1
|
De S, Mitra D, Pandit R. Dynamic multiscaling in stochastically forced Burgers turbulence. Sci Rep 2023; 13:7151. [PMID: 37130867 PMCID: PMC10154400 DOI: 10.1038/s41598-023-29056-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/14/2022] [Accepted: 01/30/2023] [Indexed: 05/04/2023] Open
Abstract
We carry out a detailed study of dynamic multiscaling in the turbulent nonequilibrium, but statistically steady, state of the stochastically forced one-dimensional Burgers equation. We introduce the concept of interval collapse time, which we define as the time taken for a spatial interval, demarcated by a pair of Lagrangian tracers, to collapse at a shock. By calculating the dynamic scaling exponents of the moments of various orders of these interval collapse times, we show that (a) there is not one but an infinity of characteristic time scales and (b) the probability distribution function of the interval collapse times is non-Gaussian and has a power-law tail. Our study is based on (a) a theoretical framework that allows us to obtain dynamic-multiscaling exponents analytically, (b) extensive direct numerical simulations, and (c) a careful comparison of the results of (a) and (b). We discuss possible generalizations of our work to higher dimensions, for the stochastically forced Burgers equation, and to other compressible flows that exhibit turbulence with shocks.
Collapse
Affiliation(s)
- Sadhitro De
- Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science, Bangalore, 560012, India
| | - Dhrubaditya Mitra
- NORDITA, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, 10691, Stockholm, Sweden.
| | - Rahul Pandit
- Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science, Bangalore, 560012, India
| |
Collapse
|
2
|
Generalized Description of Intermittency in Turbulence via Stochastic Methods. ATMOSPHERE 2020. [DOI: 10.3390/atmos11091003] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
We present a generalized picture of intermittency in turbulence that is based on the theory of stochastic processes. To this end, we rely on the experimentally and numerically verified finding by R. Friedrich and J. Peinke [Phys. Rev. Lett. 78, 863 (1997)] that allows for an interpretation of the turbulent energy cascade as a Markov process of velocity increments in scale. It is explicitly shown that phenomenological models of turbulence, which are characterized by scaling exponents ζn of velocity increment structure functions, can be reproduced by the Kramers–Moyal expansion of the velocity increment probability density function that is associated with a Markov process. We compare the different sets of Kramers–Moyal coefficients of each phenomenology and deduce that an accurate description of intermittency should take into account an infinite number of coefficients. This is demonstrated in more detail for the case of Burgers turbulence that exhibits pronounced intermittency effects. Moreover, the influence of nonlocality on Kramers–Moyal coefficients is investigated by direct numerical simulations of a generalized Burgers equation. Depending on the balance between nonlinearity and nonlocality, we encounter different intermittency behavior that ranges from self-similarity (purely nonlocal case) to intermittent behavior (intermediate case that agrees with Yakhot’s mean field theory [Phys. Rev. E 63 026307 (2001)]) to shock-like behavior (purely nonlinear Burgers case).
Collapse
|
3
|
Margazoglou G, Biferale L, Grauer R, Jansen K, Mesterházy D, Rosenow T, Tripiccione R. Hybrid Monte Carlo algorithm for sampling rare events in space-time histories of stochastic fields. Phys Rev E 2019; 99:053303. [PMID: 31212557 DOI: 10.1103/physreve.99.053303] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/05/2018] [Indexed: 11/07/2022]
Abstract
We introduce a variant of the Hybrid Monte Carlo (HMC) algorithm to address large-deviation statistics in stochastic hydrodynamics. Based on the path-integral approach to stochastic (partial) differential equations, our HMC algorithm samples space-time histories of the dynamical degrees of freedom under the influence of random noise. First, we validate and benchmark the HMC algorithm by reproducing multiscale properties of the one-dimensional Burgers equation driven by Gaussian and white-in-time noise. Second, we show how to implement an importance sampling protocol to significantly enhance, by orders of magnitudes, the probability to sample extreme and rare events, making it possible to estimate moments of field variables of extremely high order (up to 30 and more). By employing reweighting techniques, we map the biased configurations back to the original probability measure in order to probe their statistical importance. Finally, we show that by biasing the system towards very intense negative gradients, the HMC algorithm is able to explore the statistical fluctuations around instanton configurations. Our results will also be interesting and relevant in lattice gauge theory since they provide unique insights into reweighting techniques.
Collapse
Affiliation(s)
- G Margazoglou
- Department of Physics, University of Rome Tor Vergata and INFN-Tor Vergata, 00133 Rome, Italy.,Computation-based Science and Technology Research Center, Cyprus Institute, 2121 Nicosia, Cyprus
| | - L Biferale
- Department of Physics, University of Rome Tor Vergata and INFN-Tor Vergata, 00133 Rome, Italy
| | - R Grauer
- Institut für Theoretische Physik I, Ruhr-University Bochum, 44780 Bochum, Germany
| | - K Jansen
- NIC, DESY, 15738 Zeuthen, Germany
| | - D Mesterházy
- Institute for Theoretical Physics, University of Bern, 3012 Bern, Switzerland
| | - T Rosenow
- Institut für Physik, Brandenburg University of Technology Cottbus-Senftenberg, 03046 Cottbus, Germany
| | - R Tripiccione
- Department of Physics, Università di Ferrara and INFN-Ferrara, 44122 Ferrara, Italy
| |
Collapse
|
4
|
Ebener L, Margazoglou G, Friedrich J, Biferale L, Grauer R. Instanton based importance sampling for rare events in stochastic PDEs. CHAOS (WOODBURY, N.Y.) 2019; 29:063102. [PMID: 31266309 DOI: 10.1063/1.5085119] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/09/2018] [Accepted: 05/10/2019] [Indexed: 06/09/2023]
Abstract
We present a new method for sampling rare and large fluctuations in a nonequilibrium system governed by a stochastic partial differential equation (SPDE) with additive forcing. To this end, we deploy the so-called instanton formalism that corresponds to a saddle-point approximation of the action in the path integral formulation of the underlying SPDE. The crucial step in our approach is the formulation of an alternative SPDE that incorporates knowledge of the instanton solution such that we are able to constrain the dynamical evolutions around extreme flow configurations only. Finally, a reweighting procedure based on the Girsanov theorem is applied to recover the full distribution function of the original system. The entire procedure is demonstrated on the example of the one-dimensional Burgers equation. Furthermore, we compare our method to conventional direct numerical simulations as well as to Hybrid Monte Carlo methods. It will be shown that the instanton-based sampling method outperforms both approaches and allows for an accurate quantification of the whole probability density function of velocity gradients from the core to the very far tails.
Collapse
Affiliation(s)
- Lasse Ebener
- Institut für Theoretische Physik I, Ruhr-Universität Bochum, Universitätsstraße 150, 44780 Bochum, Germany
| | - Georgios Margazoglou
- Department of Physics and INFN, University of Rome "Tor Vergata," Via della Ricerca Scientifica 1, I-00133 Roma, Italy
| | - Jan Friedrich
- Institut für Theoretische Physik I, Ruhr-Universität Bochum, Universitätsstraße 150, 44780 Bochum, Germany
| | - Luca Biferale
- Department of Physics and INFN, University of Rome "Tor Vergata," Via della Ricerca Scientifica 1, I-00133 Roma, Italy
| | - Rainer Grauer
- Institut für Theoretische Physik I, Ruhr-Universität Bochum, Universitätsstraße 150, 44780 Bochum, Germany
| |
Collapse
|
5
|
Friedrich J, Margazoglou G, Biferale L, Grauer R. Multiscale velocity correlations in turbulence and Burgers turbulence: Fusion rules, Markov processes in scale, and multifractal predictions. Phys Rev E 2018; 98:023104. [PMID: 30253572 DOI: 10.1103/physreve.98.023104] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/24/2018] [Indexed: 11/07/2022]
Abstract
We compare different approaches towards an effective description of multiscale velocity field correlations in turbulence. Predictions made by the operator-product expansion, the so-called fusion rules, are placed in juxtaposition to an approach that interprets the turbulent energy cascade in terms of a Markov process of velocity increments in scale. We explicitly show that the fusion rules are a direct consequence of the Markov property provided that the structure functions exhibit scaling in the inertial range. Furthermore, the limit case of joint velocity gradient and velocity increment statistics is discussed and put into the context of the notion of dissipative anomaly. We generalize a prediction made by the multifractal model derived by Benzi et al. [R. Benzi et al., Phys. Rev. Lett. 80, 3244 (1998)PRLTAO0031-900710.1103/PhysRevLett.80.3244] to correlations among inertial range velocity increment and velocity gradients of any order. We show that for the case of squared velocity gradients such a relation can be derived from first principles in the case of Burgers equations. Our results are benchmarked by intensive direct numerical simulations of Burgers turbulence.
Collapse
Affiliation(s)
- Jan Friedrich
- Institut für Theoretische Physik I, Ruhr-Universität Bochum, Universitätsstraße 150, 44780 Bochum, Germany
| | - Georgios Margazoglou
- Department of Physics and INFN, University of Rome "Tor Vergata," Via della Ricerca Scientifica 1, 00133 Roma, Italy.,Computation-based Science and Technology Research Center, Cyprus Institute, 20 Kavafi Street, 2121 Nicosia, Cyprus
| | - Luca Biferale
- Department of Physics and INFN, University of Rome "Tor Vergata," Via della Ricerca Scientifica 1, 00133 Roma, Italy
| | - Rainer Grauer
- Institut für Theoretische Physik I, Ruhr-Universität Bochum, Universitätsstraße 150, 44780 Bochum, Germany
| |
Collapse
|
6
|
Guo ZB, Diamond PH. Zonal Flow Patterns: How Toroidal Coupling Induces Phase Jumps and Shear Layers. PHYSICAL REVIEW LETTERS 2016; 117:125002. [PMID: 27689282 DOI: 10.1103/physrevlett.117.125002] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/16/2016] [Indexed: 06/06/2023]
Abstract
A new, frequency modulation mechanism for zonal flow pattern formation is presented. The model predicts the probability distribution function of the flow strength as well as the evolution of the characteristic spatial scale. Magnetic toroidicity-induced global phase dynamics is shown to determine the spatial structure of the flow. A key result is the observation that global phase patterning can lead to zonal flow formation in the absence of turbulence inhomogeneity.
Collapse
Affiliation(s)
- Z B Guo
- University of California, San Diego, California 92093, USA
| | - P H Diamond
- University of California, San Diego, California 92093, USA
| |
Collapse
|
7
|
Bhattacharyya SN. High Rayleigh number convection in a one-dimensional model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:033006. [PMID: 26465555 DOI: 10.1103/physreve.92.033006] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/19/2014] [Indexed: 06/05/2023]
Abstract
A model for one-dimensional convection is proposed by adding a buoyancy term to the Burgers' equation and including an equation for the temperature perturbation. A linear stability analysis shows onset of instability at a critical Rayleigh number. Computation in the unstable region shows steady convection with only one convection cell. Computations up to 10^{5} times the critical Rayleigh number do not show transition to an oscillatory state or to turbulence. Using a large Rayleigh number approximation, closed form solutions for the spectrum and the scaling for the heat transport due to nonlinear convection are obtained up to two orders. These are shown to be in good agreement with numerical results at high Rayleigh number.
Collapse
Affiliation(s)
- S N Bhattacharyya
- Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur 721 302, West Bengal, India
| |
Collapse
|
8
|
Basu A, Naji A, Pandit R. Structure-function hierarchies and von Kármán-Howarth relations for turbulence in magnetohydrodynamical equations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:012117. [PMID: 24580182 DOI: 10.1103/physreve.89.012117] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/02/2013] [Indexed: 06/03/2023]
Abstract
We generalize the method of A. M. Polyakov, [ Phys. Rev. E 52 6183 (1995)] for obtaining structure-function relations in turbulence in the stochastically forced Burgers equation, to develop structure-function hierarchies for turbulence in three models for magnetohydrodynamics (MHD). These are the Burgers analogs of MHD in one dimension [ Eur. Phys. J. B 9 725 (1999)], and in three dimensions (3DMHD and 3D Hall MHD). Our study provides a convenient and unified scheme for the development of structure-function hierarchies for turbulence in a variety of coupled hydrodynamical equations. For turbulence in the three sets of MHD equations mentioned above, we obtain exact relations for third-order structure functions and their derivatives; these expressions are the analogs of the von Kármán-Howarth relations for fluid turbulence. We compare our work with earlier studies of such relations in 3DMHD and 3D Hall MHD.
Collapse
Affiliation(s)
- Abhik Basu
- Condensed Matter Physics Division, Saha Institute of Nuclear Physics, Calcutta 700064, India
| | - Ali Naji
- School of Physics, Institute for Research in Fundamental Sciences (IPM), Tehran 19395-5531, Iran
| | - Rahul Pandit
- Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science, Bangalore 560012, India
| |
Collapse
|
9
|
Ni Q, Chen S. Statistics of active and passive scalars in one-dimensional compressible turbulence. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:066307. [PMID: 23368038 DOI: 10.1103/physreve.86.066307] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/09/2012] [Revised: 11/13/2012] [Indexed: 06/01/2023]
Abstract
Statistics of the active temperature and passive concentration advected by the one-dimensional stationary compressible turbulence at Re_{λ}=2.56×10^{6} and M_{t}=1.0 is investigated by using direct numerical simulation with all-scale forcing. It is observed that the signal of velocity, as well as the two scalars, is full of small-scale sawtooth structures. The temperature spectrum corresponds to G(k)∝k^{-5/3}, whereas the concentration spectrum acts as a double power law of H(k)∝k^{-5/3} and H(k)∝k^{-7/3}. The probability distribution functions (PDFs) for the two scalar increments show that both δT and δC are strongly intermittent at small separation distance r and gradually approach the Gaussian distribution as r increases. Simultaneously, the exponent values of the PDF tails for the large negative scalar gradients are q_{θ}=-4.0 and q_{ζ}=-3.0, respectively. A single power-law region of finite width is identified in the structure function (SF) of δT; however, in the SF of δC, there are two regions with the exponents taken as a local minimum and a local maximum. As for the scalings of the two SFs, they are close to the Burgers and Obukhov-Corrsin scalings, respectively. Moreover, the negative filtered flux at large scales and the time-increasing total variance give evidences to the existence of an inverse cascade of the passive concentration, which is induced by the implosive collapse in the Lagrangian trajectories.
Collapse
Affiliation(s)
- Qionglin Ni
- State Key Laboratory of Turbulence and Complex Systems, Center for Applied Physics and Technology, and Key Laboratory of High Energy Density Physics Simulation, Ministry of Education, College of Engineering, Peking University, 100871 Beijing, People's Republic of China.
| | | |
Collapse
|
10
|
Zhang ZX, She ZS. Subensemble decomposition and Markov process analysis of Burgers turbulence. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:026326. [PMID: 21929109 DOI: 10.1103/physreve.84.026326] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/27/2011] [Indexed: 05/31/2023]
Abstract
A numerical and statistical study is performed to describe the positive and negative local subgrid energy fluxes in the one-dimensional random-force-driven Burgers turbulence (Burgulence). We use a subensemble method to decompose the field into shock wave and rarefaction wave subensembles by group velocity difference. We observe that the shock wave subensemble shows a strong intermittency which dominates the whole Burgulence field, while the rarefaction wave subensemble satisfies the Kolmogorov 1941 (K41) scaling law. We calculate the two subensemble probabilities and find that in the inertial range they maintain scale invariance, which is the important feature of turbulence self-similarity. We reveal that the interconversion of shock and rarefaction waves during the equation's evolution displays in accordance with a Markov process, which has a stationary transition probability matrix with the elements satisfying universal functions and, when the time interval is much greater than the corresponding characteristic value, exhibits the scale-invariant property.
Collapse
Affiliation(s)
- Zhi-Xiong Zhang
- State Key Laboratory of Turbulence and Complex Systems and College of Engineering, Peking University, Beijing 100871, People's Republic of China
| | | |
Collapse
|
11
|
Abstract
A Lagrangian approach to Burgers turbulence is carried out along the lines of the field theoretical Martin-Siggia-Rose formalism of stochastic hydrodynamics. We derive, from an analysis based on the hypothesis of unbroken Galilean invariance, the asymptotic form of the probability distribution function of negative velocity differences. The origin of Burgers intermittency is found to rely on the dynamical coupling between shocks, identified to instantons, and noncoherent background fluctuations, which-then-cannot be discarded in a consistent statistical description of the flow.
Collapse
Affiliation(s)
- L Moriconi
- Instituto de Física, Universidade Federal do Rio de Janeiro, C.P. 68528, Rio de Janeiro 21945-970, RJ, Brazil
| |
Collapse
|
12
|
Movahed MS, Bahraminasab A, Rezazadeh H, Masoudi AA. Level crossing analysis of Burgers equation in 1 + 1 dimensions. ACTA ACUST UNITED AC 2006. [DOI: 10.1088/0305-4470/39/15/004] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
|
13
|
Tabei SMA, Bahraminasab A, Masoudi AA, Mousavi SS, Reza Rahimi Tabar M. Intermittency of height fluctuations in stationary state of the Kardar-Parisi-Zhang equation with infinitesimal surface tension in 1+1 dimensions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:031101. [PMID: 15524500 DOI: 10.1103/physreve.70.031101] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/24/2003] [Indexed: 05/24/2023]
Abstract
The Kardar-Parisi-Zhang (KPZ) equation with infinitesimal surface tension, dynamically develops sharply connected valley structures within which the height derivative is not continuous. We discuss the intermittency issue in the problem of stationary state forced KPZ equation in 1+1 dimensions. It is proved that the moments of height increments C(a) = <|h(x(1)) - h(x(2))|(a)> behave as |x(1) - x(2)|(xi(a)) with xi(a) = a for length scales |x(1) - x(2)|<<sigma . The length scale sigma is the characteristic length of the forcing term. We have checked the analytical results by direct numerical simulation.
Collapse
Affiliation(s)
- S M A Tabei
- Department of Physics, Sharif University of Technology, P.O. Box 11365-9161, Tehran, Iran
| | | | | | | | | |
Collapse
|
14
|
Shchur LN, Novotny MA. Evolution of time horizons in parallel and grid simulations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:026703. [PMID: 15447616 DOI: 10.1103/physreve.70.026703] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/12/2004] [Indexed: 05/24/2023]
Abstract
We analyze the evolution of the local simulation times (LST) in parallel discrete event simulations. The new ingredients introduced are (i) we associate the LST with the nodes and not with the processing elements, and (ii) we propose to minimize the exchange of information between different processing elements by freezing the LST on the boundaries between processing elements for some time of processing and then releasing them by a wide-stream memory exchange between processing elements. The highlights of our approach are (i) it keeps the highest level of processor time utilization during the algorithm evolution, (ii) it takes a reasonable time for the memory exchange, excluding the time consuming and complicated process of message exchange between processors, and (iii) the communication between processors is decoupled from the calculations performed on a processor. The effectiveness of our algorithm grows with the number of nodes (or threads). This algorithm should be applicable for any parallel simulation with short-range interactions, including parallel or grid simulations of partial differential equations.
Collapse
Affiliation(s)
- L N Shchur
- Department of Physics and Astronomy and ERC Center for Computational Sciences, Mississippi State University, Mississippi State, Mississippi 39762-5167, USA.
| | | |
Collapse
|
15
|
Bec J, Iturriaga R, Khanin K. Topological shocks in Burgers turbulence. PHYSICAL REVIEW LETTERS 2002; 89:024501. [PMID: 12096997 DOI: 10.1103/physrevlett.89.024501] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/02/2002] [Indexed: 05/23/2023]
Abstract
The dynamics of the multidimensional randomly forced Burgers equation is studied in the limit of vanishing viscosity. It is shown both theoretically and numerically that the shocks have a universal global structure which is determined by the topology of the configuration space. This structure is shown to be particularly rigid for the case of periodic boundary conditions.
Collapse
Affiliation(s)
- J Bec
- Laboratoire G.-D. Cassini, Observatoire de la Côte d'Azur, B.P. 4229, 06304 Nice Cedex 4, France
| | | | | |
Collapse
|
16
|
Masoudi AA, Shahbazi F, Davoudi J, Tabar MRR. Statistical theory for the Kardar-Parisi-Zhang equation in (1+1) dimensions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:026132. [PMID: 11863612 DOI: 10.1103/physreve.65.026132] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/14/2001] [Indexed: 05/23/2023]
Abstract
The Kardar-Parisi-Zhang (KPZ) equation in (1+1) dimensions dynamically develops sharply connected valley structures within which the height derivative is not continuous. We develop a statistical theory for the KPZ equation in (1+1) dimensions driven with a random forcing that is white in time and Gaussian-correlated in space. A master equation is derived for the joint probability density function of height difference and height gradient P(h-h*, partial differential(x)h,t) when the forcing correlation length is much smaller than the system size and much larger than the typical sharp valley width. In the time scales before the creation of the sharp valleys, we find the exact generating function of h-h* and partial differential(x)h. The time scale of the sharp valley formation is expressed in terms of the force characteristics. In the stationary state, when the sharp valleys are fully developed, finite-size corrections to the scaling laws of the structure functions left angle bracket(h-h*)(n)(partial differential(x)h)(m)right angle bracket are also obtained.
Collapse
Affiliation(s)
- A A Masoudi
- Department of Physics, Sharif University of Technology, P.O. Box 11365-9161, Tehran, Iran
| | | | | | | |
Collapse
|
17
|
Bec J. Universality of velocity gradients in forced Burgers turbulence. PHYSICAL REVIEW LETTERS 2001; 87:104501. [PMID: 11531481 DOI: 10.1103/physrevlett.87.104501] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/20/2001] [Indexed: 05/23/2023]
Abstract
We demonstrate that Burgers turbulence subject to large-scale white-noise-in-time random forcing has a universal power-law tail with exponent -7/2 in the probability density function of negative velocity gradients, as predicted by E, Khanin, Mazel, and Sinai [Phys. Rev. Lett. 78, 1904 (1997)]. A particle and shock tracking numerical method gives about five decades of scaling. Using a Lagrangian approach, the -7/2 law is related to the shape of the unstable manifold associated to the global minimizer.
Collapse
Affiliation(s)
- J Bec
- Observatoire de la Côte d'Azur, Laboratoire G.-D. Cassini, B.P. 4229, 06304 Nice Cedex 4, France.
| |
Collapse
|
18
|
Davoudi J, Masoudi AA, Rahimi Tabar MR, Rastegar AR, Shahbazi F. Three-dimensional forced Burgers turbulence supplemented with a continuity equation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:056308. [PMID: 11415007 DOI: 10.1103/physreve.63.056308] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/20/2000] [Indexed: 05/23/2023]
Abstract
We investigate turbulent limit of the forced Burgers equation supplemented with a continuity equation in three dimensions. The scaling exponent of the conditional two-point correlation function of density, i.e., <rho(x1)rho(x2)/delta u> approximately /x1-x2/(-alpha3), is calculated self-consistently in the nonuniversal region from which we obtain alpha3=3. Also we derive an equation governing the evolution of the probability density function (PDF) of longitudinal velocity increments in length scale, from which a possible mechanism for the dependence of the inertial PDF to one-point u(rms) is developed.
Collapse
Affiliation(s)
- J Davoudi
- Max-Planck Institute for Complex Systems, Noethnitzer Strasse 38, 01187 Dresden, Germany
| | | | | | | | | |
Collapse
|
19
|
Fleischer J, Diamond PH. Burgers' turbulence with self-consistently evolved pressure. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 61:3912-3925. [PMID: 11088172 DOI: 10.1103/physreve.61.3912] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/04/1999] [Revised: 08/23/1999] [Indexed: 05/23/2023]
Abstract
The Burgers' model of compressible fluid dynamics in one dimension is extended to include the effects of pressure back-reaction. The system consists of two coupled equations: Burgers' equation with a pressure gradient (essentially the one-dimensional Navier-Stokes equation) and an advection-diffusion equation for the pressure field. It presents a minimal model of both adiabatic gas dynamics and compressible magnetohydrodynamics. From the magnetic perspective, it is the simplest possible system which allows for "Alfvenization," i. e., energy transfer between the fluid and magnetic field excitations. For the special case of equal fluid viscosity and (magnetic) diffusivity, the system is completely integrable, reducing to two decoupled Burgers' equations in the characteristic variables v+/-v(sound) (v+/-v(Alfven)). For arbitrary diffusivities, renormalized perturbation theory is used to calculate the effective transport coefficients for forced "Burgerlence." It is shown that energy equidissipation, not equipartition, is fundamental to the turbulent state. Both energy and dissipation are localized to shocklike structures, in which wave steepening is inhibited by small-scale forcing and by pressure back reaction. The spectral forms predicted by theory are confirmed by numerical simulations.
Collapse
Affiliation(s)
- J Fleischer
- Physics Department, University of California at San Diego, La Jolla, California 92093, USA
| | | |
Collapse
|
20
|
Chattopadhyay AK. Nonlocal Kardar-Parisi-Zhang equation with spatially correlated noise. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:293-6. [PMID: 11969762 DOI: 10.1103/physreve.60.293] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/10/1999] [Indexed: 04/18/2023]
Abstract
The effects of spatially correlated noise on a phenomenological equation equivalent to a nonlocal version of the Kardar-Parisi-Zhang (KPZ) equation are studied via the dynamic renormalization group (DRG) techniques. The correlated noise coupled with the long ranged nature of interactions prove the existence of different phases in different regimes, giving rise to a range of roughness exponents defined by their corresponding critical dimensions. Finally self-consistent mode analysis is employed to compare the non-KPZ exponents obtained as a result of the long-range interactions with the DRG results.
Collapse
Affiliation(s)
- A K Chattopadhyay
- Department of Theoretical Physics, Indian Association For the Cultivation of Science, Jadavpur, Calcutta 700 032, India.
| |
Collapse
|
21
|
Hayot F, Jayaprakash C. Multifractality in the stochastic Burgers equation. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:4681-4684. [PMID: 9965646 DOI: 10.1103/physreve.54.4681] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
|
22
|
Bouchaud JP, Mézard M. Velocity fluctuations in forced Burgers turbulence. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:5116-5121. [PMID: 9965691 DOI: 10.1103/physreve.54.5116] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
|
23
|
Gurarie V, Migdal A. Instantons in the Burgers equation. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:4908-4914. [PMID: 9965671 DOI: 10.1103/physreve.54.4908] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
|
24
|
Yakhot V, Chekhlov A. Algebraic Tails of Probability Density Functions in the Random-Force-Driven Burgers Turbulence. PHYSICAL REVIEW LETTERS 1996; 77:3118-3121. [PMID: 10062138 DOI: 10.1103/physrevlett.77.3118] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
|
25
|
Falkovich G, Meyer M. Two-dimensional acoustic turbulence. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:4431-4434. [PMID: 9965597 DOI: 10.1103/physreve.54.4431] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
|
26
|
Polyakov AM. Turbulence without pressure. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 52:6183-6188. [PMID: 9964135 DOI: 10.1103/physreve.52.6183] [Citation(s) in RCA: 18] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
|
27
|
Chen S, Cao N. Inertial range scaling in turbulence. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 52:R5757-R5759. [PMID: 9964224 DOI: 10.1103/physreve.52.r5757] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
|
28
|
Chekhlov A, Yakhot V. Kolmogorov turbulence in a random-force-driven Burgers equation: Anomalous scaling and probability density functions. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 52:5681-5684. [PMID: 9964071 DOI: 10.1103/physreve.52.5681] [Citation(s) in RCA: 66] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
|