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A novel semi-Lagrangian method is introduced to solve numerically the Euler equation for ideal incompressible flow in arbitrary space dimension. It exploits the time-analyticity of fluid particle trajectories and requires, in principle, only limited spatial smoothness of the initial data. Efficient generation of high-order time-Taylor coefficients is made possible by a recurrence relation that follows from the Cauchy invariants formulation of the Euler equation (Zheligovsky & Frisch, J. Fluid Mech. 2014, 749, 404-430). Truncated time-Taylor series of very high order allow the use of time steps vastly exceeding the Courant-Friedrichs-Lewy limit, without compromising the accuracy of the solution. Tests performed on the two-dimensional Euler equation indicate that the Cauchy-Lagrangian method is more - and occasionally much more - efficient and less prone to instability than Eulerian Runge-Kutta methods, and less prone to rapid growth of rounding errors than the high-order Eulerian time-Taylor algorithm. We also develop tools of analysis adapted to the Cauchy-Lagrangian method, such as the monitoring of the radius of convergence of the time-Taylor series. Certain other fluid equations can be handled similarly., Comment: 30 pp., 13 figures, 45 references. Minor revision. In press in Journal of Scientific Computing

It has been known for some time that a 3D incompressible Euler flow that has initially a barely smooth velocity field nonetheless has Lagrangian fluid particle trajectories that are analytic in time for at least a finite time (Ph. Serfati C.R. Acad. Sci. S\'erie I 320, 175-180 (1995); A. Shnirelman arXiv:1205.5837 (2012)). Here an elementary derivation is given, based on Cauchy's form of the Euler equations in Lagrangian coordinates. This form implies simple recurrence relations among the time-Taylor coefficients of the Lagrangian map, used here to derive bounds for the C^{1,\gamma} H\"older norms of the coefficients and infer temporal analyticity of Lagrangian trajectories when the initial velocity is C^{1,\gamma}., Comment: 8 pp., 19 refs. Communications in Mathematical Physics, in press

We present three novel forms of the Monge-Ampere equation, which is used, e.g., in image processing and in reconstruction of mass transportation in the primordial Universe. The central role in this paper is played by our Fourier integral form, for which we establish positivity and sharp bound properties of the kernels. This is the basis for the development of a new method for solving numerically the space-periodic Monge-Ampere problem in an odd-dimensional space. Convergence is illustrated for a test problem of cosmological type, in which a Gaussian distribution of matter is assumed in each localised object, and the right-hand side of the Monge-Ampere equation is a sum of such distributions., Comment: 24 pages, 2 tables, 5 figures, 32 references. Submitted to J. Computational Physics. Times of runs added, multiple improvements of the manuscript implemented.

We show that, for two-dimensional space-periodic incompressible flow, the solution can be evaluated numerically in Lagrangian coordinates with the same accuracy achieved in standard Eulerian spectral methods. This allows the determination of complex-space Lagrangian singularities. Lagrangian singularities are found to be closer to the real domain than Eulerian singularities and seem to correspond to fluid particles which escape to (complex) infinity by the current time. Various mathematical conjectures regarding Eulerian/Lagrangian singularities are presented., Comment: 5 pages, 2 figures, submitted to Physica D

Given a Taylor series with a finite radius of convergence, its Borel transform defines an entire function. A theorem of P\'olya relates the large d istance behavior of the Borel transform in different directions to singularities of the original function. With the help of the new asymptotic interpolation method of van der Hoeven, we show that from the knowledge of a large number of Taylor coefficients we can identify precisely the location of such singularities, as well as their type when they are isolated. There is no risk of getting artefacts with this method, which also gives us access to some of the singularities beyond the convergence disk. The method can also be applied to Fourier series of analytic periodic functions and is here tested on various instances constructed from solutions to the Burgers equation. Large precision on scaling exponents (up to twenty accurate digits) can be achieved., Comment: 14 pages, 5 figures

A detailed study of complex-space singularities of the two-dimensional incompressible Euler equation is performed in the short-time asymptotic r\'egime when such singularities are very far from the real domain; this allows an exact recursive determination of arbitrarily many spatial Fourier coefficients. Using high-precision arithmetic we find that the Fourier coefficients of the stream function are given over more than two decades of wavenumbers by $\hat F(\k) = C(\theta) k^{-\alpha} \ue ^ {-k \delta(\theta)}$, where $\k = k(\cos \theta, \sin \theta)$. The prefactor exponent $\alpha$, typically between 5/2 and 8/3, is determined with an accuracy better than 0.01. It depends on the initial condition but not on $\theta$. The vorticity diverges as $s^{-\beta}$, where $\alpha+\beta= 7/2$ and $s$ is the distance to the (complex) singular manifold. This new type of non-universal singularity is permitted by the strong reduction of nonlinearity (depletion) which is associated to incompressibility. Spectral calculations show that the scaling reported above persists well beyond the time of validity of the short-time asymptotics. A simple model in which the vorticity is treated as a passive scalar is shown analytically to have universal singularities with exponent $\alpha =5/2$., Comment: 22 pages, 24 figures, published version; a version of the paper with higher-quality figures is available at http://www.obs-nice.fr/etc7/euler.pdf

Using a very high precision spectral calculation applied to the incompressible and inviscid flow with initial condition $\psi_0(x_1, x_2) = \cos x_1+\cos 2x_2$, we find that the width $\delta(t)$ of its analyticity strip follows a $\ln(1/t)$ law at short times over eight decades. The asymptotic equation governing the structure of spatial complex-space singularities at short times (Frisch, Matsumoto and Bec 2003, J.Stat.Phys. 113, 761--781) is solved by a high-precision expansion method. Strong numerical evidence is obtained that singularities have infinite vorticity and lie on a complex manifold which is constructed explicitly as an envelope of analyticity disks., Comment: 19 pages, 14 figures, published version

It is shown that fractional derivatives of the (integrated) invariant measure of the Feigenbaum map at the onset of chaos have power-law tails in their cumulative distributions, whose exponents can be related to the spectrum of singularities $f(\alpha)$. This is a new way of characterizing multifractality in dynamical systems, so far applied only to multifractal random functions (Frisch and Matsumoto (J. Stat. Phys. 108:1181, 2002)). The relation between the thermodynamic approach (Vul, Sinai and Khanin (Russian Math. Surveys 39:1, 1984)) and that based on singularities of the invariant measures is also examined. The theory for fractional derivatives is developed from a heuristic point view and tested by very accurate simulations., Comment: 20 pages, 5 figures, J.Stat.Phys. in press

We show that the deterministic past history of the Universe can be uniquely reconstructed from the knowledge of the present mass density field, the latter being inferred from the 3D distribution of luminous matter, assumed to be tracing the distribution of dark matter up to a known bias. Reconstruction ceases to be unique below those scales -- a few Mpc -- where multi-streaming becomes significant. Above 6 Mpc/h we propose and implement an effective Monge-Ampere-Kantorovich method of unique reconstruction. At such scales the Zel'dovich approximation is well satisfied and reconstruction becomes an instance of optimal mass transportation, a problem which goes back to Monge (1781). After discretization into N point masses one obtains an assignment problem that can be handled by effective algorithms with not more than cubic time complexity in N and reasonable CPU time requirements. Testing against N-body cosmological simulations gives over 60% of exactly reconstructed points. We apply several interrelated tools from optimization theory that were not used in cosmological reconstruction before, such as the Monge-Ampere equation, its relation to the mass transportation problem, the Kantorovich duality and the auction algorithm for optimal assignment. Self-contained discussion of relevant notions and techniques is provided., Comment: 26 pages, 14 figures; accepted to MNRAS. Version 2: numerous minour clarifications in the text, additional material on the history of the Monge-Ampere equation, improved description of the auction algorithm, updated bibliography. Version 3: several misprints corrected

A method for the reconstruction of the primordial density fluctuation field is presented. Various previous approaches to this problem rendered {\it non-unique} solutions. Here, it is demonstrated that the initial positions of dark matter fluid elements, under the hypothesis that their displacement is the gradient of a convex potential, can be reconstructed uniquely. In our approach, the cosmological reconstruction problem is reformulated as an assignment problem in optimisation theory. When tested against numerical simulations, our scheme yields excellent reconstruction on scales larger than a few megaparsecs., Comment: 14 pages, 10 figures

Does three-dimensional incompressible Euler flow with smooth initial conditions develop a singularity with infinite vorticity after a finite time? This blowup problem is still open. After briefly reviewing what is known and pointing out some of the difficulties, we propose to tackle this issue for the class of flows having analytic initial data for which hypothetical real singularities are preceded by singularities at complex locations. We present some results concerning the nature of complex space singularities in two dimensions and propose a new strategy for the numerical investigation of blowup.(A version of the paper with higher-quality figures is available at http://www.obs-nice.fr/etc7/complex.pdf), Comment: RevTeX4, 10 pages, 9 figures. J.Stat.Phys. in press (updated version)

It is shown phenomenologically that the fractional derivative $\xi=D^\alpha u$ of order $\alpha$ of a multifractal function has a power-law tail $\propto |\xi| ^{-p_\star}$ in its cumulative probability, for a suitable range of $\alpha$'s. The exponent is determined by the condition $\zeta_{p_\star} = \alpha p_\star$, where $\zeta_p$ is the exponent of the structure function of order $p$. A detailed study is made for the case of random multiplicative processes (Benzi {\it et al.} 1993 Physica D {\bf 65}: 352) which are amenable to both theory and numerical simulations. Large deviations theory provides a concrete criterion, which involves the departure from straightness of the $\zeta_p$ graph, for the presence of power-law tails when there is only a limited range over which the data possess scaling properties (e.g. because of the presence of a viscous cutoff). The method is also applied to wind tunnel data and financial data., Comment: LaTeX 11 pages, 11 figures, published version

Extending work of E, Khanin, Mazel and Sinai (1997 PRL 78:1904-1907) on the one-dimensional Burgers equation, we show that density pdf's have universal power-law tails with exponent -7/2. This behavior stems from singularities, other than shocks, whose nature is quite different in one and several dimensions. We briefly discuss the possibility of detecting singularities of Navier-Stokes turbulence using pdf tails., Comment: 4 pages, 3 figures, Proceedings ETC-8, Barcelona, June 2000

This is a review of selected work on the one- and multi-dimensional random Burgers equation (burgulence) with emphasis on questions generally asked for incompressible Navier--Stokes turbulence, such as the law of decay of the energy and the pdf of velocity gradients. Most of the material is devoted to decaying (unforced) burgulence. For more details see the Table of Contents., Comment: 41 pages, 33 figures, to appear in Proceedings Les Houches 2000 "New Trends in Turbulence"

It is shown that non-helical (more precisely, parity-invariant) flows capable of sustaining a large-scale dynamo by the negative magnetic eddy diffusivity effect are quite common. This conclusion is based on numerical examination of a large number of randomly selected flows. Few outliers with strongly negative eddy diffusivities are also found, and they are interpreted in terms of the closeness of the control parameter to a critical value for generation of a small-scale magnetic field. Furthermore, it is shown that, for parity-invariant flows, a moderate separation of scales between the basic flow and the magnetic field often significantly reduces the critical magnetic Reynolds number for the onset of dynamo action., Comment: 44 pages,11 figures, significantly revised version

The asymptotic decay of passive scalar fields is solved analytically for the Kraichnan model, where the velocity has a short correlation time. At long times, two universality classes are found, both characterized by a distribution of the scalar -- generally non-Gaussian -- with global self-similar evolution in time. Analogous behavior is found numerically with a more realistic flow resulting from an inverse energy cascade., Comment: 4 pages, 3 Postscript figures, submitted to PRL

We are interested in the tail behavior of the pdf of mass density within the one and $d$-dimensional Burgers/adhesion model used, e.g., to model the formation of large-scale structures in the Universe after baryon-photon decoupling. We show that large densities are localized near ``kurtoparabolic'' singularities residing on space-time manifolds of codimension two ($d \le 2$) or higher ($d \ge 3$). For smooth initial conditions, such singularities are obtained from the convex hull of the Lagrangian potential (the initial velocity potential minus a parabolic term). The singularities contribute {\em \hbox{universal} power-law tails} to the density pdf when the initial conditions are random. In one dimension the singularities are preshocks (nascent shocks), whereas in two and three dimensions they persist in time and correspond to boundaries of shocks; in all cases the corresponding density pdf has the exponent -7/2, originally proposed by E, Khanin, Mazel and Sinai (1997 Phys. Rev. Lett. 78, 1904) for the pdf of velocity gradients in one-dimensional forced Burgers turbulence. We also briefly consider models permitting particle crossings and thus multi-stream solutions, such as the Zel'dovich approximation and the (Jeans)--Vlasov--Poisson equation with single-stream initial data: they have singularities of codimension one, yielding power-law tails with exponent -3., Comment: LATEX 11 pages, 6 figures, revised; Physica D, in press

Burgers turbulence subject to a force $f(x,t)=\sum_jf_j(x)\delta(t-t_j)$, where the $t_j$'s are ``kicking times'' and the ``impulses'' $f_j(x)$ have arbitrary space dependence, combines features of the purely decaying and the continuously forced cases. With large-scale forcing this ``kicked'' Burgers turbulence presents many of the regimes proposed by E, Khanin, Mazel and Sinai (1997) for the case of random white-in-time forcing. It is also amenable to efficient numerical simulations in the inviscid limit, using a modification of the Fast Legendre Transform method developed for decaying Burgers turbulence by Noullez and Vergassola (1994). For the kicked case, concepts such as ``minimizers'' and ``main shock'', which play crucial roles in recent developments for forced Burgers turbulence, become elementary since everything can be constructed from simple two-dimensional area-preserving Euler--Lagrange maps. One key result is for the case of identical deterministic kicks which are periodic and analytic in space and are applied periodically in time: the probability densities of large negative velocity gradients and of (not-too-large) negative velocity increments follow the power law with -7/2 exponent proposed by E {\it et al}. (1997) in the inviscid limit, whose existence is still controversial in the case of white-in-time forcing. (More in the full-length abstract at the beginning of the paper.), Comment: LATEX 30 pages, 11 figures, J. Fluid Mech, in press

A Lagrangian method is used to show that the power-law with a -7/2 exponent in the negative tail of the pdf of the velocity gradient and of velocity increments, predicted by E, Khanin, Mazel and Sinai (1997 Phys. Rev. Lett. 78, 1904) for forced Burgers turbulence, is also present in the unforced case. The theory is extended to the second-order space derivative whose pdf has power-law tails with exponent -2 at both large positive and negative values and to the time derivatives. Pdf's of space and time derivatives have the same (asymptotic) functional forms. This is interpreted in terms of a "random Taylor hypothesis"., Comment: LATEX 8 pages, 3 figures, to appear in Phys. Rev. E

It is shown by perturbation techniques and numerical simulations that the inverse cascade of kink-antikink annihilations, characteristic of the Kolmogorov flow in the slightly supercritical Reynolds number regime, is halted by the dispersive action of Rossby waves in the beta-plane approximation. For beta tending to zero, the largest excited scale is proportional to the logarithm of one over beta and differs strongly from what is predicted by standard dimensional phenomenology which ignores depletion of nonlinearity., Comment: 4 pages, LATEX, 3 figures. v3: revised version with minor corrections

A Lagrangian method is introduced for calculating simultaneous n-point correlations of a passive scalar advected by a random velocity field, with random forcing and finite molecular diffusivity kappa. The method, which is here presented in detail, is particularly well suited for studying the kappa tending to 0 limit when the velocity field is not smooth. Efficient Monte Carlo simulations based on this method are applied to the Kraichnan model of passive scalar and lead to accurate determinations of the anomalous intermittency corrections in the fourth-order structure function as a function of the scaling exponent xi of the velocity field in two and three dimensions. Anomalous corrections are found to vanish in the limits xi tending to 0 and xi tending to 2, as predicted by perturbation theory., Comment: 11 pages, LATEX, 8 figures. To appear in Phys. Fluids (revised version, minor corrections) Special issue Proceedings of the May 1998 Los Alamos workshop in honor of Robert H. Kraichnan

A Lagrangian method for the numerical simulation of the Kraichnan passive scalar model is introduced. The method is based on Monte--Carlo simulations of tracer trajectories, supplemented by a point-splitting procedure for coinciding points. Clean scaling behavior for scalar structure functions is observed. The scheme is exploited to investigate the dependence of scalar anomalies on the scaling exponent $\xi$ of the advecting velocity field. The three-dimensional fourth-order structure function is specifically considered., Comment: 4 pages, 5 figures

This work is devoted to the decay ofrandom solutions of the unforced Burgers equation in one dimension in the limit of vanishing viscosity. The initial velocity is homogeneous and Gaussian with a spectrum proportional to $k^n$ at small wavenumbers $k$ and falling off quickly at large wavenumbers. In physical space, at sufficiently large distances, there is an ``outer region'', where the velocity correlation function preserves exactly its initial form (a power law) when $n$ is not an even integer. When $11$. A systematic derivation is given in which both the leading term and estimates of higher order corrections can be obtained. High-resolution numerical simulations are presented which support our findings., Comment: In LaTeX with 11 PostScript figures. 56 pages. One figure contributed by Alain Noullez (Observatoire de Nice, France)

Stretched exponential probability density functions (pdf), having the form of the exponential of minus a fractional power of the argument, are commonly found in turbulence and other areas. They can arise because of an underlying random multiplicative process. For this, a theory of extreme deviations is developed, devoted to the far tail of the pdf of the sum $X$ of a finite number $n$ of independent random variables with a common pdf $e^{-f(x)}$. The function $f(x)$ is chosen (i) such that the pdf is normalized and (ii) with a strong convexity condition that $f''(x)>0$ and that $x^2f''(x)\to +\infty$ for $|x|\to\infty$. Additional technical conditions ensure the control of the variations of $f''(x)$. The tail behavior of the sum comes then mostly from individual variables in the sum all close to $X/n$ and the tail of the pdf is $\sim e^{-nf(X/n)}$. This theory is then applied to products of independent random variables, such that their logarithms are in the above class, yielding usually stretched exponential tails. An application to fragmentation is developed and compared to data from fault gouges. The pdf by mass is obtained as a weighted superposition of stretched exponentials, reflecting the coexistence of different fragmentation generations. For sizes near and above the peak size, the pdf is approximately log-normal, while it is a power law for the smaller fragments, with an exponent which is a decreasing function of the peak fragment size. The anomalous relaxation of glasses can also be rationalized using our result together with a simple multiplicative model of local atom configurations. Finally, we indicate the possible relevance to the distribution of small-scale velocity increments in turbulent flow., Comment: 26 pages, 1 figure ps (now available), addition and discussion of mathematical references; appeared in J. Phys. I France 7, 1155-1171 (1997)

It is shown that the inverse Lagrangian map for the solution of the Burgers equation (in the inviscid limit) with Brownian initial velocity presents a bifractality (phase transition) similar to that of the Devil's staircase for the standard triadic Cantor set. Both heuristic and rigorous derivations are given. It is explained why artifacts can easily mask this phenomenon in numerical simulations., Comment: 12 pages, LaTeX

We define a (chaotic) deterministic variant of random multiplicative cascade models of turbulence. It preserves the hierarchical tree structure, thanks to the addition of infinitesimal noise. The zero-noise limit can be handled by Perron-Frobenius theory, just as the zero-diffusivity limit for the fast dynamo problem. Random multiplicative models do not possess Kolmogorov 1941 (K41) scaling because of a large-deviations effect. Our numerical studies indicate that deterministic multiplicative models can be chaotic and still have exact K41 scaling. A mechanism is suggested for avoiding large deviations, which is present in maps with a neutrally unstable fixed point., Comment: 14 pages, plain LaTex, 6 figures available upon request as hard copy (no local report #)