Cascade of Random Rotation and Scaling in a Shell Model Intermittent Turbulence

Commun. Theor. Phys. (Beijing, China) 46 (2006) pp. 1074 1080 c International Academic Publishers Vol. 46, No. 6, December 15, 2006 Cascade of Random Rotation and Scaling in a Shell Model Intermittent Turbulence SUN Peng, 1,2, CHEN Shi-Gang, 3 and WANG Guang-Rui 3 1 Graduate School of China Academy of Engineering Physics, P.O. Box 2101, Beijing 100088, China 2 Physics Department, Anshan Normal College, Anshan 114005, China 3 Institute of Applied Physics and Computational Mathematics, P.O. Box 8009 (28), Beijing 100088, China (Received February 15, 2006) Abstract The time behaviors of intermittent turbulence in Gledzer Ohkitani Yamada model are investigated. Two kinds of orbits of each shell which is in the inertial range are discussed by portrait analysis in phase space. We find intermittent orbit parts wandering randomly and the directions of unstable quasi-periodic orbit parts of different shells form rotational, reversal and locked cascade of period three with shell number. We calculate the critical scaling of intermittent turbulence and the extended self-similarity of the two parts of orbit and point out that nonlinear scaling in inertial-range is decided by intermittent orbit parts. PACS numbers: 47.27.Gs, 47.52.+j, 05.45.Jn, 05.45.Pq Key words: intermittent orbit, unstable quasi-periodic orbit, critical scaling, extended self-similarity (ESS) 1 Introduction One of the intriguing problems in three-dimensional turbulence is related to the understanding of the dynamical mechanism triggering and supporting the energy cascade from large to small scales. The Gleder Ohkitani Yamada (GOY) [1,2] model can be seen as a truncation of Navier Stokes equation. The most outstanding property of the GOY model is that, for a suitable choice of free parameter, the set of scaling exponents ζ p coincides with that measured in true turbulence flows. It is still interesting to investigate time behaviors of intermittent turbulence with GOY model, because it may be important to simulate the fast course of laser beam propagation in the atmosphere. Recently, Kato and Yamada [3] studied unstable periodic solution and turbulence solution of system of 12 shells in GOY model. They described statistics similarity between two solutions and found that the scaling exponents of structure function of two solutions have a similar nonlinear scaling at the same parameter values. But they didn t give detailed structure and analysis of unstable period solution in phase space. In this paper, we study the phase portrait of shell velocity in inertial range of system of 22 shells in GOY model and find that the random wandering of intermittent orbit (IO) parts and the direction of unstable quasi-periodic orbit (UQO) parts in phase space form the rotational, reversal and locked cascade of period three with shell number. We also calculate the critical scaling of intermittent turbulence and the extended self-similarity (ESS) of IO and UQO respectively. GOY model describes a one-dimensional cascade of energies among a set of complex velocities. It is an ordinary differential equation system, du n dt = νk 2 nu n +fδ n,4 + ik n (U n+1u n+2 δ λ U n 1U n+1 1 δ λ 2 U n 1U n 2 ), (1) where ν is viscosity, wave numbers k n are given by k n = k 0 λ n (n = 1,..., N), and N is shell number. External forcing f is applied to the fourth shell, stands for complex conjugate, and δ stands for nonlinear coupling parameter, which is an adjustable parameter. This model holds the main symmetry of Navier Stocks equation. The time-average energy flux through the nth shell is Π n = Im n+1 + (1 δ) n, (2) where triple products n = k n 1 U n 1 U n U n+1. We then can picture a steady state of the dynamical system as cascade of energy from large eddies to smaller ones, where the energy is dissipated through viscous diffusion. It is in this sense that the dynamics may simulate real turbulence. In the numerical implementation, we used a slaved Adams Bashforth-schems. [4] The result is qualitatively consistent with Runge Kutta method. Throughout this paper we make choices N = 22, k 0 = 2 4, λ = 2, ν = 1 10 7, f = 5 (1+i) 10 3 δ n,4, and step h = 10 4. The course of transient 3000 n.u is cutoff, when we calculate scaling law. 2 Property of Orbit in Phase Space and Critical Scaling 2.1 Property of Stable Quasi-periodic Orbits in Phase Space In inertial range, Π n = 2νQ, where Q = n k2 n u n 2 /2, inertial range is adopted n = 5 13. For each shell in the inertial range, the orbit of solution in phase space is plotted by taking Re (U n ) and Im (U n ) as coordinates. The project supported by National Natural Science Foundation of China, the Science Foundation of China Academy of Engineering Physics under Grant No. 10576076, the Major Projects of National Natural Science Foundation of China under Grant No. 10335010, and the Science Foundation of China Academy of Engineering Physics under Grant No. 20040430 E-mail: sunpeng169@sina.com

No. 6 Cascade of Random Rotation and Scaling in a Shell Model Intermittent Turbulence 1075 When δ < δ c = 0.375 02, the orbits of solution correspond to stable quasi-periodic time series. Here δ c is critical point from steady quasi-periodic time series to intermittent time series. At δ = 0.371, the orbits of shells n = 5, 8, 11 and n = 6, 9, 12 are envelope cycles which are shaped quasi-periodically by unclosed loop curves precessing clockwise and counterclockwise around (0, 0) respectively (see Fig. 1(a)). Those envelope cycles are delayed slightly from big shells to small ones (see Fig. 1(d)). For example, envelope cycle of shell 5 includes about 172.002 whole precessions with the average precession angle being about 2.093, and the average period of each precession is T = 556.44 with the relative error being about 8.0 10 4. The eddy turnover time of shell 5 is about 7.47. Similarly, envelope cycle of shell 6 includes about 169.891 whole precessions with the average precession angle being about 2.119, and the average period of each precession is about T = 556.45 with the relative error being about 9.5 10 4 (see Fig. 1(b)). The eddy turnover time of shell 6 is about 2.27. On the other hand, the phase orbits of shells n = 7, 10, 13 form successively quasi-periodic circles which are all locked at the same direction (see Fig. 1(c)). These phenomena have characters of periodic three with shell number, which can be explained as the fact that the GOY model (1) in the case of infinite number of shells with inviscid and unforced limit has static solution for 0 < δ < 1, Un K41 = kn 1/3 h 1 (n). (3) Here h 1 (n) is any periodic function of periodic three. [5] Fig. 1 Phase portraits at δ = 0.371. (a) The unclosed loop curves of shells 5 and 6 precess around (0, 0) with average period 556.44 and 556.45 clockwise and counterclockwise respectively. The arrowhead is the initial position of envelope cycle; (b) Shells 5 and 6 form the small and big envelope cycle respectively; (c) Stable quasi-periodic cycle locked at the same direction of shells 7, 10, and 13. (d) At δ = 0.371, the time series of velocity modulus of shells 5, 6, and 7 are delayed slightly. At δ = 0.374, the envelope cycles of shells n = 5, 8, 11 and n = 6, 9, 12 change into stable quasi-periodic cycle. The direction of quasi-periodic cycle is defined in terms of the phase of intersection of quasi-periodic cycle. For the direction of stable quasi-periodic cycle of shell n = 5, 6, the results are about 121.42 and 76.71 with relative error being smaller than 1.5 10 3 (see Fig. 2). The power spectrum of each shell shows that it has the same fundamental and multiple frequencies.

1076 SUN Peng, CHEN Shi-Gang, and WANG Guang-Rui Vol. 46 Fig. 2 Phase portraits at δ = 0.374. (a) Shells n = 5, 8, and 11; (b) Shells n = 6, 9, and 12. 2.2 Random Wandering of Intermittent Orbit and Property of Rotation of Unstable Quasi-periodic Orbit At δ > δ c, the time series of the velocity modulus of each shell can be divided into intermittent time series (chaos part) and unstable quasi-periodic time series. Similarly, for the orbits in the phase space, the orbits can be divided into intermittent orbit (IO) parts corresponding to the intermittent time series and unstable quasiperiodic orbit (UQO) parts corresponding to the unstable quasi-period time series. UQO parts in our situation are unstable quasi-period cycles. For the same shell, one quasi-periodic cycle which has a direction around (0, 0) gradually processes to the intermittent orbit. Since the phases of intermittent orbit part wander randomly around (0, 0) in the interval [0,2π], the direction of quasi-periodic cycle rotates randomly around (0, 0) after the intermittence (see Fig. 3). The two part orbits evolve with time alternately. Fig. 3 At δ = 0.378, time series of velocity modulus evolving from unstable quasi-period to chaos and their corresponding phase portrait. (a) Shell 5; (b) Shell 6.

No. 6 Cascade of Random Rotation and Scaling in a Shell Model Intermittent Turbulence 1077 The difference of rotation direction can be defined in terms of the direction difference of unstable quasi-periodic cycle at the front and back of intermittence. At δ = 0.378, the difference of rotation direction of unstable quasi-periodic cycle of shells n = 5, 8, 11 is about 21.88, 21.51, and 20.08 with the relative error being smaller than 1.6 10 3, and the difference of rotation direction of quasi-periodic cycle of shells n = 6, 9, 12 is about 21.71, 21.25, and 20.86 with the relative error being smaller than 2.0 10 3. The difference of rotation direction of unstable quasi-periodic cycle of shells n = 7, 10, 13 is locked at 45. The phenomena also have characters of periodic three with shell number (see Fig. 4). Fig. 4 Phase portraits of unstable quasi-periodic cycle of shells before intermittence. (a) n = 5, 8, and 11; (b) n = 6, 9, and 12; (e) n = 7, 10, and 13. Phase portraits of unstable quasi-periodic cycle of shells after intermittence: (c) n = 5, 8, and 11; (d) n = 6, 9, and 12; (f) n = 7, 10, and 13. Benzi and Gat et al. [6,7] pointed out that shell model without external forcing has phase symmetry corresponding to the space translation symmetry of Navier Stokes equation. Let U n = kn 1/3 ρ n exp(iθ n ). (4) At f = 0, with this choice, equation (1) becomes ( d [ dt n) + νk2 ρ n = kn 2/3 ρ n+2 ρ n+1 sin( n+2 ) δ 2 ρ n+1ρ n 1 sin( n+1 ) 1 δ ] 2 ρ n 1ρ n 2 sin( n ). (5) Here n = θ n 2 + θ n 1 + θ n and the probability distribution of θ n is uniform in the interval [0, 2π]. By simply taking θ n θ n, θ n 1 θ n 1 α, θ n 2 θ n 2 + α. (6)

1078 SUN Peng, CHEN Shi-Gang, and WANG Guang-Rui Vol. 46 For any n, one solution of the equation can be transformed into another solution. In our case, the external forcing in Eq. (1) acts on shell n = 4. If we interpret θ n as the rotation direction of the unstable quasi-periodic cycle, the rotation direction of shell n = 4 locks and so do for shells n = 7, 10, and 13 because of character of periodic three with shells number. According to Eq. (6), the difference of rotation direction at the front and back of intermittence of shells 7, 10, and 13 is θ 7 = θ 10 = θ 13 = 0, while for shells 5, 8, and 11 and shells 6, 9, and 12, θ 5 θ 6, θ 8 θ 9, θ 11 θ 12. This means that the direction of unstable quasi-periodic cycle of different shells in inertial range forms the rotational, reversal and locked cascade of period three with shell number. 2.3 Critical Scaling of Intermittent Turbulence With δ increasing, the parts of the intermittent time series of the velocity modulus increase gradually, while unstable quasi-periodic time series decrease correspondingly. We can calculate the statistical average of unstable quasi-periodic time series length T UQO. The form of critical scaling is T UQO = C(δ δ C ) α. (7) We can obtain C = 2.2671 10 4, critical point δ c = 0.375 02, and critical exponent α = 1.83 with the method of least squares (see Fig. 5). Fig. 5 Critical scaling fit curve between δ = 0.3764 and 0.3779. Fig. 6 Phase portraits of the first 16 UQO at δ = 0.38. (a) n = 5; (b) n = 6. By the long-time numerical evolving, for the same shell, the direction of UQO rotates randomly around (0, 0) clockwise or counterclockwise in an annular range (see Fig. 6). The phenomenon is consistent with chaos orbit including infinite numbers of UQO. The width of the annular range is equal to the biggest amplitude of unstable quasi-periodic time series. These orbits are the projects in high dimensional torus. 3 Character of ESS in Intermittent Turbulence The extended self-similarity (ESS) can be defined as structure function of order p against structure function of order q (q = 3), [8 11] S p (K n ) S 3 (K n ) β(p,3). (8)

No. 6 Cascade of Random Rotation and Scaling in a Shell Model Intermittent Turbulence 1079 Here S p (k n ) = Π n /k n p/3, and β(p, 3) = ζ p /ζ 3. If we adopt this structure function to calculate scaling exponent, period three oscillation can be eliminated and inertial range can be extended to n = 5 16. Numerically, we can obtain ESS from the log-log plot of the sixth order structure function against the third order structure function. If the time series of intermittent turbulence can be divided into intermittent time series parts and unstable quasi-periodic time series parts, we can obtain the ESS of IO and ESS of UQO respectively. 3.1 Extended Self-similarity of Intermittent Turbulence At δ 0.4, the ESS of intermittent turbulence is very similar to linear case. The ESS of intermittent turbulence at δ = 0.3796, 0.38, and 0.384 are all smaller than K41 (ζ p = p/3) (see Fig. 7). Fig. 7 Scaling exponent of structure function. The line stands for K41 and square stands for the anomalous scaling of fully developed turbulence at δ = 0.5; Circles, triangles, and diamonds show the results for the ESS of intermittent turbulence at δ = 0.3796, 0.38, and 0.384 respectively. 3.2 Extended Self-similarity Scaling of IO and UQO At δ 0.384, the UQO and IO can be separated definitely. For the parameter range δ = 0.3796 0.384, the ESS of UQO is extremely near K41 and the slopes of the ESS of UQO are 0.335 ± 0.001 at δ = 0.3796, 0.38, and 0.384 (Fig. 8(a)). The ESS of IO is nonlinear (Fig. 8(b)). This means that the ESS of IO reveals the nonlinear character of ESS. Fig. 8 Scaling exponent of structure function. The lines stand for K41; squares, circles and triangles stand for ESS of UQO (a) and ESS of IO (b) at δ = 0.3796, 0.38, and 0.384 respectively. Diamonds of (b) stand for the anomalous scaling at δ = 0.5. Squares and circles of (c) stand for the log-log portrait of sixth order structure function against the third order structure function of IO (slopes is 1.87) and UQO (slopes is 2.00) at δ = 0.384, and the lines stand for K41 which slope is 2.0. 4 Conclusion In the phase space consisting of Re (U n ) and Im (U n ), the phase of IO of each shell shows the random wandering around (0, 0) in the interval [0, 2π], while the direction of UQO of different shells form the rotational, reversal and

1080 SUN Peng, CHEN Shi-Gang, and WANG Guang-Rui Vol. 46 locked cascade of periodic three with shell number in the annular range around (0, 0). The critical scaling and critical point are obtained from the statistical average of unstable quasi-periodic time series length. In δ = 0.3796 0.384, the ESS of UQO is extremely near K41, but the ESS of IO is nonlinear. In the course of laser beam propagation in the atmosphere, laser beam can propagate both in intermittence condition and in non-intermittence condition, so that the corresponding scaling may also be divided into the scaling in intermittent course and in non-intermittent course. In this paper, we discuss the intermittent property changing with parameter δ, but it is more interesting to discuss the intermittent property changing with external force which will be investigated. Acknowledgments We would like to thank Guo-Yong Yuan and Li-Bin Fu, for their useful discussions. References [1] E.B. Gledzer, Dokl. Akad. Nauk SSR 209 (1973) 1046 (Sov. Phys. Dokl. 18 (1973) 216). [2] M. Yamada and K. Ohkitani, J. Phys. Soc. Jpn. 56 (1987) 4210. [3] S. Kato and M. Yamada, Phys. Rev. E. 68 (2003) 025302. [4] D. Pisarenko, L. Biferale, D. Courvoisier, U. Frisch, and M. Vergassoia, Phys. Fluids A 5 (1993) 2533. [5] R. Biferale, A. Lambert, R. Lima, and G. Paladin, Physica D 80 (1995) 105. [6] R. Benzi, L. Biferale, and G. Parisi, Physica D 65 (1993) 163. [7] O. Gat, I. Procaccia, and R. Zeitak, Phys. Rev. E 51 (1995) 1148. [8] R. Benzi, S. Ciliberto, R. Tripiccione, C. Baudet, F. Massaioli, and S. Succi, Phys. Rev. E 48 (1993) 29. [9] R. Benzi, S. Ciliberto, G. Ruiz Chavarria, and R. Tripiccione, Europhys. Lett. 24 (1993) 275. [10] M. Briscolini, P. Santangelo, S. Succi, and R. Benzi, Phys. Rev. E 50 (1994) 1745. [11] R. Benzi, L. Biferale, S. Giliberto, M.V. Struglia, and R. Tripiccione, Physica D 96 (1996) 162.