The probability density function (PDF) of flux R is computed in systems with logarithmic non-linearity using a model non-linear dynamical equation. The PDF tails of the first moment flux are analytically predicted to be power law. These PDF tails are shown to be broader than a Gaussian distribution and are a manifestation of intermittency caused by short lived coherent structures (instantons). (c) 2010 Elsevier B.V. All rights reserved

Anderson, J.

Kim, E.J.

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arXiv

Kim, E.-j.

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Predicting PDF tails in systems with logarithmic non-linearity

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promoting access to White Rose research papers    White Rose Research Online eprints@whiterose.ac.uk    Universities of Leeds, Sheffield and York http://eprints.whiterose.ac.uk/    This is an author produced version of a paper published in Physics Letters A.    White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/10828     Published paper  Anderson, J., Kim, E.J. (2010) Predicting PDF tails in systems with logarithmic non-linearity, Physics Letters A., 374 (15-16), pp. 1621-1624 http://dx.doi.org/10.1016/j.physleta.2010.02.017    Predicting PDF tails of flux in plasma sheathregionJohan Anderson1 and Eun-jin Kim?Max-Planck-Institut für Plasmaphysik, IPP-Euratom Association, Teilinsti-tut Greifswald, 17491 Greifswald, Germany? University of Sheffield, Department of Applied Mathematics, Hicks Build-ing, Hounsfield Road, Sheffield, S3 7RH, UKAbstractThis letter provides the first prediction of the probability density func-tion (PDF) of flux R in plasma sheath region in magnetic fusion de-vices which is characterized by dynamical equations with exponentialnon-linearities. By using a non-perturbative statistical theory (in-stantons), the PDF tails of first moment are shown to be a modifiedGumbel distribution which represents a frequency distribution of theextreme values of the ensemble. The non-Gaussian PDF tails that maybe enhanced over Gaussian predictions are the result of intermittencycaused by short lived coherent structures (instantons).1anderson.johan@gmail.com1In the edge region of magnetically confined fusion experiment the plasmaexhibits large bursty fluctuations, often called intermittency [1]- [9]. Inter-mittent turbulence is characterized by patchy spatial structure that is burstyin time. The likelihood of these events are described by probability den-sity functions (PDFs) which significantly deviate from Gaussian predictions.More specifically, exponential scalings are often observed in the PDF tailsin magnetic confinement experiments [6]- [9], and intermittency at the edgestrongly influences the overall global particle and heat transport. In par-ticular it may for instance influence the threshold for the high confinementmode (H-mode) in tokamak experiments [10]. In the view of these experi-mental results, theories built on average transport coefficients and Gaussianstatistics fall short in predicting vital transport processes. The high possibil-ity of confinement degradation by intermittency strongly calls for a powerfulpredictive theory.The purpose of this letter is to provide the theoretical prediction for thePDF tails of flux in plasma sheath by using a non-perturbative statisticalmethod. The basic idea is to associate the bursty event with the creationof a coherent structure (e.g. blobs, streamers etc.). The candidate thatcould describe the creation process is the instanton which is localized intime, existing during the formation of the coherent structure. This idea isembedded in our theoretical method - the so-called instanton method - whichis a non-perturbative way of calculating the PDF tails [11]- [22]. Note that,the instanton method has been adopted from quantum field theory and thenmodified to classical statistical physics for Burgers turbulence [12]- [13] andin the model by Kraichnan [23].Our previous papers using this instanton method have shown that thePDF tails of momentum flux and heat flux are significantly enhanced over2the Gaussian prediction [14]- [18], providing a novel explanation for expo-nential PDF tails of momentum flux found in recent experiments at CSDXat UCSD [9]. PDF tails of forced shear flow R in a non-linear diffusion modelwere shown to be ∼ exp{−c(R/R0)4} that falls off much faster than a Gaus-sian by both analytical (e.g. instanton) and numerical investigations [21].In the following we study the PDFs of flux of 1st moment variables suchas potential, density etc in plasma sheath where electric potential is governedby exponential non-linear interaction. In Ref. [20], we computed the PDFtails of blob density by using a simplified model for blobby transport withdifferent non-linear interaction. Note that in the instanton analysis a carefultreatment of the dominant non-linear interaction is necessary so that keepingonly leading order non-linear interaction by Taylor expansion may lead to theerroneous predictions for the PDF tails [22]. By using the instanton methodwe show that the PDF tails of the first moment satisfy modified Gumbeldistributions [24]- [25]. The Gumbel distribution is used to model extremevalue distributions, i.e. a frequency distribution of the largest or smallestvalues of the ensemble. This is due to the fact that Taylor expansion ofthe exponential contains infinite sum of all powers. In a comparison with aGaussian distribution it is shown that the flux in plasma sheath may exhibitstretched PDF tails. We emphasise that the predicted Gumbel distributionis new and has not been found in any related previous work.As a physical model of plasma sheath, we consider a generic simplifiedmodel of turbulence with a dominant exponential non-linear interaction. Themodel is a simplified version of the three field Braghinskii fluid model pre-sented in Ref. [26] where we have retained the exponential non-linear inter-action. Note that this model can be reduced to the model derived in Ref. [27]by linearization and keeping only non-linear terms up to second order. This3model is sufficient for the prediction of the PDF tails since our previous pa-pers showed that the PDF tails are rather insensitive to the details of the dy-namics and dependent only on the dominant non-linear interaction term [22].We model the sheath region with a general non-linear equation describing thetime evolution of the variable φ for fluctuations with exponential interactionKecφ [7], [27]- [28],∂φ∂t+ Kecφ = f. (1)Here f is the forcing. The statistics of the forcing is assumed to be Gaussianwith a short correlation time modeled by the delta function as〈f(x, t)f(x′, t′)〉 = δ(t− t′)κ(x− x′), (2)and 〈f〉 = 0. The angular brackets denote the average over the statistics ofthe forcing f . The delta correlation in time is chosen for the sake of simplicityof the analysis. Note that in the case of a finite correlation time non-localintegral equations in time are needed.We calculate the PDF tails of local flux M(φ(x = x0)) that is the first mo-ment of φ (e.g. density, potential) by using the instanton method. The PDFtails are expressed in terms of a path-integral using the Gaussian statisticsof the forcing [11]. The optimum path is then associated with the creation ofa short lived coherent structure (among all possible paths) and the action isevaluated using the saddle-point method on the effective action. The saddle-point solution of the dynamical variable φ(x, t) of the form φ(x, t) = F (t)v(x)is called an instanton if F (t) = 0 at t = −∞ and F (t) 6= 0 at t = 0. Notethat, the function v(x) here represents the spatial form of the coherent struc-ture. The probability density function of the flux R can be defined asP (R) = 〈δ(M(φ(x = x0))−R)〉 =∫dλeiλRIλ, (3)4whereIλ = 〈exp(−iλM(φ(x = x0)))〉. (4)The integrand can then be rewritten in the form of a path-integral asIλ =∫DφDφ̄e−Sλ . (5)Here the effective action Sλ in Eq. (5) is expressed as,Sλ = −i∫dxdtφ̄(∂φ∂t+ Kecφ)+12∫dxdx′dtφ̄(x, t)κ(x− x′)φ̄(x′, t)+ iλ∫dxdtM(φ)δ(x− x0)δ(t). (6)We make use of a logarithmic transform φ = ln u and u = v(F +1) where v isthe spatial structure of the instanton and F is the time dependent function.The action can now be recast into,Sλ = −i∫dtc1F̄1(∂ ln(F + 1)∂t+ c2(F + 1)c)+12c1c4∫dtF̄ 21+ iλ∫dtc1c3 ln(F + 1)δ(t). (7)Here we have used the definitions,c1 =∫dxv̄(x), (8)c1c2 =∫dxv̄(x)Kv(x)c, (9)c1c3 =∫dxδ(x− x0), (10)c1c4 =∫dxdyv̄(x)v̄(y)K(x− y), (11)In order to find the path with highest probability identified by the instan-ton or the extremum of the action we compute the first variational derivatives5to minimize Sλ with respect to F and F̄1,δSλδF̄1= −i(∂∂tln(F + 1) + c2(F + 1)c)+ c4F̄1 = 0, (12)δSλδF= −i − ˙̄F 1F + 1+ c2c(F + 1)c−1F̄1 + iλc31F + 1δ(t) = 0. (13)Eq. (12) gives+i(Ḟ + c2(F + 1)c+1)= c4(F + 1)F̄1 (14)Differentiating Eq. (14) w.r.t time gives,(F̈ + c2(c + 1)(F + 1)cḞ − Ḟ2F + 1− c2(F + 1)cḞ)= −ic4(F + 1) ˙̄F 1. (15)We solve Eq. (15) for t < 0 and match the solution at t = 0. Note thatthe instanton solution F rapidly grows at t = 0 with increasing λ while itvanishes as t → −∞. By using Eqs. (13) to (15),− d2dt2ln(F + 1) + cc22(F + 1)2c = 0, (16)we find the solution,ddtln(F + 1) = c2(F + 1)c, (17)F (t) = −1 +(α1c22+√α1c22tanh(c√α1t− α2/2))1/2c. (18)The constants α1 and α2 are to be determined using the boundary conditionsF (0) = F0 and F (−∞) = 0. To find the value of F at t = 0 we integrate Eq.(13) over (−ε, ε) and use Eq. (14) to obtain,F̄1(0) ≈ c3λ. (19)6We then use Eq. (12) at t = 0 to obtainF (0)c = F c0 ≈ −iλc4c32c2. (20)Using Eq. (20) and the condition that F (−∞) = 0, we find the values ofα1 =14(1±√1− 4c22)2 and α2 = −2 tanh−1( c22(F0+1)+α1√α1).We now find an approximate value of the PDF in the limit of λ → ∞which is expressed by a path-integral, using the saddle point method. Firstwe compute the λ-dependence of the action Sλ and then proceed to determinethe PDF tails. In the limit of λ →∞, Sλ becomesSλ = −i∫dtc1(F̄1(∂∂tln(F + 1) + c2(F + 1)c))+12c1c4∫dtF̄ 21+ iλ∫dtc1c3 ln(F + 1)δ(t)=2c21c2c4cF c(0) + iλc3 ln(F (0) + 1)= −ik1λ + ik1λ ln(−ik2λ), (21)where the coefficient k1 and k2 are defined ask1 =c1c3c, (22)k2 =c3c42c2. (23)The tail of the PDF is then found by performing the λ-integral in Eq. (3)by the saddle point method in the limit R →∞. It is later shown that thiscorresponds to λ →∞,P (R) ∼∫dλeiλR+ik1λ−ik1λ ln(−ik2λ) ∼ 1√f ′′(λ0)ef(λ0) (24)≈ e−k1k2eR/k1+R/(2k1). (25)7Here, we have evaluated the λ-integral using the extreme pointf ′(λ0) = 0 and λ0 = i 1k2 eR/k1 of f(λ) = −iλR + ik1λ + ik2λ ln λ by usingthe saddle-point method. Note that the first term (e− k1k2eR/k1) in Eq. (25)is the zeroth order term and (eR/(2k1)) is the correction term. The resultingPDF tail in Eq. (25) satisfies a Gumbel distribution [24]- [25]. The Gumbeldistribution is one type of an extreme value distribution that is the limitdistribution of maxima/minima of independent and identically distributedrandom variables. Thus the Gumbel distribution is one model of approximatemaxima/minima of finite sequences of random variables. This is due to thefact that Taylor expansion of the exponential contains an infinite sum of allpowers.We now show explicitly that systems with exponential non-linearities mayhave enhanced PDF tails compared to Gaussian distributions. In figure 1, acomparison of the PDF tails of flux with an exponential non-linearity foundin Eq. 25 with k1 = 2.0 and k2 = 1.0 (black line), k1 = 1.0 and k2 = 1.0(green line), k1 = 0.5 and k2 = 1.0 (blue line) and a Gaussian distribution(red line) are shown. We have normalized all the distributions asone-sided PDFs. The figure shows that the PDF tails with exponentialnon-linearity may be enhanced over a Gaussian distribution depending onthe parameters k1 and k2. It is important to note that as R → ∞ theGaussian distribution takes larger values than the Gumbel distribution theprobability of taking such a large value of R may become extremely small(∼ 10−10). However, if the exponential non-linear interaction is weak andcan be expanded in a truncated Taylor series up to second order the plasmasheath region is governed by a power-law non-linearity, a slightly differentPDF tail ∼ exp{−c(R/R0)s} with s = (n+1)/m can be expected [22]. Here,n = 2 and m are the order of the highest dominant nonlinear interaction8term and moments for which the PDFs are computed, respectively.Note that the usefulness of the found PDF tails is limited by the spatialstructure of the specific solutions to the non-linear equations. The spatialstructure determines the magnitude and sign of the coefficients k1 and k2.Furthermore, note that to evaluate the accuracy of the Gumbeldistribution in representing the PDF would require the exact com-putation of our path integral and λ-integral in Eq. (3), which is arather formidable task beyond the scope of the present paper. Wecan however obtain rough estimate on errors in our predictions byconsidering the two major, leading order contributions that wereneglected - the first one in using the saddle-point method for thepath integral [Eq. (5)], and the second in the λ-integral in Eq.(24). We note that we computed the PDFs keeping zeroth, firstand second order terms in the λ-integral and up to first order inusing the saddle-point path integral. For instance, for R = 0.6 anerror of the same order as the expansion in inverse aspect ratio≈ 30% (at R = 1 the error is only a few percent) is found fromthe sub-leading order terms. Therefore, the Gumbel distributionis a reasonable estimate for R > 0.6 − 0.7. The exact calculationof corrections to our model is however very involved, and will beaddressed in future publications.In summary, in the present letter we have presented a statistical theory ofturbulence and intermittency due to coherent structures in the sheath regionin fusion devices. The use of coherent structures is motivated by various ex-perimental results in the sheath region that bursty events cause a significanttransport which are linked to coherent structures. To investigate intermit-tency caused by these coherent structures, we have computed the PDF tails9of 1st moment flux (e.g. density, potential) using the non-perturbative tech-nique called the instanton method. The PDF tails are found to be in theform of modified Gumbel distributions. Interestingly, the Gumbel PDF tailsmay be enhanced in comparison with a Gaussian distribution. It is impor-tant to note that PDF tails of flux were shown to be far from Gaussianalthough Gaussian forcing was used. This result could guide in interpretingexperimental results. Specifically, the PDF tails derived here and in previouspapers may directly be compared with experimental results using log-log orlog-lin plots. Furthermore this model could elucidate the very complex in-termittent behavior at the edge of magnetic confinement systems.AcknowledgmentThis research was supported by the Engineering and Physical Sciences Re-search Council (EPSRC) EP/D064317/1.10References[1] Stangeby P C 2000 The plasma boundary of magnetic fusion devices(IOP publishing, Bristol)[2] Sattin F, Vianello N and Valisa M 2004 Phys. Plasmas 11 5032[3] Carter T A 2006 Phys. Plasmas 13 010701[4] Saha S K and Chowdhury S 2006 Phys. Plasmas 13 092512[5] Agostini M, Zweben S J, Cavazzana R, Scarin P, Serianni G, MaquedaR J and Stotler D P 2007 Phys. Plasmas 14 102305[6] Zweben S J, Boedo J A, Grulke O, Hidalgo C, LaBombard B, MaquedaR J, Scarin P, and Terry J L 2007 Plasma Phys. Controlled Fusion 49S1[7] Sattin F, Scarin P, Agostini M, Cavazzana R, Serianni G, Spolaore M,and Vianello N 2006 Plasma Phys. Controlled Fusion 48 1033[8] Myra J R, Russell D A, and DIppolito D A 2008 Phys. Plasmas 15032304[9] Yan Z, Tynan G R, Yu J H, Holland C, Muller S, and Xu M, 2007Bull. Am. Phys. Soc. 52 74[10] Connor J W and Wilson H R 2000 Plasma Phys. Contr. Fusion 421408[11] Zinn-Justin J 1989 Field Theory and Critical Phenomena (Clarendon,Oxford)[12] Gurarie V and Migdal A 1996 Phys. Rev. E 54 490811[13] Falkovich G, Kolokolov I, Lebedev V and Migdal A 1996 Phys. Rev. E54 4896[14] Kim E and Diamond P H 2002 Phys. Plasmas 9 71[15] Kim E and Diamond P H 2002 Phys. Rev. Lett. 88 225002[16] Kim E, Diamond P H, Malkov M, Hahm T S, Itoh K, Itoh S, Cham-peaux S, Gruzinov I, Gurcan O, Holland C, Rosenbluth M N andSmolyakov A 2003 Nucl. Fusion 43 961[17] Anderson J and Kim E 2008 Phys. Plasmas 15 052306[18] Anderson J and Kim E 2008 Phys. Plasmas 15 082312[19] Anderson J and Kim E 2009 Nuclear Fusion 49 075027[20] Anderson J and Kim E 2008 Phys. Plasmas 15 122303[21] Kim E, Liu HL and Anderson J 2009 Phys. Plasmas 16, 052304[22] Kim E and Anderson J 2008 Phys. Plasmas 15 114506[23] Balkovsky E and Lebedev V 1998 JETP Lett. 68 616[24] Fisher R A and Tippett L H C 1928 Proc. Cambridge PhilosophicalSociety 24 190[25] Gumbel E J 1958 Statistics of Extremes (Columbia University Press,New York)[26] D’Ippolito D A, Myra J R and Krasheninnikov S I 2002 Phys. Plasmas9 222[27] Krasheninnikov S I 2001 Phys. Lett. A 283 36812[28] Sarazin Y and Ghendrih P 1998 Phys. Plasmas 5 421413Figure captionsFigure 1. The PDF tails of flux in a system with exponential non-linearitywith k1 = 2.0 and k2 = 1.0 (black line), k1 = 1.0 and k2 = 1.0 (green line),k1 = 0.5 and k2 = 1.0 (blue line) and a Gaussian distribution (red line) areshown.14

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Predicting PDF tails in systems with logarithmic non-linearity

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