Direct Numerical Simulation Tests of Eddy Viscosity in Two Dimensions

Two-parametric eddy viscosity (TPEV) and other spectral characteristics of two-dimensional (2D) turbulence in the energy transfer sub-range are calculated from direct numerical simulation (DNS) with 512$^2$ resolution. The DNS-based TPEV is compared with those calculated from the test field model (TFM) and from the renormalization group (RG) theory. Very good agreement between all three results is observed.Comment: 9 pages (RevTeX) and 5 figures, published in Phys. Fluids 6, 2548 (1994

Symmetries of the stochastic Burgers equation

All Lie symmetries of the Burgers equation driven by an external random force are found. Besides the generalized Galilean transformations, this equation is also invariant under the time reparametrizations. It is shown that the Gaussian distribution of a pumping force is not invariant under the symmetries and breaks them down leading to the nontrivial vacuum (instanton). Integration over the volume of the symmetry groups provides the description of fluctuations around the instanton and leads to an exactly solvable quantum mechanical problem.Comment: 4 pages, REVTeX, replaced with published versio

Large eddy simulation of two-dimensional isotropic turbulence

Large eddy simulation (LES) of forced, homogeneous, isotropic, two-dimensional (2D) turbulence in the energy transfer subrange is the subject of this paper. A difficulty specific to this LES and its subgrid scale (SGS) representation is in that the energy source resides in high wave number modes excluded in simulations. Therefore, the SGS scheme in this case should assume the function of the energy source. In addition, the controversial requirements to ensure direct enstrophy transfer and inverse energy transfer make the conventional scheme of positive and dissipative eddy viscosity inapplicable to 2D turbulence. It is shown that these requirements can be reconciled by utilizing a two-parametric viscosity introduced by Kraichnan (1976) that accounts for the energy and enstrophy exchange between the resolved and subgrid scale modes in a way consistent with the dynamics of 2D turbulence; it is negative on large scales, positive on small scales and complies with the basic conservation laws for energy and enstrophy. Different implementations of the two-parametric viscosity for LES of 2D turbulence were considered. It was found that if kept constant, this viscosity results in unstable numerical scheme. Therefore, another scheme was advanced in which the two-parametric viscosity depends on the flow field. In addition, to extend simulations beyond the limits imposed by the finiteness of computational domain, a large scale drag was introduced. The resulting LES exhibited remarkable and fast convergence to the solution obtained in the preceding direct numerical simulations (DNS) by Chekhlov et al. (1994) while the flow parameters were in good agreement with their DNS counterparts. Also, good agreement with the Kolmogorov theory was found. This LES could be continued virtually indefinitely. Then, a simplifiedComment: 34 pages plain tex + 18 postscript figures separately, uses auxilary djnlx.tex fil

Universality of Velocity Gradients in Forced Burgers Turbulence

It is demonstrated 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 (1997, Phys. Rev. Lett. 78, 1904). 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.Comment: 4 pages, 2 figures, RevTex4, published versio

Generation and Structure of Solitary Rossby Vortices in Rotating Fluids

The formation of zonal flows and vortices in the generalized Charney-Hasegawa-Mima equation is studied. We focus on the regime when the size of structures is comparable to or larger than the deformation (Rossby) radius. Numerical simulations show the formation of anticyclonic vortices in unstable shear flows and ring-like vortices with quiescent cores and vorticity concentrated in a ring. Physical mechanisms that lead to these phenomena and their relevance to turbulence in planetary atmospheres are discussed.Comment: 3 pages in REVTeX, 5 postscript figures separately, submitted to Phys. Rev.

Viscous Instanton for Burgers' Turbulence

We consider the tails of probability density functions (PDF) for different characteristics of velocity that satisfies Burgers equation driven by a large-scale force. The saddle-point approximation is employed in the path integral so that the calculation of the PDF tails boils down to finding the special field-force configuration (instanton) that realizes the extremum of probability. We calculate high moments of the velocity gradient $\partial_xu$ and find out that they correspond to the PDF with $\ln[{\cal P}(\partial_xu)]\propto-(-\partial_xu/{\rm Re})^{3/2}$ where ${\rm Re}$ is the Reynolds number. That stretched exponential form is valid for negative $\partial_xu$ with the modulus much larger than its root-mean-square (rms) value. The respective tail of PDF for negative velocity differences $w$ is steeper than Gaussian, $\ln{\cal P}(w)\sim-(w/u_{\rm rms})^3$, as well as single-point velocity PDF $\ln{\cal P}(u)\sim-(|u|/u_{\rm rms})^3$. For high velocity derivatives $u^{(k)}=\partial_x^ku$, the general formula is found: $\ln{\cal P}(|u^{(k)}|)\propto -(|u^{(k)}|/{\rm Re}^k)^{3/(k+1)}$.Comment: 15 pages, RevTeX 3.

Quantized Scaling of Growing Surfaces

The Kardar-Parisi-Zhang universality class of stochastic surface growth is studied by exact field-theoretic methods. From previous numerical results, a few qualitative assumptions are inferred. In particular, height correlations should satisfy an operator product expansion and, unlike the correlations in a turbulent fluid, exhibit no multiscaling. These properties impose a quantization condition on the roughness exponent $\chi$ and the dynamic exponent $z$. Hence the exact values $\chi = 2/5, z = 8/5$ for two-dimensional and $\chi = 2/7, z = 12/7$ for three-dimensional surfaces are derived.Comment: 4 pages, revtex, no figure

Towards a Simple Model of Compressible Alfvenic Turbulence

A simple model collisionless, dissipative, compressible MHD (Alfvenic) turbulence in a magnetized system is investigated. In contrast to more familiar paradigms of turbulence, dissipation arises from Landau damping, enters via nonlinearity, and is distributed over all scales. The theory predicts that two different regimes or phases of turbulence are possible, depending on the ratio of steepening to damping coefficient (m_1/m_2). For strong damping (|m_1/m_2|<1), a regime of smooth, hydrodynamic turbulence is predicted. For |m_1/m_2|>1, steady state turbulence does not exist in the hydrodynamic limit. Rather, spikey, small scale structure is predicted.Comment: 6 pages, one figure, REVTeX; this version to be published in PRE. For related papers, see http://sdphpd.ucsd.edu/~medvedev/papers.htm

Burgers' Flows as Markovian Diffusion Processes

We analyze the unforced and deterministically forced Burgers equation in the framework of the (diffusive) interpolating dynamics that solves the so-called Schr\"{o}dinger boundary data problem for the random matter transport. This entails an exploration of the consistency conditions that allow to interpret dispersion of passive contaminants in the Burgers flow as a Markovian diffusion process. In general, the usage of a continuity equation $\partial_t\rho =-\nabla (\vec{v}\rho)$, where $\vec{v}=\vec{v}(\vec{x},t)$ stands for the Burgers field and $\rho$ is the density of transported matter, is at variance with the explicit diffusion scenario. Under these circumstances, we give a complete characterisation of the diffusive transport that is governed by Burgers velocity fields. The result extends both to the approximate description of the transport driven by an incompressible fluid and to motions in an infinitely compressible medium. Also, in conjunction with the Born statistical postulate in quantum theory, it pertains to the probabilistic (diffusive) counterpart of the Schr\"{o}dinger picture quantum dynamics.Comment: Latex fil

Statistical Theory for the Kardar-Parisi-Zhang Equation in 1+1 Dimension

The Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimension dynamically develops sharply connected valley structures within which the height derivative {\it is not} continuous. There are two different regimes before and after creation of the sharp valleys. We develop a statistical theory for the KPZ equation in 1+1 dimension driven with a random forcing which 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 $P(h-\bar h,\partial_{x}h,t)$ when the forcing correlation length is much smaller than the system size and much bigger than the typical sharp valley width. In the time scales before the creation of the sharp valleys we find the exact generating function of $h-\bar h$ and $\partial_x h$. Then we express the time scale when the sharp valleys develop, in terms of the forcing characteristics. In the stationary state, when the sharp valleys are fully developed, finite size corrections to the scaling laws of the structure functions $<(h-\bar h)^n (\partial_x h)^m>$ are also obtained.Comment: 50 Pages, 5 figure