Academic literature on the topic 'Nonlinear isotropic scale-space'

The existence of two-dimensional flows with an isotropic and negative eddy viscosity is demonstrated. Such flows, when subject to a very weak large-scale perturbation of wavenumberkwill amplify it with a rate proportional tok2, independent of the direction.Specifically, it is assumed that the basic (unperturbed) flow is space-time periodic, possesses a centre of symmetry (parity-invariance) and has three- or six-fold rotational invariance to ensure isotropy of the eddy-viscosity tensor.The eddy viscosities emerging from the multiscale analysis are calculated by two different methods. First, there is an expansion in powers of the Reynolds number which can be carried out to large orders, and then extended analytically (thanks to a meromorphy property) beyond the disk of convergence. Secondly, there is a spectral method. The two methods typically agree within a fraction of 1%.A simple example, the ‘decorated hexagonal flow’, of a time-independent flow with isotropic negative eddy viscosity is given. Flows with randomly chosen Fourier components and the required symmetry have typically a 30% chance of developing a negative eddy viscosity when the Reynolds number is increased.For basic flow driven by a prescribed external force and sufficiently strong largescale flow, the analysis is extended to the nonlinear régime. It is found that the largescale dynamics is governed by a Navier-Stokes or a Navier-Stokes-Kuramoto-Sivashinsky equation, depending on the sign and strength of the eddy viscosity. When the driving force is not mirror-symmetric, a new ‘chiral’ nonlinearity appears. In special cases, the large-scale equation reduces to the Burgers equation. With chiral forcing, circular vortex patches are strongly enhanced or attenuated, depending on their cyclonicity.

Abstract. Temperature or pressure anisotropies are characteristic of space plasmas, standard magnetohydrodynamic (MHD) model for describing large-scale plasma phenomena however usually assumes isotropic pressure. In this paper we examine the characteristics of MHD waves, fire-hose and mirror instabilities in anisotropic homogeneous magnetized plasmas. The model equations are a set of gyrotropic MHD equations closed by the generalized Chew-Goldberger-Low (CGL) laws with two polytropic exponents representing various thermodynamic conditions. Both ions and electrons are allowed to have separate plasma beta, pressure anisotropy and energy equations. The properties of linear MHD waves and instability criteria are examined and numerical examples for the nonlinear evolutions of slow waves, fire-hose and mirror instabilities are shown. One significant result is that slow waves may develop not only mirror instability but also a new type of compressible fire-hose instability. Their corresponding nonlinear structures thus may exhibit anticorrelated density and magnetic field perturbations, a property used for identifying slow and mirror mode structures in the space plasma environment. The conditions for nonlinear saturation of both fire-hose and mirror instabilities are examined.

We discuss a new mechanism for the onset of scale-invariant sidebranching in quasi-stable cracking of 2D continuous media. We argue that oscillating terms in the stress field lead to nucleation of defects spaced periodically in lnr space, which are unstable and grow into sidebranches. Requiring that the growth rate be slower than the speed of sound defines the range of validity of this analysis. For an isotropic growth we find the marginally stable angle between major arms of the pattern.

The physical mechanisms underlying the dynamics of the dissipation of passive scalar fluctuations with a uniform mean gradient in stationary isotropic turbulence are studied using data from direct numerical simulations (DNS), at grid resolutions up to 5123. The ensemble-averaged Taylor-scale Reynolds number is up to about 240 and the Schmidt number is from ⅛ to 1. Special attention is given to statistics conditioned upon the energy dissipation rate because of their important role in the Lagrangian spectral relaxation (LSR) model of turbulent mixing. In general, the dominant physical processes are those of nonlinear amplification by strain rate fluctuations, and destruction by molecular diffusivity. Scalar dissipation tends to form elongated structures in space, with only a limited overlap with zones of intense energy dissipation. Scalar gradient fluctuations are preferentially aligned with the direction of most compressive strain rate, especially in regions of high energy dissipation. Both the nature of this alignment and the timescale of the resulting scalar gradient amplification appear to be nearly universal in regard to Reynolds and Schmidt numbers. Most of the terms appearing in the budget equation for conditional scalar dissipation show neutral behaviour at low energy dissipation but increased magnitudes at high energy dissipation. Although homogeneity requires that transport terms have a zero unconditional average, conditional molecular transport is found to be significant, especially at lower Reynolds or Schmidt numbers within the simulation data range. The physical insights obtained from DNS are used for a priori testing and development of the LSR model. In particular, based on the DNS data, improved functional forms are introduced for several model coefficients which were previously taken as constants. Similar improvements including new closure schemes for specific terms are also achieved for the modelling of conditional scalar variance.

Results from investigations are summarized into: (1) transient refractive and absorptive (two-photon) nonlinearities at 0.532 μm by the Z-scan method, and (2) reflective nonlinearity in the near-IR, of linearly nonabsorbing cyanobiphenyl liquid crystals under nanosecond laser irradiation. (1) For isotropic liquid crystals at the several-nanosecond time scale and several tens-micrometers beam-waist-diameter, transient molecular-reorientation and thermal/density refractive nonlinearities compete in changing the sign of the total transient refractive nonlinearity. For the different, given pulse durations, the influence of coupled thermal and density effects on nonlinear refraction depends, through buildup time, on the beam-waist diameter. Nonlinear absorption coefficients depend on the incident intensity. For planar nematic layers, cumulative effects in heating (and in refractive nonlinearity) were observed even at low, 2–10 Hz pulse repetition rate. These results are useful for optical power limiting applications, and for intensity and beam-quality sensors of pulsed, high-power lasers. (2) Reflective nonlinearity of chiral-nematic (cholesteric) mirrors near selective reflection conditions for circular polarized light at λ=1.064 μm was studied both under free space irradiation and inside a laser resonator. Specially chosen experimental irradiation conditions make it possible to attribute the observed changing of reflectivity to athermal helix unwinding by the optical field. The results can find applications in laser-resonator mirrors, Q-switches and soft apertures for beam-profile formation, and also in showing the limits of use cholesteric optical elements in high-power laser beams.

Abstract Zonostrophic instability leads to the spontaneous emergence of zonal jets on a β plane from a jetless basic-state flow that is damped by bottom drag and driven by a random body force. Decomposing the barotropic vorticity equation into the zonal mean and eddy equations, and neglecting the eddy–eddy interactions, defines the quasilinear (QL) system. Numerical solution of the QL system shows zonal jets with length scales comparable to jets obtained by solving the nonlinear (NL) system. Starting with the QL system, one can construct a deterministic equation for the evolution of the two-point single-time correlation function of the vorticity, from which one can obtain the Reynolds stress that drives the zonal mean flow. This deterministic system has an exact nonlinear solution, which is an isotropic and homogenous eddy field with no jets. The authors characterize the linear stability of this jetless solution by calculating the critical stability curve in the parameter space and successfully comparing this analytic result with numerical solutions of the QL system. But the critical drag required for the onset of NL zonostrophic instability is sometimes a factor of 6 smaller than that for QL zonostrophic instability. Near the critical stability curve, the jet scale predicted by linear stability theory agrees with that obtained via QL numerics. But on reducing the drag, the emerging QL jets agree with the linear stability prediction at only short times. Subsequently jets merge with their neighbors until the flow matures into a state with jets that are significantly broader than the linear prediction but have spacing similar to NL jets.

The micropipette aspiration test has been used extensively in recent years as a means of quantifying cellular mechanics and molecular interactions at the microscopic scale. However, previous studies have generally modeled the cell as an infinite half-space in order to develop an analytical solution for a viscoelastic solid cell. In this study, an axisymmetric boundary integral formulation of the governing equations of incompressible linear viscoelasticity is presented and used to simulate the micropipette aspiration contact problem. The cell is idealized as a homogenous and isotropic continuum with constitutive equation given by three-parameter E,τ1,τ2 standard linear viscoelasticity. The formulation is used to develop a computational model via a “correspondence principle” in which the solution is written as the sum of a homogeneous (elastic) part and a nonhomogeneous part, which depends only on past values of the solution. Via a time-marching scheme, the solution of the viscoelastic problem is obtained by employing an elastic boundary element method with modified boundary conditions. The accuracy and convergence of the time-marching scheme are verified using an analytical solution. An incremental reformulation of the scheme is presented to facilitate the simulation of micropipette aspiration, a nonlinear contact problem. In contrast to the halfspace model (Sato et al., 1990), this computational model accounts for nonlinearities in the cell response that result from a consideration of geometric factors including the finite cell dimension (radius R), curvature of the cell boundary, evolution of the cell–micropipette contact region, and curvature of the edges of the micropipette (inner radius a, edge curvature radius ε). Using 60 quadratic boundary elements, a micropipette aspiration creep test with ramp time t*=0.1 s and ramp pressure p*/E=0.8 is simulated for the cases a/R=0.3, 0.4, 0.5 using mean parameter values for primary chondrocytes. Comparisons to the half-space model indicate that the computational model predicts an aspiration length that is less stiff during the initial ramp response t=0-1 s but more stiff at equilibrium t=200 s. Overall, the ramp and equilibrium predictions of aspiration length by the computational model are fairly insensitive to aspect ratio a/R but can differ from the half-space model by up to 20 percent. This computational approach may be readily extended to account for more complex geometries or inhomogeneities in cellular properties. [S0148-0731(00)00503-3]

The spatial evolution of the velocity in a turbulent fluid is shown to consist of a mean strain rate plus a noise term whose mean-square amplitude is dependent upon the local velocity. Assuming the independence of different length scales allows one to eliminate the effect of cubic and higher moments of the velocity upon the evolution of the two-point probability density function for the velocity, analogous to the system-size expansion of the master equation (N. G. van Kampen. Adv. Chem. Phys. 34, 245 (1976); R. Kubo, K. Matsuo, and K. Kitahara. J. Stat. Phys. 9, 51 (1973); M. Suzuki. Adv. Chem. Phys. 46, 198 (1981)). Furthermore, for the velocity structures that exhibit the most internal correlation (i.e., the flow fields that are most likely to be observed) the mean strain rate drops out, leaving only the nonlinear diffusion term. In other words, the dependence of the two-point velocity statistics upon length scale or separation is found to be governed by the gradients of the stress (rather than of the velocity, as is usually assumed.) This provides a new approach for predicting the dominant patterns observed in turbulent flows and a new means of characterizing or classifying different structures, namely, by their nonlinear diffusion coefficient. As a physically relevant example, the isotropic case with algebraic nonlinearity is considered.

Observations of the expansion rate of the universe at late times disagree by a factor of 1.5–2 with the prediction of homogeneous and isotropic models based on ordinary matter and gravity. We discuss how the departure from linearly perturbed homogeneity and isotropy due to structure formation could explain this discrepancy. We evaluate the expansion rate in a dust universe which contains nonlinear structures with a statistically homogeneous and isotropic distribution. The expansion rate is found to increase relative to the exactly homogeneous and isotropic case by a factor of 1.1–1.3 at some tens of billions of years. The time scale follows from the cold dark matter transfer function and the amplitude of primordial perturbations without additional free parameters.

We present a study of the turbulence cascade on the centreline of an inhomogeneous and anisotropic near-field turbulent wake generated by a square prism at a Reynolds number of$Re=3900$using the Kármán–Howarth–Monin–Hill equation. This is the fully generalised scale-by-scale energy balance which, unlike the Kármán–Howarth equation, does not require homogeneity or isotropy assumptions. Our data are obtained from a direct numerical simulation and therefore enable us to access all of the processes involved in this energy balance. A significant range of length scales exists where the orientation-averaged nonlinear interscale transfer rate is approximately constant and negative, indicating a forward turbulence cascade on average. This average cascade consists of coexisting forward and inverse cascade behaviours in different scale-space orientations. With increasing distance from the prism but within the near field of the wake, the orientation-averaged nonlinear interscale transfer rate tends to be approximately equal to minus the turbulence dissipation rate even though all of the inhomogeneity-related energy processes in the scale-by-scale energy balance are significant, if not equally important. We also find well-defined near$-5/3$energy spectra in the streamwise direction, in particular at a centreline position where the inverse cascade behaviour occurs for streamwise oriented length scales.

The usage of automatic fingerprint identification systems as a means of identification and/or verification have increased substantially during the last couple of years. It is well known that small deviations may occur within a fingerprint over time, a problem referred to as template ageing. This problem, and other reasons for deviations between two images of the same fingerprint, complicates the identification/verification process, since distinct features may appear somewhat different in the two images that are matched. Commonly used to try and minimise this type of problem are different kinds of fingerprint image enhancement algorithms. This thesis tests different methods within the scale-space framework and evaluate their performance as fingerprint image enhancement methods.

The methods tested within this thesis ranges from linear scale-space filtering, where no prior information about the images is known, to scalar and tensor driven diffusion where analysis of the images precedes and controls the diffusion process.

The linear scale-space approach is shown to improve correlation values, which was anticipated since the image structure is flattened at coarser scales. There is however no increase in the number of accurate matches, since inaccurate features also tends to get higher correlation value at large scales.

The nonlinear isotropic scale-space (scalar dependent diffusion), or the edge- preservation, approach is proven to be an ill fit method for fingerprint image enhancement. This is due to the fact that the analysis of edges may be unreliable, since edge structure is often distorted in fingerprints affected by the template ageing problem.

The nonlinear anisotropic scale-space (tensor dependent diffusion), or coherence-enhancing, method does not give any overall improvements of the number of accurate matches. It is however shown that for a certain type of template ageing problem, where the deviating structure does not significantly affect the ridge orientation, the nonlinear anisotropic diffusion is able to accurately match correlation pairs that resulted in a false match before they were enhanced.