A macroscopic multifractal analysis of parabolic stochastic PDEs

It is generally argued that the solution to a stochastic PDE with multiplicative noise---such as $\dot{u}=\frac12 u"+u\xi$, where $\xi$ denotes space-time white noise---routinely produces exceptionally-large peaks that are "macroscopically multifractal." See, for example, Gibbon and Doering (2005), Gibbon and Titi (2005), and Zimmermann et al (2000). A few years ago, we proved that the spatial peaks of the solution to the mentioned stochastic PDE indeed form a random multifractal in the macroscopic sense of Barlow and Taylor (1989; 1992). The main result of the present paper is a proof of a rigorous formulation of the assertion that the spatio-temporal peaks of the solution form infinitely-many different multifractals on infinitely-many different scales, which we sometimes refer to as "stretch factors." A simpler, though still complex, such structure is shown to also exist for the constant-coefficient version of the said stochastic PDE.Comment: 41 page

Kahler Geometry and Burgers' Vortices

We study the Navier-Stokes and Euler equations of incompressible hydrodynamics. Taking the divergence of the momentum equation leads, as usual, to a Poisson equation for the pressure: in this paper we study this equation in two spatial dimensions using Monge-Ampere structures. In two dimensional flows where the Laplacian of the pressure is positive, a Kahler geometry is described on the phase space of the fluid; in regions where the Laplacian of the pressure is negative, a product structure is described. These structures can be related to the ellipticity and hyperbolicity (respectively) of a Monge-Ampere equation. We then show how this structure can be extended to a class of canonical vortex structures in three dimensions

Length-scale estimates for the LANS-alpha equations in terms of the Reynolds number

Foias, Holm & Titi \cite{FHT2} have settled the problem of existence and uniqueness for the 3D \lans equations on periodic box $[0,L]^{3}$. There still remains the problem, first introduced by Doering and Foias \cite{DF} for the Navier-Stokes equations, of obtaining estimates in terms of the Reynolds number \Rey, whose character depends on the fluid response, as opposed to the Grashof number, whose character depends on the forcing. \Rey is defined as \Rey = U\ell/\nu where $U$ is a bounded spatio-temporally averaged Navier-Stokes velocity field and $\ell$ the characteristic scale of the forcing. It is found that the inverse Kolmogorov length is estimated by \ell\lambda_{k}^{-1} \leq c (\ell/\alpha)^{1/4}\Rey^{5/8}. Moreover, the estimate of Foias, Holm & Titi for the fractal dimension of the global attractor, in terms of \Rey, comes out to be d_{F}(\mathcal{A}) \leq c \frac{V_{\alpha}V_{\ell}^{1/2}}{(L^{2}\lambda_{1})^{9/8}} \Rey^{9/4} where $V_{\alpha} = (L/(\ell\alpha)^{1/2})^{3}$ and $V_{\ell} = (L/\ell)^{3}$. It is also shown that there exists a series of time-averaged inverse squared length scales whose members, \left, %, are related to the $2n$th-moments of the energy spectrum when $\alpha\to 0$. are estimated as $(n\geq 1)$ \ell^{2}\left \leq c_{n,\alpha}V_{\alpha}^{\frac{n-1}{n}} \Rey^{{11/4} - \frac{7}{4n}}(\ln\Rey)^{\frac{1}{n}} + c_{1}\Rey(\ln\Rey) . The upper bound on the first member of the hierarchy \left coincides with the inverse squared Taylor micro-scale to within log-corrections.Comment: 16 pages, no figures, final version accepted for Physica

Role of surface roughness in hard x-ray emission from femtosecond laser produced copper plasmas

The hard x-ray emission in the energy range of 30-300 keV from copper plasmas produced by 100 fs, 806 nm laser pulses at intensities in the range of 10$^{15}-10^{16}$ W cm$^{-2}$ is investigated. We demonstrate that surface roughness of the targets overrides the role of polarization state in the coupling of light to the plasma. We further show that surface roughness has a significant role in enhancing the x-ray emission in the above mentioned energy range.Comment: 5 pages, 4 figures, to appear in Phys. Rev.

Coherent vortex structures and 3D enstrophy cascade

Existence of 2D enstrophy cascade in a suitable mathematical setting, and under suitable conditions compatible with 2D turbulence phenomenology, is known both in the Fourier and in the physical scales. The goal of this paper is to show that the same geometric condition preventing the formation of singularities - 1/2-H\"older coherence of the vorticity direction - coupled with a suitable condition on a modified Kraichnan scale, and under a certain modulation assumption on evolution of the vorticity, leads to existence of 3D enstrophy cascade in physical scales of the flow.Comment: 15 pp; final version -- to appear in CM

Analysis of a General Family of Regularized Navier-Stokes and MHD Models

We consider a general family of regularized Navier-Stokes and Magnetohydrodynamics (MHD) models on n-dimensional smooth compact Riemannian manifolds with or without boundary, with n greater than or equal to 2. This family captures most of the specific regularized models that have been proposed and analyzed in the literature, including the Navier-Stokes equations, the Navier-Stokes-alpha model, the Leray-alpha model, the Modified Leray-alpha model, the Simplified Bardina model, the Navier-Stokes-Voight model, the Navier-Stokes-alpha-like models, and certain MHD models, in addition to representing a larger 3-parameter family of models not previously analyzed. We give a unified analysis of the entire three-parameter family using only abstract mapping properties of the principle dissipation and smoothing operators, and then use specific parameterizations to obtain the sharpest results. We first establish existence and regularity results, and under appropriate assumptions show uniqueness and stability. We then establish results for singular perturbations, including the inviscid and alpha limits. Next we show existence of a global attractor for the general model, and give estimates for its dimension. We finish by establishing some results on determining operators for subfamilies of dissipative and non-dissipative models. In addition to establishing a number of results for all models in this general family, the framework recovers most of the previous results on existence, regularity, uniqueness, stability, attractor existence and dimension, and determining operators for well-known members of this family.Comment: 37 pages; references added, minor typos corrected, minor changes to revise for publicatio

Kahler geometry and Burgers' vortices

We study the Navier-Stokes and Euler equations of incompressible hydrodynamics in two spatial dimensions. Taking the divergence of the momentum equation leads, as usual, to a Poisson equation for the pressure: in this paper we study this equation using Monge-Amp`ere structures. In two dimensional flows where the laplacian of the pressure is positive, a K¨ahler geometry is described on the phase space of the fluid; in regions where the laplacian of the pressure is negative, a product structure is described. These structures can be related to the ellipticity and hyperbolicity (respectively) of a Monge-Amp`ere equation. We then show how this structure can be extended to a class of canonical vortex structures in three dimensions

Finite dimensionality in the complex Ginzburg-Landau equation

Finite dimensionality is shown to exist in the complex Ginzburg-Landau equation periodic on the interval (0,1). A cone condition is derived and explained which gives upper bounds on the number of Fourier modes required to span the universal attractor and hence upper bounds on the attractor dimension itself. In terms of the parameter R these bounds are not large. For instance, when vertical bar ..mu.. vertical bar less than or equal to ..sqrt..3, the Fourier spanning dimension is 0(R/sup 3/2/). Lower bounds are estimated from the number of unstable side-bands using ideas from work on the Eckhaus instability. Upper bounds on the dimension of the attractor itself are obtained by bounding (or, for vertical bar ..mu.. vertical bar less than or equal to ..sqrt..3, computing exactly) the Lyapunov dimension and invoking a recent theorem of Constantin and Foias, which asserts that the Lyapunov dimension, defined by the Kaplan-Yorke formula, is an upper bound on the Hausdorff dimension. 39 refs., 7 figs