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Transport in hydrodynamic flows: analytical and numerical approaches

Journée franco-russe / Ôðàíöóçñêî-ðóññêèé ñåìèíàð / French-Russian workshop

Centre du calcul scientifique, Université de Moscou, 19 septembre 2008

Co-supported by RFBR under project 07-01-92217-CNRSL-a.

Programme

• Lagrangian structure function of fully-developed turbulence (K. Zybin, V. Sirota, A. Il'in, A. Gurevich)
• Solar dynamo wave with meridional circulation: a WKB approach (H. Popova, D. Sokoloff)
• Moment equations for linear and nonlinear equations with random coefficients (D. Grachev, D. Sokoloff)
• The cross-Burgers equation (W. Pauls, S. S. Ray)
• Surprises in the Galerkin-truncated Burgers equation (S. S. Ray)
• Weighted nearest neighbor search in numerical transport optimization (A. Andrievskii, A. Sobolevskii)
• A fast unstable solver for the Monge-Ampere equation: a total failure? (V. Chertovskih, V. Zheligovsky)

Résumés

Lagrangian structure function of fully-developed turbulence (K. Zybin, V. Sirota, A. Il'in, A. Gurevich)

The Lagrangian velocity structure functions in the inertial range of fully developed fluid turbulence are derived basing on the Navier-Stokes equations. For time τ much smaller than the correlation time, the structure functions are shown to obey the scaling relations K_n(τ) ∝ τ^ζ_n. The scaling exponents ζ_n are calculated analytically. The obtained values are in amazing agreement with the unique experimental results of E. Bodenschatz's group. A new notion, the Lagrangian position structure functions R_n(τ), is introduced. All the R_n of the order n>3 are shown to have a universal scaling.

Solar dynamo wave with meridional circulation: a WKB approach (H. Popova, D. Sokoloff)

The effect of meridional circulation on spherical shell dynamos is considered in the Parker approximation. We demonstrate that the type of the exited solution crucially depend on the intensity of the meridional circulation. If the circulation is equatorward or if it is polarward however do not exceed some critical value, an oscillating solution in form of an equatorward traveling wave is excited. If the polarward meridional circulation becomes too intensive the solution becomes steady growing.

Moment equations for linear and nonlinear equations with random coefficients (D. Grachev, D. Sokoloff)

We consider the following simplest ordinary differential equations: the Jacobi equation y'' + K(x)y = 0 with the random coefficient K(x) = K(x,ω) and the equation y' = a(x)y with the random coefficient a(x) = a(x,ω). A relation between numerical and analytical approaches to the study of solutions to these equations is examined. The advantages of these approaches are discussed. Possible generalizations of the results obtained on nonlinear equations are discussed.

Weighted nearest neighbor search in numerical transport optimization (A. Andrievskii, A. Sobolevskii)

We introduce WANN (for Weighted Approximate Nearest Neighbors), a library of C++ classes for weighted nearest neighbor search: given the set P of points p₁, p₂, …, p_N endowed with weights w₁, w₂, …, w_N and a query point q, find p_k such that |q - p_k|² + w_k is minimal. This problem arises as a building block in solution of the discrete optimal transport problem, where weights appear as dual variables of the corresponding linear program, but it is general enough to warrant a separate piece of software. The WANN library is based on the ANN library of D. Mount and S. Arya and generalizes their implementation of nearest neighbor search (all w_k = 0) to the weighted case. WANN is part of a multipurpose library for numerical transport optimization developed in the OTARIE project.