 6 8 4 6 R/L T Figure 3. Real frequency as a function of the temperature gradient for different τ. CBC parameters. γ/cs/r.3.5..5. τ =. τ =.3 τ =.5 τ =. τ =.5 τ =.5.5 4 6 8 R/L T Figure 4. Imaginary part of eigenvalue as a function of the temperature gradient for different τ. Parameters like in Fig but ŝ =., ǫ =.3. The pedestal at marginality is evident.
6 A. Zocco, et al..4. τ =. τ =.3 τ =.5 τ =. τ =.5 τ =.5 /cs/r.8.6.4. 4 6 8 R/L T Figure 5. Real part of eigenvalue as a function of the temperature gradient for different τ. Parameters like in Fig. 3 but ŝ =., ǫ =.3. A jump is evident..5 3 µ.4.3...5.5.5 - -.5.5 v Figure 6. Ion distribution function in phase space. τ =, R/L T = 3, weak mode that belongs to the pedestal of Fig. 4 Resonance occurs inside the trapped-passing boundary.
7 Resonant destabilization of ITG mode 5.4.3 5.. 5 µ.5 - -.5.5 vk Figure 7. Ion distribution function in phase space. τ =, R/LT = 5, strong mode. Resonance occurs outside the trapped-passing boundary. 4.5 R/Ln =.5 R/Ln = R/LT crit 4 3.5 3.5.5.5.5.5 τ Figure 8. Critical threshold from GENE data of Fig. 4 for several density gradients R/Ln.
8 A. Zocco, et al...8 W7X low mirror.6.4 γ/cs/a...8.6.4..5.5.5 3 a/l T Figure 9. Growth rate of microinstability in Wendelstein7X from the GENE/GIST code flux tube. Flat density, low mirror configuration, adiabatic electrons, radial position r/a =.7. The pedestal is evident, and more pronounced than in the tokamak case... New Frontiers in Plasma Physics: the Stellarator case While analysing the data that generated Fig., and trying to recover an asymptotic limit where the local theory would be valid, we discovered a family of unstable modes below the critical threshold of the ITG [See Fig. 6]. These modes are present at small global shear ŝ, 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 such modes for turbulence in complex geometries has parhaps been overlooked.. Essential literature and preliminary discussion The first study of the resonant toroidal ITG was presented by Terry et al. 98. The full dispersion relation is studied numerically. Our integral I i is presented in Eq. A9, but the authors say While this form is exact, it is cumbersome analytically. The authors then focus their analysis on the so-called grad-b drift model and on ther/l n limit. Romanelli 989, instead, focuses on the more relevant flat density limit R/L n. In 988, Romanelli also used the grad-b drift model in order to derive the mode frequency at marginality [Eq. 5 of Ref. Romanelli 989], which in turns determines the critical gradient length for destabilization. The same year, Biglari et al. 989 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] = equation of page of Ref. Biglari et al. 989 [See Appendices C and D]. The impossibility of determining analytically the frequency at marginality can be traced to the use of the full expression of the magnetic drifts. In all cases, the frequency at marginal stability does not depend on the temperature ratio τ, at odds with the numerical results here reported. Therefore,
Resonant destabilization of ITG mode 9.6.55 W7X low mirror.5 /cs/a.45.4.35.3.5..5.5.5.5 3 a/l T Figure. Frequency associated with the unstable modes of Fig. 9 evaluated for Wendelstein7X from the GENE/GIST code flux tube. Low mirror configuration, adiabatic electrons, radial position r/a =.7. A sharp jump is present. the connection of the fitting formula for the critical gradient of Jenko et al. to previous local resonant theories seems to be weakened by our results. The key original result of this study is the perhaps surprising presence of a family of unstable trapped-ionmodes below the critical threshold for ITG destabilisation. These modes modify what one would 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 a factor of two lower than then in the ITG case. 3. Open Discussion [Questions] [Answers]
A. Zocco, et al. 4. Complementary material: local kinetic limit formulation We study the following dispersion relation +τ = I i d 3 v T i J k ρ iˆv F i, 4. di n where τ = T i /T e, is the mode complex frequency, k is the wave vector, ˆv = v + v /v thi, with v thi = T i /m i the ion thermal speed, and m i and n the ion mass and density, respectively. The equilbrium distribution function is taken to be Maxwellian, F i = Two characteristic frequencies are present in Eq. 4. with i =.5k y ρ i v thi /L N, η i = L N /L T, and n 3/e ˆv. 4. πvthi T i = i [ +ηi ˆv 3/ ], 4.3 d = ˆv + Bˆv /, 4.4 where ρ i = v thi /Ω ci is the ion Larmor radius, =, B =, L N = n dn /dx, and L T = T i dt i /dx. Equation 4. [explain]. We proceed with our analysis. Some preliminary manipulations are in order. We add and subtract ˆv + Bˆv / at the numerator of I i to obtain I i = i dˆv ˆv J 3 η i J + i η i J + B/ i η i J, k ρ iˆv e ˆv 4.5 with, = dˆv ˆv dˆv ˆv n J k ρ iˆv e, κ ˆv + B ˆv / J n ˆv +ˆv π. 4.6 The first line of Eq. 4.5 gives the well known result of gyrokinetic theory dˆv ˆv J k ρ iˆv e ˆv = Γ b, 4.7 where b = k ρ i /, and Γ b = I bexp b, with I the modified Bessel function. We now integrate the resonances in a way similar to that presented by Biglari Diamond and Rosenbluth Biglari et al. 989 BDR in the following. For J we have
Resonant destabilization of ITG mode J = i = i = i = i dˆv ˆv J dλe iλ dˆv ˆv J dλe iλ dˆv ˆv J dλ e iλ +iλ / with ˆb = b/[+iλ B /]. We need dˆv dλe iλ[ κ ˆv + B ˆv /] ˆv +ˆv π dˆv π e ˆv +iλ B e ˆv +iλ κ e ˆv +iλ B +iλ Γ ˆb +iλ B, / 4.8 and I[λ ] <, 4.9 I[λ B ] <, 4. I[λ] > 4. in order to guarantee convergence of the velocity-space and dλ integrals, respectively. Conditions 4.9-4. thus define an abscissa of convergence in the complex λ plane R[λ] > α c I[] I[λ]R[] /. 4. Here we are using B valid in the electrostatic case and taking the most restrictive of conditions 4.9-4.. Similarly, we obtain Γ ˆb and J J = i = i dλ e iλ dλ +iλ / We now introduce the Padé approximants and Γ ˆb e iλ +iλ 3/ Γ ˆb [ ] Γ ˆb +ˆb Γ ˆb Γ ˆb +iλ B, 4.3 [ ] +ˆb Γ ˆb Γ ˆb +iλ B. 4.4 + b +iλ B, 4.5, 4.6 b + +iλ B which will allow us to perform the dλ integration. The integral J becomes J = i dλ For λ λ/, b, and B, e iλ +iλ / +iλ B +b. 4.7
A. Zocco, et al. lim b J = i = F,, dλ e i κ λ +iλ / +i λ 4.8 with F, defined in Eq. A of BDR. We notice that the change of variables λ λ/ requiresr[λ]i[]+i[λ]r[]/ > for the convergence of the dλ integral. At marginal stability, for < and <, this is condition 4.. 5. Arbitrary Larmor radii solution For b, the analytical technique used by BDR to solve for F, cannot be used to perform the integration. We prefer to proceed in a different way. Let us consider R = i with Ω = /. We change variables and obtain e iωλ dλ +iλ / +iλ B +b, 5. iλ = t, 5. R = 4 B e i π 4 dt = 4 e i π 4 κ e Ω B / g e Ωt κ B +b +t du e Ωgu g +u, 5.3 with g = B +b. 5.4 In the case = B, g = +b > b. We split the domain of integration and define R = 4 e i π 4 κ e Ω du / g du e Ωgu B g g +u 5.5 4 B e Ω I I g. In appendix A and B, we show that R = 4 e Ω { i πz gω πe gω T B g [ i gω; ]}, 5.6 g where Z is the plasma dispersion function Fried & Conte 96, and T the Owen T function Owen 956. By direct comparison of Eqs. 4.7 and 5., we have J = R. 5.7
The integral J Resonant destabilization of ITG mode 3 now becomes J = i dλ = d db J. e iλ +iλ / +iλ B +b 5.8 To evaluate the integral J, it is convenient to consider a simple generalization of the auxiliary integral R, namely Then R ν = i e iωλ dλ ν +iλ / +iλ B +b. 5.9 J = lim Ω d ν dν R ν. 5. The integral R ν can be related to R after some straightforward algebra. We find with R ν = 4 e i π 4 κ e νω du e gνωu B ν/gν gν +u = 4 e νω { i πz gν Ω πe B gν gνω T [ i ]}, 5. ν g ν Ω; g ν g ν = B +b ν. 5. Summarizing, the local kinetic dispersion relation is with +τ = Γ b i 3 η i + i + i κ η i i B η i i J J J, 5.3 J = lim R ν, 5.4 ν J = d db J, 5.5 J = lim Ω d ν dν R ν, 5.6 with R ν defined in Eq. 5., Z is the plasma dispersion function, Ω = /, g ν = +b / B ν, and T [ i ν g ν ; g ν ] = egν κ π g ν n= ν/g ν n n+ n g ν m. 5.7 m! m= This expression for the Owen T function is derived in Appendix B.
4 A. Zocco, et al. Appendix A. The integral I For I, we change variables to gωu ξ, A so that, when u e iπ 4, ξ e i3 4 π+arggω. A closed form of the integral can be found for an unstable mode argωg = arg{[ +b B ]/ B } π/, for R[]. Then, ξ e i 3 4 π+arggω e i 3 4 π+π 4 =, and I becomes I = i Ω Ω = i = i dξ e ξ gω π g Z gω, e ξ dξ gω ξ { } ξ + Ω ξ Ω A where Z is the plasma dispersion function Fried & Conte 96. Analytic continuation is needed for damped modes. Appendix B. The integral I g and the Owen T function In the case of I g, we have I g = / g du e Ωgu g +u. B This integral can be written in terms of the Owen T function Owen 956. I g = / g / g = e gω = π e gω g T du g e Ωgu +u du e i g [ i gω; gω u + +u ]. g For analytical manipulations and numerical implementation, the most convenient way of writing the Owen function is perhaps the following. Since T [ ] ν i g ν ; = ν/g ν g π dt egν κ +t +t, B
if we rewrite Resonant destabilization of ITG mode 5 [ ] ν T i g ν ; g = ν/g ν dξ π = = n= π egν π egν κ κ = ν/g ν π dt egν κ +t +t dte ξ+t e gν κ +t dξe ξ ν/g ν dte ξ g κ t dξe ξ π/4 ξ Erf ν ξ g ν g ν, B 3 we are then able to introduce the series representation of the Error function, to obtain [ ] ν T i g ν ; = egν κ g π dξe ξ ν/g ν n n ξ g ν g ν n!n+ n= κ = egν κ π ν/g ν n n dξe ξ ξ g ν g ν n!n+ = π g ν = π g ν = π g ν n= n= n= ν/g ν n n!n+ ν/g ν n n!n+ ν/g ν n n!n+ g ν κ dy dye y y n gν κ dy e y y n [ Γn+ γn+, g ν ], B 4 where γ n+, g ν B 5 is the incomplete gamma function Tricomi 95b,a. Tricomi shows that, for n integer γ n+, g ν = n! [ e gν κ n= n m= g ν m m! ], B 6 therefore, our series representation of the Owen function is [ ] ν T i g ν ; = egν κ g ν π ν/g ν n n g ν m. B 7 g ν n+ m! m= Appendix C. Proof of some identities We derived the following result R = 4 e Ω { i πz gω πe gω T B g which is valid b. We now prove that, for B, [ i gω; ]}, C g lim R = F,, C b
6 A. Zocco, et al. where lim R = i πe Ω Z b F /, = e Ω Ω is the result found by BDR Biglari et al. 989. Since for b, g, then we have Ω +4πe Ω Φ where It is easy to see that Then lim R = i πe Ω Z Ω b = i πe Ω{ Z Ω Φx = π x Φx = i Ω dz e z π z / = i π eω F /, +F /, +4πe Ω Φ } +F,. dz e z z / C 3 i [ Ω Φ i ] Ω, C 4 dte t /. C 5 i [ Ω Φ i ] Ω, C 6 C 7 However, it is also true that F /, = e Ω Ω iω / = ie Ω = Z Ω, dz e z z / dte t C 8 then the first two terms in equation C 7 cancel, leaving us with which is what we wanted to prove. lim R = F,, C 9 b Appendix D. Real frequency at marginality, long wavelength limit b In the light of the identities just proved in Appendix C, we are in the position to say that, for b, D +τ+i i = +τ+ i ΩZ Ω i +η i { ΩΩZ } Ω Ω 3/ Z Ω, D which corresponds to D in Eq. 3 of BDR Biglari et al. 989. The real frequency at marginality is evaluated by solving I[D r ] D i r =, where we are setting
Resonant destabilization of ITG mode 7 = r +iγ, and taking γ. This comes directly from Eq. D { i i +η i } [ ] r i R Z = η i. r r r D The solution of this equation cannot possibly be given Eq. 5 or Romanelli 989, which was evaluated by using the grad-b drift model. Indeed, Eq. D is the third equation on the second column of page of BDR Biglari et al. 989. Since the condition that determines destabilization is R[D r ] >, we see how important is to obtain the frequency correctly. As a sanity check, one can set J in the definition of I i and integrate the resonances in yet another way. It shall be ] [+η i ˆv + ˆv 3/ I i = dˆv ˆv e ˆv dˆv e ˆv π Ω Ω = dˆv ˆv e ˆv Ω Ω i 3 η i Ω Z ˆv +η i Ω i + dˆv ˆv 3 e ˆv +η i Ω i It is easy to see that and Then dˆv ˆv e ˆv dˆv ˆv 3 e ˆv dˆv ˆv e ˆv Ω ˆv + ˆv Ω Z ˆv / Ω ˆv / / Ω ˆv / Ω ˆv Ω /Z ˆv /. Ω Z ˆv / = 4ie Ω dζe ζ Ziζ I Ω ˆv, / iω / Ω Z ˆv / = Ω ˆv / dˆv ˆv ˆv Ω+Ω e ˆv = ΩI + 4ie Ω dζζ e ζ Ziζ iω / ΩI +I, D3 D4 Z Ω ˆv / Ω ˆv / D5 dˆv ˆv e ˆv Ω ˆv Ω /Z ˆv / = I. D6 [ +τ η i Ω i = {Ω Ω i 3 η i 43 ]} Ω I η i Ω i I. D7 By writing the plasma dispersion function as an Error function of imaginary argument, one sees that the integrals I, and I can be performed analytically to obtain +τ = { Ω Ω i η i Ω i Ω}πe Ω[ Erfi ] Ω ierfi Ω [ ] +η i Ω i πωe Ω Erfi Ω +i, D8
8 A. Zocco, et al. whose imaginary part set to zero gives Eq. D. REFERENCES Biglari, H., Diamond, P. H. & Rosenbluth, M. N. 989 Toroidal ion-pressure-gradientdriven drift instabilities and transport revisited. Phys. of Fluids B, 9 8. Fried, B.D. & Conte, S.D. 96 The Plasma Dispersion Function. Academic Press. Hastie, R.J. & Hesketh, K.W. 98 Kinetic modifications to the mhd ballooning mode. Nucl. Fusion 6, 65. Jenko, F., Dorland, W. & Hammett, G. W. Critical gradient formula for toroidal electron temperature gradient modes. Phys. Plasmas 8 9, 496 44. Owen, Donald B. 956 Tables for computing bivariate normal probabilities. The Annals of Mathematical Statistics 7 4, 75 9. Romanelli, F. 989 Ion temperature-gradient-driven modes and anomalous ion transport in tokamaks. Phys. Fluids B 5, 8. Terry, P., Anderson, W. & Horton, W. 98 Kinetic effects on the toroidal ion pressure gradient drift mode. Nucl. Fusion 4, 487. Tricomi, F G 95a Asymptotische eigenschaften der unvollständigen gammafunktion. Mathematische Zeitschrift 53, 36 48. Tricomi, F G 95b Sulla funzione gamma incompleta. Annali di Matematica Pura ed Applicata 3, 63 79.