Search

We study a correspondence between the multifractal model of turbulence and the Navier-Stokes equations in $d$ spatial dimensions by comparing their respective dissipation length scales. In Kolmogorov's 1941 theory the key parameter $h$, which is an exponent in the Navier-Stokes invariance scaling, is fixed at $h=1/3$ but is allowed a spectrum of values in multifractal theory. Taking into account all derivatives of the Navier-Stokes equations, it is found that for this correspondence to hold the multifractal spectrum $C(h)$ must be bounded from below such that $C(h) \geq 1-3h$, which is consistent with the four-fifths law. Moreover, $h$ must also be bounded from below such that $h \geq (1-d)/3$. When $d=3$ the allowed range of $h$ is given by $h \geq -2/3$ thereby bounding $h$ away from $h=-1$. The implications of this are discussed., Comment: 9 pages, 1 figure

Martin David Kruskal was one of the most versatile theoretical physicists of his generation and is distinguished for his enduring work in several different areas, most notably plasma physics, a memorable detour into relativity and his pioneering work in nonlinear waves. In the latter, together with Norman Zabusky, he invented the concept of the soliton and, with others, developed its application to classes of partial differential equations of physical significance.

Bibliography

An investigation of lower bounds on the quantities Wn(t)=∫Ω|ω|2n dV for n⩾1 for the incompressible three-dimensional (3D) Euler equations has led us to consider a set of spatially averaged weighted “eigenvalues,” λS(n)(t) and λP(n)(t), of the strain matrix S and the Hessian matrix of the pressure P={p,ij}, respectively. It is shown that these obey the simple inequality, λS(n)+f(θn)(λS(n))2+λP(n)⩾0, where f(θn)=1−tan2 θn. The θn are spatially averaged weighted angles between the vorticity vector ω and the vortex stretching vector σ=ω⋅∇u. The weighting in the averaging process highlights regions of large vorticity. This is the angle considered by Tsinober, Kit, and Dracos in their analysis of data from turbulent grid flow experiments in which they noted a tendency toward alignment between ω and σ. The Burgers vortex turns out to be a sharp solution of this inequality with a corresponding angle θn=0, giving rise to exponential growth in Wn. Some special solutions for cases where θn moves between θn=0 and θn...

Inverse length scales for the 2d and 3d thermal convection equations on periodic boundary conditions are estimated directly from the PDEs using a ladder theorem. To within logarithmic dependence on the Rayleigh number RT, the 2d estimate approximately agrees with the inverse length scale estimate found by Manley, Foias and Temam(1987) which was based on the attractor dimension. In keeping with numerical experiments, it shows that RT1/2*RT1/2 grid points should be roughly enough to resolve the smallest features in the flow. Similar methods applied to the 3d case produce a curious result where an inverse length scale for the flow can be estimated rigorously for weak solutions. This estimate is proportional to RT2 which has an associated length scale which is unfortunately much too small to be of any practical use. The same methods also apply to double diffusive convection.

Two possible diagnostics of stretching and folding (S&F) in fluid flows are discussed, based on the dynamics of the gradient of potential vorticity ($q = \bom\cdot\nabla\theta$) associated with solutions of the three-dimensional Euler and Navier-Stokes equations. The vector $\bdB = \nabla q \times \nabla\theta$ satisfies the same type of stretching and folding equation as that for the vorticity field $\bom $ in the incompressible Euler equations (Gibbon & Holm, 2010). The quantity $\theta$ may be chosen as the potential temperature for the stratified, rotating Euler/Navier-Stokes equations, or it may play the role of a seeded passive scalar for the Euler equations alone. The first discussion of these S&F-flow diagnostics concerns a numerical test for Euler codes and also includes a connection with the two-dimensional surface quasi-geostrophic equations. The second S&F-flow diagnostic concerns the evolution of the Lamb vector $\bsD = \bom\times\bu$, which is the nonlinearity for Euler's equations apart from the pressure. The curl of the Lamb vector ($\boldsymbol{\varpi} := \bsD$) turns out to possess similar stretching and folding properties to that of the $\bdB$-vector.

More than 150 years after their invention by Hamilton, quaternions are now widely used in the aerospace and computer animation industries to track the paths of moving objects undergoing three-axis rotations. It is shown here that they provide a natural way of selecting an appropriate ortho-normal frame -- designated the quaternion-frame -- for a particle in a Lagrangian flow, and of obtaining the equations for its dynamics. How these ideas can be applied to the three-dimensional Euler fluid equations is then considered. This work has a bearing on the issue of whether the Euler equations develop a singularity in a finite time. Some of the literature on this topic is reviewed, which includes both the Beale-Kato-Majda theorem and associated work on the direction of vorticity by both Constantin, Fefferman & Majda and Deng, Hou and Yu. It is then shown how the quaternion formulation provides a further direction of vorticity result using the Hessian of the pressure., 22 pages. This review is based on an invited lecture given at the meeting Mathematical Hydrodynamics held at the Steklov Institute Moscow, in June 2006

The Navier–Stokes equations are a set of nonlinear partial differential equations comprising the fundamental dynamical description of fluid motion. They are applied routinely to problems in engineering, geophysics, astrophysics, and atmospheric science. This book is an introductory physical and mathematical presentation of the Navier–Stokes equations, focusing on unresolved questions of the regularity of solutions in three spatial dimensions, and the relation of these issues to the physical phenomenon of turbulent fluid motion. Intended for graduate students and researchers in applied mathematics and theoretical physics, results and techniques from nonlinear functional analysis are introduced as needed with an eye toward communicating the essential ideas behind the rigorous analyses.

To many fluid dynamicists, the only real turbulence is the fine scale 3-dimensional turbulence which occurs at high Reynolds numbers, with an energy cascade and an inertial subrange. The number of degrees of freedom in 3d strong turbulence is clearly many orders of magnitude greater than in such phenomena as convection in a box (see references in [1]) where perhaps only a few spatial modes govern the dynamics.

We present analytical methods whereby weak and strong turbulence are predicted in the D dimensional complex Ginzburg Landau (CGL) equation \(A_{t}=RA+(1+i\nu)\Delta A-(1+i\mu)A\vert A\vert^{2q}\) on a periodic domain [0,1]. Strong (hard) turbulence is characterised by large fluctuations away from space & time averages while no such fluctuations occur in weak (soft) turbulence. In the Δ-ν plane, there are different areas where weak & strong behaviour can occur. In the strong case (Δ, ν → ±∞, ‡∞), the corresponding areas go out to the inviscid limit where the CGL equation becomes the NLS equation in which a finite time singularity occurs when qD ≥ 2. A new infinite set of differential inequalities for the “lattice” of functionals \(F_{n,m}=\int\left[{\vert\nabla^{n-1}A\vert}^{2m}+\alpha_{n,m}\vert A\vert^{2m\left[q(n-1)+1\right]}\right]d\underline{x}\) enables us to construct large time upper bounds on the F n,m . The occurrence of strong spiky turbulence is predicted for qD = 2 by showing that exponents of R in the upper bounds of F n,m & ‖A‖∞ in the strong regions are dependent on the quantity |ν| which gets large in the inviscid limit. The critical value qD = 2 plays an important role: when qD > 2 the CGL equation has some similarities with the 3D Navier equations. A comparison is made between the two & the possibility of having a 1D system which mimics some limited features of the Navier Stokes equations is discussed.

Engineers tend to be interested mainly in results on real fluids and therefore have less interest in theoretical models other than the Navier Stokes equations. In contrast, physicists and applied mathematicians are more interested in the underlying mechanisms which cause and govern turbulence. This sometimes leads them to study idealised systems which are not wholly physical but which give a degree of insight into the mechanisms behind it. Only in 2 spatial dimensions are the incompressible Navier Stokes equations understood analytically in the sense that there is a rigorous proof of the existence of a finite dimensional global attractor. On a finite periodic domain, if G is the Grashof number, then it turns out1,2,3 that the dimension of the global attractor for the 2D Navier Stokes equations is bounded above by cG2/3(1 + logG)1/3 & below by cG2/3. Computational methods4 are generally good enough to resolve the smallest scale in a 2D flow and, for 2D homogeneous decaying turbulence, the vorticity obeys a maximum principle. No such maximum principle is known to exist in 3D & regularity remains to be proved in this case. Furthermore, numerical resolution of the smallest scale in a 3D flow is still a long way off.

The magnitude of the vorticity ω=|ω| for the 3d incompressible Euler equations on the domain Ω=[0,L]3 with boundary conditions un|∂Ω=0 is shown to satisfy the inequality ‖logω(t)‖2−‖logω(0)‖2⩽∫0t‖ω(τ)‖2 dτ, for smooth initial data with no zeros in ω. The notation is ‖ω‖22=∫Ωω2dV and t is time. The case when initial data have zeros in ω is also discussed.

A method is described which enables multiple three-dimensional soliton-like solutions to be found for non-linear Klein-Gordon equations of the type Square Operator phi =F( phi ). The method is extendable to coupled equations Square Operator phi =F( phi , psi ); Square Operator psi =G( phi , psi ).

We consider solutions of the equation\(\psi \), where\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{L} = \alpha ^{ij} \partial _i \partial _j (i,j = 0,...,n)\) and where α is any constant symmetric matrix. The solutions behave as soliton-shaped wave fronts inn dimensions. These interact but keep their identity as in one dimension. We give solutions for the interaction of up to four of these waves for any matrix α and suggest a solution forN of them. There is only one essential difference between these solutions and those of the one-dimensional system;i.e. conditions are found which restrict the motion of then-dimensional waves and it is these conditions we find the most interesting. If we choose α=diag (−1, 1, 1), then it is found that any three interacting waves intersect and form a triangle. For this α we find that the area of this triangle must remain constant in time and hence the wave fronts are forced to move in a specific fixed configuration. We also briefly consider a type of two-dimensional Korteweg-de Vries equation which, although artificially constructed from the one-dimensional case, has similar solutions to the 2-dimensional sine-Gordon equation.

A comprehensive perturbation theory of laser light modulating an electron beam within a crystal shows that the finite size of the crystal plays an essential role. In particular the theory of the central diffraction spot supports the conclusions of Van Zandt and Meyer (1970). Bremsstrahlung provides an explanation of the light emitted from a central spot if but only if the incident electron beam is sufficiently monochromatic. The intensity of the light emitted by other processes is too weak to be observed.

The equations of motion of a relativistic electron in a spatially modulated magnetic field are solved for the two limiting cases of a strong or a weak field. The microwave frequency spectrum emitted in the strong field limit depends on a critical field parameter Bcr, in qualitative agreement with some recent experimental results of Friedman and Herndon. (see abstr. A17480 of 1972).

The main aim of this article is to consider, in the weakly nonlinear sense, the evolution of systems which become unstable as a parameter (μ) is varied but in which the instability is caused not by dissipation but by dispersive effects. We use the method first developed by STUART [1] in hydrodynamic stability to consider the development of waves at the critical or bifurcation point.