Numerical study on the incompressible Euler equations as a Hamiltonian system: Sectional curvature and Jacobi field

We study some of the key quantities arising in the theory of [Arnold "Sur la geometrie differentielle des groupes de Lie de dimension infinie et ses applications a l'hydrodynamique des fluides parfaits," Annales de l'institut Fourier 16, 319 (1966)] of the incompressible Euler equations both in two and three dimensions. The sectional curvatures for the Taylor-Green vortex and the ABC flow initial conditions are calculated exactly in three dimensions. We trace the time evolution of the Jacobi fields by direct numerical simulations and, in particular, see how the sectional curvatures get more and more negative in time. The spatial structure of the Jacobi fields is compared to the vorticity fields by visualizations. The Jacobi fields are found to grow exponentially in time for the flows with negative sectional curvatures. In two dimensions, a family of initial data proposed by Arnold (1966) is considered. The sectional curvature is observed to change its sign quickly even if it starts from a positive value. The Jacobi field is shown to be correlated with the passive scalar gradient in spatial structure. On the basis of Rouchon's physical-space based expression for the sectional curvature (1984), the origin of negative curvature is investigated. It is found that a "potential" alpha(xi) appearing in the definition of covariant time derivative plays an important role, in that a rapid growth in its gradient makes a major contribution to the negative curvature. (C) 2010 American Institute of Physics. [doi: 10.1063/1.3407673

Near-invariance under dynamic scaling for the Navier-Stokes equations in critical spaces: a probabilistic approach to regularity problems

We make a detailed comparison between the Navier-Stokes equations and their dynamically-scaled counterpart, the so-called Leray equations. The Navier-Stokes equations are invariant under static scaling transforms, but are not generally invariant under dynamic scaling transforms. We will study how close they can be brought together using the critical dependent variables and discuss the implications on the regularity problems. Assuming that the Navier-Stokes equations written in the vector potential have a solution that blows up at t = 1, we derive the Leray equations by dynamic scaling. We observe: (1) The Leray equations have only one term extra on top of those of the Navier-Stokes equations. (2) We can recast the Navier-Stokes equations as a Wiener path integral and the Leray equations as another Ornstein-Uhlenbeck path integral. By the Maruyama-Girsanov theorem, both equations take the identical form modulo the Maruyama-Girsanov density, which is valid up to t = 2√ 2 by the Novikov condition. (3) The global solution of the Leray equations is given by a finite-dimensional projection R of a functional of an Ornstein-Uhlenbeck process and a probability measure. If R remains smooth beyond t = 1 under an absolute continuous change of the probability measure, we can rule out finite-time blowup by contradiction. There are two cases: (A) R given by a finite number of Wiener integrals, and (B) otherwise. Ruling out blowup in (A) is straightforward. For (B), a condition based on a limit passage in the Picard iterations is identified for such a contradiction to come out. The whole argument equally holds in R d for any d ≥ 2

Study of the Navier–Stokes regularity problem with critical norms

We study the basic problems of regularity of the Navier–Stokes equations. The blowup criteria on the basis of the critical ${H}^{1/2}$-norm, is bounded from above by a logarithmic function, (Robinson et al 2012 J. Math. Phys. 53 115618). Assuming that the Cauchy–Schwarz inequality for the ${H}^{1/2}$-norm is not an overestimate, we replace it by a square-root of a product of the energy and the enstrophy. We carry out a simple asymptotic analysis to determine the time evolution of the energy. This generalises the (already ruled-out) self-similar blowup ansatz. Some numerical results are also presented, which support the above-mentioned replacement. We carry out a similar analysis for the four-dimensional Navier–Stokes equations

Self-similar solutions to the hypoviscous Burgers and SQG equations at criticality

After reviewing the source-type solution of the Burgers equation with standard dissipativity, we study the hypoviscous counterpart of the Burgers equation. 1) We determine an equation that governs the near-identity transformation underlying its self-similar solution. 2) We develop its approximation scheme and construct the first-order approximation. 3) We obtain the source-type solution numerically by the Newton-Raphson iteration scheme and find it to agree well with the first-order approximation. Implications of the source-type solution are given, regarding the possibility of linearisation of the hypoviscous Burgers equation. Finally we address the problems of the incompressible fluid equations in two dimensions, centering on the surface quasi-geostrophic equation with standard and hypoviscous dissipativity

Remarks on the principles of statistical fluid mechanics

This is an idiosyncratic survey of statistical fluid mechanics centering on the Hopf functional differential equation. Using the Burgers equation for illustration, we review several functional integration approaches to the theory of turbulence. We note in particular that some important contributions have been brought about by researchers working on wave propagation in random media, among which Uriel Frisch is not an exception. We also discuss a particular finite-dimensional approximation for the Burgers equation

Burgers equation with a passive scalar: Dissipation anomaly and Colombeau calculus

A connection between dissipation anomaly in fluid dynamics and Colombeau’s theory of products of distributions is exemplified by considering Burgers equation with a passive scalar. Besides the well-known viscosity-independent dissipation of energy in the steadily propagating shock wave solution, the lesser known case of passive scalar subject to the shock wave is studied. An exact dependence of the dissipation rate ϵθϵθ of the passive scalar on the Prandtl number PrPr is given by a simple analysis: we show, in particular, ϵθ∝1/Pr−−√ϵθ∝1/Pr for large PrPr. The passive scalar profile is shown to have a form of a sum of tanh2n+1 xtanh2n+1 x with suitably scaled xx, thereby implying the necessity to distinguish HH from HnHn when PrPr is large, where HH is the Heaviside function and nn is a positive integer. An incorrect result of ϵθ∝1/Prϵθ∝1/Pr would otherwise be obtained. This is a typical example where Colombeau calculus for products of weak solutions is required for a correct interpretation. A Cole–Hopf-type transform is also given for the case of unit Prandtl number

Numerical study on comparison of Navier-Stokes and Burgers equations

We compare freely decaying evolution of the Navier-Stokes equations with that of the 3D Burgers equations with the same kinematicviscosity and the same incompressible initial data by using direct numerical simulations. The Burgers equations are well-known to be regular by a maximum principle [A. A. Kiselev and O. A. Ladyzenskaya, “On existence and uniqueness of the solutions of the nonstationary problem for a viscous incompressible fluid,” Izv. Akad. Nauk SSSR Ser. Mat.21, 655 (1957); A. A. Kiselev and O. A. Ladyzenskaya, Am. Math. Soc. Transl.24, 79 (1957)] unlike the Navier-Stokes equations. It is found in the Burgers equations that the potential part of velocity becomes large in comparison with the solenoidal part which decays more quickly. The probability distribution of the nonlocal term −u⋅∇p−u·∇p, which spoils the maximum principle, in the local energy budget is studied in detail. It is basically symmetric, i.e., it can be either positive or negative with fluctuations. Its joint probability density functions with 12|u|212|u|2 and with 12|ω|212|ω|2 are also found to be symmetric, fluctuating at the same times as the probability density function of −u⋅∇p−u·∇p. A power-law relationship is found in the mathematical bound for the enstrophy growth dQdt+2νP∝(Qa,Pb)α,dQdt+2νP∝QaPbα, where Q and P denote the enstrophy and the palinstrophy, respectively, and the exponents a and b are determined by calculus inequalities. We propose to quantify nonlinearity depletion by the exponent α on this basis