Geophysical & Astrophysical Fluid Dynamics ISSN: 0309-1929 (Print) 1029-0419 (Online) Journal homepage: http://www.tandfonline.com/loi/ggaf20 Magnetic helicity generation in the frame of Kazantsev model Egor V. Yushkov & Alexander S. Lukin To cite this article: Egor V. Yushkov & Alexander S. Lukin (2017): Magnetic helicity generation in the frame of Kazantsev model, Geophysical & Astrophysical Fluid Dynamics, DOI: 10.1080/03091929.2017.1388376 To link to this article: http://dx.doi.org/10.1080/03091929.2017.1388376 Published online: 23 Oct 2017. Submit your article to this journal View related articles View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=ggaf20 Download by: [UAE University] Date: 25 October 2017, At: 15:52 GEOPHYSICAL & ASTROPHYSICAL FLUID DYNAMICS, 2017 https://doi.org/10.1080/03091929.2017.1388376 Magnetic helicity generation in the frame of Kazantsev model Egor V. Yushkova,b and Alexander S. Lukina,b,c Downloaded by [UAE University] at 15:52 25 October 2017 a Faculty of Physics, Department of Mathematics, Lomonosov University, Moscow, Russia; b Space Research Institute RAS (IKI), Moscow, Russia; c National Research University Higher School of Economics, Moscow, Russia ABSTRACT ARTICLE HISTORY Using a magnetic dynamo model, suggested by Kazantsev (J. Exp. Theor. Phys. 1968, vol. 26, p. 1031), we study the small-scale helicity generation in a turbulent electrically conducting fluid. We obtain the asymptotic dependencies of dynamo growth rate and magnetic correlation functions on magnetic Reynolds numbers. Special attention is devoted to the comparison of a longitudinal correlation function and a function of magnetic helicity for various conditions of asymmetric turbulent flows. We compare the analytical solutions on small scales with numerical results, calculated by an iterative algorithm on non-uniform grids. We show that the exponential growth of current helicity is simultaneous with the magnetic energy for Reynolds numbers larger than some critical value and estimate this value for various types of asymmetry. Received 8 June 2017 Accepted 2 October 2017 KEYWORDS Small-scale dynamo; Kazantsev model; asymptotic and numerical analysis; large magnetic Reynolds number 1. Introduction The motion of a conducting ﬂuid with frozen-in magnetic ﬁeld can support the growth of average ﬁeld characteristics such as magnetic energy B 2 and current helicity B·[∇ × B] (Landau et al. 1960, Berger 1999, Brown et al. 1999). This process, usually called a magnetic dynamo, is assumed as the most probable mechanism for the formation of the magnetospheres of planets, stars and galaxies (see e.g. Molchanov et al. 1985). The traditional method of dynamo analysis for these astronomical objects is connected with spatial averaging on a typical correlation length of a random velocity ﬁeld. Of course, such averaged problems can describe only large-scale eﬀects, however a general analysis without spatial averaging is also possible (Kraichnan 1968, Moﬀatt 1978, Subramanian 1997, Subramanian 2003). This approach allows us to study the dynamo not only on large scales but also for small dissipation. However, as this is more complex and problematical, in our work we consider only the simplest case of turbulent isotropic ﬂows with very short velocity correlation times. The use of a delta-correlated velocity ﬁeld model with small, but equal (for all scales) correlation time intervals was suggested by Kazantsev (1968). The equation derived within this framework, describes the small-scale turbulent dynamo. For a long time it was considered as fundamental and was studied by both analytical and numerical methods (see e.g. Novikov et al. 1983, Molchanov et al. 1985). In particular, for a mirror-symmetric CONTACT Egor V. Yushkov email@example.com © 2017 Informa UK Limited, trading as Taylor & Francis Group 2 E. V. YUSHKOV AND A. S. LUKIN isotropic velocity ﬁeld the exponential temporal growth of the magnetic energy had been demonstrated for large enough Reynolds numbers (see Zeldovich et al. 1990, and references therein). Meanwhile, the question about the small-scale generation of the averaged magnetic ﬁeld in asymmetric ﬂows had not been considered until Vainshtein and Kichatinov (1986) (and a little bit later, but independently by Subramanian 1997), who deduced the system, which describes not only symmetric but also an asymmetric part of correlation tensor: Downloaded by [UAE University] at 15:52 25 October 2017 where Mt = 2r −4 (r 4 ηMr )r + 2Mr −4 (r 4 ηr )r − 4αK , Kt = r −4 r 4 (αM + 2ηK)r r , (1b) α(r) = G0 − G(r) . (1c) (1a) The system of Equations (1a,b) unequivocally deﬁnes the evolution of a longitudinal correlation function M(r, t) and a helicity correlation function K(r, t) depending on the magnetic viscosity η(r) = 1/Rm + F0 /3 − F(r)/3 and on the correlation functions F(r) and G(r) for a turbulent incompressible liquid: i rirj rirj v (r, t) v j (0, 0) = F + 12 rFr δ ij − 2 + F 2 + G ijk r k δ(t) . r r (2) However its analysis had encountered many obstacles, which have not been resolved yet. For example, we are unsure whether or not growth of anti-symmetric part of magnetic tensor is possible. Does the critical value of Reynolds number (when ﬁeld a generation begins) change signiﬁcantly with growth of asymmetry? Is the current helicity comparable with magnetic energy on small-scales or can it be neglected? etc. Here we show that by solving the system (1a,b) and calculating the magnetic energy characteristics B 2 and B·[∇ × B] from the functions M(r, t) and K(r, t) that deﬁne i rirj rirj B (r, t) Bj (0, 0) = M + 12 rMr δ ij − 2 + M 2 + K ijk r k , r r (3) we can obtain answers not only to the question about the possibility of magnetic helicity generation, but also about its dependence on the ﬂow helicity and magnetic viscosity. The cornerstone of our approach is the large magnetic Reynolds number realised by almost all physical dynamo-systems (e.g. the value of Rm for the Earth is about 102 −104 , for the galaxies about 105 −106 , and for the Sun 107 −108 ). Therefore, the direct solution of the Kazantsev system can be accomplished by standard large parameter asymptotic analysis. In the mirror-symmetric case such asymptotics has been undertaken by Molchanov et al. (1985), Artamonova and Sokolov (1986) and Rogachevskii and Kleeorin (1997), while for asymmetric motion the problem had been studied by Boldyrev et al. (2005), Malyshkin and Boldyrev (2007) and Yushkov and Lukin (2017). Without recalling the speciﬁc details of the previous work, we simply note that earlier results do not allow analysis of the functions M(r, t) and K(r, t) in the regime, where their behaviour is the most interesting and obscure. It was shown by Yushkov and Lukin (2017) that for large Rm the transformed correlation functions can grow on the interval r ∈ [Rm−1/2 , 1], but near zero and for large scales the answer to the question remained unclear. Moreover we did not analyse exactly GEOPHYSICAL & ASTROPHYSICAL FLUID DYNAMICS 3 the functions M(r, t) and K(r, t) (only transformed ones), although some ideas about their behaviour had been proposed. Below we show that the developed ideas and asymptotic accuracy of the previous studies can be improved signiﬁcantly, and analytical analysis of magnetic characteristics near r = 0 can be achieved. Moreover, corrected numerical schemes on non-uniform grids can conﬁrm the constructed asymptotics. They can also be used in the cascade models and for comparison with experimental data for each particular turbulent conditions. 2. Transformation of Kazantsev system and previous results We rewrite the Kazantsev system (1a,b), using new notation and by assuming the exponential ∼ exp (2γ t) time dependence for the correlation functions: Downloaded by [UAE University] at 15:52 25 October 2017 M(r, t) = e2γ t r −2 η−1/2 φ(r) and K(r, t) = 1 2γ t −4 r 2e 4 −2 r r θ(r) r r . (4a,b) This traditional approach leads us to the Sturm–Liouville problem (see, e.g. Hazewinkel 2001) for the system γ φ = ηφrr + U(r)φ − δη θrr − 2r −2 θ , γ θ = η θrr − 2r −2 θ + δφ + C1 r 2 + C2 r −1 , (5a) (5b) where U(r) = 2ηr 2η η2 ηrr + − 2 + r 2 r r 4η and δ(r) = G0 − G(r) η−1/2 . (5c,d) The solution of (5) deﬁnes the non-zero eigenvalue functions φ(r), θ(r) and correlation functions M(r, t), K(r, t), which increase in time with the maximum rate 2γ and dominate within magnetic correlation tensor after a suﬃciently long time. Note that the (5b) follows from (1b) after double integration, and the constants of integration C1 and C2 can be removed by adding contributions to the function θ(r) proportional to r 2 and r −1 , which cannot change the helicity function K(r, t) (see the transformation (4b)). We assume that a symmetric part of correlation tensor for the velocity ﬁeld in a random ﬂow has the classical Gaussian form (for more details see Novikov et al. 1983, Artamonova and Sokolov 1986): (r 3 F)r (6a) f (r) = 3r 2 with either (6b,c) f (r) = exp ( − r 2 ) or F(r) = exp ( − 3r 2 /5) . Then on only retaining logarithmic accuracy order, we consider problem (1a) on three √ √ distinct intervals r ∈ [0, ε ], r ∈ [ ε, 1] and r ∈ [1, ∞), where ε = 1/Rm. Such intervals are convenient for asymptotic expansion of the functions F(r) and η(r). Indeed, for large r 1 (6) deﬁne rapid decreasing functions F(r) and, consequently, η ∼ 1/3. By contrast, for small r 1 the function η(r) increases and can be expanded near r = 0 by the Taylor series 3r 4 r2 − + o(r 4 ) . (7) η = ε+ 5 50 4 E. V. YUSHKOV AND A. S. LUKIN Using this expansion, we rewrite the potential U(r) (see (5c)) in the form U(r) = − 2ε 3 48r 2 1/5 − 2r 2 /50 − + + + o(r 4 ) . r2 5 50 1 + 5ε/r 2 For a random isotropic and reﬂection-invariant (δ = 0) velocity ﬁeld the system (5a,b) decays into two independent equations and the problem becomes much easier. Equation (5a) containing the potential U(r) can be considered equivalent to the stationary Schrodinger problem with variable mass. Its solution gives the asymptotics of the longitudinal correlation function M(r, t) (studied earlier by Artamonova and Sokolov 1986): Downloaded by [UAE University] at 15:52 25 October 2017 2π ln r exp (2γ t) . M(r, t) = 3/2 1/2 sin r η ln ε (8) Solving (8) subject to zero boundary conditions, we ﬁnd that the dynamo rate γ depends only on magnetic Reynolds number Rm = ε−1 in the form 3 1 γ = − 4 5 2π ln ε 2 with Rmcr = exp 2π 4 15 . (9) We obtain the growth rate γ and the critical value of Reynolds number Rmcr , from which the dynamo-process begins (γ = 0) within the asymptotic framework of weak logarithmic convergence. However, we have successfully veriﬁed this asymptotic result by numerical methods, which use the algorithm proposed by Novikov et al. (1983) for this problem. Note also that the obtained asymptotic function (8) is non-zero only in the middle interval √ r ∈ [ ε, 1] (outside this interval φ(r) ≡ 0). Thus the correlation function M(r, t) is undeﬁned for small r, and so the function B(0)·B(r) can not be constructed and the accuracy of solution is insuﬃcient for magnetic energy analysis. Besides, the solution of the second Equation (5b), with δ = 0 and for positive γ , is zero. Therefore both the helicity function K(r, t) and current helicity B·[∇ × B] in the mirror-symmetric ﬂow decays. Below we show that, even for a weakly-asymmetrical ﬂows δ = 0, the situation totally reverses. 3. Solutions of Kazantsev system at three intervals We consider the large r interval r ∈ [1, +∞). We assume that γ 2r −2 , η = 1/3, and rewrite the system (5a,b) in the form φrr = 3(γ − δ 2 )φ + 3γ δθ , (10a) θrr = − 3δφ + 3γ θ . (10b) For each particular random velocity ﬁeld the ﬂow helicity function G(r) has its own speciﬁc form. However the √ idea of rapid decay for r 1 seems reasonable, and so √the asymmetry parameter δ ∼ 3(G0 − G(r)) can be equated here to the constant 3G0 . Then the solution of the system (10) can be written as a combination of exponentials exp (λr), in GEOPHYSICAL & ASTROPHYSICAL FLUID DYNAMICS 5 which the values of λ are the roots of the characteristic equation (see, e.g. Kamke 1976) λ4 + (3δ 2 − 6γ )λ2 + 9γ 2 = 0 . (11) It has four roots, two with positive and two with negative real parts: 3δ 2 λ2 = 2 2γ −1±i δ2 4γ −1 δ2 =⇒ ⎞ ⎛ 2 2 δ δ ⎠. ±i λ = ± 3γ ⎝ 1 − 4γ 4γ Downloaded by [UAE University] at 15:52 25 October 2017 (12a,b) Obviously we consider the solutions decaying at inﬁnity. For that reason, the asymmetry √ parameter needs to be suﬃciently small: δ 2 < 4γ . The exponents with λ = − 3γ ( · ± i · ) from (12b) determine solutions r 4γ r √ + F , cos − 1 φ = E γ exp − δ2 l l ⎤ ⎡ r r r ⎣ δ2 δ2 4γ cos +F + 1− sin + F ⎦, θ = E exp − −1 δ2 l 4γ l 4γ l (13a) (13b) where l= 4 . 3δ 2 (13c) The equalities (13a,b) show that, by contrast to the mirror-symmetric case, the monotonic decay of function φ(r) is absent and the dynamo oscillates in throughout entire space with the large wavelength 2πl, where l = 2/(3G0 ). Substituting the value η = 1/3 for the magnetic diﬀusivity, we see that this results remarkably coincides with a typical scale of large-scale dynamo process = 2η/G0 (for more details, see Vainshtein 1970, Moﬀatt 1978). However, further for simplicity we neglect terms proportional to 3δ 2 in (11) in comparison with terms proportional to γ . The approximation leaves us with non-oscillatory exponentially decaying solutions φ = E1 exp − 3γ r and θ = E2 exp − 3γ r . (14a,b) √ We consider the middle interval r ∈ [ ε, 1]. For that we use the expansion (7) for the magnetic diﬀusivity and note that r 2 /5 ε. Then from(5a,b) we obtain the system δ 4 r2 φrr + φ − r 2 θrr − 2θ = γ φ , 5 5 5 1 2 r θrr − 2θ + δφ = γ θ . 5 (15a) (15b) 6 E. V. YUSHKOV AND A. S. LUKIN Upon making the substitutions φ = ey/2 ϕ, θ = ey/2 ϑ and y = ln r the system (15) becomes 15 ϕyy = 5γ − 5δ 2 − ϕ + 5γ δϑ , (16a) 4 9 ϑyy = − 5δϕ + 5γ + ϑ. (16b) 4 Downloaded by [UAE University] at 15:52 25 October 2017 The coeﬃcients in the system depend on the helicity function, because δ(r) = (G0 − √ 2 G(r))/ η. Upon taking into account √ that G(r) ∼ G0 +√G0 r and η ∼ r /5, we ﬁnd that the asymmetry parameter δ ∼ 5 G0 − G(r) /r ∼ − 5G0 can be assumed constant. So in the ﬁrst approximation the solution of (16) can be presented as the combination of exponentials exp (λr), where each λ is a root of the characteristic equation 45δ 2 135 3 15 λ2 + 25γ 2 − γ − − = 0, λ4 − 10γ − 5δ 2 − 2 2 4 16 (17a) which determines 5 3 λ2 = 5γ − δ 2 − ± 2 4 5 2 δ +3 2 2 − 25γ δ 2 . (17b) For small mirror-asymmetry γ δ 2 < (3/5 + δ 2 /2)2 the square root in (17b) can be expanded to give two roots with diﬀerent signs: λ2 = 5γ 6 9 + > 0 2 6 + 5δ 4 and λ2 = 5γ 6 + 10δ 2 15 + 20δ 2 < 0. − 6 + 5δ 2 4 (18) From the previous result (8) and numerical solutions we know that the eigenfunction φ(r) is nonmonotonic. So we choose λ2 < 0 and, consequently, obtain the solution of the system (16) in the form 5 3 1/2 φ ∼ 5γ δr sin λ ln r + D , λ = 2 − 25γ δ 2 − 5γ + δ 2 + , (19a,b) 2 4 2 1 φ , = 5δ + 6 . (19c,d) θ ∼ − 2 − 25γ δ 2 5γ δ 2 For weak asymmetry (δ 2 9/25γ ) the functions φ(r) and θ(r) can be approximated by φ = Cr 1/2 sin λ ln r + D , (20a) 5δ 1/2 θ = C r sin λ ln r + D . (20b) 6 √ Finally, we consider the interval r ∈ [0, ε] near r = 0. Inspection of the asymptotics (20) reveals that for small δ the ratio θ/φ is insigniﬁcant. Thus, the contribution of the last term in (5a) can be ignored and for small r (γ 2r −2 ) the function φ(r) should satisfy the equality GEOPHYSICAL & ASTROPHYSICAL FLUID DYNAMICS 0 = εφrr − 2εr −2 φ . 7 (21) After repeated integration of (21), we obtain φ = Ar 2 , where A is a constant of integration. To determine θ, we consider (5b) by expanding the functions φ(r) and θ(r) in terms of power series. We ﬁnd that the ﬁrst correction to θ(r) is proportional to ∼ r 4 , and that the correction to φ(r) is smaller than ∼ r 5 . Thus, ignoring φ(r) in (5b) and assuming that η ∼ ε near r = 0, we obtain the equation γ θ = ε(θrr − 2r −2 θ) , which can be solved by standard methods (see, e.g. Kamke 1976, p. 401): Downloaded by [UAE University] at 15:52 25 October 2017 θ = 1 1 γ γ γ γ − exp r + + exp − r . ε r ε ε r ε Saving only ﬁrst two terms and approximating ε by η, we obtain the asymptotic result γ r2 2 θ = Br 1 + + o(r ) . 10 η 2 (22) Near r = 0 the ﬁrst term dominates. So on the interval r ∈ [0, solutions in the form φ = Ar 2 , √ ε] we can rewrite the θ = Br 2 . (23a,b) These solutions describe the general features of the helicity and longitudinal correlation functions, but they contrast in an essential way with numerical results (ﬁgure 1, right panel), as we will discussed below. 4. Conjunction at the boundaries The simplest way to realise solution conjunction is to use the asymptotics (14), (20) and (23). Accordingly we demand continuity of ⎧ 2 ⎪ ⎨Ar , φ(r) = Cr 1/2 sin λ ln r + D , ⎪ √ ⎩ E exp − 3γ r , √ for r ∈ [0, ε], √ for r ∈ [ ε, 1], for r ∈ [1, ∞) (24) and its ﬁrst derivations φ (r) at the boundaries of the intervals identiﬁed. The conditions on the function θ(r) will be automatically fulﬁlled due to their proportionality: θ 5 = δ φ 6 √ [ ε,1] B = A √ [0, ε] E2 = E1 [1,∞) = 5 − G0 √ 6 5 . (25) If we take into account from our previous approximations that the constants A, E, D and ! λ, where λ = 2! λ + 2π/ ln ε, should be small, then from the boundary conditions we determine the following equalities 8 E. V. YUSHKOV AND A. S. LUKIN A = − Cε−3/4 ! λ ln ε + D , E = C exp ( 3γ )D, 4π 4π = 3 ! = − (2 3γ + 1)D. 4! λ+ λ ln ε + D , 4! λ+ ln ε ln ε (26) (27) We solve (27) to obtain −6π , ! λ = − D = √ √ (3 3γ + 3/2) ln ε − (4 3γ + 8) √ 3γ + 2 3 2D . ln ε (28) Note that these formulas follow strictly from the boundary conditions of function φ(r). From (4a,b) and (5b) the correlation functions can now be expressed as φ(r) (29a) √ , r2 η √ φ(r) γ θ φ(r) G0 r 5 5 K(r, t) = exp (2γ t) 2 − δ = exp (2γ t) 2 . (29b) √ − γ G0 2r η φ 2r η η 6 Downloaded by [UAE University] at 15:52 25 October 2017 M(r, t) = exp (2γ t) A delicate matter arises here, because the equalities (25) show that the ratio θ/φ is approximately constant on all intervals, while the numerical calculations demonstrate √ diﬀerent constants near r = 0 and on the interval r ∈ [ ε, 1]. This problem follows from √ the conjunction condition at the point r = ε for two asymptotics, which are not exactly √ √ √ correct at r = ε, but rather correct for small r ( ε) and for large r ( ε). The disparity can be rectiﬁed by using the improved asymptotic result (22), in which the second term will be retained. This improved result does not change the values A, C, E, λ, because they were obtained only from φ-conditions, but it leads to complex questions about the conjunctions for the derivatives of the function θ(r). Here, we ignore these additional corrections due to their insigniﬁcance at large Rm and small asymmetry. Analysing (22) √ at large r we see, that the ratio θ/φ converges to a constant for r ε. Thus we can choose parameter B in such way to make this constant equal to the value 5δ/6 in the point of conjunction (see the ratio of (20a,b)). In this way, we obtain the equality √ B γ r2 B 5γ θ 5 5 = 1+ =⇒ 1+ ∼ − G0 (30) φ A 10 η A 10 6 √ together with the corresponding constant B = −AG0 5 5/(6 + 3γ ) (cf. (25)). Thus, on √ √ the both intervals r ∈ [0, ε] and r ∈ [ ε, 1] we can write the ratio in the form √ γ r2 θ 5 5 = − G0 1+ , (31) φ 6 + 3γ 10 η but disregarding it beyond these intervals due to the rapid exponential decay of θ and φ. Therefore, for r ∈ [0, 1], the correlation functions take the asymptotic forms φ(r) √ , r2 η √ " # G0 5 5 φ(r) G0 r γ r2 K(r, t) = exp (2γ t) 2 √ − γ 1+ , 2r η η 6 + 3γ 10 η M(r, t) = exp (2γ t) (32a) (32b) Downloaded by [UAE University] at 15:52 25 October 2017 GEOPHYSICAL & ASTROPHYSICAL FLUID DYNAMICS 9 Figure 1. On the left panel the eigenfunctions ψ(r) of problem (5) calculated from the initial approximation (marked by points) on the half line [0, ∞] for Rm = 104 . On the right panel are the longitudinal M(r) and helicity K(r) functions, following from the numerical results (marked by points), and the analytical rough and improved asymptotics, obtained in (29), for Rm = 104 and marked by dashed and solid lines respectively. and much smaller oscillating functions for r ∈ [1, ∞) with periods typical for largescale dynamo. The constants in (29) and (32) are deﬁned by the formulas (26)–(28). The magnetic energy characteristics can be calculated as B(r)·B(0) = 3M(r) + rM (r) , B(0)·[∇ × B](r) = 2 3K(r) + rK (r) . (33a) (33b) Approximating λ = 2! λ/ ln ε in (19a) and retaining accuracy up to order δ 2 , we can write the growth rate γ in the form 6 + 5δ 2 γ = 6 + 10δ 2 3 1 + δ2 − 4 5 2π ln ε 2 , (34) and estimate the critical Reynolds number as 2π Rmcr = exp 15/4 + δ 2 , where δ 2 = 5(G0 )2 , (35) Of course, these estimates are not very accurate, because they were obtained for small values of the asymmetry parameter δ and for very large magnetic Reynolds numbers (log Rm 5, in other words Rm 10000). Moreover, for γ close to zero the asymptotics fails due to the loss of solution localisation and fast exponential decay for r → ∞, so the more precise result can be obtained from (19b) numerically or from the numerical experiments. However, the formulas (34) and (35) demonstrate the main features of the dependence on the ﬂow conditions: the saturation level of the dynamo rate increases and the critical value decreases simultaneously with the mirror-asymmetry growth. Downloaded by [UAE University] at 15:52 25 October 2017 10 E. V. YUSHKOV AND A. S. LUKIN Figure 2. On the left panel the first and the second eigenvalues of problem (5) for various Rm obtained for mirror-symmetric (9) and asymmetric (34) cases and marked by dashed and solid lines (points show the numerical results). On the right panel are the first eigenvalues calculated for various helicity functions G(r) of the velocity random field (solid and dashed lines show the mirror-symmetric (9) and asymmetric (34) asymptotics and points show the numerical results: linear (large points), exponential (small points) and gauss (circles). Figure 3. On the left panel the energy B(0)·B(r) and helicity B(0)·[∇ × B(r)] characteristics of generated magnetic field calculated by (42) from analytical (solid lines) and numerical (points) results for Rm = 104 . On the right panel are the dependencies of the ratio of helicity and energy B(0)·[∇ × B(0)]/B(0)2 for magnetic field plotted against the same diagnostic for the velocity field, normalised by typical length Rm−1/2 and 1 respectively. Bold and thin solid lines show two cases of Reynolds numbers Rm = 102 and Rm = 106 , and points show the numerical results. 5. Numerical analysis of Kazantsev system The numerical analysis of the system (5) is based on the classical iterative schemes for Sturm–Liouville problems. This method for dynamo models in mirror-symmetric ﬂows was proposed by Novikov et al. (1983), and later in Yushkov (2015), Yushkov and Lukin (2017). It was developed for non-uniform grids and for weak mirror-asymmetry. Here we use the reverse iteration method, presented in details in, e.g. Kalitkin et al. (2005), GEOPHYSICAL & ASTROPHYSICAL FLUID DYNAMICS 11 Wilkinson (1965), and realised on the half-line with knots at the points ri∗ ∈ [0, ∞]: ri∗ = 0.05ri , (1 − ri )2 where ri ∈ [0, 1] is a uniform grid. (36) Following the general idea of iterative approach, we chose the analytical results (8) and (9) as the zero-step approximation (we have also tried to use the new asymptotics on the initial step, but there were no signiﬁcant diﬀerences), and then apply the iterative formula (A − γ̄ E)ψ (s+1) = ψ (s) , (37) Downloaded by [UAE University] at 15:52 25 October 2017 in which the meaning of the functions on (s+1)- and (s)-step is deﬁned by the unit matrix E and composite matrix of diﬀerential operator in (5a,b): A11 A12 , A= A21 A22 (38a) where A11 d2 = η 2 + U(r) , dr A21 = δ , A12 A22 d2 −2 = − δη − 2r , dr 2 2 d =η − 2r −2 . dr 2 (38b,c) (38d,e) The diﬀerence derivatives are used with shifted steps common for non-uniform grids and unbounded intervals (Kalitkin et al. 2005): un+1 − un−1 , 2(xn+1/2 − xn−1/2 ) un+1 − un 1 un − un−1 un = − . 2(xn+1/2 − xn−1/2 ) xn+3/4 − xn+1/4 xn−1/4 − xn−3/4 un = Using the eigenfunction ψ (s+1) calculated by (37), we obtain the eigenvalue γ (s+1) after a substitution of Aψ (s+1) = γ (s+1) ψ (s+1) into (37) and minimisation of the norm $ $ (s+1) $(γ − γ̄ )ψ (s+1) − ψ (s) $. (39) γ̄ ) (see$ (39)), so to Note that at each step we derive ψ (s) via the very small value (γ (s+1) −%$ (s+1) = ψ (s) $ψ (s) $. The norm avoid the unlimited growth, we make the normalisation ψ $$ $·$ can be chosen for any linear space, e.g. in Novikov et al. (1983) the space of continuous functions was used, while here we use the norm for integrable functions: $ $2 $u$ = (u, u ), & where (u, w) = u(r)w(r)dr , (40) 12 E. V. YUSHKOV AND A. S. LUKIN because for our examples this choice exhibits better numerical convergence and stability. The minimisation (39) gives the eigenvalue at the next step γ s+1 = γ̄ + (ψ (s+1) , ψ (s) ) (ψ (s+1) , ψ (s+1) ) (41) and allows us to repeat the iterative process from the beginning. Conﬁning ourselves only by ten iterations, we obtain and present results of numerical calculations for various random velocity ﬁelds and wide range of Reynolds numbers on the ﬁgures 1–3 and speculate on them in the next section. Downloaded by [UAE University] at 15:52 25 October 2017 6. Discussion and conclusions The studying of small-scale dynamo processes in the frame of the Kazantsev model shows that in mirror-asymmetric ﬂow the current helicity can grow simultaneously with the averaged magnetic energy. This dynamo eﬀect can be observed only for large Reynolds numbers Rm and can be analysed by both numerical and analytical methods. By constructing the analytic asymptotics for Rm 1 and considering weak asymmetry, we verify the analytic results by the numerical approach. The clear coincidence conﬁrms the obtained dependencies on random velocity ﬁeld parameters. For asymptotic analysis of Kazantsev problem we use ideas suggested for the symmetric case by Artamonova and Sokolov (1986). Using special substitutions (4), we rewrite √ √ the Kazantsev system (1a,b) on the three intervals [0, ε], [ ε, 1] and [1, ∞), obtain solutions and blend them on the boundaries. This conjunction gives the asymptotics of the dynamo growth rate γ and eigenfunctions, from which the magnetic correlation tensor Bi (r, t)·Bj (0, 0) can be restored. Of course, the results depend on the longitudinal and helicity correlation functions of the random velocity ﬁeld, so we consider analytically a special case, namely the longitudinal function is chosen to be of Gaussian form (this is a general way, used by Novikov et al. (1983), Artamonova and Sokolov (1986), Yushkov (2015), and for helicity we take a linear dependence on r as the ﬁrst approximation. More complicated and particular cases were examined only by the numerical approach. Our numerical method, adapted for the modiﬁed Kazantsev system (5), is based on the iterative schemes for Sturm–Liouville problems, earlier applied by Novikov for analogous but symmetric cases. Using the non-uniform grid (36) and shifted derivatives, we reveal fast convergence of the process for a wide range of initial approximations. The example of convergence (for ten iterations) in asymmetric case from the initial approximation in the form of the old asymptotics (8) is presented on the ﬁgure 1 (left panel). Using numerically obtained eigenfunctions, the longitudinal and helicity correlation functions, M(r) and K(r), can be calculated by (4) and compared with the analytical curves. On ﬁgure 1 one can see that the symmetric part of the correlation tensor ∼ M(r) monotonically √ decreases on the typical scale r ∼ ε and the asymmetric part ∼ K(r) changes sign on this interval; that probably follows from the closing of magnetic ﬁeld line loops. Comparing the right panel of ﬁgure 1 and the left panel of ﬁgure 2, we see that the agreement between numerical and analytical results is quite good not only for correlation functions, but also for the magnetic energy growth rate, especially at large Rm. Analysing a wide range of asymmetry parameters, we see that, just as in the symmetric cases, generation is possible Downloaded by [UAE University] at 15:52 25 October 2017 GEOPHYSICAL & ASTROPHYSICAL FLUID DYNAMICS 13 only at large Rm > Rmcr . This critical value Rmcr decreases with asymmetry growth (35) and simultaneously the maximum value of the dynamo growth rate for Rm → ∞ increases (34). So we can claim that the small-scale dynamo becomes more active for more asymmetric ﬂows. Note also that the small-scale Kazantsev model for asymmetric ﬂow has many features, which are intrinsic to the mean-ﬁeld dynamo. For example, the typical scale of oscillations δ = 2η/G0 (see (13a)) and the typical growth rate γ = G02 /4η (when the process loses its localisation near the dissipative scales, see (13b)) precisely correspond to the typical times and distances of the large-scale Steenbeck–Krause–Radler model (more details for this mean-ﬁeld dynamo model in the 3D space can be found in Yushkov (2014)). Thus, we can suppose that the advocated Equations (1a,b) allow us to study not only the helicity growth near the small scales, but also to investigate the role of this helicity in large-scale processes. The constructed asymptotic dynamo rate (34) is deﬁned by two parameters: the magnetic Reynolds number Rm and the inclination of ﬂow helicity function G (0), where G(r) ∼ V ·[∇ × V ]. In our analytical case this function was linear with constant tilt from r = 0 until r = 1, however the numerical approach allows us to consider other cases, for example, exponential G(r) ∼ exp ( − r) or Gaussian G(r) ∼ exp ( − r 2 ) functions and make sure that the main contribution in helicity generation is provided by the behaviour √ of G(r) only near zero r < ε. Indeed, there are almost no diﬀerences on the right panel of ﬁgure 2 between linear and exponential form of G(r), which are close to each other on √ √ the interval [0, ε], while for Gaussian form of G(r) almost zero, meaning on [0, ε] the growth rate is close to symmetric case even for large Rm. Therefore, we must emphasise that for small scales the dynamo should be analysed in details for each particular random ﬁelds, and our results are correct only as the ﬁrst step. The restored correlation tensor of magnetic ﬁeld (3) and its derivatives allow us to estimate by (33) the natural physical characteristics, for example, energy and current helicity dependencies B(0)·B(r) and B(0)·[∇ × B(r)]. The left panel of ﬁgure 3 shows that except of points of conjunction (where the small “peaks” arise from the nonideal nature of matching conditions and calculated derivatives) they repeat the correlation √ functions M(r) and K(r), localised on the dissipative scale r ∼ ε. However the inﬂuence of this generated current helicity can be important not only in the vicinity of the dissipative scales, but also for the large-sale dynamo. The results obtained show that the ratio of magnetic helicity and energy is quite signiﬁcant √ B·[∇ × B] −1/2 5 5γ . (42) = − G0 Rm 6 + 3γ B 2 Indeed, the value (42), multiplied by the scale of the correlation function localisation √ ε = Rm−1/2 , is dimensionless and it is only twice smaller than the same ratio for velocity ﬁeld normalised on the typical scale of the velocity random ﬁeld r = 1 (see the curves on the right panel of ﬁgure 3). Moreover, knowing that there are no limitations on the helicity transport toward the larger scales, we should suspect that the small-scale contribution in the total magnetic dynamo can play a serious role in the large-scale helicity balance and nonlinear mean-ﬁeld dynamo suppression, so it should be additionally studied both in the numerical cascade models and by the real laboratory experiments. 14 E. V. YUSHKOV AND A. S. LUKIN Acknowledgements The authors are grateful to D.D. Sokoloﬀ for valuable help in the discussion of general ideas and technical details. Also we appreciate the attempts of T. Postnikova to solve the problem. 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